On the potential of data assimilation to generate SMOS-Level 4 maps of sea surface salinity

Remote Sensing of Environment 146 (2014) 188–200

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Remote Sensing of Environment

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On the potential of data assimilation to generate SMOS-Level 4 mapsof sea surface salinity

Nina Hoareau ⁎, Marta Umbert, Justino Martínez, Antonio Turiel, Joaquim Ballabrera-PoyInstitut de Ciències del Mar, CSIC, Barcelona, Spain

⁎ Corresponding author at: Departament d'OceaCiències del Mar, CSIC, Passeig Marítim de la BarcelonSpain. Tel.: +34 93 230 95 00.

E-mail address: [email protected] (N. Hoareau).

0034-4257/$ – see front matter © 2013 Elsevier Inc. All rihttp://dx.doi.org/10.1016/j.rse.2013.10.005

a b s t r a c t

Article history:Received 31 October 2012Received in revised form 25 September 2013Accepted 2 October 2013Available online 30 October 2013

Keywords:Data assimilationNudgingSea surface salinitySMOSSingularity analysisEastern subtropical North-Atlantic Ocean

The Soil Moisture/Ocean Salinity (SMOS) satellite, launched in November 2009, measures visibilities at L-band, from which brightness temperatures are computed. This information is used to retrieve values ofthe sea surface salinity (SSS) and soil moisture; two variables whose observation is a key to betterunderstand the oceanic component of the water cycle. A hierarchy of SSS products has been defined inthe SMOS data processing chain. This work focuses on the so-called Level 3 (binned maps of SSS) andLevel 4 (products combining SMOS data with any other source of information). The objective is toillustrate the feasibility of using data assimilation to produce Level 4 maps of sea surface salinity. Thenumerical model will increase the geophysical coherence of SMOS data as a dynamical interpolator.Here, the employment of data assimilation differs from its usual applications (improving model outputsfor example). Indeed, the numerical model will interpolate the observations according to the generallaws of fluid mechanics and, if possible, reduce the error contained in the original observations. Thedata assimilation method analyzed is a nudging algorithm. The domain of application for this feasibilitystudy is the Northeast subtropical Atlantic gyre, a challenging region due to the presence of a largeamount of noise that deteriorates the SMOS data. The main sources of errors are the vicinity of largelandmasses that introduce a spurious bias, and the presence of a significant amount of artificial radiofrequency interferences (RFI). While the Quality Controls already set up in the SMOS processing chaindo filter the retrievals containing too large errors, wrong data are still present in Level 3 maps. Despitethis difficulty, the results provide meaningful SMOS SSS Level 4 products in terms of their geophysicalcoherence (estimated using singularity analysis) and better agreement with in-situ data than Level 3product.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

Sea surface salinity (SSS) is a fundamental ocean variable because itcontributes, together with sea surface temperature (SST), to sea surfacedensity that modulates the mixing in the upper layer of the oceanand the formation of water masses. Despite its importance, salinitywas often disregarded in comparison with the efforts to observe andmodel ocean temperature. However during the last years, the scientificcommunity has started to stress out the need of a better knowledgeof salinity. Indeed, sequential ocean data assimilation provides unre-alistic results when temperature is updated, but not salinity (Cooper,1988); vertical salinity gradients may give rise due to the presence ofbarrier layers that modify the ocean-atmosphere coupling (Lukas &Lindstrom, 1991); horizontal salinity gradients do provide potentialenergy to the fresh equatorial jets (Roemmich, Morris, Young, &

nografia Física, Institut deeta, 37-49, Barcelona 08003,

ghts reserved.

Donguy, 1994); the absence of sea surface salinity observations oftentranslates to erroneous ocean surface currents in the models (Acero-Schertzer, Hansen, & Swenson, 1997); upper ocean salinity variationsmodulate high-latitude convection (Dickson et al., 2002); salinity andfreshwater transports can enhance, or reduce, heat transport(Ganachaud, 2003); tropical salinity anomalies have been found toprovide independent, significant information for statistical predictionsof El Niño (Ballabrera-Poy, Murtugudde, & Busalacchi, 2002); salinitymodulates ocean ventilation (U.S. CLIVAR office, 2007); and salinityprovides information to better understand the ocean branch of thewater cycle (Hosoda, Sugo, Shikama, & Mizuno, 2009), and carboncycle (Lefèvre, 2009).

