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Showing papers in "Geophysics in 1982"


Journal ArticleDOI
TL;DR: A method for rapidly making depth estimates from large amounts of magnetic data is described, based upon Euler’s homogeneity relationship, which can be applied in a wider variety of geologic situations than can model‐dependent techniques.
Abstract: 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

1,161 citations


Journal ArticleDOI
TL;DR: In this paper, a Gabor expansion involving basic wavelets with a constant time duration/mean period ratio was proposed for normal incidence propagation of plane waves through perfectly elastic multilayered media.
Abstract: From experimental studies in digital processing of seismic reflection data, geophysicists know that a seismic signal does vary in amplitude, shape, frequency and phase, versus propagation time To enhance the resolution of the seismic reflection method, we must investigate these variations in more detail. We present quantitative results of theoretical studies on propagation of plane waves for normal incidence, through perfectly elastic multilayered media. As wavelet shapes, we use zero-phase cosine wavelets modulated by a Gaussian envelope and the corresponding complex wavelets. A finite set of such wavelets, for an appropriate sampling of the frequency domain, may be taken as the basic wavelets for a Gabor expansion of any signal or trace in a two-dimensional (2-D) domain (time and frequency). We can then compute the wave propagation using complex functions and thereby obtain quantitative results including energy and phase of the propagating signals. These results appear as complex 2-D functions of time and frequency, i.e., as “instantaneous frequency spectra. ’ ’ Choosing a constant sampling rate on the logarithmic scale in the frequency domain leads to an appropriate sampling method for phase preservation of the complex signals or traces. For this purpose, we developed a Gabor expansion involving basic wavelets with a constant time duration/mean period ratio. For layered media, as found in sedimentary basins,

1,135 citations


Journal ArticleDOI
TL;DR: In this paper, a 2-dimensional (2-D) Gabor expansion on a set of basic wavelets adapted to phase preservation is proposed to handle instantaneous frequency spectra.
Abstract: Morlet et al (1982, this issue) showed the advantages of using complex values for both waves and characteristics of the media. We simulated the theoretical tools we present here, using the Goupillaud-Kunetz algorithm. Now we present sampling methods for complex signals or traces corresponding to received waves, and sampling methods for complex characterization of multilayered or heterogeneous media. Regarding the complex signals, we present a twodimecsional(2-D) method of sampling in the time-frequency domain using a special or “extended” Gabor expansion on a set of basic wavelets adapted to phase preservation. Such a 2-D expansion permits us to handle in a proper manner instantaneous frequency spectra. We show the differences between “wavelet resolution” and “sampling grid resolution.” We also show the importance of phase preservation in high-resolution seismic. Regarding the media, we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution. Analytical studies of wave propagation in periodic structures are well known in solid state physics, and lead to the so-called “Bloch waves.” The introduction of complex waves leads to replacing the classical wave equation by a Schriidinger equation. Finally, we show that complex wave equations, Gabor expansion, and Bloch waves are three different ways of ‘introducing the tools of quantum mechanics in highresolution seismic (Gabor, 1946; Kittel, 1976, Morlet, 1975). And conversely, the Goupillaud-Kunetz algorithm and an extended Gabor expansion may be of some use in solid state physics.

595 citations


Journal ArticleDOI
TL;DR: In this paper, a pseudospectral forward-modeling algorithm for solving the two-dimensional acoustic wave equation is presented, which utilizes a spatial numerical grid to calculate spatial derivatives by the fast Fourier transform.
Abstract: A Fourier or pseudospectral forward-modeling algorithm for solving the two-dimensional acoustic wave equation is presented. The method utilizes a spatial numerical grid to calculate spatial derivatives by the fast Fourier transform. time derivatives which appear in the wave equation are calculated by second-order differcncing. The scheme requires fewer grid points than finite-diffcrcnce methods to achieve the same accuracy. It is therefore believed that the Fourier method will prove more efficient than finitedifference methods. especially when dealing with threedimensional models. The Fourier forward-modeling method was tested against two problems, a single-layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling. The numerical results agreed with both the analytic and physical model results. Furthermore, the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account.

