Covariance regularization through a Gaussian graphical model and its application to ocean data assimilation

In data assimilation, covariance matrices are introduced to prescribe weights of the initial (background) state, model dynamics, and observation and suitable specification of the covariances is known to be essential for obtaining sensible state estimates. Covariance matrices are specified by sample covariances and modeled covariance structure. Covariance modelling consists of regularization of the sample covariance and constraint of dynamical relationship. Regularization is required for converting the singular sample covariance into non-singular one, removing spurious correlation between variables at distant points, and reducing a required number of parameters that specify the covariances. Regularization of the sample covariance has been carried out in physical (grid) space, spectral space, and wavelet space. We propose a covariance regularization method in inverse space with a Gaussian graphical model. Defining a neighbor of a variable, we assume conditional independence between the variable and those outside of the neighbor. Conditional independence is expressed by specifying zero elements in the inverse covariance matrix. We give an illustrative example using a simple matrix, and show application to sample covariance obtained from sea- surface height (SSH) observations.