bluuebunny/crossref_metadata_2025 huggingface dataset:
Variation of the electrical resistivity of large rock samples with stress
Authors: Morat P. , Le Mouel J. L.
Published: October 2002

Many experimental and theoretical efforts have been devoted to the study of the electrical properties of rocks (e.g., Brace et al., 1965; Madden, 1982; Parkhomenko, 1982; Yamazaki, 1966). Special attention has been paid to possible relationships between variations of mechanical stresses inside the rock and variations of its electrical properties, in reference to a possible technique of predicting earthquakes (Raleigh et al., 1977). These studies were generally performed on small (decimeter sized) samples, but the complexity of the physical processes involved and the heterogeneity of rocks in the field make it necessary to try to get results from larger blocks, with dimensions of several decimeters (Sobolev et al., 1983) or of meters (this study). It also appears quite desirable to look for information on the electrical properties of rocks at different scales (from a centimeter to a kilometer, say) to see if some scaling laws could be experimentally established. These scaling laws would allow inference of electrical properties at a given scale from the knowledge of the electrical properties at a smaller scale (Madden, 1976). Scaling‐law methods seem to be promising in seismogenesis studies (e.g., Allegre et al., 1984; Turcotte, 1986).
On: “Determination of crustal interface topography from potential fields” by M. Pilkington and D. J. Crossley (GEOPHYSICS, 51, 1277–1284, June 1986) and “Inversion of aeromagnetic data for multilayered models” by M. Pilkington and D. J. Crossley (GEOPHYSICS, 51, 2250–2254, December 1986).
Authors: Steenland Nelson C.
Published: October 2002

The introduction of the airborne magnetometer in the 1940s led to the largest flow of data in geophysical history. Vacquier (1951) recognized the need for a concomitant system of interpretation of these data and pioneered the utilization of a rejected category of anomalies, the intrabasement contrasts in magnetization, for the basis of the desired system. These anomalies had great acclaim in oil and gas exploration in the late 1920s, but they led to the total disrepute of magnetics when facsimiles to the coincidence of production and intrabasement magnetization at Hobbs, New Mexico, discovered in 1928, did not achieve the same relationship. But Vacquier recognized the omnipresence of these anomalies, the singularity of individual anomalies, and their source in thick bodies of relatively steep sides and induced magnetizations. These simple but powerful deductions were reached pragmatically. To repeat, anomalies were singular because they were quite separate from one another. Therefore, the sides of their causative bodies had to be quite steep. An anomaly must be attributed to one magnetization contrast because it was one anomaly, and that magnetization was induced because the ratios of their positive and negative components correlated with the inclination of the Earth’s main magnetic field. (All of this may be found in GSA Memoir 47, 1951.)
On: “Nomogram for the direct interpretation of magnetic anomalies due to long horizontal cylinders,” by T. K. S. Prakasa Rao, M. Subrahmanyam, and A. Srikrishna Murty, (GEOPHYSICS, 51, 2156–2159, November 1986.)
Authors: Green Ronald
Published: October 2002

In the article by T. K. S. Prakasa Rao, M. Subrahmanyam, and A. Srikrishna Murty, the trivial problem of interpreting the magnetic anomaly over a horizontal cylinder was examined and a set of nomograms to assist with the interpretation was presented. Prakasa Rao et al. begin their discussion with equation (1) from Gay (1965).
On: “An investigation into fundamental relationships between attenuation, phase dispersion, and frequency using seismic refraction profiles over sedimentary structures,” by R.S. Jacobson (GEOPHYSICS, 52, 72–87, January, 1987).
Authors: Schoenberger M. , Levin F.K.
Published: October 2002

Figure 2 of Dr. Jacobson’s paper is a redrafting of Figure 7 of Schoenberger and Levin’s 1978 paper.
Aberration‐free image for SH reflection in transversely isotropic media
Authors: Blair J. M. , Korringa J.
Published: October 2002

This note is intended formulate and prove a theorem about shear (S) waves in a transversely isotropic medium for which we have found no reference in the literature. The theorem states the following: SH waves emanating from a point source in a homogeneous transversely isotropic medium are reflected from a planar interface between the transversely isotropic medium and another homogeneous medium in such a way that they define a reflective image that is free of aberrations, regardless of the relative orientation of the elastic symmetry axis and the interface. It is an image for rays in the direction of the group velocity vectors, not the slowness vectors. The image is located on a line through the source point in the direction of the group velocity of a wave for which the slowness vector is perpendicular to the interface. The distance, measured along this line, of the image behind the interface is equal to that of the source point in front. An analogous theorem for slowness vectors exists only for isotropic media, where it is trivial and coincides with the above.
An effective anisotropy parameter in transversely isotropic media
Authors: Banik N. C.
Published: October 2002

An interesting physical meaning is presented for the anisotropy parameter δ, previously introduced by Thomsen to describe weak anisotropy in transversely isotropic media. Roughly, δ is the difference between the P-wave and SV-wave anisotropies of the medium. The observed systematic depth errors in the North Sea are reexamined in view of the new interpretation of the moveout velocity through δ. The changes in δ at an interface adequately describe the effects of transverse isotropy on the P-wave reflection amplitude, The reflection coefficient expression is linearized in terms of changes in elastic parameters. The linearized expression clearly shows that it is the variation of δ at the interface that gives the anisotropic effects at small incidence angles. Thus, δ effectively describes both the moveout velocity and the reflection amplitude variation, two very important pieces of information in reflection seismic prospecting, in the presence of transverse isotropy.
Boundary conditions for a fluid‐saturated porous solid
Authors: Lovera Oscar M.
Published: October 2002

Using Biot’s theory, a set of boundary conditions is presented for wave transmission and reflection at the contact surface between an elastic medium (fluid or solid) and a fluid‐saturated porous solid (Biot medium). The analysis shows the continuity of the normal component of the density energy flux vectors across the contact surfaces, so that total energy is preserved. Energy reflection and transmission coefficients are computed for each kind of Biot wave.
Computing the gravitational and magnetic anomalies due to a polygon: Algorithms and Fortran subroutines
Authors: Won I. J. , Bevis Michael
Published: October 2002

We present two algorithms for computing the gravitational and magnetic anomalies due to an n‐sided polygon in a two‐dimensional space. Both algorithms have been implemented as subroutines coded in Fortran-77, and listings are provided. Because references to trigonometric functions have been almost completely eliminated, these codes run substantially faster than mosts codes now in existence. Furthermore, anomalies can be computed at any point outside, on, or inside the polygon. Unlike other codes, these algorithms can be used to model subsurface observations.
On: “Interpretation of some two‐dimensional magnetic bodies using Hilbert transforms, by N. L. Mohan, N. Sundararajan, and S. V. Seshagiri (GEOPHYSICS, 47, 376–387, March, 1982)
Authors: Nagendra R. , Ram Babu H. V.
Published: October 2002

We read Mohan et al.’s paper on interpretation of magnetic anomalies using Hilbert transforms and the discussion raised on it by Pauls (1985) and the authors’ reply (Mohan et al., 1985). While critically going through the authors’ reply (Mohan et al., 1985), we noticed a serious error in the subject paper related to the incompatibility of its equations (1) and (2).
Full transmitter waveform transient electromagnetic modeling and inversion for soundings over coal measures
Authors: Asten Michael W.
Published: October 2002

Modeling of transient electromagnetic soundings requires consideration of the full bipolar transmitter waveform in order to obtain full measurement accuracy from field instruments. The full‐waveform transient may be computed using existing impulse‐response forward algorithms followed by a convolution procedure with truncation and residual estimation after an even number of terms, which results in average increases in run time of about 80 percent. Simple calculations based on approximations for the behavior of transient decay over (a) a conductive layer over a resistive basement, (b) a homogeneous half‐space, and (c) a resistive layer over a conductive basement show that the effect of the full waveform is greatest at sample times near the end of the transmitter off‐time, and for the case of a conductive basement. Failure to consider the full waveform is expected to yield errors in the amplitude of the computed transient in the range of 4 to 100 percent or more, depending upon the sample time and type of earth model. Examples of soundings over moderately conductive coal measures and highly conductive cindered coal show that failure to use the full waveform yields depths and resistivities biased to erroneously high values. The bias is small in the former case, but produces substantial errors in the latter case. Depth to the electrical basement of 300 to 400 m may be estimated to accuracies of approximately 5 percent by computer inversion to three‐layer or four‐layer models.
Occam’s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data
Authors: Constable Steven C. , Parker Robert L. , Constable Catherine G.
Published: October 2002

The inversion of electromagnetic sounding data does not yield a unique solution, but inevitably a single model to interpret the observations is sought. We recommend that this model be as simple, or smooth, as possible, in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models. To obtain smooth models, the nonlinear forward problem is linearized about a starting model in the usual way, but it is then solved explicitly for the desired model rather than for a model correction. By parameterizing the model in terms of its first or second derivative with depth, the minimum norm solution yields the smoothest possible model. Rather than fitting the experimental data as well as possible (which maximizes the roughness of the model), the smoothest model which fits the data to within an expected tolerance is sought. A practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models, resulting in a stable and rapidly convergent algorithm. The inversion of both magnetotelluric and Schlumberger sounding field data, and a joint magnetotelluric‐resistivity inversion, demonstrate the method and show it to have practical application.
Downhole magnetometric resistivity response of a half‐plane conductor
Authors: Pai D. M. , Yeoh J. P.
Published: October 2002

