Prestack inversion with plane‐layer point source modeling

Geophysics ◽  
1985 ◽  
Vol 50 (1) ◽  
pp. 77-89 ◽  
Author(s):  
Alastair D. McAulay
Keyword(s):  
Velocity Profile ◽  
Point Source ◽  
Spherical Wave ◽  
Normal Incidence ◽  
Plane Layer ◽  
Source Modeling ◽  
Low Frequencies ◽  
Prestack Inversion ◽  
Band Limited

Prestack inversion with point‐source plane‐layer modeling has many advantages over poststack or normal incidence inversion. For example, it permits the determination of absolute compressional and shear velocities, density variations, and the accurate accounting of interbed and surface multiples. I neglect shear effects in this paper by assuming that they are adequately suppressed by velocity filtering. In the forward modeling step, a spherical wave expansion into plane waves is used to account for the point source. The plane‐wave reflection response for a set of plane layers is extended to the nonnormal incidence case. I use a Hankel transform to account for cylindrical symmetry. Generalized linear inversion is used because the fast recursive approaches available for normal incidence inversion are no longer applicable. I provide the derivation for the required derivative matrix, and I take into account the band‐limited nature of the data in frequency, time, and space. I demonstrate that moveout of events on realistic simulated prestack data enables the determination of absolute compressional velocity in the velocity‐depth profile, even though the data are band‐limited in frequency. I assume that preprocessing has adequately removed the shear and surface effects and that density is constant. Low frequencies in the velocity profile may be obtained more accurately than with velocity analysis used for stacking, because interbed multiples and other modeling phenomena are accounted for in the computation. Autoregressive modeling procedures that predict into the low frequencies of the velocity profile are also less accurate and cannot generate absolute velocity. I suggest future research leading to cost‐effective inversion of real data.

Scattering attenuation in sediments modeled by ARMA processes—Validation of simple Q models

Geophysics ◽  
1994 ◽  
Vol 59 (12) ◽  
pp. 1813-1826 ◽  
Author(s):  
Claudia Kerner ◽  
Peter E. Harris
Keyword(s):  
Point Source ◽  
Moving Average ◽  
Normal Incidence ◽  
Distribution Functions ◽  
Depth Interval ◽  
Elastic Plane ◽  
Depth Range ◽  
Source Modeling ◽  
Scattering Attenuation ◽  
Sedimentary Sequences

We investigate the data requirements for a reliable analysis of frequency‐dependent Q caused by scattering in a finely layered geological structure. Numerical wave propagation experiments in stochastic models were performed. We set up autoregressive‐moving average [ARMA(1,1)] models for the reflection coefficients with non‐Gaussian distribution functions and used published parameter sets estimated for sedimentary sequences from real log data. For ARMA models, analytical expressions for the scattering attenuation α and the quality factor Q can be derived from the O’Doherty‐Anstey formula. The aim of this study is to investigate whether scattering attenuation as derived from the O’Doherty‐Anstey formula is measurable with sufficient accuracy with a traditional vertical seismic profile (VSP) configuration in realistic sedimentary sequences, and if so, whether the data can be inverted to yield the statistics of the sediment sequence. The main result is that reliable estimation of scattering attenuation requires VSP data over a considerable depth interval, depending on the magnitude of the attenuation with errors in the estimates increasing inversely as the depth range increases. Extensions of the O’Doherty‐Anstey theory to non‐normal incidence have been given in the literature. We examine the angle dependence of the results using both elastic plane‐wave modeling and acoustic point‐source modeling. For the weak medium variations considered, elastic effects (e.g., mode conversions) and point‐source effects are negligible at angles up to about 25 degrees.

