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Data-driven Models for the Spatio-Temporal

Interpolation of satellite-derived SST Fields

Ronan Fablet, Senior Member, IEEE, Phi Huynh Viet and Redouane Lguensat, Student Member, IEEE

Abstract

Satellite-derived products are of key importance for the high-resolution monitoring of the ocean surface on a

global scale. Due to the sensitivity of spaceborne sensors to the atmospheric conditions as well as the associated

spatio-temporal sampling, ocean remote sensing data may be subject to high-missing data rates. The spatio-temporal

interpolation of these data remains a key challenge to deliver L4 gridded products to end-users. Whereas opera-

tional products mostly rely on model-driven approaches, especially optimal interpolation based on Gaussian process

priors, the availability of large-scale observation and simulation datasets calls for the development of novel data-

driven models. This study investigates such models. We extend the recently introduced analog data assimilation to

high-dimensional spatio-temporal ﬁelds using a multi-scale patch-based decomposition. Using an Observing System

Simulation Experiment (OSSE) for sea surface temperature, we demonstrate the relevance of the proposed data-driven

scheme for the real missing data patterns of the high-resolution infrared METOP sensor. It has resulted in a signiﬁcant

improvement w.r.t. state-of-the-art techniques in terms of interpolation error (about 50% of relative gain) and spectral

characteristics for horizontal scales smaller than 100km. We further discuss the key features and parameterizations

of the proposed data-driven approach as well as its relevance with respect to classical interpolation techniques.

Index Terms

Ocean remtote sensing data, data assimilation, optimal interpolation, analog and exemplar-based models, multi-

scale decomposition, patch-based representation

I. INTRODUCTION

Satellite-derived products are of key importance for the high-resolution monitoring of the ocean surface on a

global scale. A variety of sensors record observations of geophysical parameters, such as Sea Surface Temperature

(SST) [1], Sea Surface Height (SSH) [2], Ocean Color [3], Sea surface Salinity (SSS) [4], etc. In all cases, the

delivery of L4 gridded products for end-users involves a number of pre-processing steps from the L1 data acquired

and transmitted by spaceborne sensors. Due to both the space-time sampling geometry of satellite sensors and their

sensitivity to the atmospheric conditions (e.g., rains, aerosols, clouds), ocean remote sensing data may involve very

large missing data rates as illustrated in Fig.3. Hence, spatio-temporal interpolation is of key importance to deliver

gap-free gridded sea surface ﬁelds for further analysis.

R. Fablet, P.H. Viet and R. Lguensat are with IMT Atlantique; Lab-STICC, Brest, France, e-mail: ronan.fablet@telecom-bretagne.eu

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Optimal interpolation is certainly the state-of-the-art approach for the spatio-temporal interpolation of satellite-

derived sea surface geophysical ﬁelds [5], [6]. Optimal interpolation relies on the modeling of the covariance of the

considered spatio-temporal ﬁelds. The choice of the covariance model is a critical step [7], [6], [8], [9]. Stationary

covariance hypotheses are generally considered, though they might not be veriﬁed. For instance, frontal areas as

illustrated in Fig.5 may involve time-varying and space-varying anisotropical features. In such cases, considering

mean covariance model typically results in the smoothing out of the ﬁne-scale SST details. Data assimilation

techniques for missing data interpolation may be regarded as another important category of model-driven approaches

([10], [8], [11]). A critical aspect of their implementation lies in the choice of the dynamical model, more precisely

the trade-off between its computational complexity and its ability to correctly represent real sea surface dynamics.

The tremendous amount of satellite observation data pouring from space, along with the wider availability of

reanalysis and/or numerical simulation datasets supports the development of data-driven approaches as an alternative

to model-driven schemes. In this respect, statistical and machine learning models offer new computational means

to account for space-time variabilities that cannot be completely captured by simpliﬁed physical models. The

application of Principal Component Analysis (PCA), also referred to as Empirical Orthogonal Functions (EOF) in

the geoscience ﬁeld, to remote sensing missing data interpolation [12], [13] may be regarded as an example of

such data-driven schemes, though it proves mainly relevant for large-scale variabilities [13]. One may also cite the

development of exemplar-based models in image processing and their applications to missing data interpolation for

single-date remote sensing data [14], [15].

In this study, we investigate such data-driven and exemplar-based models for the spatio-temporal interpolation

of missing data in ocean remote sensing time series. We aim to exploit the implicit knowledge conveyed by

available multi-annual satellite-derived datasets to improve the interpolation of high-resolution spatio-temporal sea

surface geophysical ﬁelds. We rely on analog data assimilation [9], [16] and develop, to our knowledge, the ﬁrst

application of analog data assimilation to high-dimensional spatio-temporal ﬁelds. Our methodological contributions

lie in the introduction of a multiscale analog data assimilation applied to local patch-based and PCA-constrained

representations. We demonstrate the relevance of the proposed scheme through an application to SST time series.

We report signiﬁcant gain compared to state-of-the-art approaches, namely optimal interpolation [17], [7] and

PCA-based interpolation [12], [13].

This paper is organized as follows. Section II reviews the related work. Section III presents the proposed multiscale

analog data assimilation scheme. Numerical experiments are reported in Section IV and we further discuss our key

contributions and future work in Section V.

