On the use of Finite-Time Lyapunov Exponents and Vectors ... · 3 Finite-Time Lyapunov Exponents...

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On the use of Finite-Time Lyapunov Exponents andVectors for direct assimilation of tracer images into ocean

models

Olivier Titaud1,2 ?, Jean-Michel Brankart1 and Jacques Verron1,1LEGI, CNRS & Universite de Grenoble, BP53X, 38041 Grenoble Cedex, France;2CERFACS, 42 avenue Gaspard Coriolis, 31057

Toulouse Cedex 01, France

(Manuscript received January 2011)

ABSTRACTSatellite ocean tracer images, of Sea Surface Temperature (SST) and Ocean Colourimages, for example, show patterns like fronts and filaments that characterise the flowdynamics. These patterns can be described using Lagrangian tools such as Finite-Time Lyapunov Exponents (FTLE) or Finite-Time Lyapunov Vectors (FTLV). Inrecent years, several studies have investigated the possibility of directly assimilatingstructured data from satellite images into numerical models. In this paper, we exploitspecific properties of FTLE and FTLV to define observation operators that can beused in a direct ocean tracer image assimilation scheme. In an idealised context, weshow that high-resolution SST and Ocean Colour images can be exploited to correctvelocity fields using FTLE or FTLV.

1 Introduction

Data Assimilation techniques combine all the avail-able information to perform a realistic simulation of adynamic system. This information comes from differentsources: mathematical models based on physical laws, obser-vations and a priori knowledge (e.g. errors). The techniquesare widely used to simulate geophysical fluid flows. Theymay be based on sequential filtering (Brasseur and Verron,2006; Carme et al., 2001) or variational (Le Dimet and Ta-lagrand, 1986; Luong et al., 1998) approaches. Both involvean objective function that evaluates the discrepancy betweenmodel outputs and observations using a least-squares term.This term takes the following general form

J(X) = ‖H(X)− y‖2O, (1)

where X ∈ X and y ∈ O are respectively the state vari-able and the observation vectors. Assimilation schemes aredesigned to minimise the objective function with respect tothe state (or control) variable X.

The difference is evaluated in the observation space Ousing the norm ‖ · ‖O: the so-called observation operator Hmaps the state variable space X onto the observation space.The norm is usually derived from an inner product thattakes into account the observation error statistics (e.g. mea-surement and representativity errors). When observed vari-ables correspond to certain state variables, H may “simply”represent a projection in space and time. This is generallythe case of in-situ observations like temperature or salinity

? Corresponding author.

e-mail: [email protected]

profiles in the oceanographic context. Remote observationsfrom infrared or colour radiometers installed on satellites areradiances that are indirectly linked to the state variables.The corresponding observation operator may then containcomplex radiative transfer laws. Another way of assimilat-ing this kind of data is to invert radiance measurements intopseudo-observations of a state variable. Typical examplesare the infrared and microwaves radiances that are invertedrespectively into surface temperature and salinity measure-ments. But the link between radiance measurements andthe concentration of bio-geochemical tracers (such as chloro-phyll) is far more complex. Assimilation of these radiancesis still the subject of intensive research (see e.g. Carmilletet al., 2001; Gregg, 2008).

The improvement of remote sensing systems has re-sulted in to satellite images of increasingly resolution thatshow mesoscale and submesoscale patterns such as filaments,fronts and eddies (see Figure 1). The location, shape andevolution of these structures are strongly linked to flow dy-namics (Ottino, 1989). A single ocean tracer image inte-grates the past temporal evolution of the velocity field thatadvected it (Lehahn et al., 2007); also, as with passive tracergradients in a turbulent flow, ocean tracer gradients becomealigned in certain specific orientations (namely the orien-tation of backward Finite-Time Lyapunov Vectors, see Sec-tion 3) which in turn characterise the velocity field (Lapeyre,2002). However, structured information contained in oceantracer images are still underused by forecasting systems. Re-cent work has been conducted to develop methodologies forassimilating such data into numerical models, and today twodifferent approaches are under study. The first (which is notexamined in this paper) assimilates pseudo-observations ofvelocity fields produced from image sequences using the mo-

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2 O. TITAUD, J.-M. BRANKART, J. VERRON

tion estimation technique (Horn and Schunck, 1981; Herlinet al., 2004). This approach takes into account little physi-cal information about the underlying processes that are ob-served in the image. The second approach, referred to asDirect Image Assimilation (DIA) in this paper, directly as-similates the image data into the dynamic model: in Corpettiet al. (2009) the image pixel is assimilated as a tracer con-centration measurement; in Titaud et al. (2010) the struc-tured information extracted from the image are assimilatedusing an adapted observation operator. The latter does notuse the pixel basis of the image but higher levels of inter-pretation (e.g. a multi-scale decomposition or a descriptionof contours). It requires observation operators that link thestructured information from the image with the model out-put.

In this paper we adopt the second approach andshow the relevance of using Finite-Time Lyapunov Expo-nents (FTLE) (Pierrehumbert and Yang, 1993) in the DIAframework. FTLE is a Lagrangian tool that was intro-duced to characterise coherent structures in time-dependentflows. Together with Finite-Size Lyapunov Exponents (Ar-tale et al., 1997; Aurell et al., 1997) FTLEs are widelyused in oceanography to link ocean tracer distribution withmesoscale geostrophic currents in order to study stirringand mixing processes (Abraham and Bowen, 2002; d’Ovidioet al., 2004; Lehahn et al., 2007). The main objective of ourpaper is to show, in an idealised, experimental, but fairlyrealistic test case, that high-resolution ocean tracer imagesmay be exploited to correct larger scale velocity fields us-ing a DIA framework. We suggest using sophisticated obser-vation operators based on the computation of Finite-TimeLyapunov Exponents (FTLE) and Finite-Time LyapunovVectors (FTLV). Their construction does not require the nu-merical advection of a synthetic passive tracer, as was thecase in Titaud et al. (2010).

