Online wave direction and wave number estimation from surface vessel motions using distributed inertial measurement arrays and phase-time-path-differences

A common approach for finding the direction of ocean waves is to use the phase-time-path-differences (PTPDs) between a static array of various types of sensors mounted on either the sea surface or seabed. However, some practical drawbacks of such arrays are that they tend to be expensive, difficult to install, and fixed in location. We show that the PTPD approach can be generalized to a portable shipboard array of spatially distributed sensors rendering it more practical. In this respect, we derive a nonlinear PTPD model for shipboard sensor arrays and prove that the wave direction and wave number can be resolved from a minimum of three noncollinear sensors using observability analysis. Moreover, based on our PTPD model, we propose an unscented Kalman filter algorithm for online estimation of the wave direction and wave number, which offers a convenient framework for adding multiple measurements and incorporating uncertainties. Our experimental results from model basin testing with a model ship equipped with several inertial measurement units (IMUs) confirm that the wave direction and wave number can be estimated from the wave-induced motions of a surface vessel with a minimum of three noncollinear IMUs. In this study, we consider parameter estimation for regular waves and assume a dynamically positioned surface vessel with small roll and pitch angles.