Attacking the Problems of the Synthetic Seismogram

Anstey, N. A.

doi:10.1111/j.1365-2478.1960.tb01500.x

Cited by 12 publications

(2 citation statements)

References 5 publications

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“…This model has a long pedigree in exploration geophysics, and the reader is referred to the works of Anstey (1960), Baranov andKunetz (1958, 1960) For layered-earth models subject to more general wave motion the reader is referred to B"ath (1968), and for spectral analysis to B%h (1974).…”

Section: The Seismic Modelmentioning

confidence: 99%

Dynamic Predictive Deconvolution*

Robinson¹

1975

Geophysical Prospecting

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Dynamic predictive deconvolution makes use of an entire seismic trace including all primary and multiple reflections to yield an approximation to the subsurface structure. We consider plane‐wave motion at normal incidence in an horizontally layered system sandwiched between the air and the basement rock. Energy degradation effects are neglected so that the layered system represents a lossless system in which energy is lost only by net transmission downward into the basement or net reflection upward into the air; there is no internal loss of energy by absorption within the layers. The layered system is frequency selective in that the energy from a surface input is divided between that energy which is accepted over time by net transmission downward into the basement and the remaining energy that is rejected over time by net reflection upward into the air. Thus the energy from a downgoing unit spike at the surface as input is divided between the wave transmitted by the layered system into the basement and the wave reflected by the layered system into the air. This reflected wave is the observed seismic trace resulting from the unit spike input. From surface measurements we can compute both the input energy spectrum, which by assumption is unity, and the reflection energy spectrum, which is the energy spectrum of the trace. But, by the conservation of energy, the input energy spectrum is equal to the sum of the reflection energy spectrum and the transmission energy spectrum. Thus we can compute the transmission energy spectrum as the difference of the input energy spectrum and the reflection energy spectrum. Furthermore, we know that the layered system acts as a pure feedback system in producing the transmitted wave, from which it follows that the transmitted wave is minimum‐delay. Hence from the computed energy spectrum of the transmitted wave we can compute the prediction‐error operator that contracts the transmitted wave to a spike. We also know that the layered system acts as a system with both a feedback component and a feed‐forward component in producing the reflected wave, that is, the observed seismic trace. Moreover, this feedback component is identical to the pure feedback system that produces the transmitted wave. Thus, we can deconvolve the observed seismic trace by the prediction‐error operator computed above; the result of the deconvolution is the wave‐form due to the feedforward component alone. Now the feedforward component represents the wanted dynamic structure of the layered system whereas the feedback component represents the unwanted reverberatory effects of the layered system. Because this deconvolution process yields the wanted dynamic structure and destroys the unwanted reverberatory effects, we call the process dynamic predictive deconvolution. The resulting feedforward waveform in itself represents an approximation to the subsurface structure; a further decomposition yields the reflection coefficients of the interfaces separating the layers. In this work we do not make the assumptio...

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Section: The Seismic Modelmentioning

confidence: 99%

Dynamic Predictive Deconvolution*

Robinson¹

1975

Geophysical Prospecting

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“…For further detail on these and related matters, we refer the reader to the contributions of Anstey (1960), Baranov and Kunetz (1960), Wuenschel (1960), Bois, Grau, Hemon, and LaPorte (1962), Trorey (1962, Darby and Neidell (1966), Frasier (1970)~ and Riley (1974. The purpose of this paper is to revisit the whole area of the communication theory approach to seismic wave propagation through layered media.…”

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The Spectral Function of a Layered System and the Determination of the Waveforms at Depth*

Robinson¹,

Treitel²

1977

Geophysical Prospecting

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Consider the mathematical model of a horizontally layered system subject to an initial downgoing source pulse in the upper layer and to the condition that no upgoing waveforms enter the layered system from below the deepest interface. The downgoing waveform (as measured from its first arrival) in each layer is necessarily minimum‐phase. The net downgoing energy in any layer, defined as the difference of the energy spectrum of the downgoing wave minus the energy spectrum of the upgoing wave, is itself in the form of an energy spectrum, that is, it is non‐negative for all frequencies. The z‐transform of the autocorrelation function corresponding to the net downgoing energy spectrum is called the net downgoing spectral function for the layer in question. The net downgoing spectral functions of any two layers A and B are related as follows: the product of the net downgoing spectral function of layer A times the overall transmission coefficient from A to B equals the product of the net downgoing spectral function of layer B times the overall transmission coefficient from B to A. The net downgoing spectral function for the upper layer is called simply the spectral function of the system. In the case of a marine seismogram, the autocorrelation function corresponding to the spectral function can be used to recursively generate prediction error operators of successively increasing lengths, and at the same time the reflection coefficients at successively increasing depths. This recursive method is mathematically equivalent to that used in solving the normal equations in the case of Toeplitz forms. The upgoing wave‐form in any given layer multiplied by the direct transmission coefficient from that layer to the surface is equal to the convolution of the corresponding prediction error operator with the surface seismogram. The downgoing waveform in this given layer multiplied by the direct transmission coefficient from that layer to the surface is equal to the convolution of the corresponding hindsight error operator (i.e., the time reverse of the prediction error operator) with the surface seismogram.

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The Wave form of a Longitudinal Reflected Wave from a Thin High-Speed Layer

Kun¹

1969

Seismic Wave Propagation in Real Media / Modeli Real’nykh Sred I Seismicheskie Volnovye Polya / Модели Реальных Сред И Сейсмиче

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Attacking the Problems of the Synthetic Seismogram

Cited by 12 publications

References 5 publications

Dynamic Predictive Deconvolution*

Dynamic Predictive Deconvolution*

The Spectral Function of a Layered System and the Determination of the Waveforms at Depth*

The Wave form of a Longitudinal Reflected Wave from a Thin High-Speed Layer

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