bluuebunny/crossref_metadata_2025 huggingface dataset:
Numerical integration of related Hankel transforms of orders 0 and 1 by adaptive digital filtering
Authors: Anderson Walter L.
Published: October 2002

A linear digital filtering algorithm is presented for rapid and accurate numerical evaluation of Hankel transform integrals of orders 0 and 1 containing related complex kernel functions. The kernel for Hankel transforms is defined as the non‐Bessel function factor of the integrand. Related transforms are defined as transforms, of either order 0 or 1, whose kernel functions are related to one another by simple algebraic relationships. Previously saved kernel evaluations are used in the algorithm to obtain rapidly either order transform following an initial convolution operation. Each order filter is designed with identical abscissas over a large range so that an adaptive convolution procedure can be applied to a large class of kernels. Different order Hankel transforms with related kernels are often found in electromagnetic (EM) applications. Because of the general nature of this algorithm, the need to design new filters should not be necessary for most applications. Accuracy of the filters is comparable to that of single‐precision numerical quadrature methods, provided well‐behaved kernels and moderate values of the transform argument are used. Filtering errors of less than 0.005 percent are demonstrated numerically using known analytical Hankel transform pairs. The digital filter accuracy is also illustrated by comparison with other published filters for computing the apparent resistivity for a Schlumberger array over a horizontally layered earth model. The algorithm is written in Fortran IV and is listed in the Appendix along with a test driver program. Detailed comments are included to define sufficiently all calling parameter requirements.
On: “The gravimetric density formula for a spherical shell,” by J. R. Hearst and R. C. Carlson (GEOPHYSICS, December 1977, p. 1469).
Authors: Fahlquist D. A. , Carlson R. L.
Published: October 2002

According to Hammer (1950), the gravimetric density (ρ) of material between two stations in a bore‐hole is [Formula: see text], (1) where Δz is the vertical separation of the stations, Δg is the difference in gravity, and F is the free‐air gradient. Furthermore, the difference in gravity for stations above and below a Bouguer slab, owing to the attraction of the slab alone, is [Formula: see text], (2) where Δz and ρ are, respectively, the thickness and density of the slab.
Acoustic impedance logs computed from seismic traces
Authors: Becquey M. , Lavergne M. , Willm C.
Published: October 2002

Acoustic impedance, the product of seismic velocity and density, is a basic physical property of rocks. Seismic traces are converted into pseudoreflection‐coefficient time series by appropriate initial processing, then into acoustic impedance by the inversion of the time series. Such pseudologs are roughly equivalent to logs recorded in wells drilled at every seismic trace location. They yield important information concerning the nature of the rock and variations in lithology. To obtain the best quality pseudologs, careful initial processing is necessary: true‐amplitude recovery, appropriate deconvolution, common‐depth‐point (CDP) stack, wave‐shaping, wave‐equation migration, and amplitude scaling. The low frequencies from moveout velocity information are inserted. Both the short‐period information computed from reflection amplitudes and the long‐period trend computed from reflection moveout are displayed on acoustic impedance logs. Possible causes of pseudolog distortions are inaccuracies of amplitude recovery and scaling, imperfection of deconvolution and migration, and difficulties of calibrating the pseudolog to an acoustic log derived from well logs. Such calibration increases the precision; facies variations observed in well logs can be extrapoled to large distances from the wells, leading to a more accurate estimation of hydrocarbon reserves.
On: “Diurnal drift removal from aeromagnetic data using least squares,” by H. L. Yarger, R. R. Robertson, and R. L. Wentland (GEOPHYSICS, October 1978, p. 1148–1156).
Authors: Kellogg William C.
Published: October 2002

With the passage of time it is interesting to note how the present generation of geophysicists is absorbed with the mathematical development and substantiation of geophysical observations, whereas earlier, more simple and practical methods predominated. It was with particular interest that I read the paper in the October, 1978 issue of Geophysics regarding diurnal drift removal and the mathematical development of a least‐squares procedure to do this.
Some recent magnetic induced‐polarization developments—Part I: Theory
Authors: Howland‐Rose A. W. , Linford G. , Pitcher D. H. , Seigel H. O.
Published: October 2002

Additional mathematical development using vertical dikes and contacts has been carried out to provide a better understanding of the response patterns encountered in field measurements with the magnetic induced‐polarization (MIP®) method. It is shown by model studies that the presence of large conductivity contrasts in the earth may strongly influence the induced‐polarization (IP) response, at the earth’s surface, of a buried electrically anomalous body. For example, an overlying conducting layer will greatly reduce the electrical induced polarization (EIP) relative to the MIP response. Lateral conductivity changes likewise may reduce the relative EIP response amplitude and markedly distort the MIP response pattern.
Calculation of gravity and magnetic anomalies of finite‐length right polygonal prisms
Authors: Cady John W.
Published: October 2002

An equation is derived for the vertical gravity field due to a homogeneous body with polygonal cross‐section and finite strike‐length. The equation can be separated into the two‐dimensional (2-D) terms of Talwani et al. (1959) and exact terms for the contributions of the ends of the prism. Equations for the magnetic field due to a similar body were derived by Shuey and Pasquale (1973), who coined the term “two‐and‐a‐half dimensional” (2[Formula: see text]-D) to describe the geometry. Magnetic intensities are expressed as a vector sum, from which the common dot product formulation can be obtained by binomial expansion.
On: “Theory of microwave dielectric constant logging using the electromagnetic wave propagation method,” by R. Freedman and J. Vogiatzis (GEOPHYSICS, May, 1979, p. 969–986).
Authors: Pascal H. , Rankin D.
Published: October 2002

In the interesting paper by Freedman and Vogiatzis, the Schlumberger EPT dielectric well logging method is discussed. The authors pay particular attention to the analysis of the formation traveltime and calculate, for several realistic sources, the corrections to the values of the plane wave model used by Schlumberger.
SEG Delegation visit to People’s Republic of China
Authors:   Delegation Members
Published: October 2002

A delegation of 11 SEG members visited the People’s Republic of China (PRC), September 5–27, 1979, in exchange for a visit by an equal number of PRC geophysicists to the U.S., November 1–20, 1979. Represented were 7 U.S. petroleum and 4 geophysical service companies. Facilities visited were the Geophysical Research Institute and an instrument factory in Peking; Electronic Computer Center and Geophysical Research Institute in Cho Hsien, 60 km south of Peking; Shengli oil field on the Yellow River delta; Marine Branch of the Geological Exploration Corp. in Shanghai; Southwest Branch of the China Petroleum Corp. in Chengtu; and China Petroleum Corp. offices in Canton. Delegation members presented papers at each facility visited except at the instrument factory in Peking. Data processing computer systems (Chinese, French, and U.S.) were toured at the computer center and also at the Shengli oil field. The visit to the latter also included a tour of the oil field and a geochemical laboratory. A highlight was a visit to a seismic field party operating near Chungking. Group discussions followed presentations of papers by delegation members. Generally, these consisted of a description by PRC geophysicists of current seismic exploration efforts and associated problems peculiar to the areas being explored. Delegation members then endeavored to answer specific questions and offer potential solutions to problems encountered. A wide range of topics was involved, covering seismic data acquisition, processing, and interpretation. Of special interest were (1) determination of lithology, (2) reef exploration, (3) operations in areas of rugged topography, (4) deconvolution, (5) modeling and migration, and (6) static time corrections. The delegation found that PRC geophysicists generally are aware of latest techniques in seismic data acquisition and processing but are lacking in the effective application of these. Undoubtedly, this is due to incomplete acquisition of modern field equipment and data processing systems.
Electrical geophysics in the U.S.S.R.
Authors: Buselli G.
Published: October 2002

As part of the Australia‐U.S.S.R. science exchange program, the author spent four months in 1977 in the Soviet Union, observing EM and IP of geophysical exploration. The main fields covered in discussions were transient electromagnetics (TEM), induced polarization, downhole induction, radio wave shadowing, electrochemical exploration, and seismoelectricity. The performance of a prototype of Sirotem, the latest Australian TEM instrumentation developed by CSIRO, was compared with the prototype of the latest Soviet TEM instrumentation, the MPP-4.
Electrical, electromagnetic, and magnetotelluric methods
Authors: Ward S. H.
Published: October 2002

Application of electrical methods began with Robert W. Fox’s 1830 observation of self potentials associated with copper vein deposits in Cornwall. Conrad Schlumberger introduced the direct current equal potential line resistivity method in 1912. Harry W. Conklin received the first patents on the electromagnetic (EM) method in 1917. From these beginnings, the history of the development of the resistivity induced‐polarization (IP), magnetotelluric and EM methods are traced to the present time. It is of interest to note that application of electrical methods flourished from about 1920 to 1930, but then the methods were developed slowly until after World War II when a major burst of development activity took place. However, the interpretational basis of the methods remained poor until the last several years when application of numerical techniques using computers permitted forward and inverse solutions to electrical boundary value problems in two and three dimensions. Field hardware gradually evolved from 1960 through 1980 to permit precise broadband IP, electromagnetic, and magnetotelluric data acquisition with coherent detection and remote reference noise rejection, in‐field digital data processing and interpretation. The future of electrical methods is briefly sketched, with trends largely toward better numerical interpretation schemes, more broadband applications, more sophisticated use of in‐field digital acquisition processing and interpretation schemes, and accommodation of more channels of information.
The effect of source field polarization on estimates of the magnetotelluric impedance tensor
Authors: Lienert Barry R.
Published: October 2002

A technique is described which can be used to estimate magnetotelluric impedances using magnetic source fields having predetermined polarization characteristics. The technique involves digital filtering of the time‐series data, followed by estimates of the amplitudes and phases of single cycles, utilizing a simple “maximum/minimum” approach. The resulting impedance estimates are at least as consistent as estimates obtained using the more usual fast Fourier transform and bandwidth averaging technique. When magnetic source field amplitudes are subdivided into groups on the basis of their polarization directions, diagonal elements of the impedance tensors determined from individual groups show an upward bias. We interpret this as due to instability in solutions for the impedances which occurs when magnetic source fields are highly polarized. In addition, one of the tensor elements shows a consistent upward bias when the magnetic field is polarized predominantly in the east‐west direction. Since we observe this latter effect at three separate sites, we suggest that it may be due to either a noise source which is polarized in the east‐west direction, or to the assumption of vertical incidence not holding for horizontal magnetic source fields which are polarized in the east‐west sense.
Migration before stack—Procedure and significance
Authors: Jain Sudhir , Wren A. Easton
Published: October 2002

Common‐depth‐point (CDP) stacking is based on the assumption that reflection points are coincident and situated midway between the respective source and receiver locations. If the reflector is structurally deformed, the reflection points move updip from the midpoint. As the structural dip increases, the reflection points for a CDP group of traces are farther removed from each other and normal stacking procedures [i.e., reflection apparent velocities for horizontal reflectors used for normal moveout (NMO) corrections] become increasingly inaccurate. Under such circumstances prestack migration is desirable, particularly when high frequencies are to be preserved. One published approach to prestack migration (Sattlegger and Stiller, 1973) involves the generation of substacks of adjacent traces followed by migration and summation of individual substacks. While adequate in many instances, cases exist where even substacks are degraded by the reflection time differential between component traces. This paper discusses an alternative technique to prestack migration without recourse to substacks. Common‐source traces, after gain recovery and static time corrections but before NMO corrections, are migrated using Kirchhoff summation. Aperture is computed for each sample according to specified maximum dips. Traces are simultaneously migrated and stacked, then output sequentially in sets of 12. The method is economical and provides enhanced reflection continuity and reliability in comparison to poststack migration. Moreover, the collapsing of diffractions is more effective.
Absorbing boundary conditions for wave‐equation migration
Authors: Clayton Robert W. , Engquist Björn
Published: October 2002

The standard boundary conditions used at the sides of a seismic section in wave‐equation migration generate artificial reflections. These reflections from the edges of the computational grid appear as artifacts in the final section. Padding the section with zero traces on either side adds to the cost of migration and simply delays the inevitable reflections. We develop stable absorbing boundary conditions that annihilate almost all of the artificial reflections. This is demonstrated analytically and with synthetic examples. The absorbing boundary conditions presented can be used with any of the different types of finite‐difference wave‐equation migration, at essentially no extra cost.
Magnitude of anomalies in the vertical gradient of gravity
Authors: Hammer Sigmund
Published: October 2002

The maximum gradient which can be caused by a simple mass is that of a spherical body. The equation for the vertical gradient at the point P above the center of a spherical mass of density contrast σ (see insert on Figure 1) can be written in the form [Formula: see text] where G is the universal gravity constant [Formula: see text] Expressing the gradient in the Eötvös units, we have [Formula: see text] In terms of percentage of the earth’s normal vertical gradient, the anomaly is [Formula: see text] of 3086 E°. At the surface of the sphere (h = 0), we have the maximum value [Formula: see text] of 3086 E° which is independent of the radius R.
Wave field extrapolation techniques in seismic migration, a tutorial
Authors: Berkhout A. J.
Published: October 2002

The objective of this paper is to provide a general view on methods of wave field extrapolation as used in seismic modeling and seismic migration, i.e., the Kirchhoff‐summation approach, the plane‐wave method (k–f method), and the finite‐difference technique. Particular emphasis is given to the relationship between the different methods. By formulating the problem in the space‐frequency domain (x, y, ω‐domain), a systems approach can be adopted which results in simple and concise expressions. These expressions clearly show that forward extrapolation is described by a spatial convolution procedure and inverse extrapolation is described by a spatial deconvolution procedure. In the situation of lateral velocity variations, the (de)convolution procedure becomes space‐variant. The space‐frequency domain is most suitable for recursive depth migration. In addition, frequency dependent properties such as absorption, dispersion, and spatial bandwidth can be handled easily. It is shown that all extrapolation methods are based on two equations: Taylor series and wave equation. In the Kirchhoff‐summation approach all terms of the Taylor series are summed to an exact analytical expression—the Kirchhoff‐integral for plane surfaces. It formulates the extrapolation procedure in terms of a spatial convolution integral which must be discretized in practical applications. The Fourier‐transformed version of the Kirchhoff‐integral is used in the plane wave method (k–f method). This actually means that spatial (de)convolution in the x, y, ω‐domain is translated into multiplication in the [Formula: see text], [Formula: see text], ω‐domain. Of course, this is not allowed if the extrapolation operators are space‐variant. In explicit finite‐difference techniques a truncated version of the Taylor series is used with some optimum adjustments of the coefficients. For only one or two terms in the Taylor series, a spatial low‐pass filter must be applied to compensate for the amplitude errors at high tilt angles. Explicit methods are simple and most suitable for three‐dimensional (3-D) applications. In implicit finite‐difference schemes the wave field extrapolator is written in terms of an explicit forward extrapolator and an explicit inverse extrapolator. Properly designed implicit schemes do not show amplitude errors and, therefore, amplitude correction filters need not be applied. In comparison with explicit schemes, implicit schemes are more sensitive to improper boundary conditions at both ends of the data file. It is shown that the forward seismic model can be elegantly described by a matrix equation, using separate operators for downward and upward traveling waves. Using this model, inverse extrapolation involves one matrix inversion procedure to compensate for the downward propagation effects and one matrix inversion procedure to conpensate for the upward propagation effects.
Direct mapping of seismic data to the domain of intercept time and ray parameter—A plane‐wave decomposition
Authors: Stoffa Paul L. , Buhl Peter , Diebold John B. , Wenzel Friedemann
Published: October 2002

