Wednesday, 25 January 2012: 11:30 AM

A SOM-PCA Model for Inverting Shalow-Water Acoustic Data

Room 340 and 341 (New Orleans Convention Center )

Poster PDF (2.4 MB)

Variational assimilation (VA) is a data assimilation method widely used for weather and ocean state forecasting. It consists in introducing a cost function J representing the misfit between the output of a numerical model of the physical phenomena and the corresponding observations, to which one adds a restoring term to a background. VA method usually assumes that one has an initial solution cb (background) around which the unknown solutions (control parameters) are normally distributed with a covariance matrix B. The minimization of function J with this a priori information leads to a local search around the vector cb. In geophysical problems, the control parameters c belong to a high dimensional space and its components are highly correlated (like the profiles of salinity or temperature in the ocean or the profiles of humidity in the atmosphere). They present continuous variations in space and time and for that reason they are located on a non linear continuous manifold of smaller intrinsic dimension. As this manifold is continuous, one can assume that it is locally linear in the vicinity of the background cb. Thus the variational assimilation becomes a local search on a quasi linear sub manifold around vector cb. The major assumptions of VA are that the local density of the control parameters is a normal density and that the result of the assimilation is not far from the background. This is partially true for the geophysical measurements if they are limited in space and in time. Locally it becomes possible to model the local density using the PPCA (probabilistic principal component analysis) approach, which permits estimating the background error covariance matrix B and characterizing and removing the noise superimposed to the data. But quite often there is not enough information to determine, in an appropriate manner, locally and around a background vector. We must in this case, make a global optimization, seeking the optimal solution on the nonlinear manifold of the control vector. Because of the high variability at a seasonal time scale of the oceanographic and atmospheric problems, the over all data set representing the control vector is not Gaussian and the use of PPCA is not valid in this case. Taking into account the linear local structure of the whole set of control vector, it is possible to assume that its overall density function is a mixture of PPCA. Modelling by mixture of PPCA has been well formalized, but other empirical models have been proposed such as the local PCA. These methods perform a partition of the set of control vectors and each subset of the partition admits a local Gaussian structure, which allows its modelling by PPCA. The methodology (PPCA) can be applied locally, and therefore allows a local variational assimilation. It is then possible to find the global optimal solution by applying the previous methodology at each component of the partition obtained by local PCA method (or mixture) and retain the "optimal" solution among all obtained solutions. But this exhaustive search is very time consuming and therefore not operational. We must find an initial partition, close to the partition that contains the global optimal solution, and perform research on partitions of its neighbourhood in order to locate the one containing the desired control vector. Determining the neighbourhood structure between the different partitions (of local PCA) is possible by self organizing maps (SOM), which are unsupervised learning models. These maps allow to partition the space of control vector and to define a discrete distance, between the obtained subsets, which takes into account their proximity in space. It remains to define convenient probabilities on the SOM map permitting to select an initial partition and then to define a random walk on the map, with a Markov chain. The purpose of the random walk, on the map, is to restrict the number of explored partitions and therefore the number of local variational data assimilation sessions. The overall methodology will be illustrated on a shallow-water acoustic tomography (SWAT) application.

Supplementary URL: