Non-synoptic versus pseudo-synoptic data sets: an assimilation experiment

Ž .Journal of Marine Systems 29 2001 313–333www.elsevier.nlrlocaterjmarsys

Non-synoptic versus pseudo-synoptic data sets:an assimilation experiment

M. Rixen a,), J.T. Allen b, J.-M. Beckers a,1

a GHER, B5 Institut de Physique, UniÕersite de Liege, 4000 Liege, Belgium´ ` `b Southampton Oceanography Center, Southampton, S0143ZH, England, UK

Received 31 August 1999; accepted 2 August 2000

Abstract

Several first-order correction methods are implemented to compute pseudo-synoptic data sets from non-synoptic raw datasets. These include a geostrophic relocation method, a linear and a quadratic interpolation method, and a method usingspatio-temporal correlation functions. The relocation method involves analyses and geostrophic velocity computations toallow the relocation of stations in time and space to a particular analysis time. Interpolation methods involve several almostidentical and consecutive surveys interpolated in time. Temporal weighting methods are based upon a spatio-temporalfunction modifying the weight on data with respect to the time at which they have been sampled. These techniques are tested

Ž .on the OMEGA data set and are validated by simple nudging assimilation into a 3D primitive equation model PE . It isshown that, under certain hypothesis, these methods are able to correct the lack of synopticity in hydrographic data sets, andimprove the diagnosis of vertical velocities computed from the Omega equation.

These methods are of particular interest for the scientific community. They might be used together with diagnosticmodels. They might provide suitable pseudo-synoptic fields required by 3D PE models as initial conditions, boundaryconditions or assimilation data sets. They may also be useful in the design of mesoscale samplings. q 2001 Elsevier ScienceB.V. All rights reserved.

Keywords: Non-synoptic data sets; Pseudo-synoptic data sets; Assimilation

1. Introduction

Up to now, the sea does not allow synopticsampling of its state by ship cruises, because hydro-

Ž .graphic data sets are collected over a certain long

) Corresponding author. Tel.: q32-4-3663650; fax: q32-4-3662355.

Ž .E-mail address: [email protected] M. Rixen .1 Research Associate, National Fund for Scientific Research,

Belgium.

period of time. Since field surveys typically lastseveral days, weeks or even months, the sea statecannot be considered stationary over the samplingperiod. Thus, for example, due to the transient be-haviour of the ocean during a field survey, the sea

Ž . Žsurface temperature SST in situ data set non-syn-.optic might be quite different from the correspond-

ing SST satellite image. In addition, diagnosed vari-ables deducted from the in situ data set might beeven more sensitive to synopticity problems and lessconsistent with the true synoptic state of the ocean.

0924-7963r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved.Ž .PII: S0924-7963 01 00022-7

( )M. Rixen et al.rJournal of Marine Systems 29 2001 313–333314

Small changes in the density field s for examplemight result in important differences in vertical ve-locities computed from the diagnostic Quasi-Geo-

Ž . Žstrophic QG Omega equation e.g. Hoskins et al.,.1978 .

Ž .A recent work of Allen et al. 2001 investigatesthe bias induced by asynoptic sampling in the esti-mation of diagnostic vertical velocities and fluxesŽ .see Section 2 .

This critical point was also addressed during theOMEGA2 project which focused on mesoscale hy-drodynamics in the Alboran Sea: were the data setssynoptic enough to allow reliable vertical velocitiesestimations from the Omega equation?

The actual OMEGA data sets were collected dur-Žing two different cruises: the first cruise Allen et al.,

.1997 was carried out on Bio-Hesperides from30r9r96 to 14r10r96 in the Alboran Sea sampledat time and space scales that had not been addressed

Ž .in the past Gomis et al., 2000 . As the objective wasto diagnose the 3D geostrophic and ageostrophiccirculations within the frontal structure associated

Ž .with the Atlantic Jet AJ , the sampling strategy hadto be suitable for small, rapidly evolving structures.Indeed, both in situ measurement and remote sensingobservations in the past have already provided evi-dence of strong short-term variability in the Alboran

ŽSea e.g. La Violette 1986; Tintore et al., 1991;´Prieur and Sournia, 1994; Folkard et al., 1994; Viudezánd Tintore, 1995; Baldacci et al., 1998; Viudez et´ ´

.al., 1998 .The cruise involved three repeated surveys of 11,

9 and 10 respective parallel north–south legs ;10Ž .km apart in a domain ;80 km wide north–south

Ž . Žand ;110 km long east–west , using SeaSoar aversatile towed undulating vehicle used to deploy a

.wide range of oceanographic monitoring equipmentŽfor hydrographic observations and an ADCP Acous-

.tic Doppler Current Profiler for velocity measure-Ž .ments Allen et al., 1997 . Stations along sections lie

Ž .approximately 0.0368 ;4 km apart. All surveyswere carried out eastward, that is downstream with

2 MASTIII project funded by the E.C., contract MAS3-CT95-0001.

respect to the AJ mean flow. The first survey beganon 1r10r96 and ended on 4r10r96. The secondsurvey began on 6r10r96 and ended on 8r10r96.The last survey began on 9r10r96 and ended on11r10r96. A detailed description of the calibrationand preliminary data processing of the SeaSoar and

Ž .ADCP data is given in Allen et al. 1997 .The characteristic time of variability of the AJ in

Ž .the Western Alboran Gyre WAG is estimated to beŽ .5 to 9 days Perkins et al., 1990 . A priori, these

surveys seem to be synoptic since they do not lastlonger than 3 or 4 days. We will see, however, thatthe OMEGA raw data sets do contain a synopticityerror contribution and we will quantify its signifi-cance.

