On the imaging of reflectors in the earth

Geophysics ◽  
1987 ◽  
Vol 52 (7) ◽  
pp. 931-942 ◽  
Author(s):  
Norman Bleistein
Keyword(s):  
Reflection Coefficient ◽  
Sound Speed ◽  
Geometrical Optics ◽  
Opening Angle ◽  
Singular Function ◽  
Dirac Delta ◽  
Kirchhoff Migration ◽  
Inversion Operator ◽  
Band Limited ◽  
Reflecting Surface

In this paper, I present a modification of the Beylkin inversion operator. This modification accounts for the band‐limited nature of the data and makes the role of discontinuities in the sound speed more precise. The inversion presented here partially dispenses with the small‐parameter constraint of the Born approximation. This is shown by applying the proposed inversion operator to upward scattered data represented by the Kirchhoff approximation, using the angularly dependent geometrical‐optics reflection coefficient. A fully nonlinear estimate of the jump in sound speed may be extracted from the output of this algorithm interpreted in the context of these Kirchhoff‐approximate data for the forward problem. The inversion of these data involves integration over the source‐receiver surface, the reflecting surface, and frequency. The spatial integrals are computed by the method of stationary phase. The output is asymptotically a scaled singular function of the reflecting surface. The singular function of a surface is a Dirac delta function whose support is on the surface. Thus, knowledge of the singular functions is equivalent to mathematical imaging of the reflector. The scale factor multiplying the singular function is proportional to the geometrical‐optics reflection coefficient. In addition to its dependence on the variations in sound speed, this reflection coefficient depends on an opening angle between rays from a source and receiver pair to the reflector. I show how to determine this unknown angle. With the angle determined, the reflection coefficient contains only the sound speed below the reflector as an unknown, and it can be determined. A recursive application of the inversion formalism is possible. That is, starting from the upper surface, each time a major reflector is imaged, the background sound speed is updated to account for the new information and data are processed deeper into the section until a new major reflector is imaged. Hence, the present inversion formalism lends itself to this type of recursive implementation. The inversion proposed here takes the form of a Kirchhoff migration of filtered data traces, with the space‐domain amplitude and frequency‐domain filter deduced from the inversion theory. Thus, one could view this type of inversion and parameter estimation as a Kirchhoff migration with careful attention to amplitude.

Prestack Kirchhoff inversion of common‐offset data

Geophysics ◽  
1987 ◽  
Vol 52 (6) ◽  
pp. 745-754 ◽  
Author(s):  
Michael F. Sullivan ◽  
Jack K. Cohen
Keyword(s):  
Reflection Coefficient ◽  
Seismic Data ◽  
High Frequency ◽  
Constant Velocity ◽  
Three Dimensional ◽  
Singular Function ◽  
Amplitude Function ◽  
Stratigraphic Analysis ◽  
Inversion Operator ◽  
Band Limited

In trying to resolve complex geologic structures, the pitfalls in employing the CDP method become evident. Additionally, stacking multioffset traces corrupts the amplitudes necessary for stratigraphic analysis. In order to preserve whatever structural and amplitude information is in the data, prestack processing should be performed. Given common‐offset data and the velocity above a reflector, prestack acoustic Kirchhoff inversion resolves the location of the interface. When amplitude information has been preserved in the data, the method additionally calculates the reflection coefficient at each interface point. For band‐limited seismic data, the inversion operator produces a sinc‐like picture of the reflector, with the peak amplitude of this band‐limited singular function equal to the angularly dependent reflection coefficient. The inversion development is based upon high‐frequency Kirchhoff data which are inserted into a general 3-D inversion operator. Asymptotically evaluating the four resulting integrals by the method of four‐dimensional stationary phase permits an inversion amplitude function to be chosen so that the inversion operator produces a singular function of support on the reflector, weighted by the reflection coefficient. Specializing the three‐dimensional inversion operator to two and one‐half dimensions allows for processing of single lines of common‐offset data. Synthetic examples illustrate the accuracy of the method for constant‐velocity Kirchhoff data, as well as the problems in applying constant‐velocity data to multivelocity models.

