Forecasting front displacements with a satellite based ocean forecasting (SOFT) system

s 65 (2007) 299–313www.elsevier.com/locate/jmarsys

Journal of Marine System

Forecasting front displacements with a satellite based oceanforecasting (SOFT) system

A. Alvarez a,⁎, A. Orfila b, G. Basterretxea a, J. Tintoré a, G. Vizoso a, A. Fornes a

a Instituto Mediterráneo de Estudios Avanzados, IMEDEA, (CSIC-UIB), Miquel Marqués, 21, 07190 Esporles, Spainb School of Civil and Environmental Engineering, Cornell University, Hollister Hall, 14853 Ithaca, NY, United States

Received 25 October 2004; accepted 3 November 2005Available online 30 October 2006

Abstract

Relatively long term time series of satellite data are nowadays available. These spatio–temporal time series of satelliteobservations can be employed to build empirical models, called satellite based ocean forecasting (SOFT) systems, to forecastcertain aspects of future ocean states. The forecast skill of SOFT systems predicting the sea surface temperature (SST) at sub-basinspatial scale (from hundreds to thousand kilometres), has been extensively explored in previous works. Thus, these works weremostly focussed on predicting large scale patterns spatially stationary. At spatial scales smaller than sub-basin (from tens tohundred kilometres), spatio–temporal variability is more complex and propagating structures are frequently present. In this case,traditional SOFT systems based on Empirical Orthogonal Function (EOF) decompositions could not be optimal prediction systems.Instead, SOFT systems based on Complex Empirical Orthogonal Functions (CEOFs) are, a priori, better candidates to resolve thesecases.

In this work we study and compare the performance of an EOF and CEOF based SOFT systems forecasting the SST at weeklytime scales of a propagating mesoscale structure. The SOFT system was implemented in an area of the Northern Balearic Sea(Western Mediterranean Sea) where a moving frontal structure is recurrently observed. Predictions from both SOFT systems arecompared with observations and with the predictions obtained from persistence models. Results indicate that the implementedSOFT systems are superior in terms of predictability to persistence. No substantial differences have been found between the EOFand CEOF-SOFT systems.© 2006 Elsevier B.V. All rights reserved.

Keywords: Satellite data; Ocean prediction; Genetic programming; Front evolution

1. Introduction

Predicting future states of the ocean has a valuableoperational interest on human related activities. Exam-ples of these activities include warning announcements(of coastal floods, ice and storm damage, harmful algal

⁎ Corresponding author. Tel.: +34 971611730; fax: +34 971611761.E-mail address: [email protected] (A. Alvarez).

0924-7963/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.jmarsys.2005.11.017

blooms and contaminants, etc.), optimizing routes forships, prediction of seasonal or annual primary produc-tivity and ocean currents, obtaining offshore designcriteria, determining ocean climate variability etc. Theinformation required from the ocean environment variesgreatly with the operational activity.

Traditionally, systems forecasting certain aspects ofthe ocean variability are comprised of explanatorymodels supported with field observations. Explanatory

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http://dx.doi.org/10.1016/j.jmarsys.2005.11.017

Fig. 1. Flow chart of a satellite based ocean forecasting (SOFT)system.

300 A. Alvarez et al. / Journal of Marine Systems 65 (2007) 299–313

models are based on mathematical descriptions andphysical understanding of that part of the oceanvariability we are interested to predict. In this way,wave prediction systems usually involve energy balancemodels for wind–sea. Moreover, predicting oceancirculation requires the whole ocean hydrodynamical–thermodynamical model, incorporating the law ofconservation of momentum, mass and energy. Initialconditions and atmospheric forcing needed to integrateforward in time the models are obtained from observa-tions and weather forecasts, respectively. Furtherobservations are assimilated into models as the forecastsadvance in time. Assimilation of measurements intomodels has significant impact on the accuracy of theforecasts (Anderson et al., 1996; Swanson and Ward,1999). Unfortunately, ocean observations are sparse,difficult and expensive to acquire.