II. PRO BL EM S TATEM EN T AN D RE LATE D WO RK

A. Model-driven approaches

As previously mentioned, model-driven approaches are the state-of-the-art techniques for the spatio-temporal

interpolation of missing data in ocean remote sensing observations [6], [11]. In particular, optimal interpolation

relates to the following formulation:

X∝ G(Xb,Γ) (1)

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Y(t, s) = X(t, s) + (t, s),∀s∈Ωt(2)

where G(Xb,Γ) is a spatio-temporal Gaussian ﬁeld with mean background ﬁeld Xband covariance function Γ, and

the observation noise assumed to be Gaussian. Ωtrefers to the region domain for which observations are truly

available at time t. Given a series of observation ﬁelds Yand a known covariance function Γ, optimal interpolation

leads to an analytical MAP (Maximum A Posteriori) solution for ﬁeld X, equivalent to the minimization of a

reweighted least-square criterion w.r.t. the covariance of noise . The choice of the covariance function Γis a

critical step. Exponential and Gaussian covariance models [8], [7] are the most classical choices with both constant

parameters as well as space-time-varying parameterization [18].

When dealing with high-dimensional ﬁelds, such as ocean remote sensing observations, the numerical computation

of the solution of the optimal interpolation may not be feasible, as it involves the inversion of a very large covariance

matrix. Sequential approaches, such as ensemble Kalman techniques [11], are then considered. They may be restated

as data assimilation formulations. Considering a discrete setting, they amount to the following model for ﬁeld X:

X(t) = M(X(t−1), η(t−1)) (3)

where Mis referred to as the dynamical model and ηis a random perturbation. Model (1) may be restated according

to this formulation with a linear model Mand a Gaussian process ηderived from the considered Gaussian ﬁeld

with covariance Γ. Other parameterizations of the dynamical model may be derived from ﬂuid dynamics equations,

including for instance advection-diffusion models [10]. Ensemble Kalman schemes [11] are the state-of-the-art

techniques to numerically solve for the reconstruction of spatio-temporal ﬁeld Xgiven partial observation ﬁeld Y

under model (3). Using a sample-based representation of Gaussian distributions, they provide forward-backward

ﬁltering schemes to approximate the optimal interpolation solution. We let the reader refer to [11] and reference

therein for additional details on stochastic data assimilation. We may also point out variational data assimilation

[19], [10], which exploits a continuous formulations of Model (3) and involves a gradient-based minimization of

the observation error under model (3).

A typical example of the optimal interpolation of an SST ﬁeld from a series of partial observations is reported in

Fig.3. An important limitation of model-driven approaches lies in modeling uncertainties. Due to the autocorrelation

structure of sea surface geophysical structures and the observation sampling rate, optimal interpolation results to

accurate reconstruction of the spatio-temporal ﬁelds for spatial scales larger than 100km. However, ﬁner scales are

signiﬁcantly ﬁltered out (see Fig.3). This property directly relates to the correlation length of the covariance model

(here, 100km). This correlation is a trade-off between the spatial resolution of the observation ﬁelds (here, 5km)

and the size of the gaps.

As detailed below, we explore data-driven approaches to take advantage of available observation or simulation

datasets with a view to improving the reconstruction of the ﬁne-scale structures of sea surface ﬁelds.

B. Data-driven approaches

With the increasing availability of representative observation datasets, data-driven models become more and more

appealing to solve inverse image problems, including missing data interpolation. Initially mostly investigated for

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computer vision and computer graphics applications, such as synthesis, inpainting and super-resolution issues [20],

[21], they have also gained interest for applications to remote sensing data [14], [15]. Patch-based and exemplar-

based models have emerged as powerful representations to project images onto large sets of patch exemplars

and/or dictionaries. Non-local means and non-local priors [22], [23] are state-of-the-art examples of such models

for image reconstruction issues. Developments for multivariate time series have also recently been investigated,

especially exemplar-driven data assimilation referred to in the geoscience ﬁeld as analog data assimilation [9],

[16], [24]. Two main features make the direct application of these exemplar-based strategies to spatiotemporal

ﬁelds poorly efﬁcient: their computational complexity and their ability to jointly capture large-scale and ﬁne-scale

structures. Patch-based techniques generally involve small image patches (typically, from 3x3 to 11x11 patches for

2D images), which cannot resolve large structures, with a typical scale greater than the width of the patches. In

addition, the considered minimization schemes involve repeated iterations over the entire set of exemplars, which

may make them extremely computationally-demanding for applications to spatio-temporal data. By contrast, analog

data assimilation provides an efﬁcient sequential scheme, but remains limited to relatively low-dimensional space

(up to a few tens of dimensions in [9], [16]).

PCA-based models are popular in the geoscience ﬁeld. They have also gained interest for application to missing

data interpolation, especially DINEOF approaches [12], [13]. These involve two key steps: i) the estimation of basis

functions, which provide a lower-dimensional representation of the variability spanned by the considered spatial or

spatio-temporal data, ii) the interpolation of the missing data from projections onto the basis functions. VE-DINEOF

[13] has recently improved compared to the original DINEOF scheme [12]. In both cases, applications to ocean

remote sensing data, especially SST, were considered. Applied on a global or regional scale, the lower-dimensional

PCA-based representation is mostly relevant to recover large-scale structures and not as appropriate to reconstruct

ﬁne-scale details. Overall, PCA-based decompositions are regarded as relevant representations to encode the spatial

patterns exhibited by geophysical ﬁelds. It may be noted that PCA representations are also often used in patch-based

image processing (see for instance [28], [25]).

