Showing papers in "Geophysics in 1980"

Connection between formation factor for electrical resistivity and fluid‐solid coupling factor in Biot’s equations for acoustic waves in fluid‐filled porous media

Abstract: The coefficient of inertial coupling between fluid and solid for elastic waves in fluid‐filled porous media is shown to be equal to or greater than the porosity times the low‐frequency electrical‐resistivity formation factor, being equal to this product when effects of fluid viscosity are minimal, as they may be for high‐frequency measurements on porous materials of high permeability and porosity. At sufficiently low frequencies, the coupling is due primarily to viscous forces. At intermediate frequencies, viscous forces increase inertial coupling, and inertial forces increase viscous coupling. Thus, porosity and formation factor measurements provide lower limits to coupling coefficients that can be used in the interpretation of acoustic data.

Estimating crack parameters from observations of P‐wave velocity anisotropy

Abstract: Cracks are a very common feature of crustal rocks, and their behavior and properties under various conditions are of growing importance to a variety of applications, including investigations of oil and water resources in cracked reservoirs, extraction of geothermal heat, and earthquake prediction by the seismic effects of dilatancy. This paper discusses mapping cracks by me$surements of velocity-anisotropy in underground refractors. The technique is demonstrated by interpreting the velocity measurements of Bamford and Nunn (1979) in shallow fractured limestone. The estimated crack distributions are t’ound to be consistent with rose-diagrams at neighboring outcrops, and the degree of saturation of the cracks has a major effect on velocity variations. Elastic constants can be derived for an equivalent purely elastic solid which has the snmc velocity variations as the cracked solid. This allows synthetic seismograms to be calculated for propagation through cracked structures, which demonstrates that use of polarization anomalies in three-component recordings is a sensitive technique for investigating cracked solids.

Topographic effects in resistivity and induced-polarization surveys

Abstract: We have made a systematic study of dipole‐dipole apparent resistivity anomalies due to topography and of the effect of irregular terrain on induced‐polarization (IP) anomalies, using a two‐dimensional (2-D), finite‐element computer program. A valley produces a central apparent resistivity low in the resistivity pseudosection, flanked by zones of higher apparent resistivity. A ridge produces just the opposite anomaly pattern—a central high flanked by lows. A slope generates an apparent resistivity low at its base and a high at its top. Topographic effects are important for slope angles of 10 degrees or more and for slope lengths of one dipole‐length or greater. The IP response of a homogeneous earth is not affected by topography. However, irregular terrain does affect the observed IP response of a polarizable body due to variations in the distance between the electrodes and the body. These terrain‐induced anomalies can lead to erroneous interpretations unless topography is included in numerical modeling. A...

Calculation of gravity and magnetic anomalies of finite‐length right polygonal prisms

Abstract: An equation is derived for the vertical gravity field due to a homogeneous body with polygonal cross‐section and finite strike‐length. The equation can be separated into the two‐dimensional (2-D) terms of Talwani et al. (1959) and exact terms for the contributions of the ends of the prism. Equations for the magnetic field due to a similar body were derived by Shuey and Pasquale (1973), who coined the term “two‐and‐a‐half dimensional” (212-D) to describe the geometry. Magnetic intensities are expressed as a vector sum, from which the common dot product formulation can be obtained by binomial expansion.

Absorbing boundary conditions for wave-equation migration

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

Aeromagnetic survey design

Abstract: Aeromagnetic surveys are flown with a wide variety of terrain clearances, sampling rates, and line spacings. The results are generally presented as contour maps, implying that the survey grid defines the continuous magnetic field sufficiently well to justify interpolation.

Algorithms for least-squares linear prediction and maximum entropy spectral analysis—Part I: Theory

Abstract: Experience with the maximum entropy spectral analysis (MESA) method suggests that (1) it can produce inaccurate frequency estimates of short sample sinusoidal data, and (2) it sometimes produces calculated values for the filter coefficients that are unduly contaminated by rounding errors. Consequently, in this report we develop an algorithm for solving the underlying least-squares linear prediction (LSLP) problem directly, without forcing a Toeplitz structure on the model. This approach leads to more accurate frequency determination for short sample harmonic processes, and our algorithm is computationally efficient and numerically stable. The algorithm can also be applied to two other versions of the linear prediction problem. A Fortran program is given in Part II.

