Showing papers in "Geophysics in 1978"

Migration by fourier transform

Abstract: Wave equation migration is known to be simpler in principle when the horizontal coordinate or coordinates are replaced by their Fourier conjugates. Two practical migration schemes utilizing this concept are developed in this paper. One scheme extends the Claerbout finite difference method, greatly reducing dispersion problems usually associated with this method at higher dips and frequencies. The second scheme effects a Fourier transform in both space and time; by using the full scalar wave equation in the conjugate space, the method eliminates (up to the aliasing frequency) dispersion altogether. The second method in particular appears adaptable to three‐dimensional migration and migration before stack.

Wave equation migration with the phase-shift method

Abstract: Accurate methods for the solution of the migration of zero-offset seismic records have been developed. The numerical operations are defined in the frequency domain. The source and recorder positions are lowered by means of a phase shift, or a rotation of the phase angle of the Fourier coefficients. For applications with laterally invariant velocities, the equations governing the migration process are solved very accurately by the phase-shift method. The partial differential equations considered include the 15 degree equation, as well as higher order approximations to the exact migration process. The most accurate migration is accomplished by using the asymptotic equation, whose dispersion relation is the same as that of the full wave equation for downward propagating waves. These equations, however, do not account for the reflection and transmission effects, multiples, or evanescent waves. For comparable accuracy, the present approach to migration is expected to be computationally more efficient than finite-difference methods in general.

Integral formulation for migration in two and three dimensions

Abstract: Computer migration of seismic data emerged in the late 1960s as a natural outgrowth of manual migration techniques based on wavefront charts and diffraction curves. Summation (integration) along a diffraction hyperbola was recognized as a way to automate the familiar point‐to‐point coordinate transformation performed by interpreters in mapping reflections from the x, t (traveltime) domain into the x, z (depth domain). We will discuss the mathematical formulation of migration as a solution to the scalar wave equation in which surface seismic observations are the known boundary values. Solution of this boundary value problem follows standard techniques, and the migrated image is expressed as a surface integral over the known seismic observations when areal or 3-D overage exists. If only 2-D seismic coverage is available, wave equation migration is still possible by assuming the subsurface and hence surface recorded data do not vary perpendicular to the seismic profile. With this assumption, the surface inte...

Mineral discrimination and removal of inductive coupling with multifrequency IP

Abstract: In‐situ complex resistivity measurements over the frequency range 10-2 to 10+5Hz have been made on 26 North American massive sulfide, graphite, magnetite, pyrrhotite, and porphyry copper deposits. The results reveal significant differences between the spectral responses of massive sulfides and graphite and present encouragement for their differentiation in the field. There are also differences between the spectra of magnetite and nickeliferrous pyrrhotite mineralization, which may prove useful in attempting to distinguish between these two common IP sources in nickel sulfide exploration. Lastly, there are differences in the spectra typically arising from the economic mineralization and the barren pyrite halo in porphyry copper systems. It appears that all these differences arise mainly from mineral texture, since laboratory studies of different specific mineral‐electrolyte interfaces show relatively small variations. All of the in‐situ spectra may be described by one or two simple Cole‐Cole relaxation mod...

Resistivity-porosity-particle shape relationships for marine sands

Abstract: A laboratory investigation has been made of formation factor‐porosity relationships (formation factor being the ratio of the resistivity of a porous medium to the resistivity of the pore‐fluid), using natural and artificial sand samples whose grains varied widely in both size and shape. All samples obeyed Archie’s law, FF=n-m (where FF is the formation factor and n is the porosity) including mixtures of two differently shaped particle types. The exponent m was dependent on the shape of the particles, increasing as they became less spherical, while variations in size and spread of sizes appeared to have little effect. The results have been combined to produce an FF/n relationship, with an error “envelope”, which may be applicable to marine sediments in general, being in agreement with published data for marine clays. It is also suggested that the exponent m may be a better measure of the “tortuosity” of porous media than the formulas quoted in the literature.