III. PROP OS ED DATA-DRIVEN MODEL

From the above review of the related work, a key issue in developing a data-driven framework is the decomposition

and representation of the spatio-temporal variabilities of the considered ﬁelds using appropriate models both in terms

of computational efﬁciency and modeling uncertainties, especially for ﬁne-scale details. This naturally calls for a

multi-scale representation. Formally, we considered a three-scale model where ﬁeld Xwas decomposed as follows:

X=¯

X+dX1+dX2+ξ(4)

where ¯

Xrefers to the large-scale (low-frequency) component of X,dX1,2to details at two scales and ξto unresolved

scales. By convention, dX1refers to the intermediate scale and dX2to the ﬁnest scale. The spatio-temporal

interpolation of ﬁeld Xresults in the estimation of components ¯

Xand dX1,2in (4) given Observation Model (2).

Following Formulation (3), this led to the deﬁnition of relevant dynamical priors for each component. For large-

scale component ¯

X, a model-driven representation associated to optimal interpolation naturally arises as a relevant

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representation given its efﬁciency in recovering large-scale structure. We thus assume that ¯

Xis a spatio-temporal

Gaussian process with mean ¯

Xband spatio-temporal covariance Γ. Here, ¯

Xbis given by the climatological mean

over the entire times series and we therefore considered a Gaussian covariance model. We let the reader refer to

[26], [17], [27] for additional details on optimal interpolation using a Gaussian prior for Γ. By contrast, for detail

components dX1,2, we introduced data-driven spatio-temporal priors as detailed in the next section.

A. Multi-scale data-driven priors

The deﬁnition of detail ﬁelds dX1,2combines patch-based and PCA-based representations. For scale i= 1 or 2,

let us consider Pi×Pipatches, such that P1> P2(typically P1= 40 and P2= 20). We proceed as follows for

the scale i= 1. Given ¯

Xin multi-scale decomposition (4), each P1×P1patch of detail ﬁeld dX1is given by the

projection of the associated patch for residual ﬁeld X−¯

Xonto a low-dimensional PCA decomposition. This PCA

decomposition is learnt from P1×P1patches of a training dataset of residual ﬁelds X−¯

X. We apply the same

procedure for detail ﬁeld dX2from residual ﬁeld X−¯

X−dX1. Formally, this leads to the following deﬁnition of

detail ﬁelds dX1,2:

dX1=P1X−¯

X

dX2=P2X−¯

X−dX1(5)

where P1,2are patch-based PCA image projection operators [28], [25]. They result in the decomposition of any

patch Psaround point sat time tof detail ﬁeld dXias a linear combination of the principal components of the

PCA for scale i:

dXi(Ps, t) =

Ki

X

k=1

αi,k(s, t)Bi,k (6)

with Bi,k the kth principal component of the PCA at scale iand αi,k(s, t)the associated coefﬁcient for patch Ps

at time t.NP CA,i refers to the number of vectors of the PCA basis at scale i. The spectral properties of PCA

decompositions along with the lower patch size at scale i= 2,i.e. P1> P2, lead to a scale-space decomposition

[31]. Contrary to a wavelet decomposition, we only implicitly set the considered scale ranges through the number

of principal components kept at each scale. The key interest here is a local adaption with point-speciﬁc PCA bases

which can also account for any image geometry (e.g., the presence of land points in the considered region).

Given these deﬁnitions for detail ﬁelds dX1,2, we considered an analog (data-driven) formulation of the associated

dynamical models (3). As stated in Section II, analog dynamical models introduced in [9], [24] do not directly apply

to high-dimensional ﬁelds and we considered patch-based models. We ﬁrst assumed that we were provided with

representative catalogs C1,2of patch exemplars of the dynamics of details ﬁelds dX1,2. Each catalog is composed

of a set of patch exemplars {dXi(Psk, tk)}k, referred to hereafter as analogs, and of their temporal successors

{dXi(Psk, tk+ 1)}k. For a given patch Psand scale i, the deﬁnition of the analog dynamical model leads to the

deﬁnition of an exemplar-driven sampling strategy for the distribution of the state at time t,dXi(Ps, t), conditionally

to the state at time t−1,dXi(Ps, t −1). Formally, we considered Gaussian conditional distributions of the form

dXi(Ps, t)|dXi(Ps, t −1) = u∝ G (µi(u, Ci),Σ (u, Ci)) (7)

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where G(·)is a Gaussian distribution. Mean µi(u, Ci)is deﬁned as a weighted function of the successors of the

Knearest-neighbor of uin catalog Ci. Similarly, covariance Σ (u, Ci)is issued from the weighted covariance of

the successors of the Knearest neighbors. Theses weights and the nearest-neighbor search involve a predeﬁned

kernel Kas detailed below. Let us denote by (Ak(u),Sk(u)) the analog-successor pair of the kth nearest-neighbor

to uin Ci. Following [9], [24], we investigate three different analog dynamical models corresponding to different

parameterizations of the above mean and covariance:

•Locally-constant analog model: mean µi(u, Ci)and covariance Σ (u, Ci)are given by the weighted mean

and covariance of the Ksuccessors {Sk(u)}k.

•Locally-incremental analog model: it proceeds similarly to the locally-constant analog model, but for the

differences between the successors and the analogs, such that mean µi(u, Ci)is given by the sum of uand of

the weighted mean of the Kdifferences {Sk(u)− Ak(u)}k.Σ (u, Ci)results in the weighted covariance of

these differences.

•Locally-linear analog model: given the K analog-successor pairs {Ak(u),Sk(u)}k, it ﬁrst comes to the

weighted least-square estimation of the linear regression of the state at time tgiven the state at time t−1.

