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Wavelet estimation and Einstein deconvolution

Columbia University, New York City, New York, U.S.

Corresponding author: ear11@columbia.edu

The first 20% of the full text of this article appears below.

Digital seismic processing transforms raw data into computer-generated images of the subsurface, and each processing method is based on a specific model that explains the propagation of seismic waves. The wavelet is a basic building block of these seismic models, and wavelet estimation plays a fundamental role in geophysics.

A most useful approach is the convolutional model, which appears in one form or another in many processing and interpretation methods. The convolutional model has three components—the wavelet as input, the unit-impulse response function, and the trace as output. The trace is equal to the convolution of the wavelet with the unit-impulse response function.

The trace represents a known quantity. The unit-impulse response represents the desired unknown quantity, and the goal of deconvolution is to obtain an estimate of this quantity. This is accomplished, during the deconvolution process, by removing the wavelet from the trace. This, in turn, requires suitable information about the wavelet. As a result, wavelet estimation becomes an important aspect in the efficacious use of deconvolution.

This, of course, is well known to most geophysicists. But at this point it is necessary to define terms carefully and to review some basic concepts because this article will describe a new method of deconvolution.

Wavelet is a generic term that can stand for many things. Depending upon the application, the wavelet can be made up of one or many components (e.g., the source signature, the receiver response, attenuation, and reverberations and other types of multiple reflections).

There are two types of deconvolution: statistical deconvolution and deterministic deconvolution. Statistical deconvolution takes the input wavelet as an unknown quantity and, as a result, the wavelet must be estimated from the trace itself. This requires a statistical hypothesis about the makeup of the trace. For example, predictive deconvolution (a form of statistical deconvolution) . . . [Full Text of this Article]