Showing papers in "Geophysics in 1981"

A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads

Abstract: We develop a theory for dielectric response of water‐saturated rocks based on a realistic model of the pore space. The absence of a percolation threshold manifest in Archie’s law, porecasts, electron‐micrographs, and general theories of formation of detrital sedimentary rocks indicates that the pore spaces within such rocks remain interconnected to very low values of the porosity ϕ. In the simplest geometric model for which the conducting paths remain interconnected, each grain is envisioned to be coated with water. The dielectric constant of the assembly of water‐coated grains is obtained by a self‐consistent effective medium theory. In the dc limit, this gives Maxwell’s relation for conductivity σ of the rock σ=2σwϕ/(3-ϕ), where σw is the conductivity of water. In order to include the local environmental effects around a grain, a self‐similar model is generated by envisioning that each rock grain itself is coated with a skin made of other coated spheres; the coating at each level consists of other coate...

Analysis of dispersive waves by wave field transformation

Abstract: The dispersive waves in a common‐shot wave field can be transformed into images of the dispersion curves of each mode in the data. The procedure consists of two linear transformations: a slant stack of the data produces a wave field in the phase slowness‐time intercept (p — τ) plane in which phase velocities are separated. The spectral peak of the one‐dimensional (1-D) Fourier transform of the p — τ wave field then gives the frequency associated with each phase velocity. Thus, the data wave field is linearly transformed from the time‐distance domain into the slowness‐frequency (p — ω) domain, where dispersion curves are imaged. All the data are present throughout the transformations. Dispersion curves for the mode overtones as well as the fundamental are directly observed in the transformed wave field. In the p — ω domain, each mode is separated from the others even when its presence is not visually detectable in the untransformed data. The resolution achieved in the result is indicated in the p — ω wave ...

Elastic wave propagation in a fluid-filled borehole and synthetic acoustic logs

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

Ultrasonic velocities in Cretaceous shales from the Williston basin

Abstract: Compressional and shear‐wave velocities were measured in the laboratory from 1 bar to 4 kbar confining pressure for wet, undrained samples of Cretaceous shales from depths of 3200 and 5000 ft in the Williston basin, North Dakota. These shales behave as transversely isotropic elastic media, the plane of circular symmetry coinciding with the bedding plane. For compressional waves, the velocity is higher for propagation in the bedding plane than at right angles to it, and the anisotropy is greater for the 5000-ft shale. For shear waves, the SH‐wave perpendicular to bedding and the SV‐wave parallel to bedding propagate with the same speed, which is about 25 percent lower than that for the SH‐wave parallel to bedding. In general, compressional and shear velocities are higher for the indurated 5000-ft shale than for the friable 3200-ft shale. All velocities increase with in‐increasing confining pressure to 4 kbar. The 3200-ft shale exhibits velocity hysteresis as a function of pressure, whereas this effect is a...

A Born-WKBJ inversion method for acoustic reflection data

Abstract: Density and bulk modulus variations in an acoustic earth are separately recoverable from standard reflection surveys by utilizing the amplitude-versus-offset information present in the observed wave fields. Both earth structure and a variable background velocity can be accounted for by combining the Born and WKBJ approximations, in a "before stack" migration with two output sections, one for density variations and the other for bulk modulus variations. For the inversion, the medium is considered to be composed of a known low-spatial frequency variation (the background) plus an unknown high-spatial frequency variation in bulk modulus and density (the reflectivity). The division between the background and the reflectivity depends upon the frequency content of the source. For constant background parameters, computations are done in the Fourier domain, where the first part of the algorithm includes a frequency shift identical to that in an F-K migration. The modulus and density variations are then determined by observing in a least-squares sense amplitude versus offset wavenumber. For a spatially variable background, WKBJ Green's operators that model the direct wave in a medium with a smoothly varying background are used. A downward continuation with these operators removes the effects of variable velocity from the problem, and, consequently, the remainder of the inversion essentially proceeds as if the background were constant. If the background is strictly depth dependent, the inversion can be expressed in closed form. The method neglects multiples and surface waves and it is restricted to precritical reflections. Density is distinguishable from bulk modulus only if a sufficient range of precritical incident angles is present in the data.

