Showing papers in "Geophysics in 1986"

Weak elastic anisotropy

Abstract: Most bulk elastic media are weakly anisotropic. -The equations governing weak anisotropy are much simpler than those governing strong anisotropy, and they are much easier to grasp intuitively. These equations indicate that a certain anisotropic parameter (denoted 6) controls most anisotropic phenomena of importance in exploration geophysics. some of which are nonnegligible even when the anisotropy is weak. The critical parameter 6 is an awkward combination of elastic parameters, a combination which is totally independent of horizontal velocity and which may be either positive or negative in natural contexts.

P-SV wave propagation in heterogeneous media: Velocity‐stress finite‐difference method

Abstract: I present a finite-difference method for modeling P-SV wave propagation in heterogeneous media This is an extension of the method I previously proposed for modeling SH-wave propagation by using velocity and stress in a discrete grid The two components of the velocity cannot be defined at the same node for a complete staggered grid: the stability condition and the P-wave phase velocity dispersion curve do not depend on the Poisson's ratio, while the S-wave phase velocity dispersion curve behavior is rather insensitive to the Poisson's ratio Therefore, the same code used for elastic media can be used for liquid media, where S-wave velocity goes to zero, and no special treatment is needed for a liquid-solid interface Typical physical phenomena arising with P-SV modeling, such as surface waves, are in agreement with analytical results The weathered-layer and corner-edge models show in seismograms the same converted phases obtained by previous authors This method gives stable results for step discontinuities, as shown for a liquid layer above an elastic half-space The head wave preserves the correct amplitude Finally, the corner-edge model illustrates a more complex geometry for the liquid-solid interface As the Poisson's ratio v increases from 025 to 05, the shear converted phases are removed from seismograms and from the time section of the wave field

Effects of porosity and clay content on wave velocities in sandstones

Abstract: The ultrasonic compressional (Vp) and shear (Vs) velocities and first‐arrival peak amplitude (Ap) were measured as functions of differential pressure to 50 MPa and to a state of saturation on 75 different sandstone samples, with porosities ϕ ranging from 2 to 30 percent and volume clay content C ranging from 0 to 50 percent, respectively. Both Vp and Vs were found to correlate linearly with porosity and clay content in shaly sandstones. At confining pressure of 40 MPa and pore pressure of 1.0 MPa, the best least‐squares fits to the velocity data are Vp(km/s)=5.59-6.93ϕ-2.18C and Vs(km/s)=3.52-4.91ϕ-1.89C. Deviations from these equations are less than 3 percent and 5 percent for Vp and Vs, respectively. The velocities of clean sandstones are significantly higher than those predicted by the above linear fits (about 7 percent for Vp and 11 percent for Vs), which indicates that a very small amount of clay (1 or a few percent of volume fraction) significantly reduces the elastic moduli of sandstones. For shaly...

A strategy for nonlinear elastic inversion of seismic reflection data

Abstract: The problem of interpretation of seismic reflection data can be posed with sufficient generality using the concepts of inverse theory. In its roughest formulation, the inverse problem consists of obtaining the Earth model for which the predicted data best fit the observed data. If an adequate forward model is used, this best model will give the best images of the Earth’s interior. Three parameters are needed for describing a perfectly elastic, isotropic, Earth: the density ρ(x) and the Lame parameters λ(x) and μ(x), or the density ρ(x) and the P-wave and S-wave velocities α(x) and β(x). The choice of parameters is not neutral, in the sense that although theoretically equivalent, if they are not adequately chosen the numerical algorithms in the inversion can be inefficient. In the long (spatial) wavelengths of the model, adequate parameters are the P-wave and S-wave velocities, while in the short (spatial) wavelengths, P-wave impedance, S-wave impedance, and density are adequate. The problem of inversion o...

