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Modélisation spatio-temporel des champs de précipitation et des couverts nuageux associés.
Applications à la propagation terre-espace
Ce manuscrit réunit les résultats d’études consacrées à la modélisation spatio-temporelle des affaiblissements de propagation dans la troposphère. La modélisation de ces affaiblissements est utile pour l’optimisation et le dimensionnement des dispositifs adaptatifs de gestion de la ressource des satellites de télécommunication opérant en bande Ka ou Q/V. Ce manuscrit est organisé autour de cinq articles de revues. Les deux premières contributions sont dédiées à la modélisation des distributions statistiques de contenus troposphériques en vapeur d’eau et en eau liquide à partir respectivement de
mesures radiométriques et de bases de données climatiques. La troisième étude touche à l’évaluation des performances d’un système utilisant la diversité de site à partir de données de radar météorologique ; une étude de l’advection des champs précipitations et de son paramétrage par des sorties de modèles de prévision météorologique est également entreprise. Le quatrième article vise à établir une expression analytique de la distribution de la fraction d’une zone, d’une taille comparable à celle d’un faisceau
satellite, affectée par la pluie ; ce modèle est établi en assimilant les champs de précipitations à des champs aléatoires. Le formalisme des champs aléatoires est repris dans le cinquième article qui propose
une modélisation spatio-temporelle des champs de précipitations et des atténuations sous-jacentes à hautes résolutions et à l’échelle d’une couverture satellite continentale. Mots clefs : Propagation troposphérique, Champs aléatoires, FMT, Modèle de canal, Modélisation des précipitations.
Spatio-temporal modelling of rain field and of associated liquid water field.
Application to earth space propagation studies This manuscript gathers together the results studies devoted to the spatio-temporal modeling of propagation impairments through the troposphere. The modelling of the propagation impairments is of
prime importance for the dimensioning and the optimisation of adaptive radio resource management
systems of telecommunication satellite operating at Ka or Q/V band. This manuscript is a compilation of five articles. The two firsts are dedicated to the study of liquid water and water vapor distribution from radiometric and climatologic data. The third article deals with the study of site diversity systems using weather radar data. A study of rain advection and of its parameterization with a meteorological reanalysis database is also presented. The fourth article strives at establishing an analytical distribution of the fraction of an area affected by rain over a given threshold using computations on random fields.
The last publication rely also on this stochastic formalism, in order to propose a space-time modelling of rain field and of the underlying attenuation at high resolution and up to the scale of a continental satellite coverage. Keywords : Tropospheric propagation, Random fields, FMT, Channel Models, Rainfall modelling.
Nic
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THÈSE
En vue de l'obtention du
DDOOCCTTOORRAATT DDEE LL’’UUNNIIVVEERRSSIITTÉÉ DDEE TTOOUULLOOUUSSEE
Délivré par l’Institut Supérieur de l’Aéronautique et de l’Espace
Spécialité : Réseaux et télécommunications
Présentée et soutenue par Nicolas JEANNIN
le 14 novembre 2008
Modélisation spatio-temporelle des champs de précipitation et des couverts nuageux associés.
Applications à la propagation terre-espace
JURY
M. Michel Bousquet, président
M. Laurent Castanet M. Laurent Féral, co-directeur de thèse
M. Antonio Martellucci M. Aldo Paraboni, rapporteur M. Henri Sauvageot, directeur de thèse
Mme Danielle Vanhoenecker-Janvier, rapporteur M. Philippe Waldteufel, rapporteur
École doctorale : Mathématiques, informatique et télécommunications de Toulouse
Unité de recherche : Équipe d'accueil ISAE-ONERA SCANR
Directeur de thèse : M. Henri Sauvageot
Co-directeur de thèse : M. Laurent Féral
Modélisation spatio-temporel des champs de précipitation et des couverts nuageux associés.
Applications à la propagation terre-espace
Ce manuscrit réunit les résultats d’études consacrées à la modélisation spatio-temporelle des affaiblissements de propagation dans la troposphère. La modélisation de ces affaiblissements est utile pour l’optimisation et le dimensionnement des dispositifs adaptatifs de gestion de la ressource des satellites de télécommunication opérant en bande Ka ou Q/V. Ce manuscrit est organisé autour de cinq articles de revues. Les deux premières contributions sont dédiées à la modélisation des distributions statistiques de contenus troposphériques en vapeur d’eau et en eau liquide à partir respectivement de
mesures radiométriques et de bases de données climatiques. La troisième étude touche à l’évaluation des performances d’un système utilisant la diversité de site à partir de données de radar météorologique ; une étude de l’advection des champs précipitations et de son paramétrage par des sorties de modèles de prévision météorologique est également entreprise. Le quatrième article vise à établir une expression analytique de la distribution de la fraction d’une zone, d’une taille comparable à celle d’un faisceau
satellite, affectée par la pluie ; ce modèle est établi en assimilant les champs de précipitations à des champs aléatoires. Le formalisme des champs aléatoires est repris dans le cinquième article qui propose
une modélisation spatio-temporelle des champs de précipitations et des atténuations sous-jacentes à hautes résolutions et à l’échelle d’une couverture satellite continentale. Mots clefs : Propagation troposphérique, Champs aléatoires, FMT, Modèle de canal, Modélisation des précipitations.
Spatio-temporal modelling of rain field and of associated liquid water field.
Application to earth space propagation studies This manuscript gathers together the results studies devoted to the spatio-temporal modeling of propagation impairments through the troposphere. The modelling of the propagation impairments is of
prime importance for the dimensioning and the optimisation of adaptive radio resource management
systems of telecommunication satellite operating at Ka or Q/V band. This manuscript is a compilation of five articles. The two firsts are dedicated to the study of liquid water and water vapor distribution from radiometric and climatologic data. The third article deals with the study of site diversity systems using weather radar data. A study of rain advection and of its parameterization with a meteorological reanalysis database is also presented. The fourth article strives at establishing an analytical distribution of the fraction of an area affected by rain over a given threshold using computations on random fields.
The last publication rely also on this stochastic formalism, in order to propose a space-time modelling of rain field and of the underlying attenuation at high resolution and up to the scale of a continental satellite coverage. Keywords : Tropospheric propagation, Random fields, FMT, Channel Models, Rainfall modelling.
N
icola
s J
EA
NN
IN
– M
odéli
sati
on
sp
ati
o-t
em
po
rell
e d
es c
ham
ps d
e p
récip
itati
on
et
des c
ou
verts
nu
ag
eu
x a
sso
cié
s.
Ap
pli
cati
on
s à
la p
ro
pag
ati
on
terre e
sp
ace
THÈSE
En vue de l'obtention du
DDOOCCTTOORRAATT DDEE LL’’UUNNIIVVEERRSSIITTÉÉ DDEE TTOOUULLOOUUSSEE
Délivré par l’Institut Supérieur de l’Aéronautique et de l’Espace
Spécialité : Réseaux et télécommunications
Présentée et soutenue par Nicolas JEANNIN
le 14 novembre 2008
Modélisation spatio-temporelle des champs de précipitation et des couverts nuageux associés.
Applications à la propagation terre-espace
JURY
M. Michel Bousquet, président
M. Laurent Castanet M. Laurent Féral, co-directeur de thèse
M. Antonio Martellucci M. Aldo Paraboni, rapporteur M. Henri Sauvageot, directeur de thèse
Mme Danielle Vanhoenecker-Janvier, rapporteur M. Philippe Waldteufel, rapporteur
École doctorale : Mathématiques, informatique et télécommunications de Toulouse
Unité de recherche : Équipe d'accueil ISAE-ONERA SCANR
Directeur de thèse : M. Henri Sauvageot
Co-directeur de thèse : M. Laurent Féral
Résumé :
Ce manuscrit réunit les résultats d’une étude consacrée à la modélisation
spatio-temporelle des affaiblissements de propagation dans la troposphère. La
modélisation de ces affaiblissements est utile pour l’optimisation et le
dimensionnement des dispositifs adaptatifs de gestion de la ressource des satellites
de télécommunication opérant en bande Ka ou Q/V. Ce manuscrit est organisé autour
de cinq contributions. Les deux premières contributions sont dédiées à la
modélisation des distributions statistiques de contenus troposphériques en vapeur
d’eau et en eau liquide à partir respectivement de mesures radiométriques et de bases
de données climatiques. La troisième étude touche à l’évaluation des performances
d’un système utilisant la diversité de site à partir de données de radar
météorologique ; une étude de l’advection des champs précipitations et de son
paramétrage par des sorties de modèles de prévision météorologique est également
entreprise. Le quatrième article vise à établir une expression analytique de la
distribution de fraction d’une zone d’une taille comparable à celle d’un faisceau
satellite affectée par la pluie ; ce modèle est établi en assimilant les champs de
précipitations à des champs aléatoires. Le formalisme des champs aléatoires est
repris dans le cinquième article qui propose une modélisation spatio-temporelle des
champs de précipitations et des atténuations sous-jacentes à hautes résolutions et à
l’échelle d’une couverture satellite continentale.
.
Mots clefs : Physique de l’atmosphère – Propagation radioélectrique – Télédétection
atmosphérique – Radar météorologique – Télécommunications spatiales – Champs
aléatoires
Organisation du manuscrit :
Le présent mémoire est constitué de la juxtaposition de cinq articles en cours
de publication dans des revues spécialisées. Ces cinq contributions constituent les
chapitres un à cinq de la deuxième partie de ce manuscrit.
La première partie présente une synthèse en français des divers articles et de
la problématique générale.
Liste des publications de l’auteur
Articles de revues
“Distribution of tropospheric water vapor in clear and cloudy conditions from
microwave radiometric profiling”,
Iassamen A. , H. Sauvageot, N. Jeannin, S. Ameur, Journal of applied
meteorology and climatology, 2009, vol. 48, no3, pp. 600-615.
“Statistical distribution of the fraction of an area affected by rain above a
given threshold”,
Jeannin N, L. Féral, H. Sauvageot, L. Castanet , J. Lemorton, Journal of
Geophysical Research, 2008, vol. 113, D21120, doi:10.1029/2008JD009780.
“Statistical distribution of integrated liquid water and water vapor content
from meteorological reanalysis”,
Jeannin N, L. Féral, H. Sauvageot, L. Castanet , IEEE Transactions on
Antennas and Propagation, 2008, vol. 56, no10, pp. 3350-3355
“Use of weather radar data for site diversity predictions and impact of rain
field advection”,
L. Luini, N. Jeannin, C. Riva, C, C. Capsoni, L. Castanet, J. Lemorton
accepté à International Journal of Satellite Communication.
“A large scale space-time stochastic model of rain attenuation for the design
and optimization of adaptive satellite communication systems operating between 10
and 50GHz”,
Jeannin N, L. Féral, H. Sauvageot, L. Castanet , J. Lemorton, F. Lacoste
soumis à IEEE Transactions on Antennas and Remote Sensing.
Conférences internationales
Jeannin N, L. Féral, H. Sauvageot, L. Castanet, J. Lemorton, K. Leconte and
F. Lacoste, “Modelling of rain fields and attenuation fields spatially correlated over
Europe and US”, in Proceedings 3rd CNES Workshop on Earth - Space Propagation,
(CNES, ed.), Toulouse, France, Sep. 2006.
Jeannin N., L. Féral, H. Sauvageot, L. Castanet, J. Lemorton, F. Lacoste and
K. Leconte, “Modeling of rain fields at large scale”, in International Workshop on
Satellite and Space Communications (IWSSC) 2006, Madrid, Spain, Sep. 2006.
Jeannin N, L. Féral, H. Sauvageot, L. Castanet, J. Lemorton, F. Lacoste
“Stochastic Spatio-Temporal Modelling of Rain Attenuation for Propagation
Studies”, in European Conference on Antennas and Propagation 2007, Edinburgh,
UK, Nov 2007.
Jeannin N., L. Luini, L. Castanet, J. Lemorton, C. Capsoni, C. Riva, A.
Paraboni: “Comparison of the advection of rain fields derived from radar data and
wind outputs from the ERA-40 database”, in Second SatNEx JA-2310 workshop,
Oberpfaffenhofen, Germany, April 2008.
Luini L., N. Jeannin, L. Castanet, J. Lemorton, C. Capsoni, C. Riva, A.
Paraboni: “Simulation of a site diversity system using radar data”, in Second SatNEx
JA-2310 workshop, Oberpfaffenhofen, Germany, April 2008.
Jeannin N., Féral L., Sauvageot H., Castanet L., Lacoste F. “A large scale,
high resolution channel model for propagation impairment techniques design and
optimization”, in International Workshop on Satellite and Space Communications
(IWSSC) 2008, Toulouse, France, Oct. 2008.
Table des matières
0H0HA. Synthèse des travaux 115H1
1H1HProblématique 116H2
2H2HI. Distribution statistique des affaiblissements de propagation 117H12 3H3H1 Introduction 118H12 4H4H2 Données utilisées 119H13 5H5H3 Distribution des intensités de précipitations 120H14
6H6H3.1 Modélisation UIT 121H15 7H7H3.2 Perspectives d’amélioration 122H18
8H8H4 Distribution des contenus intégrés en eau liquide 123H18 9H9H5 Distribution des contenus intégrés en vapeur d’eau 124H21 10H10H6 Conclusion 125H24
11H11HII. Étude de la variabilité spatiale et temporelle des affaiblissements de
propagation 126H26 12H12H1 Introduction 127H26 13H13H2 Etude d'un système de diversité de sites à partir de données radars 128H26 14H14H3 Modélisation spatiale des champs de précipitations 129H30
15H15H3.1 Besoins 130H30 16H16H3.2 Modélisation proposée 131H31 17H17H3.3 Évaluation de la pertinence des résultats obtenus 132H43 18H18H3.4 Limitations 133H48
19H19H4 Modélisation spatio-temporelle des affaiblissements de propagation 134H48 20H20H4.1 Problématique 135H48 21H21H4.2 Méthodologie 136H49 22H22H4.3 Comparaison des caractéristiques des simulations à des données
indépendantes 137H55 23H23H5 Conclusion 138H56
24H24HConclusion et perspectives 139H58
25H25HBibliographie 140H60
26H26HB. Articles 141H67
27H27HStatistical distribution of integrated liquid water and water vapor content
from meteorological reanalysis 142H69 28H28HAbstract 143H69 29H29H1 Introduction 144H69 30H30H2 The data 145H71 31H31H3 Modelling of the ILWC distribution 146H72
32H32H3.1 Analytic formulation and parameter derivation 147H72 33H33H3.2 Model accuracy 148H75
34H34H4 Modelling of the IWVC distribution 149H77 35H35H4.1 Analytic formulation and parameter derivation 150H77 36H36H4.2 Model accuracy 151H80
37H37HConclusion 152H81 38H38HAcknowledgements: 153H82 39H39HReferences 154H82
40H40HDistribution of tropospheric water vapor in clear and cloudy conditions from
microwave radiometric profiling 155H86 41H41HAbstract 156H86 42H42H1 Introduction 157H87 43H43H2 Data 158H89
44H44H2.1 Water vapor parameters 159H89 45H45H2.2 Microwave radiometric measurements 160H90 46H46H2.3 Accuracy and resolution 161H92 47H47H2.4 Site and dataset 162H94
48H48H3 Water vapor content distribution 163H99 49H49H4 Distribution of IWV 164H110 50H50H5 Comparison with operational radiosonde soundings 165H115 51H51H6 Summary and conclusions 166H119 52H52HAcknowledgements 167H120 53H53HReferences 168H120
54H54HWeather radar data for site diversity predictions and evaluation of the
impact of rain field advection 169H125 55H55HAbstract 170H125 56H56H1 Introduction 171H126 57H57H2 The weather radar datasets 172H127 58H58H3 Simulation of site diversity systems using radar images 173H128
59H59H3.1 Gain dependence on the link orientation 174H131 60H60H3.2 Gain dependence on the baseline orientation 175H132 61H61H3.3 Gain dependence on the rain event type 176H135 62H62H3.4 Comparison between the two sites 177H137
63H63H3.5 Comparison with the site diversity model currently recommended by ITU-R
178H138 64H64H4 Comparison between the rain field advection and the ERA-40 wind outputs 179H140
65H65H4.1 Determination of rain advection from radar data 180H141 66H66H4.2 Comparison between radar derived wind speed and direction and ERA-40
wind outputs 181H143 67H67H5 Impact of the wind direction on the rain field spatial anisotropy 182H147 68H68H6 Conclusion 183H149 69H69HAcknowledgements 184H150 70H70HReferences 185H150
71H71HStatistical distribution of the fractional area affected by rain 186H153 72H72HAbstract 187H153 73H73H1 Introduction 188H154 74H74H2 Theoretical considerations on stationary Gaussian random fields 189H157
75H75H2.1 Spatial average M of a Gaussian random field 190H158 76H76H2.2 Spatial variance V of a Gaussian random field 191H159
77H77H3 Model for the CDF of the fractional area over a given threshold 192H161 78H78H3.1 Analytical derivation 193H161 79H79H3.2 Model accuracy and parameter sensitivity 194H162
80H80H4 Application to rain fields 195H165 81H81H4.1 Model extension to rain field 196H165 82H82H4.2 Radar data 197H166 83H83H4.3 Results 198H168 84H84H4.4 Parameterization of the distribution 199H176
85H85H5 Concluding remarks and applications 200H177 86H86HAcknowledgment: 201H178 87H87HBibliography 202H178
88H88HA large scale space-time stochastic model of rain attenuation for the design
and optimization of adaptive satellite communication systems operating between 10
and 50GHz 203H184 89H89HAbstract: 204H185 90H90HIntroduction 205H186 91H91H1 Spatio-temporal modelling of rain fields 206H188
92H92H1.1 Generation of correlated Gaussian random field 207H189 93H93H1.2 Conversion into rain rate fields 208H191 94H94H1.3 Parameterization and refinement of the modeling 209H192 95H95H1.4 Limitations of the presented methodology 210H199
96H96H2 Extension of the model using inputs from meteorological reanalysis database 211H200 97H97H2.1 Description of the ECMWF ERA-40 dataset 212H201 98H98H2.2 Parameterization of the rain amount 213H201 99H99H2.3 Advection 214H208
100H100H3 Conversion into attenuation 215H209
101H101H3.1 Conversion of rain rate fields into attenuation fields 216H209 102H102H3.2 Over sampling of the attenuation time series 217H210
103H103H4 Preliminary validations, limitations 218H211 104H104H4.1 First order statistics 219H211 105H105H4.2 Second order statistics 220H213
106H106H5 Conclusion 221H215 107H107HAcknowledgment: 222H216 108H108HBibliography 223H217
A. Synthèse des travaux
Synthèse des travaux
Problématique
2
Problématique
La demande de plus en plus importante pour des services multimédia nécessitant
un débit et une bande passante importants conduit les systèmes de télécommunications
par satellite à utiliser des bandes de fréquences plus élevées. En effet, les bandes de
fréquences précédemment allouées pour le service fixe, C (4-6 GHz) ou Ku (11-14
GHz), sont engorgées. Les largeurs de bande correspondantes ne suffisent plus pour le
développement de nouveaux systèmes de télécommunication par satellite, qui doivent
présenter des performances comparables à celles offertes par les réseaux terrestres afin
de s’inscrire dans l’infrastructure globale des sytèmes d’information et de
télécommunication.
Par conséquent, de nouveaux systèmes opérant à des bandes de fréquence
supérieures, Ka (20-30 GHz), voire Q/V (40-50 GHz), pour les télécommunications
civiles, ou EHF (20-45 GHz) pour les télécommunications militaires, sont
progressivement mis en service.
A ces fréquences, la propagation des signaux à travers la troposphère joue un rôle
clef dans le dimensionnement des bilans de liaison. Les atténuations rencontrées à ces
fréquences constituent un frein au déploiement de ces systèmes. En effet, lors de la
traversée de la troposphère (couche inférieure de l'atmosphère correspondant en
moyenne aux altitudes 0-12 km), les ondes électromagnétiques subissent une atténuation
causée par les gaz atmosphériques (vapeurs d'eau, oxygène), les nuages et les
hydrométéores (pluie, grêle, neige…) et sont affectées par un effet de scintillation dû à
la turbulence atmosphérique. Les atténuations les plus fortes dans le domaine pour les
fréquences de 10 à 50 GHz sont rencontrées en présence de fortes pluies ou de grêle, et
elles augmentent avec la fréquence de la liaison.
Pour les liaisons terre-espace en bande Ku, ces atténuations restent relativement
faibles et ne dépassent généralement pas 10 dB pendant plus d'une heure par an (0.01%
d’une année) du moins en région tempérée. Afin de tenir compte de cette atténuation
potentielle dans le bilan de liaison des systèmes grand public, une marge de puissance
de quelques dB est suffisante.
Cette solution n'est plus satisfaisante en bande Ka et au dessus car des
atténuations de plusieurs dizaines de dB peuvent être dépassées plusieurs heures par an.
Synthèse des travaux
Problématique
3
L'utilisation d'une marge de puissance fixe pour compenser cette atténuation se
traduirait par un coût prohibitif du système et notamment des segments sols.
Toutefois, afin d'assurer une qualité de service et une disponibilité comparables à
celles des systèmes de télécommunications opérant à des gammes de fréquences
inférieures, des techniques de compensation des affaiblissements qui adaptent en temps
réel les caractéristiques du système en fonction de la variabilité spatiale et temporelle
des atténuations troposphériques sont mises en place. Différentes techniques adaptatives
(ou FMT pour Fade Mitigation Techniques) éventuellement cumulables sont
envisageables selon les configurations (liaison montante, liaison descendante, diffusion,
station sols…) du système envisagé [Castanet, 2001]. Le dimensionnement et
l’optimisation de ces systèmes de compensation adaptatifs des affaiblissements de
propagation nécessitent une connaissance fine des variabilités spatiales et temporelles de
ces affaiblissements de propagation qui ne peut se réduire à une simple étude de la
distribution statistique de ces affaiblissements [Cost 255, 2002].
Le dimensionnement des marges statiques de puissance utilisées pour compenser
les atténuations troposphériques, en bande Ku ou à des fréquences inférieures, a donné
lieu à de nombreuses études statistiques des distributions ponctuelles d’atténuation et de
chacun des paramètres radio-climatiques sous-jacents (intensité de précipitation,
contenu intégré en eau liquide, contenu intégré en vapeur d’eau…).
Ces études ont montré une grande disparité de ces distributions d’atténuation
selon les zones climatiques considérées, principalement imputable à la variabilité des
régimes de précipitations. En effet, les pluies fortement convectives des régions
tropicales ou équatoriales génèrent des atténuations supérieures à celles induites par les
précipitations des régions tempérées.
Afin de quantifier aux mieux ces distributions et de prendre en compte cette
dépendance climatique, ces études ont abouti à la création de cartes de paramètres radio-
climatiques et de méthodes permettant d’estimer ces distributions d’atténuation en tout
point du globe. Ces paramètres sont tabulés par des recommandations de l’Union
Internationale des Télécommunications (UIT-R).
Considérant une liaison avec des caractéristiques géographiques et
radioélectriques données, ces recommandations permettent d’estimer les probabilités
d’atténuation sur cette liaison dépassée pendant un certain pourcentage du temps d’une
Synthèse des travaux
Problématique
4
année moyenne. A cet effet, les recommandations UIT-R P.618, P.676 et P.840
permettent d’obtenir cette distribution cumulative d’atténuation totale en fonction des
distributions cumulatives d’intensité de précipitations (recommandation UIT-R P.837),
de contenus intégrés en eau liquide (recommandation UIT-R P.840), de contenus
intégrés en vapeur d’eau (recommandation UIT-R P.836) ainsi que de la température et
de l’hygrométrie du sol (recommandation UIT-R P.1511) et de l’altitude de l’isotherme
–2oC (recommandation UIT-R P.839).
Par conséquent, connaissant la spécification de l’indisponibilité nominale que
peut subir une liaison terre-espace compte tenu du service à assurer (généralement
moins de 1 % du temps de fonctionnement pour des terminaux utilisateurs à faible
disponibilité et parfois jusqu’à 0.001 % pour des hubs), la marge statique de puissance
minimale pour compenser les effets atmosphériques dans le bilan de liaison peut être
calculée. Elle correspond à la valeur d’atténuation dépassée pour un temps équivalant à
l’indisponibilité tolérée. La recommandation UIT-R P.618, qui permet d’estimer la
distribution cumulative de l’atténuation totale en fonction du temps quelles que soient
les caractéristiques géographiques, géométriques et radioélectriques de la liaison, ainsi
que la climatologie locale, permet ainsi d’estimer cette marge.
Cependant, l’utilisation de techniques adaptatives de compensation des
affaiblissements de propagation, rendue nécessaire par l’utilisation de fréquences plus
élevées, requiert une connaissance plus fine du canal de propagation. En effet, la seule
connaissance des distributions en un point des affaiblissements de propagation ne
permet pas d’optimiser convenablement ces dispositifs. Par exemple, dans le cas d’une
liaison entre un émetteur et un satellite en présence d’une forte atténuation, la liaison
peut être maintenue en agissant sur la puissance d’émission du terminal, sur la forme
d’onde utilisée, ou encore sur le débit de l’information. L’utilisation de ces différents
mécanismes requiert une connaissance de la dynamique temporelle du canal de
propagation [Castanet et al., 2001].
Dans le cas du contrôle de la puissance émise, il faut ajuster cette puissance aux
variations des affaiblissements de propagation tout en évitant d’interférer sur les liaisons
des spots adjacents réutilisant les mêmes fréquences. Un ajustement de puissance de 3 à
5dB pour les terminaux commerciaux et de 10dB pour les hubs est envisagé pour les
systèmes en cours de déploiement.
Synthèse des travaux
Problématique
5
Dans le cas de l’adaptation de la forme d’onde pour laquelle les derniers
standards proposent des modulations résistantes à des atténuations allant jusqu’à 20dB,
il faut également une estimation précise des variations temporelles des atténuations
troposphériques, et en particulier de celles dues à la pluie. En effet, la afin de déterminer
d’une part les différents modes de codages à implémenter et d’autre part les différentes
modulations envisageables [Bolea-Alamañac, 2004].
Par conséquent, en plus d’avoir une connaissance de la statistique des
affaiblissements de propagation affectant la liaison, il est impératif d’avoir une
connaissance des caractéristiques liées à la corrélation temporelle de ces
affaiblissements de propagation pour le dimensionnement et l’optimisation de ces
techniques de compensations adaptatives.
D’autres mécanismes de compensation adaptatifs nécessitent, quant à eux, une
connaissance statistique de la variabilité spatiale des affaiblissements. Par exemple, pour
augmenter la disponibilité d’une liaison entre une station sol et le satellite, une
deuxième station sol, généralement localisée à quelques dizaines de kilomètres et reliée
par un lien terrestre à la première, peut être implantée. Dans le cas où la liaison entre la
première station et le satellite est affectée par une atténuation ne permettant pas la
transmission du signal (en présence de fortes précipitations), l’autre station est utilisée
et le signal est relayé par le réseau terrestre ou une liaison sol dédiée. Cette technique
pour augmenter la disponibilité de la liaison, connue sous le nom de diversité de site,
repose sur l'hétérogénéité spatiale des champs de précipitations [Goldhirch et Robison
1975] .
En effet, considérant deux stations distantes de quelques dizaines de kilomètres,
la probabilité que les liaisons entre le satellite et chacune de ces stations soient
simultanément affectées par un affaiblissement ne permettant pas la transmission du
signal est plus faible que la probabilité qu’une seule liaison subisse un affaiblissement
trop élevé du fait de l’inhomogénéité spatiale des champs de précipitations. La
problématique principale pour le dimensionnement de ces systèmes est celle de la
distance minimale devant séparer les deux sites afin d’obtenir une disponibilité donnée.
La connaissance de la variabilité spatiale des affaiblissements de propagation est
également essentielle du point de vue de la gestion de la ressource embarquée [Pech,
Synthèse des travaux
Problématique
6
2003]. En effet, l’allocation des plages de fréquences se fait par groupes d’utilisateurs
situés à l’intérieur d’un même faisceau du satellite, et nécessite d’avoir une estimation
de l’atténuation encourue par chacune des liaisons à l’intérieur de ce faisceau.
Enfin, tenant compte des derniers développements technologiques sur les
antennes et s’appuyant sur le caractère multi-faisceaux des nouveaux systèmes
satellitaires, une reconfiguration du diagramme d’antenne et des gains afférents peut
permettre de contrer les affaiblissements de propagation. En focalisant plus fortement
les antennes sur les zones susceptibles de subir de fortes atténuations, et donc des
précipitations, des marges supplémentaires peuvent être dégagées dans le bilan de
liaison. A cet effet, le mécanisme sous-jacent de reconfiguration des antennes requiert
une connaissance de la répartition spatiale de l’affaiblissement sur la couverture globale
du système (généralement un continent). Cette connaissance doit être statistique pour le
dimensionnement du dispositif, voire prédictive à une échéance de quelques heures pour
la reconfiguration des antennes [Buti et al, 2008].
Certains mécanismes de rétroaction nécessitent une connaissance à la fois
temporelle et spatiale du canal de propagation [Bousquet et al., 2003]. En effet,
l’implémentation de mécanismes de codage et de modulation adaptatifs requiert une
connaissance de l’évolution du canal pour chaque terminal. L’allocation dynamique des
porteuses (dans le cas d’un accès multiple en fréquence) ou des paquets (dans le cas
d’un accès multiple dans le temps) ne peut se faire que globalement à l’échelle d’un
spot, la ressource embarquée étant limitée, et ne peut par conséquent être optimisée
qu’en prenant en considération de manière simultanée l’état du canal de propagation
pour chacun des terminaux.
Pour répondre aux problématiques citées ci-dessus, des modèles statistiques
ont été proposés par l’UIT-R pour donner des indications statistiques sur les variabilités
temporelle et spatiale des affaiblissements de propagation. Ainsi, la recommandation
UIT-R P.1623 propose une méthode d’estimation des statistiques sur la durée des
affaiblissements au-dessus d’un certain niveau d’atténuation, sur le gradient temporel
des affaiblissements ainsi que sur la durée entre deux événements d’atténuation.
D’autres parts, la recommandation UIT - R P.618 permet de calculer le gain de diversité
ou la probabilité jointe d’atténuation simultanément dépassée en fonction de la distance
Synthèse des travaux
Problématique
7
entre deux stations. Même si ces méthodes statistiques permettent l’évaluation de
différents paramètres dimensionnant des boucles de contrôle de certaines FMT, elles
s’avèrent relativement imprécises lorsque comparées à des mesures du canal de
propagation. De plus elles ne permettent qu’une description partielle des caractéristiques
du canal de propagation, insuffisante pour résoudre les problèmes liés au
dimensionnement des FMT. Un recours à des données ou méthodes de représentation
différentes doit donc être envisagé.
Diverses expériences de mesure des affaiblissements de propagation sur les
liaisons terre-espace ont été réalisées au cours des dernières décennies (SIRIO [Carassa,
1978], OLYMPUS [OPEX, 1994], ACTS [Crane, 1997], ITALSAT [Riva, 2004] et
bientôt ALPHASAT [Paraboni et al., 2007] et GSAT-4 [Katti et al., 2007]). Elles ont
permis d’obtenir des séries temporelles d’atténuation subie par une liaison terre-espace
en différents points du globe, sur des durées relativement longues, et échantillonnées à
des cadences de l’ordre de la seconde.
Si elles ont contribué à mieux caractériser la dynamique fine du canal de
propagation, ces campagnes de mesure d’atténuation troposphérique ne sont toutefois
pas suffisantes pour résoudre les problèmes posés par le dimensionnement des boucles
de contrôle FMT pour plusieurs raisons. Le nombre de stations de mesure impliquées
dans ces expériences était limité, sans synchronisation entre les stations, ce qui par
conséquent ne donne qu’une idée sommaire de la répartition spatiale des
affaiblissements de propagation. Ensuite, ces expériences ont été menées à des
fréquences, des élévations et dans des régions climatiques bien précises et il n’est pas
aisé de déduire les caractéristiques des affaiblissements pour des liaisons à d’autres
fréquences, élévations et régions climatiques. La prise en compte des affaiblissements
de propagation pour le dimensionnement des boucles FMT doit donc être réalisée par le
biais d’autres moyens.
Une des approches permettant de s’affranchir de ces limitations est l’étude de la
variabilité des affaiblissements de propagation à partir de paramètres météorologiques
causant ces affaiblissements. Comme mentionné précédemment, les affaiblissements de
propagation sont liés à la présence de pluie, de nuages et des gaz atmosphériques
(oxygène et vapeur d’eau). Connaissant les intensités de précipitations, les contenus
Synthèse des travaux
Problématique
8
intégrés en eau liquide des nuages et les concentrations en gaz le long d’une liaison
terre-espace, des modèles statistiques reposant sur une base physique ont été développés
pour estimer l’atténuation subie par l’onde électromagnétique le long de cette liaison.
De ce fait, à partir de données ou de prévisions météorologiques portant sur ces
paramètres, il est possible d’estimer l’atténuation subie par la liaison et ce quelles que
soient la configuration géométrique ou la fréquence de la liaison.
Comme les systèmes de télécommunication par satellite qui seront déployés dans
les prochaines années visent l’utilisation de la bande Ka, pour laquelle la contribution de
la pluie à l’atténuation totale est prépondérante, la plupart des études se concentrent sur
les précipitations.
Les données météorologiques permettant l’étude des précipitations sont multiples
[SatNex, 2008]. Les plus appropriées pour calculer l’atténuation sur une liaison satellite
sont celles obtenues à partir des radars météorologiques. En effet, ces derniers
fournissent, à intervalles réguliers, une cartographie des réflectivités induites par les
précipitations. Ces réflectivités radars peuvent être liées aux intensités de précipitations
par diverses relations telles que celles de Marshall-Palmer. Les intensités de
précipitations étant liées à l’atténuation spécifique, l’atténuation totale peut être
calculée à partir de ces données par intégration de l’atténuation spécifique le long de la
liaison. La couverture des réseaux radars météorologiques est cependant insuffisante
pour que ces données soient directement exploitables pour réaliser des simulations
systèmes.
Il serait également envisageable de convertir des sorties de modèle
météorologique méso-échelle contraint par des données de réanalyse en atténuation
troposphérique [Hodges et al., 2006]. Cependant, les temps de calcul et la fiabilité des
prévisions [Lascaux et al., 2003] constituent pour le moment un obstacle à la mise en
œuvre systématique d’un tel dispositif. De plus, la représentativité statistique des
atténuations générées, qui reste un des problèmes majeurs dans le contexte des
télécommunications par satellite, n’est pas assurée par par la description des processus
de microphysiques implémentée dans les modéles météorologiques. Cependant comme
mentionné dans [Hodges et al., 2006], l’utilisation de modèles de prévision méso-
échelle, rest la piste la plus prometteuse pour les prédictions des champs d’atténuations
à une échéances de quelques heures.
Synthèse des travaux
Problématique
9
Les autres sources de données météorologiques, (données de télédétection
spatiale, modèles de réanalyse), n'ont pas une résolution suffisante pour rendre compte
précisément des phénomènes donnant lieu à de l'atténuation. L'utilisation de mesures
ponctuelles issues de radio-sondages, de radiomètres ou de réseaux de disdromètres
pourrait également être pertinente, mais, comme pour les mesures de canal de
propagation, le maillage spatial des réseaux de mesures est trop lâche pour fournir une
information spatiale suffisante.