The link of Kirchhoff migration and demigration to Kirchhoff and Born modeling

Geophysics ◽  
1999 ◽  
Vol 64 (6) ◽  
pp. 1793-1805 ◽  
Author(s):  
Herman H. Jaramillo ◽  
Norman Bleistein
Keyword(s):  
Reflection Coefficient ◽  
Seismic Data ◽  
Kirchhoff Approximation ◽  
Kirchhoff Migration ◽  
Inversion Techniques

The Kirchhoff approximation provides a representation of seismic data as a summation of imaged data along isochron surfaces (demigration). The asymptotic inversion of this representation provides a migration as a summation of seismic data along diffraction surfaces. We replace Born inversion techniques with Kirchhoff inversion techniques and further show the link between the Kirchhoff and Born representations after the Born linearized reflection coefficient is replaced by the Kirchhoff reflection coefficient.

Sediment sound speed dispersion inferences from broadband reflection coefficient measurements

2017 ◽  
Vol 141 (5) ◽  
pp. 3950-3950
Author(s):  
Charles W. Holland ◽  
Samuel Pinson ◽  
Derek R. Olson
Keyword(s):  
Reflection Coefficient ◽  
Sound Speed

Computational and asymptotic aspects of velocity inversion

Geophysics ◽  
1985 ◽  
Vol 50 (8) ◽  
pp. 1253-1265 ◽  
Author(s):  
Norman Bleistein ◽  
Jack K. Cohen ◽  
Frank G. Hagin
Keyword(s):  
Asymptotic Analysis ◽  
Synthetic Data ◽  
Small Variation ◽  
Computer Time ◽  
Velocity Variation ◽  
Inversion Method ◽  
Singular Function ◽  
Velocity Inversion ◽  
Dirac Delta ◽  
Peak Value

We discuss computational and asymptotic aspects of the Born inversion method and show how asymptotic analysis is exploited to reduce the number of integrations in an f-k like solution formula for the velocity variation. The output of this alternative algorithm produces the reflectivity function of the surface. This is an array of singular functions—Dirac delta functions which peak on the reflecting surfaces—each scaled by the normal reflection strength at the surface. Thus, imaging of a reflector is achieved by construction of its singular function and estimation of the reflection strength is deduced from the peak value of that function. By asymptotic analysis of the application of the algorithm to the Kirchhoff representation of the backscattered field, we show that the peak value of the output estimates the reflection strength even when the condition of small variation in velocity (an assumption of the original derivation) is violated. Furthermore, this analysis demonstrates that the method provides a migration algorithm when the amplitude has not been preserved in the data. The design of the computer algorithm is discussed, including such aspects as constraints due to causality and spatial aliasing. We also provide O‐estimates of computer time. This algorithm has been successfully implemented on both synthetic data and common‐midpoint stacked field data.

A note on the geometrical optics of diffraction by an interface

1964 ◽  
Vol 60 (4) ◽  
pp. 1013-1022 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Wave Equation ◽  
Geometrical Optics ◽  
Cauchy Type ◽  
Order Partial Differential Equation ◽  
Normal Distance ◽  
The Cauchy Problem ◽  
Type Boundary ◽  
Image Position ◽  
Reflecting Surface ◽  
Homogeneous Media

1. It is well known that solutions of the Cauchy problem for the wave equation represent disturbances obeying the laws of geometrical optics. Specifically a solution ψ of the wave equationfor which ψ = δψ/δt = 0 initially outside a surface C0, vanishes at time t in the exterior of a surface Ct parallel to and at a normal distance ct from C0 (see e.g. (l), page 643). Analogous results hold for the solutions of any linear hyperbolic second-order partial differential equation with boundary-value conditions of the Cauchy type. Boundary conditions of the type representing reflexion have been treated by Friedlander(2). He showed that as well as the incident and reflected wavefronts, there sometimes exists a ‘shadow’ where diffraction occurs, and that the diffracted wave fronts are normal to the reflecting surface, the corresponding rays travelling along the surface and leaving it tangentially. The purpose of this paper is to extend these results to refraction, where instead of a purely reflecting surface we have an interface between two different homogeneous media.

