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Velocity and Migration

Monte Carlo automatic velocity picks (ps 1424K) (src 4127K)
Lumley D. E.
I develop a method for making automatic velocity picks from velocity semblance scans, using a Monte Carlo nonlinear fitting technique. Given a single velocity semblance scan, random interval velocity models are generated by performing many Monte Carlo random walks. For each trial random interval velocity model, the corresponding rms velocity is calculated, and the semblance is integrated along the trial rms velocity path through the velocity scan. After several random walks, one interval velocity model, and its associated rms velocity function, is retained as the global best fit to the velocity scan which maximizes semblance. The result is a nonlinearly optimal rms velocity function, and an associated geologically reasonable interval velocity model, at each surface CMP location. I tested the method on 300 marine CMP gathers, and the Monte Carlo picks gave reasonable rms and interval velocity models which enhanced the stacked and prestack migrated images.
Plumes: Response of time migration to lateral velocity variation (ps 234K) (src 11423K)
Black J. L., Bevc D., Brzostowski M. A., Mo L., and Palacharla G.
Lateral velocity variation causes time migration to misposition events. Black and Brzostowski 1993 quantify this error by developing a first order theory for the positioning error in terms of x and t. The response of a point diffractor in a medium with a simple lateral velocity variation is a distinctive cusp-like shape that we call a ``plume''. The nature of the plume is revealed by making two geometrical constructions based upon point diffractor ray-tracing: superposition of semicircles and summation along hyperbolas. The plume shapes derived by these two techniques are consistent with results obtained by migrating synthetic point diffractors using the Kirchhoff, Stolt, and Gazdag time migration methods. The shapes of these plumes resemble those predicted by Black and Brzostowski's simple first-order theory, but the actual plumes are less symmetric than the predictions. Thus a higher-order theory is needed.
Migration velocity dependence of plumes (ps 377K) (src 11383K)
Black J. L., Bevc D., and Palacharla G.
Changing the migration velocity causes the response of time migration to a point diffractor beneath a lateral velocity variation to evolve. At the extremes of migration velocity, the image is either fully undermigrated or fully overmigrated. Between these extremes, there is the mixture of overmigration and undermigration that leads to ``plume'' responses. As the velocity is changed, these plumes transform and rotate in a systematic manner. A simple first-order theory of plume velocity dependence explains some, but not all, of the observed behavior.
Migration of crosswell seismic data (ps 816K) (src 3844K)
Mo L.
Crosswell seismic survey has proven to be an effective, high-resolution method of reservoir characterization. However, its application has been mainly confined to transmission traveltime tomography, even though extensions have recently been made to reflection imaging. This paper describes the application of reverse-time wavefield extrapolation coupled with the excitation-time imaging condition to the migration of crosswell seismic common-shot profiles. I have applied fourth-order-in-time, tenth-order-in-space finite differencing to perform the crosswell seismic modeling of the two-way acoustic wave equation. During the migration wavefield backpropagation, I gradually drop the order of finite differencing to the second-order near the recording boundary for the derivative perpendicular to the recording boundary, so that the numerical computation conforms to physical wave phenomena. Modeling and migration are based on the geometry of a true field survey. This method successfully models high frequency crosswell seismic wavefields. Migration of one common-shot profile gives a very clear reflectivity image. Vertical resolution up to a half wavelength and horizontal resolution as governed by the first Fresnel zone width have been realized. At the end, the paper presents some suggestions to facilitate field data migration, which include removal of tube waves and wavefield separation.
Gaussian beam migration (ps 868K) (src 1491K)
Mo L.
This paper reports on my use of Gaussian beam wave propagation theory to downward continue all the frequency components of individual surface recorded wavefield traces (point sources) into the subsurface. According to the Huygens secondary point source principle, the surface recorded wavefields can be regarded as a series of point sources. The theory of Gaussian beam wave propagation asserts that asymptotic high frequency wavefield components propagate along rays, of which the wavefronts are parabolas and the amplitude profiles overriding the ray paths are Gaussian functions decaying away from the central ray paths. With my method, Gaussian beam wave propagation is implemented in the space-frequency domain. Poststack and prestack common-shot profile migration are accomplished by applying the corresponding imaging condition to the downward continued Gaussian beam wavefields in the frequency domain. The surface seismograms are migrated separately, and the migration process finishes by summing the contributions of all surface recorded traces to obtain the subsurface reflectivity image. This migration scheme combines the advantages of complete wave equation migration and ray tracing based migration. Tests with synthetic data show that the Gaussian beams evaluated along only seventeen rays from each surface seismogram location can image a fairly complex velocity and reflectivity model.
Numerical analysis of ${\bf \omega-x}$ migration (ps 55K) (src 6K)
Mo L.
For the constant velocity 15-degree space-frequency ($-x$) time migration algorithm, the parabolic approximation of the exact one-way wave equation causes undermigration, and the Crank-Nicolson numerical method that is used to solve the diffraction equation causes overmigration. However, the modified wave equation that the numerical solution actually satisfies for the diffraction equation shows that the trace spacing interval plays a larger role than the depth step size of downward continuation in determining the accuracy of the numerical solution.
Fast anti-aliased Kirchhoff migration and modeling (ps 260K) (src 265K)
Bevc D. and Claerbout J.
We describe an optimized Kirchhoff migration and modeling routine which handles the problem of operator aliasing. Slight modifications of this algorithm could be useful for velocity analysis, pre-stack migration and DMO. Several Kirchhoff routines are presented in SEP73. The simplest routines Claerbout (1992b) are tutorial and do not address the problem of operator aliasing. A more elaborate routine which overcomes the aliasing problem ...

