Pressure image assimilation for atmospheric motion estimation

Tellus 000, 000–000 (0000) Printed September 1, 2008 (Tellus LATEX style file v2.2)

Pressure image assimilation for atmospheric motion

estimation

By Thomas Corpetti1,5⋆, Patrick Heas2, Etienne Memin1,3, Nicolas Papadakis4,1IRISA/INRIA - Campus de Beaulieu - 35042 Rennes cedex - France

2CEMAGREF - Avenue de Cucille- 35044 Rennes cedex - France3Fac. de Ing. de la Univ. Buenos-Aires, Av. Paseo Colon 850, C1063ACV Buenos Aires - Argentina

4UPF, Barcelona Media, Carrer Ocata 1, 08003 Barcelona - Spain5CNRS / COSTEL UMR 6554, Place du Recteur Henri Le Moal, 35043 Rennes Cedex - France

September 1, 2008

ABSTRACTThe complexity of the laws of dynamics governing 3D atmospheric flows associatedwith incomplete and noisy observations make the recovery of atmospheric dynamicsfrom satellite image sequences very difficult. In this paper, we address the challengingproblem of estimating physical sound and time-consistent horizontal motion fields atvarious atmospheric depths for a whole image sequence. Based on a vertical decom-position of the atmosphere, we propose a dynamically consistent atmospheric motionestimator relying on a multi-layer dynamic model. This estimator is based on a weakconstraint variational data assimilation scheme and is applied on noisy and incompletepressure difference observations derived from satellite images. The dynamic model is asimplified vorticity-divergence form of a multi-layer shallow-water model. Average hor-izontal motion fields are estimated for each layer. The performance of the proposedtechnique is assessed using synthetic examples and using real world meteorologicalsatellite image sequences. In particular, it is shown that the estimator enables exploit-ing fine spatio-temporal image structures and succeeds in characterizing motion atsmall spatial scales.

1 Introduction

1.1 Overview

Geophysical motion characterization and image se-quence analysis are crucial issues for numerous scientificdomains involved in the study of climate change, weatherforecasting and climate prediction or biosphere analysis.The use of surface station, balloon or in-flight aircraftmeasurements has improved the estimation of wind fieldsand has been a further step for an improved understandingof meteorological phenomena. However, the network’stemporal and spatial resolutions may be insufficient forthe analysis of mesoscale dynamics. Recently, in an ef-fort to avoid these limitations, high-resolution satellitessensors have been designed to provide image sequencescharacterized by finer spatial and temporal resolutions.Nevertheless, the analysis of atmospheric motion fromimage sequences remains particularly challenging due tothe complexity of atmospheric dynamics observed at suchscales. Thus, advanced techniques are needed to exploit thelatest generation of satellite images.

⋆ Corresponding author.e-mail: [email protected]

1.2 Related works

For image-based geophysical motion analysis standardtechniques from computer vision, originally designed for bi-dimensional quasi-rigid motions with stable salient features,are not well adapted (Leese et al., 1971; Horn and Schunck,1981). The design of techniques specific to fluid flows hasbeen a step forward, towards the elaboration of reliablemethods to extract characteristic features of flows (Zhouet al., 2000; Corpetti et al., 2002; 2006; Cuzol et al., 2007;Cuzol and Memin, 2008; Yuan et al., 2007). However, forgeophysical applications, existing fluid-dedicated methodsare all limited to frame-to-frame estimation and do not relyon physical conservation laws.

Geophysical flows are quite well described by appropri-ate physical models. As a consequence, in such contexts, theinclusion of physical evolution laws should constitute a verypowerful means for the motion analysis of satellite imagedata, when compared with standard variational or statisti-cal generic image based motion estimation techniques.

Recently, a layered motion estimator based on theshallow-water mass conservation equation has been pro-posed in (Heas et al., 2007; Heas and Memin, 2008). In thesestudies, time consistency is reinforced by the introduction ofa frame-to-frame temporal regularization based on a simpli-fied vorticity-divergence form of the shallow-water momen-tum conservation equations. This dynamic model providesan additional constraint guiding the estimation process from

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2 T. CORPETTI, P. HEAS, E. MEMIN, N. PAPADAKIS

the results obtained using the previous frames. Nevertheless,it is important to outline that this constraint is not imposedglobally on the sequence of observations but only on a pairof images.

Variational data assimilation (Le Dimet and Talagrand,1986; Talagrand and Courtier, 1987), derived from optimalcontrol theory (Lions, 1971), offers a global optimal formu-lation allowing the combination of physical models and dif-ferent kinds of observations. Since its introduction, the vari-ational assimilation technique, commonly known as 4D-Var,has been widely used for several atmospheric applications(Bennet, 1992; Talagrand, 1997) and especially for numericalweather forecasting or climate numerical modeling (Bennetet al., 1996; Courtier et al., 1994; Courtier and Talagrand,1990; Ghil and Malanotte-Rizzoli, 1991; Le Dimet and Blum,2002; Rabier et al., 1998). Different variations of the originalprinciples have been proposed over the last 15 years and thistechnique is routinely used in several operational meteoro-logical centers. The exploitation of the spatial informationof satellite data have been shown to be very promising forimproving Numerical Weather Prediction (NWP – see forinstance some applications concerning assimilation of top-clouds in (Bayler et al., 2000; Benjamin et al., 2002; Kim andBenjamin, 2000) or radiances in (Andersson et al., 1994)).

Image observations possess very good spatial resolu-tions compared to spatial scales of standard data and arelargely used in assimilation systems. However, only a fewattempts have been made to date to consider their usefor the direct estimation of dynamic parameters (like thewind fields) that are indirectly observed through the im-age sequences. Even if some assimilation procedures areable to capture a wind field (viewed principally as a con-trol parameter of a dynamic system like in (Anderssonet al., 1994)), none of them assimilate this quantity di-rectly through an observation operator linking the motionto the image data. This issue is indeed particularly difficultdue to the phenomenon responsible for the image formationand to the inherent projection on the image plane of thethree-dimensional observed quantity. As a consequence, theoperational estimation of wind fields is currently done us-ing correlation techniques, such as the Atmospheric MotionVectors (AMV) product supplied by Eumetsat (the Euro-pean agency that supplies the METEOSAT satellite data).The height is then determined from the infrared temperatureand converted to pressure using an ECMWF forecast profile(Menzel et al., 1983; Holmlund, 1998; Schmetz et al., 1993;1995). Once estimated, these AMV can either be the inputof data assimilation algorithms or be assimilated themselves.In this paper, we propose the suppression od the intermedi-ary step of estimating AMV and to assimilate directly thewind field from the sequence of images. Indeed, as shownin a recent study on a turbulent 2D flow (Papadakis et al.,2007) the use of a direct image measurement operator clearlyleads to a great improvement of the quality of the estima-tion compared to an assimilation process based on motionfields given by an external process (pseudo-observations ofthe Eulerian velocity fields).

Variational data assimilation techniques relying on im-age data or on features extracted from image data have re-cently been proposed. A technique for the assimilation ofa low order dynamic system, obtain from a POD-Galerkinprojection, has been proposed in (D’Adamo et al., 2007).

This technique makes use of noisy flow motion measure-ments estimated from a set of images of particles. In (Ruh-nau and Schnorr, 2007), an optimal control strategy has alsobeen proposed for the recovery of fluid motion. This ap-proach based on a Stokes flow model nevertheless remainsan estimation technique that works only on two consecu-tive frames and no dynamic coherency can be guaranteedalong the image sequence. Taking a different approach, twovariational assimilation strategies have been proposed inthe computer vision community for the tracking of curves(Papadakis and Memin, 2008) and the estimation of fluidmotion fields (Papadakis et al., 2007). Both of these meth-ods directly assimilate image data. They do not depend onpseudo-measurements obtained by external techniques butrather propose to directly include a differential observationoperator borrowed to image features estimation techniques.

The technique proposed in this paper consists of an ex-tension of these studies for atmospheric motion estimationfrom satellite image sequences.

1.3 Contributions

The method we propose differs significantly from previ-ous stuidies on remote sensing motion analysis. In contrastto previous image-based techniques relying on two consec-utive frames, the proposed approach estimates dynamicallyconsistent motion fields through a whole image sequence.