The paper is organised as follows: in section 2 we brieflydescribe the test case used in our experiments. In section 3we review the definitions of FTLE and FTLV (FTLE-V)associated with a time-dependent flow, and point out theproperties that are relevant to this work. The use of thisLagrangian tool in a DIA framework is described in section4. In section 5 we detail the methodology of our study andthen present and discuss the main results. The conclusionsof the study are presented in section 6.

2 Test Case

Our study is based on a one-year simulation of an ide-alised bio-physical model of a sub-region of the North At-lantic Ocean. The coupled model (Kremeur et al., 2009, de-scribed in) is applied to a double-gyre configuration thatmimics the North Atlantic Ocean circulation and its west-ern boundary current system. A 1/54 high-resolution ver-sion of this configuration (referred as R54 in this paper) hasbeen carried out by Levy et al. (2009; 2010). It is of inter-est here because it emphasises in particular the importanceof submesoscale dynamics. The physical component of themodel is based on the NEMO-OPA code which is quite astandard ocean global circulation model based on primitiveequations (Madec, 2008). The biogeochemical componentcomes from the NEMO-TOP2 passive tracer engine based on

the rather simple six-compartment LOBSTER model (Levyet al., 2001). The vertical resolution of the model is non-uniform, with 5 m at the top and 250 m at the bottom.We are particularly interested in the respective dynamicaland biogeochemical tracers such as temperature and phy-toplankton. As we are concerned with the application ofsatellite images, we focus on the surface activity.

The reference domain Ω of our study is a 6×6 squaresub-domain of the full double-gyre configuration. This do-main is located between longitudes −74.62E and −68.62Eand latitudes 22.36N and 28.36N:

Ω = [−74.62,−68.62]× [22.36, 28.36]. (2)

This region is located in the southeast recirculation branchof the – idealised – Gulf Stream (see Figure 2, left panel: Ω ismarked by a dashed box). It is characterised by strong eddyactivity associated with fairly active submesoscale patterns.It is typical of the strong turbulent activity associated withthe mid-latitude non-linear and unstable jets of the worldocean. We arbitrarily choose day k0 = 99 to be the referencedate which corresponds to April, 9. SST and Mixed LayerPhytoplankton (MLP) fields show the main characteristicpatterns that can be found in ocean tracer fields (see Fig-ure 4): a mesoscale eddy is located on the northeast side ofthe domain, and a front on the east side; submesoscale fila-ments are also observed. The results presented in this paperare independent of the reference date and the sub-domainΩ: other experiments with other reference dates and sub-domains have resulted in the same conclusions (not shown).Note that MLP is closely linked with Ocean Colour imageproducts.

3 Finite-Time Lyapunov Exponents and Vectors

3.1 Definition

The transport of a tracer in a fluid is closely related toemergent patterns that are commonly referred to as coher-ent structures. For time-independant dynamic systems, theycorrespond to stable and unstable manifolds of hyperbolictrajectories (Wiggins, 1992). Coherent structures delimitregions of whirls, stretching, or contraction of tracer (Ot-tino, 1989): contraction is observed along stable manifoldswhereas unstable manifolds correspond to divergent direc-tions along which the tracer is stretched. Stable and un-stable manifolds are material curves that act as transportbarriers : they cannot be crossed by fluid particles. General-izing these concepts to flows with general time dependenceis not straightforward and is still the subject of an intenseresearch. Studies that concern theoretical and practical as-pects of coherent structures deal with their rigorous defini-tion and their detection. When coherent structures are de-fined using fluid trajectories they are often called LagrangianCoherent Structures (LCS). Haller and Yuan (2000) definea LCS as a material curve (more precisely a material sur-face in an extended phase space) which exhibits locally thestrongest attraction, repulsion or shearing in the flow overa finite-time interval. A rigorous mathematical theory thatfits with this physical concept was recently developped inHaller (2011) where quantitative and robust critera are givento identify hyperbolic (i.e. repelling and attracting) LCSs.However, and despite some caveats detailed in Haller (2011),

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FTLE-V FOR DIRECT IMAGE ASSIMILATION IN OCEAN MODELS 3

LCSs are usually indentified in a practicall manner as max-imizing ridges of Finite-Time Lyapunov Exponents (FTLE)field (Haller, 2001a;b; 2002; Shadden et al., 2005; Mathuret al., 2007). FTLE is defined as the largest eigenvalue ofthe Cauchy-Green strain tensor of the flow map (see belowfor more details). The corresponding eigenvector is calledFinite-Time Lyapunov Vector (FTLV) in this paper. FTLEand the orientation of FTLV constitute the key tools used inthis work. FTLEs are widely used to characterise transportprocesses in oceanographic flows where the velocity field isonly known as a finite data set (Beron-Vera et al., 2008;Beron-Vera and Olascoaga, 2009; Beron-Vera et al., 2010;Lekien et al., 2005; Olascoaga, 2010; Olascoaga et al., 2008;Shadden et al., 2009). Even if not used in this paper we alsomention the Finite-Size Lyapunov Exponents (FSLE) (Au-rell et al., 1997; Artale et al., 1997) which is another La-grangian tool that is commonly used in oceanographic con-texts for studying mixing processes (d’Ovidio et al., 2004;2009a;b; Lehahn et al., 2007, and references therein). Werefer to Peacock and Dabiri (2010) who review the use ofFTLE for detecting LCSs in other contexts.