Satellite remote sensing is the only observingtechnique able to systematically and continuouslymonitor some aspects of the dynamic variability ofspatially extended ocean areas. Spatio–temporal timeseries of sea surface temperature (SST), sea levelanomaly (SLA) and ocean colour are now availablefrom satellites. This data is usually employed to remotelydiagnose present ocean states or to study past oceanvariability. Besides, empirical predictive models can bebuilt on the basis of satellite data. These empiricalmodels are the so-called satellite based ocean forecasting(SOFT) systems (Alvarez et al., 2003).

The working procedure of a SOFT system, sketchedin Fig. 1, is divided in three major tasks (a more detailedexplanation can be found in Alvarez et al., 2004): First,the space–time variability of the satellite-observed datais decomposed into its spatial and time components. TheEmpirical Orthogonal Function (EOF) technique (Pre-isendorfer, 1988) is employed to accomplish this task. Inthis way, the space-and time-distributed satellite data isdecomposed into modes ranked by their temporal orspatial variance. Among both possibilities, EOF covari-ance analysis has shown to be slightly superior to EOFgradient decomposition in terms of predictability(Alvarez, 2003).

Spatial patterns and corresponding amplitude func-tions obtained from EOF decomposition show somedegree of noisy nature. Thus, the second task of theSOFT system is to reduce the degree of noise in thereconstructed spatial pattern, neglecting those EOFs ofsmall variance. Besides, Singular Spectral Analysis(SSA) or data adaptive approach (Broomhead andLowe, 1987; Elsner and Tsonis, 1996) is employed toremove noise in the time dependent amplitude functionsof the considered spatial patterns. In real-time forecasting

situations, SSA filtering must be prevented due to theappearance of spurious filtering border effects that reduceprediction capabilities (Alvarez et al., 2004).

The third task involved in a SOFT system is to obtaina predictive model for each filtered amplitude function.Various prediction techniques can be employed toaccomplish this task (Casdagli et al., 1992). Due to itsrobustness and performance, genetic programming(Szpiro, 1997; Alvarez et al., 2001) has been frequentlyemployed in previously developed SOFT systems.Finally, prediction of a satellite-observed field isachieved by adding the most relevant modes previouslymultiplied by their corresponding forecast amplitudes.

SOFT systems have been successfully implementedat different time scales and ocean regions. Specifically,accurate forecasts of monthly averaged SST patternswere obtained in the Alboran, Ligurian and Adriatic seasby SOFT systems (Alvarez et al., 2000; Alvarez, 2003;Alvarez et al., 2003). A SOFTsystem predicting the SST

301A. Alvarez et al. / Journal of Marine Systems 65 (2007) 299–313

field at weekly time scales in real-time has been recentlyimplemented in the Ligurian Sea (Alvarez et al., 2004).

Previous SOFT systems were focussed to predictquasi-static and spatially constrained features at sub-basin scale. In these cases, the purpose of EOF analysiswas to identify patterns that characterize variations inthe current state of the scalar field without taking intoaccount the time evolution of the analyzed field. Thus,EOF decomposition is not the most adequate forencoding satellite data when propagating features arepresent. Instead its extrapolation to the complex plane,the so called Complex Empirical Orthogonal Functions(CEOFs) (Horel, 1984), account for such time evolutionquantifying the time series in terms of complex numbers

Fig. 2. Geographic location and major oceanographic features of the

by adding to its Hilbert transform. This procedure givesinformation about the rate of change of the field as anartificial imaginary part. The result is that the informa-tion contained in the Hilbert scalar field is greater thanthat in the original field. Specifically, information aboutthe spatial distribution of variability, the phase fluctu-ation among various spatial locations and the amplitudeand phase of the temporal variability of each mode areobtained from the CEOF analysis. Propagating phe-nomena are represented by CEOFmodes with regions ofroughly constant spatial amplitudes and spatial andtemporal phases varying with distance and time,respectively. The variation of the spatial and temporalphase with position and time provides a measure of the

study area (adapted from Millot, 1999). Isobaths are in meters.


“local” wavenumber and “instantaneous” frequency,respectively. Phase speed can be inferred from the ratioof these magnitudes. Conversely, standing oscillationsare found in areas with maximum spatial amplitudesand roughly constant spatial and temporal phases.CEOF analysis is useful when the signal shows astationary or weakly stationary behavior. Otherwise,this technique can introduce extra modes to explainnonstationarity.