The method exploits a simplified version of a divergencevorticity formulation of a shallow-water dynamic system.This inexact dynamics is complemented by the use of un-certainty terms. The assimilation is embedded into an incre-mental optimal control scheme and relies on sparse pressuredata of atmospheric layers. The choice of this simplified dy-namics is guided by the nature of the observations we haveat our disposal. The efficiency of this process will be demon-strated using synthetic benchmarks and analyzed using realworld satellite sequences.

This article is organized as follows. After a section de-scribing the context of the study, the data and the maindynamic assumptions, the assimilation methodology is pre-sented in section 3. Section 4 is devoted to the exploitationof this framework to the present problem. Experimental re-sults, on synthetic and real examples, are provided in sec-tion 5. A summary, some perspectives as well as appendixesconclude this paper.

Before entering into the detailed description of the pro-posed technique, let us first situate more precisely the aimsof our study, the assumptions inherited from these objectivesand the nature of the available image data.

2 Context, objectives and ensuing dynamic

assumptions

In this study we aim to recover as accurately as possi-ble a dense description of horizontal wind fields in the 3Dphysical space. Nevertheless, as satellite images usually onlyprovide sparse data of partially visible 3D cloud layers fromwhich only inaccurate pressure vertical coordinates can beindirectly inferred, we will focus on the simpler task of es-timating integrated horizontal wind fields for successive at-mospheric layers.

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PRESSURE IMAGE ASSIMILATION FOR ATMOSPHERIC MOTION ESTIMATION 3

2.1 Layers decomposition of the atmosphere

The layering of atmospheric flow in the troposphere isvalid as long as horizontal scales are much greater than thevertical scale height. Thus, for layers of about 1km thick, thishypothesis is roughly valid for horizontal scales greater thanor equal to 100 km. It is thus impossible to truly character-ize a layered atmosphere with a local analysis performed inthe vicinity of a pixel representing a kilometer order scale.Nevertheless, one can still decompose the 3D space into ele-ments of variable thickness, where only sufficiently thin re-gions of such elements may really correspond to commonlayers. An analysis based on such a decomposition has theprincipal advantage of operating at different atmosphericpressure ranges and avoids the mixing of heterogeneous ob-servations.

The 3D space decomposition used here is one introducedin earlier articles (Heas et al., 2007; Heas and Memin, 2008),where the k-th (k ∈ [1, K]) layer corresponds to the volumelying in between an upper surface sk+1 and a lower surfacesk. These surfaces sk+1 are defined by the height of the topof clouds belonging to the k-th layer. They are thus onlydefined in areas where clouds belonging to the k-th layerexist, and remain undefined elsewhere. Cloud classificationmaps, as illustrated in figure 1, determine the allocation ofthe top of clouds to the different layers. The EUMETSATconsortium routinely provides such classifications, which arebased on thresholds of top of cloud pressure.

2.2 Sparse pressure difference image observations

Top of cloud pressure images of the order of one kilome-ter are also routinely provided by the EUMETSAT consor-tium1. We denote by Ck the class corresponding to the k-thlayer. It should be noted that the top of cloud pressure imageis composed of segments of top of cloud pressure functionsp(sk) related to the different layers :

S

k p(x, y, sk+1|(x, y) ∈

Ck.Thus, pressure images of top of clouds are used to con-

stitute sparse pressure maps of the upper boundaries ofthe layers p(sk+1). As with satellite images the lower cloudboundaries are always occluded, we coarsely approximatethe missing pressure observations p(sk) by an average pres-sure value p(sk) observed on the top of the clouds of thelayer underneath. Finally, for the layer k ∈ [1, K], we de-fine observations hk

obs as pressure differences in hecto Pascal(hPa) units:

1 Top of cloud pressure images are derived from a radiative trans-fer model using ancillary data, namely temperature and humidityprofiles obtained by analysis of short term forecasts. This modelsimulates the radiation by the top of an opaque cloud at dif-ferent vertical levels, which might be observed by a satellite. Thepressure level where the simulated radiation best fits with the ob-served radiation determines the pressure of the cloud top for thecorresponding pixel (Lutz, 1999). Multi-channel techniques (us-

ing a thermal IR with a water vapor or CO2 absorption channels)enable the determination of the temperature of the top of semi-transparent clouds (Menzel et al., 1983; Schmetz et al., 1993),and thus their equivalent pressure level (with the help of analyzedor forecast data) independently from foreground or backgroundeffects.

hkobs =

p(x, y, sk) − p(x, y, sk+1) if (x, y) ∈ Ck,0 else.

(1)

The resulting observations from a METEOSAT image(5-June-2004, 13h30 UTC –universal time, similar to GMT: Greenwich mean time– or 2004-06-05T13:30Z in ISO8601)are illustrated in figure 1. The scattered nature of these maps(less than 15% is exploitable) is due to the sparsity of cloudcoverage but is also enforced by the higher altitude cloudsoccultation of the lower layers. We should also note that theuse of pressure images, besides being coherent with our lay-ers definition, will enable the creation of a precise data modelrelating the observed pressure data to the unknown motionfield. In contrast to traditional atmospheric wind field ex-traction techniques, we do not rely on the assumption thatcloud luminance patterns are passive tracers of the flow (as-sumption which is known to be violated in some situations).This is a key advantage of the proposed approach.

2.3 Ensuing dynamic assumptions

In order to construct the best possible dynamic modeldescribing the evolution of the pressure layers analyzed andin accordance with the objectives of this study (i.e. the re-covering of the horizontal displacements of the set of layers),we rely on the shallow-water approximation to integrate ver-tically the horizontal primitive equations of atmospheric dy-namics. Using the hypothesis of hydrostatic equilibrium, theatmospheric hydrostatic primitive equations read:

8><

>:

dudt

− uv tan φa

+ uwa

+ px

ρ= +2Ωv sin φ − 2Ωw cos φ + Fu

dvdt

+ u2 tan φa

+ vwa

+py

ρ= −2Ωu sin φ + Fv

dpdz

+ ρg = 0,

(2)where (u, v) represents the horizontal velocity field, w its

vertical component, p the pressure (px and py denoting thespatial derivatives w.r.t x and y), (Fu,Fv) the horizontalcomponents of viscous forces at subgrid scales (Frisch, 1995),φ the latitude, ρ the density, Ω and a the Earth angular ve-locity and its radius. The continuity equation complementsthe system:

1ρ

dρdt

+ ∇ · V = 0, (3)

where V = (u, v, w)T stands for the 3D velocity.A vertical integration of the primitive equations is not

straightforward since, in our case, we only rely on pres-sure image analysis and vertical density profiles remain un-known. We circumvent this difficulty by relying on a shallow-water approximation of these primitive equations (De Saint-Venant, 1871).

The shallow-water approximation is based on two mainassumptions 1) vertical motion is negligible with respect tohorizontal motion and 2) incompressibility. For atmosphericdynamics, this is valid for the upper range of mesoscale anal-ysis in a layered atmosphere. In the following section wederive a dynamic model using these two hypotheses.

2.3.1 Negligible influence of the vertical motion Consid-ering horizontal scales of the order of 100 km, combinedwith layer depths of the order of 1 km, makes the shallow-water approximation relevant. Therefore, in order to obtain

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a valid dynamic model on a pixel grid, we filter the hydro-static primitives equations with a Gaussian kernel functionKδx of standard deviation equal to δx = 100δ−1

p , where δp

denotes the image pixel resolution in kilometers2. Followingthe scale analysis presented in appendix A, where the Rossbynumber is around 0.5, Coriolis terms have to be included inthe momentum components of the filtered hydrostatic prim-itive equations adapted to mesoscale dynamics. These latterequations are written:

(dudt

+ px

ρ− vfφ = νT ∆u,

dvdt

+py

ρ+ ufφ = νT ∆v,

(4)

where we have introduced the filtered pressure p = Kδx ∗ p,the filtered horizontal wind v = (u, v)T = Kδx ∗ (u, v)T

and where ρ0, fφ and νT denote the local mean density, theCoriolis factor depending on latitude φ and the turbulentviscosity produced at sub-grid scales. The induced turbulentdissipation can be approached by sub-grid models proposedin Large Eddy Simulation literature (Meneveau and Katz,2000). The simplest one is the well known Smagorinsky sub-grid model which is in agreement with Kolmogorov “K41”theory (Smagorinsky, 1963). For 2D flows, it results in ananisotropic diffusion with a turbulent viscosity coefficientequal to :

νT = (Cδx)2q

2(u2x + v2

y + (uy + vx)2), (5)

where C is the Smagorinsky coefficient (which is usuallyfixed to 0.17).

