Wiener estimation of the Green’s function

Abstract

I consider the problem of finding the impulse response, or Green’s function, from a measured response including noise, given an estimate of the source time function. This process is usually known as signature deconvolution. Classical signature deconvolution provides no measure of the quality of the result and does not separate signal from noise. Recovery of the earth impulse response is here formulated as the calculation of a Wiener filter in which the estimated source signature is the input and the measured response is the desired output. Convolution of this filter with the estimated source signature is the part of the measured response that is correlated with the estimated signature. Subtraction of the correlated part from the measured response yields the estimated noise, or the uncorrelated part. The fraction of energy not contained in this uncorrelated component is defined as the quality of the filter. If the estimated source signature contains errors, the estimated earth impulse response is incomplete, and the estimated noise contains signal, recognizable as trace-to-trace correlation. The method can be applied to many types of geophysical data, including earthquake seismic data, exploration seismic data, and controlled source electromagnetic data; it is illustrated here with examples of marine seismic and marine transient electromagnetic data.