The magnetometric resistivity (MMR) methods measure the magnetic field of the galvanic current flow introduced by current electrodes in the ground. The downhole MMR version overcomes some of the limitations imposed by a conductive overburden on the surface MMR methods. This paper is concerned with downhole MMR responses of buried conductors. The model considered is an infinitely conductive half‐plane conductor in a uniform whole‐space host, as an ideal representation of a plate‐like conductive orebody buried deep underground. Exact analytical downhole MMR responses are derived for arbitrary locations of current electrode sources and magnetic field sensors. Type curves are shown for various source‐sensor‐target orientations to illustrate some characteristics of the responses.
Truncated asymptotic representation of waves in a one‐dimensional elastic medium
Authors: Yedlin M. J. , Seymour B. R. , Zelt B. C.
Published: October 2002

A new time‐domain method has been developed for solving for the stress and displacement of normally incident plane waves propagating in a smoothly varying one‐dimensional elastic medium. Both the Young’s modulus E and the density ρ are allowed to vary smoothly with depth. The restriction of geometrical optics, that the wavelength be much less than the material stratification length, is not required in this new method. We truncate the infinite geometrical‐optics asymptotic expansion after n terms (n = 2 in this paper), which imposes a condition on the acoustic impedance I for exact solutions to exist. The resultant expansion is uniform and exact for three general classes of impedance functions. Results are calculated for the case of a medium with a linear velocity gradient (for which there is an exact solution in the frequency domain); the results are compared with a two‐term WKBJ approximation and the new truncated expansion method. Since a linear velocity gradient is not one of the foregoing classes of impedance functions, a curve‐fitting approach is necessary. The results show that the new method compares favorably to both the WKBJ results and the exact solution and is accurate to within the error of the required curve fit. Two classes of synthetic seismograms are then calculated for smooth velocity and density variations. The same impedance as a function of traveltime is used for both classes. In the first class the principal variation in impedance is due to velocity, while in the second it is mainly due to density. The amplitudes in both classes of synthetic seismograms are very similar, but, as expected, the traveltime curves for each class are widely separated.
Application of the modified image method to inversion of synthetic active electromagnetic bathymetric data
Authors: Bergeron Clyde J. , Ioup Juliette W. , Michel Gus A.
Published: October 2002

A modified image method is used to invert synthetic active electromagnetic data generated from shallow ocean models by means of the exact Sommerfeld theory. The modified image method offers approximate solutions to the Sommerfeld problem: the response of a layered structure of ohmic conductors to an active electromagnetic source. The image plane for the active electromagnetic source is assumed to be at a complex depth given by [Formula: see text], where [Formula: see text] is the skin depth of the first layer and [Formula: see text] is the multilayer correction factor. Two ad hoc correction factors have been introduced into the modified image method. They bring the image field [Formula: see text] of the single image at a complex depth into very good agreement with the secondary field [Formula: see text] of the exact Sommerfeld theory of a two‐layer bathymetric model. We demonstrate that these correction factors can be calculated from synthetic data. This allows for an accurate, algebraic, and fast inversion of active electromagnetic data in terms of the parameters of a two‐layer model: the conductivities of the sea and sea bottom, and the depth and ohmic skin depth of the sea. The β factor, which takes into account the departure of the primary active electromagnetic field from a plane wave form at the air‐sea interface, is incorporated into the complex two‐layer correction function [Formula: see text]. A rescaling factor F brings [Formula: see text] into good agreement with [Formula: see text] in the depth regime where the β correction factor is ineffective, i.e., where the first layer depth is greater than two skin depths.
Techniques for seismic delineation of complex structure: A case history
Authors: Morse Vance C. , Johnson James H. , Crittenden James L. , Anderson Tom D.
Published: October 2002

Techniques for exploring in structurally complex areas were derived from recording and interpreting a single seismic line across the thrust‐faulted west flank of the Casper Arch. Line 306 was recorded to delineate a suspected subthrust prospect. The line was oriented along true dip based on the attitude of outcropping beds. We used short arrays and high‐fold stack, resulting in good data quality, especially in imaging of steep dips from the hanging wall. A two‐dimensional depth model of the structure was constructed. Outcrop dips constrained the interpretation and helped model the hangingwall. A correct model was confirmed when its synthetic time response (time model) matched the real reflections of a near‐offset section. Key to this match was accurate delineation of the hangingwall, which was confirmed when the model’s time response from the subthrust beds matched the static shifts of these same reflections on the real data. A perfect depth model was not achieved, however, due to constraints of the modeling program and the complexity of the thrust‐fault zone. Not all objectives of the project were met: no prospect was found, and proper migration of the data was never achieved. However, the field techniques used did provide good record quality, and the static problems were adequately handled. The final depth model was a reasonably accurate structural picture, although greater confirmation of its accuracy might have been achieved if a crossline had been available. In addition, the model might have been constructed more accurately and in less time using a real‐time, interactive modeling program.
Digital classification of contact metamorphic rocks in Extremadura, Spain, using Landsat thematic mapper data
Authors: Rowan Lawrence C. , Anton‐Pacheco Carmen , Brickey David W. , Kingston Marguerite J. , Payas Alba , Vergo Norma , Crowley James K.
Published: October 2002