2.3.2 Incompressibility The shallow-water approximationalso assumes incompressibility. Thus, we consider a constantdensity ρk

0 within any layer k (k is the index related to thenumber of the layer). We should note that this incompress-ibility simplification which underlies the shallow-water mod-eling is reasonable in our case, while it may be erroneous forfiner horizontal scales. Layer average densities ρk

0 can be re-lated to the average pressures pk by vertical integration ofthe equation of state for dry air (p = ρRT ), combined withthe hydrostatic relation (dp = −gρdz), using the assumptionof constant lapse rate (T = T0 + γz), where g, R, T0 and γdenote the standard physical constants. Indeed, as demon-strated in appendix B, the mean density related to the k-thlayer may be expressed as :

ρk0 =

p20

(pk+1 − pk)(γRg

+ 2)RT0

h“ p

p0

” γRg

+2ipk+1

pk. (6)

This relation characterizes a barotropic atmosphere (Holton,1994). Expanding the total derivatives in the isobaric coor-dinate system (x, y, p) and using the fact that the flow isincompressible (zero local 3D divergence), equation (4) canbe rewritten as:

(∂u∂t

+ ∂u2

∂x+ ∂uv

∂y+ ∂uω

∂p+ px

ρ0− vfφ = νT ∆u,

∂v∂t

+ ∂uv∂x

+ ∂v2

∂y+ ∂vω

∂p+

py

ρ0+ ufφ = νT ∆v,

(7)

2 The Gaussian filtering operation is achieved by a convolutionproduct with a Gaussian kernel and acts as a low-pass frequencyfilter: frequencies higher than a given value (which depends on the

standard deviation) are removed. Its effect on the HPE equationsis to keep only the ”low-pass” component, i.e. the large scalestructures.

where ω = dp

dtdenotes the filtered vertical wind component

in pressure coordinates. In addition, the expression of thecontinuity equation (3) in the isobaric coordinate system(x, y, p) yields the filtered relation:

∂u

∂x+

∂v

∂y+

∂w

∂p= 0, (8)

where (u, v, w) are the components of the filtered 3Dvelocity.

We now derive the integrated shallow-water model dedi-cated to atmospheric layers3. In order to perform the verticalintegration of equation (3) and equation (7) in the pressureinterval [p(sk+1), p(sk)], we first fix the boundary conditions:

8<

:

∂p(sk)∂t

+u(sk)∂p(sk)

∂x+v(sk)

∂p(sk)∂y

= ω(sk),

∂p(sk+1)∂t

+u(sk+1)∂p(sk+1)

∂x+v(sk+1)

∂p(sk+1)∂y

= ω(sk+1).

(9)

Such boundary conditions can be interpreted as the fact thatboundary surfaces p(sk) and p(sk+1) are deformed by thevertical wind ω(sk) and ω(sk+1) in a barotropic atmosphere.To achieve such a vertical integration in the pressure interval[p(sk+1), p(sk)] varying with spatial coordinates, we employthe Leibnitz formula with the previous boundary conditions.This formula, which is valid for any integrable and derivablefunction f(x, p) and for any interval [a(x), b(x)] with bound-aries depending on x, reads:

Z b(x)

a(x)

∂f(x, p)

∂xdp =

∂

∂x

Z b(x)

a(x)f(x, p)dp

!

− f(x, b(x))∂b(x)

∂x+ f(x, a(x))

∂a(x)

∂x. (10)

Relying on equations (9) and (10), the vertical integrationof equation (7) in the pressure interval [p(sk+1), p(sk)] givesrise to a momentum conservation equation for the k-th at-mospheric layer:

∂(qk)

∂t+div(

1

hkqk⊗qk)+

1

2ρk0

∇xy(hk)2+

»0 −11 0

–

fφqk = νkT

∆(qk),

(11)

with hk = p(sk) − p(sk+1), (12)

vk = (uk, vk) =1

hk

Z p(sk)

p(sk+1)vdp, (13)

qk = hkvk , (14)

div(1

hkqk ⊗ qk) =

2

4

∂(hk(uk)2)∂x

+ ∂(hkukvk)∂y

∂(hkukvk)∂x

+ ∂(hk(vk)2)∂y

3

5 . (15)

By vertical integration of the continuity equation (relation(8)) in the pressure interval [p(sk+1), p(sk)], we supplementthe momentum conservation law of equation (11) with themass conservation law :

∂hk

∂t+div(qk) = 0, (16)

3 Note that according to Fubini’s theorem, if the motion fieldsare integrable then vertically averaged horizontally filtered fieldsare equivalent to horizontally filtered vertically averaged fields.

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and obtain the independent Saint-Venant equation systemfor any atmospheric layer k ∈ [1, K]:8>><

>>:

∂hk

∂t+div(qk) = 0,

∂(qk)∂t

+div( 1hk

qk ⊗ qk)+ 12ρk

0

∇xy(hk)2+

»0 −11 0

–

fφqk

=νkT

∆(qk).(17)

Note that as we are using the isobaric coordinate system,partial derivatives with respect to x, y and t are defined atconstant pressure p. However, according to section 2.2, pres-sure difference observations hk

obs correspond to rough con-stant pressure intervals. Therefore, such data fit the shallow-water model of equation (17) defined in the isobaric coordi-nate system. This model will be the basis of our assimilationprocess. Before describing the developed approach, the nextsection presents the data assimilation principle.

3 Data assimilation

In this section we present the main principles of vari-ational data assimilation for an imperfect dynamic model.We refer the reader to (Bennet, 1992; Le Dimet and Ta-lagrand, 1986; Lions, 1971; Talagrand and Courtier, 1987;Vidard et al., 2000) for complete methodological aspects ofdata assimilation and applications concerning geophysicalflows.

Our problem consists of recovering, from an initial con-dition, a system’s state X partially observed and driven byapproximately known dynamics. This can be formalized asfinding X(s, t), for any location s at time t ∈ [t0, tf ], thatsatisfies the system:

∂X

∂t(s, t) + M(X(s, t)) = νm(s), (18)

X(s, t0) = X0(s) + νn(s), (19)

Y(s, t) = H(X(s, t)) + νo(s, t), (20)

where M is the non-linear operator relative to the dynam-ics, X0 is the initial vector at time t0 and (νn, νm) are (un-known) additive control variables relative to noise on thedynamics and the initial condition respectively. In addition,noisy measurements Y of the unknown state are availablethrough the non-linear operator H up to νo. To estimate thesystem’s state, a common methodology is the minimizationof the cost function J :

J (X) =1

2

Z tf

t0

‖Y − H(X(νm, νn))‖2R−1dt

+1

2‖X(s, t0) − X0(s)‖

2B−1

+1

2

Z tf

t0

‖∂X

∂t(s, t) + M(X(s, t))‖2

Q−1dt,

(21)

where we have introduced the information matrices R, B, Qrelative to the covariance of the errors (νm, νn, νo). The Ma-halanobis distance that has been used reads ‖X‖A−1 =XT A−1X. The evaluation of X can be done by canceling

the gradient δJX(θ) = limβ→0J(X+βθ)−J(X)

βof (21). Un-

fortunately, the estimation of such gradient is in practiceunfeasible for a large system’s state since it would be neces-sary to compute perturbations along all the components ofX. One way to cope with this difficulty is to write an ad-joint formulation of the problem. To that end, the adjoint

variables λ that express the errors of the dynamic model areintroduced as:

λ = Q−1

„∂X

∂t+ M(X + βθ)

«

. (22)

Denoting

•`

∂M

∂X

´

and`

∂H

∂X

´

the linear tangent operators of M andH respectively4,

• (∂XM)∗ and (∂XH)∗ their adjoint operators5 ,

it can be shown that canceling the gradient δJX(θ) w.r.tthe adjoint variables λ leads to a retrograde integration ofan adjoint evolution model that takes into account the ob-servations. Once the adjoint variables λ are estimated, onecan recover the system state X using relation (22). Finally,recovering X leads to the following incremental algorithm(Bennett and Thorburn, 1992):

(i) Starting from X(s, t0) = X0(s), perform a forward in-

tegration: ∂X∂t

+ M(X) = 0

(ii) X(s, t) being available, compute the adjoint vari-ables λ(s, t) with the backward equation:

λ(tf ) = 0 ;

−∂λ

∂t(t) + (∂XM)∗ λ(t) = (∂X H)∗ R−1(Y − H(X))(t)

(23)

(iii) Update the initial condition : dX(t0) = Bλ(t0);

(iv) λ being available, compute the state space dX(t)from dX(t0) with the forward integration

∂dX

∂t(t) +

„∂M

∂X

«

dX(t) = Qλ(t) (24)

(v) Update : X = X + dX

(vi) Loop to step (ii) until convergence

Intuitively, the adjoint variables λ contain informationabout the discrepancy between the observations and the dy-namic model. They are computed from a current solution Xwith the backward integration (23) that encompasses boththe observations and the dynamic operators. This deviationindicator between the observations and the model is thenused to refine the initial condition (step (iii)) and to recoverthe system state through an imperfect dynamic model whereerrors are Qλ (step (iv)). It should be noted that if the dy-namic is perfect, the associated error covariance Q is zeroand the algorithm only refines the initial condition. How-ever from an image analysis point of view, a perfect model-ing is difficult to obtain since the different models on whichone can rely are usually inaccurate due, for instance, to 3D-2D projections, varying lighting conditions, completely un-known boundary conditions at the image boarders, etc.

4 Data assimilation for multi-layer shallow-water

model

In this section, we propose to develop an image assimila-tion technique based on (17) to estimate the mesoscale flows.

4 The linear tangent of an operator A is the directional derivative

of the operator (the Gateaux derivative): limβ→0A(X+βθ)−A(X)

β5 The adjoint A

∗ of a linear operator A on a space D is such as∀x1, x2 ∈ D, < Ax1, x2 >=< x1, A

∗x2 >

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To that end, we further discuss in section 4.1 the shallow-water model and its limitations based on the available ob-servations. The sections 4.2, 4.3, 4.4 and 4.5 are respectivelydevoted to the detailed description of the system state, theselected dynamic model, the observation system and the def-inition of the different error covariance matrices.

4.1 Dynamic Model

4.1.1 Limitation of Saint Venant’s shallow-water modelThe dynamic model of equation (17) describes accuratelythe mesoscale atmospheric flows. Therefore, one can imme-diately define an assimilation scheme relying on the systemvariables:

X =

»hk

qk

–

, (25)

where the perfect dynamic model is given by relation (17).Remembering that images of pressure differences, hk, areobservable, it is possible to define the measurement systemY = H(X) as:

Y = hk

obs = Kδx∗ hk

obs,H = [Id, 0].

(26)

This system has the advantage of assimilating both thesparse pressure differences hk

obs and the associated motionfield qk without observing this latter quantity. However, it isimportant to note that the proposed shallow-water systemdescribes the dynamics of physical quantities under a lay-ering assumption. In practice, our data come from the Me-teosat Second Generation sensor where a pixel correspondsto a square of 3000 × 3000m2. Even considering a spatialconvolution by a Gaussian kernel Kδx , a scale analysis re-ported in appendix C shows that the term 1

2ρk0

∇xy(hk)2 in

(17) locally dominates. This is all the more true in areaswhere the layering assumption breaks down at the consid-ered observation scale. In particular, this term dominates onthe borders of segments in sparse pressure images, where apixel may contain heterogeneous contributions from differ-ent pressure levels. This perfect model is hence too restric-tive and is likely to generate some numerical instabilities.We consequently prefer to develop, from this shallow-watermodel (17), an imperfect dynamic system that describes theevolution of the flow while authorizing the layering assump-tion to be violated at some locations. This is the scope ofthe next section.

4.1.2 Divergence-vorticity shallow-water dynamic modelIn order to overcome the limitations mentioned above, wedescribe the evolution of the filtered state variables froma simplified version of the previously introduced shallow-water equations and perform assimilation using an imperfectmodeling scheme. To that end, we assume filtered horizontalmotion components to be homogeneous within the layer. Inother words, we neglect their vertical derivatives and con-sider that filtered horizontal winds vk, which have been ver-tically averaged, are equal to filtered horizontal winds onlayer upper surfaces sk+1. Using such an assumption yields :

vkt + ∇(vk)vk − ρ−1

0 ∇p(sk+1) +

»0 −11 0

–

fφvk = νT ∆(vk),

(27)

with the notations ∇(vk) = (∇uk, ∇vk)⊤ and ∆(vk) =(∆uk, ∆vk)⊤. Let us denote the vorticity by ζk = curl(vk)and the divergence by Dk = div(vk). The previous systemmay be expressed in its vorticity-divergence form:

ζkt + vk · ∇ζk + (ζk + fφ)Dk = νT ∆(ζk),

Dkt +vk · ∇Dk+(Dk)2−2|J |−ρ−1

0 ∆p(sk+1)+fφζk =νT ∆(Dk),(28)

where |J | is the determinant of the Jacobian matrix of vari-ables (uk, vk). For vorticity based large eddy simulation for-mulations, we may rely on enstrophy-based sub-grid mod-els (Mansour et al., 1978), instead of using the Smagorin-sky model (equation (5)). This sub-grid dissipation model isbased on Taylor’s vorticity transfer and dissipation by smallscales theory (Taylor, 1932) and reads:

νT = (Cδx)2|ζk|. (29)

In the momentum conservation formulations of equations(27) and (28), dynamic models predict the evolution of ve-locity components (uk, vk) and of divergence and vorticity(ζk, Dk). In both models, one of the major difficulties iscreated by the dependence of the pressure variable p(sk+1)which is an unknown variable of the k-th layer state. Addingpressure as a new state variable driven by the mass con-servation law (16) constitutes a solution for achieving themodel integrations. However, in section 4.4, we will need thismass conservation law to define a proper observation oper-ator. Thus, we search instead to derive a dynamic modelwhich is independent of the pressure unknown. In contrastto the classical formulation, the vorticity-divergence equa-tions have the advantage of providing such a model for thevorticity evolution. As regards the divergence, since at largescales, it can be considered weak almost everywhere, we willrely on an approximate evolution law. We assume here thatthe divergence is advected by the flow and a noise variablethat encodes the uncertainty on the model. More preciselywe will assume that the divergence map is a function ofa stochastic process representing a particle position and isdriven by the following stochastic differential equation:

dx(t) = vk(x(t))dt +

√2νT dB(t). (30)

This equation states that the particle position is known onlyup to an uncertainty that grows linearly with time. Here Bt

denotes a standard Brownian motion of IR2. The process xt

starts at point, xo. It can be shown through the Ito formulaand Kolmogorov’s forward equation, that the expectation attime t of such a function, ξ(t, x) = E[divvk(x(t))] follows anadvection diffusion equation (Oksendal, 1998):

ξt + vk · ∇ξ + ξdivvk − νT ∆ξ = 0,

ξ(0, x0) = divvk(x0).(31)

Assuming that the divergence of the flow is given by its ex-pectation (Dk ≈ ξ) allows us to write the simplified filteredvorticity-divergence model for the layer k as:

ζkt + vk · ∇ζk + (ζk + fφ)Dk = νT ∆ζk,

Dkt +vk · ∇Dk + (Dk)2 = νT ∆Dk.

. (32)

In this model we assume that the divergence of the flowis weak and is similar to the divergence expectation. Thedivergence equation no longer describes the evolution of the

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flow divergence but the evolution of its expectation. Theexpectation of the divergence value is advected by the flowand dissipates due to a subgrid isotropic uncertainty. Thishypothesis is quite natural in large scale modeling.

4.2 State variables

We choose to represent the system state X (i.e. thevelocity field) through the curl and divergence components:

X =

»ζk

Dk

–

, (33)

as their evolution can be described by equation (32) andsince they completely determine the underlying 2D veloc-ity up to a harmonic transportation component. Denotingthe orthogonal gradient by ∇⊥ = (−∂/∂y, ∂/∂x)⊤ and the2D Green kernel G associated with the Laplacian operator(G(·) = 1

2πlog(·)), the Helmholtz decomposition of the field

can be expressed as :

vk = ∇⊥(G ∗ ζk)| z

vksol

+ ∇(G ∗ Dk)| z

vkirr

+vkhar, (34)

where vkhar is a harmonic transportation part (divvk

har =curlvk

har = 0). In our applications, we will assume that thiscomponent of the velocity is stationary within the tempo-ral interval of the analyzed image sequence. This componentmay be recovered by subtracting its solenoidal and irrota-tionnal parts from the initial field vk

0 . As a consequence, thefield can be represented by its div-curl components as :

vk = ∇⊥(G ∗ ζk)+ ∇(G ∗ Dk) =h

∇⊥G∗, ∇G∗i

| z

HG

»ζk

Dk

–

| z

X

, (35)

where the operator HG can efficiently be computed in theFourier domain.