In this study the performance of two SOFT systemsforecasting the motions of a thermal front in real-timeand at weekly time scales, is investigated. The SOFTsystems were based on EOF and CEOF decompositionsrespectively, to determine the impact of the encodingmethodology when forecasting a moving structure. Thearea of study is the Northern Balearic Sea, (WesternMediterranean Sea) (Fig. 2), where a propagatingthermal front is present during late summer and fall.The article is organized as follows: Section 2 brieflysketches the oceanography of the Northern Balearic Sea.A description of the satellite data used in this study isprovided in Section 3. Details on the implementation ofthe SOFT systems are described in Section 4. Section 5shows the results obtained from the application of theSOFT systems. Finally, discussion and conclusions arepresented in Section 6.

Fig. 3. Time average

2. Oceanographic conditions in the NorthernBalearic Sea

The Northern Balearic Sea is geographically locatedin the north-western Mediterranean, near the upperboundary of the Balearic subbasin (Fig. 2). This region isdominated by the presence of a slope current, theNorthern-Current, cyclonically flowing along the conti-nental slope towards the Channel of Ibiza (Millot, 1987).The flow presents marked seasonal variability beingnarrower in winter and wider and with reducedmesoscale variability on summer (Millot, 1999). In thesouth of the Balearic Sea, part of the flow is deflectednorth-eastwards along the slope the Balearic Islands dueto the topographic restrictions of the Channel of Ibizaand due to the flow of Modified Atlantic Water (MAW)through the Balearic Channels. The North Balearic Front(NBF) is generated to the north of the Islands as results ofthe confluence of waters from the open Algerian Basinwith waters from the Gulf of Lions. During winter thethermal difference between the MAW and waters fromthe Gulf of Lions is weak and thus the NBF is mainlydistinguishable by its salinity signal. In summer andearly fall, the front shows a strong surface thermalsignature (sometimes identified like Pyrenees Front(PF), Millot, 1999). This thermal front is a recurrent

d SST pattern.


feature that appears located between 41 and 42° N inmost sea surface temperature (SST) satellite images witha thermal gradient of up to 4 °C in tens of kilometers.This is one of the highest thermal gradients in the westernMediterranean during summer (Le Vourch et al., 1992).The origin of this front is speculated to be the shadowingeffect of the Pyrenees over theMistral jet which cools thesea surface and induces mixing, counteracting theseasonal stratification (e.g. López García et al., 1994).

The spatial structure and the evolution of the PF areclosely linked to atmospheric conditions. IntenseMistral events, which are known to generate significantmesoscale variability, often result in anticyclonicstructures along the front that are attributed to the effectof the negative wind curl downstream of the Pyrenees(e.g. Pascual et al., 2002). The presence of profusemesoscale circulation is one of the main characteristicsof the circulation in the Balearic Basin.

At early fall, seasonal cooling produces weakening ofthe front, that almost disappears by December. LópezGarcía et al. (1994) in their analysis of the summer–winter transition of the Balearic Basin based on SST,describe this reduction of the thermal gradient which isparalleled by a southward displacement of the frontmainly along the Iberian Peninsula. This latitudinal

Fig. 4. 1st (a) to 4th (d) EOF of SS

migration of the front may vary from year to yearattending to climatological variability and there are noprecise estimations of its magnitude. During SOFTcruises carried out between September and October2002 in the region, the southward evolution of the PFwas estimated to occur at nearly 2 km day−1.

3. Data

Unlike other oceanographic variables, SST has beensystematically recorded from satellite since the last twodecades. Thus, relatively long spatio–temporal timeseries of SST data are now available. This makes SST themost adequate magnitude to investigate empiricalpredictive models built on the basis of satellite data. Atime series of 459 weekly averaged SST images of theNorthern Balearic Sea, from March 1st 1993 toDecember 10th 2001, has been obtained from theGerman Aerospace Research Center-DLR. The imagesare constituted by 176×151 pixels corresponding to aspatial resolution of 1.1 km. This Advance Very HighResolution Radiometer-Multi-Channel Sea SurfaceTemperature (AVHRR-MCSST) product is a mixtureof unsupervised pre-processing steps and a supervisedparametrization of cloud test carried out at DLR. Each

T (units milli-Kelvin (mK)).


weekly image is composed using for every pixel'sposition, the average of the daily maximum images. Theweekly composition normally consists of approximately40 AVHRR passes. Several tests ensure that SST valuesare derived only for cloud free water surfaces. These testsare based on the principal characteristics of water bodies,considering typical spectral and textural parameters suchas dark, warm, and homogenous surface. They arechanged from case to case depending on the specificcharacteristics of every single scene. All pixels flagged ascloud are excluded from further processing. Data is alsomanually controlled regarding the navigation quality andthe cloud tests. Detailed information about the majorprocessing steps can be fount at http://eoweb.dlr.de.

