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An Investigation Of The Use Of VAD Analysis and UHF Profiler Data To Obtain
A 2-D Wind Field
Dean Reichheld
Deparment of Atmospheric and Oceanic Sciences McGill Universiq, Montréal
July 1997
A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements of the degree of Masters of Science
in Atmospheric and Oceanic Sciences.
O Dean Reichheld 1997
National Library of Canada
Bibliothèque nationale du Canada
Acquisitions and Acquisitions et Bibliographic Services services bibliographiques
395 Wellington Street 395. rue Wellington OttawaON K1AON4 Ottawa ON KI A ON4 Canada Canada
Your 6ie Votre rëferenœ
Our fi& Norra reférence
The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant a la National Lhrary of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sell reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/fh, de
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The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts from it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation.
Abstract
in this work, we explore the possibility of cornbining information from the VAD
technique, with data from a wind profiler, to obtain the kinematic properties of a linear 2-D
wind field. The idea is f ~ s t tested on an &cial test-bed wind field, and the results show
that the procedure is limited by the degree of non-linearity of the wind field over the wind
profiler. To overcome this problem, some modifications are made to our procedure, which
is then applied to real cases of stratiform precipitation. We find that the non-linearities are
important enough such that the linear approximation is only valid out to about 40 km fiom
the radar. These non-linearïties also affect the retrieval of the vorticity to the extent th& we
had to establish a maximum limit on the uncertainty of the vorticity. These results bring to
question the assurnptions that the non-linearities are relatively unimportant in stratiform
cases.
Résumé
Au cours de ce travail est explorée la possibilité de combiner les informations
déduites de la technique VAD avec les données collectées par un profileur de vent, afin
d'obtenir les propriétés cinématiques d'un champ de vent bidimensionnel linéaire. Cette
approche est testée en premier lieu dans le cas d'un champ de vent simulé. Les résultats
obtenus montrent que la procédure proposée est limitée par le degré de non-linéarité du
champ de vent à la verticale du profileur de vent. Pour pallier cette difficulté, certaines
modifications sont apportées à notre approche. Cette procédure est ensuite appliquée à
différents cas de pluies à caractère stratiforme. 11 est obtenu au cours de ce travail que la
composante non linéaire du vent est suffisamment significative pour que l'hypothèse de
linéarité ne puisse être appliquée que jusqu'à une distance de 40 km du radar. Cette
composante non linéaire affecte également le recouvrement du tourbillon vertical à tel point
qu'il a fallu imposer un seuil de tolérance portant sur l'incertitude sur le tourbillon vertical.
Ces résultats remettent en question l'hypothèse communément acceptée de linéarité du
champ de vent dans le cas des systèmes convectifs à caractère stratiforme.
Acknowledgments
A large thank-you goes to my supervisor, Professor Isztar Zawadzki, for his
support and advice for this project. His enthusiasm for the field, his excitement in new
discovenes, and his constant push for more, have been great inspirations to me over the
past two years.
También quisiera mostrar mi aprecio de todo coraz6n a Ramon de Elia por su ayuda.
Algunos de los resultados de nuestras discusiones me han ayuadado a crear O refinar
algunos de los argumentos que aparecen en esta tesis. Y sus ideas iluminaron algunas
soluciones a mis probiernas m h complejos.
Pour tous ses renseignements et son support, je voudrais remercier Dr. Fred Fabry.
Il m'a donné une perspective fraîche quand j'en avais le plus besoin. Aussi, pour son aide
avec les traductions en Français, je voudrais dire un grand "merci" à Dr. Alain Protat.
1 also owe a large part of my thesis to the efforts of Mrs. Alumulu Kilambi of the
J.S. Marshali radar Observatory at McGill University. She not only provided the data used
in Chapter 4, but was extrernely helpful with the data storage and retrieval systems at the
observatory. The information on the methods used at the observatory to calculate the VAD
c ~ e ~ c i e n t s , dong with some discussions on the results of this work were kindly provided
by Dr. Aldo Bellon. and are dso greatly appreciated.
A lot of valued help with the details of the UHF wind profiler used for this work
came fiom Dr. William Brown. AIso, several good suggestions on the error analysis and
other aspects of the thesis came from Mark Shephard. 1 would like to thank these guys,
and the rest of my peers in the Department of Ahnospheric and Oceanic Sciences at McGill
University, for putting up with me for the past two years.
Finally, I wish to give a special thanks to Dr. Owen Hertzman for persuading me to
go ïnto the Master's program in the f rs t place. AIso 1 wish to express my eternal gratitude
to my parents, Gerald and Lynda Reichheld, for their undying support that eventually saw
me through to this point.
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . Résumé . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . Statement of Originality . . . . . . . . . . . . .
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1 . I Introduction
12 Review of VAD Andysis . . . . . . . . . 1.2.1 Use of VAD In DeveIoprnent of VVP Methods . 1.2.2 Development of EVAD Methods . . . . . 1 .2.3 Other Developments of VAD . . . . . . 1.2.4 Characteristics and Cornparisons of VAD Analysis
1.3 Discussion of UIHF Wind Profiler . . . . . . . 1.4 Discussion of Project . . . . . . . . . .
Chapter 2 Linear Wind Retrieval Method . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction
. . . . . . . . . . . . . 2.2 General Equations
. . . . . . . . . 2.3 Set Up of The Retrieval Procedure
. . . . . . . . . . . . . . . . 2.4 Surnmary
Chapter 3 Analysis of a Test-Bed Wind Field . . . . . . . 19
3 -1 Wind FieId Description . . . . . . . . 19
3.2 Retrieval of Aaificial Wind Field . . . . . . 20
3.3 Discussion o f Results . . . . . . 21
3.4Summary . . . . . . . . . . . . . . . . 24
. . . . . . . . . . . Chapter 4 Andysis of Real Data 28
4.1 Weather Synopsis . . . . . . . . . . 28
4.2 Wind Retrievai for December 14 1995 . . . . . . . 28
4.3 Error Analysis of Retrieval . . . . . . . 30
LV
4-4 Wind Retrieval for December 9 1995 . . . . 34
4.5 Further Analysis and S u m q - . - . . - . - . . 35
Chapter 5 Conclusion . . . . . . . . . . . . . . 49
5.1 Summary and Conclusions . . . . . 49
5 2 Suggestions and Future Work . . . . . . . . . . 52
Appendix A Equations In Error AnaIysis . . . . . . . . 53
AppendU: B Error Analysis of VAD Coefficients . . . . . . 55
Bibliography . . . . . . . . . . . . . . . . . 58
Statement of Originality
Aspects of this thesis that represent original contributions to knowledge are the foilowing:
A method to separate the kinematic properties of the linear wind from the non-
linear terms, using the range dependence of the VAD coefficients.
A procedure that combines the kinematic properties obtained by the method above,
with information from a UHF wind profiler, to obtain a two-dimensional linear
wind field.
The application of the above procedures to real stratiform snow cases, with the
indication that such cases may be more non-linear than originally thought.
Chapter I
Introduction
1.1 Introduction
Since the advent of Doppler radar, there have been many studies into how to
process, use, and expand upon the data received from it. Some of these studies examined
various techniques to complete the information on the wind field. The most commonly
used is the Velocity Azimuth Display (VAD). The VAD technique, as presently used, is a
diagnostic tool used to obtain kinematic properties of a wind field such as, divergence,
deformation, and the translation wind, ftom single Doppler radar. However, it cannot
determine the voriticity, thus in this work we explore the possibility of cornbining the VAD
technique with data from a wind profder, to obtain a linear approximation to a 2-D wind
field.
In this Chapter, we review the VAD technique and some of its variants. We begin
with a description of the technique as developed by Browning and Wexler (1968). Then
we discuss the papers which rnodify the VAD to obtain two and three dimensional winds
from single Doppler radar. We also review papers that deal with the separation of the
vertical velocity from the divergence, followed by a review of studies analyzing the
characteristics of the VAD analysis. We then briefly discuss papers of the low level UHF
wind profüer, to introduce the reader to the data, and data analysis that is used for this
instrument. Finally we sum up with an outLine for the rest of the thesis.
1.2 Review of VAD Analysis
The use of the VAD technique to determine the kinematic properties of a wind field
was first developed by Browning and Wexler (2968). They used two dimensional Taylor
senes expansions, tnincated at the linear terms, to represent the Cartesian components of
the wind (u, and v), and then transformed them into polar coordinates, to obtain an
equation for the radial velocity (V,). The resultant equation is in the form of a two
harmonic Fourier series, whose coefficients are related to the kinernatic properties of the
wind field (with the notable exception of the vorticity). The procedure was limited by
factors like vertical wind shear, and vertical fail speeds. Since then, studies have expanded
2 upon the VAD, in order to obtain more information about the wind field with single
Doppler radar data.
12.1 Use of VAD In Development Of VVP Methods
One of the earlier modifications to the VAD was what has been since known as the
VARD or VeIocity ARea Display, (Easterbrook, 1975). Easterbrook kept the linearity
assurnption from the VAD, but instead of perfonning the analysis over all azimuth angles
for one range, he fitted two curves within a conical section. Unlike Browning and Wexler,
who set the ongin of their coordinate system at the radar, Easterbrook set the origin at the
center of the conicd section in which he performed his retrievai. In doing this, he found
that he could obtain the divergence and deformations from fitting a curve (similar to the
VAD curve) to the data. However, he also found that the wind components were affected
by the vortÏcity. Thus, he developed a second equation to fit the data in order to determine
the two dimensional wind field within the conical sector.