Of course, expensive data assimilation schemes inŽ . Ž3D Primitive equation PE models e.g. Moore,

1991; Walstad et al., 1991; Milliff and Robinson,.1992; Lermusiaux, 1997; Pham et al., 1998 or ro-

Žbust 3D PE diagnostics e.g. Brasseur, 1994; Viudez´.et al., 1996 might be used to map a given observa-

tional data set in space and time to a spatial grid at agiven time.

Such higher order correction methods are, how-ever, difficult to implement and are strongly depen-dent upon the knowledge of model and data errors,which may be difficult to assess in general. We arethus looking for a simpler, first-order correctionmethod to obtain pseudo-synoptic data sets fromnon-synoptic data sets.

We have shown recently that a geostrophic reloca-Žtion method Rixen and Beckers, 2001; Rixen et al.,

.2001 might significantly improve accuracy in esti-mating vertical velocities and fluxes by partiallycorrecting the bias of non-synoptic sampling, even ifthe flow is not purely geostrophic and the Rossby

Ž .number quite high up to 0.2 .Here, we will analyse the performance of this

procedure and other methods on the OMEGA dataset. The rest of the paper is organised as follows.Different first-order correction methods to obtainpseudo-synoptic data sets from non-synoptic datasets are developed in Section 2 and are tested on theOMEGA data set in Section 3. The methods are thenvalidated through a simple 3D primitive equationŽ .PE assimilation experiment in Section 4. Finally,Section 5 contains conclusions and perspectives inthe framework of operational oceanography.

( )M. Rixen et al.rJournal of Marine Systems 29 2001 313–333 315

2. Methodology

2.1. Geostrophic relocation - method 1

The relevant terms for evaluating synopticity areŽ . Ž .the local changes of velocity z temperature T ,

Ž .salinity S , . . .

Ez ET ES, , , . . . 1Ž .

Et Et Et

If these terms are small enough, the field experi-ment can be considered synoptic.

The assumption of synopticity is not necessarilyrelated to the distance covered by a fluid particleduring a given time interval, but this displacementcan be used to build pseudo-synoptic data sets, asdescribed below.

Let us assume that we have a series of measure-ments is1 . . . N at locations x taken at instants t ,d i i

with measurements of temperature T and salinity Si iŽ .and possibly any other variable Y . If we assume ai

system without mixing, water parcels are advectedwithout changing their T ,S characteristics. In a La-grangian sense we have

w xd T ,S ,Yi i is0 2Ž .

d t

meaning T , S and Y remain constant along trajec-i i iŽ .tories drd t is the material derivative .

If we knew the 3D velocity field z at eachmoment and location, we could compute the dis-placement of each data point from

d x isz x ,t 3Ž . Ž .

d t

To obtain water parcel trajectories and subse-quently a pseudo-synoptic data set at a given mo-

Ž .ment t , it is sufficient to integrate Eq. 3 from tref iŽ .to t which could be posterior or prior to t . Dataref i

points sampled before t are then displaced down-ref

stream whereas data acquired past t are displacedref

upstream according to the velocity field. ClassicalŽ .Objective Analysis OA can then be used with the

data values T , S or Y now located at their newi i i

positions at the time t .ref

Since the velocity field is usually not known, thegeostrophic assumption is a simple and easy way ofestimating the velocity field from the TrS fields. In

this case, the vertical velocity is negligible, and dataare only moved on their initial standard levels if weuse geostrophy to advect parcels according to Eq.Ž .3 . So we assume that we only know the horizontalgeostrophic velocity u , computed from the thermalg

wind equation, moving the points according to

d x isu x ,t 4Ž . Ž .gd t

provided a level of no motion exists or the velocityfield at a given level is known.

However, the T ,S analyses and the geostrophicvelocity field are not independent and one needs toiterate the relocation procedure until convergence,where the density and the velocity fields are mutu-ally adjusted. Convergence is achieved when thedifference between the hydrodynamic fields of twoconsecutive iterations is less than a specific relativeerror. In practice, the adjustment was satisfactoryafter less than 10 iterations.

Relocation to the beginning or the end of a surveyis dangerous. Also, increasing the integration timeincreases the length of the trajectory of each stationand the number of data points leaving the domain.Statistically, relocating to the middle of the surveywill, on average, lead to the shorter displacementsand less mixing during advection. Consequently, we

Ž .compute a modal sampling time t s t q t r2,ref 1 Nd

where t is the time of the first measurement at1

station x and t the time of the last measurement1 Nd

of the survey at station x . More details on theNd

Ž .method may be found in Rixen et al. 2001 .In this iterative procedure, one could also use a

diagnostic vertical velocity w to relocate parcels inthe 3D space. In the present case study, w will beneglected in the advection of parcels and it will bedemonstrated that this contribution is not significant.

METHOD 1 thus works with a whole data set or asingle survey, and might be helpful provided a validŽ .geostrophic velocity field is known: this is the caseif T ,S measurements are available and the Rossbynumber is small.

2.2. Linear quadratic interpolation— METHOD 2aand 2b

At short time scales, the local time evolutionmight be approximated by a linear function if two


successive measurements are available at the samepoint. This technique has proved to be useful in

Žcorrecting AVHRR Advanced Very High Resolu-.tion Radiometer cloud-contaminated images, by in-

Žterpolating temporally successive images e.g. Farag. Ž .et al., 1998 . Likewise Gomis et al., 2000 , having

more than one survey allows to obtain a pseudo-syn-optic survey by simple linear time interpolation be-

Ž . Ž .tween two successive surveys. Let Y x and Y x1 2

be the raw state variable of surveys 1 and 2 withŽ . Ž .t x and t x and the corresponding raw sampling1 2

times. Assuming a linear local time evolution, weŽ . Ž .can estimate Y x, t at any time t not far from t x1

Ž .and t x by2

Y x t y t x yY x ty t xŽ . Ž . Ž . Ž .Ž . Ž .1 1 2 2 1Y x ,t sŽ .