Here, we briefly review the definitions of FTLE andFTLV before pointing out certain properties that we exploitin this paper. FTLE is a scalar local notion that representsthe rate of separation of initially neighboring particules overa finite-time window [t0, t0+T ], T 6= 0. Let x(t) = x(t; x0, t0)be the position of a Lagrangian particle at time t, started atx0 at t = t0 and advected by the time-dependent fluid flowu(x, t), x ∈ Ω, t ∈ [t0, t0 + T ]. Thus we have:8<:

Dx(t)

Dt= u(x(t), t),

x(t0) = x0.(3)

An infinitesimal perturbation δx(t) started at t = t0 fromδ0 = δx(t0) around x0 then verifies, for all t ∈ [t0, t0 + T ],8<:

Dδx(t)

Dt= ∇u(x(t), t).δx(t),

δx(t0) = δ0, x(t0) = x0.(4)

The Forward Finite-Time Lyapunov Exponent at a pointx0 ∈ Ω and for an advection time T is defined as the growthfactor of the norm of the perturbation δx0 started aroundx0 and advected by the flow after the finite advection timeT . Maximal stretching occurs when δx0 is aligned with theeigenvector associated with the maximum eigenvalue λmax

of the Cauchy-Green strain tensor:

∆ =h∇φt0+T

t0(x0)

i∗ h∇φt0+T

t0(x0)

i, (5)

The flow map of the system (3) φtt0 : x0 7→ x(t; x0, t0)links the location x0 of a Lagrangian particle at t = t0 toits position x(t; x0, t0) at time t 6= t0. This eigenvector isreferred to as the forward Finite-Time Lyapunov Vector andwe denote it by ϕt0+T

t0(x0). We thus have

maxδx0‖δx(T )‖ =

pλmax(∆)‖δx0‖, (6)

where δx0 is aligned with ϕt0+Tt0

(x0). Finally, the forwardFTLE at the point x0 ∈ Ω and for an advection time Tstarting at t = t0 is defined as

σt0+Tt0

(x0) =1

|T | lnpλmax(∆). (7)

We are particularly interested in backward FTLE fieldwhose ridges approximate attracting LCSs (Haller, 2011).Backward FTLE-Vs are similarly defined, with the time di-rection being inverted in (3). For more details on the prac-tical computation of FTLE-V see e.g. Shadden et al. (2005;2009) and Ott (1993) for any types of flows and d’Ovidioet al. (2004) for oceanic flows.

As previously mentioned, FTLE (FTLV) is a scalar(vector) that is computed at a given location x0. Seed-ing a domain with particules initially located on a gridleads to the computation of a discretized scalar (FTLE)and vector (FTLV) fields. Our study focuses on the sen-sibility of the distribution of FLTE-V over the test case do-main Ω with respect to small perturbations of the time-dependent flow u along a fixed time period [t0, t0 +T ]. Con-sistently with this variational approach, we will later con-sider the operators Σ[u] : x ∈ Ω → σt0+T

t0(x) ∈ R and

Φ[u] : x ∈ Ω → ϕt0+Tt0

(x) ∈ R2. These operators map theflow u with the FTLE-V distribution over Ω. The depen-dence of this map to the window [t0, t0 +T ] is not studied inthis paper: it is fixed in our experiments as it should be in anoperational context (see section 5.1.1). We then chose notto mention this dependence explicitly in order to simplifynotations. Shadden et al. (2005) discuss the dependence ofFTLE to the time advection T and its consequence on theproperties of LCS marked by FTLE ridges.

An example of backward FTLE and correspond-ing backward FTLV orientation maps, computed from amesoscale time-dependent velocity field, is given in Figure3.

3.2 Connection between FTLE-V and tracers fields

In this paper we exploit two properties of backwardFTLE-V to construct adapted observation operators for di-rect ocean tracer image assimilation (Titaud et al., 2010).

The first property concerns the contours of FTLE maps.Backward FTLE fields show contours that correspond rea-sonably well to the main structures such as filaments, frontsand spirals that appear in geophysical and bio-geochemicaltracer fields (Beron-Vera et al., 2010; Shadden et al., 2009;Olascoaga et al., 2008; 2006). Figure 3 (left panel) showsa backward FTLE field (7) in the reference domain andat the reference date of our study. This field is computedfrom a sequence of mesoscale (1/4) time-dependent velocityfields filtered from the R54 test case simulation (see Section5.1.1 for more details of the computation settings for our ex-periments). The main geometrical patterns of this field areclearly visible in the corresponding SST and MLP fields ofthe R54 simulation (i.e. same reference domain and date):see Figure 4. In this study, we adopt the direct image assim-ilation approach described in Titaud et al. (2010): first, wewill consider FTLE field as it were a tracer field which inturn can be considered as an image. This makes sense be-cause of the almost-Lagrangian nature of FTLE (Shaddenet al., 2005):

dσt0+Tt0

(x0)/dt0 = O(1/|T |). (8)

Second, we are exclusively interested in the location andshape of the structures that appear in the FTLE and tracerfields: we do not exploit the value (pixel intensity) of theexponents or the concentration.

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The second property concerns the orientations ofFTLVs. Lapeyre (2002) shows that for a freely decaying 2Dturbulence flow, the orientation of the gradient of of a pas-sive tracer converges to that of backward FTLVs. This dy-namic behaviour of the tracer concentration gradient ∇q isexplained by the fact that k × ∇q verifies the same equa-tion (4) as δx does (k being the unit vertical vector). Suchalignment properties have also been observed for realisticoceanic flows and tracers (d’Ovidio et al., 2009b). Figure 5shows the dynamic alignment of SST and MLP gradientswith FLTV: the root mean square angular error betweenbackward FTLV orientations computed at a given date andtracer gradient field orientations decreases with time. Fig-ure 3 (right panel) shows the orientations of the backwardFTLV that correspond to the aforementioned FTLE field(left panel). In our idealised experiment, SST and MLP gra-dients also show similar orientations: see Figure 6.

