Some quantitative characteristics of error covariance for Kalman filters

Some quantitative characteristics of error covariance are studied for linear Kalman filters. These quantitative characteristics include the peak value and location in the matrix, the decay rate from peak to bottom, and some algebraic constraints of the elements in the covariance matrix. We mathematically prove a matrix upper bound and its quantitative characteristics for the error covariance of Kalman filters. Computational methods are developed to numerically estimate the elements in a matrix upper bound and its decay rate. The quantitative characteristics and the computational methods are illustrated using three examples, two linear systems and one nonlinear system of shallow water equations.