Least-Squares Migration

Unitary operators in inversion (ps 39K) (src 4K)
Claerbout J. F.
Geophysical model building springs from a variety of related formulas ranging from inversion to back projection. When the number of unknowns is huge, as when we seek an image of the earth's reflectivity, then using the conjugate operator (also known as back projection or migration) is always a possible starting technique. The conjugate operator can generally be improved by any transformation that pushes it towards a unitary operator. Unitary operators play an important role in imaging practice ...
Least-squares Kirchhoff migration (ps 181K) (src 222K)
Cole S. and Karrenbach M.
Slant stacks of finite-aperture data contain artifacts that are a result of the limited aperture. Resolution in the transform domain is also adversely affected by the finite aperture. Clement Kostov (Kostov, 1990), using least-squares inverse operators formulated by Beylkin (1987), developed least-squares slant stack transforms that compensate for the limited aperture, reducing artifacts and improving resolution. These least-squares transforms are best implemented in the frequency domain, where a separate, small least-squares problem can be constructed for each frequency. ...
Least squares imaging, datuming, and interpolation using the wave equation (ps 765K) (src 3066K)
Ji J.
Though the continuous wave equation defines a unitary operator, many operators based on it are not unitary because many approximations are made to be implemented numerically in finite and discrete space. The nonunitary property of the operators produces many artifacts in the solution space when is applied to data that was generated by a continuous unitary operator, such as real seismic data. This paper presents a method of reducing these kinds of artifacts in migration, datuming, and interpolation by using the technique of a least-squares optimization.
Reverse-time migration as the conjugate of forward modeling (ps 224K) (src 4934K)
Ji J.
This paper presents a reverse-time migration algorithm as the conjugate of the forward time-extrapolation modeling. The conjugate operator is obtained by formulating the forward operator explicitly and by transposing the forward operator. The algorithm is tested by a numerical experiment for a 1-D layered model. A method to suppress artifacts, which are caused by the intrinsic non-unitary property of the operator, is introduced using the least-squares optimization technique.
A Gazdag routine passing the dot-product test (ps 40K) (src 3K)
Claerbout J. F.
We examine differential equations of the form $du/dz+iu=s(z)$ and find the pair of conjugate operators relating s(z) and u(z=0). This leads up to a Gazdag subroutine that passes the dot-product test.

