Towed array shape estimation using Kalman filters-theoretical models

The dynamical behavior of a thin flexible array towed through the water is described by the Paidoussis equation. By discretizing this equation in space and time a finite-dimensional state-space representation is obtained where the states are the transverse displacements of the array from linearity in either the horizontal or vertical plane. The form of the transition matrix in the state-space representation describes the propagation of transverse displacements down the array. The outputs of depth sensors and compasses located along the array are shown to be related in a simple, linear manner to the states. From this state-space representation a Kalman filter which recursively estimates the transverse displacements and hence the array shape is derived. It is shown how the properties of the Kalman filter reflect the physics of the propagation of motion down the array. Solutions of the Riccati equation are used to predict the mean square error of the Kalman filter estimates of the transverse displacements