It is known that in-situ observing systems are not adequate, bythemselves, to measure the ocean in all space and time scales(Busalacchi, 1997). As an alternative, remote sensing provides synopticmeasurements over large areas. Despite the fact that satellites observeonly the surface due to the opacity of the ocean to electromagneticwaves, they have become the prominent component of the oceanobserving system. Remote sensing of SSS is possible because theocean thermal emission at low frequencies, including microwaves, is

189N. Hoareau et al. / Remote Sensing of Environment 146 (2014) 188–200

proportional to its physical temperature and because the propor-tionality coefficient (the emissivity) is a function of the dielectricconstant that depends on conductivity, i.e. salinity. The satellites SoilMoisture Ocean Salinity (SMOS), launched in 2009, and Aquarius,launched in 2011, measure the brightness temperature (Tb) at the L-band (1.4GHz). After processing the first year and a half of SMOS data,it has become clear that the retrieval of SSS information is verychallenging (Font et al., 2010) because of the low sensitivity of Tb tosalinity, and also because of the persistence of large amplitude errorsin Tb. The main sources of these errors are the imperfect imagereconstruction algorithms, and the contamination by artificial or naturalradio frequency interferences (RFI). As a result, the current accuracy ofSSS retrievals for a single overpass (the so called Level 2 product) isabout 1 and 2 (Boutin et al., 2012; Font, Boutin, et al., 2012) in thepractical salinity scale (Lewis, 1980). Various strategies are imple-mented to reduce SSS retrieval error. For example, SSS errors arereduced to 0.2–0.6 through spatial and temporal averages (Font,Ballabrera-Poy, et al., 2012; Reul et al., 2012). Maps produced by spatialand temporal averages of Level 2 SSS are called Level 3 in the SMOS dataprocessing chain. Alternatively, some approaches are being imple-mented to blend SMOS data with other sources of information to createLevel 4 products as in Umbert, Hoareau, Turiel, and Ballabrera-Poy(2013).

The objective of this manuscript is to investigate the potential use ofdata assimilation to produce Level 4 salinity maps by combining SMOSLevel 3 data with an ocean general circulation model.

Data assimilation is a broad concept that refers to differentapproaches combining numerical models and observations (Ghil &Malanotte-Rizzoli, 1991). In general the applications of data assim-ilation are devoted to find the initial conditions for numericalprediction; to perform reconstructions of the state of the system(i.e. reanalysis); and to improve the numerical models through theoptimal identification of their physical parameters. This manuscriptfocuses on the ability of a simple, but robust, data assimilationmethod, the Newtonian relaxation or nudging (Anthes, 1974), toperform the dynamical interpolation of SMOS data. With thisapproach, the resulting data are being infused with the physicalcoherence inherent to the ocean model. The realism of the spatialcoherence will be assessed using the singularity analysis, a techniquefirst developed to perform pattern recognition in complex images(Turiel, Solé, Nieves, Ballabrera-Poy, & García-Ladona, 2008).

The region where this approach will be tested is the North-EastAtlantic subtropical gyre, shown in Fig. 1, where a numerical oceansimulation has already been implemented and evaluated by Mourre,Ballabrera-Poy, García-Ladona, and Font (2008) and Mourre andBallabrera-Poy (2009). As shown in Fig. 1 (top), the surface subtropicalNorth-Atlantic Ocean is characterized by a tongue of salty watercentered around 24 N. The maximum of evaporation minus precip-itation is located around 18N, revealing a role of advective and diffusiveprocesses in the evolution of the subtropical surface salinity in thisbasin. The annual SSS variability is of the order of 0.1 (Fig. 1, bottom).This zone was chosen because of this low annual variability to calibratethe SMOS satellite. But after launch, it has been found that the amount ofnoise in the SMOS brightness temperatures in this region deterioratessalinity retrievals. The estimated error of SSS in the region is 0.55(BEC-SMOS, 2013b). For comparison, the error at the equatorial andtropical bands is 0.30 and 0.43 respectively. The noise in brightnesstemperature arises from the existence of intense RFI and the vicinityof large continental masses. Another objective of this study is toinvestigate the potential of data assimilation to produce Level 4 salinitymaps even in region with large estimated error of SSS.

The outline of the manuscript is the following. All data used in thiswork are described in Section 2. The numerical simulation based onthe Nucleus for European Modelling of the Ocean (NEMO) is describedin Section 3. The data assimilation scheme is described in Section 4,and singularity analysis is cursorily presented in Section 5. Section 6

describes main results and conclusions. Finally, Section 7 provides asummary and final discussion.

2. The data

2.1. Level 3 SMOS SSS

The SMOS data used in this study come from the SMOS reprocesseddata campaign of July 2012. The period under consideration is January2011 to December 2011. Level 2 SSS data go through a quality controlprocedure that removes suspect data following a set of filtering criteria.Those criteria come from the Level 2 Ocean Salinity User Data Product(UDP) and Ocean Salinity Data Analysis Product (DAP) generated byESA. The UDP and DAP files include geophysical information and atheoretical estimate of accuracy. The pixels entering in the SSS retrievalalgorithm are discarded if the distance between the pixel and the centerof the satellite swath is larger than 360km. They are also discarded if thealgorithms detect RFI, ice, or rain, or if winds are larger than 12 m/s.After retrieval, salinity values are flagged as bad if the inversionalgorithm has returned an error flag (due to the lack of enough numberof valid pixels, excessive number of iterations, or failure to reduce thecost function). An exhaustive description of the data quality control(QC) is given in BEC-SMOS (2013a). Finally, Level 3 SSS maps (Fig. 2)are generated by binning the Level 2 data in 1/4degree bins to providemonthly average maps. The data used here can be obtained fromthe SMOS Barcelona Expert Centre on Radiometric Calibration andOcean Salinity (SMOS-BEC, http://www.smos-bec.icm.csic.es). Finally,SMOS Level 3 SSS maps are interpolated to the numerical simulationgrid (1/3° of resolution).