These two properties (pattern matching between tracerand FTLE scalar fields and tracer gradient orientation align-ment with FTLV orientation) are the first interesting prop-erties that can be exploited in a DIA framework. Further-more, Beron-Vera et al. (2010); Beron-Vera (2010) showed,using real data, that these properties remain valid with amesoscale advection, i.e. when the resolution of the velocityfield — from which FTLE-V are computed — is much lowerthan the resolution of the observed tracer field. This be-haviour was also mentionned using another Lagrangian tool(FSLE) by Lehahn et al. (2007); d’Ovidio et al. (2009b) .Our synthetic data exhibit the same behaviour: FTLE fieldof Figure 3 is computed from a mesoscale 1/4 velocity fieldon the same high-resolution (1/54) grid as that used tocompute the SST and MLP fields of Figure 4. This featureis crucial from both a practical and physical point of view:first, velocity fields obtained from ocean global circulationmodels do not often provide more than mesoscale informa-tion, whereas tracer images contain submesoscale informa-tion; second, FTLE-V may be used to quantify and charac-terise the link between scales.

4 Direct image assimilation using FTLE-V

4.1 General framework

From a practical point of view, an image is an array ofnumbers known as pixels, but from a mathematical point ofview it is more convenient to model an image as a boundedreal valued function of two bounded continuous variables.This provides a powerful and rigorous framework with whichto process or analyse them (Chan and Shen, 2005; Aubertand Kornprobst, 2006). In the following, in order to sim-plify our discussion, we employ the same notation for thediscretised and the continuous version of the operators werespectively apply to the numerical and modelled image. Inparticular, the gradient operator ∇ is approximated by afirst-order finite difference operator. In the following, IΞ de-notes the set of discrete real positive valued functions definedon a rectangular grid Ξ = [[1, n]]× [[1,m]] ⊂ N2 (more simply,IΞ is the set of n × m numerical images). The continuousversion of IΞ is denoted by I.

As in the case of human vision, the relevant informationcontained in an image is mainly provided by its discontinu-ities (Marr, 1982). This is why image assimilation focuses

on the characterisation of edges (or other structured infor-mation from the image) instead of the pixel values of theimage. In our context, the pixel value of a satellite image (ormore precisely a satellite product given in an image form)may correspond to the concentration of certain tracers. Weemphasise that we are not interested in the assimilation ofsuch measurements as in the classical cost function (1), butin the assimilation of the structured data that can be ex-tracted from the image. With this in mind Titaud et al.(2010) suggest adding an image term to the classical costfunction. This takes the general form

JS(X) = ‖HS(X)− ys‖2S , (9)

where ys ∈ S represents the structures extracted from theobserved image c ∈ I, ‖ · ‖S is a measure of the discrepancybetween the elements of S, and HS is the correspondingstructure observation operator. It maps the state variablespace X onto the observation space S. JS is the DIA termassociated with the triplet (HS ,S, ‖ · ‖S). In practice, ys isobtained using a structure extraction operator S that actson the actual observed image c: ys = S(c). This operator isnot part of the assimilation procedure and may be definedusing image analysis techniques.

In the following subsections, we define two DIA triplets(HS ,S, ‖ · ‖S) that rely respectively on the aforementionedproperties of FTLE and FTLV. In each case, we first definethe structure space S and an appropriate discrepancy mea-sure ‖ · ‖S . The structure extraction operator S is definedafterwards. We finish with the definition of the structureobservation operator HS . We show the relevance of thesetriplets in section 5.

4.2 FTLE-based observation operator

We saw in section 3.2 that FTLE fields exhibit similarcontours to those in the corresponding ocean tracer fields.These contours mark the edges of the most relevant featuresin these images, such as filaments, fronts, and eddies. Thisproperty (pattern matching) motivates the use of a spaceof contours E (which stands for “Exponent”) to constructa DIA triplet (HE , E , ‖ · ‖E). This space is simply definedas the set of binarised images, that is, the subspace of IΞ

whose elements take one of only two values (0 or 1):

E =nf : Ξ → 0, 1

o. (10)

The distance between two elements f and g of E is given bythe Root Mean Square (RMS) norm:

‖f − g‖E =

s1

n×mX

(i,j)∈Ξ

|f(i, j)− g(i, j)|2. (11)

A basic technique for a rough extraction of edges from animage is based on the binarisation of its gradient norm usinga hard threshold. We define the structure extraction opera-tor E applied to an image c ∈ IΞ by: for all (i, j) ∈ Ξ

E(c)(i, j) =

1 if ‖∇c(i, j)‖ > ε,0 else,

(12)

where ε > 0 is a given threshold parameter. Structures ex-tracted from the observed tracer image involved in (9) arethen yE = E(c) in this case. Once we have defined the struc-ture space and its associated discrepancy measure between

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two elements, we need to define the corresponding obser-vation operator HE that maps the state variable space Xonto the observation space E . In this case, we are interestedin comparing the contours in the tracer and the backwardFTLE fields (7) at a reference date t0. We then define theobservation operator as the binarisation of the backwardFTLE gradient norm field, computed from the modelledtime-dependent velocity field u, at the reference date t0:

HE(X) = E(Σ[u]), (13)

where the backward FTLE is viewed as an image operatoracting on the flow field onto the image space I:

Σ[u] : Ω → Rx 7→ σt0+T

t0(x).

(14)

Finally, the FTLE-based DIA cost function (9) associatedwith the triplet (HE , E , ‖ · ‖E) is written as

JE(u) = ‖Eε′`Σ[u]

´− Eε(c)‖2E , (15)

in which we make the dependence of E on ε in (12) explicit,because it does not necessarily take the same value for thetracers and the FTLE fields.

Technical details on the computation of the FTLE fieldsand the choice of the threshold parameter ε are given in sec-tion 5.1.1. The edge extraction technique (12) together withdistance (11) can be used to obtain a rough comparison ofthe location of fronts and patches on two images. Thoughmore sophisticated image contour extraction techniques ex-ist (see review of Chan and Shen, 2005), the results obtainedwith this basic method are nevertheless promising. Figure 7shows the corresponding binarisation of the gradient of SST,MLP and backward FTLE fields presented in Figures 4 and3 respectively. We can clearly see the relatively good matchbetween the binarised FTLE and tracer gradient fields.