484 citations


Journal ArticleDOI
TL;DR: In this article, the effects of confining pressure, pore pressure, degree of saturation, strain amplitude, and frequency on seismic attenuation in sandstone bars were studied experimentally.
Abstract: Seismic wave attenuation in rocks was studied experimentally, with particular attention focused on frictional sliding and fluid flow mechanisms. Sandstone bars were resonated at frequencies from 500 to 9000 Hz, and the effects of confining pressure, pore pressure, degree of saturation, strain amplitude, and frequency were studied. Observed changes in attenuation and velocity with strain amplitude are interpreted as evidence for frictional sliding at grain contacts. Since this amplitude dependence disappears at strains and confining pressures typical of seismic wave propagation in the earth, we infer that frictional sliding is not a significant source of seismic attenuation in situ. Partial water saturation significantly increases the attenuation of both compressional (P) and shear (S) waves relative to that in dry rock, resulting in greater P‐wave than S‐wave attenuation. Complete saturation maximizes S‐wave attenuation but causes a reduction in P‐wave attenuation. These effects can be interpreted in term...

439 citations


Journal ArticleDOI
TL;DR: In this paper, the temporal resolution of a broadband wavelet with a white spectrum is controlled by its highest terminal frequency f, and the resolution limit approximates I/ ( I.5 fU) provided the wavelet's band ratio exceeds two octaves.
Abstract: This investigation deals with resolving reflections from thin beds rather than the detection of events that may or may not be resolved. Resolution is approached by considering a thinning bed and how accurately measured times on a seismic trace represent actual, vertical two-way traveltimes through the bed. Theoretical developments are in terms of frequency and time rather than wavelength and thickness because the latter two variables require knowledge of interval velocities. These results are compared with similar studies by Rayleigh, Ricker (19.53), and Widess (1973, 1980). We show that the temporal resolution of a broadband wavelet with a white spectrum is controlled by its highest terminal frequency f,,, and the resolution limit approximates I/ ( I .5 fU), provided the wavelet’s band ratio exceeds two octaves. The practical limit of resolution, however, occurs at a one-quarter wavelength condition and approximates I /( 1.4 fJ. The resolving power of zero-phase wavelets can be compared quantitatively once a wavelet is known in the time domain.

357 citations


Journal ArticleDOI
TL;DR: In this article, the authors proposed a system to enable the signature of an air gun array to be calculated at any point in the water from a number of simultaneous independent measurements of the near-held pressure held.
Abstract: We designed a system to enable the signature of an air gun array to be calculated at any point in the water from a number of simultaneous independent measurements of the near-held pressure held [subject of a patent application]. The number of these measurements must not be less than the number of guns in the array. The underlying assumption in our method is that the oscillating bubble produced by an air gun is small compared with the wavelengths of seismic interest. Each bubble thus behaves as a point source, both in the generation of seismic waves and in its response to incident seismic radiation produced by other nearby bubbles. It follows that the intcraction effects between the bubbles may be described in terms of spherical waves. The array of interacting guns is equivalent to a notional array of noninteracting guns whose combined seismic radiation is identical. The seismic signatures of the equivalent independent elements of this notional array can be determined from the near-held measurements. The seismic radiation pattern emitted by the whole array can be computed from these signatures by linear superposition, with a spherical correction applied. The method is tested by comparing far-field signatures computed in this way with field measurements made in deep water. The computed and measured signatures match each other very closely. By comparison, signatures computed neglecting this interaction are a poor match to the

280 citations


Journal ArticleDOI
Robert H. Tatham1
TL;DR: In this article, the empirical relations between seismic compressional and shear-wave velocities and sedimentary rock lithology were investigated and two published theoretical models of cracked elastic media were examined.
Abstract: Published laboratory investigations suggest an association exists between the ratio of seismic compressional and shear‐wave velocities (Vp/Vs) and sedimentary rock lithology. Comparisons of some theoretical models with these laboratory studies suggest that crack, or pore, geometry has a stronger effect on observed Vp/Vs values than elastic constants of the minerals comprising the matrix. Further, it can be inferred that the observed association between lithology and Vp/Vs is a result of an association between lithology and distribution of pore and crack shapes. Direct observation of crack shapes for a variety of lithologies is a next step in strengthening these inferences. The present study reviews the empirical relations between Vp/Vs and lithology and examines two published theoretical models of cracked elastic media. The models suggest that seismic velocities of sandstones may be controlled by cracks and pores with aspect ratios in the range of 10-1 to 1, dolomite in the range of 10-2 to 10-1, and dens...

