Showing papers in "Geophysics in 1984"

Inversion of seismic reflection data in the acoustic approximation

Abstract: The nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation. The method is based on the generalized least‐squares criterion, and it can handle errors in the data set and a priori information on the model. Multiply reflected energy is naturally taken into account, as well as refracted energy or surface waves. The inverse problem can be solved using an iterative algorithm which gives, at each iteration, updated values of bulk modulus, density, and time source function. Each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation (backward in time) of the data residuals. The correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models. This shows, in particular, that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data, and commerc...

Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations

Abstract: Numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media, and they promise to be invaluable in solving the full inverse problem. This paper quantitatively compares finite-difference and finite-element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time-domain and frequency-domain techniques. It is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy. To be of value, a solution technique must be able to minimize (1) numerical attenuation or amplification, (2) polarization errors, (3) numerical anisotropy, (4) errors in phase and group velocities, (5) extraneous numerical (parasitic) modes, (6) numerical diffraction and scattering, and (7) errors in reflection and transmission coefficients. This paper shows that in homogeneous media the explicit finite-element and finite-difference schemes are comparable when solving the scalar wave equation and when solving the elastic wave equations with Poisson's ratio less than 0.3. Finite-elements are superior to finite-differences when modeling elastic media with Poisson's ratio between 0.3 and 0.45. For both the scalar and elastic equations, the more costly implicit time integration schemes such as the Newmark scheme are inferior to the explicit central-differences scheme, since time steps surpassing the Courant condition yield stable but highly inaccurate results. Frequency-domain finite-element solutions employing a weighted average of consistent and lumped masses yield the most accurate resuls, and they promise to be the most cost-effective method for CDP, well log, and interactive modeling.--Modified journal abstract.

SH-wave propagation in heterogeneous media; velocity-stress finite-difference method

Abstract: A new finite-difference (FD) method is presented for modeling SH-wave propagation in a generally heterogeneous medium. This method uses both velocity and stress in a discrete grid. Density and shear modulus are similarly discretized, avoiding any spatial smoothing. Therefore, boundaries will be correctly modeled under an implicit formulation. Standard problems (quarter-plane propagation, sedimentary basin propagation) are studied to compare this method with other methods. Finally a more complex example (a salt dome inside a two-layered medium) shows the effect of lateral propagation on seismograms recorded at the surface. A corner wave, always in-phase with the incident wave, and a head wave will appear, which will pose severe problems of interpretation with the usual vertical migration methods.

Plane-wave reflection coefficients for gas sands at nonnormal angles of incidence

Abstract: The P-wave reflection coefficient at an interface separating two media is known to vary with angle of incidence. The manner in which it varies is strongly affected by the relative values of Poisson’s ratio in the two media. For moderate angles of incidence, the relative change in reflection coefficient is particularly significant when Poisson’s ratio differs greatly between the two media. Theory and laboratory measurements indicate that high-porosity gas sands tend to exhibit abnormally low Poisson’s ratios. Embedding these low-velocity gas sands into sediments having “normal” Poisson’s ratios should result in an increase in reflected P-wave energy with angle of incidence. This phenomenon has been observed on conventional seismic data recorded over known gas sands.

Toward a three‐dimensional automatic interpretation of potential field data via generalized Hilbert transforms: Fundamental relations

Abstract: The paper extends to three dimensions (3-D) the two‐dimensional (2-D) Hilbert transform relations between potential field components. For the 3-D case, it is shown that the Hilbert transform is composed of two parts, with one part acting on the X component and one part on the Y component. As for the previously developed 2-D case, it is shown that in 3-D the vertical and horizontal derivatives are the Hilbert transforms of each other. The 2-D Cauchy‐Riemann relations between a potential function and its Hilbert transform are generalized for the 3-D case. Finally, the previously developed concept of analytic signal in 2-D can be extended to 3-D as a first step toward the development of an automatic interpretation technique for potential field data.

Migration of seismic data by phase shift plus interpolation

Abstract: Under the horizontally layered velocity assumption, migration is defined by a set of independent ordinary differential equations in the wavenumber‐frequency domain. The wave components are extrapolated downward by rotating their phases. This paper shows that one can generalize the concepts of the phase‐shift method to media having lateral velocity variations. The wave extrapolation procedure consists of two steps. In the first step, the wave field is extrapolated by the phase‐shift method using l laterally uniform velocity fields. The intermediate result is l reference wave fields. In the second step, the actual wave field is computed by interpolation from the reference wave fields. The phase shift plus interpolation (PSPI) method is unconditionally stable and lends itself conveniently to migration of three‐dimensional data. The performance of the methods is demonstrated on synthetic examples. The PSPI migration results are then compared with those obtained from a finite‐difference method.