The traveltime equation, tau‐p mapping, and inversion of common midpoint data

Abstract: Seismic traveltime data can be described in terms of instantaneous slope (dT/dX) and intercept time (τ). With this parameterization, the most general form of the traveltime equation is developed and found to be valid for reflections and refractions, all source‐receiver offsets, and all commonly used experimental geometries. For common source‐receiver experiments, the instantaneous slope of the traveltime data represents the true horizontal ray parameter p, and the intercept time is the sum along the ray of thickness‐vertical slowness products referenced to the fixed surface station. For common midpoint (CMP) experiments, we show that the instantaneous slope is the average slowness of upgoing and downgoing rays at the surface, and the intercept time is again the sum of thickness‐vertical slowness products, but referenced to the common midpoint. This inherent averaging immediately explains the superiority of the CMP experiment in the presence of dip. In addition to providing a general traveltime equation, w...

Direct mapping of seismic data to the domain of intercept time and ray parameter—A plane‐wave decomposition

Abstract: Marine seismic data recorded as a function of source‐receiver offset and traveltime are mapped directly to the domain of intercept or vertical delay time and horizontal ray parameter. This is a plane‐wave decomposition based on beam forming of wide‐aperture seismic array data to determine automatically the loci of coherent seismic reflection and refraction events. In this computation, semblance, in addition to the required slowness or horizontal ray parameter stack, is found for linear X — T trajectories across subarrays. Subsequently, semblance is used to derive a windowing filter that is applied to the slowness stack to determine the points of stationary phase and eliminate aliasing. The resulting filtered slowness stacks for multiple subarrays can then be linearly transformed and combined according to ray parameter, range, and time. The resulting function of intercept time and horizontal ray parameter offers significant computational and interpretational advantages for the case of horizontal homogeneou...

Reconstruction of a sparse spike train from a portion of its spectrum and application to high-resolution deconvolution

Abstract: An algorithm is proposed for the reconstruction of a sparse spike train from an incomplete set of its Fourier components. It is shown that as little as 20–25 percent of the Fourier spectrum is sufficient in practice for a high‐quality reconstruction. The method employs linear programming to minimize the L1‐norm of the output, because minimization of this norm favors solutions with isolated spikes. Given a wavelet, this technique can be used to perform deconvolution of noisy seismograms when the desired output is a sparse spike series. Relative reliability of the data is assessed in the frequency domain, and only the reliable spectral data are included in the calculation of the spike series. Equations for the unknown spike amplitudes are solved to an accuracy compatible with the uncertainties in the reliable data. In examples with 10 percent random noise, the output is superior to that obtained using conventional least‐squares techniques.

Measurements of attenuation from vertical seismic profiles

Abstract: Values of attenuation have been extracted from five detailed velocity surveys. Significant amounts of attenuation are observed in all five wells. Measured values of attenuation vary by a factor of 10, ranging from less than 0.1 to 0.9 dB/wavelength. Correlation of attenuation with lithology is good, considering the resolution limits of the surveys. One of the surveys was performed in West Texas, and the other four were carried out in the Gulf Coast area. The total number of recording depths in each survey varied from 31 to 208, with downhole recordings taken every 10 or 20 ft over key stratigraphic zones. In all five surveys, a monitor geophone was placed near the well to check for source consistency. Our procedure for measuring attenuation involved constructing the spectral ratio between each downhole pulse and one particular reference pulse. A value of cumulative attenuation (CA) was then extracted from each spectral ratio and plotted as a function of geophone depth. This method allowed us to use many m...