Plane‐layer prestack inversion in the presence of surface reverberation

Geophysics ◽  
1986 ◽  
Vol 51 (9) ◽  
pp. 1789-1800 ◽  
Author(s):  
Alastair D. McAulay
Keyword(s):  
Velocity Profile ◽  
Reference Data ◽  
Jacobian Matrix ◽  
High Surface ◽  
Synthetic Data ◽  
Surface Effects ◽  
Plane Layer ◽  
Linear Inversion ◽  
Band Limited ◽  
Shallow Strata

Experiments with synthetic data have indicated that generalized linear inversion may be used to estimate compressional velocities as a function of depth with high resolution directly from band‐limited, unstacked data. The ocean surface was not included in these experiments. In the presence of strong surface multiples, inversion is expected to take longer and be less accurate, because events from multiple surface reflections overlie primary events and normally have differing moveout. Existing velocity‐analysis techniques rely on the ability of an observer to make the difficult distinction between multiples and primaries. Equations are provided for adding the surface to the inversion procedure. This involves adding the surface effects to the Jacobian matrix as well as to the forward modeling procedure. To speed computation, the addition of the surface effects to the Jacobian matrix is delayed until after the matrix has been multiplied by a vector in the linear‐equation solution. Absorption is added to the inversion to represent the real world more closely and to improve computation speed by reducing sampling requirements. Realistic synthetic band‐limited data with high surface reverberation content were generated from a well‐log velocity profile. The inversion recovered the velocity profile to within 3 percent when a velocity increasing linearly with depth was used as a starting profile. The error in the model‐generated seismogram converges from 100 percent to within 2 percent of the reference data. The positions of interfaces are located more accurately at greater depths than at shallower depths because more sensors are observing deep strata than shallow strata. Convergence is to within 0.1 percent of that of the reference data at the maximum depth. The computation required 25 iterations and a total time of 66 hours on a DEC VAX 11/780. Reducing this time should be possible. In a preliminary study of the effects of noise, additive Gaussian noise was seen to limit the accuracy of the velocity estimate monotonically as the variance of the added noise was increased.

Recovery of the acoustic impedance from reflection seismograms

Geophysics ◽  
1983 ◽  
Vol 48 (10) ◽  
pp. 1318-1337 ◽  
Author(s):  
D. W. Oldenburg ◽  
T. Scheuer ◽  
S. Levy
Keyword(s):  
Acoustic Impedance ◽  
Recursion Formula ◽  
Low Frequency ◽  
Normal Incidence ◽  
Computationally Efficient ◽  
Low Frequencies ◽  
Frequency Information ◽  
Reflectivity Function ◽  
Band Limited ◽  
Common Signal

This paper examines the problem of recovering the acoustic impedance from a band‐limited normal incidence reflection seismogram. The convolutional model for the seismogram is adopted at the outset, and it is therefore required that initial processing has removed multiples and recovered true amplitudes as well as possible. In the first portion of the paper we investigate the effect of substituting the deconvolved seismic trace (that is, the band‐limited version of the reflectivity function) into the standard recursion formula for the acoustic impedance. The formalism of linear inverse theory is used to show that the logarithm of the normalized acoustic impedance estimated from the deconvolved seismogram is approximately an average of the true logarithm of the impedance. Moreover, the averaging function is identical to that used in deconvolving the initial seismogram. The advantage of these averages is that they are unique; their disadvantage is that low‐frequency information, which is crucial to making a geologic interpretation, is missing. We next present two methods by which the missing low‐frequency information can be recovered. The first method is a linear programming (LP) construction algorithm which attempts to find a reflectivity function made of isolated delta functions. This method is computationally efficient and robust in the presence of noise. Importantly, it also lends itself to the incorporation of impedance constraints if such geologic information is available. A second construction method makes use of the fact that the Fourier transform of a reflectivity function for a layered earth can be modeled as an autoregressive (AR) process. The missing high and low frequencies can thus be predicted from the band‐limited reflectivity function by standard techniques. Stability in the presence of additive noise on the seismogram is achieved by predicting frequencies outside the known frequency band with operators of different orders and extracting a common signal from the results. Our construction algorithms are shown to operate successfully on a variety of synthetic examples. Two sections of field data are inverted, and in both the results from the LP and AR methods are similar and compare favorably to acoustic impedance features observed at nearby wells.

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