Induced polarization of shaly sands

Abstract: Equations have been developed that relate induced polarization (IP) in shaly sands to measurable petrophysical parameters. The induced-polarization process has been modeled in terms of two mechanisms: clay counterion displacement and membrane blockage. The resulting equations can be used to determine shaliness, brine conductivity, and oil saturation from in-phase and out-of-phase conductivities. Laboratory measurements have confirmed the IP dependence on these variables, as well as on temperature.

Magnetotelluric responses of three-dimensional bodies in layered earths

Abstract: The electromagnetic fields scattered by a three‐dimensional (3-D) inhomogeneity in the earth are affected strongly by boundary charges. Boundary charges cause normalized electric field magnitudes, and thus tensor magnetotelluric (MT) apparent resistivities, to remain anomalous as frequency approaches zero. However, these E‐field distortions below certain frequencies are essentially in‐phase with the incident electric field. Moreover, normalized secondary magnetic field amplitudes over a body ultimately decline in proportion to the plane‐wave impedance of the layered host. It follows that tipper element magnitudes and all MT function phases become minimally affected at low frequencies by an inhomogeneity. Resistivity structure in nature is a collection of inhomogeneities of various scales, and the small structures in this collection can have MT responses as strong locally as those of the large structures. Hence, any telluric distortion in overlying small‐scale extraneous structure can be superimposed to ar...

Semblance processing of borehole acoustic array data

Abstract: A new method of processing borehole acoustic array data is described. The method detects arrivals by computing the scalar semblance for a large number of possible arrival times and slownesses. Maxima of semblance are interpreted as arrivals. Their associated slownesses are plotted on a graph whose axes are slowness and depth. The processing makes few prior assumptions about data and the algorithm is uncomplicated. Results of the processing applied to data from a 12-receiver device are presented for both open and cased holes.--Modified journal abstract.

Wave‐equation datuming before stack

Abstract: This note describes the extension to unstacked seismic data of a computationally efficient form of the Kirchhoff integral published several years ago. In the previous paper (Berryhill, 1979), a wave‐equation procedure was developed to change the datum of a collection of zero‐offset seismic traces from one surface of arbitrary shape to another, even when the velocity for wave propagation is not constant. This procedure was designated “wave‐equation datuming,” and its applications to zero‐offset data were shown to include velocity‐replacement datum corrections and multilayer forward modeling. Extending this procedure to unstacked data requires no change in the mathematical algorithm. It is necessary only to recognize that operating on a common‐source group of seismic traces has the effect of extrapolating the receivers from one datum to another, and that, because of reciprocity, operating on a common‐receiver group changes the datum of the sources. Two passes through the data, common‐source computations, th...

Electromagnetic modeling of three‐dimensional bodies in layered earths using integral equations

Abstract: We have developed an algorithm based on the method of integral equations to simulate the electromagnetic responses of three‐dimensional bodies in layered earths. The inhomogeneities are replaced by an equivalent current distribution which is approximated by pulse basis functions. A matrix equation is constructed using the electric tensor Green’s function appropriate to a layered earth, and it is solved for the vector current in each cell. Subsequently, scattered fields are found by integrating electric and magnetic tensor Green’s functions over the scattering currents. Efficient evaluation of the tensor Green’s functions is a major consideration in reducing computation time. We find that tabulation and interpolation of the six electric and five magnetic Hankel transforms defining the secondary Green’s functions is preferable to any direct Hankel transform calculation using linear filters. A comparison of responses over elongate three‐dimensional (3-D) bodies with responses over two‐dimensional (2-D) bodie...

Rock lithology and porosity determination from shear and compressional wave velocity

Abstract: Data examined in this study are previously published laboratory shear (S) and compressional (P) wave velocity measurements on water‐saturated sandstone, calcareous sandstone, dolomite, and limestone cores, as well as laboratory porosity measurements on the sandstone and limestone cores. Sandstone and limestone porosities range from .092 to .299 and from .006 to .229, respectively. Differential pressure was varied from 500 to 6000 psi, corresponding to approximate burial depths from 290 to 3460 m, respectively. Sandstone, limestone, and dolomite are effectively separated by Poisson’s ratio σ or, equivalently, by the ratio of P- to S-wave velocity. Separation of sandstone and limestone appears to result from the difference in σ of the matrix material, namely, quartz (.056) and calcite (.316), respectively. An empirical function, 1/V=A+Bϕ, was fit by regression analysis to sandstone and limestone velocity (Vs and Vp) versus porosity (ϕ) values at each differential pressure. In this equation A and B are const...