2.2. OSTIA SST

The SST data used here come from the Operational SST and Sea IceAnalysis (OSTIA) described by Donlon et al. (2011). The OSTIA systemcombines satellite data with in-situ observations to produce high-resolution SST maps. The analysis is performed using a variant ofoptimal interpolation (OI) described by Martin, Hines, and Bell (2007).The analysis is produced daily at a resolution of 1/20° (approx. 5 km).OSTIA data is provided by the GODAE High Resolution Sea SurfaceTemperature (GHRSST, http://ghrsst-pp.metoffice.com). A brief de-scription of the OSTIA system can be found in Stark, Donlon, Martin,and McCulloch (2007).

2.3. Argo SSS

An in-situ estimate of the SSS is derived from the Argo profilesdistributed by the CORIOLIS data center (http://www.coriolis.eu.org).For the sake of simplicity this estimate will be called the Argo SSS. Thefollowing selection rules are applied to obtain a set of satisfactoryprofiles: (i) the Argo profiler should not be in the Grey List, i.e. the listof floats which may have problems with one or more sensors that ismaintained at GDACs (http://www.usgodae.org/pub/outgoing/argo/ar_greylist.txt); (ii) at each provided level, temperature, salinity,and pressure must have valid values; (iii) the number of valid levelsmust be larger than 40% of the total number of levels in the profile;and (iv) data have been flagged as “Good”, i.e. set to 1 (good) or to 0(not checkedbecause of real time transmission) by theUSGODAE services(http://www.usgodae.org/argo/argo-dm-user-manual.pdf). Finally theestimation of SSS is done interpolating the profile at the depth of7.5m. This depth has been chosen because most of Argo profilers havethe first valid measurement in the first 10 m, and also because theconductivity measurements obtained by SOLO and PROFOR profilersare not reliable in the first 5 m. The use of a fix depth allows theimplementation of an automatic algorithm that compares threedifferent interpolation methods. Only those profiles providing a robustinterpolation at the target depth are used here (see Ballabrera-Poy,

Fig. 1. Top: Annual mean of SSS from WOA98 atlas. Bottom: standard deviation associated. The box corresponds to the region of study that includes the Macaronesian region.

190 N. Hoareau et al. / Remote Sensing of Environment 146 (2014) 188–200

Fig. 2. SMOS SSS map for December 2011, from monthly L3 Binned product.

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Mourre, García-Ladona, Turiel, & Font, 2009). A set of 4112 profiles isavailable for year 2011 (Fig. 3).

3. The model

As stated before, a numerical simulation of the North East Atlanticgyre has been already developed and validated by Mourre et al.(2008) and Mourre and Ballabrera-Poy (2009). The ocean model isthe free surface OPA9.0 ocean general circulation model (Madec,Delecluse, Imbard, & Lévy, 1998; Roullet & Madec, 2000), which is partof the NEMO modeling system. It solves the primitive equationsdiscretized on a C-grid centered at the tracer points. The horizontalresolution is 1/3 of a degree with some refinement in the Gulf of

Fig. 3. Argo SSS data for 2

Cadiz, where the distance between two grid-points is around 10 km(see Mourre et al., 2008). The simulation uses 31 z-coordinate levelsin vertical, and partial cells are used to adjust the bottom cell size tothe ocean real depth (Pacanowski & Gnanadesikan, 1998). The verticalresolution in the upper ocean is 10 m. The bathymetry is extractedfrom Etopo2 database (National Geophysical Data Center). The modelregional configuration has four open boundaries, in which the modelsolution is relaxed towards the annual climatology of the MercatorMERA 11 reanalysis (Greiner, Benkiran, & Pergaud, 2006). Thisrelaxation time scale is of one day when currents enter the domain,and 15days otherwise.

The vertical eddy coefficients are computed with the 1.5 tur-bulent kinetic energy (TKE) closure model (Blanke & Delecluse,

011 (4112 profiles).

Table 2Description of the different experiments of SMOS data assimilation.

Nudging coefficient Gain ratio map

FREE-Run – –

EXP0 5 × 10−6 NoEXP1 1.4 × 10−5 YesEXP2 5 × 10−6 Yes

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1993). The eddy-induced velocity parameterization is set to zero topermit the development of turbulence. The double diffusive mixingparameterization from Merryfield, Holloway, and Gargett (1999)modifies the vertical salinity and temperature diffusivities accordingto the potential presence of enhanced diapycnal mixing due to saltfingering or diffusive convection. A laplacian isopycnal diffusion isapplied for tracers (1000m2s−1), and a horizontal bilaplacian viscosityis used for momentum (−1.2 ·1011m4 s−1).