The VARD was expanded upon by Waidteufel and Corbin (1979), where they
developed the VVP (Volume Velocity Processing). The VVP stiu had the VAD as its base,
but the linearity hypothesis was applied to the whole volume rather than just a range circle
at a single height. This rneans that instead of retrieving a two dirnensionai wkd for a circle,
or a conical sector, they retneved a three dimensional Linear wind within a cylioder. They
found, after testing the procedure on both simulated and real data, that the VVP was
significantly affected by deparhues from linearity, and in most cases the effect was similar
to that of the VAD. However, in the case of the divergence, the VAD obtains the exact
mean divergence within a range circle and so does not depend on the linearity assumption,
whereas, the divergence obtained by the WP is dependent on the linearïty assumption in
b~ are the same manner as the other kinematic properties. The result is that the smaller f e a w
filtered out by the VVP, so that the resultant wind is a rnesosca~e flow. Koscielny et al.
(1 982)- developed a modified VVP (MVVP), which was a combination of the VVP and the
VARD, such that the VVP was applied to sectors whose widths were only 30' of azimuth.
This method provided some more information on divergence within smaller volumes, but
was very sensitive to strong s m d scde eddies.
Johnston et al. (1990) developed an extension of the MVVP and the RH1 (Range
Height Indicator) methods specifically for rainbands or any 2-dimensional system, called
BVP (Band Velocity Processing). Like the MVVP, the BVP method retrieves the wind
3 field within a selected volume so as to examine the fine scde structures. However, as the
name suggests this procedure is limited only to features that are approximately two
dimensional, or in band structures.
1.2.2 Development of EVAD Methods
Another evolution of the VAD analysis was the EVAD (Extended Velocity Az.imuth
Display) method developed by Srivastava et al, (1986). As opposed to the VVP related
methods mentioned above, the EVAD technique was developed to separate the
contamination of the vertical fail speeds from the divergence determined from the VAD. To
do this, they determined a relationship between the first VAD coefficient and the height and
range. By assurning the vertical velocity was negligible, they could then determine the
divergence and the vertical fall speeds separately, then use the continuity equation to
determine what the vertical air motion should be. However, this technique required that the
wind field be steady over a larger set of ranges since it determines the vertical fall speed
fiom a sample of many VAD over many ranges.
The EVAD was then rnodified by Matejka and Srivastava (1991) so that the Fourier
series coefficients are weighted accordingly to their estimated error. They also developed a
method of removing the outliers frorn the fits, so as to reduce the effect of the srnall scale
perturbations, then they obtained the vertical motion through a variational method that they
developed, simila to that developed by O'Brien (1970). These modifications were made
so as to allow the EVAD to obtain finer structure to the vertical air motion, instead of
requiring the entire field to be unifoxm. However, in order for the EVAD to perform its
best, it requires a scanning procedure within guidelines the authors have set up and even
with these, the features of divergence and vetical motions Iess than the synoptic scale aie
fitered out.
Finally, Matejka (1993) created a fuaher modification on the EVAD, known as the
CEVAD (Concurrent Extended Velocity Azimuth Display). The main daerence in this
method, is the fact that the divergence and vertical motion profiles are solved concurrently.
This means that for each height, the values for divergence and the vertical motion are
required to confonn to the additional constraint of certain boundary conditions. It also has the advantage of imposing these boundary conditions during the calculation of the
divergence and vertical motion, as opposed to the EVAD where these are applied after
values for each height are determined separately.
1.23 Other Developments of VAD
Fuaher modifications to the VAD analysis were proposed by Hanis (1975), and
later used by Testud et al. (1980). Both of these papers used the standard VAD analysis to
obtain a linear wind, however, after they performed the VAD Fourier fit, they subtracted it
from the data that they fitted it to. They observed that the remainder appeared as an
organized wave pattern, and when they performed a perturbation analysis to the radial
velocity, they found that this remainder could be considered as wave perturbations to the
mean horizontal wind (Iinear wind), thus they could represent some of the finer scale
features of the wind field they were studying.
The VAD method has also been rnodified for Dual-Doppler analysis by Scidom and
Testud (I986), to obtain the DVAD (Double Velocity Azimuth Display). In this case a
VAD analysis is used for each of the two radars, then the results are merged to retrieve the
three dimensional wind. The DVAD, however, still employs the Linearity hypothesis for
the analysis from each radar, so Scialom and Lemaître (1994) proposed the QVAD
(Quadratic Velocity Azirnuth Display). The main assurnption in the QVAD is that the wind
is assumed to be quadratic, instead of linear. They then could perform the VAD analysis on
the two separate radars and solve for the quadratic terms when the results were rnerged.
They d s o proposed that the method could be applied with a synthetic Duai-Doppler (single
Doppler scans at two different tirnes).
1.2.4 Characteristics and Cornparisons of VAD Analysis
Other studies have concentrated more on the characteristics of the VAD, and also
comparing it with data obtained by other methods. One such study done by Larsen et al.
(199 1) did a cornparison of vertical velocities inferred by a VAD analysis to those measured
by a VHF profder. Since this study used cases with no precipitation, there was no effect of
the fall speed of the hydrometeors, thus they could use the divergence from the VAD to
infer a mean vertical velocity (in cases of precipitation, refer to the EVAD analyses above).
They found that the VAD results compared well to those fkom the VHF profiler, but only in
situations where the vertical motions were reasonably uniform. Similar results were
obtained by Boccippio and Matejka (1996), where they performed the cornparisons in the
stratiform region of an MCS. However, since their case involved precipitation, they
compared the VHF profiler vertical velocities to those obtained fiom the EVAD, CEVAD, and VVP analyses, rather than the simple VAD. What they found was that the generd
5 structures where sirnilar in all of the single Doppler techniques, with some varïability in
profde heights and velocity magnitudes.
Finally, since this project involves utilizing the charactenstics of the VAD andysis,
it is important to discuss the previous work that examines what is, exactly, obtained from
the VAD analysis. The e s t such study was done by Pasarelli (1983)- where he examined
the mathematics of the VAD technique, to determine the minimum restrictions needed to
obtain the vorticity. What he did, was to not only represent the radial velocity in terms of a
Fourier series as in the VAD analysis, but the tangential velocity, vorticity, and the
divergence as well. He then obtained a relationship between each of the coefficients from
al1 four Fourier senes, finally giving him four equations with six unknowns. He then
discussed various closure schemes to these equations that could be used to obtain the
vorticity. However, he did point out that with this set-up, the linearity hypothesis seemed
to be a unduly restrictive assumption.
Another investigation of the VAD technique was performed by Rabin and Zawadzki
(1984), where they exarnined the behaviour of the only well defined quantity of the VAD
retrieval, the divergence. Some of the things they examined were the effects of beam
smoothing, reflectivity weighting, and the effect due to the geometry a f the conical scan.
To overcome these effects, they developed a deconvolution technique that they then applied
to the divergence profiles that they had obtained. They found that the shape of the profiles
changed very littie after deconvolution, thus they concluded that an average proNe could
provide useful information about the divergence.
Finally, as an expansion on the work of Rabin and Zawadzki (1984), Caya and
Zawadzki (1992) examined the characteristics of the VAD when applied to a non-linear
wind field. They discovered that the coefficients frorn a two harmonic Fourier fit did not
give the linear, or mean, kinematic properties (with the exception of the divergence). In
fact, they found that what is obtained is the linear kinernatic property plus higher order
terms. They noticed that with this, the coefficients varied as even (odd) polynornials for the
odd (even) Fourier coefficients, and that if these polynornids were extrapolated to zero
range (over the radar), one could obtain the Iinear kinernatic properties. Et is from this
study that we will continue, with the addition of the UHF wind profiler data, to obtain an
estimate for the linear vorticity.
13 Discussion of UHF Wind Profiler
Since this project also involves work with a UKF wind profiler, we feel it is
important to introduce some of its basic concepts that have been studied. We begin with the
study by Balsley and Gage (1982), where they review the profiling methods and
instruments of that time and discuss possible future developments. In their study, they
mention some possible developments of the available technology, so as to use it for
operational applications. This lead to the developrnent of a lower Troposphere UHF wind
profiler (Ecklund et al. 1988), on which Our wind profiler is based.
The method we use to calculate the horizontal winds from the wind profiler, was
developed by Wuertz et al. (1988). For cases that we wiLl look at (stratiform precipitation
with relatively uniform horizontal winds), they suggested a 3 beam profiler, where one
beam is pointed to the vertical, and the other two are pointed at small angles from the
vertical and perpendicular to each other. The vertical beam detemines the vertical fall
speed, which can then be subtracted from the other beams, from which the horizontal
velocities are obtained. Some examples of the application of the wind profiler to
precipitation cases can be found in Rogers et al. (2993 and MM) .
1.4 Discussion of Project
As mentioned before, the objective of this project is to obtain a h e a r approximation
to the two dimensional wind using information from the VAD analysis, and a UHF wind
profiler. One motivation for this project is to obtain all of the kinematic properties valid
over the scanning radar, and use them for mode1 evaluation, and initialization. The
procedure could also be beneficial for operational use in terms of aiding nowcasts for some
precipitation events. in order to do this we wiU continue from the work by Caya and
Zawadzki (1992), by accepting that the wind field is non-linear, but still using the
assumption that we can obtain the linear components of the kinematic properties, and that
these are close approximations to the larger scale values.
A mathematical description of the method we are proposing will be developed in
Chapter 2, dong with the set up for the steps in the procedure. In Chapter 3, we wiu apply
the procedure to a non-linear test-bed wind field in order to determine the potential
problems due to the non-linearities themselves. Chapter 4, will concentrate on the
application of the procedure to two precipitation cases and the resulting two dimensional
7 winds- As well we will show our results h m the error analysis performed on the procedure. In Chapter 5, we will discuss the implication of the results fiom Chapter 4, and
describe future work to be done to enhance the procedure, or study its limitations.
Chapter 2
Linear Wind Retrieval Method
2.1 Introduction
In the previous chapter, the VAD analysis technique to retrieve bernatic properties
(Browning and Wexler, 1968), was reviewed briefly. The objective of the present work is
to explore the possibility of using the VAD technique, and data from a UHF wind profiler,
to obtain a 2-dimensional iinear approximation to the measured wind field. Thus the theory
of VAD will be studied in more detail so that the equations can be set up to caiculate the
average vorticity, and then the b e a r wind.