Landsat Thematic Mapper (TM) images of an extensively cultivated part of Extremadura, Spain, have been used to distinguish soil developed on contact metamorphic rocks in aureoles around late Hercynian granitic plutons from soil formed on stratigraphically equivalent Late Proterozoic slate and metagraywacke that has been regionally metamorphosed to the greenschist facies. Reflectance spectra of contact metamorphic soil have lower reflectance, especially in the 1.6 μm wavelength region, and weaker Al‐OH, Mg‐OH, and [Formula: see text] absorption features than do spectra of the slate‐metagraywacke soil. These spectral differences are attributed to highly absorbing carbonaceous material in the contact metamorphic soil that was subjected to high temperatures during emplacement of the plutons. These spectral reflectance differences are evident in a density‐sliced TM band 5 image, in color‐ratio composite images that incorporate TM 4 : 5, 4 : 3, and 3 : 1 ratios, and in principal‐component composite images. Digital classification of the numerous tilled, vegetation‐free fields was used to map the contact metamorphic soil in an August, 1984, TM scene of the Caceres study area. First, TM 4 : 3 was used to identify these fields. Then ranges of TM 5 and TM 3 : 1 values were determined for selected tilled fields within and outside the contact aureoles. Field evaluation of a classification map based on TM 5 plus TM 3 : 1 shows more extensive aureoles than published geologic maps and few misclassified areas. Similar results were achieved using TM 4 : 5 instead of TM 5. This approach was subsequently used to map two linear zones of contact metamorphic rocks in the San Nicolas mine area where only two small exposures of granite have been documented. Landsat Multispectral Scanner (MSS) images can also be used to map contact aureoles in the study areas, but extensive field evaluation is required because of more frequent misclassification.
An in‐hole shotgun source for engineering seismic surveys
Authors: Pullan S. E. , MacAulay H. A.
Published: October 2002

We have conducted several source comparisons involving 12‐gauge and 8‐gauge Buffalo guns, a 7.3 kg sledgehammer, and a 75 kg weight drop. The results are strongly site‐dependent. We found that, when the near‐surface consisted of fine‐grained, water‐saturated sediments, the 12‐gauge Buffalo gun produced up to two orders of magnitude more energy than the conventional hammer across a broad frequency range. Under such conditions the gun produced the greatest improvement in energy between 200 and 400 Hz, where it yielded up to ten times more energy than the 75 kg weight drop. This indicates that the Buffalo gun may be particularly useful as a shallow reflection seismic source. However, at sites where the near‐surface materials were coarse‐grained and the water table was well below the ground surface, the advantages of using an in‐hole shotgun source as opposed to a hammer or weight drop were minimal. Nevertheless, in many areas we believe that the Buffalo gun is an excellent source for engineering seismic surveys. It is lightweight and portable (<5 kg), inexpensive to build (<$100 US), simple to use and maintain, and a good source of high‐frequency energy.
Tomographic determination of three‐dimensional seismic velocity structure using well logs, vertical seismic profiles, and surface seismic data
Authors: Chiu Stephen K. L. , Stewart Robert R.
Published: October 2002

A tomographic technique (traveltime inversion) has been developed to obtain a two‐ or three‐dimensional velocity structure of the subsurface from well logs, vertical seismic profiles (VSP), and surface seismic measurements. The earth was modeled by continuous curved interfaces (polynomial or sinusoidal series), separating regions of constant velocity or transversely isotropic velocity. Ray tracing for each seismic source‐receiver pair was performed by solving a system of nonlinear equations which satisfy the generalized Snell’s law. Surface‐to‐borehole and surface‐to‐surface rays were included. A damped least‐squares formulation provided the updating of the earth model by minimizing the difference between the traveltimes picked from the real data and calculated traveltimes. Synthetic results indicated the following conclusions. For noise‐free cases, the inversion converged closely from the initial guess to the true model for either surface or VSP data. Adding random noise to the observations and performing the inversion indicated that (1) using surface data alone allows reconstruction of the broad velocity structure but with some inaccuracy; (2) using VSP data alone gives a very accurate but laterally limited velocity structure; and (3) the integration of both data sets produces a more laterally extensive, accurate image of the subsurface. Finally, a field example illustrates the viability of the method to construct a velocity structure from real data.
Nonlinear two‐dimensional elastic inversion of multioffset seismic data
Authors: Mora Peter
Published: October 2002