Nomogram for Fresnel-zone calculation

Abstract: Geophysicists commonly recognize that a sizeable portion of a reflector is involved in causing a reflection as seen on a seismic trace, but the areal extent is usually not calculated and hence not appreciated. Most commonly, concepts are simply transferred from classical physical optics and called Fresnel-zone effects. Confusion sometimes results due to the failure to distinguish whether incident waves are plane or spherical. An excellent explanation (but one which does not use the term “Fresnel zone”) is given by Hagedoorn (1954), from which Figure 1 is taken.

A generalized multibody model for inversion of magnetic anomalies

Abstract: The height of the observation surface above a magnetized region primarily determines the critical dimension of the smallest inhomogeneity in magnetization that can be resolved from magnetic survey data. When a rectangular block is smaller in size than this critical dimension, it appears homogeneously magnetized in the observed magnetic field. This consideration leads to the selection of a unit rectangular block of suitable dimensions with homogeneous magnetization. The magnetized region creating the anomalous field values in the area of observation can, therefore, be broken up into several blocks having different magnetizations, each block being equal in size and uniformly magnetized.The iterative method described here assumes initially that the anomalous field values are caused by a three-dimensional (3-D) distribution of magnetized rectangular blocks. The optimum orientation of these blocks with respect to geographic north is then determined. This orientation is particularly insensitive to adjustments in the dimensions of the blocks. The top and bottom surfaces of each of the blocks in one or more layers are adjusted in a least-squares sense to minimize the difference between observed and calculated field values. A method is also described for constraining the magnetization vector of each block to lie within a specified angle of the normal or reversed direction of the geomagnetic field vector. The procedure for analysis of data can also be extended to the case of anomalies over a draped surface. At the conclusion of the iterations, a 3-D distribution of magnetization is generated to delineate the magnetized region responsible for the observed anomalous magnetic field. Examples including model and aeromagnetic data are provided to demonstrate the usefulness of a generalized multibody model for inversion of magnetic anomalies.

Discrete inverse methods for elastic waves in layered media

Abstract: The seismic inverse problem for waves at normal incidence on horizontally layered media is discussed The emphasis is theoretical rather than practical, but some long‐standing questions concerning the general applicability of the often taught Goupillaud inverse method are answered The main purpose is to demonstrate in detail the equivalence between the Goupillaud method of inversion for the wave equation and the Marchenko integral equation (inverse scattering) method for the Schroedinger equation We show that the very simple method of solution due to Goupillaud for a specialized model (layers of equal traveltime) actually has a much wider significance If seismic data are smoothed before sampling using a type of antialiasing filter, the Goupillaud method gives a valid approximate inversion for models with arbitrary layer thicknesses (or continuous impedance variation) when the “reflection coefficients” are appropriately reinterpreted In all, three inverse methods are considered: (1) the Goupillaud meth

The development and applications of a wide band electromagnetic sounding system using a pseudo-noise source

Abstract: A very wide band (0.03 Hz to 15 kHz) electromagnetic (EM) system has been developed. It was used over the band 1 Hz to 10 kHz to determine the electrical structure of the earth’s crust from depths of a few meters to over 40 km. A direct current of from 1 to 5 A was reversed through a long wire bipole transmitter in a pseudo‐random binary sequence (PRBS). Depending upon the frequencies selected, a sensitive one‐component flux gate magnetometer or an air‐core coil was used to monitor temporal changes in the vertical component of the magnetic field at a recording site. The measured signal was crosscorrelated digitally in real time with an exact copy of the transmitted waveform in order to obtain a good signal‐to‐noise ratio at distances up to 5 times the length of the bipole. The output crosscorrelogram was deconvolved from the system input, the autocorrelogram of the transmitted waveform, using a Wiener least‐squares filter to give the impulse response of the earth. This was then transformed into frequency ...