213 citations


Journal ArticleDOI
TL;DR: In this article, the effect of particulate anisotropy on the electrical properties of sedimentary rocks has been examined by generalizing the treatment of Sen et al. to the case of ellipsoidal grains with a distribution of orientations and depolarizing factors.
Abstract: 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, σ=σwϕm, with σw 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 1.2≲m≲4. For aligned grains the conductivity is a tensor with principal values that satisfy a generalized Archie’s law of the form σj=aj(ϕ)σwϕm, where σj is the jth principal value of the conductivity and aj(ϕ) ...

211 citations


Journal ArticleDOI
TL;DR: In this article, an iterative algorithm for the inversion of the one-dimensional (1-D) wave equation, together with a stabilizing constraint on the sums of the jumps of the desired impedance, is proposed.
Abstract: The well‐known instability of Kunetz’s (1963) inversion algorithm can be explained by the progressive manner in which the calculations are done (descending from the surface) and by the fact that completely different impedances can yield indistinguishable synthetic seismograms. Those difficulties can be overcome by using an iterative algorithm for the inversion of the one‐dimensional (1-D) wave equation, together with a stabilizing constraint on the sums of the jumps of the desired impedance. For computational efficiency, the synthetic seismogram is computed by the method of characteristics, and the gradient of the error criterion is computed by optimal control techniques (adjoint state equation). The numerical results on simulated data confirm the expected stability of the algorithm in the presence of measurement noise (tests include noise levels of 50 percent). The inversion of two field sections demonstrates the practical feasibility of the method and the importance of taking into account all internal a...

156 citations


Journal ArticleDOI
TL;DR: In this article, the Laplace transform of the solution for TEM soundings over an N-layer earth was derived and used to invert it numerically using the Gaver-Stehfest algorithm.
Abstract: Calculations for the transient electromagnetic (TEM) method are commonly performed by using a discrete Fourier transform method to invert the appropriate transform of the solution. We derive the Laplace transform of the solution for TEM soundings over an N-layer earth and show how to use the Gaver-Stehfest algorithm to invert it numerically. This is considerably more stable and computationally efficient than inversion using the discrete Fourier transform.

Journal ArticleDOI
TL;DR: In this paper, a point-source seismic recording can be decomposed into a set of plane-wave seismograms for arbitrary angles of incidence, which make them amenable to existing inversion methods such as predictive deconvolution.
Abstract: A point-source seismic recording can be decomposed into a set of plane-wave seismograms for arbitrary angles of incidence. Such plane-wave seismograms possess an inherently simple structure that make them amenable to existing inversion methods such as predictive deconvolution. Implementation of plane-wave decomposition (PWD) takes place in the frequency-wavenumber domain under the assumption of radial symmetry. This version of PWD is equivalent to slant stacking if allowance is made for the customary use of linear recording arrays on the surface of a three-dimensional medium. An imaging principle embodying both kinematic as well as dynamic characteristics allows us to perform time migration of the plane-wave seismograms. The imaging procedure is implementable as a two-dimensional filter whose independent variables are traveltime and angle of incidence.

Journal ArticleDOI
TL;DR: In this paper, a plane-geometry model involving a layer of fluid between two elastic half-spaces is shown to provide velocity dispersion curves for propagating modes that are very similar to those for the fluid-filled borehole.
Abstract: Acoustic waveform measurements in boreholes have important applications in fracture hydrology and radioactive waste disposal, but ambiguities in existing interpretation techniques remain a problem. We have addressed the problem by using residue theory to predict the relative excitation of various modes contained in experimental waveforms. A plane‐geometry model involving a layer of fluid between two elastic half‐spaces is shown to provide velocity dispersion curves for propagating modes that are very similar to those for the fluid‐filled borehole. We use the plane‐geometry model to illustrate the effects of the confined borehole fluid on surface and body waves traveling along the borehole in the elastic solid. We also computed excitation functions for some of the lowest‐order symmetric modes, calculated the time‐domain response of the trapped modes following the shear head waves, and compared them to waveforms recorded in boreholes through several homogeneous formations. The insight into the mode composit...