Pour pallier ce manque de données directement et systématiquement utilisables
pour le dimensionnement des FMT, des modèles permettant de générer des conditions de
propagation réalistes ont été développés. Leur élaboration a été permise par l'analyse
statistique des données de propagation issues des campagnes de mesures [Audoire, 2001,
Van de Kamp, 2003, Lacoste, 2005], mais aussi des divers types de données
météorologiques évoquées précédemment. Les efforts se sont dans un premier temps
portés sur des modélisations temporelles du canal de propagation et il existe désormais
des modèles reproduisant de manière assez réaliste la dynamique observée sur des
données expérimentales [Maseng et Bakken, 1981, Gremont et Filip, 2004, Lacoste et
al., 2006, Lemorton et al., 2007].
Des modèles portant sur l'aspect variabilité spatiale des propagations ont ensuite
été étudiés [Casponi et al., 1987a, 1987b], [Féral et al, 2003a, 2003b, 2006],
[Callaghan, 2006]. Ils se sont plus particulièrement attachés à donner des
représentations réalistes des champs de précipitation observés par les radars
météorologiques. Pour le développement de ces modèles, proches de problématiques
relatives à l'hydrologie, différentes pistes ont été explorées. Des modélisations reposant
sur l'individualisation des cellules de précipitation [Féral et al, 2003a], ou sur le
caractère fractal des champs de précipitation ont ainsi été proposées. Des modélisations
de champs de précipitations par des champs aléatoires corrélés dans l'espace ont
également vu le jour.
Le potentiel de ces modèles dépend toutefois de la possibilité de simuler des
champs à des résolutions suffisamment fines (kilomètriques) et sur des étendues
suffisantes (couverture continentale) pour répondre aux besoins exprimés pour les
systèmes de télécommunication par satellite. C'est pourquoi des méthodologies de
généralisation de ces modèles combinant plusieurs approches ou prenant également en
Synthèse des travaux
Problématique
10
compte des observations ont été récemment développées pour répondre plus
spécifiquement à ces besoins.
A un degré plus marqué encore, les modèles de propagation troposphérique
prenant en compte les aspects à la fois temporels et spatiaux ne permettent pas pour
l'heure de répondre de manière globale aux problématiques posées par le
dimensionnement des boucles FMT des techniques de gestion de la ressource. Pour
pallier ce manque, l'objectif de cette thèse a été le développement d'un générateur de
champs d'atténuation totale corrélé dans l'espace et dans le temps.
Le présent mémoire réunit différents travaux publiés dans des revues
internationales, et dont l’objectif général est d’arriver à une meilleure connaissance du
canal de propagation dans le temps et dans l'espace en vue du dimensionnement optimal
des systèmes de télécommunications par satellite opérant aux fréquences supérieures à
10 GHz. Ils s’appuient sur une analyse statistique des paramètres météorologiques
influant sur les conditions de propagation.
Le premier article vise à renforcer la connaissance statistique des distributions de
contenus intégrés en eau liquide et en eau vapeur en utilisant des données de réanalyse
météorologique. Cette étude a principalement été motivée par le peu de données et leur
faible qualité concernant la distribution de ces paramètres météorologiques.
Le second article présente une modélisation des distributions de contenus
intégrés en vapeur d'eau à partir de profils radiométriques. Différentes modélisations de
ces distributions en fonction de la présence de nuages ou non et de la température sont
proposées. L'étude de ces distributions peut permettre de mieux connaître les couplages
existant entre les atténuations dues aux nuages et à la vapeur d’eau.
Le troisième article est une étude d’un système de diversité de sites à partir de
données de radars météorologiques. Cette étude montre l’avantage de l’utilisation de
données radars par rapport aux statistiques de l’UIT-R pour le dimensionnement optimal
de ce dispositif. Cette étude évalue également l’influence de différents paramètres sur
les performances du système. En particulier, il est montré que la direction prédominante
du vent a un impact sur l’orientation optimale des stations l’une par rapport à l’autre.
La contribution suivante propose un modèle de distribution statistique de la
fraction d’une zone affectée par des intensités de précipitation dépassant un certain
Synthèse des travaux
Problématique
11
seuil, et dont la taille est comparable à celle d’un faisceau satellite. Cette étude permet
d’une part, à l’intérieur d’un faisceau, d’estimer le nombre de récepteurs touchés par une
atténuation au-delà d’un certain seuil, et d’autre part de poser les fondements théoriques
d’un modèle de génération de champs de précipitations reposant sur l’utilisation de
champs aléatoires développés dans le dernier article.
Dans ce dernier article, un modèle de génération de champs d’atténuation due à la
pluie corrélés dans l’espace et dans le temps est proposé. Il repose sur une modélisation
existante des champs de précipitation par le biais de champs aléatoires corrélés dans
l’espace et dans le temps. Une méthode de paramétrage de ce modèle est décrite, et le
modèle lui-même est amélioré afin d’étendre son domaine de validité dans l'espace et
dans le temps. Cette extension du domaine de validité est obtenue en couplant le modèle
avec des données de réanalyse météorologique. Il génère des sorties directement
utilisables pour simuler la couche propagation dans les simulations de systèmes de
télécommunication par satellite.
La première partie de ce mémoire est une synthèse des différentes
problématiques afférentes à la prise en compte des affaiblissements de propagation dans
la troposphère, en vue de dimensionner les systèmes de télécommunication par satellite.
Un premier chapitre est consacré à la modélisation des distributions statistiques
d'atténuation et des paramètres météorologiques sous-jacents. Le deuxième chapitre
traite de la modélisation des variabilités spatiales et spatio-temporelles des
affaiblissements de propagation qui a constitué la plus grande partie de mes travaux de
recherche
La seconde partie du manuscrit reprend le texte des cinq articles, publiés ou en
cours de publication, écrits au cours de la thèse.
Synthèse des travaux
I. Distribution statistique des affaiblissements de propagation
12
I. Distribution statistique des affaiblissements de propagation
1 Introduction
Les études de bilan de liaison en vue du déploiement des systèmes de
télécommunication par satellite ont conduit à de nombreux travaux sur la distribution
statistique des affaiblissements de propagation à travers l’atmosphère. Ces études ont
débouché sur des modélisations communément acceptées au travers de recommandations
de l’UIT-R qui permettent d’obtenir une estimation de la probabilité d’occurrence d’une
atténuation pour une liaison donnée comme illsutré figure 1.
Figure 1: Affaiblissement dépassé durant P% d’une année moyenne à Paris calculé à
partir de la Recommandation UIT-R P.618-9. Fréquence = 40 GHz, élévation = 30°
Ces distributions d’atténuation sont obtenues en établissant un lien physique ou
statistique avec les distributions des paramètres météorologiques sous-jacents.
Synthèse des travaux
I. Distribution statistique des affaiblissements de propagation
13
Néanmoins, une amélioration de ces modélisations est encore nécessaire. En effet, il
serait intéressant de pouvoir estimer les variabilités de ces distributions (saisonnières,
diurnes, interannuelles) lesquelles ne sont pas quantifiées à l’heure actuelle. D’autre
part, une attention toute particulière doit être portée à l’atténuation induite par les gaz et
les nuages, qui, aux bandes Q/V ne dépasse certe pas quelaues dB et pour lesquels les
modélisations actuelles sont imprécises. Les articles “Statistical distribution of
integrated liquid water and water vapor content from meteorological reanalysis” et
“Distribution of tropospheric water vapor in clear and cloudy conditions from
microwave radiometric profiling” sont consacrés à cette problématique.
La connaissance de ces distributions est un point de départ incontournable non
seulement pour le dimensionnement du bilan de liaison des satellites de
télécommunication opérant aux fréquences supérieures à 5 GHz, mais aussi pour les
modèles permettant de générerle comportement fin du canal de propagation dans
l’éspace et dans le temps. L’amélioration de la description statistique des
affaiblissements a, par conséquent, un effet bénéfique sur le réalisme des modèles s’y
référant.
2 Données utilisées
La modélisation des distributions d'atténuation telle que proposée par l’UIT-R
repose sur l’utilisation de deux sortes de données :
- des données expérimentales de référence, ponctuelles, collectées en différents
points du globe mais dont le nombre est relativement restreint ;
- des bases de données mondiales de paramètres radio-climatiques qui servent
d’entrée au modèle proposé.
La démarche générale a consisté à élaborer des modèles statistiques, prenant en
entrée les paramètres radio-climatiques des bases de données mondiales, et à optimiser
les paramètres de ces modèles afin qu’ils reproduisent le plus fidèlement possible les
statistiques obtenues à partir des données expérimentales de référence.
Différents types de données de référence ont été considérés pour la dérivation des
distributions de paramètres météorologiques ou d’atténuation. Les données des
Synthèse des travaux
I. Distribution statistique des affaiblissements de propagation
14
campagnes de mesures de propagation sont évidemment considérées comme les données
expérimentales de référence pour les effets de propagation à savoir l’atténuation, la
scintillation ou la dépolarisation. Les données expérimentales utilisées pour la
modélisation des précipitations sont issues de mesures disdrométriques ou
pluviométriques, celles pour la modélisation des contenus en eau liquide ou en vapeur
d’eau sont issues de mesures effectuées à partir de radiomètres micro-ondes ou de radio-
sondages. Les mesures servant de référence ont été recueillies sur plusieurs décennies à
partir de différentes instrumentations dont les résultats ne sont pas nécessairement
comparables entre-elles.
Les bases de données mondiales sont quant à elles généralement dérivées de
données de réanalyse météorologique. Ces données, représentant l’état passé de
l’atmosphère à l’échelle mondiale, sont issues de modèles de prévision numérique,
contraints par des observations (radio-sondages, données satellitaires, barométriques,
anémométriques…).
Dans le cadre des études de propagation menées jusqu’ici, les bases de données
de réanalyse utilisées émanent majoritairement de l’ECMWF (Centre Européen de
Prévision Météorologique à Moyen Terme). Le processus de réanalyse porte uniquement
sur certains paramètres atmosphériques, à savoir : la pression, la température, l’humidité
relative et le vent horizontal. Les paramètres importants pour l’étude de la propagation,
comme les précipitations ou les contenus intégrés en eau liquide des nuages, sont
déduits de ces paramètres internes et ne sont pas recalibrés sur des observations.
3 Distribution des intensités de précipitations
De nombreuses hypothèses et pistes ont été explorées pour modéliser les
statistiques des intensités de précipitation. Ainsi des lois, hyperbolique, Weibull, log-
normale, Gamma, ont été proposées, pour modéliser cette distribution, principalement
dans le domaine de l’hydrologie. Aucun consensus absolu ne se dégage, bien que
certaines modélisations semblent plus adaptées que d’autres pour décrire la majorité des
distributions d’intensités de précipitation observées et du fait des hypothèses physiques
sous-jacentes. La modélisation ayant cours dans le domaine des télécommunications par
Synthèse des travaux
I. Distribution statistique des affaiblissements de propagation
15
satellite ne repose sur aucun de ces modèles. Toutefois, elle présente l’intérêt de fournir
une modélisation de l’intensité des précipitations en tout point du globe.
3.1 Modélisation UIT
La recommandation UIT-R P.837-5 ([Poaires-Baptista and Salonen, 1997] mis à
jour dans [Castanet et al., 2007]) propose le modèle suivant pour la CCDF (fonction de
répartition complémentaire) d’intensité de précipitations en tout point du globe :
*
**
11
0* )( Rc
RbRa
ePRRP +
+−
=> (1)
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−=
−−
6
)1(0079.0
60 1 r
T
PM
r ePPβ
(2)
où P0 représente la probabilité de pluie en un point, avec :
bcP
Mb
a
T
02.2621797
09.1
0
=
=
=
(3)
Trois paramètres calculés à partir de la réanalyse météorologique ERA-40 de
l’ECMWF [Uppala et al., 2005], servent d’entrée au modèle (1). La base de données
ERA-40 couvre une quarantaine d’années (1957-2001). Comme ces données comportent
des erreurs d’estimation, elles ont été recalibrées en utilisant les bases de données
GPCC [Rudolf et al., 2003] et GPCP [Huffman et al., 1997] élaborées à partir de
données pluviométriques mensuelles.
Les paramètres de cette distribution sont :
- MT, le cumul annuel moyen de précipitation obtenu en un point de la base de
donnée ERA-40 ;
- β, le rapport entre le cumul annuel moyen de précipitations convectives et le
cumul annuel moyen de précipitation totale déduit de la base de données ERA-40 ;
- Pr6, la probabilité (en pourcentage) en un point d’avoir un cumul de
précipitation supérieur à 0.1 mm par tranches de 6 heures.
Synthèse des travaux
I. Distribution statistique des affaiblissements de propagation
16
Les cartes mondiales de ces paramètres sont présentées respectivement aux
figures 1, 2 et 3. Les coefficients a, b et c sont obtenus en minimisant l’écart
quadratique entre la base de données DBSG-3 de l’UIT-R de CCDF d’intensité de
précipitations issue de données pluviométriques et le modèle appliqué aux endroits où
les mesures ont été effectuées.
Figure 1: Carte mondiale du paramètre Mt
Figure 2: Carte mondiale du paramètre β
La prise en compte des cartes de paramètres géo-climatiques permet de modéliser
de manière réaliste la disparité des intensités de précipitations et de ce fait d’identifier
les zones climatiques les plus sujettes aux affaiblissements dus à la pluie, comme illustré
figure 4.
Synthèse des travaux
I. Distribution statistique des affaiblissements de propagation
17
Figure 3: Carte mondiale du paramètre Pr6
Figure 4: Carte mondiale d’intensité de précipitations dépassée durant, 0.01 % d’une
année moyenne
Synthèse des travaux
I. Distribution statistique des affaiblissements de propagation
18
3.2 Perspectives d’amélioration
Si la recommandation actuelle proposée par l’UIT-R est le modèle de référence
car simple d’utilisation, il présente quelques limitations dont certaines pourraient être
levées en exploitant mieux le potentiel de la base de données ERA-40.
Par exemple, les statistiques de précipitations de l’UIT-R sont données en termes
de « pourcentage du temps d’une année moyenne ». Ce concept d’année moyenne est
assez vague compte tenu des variabilités inter-annuelles annuelles des cumuls de
précipitations qui sont de l’ordre de 30%. Certains cycles observés pour les régimes de
précipitations présentent des durées de l’ordre d’une dizaine d’années [Poiares Batista
et al., 1987], ce qui est à rapprocher de la durée de vie des satellites géostationnaires. Il
n’est dès lors pas évident que les statistiques de précipitations calculées sur la durée de
vie du satellite soient représentatives de cette année moyenne avec les implications que
cela peut avoir sur la disponibilité résultante des liaisons, et cela sans tenir compte
d’éventuelles tendances long terme. Une analyse plus fine des variabilités inter-
annuelles des précipitations rendrait possible l'élaboration d'une marge de confiance par
rapport aux distributions calculées pour une « année moyenne ». Les quarante-trois
années de données de la base de réanalyse ERA-40 devraient permettre de quantifier
plus précisément cette variabilité, et de donner un intervalle de confiance associé à la
distribution moyenne des intensités de précipitations. Il devrait également être possible
de donner une indication sur les variations mensuelles voire diurnes de ces distributions.
4 Distribution des contenus intégrés en eau liquide
Jusqu’à présent, il n’existe pas de modèle similaire à celui proposé par l’UIT-R
pour les distributions d’intensité de précipitations, pour quantifier les distributions de
contenus intégrés en eau liquide. L’UIT-R propose des cartes statistiques de contenus
intégrés en eau liquide dépassés pour différents pourcentages du temps, issues de la base
de données de réanalyse NA-4 de l’ECMWF portant sur 2 années. Cette
recommandation UIT-R présente deux carences importantes :
- La représentativité statistique des cartes calculées sur deux années de données
est insuffisante,
Synthèse des travaux
I. Distribution statistique des affaiblissements de propagation
19
-L’absence de modèle analytique des distributions de contenus intégré en eau
liquide constitue une entrave au développement de modèles tentant de représenter des
champs de contenus intégrés en eau liquide corrélés dans l’espace et dans le temps, qui
ont besoin d’une description analytique de ce paramètre.
Pour remédier à ces défauts, la première partie de l’article « Statistical
distribution of integrated liquid water and water vapor content from meteorological
reanalysis » propose un modèle analytique de cette distribution du contenu intégré en
eau liquide reposant sur des cartes de paramètres géoclimatiques dérivés des quinze
années de données de la base de réanalyse ERA-15 de l’ECMWF. Parmi plusieurs
modèles de distributions testés, la distribution mixte Dirac log-normale est la mieux
appropriée. L’utilisation de cette distribution suppose que le contenu en eau liquide est
distribué de manière log-normale avec une probabilité Pcloud et nul avec une
la probabilité 1-Pcloud . L’expression analytique de cette distribution est la suivante :
( )
1)0(
0L if ²2
²lnexp21)( **
*
=≥
>⎥⎦⎤
⎢⎣⎡ −−=≥ ∫
+∞
LP
dlll
PLLPL
cloud σμ
σπ , (4)
où L représente le contenu intégré en eau liquide, μ, σ sont les paramètres de la
loi log-normale qui représente la distribution des contenus en eau liquide strictement
positifs. Pcloud représente la probabilité en un point d’avoir un contenu intégré en eau
liquide strictement positif. Les cartes mondiales des paramètres obtenus en régressant le
modèle (4) sur les cartes statistiques dérivées de la base de données ERA-15 sont
représentées aux figures 5, 6, 7.
Synthèse des travaux
I. Distribution statistique des affaiblissements de propagation
20
Figure 5 : Carte mondiale de Pcloud régressée à partir des données ERA-15.
Figure 6 : Carte mondiale de μ régressée à partir des données ERA-15.
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I. Distribution statistique des affaiblissements de propagation
21
Figure 7: Carte mondiale de σ regressée à partir des données ERA-15.
Le modèle de contenu intégré en eau liquide (4) combiné avec les cartes des
figures 5, 6 et 7 permet de caractériser en tout point du globe les distributions de
contenus intégrés en eau liquide et d’en déduire les atténuations sous-jacentes en
utilisant la Recommandation UIT-R P. 840-3.
5 Distribution des contenus intégrés en vapeur d’eau
La problématique rencontrée pour les distributions du contenu intégré en vapeur
d’eau est exactement la même que celle pour les distributions du contenu intégré en eau
liquide. En effet, la recommandation UIT-R P.836-3 actuelle ne propose pas de modèle
analytique et les cartes statistiques proposées ont une représentativité limitée du fait de
l’utilisation de la base de données NA-4 de l’ECMWF. La deuxième partie de
l’article “Statistical distribution of integrated liquid water and water vapor content from
meteorological reanalysis” propose un modèle de distribution pour ces contenus intégrés
en vapeur d'eau reposant sur l’utilisation de statistiques dérivées de la base de réanalyse
ERA-15. Cet article montre que la distribution la plus appropriée est une distribution de
Weibull dont la CCDF est donnée par :
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−=≥
kVVVPλ
** exp)( (5)
Synthèse des travaux
I. Distribution statistique des affaiblissements de propagation
22
où V représente le contenu intégré en vapeur d’eau, et k et λ sont les paramètres
de la loi.
Les cartes mondiales de ces paramètres ont été obtenues en minimisant l’erreur
quadratique du modèle (5) avec les distributions statistiques dérivées d'ERA-15.
Figure 8 : Cartes mondiales du paramètre λ de la loi de Weibull représentant les
contenus intégrés en vapeur d'eau.
Figure 9: Cartes mondiales du paramètre k de la loi de Weibull représentant les
contenus intégrés en vapeur d'eau
Synthèse des travaux
I. Distribution statistique des affaiblissements de propagation
23
Combinés avec les paramètres représentés sur les figures 8 et 9, le modèle (5)
permet d'obtenir en tout point du globe une estimation de la distribution des contenus
intégrés en vapeur d'eau.
La connaissance des distributions de chacun des paramètres dont découle
l'atténuation n'est pas suffisante pour décrire la combinaison des effets. A l'heure
actuelle, des méthodes statistiques approchées permettent de combiner les différents
effets afin d'obtenir la distribution de l'atténuation totale [Castanet et al., 2001].
Cependant, une estimation plus précise de cette distribution nécessiterait une meilleure
connaissance des corrélations entre les différents paramètres. Ainsi, une partie de
l'article "Distribution of tropospheric water vapor in clear and cloudy conditions from
microwave radiometric profiling", se propose d'étudier la distribution des contenus
intégrés en vapeur d'eau en présence de nuages et en air clair. Ces distributions sont
obtenues à partir de profils radiométriques pour différents sites situés dans le Sud-ouest
de la France. Elles soulignent la forte dépendance entre distribution des contenus
intégrés en vapeur d'eau et présence ou non de nuages, conformément à la figure 10.
Synthèse des travaux
I. Distribution statistique des affaiblissements de propagation
24
Figure 10 : Distribution de contenus intégrés en vapeur d'eau dérivés des données
radiométriques conditionnée à la présence de nuages, et en air clair.
L'étude de la distribution des contenus intégrés en vapeur d'eau à partir des
données du radiomètre micro-ondes de l’ONERA montre également que le modèle (5) et
le paramétrage associé est valable dans le sud-ouest de la France.
6 Conclusion
La modélisation statistique des distributions d’atténuation troposphérique et des
quantités météorologiques sous-jacentes est une thématique qui a été largement abordée
au cours des études de propagation pour les liaisons Terre-satellite opérant aux
fréquences supérieures à 10 GHz. Des modèles performants ont été élaborés pour les
Synthèse des travaux
I. Distribution statistique des affaiblissements de propagation
25
précipitations. Ils permettent de caractériser de manière relativement satisfaisante les
distributions d’intensité de précipitation et d’atténuation due à la pluie, même si la
quantification de l'incertitude sur ces distributions serait une amélioration considérable.
En effet, cela permettrait d’en déduire un intervalle de confiance pour les valeurs
d'indisponibilité des liaisons.
Les distributions statistiques des contenus intégrés en vapeur d’eau et en eau
liquide, rendues indispensables pour les études de propagation en bande Ka, mais
surtout en EHF ou dans la bande Q/V, doivent être améliorées. Un modèle, plus
pertinent que la recommandation UIT-R actuelle, a par conséquent été développé et testé
dans le cadre de ce travail de thèse. Cependant, la question de la corrélation des
différents paramètres et de leur combinaison reste ouverte.
L'amélioration des modèles décrivant les distributions des paramètres
météorologiques reste intéressante pour le dimensionnement des liaisons terrestres ou
terre-espace aux fréquences supérieures à 20 GHz. La majorité des modèles de canal
développés pour optimiser la gestion de la ressource prennent en entrée ces statistiques.
Aussi une plus grande précision sur ces distributions améliorera la précision des
modèles de canal.
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26
II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
1 Introduction
Comme expliqué dans la problématique générale, le dimensionnement et
l’optimisation des systèmes de gestion adaptatifs de la ressource, rendus nécessaires par
l’utilisation de fréquences supérieures à 20 GHz, requièrent une connaissance des
variabilités spatiales et temporelles des affaiblissements de propagation. Considérant la
diversité des techniques qui peuvent être employées pour contrer ces affaiblissements,
les besoins en termes de modélisation du canal de propagation peuvent prendre
différentes formes. Ce chapitre regroupe une description des réponses envisagées pour
répondre aux diverses problématiques posées par le dimensionnement des techniques de
compensation. Ces réponses peuvent englober l'utilisation directe de données, de
modèles statistiques ou de modélisations. Ces différents aspects sont détaillés dans les
sections suivantes. L’utilisation de données radars pour le dimensionnement d’un
système de diversité est dans un premier temps présentée. Une modélisation spatiale des
champs d’atténuation totale et une modélisation spatio-temporelle des affaiblissements
de propagation sont ensuite détaillées.
2 Etude d'un système de diversité de sites à partir de données radars
Certains problèmes liés au dimensionnement des systèmes adaptatifs de
compensation des affaiblissements peuvent être résolus par une exploitation directe des
données météorologiques existantes. Par exemple, pour le dimensionnement des
systèmes utilisant la diversité de sites, les données radars peuvent constituer une
solution relativement simple si une archive de données radars couvre la zone d'intérêt
sur une période relativement longue (au moins une année). La diversité de sites est
utilisée dans le cas où il est nécessaire d'avoir des liaisons à très haute disponibilité
(stations d’ancrage, stations de contrôle, gateways).Une ou plusieurs autres stations sont
Synthèse des travaux
II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
27
implantées, généralement à quelques dizaines de kilomètres les unes des autres, comme
illustré figure 11, si le bilan de liaison d'une station ne permet pas d'assurer la
disponibilité requise. Le but est de tirer partie de l'inhomogénéité spatiale des champs de
précipitation.
Figure 11: Principe des systèmes de diversité de sites. Dans le cas où une des stations
subit une atténuation trop forte pour assurer la liaison, l'autre station assure le relais
de l'information.
En effet, plus les stations sont éloignées les unes des autres, plus les atténuations
affectant les différentes liaisons sont décorrélées. En cas de décorrélation complète de
deux liaisons, la probabilité d'indisponibilité (probabilité que les deux liaisons subissent
une atténuation supérieure à un seuil A* du système de diversité résultant) s'exprimerait
en fonction des indisponibilités affectant les deux liaisons par :
P(A1>A*,A2>A*) = P(A1>A*)P(A2>A*) (6)
Ainsi, supposant que les deux liaisons aient une probabilité de 0.1 %, la
probabilité que les deux stations soient simultanément affectées serait de 0.0001 %. En
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28
pratique, les champs d’atténuation due aux précipitations présentent une corrélation
spatiale décroissant avec la distance. La problématique de dimensionnement se résume
donc à trouver la distance minimale entre les deux sites permettant d'assurer la
disponibilité requise. L'augmentation de disponibilité induite par l'utilisation de deux
sites (par rapport à l'utilisation d'un seul), est quantifiée par l’avantage de diversité
comme illustré figure 12. Un autre paramètre, le gain de diversité permet aussi d’estimer
les performances de la diversité de site. Il se definit comme le différentiel d’atténuation,
pour une disponibilité donnée, entre l’atténuation affectant un site et l’atténuation
affectant simultanément deux sites:
),()(),( PDAPAPDG js −= (7)
où As est l’atténuation dépassée pendant le pourcentage P du temps et Aj est
l’atténuation jointe dépassée par les deux station en diversité séparées par une distance
D pendant le pourcentage P du temps, comme illustré figure 12. Il est également
possible d’exprimer le gain de diversité G(D,As) en fonction de la distance et de
l’atténuation As dépassée pendant un pourcentage du temp P pour un des sites comme
illustré à la figure 13 du fait de la bijectivité de la relation liant les deux quantitées.
0 5 10 15 20 25 30 35 40
10-1
100
Attenuation [dB]
Per
cent
age
[%]
Spino d'Adda - f = 30 GHz - distance = 8 km
Single-site attenuation CDFJoint two-site attenuation CDF (D = 8 km)
PG
AsAj
Figure 12 : Lien entre gain de diversité, loi jointe d’atténuation et loi d’atténuation sur
un site
Avantage
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29
La connaissance du gain de diversité en fonction de la distance permet
d'optimiser la distance nécessaire entre deux sites afin d'assurer la disponibilité requise.
Les données radar converties en atténuation grâce à la recommandation UIT-R
P.618 permettent d'estimer la loi jointe d'atténuation simultanément dépassée pour deux
sites séparés d'une distance D, ainsi que la distribution d'atténuation pour un seul site.
Par conséquent, il est possible de calculer le gain de diversité en fonction de la distance
comme illustré sur la figure 13 où le gain de diversité a été calculé à partir d'une année
de données du radar météorologique du réseau ARAMIS de Météo-France situé à
Bordeaux.
0 10 20 30 40 50 600
5
10
15
20
25
30
Distance [km]
Gai
n [d
B]
Bordeaux
4 dB 8 dB12 dB16 dB20 dB24 dB28 dB32 dB
Figure 13: Gain de diversité fonction de la distance, calculé à partir des données radars
(lignes pleines) et à partir de la recommandation UIT-R P.618 (lignes pointillées) pour
différentes marges d’atténuation sur une liaison.
La figure 13 montre qu'il existe des différences importantes entre le gain de
diversité dérivé des données radars et le modèle statistique de l'UIT-R. Les possibilités
offertes par les données de radars météorologiques sont détaillées dans l'article “Use of
weather radar data for site diversity predictions and impact of rain field advection”,
présent dans la deuxième partie de ce manuscrit.
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L'utilisation d'une base de données de champs de précipitation observés par des
radars météorologiques constitue certainement la meilleure solution pour optimiser les
mécanismes de gestion adaptatifs de la ressource, même si les problèmes inhérents à la
mesure radar peuvent donner lieu à des pré-traitements relativement complexes.
Cependant, la couverture des réseaux radars est relativement restreinte à l'échelle
mondiale, et de qualité inhomogène (présence d’échos de sols, méthodes de calibration
différentes, résolutions différentes…). Cela implique un recours à la modélisation pour
pallier ce manque de données disponibles. Le réseau américain NEXRAD [Klazura et
Imy, 1993] et, dans une moindre mesure, les réseaux européens homogénéisés dans le
cadre du projet en cours OPERA2 [Huuskonen A. et al., 2007], constituent néanmoins
des entrées de première importance pour la modélisation.
3 Modélisation spatiale des champs de précipitations
3.1 Besoins
Les données radars existantes étant trop peu nombreuses, il est nécessaire de
modéliser les champs d'atténuation afin de servir d'entrée aux simulateurs systèmes. Ces
champs doivent être générés sur la couverture continentale du satellite de
télécommunication afin de simuler de manière aussi réaliste que possible l’allocation de
la ressource embarquée à l’intérieur d’un groupe d’utilisateurs ou d’un spot, mais aussi
entre les différents groupes d’utilisateurs ou entre spots. Par conséquent, il est important
que les champs d’atténuation et de précipitation simulés soient représentatifs des
caractéristiques statistiques déduites des données ayant servi de base à la modélisation.
Ainsi, il est nécessaire que les distributions d'atténuation en un point, calculées à partir
d'un grand nombre de simulations, soient aussi proches que possible de celles évaluées à
partir de jeux de données indépendants. La répartition spatiale des affaiblissements doit
elle aussi être représentative de celle observée sur les données.
D’autre part, considérant l'évolution des systèmes de télécommunication vers
l’utilisation de bandes de fréquences de plus en plus élevées, il n'est plus possible de
s'intéresser uniquement aux atténuations dues aux précipitations car l'importance des
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atténuations dues aux nuages et à la vapeur d'eau est croissante. Même si comme illustré
figure 1, l’atténuation due aux nuages et à la vapeur d’eau en l’absence de précipitation
est de l’ordre de 3-4 dB à 40GHz, la compensation adaptatives de ces effets, eux aussi
extrêmement variables dans l’espace, peut permettre d’augmenter de manière
significative les performances des systèmes de télécommunications à ces bandes de
fréquence. En effet l’utilisation de modulations efficaces en air clair peut augmenter
sensiblement les capacités de transport globales du système.
Aussi, une méthode de génération de champs d'atténuation totale à l'échelle
continentale est présentée dans cette section.
3.2 Modélisation proposée
La modélisation proposée [Jeannin et al., 2006a, 2006b] pour répondre aux
besoins s'appuie sur la méthodologie HYCELL de génération de champs de
précipitations spatialement corrélés à l’échelle de la France [Féral et al. 2003a, 2003b,
2006]. Le modèle développé permet de générer des champs de paramètres
météorologiques (précipitation, eau liquide, eau vapeur) à l’échelle continentale sur
l’Europe, les États-Unis, ou l’Afrique respectant la climatologie locale et de les
transformer, considérant une liaison radioélectrique donnée, en champs d’atténuation
totale. Cette section reprend les diverses étapes ayant permis de s’acheminer vers
l’élaboration de ce modèle.
3.2.1 HYCELL
Les principales hypothèses du modèle HYCELL développé dans [Féral et al.
2003a, 2003b] sont rappelées dans cette section. Cette méthodologie, établie à partir de
l'étude de données radars en région tempérée, permet de générer des champs de
précipitations spatialement corrélés à moyenne échelle (~150×150 km²), connaissant la
proportion de la zone de moyenne échelle affectée par des précipitations.
Cette méthode repose sur la modélisation des champs de précipitations comme
une conglomération de cellules convectives. Les cellules de précipitation sont
modélisées par un cœur convectif à décroissance Gaussienne et un étalement stratiforme
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à décroissance exponentielle comme illustré figure 14. Les cellules de précipitation sont
supposées de section horizontale circulaire. En effet, si d’un point de vue
observationnel, la majorité des cellules de pluie présente plutôt une forme elliptique,
l’équiprobabilité de leur distribution conduit finalement à considérer des cellules de
forme circulaire.
Figure 14 : Modélisation d'une cellule de précipitation par HYCELL.
Comme illustré sur la figure 14, la décroissance de la partie Gaussienne de la
cellule est caractérisée par le paramètre aG. La décroissance de la partie exponentielle
est quant à elle caractérisée par le paramètre aE. L’intensité de précipitations maximale
de la cellule est atteinte pour l’intensité pic de la partie gaussienne RG. L’intensité de
précipitation notée R1 sur la figure 14 est l’intensité de précipitation limite entre zones
de définition Gaussienne et exponentielle. Les intensités de précipitation inférieures à
Rm=1 mm h-1 (valeur conforme à la littérature permettant d’isoler les cellules de pluie de
leur environnement stratiforme) ne sont pas représentées par HYCELL. Le paramètre RE
correspond à l’intensité de précipitation maximale de la cellule si la cellule était
uniquement modélisée par une exponentielle.
Une étude des cellules de précipitation observées par radar en zone tempérée a
permis d’établir un lien statistique entre leur surface à Rm=1 mm h-1 et leur intensité pic.
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Par ailleurs, une dépendance statistique entre les paramètres RG et R1 de la modélisation
HYCELL a également été mise en évidence. Compte tenu des relations existant entre les
différents paramètres (équation de continuité en R=R1), un paramétrage générique des
cellules a ainsi pu être défini à partir de la connaissance de la surface de la cellule à Rm
et de l’intensité de précipitation maximale RG.
En utilisant cette modélisation générique de la cellule de précipitation, une
méthode de génération de champs de précipitation sur une zone de simulation de
moyenne échelle (150×150 km²) a été développée [Féral et al., 2003b]. Connaissant la
distribution cumulative (CDF) locale de l’intensité de précipitation ainsi que la fraction
de la zone de moyenne échelle affectée par les précipitations, les cellules de pluie sont
générées sur cette zone de simulation de manière itérative, par intensité de précipitation
décroissante. La surface occupée à Rm=1 mm.h-1 par chacune des cellules est tirée
suivant une distribution exponentielle conformément aux études déjà menées [Denis et
Fernald, 1963], [Goldhirsh et Musiani, 1986], [Mesnard et Sauvageot, 2003]. La surface
totale occupée à la fin du processus par l’ensemble des cellules générées correspond à la
surface de la zone de moyenne échelle affectée par des précipitations supérieures à 1
mm.h-1. La modélisation HYCELL de chaque cellule est conduite de manière à ce que la
distribution spatiale des intensités de précipitation reproduise la distribution d’intensité
de précipitation temporelle locale conformément à l’hypothèse d’ergodicité des champs
de pluie formulée dans [Nzeukou et Sauvageot, 2002].
Afin de localiser les cellules ainsi générées à l’intérieur de la zone de moyenne
échelle, les cellules sont positionnées par une marche aléatoire comme illustré figure 15.