Attenuation by squirt flow in undersaturated gas sands

Abstract: In unconsolidated sand at low saturations, water exists in pendular rings at grain contact points. As the grains are pressed together by a passing wave, the squirt flow will contribute to wave attenuation. In a model which consists of a sphere-pack framework, fluid flow equations are solved under certain approximations to calculate the viscous losses and, hence, attenuation. Attenuation is negligible for a sphere-pack composed of equal-sized spherical grains. The model is extended to include a spectrum of grain contact geometries, in particular a log-normal spectrum of aspect ratios. The attenuation is substantial and depends markedly upon a lower limit imposed on the aspect ratios. For contacts of Type B (water separating two grains) only, results at 1 Hz are consistent with those of Mavko and Nur (1979) based on two-dimensional (2-D) cracks. If contacts of Type A (two grains in contact) and Type B are present, the attenuation is greater by 2-3 orders of magnitude.The model with Type B contacts displays a dependence of attenuation on water saturation in the range 1-10 percent; the model with Type A and B contacts does not. When the results were fitted to the measurements of absolute attenuation and saturation dependence at kHz frequencies by Gardner et al (1964), only the model with Type B contacts with aspect ratios > or =10 (super -2) could fit the data.

Some aspects of regional‐residual separation of gravity anomalies in a Precambrian terrain

Abstract: The Bouguer gravity field measured over two Archean greenstone belts of northwestern Ontario is analyzed using three different regional-residual separation techniques. The purpose of the analysis is to obtain a residual map suitable for gravity modeling studies to help define the subsurface characteristics of the greenstone belts and associated granitic areas.The methods used to derive the regional and residual maps are spectral factorization, upward continuation, and graphical smoothing. The substantial differences in the three sets of maps emphasize the ambiguity and subjectivity of the separation process. Each method may yield nonunique results. For example, in the spectral factorization technique, the filter design is dictated by the clarity of the slope change between the short- and long-wavelength features and, in the case of the upward continuation technique, by the choice of the continuation height. The graphical method is empirical and clearly nonunique.The regional map obtained through graphical smoothing is the most satisfactory for the purpose stated since it has been designed to have minimal contributions from the shallow and broad greenstone masses outcropping at the surface. These features are clearly visible in the spectrum-based regional map and to a lesser extent in the upward-continued regional map.All three types of residual maps follow the general outline of the geologic units and thus are probably equally useful for a qualitative study of the anomaly shapes. However, for quantitative modeling purposes, the graphically produced residual is most suitable, since it can be successfully fitted by models that are consistent with the known surface geology and measured density values. The location, spatial extent, and the amplitudes of the analytically produced residual anomalies, in many areas, show poor correlation with the surface measurements, rendering these maps less satisfactory.

A new method for shear‐wave logging

Abstract: Reliable evaluation of shear (S) wave characteristics in boreholes may be facilitated by the system proposed here, in a wide variety of geologic conditions and depths. The measurements can be done with a sonde, suspended freely in water contained in a borehole. The main part of the sonde consists of a source, a filter tube, and receivers. In this system, the wave field is treated approximately as that in an infinite homogeneous solid medium, because the wavelength is sufficiently longer than the borehole diameter. The source behaves as a single point force. The direct S‐wave is detected on a line (borehole axis) perpendicular to the force axis, in which pre‐dominant radiation of shear wave is expected. This fact is completely different from some modified systems of sonic log, in which (shear wave) is the converted‐refracted wave propagating as the shear wave along a borehole wall. The proposed source system is the (indirect‐excitation type), wherein the force is applied to a borehole wall indirectly throu...