4.3 FTLV-based observation operator

In section 3.2 we noticed that the orientations of thegradients of an ocean tracer image (or concentration field)are almost aligned with the backward FTLV orientations ofthe corresponding flow where the tracer is advected. Thisproperty motivates the use of backward FTLV as a tool forbuilding an observation operator that links the velocity fieldof an ocean model to ocean tracer images. We will constructthe corresponding FTLV-based DIA triplet in the same wayas we constructed the FTLE-based DIA triplet (HE , E , ‖ ·‖E) in the previous section, starting with the definition ofthe structure space: as we want to compare the orientationsof tracer image gradients with the FTLV orientations, wedefine the structure space V (which stands for “Vector”) asthe space of functions on Ξ with values in S2, the Euclideanunit sphere of R2:

V = f : Ξ→ S2. (16)

The discrepancy between two elements f and g of V is de-fined by

‖f − g‖V =

s1

n×mXi,j∈Ξ

sin2ˆΘ`f(i, j)

´−Θ

`g(i, j)

´˜,

(17)

where Θ(w) represents the orientation of an element of w =(w1, w2) ∈ S2:

Θ(w) = arctan(w2/w1) ∈h−π

2,π

2

i. (18)

‖f − g‖V measures the angular discrepancy between thestraight lines whose normal directions are respectively f andg. We therefore emphasise that we are interested in compar-ing the orientations of two vectors, independently of theirdirections. We define the observed structures in the tracerimage c as its normalised gradient field. We define for all(i, j) ∈ Ξ

yV(i, j) = V(c)(i, j) =∇c(i, j)‖∇c(i, j)‖ ∈ S

2. (19)

As we want to compare the fields of tracer gradientorientations with the backward FTLV orientations, we definethe observation operator as

HV(X) = Φ[u], (20)

where the backward FTLV field is computed at the referencedate t0. It is considered as an operator acting on the time-dependent flow field u onto V:

Φ[u] : Ω → S2

x 7→ ϕt0+Tt0

(x).(21)

Examples of FTLV orientation and tracer orientation fieldsare shown in Figures 3 (right panel) and 6 respectively. Fi-nally, the FTLV-based DIA cost function (9) associated withthe triplet (HV ,V, ‖ · ‖V) is written as:

JV(u) = ‖Φ[u]− V(c)‖2V . (22)

5 Sensitivity studies

In this section, we experimentally show that backwardFTLE-V can be used in a direct image assimilation frame-work to correct a larger scale velocity field from a high-resolution ocean tracer field showing submesoscale patterns.

5.1 Methodology

5.1.1 Computation settings We want our framework tobe close as possible to that of direct assimilation of high-resolution satellite images into eddy-resolving ocean models.In order to construct a consistent sequence of mesoscale ve-locity fields from the R54 simulation described in section 2,we apply a space Lanczos filter to the surface model velocityfield. The filter length is chosen so as to preserve mesoscalefeatures. The result is interpolated on a 1/4 horizontal res-olution grid. We thus obtain a velocity sequence that rep-resents a mesoscale view of the corresponding original sub-mesoscale simulation. The mesoscale velocity field obtainedcan also be viewed as a mesoscale simulation with an ideallyparameterised 1/54 submesoscale physics. We should em-phasise that the sequence of velocity fields u is only knownas a finite-time data set: it represents a finite sequence ofdiscrete velocity fields (uk)k0k=k0−T , k0 being the referencedate and T the advection time.

FTLE-V of the above sequence of filtered mesoscale ve-locity fields (uk)k0k=k0−T are computed backward in time,

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from the reference date k0 = 99, on the same 1/54 horizon-tal resolution grid as the tracers fields. The backward advec-tion of the particles is performed using a fourth order Runge-Kutta scheme during a time period T = 10 days which corre-sponds to an eddy turnover time scale. The Cauchy-Greenstrain tensor (5) is approximated using a finite differencescheme. The initial separation parameter δ0 in Equation (4)is chosen so that it is equal to the mesh size of the tracergrid, as suggested in d’Ovidio et al. (2004): δ0 = 1/54.

The threshold parameter ε involved in the definitionof the structure extraction operator E in (12) is implicitlychosen such that a given percentage p of the values of theimage gradient norm are larger than ε. This is a classicaltechnique that allows p to be set regardless of the image c.In our experiments, we set p = 0.8.

5.1.2 Velocity perturbations modelling A prerequisite fora triplet (H,O, ‖ · ‖O) to be used to assimilate a data sety ∈ O is that the corresponding cost function (1) has tobe sensitive to perturbations of the state variable X ∈ X .More specifically, we expect that the cost function admits aminimum value at the non-perturbed state. In our experi-mental framework, we chose to study the behaviour of thecost function (9) with respect to the amplitude λ ∈ R of aperturbation δX. In other words, we are interested in thevariation of the following sensitivity function

λ ∈ R 7→ ‖HS(X + λδX)− yS‖. (23)

We expect it to be sufficiently smooth because standarddescent-type minimisation algorithms need such conditionsto be efficient. Defining a set of perturbations is in generalnot a straightforward task and this is why we will consider aspecific perturbation model. As this study is a preliminarystep toward data assimilation, we construct a set of pertur-bations that is fairly consistent with the errors dealt withby such methods. We focus in particular on velocity errorsbecause we want to show that structured information fromocean tracer images can be used to correct velocity fields.

