Aperture effects in 2.5D Kirchhoff migration: A geometrical explanation

Geophysics ◽  
2003 ◽  
Vol 68 (5) ◽  
pp. 1673-1684 ◽  
Author(s):  
Thomas Hertweck ◽  
Christoph Jäger ◽  
Alexander Goertz ◽  
Jörg Schleicher
Keyword(s):  
Stationary Phase ◽  
Amplitude Variation ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Strong Amplitude ◽  
Zero Offset ◽  
The Time Domain ◽  
Geometrical Explanation ◽  
Seismic Images

Seismic images obtained by Kirchhoff time or depth migration are always accompanied by some artifacts known as migration noise, migration boundary effects, or diffraction smiles, which may severely affect the quality of the migration result. Most of these undesirable effects are caused by a limited aperture if the algorithms make no special disposition to avoid them. Strong amplitude variation along reflection events may cause similar artifacts. All of these effects can be explained mathematically by means of the method of stationary phase. However, such a purely theoretical explication is not always easily understood by applied geophysicists. A geometrical interpretation of the terms of the stationary‐phase approximation in relation to the diffraction and reflection traveltime curves in the time domain can help to develop a more intuitive understanding of the migration artifacts. A simple numerical experiment for poststack (zero‐offset) data indicates the problem and helps to demonstrate the effects and the methods to avoid them.

Adaptive focusing window for seismic angle migration

Geophysics ◽  
2009 ◽  
Vol 74 (1) ◽  
pp. S1-S10 ◽  
Author(s):  
Mathias Alerini ◽  
Bjørn Ursin
Keyword(s):  
Synthetic Data ◽  
Main Idea ◽  
Three Dimensions ◽  
User Friendliness ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Migration Operator ◽  
Local Geology ◽  
Original Approach

Kirchhoff migration is based on a continuous integral ranging from minus infinity to plus infinity. The necessary discretization and truncation of this integral introduces noise in the migrated image. The attenuation of this noise has been studied by many authors who propose different strategies. The main idea is to limit the migration operator around the specular point. This means that the specular point must be known before migration and that a criterion exists to determine the size of the migration operator. We propose an original approach to estimate the size of the focusing window, knowing the geologic dip. The approach benefits from the use of prestack depth migration in angle domain, which is recognized as the most artifact-free Kirchhoff-type migration. The main advantages of the method are ease of implementation in an existing angle-migration code (two or three dimensions), user friendliness, ability to take into account multiorientation of the local geology as in faulted regions, and flexibility with respect to the quality of the estimated geologic dip field. Common-image gathers resulting from the method are free from migration noise and can be postprocessed in an easier way. We validate the approach and its possibilities on synthetic data examples with different levels of complexity.

JOINT INTERPRETATION OF AIRBORNE ELECTROMAGNETIC DATA IN THE TIME DOMAIN AND FREQUENCY DOMAIN

2018 ◽  
Vol 12 (7-8) ◽  
pp. 76-83
Author(s):  
E. V. KARSHAKOV ◽  
J. MOILANEN
Keyword(s):  
Fourier Transform ◽  
Frequency Domain ◽  
Time Domain ◽  
Frequency Interval ◽  
Single Spectrum ◽  
Simultaneous Inversion ◽  
Homogeneous Half Space ◽  
Time Domain Data ◽  
The Time Domain

Тhe advantage of combine processing of frequency domain and time domain data provided by the EQUATOR system is discussed. The heliborne complex has a towed transmitter, and, raised above it on the same cable a towed receiver. The excitation signal contains both pulsed and harmonic components. In fact, there are two independent transmitters operate in the system: one of them is a normal pulsed domain transmitter, with a half-sinusoidal pulse and a small "cut" on the falling edge, and the other one is a classical frequency domain transmitter at several specially selected frequencies. The received signal is first processed to a direct Fourier transform with high Q-factor detection at all significant frequencies. After that, in the spectral region, operations of converting the spectra of two sounding signals to a single spectrum of an ideal transmitter are performed. Than we do an inverse Fourier transform and return to the time domain. The detection of spectral components is done at a frequency band of several Hz, the receiver has the ability to perfectly suppress all sorts of extra-band noise. The detection bandwidth is several dozen times less the frequency interval between the harmonics, it turns out thatto achieve the same measurement quality of ground response without using out-of-band suppression you need several dozen times higher moment of airborne transmitting system. The data obtained from the model of a homogeneous half-space, a two-layered model, and a model of a horizontally layered medium is considered. A time-domain data makes it easier to detect a conductor in a relative insulator at greater depths. The data in the frequency domain gives more detailed information about subsurface. These conclusions are illustrated by the example of processing the survey data of the Republic of Rwanda in 2017. The simultaneous inversion of data in frequency domain and time domain can significantly improve the quality of interpretation.