t x y t xŽ . Ž .1 2

E2 Y2qO D t 5Ž .2ž /Et

where the last right-hand side term represents thelinear interpolation error with D ts t y t . In partic-2 1

ular, if t is a common time for all stations, onethereby obtains a pseudo-synoptic data set. The above

Ž .procedure implies interpolating data Y x, t if t liesw Ž . Ž .xin the range t x , t x and extrapolation of the1 2

data outside of that range. In practice, successivesurveys do not have exactly the same locations alongthe legs or not even the same legs. One way toovercome this problem is to associate proximal sta-

Žtions with a single ‘mean’ location Gomis et al.,.2000 . But, more generally, to avoid solving the

problem of correspondence between stations of suc-Žcessive surveys especially for surveys that are not

.similar , one performs analyses of the hydrographicdata set and its sampling time onto a regular grid for

Ž .both surveys before the time interpolation. Eq. 5 isthen applied to hydrographic analysed gridded fieldsand sampling time analysed gridded fields. One di-rect extension of the above method is quadraticinterpolation when three successive surveys areavailable.

The linear and quadratic interpolation methodsmight lead to unrealistic hydrodynamic fields, firstbecause extrapolation occurs outside of the interpola-tion range, and secondly because quadratic or higher

order polynomial interpolations are quite dangerouseven in the interpolation range, when the time inter-val D ts t y t is not small compared to the time2 1

scale of the phenomena measured.Ž . ŽMETHODS 2a resp. 2b thus work if two resp.

.three almost identical successive surveys are avail-able and if the time evolution of the fields to becorrected may be approximated by a simple linearŽ .resp. quadratic function. This is the case if the timelapse between surveys is short.

2.3. Spatio-temporal correlation function—METHOD 3

The relative importance of each data from thedifferent surveys may be modified in the analysisaccording to the time lapse between the referencesampling time and the sampling time of each T ,Smeasurement. This can be achieved by strongly at-tracting the field towards data for which t is close toi

t , and reducing the attraction artificially when t isref i

far from t , assuming a Gaussian distribution ofref

variables in time. This method has already been usedŽ .by Brankart and Pinardi 1998 in the case of clima-

tological, seasonal and inter-annual variability analy-ses. Instead of analysing each data set separately fordifferent years or seasons, one may weight the dataaccording to the season of interest. For example, asummer analysis would take into account all the dataavailable from the database, but the weight on winterŽ .and to some extent fall and spring data should bedecreased. This technique revealed finer details inthe climatological analyses and has been naturallytransposed to synoptic scale analyses, differently tak-ing into account data from different surveys.

Thus one modifies the weight applied to each datain the analysis by a factor

2 2exp y t y t rT 6Ž . Ž .Ž .i ref car

where T s4 days, typically the time scale of phe-car

nomena observed in the Alboran Sea.METHOD 3 thus works with any kind of data set.

With two similar overlapping surveys, it is expectedto work if the time lapse between surveys is short. Itseems a priori more robust than simple interpolationbetween surveys.


2.4. The Omega equation

The effect of applying first-order correction meth-ods to the OMEGA data set will be tested on diag-nostic vertical velocities obtained from the Omegaequation.

ŽThe QG Omega equation Hoskins and Draghici,.1977; Hoskins et al., 1978 allows the derivation of

diagnostic vertical velocities w from the knowledgeof the density field only, provided boundary condi-tions for w and a realistic reference level for com-puting the dynamic height field from the densityfield are known.

Ž .Noting f the Coriolis parameter, u s u , Õ , 0g g g

the geostrophic velocities, N the Brunt–Vaisala fre-¨ ¨ ¨quency, on an f-plane and neglecting diffusion ef-fects, the Omega equation reads

E2 w E2 E22 2f q q N w s= PQ 7Ž . Ž .h2 2 2ž /Ez Ex E y

where

EÕ Eu EÕ EÕg g g gQs Q ,Q s 2 f q ,Ž .x y ž /Ex Ez E y Ez

Eu Eu Eu EÕg g g gy2 f q 8Ž .ž /Ex Ez E y Ez

is known as the Q-Õector.ws0 are used at all boundaries. The null gradi-

ent condition was also tested but differences betweenthe two approaches were confined to a few boundarygrid points. Geostrophic velocities are obtained fromthe hydrographic analyses with a level of no-motionat 197 m. Our reference level is very similar to thosetraditionally used to compute geostrophic velocities

Žin the Alboran Sea Pollard and Regier, 1992; Allen.and Smeed, 1996; Gomis et al., 2000 . Viudez et al.´

Ž .2000 show for the same data set that the magnitudeand shear of geostrophic velocities become negligi-ble below 200 m and suggests that any level below200 m might be selected as the level of no-motion. Ithas also been verified that the variability between

Ž² : .surveys denotes the mean operator

2² :s ysŽ .i j/ i9Ž .2² ² : :s y sŽ .i i

Ž 2 .is minimum at levels close to 200 m ;0.4 .Similarly, the variability between raw and pseudosynoptic fields

2² :s ysŽ .i Method i 10Ž .2² ² : :s y sŽ .i i

is found to be minimum at levels close to 200 mŽ w x2 .; 0.1–0.3 .

The ADCP analysed data could have been com-bined with the relative geostrophic velocity field toobtain absolute geostrophic velocities but for consis-tency in comparing the different methods, we usedrelative geostrophic velocities, considering that the

Žreference level is a ‘true’ level of no motion see aŽ ..detailed justification in Viudez et al. 2000 .´

A study regarding the sensitivity to the referencelevel and the use of ADCP data in a multivariateanalysis approach for the same data set may de

Ž .found in Gomis et al. 2000 .