4. Implementation of the soft system

The period of time ranging from March 1st 1993 toJanuary 10th 2000 was employed to build two SOFTsystems: a SOFT system employing the EOF-basedencoding and a second one based on CEOF datadecomposition. Thus, EOFs, CEOFs and correspondingamplitude functions were first computed for this period.Determination of predictive laws for the amplitudefunctions of the most relevant EOFs and CEOFs was

Fig. 5. Amplitude functions (black) and one week ahead predictions (gray) du10th 2001 for the amplitude functions corresponding to the first (a) to fourth

then attempted. The genetic program called DARWIN(Alvarez et al., 2001) was directly applied to theamplitude functions to obtain predictive laws of theamplitude functions. Besides, a linear autoregressivepredictor was also considered.

To estimate the forecast skill of the time seriespredictors, a retroactive method (Barnston et al., 1999)was implemented. Thus, the predictor DARWIN wastrained with data ranging from March 1st 1993 toJanuary 10th 2000 (359 samples), and validated in asubsequent period from January 17th 2000 to December10th 2001 (100 samples). This validation periodcompletely covers the genesis, displacements anddisappearance of the frontal system we are interestedto forecast. During validation, a real-time modusoperandi was emulated. To do that, the present valueof the amplitude function at each simulation time wasprovided to the predictor to obtain forecast of the nextvalue. Thus, knowledge about the time evolution of theamplitude function is provided to the predictor up to thepresent simulation time. To obtain the prediction laws,DARWIN was configured in such a way that themaximum number of symbols (variables, numericalconstants and arithmetic operators) allowed for eachtentative equation is 20. Each generation consisted of a

ring the validation period ranging from January 17th 2000 to December(d) EOF of SST.

http://eoweb.dlr.de

Table 1One week ahead predictability of the EOF amplitude functionsobtained with DARWIN, the autoregressive model and persistenceduring the validation period ranging from January 17th 2000 toDecember 10th 2001

Amplitude function 1st 2nd 3rd 4th

Amplitude predictabilityDARWIN (R2×100)

98 44 48 56

Amplitude predictabilityAutoregresive (R2×100)

97 44 49 57

Amplitude predictabilityPersistence (R2×100)

95 34 43 54


population of 120 randomly generated equations. As aresult of applying an evolutionary process to the initialgeneration, a new set of equations (generation) withbetter fitness properties is obtained. The evolutionaryprocess is then repeated a large number of times, toimprove the fitness of the evolving population. In thiswork, the number of generations considered in eachsimulation was 10,000.

Unlike the DARWIN predictor, the autoregressivemodel was not unique for the whole validation period,but it was recomputed each time a new value of thevalidation set was provided. Specifically, the autore-gressive model that best fitted the data up to theconsidered time was searched. To do that, the order ofthe linear model was varied from one to twelve. Finallya persistence model defined for given amplitudefunction at time t (A(t)) like Â(t+1)=A(t), was employedto compare with.

The forecast skill of the time series predictors as wellas the predictability of given amplitude functions weremeasured by the explained variance:

R2 ¼ 1−

XN

t¼1

ðAðtÞ−AðtÞÞ2

PN

t¼1ðAðtÞ−A Þ2

ð1Þ

where N is the number of points in the validation set,Â(t) is the predicted value of the amplitude function attime t and Ā is the mean value of the amplitude functionA(t) in the validation period. Values of R2 close to oneindicate good performance of the prediction system orhigh predictable amplitude functions, while values ofR2 equal or less than zero describe systems with poorprediction performance or unpredictable amplitudes.Only the performance of the SOFT systems fromJanuary 17th 2000 to December 10th 2001 will beconsidered.