Marine seismic data recorded as a function of source‐receiver offset and traveltime are mapped directly to the domain of intercept or vertical delay time and horizontal ray parameter. This is a plane‐wave decomposition based on beam forming of wide‐aperture seismic array data to determine automatically the loci of coherent seismic reflection and refraction events. In this computation, semblance, in addition to the required slowness or horizontal ray parameter stack, is found for linear X — T trajectories across subarrays. Subsequently, semblance is used to derive a windowing filter that is applied to the slowness stack to determine the points of stationary phase and eliminate aliasing. The resulting filtered slowness stacks for multiple subarrays can then be linearly transformed and combined according to ray parameter, range, and time. The resulting function of intercept time and horizontal ray parameter offers significant computational and interpretational advantages for the case of horizontal homogeneous layers and leads directly to the derivation of a detailed velocity‐depth function.
On: “Gravity interpretation with the aid of quadratic programming,” by N. J. Fisher and L. E. Howard (GEOPHYSICS, v. 45, March 1980, p. 403–419).
Authors: Anfiloff W.
Published: October 2002

The inversion and forward modeling approaches to interpretation are separated by a vast philosophical gulf, and mathematicians and geologists standing on opposite sides of this gulf find it difficult to see the other side. The use by Fisher and Howard of the gravity profile across the Donnybrook structure to demonstrate an inversion method provides an opportunity to narrow the gulf at the expense of the inversion school.
A self‐similar model for sedimentary rocks with application to the dielectric constant of fused glass beads
Authors: Sen P. N. , Scala C. , Cohen M. H.
Published: October 2002

We develop a theory for dielectric response of water‐saturated rocks based on a realistic model of the pore space. The absence of a percolation threshold manifest in Archie’s law, porecasts, electron‐micrographs, and general theories of formation of detrital sedimentary rocks indicates that the pore spaces within such rocks remain interconnected to very low values of the porosity ϕ. In the simplest geometric model for which the conducting paths remain interconnected, each grain is envisioned to be coated with water. The dielectric constant of the assembly of water‐coated grains is obtained by a self‐consistent effective medium theory. In the dc limit, this gives Maxwell’s relation for conductivity σ of the rock [Formula: see text], where [Formula: see text] is the conductivity of water. In order to include the local environmental effects around a grain, a self‐similar model is generated by envisioning that each rock grain itself is coated with a skin made of other coated spheres; the coating at each level consists of other coated spheres. The self‐consistent complex dielectric constant [Formula: see text] is given in this model in terms of that of water [Formula: see text] and of rock [Formula: see text], by [Formula: see text] for spherical particles. This gives, in the dc limit, [Formula: see text]. For nonspherical particles, the exponent m in Archie’s law [Formula: see text] is greater than 3/2 for the plate‐like grains or cylinders with axis perpendicular to the external field and smaller than 3/2 for plates or cylindrical particles with axis parallel to the external field. Artificial rocks with a wide range of porosities were made from glass beads. We present data on the glass bead rocks for dc conductivity and the dielectric constant at 1.1 GHz. The data follow the conductivity and the dielectric responses given by the self‐similar model. The present theory fails to explain the salinity dependence of [Formula: see text] at lower frequencies.
Three‐dimensional velocity profile inversion from finite‐offset scattering data
Authors: Raz S.
Published: October 2002

The reconstruction of three‐dimensional (3-D) velocity variations from finite‐offset scattering data is formulated. Reduction to the limiting cases of zero and small offset distances as well as the case of one‐dimensional (1-D) stratification is given. An inherent increase in complexity is cited and interpreted. The relationship of the proposed inversion to the F-K migration is discussed.
Inversion of refraction data by wave field continuation
Authors: Clayton Robert W. , McMechan George A.
Published: October 2002