2.5. The bias arising from asynoptic sampling

Defining c as the complex phase speed of thewave j to be sampled and Õ as the overall shipf

speed along the front, one writes

jsj e i kŽ xyct . 11Ž .0

The observed distribution of j is

j sj e i ka x 12Ž .a 0

where

c cr ik sk 1y y i 13Ž .a ž /Õ Õf f

Thus, there are two effects of the asypnotic sam-pling. First, the apparent wavenumber of oscillation

Ž . Žwill be increased resp. decreased by a factor 1y. Žc rÕ if the ship is travelling upstream resp. down-r f. Ž .stream . Secondly, for positive resp. negative c ,i

the amplitude of oscillation will appear to increaseŽ .resp. decrease in direction in which the ship istravelling.

QG vertical velocities are also affected by theway the sampling has been planned. Using the QGOmega equation for a two-layer flow, to first order

5 5in crÕ it can be shown that the apparent QGfŽ .vertical velocity w obtained by solving Eqs. 7 anda

Ž . Ž .8 with the apparent raw density field is related to


Žthe true QG vertical velocity w by see Allen et al.,.2001 for details

w sw 1yK e ix 14Ž . Ž .a

where

c cr iK; , x;y , c sR c , c sFF c 15Ž . Ž . Ž .r i

Õ Õf f

Thus, the asypnotic sampling gives rise to differenterrors in w.

The apparent wavenumber of oscillation of thewave and the amplitude of w will be changed by a

Ž .factor 1yK . If the ship and the wave move in thesame direction, the apparent amplitude of w will bedecreased. On the contrary, if the ship and the wavemove in the opposite direction, the apparent ampli-

Ž .tude of w will increase Doppler effect .Ž .A positive resp. negative value of c will noti

change the apparent wavelength but will give rise toŽ .an apparent increase resp. decrease in the ampli-

tude of oscillation of the wave and in the amplitudeof w in the direction of advancement of the ship.

These are two effects of asynoptic sampling thatmight be correct by the different methods.

First, downstream sampling causes weaker den-sity gradients, weaker advection and therefore weakerQG w. Correcting this bias by any method shouldintensify density gradients and w. In the case ofupstream sampling, correction methods shouldweaken gradients and w.

Secondly, compared to the synoptic field corre-sponding to the middle of the cruise, for a temporally

Ž .growing wave c positive , the correction methodsiŽ .should increase resp. decrease gradients at the be-

Ž .ginning resp. end of the cruise. The reverse effectŽis obtained if the amplitude of the wave decreases ci

.negative .Assuming a typical survey involving across-front

sampling legs, one expects METHOD 1 to correctthe first effect by modifying the distance betweenlegs and the second effect by modifying the distancebetween stations along a leg. METHODS 2 and 3 areexpected to correct both effects together through theapproximation of the temporal evolution of the field.

3. Analyses

A common grid was adopted within the OMEGAgroup for comparison purposes. The analysis domainextends from 58W to 3.718W, and from 35.78N to36.48N resulting in a 44=29 horizontal grid with arespective grid resolution of 0.038 and 0.0258. In thevertical, 38 equally spaced levels were considered

Ž .from 5 to 301 m one level every 8 m . Finer verticalresolutions were tested but changes were not signifi-cant.

The data sets of the three surveys have beenobjectively analysed by means of the Variational

Ž . ŽInverse Model VIM Brasseur, 1994; Brankart and.Brasseur, 1996, 1998; Rixen et al., 2001 , equivalent

Ž . Žto classical optimal interpolation scheme OI see.for example Gandin, 1965; Bretherton et al., 1976

in an infinite space. The reference field has beencomputed by means of a semi-normed analysisŽ .Brankart and Brasseur, 1996 in order to reduceunrealistic features in regions void of data, speciallyin the application of METHOD 1.

The characteristic length L and the signal-to-noiseratio SrN, the free parameters of the analysis, havebeen calibrated by Generalised Cross-ValidationŽ . ŽGCV e.g. Wahba and Wold, 1975; Craven andWahba, 1979; Golub et al., 1979; Wahba, 1990;

.Brankart and Brasseur, 1996; Rixen et al., 2001 ,independently for the three surveys. Different valuesfor L and SrN were found between the surveys butthese were considered rather insignificant.

The characteristic lengths for temperature andsalinity are quite similar, ranging between 5 and 40

Žkm with low values at the surface and the bottomŽ . .301 m and maxima between 100 and 150 m . Thesignal-to-noise ratio is close to 100, but rangingbetween 10 and 1000.

3.1. Raw data set density fields

The non-synoptic raw data sets have been anal-Žysed for the three surveys. The density fields see

.Fig. 1 reflect the strong variability present in theŽ .Western Alboran Basin WAB . Note that there are

only 9 legs in survey 2 and 10 legs in survey 3. Inour interpretations, extreme eastern and western ar-eas will thus not be considered. Both T ,S fieldssuggest a time evolution with an intensification of


Ž . Ž . Ž . Ž .Fig. 1. Potential density and geostrophic velocity field at 101 m of surveys 1 top , 2 middle and 3 bottom raw data set .

Ž . Ž .the Atlantic Jet AJ increase of gradients at thewestern boundary and a weakening of the jet at the

eastern boundary together with a southward displace-ment of the front from survey 1 to survey 3. Note


that the core of the WAG has almost disappeared insurvey 3. The entrance latitude of the AJ has movedsouthward by almost 0.18 whereas the output latitudehas shifted northward by approximately 0.28. Thesemi-circular front latitude has shifted northward byapproximately 0.28. The semi-circular front observedin survey 1 evolves in a more complicated meanderin survey 3. These analyses are consistent with theAdvanced Very High Resolution RadiometerŽ . ŽAVHRR SST satellite images Baldacci et al.,

.1998 , setting aside the issue of synopticity.

3.2. Pseudo-synoptic density fields

ŽThe three methods METHOD 1—geostrophicrelocation, METHOD 2a and 2b—linear andquadratic interpolation, and METHOD 3—spatio-

.temporal correlation function have been tested onthe OMEGA data set. Fig. 2 shows the pseudo-syn-optic fields relative to t , the modal sampling timeref1

of survey 1.

3.2.1. METHOD 1ŽAs the cruise was conducted downstream see

.Fig. 2, top , the relocated stations are now concen-trated in the middle of the area. Stations sampled

Žbefore t have been displaced eastward down-ref1

.stream and stations sampled after t have beenref1

Ž .displaced westward upstream . There is a clear re-dundancy of information: a large fraction of the

Ž .observations have sampled the same central watermass. The uncovered areas highlight the semi-normedreference field, computed from the raw data set, inorder to avoid unrealistic relocation velocities. Thefact that a portion of the domain is void of data afterrelocation will have some consequences in the vali-dation of results.

3.2.2. METHODS 2a, 2b and 3Ž . ŽMETHOD 2a not shown , METHOD 2b see

. Ž .Fig. 2, middle and METHOD 3 see Fig. 2, bottomexhibit only relatively small differences compared to

Ž .the raw data analyses Fig. 1, top . Some differencesmay be found west and east of the domain and in thecore of the WAG. This is not surprising since for

Ž .METHODS 2a and 2b we interpolate or extrapolatetowards the mean sampling time of survey 1, t .ref1

Interpolation occurs where the sampling time of

Ž .stations of survey 1, t - t west because thei1 ref1

corresponding sampling time of stations in survey 2,t ) t everywhere. On the contrary, extrapolationi2 ref1

occurs where the sampling time of stations of surveyŽ .1, t ) t east , because the mean sampling timei1 ref1

of survey 1 satisfies t - t - t , which is outsideref i1 i21

of the interpolation range.One should bear in mind that the missing legs east

and west in the second and third surveys are alsoresponsible for the large discrepancies between theraw and pseudo-synoptic fields at the boundaries ofthe domain.

Results at t and t are qualitatively similar toref ref2 3

their corresponding raw analysis and lead to similarŽ .conclusions see Fig. 3 .

3.3. WaÕe eÕolution

Fields obtained with METHODS 1, 2 and 3 areŽ .thus hoped to be synoptic ‘pseudo-synoptic’ and

might be used to study the evolution of the frontalstructure that was already partially depicted for theraw data analysis. However, METHOD 1 leavesareas void of data at the boundaries and METHODS2 might extrapolate the field. A priori, METHOD 3seems better suited for this investigation since itmight be more robust than any other method. Toidentify the characteristics of the wave, the mean ofthe three fields obtained by METHOD 3 was sub-tracted from these fields in order to focus on theperturbation to the mean hydrographic field at t ,ref1

t and t . These fields are shown in Fig. 4.ref ref2 3

One may obtain the characteristics of the wave byŽidentifying the corresponding extrema represented

.by the isoline Dssy0.1 between successive fig-Žures of Fig. 4 excluding the extrema at the bound-

aries because of the possible bias induced by the.missing legs in surveys 2 and 3 . For example, the

Ž .minimum at t in the western part Fig. 4, topref1

might correspond to the minimum in the northeasternŽ .part at t middle figure , because this minimum isref 2

Ž .located somewhat further east at t bottom figure .ref 3

The displacement of this perturbation allows a roughestimation of the wave speed. The growth rate of thewave might be estimated as the e-folding timescalefor increasing RMS. Using the same notations as

Ž . Ž .Allen et al. 2001 see Section 2 and consideringŽ .successively surveys 1,2 subscript A1,2B and 2,3


Ž . Ž .Fig. 2. Potential density and geostrophic velocity field at 101 m at time t obtained from METHOD 1 top , METHOD 2b middle andref1Ž . Ž .METHOD 3 bottom pseudo-synoptic data set .

Ž .subscripts A2,3B , we have u s0.53 mrs, c sf r1,2 1,2

0.13 mrs, c sy0.086 mrs, K s0.24, x si 1,2 1,21,2

0.16 and Õ s0.63 mrs, c s0.09 mrs, c sf r i2,3 2,3 2,3

0.043 mrs, K s0.15, x sy0.06, respectively.2,3 i1,2


Ž . Ž . Ž .Fig. 3. Potential density and geostrophic velocity field at 101 m at time t left and t right obtained from METHOD 1 top ,ref ref2 3Ž . Ž . Ž .METHOD 2b middle and METHOD 3 bottom pseudo-synoptic data set .

These parameters suggest that the wave is aneastward propagating wave. This may explain therather smaller wave length observed in the perturba-

Žtions of the corrected fields about 5–30%, consis-.tent with theoretical K , not shown and the stronger

density gradients found in the pseudo-synoptic fieldsŽcompared to the raw fields expect perhaps for

.METHOD 1 at t .ref 3

These parameters also suggest that the wave am-plitude decreases between t and t and in-ref ref1 2

Žcreases between t and t . outside of the tempo-ref ref2 3

w xral range t t , no direct conclusion can beref ref1 3

.drawn . This may explain why the strongest densitygradients are found in the eastern part at t and inref1

the western part at t in the pseudo-synoptic fieldsref 3

Žexpect for METHOD 1, where the interpretation is


Ž . Ž . Ž .Fig. 4. Evolution of the difference between the potential density field for survey 1 top , survey 2 middle and survey 3 bottom and theirmean, computed from the pseudo-synoptic results obtained from METHOD 3 at 101 m. Empirical parameters have been obtained byconsidering the ‘center of mass’ of areas delimited by the contours at Dssy0.1.


.more difficult . At t , changes in density gradientsref 2

are small: the downstream sampling of the waveŽleading to an expected increase of density gradients

.in the pseudo-synoptic fields is compensated by thelow amplitude of the wave at this time.