An investigation of finite-element modeling for electrical and electromagnetic data in three dimensions

Abstract: The finite-element method can be used to solve the differential equations which describe electrical and electromagnetic (EM) field behavior. The equations are, respectively, Poisson's equation and the vector, damped wave equation. The finite-element equations are derived, in both cases, using the minimum theorem. While both tetrahedral and hexahedral elements may be used for the modeling of the resistivity problem, only hexahedral elements give satisfactory results for the EM problem. A disadvantage of the relatively simple mesh design used in the approach described here is the presence of long thin elements. Such elements have very poor interpolating properties, and they adversely affect the rate of convergence of the overrelaxation technique used in solving the resulting system of linear equations.For the modeling of resistivity data over an earth with one plane of symmetry, the system of equations typically has about 9000 unknowns. About 50,000 unknowns are needed to give a satisfactory solution to an EM problem where the earth has one plane of symmetry. The advantage of solving these problems with a technique such as the finite-element method is that earths with an almost arbitrary distribution of conductivity can be modeled. On the other hand, an integral-equation method can be far more cost effective for small inhomogeneities. The results from the resistivity algorithm show the adverse effect of an irregular, conducting, and polarizable overburden on dipole-dipole, induced polarization surveys. Modeling of a horizontal loop EM survey illustrates the importance of assessing the host rock conductivity before attempting to interpret inhomogeneity responses.

An Introduction to the generalized reciprocal method of seismic refraction interpretation

Abstract: The generalized reciprocal method (GRM) is a technique for delineating undulating refractors at any depth from in-line seismic refraction data consisting of forward and reverse traveltimes.The traveltimes at two geophones, separated by a variable distance XY, are used in refractor velocity analysis and time-depth calculations. At the optimum XY spacing, the upward traveling segments of the rays to each geophone emerge from near the same point on the refractor. This results in the refractor velocity analysis being the simplest and the time-depths showing the most detail. In contrast, the conventional reciprocal method which has XY equal to zero is especially prone to produce numerous fictitious refractor velocity changes, as well as producing gross smoothing of irregular refractor topography.The depth conversion factor is relatively insensitive to dip angles up to about 20 degrees, because both forward and reverse data are used. As a result, depth calculations to an undulating refractor are particularly convenient even when the overlying strata have velocity gradients.The GRM provides a means of recognizing and accommodating undetected layers, provided an optimum XY value can be recovered from the traveltime data, the refractor velocity analysis, and/or the time-depths. The presence of undetected layers can be inferred when the observed optimum XY value differs from the XY value calculated from the computed depth section. The undetected layers can be accommodated by using an average velocity based on the optimum XY value. This average velocity permits accurate depth calculations with commonly encountered velocity contrasts.

Modeling of the acoustic wave equation with transform methods

Abstract: Numerical methods are described for the simulation of wave phenomena with application to the modeling of seismic data. Two separate topics are studied. The first deals with the solution of the acoustic wave equation. The second topic treats wave phenomena whose direction of propagation is restricted within ±90 degrees from a given axis. In the numerical methods developed here, the wave field is advanced in time by using standard time differencing schemes. On the other hand, expressions including space derivative terms are computed by Fourier transform methods. This approach to computing derivatives minimizes truncation errors. Another benefit of transform methods becomes evident when attempting to restrict propagation to upward moving waves, e.g., to avoid multiple reflections. Constraints imposed on the direction of the wave propagation are accomplished most precisely in the wavenumber domain. The error analysis of the algorithms shows that truncation errors are due mainly to time discretization. Such er...

Integral equation modeling of three-dimensional magnetotelluric response

Abstract: We have adapted a three‐dimensional (3-D) volume integral equation algorithm to magnetotelluric (MT) modeling. Incorporating an integro‐difference scheme increases accuracy somewhat. Utilizing the two symmetry planes of a buried prismatic body and a normally incident plane wave source greatly reduces required computation time and storage. Convergence checks and comparisons with one‐dimensional (1-D) and two‐dimensional (2-D) models indicate that our results are valid. We show theoretical surface anomalies due to a 3-D prismatic conductive body buried in a half‐space earth. Instead of studying the electric and magnetic fields, we have obtained impedance tensor and magnetic transfer functions by imposing two different source polarizations. Manipulation of the impedance tensor and magnetic transfer functions yields the following MT quantities: apparent resistivity and phase, impedance polar diagrams, tipper direction and magnitude, principal directions, skew, and ellipticity. With our preliminary analyses of...