3-D prestack Kirchhoff beam migration for depth imaging

Geophysics ◽  
2000 ◽  
Vol 65 (5) ◽  
pp. 1592-1603 ◽  
Author(s):  
Yonghe Sun ◽  
Fuhao Qin ◽  
Steve Checkles ◽  
Jacques P. Leveille
Keyword(s):  
Distributed Environment ◽  
Small Surface ◽  
Depth Imaging ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Surface Patches ◽  
Image Volume ◽  
Kirchhoff Depth Migration ◽  
Full Volume ◽  
Single Set

A beam implementation is presented for efficient full‐volume 3-D prestack Kirchhoff depth migration of seismic data. Unlike conventional Kirchhoff migration in which the input seismic traces in time are migrated one trace at a time into the 3-D image volume for the earth’s subsurface, the beam migration processes a group of input traces (a supergather) together. The requirement for a supergather is that the source and receiver coordinates of the traces fall into two small surface patches. The patches are small enough that a single set of time maps pertaining to the centers of the patches can be used to migrate all the traces within the supergather by Taylor expansion or interpolation. The migration of a supergather consists of two major steps: stacking the traces into a τ-P beam volume, and mapping the beams into the image volume. Since the beam volume is much smaller than the image volume, the beam migration cost is roughly proportional to the number of input supergathers. The computational speedup of beam migration over conventional Kirchhoff migration is roughly proportional to [Formula: see text], the average number of traces per supergather, resulting a theoretical speedup up to two orders of magnitudes. The beam migration was successfully implemented and has been in production use for several years. A factor of 5–25 speedup has been achieved in our in‐house depth migrations. The implementation made 3-D prestack full‐volume depth imaging feasible in a parallel distributed environment.

Velocity estimation in laterally varying media

Geophysics ◽  
1982 ◽  
Vol 47 (6) ◽  
pp. 884-897 ◽  
Author(s):  
Walter S. Lynn ◽  
Jon F. Claerbout
Keyword(s):  
Taylor Series Expansion ◽  
Velocity Estimation ◽  
Lateral Resolution ◽  
Velocity Variation ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Derivative Term ◽  
Fourth Derivative ◽  
Zero Offset ◽  
Lateral Variations

In areas of large lateral variations in velocity, stacking velocities computed on the basis of hyperbolic moveout can differ substantially from the actual root mean square (rms) velocities. This paper addresses the problem of obtaining rms or migration velocities from stacking velocities in such areas. The first‐order difference between the stacking and the vertical rms velocities due to lateral variations in velocity are shown to be related to the second lateral derivative of the rms slowness [Formula: see text]. Approximations leading to this relation are straight raypaths and that the vertical rms slowness to a given interface can be expressed as a second‐order Taylor series expansion in the midpoint direction. Under these approximations, the effect of the first lateral derivative of the slowness on the traveltime is negligible. The linearization of the equation relating the stacking and true velocities results in a set of equations whose inversion is unstable. Stability is achieved, however, by adding a nonphysical fourth derivative term which affects only the higher spatial wavenumbers, those beyond the lateral resolution of the lateral derivative method (LDM). Thus, given the stacking velocities and the zero‐offset traveltime to a given event as a function of midpoint, the LDM provides an estimate of the true vertical rms velocity to that event with a lateral resolution of about two mute zones or cable lengths. The LDM is applicable when lateral variations of velocity greater than 2 percent occur over the mute zone. At variations of 30 percent or greater, the internal assumptions of the LDM begin to break down. Synthetic models designed to test the LDM when the different assumptions are violated show that, in all cases, the results are not seriously affected. A test of the LDM on field data having a lateral velocity variation caused by sea floor topography gives a result which is supported by depth migration.

Introduction to this special section: Acquisition and sensing

2019 ◽  
Vol 38 (9) ◽  
pp. 670-670
Author(s):  
Margarita Corzo ◽  
Tim Brice ◽  
Ray Abma
Keyword(s):  
Special Section ◽  
Seismic Survey ◽  
Ocean Bottom ◽  
Total Cost ◽  
Small Boat ◽  
Air Gun ◽  
Seismic Acquisition ◽  
The Cost ◽  
Seismic Images

Seismic acquisition has undergone a revolution over the last few decades. The volume of data acquired has increased exponentially, and the quality of seismic images obtained has improved tremendously. While the total cost of acquiring a seismic survey has increased, the cost per trace has dropped precipitously. Land surveys have evolved from sparse 2D lines acquired with a few dozen receivers to densely sampled 3D multiazimuth surveys. Marine surveys that once may have consisted of a small boat pulling a single cable have evolved to large streamer vessels pulling multiple cables and air-gun arrays and to ocean-bottom detectors that require significant fleets to place the detectors, shoot the sources, and provide support. These surveys collect data that are wide azimuth and typically fairly well sampled.