The model is forced with monthly fields of precipitation rate, cloudcover and humidity, and daily fields of wind stress, 10-m wind speedand 2-m air temperature. All forcing come from NCEP-NCAR (Kalnayet al., 1996). The heat and evaporation fluxes at ocean surface arederived from semi-empirical bulk formulae linking heat fluxes withthe state of ocean surface and lower atmosphere. These formulae followthe work of Oberhuber (1988) with some adjustments according to thecoupled large-scale ice ocean (CLIO) model parameterization (Goosse,1997; Mourre et al., 2008). There is no surface temperature or salinityrelaxation to climatological values when calculating surface heat andfreshwater fluxes in this configuration. Runoffs are included, withmonthly data from rivers Senegal, Sebou, Guadalquivir, Guadiana,Tagus and Douro taken from the Global Runoff Data Center (http://www.bafg.de/GRDC). A summary of the model configuration is givenin Table 1.

Initial conditions come from an eight years simulation (2002–2010)that nudged the model towards SST data coming from Reynolds andSmith (1994) and towards a 3D field of pseudo observations oftemperature and salinity. These pseudo observations were constructedby fitting multivariate empirical orthogonal functions of the model toArgo data. The nudging coefficient used to assimilate the SST data was5.10−6 s−1. The nudging coefficient of the temperature and salinity3D-fields was a function of depth, derived from a cross-validation ofthese fields towards a subset of Argo data withheld before fitting.

The numerical simulation carried out for the year 2011, and startingfrom the above described initial conditions, is called FREE-Run (seeTable 2).

4. The data assimilation

The main focus of this work is to test the ability of data assimilationto fill gaps in the SMOS-SSS products and to reduce observational noise.Themethod used here is nudging, used for the first time inmeteorologyby Anthes (1974). It was introduced in ocean data assimilation byVerron and Holland (1989) and Holland and Malanotte-Rizzoli (1989).The method proved to be able to reconstruct the sea surface circulationwhen altimetry data is being assimilated. Because of its easy imple-mentation and robustness, nudging was the first operational dataassimilation method used in oceanography (Lyne, Swinbank, & Birch,1982).

The basic implementation of nudging requires adding, in theprognostic equations of the model, a term that pull the model solution

Table 1Basic characteristics of the regional ocean simulation.

Physical domain

Boundaries 45°W–5°W, 15°N–44°NGrid size 128 × 100× 31Spatial resolution 1/3° (33 km at the equator)Time step 1800 s (48 time steps/day)

Parameterization

Horizontal turbulent diffusivity Laplacian, 300m2 s−1

Horizontal turbulent viscosity Bilaplacian, 1.2 × 10−11 m4 s−1

Deep vertical diffusion Laplacian, 1.0 × 10−6m2 s−1

Surface vertical diffusion Laplacian, 1.0 × 10−4m2 s−1

Vertical turbulent mixing TKE model

towards the observations. In an idealized one-variable model, thenudging may be applied as

∂x∂t ¼ F xð Þ þ μ xo−x

� �: ð1Þ

In this expression, x is the variable of interest, F(x) represents itsbehavioral law (i.e. the physics of the model), xo is the observed value,and μ is the nudging coefficient. Mathematical justification of nudging iseasily illustrated in the case of linear, perfect model and perfectobservations (generalization to imperfect observations is straight-forward). In this case, if xt represents the true state of the system, then

∂x∂t ¼ F xþ μ xt−x

� �;

∂xt

∂t ¼ F xt :ð2Þ

The difference between the two equations in Eq. (2) allows writingthe time-evolution equation of the error ε= x− xt:

∂ε∂t ¼ F−μð Þε: ð3Þ

Eq. (3) indicates that the error decreases exponentially as soon asthe nudging coefficient attains some threshold value. In Eq. (3), theerror decreases if μN F.

In matricial form, Eq. (1) can be generalized as:

∂x∂t ¼ F xð Þ þ K xo−Hx

� � ð4Þ

where the matrix K is a gain matrix projecting the informationcontained in the innovation vector (the part of the observation datanot accounted for by the model) to the model state variables. Thearray H is a p × n linear operator projecting the n-dimensional modelgrid points onto the p available observations (as, for example a matrixof ones and zeros to indicate if a grid point has been observed or not,respectively). In its simplest form, K= μHT corresponds to the use of aconstant nudging coefficient for all the observed points.

The various data assimilation experiments done here are describedin Table 2. A preliminary data assimilation experiment (called EXP0) isdone applying a spatially constant nudging coefficient (5 · 10−6 s−1)on the numerical model described in Section 3. Two additional nudgingexperiments, using a spatially dependent nudging coefficient, will bedescribed in Section 6.

5. Singularity analysis

Singularity analysis refers to any technique capable of assigning asingularity exponent to each point in a map of a given variable. Thesingularity exponent of a given point is a measure of the regularity orirregularity of the variable around that point, and it generalizes theusual concept of differentiability to the case in which no integerdifferential degree can be assigned, or even if the function is completelyirregular at that point (Turiel, Yahia, & Pérez-Vicente, 2008). Singularityexponents are dimensionless quantities and they are unaffected byglobal or local changes of amplitude. Following Turiel, Yahia et al.

