ASSIMILATION OF IMAGES FOR GEOPHYSICAL FLUIDS

1. ASSIMILATION OFIMAGES FOR GEOPHYSICAL FLUIDS François-Xavier Le Dimet Arthur Vidard Innocent Souopgui Université Joseph Fourier and INRIA, Grenoble NASA, JPL,January 2011

2. ADDISA Research Group • CLIME INRIA Paris • Météo France • Institut de Mathématiques, Université de Toulouse • LEGI, Grenoble • MOISE INRIA Grenoble and Université de Grenoble

3. SUMMARY • Observing the Earthwith satellites. • Data Assimilation. • Images. • Plugging images intonumericalmodels. • Variationalapproach. • Pseudo-Observationsmethods. • Direct Assimilation of Images Sequences. • Operational Applications • Perspectives

4. Data Assimilation • Numericalmodels are not sufficient to carry out a prediction. • Numericalmodels are based on non linearPDE’s and after a spatial discretisation a system of first orderODE’s of hugedimensionnality (at the present time around one billions of equations for operationalmodels. • Predictionisobtained by an integration of the model startingfrom an initial condition. • The processnecessary for obtaining an initial condition from data isnamed Data Assimilation

5. Data Assimilation (2) • Basicallyit’s a ill-posedproblem : about 10 millions of daily data to retrieve 1 billions of unknowns • Interpolation methods are not sufficient to obtain consistant fields (with respect to fluiddynamics) • VariationalMethods are based on Optimal Control Methods and are presentlyused by the main meteorologicalcenters • Kalmanfilterapproachisused in mainly in a researchcontext

6. Assimilation of Images • Images provided by the observation of the earthquantity a large amount of information • This information isused in a qualitative wayratherthan in a quantitative one. • How to couple this source of information withmathematicalmodels in order to improveprediction?

7. 27 satellite data sources used in 4D-Var DMSP SSM/I NOAA AMSUA/B HIRS, AQUA AIRS SCATTEROMETERS GEOS TERRA / AQUA MODIS OZONE ICTMA 13

8. Number ofData used per Day ICTMA 13

13. Images • Images are defined by pixels • For black and white image each pixel isassociatedwith a greylevel (0<gl<1). For Meteosat 256 greylevels • For color images each pixel isassociated to 3 numbers. • Each image (Meteosat) has around 25 millions of pixels. • A full sequence of images is a very large data set and cannotbedirectlyused in an operationalcontext

14. Whatisseen ? • The basic variables of meteorologicalmodels are : wind, temperature, humidity, atmospheric pressure onlyhumiditycanbeseen on some satellites. • For oceanicmodels : stream, temperature, salinity, surface elevation. Onlysalinity and temperaturegive 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 oceancanbeocculted by clouds.

17. Experimentalframework: Coriolis Rotating Platform

19. MathematicalModels for GeophysicalFlows • Based on laws of conservation (mass, energy) • NonlinearPDE’slinking the state variables of the model • To use images itisnecessary to introduce the evolution of the quantitiesdispalyed by images: • Humidity (for meteorologicalmodels) • Salinity (for oceanicmodels) • Conservation of a (supposed to be) passive tracer (e.g. phyloplakton in oceanicmodels) • Conservation of luminance • In any case a complexification of the models if images are takenintoaccount.

20. Variationalapproach for Data Assimilation.

21. Observation Operator

22. Optimality System: using adjoint variable

23. Two basic approaches for assimilating images. • Pseudo Observations Methods. • From images velocities are extracted, thenused as regular observations. • Direct Assimilation of Images. • An extra termisadded in the costfunctionevaluating the discrepancybetween the pseudo-imagesissuedfrom the numerical model and the oberved image, then the usualtools of VDA are used

24. Using Image model

25. Image Model Approach (1) • The temporal coherency of a sequence of image isobtained by a law of conservation of brightness • If the gradient of brightness and the velocity are orthogonal then no information isadded (if an image isuniformthenitcan’tprovide information on velocity) • How to isolate structures suchthatthisequationisrepresentative of the flow?

26. Image Model Approach (2) • Recovering U from I is a illposedproblem • Introducing a problem of optimization

27. Zoology of Regularization

28. MultiscaleApproach • The minimization of the costfunctionisperformed in nestedsubspaces of admissible deplacementsfieldsatscale q. It containspiecewise affine vectorfieldswith respect to eachspace variables on a square of size qxq pixels. • In practice with 2 successive time steps

29. Example : Shallow-Waterequation

37. Object Tracking

38. Experimental Data (Coriolis Rotating Platform)

40. Extended Image Model

41. Extended Model Image • Model Image methods do no takeintoaccount the physicalproperties of the fieldsissuedfromfluidmechanics. • The retrievedfieldcanbecoherent but with few physicalsense. • The ideais to add to the optimizationproblem a physicalconstraintissuedfrom the equationgoverning the fields.

42. Example : Shallow Water model + Thermodynamics.

43. OptimizationProblem

44. Comparison of IM and EIM (SST)

45. Comparison of IM and EIM (Velocity)

46. TwinExperiments : Images ( 1 hourbetween 2 images

47. Initial Condition for the optimization

48. RetrievedfieldsafterOptimization

49. Direct Assimilation of Images : principle

50. Direct Assimilation of Images : CostFunction