Surface consistent corrections

Abstract: Amplitudes of seismic reflections have been of interest since the first days of exploration seismology. Any change of amplitude or anomalous behavior may be significant, so it is important that the zones of interest be free from outside disturbances, such as those caused by the near‐surface layers. Surface consistent factors may be divided into source, receiver, offset, and subsurface components, and these may be divided further into amplitude and phase (or time shift) factors. Correction of trace amplitudes using multiplication by a scale factor is similar to correction of phase distortions by a static shift, and both corrections enhance seismic data. Displays of surface consistent components for time and amplitude corrections provide an additional diagnostic for the geophysicist.

Analysis of electrical conductivity imaging

Abstract: We investigate the feasibility of imaging the electrical conductivity in a cross-section of an object (such as a core sample) by numerical inversion of low-frequency, electromagnetic (EM) boundary data. Current flow is assumed to be confined to the cross-section, which is modeled as a network of resistors. The network serves as a discrete approximation of the distributed-parameter system that is described mathematically by Maxwell9s equations for steady current flow in a nonhomogeneous medium. A complete set of linearly independent voltage vectors is applied to the peripheral nodes, and the resulting node currents serve as the measured data for estimating the internal conductivity pattern (image). We generate estimates of this conductivity image by using an iterative process on network equations that are linearized in the unknown conductance variables.The mathematical feasibility of this approach is demonstrated by computer simulation studies using data generated from the network model. Reconstructed images are presented for sample conductance patterns under both ideal and noisy data conditions. An error analysis is performed to relate data noise to image-estimation error.

Relation of certain geometrical features to the dielectric anomaly of rocks

Abstract: We show that the real part of the dielectric constant e′ of rocks at low frequencies can be anomalously high due to the presence of a small concentration η of high aspect ratio particles. For oblate spheroidal grains (a≪b=c) with depolarization factor along symmetry (-a) axis, Ls≅1-δ, δ=πa/2b, the static value of the dielectric constant of rock es and dc conductivity σ(0) are given for (1)δ ηbyσ(0)≅σR(1-η/δ), es≅ηe′m/δ2. Here e′m is the dielectric constant of the grain; σR is the dc conductivity of the host rock. Case (1) corresponds to the well known Maxwell‐Wagner effect with es diverging as η → 0, and σ(0)→0. Case (2) gives a novel result that es may diverge for δ>η≫δ2, with a nonvanishing σ(0). Case (2) is applied to explain frequency and salinity dependences and the giant values (∼104) of the dielectric constant of conducting sedimentary rocks. For η∼10-4, δ∼10-3, e′m=10, we find es∼1000, which is large compared to e′m or the dielectric constant of water e′w(∼80).

Wave field extrapolation techniques in seismic migration; a tutorial

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

Inversion of refraction data by wave field continuation

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

Algebraic reconstruction of a two‐dimensional velocity inhomogeneity in the High Hazles seam of Thoresby colliery

Abstract: A high proportion of faces started in mechanized coal mines run into underground faults The faults take many forms, from the splitting of a seam through a hidden stress pattern caused by subsidence or folding, to a washout or a vertical throw All faults reduce face output A throw of only 15 m can lead to a face being abandoned Faults of this order cannot be mapped reliably from the surface They may be mapped in an underground seismic surveyRoadways of a mine may give access to fast refracting horizons above and below a coal seam Waveguiding in the plane of the seam, if it occurs, simplifies migration of wave trains reflected from discrete faults The reduction problem involved in a fault mapping is only two-dimensional Given guiding, whether leaky or not, it is possible to map distributed faults of low reflectivity by shooting in transmission across a panel of coalAlgebraic reconstruction techniques are used here to reduce first break times-of-flight through a 425 X 950 m rectangular block of coal into the profile of a velocity inhomogeneity Input data are derived by static correction from hand-picked arrival times The reduction itself is effected using an algorithm which accommodates underground site access restrictions In back projecting first break velocities, a truncated cosine is used to weight the relative contributions of rays passing at different distances from any given mapping pointThe reconstructed velocity field suggests that the coal panel is bisected by a ridge of higher velocity The suspicion of a ridge is reinforced by results of an aberration test based on a standard Huygens-Kirchhoff migration The ridge is found to follow the general line of a system of pillars left in place during the mining of a lower horizon It is concluded that channel waves may be used to map subsidence into old workings underground Coal seams apparently share, with other sedimentary rocks, the property of a pressure-sensitive seismic velocity

Fundamentals of frequency domain migration

Abstract: Frequency domain migration is founded upon the wave equation, and so includes diffractions and other effects. This paper seeks to motivate and illuminate frequency domain migration using straightforward geometric techniques and simple frequency domain observations.