Denoting by Ai(s, t)the estimated local linear operator, mean µi(u, Ci)is given by Ai(s, t)·dXi(Ps, t −1)

and covariance Σ (u, Ci)by the weighted covariance of the residuals of the ﬁtted linear regression.

KG(u(t), v(t)) = exp −ku(t)−v(t)k2

σ,(8)

and a cone kernel KC, recently introduced for dynamical systems in [30]. For any pair of states u(t), v(t), it leads

to

KC(u(t), v(t)) = exp −Lζ(u(t), v(t))

σ(9)

Lν(u, v) = kω(t)k21−ζcos2θ1−ζcos2φ1/2

k∂tu(t)kk∂tv(t)k(10)

where ω(t) = u(t)−v(t),∂tu(t) = u(t)−u(t−1),∂tv(t) = u(t)−u(t−1),cosθ =hω(t), du(t)iand

cosφ =hω(t), dv(t)i. Compared to a classical Gaussian kernel, the cone kernel takes into account not only the

distance between the two states, but also the alignment of their instantaneous velocities with the difference between

the two states. It has been shown in [30] that the cone kernel may be more appropriate for analog forecasting

schemes. For the Gaussian (resp. cone) kernels, scale parameter σis locally-adapted to the median value of the

distances ku(t)−v(t)k2(resp. Lν(u(t), v(t))) to the K nearest neighbors in the catalogs of exemplars. Parameter

νis set empirically between 0 and 1. In all cases, we take advantage of the considered PCA-based representation

of the patches to compute patch similarities within the associated low-dimensional spaces, and not in the original

patch space.

B. Numerical resolution

Given the data-driven priors M1,2, we proceed to the resolution of model (4) according to a MAP criterion.

We might consider a direct discrete gradient-based numerical resolution as the considered parameterization for

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model (4) can be regarded as a spatio-temporal Markov Random ﬁeld [14], [29]. This would however lead to

an extremely-demanding computational scheme. We prefered to exploit the multi-scale nature of our model to

develop a coarse-to-ﬁne strategy and cast the global minimization problem as series of smaller problems, which

can be solved more efﬁciently. More precisely, we proceeded as follows. We ﬁrst solved for the reconstruction of

large-scale component ¯

Xusing optimal interpolation with covariance model Γ. We then successively solved for

the reconstruction of detail ﬁelds dX1,2. This step runs independent resolution along the temporal dimension for

each patch position using sequential data assimilation algorithms, namely an analog Ensemble Kalman Smoother

(AnEnKS) and an HMM-based analog smoother (AnHMM). We refer the reader to [24] for additional details on the

implementation of the analog EnKS. The independent solutions computed for each patch position were recombined

using averaging. To reduce the computational complexity, we did not process all possible patch positions, but only

overlapping patches with a 35x35 (resp. 15x15) spatial sampling for P1×P1patches (resp. P2×P2). To remove

potential block artifacts, we apply a PCA-based ﬁltering step onto 10 ×10 patches. As initialization for the analog

data assimilation iterations, we use a VE-DINEOF solution [13].

All implementations were run under Python. We used [5] for optimal interpolation, and Analog Data Assimilation

toolbox [24]. The Python code of the proposed patch-based and multi-scale analog data assimilation is made available

(github.com/rfablet/PB ANDA).

IV. RES ULT S

A. Experimental setting

Considered case-study: To perform a qualitative and quantitative evaluation of the proposed framework, we used

a reference gap-free L4 SST time series from which we create a SST with missing data using real missing data

masks. As reference SST, we used OSTIA product delivered daily by the UK Met Ofﬁce[6] with a 0.05◦spatial

resolution (approx. 5km) from January 2007 to April 2016. The OSTIA analysis combines satellite data provided

by infrared sensors (AVHRR, AATSR, SEVIRI), microwave sensors (AMSRE, TMI) and in situ data from drifting

and moored buoys. For the missing data mask series, we studied an infrared sensor, more speciﬁcally METOP,

which may involve very high missing data rates as illustrated in Fig.3 & 5.

As a case-study region, we selected an area off South Africa. This highly dynamic ocean region involves complex

ﬁne-scale SST structures (e.g., ﬁlaments, fronts) as shown in Fig.3. Our evaluation focused on the interpolation of

the SST ﬁelds for year 2015, other years being used to build a catalog of exemplars for the analog frameworks. The

Python code used for the creation of the considered SST data is available (github.com/rfablet/SSTData TCI rfablet).

Parameter setting of the proposed approaches: We performed interpolation experiments with both AnHMM

and AnEnKF/KS schemes (see Section III for details). We exploited a three-scale model: the global scale (entire

region), 40x40 patches and 20x20 patches. At each scale, each patch was encoded by its PCA-based decomposition

using a 10-component PCA. As initialization for missing data areas, we used an optimal interpolation on the

global scale. The parameterizations of the optimal interpolation and of the DINEOF scheme were those used for

comparison purposes as detailed below. In the analog setting, the number of neighbors was varied from 10 to 110

and we compared Gaussian and Cone kernels.