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(2008), singularity exponents are derived from the wavelet projectionsof the gradient of the signal; so that, the value of singularity exponentsranges from a minimum of −1 (given by physical constraints) toinfinity, although in practice values beyond two are rarely observed.Thus, the singularity exponents associated to a scalar θ are computedby analyzing the projections of its gradient over an appropriate wavelet,ψ, at a given point, x, and for a given scale, r, namely:

Tψ ∇θj j x; rð Þ ¼Z

dy ∇θj j yð Þ 1r2

ψx−yr

� �: ð5Þ

The point x can be assigned a singularity exponent h(x) if thefollowing relation holds:

Tψ ∇θj j x; rð Þ ¼ α xð Þ rh xð Þ þ o rh xð Þ� �; ð6Þ

where the term o(rh(x)) means a termwhich is negligible in comparisonwith rh(x) when the scale r goes to zero.

Fig. 4. Global maps of Singularity Analysis exponents. On top, exponents from OSTIA SST dataspatial resolution is 0.25°.

The interest of the singularity exponents in oceanographic appli-cations of remote sensing arises from the fact that they are acharacteristic property of the flow (Turiel, Isern-Fontanet, García-Ladona, & Font, 2005). In fact, singularity exponents allow thedetermination of the streamlines of the flow (Turiel, Solé, et al., 2008;Turiel et al., 2009). It has been shown that different scalar variablessuch as SST and chlorophyll concentration have the same singularityexponents (Nieves et al., 2007). That means that singularity exponentsare connected to the particularities of the surface velocity field(currents) and are not specific to any scalar. In this sense, singularityexponents represent the part that all scalar fields share.

The resemblance between the structures in the singularity maps ofSST and SSS is discussed in Umbert et al. (2013) using the outputs ofhigh-resolution numerical simulations. Fig. 4 (top) shows the map ofthe singularity exponents of the global, daily OSTIA SST, while Fig. 4(bottom) presents the singularity exponents of the global, monthlyLevel 3 SMOS SSS, for December 2011 (1/4 degree resolution in bothcases). In contrast to the case of SST, the map of SMOS singularity

(daily). On bottom exponents from Level 3 SMOS SSS (monthly) for December 2011. The

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exponents appears unstructured; only some features associated to largeWestern boundary currents can be recognized. One reason for this lackof structure might be the overall failure of the retrieval salinityalgorithm. Other possible reason could be that the amplitude of noisein SMOS data is so large that the calculated singularity exponentscomprise large random errors, masking out the underlying physicalstructures. A useful tool to help answering this question is the use ofconditioned histograms.

A conditioned histogram is a 2D representation of the probability ofoccurrence of a given variable Y once a second variable X has beenobserved. If there is a functional relation between Y and X (namely,Y = f(X)) then the histogram of Y conditioned by X will have amaximum probability line around the curve Y=f(X), with a dispersiongiven by the noise on X and Y. In fact, all the columns in the conditionedhistogram would be identical except for a shift given by f(X). Whenthere is no single global functional relation between Y andX, but severallocal relations (i.e., Y= fi(X) valid on a given subdomain Ωi), multiplemaximum probability lines should be expected.

In Fig. 5 (top) the histogram of SMOS SSS conditioned by the valuesof OSTIA SST is shown. The line of maximum probability is formed byseveral segments with different slopes, indicating the existence oflocal relations between SSS and SST. On the contrary, the histogram of

Fig. 5. Top: Histogramofmonthly SMOS SSS conditioned by the values ofmonthly OSTIA SST (NHistogram of singularity exponents of monthly SMOS SSS conditioned by the values of singula

SSS singularity exponents conditioned by SST singularity exponents(Fig. 5 bottom) presents a single slope (which is saturated on theright hand, forming a horizontal line, due to the effect of noise).This means the existence of a unique, global functional relationbetween both types of singularity maps, similarly to what wasevidenced in Umbert et al. (2013). This maximum probability linehas a slope close to one in the left part of the histogram, indicatingthat singularities coming from SST and SSS coincide when the effectof noise is not too large. In Fig. 2 of Umbert et al. (2013) thedispersion around such a line is small in numerical models indicatinga low uncertainty level. In the case of SMOS and OSTIA data, theaverage dispersion around themaximumprobability line is significantlywider (0.15 for numerical models, but 0.25 for OSTIA-SMOS). Theseresults indicate that the SMOS salinity retrieval process is notresponsible for the lack of geophysical structure, but that the amountof noise is large enough to prevent the singularity structures to bevisually evident.

6. Results

Fig. 6 (top) displays the differences between SMOS data (monthlyaveraged) and Argo SSS. Salinity differences as large as 0.5 are found

ovember 2011, global scale). Brightest colors correspond to the largest probability. Bottom:rity exponents of monthly OSTIA SST.