2.2 General equations
In their work, Browning and Wexler (1968) first expressed the radial velocity (V' )
in terms of the standard Cartesian wind components, u, v, and w :
Where, /3 is the azimuth angle, measured clockwise fiom north, a is the elevation angle of
the scanning radar (Figs. 2.1, and 2 2 ) , and thz overbars indicate expandable terms (this
will be explained later).
Fig. 2.1 Definition of the azirnuth angle,
Fig. 2.2 Definition of the elevation angle, a
A similar expression can be written for the tangentid velocity, Vt :
In turn, the u and v components of the horizontal wind can be represented by a Taylor
Series expmsion:
where uo, and vo are the components of the horizontal wind over the scanning radar, taken
to be at the origin of the coordinate system, ux, uy, vx, and vy are the derivatives of the
two wind components with respect to x and y , , and EN are the random non-Iinearities
of the u and v components, and , and ESV are the systematic non-linearities of the u
and v components (the reasons for the distinction will become c1ea.r later on). Strictly
speaking, Browning and Wexler only worked with the linear truncation of the Taylor
Series, but for our purposes it is important that the other terrns are represented.
In (2.3) and (2.4), the Taylor series have been divided into linear and non-linear
parts, where the non-linear parts thernselves, have then been divided into random and
systernatic components. The systematic non-linearities are simply the higher order terrns to
the Taylor Series, and the random non-linearities represent rnainiy small scale eddies not
representable by a finite number of terms in the Taylor expansion. To deal with the random
non-linearities, we appty Iinear perturbation theory, beginning with the separatirig of u and
v into a mean plus a perturbation such as:
- u = u + u r
and -
v = v + v f
where u t , and v' are the random non-linearities, Ü, and ; are the terrns thst can be
represented by a finite number of terms of a Taylor Series. Therefore, to simplify things
we assume that the random non-linearities are fdtered out by the Fourier fitting to the actual
data, and so only substitute 2, and ; into (2.3) and (2.4).
An expression for Vr, as a fùnction of the Taylor Series denvatives, is then
obtained by substituting (2.3a) and (2.4a) into (2.1). For clarity, (2.3a) and (2.4a) are expanded to the third order before the substitution (thus, only two ternis of &su and
are retained) . With some trigonometry , the follo w ing equation is obtained:
This gives a Fourier Series of the form:
Where an and bn are the Fourier Series coefficients for the series. We find from
comparing (2.5) and (2.6) that the Fourier coefficients of a two harmonic Fourier fit to a
non-linear wind, will not give the linear kinernatic properties. This can be seen if we wnte
out the coefficients fkom (2.5) as follows:
r cos a % = 3 ( ~ i v ) + w - sin a
1 - + -unr21 cos a 8 1
where Div is the divergence, Shr is the shearing, and Sr is the stretching of the linear
component of the wind field.
It has been standard practice to assume that the higher order terms in (2.7) - (2.1 1)
were negligible, however, due to the dependance on range in these equations we c m see
that this assumption will become invalid at large ranges. Another interpretation has been to
take the Fourier coefficients as the kinematic properties of the mean wind field, however,
we can show that this is also a misinterpretation for al1 except the divergence, by using the
proof offered by Caya and Zawadzki (1992).
First, we rewrite (2.7) - (2.1 l), neglecting the effects of verticai velocity and the elevation
angle, a .
We can then show that (2.12) gives the mean divergence within a circle of range r, with the
followillg:
which, for a cubic wind gives:
Recall that we represent the divergence of the linear component of the wind, (ux+vy) , in
(2.12) with Div , thus we see that (2.1 8) is exactIy (2.121, which rneans that 2ao/r c m be
interpreted as the mean divergence within the circle of range r. If we repeat this for the
other kinematic propenties, we fhd the following:
When we compare (2.19) - (2.22) to (2.13) - (2.16), we see that the coefficients obtained
h m a 2-harmonic VAD fit do not correspond to the mean kinematic properties. Thus the
most we can Say about the coefficients fiom VAD fi& at any given range is that they contain
the kinematic properties associated with the linear component of the wind, but they also
contain contamination from the non-iinear terms-
To solve the problem of the non-linear contamination, we examine a general form of
(2-12) - (2.16):
~ ( r ) = k + f ( r ) (2.23)
where c(r) is the Fourier coefficient, k is the kinematic property for the linear component
of the wind, and f(r) is an even fünction of range. The implication of (223) is that if we
14 perform VAD fits at many ranges, and plot the coefficients fiom these fits as a fimction of
range, then a polynornial fit of (2.23) to this plot should provide k . If we include the effects of the vertical motion (V+w ) and the elevation angle, a, the relationship will still
hold, however the lefi hand sides of (2.12) - (2.16) are multiplied by cos(a), and w sin(a)
is added to the left hand side of (2.12).
The inclusion of the elevation angle, a, in the above equations means that we
cannot solve for the divergence, (Div), without knowing the fall speed of the particles,
(V+w ) (this was the motivation for the creation of the EVAD, Srivastava et al. 1986). To
determine this we will use the information from the vertical wind profder, which provides
the horizontal wind plus a vertical profde of (V+w ) over a point. For this to be useful, the
wind profiler must be located within the domain of the scanning radar (Fig. 2.3), and we
will assume that in stratiform precipitation, the fall speeds are the same throughout the
domain.
Domain of the scanning
Fig 2 3 Location of the wind profiler (W.P.)
with respect to the scanning radar (R).
Using the profiler to determine the vorticity involves solving for the tangential velocity, VI ,
using a similar method as the one used for the radiai velocity, V , .
We begin by substituting (2.3a) and (2.4a) into (2.2), and applying some
trigonometry, to obtain the following equation:
As with the radial velocity, (2.24) has the f o m of a Fourier series:
Where cn and dn are the Fourier coefficients. As was done before, we equate the
coefficients of (2.25) with the values in (2.24) to obtain the foilowing:
16 Where Vor is the vorticity of the wind field. These equations demonstrate that a linear
approximation of the vorticity is possible, if values of uo, vo, Stretching, Shearing, and a
tangential velocity are known. This is possible at the wind profiler location by using the
kinematic properties obtained fkom the VAD analysis descnbed above, and the tangential
velocity measured by the profiler. This is shown in the following denvations:
Rewriting (2.24):
- ' -~ tr -s in2/3 3 -'- 3 Shr - cos 2B
Since we are interested in determinhg the vorticity for the linear component of the wind, all
t ems in (2.3 1) refer to those associated with the linear wind. However since the profder
measures the true tangential velocity, which may not necessarily be Iinear, we will be
introducing a bias by fncluding this information. This can be seen with the following
relations hip:
Where, VmP is the tangential wind measured by the wind profder, and V, is the tangential
velocity of the lrnear wind.
We then solve (2.3 1) for the vorticity, and subshtute in the values from the wind profiler to
obtain:
Where, is the aziinuth angle of the wind profüer, and rwp is the range of the wind
profder. With the assumption that the vorticity cdculated is valid for the entire domain, the
linear horizontal wind field becomes completely defined.
To obtain an idea of how the precision of this equation will be affected by the enors
in the measured values, an g e n e d error analysis is needed on (233):
Rewriting (2.33):
2 2 - sin p, 2 - cos ,6, Vor = - - Vw - -vo +
Y Y 'Uo
- Sm - sin(2BWp) + Shr cos(2Bwp)
Taking the partial derivative of Vor with respect to; u,, v,, Str , and Shr , we get the
following:
Equation (2.34) shows that the uncertainty in the vorticity is a two-harmonic sine function
of the position of the profiler and the uncertainties in VAD derived kinematic properties,
and proportional to the uncertainty in the tangentid velocity over the wind profder. The
implication of this equation is that the vorticity wili have an uncertainty that is essentially the
sum of al1 the uncertainties from each kinematic property, as well as the degree of non-
Iinearity over the wind profder.
2 3 Set Up Of The Retrieval Procedure
With the mathematical ground work in place, the next step is to combine everything
into a coherent procedure that can be applied to a set of data. First, since Our aim is to
calculate properties of the horizontal wind, only data at constant heights will be used in the
procedure. In other words, to obtain a 2-Dimensional wind at a given height, the only
ranges to be used are those at which the elevation angles intersect the desired height (Fig.
2.4).
Fig. 2.4 Ranges rl , r2, and r3 indicate the ranges used from elevation angles a l ,az , and a3 for a retrieval at height , Z .
The VAD analysis is then performed on the data at each range used. A plot can then
be made to determine the variation of the Fourier coefficients as a fu~ct ion of range, to
which we c m fit ( 2 2 3 ) and therefore find the kinemaùc properties of the linear wind, k . To incorporate the wind profiler data, a fine is fitted to a time series of the u, and v
components of the horizontal wind in order to fiter out the small scale non-linearities. The
values of these linear fits, at the desired thne of the retrieval, are then used in equation
(233) to obtain an estimate for the vorticity. With this full set of kinematic properties, we
can calculate the two dimensional h e a r wind using the Taylor series expansions of u, and
v , truncated at the linear terms.
2.4 Summary
Some difficulties with this retrieval procedure can be anticipated. First, there is the
problem of the non-linearity of the wind field. As mentioned eadier, the coefficients from
the VAD analysis Vary with range for a non-linear wind field, thus, the linear
approximation will have a limited range of applicability . f iom this, it can be concluded
that the procedure should only be applied to wind fields that are reasonably close to linear.
Another concern is the effect of non-linearities over the wind profiler. In other words, how
strongly non-linear does the tangentid velocity have to be, to give an unreasonable error in
the vorticity. To answer this, we will test the method on an artificial two-dimensional
wind, so as to determine the sensitivity of the rnethod. The next chapter will briefly
describe the artScid wind field used, how well the procedure worked, and the conclusions
drawn from the results.