Depth migration before stack

Abstract: When seismic data are migrated using operators derived from the scalar wave equation, an assumption is normally made that the seismic velocity in the propagating medium is locally laterally invariant. This simplifying assumption causes reflectors to be imaged incorrectly when lateral velocity gradients exist, irrespective of the degree of accuracy to which the subsurface velocity structure is known. A finite‐difference method has been implemented for migration of unstacked data in the presence of lateral velocity gradients, where the operation of wave field extrapolation is done in increments of depth rather than time. Performing this depth migration on unstacked data results in the imaging of reflectors on the zero‐offset trace, whereupon a zero‐offset section becomes a fully imaged‐in‐depth seismic section. Such a section, in addition to being a correctly migrated depth section, shows the same order of signal amplitude enhancement as in a normal stacking process.

The influence of the planted geophone on seismic land data

Abstract: Characteristics of the seismic data acquisition system that previously have been ignored become important as more sophisticated interpretive methods based on broader frequency bandwidths are developed to extract stratigraphic information from land data in hydrocarbon and mineral exploration. Theoretical and experimental results indicate that the geophone plant can be approximated by a damped oscillatory coupling, properties dependent upon the geophone mass, dimension of earth contact, and local soil consolidation.A massive geophone with minimal earth contact exhibits a low-frequency plant resonance with weak damping and acts as a low-pass filter to eliminate the high-frequency components of a recorded signal.A lightweight geophone with large earth contact exhibits a high-frequency plant resonance with strong damping and, consequently, filtering effects are minimal if the plant resonance is well above the signal bandwidth. Although signal filtering influences are weak for strong damping, phase delays can introduce errors of several milliseconds which resemble static errors. Additional complications arise since these time shifts are frequency dependent and, consequently, not identical for all reflection events in a seismic trace.The resonant frequency of the geophone plant increases with increased soil consolidation; however, damping demonstrates only a weak dependence upon consolidation.All of these factors can limit the effectiveness of common-depth-point (CDP) stacking methods if the proper technique is not practiced in the acquisition of broad-bandwidth seismic data.

Modeling seismic impedance with Markov chains

Abstract: Acoustic impedance is modeled as a special type of Markov chain, one which is constrained to have a purely exponential correlation function. The stochastic model is parsimoniously described by M parameters, where M is the number of states or rocks composing an impedance well log. The probability mass function of the states provides M-1 parameters, and the “blockiness” of the log determines the remaining degree of freedom. Synthetic impedance and reflectivity logs constructed using the Markov model mimic the blockiness of the original logs. Both synthetic impedance and reflectivity are shown to be Bussgang, i.e., if the sequence is input into an instantaneous nonlinear device, then the correlation of input and output is proportional to the autocorrelation of the input. The final part of the paper uses the stochastic model in formulating an algorithm that transforms a deconvolved seismogram into acoustic impedance. The resulting function is blocky and free of random walks or sags. Low‐frequency information,...

Design parameters for aerial gamma‐ray surveys

Abstract: The initial parameter to consider when planning an aerial gamma‐ray survey is whether the survey will obtain total‐count or spectrometric data. This decision is often controlled by the money available, because spectrometric surveys are at least three times more expensive than total‐count surveys. Other parameters important in the planning of an aerial survey are flight‐line spacing, survey altitude, and detector volume. These parameters are shown in graphs that provide a better understanding of their interaction and can be used in survey planning. The graphs are based upon the concepts of the infinite source yield and the circle of investigation. These concepts relate an aerial gamma‐ray detector to the ground area viewed by the detector. Concentric rings that show the ground area viewed by a stationary detector are constructed using the radius of the circle of investigation for fixed percentages of the infinite source yield as a function of detector altitude above ground level. The data permit calculatio...

Channel wave mapping of coal seams in the United Kingdom

Abstract: We present an analysis of errors contributing to the inaccuracy of aneroid altimeter observations under tropical conditions. Instrumental errors were investigated experimentally, while errors due to meterological factors were analyzed using an autocorrelation analysis of actual pressure observations. For an estimated accuracy of ±1.2 to ±2.5 m, it is recommended that a diurnal pressure variation be eliminated from the altimeter field readings and that measurements be restricted to 0500 to 1400 hours LST in the wet season and from 0500 to 1100 hours LST or 1500 to 1830 hours LST in the dry season.