5. Results

5.1. EOF based SOFT system

The first four EOFs from the decomposition of the SSTtime variability were selected for physical interpretationbased on their percentage of the total temporal variance.Fig. 3 displays the mean SST field averaged from March1st 1993 to January 10th 2000 and subtracted from theimages during the EOF computation. The mean SST fieldreveals a southward temperature gradient generated by thethermal differences between the Northern Balearic Sea

and Southern Gulf of Lion. The spatial patterns associatedwith the selected EOFs are shown in Fig. 4a–d. The firstEOF, Fig. 4a, accounts for 97.89% of the total temporalvariance. It is characterized by a north–south gradientassociated to the seasonal modulation of thermaldifferences existing between the Northern Balearic Seaand the southern area of the Gulf of Lion. SST differencesare smaller during winter than in summer when a relevantdifferential heating exists among the two sub-basins. Thesecond EOF, Fig. 4b, represents 0.67% of the temporalvariance. The spatial pattern associated to this EOFdescribes a mode of variability characterized by a south-westward gradient. This pattern of variability can beinterpreted as a combination of the southward gradient oftemperature generated by the SST differences between theBalearic Sea and Gulf of Lion, plus the westwardtemperature gradient existing between coastal and opensea waters, the former being warmer in summer. The thirdEOF, in Fig. 4c, resembles the structure of the first one. Itaccounts for 0.29% of the total temporal variance and itsvariability might be related with the entrance of coldwaters from the Gulf of Lion during late summer and fall.Finally, the fourth EOF, accounting for 0.14% of thevariability, represents a quadruple structure of variability.Assignment of known oceanographic processes to thevariability displayed by this EOF is more difficult. EOFsare mathematically constrained to be orthogonal and thus,their patterns are not always easily related to physicalfeatures or processes (P. F. J. Lermusiaux, 2003, personalcommunication). This is the case for the present EOF.

Fig. 5a–d show original values and one week aheadpredictions of the considered amplitude functions fromJanuary 17th 2000 to December 10th 2001. From thesefigures it is possible to infer qualitatively the performanceof the developed time series predictors. A morequantitative measure of this performance is provided inTable 1. First amplitude function, Fig. 5a, is almost fullypredictable as it is shown from the high explainedvariance obtained from the considered predictionmodels.

Fig. 6. a) Spatial average SST error in one week ahead predictions obtained during the validation period by the EOF–SOFT system (gray) andpersistence model (black). b) Spatial correlation between EOF–SOFT (gray) and persistence (black) models with observations during the validationperiod.

Fig. 7. Spatial amplitude (a) and phase (b) functions of the first CEOF. c) and d) the same that a) and b) respectively but for the second CEOF.



This result is expected considering that this EOF ismainly associated to seasonal variability. Predictabilitysubstantially decreases for remaining amplitude func-tions where values for explained variances are around 0.5(50%). Decrease in predictability corresponds to anincrease in the complexity shown by these amplitudefunctions. In the case of the second amplitude function,the nonlinear predictor Darwin model as well as the linearautoregressive model substantially improves persistenceforecasts. Conversely, forecast skills of the nonlinear andlinear predictors are similar to persistence for the third andfourth amplitude functions. For these amplitude func-tions, a red noise hypothesis is sufficient to describe theirtime behaviour. Notice that looking at EOFs with fewpercentage of the total temporal variance implies lookingat smaller and faster scales the variability which physicalmeaning is doubtful due to the orthogonal constraint.

A SOFT system has been built using the bestpredictor for each amplitude. The global performanceof the SOFT system is measured by two criteria: First,the spatial averaged SST error ΔT in the prediction andsecond, the spatial correlation between the predicted andobserved SST fields, Fig. 6a and b. The first estimates

Fig. 8. Spatial amplitude (a), and phase (b) functions of the third CEOF. c)

the temperature error at each pixel while the secondprovides an idea on the success of the SOFT system topredict the spatial SST structures. Fig. 6a and b comparethe temperature error and spatial correlation obtainedfrom the SOFT system with those obtained from thepersistence model. Concerning the temperature error,results indicate that the SOFT system performs better orsimilar than the persistence model during almost all thevalidation period, except during mid September and thefirst three weeks of October—2001. As depicted fromFig. 6a, during these weeks the persistence model isclearly superior in terms of predictability to the SOFTsystem. This fact is due to the inability of the SOFTsystem to correctly describe the dynamical changes thatare occurring during that time period. This dynamicalvariability is characterized by homogeneous and suddenchanges of the SST structure existing in the previousweek plus modifications at relatively small scales. Theresulting spatial structure of the SST field closelyresembles the one existing in the previous week exceptfrom small scale variations. These small scale variationsare unpredictable by the SOFT system due to the EOFtruncation carried out during reconstruction.

and d) the same that a) and b) respectively but for the fourth CEOF.