The process of wave equation continuation (migration) is adapted for refraction data in order to produce velocity‐depth models directly from the recorded data. The procedure consists of two linear transformations: a slant stack of the data produces a wave field in the p — τ plane which is then downward continued using τ = 0 as the imaging condition. The result is that the data wave field is linearly transformed from the time‐distance domain into the slowness‐depth domain, where the velocity profile can be picked directly. No traveltime picking is involved, and all the data are present throughout the inversion. The method is iterative because it is necessary to specify a velocity function for the continuation. The solution produced by a given iteration is used as the continuation velocity function for the next step. Convergence is determined when the output wave field images the same velocity‐depth function as was input to the continuation. The method obviates the problems associated with determining the envelope of solutions that are consistent with the observations, since the time resolution in the data is transformed into a depth resolution in the slowness‐depth domain. The method is illustrated with several synthetic examples, and with a refraction line recorded in the Imperial Valley, California.
Elastic wave propagation in a fluid‐filled borehole and synthetic acoustic logs
Authors: Cheng Chuen Hon , Toksöz M. Nafi
Published: October 2002

The propagation and dispersion characteristics of guided waves in a fluid‐filled borehole are studied using dispersion curves and modeling full‐wave acoustic logs by synthetic microseismograms. The dispersion characteristics of the pseudo‐Rayleigh (reflected) and Stoneley waves in a borehole with and without a tool in the center are compared. Effects of different tool properties are calculated. The effect of a rigid tool is to make the effective borehole radius smaller. As an approximation, dispersion characteristics of the guided waves in a borehole with a tool can be calculated as a purely fluid‐filled borehole with a smaller effective radius. Theoretical waveforms (microseismograms) of elastic waves propagating in a borehole are calculated using a discrete wavenumber integration. With an appropriate choice of parameters, our results look similar to the acoustic waveforms recorded in a limestone and a shale formation. Several factors affect the shape of an acoustic log microseismogram. The effective radius of the borehole determines the relative amplitudes of the modes generated. Poisson’s ratio of the formation is the primary factor determining the relative amplitude of the leaky mode following the compressional arrival. Attenuation affects the duration and decay rate of the guided waves.
Inversion of reflection times in three dimensions
Authors: Gjøystdal Håvar , Ursin Bjørn
Published: October 2002

When reflection data are available from a grid of crossing seismic lines, it is possible to construct normal incidence time maps from interpreted stacked sections and then apply three‐dimensional (3-D) ray‐tracing techniques following the normal‐incidence raypaths down to the various reflectors. The main disadvantage of this well‐known “time map migration” procedure is that interval velocities must be known a priori, and they must be estimated in advance by some approximate method. A technique is presented here which combines the above procedure with an inversion algorithm, providing direct calculations of interval velocities from the additional use of nonzero offset traveltime observations. A generalized linear inversion scheme is used, making possible a complete calculation of interval velocities and reflection interfaces, the latter represented by bicubic spline functions. To test the method in practice, we have applied it to (1) synthetic data generated from a constructed model, and (2) real data obtained from marine seismic sections. In the latter case, velocities and reflector depths obtained were compared to those obtained directly from a well log in the area. These results show a reasonably good resolution for layers that are not too deep relative to the shot/receiver offsets used. For deep and/or thin layers, the results are not satisfactory. This indicates the general limitation of seismic reflection data to resolve interval velocity, even in the presence of horizontally layered structure.
The telluric‐magnetotelluric method in two‐ and three‐dimensional environments
Authors: Stodt John A. , Hohmann Gerald W. , Ting Sam C.
Published: October 2002

The assumption of spatial uniformity of the horizontal magnetic field, which is an implicit assumption made in straightforward applications of the telluric‐magnetotelluric (T-MT) method, is not always valid near conductivity inhomogeneities. For a two‐dimensional (2-D) case, the transverse electric mode horizontal magnetic field may vary more than a factor of three. The spatial variation of the horizontal magnetic field is not as great over three‐dimensional (3-D) inhomogeneities, but it may still contribute significantly to impedance magnitude and phase over shallow inhomogeneities at higher frequencies. Spatial variation of the horizontal magnetic field can cause T-MT impedances to differ significantly from magnetotelluric (MT) impedances. Consequently, MT modeling of T-MT field data could result in a misleading interpretation of conductivity structure. To avoid erroneous interpretation, numerical modeling programs should calculate actual T-MT responses, rather than MT responses, for comparison with T-MT data. Resolution of a conductivity structure which produces significant spatial variation of the magnetic field is less with T-MT data than with an equal number of MT data. However, the increased number of data obtained from a T-MT survey may offset this shortcoming if the data are properly interpreted.
Reconstruction of a sparse spike train from a portion of its spectrum and application to high‐resolution deconvolution
Authors: Levy Shlomo , Fullagar Peter K.
Published: October 2002

An algorithm is proposed for the reconstruction of a sparse spike train from an incomplete set of its Fourier components. It is shown that as little as 20–25 percent of the Fourier spectrum is sufficient in practice for a high‐quality reconstruction. The method employs linear programming to minimize the [Formula: see text]‐norm of the output, because minimization of this norm favors solutions with isolated spikes. Given a wavelet, this technique can be used to perform deconvolution of noisy seismograms when the desired output is a sparse spike series. Relative reliability of the data is assessed in the frequency domain, and only the reliable spectral data are included in the calculation of the spike series. Equations for the unknown spike amplitudes are solved to an accuracy compatible with the uncertainties in the reliable data. In examples with 10 percent random noise, the output is superior to that obtained using conventional least‐squares techniques.
EULDPH: A new technique for making computer‐assisted depth estimates from magnetic data
Authors: Thompson D. T.
Published: October 2002