Journal ArticleDOI
TL;DR: In this paper, Radon measurements in the ground surface over the Laisvall lead mine have given evidence of radon transport through rock exceeding a distance of 100 m, which is possible only if the migration is a flow transport with a characteristic transport length larger than about 10 m/day.
Abstract: Concentration of radon has been measured in the soil near the ground surface with solid‐state, nuclear track detectors with the inverted cup technique. Measurements were made in the overburden at depth intervals 0.1–0.7 m, at 0.1–6 m, and at a constant depth of 0.2 m, in a narrow rectangular matrix. The results disagree with the hypothesis that radon concentration only depends upon local production and migration by diffusion with a diffusion length of about 1 m. A transport length of 0.1–0.2 m is observed near the ground surface and the transport is dominated by a flow component. Radon measurements in the ground surface over the Laisvall lead mine have given evidence of radon transport through rock exceeding a distance of 100 m, which is possible only if the migration is a flow transport with a characteristic transport length larger than about 10 m/day. To explain the radon transport in the overburden and through the rock with a common transport system, the existence of a general upward flow of geo‐gas is...

Journal ArticleDOI
TL;DR: Cohen et al. as discussed by the authors found that the interaction between clay minerals and groundwater can produce polarization phenomena, decreases in resistivity, and supposed nonlinear behavior, which cannot be predicted by bulk electrical properties of the constituents.
Abstract: Rocks which contain clay minerals often display electrical properties which cannot be predicted by bulk electrical properties of the constituents (Cohen, 1981). Interactions between clay minerals and groundwater can produce polarization phenomena, decreases in resistivity, and supposed nonlinear behavior.

Journal ArticleDOI
TL;DR: In this article, the basic problems encountered in extracting estimates of seismic dissipation from data recorded on vertical seismic profiles are analyzed, and a statistical perspective is introduced to simulate multiple measurements in an inhomogeneous interval and to draw conclusions which apply to an entire class of impedance structures.
Abstract: The basic problems encountered in extracting estimates of seismic dissipation from data recorded on vertical seismic profiles are analyzed. Because anomalous dissipation in the subsurface is likely to be associated with conditions or lithologies of limited vertical extent, a knowledge of the factors which influence the spatial resolution of an attenuation measurement is of considerable importance. By introducing a statistical perspective, it is possible to simulate multiple measurements in an inhomogeneous interval and to draw conclusions which apply to an entire class of impedance structures. Theoretical seismograms are analyzed to demonstrate that for small receiver separations neither a single measurement nor the mean value determined from multiple measurements is likely to give a good estimate of the attenuation for an inhomogeneous depth interval. For small receiver separations, the attenuation computed from the amplitude ratios method is much more strongly influenced by the local stratigraphy in the...

Journal ArticleDOI
TL;DR: In this article, it was shown that for any physically reasonable frequency-independent distribution of electrical conductivity and magnetic permeability, the voltage response to a step function driving current is of one sign only.
Abstract: The occasional occurrence of persistent sign reversals in coincident loop transient electromagnetic (TEM) measurements stimulates an investigation of possible causes for this effect. By examining the response in the complex frequency plane near the spectrum of freely decaying current modes, it is shown that for any physically reasonable frequency‐independent distribution of electrical conductivity and magnetic permeability the voltage response to a step function driving current is of one sign only. Moreover, under the conditions mentioned above, the logarithm of the induced voltage is a decreasing convex function of time. These characteristics are retained for more general time functions of the driving current. The conservation of sign for frequency‐independent material parameters supports the assumption of IP effects as a possible mechanism for sign reversals. The latter point is illustrated by a simplified example.

