Horizontal expansion function in OAT

The perturbation of sound-speed δc is modeled by using some orthogonal functions in the analysis of ocean acoustic tomography. However, no suitable horizontal functions have been found previously. The reason is that there is no way of calculating the covariance matrix for an arbitrary function. The method for calculating the covariance is presented. Its outline is as follows: 1) Calculate the covariance for the model with the expanded rectangular function horizontally, under the condition that the number of the modes is sufficiently larger than those of arbitrary functions; 2) Calculate the value of arbitrary function at the center of each box and form column vectors; 3) Compose a matrix whose elements are the column vectors; 4) Transform the covariance matrix of rectangular function linearly with a Moore-Penrose matrix of the matrix in 3). Using this method, the authors compare the analysis results with three types of horizontal functions (rectangular, trigonometric, and Gaussian) by numerical simulations. It is concluded that the method presented is rigid