A partir d’observations radar, il a été proposé de modéliser la localisation spatiale des
cellules de pluie à moyenne échelle par agrégats. Leurs caractéristiques statistiques en
termes de nombre de cellules par agrégat, de distances intercellulaires ou de distance
inter-agrégats ont été mises en évidence à partir des données du radar de Bordeaux
[Féral et al., 2003b]. Dés lors, la marche aléatoire permet de localiser les agrégats et les
différentes cellules qui les composent dans une zone de simulation de moyenne échelle.
Considérant un grand nombre de champs de précipitation générés avec la marche
aléatoire, la distribution des distances intercellulaires obtenue reproduit celle dérivée des
observations radar. Cette procédure de localisation des cellules introduit donc une
corrélation spatiale sur les champs de précipitations simulés.
Synthèse des travaux
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34
Figure 15 : A gauche, champ de précipitation modélisé par Hycell avec une localisation
complètement aléatoire des cellules. A droite, champ de précipitation dont les cellules
ont été localisées par la marche aléatoire.
3.2.2 Extension de la zone de simulation à l’échelle de la France
Afin d’étendre cette modélisation à l’échelle de la France métropolitaine, [Féral
et al., 2006] proposent de coupler cette modélisation cellulaire du champ de
précipitation à moyenne échelle avec une modélisation spatiale des zones précipitantes
(R>0 mm.h-1) à l’échelle de la France. Considérant les observations du réseau de radar
ARAMIS de Météo France, il a été montré que les zones affectées par les précipitations
(R>0 mm.h-1) à l’échelle de la France pouvaient être correctement représentées par un
champ aléatoire stationnaire Gaussien G seuillé, de corrélation :
)))sin()cos(())sin()cos((exp(),( 22
XYG L
xyL
yxyxc θθθθ −+
+−= , (8)
avec LX =200 km, LY=100 km et θ est l'orientation de la structure frontale associée.
Cette fonction de corrélation a été déterminée en définissant une relation entre la
corrélation des champs binaires de précipitation (pluie/non-pluie) calculée à partir des
données radar et la corrélation du champ Gaussien sous-jacent. A partir des observations
ARAMIS sur la France, la distribution de probabilité de la fraction f du territoire
national affectée par la pluie a été dérivée. Celle-ci a été modélisée par une loi
exponentielle de moyenne 8.8%. Afin de localiser les zones affectées par des
Synthèse des travaux
II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
35
précipitations sur les simulations, le champ Gaussien G est transformé par seuillage en
un champ binaire de précipitation B de manière à ce que la fraction de la zone affectée
par des précipitations soit f conformément à la figure 16.
Figure 16 : Champ Gaussien généré sur la France et Champ binaire de précipitation
seuillé afin d’obtenir une occupation des précipitations de f=15%.
Le couplage avec la modélisatin HYCELL est immédiat dès lors que le champ de
précipitation binaire à large échelle est découpé en sous-zones de moyenne échelle
conformément à la figure 17. Finalement, cette approche permet de générer des champs
de précipitation corrélés dans l’espace à l’échelle de la France ( 224HFigure 17).
Figure 17 : Division de la couverture en sous zones de moyenne échelle et couplage
avec la modélisation cellulaire pour chaque zone de moyenne échelle
θ
x
O
x’
y
y’
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II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
36
Toutefois, la modélisation des zones affectées par les précipitations n’est pas
entièrement satisfaisante car elle ne permet pas de prendre en compte la variabilité de la
probabilité d’apparition de la pluie à l’échelle de la France laquelle présente pourtant
des disparités importantes. De ce fait, il n’est pas possible d’étendre cette modélisation à
l’échelle d’une couverture satellitaire (i.e. échelle continentale).
3.2.3 Extension de la zone de simulation à l’échelle d’une couverture satellitaire
L’inhomogénéité spatiale des champs de précipitation (illustrée figure 18) ne
permet pas d’appliquer la modélisation proposée ci-dessus à l’ensemble d’une
couverture satellitaire. Une modification de cette modélisation a par conséquent été
proposée [Jeannin et al., 2006a] afin de tenir compte de cette variabilité.
Figure 18: Probabilité d’apparition de la pluie sur l’Europe dérivée de la Recommandation
UIT-R P.837
La conversion du champ aléatoire Gaussien G en un champ binaire de précipitation B est
maintenant réalisée en utilisant un seuil dépendant en tout point de la probabilité locale
d’apparition de la pluie. Ainsi, si de nombreux champs sont générés, la probabilité
d’obtenir des précipitations en un point des simulations tendra vers la probabilité
d’apparition de la pluie de la Rec UIT-R P.837. Si ce seuil tenant compte de la
probabilité d’apparition de la pluie permet de s’affranchir du problème posé par
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II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
37
l’inhomogénéité des probabilités d’apparition de la pluie, la fraction de la zone affectée
par les précipitations n’est elle plus contrôlée. En utilisant la fonction de corrélation
donnée par (8), il apparaît que la fraction de la zone de couverture affectée par la pluie
sur l’ensemble de la couverture est très peu variable, proche de la moyenne des
probabilités d’apparition de la pluie sur la zone, alors qu’une étude des champs de
précipitations à partir de données TRMM (Tropical Rainfall Measurement Mission)
[Kummerow et al., 1998] montre qu’il existe une variabilité importante de ce paramètre
[Jeannin et al., 2006b]. Afin d’une part de reproduire la probabilité d’apparition de la
pluie en tout point et d’autre part d’approcher la distribution de la fraction de la
couverture occupée par les précipitations, une modification de la fonction de corrélation
(8) a été proposée :
2
2222
1
)800
exp())130
)sin()cos(()200
)sin()cos((exp(),(
Z
Z
GL
yxLxyyx
yxc+
+−+
−+
+−
=
θθθθ
, (9)
.
avec LZ valant 0.25 sur l’Europe et 0.20 sur l’Amérique du Nord. L’introduction d’une
composante de corrélation à décroissance lente permet d’augmenter la variabilité de la
distribution de la fraction de la zone de couverture affectée par les précipitations comme
expliqué plus en détail dans l’article « Statistical Distribution of the fractional area
affected by rain » inclus dans la deuxième partie de ce manuscrit. Le paramètre LZ a été
ajusté de manière à reproduire aussi fidèlement que possible la distribution de la fraction
de la zone affectée par la pluie déduite des données TRMM. Ainsi l’utilisation de la
forme de corrélation (9) permet à la fois de reproduire en tout point la probabilité
d’apparition de la pluie dérivée de la recommandation ITU-R P.837 ainsi que la
distribution de la fraction de la zone de couverture affectée par les précipitations déduite
des observations TRMM.
Une fois le champ binaire de précipitation généré, la méthodologie Hycell est appliquée
sur chaque sous zone de moyenne échelle de (150×150 km2) conformément à la 225HFigure
17, ce qui permet de générer des champs de précipitation sur l’ensemble de la couverture
satellite comme illustré figure 19.
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II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
38
Figure 19: Champ de précipitation généré sur l’Europe avec la méthodologie HYCELL.
Cette modélisation a été utilisée pour générer des champs de précipitation
spatialement corrélés sur l’Europe, l’Amérique du Nord et l’Afrique. Afin de tenir
compte d’une possible dépendance climatique de la modélisation cellulaire, les relations
statistiques utilisées pour définir la forme générique des cellules ainsi que pour le
paramétrage de la marche aléatoire ont été étudiées sur différents jeux de données radar.
Des études ont ainsi été menées sur des couvertures radar en Europe (France,
Scandinavie, Europe centrale), aux États-Unis, (Louisiane, Floride, Californie, New
Jersey, Dakota), et en zone tropicale (Martinique, Sénégal). Une modélisation un peu
différente pour prendre en compte la saisonnalité extrêmement marqué des régimes de
précipitation sur l’Afrique a été proposée. Ces différents points sont présentés à la
section 226H 3.3.
3.2.4 Génération des champs de contenus intégrés en vapeur d’eau et en eau liquide
Afin de prendre en compte également les affaiblissements dus à la vapeur d'eau et
aux nuages, importants aux bandes de fréquences supérieures à 20 GHz, des champs de
contenus intégrés en eau liquide et en vapeur d’eau corrélés aux champs de précipitation
doivent être générés. La solution retenue, développée dans le cadre de ce travail de
Synthèse des travaux
II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
39
thèse, repose sur une modélisation des champs météorologiques d’intérêt par des champs
aléatoires stationnaires. La méthodologie de génération des champs est similaire à celle
utilisée pour générer les champs binaires de précipitation. Un champ gaussien
stationnaire est généré dans l'espace et, en tout point, la variable gaussienne tirée dans
une loi normale centrée réduite est convertie, en utilisant une équivalence des fonctions
de répartition, en une variable suivant la distribution des contenus intégrés en eau
liquide des nuages ou en vapeur d'eau, présentées au chapitre précédent. Dès lors, reste à
déterminer la corrélation spatiale de ces champs. Des données de contenus intégrés en
vapeur d'eau issues des réanalyses ERA-40 ont été utilisées pour déterminer la fonction
de corrélation du champ aléatoire gaussien générateur des champs de contenus intégrés
en vapeur d'eau. Quant à la fonction de corrélation des contenus en eau liquide, elle a été
obtenue à partir de données infrarouges de Météosat seuillées à une température de
brillance de -38°C. Une forme analytique approchée de ces fonctions de corrélation a été
déterminée. Pour les contenus en vapeur d'eau la fonction de corrélation du champ
Gaussien utilisée est:
))385
)sin()cos(()565
)sin()cos((exp(),( 22 θθθθ xyyxyxcG−
++
−= , (10)
où l'angle θ représente l'orientation de la structure frontale associée. Pour les champs de
contenu intégré en eau liquide des nuages, la fonction de corrélation déterminée a été
trouvée identique à celle déterminée pour la génération des champs gaussiens
générateurs des champs binaires de précipitation (9).
Pour coupler les champs de contenus intégrés en vapeur d'eau et les champs de
contenus en eau liquide aux champs de précipitations, il est supposé qu'il existe une
corrélation maximale entre contenus en eau liquide des nuages, contenus en vapeur d'eau
et intensité de précipitation. Il n’existe pas de données pour étayer cette hypothèse.
Cependant les mesures radiométriques simultanées, de contenu en eau nuageuse, de
contenu en vapeur d’eau et de présence de précipitation présentées sur la figure 1 de
l’article «Distribution of tropospheric water vapor in clear and cloudy conditions from
microwave radiometric profiling » montre que cette corrélation est importante, puisque
l’apparition de précipitation est associée à une augmentation de contenu en eau liquide
et de contenu en vapeur d’eau. Faute d’autres données, cette modélisation simpliste a été
adoptée.
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II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
40
Figure 20 : Génération des champs aléatoires générateurs des divers champs
météorologiques d’intérêt.
Ce couplage est obtenu en utilisant la même graine aléatoire pour la génération
des différents champs gaussiens comme illustré à la figure 20. Les champs de contenu
intégré en eau liquide englobent par construction les zones de précipitations (figure 21)
car la probabilité d’avoir un contenu en eau liquide positif est supérieure à celle d’avoir
des précipitations.
Graine aléatoire
Corrélation (10)
Corrélation (9)
Champ gaussien générateur d’un champ de contenu intégré en
vapeur d’eau
Champ gaussien générateur d’un champ binaire de précipitation et d’un
champ de contenu intégré en eau liquide
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41
Figure 21: Champ de précipitation (rouge) et couvert nuageux avec eau liquide(bleu)
Le couplage des différents champs permet d'obtenir des champs d'atténuation
totale comme illustré à la figure 22.
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42
c
Figure 22 : Exemple de champs météorologiques générés sur une zone de moyenne
échelle et champs d’atténuation résultant. De gauche à droite et de bas en haut: champs
de précipitation, de contenu intégré en eau liquide, de contenu intégré en vapeur d'eau,
d’atténuation due aux précipitations, d'atténuation due aux nuages et à la vapeur d’eau,
champ d’atténuation due à l’oxygène et enfin, champ d’atténuation totale.
La méthodologie présentée permet de générer des champs d’atténuation totale
statistiquement réalistes à l’échelle de la couverture continentale d’un satellite de
télécommunication. Le réalisme de ces simulations est évalué dans la section suivante
permettant ainsi d’identifier les limites d’utilisation de ce modèle.
Champs d’atténuation
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II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
43
3.3 Évaluation de la pertinence des résultats obtenus
Pour évaluer le réalisme de la méthodologie de simulation employée, les
caractéristiques statistiques des divers champs générés ont été comparées à celles de
données réelles issues de radars météorologiques ou de données de télédétection. Pour
cela une base de données de 100 000 simulations a été générée sur diverses zones
climatiques [Jeannin et al., 2006b].
Les hypothèses relatives au modèle cellulaire ont été évaluées sur des données
radar européennes, américaines, africaines et de Martinique. Elles se révèlent
appropriées quelle que soit la zone climatique considérée. Ceci est illustré à la figure 23
pour la distribution des distances intercellulaires et à la figure 24 pour la distribution des
diamètres des cellules de précipitation.
Figure 23 : Distribution des distances intercellulaires pour la Martinique, Dakar et Bordeaux
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Figure 24 : Distribution du diamètre des cellules observées sur différentes couvertures
radar des Etats-Unis
La distribution d'atténuation totale en chaque point approche précisément la
distribution d'atténuation totale dérivée de la recommandation ITU-R P.618-9. Un
exemple est donné à la figure 26.
Figure 25: CDF d’atténuation totale 30 GHz stimulant une liaison avec OLYMPUS.
Synthèse des travaux
II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
45
La description de la variabilité spatiale des champs d’atténuation par le modèle
semble cohérente comme l’atteste la figure 26.
Figure 26: Gain de diversité en fonction de la distance calculé sur New York en
simulant une liaison avec ACTS à 27.5 GHz à partir de données radars, de simulations,
et de différents modèles pour une 0.03% du temps
Des validations de la fonction de corrélation utilisée pour la génération des
champs binaires de précipitation ont aussi été entreprises. Les distributions
d'occupations spatiales des précipitations ont également été comparées à celles dérivées
des données radars. Toutes ces validations sont expliquées plus en détail dans [Jeannin
et al., 2006b].
Afin de s'adapter aux caractéristiques des pluies tropicales, des probabilités
mensuelles de précipitations ont été calculées sur l'Afrique à partir des données TRMM
(voir figure 27). La prise en compte de ces probabilités mensuelles de précipitation
permet de modéliser le phénomène de mousson, caractéristique des régimes de
précipitation des zones équatoriales et tropicales.
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46
Synthèse des travaux
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47
Figure 27: Cartes mensuelles de probabilité de précipitation déduites des données TRMM et de la Recommandation ITU-R P.837
Synthèse des travaux
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48
3.4 Limitations
Diverses limitations sont apparues lors de l’utilisation et de la validation de ce
modèle.
-La première est inhérente à la modélisation cellulaire employée: les intensités de
précipitation générées sont uniquement celles supérieures à 1 mm.h-1, faussant donc le
réalisme des simulations pour les faibles valeurs d’atténuation.
-L'hypothèse de stationnarité pour des champs aléatoires sur de larges étendues
est peu réaliste
-La complexité du modèle cellulaire induit des temps de calcul très importants.
-Le nombre de paramètres du modèle cellulaire HYCELL rend difficile une
inclusion de la dimension temporelle dans la modélisation.
Du fait de ces diverses limitations, l'approche suivie pour la modélisation spatio-
temporelle des affaiblissements de propagation dus aux précipitations a reposé sur
d'autres fondements théoriques.
4 Modélisation spatio-temporelle des affaiblissements de propagation
4.1 Problématique
Le dimensionnement des dispositifs de gestion adaptative de la ressource
embarquée utilisant des codages et modulations adaptatifs, nécessite d'avoir une
représentation finedu canal de propagation d’une part dans le temps, afin de pouvoir
commuter d'un mode à l'autre et d’autre part dans l'espace, afin de répartir les porteuses
ou les paquets de données aux utilisateurs, au mieux des capacités du satellite et du
réseau. Peu de modèles existent dans la littérature pour générer des conditions de
propagation corrélées dans l'espace et dans le temps [Gremont et Filip, 2004],
[Bertorelli et Paraboni, 2005]. De plus ces modèles ne permettent pas de simuler des
conditions de propagation réalistes sur de larges étendues et des périodes suffisemment
longues pour obtenir des statistiques stables. Le modèle de génération de champs
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49
d'atténuation due à la pluie corrélés dans l'espace et dans le temps présenté en détail
dans "A large scale space-time stochastic model of rain attenuation for the design and
optimization of adaptive satellite communication systems operating between 10 and
50GHz " a pour objet de simuler des conditions de propagation corrélées dans le temps
et dans l'espace à l'échelle d’une couverture satellitaire continentale. La modélisation
s'appuie sur une modélisation existante [Bell, 1987] des champs de précipitation par des
champs aléatoires corrélés dans l'espace et dans le temps. L'évolution majeure apportée
à cette modélisation est l'extension du domaine de validité dans l'espace (à l’échelle
continentale putôt que sur des zones de 100x100 km2) et dans le temps (jusqu’à
plusieurs années plutôt que sur des durées de quelques heures) en contraignant le modèle
à reproduire les sorties du modèle de réanalyse météorologique ERA-40. Une
méthodologie de paramétrage de la corrélation du modèle à partir de jeux de données
radar est également présentée.
4.2 Méthodologie
4.2.1 Modélisation des champs de précipitations par des champs aléatoires
La modélisation proposée dans [Bell, 1987], repose sur la modélisation de
champs de précipitation corrélés dans l'espace et dans le temps R(x,t) par une
transformation non linéaire d'un champ gaussien stationnaire G(x,t) lui-même corrélé
dans l'espace et dans le temps tel que :
)),((),( txGtxR ψ= . (11)
La transformation ψ est paramétrée de telle manière que la distribution statistique
des intensités de précipitations strictement positives générées soit log-normale.
L'évolution temporelle du champ est régie par une évolution des fréquences du
spectre spatial des précipitations suivant un processus de Markov. Le champ de
précipitation est également soumis à une advection prise en compte par une rotation de
phase dans le domaine de Fourier du champ aléatoire Gaussien. Le paramétrage du
modèle a été effectué à partir des données du radar de Bordeaux.
Cette modélisation répond aux besoins exprimés de génération de champs de
précipitations corrélés dans l’espace et dans le temps. Cependant, l'hypothèse de
stationnarité des champs limite le réalisme de la modélisation à des zones de quelques
Synthèse des travaux
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50
centaines de kilomètres de côté. L'évolution temporelle des champs de précipitation
manque elle aussi de réalisme au-delà de quelques heures. En effet, la fraction de la
zone affectée par la pluie reste aproximativement constante au cours des simulations ce
qui empèche de modéliser l'alternance d'évènements pluvieux et de périodes de ciel
clair. De plus la direction et la vitesse d'advection restent constantes contrairement aux
observations. Ces limitations sur l’extension spatiale et temporelle du domaine de
simulation, restreignent en l'état l'intérêt de ce modèle pour les études de propagation.
4.2.2 Couplage de la modélisation avec des sorties de modèles de réanalyse
Afin de lever les limitations sur le domaine de validité des simulations, dans
l'espace et dans le temps, une méthode pour contraindre l'évolution du modèle par des
sorties de données de réanalyses météorologiques a été développée. Elle s'articule autour
de deux axes:
-Une relation statistique connue sous le nom d'aires fractionnaires [Donneaud et
al., 1984, Chiu 1988, Eltahir et Bras 1993, Sauvageot 1994] est utilisée pour déduire la
fraction d'une cellule de résolution du modèle de réanalyse affectée par la pluie
-Une relation théorique est établie en utilisant la modélisation de [Bell, 1987]
entre la fraction d'une zone de simulation affectée par la pluie et la moyenne du champ
générateur Gaussien. La justification de cette relation est dévelopée dans l'article
"Statistical distribution of the fractional area affected by rain".
Ainsi, considérant une série temporelle de cumul de précipitation, il est possible
de déduire une série temporelle de la valeur moyenne du champ aléatoire comme illustré
à la figure 28. En utilisant la méthodologie de [Bell, 1987] et en contraignant la
moyenne du champ aléatoire, il est possible de générer des champs d'atténuation corrélés
dans l'espace à l’échelle continentale et dans le temps sur de très longues durées
(plusieurs années).
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51
Figure 28: Déduction de la série temporelle de la moyenne du champ Gaussien à partir
des cumuls de précipitation de la base de donnée ERA-40
Les simulations ont alors la taille de la cellule de résolution des données ERA-40
utilisées à savoir 2.5°x2.5°. Pour obtenir une simulation sur de larges étendues la
méthodologie est appliquée à plusieurs cellules de résolution de la base de données
ERA-40. Une méthode pour assurer la continuité des champs générés sans en modifier
les caractéristiques statistiques a été dévelopée et permet de générer des champs sur
l'ensemble d'une couverture satellite comme montré à la figure 29. L'advection des
champs de précipitation est également paramétrée à partir des données ERA-40.
Effectivement, il est montré dans l'article "Use of weather radar data for site diversity
predictions and impact of rain field advection" que le vent horizontal extrait au niveau
700 hPa de la base de donnée ERA-40 est fortement corrélé en terme de vitesse et de
direction à l'advection des précipitations.
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52
Figure 29: Exemple de champ de précipitation généré sur la couverture Européenne et
sorties concurrentes du modèle ERA-40.
4.2.3 Génération de séries temporelles d'atténuation corrélées dans l'espace et dans le temps
La procédure de conversion des champs de précipitations en champ d'atténuation
est détaillée dans [Féral et al., 2006]. Cette conversion en atténuation se fait par
intégration de l'atténuation spécifique le long du trajet de la liaison sous la pluie. Celui-
ci dépend de la hauteur de pluie qui est assimilée à la hauteur de l'isotherme -2oC
(Conformément à la Recommandation UIT-R P.839) qui peut être déduite des profils de
température de la base de donnée ERA-40 concurrents au champ de précipitation généré.
A ce stade les champs d'atténuation dus aux précipitations sont générés avec une
résolution spatiale de 1 km et une résolution temporelle de 0.1 h, comparable à la
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II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
53
résolution des jeux de données radar utilisés pour développer le modèle. Toutefois, cette
résolution temporelle est insuffisante pour certaines applications puisque les fluctuations
rapides du canal de propagation peuvent avoir une influence significative sur le
dimensionnement des dispositifs de codage ou de modulation adaptatifs ou sur les règles
de commutation d’un système utilisant la diversité de sites. Pour cette raison le modèle
d'interpolation stochastique "on demand" [Lacoste et al., 2006, Satnex 2008] est utilisé
pour améliorer la résolution temporelle des simulations. Comme illustré à la figure 30,
ce modèle est appliqué aux séries temporelles d'atténuation à basse résolution (0.1 h).
Sur la série temporelle basse résolution, les évènements de précipitation (échantillons
consécutifs d'atténuation strictement positive) sont isolés. Pour chacun de ces
événements, la méthodologie d'interpolation stochastique du modèle on-demand est
appliquée comme illustré à la figure 30. Cette méthode d’interpolation stochastique
repose sur une hypothèse de Markovianité du canal de propagation.
Figure 30 : Interpolation haute résolution des séries temporelles d’atténuation faible
résolution (points rouges)
En localisant plusieurs points dans la zone de simulation, correspondant chacun à
la position d'un terminal, et en appliquant la procédure décrite ci dessous, des séries
temporelles corrélées dans l'espace et dans le temps sont générées, comme illustré à la
figure 31.
Synthèse des travaux
II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
54
Figure 31: Exemple de simulation de champs sur plusieurs heures et couplage des
séries temporelles obtenues sur différents sites à faible résolution avec le modèle
stochastique de canal
Synthèse des travaux
II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
55
4.3 Comparaison des caractéristiques des simulations à des données indépendantes
Les caractéristiques statistiques des simulations ont été confrontées à des
statistiques déduites soit de campagnes de mesures d’affaiblissement, soit de données
radars.
Par exemple, considérant la campagne de mesure de diversité sur deux sites
effectuée par le CNET (Ex France Télécom R&D) près de Paris [OPEX, 1994], des
séries temporelles de 9 mois d’affaiblissements dus à la pluie ont été générées avec les
mêmes caractéristiques (durée, localisation, caractéristiques de la liaison). Les CCDF
d’atténuation obtenues à partir des simulations et des données radar sont présentées à la
figure 32.
Figure 32: CCDF d’atténuation obtenues lors des mesures de la campagne OLYMPUS
et à partir des simulations (Fréquence 20GHz)
La figure 32 montre que les simulations reproduisent de manière convenable les
distributions d’atténuation expérimentales. Afin d’évaluer la capacité de la modélisation
à reproduire la variabilité spatiale des affaiblissements, la loi jointe d’atténuation
simultanément dépassée sur les deux sites déduite des séries temporelles simulées est
Synthèse des travaux
II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
56
comparée à celle dérivée de la campagne de mesure. La figure 33 présente les résultats
obtenus et montre l’adéquation entre simulation et expérimentation.
Figure 33: CCDF d’atténuation simultanément dépassée obtenues à partir des données
de la campagne OLYMPUS et à partir des simulations
Des comparaisons simulations / expérimentations sont présentées de manière plus
exhaustive dans la dernière section de l'article "A large scale space-time stochastic
model of rain attenuation for the design and optimization of adaptive satellite
communication systems operating between 10 and 50 GHz".
5 Conclusion
Le modèle de génération de champs d'atténuation corrélés dans l'espace et dans le
temps fournit les entrées requises pour les simulations des dispositifs de gestion
adaptative de la ressource. En effet, il permet de générer des séries temporelles
d'atténuation sur de longues durées (plusieurs années), en reflétant les fluctuations
rapides du canal (résolution temporelle de 1 s), et sur l'étendue d'une couverture satellite
(échelle continentale). La modélisation proposée repose sur une modélisation existante
Synthèse des travaux
II. Étude de la variabilité spatiale et temporelle des affaiblissements de propagation
57
des champs de précipitation à moyenne échelle contrainte par des sorties de modèles de
réanalyses météorologiques. Cette utilisation des données de réanalyse peut permettre
une étude des champs d’atténuation en fonction des conditions météorologiques. Il est
ainsi envisageable de simuler les cas les plus défavorables, d'étudier les variations
diurnes ou saisonnières des précipitations.
Diverses caractéristiques statistiques de simulation ont été comparées à celles de
jeux de données indépendants, accréditant ainsi le réalisme des simulations effectuées.
Différents points restent à approfondir pour fournir une modélisation complètement
satisfaisante. En particulier la sensibilité du paramétrage des corrélations vis-à-vis de la
climatologie locale n'est à l'heure actuelle pas évaluée de manière satisfaisante. Il de
plus est vraisemblable que la méthodologie proposée ne permettra pas de modéliser
convenablement les systèmes précipitants aux faibles latitudes. En effet les cyclones
tropicaux ou les lignes de grains caractéristiques des régimes de précipitation à ces
latitudes présentent une organisation spatiale qui ne peut pas être représentée de manière
réaliste par un processus aléatoire stationnaire. L’impact de ce manque de réalisme des
simulations aux basses latitudes doit être évalué.
D'autre part, des efforts restent à fournir afin d'inclure la scintillation, les
couverts nuageux et la vapeur d'eau dans la modélisation afin d'avoir une représentation
de l'affaiblissement total. Le point délicat reste la corrélation entre les différents effets
qui pourra vraisemblablement être étudiée à partir de l'observation simultanée de radar
et de radiomètres spatio-portés comme ceux embarqués sur le satellite MODIS.
Synthèse des travaux
Conclusion et perspectives
58
Conclusion et perspectives
Cette thèse a été centrée sur la modélisation spatio-temporelle des
affaiblissements de propagation à partir de diverses données météorologiques. Cette
modélisation a requis dans un premier temps l'extraction des différentes caractéristiques
statistiques des affaiblissements de propagation. Après une étude de la variabilité
spatiale des affaiblissements, un couplage entre la dimension spatiale et la dimension
temporelle a été entrepris. Comme le modèle de représentation des variabilités spatiales
des affaiblissements développés dans un premier temps a été jugé inadapté pour
l'adjonction de la dimension temporelle, une modélisation couplant une représentation
stochastique des champs d'affaiblissement avec des données de réanalyse
météorologique a par la suite été développée. Les sorties de ce modèle peuvent être
directement utilisées pour des simulations de dispositifs de gestion adaptative de la
ressource même si l'atténuation due à l'eau liquide et à la vapeur d'eau font encore
défaut. Le modèle développé devrait ainsi permettre de répondre à certains problèmes
posés par le dimensionnement des systèmes de gestion adaptative de la ressource
embarquée.
L'utilisation de données de réanalyse météorologique ouvre des perspectives
intéressantes pour la simulation des conditions de propagation sur des situations ciblées
qui peuvent être intéressantes d'un point de vue de la gestion de la ressource ainsi que
pour la modélisation des systèmes précipitants.
Dans la continuité des travaux présentés, différents axes d'étude pourraient
enrichir les modélisations déjà développées et constituer des compléments intéressants.
Toujours orienté vers la même problématique des télécommunications par
satellite, l'exploitation de nouvelles données issues de campagnes de mesures de
propagation rendra possible un affinement des modèles.
- L'expérience GSAT-4 [Katti et al., 2007] (charge utile de propagation en bande
Ka prévue pour 2009), mettant en jeu cinq stations de réception centrées sur l'Inde et des
moyens météorologiques abondants (radars, pluviomètres, radiomètres), devrait
permettre de mieux caractériser le canal de propagation dans les régions tropicales
Synthèse des travaux
Conclusion et perspectives
59
- Le satellite Alphasat [Paraboni et al., 2007] (charge utile de propagation en
bande Ka et Q/V prévue pour 2011), devant impliquer plusieurs dizaines de terminaux
sur l'Europe, a pour but d’effectuer directement des mesures synchronisées des
affaiblissements de propagation et devrait donc permettre d'améliorer la connaissance de
la dynamique spatio-temporelle des affaiblissements de propagation, tout en
expérimentant des méthodes de compensation adaptatives.
Néanmoins, avant de chercher des améliorations aux modélisations de canal
proposées, il serait utile de quantifier l'impact des imprécisions dans la modélisation du
canal de propagation, sur les performances sur des systèmes de télécommunication, afin
d’apporter une réponse pertinente aux problèmes posés par le dimensionnement des
mécanismes de gestion adaptative de la ressource.
Si les modélisations présentées ont été développées pour des applications dans le
domaine des télécommunications par satellite, elles peuvent aussi répondre à des besoins
exprimés dans d'autres champs d'applications. En effet, le modèle de génération de
champs d'atténuation corrélés dans l'espace peut être utilisé pour calculer les
déformations des échos des radars altimétriques spatio-portés en bande Ka, ou simuler
l'atténuation subie sur une liaison micro-ondes entre un drone et une station sol.
Ce type de modélisation pourrait également trouver des applications dans les
domaines de l'hydrologie ou de la modélisation climatique, pour lesquels la modélisation
des systèmes précipitants est encore très approximative.
Synthèse des travaux
Bibliographie
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B. Articles
Articles
Statistical distribution of integrated liquid water and water vapor content from meteorological
reanalysis
68
Articles
Statistical distribution of integrated liquid water and water vapor content from meteorological
reanalysis
69
Statistical distribution of integrated liquid water and water vapor content from meteorological reanalysis
Nicolas Jeannin (1), Laurent Feral (2), Henri Sauvageot (3), Laurent Castanet (1) (1)ONERA, Département ElectroMagnétisme et Radar, Toulouse, France.
(2) Université Paul Sabatier, Laboratoire Antennes Dispositifs et Matériaux
Micro-ondes (AD2M), Toulouse, France. (3) Université Paul Sabatier, Observatoire Midi-Pyrénées, Laboratoire
d’Aérologie, Toulouse, France.
Abstract
In this paper a worldwide modelling of integrated liquid water and water vapor
content distributions is proposed and evaluated. The knowledge of those distributions is
valuable to predict attenuation for Earth-space communication systems operating at
frequencies higher than 10GHz.
1 Introduction
The prediction of tropospheric attenuation is of major interest for satellite
communication purposes. Up to now, most of the studies have focused on the dominant
effect: the attenuation due to rain. Nevertheless, with the use of higher frequencies such
as Ka band (20-30 GHz) and Q/V band (40-50 GHz), tropospheric attenuation can not be
limited to rain any longer.. For instance, as shown by Liebe (1985), Liebe et al. (1989)
or Pujol et al. (2006), attenuation due to clouds (i.e. liquid water) or water vapour can
exceed 5 dB and 6 dB, respectively, at 50 GHz, for 0.01% of the time. Therefore, system
engineers have now to consider these tropospheric components in addition to rain to
optimally design the next generation of communication systems.
Articles
Statistical distribution of integrated liquid water and water vapor content from meteorological
reanalysis
70
the field of radio-communications, the Cumulative Distribution Function (CDF)
of tropospheric attenuation is usually computed from statistical distributions of
meteorological parameters. The latter are derived from worldwide meteorological
databases obtained from reanalysis models. Obviously, it is not convenient to handle
with these huge datasets that are moreover not always available to the community.
Therefore models have been proposed to derive worldwide distributions of
meteorological parameters from few inputs. For instance, the local rain rate CDF is
modelled by Rec. ITU-R P.837-3 from three inputs given by ECMWF (European Centre
for Medium-Range Weather Forecast) reanalysis meteorological database ERA-15
(Salonen et al., 1994): the probability of rain during 6 hour periods, the total yearly
amount of convective rain and the total yearly amount of stratiform rain. A double
exponential prediction method is then used to derive rainfall rate CDF characteristic of
the geographical area considered (Baptista and Salonen, 1998). Such a model is of prime
importance as it provides inputs to rain field and rain attenuation field simulators
anywhere in the world, while accounting for the local characteristics of rain (Capsoni et
al., 1987a, 1987b, Féral et al., 2003a, 2003b, 2006, Grémont and Filip, 2004).
Unfortunately, such a model and its associated worldwide parameterization are not
currently available for the CDFs of the Integrated Liquid Water Content (ILWC, i.e. the
mass of liquid water contained in a columnar section of 1 square meter) and the
Integrated Water Vapor Content (IWVC, i.e. the weight of water vapor contained in a
columnar section of 1 square meter).
This paper aims at providing the community with worldwide maps of parameters
that allow the modelling of ILWC and IWVC CDFs anywhere in the world. To achieve
this, 15 years of reanalysis meteorological data (ERA-15) are considered. They are
presented in Section 2. Section 3 is devoted to the analytical modelling of the ILWC
CDF, to the assessment of the parameters and to the model accuracy. The same work is
performed in Section 4 to model the IWVC CDF.
Articles
Statistical distribution of integrated liquid water and water vapor content from meteorological
reanalysis
71
2 The data
So as to derive the analytic formulations of the ILWC and IWVC CDFs, Rec.
ITU-R P.840-3 (1999) and P.836-3 (2001) should have been considered. Indeed, the
latter give the time percentages during which a given value of ILWC and IWVC is
exceeded respectively.