True-amplitude Gaussian-beam migration

Geophysics ◽  
2009 ◽  
Vol 74 (2) ◽  
pp. S11-S23 ◽  
Author(s):  
Samuel H. Gray ◽  
Norman Bleistein
Keyword(s):  
Wave Equation ◽  
Gaussian Beam ◽  
Opening Angle ◽  
Amplitude Wave ◽  
Amplitude Variation With Offset ◽  
Imaging Condition ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Beam Depth ◽  
Gaussian Beam Migration

Gaussian-beam depth migration and related beam migration methods can image multiple arrivals, so they provide an accurate, flexible alternative to conventional single-arrival Kirchhoff migration. Also, they are not subject to the steep-dip limitations of many (so-called wave-equation) methods that use a one-way wave equation in depth to downward-continue wavefields. Previous presentations of Gaussian-beam migration have emphasized its kinematic imaging capabilities without addressing its amplitude fidelity. We offer two true-amplitude versions of Gaussian-beam migration. The first version combines aspects of the classic derivation of prestack Gaussian-beam migration with recent results on true-amplitude wave-equation migration, yields an expression involving a crosscorrelation imaging condition. To provide amplitude-versus-angle (AVA) information, true-amplitude wave-equation migration requires postmigration mapping from lateral distance (between image location and source location) to subsurface opening angle. However, Gaussian-beam migration does not require postmigration mapping to provide AVA data. Instead, the amplitudes and directions of the Gaussian beams provide information that the migration can use to produce AVA gathers as part of the migration process. The second version of true-amplitude Gaussian-beam migration is an expression involving a deconvolution imaging condition, yielding amplitude-variation-with-offset (AVO) information on migrated shot-domain common-image gathers.

scholarly journals Modification of Kirchhoff migration with variable sound speed and attenuation for acoustic imaging of media and application to tomographic imaging of the breast

2011 ◽  
Vol 38 (2) ◽  
pp. 998-1007 ◽  
Author(s):  
Steven Schmidt ◽  
Nebojsa Duric ◽  
Cuiping Li ◽  
Olivier Roy ◽  
Zhi-Feng Huang
Keyword(s):  
Sound Speed ◽  
Tomographic Imaging ◽  
Acoustic Imaging ◽  
Kirchhoff Migration

Homomorphic deconvolution

Geophysics ◽  
1983 ◽  
Vol 48 (7) ◽  
pp. 1014-1016 ◽  
Author(s):  
D. J. Jin ◽  
J. R. Rogers
Keyword(s):  
Reflection Coefficient ◽  
Additive Noise ◽  
Critical Role ◽  
Minimum Phase ◽  
Integration Algorithm ◽  
The Public ◽  
Band Limited ◽  
Frequency Transformations ◽  
Analytical Expressions ◽  
Phase Reflection

The advantages of homomorphic deconvolution are that it does not require the assumptions of minimum‐phase wavelet and of a white random reflection coefficient series. Disadvantages of the method which have been recognized in the public domain are difficulties in unwrapping the phase, in dealing with band‐limited signals, and in handling mixed‐phase reflection coefficient series. These difficulties may be respectively overcome by using an “adaptive numerical integration algorithm” (Tribolet, 1977), frequency transformations (Tribolet, 1979), and exponential weighting of the signal (Tribolet, 1979). There seems to have been some understanding in the literature and among exploration researchers that additive noise would affect the performance of homomorphic deconvolution. However, to the best of our knowledge there have not appeared in the literature any analytical expressions or experiments conclusively showing how additive noise affects homomorphic deconvolution. Analytic and experimental analyses demonstrated that additive noise plays a critical role in homomorphic deconvolution such that homomorphic deconvolution is unreliable whenever the spectral amplitudes of the signal are very small over certain frequency bands and even a small amount of noise is present. This unreliability of the method overshadows its advantages.

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