Velocity analysis in transversely isotropic media

Abstract: In transversely isotropic media, the moveout velocity obtained from common‐depth‐point (CDP) analysis may be significantly different from the horizontal velocity of the pseudo‐P wave. In Levin’s (1978) paper, he discusses, among other things, the problem of velocity determination in a medium in which the pseudo‐P wave surface produced by a point source is an ellipsoid of revolution. He points out that one would expect many sedimentary rocks to be transversely isotropic with a vertical axis of symmetry. In his Appendix he proves that an ellipse (using two dimensions for convenience) is one possible shape for the wave surface in such a medium. He also shows, as have others, that in this case CDP velocity analysis measures the velocity of horizontal propagation.

Modeling resistivity anomalies from localized voids under irregular terrain

Abstract: In applying earth resistivity methods to the problem of locating and delineating subsurface structures, surface elevation variations along the surveyed terrain introduce distortions in the soundings. The analysis presented here is aimed at characterizing such terrain variations in the detection of relatively small subsurface targets such as caves, sinks, and tunnels in otherwise homogeneous earth materials. The analytical approach involves, first, the development of a suitable earth resistivity model for localized three‐dimensional subsurface anomalies in a homogeneous flat half‐space. Next, in order to apply the half‐space resistivity model to irregular terrain, a Schwarz‐Christoffel transformation is utilized to map the terrain surface variations into an equivalent flat half‐space. The technique is illustrated by calculating the resistivity response of three tunnels located below a hill with 40-m valleys on either side.

The linear properties of thin layers, with an application to synthetic seismograms over coal seams

Abstract: Thin layers are considered from the point of view of the quasi-linear relation that exists between their thickness and their reflection response to a seismic signal. The range within which this quasi-linearity exists is investigated; for a continuous sine wave, this is done by means of the equation for the response given in Rayleigh (1945), and for a seismic wavelet by means of a synthetic seismogram program. For a wavelet, the limiting value of the dominant frequency is found to be smaller than that for a continuous sine wave, the difference being in the order of magnitude of 15 percent.Within the linearity range, a thin layer may be replaced by an equivalent layer which gives the same reflection response but differs in thickness and in acoustic impedance. In the construction of synthetic seismograms over coal seams, this equivalent replacement may be utilized to replace the seams by layers, for which the two-way traveltime is equal to an integer number of sampling intervals; by this procedure the usual rounding-off errors are avoided. The method of equivalent replacement is also applicable when the host rock above and below the seam have different velocities.

A wide-band electromagnetic exploration method—Some theoretical and experimental results

Abstract: Theoretical electromagnetic (EM) responses of a conductive circular cylinder in a conductive half‐space are obtained for a frequency range between 10 Hz and 30 kHz. The results show that the spectral characteristics can be used to determine the subsurface configuration of a target body, particularly in terms of its depth. Laboratory experiments using a wide‐band sweep‐frequency source provide support for the theoretical prediction of the spectral behavior. The experiment employed a harmonic EM source continuously sweeping logarithmically from 4 kHz to 4 MHz over a graphite slab submerged in a conductive solution. The results suggest that a wide‐band EM spectral profile displayed in a frequency‐distance space may provide many intuitive interpretation schemes which may not be possible with a single or discrete frequency profile.

Variation of density with rock type, depth, and formation in the Western Canada basin from density logs

Abstract: Gamma‐gamma logs were used to study the density of sedimentary rocks and formations in the Western Canada basin. The frequency distributions of density of all the main rock types have negative skewness, a characteristic that seems to be a general feature of density distributions of rocks based on well logging. The frequency distributions of density for sandstone and limestone were not found to be different from those of shale and dolomite, respectively, at the 0.01 level of significance. Several density‐depth functions were applied to the shale samples. Athy's (1930) exponential function provided the best fit to observed values. Using this function, the average thickness of sediments removed by denudation in south and central Alberta was calculated to be 4400 or 3600 ft on the assumption that the density of the original surface clay was 1.4 or 1.6g/cm3, respectively. Maps of average density for six formations in Alberta and Saskatchewan indicate only a partial correlation with lithology and depth. A large...