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in different places over the domain. The difference median is −0.27,and the interquartile range (IQR, i.e. the difference between percentiles75% and 25%) is 0.50. The empty areas seen off the continental coastsreveal the filtering role of the SMOS QC process that removes dataretrievals with excessive noise. Contrarily of what could be expected,the amplitude of the differences between SMOS and Argo is not reducedtowards the west edge of the domain (i.e., getting away from thecontinents). This is due to the spurious presence of artificial RFI sourcesin most of the North Atlantic Ocean, not properly removed by the QCalgorithms. On the other hand, Fig. 6 (bottom) displays the differences

Fig. 6. Top: Difference between Level 3 SMOS SSS and Argo SSS. Bottom: Difference be

between the FREE-Run experiment (see Table 2) and Argo. Thedifference median is−0.05, and the IQR is 0.24 for all 2011. The largestdiscrepancies of the free simulation are found near the northwesternopen boundary. As stated in Section 3, the model is relaxed toclimatology at the open boundaries. Thus, the model is expected to failto reproduce the interannual variability there, especially in the North-western edge, where the Gulf-Stream intersects the domain. To avoidthe incompatibility between the abated variability of the model andthe interannual information present in the data, no assimilation ofobservations will be performed at or near the open boundaries. This is

tween FREE-Run SSS and Argo SSS. All 2011 match-ups are shown in these plots.

196 N. Hoareau et al. / Remote Sensing of Environment 146 (2014) 188–200

implemented by setting the nudging coefficient equal to zero at theboundary, and increasing it in a linear manner towards its nominalvalue over 18 grid points (i.e. 6°).

If the Argo SSS is considered to be the truth, the differences shown inFig. 6 can be interpreted as errors. Thus, these results imply that SMOSdata have a larger error than the straightforward model simulation ofthe region. Assimilation of such kind of observations would deterioratethe model output. However, remind that the objective of this workdiffers from the usual application of data assimilation. The purposehere is to use the numerical model as a dynamical interpolator of thedata being assimilated. In other words, the nudging is applied toincrease the geophysical coherence of the observations (SMOS) and toreduce their error.

In a preliminary set of nudging experiments, SMOS data wereassimilated using different nudging coefficients. The coefficient thatprovided the best results was μ0 = 5 ⋅ 10−6 s−1, valued that will beused in the experiment called EXP0 (see Table 2). However, inspectionof the results of those experiments (not shown here) displaysunrealistic patterns near the African Coast, as the model is being pulledtowards data contaminated with errors larger than 0.5.

To avoid these unrealistic features, and to explicitly take into accountthe non-homogeneity of SMOS errors (Fig. 5), a new set of dataassimilation experiments is done in which a space-dependent nudgingcoefficient will be used. The rationale to construct a spatially dependentnudging coefficient starts by replacing the gain matrix in Eq. (4) by theone from the Kalman Filter (Kalman & Bucy, 1960), i.e. K = PHT

[HPHT + R]−1, where P is the error covariance of the backgroundinformation (the numerical model) and R is the error covariance ofthe observations. Next, it is supposed that all error covariance matricesare diagonal (the error at one point is independent of the error at anyother point), and that the observation system is complete (i.e., H= I).In this case, the elements of the gain matrix become Ki = Pi/(Pi + Ri),where the index i refers to the i-th model grid point. Using thisapproach, the spatial dependency of the nudging coefficient can bederived from Fig. 6. For each model grid point where SMOS and Argodata are both available, the observational error variance (Ri) isestimated from the root mean square (RMS) of their difference. Onthe other hand, at every grid point visited by Argo, the backgrounderror variance (Pi) is estimated from the RMS of the difference betweenthe model (FREE-Run) and Argo. Next, wherever possible, thecorresponding gain coefficient Ki will be estimated from the error

Fig. 7. Map of the spatially dependent gain factor: P / (P + R), as described in the text. P corobservation error variance (SMOS).

variances. Finally, these estimates (calculated only at those grid pointswhere SMOS and Argo are both available) are optimally interpolatedto create a map for the entire domain. The longitudinal and latitudinaldecorrelation scales were chosen as 12 and 6° respectively. Theresulting map is plotted in Fig. 7, and it will be hereafter called thegain map. The main feature of the gain map is the rapid decrease ofthe gain near the coasts, reflecting the large observational errorvariance. The highest gains are found on the meridional edges of thesalty tongue. Various patches of low gain reflect large errors of SMOSin this region.

As already said, the next set of assimilation experiments will use thegain map to build a nudging coefficient that varies from point to point,according to the spatial dependency of the accuracy of the data beingassimilated. In EXP1, the nudging coefficient is μi1 = 1.4 ⋅ 10−5 Ki s−1

and in EXP2, the nudging coefficient is μi2 = μ0 ⋅ Ki. The EXP1 nudgingcoefficient has been chosen to ensure that μ0=⟨μi1⟩, where ⟨·⟩ indicatesthe spatial average.