4.3 Dynamic operator and its adjoint

The dynamic model required for the assimilation in (18)is defined by relation (32). This model is associated with animperfect dynamic modeling where the model errors are hererelated to the Coriolis effect, to the pressure difference dis-sipation of the divergence component and to the sub-gridstress tensor error of the vorticity and divergence compo-nent.

The associated tangent linear system (equation (24))

around the current solution X⋆ = [ζ⋆, D⋆]T is:

∂t

»ζk

Dk

–

+

»v⋆ · ∇+D⋆−νT ∆ ζ⋆+fφ

0 v⋆ · ∇+D⋆ −νT ∆

–

| z

∂X⋆ M

»ζk

Dk

–

= Qkλ,

(36)

where νT = (Cδx)2|ζ⋆|. This equation includes an error co-variance matrix Qk that will be further defined in section4.5. At each time increment dt, the current velocity solutionv⋆ is updated according to the vorticity ζk, the divergence

Dk and the harmonic transportation vkhar using the opera-

tor defined in equation (35). The adjoint variables [λkζ , λk

D]T

are obtained through the following adjoint system:

8>>>>>>>><

>>>>>>>>:

λkζ (tf ) = 0,

λkD(tf ) = 0,

−∂t

»λk

ζ

λkD

–

+

»−v⋆ · ∇+D⋆−νT ∆ ζ⋆+fφ

0 −v⋆ · ∇+D⋆−νT ∆

–

| z

(∂X⋆ M)∗

»λk

ζ

λkD

–

= ∂Xk H∗R−1(Y − H(X)).

(37)

We should note that the turbulent viscosity νT and thevelocity v⋆ are computed during the direct integration andmaintained in the backward integration. The discretizationof the bi-dimensional advection system must be done care-fully to prevent numerical explosions. In this application,it has been achieved using the non-oscillatory schemesproposed in (Xu and Shu, 2006). The temporal integrationhas been implemented through a third order Runge-Kuttascheme and the time step dt is defined in order to respectthe total variation diminishing property (Kurganov andTadmor, 2000). Once the direct dynamic model has beendiscretized, the matrix (∂X⋆M) is available and the adjointsystem involved in (37) corresponds to the transpose ofthis matrix. More details about the construction of adjointmodels can be found in (Talagrand and Courtier, 1987).

The observation components Y = H(X) and the error co-variance matrices are defined in the next sections.

4.4 Image observation operator

4.4.1 Extracting motion fields from pressure images In or-der to define an observation operator that links the unknownmotion field vk to images of pressure differences hk

obs(s, t),we use the mass conservation law of equation (16):

∂hkobs(s, t)

∂t+∇hk

obs(s, t)·vk(s, t)+hk

obs(s, t)divvk(s, t) ≈ 0, (38)

However, this formulation alone cannot be used to estimatevk as it provides only one equation for two unknowns ateach spatio-temporal location (s, t). To remove such ambi-guities, a common approach is to assume a spatial coherenceof wind field estimates in a given neighborhood (similar tothe well-known approach of (Lucas and Kanade, 1981) usedin computer vision). In the present case, this assumption iscoherent since unknowns are velocity vectors vk(s, t) thatare spatially filtered within a neighborhood of size δx. Themeasured motion field should thus obey the following con-straint:

Kδx∗

∂hkobs

∂t+ ∇hk

obs · v + hkobsdivvk

!

≈ 0, (39)

where Kδx is the Gaussian kernel of standard deviationδx. We should note that in practice, to prevent the influ-ence of areas without any data in the border of the re-gions of interest Ck, all the points without pressure infor-mation have been removed from the Gaussian smoothing.From the relation (39) and recalling that vk = HGX (with

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X = [ζk, Dk]T ), one can easily define our observation systemY = H(X) as:

8>>>>><

>>>>>:

Y = Kδx∗

∂hkobs(s, t)

∂t

H(X) = −“

Kδx∗ ∇hk

obs

”THG(X)

−“

Kδx∗ hk

obs

”

∇THG(X)

(40)

The observation operator H is linear w.r.t the state variableX = [ζk, Dk]T and is then identical to its linear tangentoperator. Noting that the adjoint of HG is −HG (as shownin (Papadakis and Memin, 2007)), the adjoint ∂X⋆H

∗(Y ) ofH(X) therefore reads :

∂X⋆ H∗(Y ) =HG

“

(Kδx∗ ∇hk

obs)Y”

− HG

“

∇((Kδx∗ hk

obs)Y )” (41)

4.4.2 Observation retrieval and representativeness Mostof the motion estimation techniques, which aim at extractingwind fields from satellite images (like AMV –AtmosphericMotion Vectors– from Eumetsat), are based on cloud lumi-nance pattern matching over consecutive frames. Such tech-niques assume that clouds behave like tracers of the under-lying motion field and that the cloud brightness functionremains constant over time. It is unfortunately well knownthat these hypotheses are not always valid for satellite im-ages (Corpetti et al., 2002; Schmetz et al., 1993). A worth-while benefit of the present study is to dispense with theseassumptions since the observed pressure differences are di-rectly linked to the local Eulerian wind fields through a re-lation derived from the continuity equation.

However, these pressure difference maps have been ob-tained through a sequence of manipulations (cloud classi-fication, conversion from pixel to temperature, from tem-perature to pressure) and simplifications (the pressure valueof the unobserved “lower” level of the layer is arbitrarilyassigned to a mean value). These manipulations are likelyto generate some errors in the input data. These errors arequantifiable from Eumetsat products through 3 possible val-ues of a quality parameter η of the cloud height assignment.These values are set to η(s) = 0 (no pressure in s), η(s) = 1(pressure of good quality) or η(s) = 2 (poor quality). Thisinformation will be taken into account for the definition ofthe error covariance matrices in the next section.

4.5 Error covariance matrices and initialization issues

4.5.1 Observations The inverse error covariance matrixrelated to the observations (Rk)−1 is diagonal and definedusing the mask of observation Ck :

(Rk(s, s′))−1 =

(Zk)−1(P k)−1 if s = s′ and s ∈ Ck

0 else.(42)

The inverse error covariance matrices (Zk)−1 and (P k)−1

are respectively related to a confidence measure related tothe observation operator and on the pressure height assign-ment errors. We chose to define (Zk)−1 as

(Zk(s, s′))−1 = αobs

(

exp −[Y−H(X)]2

σ2 if s = s′ and s ∈ Ck,

0 else.

(43)

where σ and αobs are parameters to be fixed. As shown in(Corpetti et al., 2008), this penalization amounts to consid-ering a robust norm on the first term of the cost-functionin (21). Such a robust function allows the discarding ofpoints having large ”residual” values of the observation error[Y − H(X)] (called outliers in the Robust Statistics litera-ture (Huber, 1981)). Compared to a quadratic cost function(i.e. R = Id), the use of such robust semi-norms permitslimiting the influence of those erroneous points. The robustnorm, Φ(·), is here expressed in an equivalent (with respectto its minimization) weighted quadratic expression in whichthe weight function involved, z(.) = Φ′(

√.) = (Zk)−1(x, y),

is iteratively updated at every grid point. Different types ofcost functions (convex or non convex) that obey this semi-quadratic reformulation can be chosen under mere assump-tions (mainly strict concavity of Φ′(

√·)). Interested readersmay consult a number of courses to obtain deeper insightsinto the use of robust functions and on the convergenceof the associated semi-quadratic minimization process (Hu-ber, 1981; Geman and Reynolds, 1992; Delanay and Bresler,1998). Such robust estimators have demonstrated significantimprovements for computer vision applications (Geman andReynolds, 1992; Memin and Perez, 1998). In our application,it enables us to properly deal with corrupted areas that donot fit our data model exactly.

As for matrix (P k)−1, it is directly defined from thequality index η:

(P k)−1(s, s) =

8>><

>>:

0 if η(s) = 0, s = s′ and s ∈ Ck

1 if η(s) = 1, s = s′ and s ∈ Ck

0.5 if η(s) = 2, s = s′ and s ∈ Ck

0 else.