Chapter 3
Analysis of a Test-Bed Wind Field
3.1 Wind field description
In Chapter 2, a procedure was developed to retrieve a linear wind field using the
VAD analysis and wind profiler data, and it was shown that this procedure could be
severely lirnitea by a highly non-hear wind field. To examine the sensitivity of the
procedure to the non linearities, a cubic wind field was created whose non-Linear terms
were relatively important compared to the linear te=, and the procedure was applied to it.
The artificial wind was created using (3.1) and ( 3 3 , which are Taylor Series in ci,
and v , that are truncated at the third order tenns.
It can be seen in (3.1) and (3.2) that the cross derivatives were neglected (as was the
vertical velocity , (V+w) ), so as to sirnpl* the calculations. The constant terms of the
velocity field were taken to be: uo = 5.0 m s-1, and vo = 4.0 m s-l , (the other terms are
given in Table 3 -1). Values were then calculated for c i , and v on a 80 km x 80 km grid,
centered on the scanning radar, For the purposes of visual cornparison, this wind field is
plotted over a map of the Montréal region (Fig. 3 -1).
TabIe 3.1
Numerical Values used for the
Taylor Senes' derivatives in cubic wind
Taylor derivatives u component (s - l ) v component (s-1)
3.2 Retrieval of Artificial Wind field
Before applying the procedure fiom Chapter 2, we frrst calculate the radial velocity (Vr ) that would be detected by a radar scanning this wind field. This means that the
horizontal radial velocity is calculated only for the ranges that intersect a given height (Fig.
2.4), that is arbitrarily chosen to be 2.75 km above the Radar. Also, since the radial
velocity seen by the radar is not along a horizontal plane, but along the planes of the elevation angles, av, the horizontal radial velocity is then projected on to these planes.
With these calculations, we obtained a data set that simulates what the Scanning Radar
would "see" under ideal conditions (without random noise or ground clutter).
The first test to the procedure is verifying the range dependence of the VAD
coefficients- As was described in § 2.3, the VAD coefficients of a non-linear wind field
should vary as a function of range that is dependent on the degree of non-linearity of the
wind field. This dependence allowed us to create a general equation (2.23), that could
provide us with the kinematic properties of the Linear wind field. In the case of o u test-bed
cubic wind field, the theory from Chapter 2 tells us that we should expect that the fitted
polynomial, (2.23), to be a second order even polynomial whose fiist coefficient is the
kinematic property for the linear component of the wind field. We see from Figs. 3.2(a-e)
that a second order even polynomial fits the range dependence of the VAD coefficients
exactly, codrrming the mathematics of Our theory. Given this, we can then use the first
coefficients of these polynomials as the VAD derived kinematic properties.
The next step is to determine the vorticity of the linear wind using the data fiom the
wind profiler. As we determined in Chapter 2, it is necessary to do some smoothing to the
profiler data, in order to elimïnate contamination due to non-linearities. What we proposed
was a time senes of u, and v from the profiler (approximately one hour), to which we fit
straight lines, we then use the values from these lines in the retrieval so as to reduce (or
eliminate) the degree of non-lineari~ of the measurement. The underlying assumption in
thïs procedure is that the wind field does not evolve significantly while it is advected over
the wind profùer, so that a time series can be translated to a spatial cross-section (the so-
cded "Frozen Turbulence Assumption"). For our test-bed wind, we imposed an advection
wind of: U = 5.0 m s-1, and V = 4.0 m s-1, to give us a time senes of u, and v in Fig.
(3.3), where tirne t = O is the time of the original wind field, and time t = 60 is the values of
u and v , 60 minutes later under the advection wind (U,V). After taking the tirne senes,
we fit straight lines to u and v (Fig. 3.3), and use the values fiom these fits at time t = O to
calculate VW . We then substitute the VAD denved kinernatic properties and VW into
(2.33) to calculate the vorticity for our linear two-dimensional wind (Fig. 3.4).
3 3 Discussion of ResuIts
To veriQ that the results obtained from our retrieval, we compared them with the
values that correspond to the true linear component of the wind field. We find that the
numencal values we obtained for the VAD derived kinematic properties (Table 3.2) were
exactly the sarne as those we calculated using the linear terms from Table 3.1. However
when we compare the retrieved vorticity for the linear wind to that calculated from the
values in Table 3 -1, we find that they dBer by about 2x10-5 s-1.
Table 3.2
Comparison of Kïnematic Properties
of renieval and true wind fields.
Kinematic Property True Wind field Retrieved Wind field
U O 5.0 rn s-1 5.0 rn s-1
Vo 4.0 m s-1 4.0 rn s-1
Divergence 9.0~10-4s-1 9 -0x 1 0-4s-1
S tretchhg -5 . ox~o-~s - 1 -5.0~10-4s-1
S hearing -7 -7~10-4s- 1 -7 .7~ 10-4s- 1
Vorticity 5.7~10-4s-1 55x10-4s-1
V' of hear wind:
Substituting values into (3.3):
This indicates that if the tangentid velocity from the linear fit to the time series is off by less
than 0.4 m s-l from the tnie 1inea.r tangend& velocS at that point, then the expected error in
the vorticity retneval should be on the order of about 2x 10-5 s- 1. When this is compared to
the case of the artificial wind, we c m see that the retrieved and true vorticity do, in fact,
differ by about 2 x10-5 s-l. As a further test we repeated the retrieval for wind profiler
locations all around the same range circle. The error of the vortkity is then plotted against GV, , so as to venfj the relationship determined by (3 -3). From Fig. 3 5, we c m see that
the errors calculated from each retrieval (stars) match perfectly with the straight line
calculated from (3 -3) (solid line). Figure 3 -5 also gives us an idea of the degree of non-
linearity at the 30 km range, in that there are a significant number of retrievals with an error in the vorticity of 1 x 10-4 s-1 or greater, and that to get errors of this magnitude, 6VI need
only be greater than 1 5 rn s-1 .
Such sensitivity to the non-linearity of the wind requires us to be cautious in using
data fiom the wind profiler without ve-ing the validity of the linear approximation. This
will then require a rnethod of deterrnining the validity of the linear approximation, since we have no way of calculating dV, for a real case without prior knowledge of the complete
linear wind field.
3.4 Summary
In this chapter, we tested the procedure fiom Chapter 2 on an aMcial wind field to
determine its sensitivity to a non-linear wind fieId. In this idealized test, we showed that
the linear values of divergence, translation velocity, stretching, and shearing, are retrieved
exactly, and that the voaicity is obtained within a reasonable rnargin of error. However
this low error in the vorticity seems to be a lucky chance, since there are a significant
number of possible profder locations, dong the sarne range circle, that have unacceptably
large uncertainties .
Thus, we can conclude from the results of the test-bed application, that the
mathematics of the procedure are valid, given the good results when Vwp is within 1 m s-1 of the hear value (Vr ). However, given the sensitiviq of the retrievd to 6Vr, we have to
develop a method of detemiinhg the validity of the linear assurnption without prior
knowledge of the complete wind field. The next step is to test the procedure on real data
(which will have the added effects of srna.ll scale non-linearities), and v e e whether such
departures fiom lïnearity occur in stratiform precipitation, and if they do, how to overcome
them.
Cubic Wind, Projected over Montreal 4 0 t + R 1 ~ + * 6 ~ ; K 1 ~ a t 8
Fig. 3.1 Plot of the artificial cubic wind, projected over a map of the Montreal Region. North is at the top of the page, and East is to the right. X marks the Iocation of the scanning radar (R).
Fig. No te
3.2 that each
20 40 60 80 range (km)
' = VAD coeff.
- = fitted polynomial (2" order)
O 20 40 60 80 range (km)
The plots of the coefficients as a function of range for the artificial follow dong a 2nd order polynomial.
wind.
1 Hour time series over W.P.
W w W w w w W ~ w W W ~ m m m f i n m a m f i , n
0 - - + = u velocity over profiler
1 '= v velocity over profiler
20 30 40 time (minutes)
Fig. 3.3 Time senes of u and v over location of the wind profiler, for the artificial wind field with an advection wind.
Wind retrieval for artificial wind 4 0 r . B 3 r - b s ~ s 1 - ~ 8 G .
Fig. 3.4 Linear Wind retrieval on Artificial wind. WP. marks the location of profiler.
O 1 2 3 4 5 Vt diff. (m s")
Fig. 3.5 Difference of retrieved vorticity from the linear vorticity, as a function of the difference of Vwp from Vl. The stars represent individual realizations, whiie the solid line represents the fitted line.
Chupter 4
Analysis of Real Data
4.1 Weather Synopsis
The procedure described in the previous chapters was f i s t tested on data fiom 1800
to 2200 U.T., (hiversal Tirne) 14 December, 1995. The case was a stratiform snow
event, associated witb a surface warm front of a 1002 W a low pressure system centered
over Lake Huron at 1800 U.T.. What made this a good case to test our procedure was the
fact that the wind field was horizontally uniform in t ems of both radial velocity, and
reflectivity (Fig. 4.1). Also, at any given height over the wind profüer, the winds seem to
vary srnoothly over tirne, although there is a rather significant directional shear with height
at lower levels (Fig. 4.2), and this could affect the measured radial velocities due to
averaging over the width of the radar bearn.
The two instruments used to test these cases are the S-band Doppler scanning radar,
at Ste. Anne de Bellevue, and a UHF wind profder in downtown Montréal. The scanning
radar scans a total of 24 elevation angles, alternating between the even and odd angles to do
two volume scans of 12 elevation angles in a total of 5 minutes. The UHF wind profder
measures profiles every 60 seconds, however, in order to have the same temporal
resolution as the scanning radar, the time steps were set at 5 minute intervals. The
consensus used was 60% of the data within t 2 m s-1 in a 15 minute period, based on Rogers et al. (1994).
4.2 Wind Retrieval for December 14 1995
In order to get an idea of the behaviour of this wind field, we applied the VAD
analysis dong different elevation angles. The results of this analysis can be seen in Figs.