Boundary conditions for the numerical solution of wave propagation problems

Abstract: Many finite difference models in use for generating synthetic seismograms produce unwanted reflections from the edges of the model due to the use of Dirichlet or Neumann boundary conditions. In this paper we develop boundary conditions which greatly reduce this edge reflection. A reflection coefficient analysis is given which indicates that, for the specified boundary conditions, smaller reflection coefficients than those obtained for Dirichlet or Neumann boundary conditions are obtained. Numerical calculations support this conclusion.

Resistivity mapping with an airborne multicoil electromagnetic system

Abstract: Dighem is a helicopter-borne 900 Hz multicoil electromagnetic survey system. The EM device consists of a 30-ft towed bird containing a transmitter coil in the front and three mutually orthogonal receiver coils in the rear.Resistivity contour maps can be prepared from the EM data using any of several half-space models. In this paper, two such models are selected and field examples of apparent resistivity derived from them are shown.The multicoil system has encountered areas of widespread conductivity while surveying for metallic minerals. In such areas, EM anomalies can be generated by changes of less than 10 m in survey altitude. EM anomalies of apparent significance, therefore, can reflect decreases in survey altitude as well as increases in conductivity of the earth. Under such conditions, apparent resistivity contour maps can aid the interpretation of the airborne data. The advantage of the contour maps is that anomalies caused by altitude changes are substantially reduced, and the contours reflect mainly the conductive anomalies. Resistivity contour maps improve the interpreter's ability to differentiate between conductive trends in the bedrock and those patterns typical of conductive overburden.Airborne resistivity mapping can be applied to a number of engineering problems. The multicoil system has been used for permafrost delineation and gravel detection. To be useful, the geologic units being mapped should have a resistivity less than 1000 Omega -m (for the 900 Hz frequency) and a surface extent of several acres.

Velocity estimation and downward continuation by wavefront synthesis

Abstract: A “wave stack” is any stack over a common shot or geophone gather in which the moveout is independent of time. It synthesizes a particular wavefront by superposition of the many spherical wavefronts of raw data. Unlike the common midpoint stack, wave stacks retain the important property of being the sampling of a wave field and, as such, permit wave‐equation treatment of formerly difficult or impossible problems. Seismic sections of field data generated by wave stacks that synthesized slanted downgoing plane waves showed a similarity in appearance to the common midpoint stacks. In signal‐to‐noise ratio they lay between the single offset section and the midpoint stack. The angle selectivity of the slanted plane‐wave stacks permitted detection of a reflector that was not visible on either the midpoint stack or the raw gathers. Simple velocity estimation in slant frame coordinates differs only in detail from standard frame coordinates. Because of the wave field character of data that have been slant plane‐wa...

On power estimation in maximum entropy spectral analysis

Abstract: A conceptually simple method for power estimation in maximum entropy spectral analysis, based on evaluation of complex residues of the spectral density estimator, is suggested. Numerical integration of the peaks of the power density function is thus avoided. The agreement in simple cases with conventional estimates is demonstrated, and the explicit performance is analyzed in detail in a series of examples. The close connection between the residue power estimate and the estimate proposed recently by Pisarenko is pointed out. The method is particularly suitable for spectral decomposition of low noise time series with several harmonic components, because it allows a direct listing of frequency and power estimates, provides an indication of the purity of the obtained harmonic components and enhances the resolution of the maximum entropy spectral density estimator. Computing facilities with modern program libraries are required for efficient use of the method.

