# Deconvolution: Einstein or predictive?

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 9 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

Let us now address yet another deconvolution issue. In any such method discussed thus far, we always have assumed that the output signal is known. In general, deconvolution consists of two steps. The first step is to measure, estimate, or otherwise determine the input wavelet. The second step is to deconvolve the output signal by the input wavelet to yield the unit-impulse response. In predictive (spiking) deconvolution, the input wavelet is estimated from knowledge of the output signal. In signature deconvolution, the input wavelet (the signature) is measured directly.

In the present case, we have directly obtained the input signal as well as the output signal. Thus, we can apply signature deconvolution to the upgoing wave at the receiver location. The required signature is the downgoing wave at the receiver location. This special type of signature deconvolution in which both the output signal and the input signal are determined by use of a dual-sensor receiver is called Einstein deconvolution (Robinson, 1999[1], 2000[2]; Loewenthal and Robinson, 2000a[3], 2000b[4]) in honor of Albert Einstein.

The main difference between Einstein deconvolution and predictive deconvolution is in the first step — namely, in how the input signal is obtained. Predictive deconvolution can be used in cases in which the input signal cannot be measured directly or estimated directly. In such cases, the input signal must be determined indirectly.

Einstein deconvolution removes all effects introduced by layering above the receiver location. Einstein deconvolution eliminates not only the source signature but also the ghosts and reverberations that result from the layers lying above the receiver. As is the case with conventional signature deconvolution, Einstein deconvolution requires knowledge of the input signal as well as of the output signal. The input wavelet is the downgoing wave at the receiver, and the output signal is the upgoing wave at the receiver. In Einstein deconvolution, a dual geophone/hydrophone sensor records both the downgoing input signal and the upgoing output signal. We use the d’Alembert equations, so no traveling-wave assumption is required. The deconvolved signal is an estimate of the reflection response of the deep system (that is, of the earth below the receiver).

Figure 21.  Einstein deconvolution.

The Einstein deconvolution method can be described in two steps (Figure 21). The first step is to use the d’Alembert equations to convert the particle-velocity signal and the pressure signal into the downgoing wave and the upgoing wave at the receiver location. The second step is to deconvolve the upgoing wave by the downgoing wave. Einstein deconvolution removes the unknown but not necessarily minimum-delay source signature as well as the reverberations and ghosts originating from the layers above the receiver. The result of Einstein deconvolution is the unit-impulse reflection response of the deep-earth system.

Deconvolution of the upgoing wave by the downgoing wave at the receiver gives the unit-impulse reflection response of the subsystem located below the receiver. The Einstein deconvolution process strips away the multiples and ghosts created in the upper system. We emphasize that Einstein deconvolution also strips away the unknown source signature wavelet. The resulting Einstein-deconvolved record is the unit-impulse reflection response of the rock layers below the receiver, which is precisely the input required for dynamic deconvolution (Robinson, 1975)[5]. The output of dynamic deconvolution is the reflection coefficient series. Einstein deconvolution followed by dynamic deconvolution yields the series of interface reflection coefficients located below the receiver. Instead of dynamic deconvolution, conventional predictive deconvolution is also applicable here.

## References

1. Robinson, E. A., 1999, Seismic inversion and deconvolution, dual sensor technology: Elsevier.
2. Robinson, E. A., 2000, Wavelet estimation and Einstein deconvolution: The Leading Edge, 19, no. 1, 56-60.
3. Loewenthal, D., and E. A. Robinson, 2000a, On unified dual fields and Einstein deconvolution: Geophysics, 65, 293-303.
4. Loewenthal, D., and E. A. Robinson, 2000b, Relativistic combination of any number of collinear velocities and generalization of Einstein’s formula: Journal of Mathematical Analysis and Applications, 246, 320-324.
5. Robinson, E. A., 1975, Dynamic predictive deconvolution: Geophysical Prospecting, 23, 779-797.