Nevertheless, recalling that ILWC and IWVC values are derived from profiles of
pressure P, temperature T and humidity H (Salonen and Uppala, 1991), the P, T, H
profile database has to be selected with care. Particularly, Rec. ITU-R P.840-3 and
P.836-3 were derived from NA-4, an ECMWF database that provides 2920 P, T, H
profiles collected worldwide from 1992 to 1994. In the framework of the present study,
the ERA-15 database, also produced by the ECMWF, has been selected for statistical
stability reasons. Indeed, ERA-15 consists in 15 years (1979-1994) of reanalysed
meteorological data corresponding to 21900 samples (one sample every 6h) located all
around the world. Therefore, for each ERA-15 grid point (resolution 1.5°×1.5°), the
ILWC and IWVC values exceeded for 18 successive time percentages 18,..,1iiP = =99, 95,
90, 80, 70, 60, 50, 30, 20, 10, 5, 3, 2, 1, 0.5, 0.3, 0.2, 0.1 have been derived worldwide
from ERA-15 P, T, H profiles (Riva et al., 2006).
Now, when dealing with the ERA-15 database, some shortcomings have to be
kept in mind. First, the data are the outputs of a reanalysis process and do not
correspond to physical measurements. They are the result of a mathematical model
constrained to minimize the error with respect to local measurements derived from
remote sensing, radio-soundings, or ground stations. Second, it should be noticed that
the data from radio-soundings or radiometers are not available when it is raining, i.e.
precisely when the highest values of water vapor and liquid water content are reached
(Snider, 2000). Therefore, the confidence in the tail of ILWC and IWVC distributions
has to be limited. Nevertheless, as shown by Riva et al. (2006), distributions of ILWC
and IWVC derived from local radio-sounding compare satisfactorily with these derived
from ERA-15. Indeed, for 24 radio-sounding sites located in various climatic areas, the
average value of the RMS relative error between the CDFs of ILWC derived from ERA-
15 and the ones derived from radio-sounding is of 28%. For IWVC CDFs, this value is
around 10%. It has to be noticed that the confidence in the ILWC values has to be lower
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than the one in the IWVC values as they result from a semi-empirical model applied on
P, T, H profiles (Salonen and Uppala, 1991) and not from a direct integration of the
profiles.
3 Modelling of the ILWC distribution
3.1 Analytic formulation and parameter derivation
From aircraft and radiometric measurements, Gultepe and Isaac (1997) and
Fairall et al. (1990) have showed that liquid water content LWC and ILWC conditional
distributions (i.e. knowing that LWC>0 and ILWC>0, respectively) are well represented
by lognormal distributions. Consequently, the experimental CDFs of ILWC derived
from ERA-15 are first approximated by a lognormal law whenever the ILWC value is
strictly positive:
( )
1)0(
0L if ²2
²lnexp21)( **
*
=≥
>⎥⎦⎤
⎢⎣⎡ −−=≥ ∫
+∞
LP
dlll
PLLPL
cloud σμ
σπ (1)
In (1), L is the ILWC, Pcloud the probability to have a non zero ILWC, μ and σ the
lognormal parameters of the conditional distribution of ILWC )0/( * >≥ LLLP . For
each point of the ERA-15 grid, Pcloud, μ and σ are determined worldwide by minimizing
the error with respect to the local ILWC CDF derived from ERA-15 (1979-1994), for the
standard time percentages 18,..,jiiP = and associated ILWC 18,..,jiiL = defined in Section 2:
( )∑=
=⎥⎦
⎤⎢⎣
⎡ −18 2
,,
,,,min
i
ji i
icloudi
P LLPPF
cloud
σμσμ
. (2)
In (2), ( ) ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−×σ+μ=σμ −
cloud
i1cloudi P
P21erf2expP,,,PF is the inverse of the
cumulative distribution function and erf(x) is the error function. Worldwide maps of
Pcloud, μ and σ are given in Figures 1, 2 and 3, respectively. Importantly, in (2),
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subscript j (j≥1) indicates that only ILWC values Li greater than 0.1kg/m2 are considered
in the minimisation process. In some cases, it prevents Pcloud , μ and σ from reaching
unrealistic values.
Figure 1: Worldwide map of Pcloud (probability to have a non zero ILWC) regressed
from the ERA-15 database.
Figure 2: Worldwide map of parameter μ regressed from the ERA-15 database.
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Figure 3: Worldwide map of parameter σ regressed from the ERA-15 database.
According to Figs. 1, 2 and 3, Pcloud, μ and σ show a strong climatic dependency.
As expected, the probability to have non zero ILWC is higher over sea than over land. It
is also higher for equatorial areas than for arid or temperate areas. Moreover, in the
presence of cloud, μ is higher at low latitudes. These trends are confirmed by Table 1
which summarizes some ILWC values exceeded for different time percentages and
various locations. As shown in Table 1, ILWC values are much higher in equatorial and
tropical areas, than in mid-latitudes areas.
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ILWC (kg/m2) exceeded for x% of time Location 10% 1% 0.1%
ERA-15 0.25 0.75 1.20 Moscow
Model 0.25 0.76 1.18
ERA-15 0.81 2.31 3.27 Rio
Model 0.94 2.24 3.60
ERA-15 0.44 1.27 1.71 Paris
Model 0.47 1.25 1.87
ERA-15 2.10 3.50 3.89 Singapore
Model 2.12 3.41 4.09
ERA-15 0 0.31 0.60 Denver
Model 0.05 0.31 0.59
Table 1: Values of ILWC exceeded for different time percentages and various locations
from ERA-15 and from the model.
3.2 Model accuracy
In order to assess the model accuracy, the RMS error ε between the modelled
CDFs and those derived from ERA-15 has been computed according to:
( ) ( )( ) 100PL
PLPL)1j(18
1 18
ji
2
i15ERA
i15ERA
iMODEL
×⎥⎦
⎤⎢⎣
⎡ −−−
=ε ∑=
−
−
. (3)
So as to assess whether the lognormal approach is the best one to model the
conditional CDF of ILWC, Weibull and Gamma conditional distributions have been also
considered. Their optimal parameterisation has been obtained following the same
minimisation process used to derive the lognormal parameters μ and σ.
Besides, Weibull and Gamma distributions have been chosen due to their limited
number of parameters and because they may have a large tail, in compliance with ILWC
data derived from ERA-15.
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The results are given in Table 2, in terms of mean <ε>, standard deviation εstd
and maximum value of the error εmax computed for all the pixels of the ERA-15
database.
Weibull law Lognormal law Gamma law <ε> 24% 9% 19% εstd 20% 11% 23% εmax 85% 51% 70%
Table 2: Worldwide mean <ε>, standard deviation εstd and maximum value εmax of the
RMS error ε between the ILWC conditional CDFs derived from ERA-15 and the 3
models considered (lognormal, Weibull and Gamma).
In compliance with Table 2, the lognormal model minimises the error with
respect to ILWC conditional CDFs derived from ERA-15. The highest errors can be
observed to be mainly located over tropical oceans or highly mountainous areas, i.e. for
places where the ERA-15 database is known to show some inaccuracies (Allan et al.,
2001). Over temperate areas and over land, the error is low. To conclude, the lognormal
distribution is appropriate to model the ILWC conditional CDF worldwide.
As an example, Fig. 4 shows the ILWC CDF derived from ERA-15 for Dublin
(Ireland) and the associated lognormal modelling. The RMS error (3) is equal to 6%.
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Figure 4: ILWC CDF derived from ERA-15 for Dublin (Ireland) and associated
lognormal modelling.
4 Modelling of the IWVC distribution
4.1 Analytic formulation and parameter derivation
From data collected by aircraft, radio-sounding and remote sensing, Zhang et al.
(2003) have shown the possible existence of a bimodal behaviour of the IWVC
Probability Distribution Function (PDF) in tropical areas due to alternate dry and moist
regimes, disappearing when integrating over a wide area and a long time. Foster et al.
(2006) from GPS delay and radio-sounding measurements propose a log-normal model
and a reverse log-normal model in tropical oceans for precipitable water. However
IWVC CDFs derived from ERA-15 do not offer such a bimodal or log-normal
behaviour. They rather show that the IWVC CDFs can be appropriately represented, as
shown in the next section, by Weibull distributions:
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−=≥
kVVVPλ
** exp)( , (4)
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where V is the IWVC, k and λ are the shape and scale parameters of the Weibull
distribution, respectively. Considering the 18 values 18,..,1iiV = of IWVC exceeded a
fraction 18,..,1iiP = of the time, the Weibull approximations of the IWVC CDFs derived
from ERA-15 are determined by solving:
( )∑=
=⎥⎦
⎤⎢⎣
⎡ −18
1
2
,
,,min
i
i i
ii
k VVkPH λ
λ, (5)
where ( ) kii PkPH /1)(ln,, λλ −= is the inverse of the cumulative distribution
function. The regression is performed on each grid pixel of the ERA-15 database, using
the same minimization algorithm as the one used to derive ILWC CDF parameters.
Worldwide maps of scale parameter λ and shape parameter k are presented in Figures 5
and 6, respectively.
Figure 5: Regressed values of the scale parameter λ of the Weibull distribution of IWVC
from the ERA-15 database (1979-1994).
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Figure 6: Regressed values of the shape parameter k of the Weibull distribution of
IWVC from the ERA-15 database (1979-1994).
The parameters show a significant latitude dependency. Indeed, as can be seen in
Fig.5, the scale parameter λ, that drives the mean of the distribution, is higher at low
latitudes because the saturation water vapor content and the evaporation becomes higher
with higher temperatures. In other respects, the shape parameter k that drives the
skewness is higher in places where the troposphere saturation in water vapor is frequent.
As can be observed in Fig.1, saturation occurs more often in areas where there is a high
probability to have liquid clouds. Since values of IWVC can not exceed the saturation,
the distribution becomes negatively skewed (large λ).
Moreover, λ and k are also affected by orography (the atmosphere is thinner) and
land-sea transitions. Table 3 gives IWVC values exceeded for different time
percentages, various locations and computed from the model or derived from ERA-15.
According to Table 3 and as expected, the values of IWVC are higher in equatorial or
tropical areas than in temperate areas.
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IWVC (kg/m2) exceeded for x% of time
Location 10% 1% 0.1%
ERA-15 25.3 33.3 38.2 Moscow
Model 23.9 32.6 40.9
ERA-15 48.2 58.6 63.4 Rio
Model 50.1 58.8 64.5
ERA-15 28.4 35.4 41.2 Paris
Model 27.8 35.4 40.7
ERA-15 53.4 63.4 69.6 Singapore
Model 55.1 63.1 67.5
ERA-15 16.1 25.2 28.4 Denver
Model 15.9 24.3 30.1
Table 3: Values of IWVC exceeded for different time percentages, various locations and
derived from ERA-15 or computed from the model.
4.2 Model accuracy
The RMS error ε committed by approximating the IWVC CDF derived from
ERA-15 by a Weibull distribution has been computed according to (3). As in section 3,
so as to quantitatively assess whether the Weibull distribution is the best one to model
the IWVC CDF, the RMS error ε has also been computed considering 3 other
distributions potentially able to fit the CDFs derived from ERA-15, namely normal,
lognormal and gamma distributions. The results are given in Table 4, in terms of mean
<ε>, standard deviation εstd and maximum value of error εmax computed for all the pixels
of the ERA-15 database. As shown in Table 4, the Weibull model minimises the error
with respect to IWVC CDFs derived from ERA-15, with a mean error <ε> lower than
5% worldwide. Maximum errors are reached in specific places such as the Andes
(~30%), around the Bering sea (~20%) and in southern Indian Ocean (~15%), i.e. in
places where problems have been reported with reanalysis data (Allan et al., 2001).
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Weibull law Normal law Lognormal law Gamma law <ε> 4.3% 23% 27% 24% εstd 5.1% 19% 12% 14% εmax 37% 198% 152% 77%
Table 4: Worldwide mean <ε>, standard deviation εstd and maximum value εmax of the
RMS error ε between the IWVC CDFs derived from ERA-15 and the 4 models
considered (Weibull, normal, lognormal and Gamma).
As an example, Fig. 7 shows the IWVC CDF derived from ERA-15 for Dublin
(Ireland) and the associated lognormal modeling. The RMS error (3) is equal to 4%.
Figure 7: IWVC CDF derived from ERA-15 for Dublin (Ireland) and the associated
Weibull modelling.
Conclusion
Worldwide maps of parameters that allow the modelling of ILWC and IWVC
CDFs anywhere in the world have been proposed. They allow the computation of cloud
and water vapour attenuation CDFs worldwide from a few numbers of parameters.
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To do so, empirical CDFs of ILWC and IWVC have been derived from the
ECMWF ERA-15 database. The ILWC conditional CDFs derived from ERA-15 are
showed to be best modelled by a lognormal law anywhere in the world, with an average
RMS error lower than 10%. Highest errors are found over tropical oceans and highly
mountainous areas, i.e. in places where ERA-15 database is known to have some
inaccuracies. Concerning the IWVC CDFs, the Weibull distribution is shown to be the
one that best describes the worldwide IWVC CDFs derived from ERA-15, with an
average RMS error lower than 5%.
To model the IWVC and ILWC CDFs anywhere in the world, the worldwide
maps of parameters obtained in the framework of the present paper can be downloaded
at the URL: 111H111Hhttp://www-mip.onera.fr/ILWC_IWVC_distributions/IWVC_ILWC_map.
zip. Further studies will be possible with the forthcoming ERA-40 database whose
duration (40 years) ensures a sufficient amount of data to study the monthly dependency
of many meteorological parameters of interest.
Acknowledgements:
This research was partly supported by CNES (contract noA88550) and has been
partly carried out in the framework of the European Network of Excellence SatNex. The
authors are very grateful to Antonio Martellucci from ESA (The Netherlands) and Carlo
Riva from Politecnico di Milano (Italy) for kindly providing the statistical distributions
of ILWC and IWVC derived from ERA-15 and computed in the framework of
ESA/ESTEC Contract N°17760/03/NL/JA
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Distribution of tropospheric water vapor in clear and cloudy conditions from microwave
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Distribution of tropospheric water vapor in clear and cloudy conditions from microwave radiometric
profiling
Alia IASSAMEN(1)(3), Henri SAUVAGEOT(1),
Nicolas JEANNIN(2), and Soltane AMEUR(3)
(1) Laboratoire d'Aérologie, Observatoire Midi-Pyrénées, Université Paul
Sabatier, Toulouse, France.
(2) Département Electromagnétisme et Radar, Office National d'Etudes et de
Recherches Aérospatiales, Toulouse, France.
(3) LAMPA, Département d'Electronique, Université Mouloud Mammeri, Tizi
Ouzou, Algeria.
Abstract
A dataset gathered over 369 days in various midlatitude sites with a 12 frequency
microwave radiometric profiler is used to analyze the statistical distribution of
tropospheric water vapor content (WVC) in clear and cloudy conditions. The WVC
distribution inside intervals of temperature is analyzed. WVC is found to be well fitted
by a Weibull distribution. The two Weibull parameters, the scale (λ) and shape (k), are
temperature (T) dependent. k is almost constant, around 2.6, for clear conditions. For
cloudy conditions, at T < -10°C, k is close to 2.6. For T > -10°C, k displays a maximum
in such a way that skewness, which is positive in most conditions, reverses at negative
in a temperature region approximately centred around 0°C, i.e. at a level where the
occurrence of cumulus clouds is high. Analytical λ(T) and k(T) relations are proposed.
The WVC spatial distribution can thus be described as a function of T. The mean WVC
vertical profiles for clear and cloudy conditions are well described by a function of
temperature of the same form as the Clausius-Clapeyron equation. The
WVCcloudy/WVCclear ratio is shown to be a linear function of temperature. The
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vertically integrated WV (IWV) is found to follow a Weibull distribution. The IWV
Weibull distribution parameters retrieved from the microwave radiometric profiler agree
very well with the ones calculated from the European reanalysis meteorological database
ERA 15. The radiometric retrievals compare fairly well with the corresponding values
calculated from an operational radiosonde sounding dataset.
1 Introduction
Water vapor is a component of paramount importance of the atmosphere in
relation with weather and climate sensitivity research. It is the main greenhouse gas and
plays a predominant role in cloud dynamics through the release of wide amounts of
latent heat with condensation. Water vapor content distribution (WVCD) is also very
important in scientific fields related with the electromagnetic propagation in the
microwave domain. Knowing the WVCD is thus essential for numerous environmental
sciences (e.g. Peixoto et al. 1992, Chahine 1992, Webster 1994, IPCC 2001, Soden and
Held 2006, Sherwood et al. 2006, among others). Points of particular interest concern
the parametrization of WVCD and the quantitative differences of WVCD between clear
and cloudy atmosphere. Climatological column water vapor content varies only slightly
with the cloud cover in tropical regions but is significantly lower in clear sky than in
cloudy sky at midlatitudes and the variation is not simply due to differences in
atmospheric temperature distribution, as shown by Gaffen and Elliott (1993). The shape
of WVCD has not been discussed and that of integrated water vapor (IWV) is, again, a
point in discussion (e.g. Zhang et al. 2003, Foster et al. 2006).
Atmospheric water vapor is very difficult to quantify and describe due to the
various physical and dynamical processes affecting its distribution, notably on short
time and space scales. The in situ data used to analyze the tropospheric WVCD were
mainly collected by radiosonde observations (Liu et al. 1991, Gaffen and Elliott 1993,
Foster et al. 2006, Miloshevich et al. 2006). However, radiosonde observations do not
permit to associate tightly the presence of clouds and the WVC profiles. Some
fragmentary, in space and time, in situ observations were also gathered from aircraft
(Spichtinger et al. 2004, Korolev and Isaac 2006). In clear air and in the daytime, the
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total column water vapor from the surface was measured with simple near-infrared sun
photometer (Brooks et al. 2007). Water vapor in upper atmospheric layers was observed
by remote sensing from geostationary operational meteorological satellite using the 6.7
μm channel (e.g. Soden and Bretherton 1993) and from infrared sounder such as AIRS
(Atmospheric InfraRed Sounder). Observations were also made using microwave
sensors such as UARS (Upper Atmospheric Research Satellite) microwave limb sounder
(Stone et al. 2000) or from AMSU (Advanced Microwave Sounding Unit) instruments
and SSM/I (Special Sensor Microwave Imager) instruments or by combining the two
kinds of data (Kidder and Jones 2007). Vertically integrated (column) water vapor
(IWV), or precipitable water, data were obtained from Global Positioning System (GPS)
signals (e.g. Rocken et al. 1997, Liou et al. 2001, Van Baelen et al. 2005, Foster et al.
2006). Standard dual-channel ground-based microwave radiometers, with a frequency
close to the 22.235 GHz water vapor line and the other in a window at higher frequency,
were used to measure IWV and LWP (Liquid Water Path) simultaneously (e.g. Hogg et
al. 1983, Snider 2000, Westwater et al. 2004, Meijgaard and Crewell 2005).
Tropospheric vertical profile of water vapor mixing ratio was also observed by
active remote sensing, notably by Raman lidar by Sakai et al. (2007) in the night-time
from a ground-based device, and by Whiteman et al. (2007) in the daytime with an
airborne spectroscopic lidar. Upper-troposphere water vapor was measured by
differential absorption lidar technique (Ferrare et al. 2004). However, lidar
measurements work only in the absence of optically thick clouds. Information on
humidity profiles have been retrieved with wind profiling radar from refractive index
gradient inferred from the echo power with variational assimilation (Furumoto et al.
2007) and by combining wind profiler and ground-based radiometer (Bianco et al. 2005)
or with GPS (Furumoto et al. 2003). The above quoted techniques are not able to
provide the data required to analyse the WVC distribution on short time and space
scales, in clear and cloudy conditions.
More recently, a ground-based multi-channel microwave radiometric technique
was proposed for the retrieval of temperature, water vapor, and liquid water profiles
(Solheim et al. 1998, Solheim and Godwin 1998, Güldner and Spänkuch 2001, Ware et
al. 2003). This new technique has not yet gained the confidence of long time used
methods and its accuracy is not that of in situ measurements, but it offers some
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opportunities to capture the differences between clear and cloudy vertical profiles which
are still poorly known and documented, due to a lack of relevant data.
The object of the present paper is to discuss the form of the WVCDs and to
propose their parametrization as a function of the temperature in order to quantify the
differences between clear and cloudy sky conditions. The paper is based on a set of data
collected with a multichannel microwave radiometric profiler.
2 Data
2.1 Water vapor parameters
The quantity retrieved by the radiometer (cf. Section 2b) is the water vapor
density of the moist air, or absolute humidity, ρ. Water vapor approximately behaves as
an ideal gas, with each mole of gas obeying an equation of state that can be written:
TRe
w
ww =ρ , (1)
where ew is the partial pressure of the water vapor in hPa, Rw is the gas constant
for water vapor (that is Rw = R/mw, where R is the universal gas constant and mw is the
mass of one mole of vapor), and T is the absolute temperature in K. Usually ρw is
multiplied by 106 and expressed in g m-3.
At saturation, (1) is:
sw
w,sw,s TR
e=ρ , (2)
where the subscript s refers to saturation. The temperature variation of the
pressure at which the phase transition takes place is given by the Clausius-Clapeyron
equation which can be written for equilibrium over water and over ice. A large number
of equations were proposed to calculate the saturation vapor pressure over a surface of
liquid water or ice. Murphy and Koop (2005) review the vapor pressure of ice and
supercooled water for atmospheric applications. They conclude that all the commonly
used expressions for the vapor pressure of water and ice are very close to each other for
tropospheric temperature. In the present work, a formulation derived from the Magnus
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90
equation, convenient for routine computation, is used (cf. Pruppacher and Klett 1997, p.
117, Eq. 4-83 and 4-84 and Appendix A-4-8). No enhancement factor (defined as the
ratio of the saturation vapor pressure of moist air to that of pure water vapor (eg. Buck
1981) is used. This factor is very small and depends on air pressure. It gives:
For water
( )⎥⎦
⎤⎢⎣
⎡−
−=
w
0w0w,sw,s BT
TTAexp)T(e)T(e , (3)
with es,w in hPa, T in K, es,w(T0) = 6.1070 hPa, T0 = 237.15 K,
Aw = 17.15 and Bw = 38.25 K.
For ice
( )⎥⎦
⎤⎢⎣
⎡−
−=
BiTTTA
exp)T(e)T(e 0i0i,si,s , (4)
with es,i in hPa, T in K, es,i(T0) = 6.1064 hPa, T0 = 273.15 K, Ai = 21.88 and Bi =
7.65 K.At the triple point, for Ts = T0, we have Pa154.611ee i,sw,s == ,with
3
w
s 1032.1Re
×= and 11w gKJ463.0R −−= .
Using (3) and (4) in (2) gives the vapor density at saturation over water and ice
respectively, namely, with the numeric coefficients:
( )⎥⎦
⎤⎢⎣
⎡−
−×=ρ
25.38T15.273T15.17
expT
1032.1
s
s
s
3w,s (5)
and
( )⎥⎦
⎤⎢⎣
⎡−
−×=ρ
65.7T15.273T88.21
expT
1032.1
s
s
s
3i,s (6)
with ρ in g m-3 and T in K.
2.2 Microwave radiometric measurements
The data were collected with a microwave radiometric profiler TP/WVP-3000
manufactured by Radiometrics. This radiometer is described in several papers (e.g.
Solheim and Godwin 1998, Ware et al. 2003, Liljegren 2004) and on the web site
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113H113Hwww.radiometrics.com. In short, this instrument measures 12 radiometric brightness
temperatures through a sequential scan of 12 frequencies inside the microwave
spectrum, 5 in the K band, between 22.035 and 30.000 GHz, on the flank of the 22 GHz
water vapor absorption line, and 7 in the V band, between 51.250 and 58.800 GHz, on
the flank of the 60 GHz molecular oxygen complex. The bandwidth for each channel is
300 MHz. The beamwidth is 5 to 6 degrees at 22-30 GHz and 2 to 3 degrees at 51-59
GHz.
The radiometer also includes some in situ sensors for the ground level
measurement of temperature, pressure, and humidity, as well as a zenith looking infrared
radiometer with a 5° beamwidth and a liquid water (precipitation) detector. The
combination of the cloud base temperature provided by the IR sensor with the retrieved
temperature profile gives an estimate of the cloud base height. The cloud base height
can be associated with warm liquid or ice (e.g. cirrus), or with mixed clouds. The
combination of temperature and WVC enables the calculation of the relative humidity at
each level.
The 22-30 GHz channels are calibrated to 0.3 K rms by automated tipping
procedures. The 51-59 GHz channels are calibrated to 0.5 K RMS (Root Mean Square)
with a liquid nitrogen target (Han and Westwater 2000, Westwater et al. 2001). Liquid
nitrogen calibration is performed regularly (about each month) and when the radiometer
is moved.
The time cycle of the whole 12-channel scanning process is about 23 s. However
for the present study, the time interval between profiles is only 92 s, which means that
only one profile out of four was retained to reduce statistical redundancy.
Inputs include the 12 microwave brightness temperature measurements, one
infrared temperature, as well as temperature, humidity, and pressure at surface level,
representing 16 inputs. By inversion of the radiances measured at the different channels
through a neural network application, the radiometer retrieves, up to a height of 10 km,
the vertical profiles of temperature and water vapor, cloud base temperature and height,
vertically integrated water vapor (IWV) and liquid water (ILW). The neural network
profile retrieval is based on a training from historical series of local radiosondes
(Schroeder and Westwater 1991, Solheim et al. 1998, Liljegren 2004). Brightness
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temperatures in the five K band channels are calculated with the revised absorption
model proposed by Liljegren et al. (2004).
2.3 Accuracy and resolution
The profiler retrieval accuracy was discussed and estimated from a comparison
with simultaneous radiosonde data (Güldner and Spänkuch 2001, Liljegren et al. 2001,
Ware et al. 2003, Liljegren 2004). Such comparisons are approximate due to spatial and
volume sampling differences between radiosonde and radiometer measurements. The
neural network method is found slightly less accurate than a statistical regression
method (see Güldner and Spänkuch, 2001, for the resulting vertical profiles of
accuracy). The information content of the measurements degrades with altitude. The
vertical resolution in retrieved temperature linearly changes from about 0.1 km near the
surface to ~ 6 km at a height of 8 km. For vapor density, the resolution falls from 0.3 km
near the surface to 1.5 km at a height of 6 km and 3.5 km at 10 km (Güldner and
Spänkuch 2001, Liljegren 2004).
Accuracy is degraded when there is liquid water on the radome of the radiometer
and in the presence of heavy precipitation above the radiometer, like with other
microwave radiometers. Retrievals are not applicable in this case. Light precipitation
aloft with a dry radome is acceptable. A field campaign was performed in March and
April 2007 at Aire-sur-Adour (Table 1 and Fig. 1) in order to compare the TP/WVP-
3000 radiometric profiler with radiosounding. The analysis of the results is out of the
scope of the present paper and will be discussed elsewhere. However, it can be briefly
stated that, for accuracy and resolution, the results for the TP/WVP 3000 radiometer are
in agreement with the conclusions of the previous studies quoted above.
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Figure 1: Radiometer-observed time series of vertically integrated liquid water (ILW)
and water vapor (IWV), rain detection (binary), cloud base height, and surface air
temperature for 9 days, from 8 to 16 April 2004.
Figure 1 shows, as an example, the temporal series of ILW, IWV, binary
detection of rain presence at the surface, cloud base height as estimated by the IR
temperature combined to temperature profile, and air temperature measured at surface
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level for 9 days, from 8 to 16 April 2004 at Toulouse (cf. Section 2.d for location). The
air temperature displays the usual diurnal variations with a trend towards warming from
day 4 to day 8. For day 9, the diurnal oscillation is very small because the air
temperature is dominated by a wide rainy system sweeping the site during the last 14
hours of the series.
In the presence of rain reaching the surface in Fig. 1, the microwave radiometer
is found not to work correctly. ILW, notably, displays peaks which always exceed 2 mm
(this threshold is never reached in the absence of rain), and the cloud base height is seen
near or at the surface. The IWV is less sensitive to the presence of rain. All that justifies
the removal of the profiles with ILW > 2 mm (cf. Section 2d). In clear sky, the IR
thermometer displays C8.49 °− , an arbitrary rounded down value meaning that there is
no cloud above. The height of the corresponding level can be seen to increase from day
4 to day 9. ILW and IWV are strongly correlated but neither of them is correlated with
surface air temperature. In other words, there is no significant diurnal variation of ILW
and IWV, which are dominated by the general circulation of the synoptic scale
structures.
To assess how the results presented in the present paper are biased by the
retrieval uncertainty, we need to know the RMS error vector affecting the retrieved
profiles. Because we do not know this vector, we have used the RMS error vectors
computed from a similar 12 channel microwave radiometer operated by the Atmospheric
Radiation Measurement (ARM) Program in Germany between March and December
2007, namely the error vector for "Black Forest Germany" available on the site
114H114Hhttp://www.archive.arm.gov. We believe that this error vector is representative for our
west European climatic area and convenient for the use done in the present paper. To
compute the RMS error on the retrievals, we have added to each retrieved profile a
Gaussian noise with zero mean and a standard deviation equal to the RMS error for each
layer.
2.4 Site and dataset
The data were collected at three different sites located in the South-West of
France during six periods distributed over the four seasons as specified in Fig. 2 and
Table 1. This dataset is considered to be representative of coastal oceanic mid-latitude
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climate (e.g. Martyn 1992, see also arguments in Section 4). During the data acquisition
at the various sites, it rained for 7.5% of the time (the long term mean in this area is
8%), clouds were present 72% of the time and clear sky 28% of the time.
Figure 2: Map showing the location of the data collection sites and radiosounding
station (Bordeaux).
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Table 1. Dataset.
Subset location and designation Observation periods Duration
(day)
Profiles rejected
(%) Number of clear sky profiles Number of cloudy sky profiles
2007 March 13-15
2007 March 29-31 Aire-sur-l'Adour
2007 April 1-15
21 4.81 6166
32.85%
12603
67.15
2006 December 1-18 Toulouse 1
2007 January 1-20 38 9.31
6650
20.55%
25710
79.45%
2008 January 21-31
2008 February 1-29
2008 March 1-31 Toulouse 2
2008 April 01-06
77 9.72 23754
36.39%
41521
63.61%
2007 January 26-31 Lannemezan 1
2007 February 1-27 32 7.35
13970
50.18%
13869
49.82%
2007 April 24-30
2007 May 1-29 Lannemezan 2
2007 June 5-28
60 11.78 8131
16.36%
41572
83.64%
2007 July 2-31
2007 August 1-31
2007 September 1-30
2007 October 1-31
Lannemezan 3
2007 November 1-19
141 12.40 31384
27.06%
84597
72.94%
Total 369 10.55 90055
29.39%
219872
70.61%
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Profile retrievals are obtained at 250 m intervals from the surface to 10 km,
which represents 41 levels (or range bins) including the surface level where temperature,
humidity, and pressure are measured by in situ sensors.
The training of the neural network was performed with 10 years of radiosonde
data (2 per day) from Bordeaux-Mérignac, a meteorological station located on the
Atlantic Coast (Fig. 2). Upper air data at Bordeaux-Mérignac are assumed to be
representative of the atmosphere over the coastal, oceanic, midlatitude Western Europe.
One profile, corresponding to one 23 s measurement cycle, was retained each 92
s, which gives 309927 profiles for the whole 369 days of the observing period. One
profile includes in fact the three 41 level basic profiles of temperature, WVC, and RH,
the two integrated values IWV and ILW, and the surface parameters. Profiles associated
with the presence of rain as detected by the rain detector of the radiometer were
removed, as well as the profiles associated with an ILW higher than 2 mm. ILW > 2 mm
is due to the presence of precipitation aloft that can bias the retrievals (Snider 2000).
The remaining profiles (89.45%) were classified in two subsets considering the cloud
base temperature (tbase in °Celsius) data from the IR radiometer, namely:
- clear sky profiles when tbase ≤ -49.8°C. 90055 profiles were included in this
subset.
- cloudy sky profiles, when tbase > -49.8°C, made up of 219872 profiles.
When a profile was removed, for example for not satisfying the ILW condition,
all the other data of the profile were rejected.
Differences between beamwidth of the IR radiometer and microwave channels
may cause a classification in clear category for profiles with microwave beam partly
filled with cloud. This case in encountered in the presence of a non uniform cloud cover,
notably with cumulus clouds, when clouds go into or out of the beam while they are
advected over the radiometer. The profiles with tbase ≤ -49.8°C and ILW higher than the
minimum detectable value of the radiometer, have thus been looked for. Less than 1% of
profiles were found to be concerned. They were classified as cloudy.
Mean vertical profiles of WVC and relative humidity as a function of
temperature, for clear and cloudy conditions, over the subset Toulouse 2 (winter), are
presented in Fig. 3 with the standard deviation bars. For vertical coordinate, temperature
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was preferred to altitude because water vapor content depends only on temperature.
Using altitude as vertical coordinate blurs the representation of vertical profiles of WV
parameters. The upper limit of the profiles has been taken at -45°C because at colder
temperature WVC can fall below the minimum detectable which is about 0.02 g m-3. The
mean vertical profile of temperature versus altitude averaged over the dataset (almost
one year) is not presented because devoid of interest. In fact it shows a quasi linear
decrease of T from 10°C at the surface to -55°C at 10 km of altitude, that is the standard
atmospheric lapse rate of -6.5°C km-1. At a given altitude, temperature for cloudy
conditions is slightly warmer (about 0.5°C) than the one for clear conditions.
Figure 3 shows that the clear and cloudy profiles for WVC present a monotonic
decreases from 6-7 g m-3 near the surface to < 0.1 g m-3 at temperature < -40°C, with
some irregularities for temperature > 20°C. These irregularities (inversion of the vertical
gradient) are thought to be associated with the convective boundary layer structure.
Profiles of RH increase from the surface up to about 0°C then decreases above. This
variation with T can be seen as resulting from the combination of the effect of cloud
liquid water vertical distribution, which displays a maximum at mid tropospheric levels
(e.g. Mazin 1995, Gultepe and Isaac 1997), and of the monotonic decrease of WVC with
temperature.
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Figure 3: Mean vertical profiles of water vapor content (WVC) and relative humidity
retrieved from the microwave radiometric profiler for clear and cloudy conditions for
the subset Toulouse 2 (2008 January 21 to 2008 March 31).
3 Water vapor content distribution
To analyse the WVC distribution, three datasets are considered: clear sky, cloudy
sky, and all – i.e. the sum of clear and cloudy sky subsets – in order to emphasize the
differences between clear and cloudy. Probability density functions (pdf) were
calculated by temperature class of width ΔT = 10°C, WVC bin of 0.2 g m-3, between 20
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and -45°C, as was the fitting for two analytical forms: lognormal and Weibull. These
forms were selected among others from empirical tests and from physical and
bibliographical considerations. They are frequently used for atmospheric and cloud
characteristic representation because they are related to physical processes relevant for
atmospheric phenomena. The lognormal distribution is associated with the statistical
process of proportionate effects (see Aitchison and Brown 1966, p. 22, and Crow and
Shimizu 1988): the change in the variate at any step of the process is a random
proportion of the previous value of the variate. The lognormal form is found to be
convenient for many cloud characteristics such as rain cell size distributions (Mesnard
and Sauvageot 2003), rain rate distributions (Atlas et al. 1990, Sauvageot 1994),
precipitable water (Foster et al. 2006), and relative humidity (Soden and Bretherton
1993, Yang and Pierrehumbert 1994). The Weibull form is found convenient for
variables whose distribution is limited by extreme values, for example life variables or
wind when the velocity is limited by turbulence. For the WVC distribution the upper
limit is the vapor density at saturation ρs in the presence of condensation (cloudy case).
In clear air there is no upper limit.