Inversion of two‐dimensional resistivity and induced‐polarization data

Abstract: A fast ridge regression inversion technique has been devised for the interpretation of simple two-dimensional resistivity and induced-polarization data. The program will determine the rectangular source under a single layer of overburden which best fits the observed data.Several advantages are derived from using the ridge regression method; they include convergence from very poor initial guesses, stability in the presence of high-frequency geologic noise, readily obtained estimates of parameter statistics, and the ability for simultaneous inversion of multiple data sets. Unfortunately, each ridge regression inversion requires a great many forward problem evaluations; thus in order to achieve speed and reasonable cost, it is essential to reduce the calculation time for the forward problem to an absolute minimum. One method of achieving this is to store in the computer a data bank containing solutions for the entire range of expected parameter combinations. The forward problem then reduces to numerical interpolation between these precalculated data sets.For compilation of the data bank of forward solutions, two main numerical methods were investigated: the finite element and transmission-surface algorithms. Although these algorithms are conceptually quite different, the resulting matrix equations are very similar. The efficiency of either method depends mainly on the scheme chosen for solving the resultant large system of linear equations.Once the data bank has been created, it is possible to obtain inverse solutions for less cost than the computation of one finite element or transmission-surface forward problem. Tests on theoretical data and field data show the inversion technique to be reasonably accurate, stable, and fast.The statistics estimated by the inversion program provide additional useful information on the uncertainty in the parameters of the derived model and on high correlations between parameters. The most highly correlated parameters are, as might be anticipated, the resistivity and the width of thin conductive bodies. Two practical methods for carrying out inversion in spite of highly correlated parameters are, preferably, to add extra data sets which provide more information on some of the parameters or, alternatively, to fix some of the parameters at geologically reasonable values and invert to a more restricted model.

Magnetotelluric data analysis; removal of bias

Abstract: Two new techniques for analyzing 4-channel magnetotelluric (MT) data are described. These techniques produce estimates of the elements, Z/sub ij/, of the impedance tensor that are unbiased by noise in the autopowers of the electric and magnetic fields. Effectively, each technique uses one field channel as a reference signal that can be correlated with the other three channels. Method 1 obtains estimates for the Z/sub ij/ in terms of crosspowers of the Fourier components of the electric and magnetic fields E/sub x/(..omega..), E/sub y/(..omega..), H/sub x/(..omega..), and H/sub y/(..omega..). Method 2 is a generalization of method 1, and obtains estimates for Z/sub ij/ in terms of weighted crosspowers. Both methods fail when the geology is one-dimensional, or two-dimensional with one electrode oriented along the strike direction. To obtain results that are stable for any geology and that are unbiased by autopower noise, at least five channels of data are required. To also minimize bias by correlated noises, one needs six channels of data, two channels of which are for fields measured at a site that is remote from the base MT station. The analysis of MT data using a remote magnetometer as a reference is discussed. 2 figures. 3 tables.

The reflection, refraction, and diffraction of waves in media with an elliptical velocity dependence

Abstract: Assuming media having a velocity dependence on angle which is an ellipse, we have confirmed previously reported time‐distance relations for reflections from single interfaces, for reflections from sections of beds separated by horizontal interfaces, for refraction arrivals, and added the expression for diffractions. We also have derived expressions for plane‐wave reflection and transmission coefficients at an interface separating two transversely isotropic media. None of the properties differs greatly from those for isotropic media. However, velocities found from seismic surface reflections or refractions are horizontal components. There seems to be no way of obtaining vertical components of velocity from surface measurements alone and hence no way to compute depths from surface data.

On the theory of magnetometric resistivity (MMR) methods

Abstract: The Magnetometric Resistivity (MMR) method is based on the measurement of the low-level, low-frequency magnetic fields associated with noninductive current flow in the ground. A component of the magnetic field is measured in the vicinity of one or more grounded electrodes. Recently, the method was tested successfully in the field.The present paper presents the theoretical basis of the method in a unified format. Part of the material is derived from valuable published papers which are difficult to obtain. The remainder of the paper contains original unpublished theoretical results.It is shown that a horizontally layered earth yields no MMR anomaly. The characteristic anomalies for an anisotropic earth, vertical and dipping contacts, thin and thick dikes, and semicylindrical and hemispherical depressions, as well as alpha media are presented in detail.There are two factors which influence the MMR anomaly; geometry and conductivity contrast. For many models, it is possible to separate these two effects. Type curves are presented for very large conductivity contrasts to illustrate the effect of geometry only. Ancillary curves enable finite conductivity contrasts to be deduced from field data.

Choice of operator length for maximum entropy spectral analysis

Abstract: Empirical evidence based on maximum entropy spectra of real seismic data suggests that M = 2N/ln 2N is a reasonable a priori choice of the operator length M for discrete time series of length N. Various examples support this conclusion.

