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# Data-Driven Models for the Spatio-Temporal Interpolation of Satellite-Derived SST Fields

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## Abstract and Figures

Satellite-derived products are of key importance for the high-resolution monitoring of the ocean surface at 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 involve high-missing data rate. The spatio-temporal interpolation of these data remains a key challenge to deliver L4 gridded products to end-users. Whereas operational products mostly rely on model-driven approaches, especially optimal interpolation based on Gaussian process priors, the availability of large-scale observation and simulation datasets advocate 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 Expriment (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 resorts to 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.
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1
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),st(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. trefers 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(t1), η(t1)) (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¯
XdX1. Formally, this leads to the following deﬁnition of
detail ﬁelds dX1,2:
dX1=P1X¯
X
dX2=P2X¯
XdX1(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 t1,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 t1.
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)k21ζcos2θ1ζcos2φ1/2
ktu(t)kktv(t)k(10)
where ω(t) = u(t)v(t),tu(t) = u(t)u(t1),tv(t) = u(t)u(t1),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 0.05spatial
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
September 6, 2017 DRAFT
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
September 6, 2017 DRAFT
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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
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].
September 6, 2017 DRAFT
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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
September 6, 2017 DRAFT
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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
September 6, 2017 DRAFT
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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
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
September 6, 2017 DRAFT
16
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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September 6, 2017 DRAFT
... Satellite-based SST often loses a large amount of data for various reasons mentioned above, limiting the production of seamless daily SST from single satellite data sources. Therefore, studies to reconstruct missing SST data have been actively conducted [11][12][13][14][15][16][17][18]. Several widely used seamless daily SST products are based on Optimal Interpolation (OI) with data assimilation of multiple infrared and passive microwave satellite observations and in situ measurements, which include global products from Danish Meteorological Institute (DMI) on a grid scale of 0.05 • and Operational Sea Surface Temperature and Sea Ice Analysis (OSTIA) on a grid scale of 0.054 • [16][17][18][19]. ...
... Therefore, studies to reconstruct missing SST data have been actively conducted [11][12][13][14][15][16][17][18]. Several widely used seamless daily SST products are based on Optimal Interpolation (OI) with data assimilation of multiple infrared and passive microwave satellite observations and in situ measurements, which include global products from Danish Meteorological Institute (DMI) on a grid scale of 0.05 • and Operational Sea Surface Temperature and Sea Ice Analysis (OSTIA) on a grid scale of 0.054 • [16][17][18][19]. However, since such operational products are based on the OI approach resulting in smoothed surface, they have a relatively limited feature resolution [17]. ...
... Several widely used seamless daily SST products are based on Optimal Interpolation (OI) with data assimilation of multiple infrared and passive microwave satellite observations and in situ measurements, which include global products from Danish Meteorological Institute (DMI) on a grid scale of 0.05 • and Operational Sea Surface Temperature and Sea Ice Analysis (OSTIA) on a grid scale of 0.054 • [16][17][18][19]. However, since such operational products are based on the OI approach resulting in smoothed surface, they have a relatively limited feature resolution [17]. Several processes such as outlier removals and smoothing are conducted to improve the accuracy of the operational SST products, which often degrade the spatial pattern of the products, even though 1 km SST data are assimilated in the products [18]. ...
Full-text available
Article