Approximating edges of source bodies from magnetic or gravity anomalies

Abstract: Cordell and Grauch (1982, 1985) discussed a technique to estimate the location of abrupt lateral changes in magnetization or mass density of upper crustal rocks. The final step of their procedure is to identify maxima on a contoured map of horizontal gradient magnitudes. We attempt to automate their final step. Our method begins with gridded magnetic or gravity anomaly data and produces a plan view of inferred boundaries of magnetic or gravity sources. The method applies to both local surveys and to continent-wide compilations of magnetic and gravity data (e.g., Zietz, 1982; Simpson et al., 1983a; Kane et al., 1982).

The application of high-order differencing to the scalar wave equation

Abstract: The second‐order central difference is often used to approximate the derivatives of the wave equation. It is demonstrated that gains in computational efficiency can be made by using high‐order approximations for these derivatives. A one‐dimensional model is used to illustrate the relative accuracy of O(Δt2,Δx2),O(Δt2,Δx4),O(Δt4,Δx4),andO(Δt4,Δx10) central‐difference schemes. For comparison, O(Δt2)andO(Δt4) pseudospectral schemes are used as an additional measure of performance. The results indicate that O(Δt4,Δx10) differencing can achieve similar accuracy as the O(Δt4) spectral scheme. For practical illustration, a two‐dimensional form of the O(Δt4,Δx10) algorithm is used to compute the exploding reflector response of a salt‐dome model and compared with a fine‐grid O(Δt2,Δx4) result. Transmissive sponge‐like boundary conditions are also examined and shown to be effective.

Two‐dimensional nonlinear inversion of seismic waveforms: Numerical results

Abstract: The nonlinear problem of inversion of seismic waveforms can be set up using least‐squares methods. The inverse problem is then reduced to the problem of minimizing a lp;nonquadratic) function in a space of many (104to106) variables. Using gradient methods leads to iterative algorithms, each iteration implying a forward propagation generated by the actual sources, a backward propagation generated by the data residuals (acting as if they were sources), and a correlation at each point of the space of the two fields thus obtained, which gives the updated model. The quality of the results of any inverse method depends heavily on the realism of the forward modeling. Finite‐difference schemes are a good choice relative to realism because, although they are time‐consuming, they give excellent results. Numerical tests performed with multioffset synthetic data from a two‐dimensional model prove the feasibility of the approach. If only surface‐recorded reflections are used, the high spatial frequency content of the ...

Transient Electromagnetic Sounding for Groundwater

Abstract: The feasibility of using the transient electromagnetic sounding (TS or TDEM) method for groundwater exploration can be studied by means of numerical models. As examples of its applicability to groundwater exploration, we study four groundwater exploration problems: (1) mapping of alluvial fill and gravel zones over bedrock; (2) mapping of sand and gravel lenses in till; (3) detection of salt or brackish water interfaces in freshwater aquifers; and (4) determination of hydrostratigraphy. These groundwater problems require determination of the depth to bedrock; location of resistive, high‐porosity zones associated with fresh water; determination of formation resistivity to assess water quality; and determination of lithology and geometry, respectively. The TS method is best suited for locating conductive targets, and has very good vertical resolution. Unlike other sounding techniques where the receiver‐transmitter array must be expanded to sound more deeply, the depth of investigation for the TS method is a...

Acoustic relaxation in sedimentary rocks; dependence on grain contacts and fluid saturation

Abstract: Partial fluid saturation affects absorption and dispersion in sandstones. The proposed theoretical model describes acoustic relaxation due to local fluid flow. Previously proposed models of local flow were based on microgeometries not representative of sedimentary rocks; they were unable to describe the behavior of partially saturated sandstones. The new model is based upon observed microstructures in sandstones. A fraction of the grain contacts in sandstones are permeated by sheet‐like gaps. The incomplete solid‐solid contact allows an interconnected fluid film to exist between the grain surfaces. The model consists of a narrow gap connected to a finite annular pore. An acoustic stress wave drives the film out of the narrow contact region and into the adjacent pore. The viscous flow results in a dissipation of energy. The model predicts the real and imaginary parts of the complex frame moduli as a function of frequency and fluid saturation. The predictions agree well with experimental results.

Automatic estimation of large residual statics corrections

Abstract: Conventional approaches to residual statics estimation obtain solutions by performing linear inversion of observed traveltime deviations. A crucial component of these procedures is picking time delays; gross errors in these picks are known as "cycle skips" or "leg jumps" and are the bane of linear traveltime inversion schemes.This paper augments Rothman (1985), which demonstrated that the estimation of large statics in noise-contaminated data is posed better as a nonlinear, rather than as a linear, inverse problem. Cycle skips then appear as local (secondary) minima of the resulting nonlinear optimization problem. In the earlier paper, a Monte Carlo technique from statistical mechanics was adapted to perform global optimization, and the technique was applied to synthetic data. Here I present an application of a similar Monte Carlo method to field data from the Wyoming Overthrust belt. Key changes, however, have led to a more efficient and practical algorithm. The new technique performs explicit crosscorrelation of traces. Instead of picking the peaks of these crosscorrelation functions, the method transforms the crosscorrelation functions to probability distributions and then draws random numbers from the distributions. Estimates of statics are now iteratively updated by this procedure until convergence to the optimal stack is achieved.Here I also derive several theoretical properties of the algorithm. The method is expressed as a Markov chain, in which the equilibrium (steady-state) distribution is the Gibbs distribution of statistical mechanics.

Experimental determination of elastic anisotropy of Berea Sandstone, Chicopee Shale, and Chelmsford Granite

Abstract: We used the ultrasonic transmission method to measure P-, SH-, and SV-wave velocities for Chelmsford granite, Chicopee shale, and Berea sandstone in different directions up to 1 000 bars confining pressure. The velocity measurements indicate these three rocks are elastically anisotropic. The stiffness constants, dynamic Young’s moduli, dynamic Poisson’s ratios, and dynamic bulk moduli of the three rocks were also calculated. The elastic constants, together with velocity measurements, suggest that: (1) elastic anisotropy is due to the combined effects of pores or cracks and mineral grain orientation, and (2) elastic anisotropy decreases with increasing confining pressure. The residual anisotropy at higher confining pressure is due to mineral grain orientation.

Transient electromagnetic response of a three-dimensional body in a layered earth

Abstract: The three‐dimensional (3-D) electromagnetic scattering problem is first formulated in the frequency domain in terms of an electric field volume integral equation. Three‐dimensional responses are then Fourier transformed with sine and cosine digital filters or with the decay spectrum. The digital filter technique is applied to a sparsely sampled frequency sounding, which is replaced by a cubic spline interpolating function prior to convolution with the digital filters. Typically, 20 to 40 frequencies at five to eight points per decade are required for an accurate solution. A calculated transient is usually in error after it has decayed more than six orders in magnitude from early to late time. The decay spectrum usually requires ten frequencies for a satisfactory solution. However, the solution using the decay spectrum appears to be less accurate than the solution using the digital filters, particularly after early times. Checks on the 3-D solution include reciprocity and convergence checks in the frequenc...

Reverse-time migration of offset vertical seismic profiling data using the excitation-time imaging condition

Abstract: To apply reverse-time migration to prestack, finite-offset data from variable-velocity media, the standard (time zero) imaging condition must be generalized because each point in the image space has a different image time (or times). This generalization is the excitation-time imaging condition, in which each point is imaged at the one-way traveltime from the source to that point.Reverse-time migration with the excitation-time imaging condition consists of three elements: (1) computation of the imaging condition; (2) extrapolation of the recorder wave field; and (3) application of the imaging condition. Computation of the imaging condition for each point in the image is done by ray tracing from the source point; this is equivalent to extrapolation of the source wave field through the medium. Extrapolation of the recorded wave field is done by an acoustic finite-difference algorithm. Imaging is performed at each step of the finite-difference extrapolation by extracting, from the propagating wave field, the amplitude at each mesh point that is imaged at that time and adding these into the image space at the same spatial locations. The locus of all points imaged at one time step is a wavefront [a constant time (or phase) trajectory]. This prestack migration algorithm is very general. The excitation-time imaging condition is applicable to all source-receiver geometries and variable-velocity media and reduces exactly to the usual time-zero imaging condition when used with zero-offset surface data. The algorithm is illustrated by application to both synthetic and real VSP data. The most interesting and potentially useful result in the processing of the synthetic data is imaging of the horizontal fluid interfaces within a reservoir even when the surrounding reservoir boundaries are not well imaged.

Pore fluids and frequency-dependent wave propagation in rocks

Abstract: Attenuation is the anelastic process which dissipates seismic energy by conversion to heat, thus decreasing the amplitude and modifying the frequency and phase content of a propagating wavelet. Laboratory measurements show seismic phase velocity and attenuation are dependent upon the fluid saturation and the product of frequency and pore‐fluid viscosity, with a peak in attenuation between the seismic and sonic bands. The dominant mechanism by which seismic energy is dissipated in the upper crust is local viscous fluid flow in pores of small aspect ratio. Phenomenologically, this behavior is modeled as a series of linear viscoelastic elements with a narrow distribution of relaxation times, where velocity and attenuation are related through the Hilbert transform. This model may be generalized to include constant-Q behavior, as observed in dry rocks. Solutions to the wave equation may be generated for an arbitrary frequency dependence of phase velocity and Q. When Q is nearly independent of frequency, the im...

Acoustic multipole sources in fluid-filled boreholes

Abstract: Acoustic well logging methods historically have been based on the excitation and reception of axisymmetric wave phenomena in a fluid-filled wellbore. We consider the reception of nonsymmetric wave phenomena excited by acoustic multipole sources. The first three orders of multipoles are the monopole, dipole, and quadrupole, and we examine these particular sources in detail. Existing sonic tools make use of a monopole source, while more recently, both dipole and quadrupole sources have been explored. An exact frequency-wavenumber domain representation of the acoustic field in the borehole due to a multipole source is formulated and numerical methods are used to compute synthetic space-time domain waveforms. We consider wideband monopole, dipole, and quadrupole excitations with center frequencies of 1, 4, and 12 kHz, and treat both slow and fast formation models. Finally, we derive low-frequency, far-field asymptotic expressions for the monopole, dipole, and quadrupole waveforms. At frequencies such that the shear wavelength is on the order of the borehole diameter or less, the difference between the monopole, dipole, and quadrupole waveforms is primarily in the nature of the surface wave mode which they excite: the monopole excites a Stoneley, or tube mode; the dipole excites a flexural mode; and the quadrupole excites a screw mode. By comparison, the compressional and shear head waves and the trapped waveguide modes do not change as much as the order of the multipole is changed.At low frequencies, where the shear wavelength is much longer than the diameter of the hole, the monopole excites a dominating tube mode, while the dipole and quadrupole excites dominating shear waves. Low-frequency asymptotic expressions for the waveforms agree well with the numerically computed waveforms.

Ground-penetrating radar surveys used in archaeological investigations

Abstract: During 1982 and 1983, two ground‐penetrating radar surveys were carried out in conjunction with archaeological investigations in Canada. The first survey was a detailed, high‐resolution radar survey at the site of a sixteenth century Basque whaling station on the Labrador coast designed to locate the graves of the Basques. The second was a rapid, low‐resolution reconnaissance survey as part of a prehistory impact assessment program at the site of the new National Museum of Man in Hull, Quebec. Both surveys were experimental and were designed to see whether ground‐penetrating radar would be useful for identifying and locating anomalies of archeological significance. Radar was successful in detecting archeological anomalies several meters in size at both locations, and the high‐resolution survey was moderately successful in identifying Basque graves. Ongoing work involves comparing radar results with the archaeological investigations to increase the understanding of how radar can be applied to archaeology a...

