Theoretical Analyses and Numerical Tests of Variational Data Assimilation with Regularization Methods

1. Theoretical Analyses and Numerical Tests of Variational Data Assimilation with Regularization Methods Huang Sixun P.O.Box 003, Nanjing 211101,P.R.China Email: huangsxp@yahoo.com.cn Canada-China Workshop on Industrial Mathematics HongKong Baptist University, 2005

2. It is well known that numerical prediction of atmospheric and oceanic motions is reduced to solving a set of nonlinear partial differential equations with initial and boundary conditions, which is often called direct problems. In the recent years, a variety of methods have been proposed to boost accuracy of numerical weather prediction, such as variational data assimilation(VAR), etc. VAR is using all the available information (e.g., observational data from satellites, radars, and GPS, etc.) to determine as accurately as possible the state of the atmospheric or oceanic flow.

3. Contents Part ATheoretical aspects • A.1 What’s the variational data assimilation? • A.2 Idea of adjoint method of VAR • A.3 3D-VAR • A.4 4-D VAR

4. Part B Applications • B.1 variational assimilation for one-dimensional ocean temperature model • B.2 ENSO cycle and parameters inversion • B.3 Assimilation of tropical cyclone(TC) tracks • B.4 Inversion of radar • B.5 Inversion of satellite remote sensing data and its numerical calculation • B.6 Generalized variational data assimilation with non- differential term • B.7 Variational adjustment of 3-D wind field • B.8 The model of GPS dropsonde wind-finding system

5. A.1 What’s the variational data assimilation? Talagrand 1995 Assimilation: using all the available information, determine as accurately as possible the state of the atmospheric or oceanic flow Variational Data Assimilation: study assimilation through variational analytical method(adjoint method)

6. Data assimilation undergoes the following stages Stage 1 Objective AnalysesInterpolating observational data at irregular observational points to regular grid points by statistical methods, which would be taken as initial fields Stage 2 Initialization Filtering high frequency components in initial fields so as to reduce prediction errors

7. Stage 3 3D –VAR Adjusting initial field x0so that x0is compatible with observations yand background xb , i.e. to make the following cost function minimum H----observation operator( nonlinear operator) y---observational field xb- --- background field B---covariance matrix of background O---covariance matrix of observation

8. Stage 4 4D-VAR • Case 1 State equations F is the classical PDO Observation Xobs [0,T] Cost functional C---linear operator It means that gives the “true value of the field at the point (in space and /or in time) of observation This is optimal control of PDEs

9. Case 2 • Model w(t) is assumed to have 0 mean and covariance matrix error Q(t) • information background fields xb covariance matrix of background error

10. observational data y e(t) is assumed to have 0 mean and covariance matrix O(t). e(t) is white process, and also assumed to be uncorrelated with the model error w(t). • cost functional

11. A.2 Idea of adjoint method of VAR • As an example, we consider the inversion of IBVC for the following problem ----- observational data the cost functional is

12. Idea: solving an optimization problem by descent algorithm iteration Approximate solutions convergence

17. Some key difficulties of adjoint method of VAR (1) Ill-posedness During iteration, the cost functional oscillates, and decreases slowly so as to lead too low accuracy. The reason: ill-posedness (2) Error of BVC

19. (3) Local observations In some cases, especially in the oceans, observations are not incomplete, e.g., observations are obtained from ships, sounding balloons, which will lead to calculation unstable, and therefore is worth studying further. (4) Variational data assimilation with non-differentiable term (on-off problem) The adjoint method holds only with differentiable term; for systems containing non-differentiable physical processes( called as “on-off” ) , a new method must be developed.

20. A3 3D -VAR If H is linear operator , we obtain the optimal estimate And the error estimate matrix is

21. Some key difficulties in 3D-VAR • H is an on observational operator Prob.1: How to find H ? Prob.2: H is not a surjection. How to deal with it ? • B is non-positive • O is non-positive • The hypothesis of unbiased errors is a difficult one in practice, because there often as significent biases in the background fields(caused by biases in the forecast model) and in the observations ( or in the observational operators) • The hypothesis of uncorrelated errors • H is a nonlinear operator, which leads to J = min! is not unique, i.e. ill-posedness

22. A4 4D-VAR • Model

24. If we suppose ,then the direct equations and adjoint equations are not coupled, except at the initial time t0

25. Part B Applications • B.1 variational assimilation for one-dimensional ocean temperature model • B.2 ENSO cycle and parameters inversion • B.3 Assimilation of tropical cyclone(TC) tracks • B.4 Inversion of Radar • B.5 Inversion of satellite remote sensing data and its numerical calculation • B.6 Generalized Variational Data Assimilation for Non- Differential System • B.7 Variational Adjustment of 3-D Wind Field • B.8 The model of GPS Dropsonde wind-finding system

26. B.1 variational assimilation for one-dimensional ocean temperature model • The one-dimensional heat-diffusion model for describing the vertical distribution of sea temperature over time is, • Here is sea temperature, is the vertical eddy diffusion coefficient, is the sea water density, is the sea water specific heat capacity, is the light diffusion coefficient, is the depth of ocean upper layer, is the transmission component of solar radiation at sea surface, is the net heat flux at sea surface. It is known that there exists the unique solution of the model if the initial boundary condition and the model parameters are known and smooth.

27. Assume , are known constants, the initial boundary conditions , and model parameters , are not known exactly, e.g., they have unknown errors and need to be determined by data assimilation. Now a set of observations of sea temperature is given on the whole domain. A convenient cost functional formulation is thus defined as Where is a stable functional and is a regularization parameter. The problem is: Find the optimal initial boundary conditions and model parameters , such that J is minimal.

28. Decreasing of the cost functional J with iteration number

30. B.2 ENSO cycle and parameters inversion • ENSO: The acronym of theEl Nino -Southern Oscillationphenomenon which is the most prominent international oscillation of the tropical climate system.

31. The phase of the Southern Oscillation on El Nino • High temperature over eastern Pacific; High surface pressure over the western and low surface pressure over the south-eastern tropical Pacific coincide with heavy rainfall, unusually warm surface waters, and relaxed trade winds in the central and eastern tropical pacific

32. LaNina • The phase of the Southern Oscillation on La Nina • Surface pressure is high over the eastern but low over the western tropical Pacific, while trades are intense and the sea surface temperature and rainfall are low in the central and eastern tropical Pacific

34. A nonlinear dynamical system for ENSO: : Sea Surface Temperature Anomaly (SSTA) : thermocline depth anomaly : a monotone function of the air-sea coupling coefficient : external forcing : constants .

35. Observation: Obtain the time series of T and h(denoted by and from the observational data set TAO (Tropical Atmosphere and Oceans)

36. The time series of T (solid line) and h (dotted line); The phase orbit of T and h (Running clockwise as the time goes on)

37. Now, we seek optimal parameter and external forcing , such that the solution satisfies :the terminal control term : the control parameter.

38. Blue : the observed valuered : the value predicted by the original modelblack: the value predicted by the improved model whengreen: the value predicted by the improved model when

39. B. 3 Assimilation of tropical cyclone(TC) tracks • A TC is regarded as a point vortex, whose motion satisfies • Here , , are the velocity and coordinates of TC center respectively, and is the force exerted on TC, but don’t include the Coriolis force . Suppose that over the interval, the observational TC track is .

40. Now, the goal is: to determine the optimal initial velocity and forces , such that the corresponding solution makes the functional minimal. are referred to as the regularization parameters, is the restraint parameter at the terminal.

41. Table1. Main Characteristics of 4 TCs

42. Inversion of TC 9804 track

43. Retrieved forces for TC 9804

44. B.4 Inversion of Radar

45. The Definition of Radar Radar is an acronym for “Radio Detecting And Ranging”. Radar systems are widely used in air-traffic control, aircraft navigation, marine navigation and weather forecasting.

46. The Definition of Doppler Radar Doppler radar：the radar can detect both reflectivity intensity and radial velocity of the moving objects with the “Doppler effect”. The right graphic show： The forming process of reflectivity

47. The forming process of radial velocity

48. The 2-D horizontal wind is governed by the following conservation of reflectivity factor of Radar and of mass in the polar coordinates • where are time , redial distance and azimuth respectively, is the reflectivity factor of Radar, are redial and azimuthal velocity respectively. is eddy diffusion coefficient. is given by diagnosis. The inversion domain is

49. Suppose that the observational data are known, the aim is to determine 2-D wind and . This is a ill-posed problem. We introduce the following functional • where 、 、 、 and are weight coefficients.

50. true true vortex wind field retrieved retrieved vortex wind field