The assimilation experiment outputs, for December 22 of 2011, areshown in Fig. 8. The amount of small structures in the resulting SSSfields differs between EXP1 (Fig. 8c) and EXP2 (Fig. 8d), in agreementwith the different values of nudging coefficients, as μi1 N μi2. Comparingthe salty tongue of SMOS (Fig. 2) and the one of the FREE-Run(Fig. 8a), it appears that SMOS is fresher than themodel. This differencein the salty tongue is captured by both EXP1 and EXP2. In EXP1, thesmall features present in SMOS (Fig. 2) pervade the solution. With aweaker nudging coefficient (EXP2) the data assimilation filters mostof that small-scale noise. On the other hand, most of the unrealisticpatterns present in EXP0 (Fig. 8b) are mitigated in EXP1 and EXP2,although some are still present in the former. This can be seen inFig. 9, depicting a zonal SSS transection at 26°N into the salty tongue.In this plot, the red and black lines represent the SMOS data and theFREE-Run simulation respectively. The gray line represents the EXP0,while blue and green lines represent EXP1 and EXP2 respectively. Thegray line follows the unrealistic SMOS data in the eastern part of thebasin. The model and the SMOS data diverge as they approach thecontinental land, and the blue and green lines are less responsive tothe SMOS data. In the western part of the transect, the impact of theassimilation is clear: the resulting surface salinity moves away fromthe FREE-Run and gets closer to the SMOS data. Even in this region,the SMOS data appears to be too noisy with large spatial oscillations.On the other side, the FREE-Run surface salinity displays too little spatial

responds to the background error variance (FREE-Run simulation) and R corresponds to

Fig. 8. Snapshots of SSS corresponding to December 22, 2011: FREE-Run (a), EXP0 (b), EXP1 (c) and EXP2 (d).

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variability as compared with the Argo data (black dots). The salinityfrom the assimilation experiments demonstrates the filtering role ofthe data assimilation by strongly reducing the amplitude of the highwave-number oscillations present in the SMOS product. Moreover, theassimilation salinity displays a spatial modulation that was missingfrom the FREE Run, but present in Argo.

Fig. 10 shows the RMS against SMOS and Argo. In the top plots theRMS is calculated over the whole domain. In the bottom plots the RMS

Fig. 9. Plot of the SSS along transect at 26°N for December 22 of 2011.

is calculated over the red box shown in Fig. 7, i.e. the region where theSMOS data are less noisy. The black line corresponds to the FREE-Runsimulation and is the most different from SMOS. As expected, theassimilation solutions become closer to the data being assimilated.This indicates that, on average, the nudging approach is not rejectingthe SMOS data, despite its large error. To assess if SMOS is providingany useful information to themodel, the solutions are compared againstindependent Argo data. In general, the model is the closest to Argo,especially when the statistics are calculated over the whole region. Buteven in this case, the EXP2 salinity is as good as the FREE run(Fig. 10c). That is, the assimilation does make the model to becomecloser to SMOS without taking away the compatibility with Argo.However, it must be remembered that the noise of SMOS in the wholeregion is large. When focusing in the region where SMOS has thesmallest noise, the assimilation results eventually come closer to Argothan the FREE Run (since the month of May). Notice that afterSeptember, the error of the data assimilation (with respect to Argo)slightly increases. This is due to the increase of the error in SMOS datasince July.

From the various assimilation experiments, the one being the morerobust with respect to the SMOS noise is EXP2, which remains slightlybetter than the FREE-Run until December. This demonstrates the needto use a spatially dependent nudging coefficient in the case of datacontaminated with heterogeneous error. The benefit of using nudgingof SMOS data into a numerical model to potentially create a Level 4SSS product is justified by Fig. 10b and d. They show that theassimilation results get closer to both SMOS and Argo data as soon asthe error in SMOS is small enough. Finally, the amplitude of the nudgingcoefficient does provide a control about the degree of smoothness of the

Fig. 10. Time series of RMS between SSS fields from the numerical experiments and the Level 3 SMOS data (a,b) and against Argo (c,d). On top (a, c) the RMS is calculated over the wholedomain. On bottom (b,d), the RMS is calculated over the red box shown in Fig. 7.

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Level 4 product of SSS by giving more or less weight to the backgroundnumerical simulation.

Moreover, an added value of nudging in generating Level 4 maps ofSSS can be seen in Fig. 11, which displays maps of the singularityexponents for the FREE-Run (Fig. 11a), the SMOS data (Fig. 11b), andthe EXP2 (Fig. 11c). The similarities between the singularity structuresof the original simulation (FREE-Run) and those of the OSTIA data(Fig. 4, Top) indicate that the dynamical core of themodel does simulatethe geophysical coherence present in the ocean. This justifies using thenumerical model as a dynamical interpolator to fill the voids in SMOSdata. On the other hand, the lack of coherent spatial structures inSMOS (Figs. 4 and 11b) might seem inconsistent with the fact thatSMOS data have a singularity spectrum close to the one of OSTIA(cyan line) or the numerical simulations (Fig. 11d). The singularityspectrum D(h) is a scale-invariant function representing the dis-tribution of the singularity exponents and is linked to the cascade ofscalar dissipation (Turiel, Yahia, et al., 2008). The right part of thespectrum is linked to the less singular values (positive), which areshown in black in themaps of the singularity exponents. All simulationsdiffer in that part of the spectrum because the distribution of the lesssingular values is highly dependent on how boundaries and land–sea

