Wavelets, well logs and Wiener filters

The deconvolution of source wavelets from seismic traces can provide useful estimates of the earth 's impulse response and thereby aid in geological interpretation. This signal analysis tool has been especially useful in the resolution of thin rock layers and has received widespread application since its development in the 1950s. The deconvolution problem often involves the estimation of a wavelet before removing it by digital filtering. Even in the noiseless case, this task is difficult since the seismic trace can be viewed as the convolution of an unknown wavelet coefficient sequence with an unknown earth impulse response. In cases where multiples have been eliminated, this impulse response becomes transformed into a sequence of reftection coefficients. As pointed out by Ziolkowski (1982), this deconvolution will yield non-unique solutions since it essentially involves solving one equation for two unknown sequences: the wavelet coefficients and the reflection coefficients. However, this inverse problem is not hopeless since the two sequences generally have different statistical properties. Moreover, well log data may be used to sort out the non-uniqueness. The Earth's impulse response derived from well log information and the statistical estimate of the source wavelet can be modified together by least squares inversion in order to model the seismic trace and provide reliable wavelet estimates. Recent wavelet estimation research has shown a similarity between wavelet estimates obtained by least squares inversion and those found by deconvolution of the earth's impulse response from the seismic trace. Upon examination of these procedures, it turns out that least squares inversion with a constrained earth's impulse response is equivalent to deriving the Wiener filter which shapes the earth's impulse response to the recorded seismic trace. When reflection coefficients derived from a well log are simply used as an initial guess for model, parameters, the unconstrained inversion procedure provides wavelet estimates which are similar to wavelet estimates obtained by deconvolution of the earth's impulse response. The similarities are demonstrated for synthetic and real data cases, and explained in heuristic terms.