Interpretation of magnetic anomalies due to dikes; the complex gradient method

Abstract: A method to interpret the magnetic anomaly due to a dipping dike using the resultant of the horizontal and vertical gradients of the anomaly is suggested. The resultant of both the gradients is a vector quantity and is defined as the “complex gradient.” A few characteristic points defined on the amplitude and phase plots of the complex gradient are used to solve for the parameters of the dike. For a dike uniformly magnetized in the earth’s magnetic field, the amplitude plot is independent of θF, the index parameter, which depends upon the strike and dip of the dike and the magnetic inclination of the area. The phase plot of the complex gradient is an antisymmetric curve with an offset value equal to -θF. For a dike whose half‐width is greater than its depth of burial, two maxima at equal distances on either side of a minimum value appear on the amplitude plot. For a dike whose half‐width is equal to or less than its depth of burial, the amplitude plot is a bell‐shaped symmetric curve with its maximum appe...

A method for calculating synthetic seismograms which include the effects of absorption and dispersion

Abstract: A method is outlined for the calculation of synthetic seismograms which include the effects of absorption and dispersion. The absorption model used is the usual model of exponential decay of amplitude with distance given by A=A0e-αz, where α is a linear function of frequency. This attenuation is accounted for mathematically by allowing the elastic modulus to be a complex function of frequency. This results in a complex velocity and wavenumber, and the reflection and transmission coefficients also become complex functions of frequency. The method is based upon the communication theory approach and is applicable to plane waves in a flat layered model. The source can be placed at an arbitrary depth. The equations are outlined in detail for a particular absorption‐dispersion pair taken from Futterman (1962). An example with a surface synthetic seismogram and synthetic traces at several depths is presented.

Inversion of reflection times in three dimensions

Abstract: When reflection data are available from a grid of crossing seismic lines, it is possible to construct normal incidence time maps from interpreted stacked sections and then apply three‐dimensional (3-D) ray‐tracing techniques following the normal‐incidence raypaths down to the various reflectors. The main disadvantage of this well‐known “time map migration” procedure is that interval velocities must be known a priori, and they must be estimated in advance by some approximate method. A technique is presented here which combines the above procedure with an inversion algorithm, providing direct calculations of interval velocities from the additional use of nonzero offset traveltime observations. A generalized linear inversion scheme is used, making possible a complete calculation of interval velocities and reflection interfaces, the latter represented by bicubic spline functions. To test the method in practice, we have applied it to (1) synthetic data generated from a constructed model, and (2) real data obta...

Depth migration of imaged time sections

Abstract: None of the leading approaches to the migration of seismic sections—the Kirchhoff‐summation method, the finite‐difference method, or the frequency‐domain method—readily migrates seismic reflections to their proper positions when overburden velocities vary laterally. For inhomogeneous media, the diffraction curve for a localized, buried scatterer is no longer hyperbolic and its apex is displaced laterally from the position directly above the scatterer. Hubral observed that the Kirchhoff‐summation method images seismic data at emergent “image ray” locations rather than at the desired positions vertically above scatterers. In addition, distortions in diffraction shapes lead to incorrect imaging (i.e., incomplete diffraction collapse) and, hence, to further displacement errors for dipping reflections. The finite‐difference method has been believed to continue waves downward correctly through inhomogeneous media. In conventional implementations, however, both the finite‐difference method and frequency‐domain a...