Let (uk)k=1,m+1 be the sequence of the one-year simu-lation mesoscale surface model velocity fields obtained fromour test case, as described in the previous subsection. Inthis experiment, m = 209. We perform a principal compo-nents analysis using Empirical Orthogonal Functions (EOF)to decompose the signal, as used in Carme et al. (2001):

uk = u +

mXl=1

α(l)k u(l), (24)

where u is the time average velocity map. We define theclimatological covariance matrix as follows:

P =1

m

m+1Xk=1

(uk − u)(uk − u)∗. (25)

The EOFs u(l), l = 1,m, are equal to the normalised eigen-vectors of P times the square root of the correspondingeigenvalues. The time amplitudes α(l) are normalised vec-tors. Figure 8 shows the inertia of the signal: the first EOFaccounts for 10% of the total variance and 100 EOFs arerequired to account for 98% of total variance. Let S be thecolumn matrix of the first r = 100 EOFs; SS∗ represents

the reduced rank square root representation of the covari-ance matrix P. We consider the normally distributed veloc-ity perturbations with zero mean and covariance SS∗ of thesequence:

δu ∼ N (0,SS∗). δu =

rXl=1

u(l)δγl with δγl ∼ N (0, 1). (26)

Assimilation schemes based on linear analysis (whichare the most widely used and studied at the moment) can beshown to be optimal if the error sources (on the backgroundstate and on the observations) are assumed to be Gaussian,with known covariances. In the absence of more accurateinformation, the background error covariance is often mod-elled using the covariance of system variability (Pham et al.,1998). The Gaussian velocity perturbations in (25) thus rep-resent a reasonable model of the errors that such a classicalassimilation scheme can rectify. Two examples of random ve-locity perturbations δu with the above Probability DensityFunction (PDF) (26) are illustrated in Figure 9. In a sequen-tial assimilation scheme, the state variable is updated eachtime an observation is available. Let us consider that an ob-servation (an ocean tracer image in our case) is available forour reference date k0 and that the state variable has alreadybeen optimally evaluated. We may thus focus exclusively onperturbations of the velocity field uk0 at this reference date,that is, when this field needs to be corrected by the filter-ing algorithm. As backward FTLE-V are computed from atime-dependent velocity field, and in order to keep the nota-tions consistent with those used in (14) and (21), we definea perturbed velocity field sequence uλ = (uλk)k0k=k0−T by

uλk =

uk if k < k0,uk0 + λδu else,

(27)

where δu is a random perturbation with PDF (26) and λ ∈Λ ⊂ R is a bounded amplitude. A given realisation of therandom variable (26) is amplified and then applied to thereference date velocity field to form a realisation of a randomperturbed velocity field sequence.

5.2 Results and discussion

In this section we present and discuss the variation ofthe sensitivity cost function (23) associated with each of theFTLE-V based DIA triplets

E? = (HE , E , ‖ · ‖E), (28)

V? = (HV ,V, ‖ · ‖V), (29)

whose elements are defined respectively in section 4.2and 4.3. The variation is studied with respect to a boundedamplitude λ ∈ Λ ⊂ R of the perturbed state uλ given by(27). Data are provided respectively by the correspondingimage structure operators (12) and (19) acting on the ob-served tracer image c at the reference date. We study theparticular cases of SST and MLP concentration fields, de-noted respectively by cT and cP . This leads to the definitionof the following four sensitivity functions:

JTE (λ) = ‖HE [uλ]− E(cT )‖2E , (30)

JTV (λ) = ‖HV [uλ]− V(cT )‖2V , (31)

JPE (λ) = ‖HE [uλ]− E(cP )‖2E , (32)

JPV (λ) = ‖HV [uλ]− V(cP )‖2V , (33)

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defined for a given realisation of the perturbed state uλ.

Figure 10 shows the variation of the sensitivity functions(30) and (31) corresponding respectively to the FTLE-basedtriplet E? (28) and the FTLV-based triplet V? (29). Resultsfor nine realisations of the random perturbations (27) areplotted with amplitudes in the range λ ∈ [−4, 4]. Data comefrom the corresponding structured observations E(cT ) andV(cT ) of the SST fields shown in Figure 4 (left panel). Fig-ure 11 shows the corresponding results when data come fromthe MLP concentration field shown in Figure 4 (right). Wemay first remark that each of the sensitivity functions (30 –33) admits a global minimum for the nine random perturba-tions. Moreover, minima are generally reached around λ = 0,which corresponds to the non-perturbed state u0. These twobasic aspects clearly show that FTLE-V based triplets aregood candidates for use in ocean tracer image assimilationschemes. More careful examination of these results revealssome important points that merit discussion. Firstly, we maynote that each realisation of a given sensitivity function ex-hibits a convex shape around the minimum, which is knownto be a good situation for minimisation algorithms. Fromthis point of view, the FTLV-based triplet shows smoothervariations than the FTLE-based triplet. This is due to theregular behaviour of the corresponding angular mismatchmeasure (17). It may thus offer better conditions for thedescent-type minimisation algorithms widely used in practi-cal applications. One further advantage of the FTLV-basedtriplet is that it does not depend on any additional “tuning”parameter like the FTLE-based triplet with the threshold in-volved in the structure extraction operator (12). Even if theresults associated with the FTLE-based triplet depend verylittle on this parameter — if taken in a reasonable range —this feature should make FTLV-based triplets more robustin practical applications. Secondly, we may observe that theminimum values are not zero. This is not surprising becausethe Lagrangian tool is known to provide only an incom-plete representation of the SST and MLP dynamics. Note,however, that for our application, this is not unsatisfactory.Several reasons can be put forward to explain why FTLE-Vstructures are not an exact match with the correspondingones in tracer images for the non-perturbed state. The mainreason is probably because ocean tracers such as SST andMLP have their own dynamics that cannot be observed bythe Lagrangian tool. The high-resolution tracer gradientsalso depend on submesoscale dynamics; these dynamics arenot taken into account in the computation of FTLE-V be-cause they are computed from a mesoscale field. In addi-tion, FTLE-Vs have been computed at the ocean surfaceand we know that patterns in ocean colour images (MLPfield) are a surface signature of a three-dimensional process.The underlying dynamics also intervene in the formationof these patterns. These aspects, however, are beyond thescope of this paper. Nevertheless, we may note that a zeroglobal minimum value is effectively reached if FTLE-Vs areobserved in a precise manner. This is shown in Figure 12where sensitivity functions (23) have been computed withexact model counterpart data, respectively yS = HE(u)and yS = HV(u). Finally, we should point out that somerealisations of the sensitivity functions do not reach theirminimum at zero (i.e. at the non-perturbed state). This isparticularly the case for the FTLE-based triplet, the worsebeing with MLP data (see Figure 11, left panel). We also ob-