Sea SurfaceTemperature (SST) is a critical parameter for monitoring the marine environment and understanding various ocean phenomena. While SST can be regularly retrieved from satellite data, it often suffers from missing data due to various reasons including cloud contamination. In this study, we proposed a novel two-step data fusion framework for generating high-resolution seamless daily SST from multi-satellite data sources. The proposed approach consists of (1) SST reconstruction based on Data Interpolate Convolutional AutoEncoder (DINCAE) using the SSTs derived from two satellite sensors (i.e., Moderate Resolution Imaging Spectroradiometer (MODIS) and Advanced Microwave Scanning Radiometer 2(AMSR2)), and (2) SST improvement through data fusion using random forest for consistency with in situ measurements with two schemes (i.e., scheme 1 using the reconstructed MODIS SST variables and scheme 2 using both MODIS and AMSR2 SST variables). The proposed approach was evaluated over the Kuroshio Extension in the Northwest Pacific, where a highly dynamic SST pattern can be found, from 2015 to 2019. The results showed that the reconstructed MODIS and AMSR2 SSTs through DINCAE yielded very good performance with Root Mean Square Errors (RMSEs) of 0.85 and 0.60 °C and Mean Absolute Errors (MAEs) of 0.59 and 0.45 °C, respectively. The results from the second step showed that scheme 2 and scheme 1 produced RMSEs of 0.75 and 0.98 °C and MAEs of 0.53 and 0.68 °C, respectively, compared to the in situ measurements, which proved the superiority of scheme 2 using multi-satellite data sources. Scheme 2 also showed comparable or even better performance than two operational SST products with similar spatial resolution. In particular, scheme 2 was good at simulating features with fine resolution (~50 km). The proposed approach yielded promising results over the study area, producing seamless daily SST products with high quality and high feature resolution.
... s depends on the orbit of the satellite. The wind quality depends on multiple parameters related to the inversion algorithm (Chapron et al. 2001;Lin et al. 2008) structures by improving the optimal interpolation (Ubelmann et al. 2015), or through exploiting data-driven representations (Tandeo et al. 2015;Lguensat et al. 2017b;Lguensat et al. 2017a;R. Fablet et al. 2017;Barth et al. 2020;Beauchamp et al. 2020) as investigated in the next sections. ...
... Recently, data-driven approaches (Tandeo et al. 2015;Lguensat et al. 2017b;Lguensat et al. 2017a;R. Fablet et al. 2017;Barth et al. 2020;Beauchamp et al. 2020) have emerged as relevant alternatives to model-driven schemes. They take benefit from the increasing availability of remote sensing observations and simulation data to derive computationally efficient representations. Analog methods are one of the first data-driven techniques developed within a da ...
... to model-driven schemes. They take benefit from the increasing availability of remote sensing observations and simulation data to derive computationally efficient representations. Analog methods are one of the first data-driven techniques developed within a data assimilation framework (Tandeo et al. 2015;Lguensat et al. 2017b;Lguensat et al. 2017a;R. Fablet et al. 2017). Combining analog data assimilation (AnDA) with a patch-based representation have shown great results with respect to the state-of-theart OI and EOF-based schemes. However, the parametrization of the proposed framework involves tuning several parameters principally due to the data-driven formulation of the dynamical prior based on analo ...
Thesis
This thesis focuses on the data-driven identification of dynamical representations of upper ocean dynamics for forecasting, simulation and data assimilation applications. We focus on practical considerations regarding the provided observations and tackle multiple issues, ranging from the parametrization of the models, their time integration, the space in which the models should be definedand their implementation in data assimilation schemes.The core of our work resides in proposing a new data-driven embedding technique. This framework optimises an augmented space as a solution of an optimization problem, parametrised by a trainable Ordinary Differential Equation (ODE) that can be used for several applications such as forecasting and data assimilation. We discuss the effectiveness of the proposed framework within two different parametrizations of the trainable ODE. Namely, the Linear-quadratic and Linear ones and show that both formulations lead to interesting applications and most importantly, connect with interesting state-of-theart theory that helps understanding and constraining the proposed architecture. Regarding data assimilation applications, we explore two distinct methodologies. The first technique can be seen as an alternative to the ensemble Kalman filtering and the second one relates to the proposed dynamical embedding technique and can be extended to match recent advances of state-of-the-art filtering techniques.
... For the other types of energy forms, their parameterizations are generally set a priori and not learned from the data. Regarding more particularly datadriven and learning-based approaches, most previous works (Peyr et al., 2011;Alvera-Azcarate et al., 2016;Fablet et al., 2017) have addressed the learning of interpolation schemes under the assumption that a representative gap-free dataset is available. This gap-free dataset may be the image itself (Criminisi et al., 2004;Lorenzi et al., 2011;Peyr et al., 2011). ...
