Seismic wavelet estimation: a frequency domain solution to a geophysical noisy input-output problem

In seismic reflection prospecting for oil and gas a key step is the ability to estimate the seismic wavelet (impulse response) traveling through the earth. Such estimation enables filters to be designed to deblur the recorded seismic time series and allows the integration of “downhole” and surface seismic data for seismic interpretation purposes. An appropriate model for the seismic time series is a noisy-input/noisy-output linear model. We tackle the estimation of the impulse response in the frequency domain by estimating its frequency response function. We use a novel approach where multiple coherence analysis is applied to the replicated observed output series to estimate the output signal-to-noise ratio (SNR) at each frequency. This, combined with an estimate of the ordinary coherence between observed input and observed output, and with the spectrum of the observed input and cross-spectrum of the observed input and output, enables estimation of the frequency response function. The methodology is seen to work well on real and synthetic data.