Journal ArticleDOI
TL;DR: In this article, the authors proposed an approach for the extraction of scalar parameters from the impedance tensor Z, the transfer function which relates the observed horizontal magnetic and electric fields.
Abstract: An important step in the interpretation of magnetotelluric (MT) data is the extraction of scalar parameters from the impedance tensor Z, the transfer function which relates the observed horizontal magnetic and electric fields. The conventional approach defines parameters in terms of elements of a coordinate‐rotated tensor. The rotation angle is chosen such that Z′(θ) approximates in some sense the form for a two‐dimensional (2-D) subsurface conductivity distribution, with zero elements on the diagonal. There are two major problems with this approach. (1) Apparent resistivities, defined from the off‐diagonal elements of the rotated tensor, are independent of the trace of Z. It is problematic that apparent resistivities, the parameters for which we have physical analogs and which are most heavily used in interpretation, are insensitive to the addition of an arbitrary constant on the diagonal of Z. (2) The conventional parameter set is incomplete; there are two degrees of freedom in Z which are transparent t...

Journal ArticleDOI
TL;DR: In this paper, the main cause of the anomalous transient has been identified as the response of superparamagnetic material in the lateritic soil cover, which causes apparent resistivity values derived from the measured transient decay to decrease at late delay times in areas where the known geology indicates the values should increase with delay time toward the basement.
Abstract: The development of an instrument that enables transient electromagnetic (TEM) measurements to be made to voltage levels of 1 μV/A and less has enabled the detection of an anomalous transient response in some areas with lateritic soil cover. This anomalous transient causes apparent resistivity values derived from the measured transient decay to decrease at late delay times in areas where the known geology indicates the values should increase with delay time toward the resistivity value of the basement. The main cause of the anomalous transient has been identified as the response of superparamagnetic material in the lateritic soil cover. Both field and laboratory measurements of the voltage M induced by this transient, show a t-1 time dependence. This is the same behavior reported previously for magnetic viscosity over a longer time scale. Measurements of magnetic susceptibility of material separated magnetically from soil samples taken at areas where a residual response is measured, show that over a wide t...

Journal ArticleDOI
TL;DR: In this paper, a derivative-free Levenberg-Marquardt algorithm is used to estimate values for the depth, vertical extent, length, and intensity of the source region.
Abstract: Self‐potential (SP) data from the Cerro Prieto geothermal field in Baja California, Mexico have been inverted using a model consisting of a vertical contact separating regions of different electrical properties. A temperature source is assumed to coincide with the vertical contact between materials with different thermoelectric coupling coefficients. A derivative‐free Levenberg‐Marquardt algorithm is used to estimate values for the depth, vertical extent, length, and intensity of the source region. The depth to the top of the source is estimated to be about 1.3 ± 0.2 km, which agrees quite well with the depth to the top of the production zone determined from drilling. The vertical extent and length of the source region are estimated to be 11 ± 3 km and 9.9 ± 0.4 km, respectively. There appears to be geologic evidence for the presence of a fault or fault zone within the geothermal field that roughly coincides in location with the self‐potential source region. The conductivity on the east side of the produc...

Journal ArticleDOI
TL;DR: In this paper, the effect of the borehole fluid on the P and S wave radiation pattern and the S-wave radiation pattern was investigated using a low-frequency approximation, using a volume displacement point source acting on the axis of a borehole.
Abstract: This paper concerns far‐field radiation of compressional P and shear S waves into the surrounding medium from a fluid‐filled borehole in an infinite medium and tube waves propagating along a borehole, using a low‐frequency approximation. Two kinds of sources are considered: (1) a volume displacement point source acting on the axis of a borehole, and (2) a uniform radial stress source acting on the wall of a borehole. When the tube‐wave velocity is close to the shear‐wave velocity, the effect of the borehole fluid on the P‐wave radiation pattern and on the S‐wave radiation pattern is substantial.

Journal ArticleDOI
TL;DR: In this article, a controlled source audiomagnetotelluric (CSAMT) method was proposed to delineate the shallow resistivity pattern above a hydrothermal system.
Abstract: Theoretical and field tests indicate that the controlled‐source audiomagnetotelluric (CSAMT) method provides an efficient means of delineating the shallow resistivity pattern above a hydrothermal system. Utilizing a transmitter overcomes the main limitation of conventional audiomagnetotellurics—variable and unreliable natural source fields. Reliable CSAMT measurements can be made with a simple scalar receiver. Our calculations for a half‐space show that the plane‐wave assumption is valid when the transmitter is more than 3 skin depths away in the broadside configuration and more than 5 skin depths away in the collinear configuration. Three‐dimensional (3-D) numerical modeling results for a bipole source 5 skin depths away compare well with those for a plane‐wave source, showing that the method is valid. A CSAMT survey at the Roosevelt Hot Springs geothermal area in Utah produced apparent resistivity contour maps at four frequencies: 32, 98, 977, and 5208 Hz. These maps show the same features as those of a...