... The AnDa scheme combines the exploitation of a gap-free reference dataset to design an analog dynamical model with a state-of-the-art ensemble Kalman filter to address the interpolation issue. Here, we implement the patch-based version of AnDA as presented in Fablet et al. (2017). As AnDA and DINEOF schemes rely on some training on a gap-free dataset, we use for evaluation purposes a 20-day test period in December, with the remaining data being used as training data except 10 days before and after the 20-day test period. ...
Full-text available
Article
Earth observation satellite missions provide invaluable global observations of geophysical processes in play in the atmosphere and the oceans. Due to sensor technologies (e.g., infrared satellite sensors), atmospheric conditions (e.g., clouds and heavy rains), and satellite orbits (e.g., polar-orbiting satellites), satellite-derived observations often involve irregular space–time sampling patterns and large missing data rates. Given the current development of learning-based schemes for earth observation, the question naturally arises whether one might learn some representation of the underlying processes as well as solve interpolation issues directly from these observation datasets. In this article, we address these issues and introduce an end-to-end neural network learning scheme, which relies on an energy-based formulation of the interpolation problem. This scheme investigates different learning-based priors for the underlying geophysical field of interest. The end-to-end learning procedure jointly solves the reconstruction of gap-free fields and the training of the considered priors. Through different case studies, including observing system simulation experiments for sea surface geophysical fields, we demonstrate the relevance of the proposed framework compared with optimal interpolation and other state-of-the-art data-driven schemes. These experiments also support the relevance of energy-based representations learned to characterize the underlying processes.
... Marullo et al. (2014) fused clearsky SSTs from the SEVIRI and Mediterranean Forecasting System to generate gap-free, hourly SSTs using diurnal optimal interpolation across a moving temporal window. Different model outputs have also been evaluated for filling invalid SST pixels (Nardelli et al., 2015); however, the resulting accuracies of cloudy-sky results were highly reliant upon simulation data, especially for continuous cloudy days, and no further correction was implemented (Fablet et al., 2017). Dumitrescu et al. (2020) fused daytime hourly SEVIRI LSTs and modeled skin temperature using multiple linear regression and generalized additive models, in addition to elevation, time, and solar radiation to improve GEO LST estimation under clouds. ...
Full-text available
Article
... At present, the advances in remote sensing technologies have enabled the acquisition of global observation to be more accessible in ocean mesoscale eddy detection, tracking, prediction tasks. Mesoscale eddies have been investigated both intensively and extensively with long-term time series (Oey et al., 2005;Wang et al., 2015;Świerczyńska et al., 2016;Fablet et al., 2017;Imani et al., 2017;Fu et al., 2019), e.g., sea surface height, salinity, temperature, SLA, etc. The initialization of ocean dynamic state through assimilation underpins prediction with the optimal statistical estimation from the numerical models and the sparse ocean interior information (Thoppil et al., 2021). ...
Full-text available
Article
Ocean mesoscale eddies are ubiquitous in world ocean and account for 90% oceanic kinetic energy, which dominate the upper ocean flow field. Accurately predicting the variation of ocean mesoscale eddies is the key to understand the oceanic flow field and circulation system. In this article, we propose to make an initial attempt to explore spatio-temporal predictability of mesoscale eddies, employing deep learning architecture, which primarily establishes Memory In Memory (MIM) for sea level anomaly (SLA) prediction, combined with the existing mesoscale eddy detection. Oriented to the western Pacific ocean (125�􀀀137.5�E and 15�􀀀27.5�N), we quantitatively investigate the historic daily SLA variability at a 0.25� spatial resolution from 2000 to 2018, derived by satellite altimetry. We develop the enhanced MIM prediction strategies, equipped with Gated Recurrent Unit (GRU) and spatial attention module, in a scheduled sampling manner, which overcomes the gradient vanishing and complements to strengthen spatio-temporal features for long-term dependencies. At the early stage, the real value SLA input guides the model training process for initialization, while the scheduled sampling intentionally feeds the newly predicted value, to resolve the distribution inconsistency of inference. It has been demonstrated in our experiment results that our proposed prediction scheme outperformed the state-of-art approaches for SLA time series, with MAPE, RMSE of the 14-day prediction duration, respectively, 5.1%, 0.023 m on average, even up to 4.6%, 0.018 m for the effective sub-regions, compared to 19.8%, 0.086 m in ConvLSTM and 8.3%, 0.040 m in original MIM, which greatly facilitated the mesoscale eddy prediction. This proposed scheme will be beneficial to understand of the underlying dynamical mechanism behind the predictability of mesoscale eddies in the future, and help the deployment of ARGO, glider, AUV and other observational platforms.