Two-dimensional topographic responses in magnetotellurics modeled using finite elements

Abstract: We simulate the magnetotelluric response to two‐dimensional earth topography using finite elements. Linear interpolation of the secondary field parallel to strike over triangular elements allows accurate modeling of inclined resistivity boundaries, including topographic surfaces. To avoid discontinuities in field derivatives or resistivity, care must be taken that the nodal values of the field parallal to strike used to obtain the auxiliary secondary fields are kept within uniform earth media. The nodal locations may be shifted, but the derivatives still are evaluated at the field points of interest. Correct values may be returned at gentle breaks in slope as well as along straight surfaces. The finite‐element program is verified by comparison with the analytic transverse magnetic response of a hemicylindrical depression and with Rayleigh scattering and transmission surface results for transverse electric and transverse magnetic polarization. Agreement with the other methods generally is excellent, with t...

Fluid pressure response to undrained compression in saturated sedimentary rock

Abstract: The pore pressure response of saturated porous rock subjected to undrained compression at low effective stresses are investigated theoretically and experimentally. This behavior is quantified by the undrained pore pressure buildup coefficient, B = (dPf/dPclld,"f~O' where Pf is fluid pressure, P, is confining pressure, and mf is the mass of fluid per unit bulk volume. The measured values for B for three sandstones and a dolomite are near 1.0 at zero effective stress and decrease with increasing effective stress. In one sandstone, B is 0.62 at 13 MPa effective stress. These results agree with the theories of Gassmann (1951) and Bishop (1966), which assume a locally homogeneous solid framework. The decrease of B with increasing effective stress is probably related to crack closure and to high-compressibility materials within the rock framework. The more general theories of Biot (1955) and Brown and Korringa (1975) introduce an additional parameter, the unjacketed pore compressibility, which can be determined from induced pore pressure results. Values of B close to 1 imply that under appropriate conditions within the crust, zones of low effective pressure characterized by low seismic wave velocity and high wave attenuation could exist. Also, in confined aquifer-reservoir systems at very low effective stress states, the calculated specific storage coefficient is an order of magnitude larger than if less overpressured conditions prevailed.

A least-squares smooth fitting for irregularly spaced data; finite-element approach using the cubic B-spline basis

Abstract: A new method of multivariate smooth fitting of scattered, noisy data using cubic B-splines was developed. An optimum smoothing function was defined to minimize the 𝓁2 norm composed of the data residuals and the first and the second derivatives, which represent the total misfit, fluctuation, and roughness of the function, respectively. The function is approximated by a cubic B‐spline expansion with equispaced knots. The solution can be interpreted in three ways. From the stochastic viewpoint, it is the maximum‐likelihood estimate among the admissible functions under the a priori information that the first and second derivatives are zero everywhere due to random errors, i.e., white noise. From the physical viewpoint, it is the finite‐element approximation for a lateral displacement of a bar or a plate under tension which is pulled to the data points by springs. From a technical viewpoint, it is an improved spline‐fitting algorithm. The additional condition of minimizing the derivative norms stabilizes the l...

Reduction to the pole as an inverse problem and its application to low-latitude anomalies

Abstract: Traditionally, reduction to the pole has been accomplished either by space- or wavenumber-domain filtering. In the two-dimensional case, this procedure is stable regardless of the latitude, as long as the source strike is not parallel to the horizontal projection of the geomagnetic field. In the three-dimensional case, however, reduction-to-the-pole filtering is stable only at high magnetic latitudes. At latitudes lower than 15 degrees, it is of no practical use due to a sharply increasing instability toward the magnetic equator.The three-dimensional instability of this filtering technique is demonstrated, and the reduction-to-the-pole problem is formulated in the context of a general linear inverse problem. As a result, stable solutions are found by using well-known stabilizing procedures developed for the inverse linear problem. The distribution of magnetization of an equivalent layer of doublets that reproduces the observed data is computed. The magnetic doublets are parallel to the magnetization direction which is assumed constant throughout the sources. The magnetic field reduced to the pole is then obtained by changing the inclinations of the geomagnetic field and the doublets to 90 degrees and recalculating the total field.The usefulness and limitations of the method at low magnetic latitudes are assessed using theoretical data. The effects of noise and anomaly truncation are also investigated for both high and low latitudes. In all cases, application of the proposed method produced meaningful results regardless of the latitude. The method is applied to field data from two different low-latitude anomalies. The first anomaly is due to a seamount in the Gulf of Guinea with reversed magnetization. The geomagnetic field at this location is about -23 degrees. The second anomaly is an intrabasement anomaly from Parnaiba Basin, Brazil, where the magnetization is assumed to be induced by a geomagnetic field with -1.4 degree inclination. The results obtained confirm that the proposed method produces stable, meaningful, reduced-to-the-pole maps.

