Spectral Estimation with Lagged Products

The estimation of S(f) and B (f₁, f₂) as the Fourier transforms of $C(\tau)$ and $T (\tau_1, \tau_2)$ have many advantages. There are numerous existing optimality studies of the estimation of S(f) from $C(\tau)$ made by statisticians. Indeed the classical book by Blackman and Tukey [2] is entirely based on this procedure. Since then, the FFT has come to dominate routine spectral estimation. However, if one is studying nonlinear features of waves, there is a real advantage to avoiding the FFT. The covariance and trivariance estimations allow the use of irregularly spaced data. The covariance procedures, and a closely related semi-variogram function, form the basis for environmental and mining geostatistics [10]. Hence, ocean engineers would not have to reinvent numerous methods, but could bring them over from other disciplines. The covariance/trivariance procedures are easy to extend to (x,y,t)-space where full wave number and time-frequency analyses can be introduced to avoid assuming the validity of linear wave theory.