Journal ArticleDOI
TL;DR: In this article, the Backus-Gilbert inverse theory was applied to the Wiener shaping filter for source wavelet deconvolution in exploration seismology, and the results provided novel insights into the performance.
Abstract: Seismic source wavelet deconvolution can be treated within the framework of the Backus‐Gilbert (BG) inverse theory. A time shift‐invariant version of this theory leads to the Wiener shaping filter, which has enjoyed widespread use for source wavelet deconvolution in exploration seismology. The model of the BG theory is the ground impulse response, the BG mapping kernel is the source wavelet, and the BG resolving kernel is the convolution between the source wavelet and the Wiener shaping filter. BG inversion involves the minimization of an optimality criterion under a set of constraints. The application of the BG “filter energy” or “noise output power” constraint to Wiener filter design leads to the familiar prewhitening parameter that stabilizes the filter on the one hand, but degrades resolution on the other. The BG “unimodular” constraint produces an unbiased estimate of the model, or ground impulse response. These constraints provide novel insights into the performance of deconvolution filters.

Journal ArticleDOI
TL;DR: In this article, a quantitative formulation of vertical resolving power of seismic exploration systems is presented and is offered as a proposed characteristic, or standard, resolving power identified with individual systems, which broadens the classical concept of resolution by taking into account the reflection waveform and the noise, in addition to the classical time variable.
Abstract: A quantitative formulation of vertical resolving power of seismic exploration systems is presented and is offered as a proposed characteristic, or standard, resolving power identified with individual systems. The formulation broadens the classical concept of resolution by taking into account the reflection waveform and the noise, in addition to the classical time variable. The principal feature in the formulation is the stipulation that the intratrace distribution of reflections and of noise be treated as random (Gaussian) distribution, which is regarded as the most general representation for seismic sections as a whole.Through this quantification of vertical resolving power and therefore of intratrace reflection quality, a number of elemental reflection properties that have been described only qualitatively in the past are expressed by simple formulas. The quantification is consistent with the concept that the resolving power of a noise-free zero-phase system with a flat spectral band response is proportional to the bandwidth. The derived basic formula for the proposed characteristic resolving power is a 2 m /E, where a m is the maximum (absolute) amplitude of the signal wavelet of a seismogram interval, E is the energy of the signal wavelet, and noise is neglected. The quantification of the reflection properties, including taking the noise into account, stems from this formula.The classical concept of resolution, which considers only the time variable, such as the dominant period of signal wavelets, is applicable essentially only in cases of two noise-free equal-strength reflections. In contrast, the proposed formulation of resolving power accommodates a wide scope of applications and might be considered basic to seismic systems. I present theoretical material for evaluating the merits of the proposal. Suitable comparisons by seismic modeling would be useful in the overall evaluation.

Journal ArticleDOI
TL;DR: In this paper, the Hilbert transform was used for interpreting vertical magnetic anomalies of sheets (finite and infinite depth extent), dikes, and horizontal circular cylinders (HULC).
Abstract: 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.

Journal ArticleDOI
Bjoern Ursin1
TL;DR: In this paper, a quadratic approximation for the travel time from a source region to a receiver region is given for a three-dimensional (3-D) medium consisting of inhomogeneous layers with curved interfaces.
Abstract: A quadratic approximation for the square of the traveltime from a source region to a receiver region is given for a three-dimensional (3-D) medium consisting of inhomogeneous layers with curved interfaces. The square of the traveltime, being a function of source and receiver coordinates, is developed in a Taylor series around a reference source and receiver point. The relationships of the traveltime parameters to the ray parameters and the wavefront curvature matrices are shown. Using midpoint, half-offset coordinates gives a simplified traveltime function compared to using source-receiver coordinates only in the case that the reference source point and the reference receiver point coincide (zero-offset approximation). For a medium consisting of homogeneous layers with plane dipping interfaces, the traveltime approximation is further simplified. The derived traveltime approximation is shown to be exact for a reflection from a plane dipping interface in a homogeneous medium.Explicit expressions for the traveltime parameters in terms of the layer parameters are derived for a horizontally layered medium. The traveltime errors of two different approximations are compared for a given layered model in a numerical example.