This tracking of extrema leads to the more likelyinterpretations: values of K and K are in good1,2 2,3

agreement and have the same order of magnitude asŽ .found by Allen et al. 2001 in theoretical but similar

configuration. Moreover, the changes in the pseudo-synoptic fields are rather consistent with this inter-pretation.

One also expects the raw fields to underestimatethe magnitude of QG vertical velocities by about

Ž . Ž25% between t and t to 15% between tref ref ref1 2 2

.and t .ref 3

3.4. Raw data set QG Õertical Õelocity fields

Before solving the Omega equation, hydrographicanalysed fields have been filtered horizontally with agaussian function in space, since they contain noise

Žat analysis and sampling grid scales Pinot etal., 1996; Gomis et al., 2000; Rixen et al., 2001;

.Rixen, 1999 . It was decided to filter out only thenoise at the numerical grid scale, bearing in mindthat the hydrodynamic fields still contain physicalstructures at sampling grid resolution which cannot

Ž .be resolved by the sampling Gomis et al., 2000 . Afiltering length of 6 km, roughly twice the numerical

Ž .grid resolution the Nyquist wavelength , producedhydrographic analyses almost free of noise.

The differences between the raw vertical fields ofŽ .surveys 1, 2 and 3 see Fig. 5 highlight the strong

variability in the Alboran Sea. In the case of surveyŽ .1 Fig. 5, top , the velocity field clearly exhibits an

upward motion cell upstream of the WAG and aŽdownward motion cell downstream of the WAG e.g.

. ŽOnken, 1992 . As the front begins to meander Fig..5, middle and bottom , structures become more com-

plicated. The results are consistent with Gomis et al.Ž .2000 : small differences may be explained in ourcase by the univariate approach, a different shape of

Žthe correlation function in the analysis e.g. Brasseur,1994; Brankart, 1996; Brankart and Brasseur, 1998;

.Rixen et al., 2001 , a different posterior filtering andthe use of relative geostrophic velocities.

3.5. Pseudo-synoptic QG Õertical Õelocities

The general patterns of the pseudo-synoptic QGvertical velocities in Fig. 6 and Fig. 7 are similar tothe raw data set QG vertical velocities in Fig. 5, butsome important discrepancies can be found and mightall be explained by the changes in the density field.

Ž .For METHOD 1 top figures , the sharpening ofgradients induced by the relocation induces increased

Ž .vertical velocities, namely about 25 resp. 40 % forŽ .the covered area at t resp. t , in agreementref ref1 2

with the estimated K , except at t , where verticalref 3

velocities w are weaker.Downstream sampling of a wave propagating in

the downstream direction results in gradient smooth-ing while upstream sampling of the same wave

Žresults in gradient sharpening Gomis et al., 2000;.Allen et al., 2001 . This error is partially correctedŽ .by the relocation reverse effect . It should be noted

that in regions void of data, the reference field in theanalysis keeps a trace of the raw data set, necessaryto provide reliable estimates of geostrophic velocitiesduring the relocation iterative adjustment procedure.The transition zone between covered and uncoveredareas could itself be responsible for some uprdown-welling features which should be interpreted cau-tiously. Nevertheless, the correspondence betweencells in the raw and relocated cases seems obvious.

We now come back to the fact that the w mightbe used together with the geostrophic velocities torelocate parcels vertically too. It has been checkedthat the magnitude of the integrated vertical displace-ment over the relocation period

trefw x ,t d t 16Ž . Ž .H

ti

of any station ranges between y11 and 18 m, withan average of 1.2 m and a standard deviation of 3.7m, less than the vertical resolution. Since an increasein the number of vertical levels did not changesignificantly the results, w might be neglected in afirst approximation in the relocation process.

Ž . ŽFor METHOD 2a not shown , 2b and 3 Figs. 6.and 7, middle and bottom , the magnitude of the

downward QG w has been increased by about 5% to


Ž y5 . Ž . Ž . Ž . Ž .Fig. 5. QG vertical velocity 10 mrs at 101 m of surveys 1 top , 2 middle and 3 bottom raw data set .

40% at t in the eastern part and at t in theref ref1 3

western part, according to the evolution of the wave,as was apparent from the hydrographic analyses.Pseudo-synoptic vertical velocities at t remainref 2


Ž y5 . Ž . Ž .Fig. 6. QG vertical velocity 10 mrs at 101 m at time t obtained from METHOD 1 top , METHOD 2b middle and METHOD 3ref1Ž . Ž .bottom pseudo-synoptic data set .

Žvery similar to the raw ones according to the almost.unchanged density field.

One might wish to state that the OMEGA rawdata sets are almost synoptic, since the sensitivity of


Ž y5 . Ž . Ž . Ž .Fig. 7. QG vertical velocity 10 mrs at 101 m at time t left and t right obtained from METHOD 1 top , METHOD 2bref ref2 3Ž . Ž . Ž .middle and METHOD 3 bottom pseudo-synoptic data set .

the hydrographic fields to the correction methods issmall. However, differences of up to 30% in themagnitude of QG vertical velocities may be foundbetween raw and pseudo-synoptic fields. Major dis-

Ž .crepancies are found at the boundaries up to 40% ,but are explained by the missing legs in surveys 2and 3. Elsewhere, the sensitivity of QG w to the

correction METHODS is limited to about 10–30%,which is consistent with the theoretical K calculated.