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Comparison to state-of-the-art approaches: For comparison purposes, we consider an optimal interpolation,

which is the interpolation technique used in most operational products (e.g., [6]), VE-DINEOF [13], a PCA-based

technique, and a direct region-level application of the analog data assimilation. Their parameter settings were as

follows:

•Optimal interpolation (OI): we used a Gaussian kernel with a spatial correlation length of 100km and a temporal

correlation length of 3 days. These parameters were empirically tuned for the considered dataset using a cross-

validation experiment. We used the optimal interpolation package from [5]. The considered parameter setting

was consistent with previous work [6], [8] and stressed the strong temporal correlation of SST ﬁeld [8]. In

our case-study, a direct implementation of the OI would have required a large memory: for a missing data

rate of ∼70%, the interpolation onto the considered 300 ×600 grid would have required the inversion of a

system of 5T.104equations with T the temporal correlation. Given the considered spatial correlation length

of 100km, we achieved an optimal interpolation onto a coarser grid with a resolution of 25km and applied a

bicubic interpolation onto the targeted high-resolution grid (5km resolution).

•VE-DINEOF interpolation: we exploited a direct implementation of VE-DINEOF scheme [13] on the regional

scale using 200 PCA components, which amounted to 99.27% of the total variance of the dataset. This

VE-DINEOF setting is referred to as G-VE-DINEOF. We also considered a multi-scale version of the VE-

DINEOF procedure using the same three-scale decomposition as the multi-scale analog data assimilation. As

for MS-AnEnKS and MS-AnHMM, we used two detail components corresponding to 40x40 patches and 20x20

patches. At each scale, i.e. the coarse region scale and the two detail scale, we exploited 10-dimensional PCA

decomposition (NP CA,1=NP C A,2= 10). The resolution of this multi-scale VE-DINEOF, referred to as MS-

VE-DINEOF, applies a coarse-to-ﬁne strategy, such that at each scale, the VE-DINEOF iteratively updated the

missing data area from the projection of overlapping patches onto the 10-dimensional PCA basis;

•Global AnEnKS interpolation: to evaluate the relevance of the proposed multi-scale decomposition, we tested

a direct application of the AnEnKS at the region scale, referred to as G-AnEnKS. Similarly to G-VE-DINEOF,

we considered 200 PCA components, which amounted to 99.27% of the total variance of the dataset. From

numerical experiments, the best parameter setting combined a locally-incremental analog forecasting with

K= 100 neighbors and a Gaussian kernel.

It may be noted that variational interpolation techniques, based on the minimization of regularization norms [32],

cannot be expected to lead to relevant results given the large missing data rates in the considered dataset (above

70% on average) and were not considered in our experiments.

Qualitative and quantitative evaluation: to assess the quality of the different interpolation schemes, we ﬁrst

achieved a quantitative analysis according to mean square error (MSE) statistics for the SST reconstructed SST

ﬁelds, the associated gradient ﬁelds, and the detail ﬁelds of a 4-scale dyadic wavelet decomposition of the SST ﬁelds.

We also computed radially-averaged power spectral densities to analyze the ﬁne-scale patterns of the reconstructed

ﬁeld. In addition, we performed a qualitative analysis of these ﬁelds with a focus on the reconstruction of ﬁne-scale

structures.

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B. Interpolation performance

We shall begin with the results of our numerical experiments. We ﬁrst present the quantitative evaluation of

interpolation performance, including a comparison to state-of-the-approaches. Second, we further illustrate this

performance using interpolation examples. Third, we report a sensitivity analysis of the best analog assimilation

setting. We also include an evaluation of interpolation performance when the creation of the catalogs of exemplars

involve observation datasets with missing data.

Quantitative comparison to state-of-the-art approaches: We ﬁrst report the overall MSE statistics of the

considered interpolation approaches, namely OI, G-VE-DINEOF, MS-VE-DINEOF, MS-AnHMM, G-AnEnKS and

MS-AnEnKS, in Tab.I. The multi-scale Analog schemes are a clear improvement over the OI and VE-DINEOF

reconstruction, with a relative gain in SST RMSE up to 50% for MS-AnEnKS at the ﬁnest scale (dX2). MS-

AnHMM also leads to a signiﬁcant improvement but is clearly outperformed by MS-AnEnKS. It may be noted

that the direct application of the analog data assimilation, G-AnEnKS, to ﬁeld Xdoes not lead to very signiﬁcant

improvement. This is regarded as a direct beneﬁt of the multi-scale decomposition, which greatly increases the

representativity of the collected catalogs of exemplars. No such difference is reported for the application of global

and multi-scale VE-DINEOF schemes, which further stresses the relevance of the analog dynamical prior exploited

by MS-AnEnKS. The analysis of the MSE statistics at different scales of a dyadic wavelet decomposition indicates

that the improvement mainly refers to the third and fourth dyadic scales (i.e., spatial scales greater than 20km).

Most of the improvement is brought about by the resolution of component dX1(about 40% of relative gain w.r.t.

OI), when component dX2accounts for about 10% of relative gain w.r.t. OI. The MSE time series (Fig.1) lead

to similar observations. Interestingly, AnEnKS depicts a lower time variability of the MSE compared to OI and

VE-DINEOF (standard deviation of 0.06 vs. 0.13), the later being more sensitive to larger missing data rates. This

is viewed as a beneﬁt of the exemplar-based time regularization conveyed by the analog framework.

Qualitative analysis of interpolation results from examples: To complement this global analysis, we report

interpolation results for two dates, corresponding to relatively low (∼60%) and greater (∼90%) missing data

rates, respectively in Fig. 5 and Fig. 3. For these two examples, we visually compare OI, MS-VE-DINEOF and

MS-AnEnKS interpolations to the groundtruth both for the SST ﬁeld and the gradient magnitude ﬁelds. In Fig.3,

MS-AnEnKS clearly outperforms OI and MS-VE-DINEOF (SST (resp. SST gradient) rmse of 0.20 (resp. 0.24) vs.