The lognormal probability density function (pdf) is written:
( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛
σ
μ−−
σπ=σμ
2
y
y
yyy
y21exp
x21,;xf , (7)
where xny l= . The two parameters of the distribution are the mean μy and
variance 2yσ of y, defined as:
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛μσ
+μ=μ
2/12
x
xxy 1nl (8)
and
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛μσ
+=σ2
x
x2y 1nl . (9)
The Weibull pdf is:
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( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
λ−⎟
⎠⎞
⎜⎝⎛
λλ=λ
− k1k xexpxk,k;xf . (10)
The two parameters of the distribution are the scale λ and the shape k. The mean
μx and variance 2xσ are defined as:
( )k11x +Γλ=μ , (11)
and
( ) 222x k
21 μ−+Γλ=σ . (12)
Γ is the gamma function.
Distribution Variable Parameter x mean μx variance 2
xσ lognormal (logn) y = ln x mean μy variance 2
yσ Weibull (wbl) x scale λ shape k
Table 2. Definition of the parameters used in the present paper.
Figure 4 shows the probability density of WVC, data points, best fitting curves,
and error bars for 4 classes of temperature out of 12 (for the sake of clarity). The
numeric values of the fitting parameters and correlation coefficients are given in Table
3. For the clear case, lognormal and Weibull distributions are almost equivalent.
Lognormal is slightly better for T < -15°C and Weibull slightly better for T > -15°C. For
the cloudy case, Weibull is better than lognormal at temperatures between –20 and 10°C
and the two distributions are almost equivalent at T < -20°C and >10°C. It is the same
for all, which is dominated by the cloudy case. The Weibull fitting gives better results
because the WVC fluctuations have limits related with the condensation process when
the WVC reaches saturation.
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Clear Cloudy All Lognormal Weibull Lognormal Weibull Lognormal Weibull
Temperature class (°C) μy σy r λ K r μy σy r λ K r μy σy r λ k R [-45; -35] -2.50 0.51 1.00 0.10 2.19 1.00 -2.38 0.57 1.00 0.12 2.07 1.00 -2.42 0.55 1.00 0.12 2.08 1.00 [-40; -30] -2.05 0.49 0.99 0.16 2.18 0.81 -1.83 0.51 0.93 0.20 2.25 0.96 -1.91 0.51 0.87 0.19 2.17 0.93 [-35; -25] -1.62 0.49 0.98 0.25 2.11 0.99 -1.34 0.48 0.99 0.30 2.34 0.98 -1.43 0.50 0.99 0.30 2.20 0.99 [-30; -20] -1.18 0.52 0.99 0.40 2.04 0.97 -0.88 0.48 0.99 0.52 2.39 0.98 -0.98 0.51 0.99 0.48 2.20 0.98 [-25; -15] -0.73 0.54 0.99 0.63 2.01 0.98 -0.40 0.49 0.97 0.80 2.41 0.98 -0.55 0.52 0.97 0.74 2.21 0.98 [-20; -10] -0.27 0.56 0.98 1.00 2.02 0.97 -0.03 0.49 0.96 1.22 2.45 0.99 -0.12 0.53 0.96 1.14 2.25 0.98 [-15; -5] 0.20 0.56 0.95 1.59 2.15 0.98 0.42 0.49 0.94 1.91 2.55 0.99 0.34 0.52 0.94 1.79 2.35 0.99 [-10; 0] 0.62 0.52 0.88 2.37 2.48 0.98 0.90 0.45 0.82 3.02 2.99 0.97 0.81 0.50 0.85 2.80 2.68 0.98 [-5; 5] 0.99 0.46 0.89 3.29 2.92 0.99 1.28 0.36 0.85 4.19 3.85 0.99 1.19 0.42 0.85 3.93 3.34 0.99 [0; 10] 1.29 0.39 0.94 4.35 3.23 0.99 1.57 0.30 0.96 5.51 4.09 0.98 1.49 0.35 0.94 5.19 3.61 0.99 [5; 15] 1.51 0.33 0.98 5.27 3.59 0.98 1.81 0.28 0.99 6.99 3.86 0.96 1.73 0.32 0.98 6.58 3.51 0.97 [10; 20] 1.67 0.32 0.97 6.14 3.71 0.97 1.98 0.29 0.99 8.34 3.89 0.98 1.93 0.32 0.98 7.98 3.59 0.98
Table 3. Fitting parameters for the water vapor content pdf (Fig. 4). μy and σy are the mean and standard deviation of the lognormal
distribution, λ and k are the scale and shape of the Weibull distribution, respectively. r is the correlation coefficient. WVC unit is g m-3
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The various coefficients of Table 3 are not very sensitive to the width of the
temperature classes ΔT. Similar results are obtained for ΔT up to 15 degrees but beyond
it degrades. On the other hand, the coefficients display large differences for clear and
cloudy cases.
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Figure 4: Probability distribution function of WVC for 4 temperature classes between 10
and -30°C and Weibull best fitting for the three datasets: clear sky, cloudy sky, and all.
The corresponding numerical values are given in Table 3. The error bars were
computed with the ARM error vectors. Note that coordinate scales are different for left
and right parts.
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Figure 5 upper part shows the variation as a function of temperature of the WVC
pdf Weibull parameters calculated for 24 temperature classes of width ΔT = 5°C,
between -45°C and 20°C. Figure 5 lower part displays the corresponding Fisher's
coefficients of skewness (γ1) and kurtosis (γ2). For γ1 > 0, skewness is on the left side
(the mode is lower than the average) and there is a tail on the right side. For γ1 < 0, it is
the reverse, that is to say skewness is on the right and tail on the left. γ1 and k convey
partly the same kind of information. For k < 2.6, the Weibull pdfs are positively skewed
(with a right side tail). For 2.6 < k < 3.6, skewness coefficient approaches zero (no tail),
that is to say pdf is quasi normal. For k > 3.6, Weibull pdfs are negatively skewed, and
the tail is on the left. It is what Fig. 5 shows. At cold temperature, WVC pdfs are
positively skewed, more for clear than for cloudy. While temperature increases,
skewness diminishes and WVC pdf evolves toward a normal shape. For temperature
between about -2°C and 12°C, k for cloudy pdfs is higher than 3.6, that is to say slightly
negatively skewed. Clear case is positively skewed everywhere. For T > 10°C, clear and
cloudy cases evolve toward a same value since the corresponding atmospheric level is
clear for the two cases (i.e. below cloud base). Kurtosis variations are small around the
normal value. Clear case is leptokurtic at cold temperature. Platykurtic character
increases with temperature up to about -5°C, then, at warmer temperature, a peak of
leptokurtic character correlated with skewness is observed.
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Figure 5: Variation as a function of the temperature of the two parameters of the
Weibull distribution, shape k and scale λ, and of skewness and kurtosis for the pdf of
WVC calculated for 24 classes of temperature of width ΔT = 5°C between -45 and 20°C.
The error bars were computed with the ARM error vectors. For the sake of clarity, only
the error bars for the curves All are drawn. In the upper panel, horizontal lines at k =
2.6 and 3.6 show the domain where pdfs are quasi normal.
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This evolution of the WVC Weibull pdf parameters with temperature can be
explained, at least in part, by considering the WVC limit of saturation with respect to
water and ice. In the absence of limit, WVC is lognormally distributed as most of
dynamically controlled atmospheric parameters (cf. Crow and Shimizu 1988). Clear
profiles are in this case. Limits linked to saturation create a truncation on the right side
of the pdfs and thus induces a reverse skewness. The bump observed on the k curve, for
-10°C < T < 15°C, corresponds tightly to the average liquid cloud water vertical
distribution associated with mid latitude cumulus clouds (e.g. Mazin 1993, Gultepe and
Isaac 1997).
Curves k(T) of Fig. 5 can be fitted with a general model of Gauss2 form, namely:
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ −−+
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ −−=
2
2
22
2
1
11 c
bTexpac
bTexpa)T(k (13)
with coefficients given in Table 4.
T > 0°C a1 b1 c1 a2 b2 c2 r clear 3.70 285 17.3 0.543 273 2.74 0.99 cloudy 3.11 274 8.21 3.68 288 11.7 0.99 all 1.91 273 7.58 3.51 288 15.1 0.97 T ≤ 0°C a1 b1 c1 a2 b2 c2 r clear 2.02 282 15.1 2.22 227 74.1 0.99 cloudy 4.71 287 13.6 2.42 252 50.2 0.99 all 1.59 279 9.6 2.23 256 100 0.99
Table 4. Fitting parameters for the shape curve of Fig. 5 (Eq. 13). r is the correlation
coefficient. Upper part T > 0°C, lower part T ≤ 0°C.
Curves λ(T) of Fig. 5 can be fitted very well with a general model similar to (5),
that is:
( )⎥⎦
⎤⎢⎣
⎡−
−=λ
3
33cT
15.273Tbexp
Ta
)T( (14)
with coefficients given in Table 5 and T in K.
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T > 0°C a3 b3 c3 r clear 884 1.81 252 0.99 cloudy 1103 2.14 249 0.99 all 1039 2.43 245 0.99
T ≤ 0°C a3 b3 c3 r clear 913 10.8 124 0.99 cloudy 1147 16.8 54.9 0.99 all 1079 17.2 51.9 0.99
Table 5. Fitting parameters for the scale curve of Fig. 5 (Eq. 14). r is the correlation
coefficient. Upper part T > 0°C, lower part T ≤ 0°C.
Because it is more representative physically, the WVC mean, μWVC, has also been
plotted and fitted as shown Fig. 6. The coefficients of the fitting with a function of the
form (5), namely:
( )⎟⎠⎞
⎜⎝⎛
−−
=μCT
15.273TBexpTA)T(WVC , (15)
are given in Table 6. Of course these coefficients are not far from those of Table
5. In Fig. 6, the saturation curves ρs,w and ρs,i given by (5) and (6) are also drawn.
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Figure 6: Variation as a function of the temperature of the mean WVC, of the ratio R =
WVC cloudy/WVC clear and best fitted curves. Curves ρsw and ρsi give the water vapor
density at saturation with respect to liquid water and ice respectively. The error bars
were computed with the ARM error vectors.
T > 0°C A B C r clear 787 1.76 253 0.99 cloudy 998 2.05 250 0.99 all 933 2.35 246 0.99
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T ≤ 0°C A B C r clear 814 11.5 117 0.99 cloudy 1035 20.1 18.9 0.99 all 968 19.6 26.3 0.99
Table 6. Fitting parameters of the mean WVC for the curves of Fig. 6 (Eq. 15). r is the
correlation coefficient. Upper part T > 0°C, lower part T ≤ 0°C.
Curves of Figs 5 and 6 can be fitted acceptably with a single function for all the
temperature range considered in the present work (-40°C, +20°C). However, similarly
with the saturation water vapor density equations (5 and 6), a fitting slightly better is
obtained when considering separately the parts <0°C and >0°C. That is why Tables 4, 5,
and 6 present these two parts.
To sum up, the mean of WVC in clear and cloudy conditions depends only on the
temperature and is described accurately by (15) with the coefficients of Table 6.
From (15), using the coefficients of Table 6, the ratio of the mean of WVC for
cloudy and clear conditions can be written:
clearWVC
cloudyWVCRμ
μ= 27.1T0061.0 += for T > 0°C (16)
27.1T0007.0 += for T ≤ 0°C.
with T temperature in °Celsius.
Using (13) and (14) in (10) enables the writing of the WVC pdf as a function of
the sole temperature.
4 Distribution of IWV
As for WVC, pdf of vertically integrated WVC (IWV) was calculated and fitted
with lognormal and Weibull distributions for clear, cloudy and all datasets. Curves are
shown in Fig. 7. Fitting coefficients and correlation coefficients are given in Table 7.
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For clear conditions, the lognormal distribution gives a correlation coefficient slightly
better than the Weibull one, but for cloudy and for all, the Weibull distribution is the
best. Using a wide radiosounding and GPS (Global Positioning System) dataset, Foster
et al. (2006) found that precipitable water (that is IWV) distribution can be well fitted
with a lognormal distribution. They emphasize that, for tropical oceanic environments
IWV tends to exhibit negatively skewed histograms for which fitting with a reverse
lognormal distribution is proposed.
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Figure 7: Pdf of vertically integrated water vapor for the three datasets and lognormal
and Weibull best fitting.
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The numerical values are given in Table 7.Using 15 years of ECMWF (European
Centre for Medium-range Weather Forecast) reanalysis meteorological database ERA
15, the Weibull parameters k and λ for the IWV distribution were calculated. Isolines of
k and λ over the western European area, where the radiometric data were collected (as
shown in Fig. 2), are presented in Fig. 8. The fields of k and λ values are rather
homogeneous showing the validity of our assumption of Section 2d that this area can be
considered as a single climatological entity for WVC distribution. Averaged values over
the area represented in Fig. 8 are k = 2.5 and λ = 1.9 which compare very well with the
values for all of Table 7, that is k = 2.8 and λ = 1.9.
lognormal Weibull
μ σ r λ k r
clear 0.10 0.34 0.93 1.30 3.45 0.92
cloudy 0.59 0.34 0.97 2.12 3.44 0.98
all 0.45 0.41 0.96 1.90 2.86 0.98 Table 7. Fitting parameters for the IWV curve of Fig. 7. r is the correlation coefficient.
This agreement can be seen as supporting the validity of the IWV radiometric
measurements. In a companion paper analysing the world wide IWV distribution from
the ERA 15 database (Jeannin et al. 2008), we found that k for tropical environment is
negatively skewed, what agrees with Foster et al. (2006) results. For latitude lower than
about 10°, k is higher than 6 with peak at 10 in some very humid equatorial areas (e.g.
Amazonia and Indonesia).
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Figure 8: Isolines of the Weibull distribution parameters for the fitting table pdf of IWV
calculated from ERA 15.
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5 Comparison with operational radiosonde soundings
Operational radiosonde soundings (RS) provide data on the vertical WVC
distribution along slanted trajectories two times a day, at 00 and 12 UTC. It is not
possible to distinguish clear and cloudy vertical RS profiles in the same way that we do
with the radiometric profiles. Besides, the RS sampling frequency does not permit to
capture some important aspect of the WVC distribution related with the WVC diurnal
cycle. However a rough comparison can be made between the radiometer retrieved
WVC pdfs and profiles for the All case and those provided by a radiosonde dataset. To
perform this comparison, we have used the two daily RSs gathered at Bordeaux during
the period of 2006, 2007, and 2008 when the radiometer was active (Table 1). The RS
data considered for the statistic are taken over the altitude domain cover by the
radiometer. For example, for the period when the radiometer was operated from
Lannemezan (altitude 600 m), the RS data from Bordeaux (altitude 50 m) are taken
above 600 m.
All Lognormal Weibull
Temperature class (°C) μy σy r λ k r
[-45; -35] -2.43 0.52 1.00 0.11 2.33 1.00 [-40; -30] -2.10 0.61 1.00 0.16 1.91 1.00 [-35; -25] -1.59 0.61 0.90 0.27 1.96 0.98 [-30; -20] -1.13 0.63 0.94 0.43 1.93 0.99 [-25; -15] -0.67 0.62 0.91 0.68 2.01 0.98 [-20; -10] -0.25 0.62 0.91 1.03 2.00 0.97 [-15; -5] 0.17 0.63 0.89 1.60 1.98 0.95 [-10; 0] 0.60 0.63 0.80 2.45 2.06 0.91 [-5; 5] 1.01 0.60 0.66 3.60 2.30 0.88 [0; 10] 1.38 0.56 0.66 5.08 2.52 0.91 [5; 15] 1.60 0.50 0.68 6.66 2.83 0.94
[10; 20] 1.90 0.44 0.77 8.13 3.24 0.97 Table 8. Same as Table 3 calculated from the radiosonde dataset for the All case.
Table 8 gives the coefficients of the WVC pdfs calculated from the RS data and
fitted with lognormal and Weibull functions. The Weibull fitting is clearly very good
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and better than the lognormal one. Coefficients λ and k of the RS pdfs versus
temperature are represented in Fig. 9.
Figure 9: The dots show the variation as a function of the temperature of the two
parameters of the Weibull distribution, shape k and scale λ, and of the skewness and
kurtosis for the pdf of WVC calculated from the RS dataset for 24 classes of temperature
of width ΔT = 5°C between -45 and 20°C. The curves and error bars are the radiometer
retrievals for the All case redrawn from Fig. 5 for comparison with the RS values.
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In Fig. 9, we have redrawn the curves of Fig. 5 for the All case with their error
bars and added the values calculated form the RS data. Scale λ from RS is very close to
the radiometer retrieval one. Shape k from RS is, as the radiometric one, almost constant
and close to the radiometric values for T < 5°C. For T > 5°C it is lower than the
radiometric values but enclosed between 2.6 and 3.6, that is quasi normal. Skewness and
kurtosis also are very similar for RS and radiometer.
Fig. 10 shows the WVC mean versus temperature. In Fig 10 the curve of Fig. 6
for All with the error bars have been redrawn and the values calculated from RS have
been superimposed. A detailed discussion of the differences between the radiometric
retrievals and the RS values is out of the scope of the present paper. That is why by way
of conclusion we limit to note that the agreement for the pdf shape and the mean WVC
variation with temperature is rather good.
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Figure 10: The dots show the variation as a function of the temperature of the mean
WVC calculated from the RS dataset for 24 classes of temperature of width ΔT = 5°C
between -45 and 20°C. The curves and error bars are the radiometer retrievals for the
All case redrawn from Fig. 6.
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6 Summary and conclusions
The distribution of the vapor phase of tropospheric water was studied from 369
days of observations collected in various sites of Western Europe with a 12 frequency
microwave radiometric profiler. Data were regrouped in two subsets, clear sky and
cloudy sky. The profiles associated with the presence of rain were rejected.
Several forms of distribution were considered to fit the pdf of the atmospheric
water vapor profiles observed with the profiler. For the sake of clarity, only lognormal
and Weibull distributions were discussed in the present paper because they proved to be
the most efficient for the pdf fitting. The lognormal function is associated with the
statistical process of proportionate effect, frequently identified in the atmosphere. The
Weilbull function is convenient for random variables whose distributions are limited by
extreme values such as water vapor with condensation.
The WVC pdf inside 10°C temperature classes between 20 and -45°C are found
to be accurately fitted by Weibull distributions. The two parameters of the Weibull
distribution, the shape k and scale λ, are shown to be well described by analytical
functions of temperature t. Using k(T) and λ(T) in the pdf enables the writing of the
WVC spatial distribution as a function of T. The mean WVC vertical profile can be
represented by a function of the temperature of the same form than the Clausius-
Clapeyron equation. The ratio of mean WVC for cloudy to mean WVC for clear
conditions is linearly dependent on T, with a mean value of 1.27.
The pdf of the vertically integrated WVC, or precipitable water, is found to be
Weibull distributed rather than lognormal. The values retrieved from the microwave
radiometric profiler compare very well with the ones calculated from the ERA 15
reanalysis meteorological database.
A rough comparison between the radiometric retrievals and the values provided
by an operational radiosonde sounding dataset shows that the agreement for the pdf
shape and the mean WVC profile as a function of temperature is fairly good.
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Acknowledgements
The authors would like to thank all the contributors to the radiometer dataset,
notably Mrs Edith Gimonet and Mr Giulio Blarzino from ONERA (Office National
d'Etudes et de Recherche Aérospatiales). Thanks are also due to one anonymous
reviewer for helpful comments and suggestions and to the ARM Program for providing
the radiometer error vector used in the present paper.
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Weather radar data for site diversity predictions and evaluation of the impact of rain field advection
Lorenzo Luini(1), Nicolas Jeannin(2), Carlo Capsoni(1), Aldo Paraboni(1), Carlo
Riva(1), Laurent Castanet(2), Joël Lemorton(2) (1)DEI – Dipartimento di Elettronica e Informazione
Politecnico di Milano
Milano, Italy (2)DEMR – Département de Électromagnétisme et radar
ONERA
Toulouse, France
Manuscript submited to International Journal of Satellite Communication
Abstract
This paper presents the analysis on two weather radar datasets, collected at Spino
d’Adda (Italy) and Bordeaux (France), for the simulation and the performance
evaluation of a site diversity system. Results from the two locations are compared and
the impact of different factors such as the baseline and link orientation is assessed and
related to the local climatologic characteristics. The results obtained are then compared
with the model currently recommended by the ITU-R for the estimation of the site
diversity performance. A linkage is then established between the preferable baseline
orientation and the predominant direction of the rain field advection. The rain field
displacement is finally shown to be well approximated by the wind speed and direction
relative to the 700 hPa isobar extracted from the ECMWF ERA-40 meteorological
database.
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1 Introduction
The evolution of telecommunication systems is always pushing towards the use
of higher transmission frequencies owing to the congestion of lower bands, to the
always increasing demand of bandwidth from the users and, sometimes, to the limits
imposed on the system dimension (especially on the antenna). As widely known,
telecommunication systems operating at Ka, Q and V bands (roughly from 20 to 50
GHz) are strongly affected by attenuation phenomena due to atmospheric impairments
[Riva, 2004]. Still, they are expected to provide real-time multimedia services, and
consequently, to be reliable and guarantee the desired system availability. In this
environment, strong signal fades can no longer be overcome making use of static power
margins, but require the application of Fade Mitigation Techniques (FMTs) as a viable
solution [Fukuchi and Saito, 2004]. To this end, telecommunication systems based on
site diversity can be envisaged [Goldhirsh et al., 1997]: in fact, the use of multiple
receiving stations permits to take advantage of the spatial variability of rain, so that a
distance of the order of tens of kilometers between the stations significantly reduces the
probability that both stations are undergoing an outage and, therefore, that the system is
unavailable [Capsoni et al., 2007]. The design of a site diversity system (of FMTs in
general), however, requires the evaluation of the advantages (for example in terms of
outage probability) deriving from the implementation of such a countermeasure to signal
fades. For that purpose, weather radar data, where available, are the most suitable source
of information, as they inherently reflect the influence of the local climatology and
topography on the rain field spatial distribution, which, on turn, determines the
effectiveness of a site diversity solution [Goldhirsh and Robison, 1975].
This paper presents the analysis performed on two weather radar datasets,
collected at Spino d’Adda (Italy) and Bordeaux (France), for the simulation and the
performance evaluation of a site diversity system with tunable characteristics. After a
brief description of the two radar datasets, the site diversity gain is assessed for the two
locations and, afterwards, the impact of different factors such as link and baseline
orientation is investigated. Moreover, as expected, the study clearly shows the
dependence of the system performance on the type of rain event (stratiform or
convective) affecting the stations. The comparison with the model currently
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recommended by the ITU-R for the estimation of the site diversity performance is shown
for both locations. Afterwards the local climatologic features, namely, the predominant
direction of the rain fields advection, are linked to the trends observed for different
baselines orientation. To this end, a comparative analysis is devised between the rain
advection speed and direction and the wind outputs extracted from the ERA-40 ECMWF
(European Center for Medium-range Weather Forecast) database [Uppala et al., 2005].
Specifically, the analysis strives at determining whether ERA-40 wind data are suitable
to determine the most advantageous baseline direction for a site diversity system, as
well as whether they are adequate to parameterize global space-time rain models such as
those proposed in [Gremont and Filip 2004] and [Jeannin et al., 2007]. Lastly, a
preliminary analysis is presented to assess the linkage between the rain field advection
direction and its spatial anisotropy.
2 The weather radar datasets
The two radar datasets utilized in this study have been collected in two temperate
sites, namely Spino d’Adda, Italy (45.4° N, 9.5° E) and Bordeaux (44.5° N, -0.34° E).
They have the following characteristics:
• Spino d’Adda: approximately 15000 CAPPI (Constant Altitude Plane
Position Indicator) radar images, extracted from rain events (in the period
from 1988 to 1992) which have proven to be fully representative of the
local yearly rainfall statistics [Capsoni et al., 2008]. The maximum
operational range of the radar considered in this study is 40 km in order to
avoid the inclusion of clutter pixels due to the surrounding mountains. The
spatial resolution of the radar scans is 0.5 km × 0.5 km and the interval
between consecutive images is 77 seconds.
• Bordeaux: approximately 30000 CAPPI radar images, extracted from one
year (1996) of continuous operation (also non-rainy images have been
recorded). The maximum operational range of the radar is 100 km, the
spatial resolution of the radar scans is 1 km × 1 km and the interval
between consecutive images is 5 minutes.
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Despite some differences between the two radar datasets (period of data
acquisition, number of images at disposal, spatial and temporal resolution), the work
presented in this paper suggests that they permit to derive fully comparable results,
obviously as long as the same image dimension is considered. For that purpose, a
circular area with a 40-km radius has been selected in the South-Western portion of each
original Bordeaux radar scan, as it is a flat zone, over which reliable data, seldom
affected by clutter and by anomalous propagation effects, are obtained.
3 Simulation of site diversity systems using radar images
The radar derived rainfall images have been used for the simulation of an Earth-
satellite site diversity system with tunable characteristics. Both in Spino d’Adda and in
Bordeaux, the rainfall snapshots have been converted into maps of attenuation
experienced by a radio link pointing to a geostationary satellite, under a fixed yearly
mean rain height Hr, derived from the ITU-R Rec. P.839-3 [ITU-R, 2001]. The path
attenuation A has been calculated through the numerical integration of:
dllRkAL∫= α)( [dB] (1)
where L = Hr/sin(θ) is the path length affected by rain, θ is the link elevation, k
and α, provided by the ITU-R Rec. P.838-3 [ITU-R, 2003], are rain-to-attenuation
conversion coefficients that depend on the link elevation and on the radio wave
frequency and polarization (always vertical in this work). Obviously, R(l), indicating the
rain intensity value at position l impairing the transmission link, is strongly dependent
on the local precipitation characteristics and therefore is expected to be tightly bound to
the system performance. In this work, all attenuation maps have been calculated at the
reference frequency of 30 GHz, although results could be easily extended to different
frequencies and link geometries with a little effort.
The gain G offered by a two-site diversity system, with separation D between the
stations, has been calculated as:
),()(),( PDAPAADG jss −= [dB] (1)
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where As and Aj are the attenuation values of the cumulative distribution functions
(CDFs) (both for the same probability level P), respectively relative to a single station
and to a two-site diversity system (depending on D), for which the minimum attenuation
value is always selected (see Figure 1).
0 5 10 15 20 25 30 35 40
10-1
100
Attenuation [dB]
Per
cent
age
[%]
Spino d'Adda - f = 30 GHz - distance = 8 km
Single-site attenuation CDFJoint two-site attenuation CDF (D = 8 km)
PG
AsAj
Figure 1 Example of site diversity gain (Spino d’Adda, 30 GHz, D = 8 km).
The calculation of G has been performed according to the following options:
• reference attenuation level As: from 4 to 32 dB with a 4-dB step, so that the minimum
percentage level of the single-site CDF is approximately 10-1 % at 30 GHz;
• distance D between the receiving stations: from 4 to 56 km in 4-km step;
• link orientation φ , defined as in Figure 2 : -20° (towards the East), 0° (coinciding with
the South) and 20° (towards the West). Each value of φ corresponds to a rain-to-
attenuation conversion procedure, as different rain rates R(l) affect the link. For a
geostationary satellite, the link elevation θ is univocally defined by φ ;
• baseline orientations, defined as in Figure 3: horizontal (H), vertical (V) and 45° (HV)
with respect to the horizontal direction;
• type of rain event: stratiform or convective.
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W E
S
N
θ
φ
L
W E
S
N
θ
φ
L
Figure 2: Geometry of the link (indicated by the red solid line); θ is the elevation angle,
is the orientation angle, L is the path length affected by rain
45°
D
DDW E
N
S
45°
D
DDW E
N
S
Figure 3: Geometry of the baseline between the receiving stations (dashed lines): horizontal (H),
vertical (V) and 45° (HV) from the West-East direction. The blue solid line indicates the reference
link, with respect to which site diversity statistics are calculated
Considering all maps, the single-site CDF has been calculated starting from the
attenuation value relative to the reference pixel (blue link in Figure 3), whereas the two-
site joint attenuation CDF (for given D, θ, φ ) has been obtained from the minimum
attenuation value between the one experienced by the reference pixel and the one
relative to the associated site diversity link. In order to improve the statistic robustness
of the analysis and to prevent results from being dependent only on the climatologic
and/or topographic peculiarities of specific areas, attenuation CDFs consist of all the
values obtained by moving the stations pattern depicted in Figure 3 across the whole
map.
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3.1 Gain dependence on the link orientation
In this section, the dependence of the system performance on the link orientation
is analyzed. 227H
lists the link characteristics as a function of φ , for both sites. Figure 4 shows the
site diversity gain comparison in Spino d’Adda, for the three selected values of φ : -20°,
0°, 20°. In this case, the baseline orientation is horizontal, but very similar results
(omitted here for brevity) have been obtained also for baselines V and HV (the same
applies to Bordeaux): in all cases, the dependence of the system gain on the link
orientation is clearly negligible. Such conclusion is a positive one, if a real system is
concerned: in fact, the link orientation usually is not a design parameter, as it obviously
depends on the relative position between the satellite and the receiving stations (e.g.,
broadcasting systems). Thus, a system designer could disregard the impact of φ in
estimating the system performance through prediction methods.
θ [°] Hr [km] L [km] φ [°]
Bordeaux Spino Bordeaux Spino Bordeaux Spino -20 34.6 33.5 2.86 3.3 5 5.9 0 38.7 37.7 2.86 3.3 4.6 5.3 20 34.6 33.5 2.86 3.3 5 5.9
Table 1: Elevation angles and rain heights relative to the considered links in Spino d’Adda and
Bordeaux.
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0 10 20 30 40 50 600
5
10
15
20
25
30
Distance [km]
Gai
n [d
B]
Spino d'Adda
4 dB 8 dB12 dB16 dB20 dB24 dB28 dB32 dB
Figure 4 : Spino d’Adda, site diversity gain comparison for different link orientations:
-20° (solid lines), 0° (dashed lines with circles), 20° (dashed lines with crosses). The
baseline orientation is horizontal.
3.2 Gain dependence on the baseline orientation
If the dependence of G on φ is negligible (therefore, hereinafter, φ will always
be set to 0°), the same conclusion can not be drawn for the baseline orientation. Figure 5
depicts the comparison of the system gain, respectively for Spino d’Adda and Bordeaux,
both for H (solid lines) and V (dashed lines with circles) baseline orientations. The gain
results relative to the HV baseline orientation have been omitted as they are less
interesting: in both cases, they are comprised between the ones relative to the horizontal
and the vertical baselines. Figure 5 clearly points out how the choice of the system
baseline may affect the achievable site diversity gain. Specifically results show that for
Bordeaux, an horizontal baseline between the stations would improve G (w.r.t an
horizontal baseline), whereas the opposite is observed for Spino d’Adda, where a
vertical baseline would assure a better performance. It is worth noting that for D = 4 km,
in both cases, a horizontal baseline provides a higher gain. In fact, making reference to
the geometry in Figure 3, when the links point towards the South, the correlation
between the associated attenuation values is higher (and therefore, the gain is obviously
lower) in the case of a vertical baseline because links tend to overlap. For longer values
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of D, this explanation no longer holds as attenuation values get uncorrelated after some
tens of kilometers. The local preferable baseline orientation can be roughly linked to the
prevalent elongation direction of the rain field. In fact, a visual inspection of radar data
suggests that, most of the time, rain structures tend to stretch orthogonally to the rain
field advection direction. More details about this effect will be given in section 4, which
also suggests how to derive the preferential direction of the rain field advection from the
ERA-40 database.
0 10 20 30 40 50 600
5
10
15
20
25
30
Distance [km]
Gai
n [d
B]
Spino d'Adda
4 dB 8 dB12 dB16 dB20 dB24 dB28 dB32 dB
0 10 20 30 40 50 600
5
10
15
20
25
30
Distance [km]
Gai
n [d
B]
Bordeaux
4 dB 8 dB12 dB16 dB20 dB24 dB28 dB32 dB
Figure 5 : Site diversity gain comparison for different baselines: H (solid lines) and V (dashed lines
with circles). φ = 0°.
In order to quantify such difference, let us define the following error figure:
s
sSNsEWsbl A
ADGADGAD ),(),(100),( −− −=ε (3)
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where GW-E and GN-S are the system gains, respectively for the W-E and the N-S
baseline, relative to the stations distance D and to the reference single-site attenuation
As. 228HFigure depicts the trend of blε with D, calculated as the average value of blε relative
to all the As values at the same distance D.
0 10 20 30 40 50 60-6
-4
-2
0
2
4
6
8
Distance [km]
E[ ε
bl] [
%]
Spino d'AddaBordeaux
Figure 6:Trend of the average value of blε relative to all the As values at the same
distance D (both for Spino d’Adda and Bordeaux). φ is set to 0°
Both Figure 5 and 229HFigure clearly point out how the choice of the system baseline
may affect the achievable gain. Specifically results show that for Bordeaux, a W-E
baseline between the stations would improve G, whereas the opposite is true for Spino
d’Adda, where a N-S baseline would assure a better performance. It is worth noting that
for D = 4 km, in both cases, a W-E baseline provides a higher gain. In fact, making
reference to the geometry in Figure 3, when the links point towards the South, the
correlation between the associated attenuation values is higher (and therefore, the gain is
obviously lower) in the case of a N-S baseline because links tend to overlap. For higher
values of D, this effect is no longer visible as
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3.3 Gain dependence on the rain event type
With the aim of assessing the impact of the rain event type on the system
performance, in both sites, rain events have been classified as stratiform or convective if
belonging respectively to “colder” (from November to April) or to “warmer” (from May
to October) months. Although this is quite a rough selection criterion, nevertheless, in
temperate zones, it can provide at least a first approximated discrimination between
weak and widespread rain phenomena and intense and spatially limited convective
precipitations. If single-site stratiform rain attenuation CDFs are considered, strong
fades tend to be less probable. For this reason, in order to deal with reliable statistics, in
this section, the reference attenuation As ranges from 4 to 14 dB using a 2-dB step.
Results are reported in 230HFigure 7, where solid and dashed lines depict G relative to
stratiform and convective events, respectively. As expected, the system performance
increases when convective rain events affect the stations: in fact, convective phenomena
are characterized by intense rain rate values but cover limited areas. Obviously, both
these characteristics concur to reduce the spatial correlation of rain. As a consequence,
the site diversity gain obtained during convective events shows a steep increase,
especially within the first 15 km, while the one relative to stratiform events definitely
increases more gradually with distance.
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0 10 20 30 40 50 600
2
4
6
8
10
12
14
Distance [km]
Gai
n [d
B]
Spino d'Adda
4 dB 6 dB 8 dB10 dB12 dB14 dB
0 10 20 30 40 50 600
2
4
6
8
10
12
14
Distance [km]
Gai
n [d
B]
Bordeaux
4 dB 6 dB 8 dB10 dB12 dB14 dB
Figure 6. Site diversity gain comparison between stratiform (solid lines) and convective
(dashed lines with circles) rain events. Ф= 0°, H baseline.
0 20 40 60 80 100 120 140 160 18010-4
10-3
10-2
10-1
100
101
Rain rate [mm/h]
Per
cent
age
[%]
Spino d'AddaBordeaux
Figure 8: Comparison between the radar-derived rainfall rate CDFs at Spino d’Adda
and Bordeaux.