... Using CubeSats, Houborg and McCabe [18] present an approach to further leverage Landsat and MODIS for further spatiotemporal improvements. Similarly, nearest neighbor analog multiscale patch-decomposition data-driven models are used as state-of-the-art interpolation techniques for developing global sea surface temperature (SST) datasets [19]. In recent years, super-resolution techniques have presented state-of-the-art results for spatial enhancement of satellite images [20]- [22]. ...
Full-text available
Article
Applications of satellite data in areas such as weather tracking and modeling, ecosystem monitoring, wildfire detection, and land-cover change are heavily dependent on the tradeoffs to spatial, spectral, and temporal resolutions of observations. In weather tracking, high-frequency temporal observations are critical and used to improve forecasts, study severe events, and extract atmospheric motion, among others. However, while the current generation of geostationary (GEO) satellites has hemispheric coverage at 10-15-min intervals, higher temporal frequency observations are ideal for studying mesoscale severe weather events. In this work, we present a novel application of deep learning-based optical flow to temporal upsampling of GEO satellite imagery. We apply this technique to 16 bands of the GOES-R/Advanced Baseline Imager mesoscale dataset to temporally enhance full-disk hemispheric snapshots of different spatial resolutions from 10 to 1 min. Experiments show the effectiveness of task-specific optical flow and multiscale blocks for interpolating high-frequency severe weather events relative to bilinear and global optical flow baselines. Finally, we demonstrate strong performance in capturing variability during convective precipitation events.
Full-text available
Preprint
Satellite altimetry is a unique way for direct observations of sea surface dynamics. This is however limited to the surface-constrained geostrophic component of sea surface velocities. Ageostrophic dynamics are however expected to be significant for horizontal scales below 100~km and time scale below 10~days. The assimilation of ocean general circulation models likely reveals only a fraction of this ageostrophic component. Here, we explore a learning-based scheme to better exploit the synergies between the observed sea surface tracers, especially sea surface height (SSH) and sea surface temperature (SST), to better inform sea surface currents. More specifically, we develop a 4DVarNet scheme which exploits a variational data assimilation formulation with trainable observations and {\em a priori} terms. An Observing System Simulation Experiment (OSSE) in a region of the Gulf Stream suggests that SST-SSH synergies could reveal sea surface velocities for time scales of 2.5-3.0 days and horizontal scales of 0.5$^\circ$-0.7$^\circ$, including a significant fraction of the ageostrophic dynamics ($\approx$ 47\%). The analysis of the contribution of different observation data, namely nadir along-track altimetry, wide-swath SWOT altimetry and SST data, emphasizes the role of SST features for the reconstruction at horizontal spatial scales ranging from \nicefrac{1}{20}$^\circ$ to \nicefrac{1}{4}$^\circ$.
Full-text available
Article
Frente a una narrativa distópica presente en los medios de comunicación y cultura popular que caracteriza el avance y desarrollo de la inteligencia artificial como una amenaza o riesgo existencial (e.g. desempleo tecnológico, sistemas de armas autónomas letales, robots asesinos, propaganda política computacional etc.) quiero valorar de manera crítica y constructiva el rol de la inteligencia artificial para el bien común (AI4SG). La tecnología digital también se puede aplicar para la solución de grandes problemas de la humanidad, como los 17 Objetivos de Desarrollo Sostenible y sus 169 metas de la agenda 2030. En este artículo, comentaré distintos casos de uso y aplicación de la inteligencia artificial y la robótica encaminados a conseguir los 17 Objetivos de Desarrollo Sostenible y qué principios éticos deben guiar su aplicación para que la inteligencia artificial consiga la ambiciosa agenda 2030. También comentaré el plan de acción de España y la estrategia nacional para cumplir la agenda 2030 y de qué manera incorpora las TIC.
Full-text available
Article
This paper addresses physics-informed deep learning schemes for satellite ocean remote sensing data. Such observation datasets are characterized by the irregular space-time sampling of the ocean surface due to sensors’ characteristics and satellite orbits. With a focus on satellite altimetry, we show that end-to-end learning schemes based on variational formulations provide new means to explore and exploit such observation datasets. Through Observing System Simulation Experiments (OSSE) using numerical ocean simulations and real nadir and wide-swath altimeter sampling patterns, we demonstrate their relevance w.r.t. state-of-the-art and operational methods for space-time interpolation and short-term forecasting issues. We also stress and discuss how they could contribute to the design and calibration of ocean observing systems.
Full-text available
Conference Paper