(44)

In this way, the impact of the measurements in areas asso-ciated with a poor quality of the pressure level is lowered.

4.5.2 Initial conditions As no guarantee of convergencetowards a global minimum can be ensured for such a non-convex functional, the quality of results depends on the ini-tial conditions given to the system. We chose to initializethe velocities vk

0 and the associated divergence and vorticity[Dk

0 , ξk0 ] by filtering wind fields provided by the optic-flow al-

gorithm dedicated to atmospheric layers proposed in (Heaset al., 2007), with the Gaussian Kernel Kδx . Two inversediagonal error covariance matrices (Bk

D)−1 and (Bkξ )−1 are

defined as :

(BkD)−1(s, s) = αD

(Bkξ )−1(s, s) = αζ

(45)

where αD and αξ are fixed parameters defining the inverseof the initial variable error covariances.

4.5.3 Dynamics A nine-diagonal inverse error covariancematrix (Qk)−1 fixing the model inaccuracy with a givenparameter αQ has been defined for the dynamic model in asimilar way as for the initial conditions.

All the components needed for the definition of our as-similation process have now been successively defined in sec-tions 4.1, 4.2, 4.3, 4.4 and 4.5. The next section presentssome experiments using synthetic and real data.

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5 Experiments

For a quantitative evaluation of the proposed image as-similation scheme, we have first performed experiments onsynthetic image observations generated by short time nu-merical simulation of atmospheric layer motion. This evalu-ation is described in section 5.1. As the dynamic model ofour assimilation process in (32) is a simplified version of theprimitive shallow-water model presented in (17), two exper-iments related to different synthetic observations have beencarried out. The first synthetic expeiment (section 5.1.1)uses images generated with the simplified model whereas theimages of the second one (section 5.1.2) have been producedfollowing the shallow-water model. Section 5.2 is devoted tothe application of the assimilation scheme on real Meteosatdata. The core of section 5.3 is a discussion of practical as-pects.

5.1 Synthetic image sequences

For the sake of clarity, we denote the proposed simpli-fied dynamic model in (32) as model-a and the primitiveshallow-water model (17) as model-b. For both cases, weselected realistic initial conditions (for images and motionfields) to integrate the models in an equivalent time periodof 2h30min and to create two different sequence benchmarkscomposed of 10 frames each, which are denoted respectivelyas seq-a (for model-a) and seq-b (for model-b) . Each imageis composed of 128 × 128 pixels and represents a spatialarea of approximately 400km2. These benchmarks havethen been deteriorated by different noises and by maskingoperations corresponding to real situations (the maskingarea represents approximately 80% of the global area). Weformed 4 synthetic sequences a1, a2, a3, a4 related to seq-a,and 4 other synthetic sequences b1, b2, b3, b4 related toseq-b. Sequences a1, b1 and a2, b2 were composed of denseobservations of hk

obs in hPa, corrupted by Gaussian noiseswith standard deviation respectively equal to 20% and 40%of the pressure amplitude. A realistic cloud classificationsequence was employed to extract regions from a1, a2, b1

and b2 in order to create the noisy and incomplete syntheticsequences a3, a4, b3 and b4. To initialize the assimilationsystems, we have used the synthetic ground truth valuesof vk(t0) deteriorated by Gaussian noises with standarddeviation equal to 20% and 40% of the variable amplitudesrespectively for experiments using synthetic sequencesa1, a3, b1, b3 and a2, a4, b2, b4.

Table 1 summarizes the characteristics of the two sequencesused. This table also presents the initial value of Root MeanSquare Error (RMSE) of state variables. These values corre-spond to the discrepancy between the ground truth and thestate variables obtained after a first direct integration fromthe noisy initial conditions. Figure 2 displays examples ofnoisy and sparse pressure difference observations and initialvalues of the state variables used in the experiments.

5.1.1 Synthetic sequence generated from the proposed sim-plified model In this experiment, the dynamic model in-volved in the generation of the image sequence exactly cor-responds to the dynamics used in the assimilation process

(model-a, eq. (32)). The goal of this experiment is to analyzethe robustness of the assimilation with respect to noisy dataand initial settings.

Comparing Table 1 and Table 2, one can note asignificant decrease induced by the assimilation system onthe RMSE between the actual and the estimated velocities;the final RMSE is reduced roughly by a factor of ≈ 85%by the assimilation process. Moreover, the stable behaviorof the RMSE (in the four cases considered) demonstratesthe robustness of the approach to cope with incompleteand noisy observations. A motion field and the associatedvorticity and divergence maps obtained for experiment a4

are plotted in the top part of figure 3 and compared to theactual solution. A comparison between those images andthe images of their initial values, displayed in Figure 2,clearly illustrates the estimation efficiency.

5.1.2 Synthetic sequence generated from the primitiveshallow-water model In this application, the dynamicmodel used for the construction of images corresponds tothe primitive shallow-water model (eq (17)). Therefore, itdiffers from the dynamic model considered in the assimila-tion process. The goal of this experiment is then twofold :1- to analyze the ability of the method to deal with an im-perfect dynamic modeling involving additional uncertaintyvariables and 2- to evaluate the accuracy of the simplifica-tion considered in section 4.1.2.

Comparing Table 1 and Table 2, a significant decreaseof the RMSE after assimilation can still be observed. Thefinal RMSE of this experiment is reduced by approximately75%. The residual errors of the divergence and vorticityrecovered are slightly above the errors obtained in theprevious experiment but remains very weak. For bothexperiments, the estimated velocity fields are of the sameorder of accuracy. This result demonstrates the capacity ofthe simplified dynamics together with a weak constraintassimilation formulation to fit an exact shallow-waterdynamic. Here again, the stable behavior of the RMSEon vorticity, divergence and motion fields for the fourexperimental cases is noticeable. Vorticity, divergence andmotion field estimates obtained from experiment b2 arepresented at the bottom of Figure 3. The efficiency of theprocess can be observed by comparing those images withthe images displayed in Figure 2 and corresponding to theinitial values of the state variables for the same benchmark.

The performance of the simplified dynamic model hasbeen assessed from two different synthetic cases. These ex-periments have shown the capacity of the proposed assimi-lation scheme to estimate accurate motion fields from syn-thetic pressure sequences. Let us now study the behavior ofthe approach on a real case.

5.2 METEOSAT Satellite image sequence

We turn now to a qualitative evaluation of our methodon METEOSAT Second Generation (MSG) meteorologicalimage sequences acquired over the Northern Atlantic Oceanon 5 June 2004 from 13H30 until 15H45 UTC at the rateof one image per 15 minutes. This benchmark data, which

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has been provided by the EUMETSAT consortium, is com-posed of 10 frames sequence of top of cloud pressure andcloud-classification images. The image spatial resolution is3 × 3 km2 at the center of the whole Earth image disk.The cloud-classifications were used to segment images intoK = 3 broad layers, at low, intermediate and high altitude6.Applying the methodology described in section 4.4, pressuredifference images for the 3 layers were derived from pressureimages.

In Figure 4 we present a sample of the motion fieldsestimated for two different layers together with the originalobservations. The motion fields estimated for the differentlayers are consistent with a visual inspection of the sequence.In particular, several motion differences between layers arevery relevant. For instance, near the bottom left corner ofthe images, the lower layer possesses a southward motionwhile the intermediate layer moves northward.

In addition, let us outline that the motion of some smallcloud structures have been well characterized by this assimi-lation scheme. This is due to the fact that the shallow-waterassumption is not assumed to be strictly respected every-where. Thus, small scale information from the image obser-vations may locally generate a motion field which signifi-cantly departs from the shallow-water solution. This consti-tutes a benefit of our approach since our preliminary experi-ments in (Corpetti et al., 2008) have shown that the motionsof small structures can not be extracted from an assimila-tion scheme combining a primitive shallow-water model (17)and pressure image measurements. Of course, at this stage,the question of the validation is crucial and at the moment,no ground truth is available (especially concerning the smallscales structures). One possible and interesting idea wouldbe to compare our estimation with AMV products. However,this would be difficult as it would require the developmentof adapted procedures to deal with the resulting sparsityof AMV data when one focuses on a particular layer. Themotion vectors from AMV are indeed assigned to a preciseheight and the way to compare this kind of data with ourmotion fields is not trivial. This issue will be the object ofa future study. Nevertheless, we have compared our estima-tions with the velocity fields obtained by the layer-dedicatedoptic-flow method of (Heas et al., 2007) and whose resultshave been validated using several Meteosat data. The mapsof divergence and vorticity corresponding to this ”reference”estimator are displayed in Figure 5. It can be observed,from a visual comparison between the lower layer vorticity-divergence components estimated by (Heas et al., 2007) andby the proposed assimilation system, that our estimationsare in accordance with these motion fields. Indeed, time-persistent large structures of the flow are accurately esti-mated by both methods while noise and time-inconsistentstructures have only been removed by the assimilation ap-proach.