4.3 (a-e), where we see that in some of the coefficients there is a darnped oscillatory
dependence on range and distance. This seems to ve r i e the findings of Rabin and
Zawadzki (1984) as weU as Caya and Zawadzki (1992). Looking at constant heights,
(represented by symbols) it c m be seen that some part of this non-linearity is i fûnction of
29 range only, so at f i s t glance we can Say that even in a stratiform precipitation event, the
linear approximation will only have a limited region of applicability. With this in mùid we
begin the test of ouï retneval procedure on the data.
As was the case in Chapter 3, our f i s t test is the validation of the variation of the
VAD coeff~cients with range. In this case we want to determine if we can fit a polynomial
of the form of (223) to the data, as was done in Chapter 3. However, unlike the test-bed
wind field (which we knew was cubic), we do not know the actual degree of non-linearity
of the wind field of the real case, thus we use (4.1) as a e s t guess, assuming the constant
term (which is equivalent to the kinematic property of the linear wind) will not change
significantly with the addition of higher orders:
The results of the fit of (4.1) to the coefficients are in Fig. (4.41, and they show that
although the frts are not perfect, (4.1) does offer a reasonable approximation to the large
scale non-Linearity of the wind field. However, we also note that there are a significant
amount of small scale non-linearities visible in each of the kinematic properties. The
problem with these is that, although they are smoothed out by the fitting of (4.1), their
presence will increase the standard deviation about the coefficients of (4.L), and thus
increase the uncertainties in Our kinematic properties for the linear wind (kg ). Another
consequence of both the large, and small scale non-linearities, is that they can increase the
degree of non-linear contamination of the profder information, and therefore we need to
find a method of determining the limit of our linear approximation.
There is no way of determining the degree of non-linearity of the wind field,
without knowing ail components, however if we make the assumption that the degree of
non-fine- is the same in the tangential and radial components, then we can use the radial
velocity as an indicator. We begin by rewriting (2.6), using the constant terms of (2.7) to
(2.1 1) for the coefficients (which, if we recall from Chapter 2, are the kinematic properties
of the linear wind), to give us an equation for the radial velocity for the linear wind.
+ (Shr) r - cos a r - c o s a
sin 2P + (S t r ) - cos 2P 2 2
We then substitute the retrieved vaIues of the kinematic properties into (42) to calculate the
radial velocity of the linear wind, and compare this with the true radial velocity . If this
cornparison is done for several ranges, then we may have a reasonable idea of the
maximum range of validity for the linear approximation. Some examples of this
cornparison for the Decernber 14 1800 U.T. data can be seen in Figs. 4.5 (a-d)- From
these plots we can see that the retrieved linear radial velocity ( V A ) , fits quite well on the
measured radial velocity up to at least 36 km from the radar as can be seen in Fig. 4 5 (d).
From such cornparisons we can then conclude that for the radial component, the linear wind
seems to be a reasonable approximation of the true wind, up to about a range of 40 km for
this pariicular case. Thus, using Our assumption that the behaviour of V, is representative
of Vr, we can then Say that the Linear approximation should be representative of the full
wind field up to 40 km. As a final check, we examine the tirne series of the winds over the
profiler for the previous hour (1700 - 1800 U.T.), and we find that a straight line fits the
data reasonably well, (Fig. 4.6) indicating that the profiler information is not contaminated
significantly by any s m d scale perturbation.
With the maximum range for the retrieval established, we apply the retrieval to the
data and find that the retrieved wind (Fig. 4.7), has a strong anticyclonic vorticity (about
-2.0 x IO-^ s-1). Considering the fact that there is an approaching synoptic scale low
pressure systern, we would expect at !east cyclonic (positive) vorticity. On the other hand,
since we are only retrieving a section of the wind field which is 80 km in diameter, perhaps
this is an accurate representation of the mesoscale around the radar. However, in order to
have ony confidence in our results, we need to perform an error analysis on the procedure
at severai times during the precipitation event.
4.3 Error analysis of Retrieval
To obtain a reasonable estimate for the uncertainty in the retrieved vorticity, we have
to analyze every step of the procedure to see what are the sources of errors for each. We
can then combine the errors of each step, and determine the overall uncertainty in the
retrieved vorticity .
Our first source of uncertainty in the retrieval cornes from the errors due to the VAD
analysis performed at each range. As we demonstrated in Chapter 2, one source of error
will be due to the non-linear terrns that are within the VAD coeffcients. However, while
31 these non-linearïties will lead to misinterpretations of the coefficients, we have developed a
method in Chapter 2 that will separate the non-linear terms fiom the linear terms, therefore
we will ignore these errors for the moment. Other sources of unceaainty in the VAD
analyses include random noise of the measurement, beam smoothing, and reflectivity
weighting . B eam smoothing , and reflec tivity weighting s hould have minimal effects since
we have picked a case where the reflectivity is reIatively unifonn, and the velocity has a
reasonably smooth gradient. On the other hand, random noise could have a significant
effect on the VAD analyses, and thus should be included in the enor estimate.
One of the methods we use to estimate the errors due to random noise is the
standard deviation about the coefficients on the two-harmonic fits, the definition of which is
given in Appendix A. In essence, it gives a quantitative value of the scatter about the
coefficient which we take to be the uncertainty. This uncertainty would be due to the
combination of the random noise and s m d scale eddies. Since this is a combination of
effects, one of which we do not want to include (that of the s m d scale non-linearïties), the
results of this error analysis would be an overestimation of the error in the VAD analysis.
However, it should still provide a reasonable estimate of the uncertainty of the coefficients
at a given range.
Another method of estimating the error in the VAD coefficients estimates the root-
mean-square error of the VAD coefficients. In this method we caiculate the difference
between the coefficients of fits from adjacent ranges, square hem, then take the mean of the
squares to determine the contribution to the error due to random noise. This procedure is
described in more detail in Appendix B. The result of this procedure is that it estirnates the
contribution of the random variations to the variation of the coefficients in range.
With the results of both of these error estimates, we can determine the overali efiect
of the random error on the VAD coefficients. The f i s t method gives us the variation of the
coefficients at a given range, due to the random error. This gives us an idea of the
uncertainty of the points to which we fit our polynomial in the next step. As we can see
from the error bars in Fig. 4.4, the errors are relatively srnall cornpared to the measured values, and thus they do not have a significanî contribution to the uncertainty in the
polynomial fit. The second method gives the contribution of the random error to the
coefficients variation with range. Again we can see from the errors obtained from this
method (Table BA), that they are an order of magnitude less than the coefficients
themselves. We can conclude fiom this that the random error does not have a signifkant
effect on the variation with range itself.
The next step of the procedure is the fitting of (4.1) to the coefficients as a function
of range. To estimate the uncertainties in the kinematic properties obtained fiom these fits
for a given time, we will use the variances about the polynomial fits. This wiil give us the
uncertainty due to the fact that (4.1) is not an exact representation of the non-linearity of the
wiod field. We fmd fÏom such an analysis on the data frorn 1800 U J. (Table U), that the
resultant errors are relatively small compared to the retrieved values.
Table 4.1
VAD denved linear bernatic properties
with uncertainties for 1800 UT.
December 14 1995
Kinematic Property Retrieved Value Uncertainty
Divergence -1.12 x 10-4 s-1 + 2.0 x 10-5 s-1
v o 11.6 m s-1 k 0.5 m s-1 S tretc hing 4.55 x 10-5 s-1 + 1.3 x 1 0 - 5 ~ - ~
It should be noted that the errors in Table 4.1 are also overestirnates, since they include the
effects of random noise, and higher order non-linearities. While overestimating the error in
the retneved kinematic properties helps to illustrate the point that it is relatively unimportant,
it will be of no value to do so when the error is used in the estimation of the uncertainty in
the vorticity. Therefore, we need to estirnate the true variance of the retrieved kinematic
properties, which we can then use to find the uncertainty in the retrieved volticity.
In order to find the variation of the retrieved kinernatic properties, we do an
autocovariance on a time senes of two hours. While the statistical sample is not very large,
we believe that with the smoothing of the VAD fitting, and the polynomial fitting, the
autocovariance should be stable enough to provide a reasonable approximation of the
random variations. To do this, we assume that the random error is completely uncorrelated
by the first time lag, so that the autocovariance at the "zeroth" lag is a combination of
random and non-random errors, while the other lags contain only the non-random error.
To find the contribution of the non-random error at the "zeroth" lag, we determine the trend
33 of the autocovariance for the other lags (1 to x ), and extrapolate this trend to the "zeroth"
lag. This is then subtracted from the total value of the autocovariance at that lag, giving us
the random error. Continuing with the same assumptions, if the random error was large,
then we would see a sharp change in the values of the autocovariance from the "zeroth" to
the firs t lag .
The autocovariance curves are shown in Fig. 4.8, and from these we can see that
the random variations are very small, since we do not see the sharp peak that is associated
with the randorn errors (as described above). To show how these errors influence the
retrieved vorticity, we modified equation (2.34), such that we ignore for the moment the
error due to the proNer then we take the absolute value of each of the remaining terms.
When equation (4.3) is caiculated for each five minute time step of the two hours of
data, we obtain the error bars on the vorticity in Fig. 4.9. What this clearly shows is that
the random errors from the VAD derived coefficients do not significantly affect the
precision of the retrieved vorticity. The next step is to investigate the uncertainty that arises
from the degree of non-linearity of the tangential veIocity measured by the UHF wind
profiler (VW ). This is important, since we assume in calculating the vorticity in (2.33),
that Ilfwp and al l of the kinematic properties (u,, v,, Stretching, and Shearing) we
retrieved, are those of the h e a r component of the wind, meaning that the vorticity that we
obtain is also the vorticity of the linear wind.
