Sparse Spectrum Model of the Sea Surface

Abstract

Traditionally statistical modeling of the sea surface is based on the Fast Fourier Transform (FFT) technique. In this approach the surface is represented as a sum of sinusoids with equally spaced frequencies and random amplitudes and phases. Statistic of the amplitudes and phases is prescribed to match the required observation-based spectral density. Higher spatial resolution and/or larger coverage domain is achieved by increasing the number of the spectral components in the model. For some remote sensing applications the required number of components can be as high as 1011. The conventional FFT technique is based on the Dense Spectrum (DS) model: namely on conjecture that at any given time moment all spectral components that are allowed by the spectral density are actually present at the surface. However it was realized by M. S. Longuet-Higgins as early as in 1956 [1], that DS is just an assumption. We propose a novel Sparse Spectrum (SS) model of the sea surface where each surface realization contains a finite, possibly random, number of sinusoidal components with random frequency, phase and amplitude. Unlike the FFT-based model, the number of spectral components forming the surface is determined by the sea state, but not the desired spatial resolution and domain size. A single constraint on the probability distributions of the wavenumbers, and the amplitude variances of the components allows the SS surface to conform to any prescribed spectral density. This affords a great flexibility in designing Monte-Carlo simulation algorithms that produce the surface samples, and provides substantial reduction of the computation efforts compare to the conventional FFT-based models. We designed and tested several versions of the Monte-Carlo surface simulation models that generate samples of the surface conforming to the Elfouhaily [2] spectral model. Visual inspection of the model-generated dynamic samples suggests that about 3000 spectral components are sufficient to generate the realistic surface samples capturing the scales from 2mm to 300m. In addition to the computational savings, the SS model is capable of providing a well-defined statistics of the individual waves that are important for the marine engineering applications.