An overview of depth imaging in exploration geophysics

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCA5-WCA17 ◽  
Author(s):  
John Etgen ◽  
Samuel H. Gray ◽  
Yu Zhang
Keyword(s):  
Structural Description ◽  
Practical Application ◽  
Depth Imaging ◽  
Seismic Migration ◽  
Depth Migration ◽  
The Past ◽  
The Earth ◽  
Seismic Images ◽  
Migration Technique ◽  
Made In

Prestack depth migration is the most glamorous step of seismic processing because it transforms mere data into an image, and that image is considered to be an accurate structural description of the earth. Thus, our expectations of its accuracy, robustness, and reliability are high. Amazingly, seismic migration usually delivers. The past few decades have seen migration move from its heuristic roots to mathematically sound techniques that, using relatively few assumptions, render accurate pictures of the interior of the earth. Interestingly, the earth and the subjects we want to image inside it are varied enough that, so far, no single migration technique has dominated practical application. All techniques continually improve and borrow from each other, so one technique may never dominate. Despite the progress in structural imaging, we have not reached the point where seismic images provide quantitatively accurate descriptions of rocks and fluids. Nor have we attained the goal of using migration as part of a purely computational process to determine subsurface velocity. In areas where images have the highest quality, we might be nearing those goals, collectively called inversion. Where data are more challenging, the goals seem elusive. We describe the progress made in depth migration to the present and the most significant barriers to attaining its inversion goals in the future. We also conjecture on progress likely to be made in the years ahead and on challenges that migration might not be able to meet.

Kirchhoff modeling, inversion for reflectivity, and subsurface illumination

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1195-1209 ◽  
Author(s):  
Bertrand Duquet ◽  
Kurt J. Marfurt ◽  
Joe A. Dellinger
Keyword(s):  
Least Squares ◽  
A Priori ◽  
Computational Cost ◽  
Image Point ◽  
Surface Sampling ◽  
Constrained Least Squares ◽  
Amplitude Variation With Offset ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Least Squares Migration

Because of its computational efficiency, prestack Kirchhoff depth migration is currently one of the most popular algorithms used in 2-D and 3-D subsurface depth imaging. Nevertheless, Kirchhoff algorithms in their typical implementation produce less than ideal results in complex terranes where multipathing from the surface to a given image point may occur, and beneath fast carbonates, salt, or volcanics through which ray‐theoretical energy cannot penetrate to illuminate underlying slower‐velocity sediments. To evaluate the likely effectiveness of a proposed seismic‐acquisition program, we could perform a forward‐modeling study, but this can be expensive. We show how Kirchhoff modeling can be defined as the mathematical transpose of Kirchhoff migration. The resulting Kirchhoff modeling algorithm has the same low computational cost as Kirchhoff migration and, unlike expensive full acoustic or elastic wave‐equation methods, only models the events that Kirchhoff migration can image. Kirchhoff modeling is also a necessary element of constrained least‐squares Kirchhoff migration. We show how including a simple a priori constraint during the inversion (that adjacent common‐offset images should be similar) can greatly improve the resulting image by partially compensating for irregularities in surface sampling (including missing data), as well as for irregularities in ray coverage due to strong lateral variations in velocity and our failure to account for multipathing. By allowing unstacked common‐offset gathers to become interpretable, the additional cost of constrained least‐squares migration may be justifiable for velocity analysis and amplitude‐variation‐with‐offset studies. One useful by‐product of least‐squares migration is an image of the subsurface illumination for each offset. If the data are sufficiently well sampled (so that including the constraint term is not necessary), the illumination can instead be calculated directly and used to balance the result of conventional migration, obtaining most of the advantages of least‐squares migration for only about twice the cost of conventional migration.