Calculation of complete seismograms for an explosive source in a layered medium

Abstract: We apply the method of discrete wavenumber representation of elastic wave fields of Bouchon and Aki (1977) to the computation of synthetic seismograms for an explosive source in a layered medium. The method is based on the representation of the source radiation by a superposition of plane waves propagating in discrete directions. This discretization is exact and results from a periodic arrangement of sources. The two-dimensional (2-D) and three-dimensional (3-D) problems are described, and some examples of calculation are presented. They show that very complex seismograms can be obtained for rather simple geologic structures.

Algorithms for least-squares linear prediction and maximum entropy spectral analysis; Part II; Fortran program

Abstract: We present a set of Fortran subroutines which implement the least‐squares algorithm described in Part I (Barrodale and Erickson, 1980, this issue) for solving three variants of the linear prediction problem. When combined with a suitable driver program, these subroutines can be used to calculate maximum entropy spectra without imposing a Toeplitz structure on the model.

Fortran routines for linear inverse problems

Abstract: A package of Fortran routines using linear programming techniques in linear inverse problems as gravimetry or geomagnetism is presented. These routines, with a complete description and examples of their applications, are available at the SEG Business Office. This paper presents a brief description of the package and the general features of the method; taking into account the nonuniqueness of the solution, the bounds of some parameters or some particular solutions, such as the “ideal body” [Parker (1975)], are computed. This package can be applied to inverse gravity or geomagnetic problems with a maximum data size on IBM 360/65 of the order such that the number of measurements ⩽ 250, number of unknowns ⩽ 600.

Depth migration after stack

Abstract: The conventional methods for migrating a seismic section, e.g., the finite‐difference method and the Kirchhoff summation method, are inadequate in the presence of significant lateral variations in velocity. For this type of velocity distribution, the basic migration output should be in true depth, although for practical purposes it may be preferable to display it with a nonlinear depth scale. A finite‐difference method has been implemented for obtaining migrated depth sections. The concept underlying this involves all the usual assumptions of a dip line and primary reflections only, with the seismic section considered as the surface measurement of an upcoming wave field which we process with downward continuation in small increments of depth, rather than the customary increments of traveltime. The specified velocity variation laterally along a thin layer results in transmission time changes which must be corrected by a small static time shift applied to each seismic trace. This additional operation within...

Electrical, electromagnetic, and magnetotelluric methods

Abstract: Application of electrical methods began with Robert W. Fox’s 1830 observation of self potentials associated with copper vein deposits in Cornwall. Conrad Schlumberger introduced the direct current equal potential line resistivity method in 1912. Harry W. Conklin received the first patents on the electromagnetic (EM) method in 1917. From these beginnings, the history of the development of the resistivity induced‐polarization (IP), magnetotelluric and EM methods are traced to the present time. It is of interest to note that application of electrical methods flourished from about 1920 to 1930, but then the methods were developed slowly until after World War II when a major burst of development activity took place. However, the interpretational basis of the methods remained poor until the last several years when application of numerical techniques using computers permitted forward and inverse solutions to electrical boundary value problems in two and three dimensions. Field hardware gradually evolved from 196...

Calculation of apparent conductivity for the transient electromagnetic (coincident loop) method using an HP-67 calculator

Abstract: The interpretation of coincident‐loop transient electromagnetic (TEM) surveys is often aided by the transformation of TEM data to apparent conductivity values. Present methods of calculating apparent conductivity have a limited usefulness or require access to a computer. This paper presents an algorithm for calculating apparent conductivity to an accuracy of better than 1 percent for values of the parameter x = σμA/4πt up to 10, where σ = conductivity, μ = magnetic permeability, A = loop area, and t = sample time. This range of x is sufficient for most situations encountered in field work. The algorithm has been programmed for an HP-67 pocket calculator, and the program steps and user instructions are given in standard format. Execution time ranges from 6 to 15 sec. The program should assist the interpretation of TEM surveys in the field.