Some results from audiomagnetotelluric investigations in geothermal areas

Abstract: Audiomagnetotelluric (AMT) surveys have been made in more than 40 geothermal areas in the western U.S. The majority of the measurements was made in the Basin and Range and Columbia Plateau physiographic provinces. Complete spectral analyses made in conjunction with the field surveys illustrate the impulsive nature of the natural signals which appear to overshadow the Schumann resonances at the low frequencies. The spectrum below 200 Hz is fairly flat and the signal strength is sufficient for operation throughout the year. Above 200 Hz and centered near 2 kHz, an absorptive band limits the signal strength making data acquisition difficult if not impossible in the mid‐band region, particularly during the winter months. The spectra also show that power line harmonics may be very strong to about 1000 Hz and can be 30 to 50 dB above the natural signal level. Histograms of the measured apparent resistivities at 7.5 and 27 Hz for all measurement locations within the Basin and Range and Columbia Plateau physiogra...

Wavenumber Domain Expressions for Potential Fields from Arbitrary 2-, 21/2-, and 3-DIMENSIONAL Bodies

Abstract: The spectral representation of potential fields has gained a great deal of importance within the last decade. There have been two main categories of application. The first one relates to the interpretation of single anomalies, which are interpreted in terms of a single homogeneous body by means of certain characteristica of the spectra. Examples have been given by Odegard and Berg (1965), Sharma et al (1970), Sengupta (1974), Bhattacharyya (1966), Bhattacharyya and Leu (1975, 1977), Bhimasankaram et al (1977), Collins et al (1974). The second category relates to the interpretation of a whole collection of anomalies, which are interpreted on a statistical basis by extracting main features from the average spectra. Examples have been given by Spector and Grant (1970), Treitel et al (1971), Cassano and Rocca (1975), and Hahn et al (1976).

A summary of the geology, geochemistry, and geophysics of the Roosevelt hot springs thermal area, Utah

Abstract: The Roosevelt Hot Springs thermal area is a newly discovered geothermal power prospect in Utah. Seven production wells have been drilled with a maximum per well flow capability averaging 4.5×105kg of combined vapor and liquid per hour at a shut‐in bottom hole temperature near 260°C. The thermal area is located on the western margin of the Mineral Mountains, which consist dominantly of a Tertiary granitic pluton 32 km long by 8 km wide. Rhyolitic tuffs, flows, and domes cover about 25km2 of the crest and west side of the Mineral Mountains within 5 km of the thermal area. The rhyolitic volcanism occurred between 0.8 and 0.5 m.y. ago and constitutes a major Pleistocene thermal event believed to be significant to the evaluation of the Roosevelt Hot Springs thermal area. Thermal waters of the (now) dry spring, a seep, and the deep reservoir are dilute (ionic strength 0.1 to 0.2) sodium chloride brines.

The interpretation of direct current resistivity measurements

Abstract: The linearized inverse theory of Backus and Gilbert has been used to invert potential difference measurements obtained from direct current resistivity soundings. The resistivity is assumed to be a continuous function of depth, hence many of the difficulties encountered when assuming that the earth is a layered half‐space are avoided. An iterative technique is used to construct a resistivity model whose calculated responses agree with the observations, and the model is then appraised to find those features which are uniquely determined by the surface observations. Also, the existence of the Frechet kernels allows direct comparisons of the resolution provided by various electrode geometries and thus the design of electrode arrays to enhance resolution becomes more feasible.

Maximum entropy spectral analysis of multiple sinusoids in noise

Abstract: An analytical technique based on the method of undetermined coefficients is applied to the problem of computing the theoretical spectral estimate by the maximum entropy method (MEM) when the autocorrelation function of the data is known exactly and corresponds to N sinusoids in additive white noise and to N sinusoids in additive 1-pole, low‐pass noise. For the white noise case, the L prediction filter coefficients are expanded directly in terms of the input sinusoids. This expansion leads to a transformation of the L × L normal equations for the prediction filter coefficients to a set of 2N × 2N equations. The transformed equations are a smaller set of equations to be solved whenever L > 2N and provide a convenient description of the interaction between the various frequency components of the sinusoids which occurs in the MEM estimate. Further, for certain cases where there is little interaction between some of the frequency components of the sinusoids, the solution of the 2N × 2N equations may be approxi...

