Optimal data-based kernel estimation of evolutionary spectra

Complex demodulation of evolutionary spectra is formulated as a two-dimensional kernel smoother in the time-frequency domain. First, a tapered Fourier transform, y_v(f, t), is calculated. Then the log-spectral estimate, is smoothed. As the characteristic widths of the kernel smoother increase, the bias from the temporal and frequency averaging increases while the variance decreases. The demodulation parameters, such as the order, length, and bandwidth of spectral taper and the kernel smoother, are determined by minimizing the expected error. For well-resolved evolutionary, spectra, the optimal taper length is a small fraction of the optimal kernel halfwidth. The optimal frequency bandwidth, w, for the spectral window scales as w²~λ/τ, where τ is the characteristic time and λ_F is the characteristic frequency scalelength. In contrast, the optimal halfwidths for the second stage kernel smoother scales as h~1/(τλ_F )¹(p+2)/ where p is the order of the kernel smoother. The ratio of the optimal-frequency halfwidth to the optimal-time halfwidth is determined