The treatment of multioffset seismic data as an acoustic wave field is becoming increasingly disturbing to many geophysicists who see a multitude of wave phenomena, such as amplitude‐offset variations and shearwave events, which can only be explained by using the more correct elastic wave equation. Not only are such phenomena ignored by acoustic theory, but they are also treated as undesirable noise when they should be used to provide extra information, such as S‐wave velocity, about the subsurface. The problems of using the conventional acoustic wave equation approach can be eliminated via an elastic approach. In this paper, equations have been derived to perform an inversion for P‐wave velocity, S‐wave velocity, and density as well as the P‐wave impedance, S‐wave impedance, and density. These are better resolved than the Lamé parameters. The inversion is based on nonlinear least squares and proceeds by iteratively updating the earth parameters until a good fit is achieved between the observed data and the modeled data corresponding to these earth parameters. The iterations are based on the preconditioned conjugate gradient algorithm. The fundamental requirement of such a least‐squares algorithm is the gradient direction which tells how to update the model parameters. The gradient direction can be derived directly from the wave equation and it may be computed by several wave propagations. Although in principle any scheme could be chosen to perform the wave propagations, the elastic finite‐ difference method is used because it directly simulates the elastic wave equation and can handle complex, and thus realistic, distributions of elastic parameters. This method of inversion is costly since it is similar to an iterative prestack shot‐profile migration. However, it has greater power than any migration since it solves for the P‐wave velocity, S‐wave velocity, and density and can handle very general situations including transmission problems. Three main weaknesses of this technique are that it requires fairly accurate a priori knowledge of the low‐ wavenumber velocity model, it assumes Gaussian model statistics, and it is very computer‐intensive. All these problems seem surmountable. The low‐wavenumber information can be obtained either by a prior tomographic step, by the conventional normal‐moveout method, by a priori knowledge and empirical relationships, or by adding an additional inversion step for low wavenumbers to each iteration. The Gaussian statistics can be altered by preconditioning the gradient direction, perhaps to make the solution blocky in appearance like well logs, or by using large model variances in the inversion to reduce the effect of the Gaussian model constraints. Moreover, with some improvements to the algorithm and more parallel computers, it is hoped the technique will soon become routinely feasible.
Diffraction and conversion of elastic waves at a corrugated interface
Authors: Paul Anne , Campillo Michel
Published: October 2002

Numerical modeling is used to investigate the effect of small‐scale irregularities of a reflecting boundary on elastic wave reflections. The scattered wave field is computed by using a discretized form of boundary integral equations and a plane‐wave decomposition of seismic wave fields. For various values of incidence angle of the P wave, we compute the distribution of diffracted energy for both P waves and S waves as a function of reflection angle. We show that corrugations with mean wavelength of the order of, or smaller than, the seismic wavelength have little effect on the reflected P wave. However, the pattern of P‐to‐S conversion is very different from that with a plane boundary. Scattered S waves appear at postcritical angles for any angle of incidence of the P wave. The amplitude of these nongeometrical shear waves decreases rapidly with decreasing amplitude of the corrugations, or when the mean wavelength of the corrugations becomes larger than the dominant seismic wavelength. The local geometry of the irregularities has a negligible effect on the scattered S waves. By analogy with perturbation theory, we propose interpreting the postcritically scattered S waves as the contribution to the shear wave field of converted inhomogeneous P waves diffracted along the boundary.
The borehole transient electromagnetic response of a three‐dimensional fracture zone in a conductive half‐space
Authors: West Richard C. , Ward Stanley H.
Published: October 2002

Borehole geophysical methods can be useful in detecting subsurface fracture zones and mineral deposits which are nearby, but not intersected by boreholes. One electrical borehole technique which can be applied to this problem is the surface‐to‐borehole transient electromagnetic (TEM) method. In this method a transmitting loop is deployed on the surface while a receiving coil is moved down a borehole. A conductive, horizontal, tabular body in a homogeneous half space was chosen to simulate a 3-D fracture zone or mineral deposit within the earth. Theoretical borehole TEM responses for several models of practical interest were computed using a direct integral‐equation formulation. The anomalous TEM response (secondary response) is the result of a complex interaction between vortex and galvanic currents within the body. Distortion of the secondary response by the conductive host does not affect the estimate of the depth to the horizontal body but it does lead to erroneous estimates of the conductivity and size of the body. Increasing the resistivity of the host decreases the host effects and increases the peak response of the body. Decreasing the separation between the body and borehole or decreasing the depth of the body increases the secondary response. The decrease in the vortex response due to the decreased coupling when a transmitting loop is offset from the body is nearly countered by an increase in the galvanic response at late times; however, this phenomenon is model‐dependent. This study indicates promise for the borehole TEM method, but the application of the technique is limited by the hardware and modest modeling capabilities presently available.
Tunisian geothermal data from oil wells
Authors: Dhia Hamed Ben
Published: October 2002

