# Surface-consistent deconvolution

Other languages:
English • ‎español
Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 11 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

Automatic statics determination can be based on a surface-consistent model (Taner and Koehler, 1981[1]). In such a model, a delay ${\displaystyle R_{i}}$ is associated with the geophone group at location i, and a delay ${\displaystyle S_{j}}$ is associated with the source at location j. Such delays are caused by varying elevations, by varying overburden depths and/or velocities, and by other factors.

The surface-consistent model is not restricted merely to determination of time shifts for statics correction. For the case of surface-consistent amplitude adjustment, for example, we assume an attenuation associated with each geophone and an attenuation associated with each source. The subsequent analysis follows basically the same procedure as does that for surface-consistent statics. Likewise, wavelet extraction, deconvolution, and other operations can be based on surface-consistent models (Levin, 1989[2]; Cambois and Stoffa, 1992[3]).

The surface-consistent conditions do not require that the static shifts, attenuation, and so on are the same when a geophone is located at point P as when the source is located at P. Although the amount of time shift to be associated with any trace might not be known, crosscorrelation methods give a way of finding the time shift of one trace relative to that of another. Such a shift produces the optimum alignment of the two traces.

The surface-consistent deconvolution model is based on the concept that a seismic wavelet can be broken down into its four components: its source, receiver, offset, and CMP components. For surface-consistent deconvolution, the following convolutional model is used: The recorded seismic signal can be considered to be the convolution of the wavelet with the response of the earth. The earth’s response represents the reflectivity function as well as some undesirable effects, such as reverberation, absorption, and ghosting. The wavelet can be made up of any or all of its four components: source wavelet, the wavelet representing the response of the instruments and geophones, a wavelet representing the offset, and a wavelet representing the CMP position. The objective of surface-consistent deconvolution is to estimate these four component wavelets and then design and apply inverse filters to remove them. For land data, all four components normally are used in the decomposition, but usually only the shot and receiver portions are applied. The receivers are in motion in marine acquisition, so the common receiver grouping is not so well suited for marine data. In areas with highly variable bottom conditions, however, the common receiver component is useful.

Originally, seismic-reflection data were deconvolved using two-channel digital operators (Wadsworth et al., 1953)[4]. Robinson (1967[5]) gave the theory and usage of multichannel operators. Today, seismic-reflection data usually are deconvolved using single-channel operators. However, multichannel operators are starting to come into general use.

Averaging the power spectra of several neighboring traces can reduce the sensitivity of single-channel spiking deconvolution to noise. A usual technique computes the arithmetic mean of the power spectra for traces belonging to a common-shot gather. The effect of the source is the common element in all the traces of the common-shot gather. Thus, the computed average power spectrum represents the source’s effect. As a result, a single deconvolution operator computed from the average power spectrum will remove the source’s effect.

The same technique can be applied to a common receiver gather to remove the receiver’s effect. In addition, the same technique can be used on a CMP gather to remove the effect of the CMP. Finally, the same technique can be used on a common-offset gather to remove the effect of the common offset. This total process is called surface-consistent deconvolution. That is, surface-consistent deconvolution is a technique that finds the average power spectra for common-shot gathers, common-receiver gathers, common-midpoint gathers, and common-offset gathers and then uses those spectra to compute deconvolution operators that remove shot, receiver, midpoint, and offset effects.

Under the surface-consistent model, the seismic wavelet is the convolution of a source wavelet, a receiver wavelet, a midpoint wavelet, and an offset wavelet. The power spectra of these wavelets are estimated by averaging the individual power spectra over the respective gathers.

There are various ways to compute the average power spectrum. Suppose that the power spectra computed from one gather are

 {\displaystyle {\begin{aligned}{\Phi }_{\rm {1}}{,\ }{\Phi }_{2}{,\ldots ,\ }{\Phi }_{n}.\end{aligned}}} (19)

One type of average is the arithmetic mean:

 {\displaystyle {\begin{aligned}\Phi =\left({\Phi }_{1}+{\Phi }_{2}+\dots +{\Phi }_{n}\right)/n.\end{aligned}}} (20)

Another type of average is the geometric mean:

 {\displaystyle {\begin{aligned}\Phi ={\sqrt {{\Phi }_{1}{\Phi }_{2}{\ldots \Phi }_{n}}}.\end{aligned}}} (21)

In the log/Fourier domain, the geometric mean becomes

 {\displaystyle {\begin{aligned}{\rm {log\ }}\Phi =\left({\rm {log\ }}{\Phi }_{1}+{\rm {\ log\ }}{\Phi }_{2}+\ldots +{\rm {log\ }}{\Phi }_{n}\right){/2}.\end{aligned}}} (22)

In conclusion, surface-consistent deconvolution not only takes into account source position and receiver position, but it also can take into account offset and CMP. In fact, any combination of source, receiver, offset, and CMP components can be used for power-spectrum calculation and deconvolution.

## References

1. Taner, M. T., and F. Koehler, 1981, Surface consistent corrections: Geophysics, 46, 17-22.
2. Levin, S. A., 1989, Surface consistent deconvolution: Geophysics, 54, 1123-1131.
3. Cambois, G., and P. Stoffa, 1992, Surface consistent deconvolution in the log/Fourier domain: Geophysics, 57, 832-840.
4. Wadsworth, G. P., E. A. Robinson, J. G. Bryan, and P. M. Hurley, 1953, Detection of reflections on seismic records by linear operators: Geophysics, 18, 539-586.
5. Robinson, E. A., 1967, Multichannel time series analysis with digital computer programs: Holden Day Press.