6 The number of K = 3 layers is due to the nature of the inputdata –based on routines from Eumetsat– but any number of levelsis supported by the approach.

5.3 Discussion on experiments

5.3.1 Comments This experimental part has been dividedinto two subsections using synthetic and real data. The twoexperiments with synthetic data have illustrated the accu-racy and the stability of our assimilation system in presenceof noisy and sparse observations. The second set of exper-iments allowed us to assess the validity of the simplifieddivergence vorticity form of the shallow-water model con-sidered in this study.

For the experiments on real situations we have checkedthat our scheme supplied motion fields that are visuallyplausible and in accordance with a dedicated atmosphericlayer motion estimator. It is obvious that the validation hasto be more complete. A complete evaluation was beyondthe scope of this study. In the near future we plan to com-pare thoroughly our results with AMV data and in situ dataprovided by dedicated probes (radiosondes for instance) toobtain reference velocities at some points and to allow aprecise statistical assessment in the vicinity of these points.Another key point of error analysis relies on the study of theadjoint variables λ defined in relation (22) when the processhas converged. These components are indeed related to theerrors between the dynamic model and the observations andare likely to exhibit some interesting properties and infor-mation. This point will be also the object of future studies.

5.3.2 Computation time The execution time for oneforward-backward computation depends on i) the size ofimages ii) the number of images and iii) the magnitudeof the vorticity and divergence components: as mentionedin section 4.3, an explicit Runge-Kutta scheme has indeedbeen used for the temporal (forward and backward) inte-gration (Kurganov and Tadmor, 2000). To ensure stabilityof this scheme, the time-step discretization is inversely pro-portional to the magnitude of the velocity. The initializationplays also a key role in the computational cost since the iter-ative process converges faster if one correctly initializes thesystem state.

As an example, in our application, the estimation ofthe complete system state takes approximately 1 min for thesynthetic sequences (128×128 pixels) whereas the estimationon a real layer (1024 × 1024 pixels) takes around 30 minfor a Matlab code, on a PC 2.33Hz with 3Go of RAM. Anefficient “C” implementation would obviously significantlyreduce the computational time.

5.3.3 Parameter values The numerical values of the dif-ferent parameters are set according to the range of variationof the quantity of interest. The values used in all our appli-cations are given in Table 3.

6 Conclusion

In this paper, we have proposed an assimilation ap-proach enabling, for the first time, a dynamically consistentestimation of a sequence of dense and layered atmosphericwind fields from entire satellite image sequences.

The motion estimator is applied to sparse pressure

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difference images corresponding to a stack of layers in astratified atmosphere. A method was proposed to obtainsuch images from top of cloud pressure images and classi-fication, which are routinely supplied by the EUMETSATconsortium.

The dynamic constraint of our system suggested relyingon the shallow-water assumption. We have shown that aperfect shallow-water modeling approach is likely to beunstable for the Meteosat data considered. We have thenderived a simplified dynamic modeling with uncertaintyterms. This model corresponds to a simplification of thevorticity divergence form of a shallow-water dynamics forwhich the momentum equations are independent of thelayer thickness. The mass conservation law complementsthese dynamics and provides the basis of an originalimage-adapted observation operator.

An evaluation performed on synthetic image sequencesand on real METEOSAT top of cloud pressure image se-quences has shown that the technique proposed enablescharacterizing time-consistent wind fields at large and smallscales.

APPENDIX A: Scale analysis of filtered

hydrostatic primitive equations

For mesoscale atmospheric motions at horizontal scale of100 km, the following scale characteristics hold :

U ∼ 10 ms−1 horizontal velocity

W ∼ 1ms−1 vertical velocity

L ∼ 105 m length

H ∼ 103 m depth

δp ∼ 103 Pa horizontal pressure variation

ρ ∼ 1 kg m−3 density

∆t ∼ 103 s temporal sampling period

ν ∼ 10−5 m2s−1 friction coefficient

a ∼ 106 m Earth radius

Ω sin φ ∼ 10−4 s−1 Coriolis coefficientU

2Ω sin φL∼ 0.5 Rossby number

The horizontal primitive hydrostatic equations of rela-tion (2) are :

8>>>>>>><

>>>>>>>:

du

dt|z

−uv tan φ

a| z

+uw

a|z

+px

ρ|z

=+2Ωv sin φ−2Ωw cos φ+ Fu

dv

dt|z

A

+u2 tan φ

a| z

B

+vw

a|z

C

+py

ρ|z

D

=−2Ωu sin φ| z

E

|z

F

+ Fv|z

G

(A1)

An analysis of these terms using the characteristic scales isgiven in Table 4.

We conclude that terms B, C, F and G can be ig-nored in comparison to the other terms. Moreover it is wellknown that, for high Reynolds numbers characteristic ofatmospheric flows, due to the energy dissipation at unob-servable scales, turbulent viscosity dissipation T can not beignored in Navier-Stokes numerical simulations (Meneveauand Katz, 2000; Sagaut, 1998; Smagorinsky, 1963). We thus

obtain the filtered momentum equations written in a vecto-rial form:

ρdv

dt= −∇p − 2Ω sinφ

»0 −11 0

–

v + T . (A2)

APPENDIX B: Calculation of layer average

densities

The integration of the hydrostatic relation between alti-tudes z0 and z (or pressure p0 and p), with a constant lapserate may be written as:

Z p

p0

dp′

p′= −

Z z

z0

g

R(T0 + γz′)dz′. (B1)

This expression gives rise after some calculation to the ex-pression of density as a function of pressure (Holton, 1994)):

ρ(p) =p0

RT0

` p

p0

´ γRg

+1. (B2)

Computing the vertical density average, the mean densityrelated to the k-th layer reads :

ρk0 =

1

pk+1 − pk

Z pk+1

pkρ(p)dp

=p20

(pk+1 − pk)(γRg

+ 2)RT0

h“ p

p0

” γRg

+2ipk+1

pk. (B3)

Note that a constant lapse rate γ, that is to say a linearvariation of temperature with altitude, is a rough approx-imation in the troposphere. However, as we are averagingthe obtained density law vertically and horizontally over thewhole layer spatial support, such an approximation shouldhave a minor impact on the accuracy of the modeling.

APPENDIX C: Scale analysis of Saint Venant’s

model with pressure image observations

For mesoscale atmospheric motions observed through a ME-TEOSAT images sequence which has been low-pass filteredto obtain a characteristic horizontal scale of 100 km, thefollowing characteristic scales hold (with :hk = pk − pk+1)

v ∼ 10 ms−1 horizontal velocity

hk ∼ 104 Pa pressure differences

ρ ∼ 1 kg m−3 density

∆t ∼ 103 s temporal sampling period

∆xy ∼ 105 m spatial scale

fφ ∼ 10−3 s−1 Coriolis coefficient

νT ∼ 104 m2 s−1 turbulent diffusion (cf (5))

The system to study reads:

∂(qk)

∂t| z

A

+div(1

hkqk ⊗ qk)

| z

B

+1

2ρk0

∇xy(hk)2

| z

C

+

»0 −11 0

–

fφqk

| z

D

=νkT

∆(qk)| z

E

,

(C1)

where (νT = (Cδx)2p

2(u2x + v2

y + (uy + vx)2)) (cf relation(5)). Noting that the Pascal unit is hPa = kg · s−2 · m−1

c© 0000 Tellus, 000, 000–000


and recalling that q = hkv), a scale analysis of this systemis given in Table 5.