In the previous section, we made the assumption that due to the stratiform nature of
the precipitation, the statistical behaviour of Vf should be the same as V,, and we used this
in a qualitative sense to determine a maximum range for the retrieval. Since we now need a
quantitative value of the degree of non-linearity of V,, we will use the same assurnption,
and determine the mean deviation of the measured V, about the linear V, . In other words,
if we find that the measured V, has a mean deviation from the linear Vr of +x m s-1, then
we WU assume that any given measurement of Vf should not deviate from hnearity by
more than +x m s-1. Thus, we estimate the deviation of Vr at the closest range to the
profiler for which we have a VAD (recall we only have VAD at certain ranges, as shown in Fig. 2.4), and use this as an approximation the value of 6Vf for (2.32). When we apply
34 this to the retrieval at 1800 U.T., we see that this gives us an uncertainty about Vwp of
20.34 m s-1, and when this error is included, we obtain an overall uncertainty in our
retrieved vorticity of k4.2 x 10-5 s-1 .
When the error anaiysis was performed over the 2 hour period, we find that the
uncertainty in the vorticity increases dramatically after 2920 UT. (Fig. 4.10). The reason
for this is that the rnean deviation of V, from its linear component also increases
dramatically, which means that Our profiler rneasurement would most likely be significantly
dBzrent frorn V, . For example, at 2000 U.T. the deviation is approximately 4 mis from
linear, which contributes to an overall uncertainty in the vorticity of about k2.9 x 1 0 - ~ s-1.
To determine the cause of this deviation, we examined the cornparisons of the linear V, with
the measured V, at 2000 U.T. in Figure 4.11. We see fiom this cornparison at the four
ranges used widiin the retrieval domain, that there is a d e f i t e systematic deparnire from
the Iinearity hypothesis at this time. To examine this further we take a time series of the
divergences obtained from the VAD at each of these ranges, for the full two hours, and
compare their evolutions (Fig. 4.1 2). We find from these evolutions that up to about 1920
UT., one value of divergence could reasonably approximate the whole dornain. However
after 1920 UT., the rnean divergence within the 21 km range smoothly increases by 2 x
10" s-1, whereas the mean divergences at the other ranges do not increase so dramatically.
The smoothness of the change, and the fact it reaches a peak and levels off, strongly
indicates that this is a naturd variability. Also, the fact that it is more noticeable at the
shorter range (where smaU perturbations have a stronger effect on the mean divergence)
seems to support this idea. In any case, this non-linearity is rather strong and quite
obviously makes the retrieved vorticity very unreliable. Thus, in order to maintain the
uncertainty in the retrieved vorticity below a certain value, a threshold of 1 rn s-1 was
imposed on the deviation from linearity, so that for any deviation beyond 1 m s-L no
retrieval will be performed (Fig. 4.13). The choice of a 1 m s-1 threshold is rather
arbitrary, and was only chosen for dernonstration purposes.
As another check to the retrieval procedure, we tried it on another stratiform snow
case, that occurred December 9 1995.
4.4 Wind Retrieval for December 9 1995
The second case chosen to test the retrieval procedure, was also a stranform snow
event that occurred over Montréal on December 9 1995, starting at around 2000 U.T.. As
35 with the previous case, it was chosen for its lack of noticeably strong small scale feanires in
both reflectivity and velocity, and because the velocity profiles appear to be relatively
uniform over tirne. Examples of the instrument output for this case are given in Figs. 4.14
and 4-15
Repeating the process described in section $2.3 for the data at 22 15 U.T. 9
December 1995, we determine the VAD coefficients at each range at 2.0 km height (Fig.
4.1 6) , find the VAD derived linear coefficients (Table U), retrieve the 2 dimensional wind
field (Fig. 4.17), and calculate the h e m vorticity (6.7 x 10-5 si). In this case, as opposed
to the 14 December case, we obtain a positive (cyclonic) vorticity. To determine how
certain we are in this value, we repeated the retrieval and error analysis from section $4.3
on a 1 hour time series (2200 - 2300 UT.). We can see from the time series of this
retrieval (Fig. 4-18}, that the bernatic properties in this case are Iess variable than those in
the 14 December case, thus reducing the randorn error contribution to the vorticity. We cm
also see fiom the small error bars on the retrieved vorticity, that the wind field is very close
to linear, since small error bars indicate small variations about the linear radial velocity.
Table 4.2
VAD derived linear kinematic properties
with uncertainties for 22 15 U.T.
December 9,1995
Kinematic Property Retrieved Value Uncertainty
uo 11.6 m s-1 k0.77 rn s-1
VO 12.7 m s-1 *O -48 m s-l
Divergence 3.4 x 103 s-r 12.7 x s-l
S tretching -1 .O x 10-5 s-1 5~2.2 x 10-5 s-1
Shearing 3.0 x 10-5 s-1 el -7 x 10-5 s-l
4 5 Further Analysis and Summary
In this chapter we have tested the procedure on two winter cases, to see how it
would perform in an actual operational situation. We have verified that the random errors
have minimal contributions to the overall uncertainty in the system, as shown by the
smooth variations in tirne of al1 the kinematic properties. However, as we discovered in
Chapter 3 with the test-bed wind field, the main contribution to the uncertainty in the
retrieved vorticity, is due to the deviation from linearîty of the wind field at the position of
36 the wind profiler. In December 14 case, this was revealed dramatically with the sudden
increase in non-linearity after 1920 U .T..
As was mentioned in section $42, we used ody second order even polynomialç to
obtain the VA.D derived Linear kinematic properties. The question is, how would Our
results improve if we used higher order polynomials, (especially in the case of December
14). To determine this we repeated the retrieval for the December 14 case, with the o d y
clifference being that we used 4th order even polynomials for the determination of the VAD derived linear kinematic properties (Fig. 4.19). The results of this retrieval are in Table
4.3, and if we compare them with those fiom Table 4.1, we find that for the most part, the
retrieved linear kinematic properties from each method are within each other's value of
uncertainty. Also, if we examine the two hour bme senes (Fig. 4-20), using the 4th order
polynomials, we find that, although there are differences from the curves in Fig. 4.13,
there are no significant irnprovernents fiom the values derived f?om the second order fits.
TabIe 4.3
VAD derived hear kinematic properties
from 4th order polynomials for 1800 U .T. Decernber 14,1995
Kinematic Property Retrieved Value Uncer tainty
"O 4.5 m s-1 a . 3 7 m s-1
VO 11.5 m s-1 4.57 m s-1 Divergence -1 -4 x 10-4 s-1 22.5 x 10-5 s-1
Stretc hing 5.8 x 10-5 s-l -t 1.3 x 10-5 s-1
Shearing 4.8 x 10-5 s-i -cl -9 x s-l
As a counter-example, we also calculated the mean values of the coefficients, and used these as the kinematic properties for the linear wind. The resulting errors from the
scatter about the means were twice those obtained from the 2nd order even polynomial fits.
In tems of the constant wind (uo,v0) this means errors that are still within 1 rn s-l, which is
still a reasonable estirnate. However, in tems of the other coefficients, this means that we
now have errors of the order of the retrieved values. This tells us that for this case, the first
and second order derivatives are of equal importance.
37 Finally, we noticed that the errors due to the VAD derived linear kinematic
properties were srnall relative to the retrieved vorticity, and therefore not the main source of
ccncem in using the procedure. However, as was discovered in Chapter 3, the uncertainty
in the degree of non-linearity of die tangential velocity used (VW ) is a big concern in
using the procedure and should be kept as smali as possible, given the data avadable.
These, and other final conclusions about the procedure WU be presented in Chapter 5.
Radial VeIocity at height = 2.75 km
Reflectivity at height = 2-75 km Ref (d Bz)
Fig. 4.1 Constant Altitude Plan Position Indicator (CAPPI) plots of Radial Velocity
and Reflectivity, for 1800 U.T. Decernber 14 1995. The CAPPI's were made according to
the description in section $2.3 (Fig 2.4).
Fig. 4.2 Time series of the horizontal wind over the profiler, from the surface to 4.2
km, for 1705 - 2000 U.T. Dec. 14 1995.
Wind Profiler data from 14/Dec/95
t2o
I I L I 1 8 1 1 I l I I 3.5-----LA-d,-J-/-/,,L///,,J,,,,,, b t I I I l i l
/ / c c / z / c r / M / / - / / / / / / ~ / f , / / / . / / / / / -
- / / / / / / / / / / / / / / / / f / / / / / / / / P P / / f - - , / / / r / / r / / / / r r / " / " f / / f / / / / / P / / / / - - / r r r / / / r r / / ~ / r r / / / / / r r / / / f / / / / -
r / / f f f / r P / / / / / / / r / / / / / f / / / / f f f - 3 * O ~ , / / . , / / / / / , , , / , , , , , , / / , / I I f f f , -
/ / / / / / / / / / . / / / / / / / / / / / f f f f f / f /
- E
I m .- al -t
/ / / / / / f / / / / / / / / / / / f f f / f f / f / f f f - / / / / / / / / / / / / / / / / / / / f f f f f f f / / f f
- f f / / / / / / / / f f f f f / / r / / f f / / / / / / f / -
2 . 5 - t r r t ~ ~ r ~ ~ ~ r r t t r t t r r / 1 / / ~ ~ / / r t r - - i i r r t . t t t t t t t t t t t / / f f / / / / / f f f f r r t t t t r r t f t t t t t t t t t t t t t t f f f / f f / - - t t t t T t r t t t t t t t t t 1 t t t t t r t r r t t f f - - t t t t t t t t t t t t t t t f t f t t t r t t t r t r r , -
s 2 . 0 - t t t t t t t t t t t t t t t t f t t t t f t t t r t t t I - - t t t t t t t t t t t t t t t t t t t t f r 1 t t t t r t t -
' t \ t t \ \ \ t t t t t t t \ \ \ t t t t t r f f f f r r f - - t t t \ t \ \ \ \ \ \ \ \ \ ~ \ ~ \ t f r r ~ ~ ~ f f ~ ~ f - - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t \ \ t t ~ f r t f f f f f r -
1 . 5 + \ \ ~ ~ ~ \ \ \ t \ \ t t t t t t I r t t t t I I r f -
- \ \ \ \ \ \ \ \ \ t t \ \ t t t t t t t r t r t I r , I r , -
- \ \ \ \ \ \ \ \ \ \ \ \ \ t t t t t T t t t t t t t t t r t t t - - \ \ \ \ \ t t t t t t t t t t t t t t t t t t r i t - - \ \ \ f \ I \ t t z t t f t t t t t t f t t t T t t t t t t -
1 * 0 - \ \ \ \ \ \ \ \ \ z ~ f l f f l t l l l 7 t t t t t t t t t - - \ \ \ \ \ \ \ \ \ \ \ t \ t \ \ \ \ \ \ \ \ \ t \ \ \ \ f f - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -
0 . 5 - . . ~ ~ - - - - - - ~ - - ~ - - - - - - - ~ ~ ~ ~ - ~ ~ . . - _ - - - - - - - - - - C C - - Z Z C & _ _ - C - - - - - - - - - - - - - - -
- / . * . 0 # / @ / / / 0 / / / M / 0 / / / 0 / - / / / / / / -
- , a . . , / / y / - - > I I I I I I , I I I , 1 I l I I l I I - - - ---
18:OO 18:30 19:OO 19:30 20:OO time (U.T.)