The use and misuse of apparent resistivity in electromagnetic methods

Abstract: Problems and misunderstandings arise with the concept of apparent resistivity when the analogy between an apparent resistivity computed from geophysical observations and the true resistivity structure of the subsurface is drawn too tightly. Several definitions of apparent resistivity are available for use in electromagnetic methods; however, those most commonly used do not always exhibit the best behavior. Many of the features of the apparent resistivity curve which have been interpreted as physically significant with one definition disappear when alternative definitions are used. It is misleading to compare the detection or resolution capabilities of different field systems or configurations solely on the basis of the apparent resistivity curve. For the in‐loop transient electromagnetic (TEM) method, apparent resistivity computed from the magnetic field response displays much better behavior than that computed from the induced voltage response. A comparison of “exact” and “asymptotic” formulas for the TE...

High-resolution common-depth-point reflection profiling: Field acquisition parameter design

Abstract: The results of a seismic reflection profiling exercise are strongly dependent upon parameters used in field recording. The choice of parameters is determined by objectives of the survey, available resources, and geolog­ ic locality. Some simple modeling and/or a walkaway noise survey are helpful in choice of field parameters. Filtering data before analog-to-digital conversion in the field can help overcome limitations in the dynamic range of the seismograph. Source and geophone arrays can be used to a limited extent in high-resolution sur­ veys to help attenuate ground roll. Proper planting of geophones can be an important factor in obtaining the flattest spectral response. Various seismic energy sources provide the flattest spectral response. Various seismic energy sources provide different spectral character and varying degrees of convenience and cost.

Estimates of velocity dispersion between seismic and ultrasonic frequencies

Abstract: It is generally accepted that acoustic velocities in fluid‐saturated rocks vary with frequency. Evidence comes from experimental measurements and from theoretical causality arguments. We have developed a simple analysis technique that gives estimates of total velocity dispersion between zero frequency and any measurement frequency. The technique requires compressional (P) and shear (S) wave velocity measurements on dry and fully saturated rock. Assuming that the dry velocities are independent of frequency, the Biot‐Gassmann equations are used to calculate the zero‐frequency velocities in the fully saturated rock. Any difference between the measured velocities and the calculated zero‐frequency velocities is interpreted as evidence of dispersion. Application of this analysis technique to a variety c ultrasonic data sets gives consistent results. In many rocks, dispersion between zero frequency and ultrasonic frequencies is on the order of 10 percent at low effective stress, and it decreases to only a few pe...

Field comparison of shallow seismic sources

Abstract: Choosing a seismic source for a shallow reflection survey can be the most pivotal decision for the engineering geophysicist. The intent of this paper is to present data that will assist in selection of a shallow seismic source best meeting the goals within the constraints of specific projects, particularly in areas where the water table is near the surface. The data were collected (and displayed as seismograms and amplitude spectra) for 15 different shallow seismic sources in October, 1985, at a single site in New Jersey; they show the different characteristics of each source. Considering the almost three orders of magnitude difference in total source energy between the largest and smallest source, we chose a display format that presented the data as objectively as possible, while still allowing direct source‐to‐source comparisons. Two strong reflections at about 100 and 130 ms probably mark the top and bottom of a clay unit 80 m below the surface at this site. Our previous work and that of our colleagues...