This corresponds to the order of magnitudes of thequantities of interest under the shallow-water approxima-tion. From Table 5, we observe that term C of eq C1 isslightly higher than all the others. In practice, as soon as theshallow-water assumption breaks, the pressure difference ofa layer hk grows and we observed that the term (∇xy(hk)2)can reach the magnitude of 107kg.s−4. The sensitivity ofthis term w.r.t the assumption generates strong numericalinstabilities. This primitive relation is therefore not adaptedto the assimilation of sparse images of pressure differenceswhen one wants to deal with corrupted input data.

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Figure legend

• Caption Fig. 1: Image observations issued from METEOSAT (Northern Atlantic Ocean on June 2004

from 13h30 until 15H45 UTC). From top to bottom : cloud top pressure image; classification into low (in

dark gray), medium (in light gray) and high (in white) clouds; pressure difference of the higher layer; pressure

difference of the intermediate layer; pressure difference of the lower layer. Black regions correspond to missing

observations and white lines represent coastal contours, meridians and parallels (every 10o).

• Caption Fig. 2: Examples of variables at initial time before assimilation with images issued

from model-a (top), model-b (middle) and the corresponding ground truth (bottom). For each

experiment, the columns exhibit blurred initial difference pressure image hk, wind vk, vorticity ζk and di-

vergence Dk respectively. The top part corresponds to the experiment a4 whereas the middle part represents

data of experiment b2• Caption Fig. 3: Variables after assimilation with images issued from model-a (top) and model-b

(bottom). The corresponding experiments are a4 (top) and b2 (bottom). The first column corresponds to

the wind vk, the second to the vorticity ζk and the third to the divergence Dk fields at t= 1h15min. For each

experiment, the first line corresponds to the estimated variables whereas the second line corresponds to the

ground truth.

• Caption Fig. 4: Horizontal wind fields estimated by assimilation with an imperfect modeling for

the intermediate layer (top part) and the lower layer (bottom part). For each layer, the first line represents 4

images of the sequences. The three first motion fields of the second line correspond to estimation at t = 0min,

t = 1h15min and t = 2h30min respectively. The last motion field is the difference vk(2h30) − vk(1h15) and

illustrates the temporal changes.

• Caption Fig. 5: Comparison of results from assimilation (top) and optical-flow (bottom) for the

lower layer at t = 0min, t = 1h15min and t = 2h30min respectively. The first line represents the vorticity

and the second line the divergence. One can conclude that both estimations are in accordance but the one

from the assimilation has a better temporal consistency.


Table 1. Initial setup used for assimilation with images issued from model-a (top) and model-b (bottom). Root MeanSquare Error (RMSE) on the vorticity ζk, divergence Dk and wind norm vk obtained at all times t of the image synthetic sequences.

Mask Noise rmse on all ζk(t) rmse on all Dk(t) rmse on all|vk(t)|% (1/frame) (1/frame) (pixel/frame)

a1 no 20 0.01119 0.01343 0.08798

a2 no 40 0.01481 0.01518 0.09512

a3 yes 20 0.01119 0.01343 0.08798

a4 yes 40 0.01481 0.01518 0.09512

b1 no 20 0.01297 0.00932 0.09352

b2 no 40 0.01670 0.01814 0.1009

b3 yes 20 0.01297 0.00932 0.09352

b4 yes 40 0.01670 0.01814 0.1009

Table 2. Final estimation errors obtained by our assimilation approach with images issued from model-a (top) andmodel-b (bottom). Root Mean Square Error (RMSE) and residual error (RE) on estimates ζk, Dk and vk obtained at all times t ofthe image synthetic sequences.

Errors on ζk(t) Errors on Dk(t) Errors on |vk(t)|

RMSE RE RMSE RE RMSE RE(1/frame) % (1/frame) % (pixel/frame) %

a1 0.00379 8.71 0.00182 9.99 0.01787 9.59

a2 0.00405 8.91 0.00214 10.87 0.01783 11.12

a3 0.00378 9.20 0.00182 11.61 0.01782 9.58

a4 0.00405 9.51 0.00214 11.74 0.01784 11.11

b1 0.00514 8.94 0.00444 13.21 0.01883 9.77

b2 0.00536 9.37 0.00450 13.62 0.02139 11.27

b3 0.00514 9.43 0.00441 13.41 0.02000 10.61

b4 0.00560 10.69 0.00481 14.74 0.02429 13.45

Table 3. Default parameters: values of the covariance parameters used in our applications

Variable Value Description

αobs 2 Error cov. observation (eq (43))σ 5 Outlier parameter (eq (43))

αD 10−3 Error cov. on initial D (eq (45))αζ 10−3 Error cov. on initial ζ (eq (45))αQ 10−3 Error cov. on dynamics (section 4.5.3)

Table 4. Scale analysis of the horizontal primitive hydrostatic equations (A1).

A B C D E F G

x-eq. dudt

uv tan φa

uwa

1ρ

∂p∂x

2Ωv sinφ 2Ωwcosφ Fu

y-eq. dvdt

u2 tan φa

vwa

1ρ

∂p∂y

2Ωu sinφ – Fv

scales U∆t

U2

aUW

aδpρL

Ωsin φU ΩcosφW νUH2

(ms−2) 10−2 10−4 10−5 10−2 10−3 10−4 10−10

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Table 5. Scale analysis of the Saint Venant’s model (C1).

A B C D E

x-eq. ∂hkuk

∂t

∂(hk(uk)2)∂x

+ ∂(hkukvk)∂y

∂(hk)2

2ρk0

∂x−fφhkv νk

T((hku)xx + (hku)yy)

y-eq. ∂hkvk

∂t

∂(hkukvk)∂x

+∂(hk(vk)2)

∂y

∂(hk)2

2ρk0

∂yfφhku νk

T((hkv)xx + (hkv)yy)

scales q/∆t hkv2/∆xy (hk)2/ρ∆xy fφhkv νT hkv/∆2xy

(kg.s−4) 102 101 103 102 102

Fig. 1. Image observations issued from METEOSAT (Northern Atlantic Ocean on June 2004 from 13h30 until 15H45 UTC). Fromtop to bottom : cloud top pressure image; classification into low (in dark gray), medium (in light gray) and high (in white) clouds;pressure difference of the higher layer; pressure difference of the intermediate layer; pressure difference of the lower layer. Black regionscorrespond to missing observations and white lines represent coastal contours, meridians and parallels (every 10o).

c© 0000 Tellus, 000, 000–000


hk(t0) vk(t0) ζk(t0) Dk(t0)

Fig. 2. Examples of variables at initial time before assimilation with images issued from model-a (top), model-b (middle)and the corresponding ground truth (bottom). For each experiment, the columns exhibit blurred initial difference pressure imagehk, wind vk, vorticity ζk and divergence Dk respectively. The top part corresponds to the experiment a4 whereas the middle partrepresents data of experiment b2

c© 0000 Tellus, 000, 000–000


vk(t) ζk(t) Dk(t)

exp a4

ground truth

exp b2

ground truth

Fig. 3. Variables after assimilation with images issued from model-a (top) and model-b (bottom). The correspondingexperiments are a4 (top) and b2 (bottom). The first column corresponds to the wind vk, the second to the vorticity ζk and the third tothe divergence Dk fields at t= 1h15min. For each experiment, the first line corresponds to the estimated variables whereas the secondline corresponds to the ground truth.

c© 0000 Tellus, 000, 000–000


vk(0min) vk(1h15min) vk(2h30min) vk(2h30) − vk(1h15)

vk(0min) vk(1h15min) vk(2h30min) vk(2h30) − vk(1h15)

Fig. 4. Horizontal wind fields estimated by assimilation with an imperfect modeling for the intermediate layer (top part)and the lower layer (bottom part). For each layer, the first line represents 4 images of the sequences. The three first motion fields of thesecond line correspond to estimation at t = 0min, t = 1h15min and t = 2h30min respectively. The last motion field is the differencevk(2h30) − vk(1h15) and illustrates the temporal changes.

c© 0000 Tellus, 000, 000–000


t = 0min t = 1h15min t = 2h30min

ζk

Dk

ζk

Dk

Fig. 5. Comparison of results from assimilation (top) and optical-flow (bottom) for the lower layer at t = 0min, t = 1h15minand t = 2h30min respectively. The first line represents the vorticity and the second line the divergence. One can conclude that bothestimations are in accordance but the one from the assimilation has a better temporal consistency.

c© 0000 Tellus, 000, 000–000

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Pressure image assimilation for atmospheric motion estimation