A method for rapidly making depth estimates from large amounts of magnetic data is described. The technique is based upon Euler’s homogeneity relationship (hence, the acronym EULDPH) and differs from similar techniques which are currently available in that no basic geologic model is assumed. Therefore, EULDPH can be applied in a wider variety of geologic situations than can model‐dependent techniques. The price paid for this increased flexibility is a heavier burden on the interpreter. Successful interpretation of EULDPH results is partially dependent upon the interpreter’s intuitive understanding of the concept of the equivalent stratum and also partially dependent upon experience with model studies. The theoretical basis, the computational algorithm, and applications of EULDPH to model and real data are presented.
A procedure for multidimensional inversion of seismic data
Authors: Raz S.
Published: October 2002

A new space‐time approach to inverting multidimensional seismic data is formulated. This work constitutes a natural, but assuredly nontrivial, extension of a previously reported one‐dimensional (1-D) Bremmer inversion procedure. In its most general format, the scheme applies to the three‐dimensional (3-D) inversion problem; however, appropriately reduced forms applicable to two‐dimensional (2-D) as well as 1-D configurations are also presented. The basic scheme and all its variants utilize exclusively zero‐offset data. Although interpretable in terms of a distorted wave Born model, the inversion procedure and subsequent algorithms differ substantially from their Born‐based predecessors. Here, the background is not only inhomogeneous but is not known a priori. The choice of a uniquely defined “phase‐corrected” background is cardinal to the proposed reconstruction scheme. The improvement in the inversion accuracy is considerable. Comparatively large discontinuities can be handled, and the phenomenon of error accumulation with depth is overcome.
A fast three‐dimensional modeling technique and fundamentals of three‐dimensional frequency‐domain migration
Authors: Herman A. J. , Anania R. M. , Chun J. H. , Jacewitz C. A. , Pepper R. E. F.
Published: October 2002

The frequency‐domain approach to migration extends to three dimensions. Application of the ray‐tracing approach to simple seismic events, together with the principle of superposition for complex events, leads to the correct equation for frequency‐domain migration in a constant velocity medium and a further understanding of the basic migration procedure in three dimensions. In the frequency domain, three‐dimensional (3-D) migration may be reduced to a series of two‐dimensional (2-D) migrations. A 3-D seismic model can be generated economically for structures with combinations of radial and 2-D symmetry. Starting from a 2-D synthetic, a 3-D model can be constructed by rotation about the axis of symmetry. Any line across the model may then be synthesized from the basic 2-D data. Certain more complex 3-D models can also be developed using the superposition of simpler models. Synthetic examples are used to illustrate the fact that 2-D migration of 3-D seismic data will not generally result in a correct section.
A method for direct estimation of interval velocities
Authors: Schultz Philip S.
Published: October 2002

The most commonly used method for obtaining interval velocities from seismic data requires a prior estimate of the root‐mean‐square (rms) velocity function. A reduction to interval velocity uses the Dix equation, where the interval velocity in a layer emerges as a sensitive function of the rms velocity picks above and below the layer. Approximations implicit in this method are quite appropriate for deep data, and they do not contribute significantly to errors in the interval velocity estimate. However, when the data are from a shallow depth (vertical two‐way traveltime being less than direct arrival to the farthest geophone), the assumption within the rms approximation that propagation angles are small requires that much of the reflection energy be muted, along with, of course, all the refraction energy. By means of a simple data transformation to the ray parameter domain via the slanted plane‐wave stack, three types of arrivals from any given interface (subcritical and supercritical reflections and critical refractions) become organized into a single elliptical trajectory. Such a trajectory replaces the composite hyperbolic and linear moveouts in the offset domain (for reflections and critical refractions, respectively). In a layered medium, the trajectory of all but the first event becomes distorted from a true ellipse into a pseudo‐ellipse. However, by a computationally simple layer stripping operation involving p‐dependent time shifts, the interval velocity in each layer can be estimated in turn and its distorting effect removed from underlying layers, permitting a direct estimation of interval velocities for all layers. Enhanced resolution and estimation accuracy are achieved because previously neglected wide‐angle arrivals, which do not conform to the rms approximation, make a substantial contribution in the estimation procedure.
Deconvolution by successive approximations
Authors: LaCoste Lucien J. B.
Published: October 2002

Although various methods of deconvolution have been known for many years, they are not generally regarded as being routinely usable. The successive approximation method described in this article should be an improvement in that respect. It operates in the time domain and is based on a simple procedure. I first discuss the mathematics involved and then give some examples to illustrate how the method works and some of the things it can do.
A fast algorithm for normal incidence seismograms
Authors: Choate W. Clay
Published: October 2002