0.42 (resp. 0.40) and 0.41 (resp. 0.40)). We also highlight areas in which the improvements in the reconstruction of

local SST details may be noticed. Visually, the improvement is more noticeable on the gradient amplitude. Whereas

OI and MS-VE-DINEOF lead to relatively coarse SST structures, MS-AnEnKS results in ﬁner front details, which

are visually more similar to the groundtruth. This is further emphasized by the analysis of the power spectral densities

of the different ﬁelds (Fig. 6, left). OI clearly underestimates the spectral energy below 100km, as expected from

the associated spatio-temporal smoothing with a spatial correlation length of 100km. A similar underestimation

is observed for MS-VE-DINEOF for scales ranging between 70km and 150km. By contrast, MS-AnEnKS nicely

matches the spectral signature of the groundtruth up to 20km. These results appear consistent with the previous

observation that the improvement brought about by the analog assimilation was mainly noticeable in terms of MSE

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10

for scales greater than 20km. The white noise plateau observed from 20km and below for the reference SST ﬁeld

may indicate that the OSTIA ﬁeld conveys little information for scales lower than 20km for this particular date.

This is further illustrated by the analysis of a one-dimensional transect at 36.525oS accross a strong SST front in

Fig.4. The MS-AnEnKS interpolation clearly leads to a better estimation of local SST variabilities, where OI and

MS-VE-DINEOF tends to oversmooth strong gradients. Overall, the same observation holds for the second example

(Fig.5), though the lower missing data rate (59%) slightly reduces the differences observed between the different

interpolation methods.

We also illustrate the relevance of the post-processing step in the AnEnKS (Fig.2). The spatially-independent

assimilation of overlapping patches may result in block artifacts at patch boundaries as clearly highlighted by

the gradient ﬁeld. The considered EOF-based ﬁltering for 10 ×10 patches successfully removes most of these

block artifacts and retrieves a visually consistent gradient ﬁeld as discussed above. It may be noted that a different

implementation of the analog assimilation using non-sequential iterative scheme for patch-based image processing

[22], [14] would be an alternative, however at the expense of an increased computational complexity. By contrast,

the independent assimilation of each spatial patch only involves one forward and one backward iteration, such

that each space-time patch is visited only twice. We evaluate more precisely the computational complexity of the

different interpolation models in Tab.VI. MS-VE-DINEOF is clearly involves the lowest computational complexity.

In this respect, given relatively similar interpolation performance, VE-DINEFO appears as a relevant alternative to

OI for the interpolation of the coarse-scale component. By contrast, even if MS-AnHMM signiﬁcantly reduces the

computational complexity of the analog assimilation, the differences in interpolation performance reported in Tab.I

clearly recommend the selection of the MS-AnHMM as the relevant ﬁne-scale analog assimilation scheme for SST

ﬁelds.

Sensitivity analysis for MS-AnEnKS: Given the overall qualitative and quantitative analysis reported above,

we further analyze the MS-AnEnKS setting, especially its sensitivity to the selected parameter setting. In Tab.II

We report relative MSE statistics while varying the number of neighbors in the analog models. Tab.III reports a

similar analysis for different kernel parameterizations. Overall, the best parameterization combines a cone kernel

[30] using 100 neighbors and a locally-incremental analog model. It might be noted that the choice of the kernel

weakly affects interpolation performance. By contrast, the locally-incremental analog model signiﬁcantly improves

the relative MSE of the locally-linear and locally-constant strategies (Tab.IV) by about 10% and 25%. This is in

accordance with the conclusions drawn in [24]. The lower performance of the locally-linear analog model may

relate to an unfavourable trade-off between estimation uncertainty and local adaption. We may point out that all

these parameterizations of the proposed interpolation framework outperforms both OI and MS-VE-DINEOF.

Creation of catalog Cfrom observation datasets: In the experiments reported above, the catalog of patch

exemplars is built from the gap-free SST time series from 2008 to 2014. This experimental setting is representative

of an application context where one aim to exploit previous reanalyses and/or numerical simulations for the

interpolation of upcoming observations. The key interest of the analog assimilation is to facilitate the implicit

synergy between possibly computationally-expensive high-resolution models and/or reanalyses and satellite-derived

observation datasets. A second application context is also investigated. We may also directly build the catalog of

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50 100 150 200 250 300 350

0

0.2

0.4

0.6

0.8

1

Day

RMSE

RMSE of SST field on missing areas

OI

MS−VE−DINEOF

MS−AnEnKS

50 100 150 200 250 300 350

0.1

0.2

0.3

0.4

0.5

0.6

Day

RMSE

RMSE of gradient magnitude

OI

MS−VE−DINEOF

MS−AnEnKS

Fig. 1: Time series of the relative MSE: OI (black,-), VE-DINEOF (blue,-) and AnEnKS (red,-) for the estimated

SST ﬁelds (left) and gradient magnitude ﬁelds (right)

Fig. 2: Illustration of the postprocessing step for the removal of blocky artifacts: gradient magnitude ﬁeld of the

an interpolated SST ﬁeld using MS-AnEnKS before (a) and after (b) the application of the considered PCA-

based postprocessing step with 10×10 patches. We also report the radially-averaged power spectral density of the

interpolated SST ﬁelds w.r.t. the true SST ﬁeld (GT, black-).

exemplars from the satellite-derived observation datasets, which involve missing data. To simulate this experiment,

we created a representative catalog from the SST time series with the METOP missing data mask from 2008

to 2014. We proceeded similarly to the scheme described for year 2015 in Section III.3. We only retained SST

patches with less than 20% of missing data. We compared the resulting interpolation performance to that of the

ﬁrst experiment in Tab.V. Although lower MSE values are reported for this second experiment (0.22 vs. 0.20 in

terms of mean relative MSE of the interpolated SST ﬁelds), the relative gain compared to OI and VE-DINEOF

is still signiﬁcant (0.22 vs. respectively 0.40 and 0.41). The qualitative analysis of the interpolated ﬁelds leads

to conclusions similar to those drawn for the ﬁrst experiment. These results further stress the relevance of the

proposed data-driven approach in order to beneﬁt either from high-resolution simulations and/or re-analyses or real

satellite-derived observation datasets. It may be noted that our multi-scale approach may also allow us to combine

observation datasets from different sensors [14].