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The differential gain is more marked in Bordeaux than in Spino d’Adda, which
may be due both to the fact that the monthly classification of rain events into stratiform
and convective is more effective in Bordeaux and/or to the higher convectivity in
Bordeaux, probably linked to the proximity of the Atlantic ocean and to its influence on
the rain structures. The higher convectivity seems to be confirmed also by Figure 8,
where the radar-derived rainfall rate CDF relative to Bordeaux denotes a higher
convective contribution (i.e. more probable intense rain rate values) if compared to the
Spino d’Adda CDF.
3.4 Comparison between the two sites
In this section, the system performance calculated for Spino d’Adda and
Bordeaux is compared. However, since G depends on the chosen baseline, as pointed out
in section 3.2, results relative to the H, V and HV orientations have been averaged and
are compared in Figure 9. The higher gain values in Bordeaux seem to confirm the
strongest convectivity of that site, as already mentioned in the previous section.
However, it is worth noting that, despite the reduced temporal (5 min) and spatial ( 11×
km2) resolution of the Bordeaux dataset, such radar scans are clearly suitable for site
diversity simulations, as they provide results that are fully comparable with those
obtained from a dataset with a finer temporal (77 sec) and spatial ( 5.05.0 × km2)
resolution.
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0 10 20 30 40 50 600
5
10
15
20
25
30
Distance [km]
Gai
n [d
B]
Spino d'Adda and Bordeaux
4 dB 8 dB12 dB16 dB20 dB24 dB28 dB32 dB
Figure 9: Site diversity gain comparison between Spino d’Adda (solid lines) and
Bordeaux (dashed lines with circles). Results refer to average values of H, V and HV
baseline orientations.
3.5 Comparison with the site diversity model currently recommended by ITU-R
The ITU-R has recently included into the ITU-R Rec. P.618-9 [ITU-R, 2007] a
new model aiming to predict the site diversity gain on a global basis. The methodology,
developed by Paraboni and Barbaliscia and whose rationale is provided in [Paraboni
and Barbaliscia, 2002], is based on the assumption that the logarithm of the rain
intensity R and the one of the associated attenuation A follow a multivariate Gaussian
distribution. The spatial cross-correlations of the logarithm of R and A are expressed by
the following double decreasing exponential forms, respectively:
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−+⎥⎦
⎤⎢⎣⎡−=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−+⎥⎦
⎤⎢⎣⎡−=
2
2
500exp06.0
30exp94.0)(
700exp3.0
60exp7.0)(
ddd
ddd
A
R
ρ
ρ
(4)
where d is the distance (in km) between the two stations.
In this section, the radar-derived diversity gains are compared with those
estimated by the above mentioned model: concerning radar data, results of the W-E, N-S
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and NW-SE baseline orientations have been averaged. The comparison is shown Figure
10, respectively for Spino d’Adda and Bordeaux.
0 10 20 30 40 50 600
5
10
15
20
25
30
Distance [km]
Gai
n [d
B]
Spino d'Adda
0 10 20 30 40 50 600
5
10
15
20
25
30
Distance [km]
Gai
n [d
B]
Bordeaux
4 dB 8 dB12 dB16 dB20 dB24 dB28 dB32 dB
Figure 10: site diversity gain comparison between radar-derived values (solid lines) and
the ones estimated by the ITU-R Rec. P.618-9. φ = 0°, average values of horizontal,
vertical and 45° baseline orientations.
The prediction model tends to underestimate the diversity gain, especially for
high attenuation levels As. This may be due to the assumption at the basis of the model
(multivariate Gaussian distribution), which is valid as long as not too high values of rain
intensity (<15-20 mm/h) are concerned. The estimation accuracy of the model has been
quantified by defining the following error figure:
),(),(),(100),(
sr
srsmsm ADG
ADGADGAD −=ε [%] (2)
where Gm and Gr are the system gains, respectively estimated by the ITU-R Rec.
P.618-9 and derived from the radar images, relative to the stations distance D and to the
reference single-site attenuation As. Figure 11 depicts the trend of mε , calculated as the
average value of mε conditioned to D.
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0 10 20 30 40 50 60-35
-30
-25
-20
-15
-10
-5
Distance [km]
E[ ε
m] [
%]
Spino d'AddaBordeaux
Figure 11: mean value of the percentage error εm conditioned to D, for Spino d’Adda
and Bordeaux.
Results indicate a particularly high gain underestimation of the prediction model
for small distances (up to 20 km), for which the average error mε ranges approximately
from -30% to -20%. For greater distances, the error gradually decreases and tends to a
stable value (between -15% and -10%). It is worth pointing out that the ITU-R Rec.
P.618-9 is independent of the baseline orientation, as it assumes an isotropic modeling
of the rainfall spatial correlation [Paraboni and Barbaliscia, 2002]. It is therefore clear,
that, even if the model provided an accurate estimation of the diversity gain, it would
not be able to determine the most appropriate baseline orientation aimed at maximizing
the system performance. From this standpoint, radar data are certainly very useful and
offer unique chances of investigating some specific properties of the site diversity gain.
4 Comparison between the rain field advection and the ERA-40 wind outputs
As underlined in section 231H 3.2, the diversity gain at a given location exhibits
significant changes with the baseline orientation. The results obtained in this section
suggest that, in both sites, a linkage can be established between the rain front orientation
and the predominant rain advection direction. Indeed, a simple visual inspection of radar
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derived rain fields shows that most of the rain structures seem to elongate orthogonally
to their motion direction. Consequently, the knowledge of the rainfall prevailing
advection direction could help in the choice of the most advantageous baseline
orientation. To this aim, in this section, the rain advection speed and direction derived
from radar data are compared with the ERA-40 wind outputs. In addition, this study can
furthermore help to determine whether ERA-40 data are suitable for the
parameterization of the rain advection in space-time channel model such as those
proposed in [Gremont and Filip, 2004] and [Jeannin et al., 2008].
4.1 Determination of rain advection from radar data
The rain field advection direction and speed has been determined from successive
radar scans at time steps t and t+dt. It was implicitly assumed that the field advection
has a single direction and magnitude on the whole radar coverage.
Let R(x,y,t) be the rain fields at a given time t and position (x,y). The advection is
considered to be the vector [xa/dt ; ya/dt] such that R(x-xa,y-ya,t+dt) and R(x,y,t) exhibit
the maximum cross-correlation. This methodology, commonly used for this kind of
analysis [Kitzmiller et al., 2002], allows the retrieval of the rain field advection from
radar images in a simple, yet very effective way. The procedure assumes that the
temporal evolution of the rain field can be decomposed into two terms: one
corresponding to the actual modification of the rain structures within a Lagrangian
referential frame and the other one corresponding to displacement of such a frame
[Gremont and Filip, 2004]. This latter term will in the following be considered as the
advection of the field. The cross-correlation ρ between two successive maps at time t
and t+dt can be expressed as:
)]([Var)](Var[)]([)]([],,(),,([),(
dttRtRdttREtREdttyyxxRtyxREyx aa
aa+
+−+−−=ρ (6)
Where [...]E and Var[…] denote respectively the average value and the variance
of the field. The cross-correlation ρ is computed for all the xa and ya values that
correspond to reasonable displacements of the rain field, i.e. such that the motion of the
field will in general be slower than 150 km/h.
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An example of the cross-correlation between two consecutive rain maps of
Bordeaux (Figure 12) is shown in 232HFigure . The peak of ρ plotted in Figure 13 was found
for ya = 5 km (North to South) and for xa = 4 km (West to East): such displacement is
thus considered to be the advection of the field in the whole 5-minute interval. The
algorithm outlined above performs accurately if the correlation peak is higher than 0.7.
This condition implies that the time interval between the two radar observations is short
enough so as to prevent the rain field modifications from degrading too much the peak
value of ρ. Empirically, with a spatial resolution of 1 km, a time interval of 10 minutes
between consecutive images was found to be the upper bound guaranteeing a reliable
estimation of the advection. Thus, the time step between two successive images used to
compute the advection is set to 5 minutes for Bordeaux and to 3.5 minutes for Spino
d’Adda (one every three maps). Moreover, in order to obtain a sufficiently robust
estimation of ρ, the fraction of the observation area affected by rain has to be
significant. For this reason, radar observations for which the rainy area is smaller than
2% of the whole radar coverage (area with a 100-km radius) have been discarded
leading thus to consider 17% of the dataset.
Figure 12: Evaluation of the displacement on two consecutive radar maps taken at time
steps t and t+dt (Bordeaux, dt = 5 minutes)
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Figure 13: Cross-correlation for different displacement vectors between two
consecutive rain maps of Bordeaux. The maximum correlation corresponds to the most
plausible advection between the two images
4.2 Comparison between radar derived wind speed and direction and ERA-40 wind outputs
The ERA-40 data considered in this study are the wind values, relative to
different isobars, that are freely available on the web with a latitude-longitude regular
grid of 2.5°×2.5° . The period of radar data acquisition (1988-1992 for Spino d’Adda
and 1996 for Bordeaux) falls within the one of the ERA-40 reanalysis database (1957-
2002), so that a comparison of the rain advection estimated from both sources is
possible by taking into account concurrent years. Data of winds (flowing both from North to South (N-S) and from West to East
(W-E)) relative to different pressure levels have been considered in this study, namely
850, 775, 700, 600, 500 hPa. Such levels roughly correspond to reasonable mid-height
values of cloudy structures typical of temperate climates.
In order to compare the rain advection speed and direction retrieved from radar
data with the ERA-40 winds outputs, the following methodology has been applied. The
ERA-40 wind data for both locations have been derived by applying a bilinear
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interpolation of the values relative to the four closest ERA-40 pixels, as indicated Figure
14.
Figure 14: Position of the observation areas and of the surrounding ERA-40 cells
As an example, Figure 15 illustrates the wind direction statistics relative to the 775
hPa isobar and calculated from one year of ERA-40 data, both for Spino d’Adda and
Bordeaux: in the latter location, that lies in the proximity of the ocean and whose area is
not characterized by a significant orography, the wind direction is mainly eastward; on
the contrary, in Spino d’Adda, air fluxes from the North are mainly blocked by the Alps
and consequently, most of the time, air streams flow from West or South.
Figure 15: Histogram of the wind direction relative to the 775 hPa isobar computed
from one year of ERA-40 data for Spino d’Adda and Bordeaux
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In order to allow a direct comparison between the two datasets, the radar derived
rain advection speeds and directions have been averaged for each 6-hour period,
according to the time resolution of ERA-40 data. Considering the averaging operation
and the constraints illustrated in the previous section for the calculation of the rain field
advection from radar data, 160 and 64 samples have been respectively obtained for
Bordeaux and Spino d’Adda. The correlation coefficient between the rain advection
derived from the radar datasets and from the ERA-40 wind outputs at several pressure
levels is listed in Table 2, both for Bordeaux and Spino.
Direction Magnitude Pressure level (hPa)
Bordeaux Spino d’Adda Bordeaux Spino d’Adda
850 0.66 0.71 0.85 0.58
775 0.82 0.83 0.89 0.77
700 0.90 0.83 0.92 0.84
600 0.84 0.77 0.85 0.85
500 0.82 0.76 0.77 0.82 Table 2. Correlation between the rain advection derived from radar data and from ERA-
40 wind outputs for Bordeaux and Spino d’Adda
Among all the levels reported in Table 2, the 700 hPa isobar presents the highest
overall correlation. This pressure level was already found to be the most representative
of the rain field advection for different general circulation models as those in [Kitzmiler
et al., 2002] and [Sokol, 2006]. Lower levels (850 hPa, 775 hPa) underestimate the rain
advection speed and the proximity to the ground perturbs the direction estimation,
whereas, on the contrary, the advection velocity is overestimated at higher layers (600
hPa, 500 hPa).
The sample by sample comparison between the rain advection speed and the
ERA-40 wind velocity at 700 hPa is presented in Figure 16, both for Bordeaux
(asterisks) and Spino d’Adda (dots). The correlation between the two types of data is
slightly poorer for Spino d’Adda, as also observed in the comparison concerning the
wind direction reported in Figure 17. This trend may be explained by the large impact of
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the mountainous area surrounding Spino (see Figure 14) that probably leads to a less
reliable estimation of wind flows provided by the ERA-40. In fact, the mean height
associated to the four ERA-40 cells surrounding Spino d’Adda is over 1000 m a.m.s.l.
whereas the site lies only at 84 m a.m.s.l.
0 20 40 60 80 100 120 1400
20
40
60
80
100
120
140
Rain advection speed from radar data averaged over 6h [km/h]
Win
d sp
eed
from
ER
A 4
0 [k
m/h
]
Wind speed from ERA-40 and advection speed from radar
BordeauxSpino d'Adda
Figure 16: ERA-40 wind speed relative to the 700hPa isobar versus rain advection
speed for Bordeaux (asterisks) and Spino d’Adda (dots)
0 50 100 150 200 250 300 3500
50
100
150
200
250
300
350
Rain advection direction deduced from radar averaged over 6h [°]
Win
d di
rect
ion
from
ER
A-4
0 [°]
Wind direction from ERA-40 and advection direction from radar
BordeauxSpino d'Adda
Figure 17: ERA-40 wind speed at 700h Pa versus rain advection speed for Bordeaux and Spino
d’Adda
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The results obtained so far show that the rain field advection (both direction and
speed) can be approximated with a relatively high confidence by the ERA-40 wind
outputs relative to the 700 hPa isobar. As a consequence, such data can serve for the
correct parameterization of global space-time rain models such as those proposed in
[Gremont and Filip, 2004] and [Jeannin et al., 2007].
5 Impact of the wind direction on the rain field spatial anisotropy
As suggested in [Goldirsh et Robison, 1975], the local predominant wind
direction seems to be linked to the preferable baseline orientation of site diversity
systems. A confirmation of this trend is also given by the results obtained in this paper:
indeed, the diversity gain computed from the radar data of Bordeaux and Spino d’Adda
for the W-E and the N-S baselines shows sizeable differences (refer to Figure 5). In the
light of the predominant wind directions depicted in Figure 5 for the two sites, this result
suggests that, on the average, rain fields are elongated towards the N-S direction in
Bordeaux, whereas in Spino they are rather stretched towards the W-E direction: as a
result, in both locations, the most advantageous baseline orientation is parallel to the
predominant wind direction.
The effect of the wind flows on rain structures is confirmed also by Figure 18,
where the overall spatial correlation has been calculated from all the rain maps of
Bordeaux using the spatial spectrum technique [Jeannin et al., 2008]: indeed, results
indicate a slight anisotropy of the field, which actually tends to be more correlated along
the N-S direction than along the W-E one.
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Figure 18: Overall spatial correlation of the rain field computed using one year of Bordeaux radar data
It can furthermore be wondered if the influence of wind flows on the rain field
spatial correlation holds also on instantaneous basis. The analysis of several rain maps
has shown that this conclusion is not always valid: consider, for instance, Figure 19,
where it can be clearly noticed that, if in most of the areas the wind is orthogonal to the
rain bands, in some other zones, especially on the South-Western edge of the field, the
rainy front and the wind are almost parallel. A more detailed analysis of pressure fields
and of the underlying air streams may explain more physically these considerations.
Figure 19: Rain field over France acquired by the French radar Network Aramis in
October 2000 and concurrent ERA-40 wind outputs relative to the 700 hPa isobar
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6 Conclusion
In this work, weather radar data collected at two sites have been used to evaluate
the performance of Earth-satellite site diversity systems with different link and baseline
orientations. Results have shown a negligible dependence of the system performance on
the link orientation, whereas, on the contrary, the system gain has proven to be tightly
linked to the choice of the baseline direction. This result seems to be related to the
overall anisotropy of rain structures which generally tend to elongate perpendicularly to
the local prevalent direction of the wind: a baseline orientation parallel to the prevailing
wind direction (i.e. from North to South for Spino d’Adda and from West to East for
Bordeaux) has shown to maximize the system gain. As a further result of this study, a
rough classification of rain events into stratiform and convective according to “colder”
and “warmer” months has allowed to confirm the strong dependence of the system
performance on the type of precipitation, due to the different spatial correlation
characteristics associated to stratiform and convective phenomena. The site diversity
results obtained from radar data have been afterwards compared with those provided by
the model currently recommended by the ITU-R for the estimation of the site diversity
performance: the ITU-R recommendation has shown to underestimate the diversity gain,
especially when low reference attenuation levels are considered.
More in general, the analysis has shown that, despite the reduced temporal (5
minutes) and spatial (1 km × 1 km) resolution of the Bordeaux dataset, those radar scans
have proven to be adequate for site diversity simulations, as they provide results that are
fully comparable with the ones obtained from a dataset with a finer temporal (77
seconds) and spatial (0.5 km × 0.5 km) resolution (Spino d’Adda data). Specifically, this
study underlines the usefulness of radar data for the simulation and the performance
evaluation of a site diversity system, as they reflect the local climatologic and
topographic characteristics, which, on turn, may reveal some properties on the diversity
hardly caught by analytical estimation models.
With the aim of establishing a linkage between the most advantageous site
diversity baseline orientation and the local predominant wind direction, the rain field
advection has been estimated from successive radar maps (both in Spino d’Adda and
Bordeaux). Results have confirmed that, in both sites, rain structures generally tend to
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elongate perpendicularly to the local prevalent direction of the wind. In addition, ERA-
40 wind outputs relative to different isobars have been extracted and compared to the
radar derived rain field advection. Data relative to the 700 hPa isobar have proven to be
the most correlated both with the rain advection direction and speed: not only ERA-40
data can provide a useful indication for the choice of the optimum baseline orientation in
the design of site diversity systems, but, in addition, they can be used for the
parameterization of space-time rainfall models such as those presented in [Gremont and
Filip, 2004] and [Jeannin et al., 2007], in which the knowledge of the correct rain field
advection is of key importance..
Acknowledgements
This work has been partially developed in the framework of the European
Network of Excellence SatNEx.
References
Capsoni, C., M. D’Amico, “Performance of small-scale multiple-site diversity
systems investigated through radar simulations”, Radio Science, 42, 2007
Capsoni, C., M. D'Amico, P. Locatelli, “Statistical properties of rain cells in the
Padana Valley,” Journal of Atmospheric and Oceanic Technology (JTECH). In Press.
Castanet L., Bousquet M., Filip M., Gallois P., Gremont B., De Haro L.,
Lemorton J., Paraboni A., Schnell M. : "Impairment mitigation and performance
restoration", COST 255 Final Report, Chapter 5.3, ESA Publications Division, SP-1252,
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Fukuchi H., T. Saito, “Novel mitigation technologies for rain attenuation in
broadband satellite communication system using from Ka- to W-band,” 6th International
Conference on Information, Communications & Signal Processing, 10-13 Dec., 2007.
Goldhirsh, J. F. L. Robison, “Attenuation and space diversity statistics calculated from radar
reflectivity data of rain”, IEEE Transactions on Antenna and Propagation, 23(2), 221-227,1975.
Goldhirsh, J., B.H. Musiani, A.W. Dissanayake, L. Kuan-Ting, “Three-site
space-diversity experiment at 20 GHz using ACTS in the Eastern United States,” Proc.
IEEE, 85, 970-980, June, 1997.
Gremont, B., and M. Filip, “Spatio-temporal rain attenuation model for
application to fade mitigation techniques”, IEEE Trans. Ant. Prop., 52, 1245-1256,
2004.
ITU-R: “Rain height model for prediction methods,” Propagation in Non-Ionized
Media, Recommendation P.839-3, Geneva, 2001.
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Propagation in Non-Ionized Media, Recommendation P.838-3, Geneva, 2003.
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Earth-space telecommunication systems,” Propagation in Non-Ionized Media,
Recommendation P.618-9, Geneva, 2007.
Jeannin N., L. Féral, H. Sauvageot, L. Castanet, J. Lemorton, F. Lacoste,
“Stochastic Spatio-Temporal Modelling of Rain Attenuation for Propagation Studies”, EUCAP
2007, Edinburgh UK, Nov 2007.
Kitzmiller, H. D., G. F. Samplatsky, C. Mello, “Probabilistic Forecasts Of Severe
Local Storms In The 0-3 Hour Timeframe From An Advective-Statistical Technique”,
19th Conf. on weather Analysis and Forecasting/15th Conf. on Numerical Weather
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Paraboni, A., F. Barbaliscia, “Multiple site attenuation prediction models based
on the rainfall structures (meso or synoptic scales) for advanced TLC or broadcasting
systems,” Proceedings of the URSI GA, Maastricht, August 2002.
Riva, C., “Seasonal and diurnal variations of total attenuation measured with the
ITALSAT satellite at Spino d’Adda at 18.7, 39.6 and 49.5 GHz,” International Journal
of Satellite Communications and Networking, 22, 449–476, 2004.
Sokol, Z., “Nowcasting of 1-h precipitation using radar and NWP data”. Journal
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Statistical distribution of the fractional area affected by rain
Nicolas Jeannin(1), Laurent Féral (2), Henri Sauvageot (2), Laurent Castanet (1), and
Joël Lemorton (1)
(1)ONERA, Département Electromagnétisme et Radar, Toulouse, France
(2)Université Paul Sabatier, Laboratoire LAME, Toulouse, France. (3)Université Paul Sabatier, Observatoire Midi-Pyrénées, Laboratoire
d’Aérologie, Toulouse, France
Abstract
The knowledge of the fraction of an area that is affected by rain (or fractional
area) is of prime interest for hydrologic studies or for rainfall field modeling. Up to now
the statistical distribution of this parameter has been poorly studied. In the present
paper, a model of the statistical distribution of the fraction of an area affected by rain
over a given rainfall rate is proposed. It takes into account at the same time the size of
the area and the local climatology. The analytic formulation of the distribution is
established considering that rainfall fields can be obtained from a non-linear filtering of
a Gaussian random field. As the analytic derivation of the distribution lies on some
assumptions, the model accuracy is first evaluated from numerical simulations. It is then
shown that the model reproduces accurately the distribution of fractional areas derived
from radar observations of rain fields, for various rain thresholds, sizes of area, and
climatologies. A generic parameterization is then proposed for areas ranging from
100×100km2 to 300×300km2.
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1 Introduction
In a typical General Circulation Model (GCM), the spatial resolution of the
outputs is generally not sufficient to describe rain at a scale of interest for hydrologic
processes such as interception and runoff [Pitman, 1991; Thomas and Henderson-
Sellers, 1991]. Therefore, a representation of rainfall spatial variability at a finer scale is
necessary. Thus, many models were developed to assess the spatial variability of rain at
coarser scales, using either fractal approaches [Lovejoy and Schertzer, 1985; Over and
Gupta, 1996], stochastic approaches [Mejia and Rodriguez-Iturbe, 1974; Bell, 1987;
Lebel et al.,1998] or cellular approaches [Le Cam, 1961; Capsoni et al., 1987a, 1987b;
Goldhirsh 2000, Feral et al., 2003a, 2003b]. For most of these approaches, a key
parameter that has to be specified is the fraction f of the simulation area affected by rain,
often called lacunarity or intermittency. Moreover, it was shown that surface hydrology
exhibits a strong sensitivity to this parameter [Pitman et al., 1990]. Indeed, if the rain
fractional coverage over an area of interest and for a given rain amount decreases, the
climatology turns from evaporation dominated to runoff dominated, thus showing the
importance of this quantity on climate simulations.
In other respects, the knowledge of this parameter is also valuable to evaluate the
performances of earth-space telecommunication systems. Indeed, the attenuation
undergone by an earth space satellite link operating at a frequency above 20 GHz is
mainly driven by the rainfall rate along the path of the link (Castanet et al. [2001]).
High attenuations due to rain will result in an unavailability of the satellite link or will
require the use of adaptive fade mitigation techniques (Castanet et al. [2002], Neely et
al. [2003]). Considering a satellite spot beam whose diameter is typically of 300 km, the
knowledge of the fraction of the area affected by a rain rate over a given threshold gives
an estimate of the fraction of the user-terminals in the satellite spot beam that will
undergo a given level of attenuation. Consequently, this parameter gives an estimate,
considering the link budget, either of the number of the terminals that will be in outage
or of the additional margin that has to be provided to ensure a sufficient quality of
service.
A convenient way to prescribe the fractional area f was addressed from the
estimation of rainfall from space. Indeed, Donneaud et al. [1984], Chiu [1988], Braud et
al. [1993], and Oki et al. [1997] observed that the rain fractional coverage over a preset
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threshold is tightly correlated with the spatially averaged rain rate over the area of
interest. Particularly, whenever the local Probability Density Function (PDF) of rain is
known, Atlas et al. [1990], Kedem et al. [1990] and Sauvageot [1994] have shown that
the proportionality coefficient between the fractional area and the spatially averaged rain
rate can be determined from the rain conditional PDF (i.e. knowing that it is raining).
Invoking an ergodicity assumption of the rainfall process, Eltahir and Bras [1993] gave
a procedure to deduce the fractional area affected by rain from GCM outputs accounting
for the GCM spatial and temporal resolutions. Nevertheless, for rain field simulation
purposes, either as an input of the model [Feral et al., 2006] or as a tool to evaluate the
model accuracy [Guillot, 1999; Guillot and Lebel, 1999], it could be interesting to have
a simple way to get the statistical distribution of the rain fractional coverage for various
thresholds, while accounting not only for the local climatology but also for the size of
the simulation area.
This was attempted by Onof and Wheater [1996] who tried, from weather radar
observations over Wales, to approximate the PDF of the fractional area affected by rain
with a parabolic distribution. In other respects, from radar observations over the French
territory, Feral et al. [2006] approximated the distribution of the fractional area affected
by rain with an exponential law with mean 8.8%. Nevertheless, in both studies, the
dependence on the local climatology and the size of the area are not evaluated.
Moreover, only the rain/no-rain threshold is considered, so that the results highly
depend on the radar sensitivity.
From an analysis of radar data on various climatic zones and for various sizes of
observation area, this paper proposes an analytical expression of the Cumulative
Distribution Function (CDF) of the fractional area affected by rain over a preset
threshold. The formulation is derived from properties of rain fields generated using a
non-linear transformation on stationary Gaussian random fields, a generation scheme
that has already been reported in the literature (Bell 1987, Guillot 1999). This approach
has been theoretically justified in Ferraris and al., [2002]. The latter is based on two
steps. First, a correlated, stationary, homogeneous Gaussian random field is generated
on a lattice. Second, on each grid point, the Gaussian random variables are turned into
rain rate values R. More specifically, in Bell [1987] or Guillot [1999], the random values
of the Gaussian field lower than a value α driven by the local probability of rain are put
to 0 while the random values greater than α are transformed so that they follow the rain
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rate local conditional PDF )0R|R(p > . Nevertheless, as underlined by Guillot [1999],
the main defect of such kind of modeling is that the fraction of area affected by rain on
one realization is not prescribed. Furthermore, Guillot [1999] pointed out that if a long
range dependence is not introduced in the correlation function of the Gaussian random
field, the fraction of the area affected by rain converges towards the local temporal
expectancy to have rain, as a second order stationary random field is ergodic. This
means that the variance of the fractional coverage of rain computed from a large number
of simulated fields tends to 0. However, and as shown by Eltahir and Bras [1993], the
fractional coverage of rain is a highly variable parameter. Therefore, rain fields
simulated with the methodology described above must have a long range dependence to
be realistic. This long range dependence can be quantified by what Lantuejoul [1991]
called the “integral range”, related to the average value of the correlation function over
the domain of interest.
Section 2 of the present paper precisely addresses the effect of the long range
dependence on the statistical properties of the spatial average and spatial variance for
homogeneous stationary Gaussian fields generated on a finite grid. Particularly, it is
shown that, under some assumptions, the spatial distribution of the samples defining one
realization of a Gaussian field generated on a finite grid is approximately normal with
mean M and variance 1-σ², where σ² is the average value of the correlation matrix that
drives the dependencies between all the points of the grid. Additionally, the spatial
mean M is shown to be a random variable that follows a centered normal distribution
with variance σ². From those considerations, in section 3, an analytical formulation is
proposed to model the CDF of the fractional area of a stationary Gaussian random field
over a preset threshold α. The model accuracy is first evaluated numerically, from
simulated Gaussian random fields. Then, in section 4, the model is compared with
fractional area CDFs derived from true rain fields observed by weather radar over
various places. A set of parameters is then proposed in order to reproduce the CDF of
the fractional coverage of rain for various climatic zones and various sizes of area,
ranging from 100×100 km² to 300×300 km².
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2 Theoretical considerations on stationary Gaussian random fields
As mentioned in the introductive part, the simulation of rain field using a non
linear transformation of a Gaussian field has been already widely studied (Bell [1987] or
Guillot [1999]) and was seen to perform satisfactorily in numerous situations. In those
works, a methodology to derive the correlation function of the underlying Gaussian field
from observed rain fields on a range of a few hundreds of kilometers was developed.
The correlation function of the Gaussian field computed from rain fall observations
(radar, rain gauges) was found to have a steep decay in the first tens of kilometers and
was then only very slowly decreasing without reaching 0 at the end of the observation
range of some hundreds of kilometers. Considering rain gauges spread across Italy,
Barbaliscia et al [1992] and Bertorelli and Paraboni [2005] identified three scales of
evolution for the correlation of rain fields:
Below 100 km the correlation was shown to have a steep decay,
From 100 to 700 km the correlation was shown to be almost steady or very
slowly decreasing with distance,
After 700 km the correlation decreases slowly until total decorrelation.
This long range dependence can be explained by the non uniform repartition in
space over wide areas of the rainy structures and of the underlying pressure and
humidity field. This shape of correlation function is awkward because it implies that the
rain fields are stationary over wide areas. As rain fields display some preferential path
due for instance to the orography or to land sea transitions, this assumption would
probably collapse for some hundreds of kilometers. The range of validity of this
stationarity hypothesis is hardly assessed as the requirements to apply a formal
stationarity test for spatial stochastic processes as described in Fuentes [2005] are not
matched by rain fields. Consequently, the areas of side greater than 300×300 km² for
which the stationary hypothesis of the rain fields is likely to be unrealistic will not be
considered in the present paper.
The aim of the rest of this section is to derive from those models an expression of
the distribution of the fraction of an area f (or fractional area) over a given rain rate
threshold using the properties of the underlying Gaussian field. The methodology
developed in the following strives at finding an approximation of the spatial distribution
of one realization of a Gaussian field on a finite grid considering that it has a correlation
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function of the shape described above. In section 3, from this spatial distribution, an
expression of the fraction of the area of one realization of a Gaussian field is first
derived. Then, the statistical distribution of the fractional area considering a large
number of independent realizations of the Gaussian field is proposed, validated and
transposed to rain fields in section 4.
2.1 Spatial average M of a Gaussian random field
A stationary, homogeneous, standard, and centered Gaussian random field g
generated on a bidimensional grid L with size N×N can be interpreted as a set of
standard centered Gaussian random variables ( ))s(g),...,s(g),s(gg 2N21= correlated with
each other, and indexed by their position si on the grid. Let Σ be the N2×N2 correlation
matrix of the Gaussian random field g with general term )]s(g)s(g[E jiij =Σ , where E[]
is the expectancy operator. The spatial average M of the random field g over the grid L
is defined by:
)s(gN1M
2N
1ii2 ∑
=
= . (1)
As all the random variables g(si) follow a standard centered normal distribution,
the random variable M is also normal with an expectation of 0. The variance of M is
defined by:
2N
1i
N
1jij4
²N
1i
N²
1jji4
2N²
1ii4
2 2
N1
)s(g)s(gEN1
)s(gEN1)M(Var
σΣ ==
⎥⎦
⎤⎢⎣
⎡=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛=
∑∑
∑∑
∑
= =
= =
=
. (2)
Equation (2) shows that the variance Var(M) of the spatial average M is the mean
σ² of the terms of the correlation matrix Σ. At this stage we demonstrate that the spatial
average M of the field is normally distributed with mean 0 and variance σ².
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2.2 Spatial variance V of a Gaussian random field
Similarly, the spatial variance V of the Gaussian random field g over the grid is
given by:
( )∑=
−=2N
1i
2i2 M)s(g
N1V . (3)
The spatial variance V is a random variable and its expectation can be defined
as :
[ ] [ ] [ ]
[ ] [ ] [ ]
2
N
1i
N
1jij4
N
1iji
N
1j2i2
2N
1ii
2i2
2i
N
1i2
1)M(Var1
N12)M(Var1
)s(g)s(g(EN2MVar)s(gVar
N1
MEM)s(gE2)s(gEN1
])M)s(g([EN1]V[E
2 2
2 2
2
2
σ
Σ
−=−=
−+=
⎭⎬⎫
⎩⎨⎧
−+=
+−=
−=
∑∑
∑ ∑
∑
∑
= =
= =
=
=
. (4)
Consequently, the samples 2N..1ii ),s(g =ω of one realization g(ω) of the Gaussian
random field g are characterized by their spatial mean M(ω) and by their spatial variance
V. In compliance with section 2.1, the spatial mean M(ω) is drawn from a normal
distribution whose mean is 0 and whose variance σ² is the mean of the correlation
function Σ. In such conditions, the expectancy E[V] of the spatial variance V is 1-σ², as
shown by (4). However, the spatial distribution of the N² random values 2N..1ii ),s(g =ω
is still unknown. Nevertheless, if the correlation function decrease sufficiently fast with
regards to the size of the lattice (ie 02 →σ see (Lantuejoul [1991]), the distribution of
the 2N..1ii ),s(g =ω tends to be normal if the size of the lattice is large enough. As
mentioned in the preliminary part of this section, the correlation function of the
underlying Gaussian field used to simulate rain fields exhibits a very slow decay with
distance. In that case, considering a lattice on which the decorrelation of the sample is
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not reached from one side to the other, the normality of 2N..1ii ),s(g =ω is not ensured
because σ² will not tend to 0. Nevertheless, if between all the points of the grid there
exists a residual positive correlation denoted by a², the correlation gc of the stationary
Gaussian field g can be rewritten:
)ss(cba)ss(c 21g~22
21g −+=− , with 1ba 22 =+ . (5)
The random field g can thus be considered as the weighted sum of a standard
centered Gaussian random field g~ whose correlation is gc ~ and weight b with a standard
centered Gaussian random variable y and weight a. Consequently, for each point s of the
grid:
ay)s(g~b)s(g += . (6)
If g~c rapidly decreases with regards to the size of the lattice, the spatial
distribution of 2N..1ii ),s(g~ =ω tends to be normal. Moreover, as y does not vary in space,
the spatial distribution of 2N..1ii ),s(g =ω also tend to be normal. Considering the
correlation functions proposed for rain field simulation by Barbaliscia et al. 1992 or
Guillot and Lebel [1999], and grid sizes ranging from 100×100 to 300×300 km2, the
steep decay g~c with regards to the size of the grid is ensured and the normality of
2N..1ii ),s(g =ω is a reasonable assumption. As no means were found to quantify the
derivation from normality of 2N..1ii ),s(g =ω due to the finiteness of the lattice, the effect
of this approximation on the accuracy of the model proposed in section 3.1 is assessed
from simulated Gaussian fields in section 3.2.
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3 Model for the CDF of the fractional area over a given threshold
3.1 Analytical derivation
As discussed in section 2.2, for one realization g(ω) of a stationary standard
centered Gaussian random field g with suitable shape of correlation function, the spatial
distribution of 2N..1ii ),s(g =ω tends to be normal if the size of the grid L is sufficiently
large. For the purpose of this study, the spatial distribution 2N..1ii ),s(g =ω is considered
as a normal distribution with mean M and variance V=1-σ². Besides, as shown in section
2.1, M is random and follows a centered normal distribution with variance σ². Therefore,
considering one realization ω of the homogeneous random field g, the fractional area
fα(ω) of g over which the threshold value α is exceeded is given by:
⎟⎟⎠
⎞⎜⎜⎝
⎛
−
−=
)1(2)(Merfc
21)(f
2σωαωα , (7)
where erfc denotes the complementary error function.