Fig. 9. Real (a) and imaginary (b) parts of the spatial amplitude function (black) corresponding to the first CEOF. One-week ahead predictions of realand imaginary parts during the validation period ranging from January 17th 2000 to December 10th 2001, are plotted in gray colour. c) and d) are thesame that a) and b) but for the second CEOF.


Persistence and SOFT system perform reasonablywell when forecasting the SST spatial structure. This isrevealed by the high spatial correlations observed inFig. 6b. Correlations seem to be seasonally modulatedwith higher values in summer–autumn than duringwinter–spring. This modulation might be explained interms of the oceanographic conditions of the area.During winter and spring, the SST field in the area canbe described by scattered small-scale patches of SSTanomalies on an underlying homogeneous background.Thus, no significant predictable oceanographic structureis found in the area. Conversely, the frontal systemprovides a well defined spatial SST structure during latesummer and autumn. Modulations of spatial correlationsassociated to oceanographic conditions have also been

Table 2One week ahead predictability of the CEOF amplitude functions obtainedvalidation period ranging from January 17th 2000 to December 10th 2001

Amplitude function 1st

Part Real Imaginary

Amplitude predictability DARWIN (R2×100) 97 94Amplitude predictability Autoregresive (R2×100) 97 97Amplitude predictability Persistence (R2×100) 96 95

found in other ocean areas (Alvarez et al., 2004).However, a longer time series would be required to fullyconfirm this modulation. Finally, it is remarked that onaverage, the SOFT system also performs better thanpersistence when forecasting the SST spatial structure.

5.2. CEOF based SOFT system

Similarly to the previous case, only the first fourCEOFs from the decomposition of the SST timevariability were selected for physical interpretation.The first CEOF accounts for 97.93% of the total timevariability. Its spatial amplitude function, Fig. 7a,resembles the spatial structure obtained in the firstEOF, Fig. 4a, showing a tongue like distribution of low

with DARWIN, the autoregressive model and persistence during the

2nd 3rd 4th

Real Imaginary Real Imaginary Real Imaginary

39 51 32 46 52 4236 51 28 50 48 4024 37 18 47 48 39

Fig. 10. Real (a) and imaginary (b) parts of the spatial amplitude function (black) corresponding to the third CEOF. One-week ahead predictions ofreal and imaginary parts during the validation period ranging from January 17th 2000 to December 10th 2001, are plotted in gray colour. c) and d) arethe same that a) and b) but for the fourth CEOF.

Fig. 11. a) Spatial average SST error in one week ahead predictions obtained during the validation period by the CEOF–SOFT system (gray) andpersistence model (black). b) Spatial correlation between CEOF–SOFT (gray) and persistence (black) models with observations during the validationperiod.



amplitude in the northern boundary. The spatial phasefunction displayed in Fig. 7b, reveals that this gradientstructure is not stationary but propagates stretching thelow amplitude area in winter and reversed in summer. Aphase speed of 3 km day−1 for the frontal structure wasestimated from this satellite data. This agrees with theoceanographic variability of the area characterized bythe seasonal entrance and retreat of cold waters at thenorthern boundary of the region. The second CEOFrepresents 0.71% of the time variance. The spatialamplitude function of this CEOF shows a clear wavelikestructure with a wavenumber vector pointing to thesouthwest–northeast direction, Fig. 7c. The spatialphase function from Fig. 7d rotates clockwise fromsouth to north. Notice that the sharp change of phaseobserved in the figure results from ending one cycle at360° and the beginning of the next at 0°. The temporalphase function is roughly constant for this CEOF,implying most of the time zero phase speed. Thus, thepattern is quasi-stationary describing weak motions