4. Validation

A twin experiment was conducted in order tovalidate the different methods developed to correct


synopticity problems. The T ,S raw and correctedanalyses of surveys 1 and 2 were assimilated into the3D PE model through a simple nudging scheme inorder to investigate the proper prediction of the thirdsurvey. The raw density analysis from the middle leg

Ž .of each survey y4.368 longitude was used ascontrol variable for validation. Indeed, a single legonly lasts a short period, typically 4 h, which we willconsider as synoptic. This assumption is justified bythe fact that, considering the maximum velocities inthis area, namely 1 mrs, the middle leg cannot berelocated further than an adjacent leg within 2 h.Furthermore, the different correction methods haveshown that the different surveys may be considered

Žsynoptic to a first approximation although we willshow that the relative quality of the different fields is

.different . At these middle legs, differences betweenraw and pseudo-synoptic analysed density fields werelimited to 0.05 s RMS and 0.998 correlation, com-pared to the variability at other adjacent or boundarylegs, reaching 0.3 s RMS and 0.97 correlation. Themiddle leg of density computed from the raw dataset of each survey might thus be considered assynoptic.

4.1. Model

The present implementation of the 3D PE modelis based on the GHER mathematical model de-

Ž .scribed, for example, in Nihoul et al. 1989 , BeckersŽ . Ž .1991 , and Beckers et al. 1997a . Details on thenumerical discretisation may be found in Deleersni-

Ž . Ž .jder 1989 and Beckers 1991 . The model uses anŽArakawa C-grid discretisation, mode-splitting Kil-

.lworth et al., 1991 , free-surface, Total VarianceŽ .Diminishing TVD advection scheme and a double

Ž .s-coordinate change Beckers, 1991 , with a totalnumber of 31 vertical levels. The 3D-model domainextends from y9.58 to y0.8158 in longitude, andfrom 33.368 to 37.58 in latitude, both in the AtlanticOcean and in the Alboran Sea. The application of themodel has effectively shown that it is easier toimpose open-sea boundary conditions far from theStrait of Gibraltar. The above-mentioned resolutionleads to a fine eddy-resolving 193=117=31 3Dgrid.

ŽTrS fields are initialised with the MODB Medi-.terranean Oceanic Data Base September climatolog-

ical fields and relaxed to the MODB climatologicalŽfields Brasseur and Brankart, 1995; Brasseur, 1995;

.Brasseur et al., 1996 at the open sea boundarieswith a characteristic time of approximately 2 months.The initial velocity field is obtained through geo-strophic approximation and provides a first estimate

Ž .of the initial surface elevation dynamic height . TheŽ .initial turbulent kinetic energy TKE field is solved

from the stationary TKE equation.We impose zero vertical scalar fluxes and a loga-

Žrithmic profile for the velocity classical turbulent. Ž .friction layer at bottom boundaries Beckers, 1991 .

At the air–sea interface, daily forcing from ECMWFanalyses are applied.

Zero normal gradients are imposed at open seaboundaries for normal, tangential and mean tangen-tial velocity. The mean normal velocity imposes a

Žnet inflow of 0.1 Sv into the Mediterranean e.g.Tchernia, 1977; Bryden and Kinder, 1991; Candela,

.1991 .To reduce the computational burden and take

advantage of the parallel computing facilities, thedomain was split into two parts at the Strait ofGibraltar with approximately the same number ofgrid points, reducing the calculation cost by a factor2 if we disregard interprocess communicationsŽ .Beckers et al., 1997b .

4.2. Assimilation

Ž .Nudging or Newtonian relaxation is a very sim-ple assimilation method. The assimilation box corre-sponds to the analysis box. The raw and correctedT ,S analyses Y from the first two surveys is1, 2oa i

are subsequently used to correct the model simula-tion Y with a nudging relaxation in time, dependingf

upon the lapse between the integration time t of themodel and the model sampling time t of eachref i

survey i

Y tqd tŽ .f c2

ty t ref1ydt ž /t y ttqd t tqd tref ref2 1sY q e Y yYŽ .f oa f1½Trelax2

ty t ref 2y ž /t y t tqd tref ref2 1qe Y yY 17Ž .Ž .oa f2 5


where subscript AcB indicates the corrected value ofY tqd t.f

The only parameter requiring calibration duringnudging is thus the relaxation time T . This pa-relax

rameter was optimised with respect to the density onthe middle leg of survey 3, considered as the ‘true’synoptic solution, as discussed previously. It wasfound that the optimal relaxation time T to assim-relax

ilate raw or pseudo-synoptic fields was close to 12 h.Stronger assimilation produced conflicting situationsŽ .artificial fronts, etc. . Weaker assimilation did notattract sufficiently the model to the assimilated data.Similarly, horizontal sub-grid scale diffusion coeffi-cients were also optimised. However, sensitivity tosuch parameters was not significant in this assimila-tion experiment. Relaxation time was increased

Ž . Ž .Fig. 8. Model potential density field at 100 m at time t without assimilation top-left , with assimilation of raw fields top-right , withref 3Žassimilation of pseudo-synoptic fields METHOD 1 middle-left, METHOD 2a middle-right, METHOD 2b bottom-left, METHOD 3

.bottom-right .


artificially near the boundaries of the domain toavoid strong gradients between assimilated and non-assimilated areas.

First simulations were rather far from in situmeasurements, both for T and S: the model wasabout 18C too cold and 1 PSU too fresh. It wasdecided to perform a first assimilation step and usethe results as definitive initial conditions for the

Žmodel for details on the first simulations, see Rixen,. Ž1999 . The model output without assimilation see

.Fig. 8, top left produces a well-developed WAGassociated with the incoming Atlantic Jet in the

Ž .Mediterranean. The Eastern Alboran Gyre EAG inthe Eastern Alboran Basin was absent but the AJseemed to detach from the coast, maybe initiating thedevelopment of a new EAG.

Ž .The raw data set assimilation Fig. 8, top rightproduces a field which is very similar to the modeloutput without assimilation. The shape of the core ofthe WAG is, however, slightly different. The centerof the gyre is now located further southeast com-pared to its original position.

