Fundamentals of seismic exploration.

*(English)*Zbl 0635.73030
Maximum-entropy and Bayesian methods in inverse problems, Pap. 2 Workshops, Laramie/Wyo. 1981/1982, Fundam. Theor. Phys. 14, 171-210 (1985).

[For the entire collection see Zbl 0566.00019.]

Today, sophisticated inversion methods are being developed for seismic data. In this paper, we describe the present state of the art in which spectral methods in the form of lattice methods in general, and maximum entropy spectral analysis in particular, play the key role. An understanding of these spectral methods will provide a firm basis for appreciating the exciting new inversion methods that the future holds.

This paper treats the one-dimensional (1-D) case of a horizontally layered earth with seismic raypaths only in the vertical direction. This model exhibits a lattice structure that corresponds to the lattice methods of spectral estimation. It is shown that the lattice structure is mathematically equivalent to the structure of the Lorentz transformation of the special theory of relativity.

A horizontally stratified half-space bounded by a perfect reflector at the top gives rise to a seismogram that, when completed by the direct downgoing pulse at zero time and by symmetry about the origin for negative time, produces an autocorrelation function. If this autocorrelation is convolved with the corresponding prediction error operators of increasing length, we obtain a “gapped function”, which deviates more and more from the perfect symmetry exhibited by the autocorrelation. This gapped function consists of the downgoing and upgoing waveforms at the top of each layer. The gap separates the two waveforms, and the gap width increases as deeper and deeper layers are reached. In particular, the width of the gap is a measure of the entropy of the seismogram at a given depth level - the deeper we go into the subsurface, the higher the entropy of the corresponding gapped function. We explore the nature of the gapped function as it relates to the Toeplitz recursion generating the prediction error operators, and we derive the synthetic seismogram in terms of wave motion measured in units proportional to the square root of energy. We obtain an explicit relationship between the partial autocorrelation function on the one hand, and the reflection coefficient sequence on the other.

The maximum entropy spectral estimate is directly related to the reflection coefficient sequence characterizing a given subsurface model, and these considerations, in turn, impinge on the philosophy of deconvolution operator design. Finally, we investigate both the physical and the mathematical foundations of seismic deconvolution, and we attempt to establish the implications of the success of this approach on spectral analysis.

Today, sophisticated inversion methods are being developed for seismic data. In this paper, we describe the present state of the art in which spectral methods in the form of lattice methods in general, and maximum entropy spectral analysis in particular, play the key role. An understanding of these spectral methods will provide a firm basis for appreciating the exciting new inversion methods that the future holds.

This paper treats the one-dimensional (1-D) case of a horizontally layered earth with seismic raypaths only in the vertical direction. This model exhibits a lattice structure that corresponds to the lattice methods of spectral estimation. It is shown that the lattice structure is mathematically equivalent to the structure of the Lorentz transformation of the special theory of relativity.

A horizontally stratified half-space bounded by a perfect reflector at the top gives rise to a seismogram that, when completed by the direct downgoing pulse at zero time and by symmetry about the origin for negative time, produces an autocorrelation function. If this autocorrelation is convolved with the corresponding prediction error operators of increasing length, we obtain a “gapped function”, which deviates more and more from the perfect symmetry exhibited by the autocorrelation. This gapped function consists of the downgoing and upgoing waveforms at the top of each layer. The gap separates the two waveforms, and the gap width increases as deeper and deeper layers are reached. In particular, the width of the gap is a measure of the entropy of the seismogram at a given depth level - the deeper we go into the subsurface, the higher the entropy of the corresponding gapped function. We explore the nature of the gapped function as it relates to the Toeplitz recursion generating the prediction error operators, and we derive the synthetic seismogram in terms of wave motion measured in units proportional to the square root of energy. We obtain an explicit relationship between the partial autocorrelation function on the one hand, and the reflection coefficient sequence on the other.

The maximum entropy spectral estimate is directly related to the reflection coefficient sequence characterizing a given subsurface model, and these considerations, in turn, impinge on the philosophy of deconvolution operator design. Finally, we investigate both the physical and the mathematical foundations of seismic deconvolution, and we attempt to establish the implications of the success of this approach on spectral analysis.

##### MSC:

74J25 | Inverse problems for waves in solid mechanics |

74L05 | Geophysical solid mechanics |

86A15 | Seismology (including tsunami modeling), earthquakes |

60G35 | Signal detection and filtering (aspects of stochastic processes) |

62M15 | Inference from stochastic processes and spectral analysis |