Since direct measurements of steady‐state temperatures are not readily available in Tunisia, a geothermal investigation has been made using 1319 values of bottom‐hole temperatures (BHTs) obtained from 217 petroleum exploration wells. An empirical relation based on the differences between BHT and DST (drill stem tests) was used to correct BHTs and estimate geothermal gradients. The estimated geothermal gradient of the country varies between 21 and 52 °C/km. A few regions with similar gradients have been identified, and similarities between gradient contours and the main structural directions are noted. Furthermore, for 25 points from 12 wells, it was possible to apply the Horner‐plot method to determine the equilibrium formation temperature (Tf). Comparison of Tf values with those calculated by the estimated gradients reveals a good correlation (r = 97 percent) between the two estimates. This agreement permits greater confidence in the statistical method used and consequently in the estimated gradients for the whole country.
Generalized f-k (frequency‐wavenumber) migration in arbitrarily varying media
Authors: Pai David M.
Published: October 2002

Migration requires one‐way wave continuation. In the spatial domain, one‐way wave equations are derived based on various approximations to an assumed dispersion relation. In the frequency‐wavenumber domain, the well known f-k method and the phase‐shift method are strictly valid only within homogeneous models and layered models, respectively. In this paper, a frequency‐wavenumber domain method is presented for one‐way wave continuation in arbitrarily varying media. In the method, the downward continuation is accomplished, not with plane waves individually as in the f-k or the phase‐shift method, but with the whole spectrum of plane waves simultaneously in order to account for the coupling among the plane waves due to lateral inhomogeneity. The method is based on a matrix integral equation. The method has the following merits: (1) The method is a generalization of the f-k and the phase‐shift methods, valid in arbitrarily varying models. (2) The method has physical interpretations in terms of upgoing and downgoing plane waves, and as such the method has well defined steps leading from full‐wave continuation (two‐way wave) to one‐way wave continuation for migration. (3) The method is rigorous; the only approximations in the method—other than the one‐way wave approximation necessary for migration—are the discretization of a continuous system (which is necessary in computer methods) and imperfections associated with the limited spatial aperture of the data; as such, the method can achieve high solution accuracy. (4) The method can be fast, since computations are mainly matrix‐vector multiplications, which are easily vectorizable. In particular, compared to spatial domain methods, I contend that the method is (1) more rigorous in one‐way wave theory, (2) more accurate in migration of high‐dip events, and (3) faster for smooth models. I applied the method to a few examples of zero‐offset data migration, including imaging a point diffractor below a dipping interface, migration with sharp lateral variations in velocity, and migration with smooth lateral variations in velocity.
Imaging reflections in elliptically anisotropic media
Authors: Dellinger Joe , Muir Francis
Published: October 2002

In an isotropic medium, waves reflected from a mirror form a virtual image of their source. This property of planar reflectors is generally not true in the presence of anisotropy. In their short note, Blair and Korringa (1987) show that for the special case of SH waves from a point source in a transversely isotropic medium, an aberration‐free image is formed for any orientation of the mirror. While their proof is mathematical, we show the same result in an intuitive, pictorial fashion and in the process discover that although the image is indeed aberration free, it is still distorted.
Three‐dimensional imaging of steeply dipping structure near the San Andreas fault, Parkfield, California
Authors: Louie John N. , Clayton Robert W. , LeBras Ronan J.
Published: October 2002

Shot gathers from the Parkfield, California, deep crustal seismic reflection line, recorded in 1977 by COCORP, reveal coherent events having horizontal to reverse moveouts. These events were migrated using a multioffset three‐dimensional Kirchhoff summation method. This method is a ray‐equation back projection inversion of the acoustic wave field, which is valid under the Born, WKBJ, and far‐field assumptions. Migration of full‐wave acoustic synthetics, having the same limitations in geometric coverage as the COCORP survey, demonstrates the utility of the imaging process. The images obtained from back projection of the survey data suggest that the Gold Hill fault carries ultramafic rocks from the surface to 3 km depth at a dip greater than 45 degrees, where it joins the San Andreas fault, which may cut through more homogeneous materials at shallow depths. To the southwest, a 2 km Tertiary sedimentary section appears to terminate against a near‐vertical fault. The zone between this fault and the San Andreas may be floored at 3 km by flat‐lying ultramafics. Lateral velocity inhomogeneities are not accounted for in the migration but, in this case, do not seriously hinder the reconstruction of reflectors.
Time‐domain electromagnetic detection of a hidden target
Authors: Bartel David C. , Becker A.
Published: October 2002

