The convolutional model within each window on a reflection seismogram has the form of a stable feedback system, which is necessarily a minimum-delay system; hence, its inverse is a causal feedforward system. This inverse is the operator with which we convolve the reflection seismogram to obtain an estimate of the reflectivity function. That is, this operator converts the observed reflection seismogram into the desired reflectivity and thus is the required deconvolution operator. The deconvolution operator removes the multiple wavelets, leaving behind the reflectivity function. Thus we have the direct (or physical) system (1) trace = reflectivity * wavelet and the inverse (or data-processing) system (2) reflectivity = trace * deconvolution operator.
The known (or observational) information is the trace, and the desired information is the reflectivity function. We must find some way to estimate the deconvolution operator from the known data (i.e., from the seismic trace). To find such a method, however, we first must introduce the random-reflection-coefficient model.
Which are the two important assumptions? We have set up the convolutional model so that it can form the basis for a method to determine the required deconvolution operator. This model requires (1) that the earth act as a stable feedback system to produce the minimum-delay interface wavelets that appear on the reflection seismogram and (2) that the reflectivity function within each selected window be a white-noise process.
Thus, this seismic model differs from an arbitrary convolutional model in that our seismic model is a minimum-delay system with a white-noise input. Because of these special features, the seismic model can be used as a basis for determining the deconvolution operator that is valid for each window. In brief, the seismic model within each window is a time-invariant minimum-delay convolutional model with random reflection coefficients.
The seismic convolutional model has two characteristic features within each time window of interest: (1) the statistical feature that the primary events are caused by a reflectivity series (i.e. a sequence of reflection coefficients) given by a random white-noise series and (2) the deterministic feature that the wavelets attached to each primary event have the same minimum-delay wavelet shape.
What is the computational procedure for deconvolution? The predictive deconvolution process separates on the basis of two fundamental criteria: “minimum-delay” and “white.” Thus, the method of predictive deconvolution removes an estimate of the predictable minimum-delay component of the trace to yield an estimate of the nonpredictable white component, which we identify with the reflectivity. The observed data are the seismic trace recorded at the surface. The computational procedure used to determine the deconvolution operator is given by steps 1 and 2 below, and the computational procedure to carry out the deconvolution is given by step 3.
1) Compute the autocorrelation function of the portion of the seismic trace lying within the specified time window.
2) Compute the coefficients of the prediction-error operator that corresponds to this autocorrelation. The calculation involves solving a set of simultaneous linear equations called the normal equations. Because of the symmetries involved in these equations, a highly efficient computational procedure, called the Toeplitz (or Levinson) recursion, can be used to solve them for the prediction-error operator. This deconvolution (or prediction-error) operator is precisely the operator required to carry out deconvolution of the trace within a specified window. (These matters were treated in greater detail earlier in this chapter.) In summary, the purpose of spike deconvolution is to remove the wavelet m from the trace while leaving the reflectivity series intact. The spike-deconvolution filter is the least-squares inverse of the minimum-delay wavelet m.
3) We implement deconvolution by convolving the deconvolution operator with the seismic trace. Note that the deconvolution of the trace is accomplished by convolving the trace with the inverse operator — that is, with the deconvolution (or prediction-error) operator. The result of the deconvolution is called the prediction-error series. If our model is in fact appropriate, the prediction-error series will approximate the reflectivity function within the given time gate.
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Also in this chapter
- Model used for deconvolution
- Least-squares prediction and smoothing
- The prediction-error filter
- Spiking deconvolution
- Gap deconvolution
- Tail shaping and head shaping
- Seismic deconvolution
- Piecemeal convolutional model
- Time-varying convolutional model
- Implementing deconvolution
- Canonical representation
- Appendix J: Exercises