ASSIMILATION OF

IMAGES FOR GEOPHYSICAL FLUIDSFrançois-Xavier Le Dimet

Arthur Vidard Innocent Souopgui

Université Joseph Fourier and INRIA, Grenoble

NASA, JPL,January 2011

ADDISA Research Group CLIME INRIA Paris Météo France Institut de Mathématiques, Université

de Toulouse LEGI, Grenoble MOISE INRIA Grenoble and Université

de Grenoble

SUMMARY Observing the Earth with satellites. Data Assimilation. Images. Plugging images into numerical models. Variational approach. Pseudo-Observations methods. Direct Assimilation of Images Sequences. Operational Applications Perspectives

Data Assimilation Numerical models are not sufficient to carry out

a prediction. Numerical models are based on non linear

PDE’s and after a spatial discretisation a system of first order ODE’s of huge dimensionnality (at the present time around one billions of equations for operational models.

Prediction is obtained by an integration of the model starting from an initial condition.

The process necessary for obtaining an initial condition from data is named Data Assimilation

Data Assimilation (2) Basically it’s a ill-posed problem : about 10

millions of daily data to retrieve 1 billions of unknowns

Interpolation methods are not sufficient to obtain consistant fields (with respect to fluid dynamics)

Variational Methods are based on Optimal Control Methods and are presently used by the main meteorological centers

Kalman filter approach is used in mainly in a research context

Assimilation of Images Images provided by the observation of

the earth quantity a large amount of information

This information is used in a qualitative way rather than in a quantitative one.

How to couple this source of information with mathematical models in order to improve prediction?

ICTMA 13

NOAA AMSUA/B HIRS, AQUA AIRS DMSP SSM/I

SCATTEROMETERS GEOS

TERRA / AQUA MODIS OZONE

27 satellite data sources used in 4D-Var

ICTMA 13

Number of Data used per Day

Images Images are defined by pixels For black and white image each pixel is

associated with a grey level (0<gl<1). For Meteosat 256 grey levels

For color images each pixel is associated to 3 numbers.

Each image (Meteosat) has around 25 millions of pixels.

A full sequence of images is a very large data set and cannot be directly used in an operational context

What is seen ? The basic variables of meteorological models are : wind,

temperature, humidity, atmospheric pressure only humidity can be seen on some satellites.

For oceanic models : stream, temperature, salinity, surface elevation. Only salinity and temperature give images.

For the atmosphere the images represent the integral of the radiative properties of the atmosphere.

For the ocean the images represent the surface values of the radiative properties of the ocean

Information in images is borne by the discontinuities in the images (e.g. fronts)

Images of the ocean can be occulted by clouds.

Experimental framework: Coriolis Rotating Platform

Mathematical Models for Geophysical Flows

Based on laws of conservation (mass, energy) Nonlinear PDE’s linking the state variables of the

model To use images it is necessary to introduce the

evolution of the quantities dispalyed by images:› Humidity (for meteorological models)› Salinity (for oceanic models)› Conservation of a (supposed to be) passive tracer (e.g.

phyloplakton in oceanic models) › Conservation of luminance› In any case a complexification of the models if images

are taken into account.

Variational approach for Data Assimilation.

Observation Operator

Optimality System: using adjoint variable

dpdt

+∂M∂x

⎡⎣⎢

⎤⎦⎥*

p =∂H∂x

⎡⎣⎢

⎤⎦⎥*

.(H(x)−y),t∈[T,0],

p(T)=0.

⎧⎨⎪

⎩⎪⎪

∇J=∇UJ∇V J

⎛⎝⎜⎜

⎞⎠⎟⎟ =

−∂M∂U

⎡⎣⎢

⎤⎦⎥*

0

T

∫ .p

−p(0)

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥.

Two basic approaches for assimilating images.

Pseudo Observations Methods.› From images velocities are extracted, then

used as regular observations. Direct Assimilation of Images.

› An extra term is added in the cost function evaluating the discrepancy between the pseudo-images issued from the numerical model and the oberved image, then the usual tools of VDA are used

Using Image model

Image Model Approach (1) The temporal coherency of a sequence of

image is obtained by a law of conservation of brightness

If the gradient of brightness and the velocity are orthogonal then no information is added (if an image is uniform then it can’t provide information on velocity)

How to isolate structures such that this equation is representative of the flow?

dIdt

=0 =∂I∂t

+u∂I∂x

+ v∂I∂y

Image Model Approach (2) Recovering U from I is a ill posed

problem Introducing a problem of optimization

J(u,v) =0

T

∫∂I∂t

+ u∂I∂x

+v∂I∂y

⎛⎝⎜

⎞⎠⎟Ω

∫2

dtdx + R(u,v)

The problem is to determ ine (u*,v*) m inim izing J

R(u,v) is a regularisation term is the sense of Tykhonov to ensure the existence and uniqueness of a m inim um .It worth noting that the physical properties of the flow are not taken into account.

Zoology of Regularization

Multiscale Approach The minimization of the cost function is

performed in nested subspaces of admissible deplacements fields at scale q. It contains piecewise affine vector fields with respect to each space variables on a square of size qxq pixels.

In practice with 2 successive time steps

Example : Shallow-Water equation

Object Tracking

Experimental Data (Coriolis Rotating Platform)

Extended Image Model

Extended Model Image Model Image methods do no take into

account the physical properties of the fields issued from fluid mechanics.

The retrieved field can be coherent but with few physical sense.

The idea is to add to the optimization problem a physical constraint issued from the equation governing the fields.

Example : Shallow Water model + Thermodynamics.

Optimization Problem

Comparison of IM and EIM (SST)

Comparison of IM and EIM (Velocity)

Twin Experiments : Images ( 1 hour between 2 images

Initial Condition for the optimization

Retrieved fields after Optimization

Direct Assimilation of Images : principle

Direct Assimilation of Images : Cost Function

Frequential Characteristics Extraction

Curvelets

Model and Structure Operators

True, Analyzed and Forecast (velocity profiles)

Experimentation on Coriolis Platform with curvelets

Coriolis : Structure operator identity

Coriolis ; Stucture Operator Curvelets+ Threshold

Direct Assimilation of Images using Finite Time Lyapunov Exponents and Vectors

FSLE is a Lagrangian tool to characterize coherent structures in time-dependant flows.

Widely used in oceanography to link ocean tracer distribution with mesoscale geostrophic currents in order to study stirring and mixing processes .

FTLE are not directly observed but extracted from the ocean tracer images

Finite-Time Lyapunov exponents and vectors

Justification For a 2D decaying turbulent flow, the

orientation of the gradient of the concentration of a passive tracer converges to backward FLTV orientation

Extracting structure from images

Principle of the assimilation of structure.

Structure are extracted by applying a threshold of the gradient of concentration.

Same operation is carried out on the results of the numerical model.

The comparison between these structures is done on the associated FTLE

Operational Applications Evolution of dry intrusion in

cyclogenesis Follow-up of convective cells in radar

meteorology : short term prediction of severe storms

Conclusions Images have a strong predictive

potential Images can be plugged in numerical

models Some new developments are underway

for the study and parametrization of turbulence

Other developments e.g. in heart modelling and other scientific fields