Horizontal Wind (U-VI
(b) i 1- /
/ f /
I 1
1 I
! / /
20 40 60 80 range (km)
- = elv. angle 1.8 - = elv. angle 2.7 - = elv. angle 4.1 -. = elv. angle 5.9 t = 2.5 km height * = 3.0 km height
O 20 40 60 80 range (km)
Fig. 4 3 Plots of the VAD coefficients dong eievation angles, for the case of Dec. 14 1995, at 1800U.T..
O 20 40 60 80 range (km)
G 20 40 60 80 range (km)
l = VAD coeff. - = fitted polynomial (2" order)
Fig . 4.4 Kinematic properties as a function of range for 1800 U.T. Dec 14 1995. Error bars i~dicate the mean variation of adjacent measurements over half an hour. Soiid lines are 2nd order even polynornial fi&.
O 100 200 300 O 100 200 300 Azimuth angle (degrees) Azimuth angle (degrees)
Fig 4.5 Cornparisons of retrieved radial velocity (solid lines), the rneasured radiai velocity (dots), and the two-harmonic Fourier fits (dashed lines), for Dec. 14 1995 1800 UT.. At; a) range of 21 km from the radar, b) range = 26 km, c) range = 3 1 km, and d) range = 36 km.
-2 - + = u velocity over profiler -
4- ' = v velocity over profiler
- 6 - 1 ~ . ~ . ~ ~ 1 ~ i ~ t ~ . . . ~ . t ~ . . . ~ . . . . . ~ ~ ~ t . t t . r ~ i ~ ~ , , . , . . , ~ . , , , , , , . , - O 10 20 30 40 50 60
time (minutes)
Fig. 4.6 Time series of u and v over profder for 1 hour, for Dec. 14 1995. From 1700 - 1800 U.T..
Wind retrieval at h=2.75 km, 14lDec195 1800 UT.
t f f i r r r
f t i : t t t
t f t 1 t : t
t t t t t t t
f t t t i i r t i t t
l ; ? l \ ! t \ : t ! r l f l t l t \ t t \
-40 -20 O 20 40 x(km)
Fig. 4.7 Plot of retneved 2-D wind field over Montréal, Dec. 14 1995 1800 UT..
Autocovariance of Kinematic Properties -- 3.0
5 20 x - 5 1.0 O
., O
m
Fig 4.8 Autocovariances of the VAD derived kinematic properties £rom a two hour time series of; a) Divergence, b) Uo, C) Vo, d) S hearing , and e) Stretching .
Time Series of Retrieved Kinematic Properties 20 k (a) - . - . = i / w p i
- - - = Vorticity
18:OO 18:30 19:OO 19:30 20:OO Time (U.T.)
Fig. 4.9 Tirne series of al1 retrieved kinematic properties, error bars on vorticity are calculated oniy fiom the random perturbations due to the VAD denved kinematic properties. The error bars on the rest of the plotted values are due to the variation about the polynomial fits at each time.
Time Series Of Divergence and Vorticitv
18:OO 18:30 19:OO 19:30 20:OO Time (U.T.)
Fig. 4.10 Two hour tirne series of Divergence and retrieved vorticity at a height of 2-75 km for Dec. 14 1995 from 18:OO to 20:OO U.T.. Error bars on vorticity are calculated Çom the random errors fiom the VAD derived kinematic properties, and the mean deviation of the measured radiai velocity fiorn the linear radiai velocity.
Time series of Divergence at different ranges
- . - . 36 km range - - - - - - - - - - -
= fitted divergence
18:OO 18:30 19:OO 19:30 20:OO Time (U.T.)
Fig. 4.11 Two hour time series of mean divergence from four ranges within the retrieval domain, and the retrieved h e a r divergence.
O 100 200 300 O 100 200 300 Azimuth angle (degrees) Azimuth angle (degrees)
Fig. 4.12 Cornparisons of retrieved radial veIocity (solid lines) to the measured radial velocity (dots), and two-hamonic fits (dashed), for Dec. 14 1995 2000 2. At; a) range of 21 km fiom the radar, b) range = 26 km, c) range = 3 1 km, and d) range = 36 km.
Time Series Of Divergence and Vorticity
18:OO 18:30 19:OO 1 9:30 20:OO Time (U.T.)
Fig. 4.13 Two hour time series of Divergence and retrieved vorticity at a height of 2.75 km for Dec. 14 1995 from 18:OO to 20:00 U.T.. No values of vorticity are retrieved at times where the degree of non-linearity of V, exceeds i m/s.
Radia1 Velocity at 2.00 km Vr (m/s)
Reflectivity at height = 2.00 km Ref (dBz)
Fig 4 -14 Constant Altitude Plan Position Indicator (CAPPI) plots of Radial Velocity and Reflectivity , for 2215 U.T. December 9 1995. The CAPPI's were made according to the description in section $2.3 (Fig 2.4).
Wind Profiler data from 9/Dec/95
-1 -2 1.1 O 20 40 60 80
range (km)
O 20 40 60 80 range (km)
I = VAD coeff.
- = fitted polynomial (2" order)
Fig 4.16 Plot of VAD denved kinernatic properties as a function of range for 2215 U.T. December 9 1995, with polynomiai fits for a) divergence, b) uo, c) vo, d) shearing, and e) stretching.
Wind retrieval at h=2.00 km, 9/Dec/95 2215 UT.
4 0 -20 O 20 40 x ( W
Fig 4.17 Vector plot of retrieved 2-dimensional wind field for 22 15 U.T. December 9 1995, projected over the island of Montréal.
Tirne Senes of Retrieved Kinernatic Properties - = vo -.-.=vwp
22XlO 2212 2224 222% a48 Time (U.T.)
Fig 4.18 One hour t h e senes (2200 - 2300 U.T. Dec. 9/95) of retrieved kinematic properties and profiler Vt ; a) uo, vo, V-. b) divergence, and vorticity (with error bars), c) saetching, and shrearing.
O 20 40 60 80 range (km)
1 = VAD coeff. - = fitted polynomial (4" order)
0.2 - O 20 40 60 80
range (km)
Fig 4.19 Plot of VAD derived kinematic properties as a function of range for 1800 U.T. December 14 1995, with fourth order polynomial fits for a) divergence, b) uo, c ) vo, d) s hearing , and e) stretching .
18:OO 18:30 19:OO 19:30 20:OO Time (U.T.)
Fig 4.20 Two hour tirne series (1800 - 2000 U.T. Dec 14/95) of retrieved kinematic properties (from 4th order polynomial fits) and profiler Vt for; a) uo, vo, Vtwp, b) divergence, and vorticity (with error bars), c) stretching , and shrearing.
Time Series of Retrieved Kinematic Properties 20 (a) -.-.=vwp i
---=uwp -- 15p -- - = vo 4
Chapter 5
Conclusion
5.1 Summary and Conclusions
Since the developrnent of the VAD analysis two assurnptions have been made about
the meaning of the coefficients. The first was that for large scale precipitation, the non-
linearities could be neglected, and therefore the coefficients fiom a two-harrnonic Fourier
series fit would correspond to the kinernatic properties of a Iinear wind. The second
assumption, is that the coefficients were the means of the kinematic properties within the
scanned circles, as is the case with the divergence. m e sorne papers descnbed concerns
about the limitations of the linear hypothesis (e-g. Pasarelli, 1983), Caya and Zawadzki
(1992) demonstrated how both interpretations of the VAD coefficients could have serious
implications.
As a continuation of the work by Caya and Zawadzki (1992), we have added a
further modification to the VAD andysis. In Chapter 2, we began with the proof fiorn
Caya and Zawadzki (1992) that the VAD coefficients could be interpreted as a kinematic
value of the linear wind plus the non-linear terms. We then use this property of the wind
field to separate the linear frorn the non-linear terms, to obtain the kinematic values of the
iinear wind. At this point, we have two things of importance: first, we have the kinematic
properties (except the vorticity) of the wind field directly over the radar itself, and second,
we have an indication of the degree of non-iinearity of the wind field. As an addition to this
information, we developed a procedure that incorporates the tangential velocity over a wind
profiler and the retneved kinematic properties, to calculate a two dimensional linear wind
field.
In Chapter 3 we tested our procedure on a test-bed cubic wind field. When the
procedure was applied, we retneved the exact values of the VAD derived iinear kinematic
properties (divergence, u,, v,, stretching, and shearing), and obtained a vorticity within 2
x 10-5 s-1 of the true linear voaicity. However, this apparently good result was due o d y to
the face that we happened to use a tangential velocity (VmP ) at a location where the wuid
50 field was very close to linear. When we chose other points around the same range circle
that were not so linear, we found that we had tangentid velocities that deviated from
Linearity by as much as 4 m s-1. When an error analysis was performed on our equations,
we found that the error in vorticity is proportional to the deviations fiom linearity by 2/r
(where r is the range of the profiler). This rneant that at the range we "measured" our
tangential velocity (30 km) we had a 4 m s-1 deviation fiorn linearity , leading to an error in
our vorticity that was as large as 2.67 x 10-4 s i . To deal with such deviations from
hearity, originating from srnall scale or random perturbations, we decided to use a one
hour time series of the wind profiler data and apply a linear fit, as a way of approximating
the linear tangential velocity. As will be discussed later, this rnay not be a helpful or even
necessary modification.
