Optimum Filtering and Smoothing of Buoy Wave Data

Abstract

The problem of estimating ocean wave heights given accelerometer and pressure measurements from a spar buoy is described. A Kalman optimum filter and a smoother are used to combine these measurements to estimate four state variables: wave height and its time integral, and buoy vertical displacement and velocity. The time history of the ocean waves is presented as a second-order system driven by white noise. The buoy dynamics in the vertical are represented as a second-order system driven by the waves with additional white noise. Numerical computations were carried out for the state vector of order four using five minutes of digitally recorded data from an actual spar buoy. One of the state variables, buoy vertical displacement, is compared directly with the accelerometer output as doubly integrated by another numerical technique and found to be slightly more stable. Inspection of the measurement residuals from the filter and smoother suggest that the model or parameters chosen are not in final form. this paper the state vector approach and optimum esti- ation theory are used to estimate the height of ocean waves. The impetus for this problem came from difficulties encountered in analyzing wave and motion data from Flordia State University's 100-ft buoy TRITON.) This buoy was primarily intended as a platform for meteorological in- struments, but as a side project it was equipped with resistan- ce wire gages. Early attempts at numerical double integration of the vertical accelerometer output diverged rapidly. While studying the buoy motion problem the author in- vestigated a work 2 which applied the Kalman filter to the study of ocean current meter dynamics, but was not able to pursue the problem further at that time. The work described here was begun two years later using data from a smaller spar buoy, Lockheed's Measurement and Comparison System (MCS).3 The particular problem considered was to compute a wave spectrum or time history from digitized records of relative wave height and apparent buoy vertical acceleration. Since the vertical motion of the buoy was a significant fraction of the wave height over much of the frequency range, it was not possible to combine the low frequency part from the ac- celerometer and the high frequency part from the wave gage. Furthermore it would be difficult to combine spectral com- ponents at each frequency because of unknown phase relations. Thus, it was necessary first to work in the time domain and construct a history of the ocean surface, which requires numerical double integration of the vertical ac- celerometer output. At least three factors make this numerical integration difficult: unknown initial conditions, non- verticality of the accelerometer, and the finite quantization step in the digitization. An ordinary numerical quadrature leads to rapid divergence. In the work reported here the problem was solved by the Kalman filter, which has been successfully applied to such areas as inertial navigation and orbit estimation. The results of its application to a single set of data from the MCS3 buoy are given below and possible future extensions of the method are outlined. Because the Kalman filter is a time domain technique, no spectral calculations are presented. The remain- der of this paper contains additional background on measurements from buoys and the Kalman filter.