Interpretation of buried electrode resistivity data using a layered Earth model

Abstract: The layered earth model is a fundamental interpretation aid for direct current resistivity data. This paper presents a solution for the layered earth problem for a buried current source and a buried receiver. The model is developed for source and receiver electrodes buried anywhere within a horizontally stratified layered earth containing an arbitrary number of resistivity layers. Model results for the normal well‐logging array indicate that large departures between true and apparent resistivity can be caused by thin beds or highly resistant layers. A true resistivity distribution from well logs can be established by modeling when the effects from borehole rugosity and fluid resistivity are negligible. The equations derived for resistivity well logs can be used to interpret hole‐to‐hole, hole‐to‐surface, and conventional surface array data. A field example demonstrates that deviations between hole‐to‐hole field data and model results, based on well logs in the receiver hole, can be accounted for by combin...

Diurnal drift removal from aeromagnetic data using least squares

Abstract: A relatively simple method of diurnal drift removal from aeromagnetic data which does not require the use of a base station is presented. The underlying assumption is that diurnal drift during flight is a smoothly varying low order polynomial in time. The polynomial coefficients are determined by minimizing residuals at flight‐line/tie‐line intersections using least squares. This procedure is applied to a conventional sensitivity aeromagnetic survey in northeastern Kansas. The results of the drift determinations compare favorably with independent knowledge of actual drifts such as the magnetic K indices and other measurements.

Single-channel white-noise estimators for deconvolution

Abstract: The Wiener filtering approach to deconvolution is limited by certain modeling assumptions, which may not always be valid. We develop a Kalman filtering approach to deconvolution which permits more flexible modeling assumptions than the Wiener filtering approach. Our approach is applicable to time‐varying or time‐invariant wavelets as well as to nonstationary or stationary noise processes. We develop equations herein for minimum‐variance estimates of the reflection coefficient sequence, as well as error variances associated with these estimates. Our estimators are compared with an ad hoc “prediction error filter,” which has recently been reported on in the geophysics literature. We show that our estimators perform better than the prediction error filter. Simulation results are included, for both time‐invariant and time‐varying situations, which support our theoretical developments.

Frequency and transient responses of electromagnetic fields created by currents in confined conductors

Abstract: This paper considers the behavior of the frequency and transient responses of the magnetic field created by currents in conducting bodies. It is assumed that the surrounding medium is an insulator. The relationship between the low‐frequency part of the frequency spectrum and the late stage of the transient response depends on the type of conductor. These responses and the distribution of the spectrum poles differ for conductors having finite and infinite dimensions. The dependence of various components of the field on conductivity differs, which is important for understanding the resolving capabilities of the inductive methods.

Synthetic seismic sections of typical petroleum traps

Abstract: Interpretation of seismic reflections remains a key problem in seismic exploration because reflections have complex behavior, especially near geologic structures. One method to gain an understanding of this complex behavior is to study synthetic seismic sections of models of typical petroleum traps as computed by zero‐offset ray tracing for primary P‐waves only. These synthetic sections have features of significant interpretative value to the practicing geophysicist, such as variations in reflection amplitudes and complexities in reflection‐time geometries. Asymptotic ray theory was applied to calculate reflection amplitudes, accounting for mode conversion and three‐dimensional geometric divergence of ray tubes in the presence of curvilinear interfaces. This suite of synthetic seismic sections illustrates the difficulties in making correct seismic interpretations of geologic structures and suggests three conclusions: (1) The customary assumption that seismic sections are simple images of geologic cross‐se...

The calculation of magnetotelluric quantities for three‐dimensional conductivity inhomogeneities

Abstract: Three‐dimensional numerical modeling is applied to magnetotelluric principal axis and tipper analysis. Calculations of the impedance, rotation angles, and tipper azimuths are made at the surface of a conducting half‐space which contains perturbed two‐dimensional models. The object is to determine which of the directions is less disturbed by the perturbations, and hence is the more reliable guide to probable strike direction. The results indicate that the principal axis direction is a marginally more stable indicator of strike direction than is the tipper azimuth.