Computation of a normal incidence synthetic seismogram for a system of vertically stacked homogeneous plane layers can be approached by methods familiar in communication theory (Robinson and Treitel, 1966). In this approach, waves in adjacent layers are related by a [Formula: see text] matrix. Waves in nonadjacent layers can be related by applying these elementary matrices recursively—a computational task equivalent to computing the ordered product of the elementary matrices for the intervening layers. In the absence of total internal reflection, the relation of waves in the first and last layers obtained in this way characterizes the propagation of seismic energy in the system. The characterization is complete in the Fourier transform domain when matrix elements are evaluated at N/2 frequencies, where N is the number of interfaces. (If the elements were represented individually, [Formula: see text] frequencies would be required. N/2 is a prorated value based on a more efficient representation.) Analysis of the computational requirements of the standard algorithm (described in the preceding) indicates computer run time scales as [Formula: see text] for N large. This paper describes a new algorithm for which the scaling is [Formula: see text] and, therefore, is faster for large systems. Examples of seismograms computed with the new algorithm are given. Comparative timings on a VAX 11/780 computer indicate the new algorithm has about a one order of magnitude speed advantage when N = 512.
The effect of grain anisotropy on the electrical properties of sedimentary rocks
Authors: Mendelson Kenneth S. , Cohen Morrel H.
Published: October 2002

We have examined the effect of particulate anisotropy on the electrical properties of sedimentary rocks by generalizing the treatment of Sen et al (1981) to the case of ellipsoidal grains with a distribution of orientations and depolarizing factors. Two distributions in orientation have been treated in detail—randomly oriented grains in three dimensions and grains with aligned principal axes in two dimensions. In the former case the conductivity is a scalar satisfying Archie’s law, [Formula: see text], with [Formula: see text] the conductivity of the pore fluid and ϕ the porosity. The exponent m has a minimum of 1.5 for spherical grains. The presence of highly oblate (disk shaped) grains raises m significantly. As long as grains with extremely large eccentricities (≳15) are not present, the exponent falls in the observed range [Formula: see text]. For aligned grains the conductivity is a tensor with principal values that satisfy a generalized Archie’s law of the form [Formula: see text], where [Formula: see text] is the jth principal value of the conductivity and [Formula: see text] can be expanded as a power series in ϕ with a constant leading term. For grain eccentricities in the range 0–0.95, the coefficients [Formula: see text] fall in the range 0.1–4. The exponent m has a minimum value of 2 for two dimensions, independent of grain shape, if all grains have the same shape, and it is larger for any distribution of grain shapes. If the distribution of grain shapes is chosen so that the rock is isotropic, m and a have the same values as for isotropic rock composed of grains with the same distribution of shapes but with random orientations. Since different distributions of grain orientation can lead to the same effective conductivity, it is clear that measurements of dc conductivity are not sufficient to determine the grain distribution. The model is also used to obtain the complex dielectric constant. If the dielectric constant of solid rock is small compared to the real part of the dielectric constant of water, the complex dielectric constant has the same dependence on porosity as the dc conductivity except at very small porosities.
Instantaneous velocities and reflection times for transversely isotropic solids
Authors: Radovich Barbara J. , Levin Franklyn K.
Published: October 2002

Whereas [Formula: see text] plots are straight lines for a reflection from a horizontal interface beneath an isotropic solid, the plots are curved for P‐waves and SV‐waves when the solid is transversely isotropic. Velocity and intercept time are normally found by fitting a least‐squares straight line to the [Formula: see text] data. However, we can define an instantaneous velocity and an instantaneous intercept time by fitting a tangent line to the [Formula: see text] plot at each X. We obtain expressions for instantaneous velocity and intercept time for a transversely isotropic solid in terms of the plane‐wave velocity v and plane‐wave angle from the vertical θ. We relate the instantaneous quantities to measurable quantities for waves from a point source. A method is suggested for finding the minimum source‐to‐geophone separation at which the velocity becomes essentially the velocity for travel in the horizontal direction.
Interpretation of some two‐dimensional magnetic bodies using Hilbert transforms
Authors: Mohan N. L. , Sundararajan N. , Seshagiri Rao S. V.
Published: October 2002

Procedures are formulated using the Hilbert transform for interpreting vertical magnetic anomalies of (1) the sheets (finite and infinite depth extent), (2) the dike, and (3) the horizontal circular cylinder. The applicability of the method is tested on theoretical models. The method is also applied on the well‐known Kursk field anomaly of a sheet (infinite‐depth extent) and the field anomaly of a dike of Karimnagar, Andhra Pradesh, India.
Telluric fields and their gradients over a step fault
Authors: Roy K. K. , Rathi O. P. , Rao K. P.
Published: October 2002