This paper addresses physics-informed deep learning schemes for satellite ocean remote sensing data. Such observation datasets are characterized by the irregular space-time sampling of the ocean surface due to sensors' characteristics and satellite orbits. With a focus on satellite altimetry, we show that end-to-end learning schemes based on variational formulations provide new means to explore and exploit such observation datasets. Through Observing System Simulation Experiments (OSSE) using numerical ocean simulations and real nadir and wide-swath altimeter sampling patterns, we demonstrate their relevance w.r.t. state-of-the-art and operational methods for space-time interpolation and short-term forecasting issues. We also stress and discuss how they could contribute to the design and calibration of ocean observing systems.
Full-text available
Article
In light of growing interest in data-driven methods for oceanic, atmospheric, and climate sciences, this work focuses on the field of data assimilation and presents the analog data assimilation (AnDA). The proposed framework produces a reconstruction of the system dynamics in a fully data-driven manner where no explicit knowledge of the dynamical model is required. Instead, a representative catalog of trajectories of the system is assumed to be available. Based on this catalog, the analog data assimilation combines the nonparametric sampling of the dynamics using analog forecasting methods with ensemble-based assimilation techniques. This study explores different analog forecasting strategies and derives both ensemble Kalman and particle filtering versions of the proposed analog data assimilation approach. Numerical experiments are examined for two chaotic dynamical systems: the Lorenz-63 and Lorenz-96 systems. The performance of the analog data assimilation is discussed with respect to classical model-driven assimilation. A Matlab toolbox and Python library of the AnDA are provided to help further research building upon the present findings.
Full-text available
Article
In this study, an improved Data INterpolating Empirical Orthogonal Functions (DINEOF) algorithm for determination of missing values in a spatio-temporal dataset is presented. Compared with the ordinary DINEOF algorithm, the iterative reconstruction procedure until convergence based on every fixed EOF to determine the optimal EOF mode is not necessary and the convergence criterion is only reached once in the improved DINEOF algorithm. Moreover, in the ordinary DINEOF algorithm, after optimal EOF mode determination, the initial matrix with missing data will be iteratively reconstructed based on the optimal EOF mode until the reconstruction is convergent. However, the optimal EOF mode may be not the best EOF for some reconstructed matrices generated in the intermediate steps. Hence, instead of using asingle EOF to fill in the missing data, in the improved algorithm, the optimal EOFs for reconstruction are variable (because the optimal EOFs are variable, the improved algorithm is called VE-DINEOF algorithm in this study). To validate the accuracy of the VE-DINEOF algorithm, a sea surface temperature (SST) data set is reconstructed by using the DINEOF, I-DINEOF (proposed in 2015) and VE-DINEOF algorithms. Four parameters (Pearson correlation coefficient, signal-to-noise ratio, root-mean-square error, and mean absolute difference) are used as a measure of reconstructed accuracy. Compared with the DINEOF and I-DINEOF algorithms, the VE-DINEOF algorithm can significantly enhance the accuracy of reconstruction and shorten the computational time.
Full-text available
Article
Methods of data assimilation are established in physical sciences and engineering for the merging of observed data with dynamical models. When the model is nonlinear, methods such as the ensemble Kalman filter have been developed for this purpose. At the other end of the spectrum, when a model is not known, the delay coordinate method introduced by Takens has been used to reconstruct nonlinear dynamics. In this article, we merge these two important lines of research. A model-free filter is introduced based on the filtering equations of Kalman and the data-driven modeling of Takens. This procedure replaces the model with dynamics reconstructed from delay coordinates, while using the Kalman update formulation to reconcile new observations. We find that this combination of approaches results in comparable efficiency to parametric methods in identifying underlying dynamics, and may actually be superior in cases of model error.
Full-text available
Article
Two general algorithms for solving constrained minimization problems are presented and discussed in the context of analysis and assimilation of meteorological observations. The first algorithm presented in the augmented lagrangian algorithm. It generalizes the more classical penalty and duality algorithms. The second algorithm, inspired from optimal control techniques, is based on an appropriate use of an adjoint dynamical equation, and seems to be particularly well adapted to the assimilation of observations distributed in time. Simple numerical examples show the ability of these algorithms to solve non-linear minimization problems of the type encountered in meteorology. Their possible use in more complex situations is discussed, in particular in terms of their computational cost. - from Authors
Full-text available
Conference Paper