Fig. 12. a) Weekly averaged SST during the week starting on November 19tweekly averaged SST for the week starting on November 19th 2001, c) same abut prediction is obtained by the CEOF–SOFT system.

focussed in the area with smaller spatial amplitude. Thethird CEOF accounts for 0.28% of the time variability.Its spatial amplitude and phase closely resemble theones of the second CEOF after rotating the pattern 90°anticlockwise, Fig. 8a and b. Mathematical constraint ofCEOF orthogonally complicates the physical interpre-tation of these CEOFs. Both CEOFs are related with thecombined effect on the SST pattern of the intrusionsfollowing the Catalan coast of cold waters from thesouthern part of the Gulf of Lion and the general coolingof the central part of the Western Mediterranean Sea.These actions can result on the existence of isolated andelongated patches of relative warm waters in the centreof the Balearic Sea. Finally, the last CEOF considered,Fig. 8c and d, accounts for 0.12% of the time variance. Itwas not possible to assign any oceanographic process tothe variability described by this CEOF.

Figs. 9a–d and 10a–d show raw values and one weekahead predictions of the real and imaginary parts of theconsidered amplitude functions. Table 2 shows the

h 2001, b) one week ahead prediction by the persistence model of thes b) but prediction is obtained by the EOF–SOFT system, d) same as b)


performance of the different predictors to forecast thecomplex amplitude functions.

Similarly to the EOF case, a SOFT system was builtusing the best predictor for each CEOF amplitude.Fig. 11a and b display the spatial averaged SST errorΔTin the prediction and the spatial correlation between thepredicted and observed SST fields for this SOFT system.Direct comparison between Figs. 6a and 11a revealsthat, the performances of the EOF and CEOF basedSOFT systems are similar concerning the spatialaveraged SST error. Similar agreement in the perfor-mance of both SOFT systems is also found whenanalysing the SST spatial structure, Figs. 6b and 11b.

Examples of the forecasts of the SOFT systems areshown in Figs. 12 and 13 with the best and worstprediction cases, respectively. Specifically, Fig. 12ashows the weekly averaged SST pattern during the weekstarting on November 19th 2001. This pattern revealsthe existence of incoming cold waters from the Gulf ofLion at the northern boundary. The area has been

Fig. 13. a) Weekly averaged SST during the week starting on October 1st 200averaged SST for the week starting on October 1st 2001, c) same as b) butprediction is obtained by the CEOF–SOFT system.

relatively cooled down with respect to the summersurface temperature. Reminiscences of these warmwaters are still present at the southern boundary. TheSST pattern found the week before is noisier, Fig. 12b.The area is filled by cold and warm patches of surfacewaters. Notice that this SST field is the forecast by thepersistence model for the pattern of the week ofNovember 19th 2001. Fig. 12c and d show one-weekahead forecasts of the SST field obtained from the EOFand CEOF based SOFT systems, respectively. BothSOFT systems succeed to predict the general aspects ofthe SST pattern. However, slight differences are foundbetween both predictions. In this way, the EOF basedSOFT system underestimates the area of influence of thecold waters, but it forecasts the surface conditions foundat the southern boundary better than the CEOF SOFTsystem.

Fig. 13 shows the case where the SOFT systems haveobtained the poorest performance in the forecast.Fig. 13a corresponds to the averaged SST during the

1, b) one week ahead prediction by the persistence model of the weeklyprediction is obtained by the EOF–SOFT system, d) same as b) but


week starting on the October 1st 2001. Strong thermalgradients are found at surface during that period. Thesegradients are generated by the encounter of cold watersfrom the Gulf of Lion and the local surface watersstrongly heated during the summer period. Cold watersintrude into the area forming a tongue-like structure.The evolution of that cold tongue can be inferred fromFig. 13a and the situation corresponding to one weekbefore, Fig. 13b. Both SOFT systems do not correctlyforecast the spatial structure of the intruding tongue.Instead, both predictions underestimate the elongationand length of the observed structure. This failure iscaused by the limitations of SOFT systems to forecastprocesses with scales of variability not represented inthe first EOFs (CEOFs).