Numerical modeling of the time‐domain electromagnetic (EM) step response of a vertical tabular target hidden beneath a thin conductive overburden reveals that the target’s presence may be detected only during a well‐defined time window. In a situation where the secondary magnetic field is sensed by an airborne system equipped with horizontal coaxial dipoles, a conductance contrast of about ten between the target and the overburden is needed to ensure target detection. This value will, of course, vary with the size and depth of the target and, to a lesser extent, with the geometry of the system. In general, the time at which the window opens is a function of the geometrical parameters of the target, the height of the system, and the conductance of the overburden. For a given target, its width (defined as the ratio of the time of closure to the time of opening) is only a function of the conductance contrast between the target and the overburden. While the target signal is visible, one observes a maximum value of the target‐to‐overburden response ratio. The time at which this occurs is mainly controlled by the conductance of the target. The presence of the overburden causes the target signal to build up gradually before decaying toward zero. However, once the target signal dominates the overburden response, the signal can be approximated by a simple exponential decay over the time range of interest. The time constant of this decay is determined by the size and conductance of the target. Using this model, it is easy to relate the magnetic field step response calculated here to the response observable with a conventional EM system that transmits a primary field pulse of finite duration and detects the time derivative of the secondary magnetic field.
An integral formulation for three‐dimensional terrain modeling for resistivity surveys
Authors: Xu Shi‐zhe , Gao Zucheng , Zhao Sheng‐kai
Published: October 2002

A new method is presented in this paper for modeling the 3-D terrain effect that, in turn, can be approximately removed by applying a terrain correction to resistivity surveys. First, the 3-D electrical boundary value problem is transformed into an integral equation problem by use of Green’s theorem. Then the boundary element method is used to solve the integral equation. The ground surface and the boundary of the anomalous body are divided into triangular elements. Linear variation of quantities is assumed within each element and a Gaussian quadrature formula is used to calculate the integral. In this way, the integral equation is converted into a set of linear equations. The potential field value on the ground surface is obtained by solving the linear equation system with Gaussian elimination. This method uses a special triangular element mesh for division of the ground surface which is moved simultaneously with the movement of the source. Therefore, the number of computational nodes is greatly reduced. To account for the influence of 3-D terrain on the apparent resistivity, only 68 nodes are required in typical situations. Since this method does not require a large computational effort, it can be conveniently run on a microcomputer. Calculation examples for two models show that the results obtained by this method compare well with an analytic solution and results from other numerical methods. An example of the 3-D terrain correction in a resistivity survey is also given; the example shows that applying the terrain correction can greatly attenuate the terrain effect.
Inductive interaction between polarizable conductors: An explanation of a negative coincident‐loop transient electromagnetic response
Authors: Smith Richard S. , West G. F.
Published: October 2002

Theoretically, the coincident‐loop transient electromagnetic (TEM) voltage responses of an earth with frequency‐invariant conductivities and permeabilities cannot change sign. In Australia, where prospecting is often done with the SIROTEM system operating in coincident‐loop mode, the response occasionally changes sign at late times from the normal positive transient to negative values. Because it has been shown that the observed negatives cannot be explained by poor instrument design, normal displacement currents, dispersive permeabilities, or geometric effects, the mechanism causing the negatives is now inferred to be a frequency dependence (dispersion) of the conductivity. To obtain negative values, the conductivity must increase with frequency over the spectral range covered by the TEM survey (∼10–1000 Hz). Slightly dispersive conductivities are observed by induced‐polarization (IP) surveys and are geologically common; however, field experience suggests that whereas some negatives are associated with IP anomalies, others are not. We present an interpretation of SIROTEM field data in which a negative response is observed between the positive responses of two steeply dipping conductors. Although no IP survey has been performed in this area, field experience in similar areas suggests that a significant IP anomaly is not likely to be directly associated with the negative anomaly. The negatives in the field data can be explained (i) if at least one of the two conductors is moderately dispersive in the frequency range 10–1000 Hz and (ii) if the inductive coupling between the conductors is taken into account. The magnitude of the dispersion is such that no significant IP anomaly would be measured in the region where the TEM negative transient is observed. The least complicated model that can fit the field data comprises two circular, polarizable wire‐filament circuits, which are inductively coupled to each other. The geometry of the field situation has been accounted for quantitatively by mutual inductance calculations, but the conductivity and polarizability of the bodies are modeled only in terms of simple circuit analogies.
Transient electromagnetic responses of high‐contrast prisms in a layered earth
Authors: Newman Gregory A. , Hohmann Gerald W.
Published: October 2002