We here address the super-resolution of a high-resolution image involving missing data given that a low-resolution image of the same scene is available. This is a typical issue in the remote sensing of geophysical parameters from different spaceborne sensors. Such super-resolution application involves large downscaling factor (typically from 10 to 20) and the super-resolution model should account for both texture patterns and specific statistical features, especially the spectral and non-Gaussian features. In this context, we propose a novel non-local approach and formally states the solution as the joint minimization of several projection constraints. We illustrate the relevance of the proposed model on real ocean remote sensing data, namely sea surface temperature fields, as well on visual textures.
Book
Mallat's book is the undisputed reference in this field - it is the only one that covers the essential material in such breadth and depth. - Laurent Demanet, Stanford University The new edition of this classic book gives all the major concepts, techniques and applications of sparse representation, reflecting the key role the subject plays in today's signal processing. The book clearly presents the standard representations with Fourier, wavelet and time-frequency transforms, and the construction of orthogonal bases with fast algorithms. The central concept of sparsity is explained and applied to signal compression, noise reduction, and inverse problems, while coverage is given to sparse representations in redundant dictionaries, super-resolution and compressive sensing applications. Features: * Balances presentation of the mathematics with applications to signal processing * Algorithms and numerical examples are implemented in WaveLab, a MATLAB toolbox * Companion website for instructors and selected solutions and code available for students New in this edition * Sparse signal representations in dictionaries * Compressive sensing, super-resolution and source separation * Geometric image processing with curvelets and bandlets * Wavelets for computer graphics with lifting on surfaces * Time-frequency audio processing and denoising * Image compression with JPEG-2000 * New and updated exercises A Wavelet Tour of Signal Processing: The Sparse Way, third edition, is an invaluable resource for researchers and R&D engineers wishing to apply the theory in fields such as image processing, video processing and compression, bio-sensing, medical imaging, machine vision and communications engineering. Stephane Mallat is Professor in Applied Mathematics at École Polytechnique, Paris, France. From 1986 to 1996 he was a Professor at the Courant Institute of Mathematical Sciences at New York University, and between 2001 and 2007, he co-founded and became CEO of an image processing semiconductor company. Includes all the latest developments since the book was published in 1999, including its application to JPEG 2000 and MPEG-4 Algorithms and numerical examples are implemented in Wavelab, a MATLAB toolbox Balances presentation of the mathematics with applications to signal processing.
Chapter
The EnKF has recently been taken into use with simulation models for oil and gas reservoirs, with the purpose of estimating poorly known parameters and to improve the predictive capability of the models. There are economical benefits of obtaining a model which best possible represents the reservoir. Optimally, it could be used for predicting the future production and to assist in the planning of new production and injection wells. A better model also provides insight and understanding regarding the properties of the reservoir.
Article
Daily gap-free Sea Surface Temperature (SST) fields at high resolution are required by several operational users working on monitoring and forecasting the status of the marine environment. Existing instruments cannot provide such fields, and careful interpolation of available data is required to provide gap-free SST estimates. In the framework of the MyOcean projects (now turning into the European Copernicus Marine Service), several satellite (and/or satellite and in situ) interpolated products are distributed in real time, and continuous research and development activities are carried out to improve their quality. In this paper, we describe the work done to improve the High Resolution (1/16°) interpolated product covering the Mediterranean Sea, that is obtained by combining all available satellite infrared images. Three Optimal Interpolation schemes based on different space–time covariance models and background fields are derived, compared and validated versus in situ drifting buoy measurements (leading to a minimum standard deviation error (STDE) of 0.46 °K and mean bias error (MBE) of − 0.15 °K), as well as by adopting a holdout validation approach with artificial clouds (with STDE ranging between 0.22 °K and 0.55 °K, and MBE always between 0.03 °K and 0.05 °K, depending on the cloud configuration considered). Almost negligible differences are found between the three products, revealing only a slight improvement when using spatially varying covariance parameters and a daily climatology as background, which is less risky in case of prolonged cloudy conditions, though attaining the same performance with small/rapid clouds. This scheme has thus been implemented in the Mediterranean SST operational chain since April 2014.