Telluric fields and their gradients are computed theoretically over a step fault at right angles to the strike of the fault planes. A closed form solution of the boundary value problem is obtained in terms of elliptic integrals of the first and third kind by applying the Schwarz‐Christoffel method of conformal transformation. Basement is assumed to be of infinite resistivity overlain by sediments of finite thickness and resistivity. Basement faults are separated by a distance S and the throws of the vertical faults are [Formula: see text] and [Formula: see text] (defined in the text). Infinitely long line source and sink are assumed at infinite distances for simulation of the telluric field. From computed data, the gradient of the telluric field appears to be a better parameter for detection and resolution of fault planes. The resolution vanishes when the depth of the structure ([Formula: see text]) is nearly equal to the separation between the fault planes in case the throw of the fault ([Formula: see text]) is only 10 percent of the total thickness of the overburden (H). For higher throws, the resolution vanishes even earlier. Locations of the epicenters of the faults in relation to the gradient peaks on the surface are dependent upon all the geometric parameters of the structure. For well‐resolved structures, quantitative interpretation for geometric parameters will be possible from the telluric profiles and their gradients, provided the total thickness of the overburden can be obtained from the magnetotelluric sounding data. Computations were done with a Burroughs‐6700.
Effects of three‐dimensional structure on magnetotelluric sounding curves
Authors: Park Stephen K. , Orange Arnold S. , Madden Theodore R.
Published: October 2002

Significant errors may result when applying one‐dimensional (1-D) or two‐dimensional (2-D) interpretation methods to magnetotelluric (MT) data collected in three‐dimensional (3-D) environments. Both depths and resistivities can be grossly incorrect if 1-D or 2-D methods are applied in a 3-D setting. We present examples of MT sounding curves generated using a 3-D modeling program (Madden and Park, 1982) and illustrate some interpretation pitfalls if 3-D effects are not considered. The 3-D effects discussed herein are attributed to a surface layer heterogeneity, and can be readily identified in MT data from a well‐designed MT survey. The MT survey must include stations chosen to yield information about regional structure. Alternatively, carefull examination of a geologic map will help the intepreter estimate the regional effects present in the data.
On: “Gravity intepretation with the aid of quadratic programming”, by N. J. Fisher and L. E. Howard (GEOPHYSICS, March 1980, p. 403–419)
Authors: Steiner F. , Zilahy‐Sebess L.
Published: October 2002

Fisher and Howard (1980) present a new procedure where the appropriate density values corresponding to the individual prisms of their model are sought via the quadratic programming method, so that the cumulative effect of these prisms should be as close as possible in the least mean square sense to the measured values. Since the two‐dimensional system of prisms consists of q (>1) rows, the authors have to use prisms of much greater horizontal size than station spacing in order to have a sufficient number of equations for the determination of the unknown densities.
Reverse time migration
Authors: Baysal Edip , Kosloff Dan D. , Sherwood John W. C.
Published: October 2002

Migration of stacked or zero‐offset sections is based on deriving the wave amplitude in space from wave field observations at the surface. Conventionally this calculation has been carried out through a depth extrapolation. We examine the alternative of carrying out the migration through a reverse time extrapolation. This approach may offer improvements over existing migration methods, especially in cases of steeply dipping structures with strong velocity contrasts. This migration method is tested using appropriate synthetic data sets.
Digital filtering with the second moment norm
Authors: Lines L. R. , Treitel S.
Published: October 2002

The Backus‐Gilbert inverse method relates model estimates to actual earth models by use of a resolving kernel. This inversion can in turn be related to various digital filter designs. If the second moment norm is used to define the resolving kernel, two types of filters are produced by an eigenvector decomposition of a time‐weighted autocorrelation matrix. The eigenvector corresponding to the largest eigenvalue of this matrix is similar to the output energy filter, while the eigenvector for the smallest eigenvalue performs more like a deconvolution filter. Synthetic and real data examples demonstrate the characteristics of these filters and compare them to the familiar square norm filters. Our experiments suggest that second moment norm filters offer no significant advantages over their Euclidean norm relatives.
Seismic characteristics of a Precambrian pluton and its adjacent rocks
Authors: Hajnal Z. , Stauffer M. R. , King M. S. , Wallis P. F. , Wang H. F. , Jones L. E. A.
Published: October 2002

Surface, borehole, and laboratory acoustic measurements all confirm the existence of a near‐surface low‐velocity zone in metavolcanic, metasedimentary, and plutonic rocks of the Flin Flon region of Canada. This zone is caused by a high frequency of open fractures and extends from the surface to depths of between 5 and 44 m, although occasional open fractures extend to at least 60 m. There is a linear decrease in sonic velocity with increasing frequency of large fractures; the details, however, vary for different sites, depending upon several geologic features including rock type and nonfracture porosity. Laboratory sonic data indicate very low microcrack densities in the volcanic and plutonic rocks. Synthetic seismograms derived from sonic log information from the center of the granitic pluton have been compared with a nearby multifold seismic profile. This shows that the near‐surface low‐velocity zone attenuates most of the high‐frequency seismic energy. However, the remaining low‐frequency portion of the seismic spectrum can be used to map some features of the pluton.