serve the same problem with this tracer for the FTLV-basedtriplet, but it is less marked. Such behaviour reveals that thedata assimilation problem is not well-posed in the Hadamardsense, a situation quite common with such inverse problems.To overcome this difficulty, assimilation procedures take intoaccount a-priori information about the system that regu-larise the problem. We should emphasize here that we onlystudied the image part of the complete data assimilation costfunction. We can reasonably expect that the correspondingfull sensitivity cost-function (which will include a regulari-sation term) will exhibit more regular behaviour.

FTLE-Vs allow us to link mesoscale information fromthe dynamic part of the model state variable with subme-soscale information from the high-resolution tracer fields(images). Our experiments show that, in an idealised case,FTLE-V may be used to directly assimilate high-resolutionocean tracer images into mesoscale eddy-resolving oceanmodels. An approximate edge detection/location techniqueused together with the FTLE-based observation operator issufficient for the sensitivity function to exhibit a minimumvalue around the reference velocity field, both for SST andMLP concentration images. Observations of gradient orien-tation show more regular behaviour if used together with theFTLV-based observation operator. Minimisation algorithmsshould be more efficient in this situation.

6 Conclusions

In this paper we have shown that, in an idealised ex-perimental test case, ocean tracer images can be exploitedto correct surface model velocities. When viewed as im-ages, SST and MLP fields show relevant features such aseddies, fronts and filaments. Such structured observationscan be assimilated using a DIA technique. For this purposewe constructed sophisticated observation operators basedon FTLE-V computation. These Lagrangian tools are wellsuited to link mesoscale flow dynamics with the subme-soscale information from ocean tracer images. LagrangianCoherent Structures represent an elegant alternative to syn-thetic passive tracer advection in constructing image assimi-lation observation operators. This kind of operator was usedin Titaud et al. (2010) as an example to show the feasibilityof direct image assimilation in numerical models. Neverthe-less, numerical advection is known to smooth the disconti-nuities. This feature could be a drawback for image assim-ilation, but FTLE-V based operators clearly overcome thisproblem. The relevance of FTLE-V in a data assimilationframework has been shown using a sensitivity study. Specificobjective functions exhibit favourable behaviours for a set ofvelocity errors. Our perturbation model based on one-yearclimatological statistics provides a reasonable representationof the errors that common assimilation schemes have to dealwith. Even if the FTLE-V structures cannot perfectly matchthe corresponding structures in the ocean tracer fields underconsideration, these results clearly indicate that it is possi-ble to assimilate high-resolution ocean tracer images intomesoscale models using FTLE-V-based observation opera-tors. A more careful examination of the results reveals thatFTLV-based observation operators seem more appropriatethan FTLE-based one for our application. Nevertheless, amore complete comparison should be conducted to provide

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a more precise answer to this question. For instance, we mayexploit the almost-Lagrangian nature of FLTE (8) to iden-tify the FTLE ridges better. Indeed, it has been shown byHaller (2011) that FTLE gradient and FTLE-V are perpen-dicular near an FTLE ridge that marks a hyperbolic LCS.Merging FTLE and FTLV properties into a single observa-tion operator may help to constrain the data assimilationsystem better. As expected in data assimilation, our prob-lem is not well-posed (in a mathematical sense), leading tocertain defects in some of the realisations of random errors,especially for the FTLE-based operator. This can be over-come (at least partially) by regularising the minimisationproblem.

This study is based on synthetic data that were notaltered to test the robustness of the method with respectto observation errors, because it is a first step towards theassimilation of ocean tracer images using FTLE-V. Settingup exhaustive experiments to check such robustness is nota straightforward task because observation errors are of dif-ferent types (measurement error, representativeness errors).For that matter, direct image assimilation is quite unusualwith respect to this issue. Indeed, images are spatially struc-tured data (and temporally structured if image sequencesare considered), which makes correlations of observation er-rors hard to model. Furthermore, even though they are cru-cial questions, they are beyond the scope of this paper andshould therefore be the subject of future research.

Finally, we found FTLE-V well suited to satellite imageassimilation and the results obtained were very promising.We believe the work presented here also enhances the valueof satellite-image products, which are becoming increasinglynumerous, with improvements in coverage, frequency andresolution.

7 Acknowledgments

We would like to thank Marina Levy who providedus with high-resolution bio-dynamical GYRE simulation re-sults of the idealised North Atlantic basin. This simulationhas been produced with the support of the Earth Simula-tor Center, Yokohama Institute for Earth Sciences, JAM-STEC, Japan. We also thank Francesco d’Ovidio who pro-vided us with the code lamta0.3 for computing FTLE-Vs.This work is part of the ADDISA project (reference ANR-06-MDCA-001) supported by the Agence Nationale de laRecherche (French National Research Agency). Conductingthis research would also not have been possible without thecontinued support of the CNES (French Space Agency). Thefirst author would also like to thank the RTRA STAE sci-entific foundation for its support in this work. We thankanonymous reviewers for their pertinent suggestions.

The first author dedicates this paper to Alain Largillier.