Journal ArticleDOI
TL;DR: In this paper, a simple data transformation to the ray parameter domain via the slanted plane-wave stack is used to organize three types of arrivals from any given interface into a single elliptical trajectory.
Abstract: 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.

Journal ArticleDOI
TL;DR: The problem with existing finite-difference formulations of the elastic wave equation is that they have a limited stability range, and the necessity of taking small time steps can result in excessively high computation costs as discussed by the authors.
Abstract: The problem with existing finite‐difference formulations of the elastic wave equation is that they have a limited stability range, and the necessity of taking small time steps can result in excessively high computation costs. It is possible to formulate a finite‐difference scheme which is stable for arbitrarily large time steps. However, the solutions obtained by the unconditionally stable scheme are unacceptably inaccurate for time steps outside the stability range of finite‐difference schemes currently in use.

Journal ArticleDOI
TL;DR: In this article, a simple, inexpensive numerical algorithm is used to analyze the asymptotic long-period behavior of magnetotelluric (MT) fields in the vicinity of lateral offsets in sedimentary basins.
Abstract: A simple, inexpensive numerical algorithm is used to analyze the asymptotic long-period behavior of magnetotelluric (MT) fields in the vicinity of lateral offsets in sedimentary basins. The model is based on the distortion or channeling of telluric currents in a horizontal thin sheet. Although a gross oversimplification of nature, the model represents a class of structures which, because of excessive computer costs, have been relatively unstudied previously. Within, and closely adjacent to, the region of the three-dimensional (3-D) offset, significant distortion of the MT parameters occurs. Skewness coefficients vary from negligible values to over 0.7. Principal resistivities vary by an order of magnitude. On the other hand, there is not a clear correlation between the degree of distortion of the parameters usually evaluated during MT surveys and the magnitude of conventional 3-D indicators (e.g., the skewness coefficient). Calculations have simulated the technique of averaging resistivity parameters from a large number of field sites in order to arrive at a regionally representative one-dimensional (1-D) model. The results indicate that unless care is taken in adapting the nature of the averaging algorithm to the class of distortions encountered, significant bias of the averaged parameters may result. These results also suggest that formore » this class of structures grave problems may be associated with using the principal resistivity perpendicular to geologic strike, the so-called transverse magnetic (TM) mode, to infer an equivalent two-dimensional (2-D) model for the region. A 2-D model would likely show significant modulations in the physical character of the basement which are, in fact, an artifact of telluric distortion caused by current channeling in the surficial heterogeneity. 10 figures, 4 tables.« less

Journal ArticleDOI
TL;DR: Fougere et al. as discussed by the authors showed that the Burg spectra display line splitting in the presence of low noise, and as the noise is increased, the multiple peaks coalesce into a single peak shifted substantially away from the correct value.
Abstract: Spectral analysis is a very useful technique for studying geophysical problems. In earlier days, the only methods available were those of Fourier analysis or the method of Blackman and Tukey (1959) based on autocorrelation function. Recently, Burg (1967, 1968) introduced maximum entropy spectral analysis (MESA) which gives good resolution even for periods comparable to the data length. Ulrych and Bishop (1975) gave a critical appraisal of Burg’s algorithm. Several workers noticed and reported some inherent shortcomings. Thus, Chen and Stegan (1974) showed that, for truncated sinusoids, the spectral maxima showed frequency shifts sometimes as large as 20 percent, depending upon the initial phase and the length of the sample. Also, under certain conditions, the Burg spectra display line‐splitting in the presence of low noise, and as the noise is increased, the multiple peaks coalesce into a single peak shifted substantially away from the correct value (Fougere et al, 1976; Fougere, 1977). These defects can ...