The CDF of the fractional area f exceeding the threshold α can then be derived
considering the CDF of M:
( ))2(erfc)1(2P
)1(2erfc
21P)(P
*12
*
2
*
fM
fMff
−−−>=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛>
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−=>
σα
σα
α. (8)
Recalling that M follows a centred normal law with variance σ², we have:
⎟⎠
⎞⎜⎝
⎛=>σ2
erfc21)(P xxM , (9)
so that:
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−=>
−
σσα
α 2)2(erfc)1(2
erfc21)(P
*12* f
ff . (10)
Derivation of (10) gives the PDF pα(f) of the fractional area f above α:
[ ] ⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−−
−−=
−−
σσα
σσ
α 2)1(2)2(erfc
exp1)2(erfcexp)(212
21 fffp (11)
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The analytical formulation of the fractional area average value <f>α is difficult to
obtain from (11). Nevertheless, it can be accurately approximated by:
)2
(erfc21)( ααα =>=>< GPf . (12)
(12) is obtained by considering that, over several realizations of the Gaussian
random field g, the fraction f of field g above the threshold value α is simply the overall
number of points above α divided by the overall number of points. As g is
homogeneous, the expectancy to exceed α is the same everywhere in the lattice so that
<f>α amounts to (12), i.e. to the probability for a Gaussian centred standard variable G
to exceed α. The numerical computation of <f>α from the PDF (11) with different
parameterizations complies satisfactorily with (12), i.e. with a maximum error lower
than 1%.
In compliance with (10) and (11), the statistical distribution of the fractional area
f depends on 2 parameters. The first one, α, is related to the average value of f by (12).
The second one, σ², is the average value of the N²×N² Gaussian field correlation matrix
Σ. Obviously, σ² strongly depends on the size N×N of the simulation grid. So as to
quantitatively assess the model accuracy and the parameter sensitivity, (10) is compared
to the fractional area CDFs derived from Gaussian field simulation.
3.2 Model accuracy and parameter sensitivity
For each configuration detailed hereafter, 6000 stationary Gaussian fields are
generated using the spectral method described by Bell [1987]. Further details on this
approach can be found in Shinozuka and Jan [1972], Meija and Rodriguez-Iturbe
[1974], and Borgman et al. [1984]. Lattice size varies from N×N=50×50 km² to
N×N=200×200 km², in compliance with the typical coverage of the operational radars
considered in section 4. Moreover, to evaluate the model sensitivity, two analytical
formulations of the correlation function are considered for the Gaussian field g: 30/d
1,g e)d(c −= and 800/d30/d2,g e5.0e5.0)d(c −− += , where d is the distance in km. Such
exponential formulations of the correlation function are commonly accepted to simulate
rain fields (Guillot and Lebel [1999], Barbaliscia et al. [1992], Feral et al. [2006]). In
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such conditions, considering )d(c 1,g and equation (2), σ varies from 0.71 to 0.34 when
N×N varies from 50×50 km² to N×N=200×200 km², respectively. Similarly, considering
)d(c 2,g , σ varies from 0.85 to 0.72 when N×N varies from 50×50 km² to N×N=200×200
km², respectively. Six threshold values α are considered successively to compute the
fractional area CDFs from the simulated Gaussian fields, namely α=0.5, 1, 1.5, 2, 2.5, 3.
The distributions thus obtained are compared with (10). The results are shown in Fig. 1
and 2. It then appears that, whatever the configuration, model (10) accurately reproduces
the fractional area CDFs derived from the simulated fields. As all the realization of the
simulated field are independent the values f obtained for each realization of the fields
are independent and common statistical test can therefore be applied to test the validity
of (10). Particularly, the null hypothesis “the CDF )(P *ff >α derived from the
simulated Gaussian fields is (10)” is never rejected by a unilateral Smirnov-Kolmogorov
test [Chakravarti et al., 1967] with a confidence level of 0.05. To conclude, model (10)
accurately reproduces the Gaussian field fractional area CDFs whatever the threshold α,
the lattice size between 50×50 km² and 200×200 km², and the standard formulations
)d(c 1,g or )d(c 2,g of the correlation function.
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0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1CDF of the fraction of the field over α
f*
P(f
<f* )
α=0.5α=1α=1.5α=2α=2.5α=3
Figure 1: CDFs of the fractional area over the threshold α computed from model (11)
and derived from the simulation of Gaussian fields: size N×N=200×200 km², correlation
function )800/exp(5.0)30/exp(5.0)( dddcG −+−= .
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1CDF of the fraction of the field over α
f*
P(f
<f* )
α=0.5α=1α=1.5α=2α=2.5α=3
Figure 2: CDFs of the fractional area over the threshold α computed from model (11)
and derived from the simulation of Gaussian fields: size N×N=50×50 km², correlation
function )800/exp(5.0)30/exp(5.0)( dddcG −+−= .
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4 Application to rain fields
4.1 Model extension to rain field
In the previous section, a mathematical framework to describe the CDF of the
fractional area of a homogeneous Gaussian field that exceeds a preset threshold α has
been developed. The same methodology applies to rain fields whenever the latter are
derived from a non-linear transformation of a stationary homogeneous Gaussian field
[Bell, 1987; Guillot, 1999]. Indeed, considering that the rain rate R conditional PDF is
lognormal (mean Rμ , standard deviation Rσ ) as it is commonly accepted [Bell, 1987;
Sauvageot, 1994; Feral et al., 2006], and assuming that it rains a fraction P0 of the time,
the transformation ψ that converts the Gaussian field g into a rain field r is:
[ ]
[ ]0
0
P0
1RR
P
)s(gfiP
)2/)s(g(erfcerfc2exp)s(gψ)s(r
)s(gif0)s(gψ)s(r
ασμ
α
≥⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+==
<==
−,(13)
where
)P2(erfc2 01
P0
−=α . (14)
By construction, the expectation for a Gaussian value to be converted into rain is
exactly P0. Moreover, the computation of the CDF of the fraction f of a rain field
affected by a rain rate R exceeding the threshold R* amounts to the computation of the
fraction of a standard Gaussian field exceeding the threshold )(ψ *1* RR
−=α . As
[ ] [ ]*R* )s(gPR)s(rP α>=> , we have:
[ ] *1R
R)s(rP2erfc2* >= −α . (15)
If we assume that the rain field r(s) is homogeneous, the probability of rain P0 is
the same all over the grid and [ ]*R)s(rP > does not depend on the location s so that (15)
reduces to:
[ ])(P2erfc2 *1* RR
R>= −α , (16)
where P(R>R*) is the rain absolute CDF intrinsic to the simulation area. (16)
gives a convenient way to determine *Rα from radar data.
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Finally, the CDF of the fractional area f of a rain field affected by a rain rate
exceeding the threshold value R* derives from (10):
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−=>
−
σσα2
)2(erfc)1(2erfc
21)(P
*12*
R
*
*
fff R , (17)
under the hypothesis made to establish (10) and (17), namely the homogeneity of
the field (ie. the pdf of rainfall rate dos not vary inside the considered area) and a
correlation function with a fast and a slow decay component with respect to the lattice
size as advocated by Barbaliscia et al., [1992] or Guillot and Lebel [1999].
4.2 Radar data
Two yearly radar datasets are considered. The first one was collected in 1996 and
comes from the weather radar of Bordeaux that is part of the French operational radar
network managed by Météo-France. The second one refers to 2004 and consists of
composite radar images from the US NEXRAD network. For both datasets, the temporal
sampling is one observation every 5 min. The data were projected on a Cartesian grid
within a uniform pixel size of 1×1 km². Images including ground clutter or melting layer
echoes were removed from the dataset so that the used radar data only refer to rainfall
fields. Lastly, reflectivity fields were converted into rain fields using the standard Z-R
relation:
baRZ = , (18)
where a=300, b=1.35, and Z is the radar reflectivity factor in mm6 m-3 and R the
rain rate in mm h-1.
As shown in Fig. 3 and 4, areas of interest offering clearly distinct climatologies
have been selected inside the radar coverage areas. Moreover, for each selected location,
five area sizes are considered successively, namely 100×100, 150×150, 200×200,
250×250 and 300×300 km².
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Figure 3: Geographical position of the 3 areas of interest selected over the US. The
background image is a rain field observation collected by the NEXRAD radar network
on 08/27/04 at 17:00 UTC.
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Figure 4: Geographical position of the area of interest over Bordeaux (France). The
background image is an example of rain field observation collected by the ARAMIS
radar network on 09/129/01 at 16:00 UTC.
4.3 Results
The CDFs of the fractional area affected by rain were computed for thresholds
R*=0.5, 1, 2.4, 5.7, 13 mm h-1 for the areas of interest over the US and R*=1, 1.4, 2.2,
3.5 mm h-1 for the ones over Bordeaux (the thresholds considered are not exactly the
same for both datasets due to some differences in the quantization steps between the
radar of Bordeaux and the NEXRAD network).
First, the rain rate absolute climatological probabilities P(R>R*) are computed by
counting the number of pixels over R* on the whole dataset and by dividing it by the
overall number of pixels. These distributions may encounter more or less large
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fluctuations on the same geographical zone for the different area sizes. This may be due
to some defects in radar measurements (particularly attenuation with increasing range)
or to some climatologic inhomogeneities of rain processes in squares of some hundreds
of kilometres of side [Onof and Wheater, 1996]. Then, for the successive values of R*,
the corresponding *Rα is deduced from (16). The results are given in table 1.
South Dakota
R*=0.5 mm h-1 R*=1 mm h-1 R*=2.4 mm h-1 R*=5.7 mm h-1 R*=13 mm h-1 Area size km2
P(R>R*) *Rα P(R>R*) *R
α P(R>R*) *Rα P(R>R*) *R
α P(R>R*) *Rα
100×100 0.49% 2.6 0.25% 2.8 0.10% 3.1 0.04% 3.3 0.01% 3.6 200×200 0.56% 2.5 0.28% 2.8 0.12% 3 0.05% 3.3 0.02% 3.5 300×300 0.65% 2.5 0.33% 2.7 0.15% 3 0.07% 3.2 0.03% 3.4
Mississipi
R*=0.5 mm h-1 R*=1 mm h-1 R*=2.4 mm h-1 R*=5.7 mm h-1 R*=13mm h-1
Area
size (km²) P(R>R*) *R
α
P(R>R*) *Rα
P(R>R*) *R
α
P(R>R*) *Rα
P(R>R*) *R
α
100×100 1.8% 2.1 1.1% 2.3 0.63% 2.5 0.29% 2.8 0.12% 3
200×200 1.8% 2.1 1.1% 2.3 0.60% 2.5 0.29% 2.8 0.13% 3
300×300 1.8% 2.1 1.1% 2.3 0.61% 2.5 0.29% 2.8 0.13% 3
Ohio
R*=0.5 mm h-1 R*=1 mm h-1 R*=2.4 mm h-1 R*=5.7 mm h-1 R*=13 mm h-1
Area
size (km²) P(R>R*) *R
α
P(R>R*) *Rα
P(R>R*) *R
α
P(R>R*) *Rα
P(R>R*) *R
α
100×100 1.8% 2.1 1.1% 2.3 0.60% 2.5 0.26% 2.8 0.10% 3.1
200×200 2.0% 2 1.3% 2.2 0.72% 2.5 0.30% 2.7 0.10% 3.1
300×300 2.1% 2 1.4% 2.2 0.73% 2.4 0.28% 2.8 0.09% 3.1
Bordeaux
R*=1 mm h-1 R*=1.4 mm h-1 R*=2.2 mm h-1 R*=3.4 mm h-1
Area size (km²) P(R>R*) *Rα
P(R>R*) *Rα
P(R>R*) *Rα
P(R>R*) *Rα
100×100 0.8% 2.4 0.5% 2.6 0.2% 3 0.1% 3.2
200×200 1.2% 2.3 0.8% 2.4 0.4% 2.7 0.2% 3
Table 1: Rain rate absolute CDFs P(R>R*) derived from radar observations for the
successive areas of interest and associated value of *Rα computed from (17).
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For R*=1 mm h-1 only, model (17) ⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−=
−
σσα
σα2
)2(erfc)1(2erfc
21),,F(f
121
1iif
is regressed with respect to the fractional area CDF P1(f>f*) derived from radar
observations to obtain parameters σ reported in table 3 (first line, area size
N×N=100×100 km² located over the US and over Bordeaux). The values of σ were
obtained solving (19) using a non linear least square algorithm:
∑ ⎥⎦
⎤⎢⎣
⎡>
>−
i i
ii
ffPffPfF
2
1
11
)()(),,(minarg σα
σ, (19)
where α1 is determined from (16) and the fi are regularly sampled from 0.01 to
fmax by step of 0.01. fmax is defined as the 30th largest value of rain occupation found on
each radar data set for each considered threshold R*. This choice of fmax is made in
order to ensure a sufficient statistical reliability to the empirical CDFs especially for the
greatest values of occupation. This regression procedure allows defining a relative error
criterion between the model (17) and the CDF regressed from radar data:
∑⎥⎥⎦
⎤
⎢⎢⎣
⎡
>
>−×=
i iR
iRRi
ffPffPfF
N
2
)()(),,(1100
*
** σαε (20)
(20) quantifies the error made when approximating the empirical occupation CDF
by model (19) in terms of mean relative RMS error.
The values of σ, obtained using this regression procedure for each dataset are
used for all the other values of R* as σ should not depend on the threshold whenever rain
fields are stationary and homogeneous. Besides, R*=1 mm h-1 is the value considered in
the regression process because, first, it is the lowest value common to both datasets, and
second, it complies with Krajewski et al., [1992] that advocates the use of a relatively
low threshold to better estimate the fractional area from real data.
For the successive threshold values R*, Fig. 5 and 6 show the fractional area
CDFs derived from model (17) and those derived from the radar observations over
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Bordeaux (France), for area sizes N×N=100×100 km² (σ regressed for R*=1 mm h-1 is
0.88) and N×N=200×200 km² (σ regressed for R*=1 mm h-1 is 0.77), respectively.
Similarly, the results obtained for Ohio (US) are shown in Fig. 7, for N×N=100×100
km² (σ regressed for R*=1 mm h-1 is 0.9). Lastly, Fig. 8 and 9 underline the fractional
area CDF dependence on the geographical location and the size of the area, respectively.
The impact of the location on the distribution is mainly driven by the local rain rate
CDF while increasing the area size lowers the average value σ² of the correlation
function.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.96
0.965
0.97
0.975
0.98
0.985
0.99
0.995
1CDF of the fraction of the area with a rain rate exceeding R
f*
P(f
<f* )
1mm/h1.4 mm/h2.2 mm/h3.4mm/h
Figure 5: CDFs of the fractional area f affected by rain over a given threshold R*
computed from (18) (markers) and derived from the radar of Bordeaux (solid lines)
N×N=100×100 km².
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0 0.1 0.2 0.3 0.4 0.50.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1CDF of the fraction of the area with a rain rate exceeding R
f*
P(f
<f* )
1mm/h1.4 mm/h2.2 mm/h3.4mm/h
Figure 6: CDFs of the fractional area f affected by rain over a given threshold R*
computed from (18) (markers) and derived from the radar of Bordeaux (solid lines)
N×N=200×200 km².
Nevertheless, in compliance with Fig. 5 to 9, model (17) satisfactorily compares
with the fractional area CDFs derived from radar observations whatever the threshold
R*, the location, or the area size N×N ranging 100×100 to 300×300 km². However, some
trends can be noticed considering the errors computed according to (20) for different
areas, rain rate thresholds and locations as shown in table 2. Firstly, the larger are the
considered areas, the higher the errors between the two distributions. Secondly, for
threshold values ≥*R 5 mm h-1 for US radar and ≥*R 3.4 mm h-1 for Bordeaux, the
model tends to overestimate the probability of having a large fraction of the area
affected by rain.
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Rain rate threshold 100x100 km2 200x200 km2 300x300 km2 Bordeaux R=1 mm h-1 0.54 0.82 1.9 R=3.4 mm h-1 2.4 2.3 8.4 Ohio R=1 mm h-1 1.5 3.2 8.5 R=5 mm h-1 7.1 14 21 Dakota R=1 mm h-1 0.65 1.2 3.9 R=5 mm h-1 6.7 14 13 Mississippi R=1 mm h-1 2.5 4.0 4.6 R=5 mm h-1 8.1 13 17
Table 2: Mean relative RMS error as defined by (21) between the model (18) and the
empirical CDF deduced from radar observations (%) for different rain rate thresholds
and different areas.
0 0.1 0.2 0.3 0.4 0.5 0.60.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
f*
P(f
<f* )
CDF of the fraction of the areaaffected by rain rates over R
0.5 mm/h1 mm/h2.4 mm/h5.7 mm/h13 mm/h
Figure 7: CDFs of the fractional area f affected by rain over a given threshold R*
computed from (18) (markers) and derived from radar observations over Ohio (solid
lines) N×N=100×100 km².
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0 0.1 0.2 0.3 0.4 0.50.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
f*
P(f
<f* )
CDF of the fraction of the areaaffected by a rain rate over 1mm/h
DakotaMississipiOhioBordeaux
Figure 8: CDFs of the fractional area with a rain rate over 1 mm.h-1 for areas of
100×100 km² located in the 4 successive geographical places of interest.
0 0.2 0.4 0.60.9
0.92
0.94
0.96
0.98
1
f*
P(f
<f* )
CDF of the fraction of the areaaffected by rain rates over 1mm/h
100km radar100km model300km radar300km model
Figure 9: CDFs of the fractional area with a rain rate over 1 mm.h-1 for areas with size
100×100 km² and 300×300 km² located over Mississippi.
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These two trends question the rain field homogeneity hypothesis used to derive
(10) and (17). Indeed, over areas of hundreds of square kilometres, the rain rate local
CDF may differ from one point to the other. In such conditions, the model no longer
holds because the stationarity and homogeneity of the field become an unrealistic
assumption. In other respects, the differences observed for increasing values of R* may
be linked to the use of a constant parameter to describe the average value σ² of the
correlation function. Indeed, high rain rates result from a convective process. Now, the
radar observation of convective rain fields shows clustered structures, whose spatial
extent is clearly lower than that observed for stratiform rain events. This point is
partially confirmed when regressing model (17) for R*=5 mm h-1 (US radar) and R*=3.4
mm h-1 (Bordeaux) to determine σ. Indeed, the results reported in table 3 show that,
whatever the location, σ is slightly lower than the value obtained for R*=1 mm h-1,
suggesting (as expected) that high rain rates are less correlated in space than lower rain
rates because the average of the correlation of the underlying Gaussian fields for rain
fields thresholded with high values is lower than the one found for lower thresholds.
This may be due to a faster decay of the correlation considering rain fields with a more
significant proportion of convective rain.
Dakota Mississipi Ohio Bordeaux R*=1 mm h-1 0.86 0.90 0.92 0.88 R*=5 mm h-1 0.83 0.87 0.90 0.85 (R*=3.4 mm h-1)
Table 3: Regressed value of σ for 2 rain rate thresholds R* and for a 100×100 km² area
of interest.
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4.4 Parameterization of the distribution
The results obtained for the parameter σ of the distribution display an interesting
stability as shown in Fig. 10.
100 150 200 250 3000.72
0.74
0.76
0.78
0.8
0.82
0.84
0.86
0.88
0.9
Length of the side
σ
MississipiOhioSouth DakotaBordeauxBest regression
Figure 10: Dependence of the parameter σ on the size of the considered area for
various locations.
Indeed, for different climatic regions, this parameter seems to have
approximately the same behaviour with regards to the evolution of the size of the area.
A generic parameterization of σ independent from the location can hence be regressed
and given as an indication for areas ranging from 100×100 to 300×300 km2:
L0007.094.0 −=σ , (21)
where L is the length of the side in km. The relatively high values found for σ
argue in favour of the existence of a long range correlation confirming the hypothesis
made in the second section of this study. (21) allows to get for an arbitrary location an
approximation of the distribution of the fraction of an area L×L affected by a rain rate
over a given threshold, whenever the local distribution of rain rates is known. Indeed,
from Rμ , Rσ and 0P where Rμ , Rσ are the parameters of the lognormal pdf that
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characterizes the conditional distribution of rain rates and 0P is the probability of rain,
the parameter α needed to compute the distribution of the fraction of the area affected
by rain can be deduced from (16).
When combined with (17), parameters α and σ obtained that way allow to
approximate the CDF of the fraction of an area affected by rain for areas ranging from
100×100 to 300×300km2. The lower bound of this domain is driven by the requirements
on the size of the area with regards to the alleged underlying correlation function. The
upper bound is set considering the hypothesis of stationarity of the rain fields and of
spatial invariance of the point rainfall rate distribution across the areas that were
considered to establish the model. It should be noticed that the data used in this study
have been projected on a grid of 1×1 km2. Considering data with a lower spatial
resolution, the parameterization proposed will no longer hold. First, the standard
deviation of the distribution of rainfall rate that is used to determine the thresholds will
change. Second, the parameter 2σ that corresponds to the spatial average of the
correlation is likely to be slightly higher as the reduction of the resolution will smooth
the data and consequently generates higher correlation.
5 Concluding remarks and applications
From considerations on stationary Gaussian random fields, a model that
reproduces the CDF of the fractional area affected by rain over a preset threshold has
been proposed for areas ranging from 100×100 to 300×300 km2. It has been developed
assuming that rain fields can be seen as a non-linear transformation of a stationary
Gaussian random field.
First, the model was shown to match very accurately the fractional area CDF
derived from simulated Gaussian random fields. Second, when confronted to
distributions obtained from radar observations of rain field, the model satisfactorily
reproduces the CDF of the fractional area above a rain threshold R*. This model
requires two inputs. The first one α is related to the local distribution of rain rates, the
second one σ to the mean of the correlation function of the underlying Gaussian field.
A parameterization accounting for the local climatology and for the size of the area was
proposed for this parameter. The optimal value of σ was found to be slightly dependent
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on the threshold considered on radar data, questioning the hypothesis of the stationarity
of the field.
The high values found in any case for the parameter σ put forward the long
range dependence that has to be introduced in the correlation function of the Gaussian
field when generating rainfall fields. Important conclusions can be drawn from those
results for simulation purposes. For models such as those described by Bell [1987] or
Guillot and Lebel [1999], the use of a correlation function that has a mean value of 2σ
allows, when generating a large collection of simulated fields, to reproduce the local
distribution of rain rates and the distribution of the fraction of the area affected by rain
over a given threshold. Additionally, the distribution of the fraction of the area affected
by rain is required by rainfall field generators such as the one described in Feral et al.,
[2006]. The methodology presented here provides a convenient mean to approximate
this distribution without computing it from radar data.
Acknowledgment:
The authors are very grateful to Météo-France and the US NEXRAD network for
providing freely radar images. This research was partly supported by CNES (contract
n°05/CNES/2038/00) and Thales Alenia Space (contract n°A88550) and has been partly
carried out in the framework of the European Network of Excellence SatNEx.
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A large scale space-time stochastic model of rain attenuation for the design and optimization of
adaptive satellite communication systems operating between 10 and 50GHz
Nicolas Jeannin (1), Laurent Féral (2), Henri Sauvageot (3),
Laurent Castanet (1), Frédéric Lacoste (4).
(1)ONERA, Département Electromagnétisme et Radar, Toulouse, France
(2)Université Paul Sabatier, LAboratoire Micro-ondes et Electromagnétisme, Toulouse,
France (3)Université Paul Sabatier, Laboratoire d’Aérologie, Toulouse, France
(4)Centre National d’Études Spatiales (CNES), Toulouse, France
Manuscript submitted to IEEE Antennas and Propagation
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Abstract:
The design and optimization of propagation impairment techniques for space
telecommunication systems operating at frequencies above 20 GHz require a precise
knowledge of the propagation channel both in space and time. For that purpose, space
time channel models have to be developed. In this paper the description of a model for
the simulation of long term rain attenuation time series correlated both in space and time
is described. It relies on the use of a stochastic rain field simulator constrained by the
rain amount outputs of the ERA-40 reanalysis meteorological database. With this
methodology, realistic propagation conditions in terms of spatio-temporal correlation
can be generated at the scale of satellite coverage (i.e. over Europe or USA). To increase
the temporal resolution, a stochastic interpolation algorithm is used to generate spatially
correlated time series sampled at 1 Hz, providing that way valuable inputs for the study
of the performances of propagation impairment techniques required for adaptive SatCom
systems operating at Ka band and above.
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Introduction
With the congestion of conventional frequency bands such as C (4-6 GHz) or Ku
(11-14 GHz) band and the need to convey higher data rates for multimedia services, new
Sat Com systems are progressively pushed towards the use of higher frequency bands
such as Ka (20-30 GHz) or Q/V band (40-50 GHz) where larger bandwidth are available.
However, at those frequencies, strong tropospheric impairments occur on Earth-space
links with a significant impact on the system quality of service. The latter are due to
rain, clouds, water vapor and oxygen. The resulting attenuation can not be mitigated
simply acting on the margin taken on the overall link budget as it will result in a major
loss of efficiency. In order to perform a good trade off between quality of service,
system capacity and link availability, adaptive resource allocation mechanisms using
Fade Mitigation Techniques (FMT) have to be implemented [Cost 255, 2002].
FMT implementation leads also to consider the system resource allocation and
therefore upper layer issues that can be studied through protocol simulations. Network
simulations for SatCom systems operating at Ka and Q/V-bands have to take into
account the influence of the propagation channel, not only in terms of dynamics but also
in terms of the spatial variations [Bousquet et al., 2003]. In order to assess FMT
efficiency or the resulting system availability, the propagation channel has to be
simulated.
To emulate channel dynamic, attenuation time series synthesizers [Masseng and
Bakken, 1981], [Gremont and Filip, 2004], [Carrier et al., 2008] have been developed.
They are crucial to design and optimize Up Link Power Control (ULPC) systems, or
data rate reduction mechanisms [Castanet et al., 2001]. Nevertheless, the knowledge of
the temporal dynamic of the propagation channel is not sufficient to implement all the
FMTs. For instance, the optimization of site diversity systems requires also the
knowledge of the spatial variability of the propagation impairments.
Several models and methods exist to depict the spatial variability of tropospheric
impairments, especially the attenuation due to rain. Indeed, attenuation fields and the
underlying rain fields can be modeled by rain cells [Capsoni et al., 1988a, 1988b],
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[Féral et al., 2003a, 2003b, 2006] [Montopoli and Marzano, 2008], by fractals
[Callaghan, 2006] or random fields [Gremont and Filip, 2004]. Even if the spatio
temporal description of the propagation channel is of great interest for the design and
optimization of the above mentioned FMTs, it is not adapted to solve the issues raised
by on-board radio resource management for which a description both in space and in
time of the propagation channel over the whole satellite coverage is required.
Up to now, few models exist to combine the temporal and the spatial description
of the propagation channel. [Gremont and Filip, 2004] have adapted the model of rain
fields correlated in space and in time developed by [Bell, 1987], while [Bertorelli and
Paraboni, 2005] have proposed to correlate scaled time series from measurements made
during the ITALSAT campaign. The main limitation of those models lies in their
temporal or spatial range of validity. Indeed, the model proposed by [Gremont and Filip,
2004] is realistic for short durations (some hours) and small areas (a few hundreds of
square kilometers) due to the stationarity assumption necessary to construct the random
field in the Fourier plane. The model of correlated time series presented in [Bertorelli
and Paraboni, 2005] is limited to duration of one hour. As network simulations, to be
statistically reliable, require attenuation data during several years and over the satellite
coverage as a whole, there is a need for propagation inputs correlated in space and in
time for long durations and large areas.
The aim of this paper is to extend the range of validity of the stochastic rainfall
field simulator described in [Bell, 1987] and adapted for rain attenuation in [Gremont
and Fillip, 2004], using rain amount time series provided by general circulation models.
Indeed, those stochastic models are shown to reproduce appropriately the spatial and
temporal characteristics of observed rain fields, but for limited areas and durations. To
overcome these limitations, a methodology to constrain the model with inputs from the
low-resolution reanalysis database ERA-40 provided by the ECMWF (European Center
for Medium-range Weather Forecast) is presented. The use of ERA-40 data ensures a
realistic long term evolution of the rain fields as they reproduce large-scale conditions
from past meteorological situations.
The basics and limitations of the stochastic modelling of rain rate field presented
in [Bell, 1987] are recalled in the first part of the paper. A set of parameters more
suitable for temperate areas is also determined from a radar dataset acquired by the
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weather radar from Bordeaux (France). In a second part of the paper, a methodology to
parameterize the model with outputs from the ECMWF ERA-40 reanalysis database is
developed. Some statistical properties of the simulations are lastly compared to statistics
derived from weather radar datasets, ITU-R Recommendations and to statistics obtained
from beacon measurements.
1 Spatio-temporal modelling of rain fields
In this section, the basics of the rain field space-time stochastic modelling firstly
presented in [Bell, 1987] are detailed. This model was originally developed to study the
performances of satellite rainfall sensors (i.e the precipitation radar of TRMM) from
simulated rain fields which statistical characteristics must be as close as possible to the
ones of real rain fields. It was constructed to mimic the correlation structure and the
local climatologic Cumulative Distribution Function (CDF) of rain rates given as input
parameter.
The rain fields R(x,t) are generated on a domain with size N×N (grid step Δx)
during a time interval T (time step Δt) where x is the spatial index and t the temporal
one. It is assumed that for each spatial location and each time index, the rain rate is the
realization of a random variable. Using this methodology, the simulation is conducted in
two main steps. First, a stationary Gaussian random field G(x) with correlation structure
cG is generated in space. Second, to account for the temporal evolution, each spatial
frequency k of G(x) is constrained to follow a first order Markov process which
innovation depends on the spatial frequency. Moreover, the field advection is modelled
by a phase shift in the frequency domain.
Then, the Gaussian field is turned into a rainfall field converting the samples
from the Gaussian field G(x,t) into rainfall rate samples R(x,t). For that purpose, the
rainfall rate distribution is assumed to be zero with probability 1-P0 and lognormal with
probability P0, where P0 denotes the local probability of rain.
The different steps of this modelling are detailed in this section. A methodology
to derive the correlation parameters from weather radar observations is also presented. It
will be applied to the radar observations of Bordeaux.
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1.1 Generation of correlated Gaussian random field
A convenient framework to simulate a stationary random field R that has an
arbitrary distribution is to define a monotonic transformation ψ that converts a Gaussian
field G into the field R [Guillot, 1999]. It allows taking advantage of efficient methods
for the simulation of Gaussian fields and obtaining the desired field R from ψ(G). If the
correlation cR is estimated from realizations of the field R, different methodologies
described in section 1.3.2 exist to define a correlation cG such that R=ψ(G) will have the
correlation cR.
1.1.1 Generation of the Gaussian field in space
A Gaussian random field can be interpreted as a collection of Gaussian random
variables correlated the ones with each other. A Gaussian random field is said to be
stationary if its correlation function cG between two points x and y depends only on the
distance between x and y:
)yx(c)y,x(c GG −= (1)
This property of stationarity of the field is essential from a computational point
of view. Indeed, it allows reducing the size of the correlation matrix from N2×N2 to N×N
to use efficient algorithms such as turning band methods [Mantoglou and Wilson, 1982]
or circular embedded correlation matrices [Kozintsev and Kedem, 2000].
Another convenient methodology to generate a stationary Gaussian field G is to
start from its spectrum s that is directly obtained from the Fourier transform of the
correlation function cG [Bell, 1987]. The N×N stationary Gaussian random field G can
be expressed for each point ),( 21 xxx = of the simulation grid by its Fourier expansion:
∑ ∑−
=
−
=
=1
0
1
01 2
)/2exp()(N
k
N
k
Tk NxkiaxG π , (2)
where ),( 21 kkk = represents the spatial frequency. Coefficients ak are shown to
be equal to:
kkk esa = , (3)
where sk is the spectral density corresponding to the spatial frequency k and the ek
are independent standard centred complex Gaussian random variables. (2) and (3) show
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that the knowledge of the power spectrum of the field or reciprocally of its correlation
function cG is sufficient to generate a spatially correlated Gaussian field.
1.1.2 Addition of the temporal dimension
The rain field temporal evolution can be decomposed into two components. The
first one is related to the advection of the rain fields due to air streams. The second one
corresponds to the rain cell evolution along their path. In [Bell, 1987], both terms are
taken into account separately in the simulation algorithm. The temporal evolution of the
rain field is introduced through the spatial spectrum of the Gaussian field. Considering
that each spatial frequency evolves according to a first order Markov process, the
relation between the spatial spectral components ak defined by (3) of the field at time
step t and at time step t+Δt is given by:
( ) ttkkkkT
tkttk estVikaa Δ+Δ+ −+Δ−= ,2
,, 1)exp( ββ , (4)
where V represents the advection vector, sk the spectral density of the spatial
process corresponding to the spatial frequency k and ek,t+Δt are uncorrelated complex
standard centered Gaussian random variables. βk represents the autocorrelation lag of the
Markov process corresponding to each spatial frequency. In (4), the term
)exp(, tVika Ttk Δ− represents the field spectral component at spatial frequency k and time
step t but with a phase shift that corresponds to the rain advection between the time step
t and the time step t+Δt. The term ttkk es Δ+, corresponds to the innovation of the
process. The variance of ak,t is assumed to be constant in time, thus preserving the
spatial correlation of the Gaussian field when performing the Fourier expansion (2). The
parameterization proposed in [Bell, 1987] (i.e. )/exp( kk t τβ Δ−= with kτ representing
the time autocorrelation-lag for each spatial frequency) complies with the definition of a
Markov process. This parameterization of the temporal evolution induces a non-
separable spatio-temporal correlation function as it can not be expressed as the product
of a spatial correlation function by a temporal one [Fuentes, 2005]. The
parameterization and the choice of this temporal evolution will be discussed in Section
2.
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1.2 Conversion into rain rate fields
In order to convert the Gaussian random fields obtained from (2) following the
temporal evolution driven by (4) into rain fields, the local statistical distribution of rain
rate has to be known. This distribution is highly dependent on the geographical area. For
that purpose, Rec. ITU-R P.837-5 aims at giving an estimation of the rainfall rate
Cumulative Distribution Function (CDF) all over the world [Poiares Baptista and
Salonen, 1998]. The methodology followed to derive this CDF can be found in and was
updated in [Castanet et al., 2007]. It relies on input parameters computed from the ERA-
40 ECMWF database. However the analytical shape of the model is rather sophisticated.
For our purpose, the use of a lognormal approximation [Sauvageot, 1994] for the
conditional rainfall rate distribution was found to be more relevant as the moments of
the distribution can be easily computed. For temperate and terrestrial areas, the
difference between the CCDF (Complementary CDF) of rainfall rate derived from Rec.
ITU-R P.837-5 and the log normal model is very weak as illustrated on 233HFigure 1.
10−2
10−1
100
101
0
5
10
15
20
25
30
35
40
45
P(R>R*)
R*
mm
h−1
CDF of rainfall rate from ITU−R Recommandation
BordeauxHelsinkiMadridLondon
Figure 1: CCDF of rain fall rate from Rec. ITU-R P.837-5 (crosses) and best lognormal
regression (solid line) for various places.