masks are treated. But, the differences among the different experimentsin this part of the spectrum are of the same order of the error bars. Thesingularity spectrum of OSTIA SST presents a small but significantdeviation (from 0.1 to 0.5). Comparison with other sources of remotesensing of SST of lower time or space resolution (MODIS, AMSR-E,Pathfinder) indicates that the problem may come from the way inwhich OSTIA is spatially interpolated from their source data. We haveanyway kept this data set, as it is the one presently being used forproducing L4 at BEC. The left part of the spectrum corresponds to themost singular values (negative, i.e. the brightest values in the maps ofsingularity exponents). These values are linked to the abrupt changesin the flow such as fronts or filaments. In this part of the spectrum,only the model differs from the spectrum of global SST data fromOSTIA. And the difference is significant when compared to the error bar.

Notice that the singularity spectrum of SMOS SSS is in goodagreement with OSTIA SST spectra in the left part. This indicates that,despite the presence of noise, singularity exponents derived fromSMOS Level 3 describe the cascade of salinity dissipation accurately. Inother words, the signal noise has no effect on salinity dynamics (itdoes not change salinity dissipation) but it is large enough to destroythe salinity front's coherence (and hence the geometry of singular

Fig. 11. Singularity analysis exponent maps of the SSS in the Macaronesian Region: FREE-Run (a), SMOS (b) and EXP2 (c). Panel (d) shows the corresponding singularity spectra.

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lines is lost as discussed in Section 5). After assimilation (Fig. 11c), thespatial geometry of the singularity lines associated to the resulting SSSis close to the one of the model, i.e. more geometrically satisfactorythan the one of Level 3 SMOS. But, at the same time, the singularityspectrum of assimilation gets closer to the one of SMOS in the leftpart, indicating that the frontal structures after assimilation are betterrepresented than in the FREE-run. That is, by modifying the singularityspectrum of the numerical model, data assimilation experiments havebeen able to extract useful information from the observations.

7. Summary

The ability of using a simple data assimilation methodology, as theNewtonian Relaxation, to generate Level 4 SSS maps from noisy andbiased Level 3 estimates has been investigated here. The region understudy is the Northeast subtropical Atlantic gyre, where a regionalocean simulation is available. Data assimilation of remotely sensedsalinity in this region is difficult due to the large errors present in

the SSS retrievals from the SMOS brightness temperature imagereconstructions. The error IQR of SMOS is 0.50, while the error IQR ofthe FREE-Run simulation is 0.24 (Fig. 6). The plots in Fig. 6 also showthat the SMOS data has a large bias in comparison with the model.The reasons of such large error amplitude are the vicinity to the coastand the presence of artificial and natural RFI. At this stage the qualitycontrol algorithms currently implemented in the SMOS processingchain fail to filter out some inaccurate estimates of SSS (at least in theregion under study).

The goal of Level 4 products of SSS is to create SSS maps thatinterpolate data from data-rich to unobserved regions, containing lessnoise, and having a reasonable geophysical coherence. This is of specialimportance in the case of SMOS data. The lack of spatial structure in theSMOS singularities (Figs. 4 and 11b), makes necessary to devisealgorithms to restore it as in Umbert et al. (2013), or through dataassimilation as proposed here. In the case of using data assimilation tocreate L4 maps, the physical coherence will come from the numericalmodel.

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Despite its large observational error, assimilation ofmonthly-binnedSMOS data (1/4degree resolution) has fulfilled the demanded requestsas soon as a spatially dependent nudging coefficient (weighting theimpact of the observations as a function of the ratio between themodel and observational error variance) is used. The spatial depen-dency has been constructed by comparing the SMOS data and themodel outputs with in-situ estimates of SSS from Argo.

The results have shown that the data assimilation is able to pull themodel towards the SMOS binned data, while producing SSSmaps closerto the in-situ data than the original SMOSobservations (i.e. reducing theamount of noise in the data). Moreover, the data assimilation infusedthe resulting fields with the dynamical geophysical links present inthe numerical model (i.e. playing the role of dynamical interpolation)as shown by the singularity analysis. Finally, the degree of smoothnessof the resulting data may be regulated via the amplitude of the nudgingcoefficient.

The results prove that NewtonianRelaxation has been able to extractuseful information from the SMOS data and has the potential to be usedto generate Level 4 products. The simplicity of implementation, therobustness, and low cost, makes this technique well suited for itsapplication in Basin-wide or global numerical simulations.

Acknowledgments

This study is supported by the Spanish Ministry of Economy andCompetitiveness (MINECO) by means of the R&D projects MIDAS-6(Ref. AYA2010-22062-C05) and MIDAS-7 (Ref. AYA2012-39356-C05-03). M. Umbert recognizes support by the Spanish MEC through theFPI grant program. The authors want to thank A. Alvera (Guest Editor)and the anonymous reviewers for the discussion and their commentsthat improved the manuscript.

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