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.

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.

Velocity estimates derived from three‐dimensional seismic data

Geophysics ◽  
1983 ◽  
Vol 48 (11) ◽  
pp. 1486-1497 ◽  
Author(s):  
Kwame Owusu ◽  
G. H. F. Gardner ◽  
Wulf F. Massell
Keyword(s):  
Seismic Data ◽  
Constant Velocity ◽  
Three Dimensional ◽  
Logarithmic Scale ◽  
Model Data ◽  
Reconstruction Process ◽  
Input Field ◽  
Scattering Center ◽  
Straight Line ◽  
The Matrix

A new computer algorithm is described by which velocity estimates can be derived from three‐dimensional (3-D) multifold seismic data. The velocity estimate, referred to as “imaging velocity,” is that which best describes the diffraction hyperboloid due to a scatterer. The scattering center is best imaged when this velocity is used in the reconstruction process. The method is based on the 3-D Kirchhoff summation migration before stack. The implementation consists of two basic phases: (1) differentiating the input field traces and resampling them to a logarithmic time scale, and (2) shifting, weighting, and summing each resampled trace to a range of depth levels also chosen on a logarithmic scale. Peak amplitudes in the resulting image matrix give a time T and depth Z from which velocity is obtained using the relation [Formula: see text] The locus of constant velocity is a slanted straight line in the coordinate system of the matrix. In the usual application of migration for velocity analysis, each input trace of N samples is migrated for each of M constant velocity functions requiring [Formula: see text] moveout shift calculations. In the new method presented here, a constant shift is calculated for a given resampled trace, for each depth into which it is summed. This reduces the number of calculations per trace to about N, resulting in a significant improvement in computing efficiency. The operation of the algorithm is illustrated using synthetic and physical model data.

Three-dimensional Reflector Imaging of in-mine High Frequency Crosshole Seismic Data

1995 ◽  
Vol 26 (2-3) ◽  
pp. 325-330 ◽  
Author(s):  
Cvetan Sinadinovski ◽  
Stewart. A. Greenhalgh ◽  
Iain Mason
Keyword(s):  
Seismic Data ◽  
High Frequency ◽  
Three Dimensional

The variation of the reflection coefficient of extensional guided waves in pipes from defects as a function of defect depth, axial extent, circumferential extent and frequency

2002 ◽  
Vol 216 (11) ◽  
pp. 1131-1143 ◽  
Author(s):  
P Cawley ◽  
M J S Lowe ◽  
F Simonetti ◽  
C Chevalier ◽  
A G Roosenbrand
Keyword(s):  
Finite Element ◽  
Reflection Coefficient ◽  
High Frequency ◽  
Guided Waves ◽  
Three Dimensional ◽  
Element Analysis ◽  
Low Frequencies ◽  
Frequency Regime ◽  
Circumferential Extent ◽  
Part Thickness

The reflection coefficients of extensional guided modes from notches of different axial, circumferential and through-thickness extent in pipes of different diameters have been studied using finite element analysis. A selection of the predictions has also been validated by experiments. For part-thickness notches of a given circumferential extent and minimal axial extent, the reflection coefficient increases monotonically with depth at all frequencies, and increases with frequency at a given depth. When the wavelength is long compared to the pipe wall thickness, the reflection coefficient from part-thickness notches of a given circumferential extent is a strong function of the defect axial extent, the reflection being a maximum at an axial extent of about 25 per cent of the wavelength and a minimum at 0 and 50 per cent. The reflection coefficient is a linear function of the defect circumferential extent at higher frequencies (with frequency-diameter products greater than about 3000 kHz mm) where a ray theory analysis explains the behaviour, while at low frequencies the reflection coefficient at a given circumferential extent is reduced. In the high-frequency regime, the axial extent of a through-thickness defect has little influence on the reflection coefficient, while it is important at lower frequencies. Three-dimensional, finite element predictions in the high-frequency regime have shown that the reflection coefficient from a part-thickness, part-circumferential defect can be predicted by multiplying the reflection coefficient for an axisymmetric defect of the same depth and axial extent by that for a through-thickness defect of the same circumferential extent.

Three‐dimensional Born inversion with an arbitrary reference

Geophysics ◽  
1986 ◽  
Vol 51 (8) ◽  
pp. 1552-1558 ◽  
Author(s):  
Jack K. Cohen ◽  
Frank G. Hagin ◽  
Norman Bleistein
Keyword(s):  
Seismic Data ◽  
High Frequency ◽  
Perturbation Methods ◽  
Three Dimensional ◽  
Jacobian Determinant ◽  
Velocity Inversion ◽  
Basic Assumptions ◽  
General Data ◽  
Feasible Algorithm ◽  
Data Surface

Recent work of G. Beylkin helped set the stage for very general seismic inversions. We have combined these broad concepts for inversion with classical high‐frequency asymptotics and perturbation methods to bring them closer to practically implementable algorithms. Applications include inversion schemes for both stacked and unstacked seismic data. Basic assumptions are that the data have relative true amplitude, and that a reasonably accurate background velocity c(x, y, z) is available. The perturbation from this background is then sought. Since high‐frequency approximations are used throughout, the resulting algorithms essentially locate discontinuities in velocity. An expression for a full 3-D velocity inversion can be derived for a general data surface. In this degree of generality the formula does not represent a computationally feasible algorithm, primarily because a key Jacobian determinant is not expressed in practical terms. In several important cases, however, this shortcoming can be overcome and expressions can be obtained that lead to feasible computing schemes. Zero‐offsets, common‐sources, and common‐receivers are examples of such cases. Implementation of the final algorithms involves, first, processing the data by applying the FFT, making an amplitude adjustment and filtering, and applying an inverse FFT. Then, for each output point, a summation is performed over that portion of the processed data influencing the output point. This last summation involves an amplitude and traveltime along connecting rays. The resulting algorithms are computationally competitive with analogous migration schemes.

scholarly journals A critical look at the prediction of the temperature field around a laser-induced melt pool on metallic substrates

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Yi Shu ◽  
Daniel Galles ◽  
Ottman A. Tertuliano ◽  
Brandon A. McWilliams ◽  
Nancy Yang ◽  
...  
Keyword(s):  
Laser Beam ◽  
Constant Velocity ◽  
Three Dimensional ◽  
Laser Beams ◽  
Poor Control ◽  
Metallic Substrates ◽  
Melt Pool ◽  
Mechanical Models ◽  
Transport Effects ◽  
Thermal Histories

AbstractThe study of microstructure evolution in additive manufacturing of metals would be aided by knowing the thermal history. Since temperature measurements beneath the surface are difficult, estimates are obtained from computational thermo-mechanical models calibrated against traces left in the sample revealed after etching, such as the trace of the melt pool boundary. Here we examine the question of how reliable thermal histories computed from a model that reproduces the melt pool trace are. To this end, we perform experiments in which one of two different laser beams moves with constant velocity and power over a substrate of 17-4PH SS or Ti-6Al-4V, with low enough power to avoid generating a keyhole. We find that thermal histories appear to be reliably computed provided that (a) the power density distribution of the laser beam over the substrate is well characterized, and (b) convective heat transport effects are accounted for. Poor control of the laser beam leads to potentially multiple three-dimensional melt pool shapes compatible with the melt pool trace, and therefore to multiple potential thermal histories. Ignoring convective effects leads to results that are inconsistent with experiments, even for the mild melt pools here.

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