Qualitatively, the model outputs with assimilationŽ .for correction methods Fig. 8, middle and bottom

are very similar to raw data assimilation, exceptperhaps for METHOD 3, where the core of the gyrelooks smaller. Although the different assimilatedfields look fairly similar, there are discrepanciesbetween raw data set assimilation and pseudo-syn-optic data set assimilation. By assimilating thepseudo-synoptic and analyses corresponding to sur-veys 1 and 2, we expect to correct the model andproduce a solution closer to the ‘true’ synoptic state

at t as represented by the raw analysed data setref 3

density at the middle leg of survey 3.Fig. 9 shows the RMS and correlation of density

Žbetween the ‘true’ solution middle leg of raw anal-. Žysed survey i and the model results withoutrwith

.assimilation at t with is1 . . . 3.ref i

First, we note that the model without assimilationŽis not able to correctly reproduce the raw field high

.RMS and low correlation . The poor correlation inthis simple model run may be explained by thewrong place and size of the WAG, located east of itsactual place. We note also as expected, that at tref1

and t the raw data assimilation is closer to theref 2

Žraw middle legs at these times low RMS and highcorrelation, except for the correlation of METHOD 1

.at t , the correction methods showing slight dif-ref 2

ferences with the raw middle legs. At t , however,ref 3

each correction method behaves better than the rawdata set assimilation: the RMS is lower and thecorrelation higher. The weakness of METHOD 1 canbe attributed to the behaviour of the relocationmethod which leaves some areas void of data: as weassimilate the whole analysis box, in uncovered ar-eas, only the reference field information is provided.The relative weakness of METHOD 2 may be ex-plained by the extrapolation that occurs in someareas of the analysis. The other correction methods,and specially the one using a spatio-temporal correla-

Ž .tion function METHOD 3 , provide better correla-tions and lower RMS.

The correlation improvement on the density fieldsŽ .is small ;0.5% correlation and might be consid-

ered marginal but the improvement in the RMS

Ž . Ž . Ž .Fig. 9. RMS left and correlation right between the potential density of the middle leg at t raw data set, ‘true’ synoptic solution andref i

the model outputs for different assimilated fields at t .ref i


Ž .about 0.05 s represents up to 20% of the RMSerror on density. The final error is measured at t ,ref 3

shortly after the assimilation process itself. Two daysare enough to notice significant differences betweenraw and pseudo-synoptic fields assimilation, despitethe fact that the raw data assimilation already par-tially corrects this error through the dynamics of themodel. The model drift is smaller when using cor-rected assimilation fields. The different correctionmethods systematically improve the accuracy of thedensity estimation and the joint probability that theseimprovements are not significant is probably verylow. A systematic synopticity error could thereforecompromise accurate forecasting.

These results suggest that the OMEGA data setmight be considered as synoptic at a first approxima-

Žtion but also that the correction methods except.perhaps METHOD 1 provide more accurate hydro-

graphic fields and QG vertical velocities estimations.

5. Conclusions

A critical point was addressed during the OMEGAproject which focused on the observation and themodeling of geostrophic and ageostrophic circulationin the Alboran Sea: were the data sets synopticenough to allow reliable vertical velocities estima-tions from the Omega equation or were these estima-tions highly dependent upon the sampling strategyrresolution?

The aim of this study was thus twofold. First,investigate the synopticity of the OMEGA data setand, secondly, test the ability of first-order correctionmethods to produce pseudo-synoptic data sets fromraw data sets.

On the basis of QG vertical velocities which arevery sensitive to changes in hydrographic fields, itwas shown that the OMEGA data set was quiterobust with respect to correction methods and did notshow significant synopticity problems although dif-ferences of up to 30% were found between raw and

Žpseudo-synoptic QG vertical velocities. Differenceof up to 40% located at the eastern or westernboundaries were explained by missing legs in the

.second and third surveys.

By means of a sample assimilation experiment,we were able to validate the quality of the pseudo-synoptic fields against raw data set analyses. Indeed,the pseudo-synoptic fields show greater consistencyand suggest an underestimation in the magnitude ofthe OMEGA raw data set QG vertical velocities,partially corrected by our methods. In this particularcase, METHOD 1 lacks in suitable data coverageafter relocation. METHODS 2 and 3 showed andecrease of up to 20% in the RMS error on the ‘true’density in the assimilation experiment. The effectmay be even more drastic in worse cases, for longersurveys or in regions with higher hydrodynamicmesoscale variability.

The first-order correction methods developed inthis study are highly relevant for the scientific com-munity. So far, we have only used hydrographic T ,Smeasurements. Other biological and geochemicaltracers could be envisaged. Indeed, the use of correc-tion methods could modify the interpretation of thecorrelation between vertical motion and nutrient con-centration, biological enrichment, etc.

The twin experiment used in this study might behelpful to assess the quality and the bias of parti-cular sampling strategies and to elaborate optimalsampling strategies. Before sampling some area,synthetic surveys could be extracted from a modeldescribing the hydrodynamics of the domain. Thedifferent methods could then be used to assess syn-opticity problems of different samplings and eventu-ally propose the more securerreliable field experi-ment strategies.

Acknowledgements

This work would not have been possible withoutthe help and expertise of J.-M. Pinot and D. GomisŽ .UIB, Spain . We thank P. Brasseur and J.-M.

Ž .Brankart IMG, Grenoble, France , and R. HaneyŽ .NPS, Monterey, USA for their precious comments.Support for this study was obtained through E.C.

ŽMASTIII programme project OMEGA MAS3-.CT95-0001 . The development of the 3D PE model

Žof the GHER GeoHydrodynamics and Environment.Research was financed during several years by the

ŽEuropean Commission EROS2000, EUROMODEL,MEDMEX, MODB, OMEGA, MATER and TOROS

.projects . IBM Belgium. IBM Europe and IBM In-


ternational Foundation provided supercomputer as-sistance, most recently in the scope of the ERP-

ŽSALMON project Sea Air Land Modelling Opera-.tional Network .

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