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Browse Issues:

### New Mathematics for Extreme-scale Computational Science?

Let’s start with the good news: Mathematics continues to be the most important contributor to any work in large-scale computational science. This is so because computational complexity becomes ever more important with faster computers. Once systems are large enough, the best algorithms will always be the ones with the best asymptotic complexity. High-quality journals, such as the SIAM Journals on Numerical Analysis and Scientific Computing, regularly publish papers in this area that advance the research frontier.

### Obituaries

Michael James David Powell, who passed away on April 19, was one of the giants who established numerical analysis as a major discipline and created its current intellectual landscape. From his life work  have emerged both mathematical foundations and practical algorithms of nonlinear optimization, as well as decisive contributions to approximation theory.

### On Markov Decision Processes

Sequential planning under uncertainty is a basic optimization problem that arises in many different settings, ranging from artificial intelligence to operations research. In a generic system, we have an agent who chooses among different actions and then receives a reward, after which the system moves on in a stochastic way. Usually the aim is to maximize the expected (discounted) reward of the system over a finite or, in certain cases, as described below, an infinite time horizon.

### A Minimalist Minimizes an Integral

In this issue we present a solution that is shorter than Johann Bernoulli’s famous optics-based idea of minimizing
 $\tag{1} \int_{\gamma OA}F(y)ds$

over smooth curves connecting two given points $$A$$ and $$B$$; here $$F(y) > 0$$ is a given function and $$ds$$ is an element of arc length. Bernoulli based his beautiful solution on the equivalence between Fermat’s principle and Snell’s law.

### The Moody’s Mega Math Challenge Marks 10th Year

Can you recall the first time you worked with a team on a significant mathematical modeling problem? For me, it was as a senior at Oberlin College, in a project for NASA in an operations research course taught by Professor Bruce Pollack-Johnson (now at Villanova). I am certain that the experience played a large role in my decision to become an applied mathematician and to join the faculty at Harvey Mudd College, which provides industrial mathematical modeling experiences through its senior capstone clinic projects. SIAM provides students with mathematical modeling experiences through the Moody’s Mega Math (M3) Challenge, which, like COMAP’s Hi-MCM, makes the experience of team-based modeling available to U.S. high school students. M3 is entirely Internet-based, and carries no entry or participation fees.

### U.S. National Academies Elect New Members

Among the most important honors accorded to scientists and engineers in the U.S. is election to the National Academies of Engineering and Sciences. The National Academies were created (NAS in 1863 by President Abraham Lincoln; NAE in 1964) to advise the federal government in matters of science and technology. At their annual meetings, each announces the names of newly elected members. Distinguished members of the SIAM community appear regularly on both lists, and 2015 is no exception.

### European Students Gather at TU Delft for Krylov Day

On February 2, the SIAM Student Chapter at TU Delft held a one-day workshop on Krylov subspace methods. The speakers, 12 PhD students in numerical linear algebra, gave overviews of their current work and its relation to Krylov subspaces. Participants came from different universities in The Netherlands and other European countries; among them were representatives of SIAM Student Chapters at Magdeburg, Manchester, and Prague.

### Numerical Notation Systems as Cultural Artifacts

BOOK REVIEW: Numerical Notation: A Comparative History. By Stephen Chrisomalis, Cambridge University Press, New York, 2010, 496 pages, \$114.99.

### AAAS & the SIAM Community

This year’s SIAM Conference on CSE featured a new attraction: “SIAM Communication Doctors,” a booth for people wishing to craft effective messages about their research—for communication to future employers, at outreach events, or for the press. Graduate students, postdocs, and faculty visited the booth, hoping that booth doctors could turn their research summaries into good stories that would appeal to the public. At the booth, reporter Flora Lichtman, whose work has appeared in The New York Times and on NPR’s Science Friday, joined conference attendees Nick Higham (University of Manchester), Jeff Humpherys (Brigham Young University), Rachel Levy (Harvey Mudd College), and Matt Parno (Massachusetts Institute of Technology) to offer feedback as people pitched their ideas.

## SIAM Connect

 Tristan van Leeuwen receives SIAM Activity Group on Geosciences Junior Scientist Prize SIAM joins call to action for American Innovation Imperative Mathematical models with complicated dynamics for disease study

See All Connect News

## Professional Opportunities

 Accenture Analytics - Global Management - Data Scientist Analyst+ - Accenture - New York, NY Staff Scientist I - Speech Recognition, NLP and Multimedia - New MS Grad - Raytheon - Cambridge, MA Microsoft: Data Scientists - Microsoft: Data Scientists - Redmond, WA

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## Conferences & Events

July 08, 2015 To July 10, 2015

SIAM Conference on Control and Its Applications (CT15)

July 06, 2015 To July 10, 2015

ICERM IdeaLab: Inverse Problems and Uncertainty Quantification

July 07, 2015 To July 23, 2015

Graduate Summer School: Games and Contracts for Cyber-Physical Security

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## Inside SIAM

#### Upcoming SIAM Events

July 08, 2015 To July 10, 2015

SIAM Conference on Control and Its Applications (CT15)

August 03, 2015 To August 07, 2015

SIAM Conference on Applied Algebraic Geometry (AG15)

August 10, 2015 To August 14, 2015

8th International Congress on Industrial and Applied Mathematics (ICIAM 2015)