Data‐adaptive polarization filters for multichannel geophysical data

Abstract: The design of data‐adaptive filters requires that the noise be defined, statistically or otherwise, by parameters which allow some means of separating the noise from the signal. We consider here multichannel data in which one knows only that the noise is less polarized than the signal in a unitary space. This description of the noise is not sufficient for designing filters which are optimum in any sense; consequently, the filters may require a number of changes in the parameters before a satisfactory design can be found. Once this design has been achieved, the filters can be used to enhance waveforms of arbitrary shape, requiring little prior knowledge of the spectral content or temporal features of the signal. In contrast to many other data‐adaptive filters which give a scalar time‐series output, the filters we describe here with vector time series input have an equal number of input and output channels. A number of examples of filtered magnetic and seismic data are given in order to emphasize the wide r...

Transient electromagnetic response of a polarizable ground

Abstract: When a uniform ground has a conductivity which may be described by a Cole‐Cole relaxation model with a positive time constant, then the transient response of such a ground will show evidence of induced polarization (IP) effects. The IP effects cause the transient initially to decay quite rapidly and to reverse polarity. After this reversal the transient decays much more slowly, the decay at this stage being about the same rate as a nonpolarizable ground.

A hybrid three-dimensional electromagnetic modeling scheme

Abstract: We present an efficient numerical method for computing electromagnetic (EM) scattering of arbitrary three‐dimensional (3-D) local inhomogeneities buried in a uniform or two‐layered earth. In this scheme the inhomogeneity is enclosed by a volume whose conductivity is discretized by a finite‐element mesh and whose boundary is only a slight distance away from the inhomogeneity. The scheme uses two sets of independent equations. The first is a set of finite‐element equations derived from a variational integral, and the second is a mathematical expression for the fields at the boundany in terms of electric fields inside the boundary. The Green’s function is used to derive the second set of equations. An iterative algorithm has been developed to solve these two sets of equations. The solutions are the electric fields at nodes inside the finite‐element mesh. The scattered fields anywhere may then be obtained by performing volume integrations over the inhomogeneous region. The scheme is used for modeling 3-D inho...

Wavelet estimation and deconvolution

Abstract: Three methods have been presented for constructing a smooth wavelet from either a (possibly poor) estimate of the reflectivity sequence or an approximate inverse filter for the source wavelet. An approximate reflectivity sequence might be derived from a velocity log at, or near, the site where the normal incidence seismogram was recorded, or it might be equated to the averages obtained from minimum entropy deconvolution (MED). The approximate inverse filter for the source wavelet is provided by MED. All methods performed well when tested on data generated from wavelets of different character, and this provides optimism that these methods will work satisfactorily in a variety of geophysical problems where the data are the convolution of a smooth wavelet and a “spikey” model. The deconvolution problem discussed here is nonunique, and satisfactory wavelet constructions require that some subjectivity be introduced by the investigator. Even so, we present one example where the computed wavelet and reflectivity...

Coincident loop transient electromagnetic master curves for interpretation of two-layer earths

Abstract: A set of apparent conductivity master curves has been calculated for the coincident loop transient electromagnetic (TEM) method used over a two-layer earth. Conductivity contrasts range from 0.001 to 1000. Loop radius/layer depth ratios range from 0.01 to 100. The time range is sufficient to see the entire shape of the curves from the early to the late time asymptotes. These curves allow the determination of the parameters of a two-layer earth for accurate data over a sufficient time range. Examples using the curves to interpret multilayered earths are given. The curves are also used to show the limitations placed on interpretation by existing TEM equipment.

The physical foundation of formation lithology logging with gamma rays

Abstract: We develop a theory for the gamma‐ray spectrum in a scattering and absorbing medium Expressions are derived for the spectrum when sources are uniformly distributed in an infinite medium We express the view that the formation of the spectrum at a point is a local phenomenon, originating from the Compton degradation of high‐energy photons which are transported from the source to the neighborhood of the point of interest This allows one to apply the theory to a point source in an infinite medium, as well as to a geometry approriate for well logging Confirming evidence via Monte Carlo results and experiment is presented We show an application to a well logging device for measurement of gamma‐ray absorption via the photoelectric effect, a parameter which is sensitive to lithotogy