Wave-Equation Datuming

Kirchhoff wave-equation datuming with irregular acquisition topography (ps 1066K) (src 4180K)
Bevc D.
Seismic data gathered on land is distorted by irregular acquisition topography. Most seismic imaging algorithms are applied to data which is shifted to a planar datum. In regions of mild topography a static shift is adequate to perform this transformation, however as the necessary shift increases in magnitude, the static approximation becomes inadequate. In this situation a procedure based on wave-equation extrapolation is more appropriate than static shift. After describing the Kirchhoff datuming algorithm I apply it to a deep-water marine data set and to synthetic land data. The land data examples demonstrate that wave-equation datuming is more appropriate than static shift in regions of large topographic relief.
Phase shift datuming and migration from an irregular surface (ps 214K) (src 561K)
Ji J. and Claerbout J.
We present a phase shift datuming and migration algorithm for the data acquired on a nonflat surface. First of all, we formulate the exploding reflector modeling scheme for a nonflat recording surface. Then, the migration scheme is found as the conjugate operator of the modeling operator. In order to get the correct migration and datuming operators, the forward modeling must be physically reasonable because the conjugate operators are strongly dependent on the forward modeling operator. The artifacts caused by the non-unitary property of the operators can be reduced by a least-squares optimization technique.
Datuming by wavefield extrapolation (ps 1399K) (src 4131K)
Mo L.
This paper proposes a method of datuming that uses the exact two-way acoustic wave equation to do the wavefield extrapolation along with depth migration. Datuming is the process used to derive the wavefields at one datum from those at another datum. Poststack and prestack datuming regard the surface recorded wavefields as containing only subsurface primary reflections. In accord with this assumption, this paper points out that the surface recorded seismograms are the wavefields recorded at the earth surface, which is regarded as an absorbing boundary. According to the Huygens secondary-point-source principle, the surface recorded wavefields can also be regarded as a series of point sources. The datuming procedure I propose is implemented in the space-time domain by the finite-difference method. The recorded wavefields are taken as boundary conditions (sources) to drive the wave equation along the time axis in a rectangular strip containing the input and output datums. The desired wavefields at the output datum are the history wavefields at the output datum, with the four sides designed as absorbing boundaries. This method can handle any velocity variation between the two datums. In practical situations, one of the two datums is an irregular surface. This paper shows both upward and downward, poststack and prestack wave equation datuming. Tests with synthetic data show good results. The method has the potential for application to corrections of land data static and to migration in complex velocity areas.
Wave-equation datuming in variable velocity media (ps 481K) (src 4680K)
Popovici A. M.
Wave equation datuming represents the process of upward or downward continuing seismic data to a new reference surface. Usually datuming algorithms are based on the Kirchhoff integral solution to the wave equation. I present a new wave-equation datuming algorithm based on wavefield continuation through Fourier methods which handles laterally and depth varying velocity models. I show datuming results for different velocity models and the migration results before and after datuming.


Green's function traveltime interpolation (ps 742K) (src 1952K)
Lumley D. E.
Green's function traveltime fields are useful for Kirchhoff depth migration and inversion applications, and for tomographic velocity inversion. However, in 2-D, and especially 3-D, traveltimes become increasingly computationally expensive. Based on the physical continuity of (first arrival) traveltimes, I derive a least squares expression from the ray theoretic eikonal equation, which allows for the interpolation of known Green's function traveltime functions from several surface source positions, to one or more Green's function traveltime functions at intermediate arbitrary source positions. The least squares equation is parameterized in terms of the traveltime gradient, and is solved by standard weighted least-squares techniques and Singular Value Decomposition, and then integrated up to traveltime directly. For a 2-D (3-D) geometry, at least two (three) traveltime fields must be known in order to interpolate the unknown source traveltime field. I test the theory and algorithm on a constant velocity model, constant velocity gradient model, and the complex 2-D structural velocity model of Marmousi. The results are very encouraging to date, although some sparse numerical instability in the calculation of the traveltime gradient fields must be overcome in order to be of practical use.
Wave equation resampling of unevenly spaced traces (ps 351K) (src 4160K)
Zhang L. and Claerbout J.
Many seismic data processing algorithms require that input data have uniform sampling intervals both in time and in space. However, in reality, difficulties in positioning the sources and receivers may cause the spatial sampling intervals to vary from place to place. If the variations of the sampling intervals are too large to be acceptable for data processing, seismic traces have to be resampled. Because seismic data are almost always sampled uniformly in time, conventional methods perform the spatial resampling along each constant ...


Modeling reflections from the Austin Chalk - a practical application of azimuthal anisotropy (ps 1069K) (src 1810K)
Karrenbach M., Nichols D., and Muir F.
If a rock has elastic properties that vary with azimuth, then reflections from such a rock will likewise. We examine a simple situation where a homogeneous model material - representing a shale - overlies another homogeneous model material which simulates an isotropic chalk matrix and a set of vertical fractures with constant azimuth. Our modeling illustrates the small variability in azimuthal reflectivity as the amount of fracturing is increased, and also how reflection strength is affected by the nature of the fluid inclusion - gas or liquid.
``Plug and Play'' wave equation modules (ps 2991K) (src 10314K)
Karrenbach M.
Prestack modeling and migration programs can share a common structure. I outline this structure in the common shot domain for scalar or vector fields of any degree of anisotropy. The generality of the code is striking, and works for any combination of space and vector dimensionality. The actual pieces of FORTRAN 90 code are shown as listings in the paper. The advantage of this modeling strategy is that there exists only one piece of code that has to be maintained. The various media types are declared in the main program as single variables (objects), and they all use the same subroutines. The mathematical formulation is apparent in the notation and the modeling code can easily handle the usual finite differences or pseudo spectral cases or any combination. Reverse time migration is accomplished by using the same set of subroutines, conjugating of the modeling operation and additionally calculating the gradient (``imaging''). There is a trade-off between performance and common structure when, for example, the Laplacian $b ^2$ could be used instead of $b $ in constant velocity models.
Wavefront propagation by local ray-tracing (ps 6344K) (src 7223K)
Zhang L.
I develop a new wavefront-propagation method for calculating the traveltimes and geometrical amplitudes of seismic waves in a two-dimensional velocity model. The method assumes that the given velocity field is specified on a grid formed by regular triangular cells and that the velocity within each triangular cell varies linearly. These assumptions lead to a continuous representation of the velocity field. Five attributes are assigned to the wavefront at each grid point. They are traveltime, traveltime gradient, take-off angle, curvature radius of wavefront and geometric spreading factor. The attributes of a wavefront at one corner of a triangular cell are calculated from the attributes of wavefronts at other two corners by propagating the local circular wavefront through the triangular cell under the guidance of local rays. Starting from a given source position, these local computations proceed on an expanding half hexagonal ring. The new method computes the traveltimes of multiple arrivals by disassembling a multi-valued traveltime field into several single-valued fields, and then computing each of them separately. The method is evaluated with several synthetic velocity models, and its accuracy is checked with both analytical solutions and wave equation modeling.