Depth‐focusing analysis using a wavefront‐curvature criterion

Geophysics ◽  
1993 ◽  
Vol 58 (8) ◽  
pp. 1148-1156 ◽  
Author(s):  
Scott MacKay ◽  
Ray Abma
Keyword(s):  
Seismic Energy ◽  
Radius Of Curvature ◽  
Large Radius ◽  
Data Set ◽  
Depth Migration ◽  
Amplitude Data ◽  
Zero Radius ◽  
Inverse Radius ◽  
Interpretation Process ◽  
Zero Offset

Depth‐focusing analysis (DFA), a method of refining velocities for prestack depth migration, relies on amplitude buildups at zero offset to determine the extrapolation depths that best focus the migrated data. Unfortunately, seismic energy from dipping interfaces, diffractions, and noise often produce spurious amplitude indications of focusing. To reduce possible ambiguity in the DFA interpretation process, we introduce a new attribute for determining focusing that is relatively independent of amplitude. Our approach is based on estimates of the radius of wavefront curvature. The estimates are derived from normal moveout analysis of nonzero‐offset data saved during migration. By relating steeper moveout to smaller radius of wavefront curvature, focusing is defined by a wavefront curvature of zero radius. Additionally, we show that applying inverse‐radius weights to the amplitude data attenuates nonfocused events due to their large radius of curvature. Using the Marmousi data set, our weighting scheme resulted in reduced spurious focusing and enhanced velocity resolution in DFA.

AVA analysis after velocity-independent DMO and imaging

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 686-691 ◽  
Author(s):  
Gerald H. F. Gardner ◽  
Anat Canning
Keyword(s):  
Reflection Coefficient ◽  
Peak Amplitude ◽  
Angle Of Incidence ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
The Past ◽  
Reflectivity Function ◽  
Velocity Of Propagation ◽  
Zero Offset ◽  
Amplitude Versus

A common midpoint (CMP) gather usually provides amplitude variation with offset (AVO) information by displaying the reflectivity as the peak amplitude of symmetrical deconvolved wavelets. This puts a reflection coefficient R at every offset h, giving a function R(h). But how do we link h with the angle of incidence, θ, to get the reflectivity function, R(θ)? This is necessary for amplitude versus angle-of-incidence (AVA) analysis. One purpose of this paper is to derive formulas for this linkage after velocity-independent dip-moveout (DMO), done by migrating radial sections, and prestack zero-offset migration. Related studies of amplitude-preserving DMO in the past have dealt with constant-offset DMO but have not given the connection between offset and angle of incidence after processing. The results in the present paper show that the same reflectivity function can be extracted from the imaged volume whether it is produced using radial-trace DMO plus zero-offset migration, constant-offset DMO plus zero-offset migration, or directly by prestack, common-offset migration. The data acquisition geometry for this study consists of parallel, regularly spaced, multifold lines, and the velocity of propagation is constant. Events in the data are caused by an arbitrarily oriented 3-D plane reflector with any reflectivity function. The DMO operation transforms each line of data (m, h, t), i.e., midpoint, half-offset, and time, into an (m1, k, t1) space by Stolt-migrating each radial-plane section of the data, 2h = Ut, with constant velocity U/2. Merging the (m1, k, t1) spaces for all the lines forms an (x, y, k, t1) space, where the first two coordinates are the midpoint location, the third is the new half-offset, and the fourth is the time. Normal moveout (NMO) plus 3-D zero-offset migration of the subspace (x, y, t1) for each k creates a true-amplitude imaged volume (X, Y, k, T). Each peak amplitude in the volume is a reflection coefficient linked to an angle of incidence.

Reply by the authors to Arthur E. Barnes

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1944-1946
Author(s):  
M. Tygel ◽  
J. Schleicher ◽  
P. Hubral
Keyword(s):  
Destructive Interference ◽  
Depth Migration ◽  
Time Migration ◽  
Pulse Distortion ◽  
Nmo Correction ◽  
Important Processing ◽  
Offset Time ◽  
Zero Offset ◽  
Seismic Reflections ◽  
Processing Steps

We highly appreciate the useful remarks of Dr. Barnes relating our work to well‐known practical seismic processing effects. This is of particular interest as normal‐moveout (NMO) correction and post‐stack time migration are still two very important processing steps. Most exploration geophysicists know about the significance of pulse distortions known as “NM0 stretch” and “frequency shifting due to zero‐offset time migration.” As a result of the discussion of Dr. Barnes, it should now be possible to better appreciate the importance of our very general formulas (27) describing the pulse distortion of seismic reflections from an arbitrarily curved subsurface reflector when subjected to a prestack depth migration in 3‐D laterally inhomogeneous media. This discussion thus relates in particular to such important questions as how to correctly sample signals in the time or depth domain in order to avoid spatial aliasing, or how to stack seismic data without loss of information due to destructive interference of wavelets of different lengths.

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