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TABLE I: Comparison of global interpolation performance: mean relative MSE of OI, G-VE-DINEOF, MS-VE-

DINEOF, G-AnEnKS and MS-AnEnKS: we report mean relative MSE statistics in terms of the SST ﬁelds, the

gradient magnitude of the SST ﬁelds and of the detail coefﬁcients for a four-level dyadic wavelet decomposition.

For MS-ANEnKS, we report both the interpolation performance at intermediate scale i= 1 (MS-ANEnKS|dX1),

i.e. with dX2= 0 in (4), and at scale i= 2 (MS-ANEnKS|dX2). We let the reader refer the main text for details

on the associated parameter setting of the different interpolation models.

Criterion SST k∇k k=1 k=2 k=4 k=8

OI 0.4157 0.3986 0.0053 0.0212 0.0897 0.1897

G-VE-DINEOF 0.4064 0.3967 0.0124 0.0221 0.0873 0.1969

MS-VE-DINEOF 0.4052 0.3765 0.0052 0.0192 0.0803 0.1697

G-AnEnKS 0.3842 0.3922 0.0120 0.0219 0.0967 0.1902

MS-AnHMM dX20.3350 0.3529 0.0057 0.0208 0.0838 0.1711

MS-AnEnKS

dX10.2536 0.3349 0.0057 0.0212 0.0848 0.1622

dX20.2009 0.2357 0.0053 0.0173 0.0579 0.1067

TABLE II: Inﬂuence of the number of analogs on MS-AnEnKS performance: mean relative MSE of MS-AnEnKS

interpolation w.r.t. the number of analogs for the three considered analog strategies (7).

Number of analogs (K)10 20 30 40 50 60 70 80 90 100 110

Locally-constant 0.2746 0.2778 0.2822 0.2852 0.2884 0.2904 0.2926 0.2948

Locally-Linear 0.2449 0.2369 0.2325 0.2301 0.2288 0.2280 0.2278 0.2271 0.2266 0.2266

Locally-incremental 0.2119 0.2113 0.2083 0.2051 0.2030 0.2028 0.2020 0.2012 0.2009 0.2009 0.2011

TABLE III: Inﬂuence of the kernel on MS-AnEnKS performance: mean relative MSE of the interpolated SST ﬁelds

using different kernel parameterizations using a Gaussian kernel and a cone kernel [30].

Gaussian Cone ζ=0.995 Cone ζ=0.5 Cone ζ=0

0.2030 0.2028 0.2036 0.2009

TABLE IV: MS-AnEnKS performance depending on the selected analog model (7): we let the reader refer to Tab.I

for the description of the considered evaluation criteria

Criterion SST k∇k k=1 k=2 k=3 k=4

Locally-constant 0.2725 0.3214 0.0063 0.0208 0.0783 0.1529

Locally-Linear 0.2245 0.2730 0.0059 0.0186 0.0637 0.1265

Locally-Incremental 0.2009 0.2357 0.0053 0.0173 0.0579 0.1067

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TABLE V: Inﬂuence of missing data in catalogs C1,2: we let the reader refer to Tab.I for the description of the

considered evaluation criteria.

Criterion SST k∇k k=1 k=2 k=3 k=4

Catalogs C1,2built from gap-free 2008-2014 data 0.2009 0.2357 0.0053 0.0173 0.0579 0.1067

Catalogs C1,2built from 208-2015 dataset with missing data 0.2230 0.2643 0.0056 0.0194 0.0653 0.1212

TABLE VI: Computational complexity of the interpolation models evaluated in Tab.I

Method OI MS-VE-DINEOF MS-AnHMM MS-AnEnKS

Exe. time ≈3.5h ≈0.5h ≈1.2h ≈3h

V. CONCLUSION

We presented a novel data-driven model for the spatio-temporal interpolation of satellite-derived SST ﬁelds. To our

knowledge, this study reported the ﬁrst application of the analog data assimilation framework to high-dimensional

satellite-derived geophysical ﬁelds. We demonstrated its relevance with respect to state-of-the-art techniques, namely

optimal interpolation [6] and a PCA-based matrix completion scheme [12], [13]. Our model signiﬁcantly outperforms

these two techniques in terms of reconstruction error, especially for ﬁne-scale structures in the range [20km, 200km].

The considered case-study involves real missing data patterns from the METOP-AVHRR sensor. It is therefore

representative of the irregular space-time sampling of the sea surface associated with infrared satellite sensors. The

relative gain in the mean interpolation MSE of about 50% stresses the potential of data-driven computational models

in the exploitation of large-scale observation datasets to improve the reconstruction of geophysical ﬁelds from partial

satellite-derived observations. We have made our case-study dataset available as a supplementary material to our

paper with a view to favoring the benchmarking of interpolation methods for satellite-derived geophysical products1.