Elastic Theory

Elastic constants of transversely isotropic media from constrained aperture traveltimes (ps 633K) (src 656K)
Michelena R. J.
The elastic constants that control P- and SV-wave propagation in a transversely isotropic media can be estimated by using P- and SV-wave traveltimes from either cross-well or VSP geometries. The procedure consists of two steps. First, elliptical velocity models are used to fit the traveltimes near one axis. The result is four elliptical parameters that represent direct and normal moveout velocities near the chosen axis for P- and SV-waves. Second, the elliptical parameters are used to solve a system of four equations and four unknown elastic constants. The system of equations is solved analytically yielding simple expressions for the elastic constants as a function of direct and normal moveout velocities. For SH-waves the estimation of the corresponding elastic constants is easier because the phase velocity is already elliptical. The procedure, introduced for homogeneous media, can be generalized to heterogeneous media by using tomographic techniques. The technique is illustrated with synthetic examples in a homogeneous medium.
Anelliptic approximations for TI media (ps 2309K) (src 4145K)
Dellinger J., Muir F., and Karrenbach M.
Homogeneous scalar isotropy can be completely specified by a single velocity. Elliptical anisotropy with a vertical symmetry axis requires two velocities, vertical and horizontal. For some problems these two velocities may not be enough. In particular, because vertical scale is unknown for surface-recorded data the vertical velocity in elliptical anisotropy gains nothing over isotropy. We introduce two successive scalar anisotropic approximations beyond elliptical anisotropy which can be used when more independent parameters are needed but the full complexity of transverse isotropy is unnecessary. Both approximations take the form of simple rational polynomials. We call them anelliptic approximations to indicate that although they are not elliptical, they do share some of elliptical anisotropy's useful properties. We show examples of how both anelliptic approximations can be used to fit transversely isotropic wave modes with acceptable accuracy. The first anelliptic approximation is specified by three parameters: vertical velocity, surface NMO velocity, and true horizontal velocity. The first two velocity parameters determine the near-vertical paraxial behavior, which is elliptic. The second anelliptic approximation is specified by four parameters: vertical velocity, surface NMO velocity, true horizontal velocity, and borehole NMO velocity. The first anelliptic approximation occurs as a special case. To a good approximation, both of the anelliptic approximations have identical forms in the group-velocity and dispersion-relation domains and converting between the two domains is trivial. The anelliptic velocity parameters are convenient and robust because of their close relationship to standard geophysical measurements. We also show how Dix's equations for stack-moveout velocities can be interpreted in terms of a paraxial layer group, and how this formulation naturally leads to an anelliptic extension of Dix's equations. The anelliptic Dix equations encompass nonhyperbolic moveout both as a result of intrinsic anisotropy and ray bending at layers.
Estimation of elastic specular reflectivity by Kirchhoff prestack depth migration: WKBJ least-squares theory (ps 85K) (src 26K)
Lumley D. E.
Estimation of elastic specular reflectivity from seismic reflection data is important in that it may provide crucial information about subsurface physical properties related to lithology, rock physics, pore-fluid content, and physical states. I derive a generalized least-squares solution for elastic reflectivity in terms of local seismic wavefield displacement vectors. Solutions for the local wavefields are derived as representation integrals over surface source and receiver displacement fields, weighted by a WKBJ approximate Green's tensor. The resulting integral solution is a vector analogy of the Kirchhoff-Rayleigh-Sommerfeld integrals for scalar waves. In particular, I give explicit formulas for the least-squares evaluation of the ${P}{P}$ and ${P}{S_1}$ elastic specular reflection coefficients.
The inertia tensor in elastic wave modeling (ps 37K) (src 2K)
Muir F.
The homogeneous equivalent to a fine-layered, heterogeneous elastic medium may need a frequency-dependent, anisotropic inertia tensor if it is to be dynamically equivalent in a useful sense. The elastic and inertial matrices that make up this equation would likely not commute, and there would be no Christoffel-like equation. This difficulty is resolved by a local transformation of the field variable from displacement to an energy form. This inertia tensor is shown, by analogy, to be physically reasonable.

General Theory

Wavelets and seismic processing (ps 262K) (src 769K)
Abma R. and Muir F.
Various wavelet transforms are compared and the fundamentals of wavelet transforms are examined. While wavelet transforms appear to provide much promise for signal processing, we found no apparent application other than some limited data compression possibilities. The absence of useful properties within the transformed domain, such as the derivative property in the Fourier transform, account for the present lack of applications.
Transformation of discrete arrays by Daubechies wavelets (ps 1100K) (src 1815K)
Schwab M.
Wavelet transformation, analogously to fast Fourier transformation, turns a discrete vector of length N into a vector of N coefficients, which scale the transformation's base functions. It is a linear operation which can be efficiently implemented as a recursion. Requiring the base function of the transformation to be orthogonal and approximative yields a set of constraining equations for discrete wavelets. Daubechies wavelet filter coefficients are the minimum phase solutions of this nonlinear equation set. I implemented a pyramidal algorithm which performs the transformation based on Daubechies wavelets of lengths from 4 to 20 samples. In the electronic version of this report, interactive figures allow readers to explore the transformation by submitting their own input and choosing the wavelet's length. By compressing and recovering a photograph of a face, I demonstrate the conditions for a successful wavelet compression; the matrix representation of the picture in the wavelet space is sparse and samples with small values can be neglected without affecting essential features of the uncompressed picture. Seismic data render rather densely in the wavelet domain. Finally, lacking clear criteria to define seismic data quality, a compression scheme based on data reduction does not seem to be practical. A more promising seismic application of wavelet transformation is interpolation. As with fast Fourier transformation, I utilize discrete wavelet transformation for interpolation of regularly sampled seismic traces by padding zeroes.
Wavefield extrapolation in heterogeneous media using the LITWEQ method (ps 55K) (src 12K)
Mujica D. L.
Time migration algorithms have the disadvantage of erroneously handling lateral velocity variations, even if the correct velocity is provided (Hatton et al., 1981). Additionally, algorithms based on the one-way wave equation cannot handle steep dips correctly. Depth migration seems to be the answer to time migration weaknesses. This explains why it is necessary to develop depth migration algorithms. Li (1986), presents a theoretical model that he calls Linearly Transformed Wave Equation (LITWEQ) to do migration in lateraly varying media. However, it was only implemented ...
Dip-Moveout processing: A Tutorial. DMO basics and DMO by Fourier transform (ps 411K) (src 364K)
Popovici A. M.
As a Teaching Assistant (TA) for Jon Claerbout's class I prepared several lectures on DMO. Two of them appear polished enough to be included in this report. The first part explains the DMO correction in simple, geometrical terms and justifies its place in the imaging processing sequence. The second part deals with DMO by Fourier methods.