The effect of the extent of freezing on seismic velocities in unconsolidated permafrost

Abstract: We develop a model to relate velocities of seismic waves in unconsolidated permafrost is idealized as an assemblage of spherical quartz grains imbedded in a matrix composed of spherical water inclusions in ice. The theory of Kuster and Toksoz, based on wave‐scattering considerations, is used to determine the effective elastic moduli, and hence the wave speeds. The Hasin-Shtrikman theoretical bounds on the elastic moduli of heterogeneous materials and considerations establish the plausibility of the model. The model predicts Vp and Vs to be decreasing functions of both the porosity and the water‐to‐ice ratio, and the ratio Vp/Vs to be an increasing function of these two parameters. The theory is then applied to laboratory measurements of shear‐ and compressional‐wave velocities in 23 permafrost samples from different sites in the Beaufort Sea, Mackenzie River Valley, and Canadian Arctic islands. Although no direct measurements were made of the extents of freezing in these samples, the data are consistent w...

Analysis of exploitation-induced gravity changes at Wairakei Geothermal Field

Abstract: Gravity changes (corrected for subsidence) of up to -1 000 (±300) μGal have occurred in the 1km2 area of the production bore field at Wairakei, and smaller decreases extend over a 50km2 surrounding area. The largest part of these decreases occurred during the 1960s; since then the net gravity change for the whole field has been zero, indicating mass flow equilibrium. The principal causes of gravity change have been deep liquid pressure drawdown which resulted in formation of a steam zone, subsequent saturation changes in the steam zone, liquid temperature decline, and groundwater level changes. Gravity models suggest saturation of the steam zone was 0.7 (±0.1) in 1962 and decreased to 0.6 by 1972. Gravity increases in the northern and eastern bore field since the early 1970s are attributed to cool water invading the steam zone.

Migration of common‐shot gathers

Abstract: Three depth migration methods which operate on common‐shot data are presented. The first migration method maps digitized horizons from the X-T domain to the X-Z domain. Because this method is based on ray tracing, its computation time is short; it is suggested for iterative velocity analysis. The other two migration methods map the entire common‐shot gather into a depth section. The common‐shot migration requires calculation of the arrival time of the direct wave from the source to all the depth points, and is done through a direct solution of the eikonal equation. All three methods are suitable for areas with both lateral and vertical velocity variation.

Deep-water peg legs and multiples: Emulation and suppression

Abstract: This paper discusses a two-step method for predicting and attenuating multiple and peg-leg reflections in unstacked seismic data. In the first step, an (observed) seismic record is extrapolated through a round-trip traversal of the water layer, thus creating an accurate prediction of all possible multiples. In the second step, the record containing the predicted multiples is compared with and subtracted from the original.The wave-equation method employed to predict the multiples takes accurate account of sea-floor topography and so requires a precise water-bottom profile as part of the input. Information about the subsurface below the sea floor is not required. The arrival times of multiple reflections are reproduced precisely, although the amplitudes are not accurate, and the sea floor is treated as a perfect reflector. The comparison step detects the similarities between the computed multiples and the original data, and estimates a transfer function to equalize the amplitudes and account for any change in waveform caused by the sea-floor reflector.This two-step wave-equation method is effective even for dipping sea floors and dipping subsurface reflectors. It does not depend upon any assumed periodicity in the data or upon any difference in stacking velocity between primaries and multiples. Thus it is complementary to the less specialized methods of multiple suppression.

Three‐dimensional Born inversion with an arbitrary reference

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

Applications and advantages of the Slingram electromagnetic method for archaeological prospecting

Abstract: The electromagnetic Slingram method can provide three types of information simultaneously: (1) the apparent resistivity of the ground; (2) its apparent susceptibility; and (3) the characteristic signatures of buried metallic objects.To build an apparatus with these three measurement capabilities requires an appropriate choice of parameters, including frequency, coil separation, and coil orientation as well as calculations which take into account the electrical resistivity and the magnetic susceptibility of the ground. The value of the Slingram method for archaeological applications is shown by two examples: (1) a Gallo-Roman pottery workshop investigated with measurements of the apparent susceptibility using a reduced sampling wide mesh grid technique; and (2) the study of a Bronze hoard site located in a marsh.Capabilities of the Slingram method may be improved by further study of an apparatus using several receivers which would increase the depth of investigation. Use of several frequencies would overcome difficulties related to resolving the magnetic quadrature susceptibility.