As demonstrated by our experimental evaluation, the ﬁrst key feature of the proposed model is the use of a multi-

scale decomposition. Whereas a classic model-driven interpolation (OI) applies to the coarse-scale component, the

reconstruction of the ﬁne-scale components exploit the analog data assimilation [24]. A critical aspect of analog

methods is the availability of a representative catalog of exemplars. In this respect, the considered multi-scale

decomposition is regarded as a crucial means to stationarize the ﬁne-scale spatial variabilities depicted by sea

surface geophysical ﬁelds and make more relevant exemplar-based representations of these variabilities. Wavelet

analysis is generally the classic scheme to derive a multi-scale decomposition [31]. Here, we exploited PCA-based

representations for different patch sizes, so that we naturally combined a multi-scale decomposition to a low-

dimensional representation of the spatial variabilities on each scale. Such PCA-based representations also efﬁciently

deal with complex image geometries (e.g., the presence of land areas in the considered ocean case-study region).

The second critical feature of the proposed multi-scale analog assimilation is the combination of a relevant analog

1The Python code used for the creation of the considered SST data is available at: https://github.com/rfablet/SSTData TCI rfablet

September 6, 2017 DRAFT

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Fig. 3: Reconstruction of a SST ﬁeld on June, 30, 2015 with a large missing data rate (87%): (a) ﬁrst row, reference

SST ﬁeld (groundtruth (GT)), its associated gradient magnitude, observed ﬁeld; second row, interpolated ﬁelds by

OI, MS-AnEnKS, MS-VE-DINEOF; third row, gradient magnitude of the ﬁelds depicted in the second row.

.

5 10 15 20 25 30

14

16

18

20

22

Longitude(°E) ( Latitude = 36.525°S )

SST (°C)

GT

OI

MS−VE−DINEOF

MS−AnEnKS

5 10 15 20 25 30

0

1

2

3

4

Longitude(°E) ( Latitude = 36.525°S )

∇

GT

OI

MS−VE−DINEOF

MS−AnEnKS

Fig. 4: Analysis of a SST transect at 36.525oS for the interpolation results depicted in Fig. 3: we depict a one-

dimensional proﬁle at latitude 36.525oS (c) for both the SST (bottom) and the SST gradient magnitude (top)

for the reference SST ﬁeld (black,-) as well as OI (magenta,-), MS-VE-DINEOF (blue,-) and MS-AnEnKS (red,-)

interpolated SST ﬁelds.

forecasting strategy, namely a locally-incremental strategy, with a generic sequential assimilation algorithm, namely

an Ensemble Kalman Smoother. By contrast, the choice of the kernel and the number of analogs seem to be only

of secondary importance. It may be noted that Model (4) could be straightforwardly extended to a greater number

of scales. For the considered case-study however, numerical experiments did not lead to signiﬁcant improvements

with 3 or 4 detail scales.

We believe that this study opens new research avenues for the development of new data-driven models for

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Fig. 5: Reconstruction of an SST ﬁeld on February, 19, 2015 with a relatively low missing data rate (56%): see

Fig.3 for details.

101

102

103

104

10−10

10−8

10−6

10−4

10−2

100

102

Wavelength (km)

Fourier power spectrum

Radially−averaged PSD

GT

OI

MS−VE−DINEOF

MS−AnEnKS

101

102

103

104

10−10

10−8

10−6

10−4

10−2

100

102

Wavelength (km)

Fourier power spectrum

Radially−averaged PSD

GT

OI

MS−VE−DINEOF

MS−AnEnKS

Fig. 6: Spectral analysis of interpolation results depicted in Fig.3 and 5: we report the radially-averaged power

spectral densities of the reference SST ﬁeld (black,-) as well as OI (magenta,-), MS-VE-DINEOF (blue,-) and

MS-AnEnKS (red,-) interpolated SST ﬁelds for June, 30, 2015 (left) and February, 19, 2015 (right).

the reconstruction of upper ocean dynamics from satellite-derived observations, in the same way that data-driven

schemes have led to major advances in other imaging domains such as photography, microscopy, astronomy....

The exploitation of analogs for interpolation may be interpreted in a climatological sense, the key idea being that

previously observed ﬁne-scale geophysical variabilities will probably occur again, though not necessarily with the

same seasonal timing. The application to other sea surface tracers, such as ocean color, is then natural [33]. The

proposed multi-scale analog assimilation also seems particularly appealing for the downscaling of low-resolution

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satellite-derived products, such as sea surface salinity [34] and sea surface height [35]. From a methodological

point of view, multimodal extensions would be of interest to account for multi-sensor observations as well synergies

between different tracers [36], [14].

Analog strategies are particularly appealing when large and representative observation datasets are available, as

illustrated in the case-study considered here. By contrast, one may question their relevance in addressing scarce

observation datasets as well as extreme events, which are by essence rare events. In this context, the creation of

catalogs of analogs from realistic high-resolution numerical simulations [37], [38], which are becoming increasingly

available, appears to be a relevant path to be further investigated in future work.

ACK NOW LE DG ME NT S

This work was supported by ANR (Agence Nationale de la Recherche, grant ANR-13-MONU-0014), Labex

Cominlabs (grant SEACS) and TeraLab (grant TIAMSEA). We thank B. Chapron, P. Ailliot, P. Tandeo and J.F.

Piolle for discussing and commenting on the development of analog assimilation models for SST data interpolation.

We are also gratefull to A. Northan for a thorough proofread of our manuscript.

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