An integral‐equation solution for transient electromagnetic (TEM) scattering by prisms in layered half‐spaces is formulated to provide meaningful results when the prisms are in highly resistive layers. A prism is replaced with an unknown scattering current, which is approximated with pulse and divergence‐free basis functions in the frequency domain. Divergence‐free basis functions model eddy currents that exist in confined bodies in a very resistive host and hence simulate the inductive responses of the prisms. A Galerkin solution for the scattering current is obtained where the dominant charge operator is eliminated from part of the solution by integrating the tensor Green’s function around rectangular paths. After the scattering current is determined, the electric and magnetic fields scattered by the prisms are calculated; and the corresponding TEM responses are obtained by inverse Fourier transformation. The resulting solution provides meaningful results over a wide range of resistivities in layered hosts including the case of free space. The masking effect of a conductive overburden delays and suppresses the three‐dimensional TEM response of a conductor. The overburden response must be removed for the conductor’s response to be fully interpretable. An interpretation of the conductor with free‐space models is a poor approximation when the basement rock is conductive. Instead of an exponential decay at late times, the conductor’s response decays in an inverse power relationship. When the basement resistivity is increased, the conductor exhibits an exponential decay at late times. For a thin dike, the time constant estimated from this decay is identical to that for a thin plate in free space. However, the response of the dike buried beneath the overburden is larger than the response of the dike in free space. This increase in the response of the dike will bias modeling it in free space with thin plates. We have used the solution to gain insight regarding the lateral resolution of two vertical conductors for the fixed‐loop and central‐loop survey configurations. The results suggest that resolution of multiple conductors is very poor in a fixed‐loop survey; but in a central‐loop survey, the resolution is much better, provided the data are interpreted at early times. At later times, multiple conductors may not be resolvable and interpretational ambiguities could arise.
Elastic wave‐field inversion of reflection and transmission data
Authors: Mora Peter
Published: October 2002

Elastic inversion of multioffset seismic data by wave‐ field fitting yields a maximum probability P-wave and S-wave velocity and density model of the Earth. Theoretically, the inversion accounts for all elastic waves including reflected and transmitted waves, mode conversions, shear waves, head waves, Rayleigh waves, etc. These different wave types tend to resolve different components of the Earth properties. By inverting two‐ component synthetic data, I show that reflection data mainly resolve high wavenumbers, while transmission data mainly resolve low wavenumbers of the P-wave and S-wave velocity model. The inversion of reflection data (shot gathers) yields a result that looks like a prestack elastic migration but the meaning of the inverted data is not simply reflectivity: it is the P-wave and S-wave velocity perturbation. The inversion of transmission data (VSPs) yields a solution that contains useful interval velocity information and is comparable to an elastic diffraction tomography result.
Reverse‐time wave‐field extrapolation, imaging, and inversion
Authors: Esmersoy Cengiz , Oristaglio Michael
Published: October 2002

The scattered wave field propagated backward in time into an arbitrary background medium is related via a volume integral to perturbations in velocity about the background, which are expressed as a scattering potential. In general, there is no closed‐form expression for the kernel of this integral representation, although it can be expressed asymptotically as a superposition of plane waves backpropagated from the receiver array. When the receiver array completely surrounds the scatterer, the kernel reduces to the imaginary part of the Green’s function for the background medium. This integral representation is used to relate the images obtained by imaging algorithms to the actual scattering potential. Two such relations are given: (1) for the migrated image, obtained by deconvolving the extrapolated field with the incident field; and (2) for the reconstructed image, obtained by applying a one‐way wave operator to the extrapolated field and then deconvolving by the incident field. The migrated image highlights rapid changes in the scattering potential (interfaces), whereas the reconstructed image can, under ideal conditions, be a perfect reconstruction of the scattering potential. “Ideal” conditions correspond to (1) weak scattering about a smoothly varying background medium, (2) a receiver array with full angular aperture, and (3) data of infinite bandwidth. Images obtained from a multioffset vertical seismic profile (VSP) illustrate some of the practical differences between the two imaging algorithms. The reconstructed image shows a much clearer picture of the target (a reef structure), in part because the one‐way imaging operator eliminates artifacts caused by the limited aperture of the receiver array.
On: “Stratigraphic filter theory: Combined effects of parallel bedding and random inhomogeneities” by I. Lerche (GEOPHYSICS, 51, 1531–1537, August 1986).
Authors: Wu Ru‐Shan
Published: October 2002

In the paper by Lerche (hereafter referred to as Lerche 86), the author compared his formula of effective attenuation (42) based on the mean field attenuation with formula (41), attributed to Wu (1982a), and claimed that Wu’s formula “explicitly assumed that the Born approximation is valid (i.e., that the scattering by the randomly sited centers is from the unperturbed incident wave with no modification to the wave from previous scattering),” while in his result “the modification to the incident wave is taken into account through the statistically sharp mean field” and therefore is “more correct.” However, neither the statement about the validity condition of Wu’s formula not the author’s formula (42) from the mean field approach is correct. In fact, the mean field formalism, when improperly applied to amplitude attenuation, can generate physically meaningless results. The impotence and fallacy of the mean field formalism in dealing with amplitude attenuation in 3-D random media have been recognized since the early 1980s (Wu, 1980, 1982a, b; Sato, 1982a, b). This recognition and the developments thereafter can be considered as one of the major advances in the field of scattering attenuation of seismic waves (Herraiz and Espinosa, 1987; Wu, 1987). Retreat from this advance seems injudicious and illogical. Therefore, I would like to take this opportunity to clarify some basic problems of scattering attenuation.