General Geophysics

Detecting gravitational waves using seismic data (ps 1636K) (src 7213K)
Abma R.
Several years of seismic data acquired by the international deployment of accelerometers (IDA) are used in an attempt to detect gravitational waves radiated from low-frequency astronomical binary sources. These data were preconditioned with various combinations of earthquake removal, gain, and tapering. Directionally weighted spectra were calculated for 0.012 to 0.556 milli-Hz (one day to one-half hour periods) in 84 directions, one year at a time, with correction for the earth's rotation. While patterns similar to the synthetic response from a gravitational-wave source were observed, these patterns showed no consistency when compared between years. This inconsistency suggests that the patterns are noise and not the result of gravitational wave excitation by binary sources.
Analysis of ultrasonic velocities in hydrocarbon mixtures (ps 49K) (src 13K)
Berryman J. G.
Ultrasonic velocity data on hydrocarbon mixtures was shown by Wang and Nur [JASA 89, 2725 (1991)] to agree quite well with the predictions of a simple mixing rule (volume average of velocity). The same data is reanalyzed using Wood's formula and the time average approximation. Wood's formula for acoustic velocity is shown to be always less than the time average formula, which is itself always less than the simple mixing rule result. Although the volume average rule agrees best with the data on binary mixtures of 1-decene and 1-octadecene, all these formulas agree with the data to within 1% and, therefore, it is not possible to distinguish among the formulas using this data set. Similar results are also found for six multicomponent hydrocarbon mixtures.


The C++ language in physical science (ps 74K) (src 66K)
Nichols D., Dunbar G., and Claerbout J.
We have designed C++ classes that are used to solve optimization problems involving linear operators. The use of C++ enables us to write code to solve these problems at a much higher level than when writing in Fortran. The ability to design classes that abstract the concepts of operators and spaces enables us to separate the task of writing new operators from the task of writing routines to solve the optimization problem.
An implementation of operators and spaces in C++ (ps 70K) (src 152K)
Dunbar G.
This paper is a companion to the previous paper Nichols et al. (1992) which gives an overview of the problems that these C++ classes are designed to solve. It is a collection of manual pages describing my C++ work this summer, with a short extra explanation preceeding each page. It is intended as a reference document for someone who wants more details about the code that we have written. It is probably most useful to a reader who is already familiar with C++. The manual pages are in an order such that every class or function is ...
Xtpanel: an interactive panel builder (ps 350K) (src 1082K)
Cole S. and Nichols D.
Xtpanel builds interactive panels - containing objects such as buttons, sliders, and dialog boxes - from a simple script file or from the command line. With xtpanel, one can quickly add a layer of interactivity to existing software. For instance, a velocity analysis tool can be created in minutes by using xtpanel to add an interactive front end, where sliders choose velocities from a user-specified range, to an existing NMO correction program. Panels can take advantage not only of existing application programs, but also the utilities built into the UNIX operating system. In one example, we use an 80-line script to build an X windows calculator by making a front end to the line-oriented UNIX calculator bc. Another utility built with xtpanel is the xtpanel generator. This is a series of panels that let users build new panels interactively, without having to learn and use the script language.

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Stanford Exploration Project