We then applied the procedure to a stratiform snow event fiom December 14 1995.
We first applied the procedure to only one specific time (1800 U.T.), and retrieved a
vorticity that was negative (anticyclonic). This was such an unexpected result, that we
performed an error analysis on each step of the procedure, to verîQ whether this was an
actual feature of the wind field, or an artifact of the collection of uncertainties in the
procedure.
We deterrnined fkom the error analysis on the retrieval of the VAD derived linear
kinematic properties, that the uncertainty in each of the properties at a given time were an
order of magnitude less (or smaller) than the retrieved value. Also, an autocovariance on
the full two hour time senes showed that the random variations were actudy even smaller,
thus convincing us of the reiiability of the retrieved values. Thus, the negative vorticity that
we observed was not due to errors in the VAD measurements, as their contribution to the
total uncertainty was negligibly small. We then came back to the data from the wind
profiler, which we already knew from the results in Chapter 3, could make a sizable
contribution to the total error in the vorticity.
The problem with detennining the degree of non-linearity of the profiler tangentid
velocity is that, unlike the test-bed wind field, we have no linear tangential velocity to
compare it to. Our solution from Chapter 3, was to simply take an hour long time series of
the profiler winds, perform a Linear fit, and use the value from the linear fit in the retrieval.
The difficulty with this is that such a procedure requires the assumption that the wind field
does not evolve as it advects over the profiler (Frozen Turbulence assumption). While this
may be a very good assurnption over short time periods, it becomes Iess realistic over
51 longer time periods. Also, it turns out that small non-linearities would most likely be
srnoothed out by the 15 minute consensus tirne used by the profiler, and thus the linearizing
would be useless, whether the Frozen Turbulence assumption was valid or not. To see this
another way, if we assume that the system progresses at a speed of about 1 km min-'. then
a non-linearity of the order of 40 km will advect over the profüer in 10 minutes, and thus be
smoothed out by the 15 minute consensus t h e .
Since we cannot eliminate the effects of the large scale non-linearities, we found a
way to determine their importance. We did this by assuming that the wind field behaved in
such a way that the magnitude of the departures fiom Linearity of the tangential velocity
would be the same as that of the radial velocity. So that when we determined the mem
deviation frorn Linearity at a given range for the radial velocity, we had an estimate of the
rnost likely deviation from linearity we would have on a given rneasurernent of tangential
velocity. In essence, we established a way of evaluating the validity of the iinear wind
hypothesis at the profiler site, that is used to calculate the vorticity. In the case of 1800
U.T. December 14 1995, we find that the mean deviation of radial velocity at the range of
the profder is only about 0.3 m s-1, and thus we conclude that the value we use for the
tangential velocity is likely very close to the Linear tangential velocity, and that the retrieved
vorticity should be reasonably accurate (within F 4.2 x s-1). On the other hand, we
discovered that by 2000 U.T. on the same day, the deviation from linearity of the radial
velocity was about 4 rn/s, indicating that our measurement of tangential velocity would add
a significant degree of uncertainty to the retrieved vorticity. Overall, we found a way of
d e t e d n i n g the degree of non-linearîty of the wind field, and if we impose a certain
threshold on the deviation fkom lineariiy (e.g. 1 m s-l) then we cm keep the accuracy of the
retneved vorticity within a certain limit, and that the anticyclonic vorticity is indeed a real
feature of the wind field that we studied.
In conclusion, we found that the procedure can give a meaningfui result, based on
the information we have to test it with, as long as the errors are kept at a minimum. To do
this, we impose a threshold on the mean deviation from linearity of the radiai velocity . This
was confmed from the results of the application to another case (2200 - 2300 U.T.
December 9 1995). However, we also discovered that non-linear terms can be very
important, even in stratiform precipitation, even though the wind field appears to be
relatively uniform. This is significant, since it easily dernonstrates the problerns that a i se
by interpreting the VAD coefficients as the kinematic properties of the linear wind.
52 Our analysis aiso exposes a problem in the method used by Harris (1975) and
Testud et al. (1980), where they use the VAD coefficients as the kinematic properties of a mean flow, and interpret the difference between the data and the two harmonic fit as a wave
perturbation. This problern lies mainiy in the fact that the VAD coefficients are not
equivalent to the mean kinematic properties, and they may not even be close
approximations, since the non-linear terms can be of equal importance as the linear terms.
This could possibly be overcome by initially assuming that a wave function, instead of a
Taylor senes, best represents a wind field, and determining the phases and amplitudes from
a VAD analysis.
5.2 Suggestions and Future Work
If this procedure is to be implernented either operationaiiy, or as a research tool, we
suggest some further work be done in evaluating its results. An ideal evaluation would be
to do a direct cornparison with output from another source of wind measurements, such as
dual Doppler, or bistatic systems. The goals of this s o a of evaluation would be to
deterrnine whether the linear wind provides a reasonable approximation to the wind field,
and if oot, would any Taylor series be able to represent it accurately with a reasonable
number of tenns.
If the previous evaluations tum out positive results, the procedure should then be
tested on stratiform rain events. In this thesis we concentrated mainly on the snow events,
due to the overall uniforrnity of both the reflectivity and velocity fields. However, the rain
cases would provide a challenge, in that there are more fluctuations within both fields, due
to things like embedded convection. Perhaps a combination of some of the modified VAD
techniques (EVAD, CEVAD , etc.) with the profiler would be able to work around this.
Appendix A
Equations In Error Analysis
A.1 Standard Deviation
In Chapter 4, we estimated the error at each range for the VAD coefficients (Two
hamonic Fourier fits), by using the standard deviation of the data about the coefficienl. In
tems of a Gaussian fùnction, the standard deviation is the region where there is a 68% chance the measured value will be. The general equation to cdculate a standard deviation of
a given set of data is:
Where, S is the standard deviation, x is the mean value of the measured data, q, and N is
the number of data. ln Our case, since we are using a least-squares method to fit a cuve to
the data, Y, is the value of the fitted curve, rather than the mean value. This was also used
to determine the uncertainty of the linear components of the VAD derived kinematic
properties .
A.2 Autocovariance
To ob6iaïn an approximation of the random error, we perfonned an autocovariance
on the twc hour time series on 14 December 1995. The equation for the autocovariance of
a data set, x, is:
n-k
A k = i=l n - k
Where, n is the number of data, k is the time lag, and x ' , is the perturbation of the data
point fiom the mean value of the complete set,
The assumption in using the autocovariance is that the random errors would be
completely decorrelated by the fmt t h e lag, (tf+r ). This means that the value of Ak at
zero lag would be due to both the random noise, and perturbations correlated over several
time lags . In this case, we determined an upper limit to the noise by taking the ciifference
of Ag, and AI, and taking the square root to the result.
In order to determine the degree of non-linearity of the radial component of the wind
at a certain range, we calculated the mean deviation of the measured values from the iïnear
values. This measurernent will give us the effect of the larger scale non-linearities while
fdtering out the smaller scale. The equation for the mean deviation of data set x is:
Where: d is the mean deviation, xi is the measured value, xfi is the linear value, and N is
the number of data points.
Error Analysis of VAD Coefficients
8.1 Description
One of the methods described in Chapter 4 to determine the uncertainty in the VAD
coeffkients, is to perform VAD fits on adjacent ranges, take the difference between the
coefficients, square the differences, and take the mean of the squares. The idea behind this
method is that it can give an estimate of how much the rândom error affects the variation in
range of the coefficients.
We begin the description with the Fourier series equation used in the VAD fit at a
single range:
Where the terms a,, and bi , are the VAD coefficients, and /3 is the azimuth angle. We then
fit (B. 1) to adjacent ranges at different heights and elevation angles, as in Fig . (B. 1).
r
Fig, B. l Ranges and heights where VAD were performed. Numbered regions indicate
adjacent ranges where differences between coefficients were taken. Elevation angles are indicated by ar,.
We then have two sets of VAD coefficients for each of the regions (1-8) in Fig. B .1. We
then take the difference between the coefficients in each region to obtain a set of differences
56
for each coefficient. We then represent the set of differences for a given coefficient by Ai,
which can be written as:
where E~ is the contribution due to the random error, and 6, is the contribution due to the
non-linearity of the coefficient in range. The obvious dficulty is the separation of the
random error from the dependence in range. To do this, we square the differences of each
of the eight regions, and take the mean of the squares, the square root of the result gives us
the rms (root-mean-square) error of the VAD coefficients (B .3).
where N is the number of differences used (in this case eight), and ErmS is the rms error.
To show how this could separate the two sources of error, we square (B.2) to obtain the
following:
Then, if we take the mean of (B.4), and assume that the two sources of error are
uncorrelated with each other, then the product of the two wiil go to zero so that we obtain:
However, since the differences used in this calculation are d l in different locations, rhen
there is no reason to believe that they are correlated with each other and thus, the mean of 6,2 should go to zero. Then the only remaining term is the square of the random enor,
which we find by taking the square root of (B S).
We can be more certain of our assumptions if we increase the number of
independent realizations. One way of doing this is to use the same eight pairs as in Fig.
57 B -1, for several tirne steps. For example, if we did this in five minute time steps for a total
of 30 niinutes, we would have a set of 56 differences for each coefficient, instead of just
eight. When this was done for the time period of 1800 - 1830 UT. 14 December 1995, we
obtained the errors given in Table B. 1 .
Table B.1 Random Error Approximations for
Kinematic Properties
14 December 1995
Kinernatic Property Uncertainty
Divergence 7.38 x 10-6 s-i
u o 0.28 m s-1
vo 0.42 rn s-l Shearlng 1.68 x 10-5 s-1
S tre tc hing; 1.89 x 10-5 s-1
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