Consequently, in the following, the rainfall rate CCDF P(R≥R*) at any location
will be approximated by:
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( )
1)0R(P
0R if dr2
²rlnexpr2
1P)RR(P *
R2R
R2R
0*
*
=≥
>⎥⎦
⎤⎢⎣
⎡ −−=≥ ∫
+∞
σμ
σπ , (5)
where Po=P(R>0) is obtained directly from Rec. ITU-R P.837-5, and where μR
and σR are regressed by least square fitting from the ITU-R conditional CCDF
P(R≥R*|R>0). In order to convert the generated Gaussian field into rainfall rate fields,
[Bell, 1987] advocates the use of the following transformation:
[ ]
[ ] ασμϕ
αϕ
≥⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+==
<==
− )x(G if P/2
)x(Gerfcerfc 2exp)x(G)x(R
)x(G if 0)x(G)x(R
01
RR
, (6)
where )2(2 01 Perfc−=α with erfc the complementary error function and erfc-1
its inverse, and where μR, σR and Po are the parameter of the lognormal law in (5). The
transformation (6) is established considering that )GG(P)RR(P *1*1 ≥≥= −− oϕ where
)RR(P *1 ≥− is the inverse of the CCDF of rainfall rate given by (5) and )GG(P *≥
represents the CCDF of a standard centered normal random variable. Therefore, (6)
allows the conversion of a Gaussian random variable into a random variable whose
distribution is given by (5). However, transformation (6) is shown not to perform
completely satisfactorily and a refinement of this transformation will be proposed in the
next section.
1.3 Parameterization and refinement of the modeling
The spatial and temporal correlations in the model proposed by [Bell, 1987] are
the key parameters that need to be accurately assessed in order to obtain a realistic
description of the statistical dependence of rain rates in space and in time. In the initial
work of [Bell, 1987] those correlations were determined from radar data sets collected
over tropical oceans during the experimental campaign GATE. As the spatial and
temporal characteristics of the rain fields over oceanic tropical areas are likely to be
different from the ones over continental temperate areas, a retrieval of those
characteristics from mid-latitude datasets is undertaken. For that purpose, the spatial
correlation function and the parameters that drive the temporal evolution are determined
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from a weather radar dataset collected over Bordeaux (France) whose main
characteristics are described hereafter in. A slight modification of transformation (6)
that turns the Gaussian fields into rain rate fields while preserving features observed on
the radar data is also presented.
1.3.1 Description of the dataset
The data come from the weather radar located at Bordeaux (44.5oN, -0.5oE),
South western France, in the vicinity of the Atlantic Ocean. The radar is a S-band radar
which is part of the French operational radar network managed by Météo France. The
polar scans acquired each 5 mn are projected into a Cartesian grid with a pixel size of
1×1 km2. The dataset contains 35286 scans acquired from January to December 1996.
Images including ground clutter or melting layer echoes were removed from the data set
so that the used radar data only refer to rainfields. The conversion of the reflectivity into
rain rate was made using the standard Z-R relation for mid-latitudes:
Z=300×R1.5,
where Z is the radar reflectivity factor in mm6.m-3 and R the rainfall rate in
mm.h-1.
1.3.2 Refinement of the conversion of Gaussian fields into rain fields
If transformation (6) proposed by [Bell, 1987] for the conversion of Gaussian
fields into rain fields allows preserving the local distribution of rain rate, it introduces
an unrealistic feature on the generated rain fields. Indeed, (6) introduces on the
generated field a non-realistic link between the fraction fR of a simulated rain field that
is affected by rain and the spatial average of the strictly positive rain rates values
<(R|R>0)> as illustrated on 234HFigure 2. This is in contradiction with the ergodicity
assumption of rain rate fields [Eltahir and Bras, 1993], [Sauvageot, 1994], [Nzeukou
and Sauvageot, 2002] that implies that <(R|R>0)> does not change with respect to the
fractional area fR affected by rain. For that purpose, a modification of (6) is proposed. It
keeps unchanged the point distribution of rainfall rates whenever a large number of
realizations are considered while suppressing this unrealistic link. Considering a
realization of the random field G, the spatial distribution of random variables
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⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−
01 P/
2)x(Gerfcerfc2 with G(x) above the threshold α is approximately normal with
mean μoff and with standard deviation σoff. Consequently, the transformation Ψ defined
by:
[ ]
[ ] ασσ
σμ
μψ
αψ
≥⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+−==
<==
− )x(G if P/2
)x(G erfcerfc2exp)x(G)x(R
)x(G if 0)x(G)y,x(R
01
off
R
off
offR
(7)
induces that the strictly positive samples of the rain rate field are lognormally
distributed with mean μR and standard deviation σR. Moreover, considering one point of
the simulation area, the rain rate R conditional distribution P(R≥R*|R>0) is also
lognormal with parameters μR and σR. (7) guarantees the statistical independence
between the fraction of the simulation area fR that is affected by rain and the conditional
mean rain rate <(R|R>0)> as illustrated on 235HFigure 2.
Figure 2: Relation between the fraction of the area that is affected by rain and
the conditional mean rain rate computed from 35 000 radar scans and from 35 000
simulations using the transformations φ and ψ.
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1.3.3 Parameterization of the spatial correlation
The spatial correlation of the Gaussian field can be determined from the analysis
of radar data. Actually, each radar image can be assimilated to a realization of a rain rate
field R(x) where x is the spatial index in the horizontal plane. Considering
transformation (6), the correlation cG of the underlying Gaussian field G can be linked to
the covariance of the rainfall rate field R. In the following, r1 and r2 denote random
variables representing rainfall rates that are separated by a distance |x|. Moreover, we
assume that )( 11 gr ψ= and )( 22 gr ψ= , where g1 and g2 are standard centered variable
whose correlation is cG(|x|). In such conditions, the joint Probability Density Function
p(g1, g2) of g1 and g2 is given by:
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−+
−−
=)x(c1
gg)x(c2gg21exp
)x(c121)g,g(p 2
21g22
21
221gg
π (8)
Recalling that )2(2 01 Perfc−=α , the cross expectation of r1 and r2 can be linked
to the correlation between the two Gaussian random variables g1 and g2 by:
∫ ∫
∫ ∫∞ ∞
∞
∞−
∞
∞−
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−+
−−
=
=>=<
α α πψψ
ψψψψ
21221g
22
21
221
2121212121
dgdg)x(c1
gg)x(c2gg21exp
)x(c121)g()g(
dgdg)g,g(p)g()g()g()g(rr
gg
. (9)
(9) allows to get a relation between the rain rate cross expectation <r1r2> and the
correlation of the underlying Gaussian variables cG(|x|). As the numerical integration of
(9) is inaccurate and requires a significant computation time, the link between the
covariance of the rain field and the correlation of the Gaussian field is established using
a method based on Hermit polynoms, detailed in [Rivoirard, 1995] or [Guillot, 1999]:
kg
k
k xck
rrrr )(!1
2
2121 ∑∞
=
=−><ψ
, (10)
where kψ are the coefficients of the Hermit polynomial expansion of ψ . In order
to reduce the inaccuracy due to the finite number of terms in the sum (10) and its slow
convergence rate [Guillot, 1999], it was chosen to evaluate the covariance of the field
G~ . G~ is obtained from field R by the transformation )(~~ 1 RG −=ψ with )()(~ 11 xx −− =ψψ
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if x>0 and αψ =− )(~ 1 x otherwise. Thus, G~ is a Gaussian field thresholded to α. It is
linked to the Gaussian field G by the relation: )G(G~ Ω= with Ω(x)=α if x≤α and
Ω(x)=x if x>α. The main advantage of this transformation is that the terms of the
Hermite expansion of φ are bounded and the series (11) converges rapidly thus allowing
a fast numerical computation:
kg
1k
2k
2121 )x(c!k
g~g~g~g~ ∑∞
=
=−><Ω . (11)
In order to estimate the covariance of the underlying Gaussian field,
transformation 1~ −ψ is first applied to the radar data, thus converting rain rate values
)(xR into values of a truncated Gaussian field )(~ xG . Afterwards, the covariance of the
truncated Gaussian field G~ is computed performing the inverse Fourier transform of the
spatial power spectrum of the field for each radar map. Finally, the correlation cG of the
Gaussian field G is computed inverting (11), considering the average covariance of field
G~ over all the radar maps. The resulting correlation function is presented on Figure 3 for
the radar data of Bordeaux. The obtained correlation is slightly anisotropic with a larger
correlation along a north south axis. Nevertheless, in the following, the correlation is
assumed to be isotropic.
Figure 3: Radar derived correlation function cG of the Gaussian field G and its
analytical approximation by (12).
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This correlation function can be appropriately approximated by:
)800/exp(41.0)31/exp(59.0)( xxxcG −+−= . (12)
The analytical expression of the correlation function is close to the one derived
by [Paraboni and Barbaliscia, 1992] from rain gauge measurement across Italy. The
slowly decaying part of this correlation function denotes the existence of a long range
dependence of rain rates.
1.3.4 Parameterization of the temporal evolution
As mentioned in section 1.1.2, the temporal evolution of the rain fields proposed
by [Bell, 1987] can be decomposed into one term related to the advection of the rain
fields and the other related to the rain rate field dynamics along their tracks. [Bell, 1987]
has proposed to parameterize the evolution of the underlying Gaussain field G(x,t) given
by (4) according to:
)exp(k
kt
τβ Δ
−= , (13)
where τk=0.24×(π/k)2/3 is the autocorrelation time of the Fourier coefficients ak,t
of G. (13) has been regressed directly by [Bell, 1987] on successive radar maps.
Therefore, it corresponds to the time auto-correlation of rain fields and not to the one of
the underlying Gaussian field. Moreover, the effect of the advection was not isolated,
biasing the analysis. However, a rigorous parameterization of the temporal evolution of
the Gaussian field G(x,t) assuming model (4) can be derived from a radar dataset. First,
the rain fields have to be corrected for advection. To do it, the average advection vector
is computed from two successive radar observations of rain field, by maximizing their
spatial cross-correlation. It hence allows defining rain fields ),(ˆ txR whose average
motion is null. This correction of the advection allows getting rid of the shift phase
)exp( tiVkΔ− in the evolution equation (4). The temporal evolution of the Gaussian fields
),(ˆ txG such that [ ] )t,x(R)t,x(G =ψ should consequently obey to:
( ) ttkkkktkttk esaa Δ+Δ+ −+= ,2
,, 1 ββ , (15)
where ak,t are the coefficients of the spatial Fourier expansion of ),(ˆ txG , that are
also the coefficients of the Fourier expansion of the field G(x,t), and where the
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coefficients βk are the one defined in (4). From (15), recalling that ek,t is normal,
standard, centered and uncorrelated, βk can be obtained considering the temporal
autocorrelation of the components of the spatial Fourier expansion of ),(ˆ txG :
ktktktkttk aaaa β,,,, =Δ+ . (16)
The term tkttk aa ,, Δ+ corresponds to the spatial cross spectral density of the
spatial processes )t,x(G and )tt,x(G Δ+ . The remaining step to retrieve the parameters
βk lies in the definition of a correspondence between the characteristics of the field
),(ˆ txR with the ones of the field ),(ˆ txG . For that purpose, an approach quite similar to
the one used to infer the spatial correlation of the Gaussian field is used. The advection
corrected rain rate fields ),(ˆ txR are transformed into truncated Gaussian fields applying
the transformation 1~ −ψ , giving truncated Gaussian fields ),(ˆ txGT corrected for
advection. The space-time correlation function ),(ˆ txcTG , can then be deduced from the
spatio-temporal power spectrum of the successive fields. (11) gives a method to define a
correspondence between the spatio-temporal covariance function of ),(ˆ txGT and the one
of ),(ˆ txG . The Fourier inversion in space of the correlation function ),(ˆ
txcG
gives the
spatial cross-spectral density tktkk aas ,,, ττ += for different time-lags τ. Its evaluation for
a time-lag τ=Δt allows the retrieval of βk for each spatial frequency and thus an estimate
of τk:
)log()log()log( ,,,, tktktkttkkk aaaa
tt−
Δ−=
Δ−=
Δ+βτ . (17)
To evaluate the value of τk from radar data, successive rainy maps corrected for
advection have been used. As the radar spatial coverage is limited, the correction of the
advection on the radar data has been restricted to areas of 100×100 km² and has not been
considered for duration longer than 2 hours. 25 rainy events R(x,t) of 2 hours corrected
for advection have been selected in the database. For each event, the value of τk has been
computed according to the methodology describes above. As shown in Figure 4, the τk
average dependency with respect to the spatial frequency k can be approximated by: 94.0)/1(06.0 kk =τ . (18)
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Figure 4 : Evolution of the parameter τk as a function of the spatial frequency k.
(18) was found assuming that τk follows a power-law function of the spatial
frequency k. In compliance with Figure 4, the dependency of τk with respect to k
suggests, as expected, that the evolution of large features (rain field stratiform
component associated to small k) is slower (higher auto-correlation time τk) that the one
of smaller features (rain field convective component with limited spatial extent,
associated to highest k values).
1.4 Limitations of the presented methodology
Several hypotheses limit the validity range of the modeling proposed by [Bell,
1987]. The most significant one is the assumption of stationarity of the random field.
Even if [Ferraris et al, 2003] have showed that the stationarity of rain fields is realistic
for areas of some hundreds of square kilometers, this assumption is not likely to hold for
larger areas as orography or climatology related effects will introduce unstationarity in
the rain field. For that reason, the model presented above should be limited to areas of
some hundreds of square kilometers. Therefore, its use for areas of the size of a typical
satellite regional coverage (such as Europe or USA) is not appropriate. The same kind of
restriction applies for the simulation duration. Indeed, considering (13) and (18), the 0
Spatial frequency k (km-1)
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frequency term of the temporal evolution coefficient β0 is equal to 1. Consequently, the
spatial average of the Gaussian field M=a0 remains constant. It hence prevents the
evolution of integrated values of the simulation such as the average rain amount during a
given duration or the fraction of the area that is affected by rain. As a consequence, the
alternation of clear sky and rainy period can not be reproduced by the simulated rain rate
fields as the amount of rain is approximately constant during the simulation. In order to
overcome those limitations a mean to constrain this stochastic model with inputs from
meteorological reanalysis database is described in Section 2. It will ensure to benefit
from the statistical robustness of this stochastic model for small scale features combined
with a realistic repartition of the generated rain amount over long durations and large
areas.
2 Extension of the model using inputs from meteorological reanalysis database
To cope with the problem of the temporal evolution of the fraction of the area
affected by rain and hence being able to generate realistic long-term correlated time
series of rain rate or rain attenuation, the use of GCM (General Circulation Model)
outputs is proposed. For that purpose, the use of ECMWF ERA-40 reanalysis database
[Uppala et al., 2005] is chosen as it offers a worldwide estimation of rainfall amounts
for durations long enough to be statistically representative. Though the spatial resolution
of this database is not fine enough to be directly converted into attenuation for earth-
space links or terrestrial links, some of its outputs are of great interest to be used as
input parameters for space time rain rate and rain attenuation field simulators. In the
following, the use of such a database is shown to overcome the main limitations of the
model presented in [Bell, 1987] or [Gremont and Filip, 2004], namely the unrealistic
temporal evolution of rain fields and the distribution of the rainy areas at large scales.
The purpose of this work is not to forecast propagation conditions from outputs of
numerical weather prediction as proposed by [Hogdes et al., 2008] but rather to generate
statistically plausible propagation scenarios from realistic past meteorological situations.
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2.1 Description of the ECMWF ERA-40 dataset
The dataset used in this study is a subset from the ECMWF ERA-40 reanalysis
database freely available on the ECMWF website. This database strives at reproducing
the state of the atmosphere in the past (the considered period ranges from 1957 to 2002),
constraining a numerical weather forecast model with observations. The model internal
variables are pressure, temperature, wind and humidity profiles from which several other
parameters such as rain amounts are deduced. This database is provided with a spatial
resolution of 2.5o×2.5o with one sample each 6 hours. Nevertheless, some limitations
have been identified, especially the rainfall estimation. Indeed, as the vertical wind
information is not directly included in the meteorological model, the estimation of
convective rain is not reliable [Uppala et al., 2005]. In addition, the low resolution of
the model over land and different assimilation procedures between land and sea led to
some bias in costal and mountainous areas.
2.2 Parameterization of the rain amount
A relation between the rain amount of one ERA-40 resolution cell with the
fraction of this resolution cell that is affected by rain is presented in Section 2.2.1. Then,
this information related to the rain fractional area is introduced in the stochastic
approach developed in Section 1. Proceeding that way, a realistic (since driven by ERA-
40 output) spatio-temporal evolution of the simulated rain fields is insured at large scale
(continental scale) and for long durations (several hours).
2.2.1 Link between the rain amount from a GCM and the fraction of the area affected by rain
Many studies on rainfall remote sensing have put forward that if a sufficiently
large number of rain cells at different stages of their life-cycle are observed over an area
(between 1000 and 100000 km2), a tight link exists between the fraction fR of the area
that is affected by rain above a given threshold (the fractional area) and the area
averaged rain rate <R> [Donneaud, 1984], [Chiu, 1988]. As an example, 236HFigure 5 shows
the linear dependency of <R> with respect to the fractional area fR deduced from the
radar data of Bordeaux.
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Figure 5 : Area averaged rain rate <R> as a function of the fractional area fR
above 0.5mm.h-1 from the radar data set of Bordeaux. The correlation coefficient with
the linear regression (red line) is 0.85.
This relation is all the tighter because a large number of rain cells are observed
simultaneously in the rain field, giving an exhaustive representation (in a statistical
sense) of the climatological rain rate distribution [Atlas and Bell, 1992].
Besides, [Eltahir and Bras, 1993] have shown that considering rainfall outputs
from low resolution GCM, the same kind of relation holds between the rain amount,
which corresponds to a rainfall rate averaged in space and in time, and the fraction of
the GCM pixel that is under rain:
TrVfR Δ
= , (19)
where V is the rain amount over the GCM pixel, r is the conditional mean rain
rate (i.e. knowing that it is raining) on the GCM considered area and TΔ is the time
interval during which the GCM rain amount is computed. From (5), the conditional
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mean rain rate r can be obtained considering the lognormal local distribution of rainfall
rates:
r = )2
exp()0RR(2R
Rσμ +>=>< . (20)
Therefore, from the rain amount time series given by ERA-40, the associated
time series of the fractional area fR can be inferred worldwide from (19) and (20), every
TΔ =6 hours, over cells with size 2.5o×2.5o.
2.2.2 Parameterization of the mean value of the Gaussian field
In order to include the information derived from the reanalysis database in the
high resolution stochastic rain model, a link has to be found between the rain fractional
area fR and the simulation parameters. Recently, [Jeannin et al., 2008] have proposed a
model to link the fraction fG of the samples of one Gaussian field realization that are
above a given threshold with the average value MG of this realization. Under some
assumptions on the shape of the correlation function, the authors have shown that the
spatial distribution of the samples of one Gaussian field realization simulated on a finite
grid with size N×N is normal, with mean MG and standard deviation 1- 2Gσ , where 2
Gσ is
the spatial average of the Gaussian field correlation function over the N×N simulation
grid. In such conditions, [Jeannin et al., 2008] have shown that the fraction fG of a
simulated Gaussian field that is over a preset threshold αG is given by:
))1(2
M(erfc21f
2G
GGG
σ
α
−
−= . (21)
From the Gaussian modeling of rain fields developed in Section 1, αG is linked to
the rain no-rain threshold through the equation )P2(erfc2 01
G−=α where P0 is the
probability of rain defined in Section 1.2. Obviously fG=fR so that the mean value MG of
the Gaussian field can be related to the rain fractional area fR by:
)f2(erfc)1(2)P2(erfc2M R12
G01
G−− ×−−= σ (22)
By definition of the Fourier transform, the 0 frequency term a0 of the Fourier
expansion of the Gaussian field G is equal to its mean value MG. Therefore, whenever a0
is set to MG, the fraction of the simulation area that undergoes rain is fR, as expected.
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In the foregoing, the spatio-temporal evolution of the simulated rain fields is
derived from ERA-40 rain amount time series which temporal sampling ΔT is 6 hours. It
is thus mandatory to find an interpolation scheme to get values of the rain fractional area
fR at a finer time resolution. This high resolution rain fractional area time series denoted
by fHR(t) and sampled at Δt (Δt<ΔT) has to respect different constraints such as
continuity. But first and foremost, its average value over ΔT=6 hours must be equal to
the low resolution rain fractional area derived each 6 hours from ERA-40 and denoted
by fLR(t) hereafter in. Therefore, considering that fLR(t) derived from ERA-40 holds in
the temporal interval [-T/2 T/2], fHR(t) is defined as a piecewise linear function such as:
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎦⎤
⎢⎣⎡∈+=
⎥⎦⎤
⎢⎣⎡−∈+=
2;0)(
0;2
)(
2
1
Ttbtatf
Ttbtatf
HR
HR
. (23)
To ensure the continuity at the edge of the interval, we set:
⎪⎩
⎪⎨
⎧
=
−=−
)0()()2
(
)0()()2
(
LRLRHR
LRLRHR
fTfTf
fTfTf, (24)
, combined with the constraint on the average value on [-T/2 T/2]:
)0()(1)(1)(1 2/
02
0
2/1
2/
2/ LR
T
T
T
T HR fdtbtaT
dtbtaT
dttfT
=+++= ∫∫∫−
− (25)
(23), (24), (25) define a set of three equations with three unknown coefficients
a1, a2 and b. The effect of this interpolation procedure is illustrated on 237HFigure 6, where it
is shown to reproduce accurately the rain fractional area derived from ERA-40.
Nevertheless, the effect on the simulations of this piecewise linear interpolation is not
well assessed. It will be shown further that it has not a significant impact on the realism
of the simulations in terms of temporal auto-correlation.
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Figure 6 : Interpolation at higher temporal resolution of the rain fractional area
time series derived from ERA-40 (ΔT=6h). The black dashed line results from the rain
fractional area computed from rain field spatio-temporal simulations with high
resolution parameters (i.e. Δt=6mn).
238HFigure 7 sums up the step by step methodology developed to parameterize the
stochastic model from ERA-40 rain amount time series. Particularly, from the ERA-40
low resolution (ΔT=6 hours) rain amount time series, a low resolution (ΔT=6 hours)
time-series of the rain fractional area is deduced from (19). The latter is then
interpolated with a lower time step using the interpolation methodology defined above
(Δt=6mn). At this stage, a time-series of the Gaussian field mean value MG(t)=a0(t) is
derived from (22). Finally, the model generates realistic rain fields with long-term
evolution at the scale of one ERA-40 resolution cell (2.5o×2.5o) with a spatial resolution
of 1×1 km² and with a temporal resolution of 6 mn .
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Figure 7: Conversion of the rain amount time series given by ERA-40 into rain
fractional area and into spatial average of the Gaussian field MG=a0.
2.2.3 Extension to several cells of the GCM
The extension of the simulation domain to areas larger than 2.5o×2.5o requires
taking into account more than one ERA-40 cell. It is not possible to juxtapose directly
the 2.5o×2.5o rain fields obtained from the methodology described above for one ERA-
40 resolution cell because the rain field continuity between two adjoining cells is not
ensured. For that purpose, an interpolation procedure that preserves the statistical
features of each 2.5o×2.5o sub-field while ensuring the continuity of the global rain field
has been developed. Particularly, for each ERA-40 cell contained in the simulation area,
a Gaussian field with size 2N×2N is simulated according to the methodology described
in Section 1. Then, for each point (x,y) of the simulation area, the value of the global
Gaussian field G results from the weighted sum of the Gaussian fields G1, G2, G3, G4
generated for each of the ERA-40 adjacent resolution cells:
dyxGxyyxGyxdyxGydxyxGydxd
yxG),(),()(),()(),())((
),( 4321 +−+−+−−= , (26)
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where d is the distance between two ERA-40 resolution cell (d=N). Figure 8
gives an overview of the interpolation scheme.
Figure 8 : Spatial extension of the modelling to several resolution cells of the
ERA-40 reanalysis database. Each gray square figures an ERA-40 resolution cell. The
interpolation procedure (26) is applied inside the white square.
Common interpolation methodologies such as bilinear or cubic spline
interpolation are not suitable in our case because they do not preserve the Gaussian field
statistical features in terms of variance and correlation. On the contrary, the
interpolation procedure (26) does not change the variance and the correlation of the
Gaussian sub-fields Gi. Moreover, by construction, the model reproduces both the rain
amount given by ERA-40 for each resolution cell (as a laplacian filtering is applied to
the coefficient a0 of each subfield) the (local) lognormal rain rate distribution given as
input parameters. Figure 9 gives an example of rain rate field simulated at large scale,
over Europe, on 24/02/1999 at 19:30 UTC (i.e. using the ERA-40 rain amount data
provided at this date).
G1 G2
G4 G3
M(x,y)
G
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Figure 9 : Example of rain rate field simulated over Europe on 24/02/1999 at 19:30
UTC (i.e. using the ERA-40 rain amount data provided at this date).
2.3 Advection
Whenever the rain field spatio-temporal simulation lasts a few hours, a common
and practical approach is to consider the advection vector as constant both in speed and
direction. However, this frozen storm hypothesis is not satisfying if long-durations or
large-scale simulations are considered. Indeed, the wind field and the resulting rain field
motion may evolve considerably with distance or with time. In such conditions, wind
data from the ERA-40 database can provide realistic inputs for this parameter. Indeed,
comparing radar derived advection with wind outputs from meteorological model,
[Kitzmiler et al., 2002], [Luini et al., 2008] have shown that the rain field motion can be
accurately derived from the ERA-40 wind data at the 700 hPa pressure level.
Consequently, for each ERA-40 resolution cell making part of the large scale simulation
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area, the advection vector V defined in (4) is driven by the ERA-40 concurrent wind data
at a pressure level of 700 hPa. Lastly, in order to get a smooth temporal evolution of the
rain advection, both component of the winds are interpolated by cubic splines with a
time step Δt of 6 mn.
3 Conversion into attenuation
3.1 Conversion of rain rate fields into attenuation fields
Rain field simulations are converted into attenuation by integration of the specific
attenuation along the link path. The attenuation (dB) endured by the electromagnetic
wave during its propagation through rain is given by:
∫=sL
0
dx)x(kRA α , (27)
where R(x) is the rain rate intensity (mm.h-1) at x and Ls is the slant path (km)
through rain. k and α are coefficients, function of the elevation angle, the frequency and
the polarization of the electromagnetic wave. Their values are given by Rec. ITU-R
P.838. The definition of the geometric (latitude, longitude, altitude of the satellite) and
radiowave (frequency, polarization) characteristics of the telecommunication link allows
the conversion of the rainfall field into an attenuation field. Here, the altitude of the
N×N points composing the simulation grid is derived from Rec. ITU-R P.1511. The rain
height is the -2 oC isotherm height given by the ERA-40 temperature profiles. The use of
ERA-40 data, sampled each 6 hours, to define the -2oC isotherm height allows a better
description of the rain height with respect to Rec. ITU-R P.839 that provides only the
yearly average of the -2 oC isotherm height as shown in Figure 10.
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Figure 10: Height of the -2oC isotherm deduced from ERA-40 temperature profiles for
Bordeaux (France) in 2000.
3.2 Over sampling of the attenuation time series
For adaptive SatCom systems using adaptive coding, adaptive modulation or
other FMTs, it is crucial to get an estimation of the evolution of the propagation channel
at a time resolution of about 1 second. However, the temporal resolution of the model
developed in Section 2 is 6 mn. Therefore, it is mandatory to simulate the fast dynamic
fluctuations of the propagation channel to increase the model temporal resolution. For
that purpose, the on-demand time series synthesizer fully described in [Lacoste et al.,
2006] and [Carrier and Lacoste, 2008] is used. The latter makes a 1 s stochastic
interpolation of the 6 mn attenuation time series extracted from the space time model,
generating that way high frequency (1 Hz) attenuation time series as illustrated in Figure
11.
Figure 11 : 1 Hz attenuation time series simulated on two sites located 7 km
apart, at 20 GHz. The green circles represent the values of the initial time-series
extracted from the spatio-temporal model (Δt=6 mn).
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The high resolution interpolation describes in [Lacoste et al., 2006] and [Carrier
et al., 2008] holds whenever attenuation time series are Markov and follow a log normal
distribution which parameters can be derived from Rec. ITU-R P.618. The temporal
autocorrelation time-lag is supposed to be 10-4 s whatever the link configuration
[Carrier et al., 2008]. Coupled with the spatio-temporal model, the stochastic
interpolation enables to simulate 1 Hz spatially correlated attenuation time series for
thousands of links disposed across the satellite coverage, for long durations (several
years).
4 Preliminary validations, limitations
4.1 First order statistics
Preliminary analysis shows that the rain amount from the input reanalysis
database is reproduced by the simulations for each 6h period with an RMS error of about
15% as illustrated in Figure 12.
Figure 12: 6h rain amount derived from ERA-40 database on 12/24/1999 at 08:00 UTC
(left) and corresponding 6h rain amount computed from 6 mn rain field spatio-temporal
simulations (right).
By construction, the model reproduces the rain rate lognormal approximation of
Rec. ITU-R P.837-5 on each point of the simulation grid. In addition, long term
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attenuation time series reproduce Rec. ITU-R P.618 CCDF of attenuation. From
OLYMPUS satellite measurements at 20 GHz led in 1992 near Paris (France), the model
ability to reproduce attenuation statistics derived from beacon measurements has been
also investigated. High resolution (1 Hz) attenuation time series have been then
synthesised from the spatio-temporal model starting from ERA-40 data concurrent to
OLYMPUS measurement period and assuming a link with the geostationary satellite
OLYMPUS (19°W) at 20 GHz. Figure 13 shows the results in terms of attenuation
CCDF derived from simulations and from OLYMPUS measurements.
Figure 13: Attenuation CCDF at 20 GHz measured during the OLYMPUS
campaign (1992) in two sites near Paris (La-Folie-Bessin and Gometz-la-ville, color
lines) and attenuation CCDF derived from simulated attenuation time series (black
lines). The two sites are located 7 km apart.
According to Figure 13, statistics derived from spatio temporal simulations of
attenuation fields are comparable to the experimental ones up to small time percentages.
Nevertheless, it is noticeable that the variability of the experimental statistics is higher
than the one derived from the simulations.
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4.2 Second order statistics
In order to assess the model ability to reproduce the spatial variability of the
attenuation due to rain, the diversity gain has been computed, first, from the radar data
of Bordeaux collected in 1996 every 5 mn [Luini et al., 2008]. As the latter refer to rain
fields, they have been converted to attenuation fields assuming a 30 GHz link with
OLYMPUS. On the other hand, the same exercise has been conducted from the rain
fields simulated by the spatio temporal model each 6 mn using ERA-40 data of 1996.
The results are presented in Figure 14, for single site attenuation values ranging from 4
dB to 32 dB. Figure 14 shows that the spatio temporal model reproduces satisfactorily
the diversity gain derived from radar data, even if a trend to slightly underestimate the
diversity gain for the highest single site attenuation values and for low distances can be
observed.
Figure 14: Diversity gain as a function of distance for different single site
attenuation values derived from radar data (blue curves) and derived from simulations
(red curves).
A similar test has been carried out with the 1 Hz over-sampled time series.
Indeed, the joint attenuation CCDF has been computed first from OLYMPUS [OPEX,
1994] measurements at La-Folie-Bessin and Gometz-La-Ville that are 7 km apart and,
second, from the 1 Hz time series derived at both locations from the spatio temporal
model. According to Figure 15, the experimental and model based distributions match
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satisfactorily, showing thus the model ability to account for the rain attenuation spatial
variability.
Figure 15: Joint attenuation CCDF derived from simulations and derived from
OLYMPUS measurements at 20 GHz, at La-Folie-Bessin and Gometz-La-Ville (7 km
apart).
To evaluate the relevance of the model temporal parameterization, the
autocorrelation function of attenuation time series has been evaluated first from yearly
attenuation time series extracted from the radar observations at Bordeaux in 1996, and,
second, from the 6 mn attenuation time series simulated over Bordeaux in 1996.
Figure 16: Temporal autocorrelation estimated from radar derived attenuation
time series and from simulations.
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The results, reported in Figure 16, show that the temporal auto-correlation
function computed from simulations is close to the one derived from radar.
Nevertheless, the temporal validation of the model has to be deepened, notably to assess
the ability of the simulated time series to reproduce fade slopes, fade duration or inter-
fade durations statistics derived from beacon measurements.
5 Conclusion
A model to generate spatially and temporally correlated rain fields or rain
attenuation fields for propagation studies has been presented. It lies on a non-linear
transformation of random Gaussian fields constructed in the Fourier plane.
After a brief description of the mathematical framework, a method to derive the
spatio temporal correlation parameters from radar data has been proposed. It has been
applied to the weather radar data of Bordeaux (France). As the initial approach proposed
by [Bell, 1987] was not intended to describe the rain rate fields at large scale and for
long durations, a theoretical framework to enlarge the validity domain of the model has
been developed. A methodology to constrain the model with rain outputs from the ERA-
40 reanalysis database has then been presented. Particularly, the rain amount generated
each 6h by the spatio temporal model is constrained to be the one given by ERA-40. The
rain advection is also modelled by ERA-40 wind data. Coupled with a large scale
interpolation scheme, the spatio temporal model is thus able to generate realistic rain
fields for long durations (several years) and large areas (i.e. over Europe or USA, the
size of a typical SatCom system coverage) with a spatial resolution of 1×1 km² and a
temporal resolution of 6 mn. Then, defining the geometric and radio electric
characteristics of a link, the rain rate fields are turned into attenuation fields considering
the wave path through rain. Here, the stochastic interpolation scheme described in
[Carrier et al., 2008] allows reducing the temporal resolution up to 1 s.
The model has been shown to reproduce several statistical characteristics
observed on different and independent sources of data. In addition to the first order
statistics of rain and rain attenuation, the model has shown its ability to reproduce the
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A large scale space-time stochastic model of rain attenuation for the design and optimization of
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216
spatial repartition of attenuation due to rain over the coverage area (diversity gain, joint
attenuation CDF). From the temporal point of view, first validations in terms of
temporal autocorrelation function are promising but further enquiries are needed to
consolidate the results.
Reliable inputs for FMTs and RRM (Radio Resource Management) design and
optimization may be obtained from the presented model, for several configurations,
since the simulation area can reach the size of a typical satellite coverage with a
temporal resolution that can be reduced down to 1 s for several years. Particularly,
considering complex network simulations that involve thousands of links dispersed over
the whole coverage, the spatio temporal model is able to provide realistic propagation
conditions, both correlated in space and in time.
However, the use of the ERA-40 reanalysis database may lead to some bias in the
simulations, especially in coastal or mountainous areas, i.e. in places where the quality
of the reanalysis database is questionable. Nevertheless, the use of reanalysis data turn
out to be promising as it allows simulating test-cases from specific past weather
conditions. For instance, this can provide a good opportunity to study diurnal cycles,
seasonal cycles or inter-annual variability but also worst cases identified on the
reanalysis data.
A logical following to this work is the inclusion, using the same approach, of the
gaseous and liquid water attenuation as they become critical for SatCom system
operating at Q or V band or even at Ka band for other applications such as radar
altimetry.
Acknowledgment:
The authors are very grateful to Météo-France for providing the radar data from
Bordeaux and to ECMWF for providing ERA-40 data.
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