Computer Modeling in Gravity and Magnetic Interpretation

Abstract: Computation of anomalous gravity and magnetic fields generated by various models is a necessary step if techniques of curve‐matching are to be used for quantitative interpretation of potential field data. Recently developed methods show that anomalous magnetic and gravity fields are completely determined by the divergence of magnetization and the first vertical derivative of density, respectively. Using these methods, efficient algorithms can be developed for computing potential field anomalies caused by arbitrary distribution of magnetization and density in an irregularly shaped body. Automatic iterative procedures are normally employed in the space domain for estimating parameters of the selected model that yield a best‐fit anomaly curve for a set of discrete observed data. Examples of application of the Newton‐Raphson method, Marquardt method, and the Powell algorithm to the interpretation of magnetic data are presented and discussed. Amplitude and energy spectra of the anomalous fields are also used c...

Magnetic anomaly due to a vertical right circular cylinder with arbitrary polarization

Abstract: A closed form solution for the total anomalous magnetic field due to a vertical right circular cylinder with arbitrary polarization is derived under the assumption that the magnetization is uniform. As expected, the computed field is similar to the field due to a “similar” prism‐shaped body.

A statistical analysis of potential fields using a vertical circular cylinder and a dike

Abstract: Analytical expressions for the Fourier transforms of potential fields caused by inclined dikes and vertical circular cylinders are evaluated. Their spectra are used for a statistical analysis along the lines proposed by Spector and Grant (1970). Special attention is paid to the term which is related to the effect of the lateral extent of the sources. This term is found to decay via an inverse power relationship. Thus it decays fairly slowly, and it may become quite important to correct the spectrum prior to interpretation. Numerical calculations of the term for the cylinder and the prism indicate that an inverse third power relationship is appropriate for correction of practical power spectra.

Electromagnetic methods for mapping shallow permafrost

Abstract: Two electromagnetic techniques, radiohm and magnetic induction, were evaluated for their ability to delineate and measure the thickness of shallow permafrost. Radiohm instruments in the VLF (15–30 kHz) and LF (200–400 kHz) band, and a magnetic induction instrument using horizontal coplanar coils, separated by a distance of 3.66 m and operating at a frequency of 39.2 kHz, were tested. Along test sites in the Mackenzie Valley, N.W.T. and in northern Alberta, boreholes were placed to verify the interpretation of the geophysical data. The results showed that both LF and the magnetic induction instrument accurately delineated permafrost. The effective depth of exploration of VLF radiation was, in general, too large for delineating boundaries of shallow permafrost. The resistivity layering over permafrost in the winter could be represented by a two‐layer model for computing the thickness of frozen ground.

Nonlinear impedance of mineral-electrolyte interfaces; Part I, Pyrite

Abstract: The impedance of the interface between an acidic electrolyte and polished electrodes of pyrite has been investigated at current densities in the nonlinear range (up to 5 mA/cm 2 ). The potential across a single interface relative to a reference electrode was measured in response to a current sinusoid of low frequency (0.002 Hz). Polarization curves, or linear plots of current density versus electrode potential, consisted of distorted Lissajous patterns. At peak current densities the interface impedance is low and is dominated by activation controlled reactions involving pyrite dissolution and hydrogen gas evolution. The polarization curves have a series of step-like features at intermediate potentials due to current-limited reactions. These secondary reactions involve solid and/or aqueous reaction products from previous reactions. The high impedance portion of each reaction step corresponds to a limit current caused by either depletion of a particular solid reactant or employment of a current larger than can be carried by diffusion of aqueous reactants.Nonlinear behavior of pyrite was found to be independent of carrier type and conductivity. The potential of the anodic pyrite dissolution reaction was weakly dependent on pH. Potentials of cathodic reactions increased with increasing pH, indicating the involvement of H (super +) , as demonstrated by the evolution of hydrogen gas and H 2 S gas.