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LIST OF FIGURES

1 Sea Surface Temperature (left panel) and Ocean Colour (right panel) images from the products of AQUAMODIS remote instrument showing structured patterns such as fronts and filaments off the coast of Argentina.Convergence of the southward flowing Brazil and northward flowing Malvinas currents, May 2, 2005. Courtesy: NASA.2 Left panel: SST (C) field of an idealised simulation of the North Atlantic Ocean using the 1/54 horizontalresolution model of Levy et al. (2009). Our study region is contained in the dashed box. Right panel: surface modelmesoscale velocity field (m/s) filtered from this simulation in the domain and at the reference date of our study.3 Backward FTLE (day−1) (left) and corresponding backward FTLV orientations (angular degree) (right) com-puted in the reference domain and at the reference date of our study. These fields are computed from a mesoscale(1/4) time-dependent velocity fields sequence filtered from the R54 test case simulation.4 SST (C) (left) and MLP concentration (mmole-N/m3) (right) fields of the R54 test case simulation in thereference domain and at the reference date of our study. These tracer fields show pronounced typical patterns suchas an eddy (North East corner), a front (East side) and filaments.5 Dynamic alignment of SST (left) and MLP (right) gradient orientations with FTLV orientations: in both cases,angular RMS between tracer gradient and FTLV orientations computed at the reference date 99 decreases with time.6 Orientations (angular degree) of the gradients of the SST (left) and MLP (right) fields shown in Figure 4.7 Binarisation of the SST (left), MLP (centre) and FTLE (right) gradient fields of Figures 4 and 3 using thethreshold based technique (12).8 Percentage of resolved variance as a function of the number of EOF.9 Example of two random velocity perturbations with the PDF given by (26).10 Behaviour of sensitivity functions for SST.Left: Variation of the sensitivity function JTE (30) associated with the FTLE-based triplet E? (28) with respect tothe amplitude λ of nine random perturbations (27) applied to the reference velocity field uk0 .Right:Variation of the sensitivity function JTV (31) associated with the FTLV-based triplet V? (29) with respect tothe amplitude λ of nine random perturbations (27) applied to the reference velocity field uk0 .11 Behaviour of sensitivity functions for MLP.Left: Variation of the sensitivity function JPE (32) associated with the FTLE-based triplet E? (28) with respect tothe amplitude λ of nine random perturbations (27) applied to the reference velocity field uk0 .Right: Variation of the sensitivity function JPV (33) associated with the FTLV-based triplet V? (29) with respect tothe amplitude λ of nine random perturbations (27) applied to the reference velocity field uk0 .12 Variation of the sensivity function (23) computed with exact model counterpart data yS = H(X). Variationare computed with respect to the amplitude λ of nine random perturbations (27) applied to the reference velocityfield uk0 .Left: FTLE-based triplet E? (28).Right: FTLV-based triplet V? (29).

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Figure 1. Sea Surface Temperature (left panel) and Ocean Colour (right panel) images from the products of AQUA MODIS remoteinstrument showing structured patterns such as fronts and filaments off the coast of Argentina. Convergence of the southward flowing

Brazil and northward flowing Malvinas currents, May 2, 2005. Courtesy: NASA.

Figure 2. Left panel: SST (C) field of an idealised simulation of the North Atlantic Ocean using the 1/54 horizontal resolution modelof Levy et al. (2009). Our study region is contained in the dashed box. Right panel: surface model mesoscale velocity field (m/s) filtered

from this simulation in the domain and at the reference date of our study.

Figure 3. Backward FTLE (day−1) (left) and corresponding backward FTLV orientations (angular degree) (right) computed in the

reference domain and at the reference date of our study. These fields are computed from a mesoscale (1/4) time-dependent velocityfields sequence filtered from the R54 test case simulation.

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Figure 4. SST (C) (left) and MLP concentration (mmole-N/m3) (right) fields of the R54 test case simulation in the reference domain

and at the reference date of our study. These tracer fields show pronounced typical patterns such as an eddy (North East corner), a front(East side) and filaments.

Figure 5. Dynamic alignment of SST (left) and MLP (right) gradient orientations with FTLV orientations: in both cases, angular RMSbetween tracer gradient and FTLV orientations computed at the reference date 99 decreases with time.

Figure 6. Orientations (angular degree) of the gradients of the SST (left) and MLP (right) fields shown in Figure 4.

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Figure 7. Binarisation of the SST (left), MLP (centre) and FTLE (right) gradient fields of Figures 4 and 3 using the threshold based

technique (12).

Figure 8. Percentage of resolved variance as a function of the number of EOF.

Figure 9. Example of two random velocity perturbations with the PDF given by (26).

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Figure 10. Behaviour of sensitivity functions for SST.

Left: Variation of the sensitivity function JTE (30) associated with the FTLE-based triplet E? (28) with respect to the amplitude λ ofnine random perturbations (27) applied to the reference velocity field uk0 .

Right:Variation of the sensitivity function JTV (31) associated with the FTLV-based triplet V? (29) with respect to the amplitude λ of

nine random perturbations (27) applied to the reference velocity field uk0 .

Figure 11. Behaviour of sensitivity functions for MLP.

Left: Variation of the sensitivity function JPE (32) associated with the FTLE-based triplet E? (28) with respect to the amplitude λ ofnine random perturbations (27) applied to the reference velocity field uk0 .

Right: Variation of the sensitivity function JPV (33) associated with the FTLV-based triplet V? (29) with respect to the amplitude λ ofnine random perturbations (27) applied to the reference velocity field uk0 .

c© 0000 Tellus, 000, 000–000


Figure 12. Variation of the sensivity function (23) computed with exact model counterpart data yS = H(X). Variation are computedwith respect to the amplitude λ of nine random perturbations (27) applied to the reference velocity field uk0 .

Left: FTLE-based triplet E? (28).

Right: FTLV-based triplet V? (29).

c© 0000 Tellus, 000, 000–000

On the use of Finite-Time Lyapunov Exponents and Vectors ... · 3 Finite-Time Lyapunov Exponents...

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