6. Discussion and conclusion

This work has explored the forecasting skill of anEOF and CEOF based SOFT systems, predicting in realtime the SST signature of a moving structure at weeklytime scales. The selected moving structure was a strongfrontal system located at the Northern Balearic Sea. Thefront results from the differential heating occurring inlate summer and fall between the Balearic and LigurianSeas. After its formation, the front propagates south-wards following the advection of cold waters into theBalearic basin. Initially, cold waters intrude the area in atongue-like shape generating high mesoscale activitycharacterized by meanders and eddies. This mesoscaleactivity distorts the front line into complex shapes. Asthe front moves southward, the upstream waters coolsdown homogenizing the SST of the area and smoothingthe front shape. Thus, the front variability can bedescribed by a southward propagation plus distortionsof its shape.

The implementation of an EOF and CEOF based SOFTsystems to predict the SST variability of the NorthernBalearic Sea served to compare the results obtained withboth SOFT systems, when a propagating structure ispresent. No significant improvement in the performance ofthe CEOF–SOFT system has been found when comparedwith the forecasts obtained with the EOF–SOFT system.CEOF encoding captures more variability than EOF for apartial reconstruction of the spatio–temporal time series. Toconfirm this point, the spatial rootmean squared of the SSTdifferences between the real field and the reconstructionwith the first fourth EOFs and CEOFs was computed. Thevalues averaged over the validation period are 0.5 °C and0.4 °C for the EOF and CEOF reconstructions, respective-ly. Thus, the SST field reconstructed with the first fourCEOFs is more similar to the real SST field than the one

obtained with the corresponding EOFs. However, thesuperior capability of CEOFs to encode information is nottranslated into better forecast skills of the CEOF SOFTsystem. Figs. 6 and 11 clearly show that differencesbetween forecasting errors of both SOFT systems isnegligible. A more quantitative estimate of this similarityis given by averaging the forecasting error for each SOFTsystem over the validation period. This mean error is0.49 °C for the EOF SOFTsystem versus an averaged errorof 0.5 °C obtained with the CEOF SOFT system.Moreover, Figs. 12 and 13 have explicitly shown examplesof this similarity in the forecasting performance. Why thesuperiority of CEOFs to encode the observed variabilitydoes not drive to substantial improvements of theforecasting performance of a CEOF SOFT system versusan EOF one, can be explained by the following hypothesis:Complex amplitude functions require separated predictionof the real and complex components of the amplitude (ormodulus and phase). Forecast errors are then increased, thecoherence between components is lost and the overallforecasting skill is degraded. To validate this hypothesis asimplified forecasting exercise is considered. Specifically,the spatial root mean squared of the SST differences in thevalidation period between the real field and the reconstruc-tionwith only the first EOF andCEOFwere analysed. Tworeconstructions have been considered for each case: firstconsidering the true amplitude and second using itspredicted value. In the EOF case, themean SST differencesare 0.56 °C for the real and predicted reconstructions.Instead for the CEOF case, differences are 0.56 and 0.84 °Crespectively. Thus, the reconstruction from the predictedcomplex amplitude is severely degraded.

Concluding, no benefits in terms of prediction perfor-mance have been obtained using a CEOF SOFT system toforecast a propagating front. Although the CEOF SOFTsystem provides better encoding capabilities and moreinformation than the EOF SOFT system, its drawbackarises in the need to forecast real and complex part of theresulting amplitude functions. Prediction errors of the realand complex parts pile up when reconstructing the forecastfield, degrading the overall prediction. In other words, amore accurate forecast of each component of the amplitudefunction is needed. Results indicate that EOF SOFTsystems would still be the most adequate to consider evenwhen propagating features are present. Concerning SOFTprediction capabilities of the front displacements, bothSOFT systems have shown better performance thanpersistence. Specifically, the mean root square differencesfound between the real and forecast fields in the validationperiod were 0.49–0.5 °C and 0.57 °C for the SOFT andpersistence predictors, respectively. Thus, SOFT systemshave provided one-week-ahead description of the front


evolution more accurate than considering the presentsituation like the forecast field.

Acknowledgements

This work has been supported by the SOFT-EVK3-CT-2000-0028 European Project and the SpanishProject REN 2001-3982-E.

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Forecasting front displacements with a satellite based ocean forecasting (SOFT) system

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