Estimation of Large Spectral Function and Its Application

Abstract

It is of fundamental interest to routinely monitor waves and currents in the nearshore seas both scientifically and to the general public, because they play an important role in coastline erosion and they have a significant effect in the nearshore recreational activities. In this work, we show the way to estimate both wave height and wave direction with the data observed from a bottom-mounted, upward-looking Acoustic Doppler Current Profiler.

One of the most challenging works is to estimate the wave-number spectra using all gathered observations of receiving antennas. The frame of observed data is 100-dimensional time series with T = 2399. Due to the fact that there is only one realization of this multivariate time series, the conventional methods are either applicable for univariate time series or appropriate in low dimensional setting. In this work, we propose a new regularization estimator for wave-number spectral density with three merits: positive definite, smoothness and sparsity. This method can also be used to regularize any complex/real tensor in order to gain a resulting estimator with the above three merits. We describe and prove the convergence of our proposed algorithm, and compare our proposed estimator with the sample wave-number spectra and the other two regularization estimators: banding and extended tapering. The numerical results show that the estimation performance of our proposed approach is overwhelming better than other estimators. The proposed estimator and the extended tapering estimator are comparable in smoothness and positive definiteness. Unlike other estimators, our approach can produce a sparse estimator which would massively reduce the computation complexity for further study.