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format: plain


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### Enhanced mutual learning artificial bee colony algorithm

LUO Hao1，2, LIU Yu1，2

1. 1.School of Software, Dalian University of Technology, Dalian, Liaoning 116024, China
2.Institute of IT Service Engineering and Management, Dalian University of Technology, Dalian, Liaoning 116024, China
• Online:2016-08-15 Published:2016-08-12

### 一种强化互学习的人工蜂群算法

1. 1.大连理工大学 软件学院，辽宁 大连 116024
2.大连理工大学 IT服务工程与管理研究所，辽宁 大连 116024

Abstract: In order to deal with the basic ABC algorithm for its slow convergence, tending to get stagnation on local optima, and further to improve its searching efficiency in exploration and exploitation, this paper proposes an improved artificial bee colony algorithm called Enhanced Mutual Learning ABC algorithm（EMLABC）, applying different kind of honey bees with distinguished strategies, firstly for employed bees, by exemplifying mutation perturbation learning frequency and basing on multi comparatively prior neighbors for learning, to enhance global exploration and avoid premature, and then applying onlooker bees with extensive mutual learning strategy, which can enable the new candidate solutions more likely to search in potential better space, thus to achieve fast convergence and accuracy. The experiments are conducted on a benchmark suite of 16 unimodal and multimodal test functions, the results demonstrate significant improvements of EMLABC when compared with the basic ABC algorithm and several recent variants of ABC algorithm.


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# Custom \bonusqformat in exam class not level with question text

I am creating a set of problems, with the bonus problems starred with an *, but when I use \bonusqformat to customize the mark, it pushes the text onto a new line. Is there any way to prevent this from happening?

MWE

\documentclass{exam}

\bonusqformat{\thequestion *.}
\begin{document}
\begin{questions}
\bonusquestion This is a question.

\end{questions}
\end{document}

-
This is the default behavior of exam. The question title (typeset according to \qformat or \bonusqformat) is on a line by itself and the question text is below it. –  egreg Jan 17 at 12:47
I understand this is the default behaviour, I'm more looking for a way to overcome it, whilst still having the starred questions. –  Nathanael Farley Jan 17 at 13:22
Thanks for pointing that out @egreg, prompted me to look elsewhere for the answer! Posted it below. –  Nathanael Farley Jan 17 at 13:33

Turns out that because of the default behaviour of \bonusqformat (pointed out by egreg), it is easier to achieve this effect by the following code in the preamble:

\newcommand\bonusquestionlabel{\thequestion{}*.}
\makeatletter
\def\question@number{%
\if@bonus
\if@bonusqformat
\makebox[\hsize][s]{\@bonusquestionformat}\hskip-\labelsep
\else
\bonusquestionlabel
\fi
\else
\if@qformat
\makebox[\hsize][s]{\@questionformat}\hskip-\labelsep
\else
\questionlabel
\fi
\fi
}
\makeatother

This code redefines the question label for bonus questions and makes a new label for them.

Result from the MWE with the extra code in the preamble

-


text:
CryptoDB

Paper: Deterministic Polynomial-Time Equivalence of Computing the CRT-RSA Secret Keys and Factoring

Authors: Subhamoy Maitra Santanu Sarkar URL: http://eprint.iacr.org/2009/062 Search ePrint Search Google Let $N = pq$ be the product of two large primes. Consider CRT-RSA with the public encryption exponent $e$ and private decryption exponents $d_p, d_q$. It is well known that given any one of $d_p$ or $d_q$ (or both) one can factorize $N$ in probabilistic poly$(\log N)$ time with success probability almost equal to 1. Though this serves all the practical purposes, from theoretical point of view, this is not a deterministic polynomial time algorithm. In this paper, we present a lattice based deterministic poly$(\log N)$ time algorithm that uses both $d_p, d_q$ (in addition to the public information $e, N$) to factorize $N$.
BibTeX
@misc{eprint-2009-18254,
title={Deterministic Polynomial-Time Equivalence of Computing the CRT-RSA Secret Keys and Factoring},
booktitle={IACR Eprint archive},
keywords={public-key cryptography / CRT-RSA, Cryptanalysis, Factorization, LLL Algorithm, RSA.},
url={http://eprint.iacr.org/2009/062},
note={ subho@isical.ac.in 14286 received 6 Feb 2009, last revised 10 Feb 2009},
author={Subhamoy Maitra and Santanu Sarkar},
year=2009
}