Shallow seismic reflection mapping of the overburden-bedrock interface with the engineering seismograph; some simple techniques

Abstract: Where unconsolidated overburden exceeds 20 m in thickness, the reflection method may be efficiently used with a 12-channel engineering seismograph to map topography on the overburden-bedrock interface as well as possible structure within the overburden. The two techniques we suggest are the simplest forms of reflection profiling which can be applied with a minimum amount of equipment which we suggest is a 12-channel enhancement seismograph, a 12-geophone array, and a hammer source. These techniques require good transmission characteristics of the overburden as well as a sharp velocity discontinuity at the overburden-bedrock interface. For data processing and display a microcomputer is essential.

Microgravimetric and gravity gradient techniques for detection of subsurface cavities

Abstract: . Microgravimetric and gravity gradient surveying techniques are applicable to the detection and delineation of shallow subsurface cavities and tunnels. Two case histories of the use of these techniques to site investigations in karst regions are presented. In the first case history, the delineation of a shallow (_ 10 m deep), airfilled cavity system by a microgravimetric survey is demonstrated. Also, application of familiar ring and center point techniques produces derivative maps which demonstrate (1) the use of second derivative techniques to produce a “residual” gravity map, and (2) the ability of first derivative techniques to resolve closely spaced or complex subsurface features. In the second case history, a deeper (-30 m deep), water-filled cavity system is adequately detected by a microgravity survey. Results of an interval (tower) vertical gradient survey along a profile line are presented in the second case history; this vertical gradient survey successfully detected shallow (< 6 m) anomalous features such as limestone pinnacles and clay pockets, but the data are too “noisy” to permit detection of the vertical gradient anomaly caused by the cavity system. Interval horizontal gradients were determined along the same profile line at the second site, and a vertical gradient profile is determined from the horizontal gradient profile by a Hilbert transform technique. The measured horizontal gradient profile and the computed vertical gradient profile compare quite well with corresponding profiles calculated for a two-dimensional model of the cavity system.

The relationship between acoustic properties and the petrographic character of carbonate rocks

Abstract: Laboratory studies of the detailed relationships between acoustic properties and the petrographic character of brine‐ and air‐saturated carbonate rocks with a wide range of facies, porosities, lithologies, and rock fabrics indicate that porosity is the major factor influencing both P- and S-wave impedance and velocity. Primary lithology and secondary mineralogy have only a small influence on impedance and velocity. Combined use of P- and S-wave velocity data discriminates porosity changes from lithologic changes. All other variables, including pore‐fluid type and petrographic fabric, have no significant influence on velocities. Laboratory measurements of P‐wave velocity under simulated in‐situ conditions reproduce well‐log velocity values reliably. Laboratory porosity‐velocity trends agree with the time‐average equation when the correct matrix velocities are used. Rock property results were used to interpret porosity/lithology variations for an inverted seismic section from the Williston basin. Where well...

Velocity anisotropy of shales and depth estimation in the North Sea basin

Abstract: It is known that in the North Sea basin the depths to major reflectors as determined from surface seismic data are often larger than the well-log depths. From a study of data sets which tie 21 wells, I found a strong correlation between the occurrence of the depth error and the presence of shales in the subsurface. Assuming that the error is caused by elliptical velocity anisotropy in shales, I measured the anisotropy from a comparison of the well-log sonic data and the interval velocity profile obtained from the surface seismic data and also from a comparison of the seismic depth and the well-log depth. It was found that the two methods of measurements agree with each other and also agree qualitatively with the previous laboratory measurements of anisotropy in shale samples. The results strongly suggest that the depth anomaly in the North Sea basin is caused by the velocity anisotropy of shales. A simple method to correct the seismic depth is given.

Dip-moveout by Fourier transform

Abstract: The conventional normal‐moveout (NMO) and common‐midpoint (CMP) stacking process enhances reflections having a particular moveout velocity, while attenuating events (such as multiple reflections) having different moveout velocities. Unfortunately, this process also acts as a dip filter applied to the CMP stack. In other words, NMO and stacking enhances reflections having a particular slope in the CMP stack, while attenuating reflections having different slopes. NMO and stacking, like any dip filter, degrades lateral resolution. Fortunately, this dip‐filtering action can be suppressed by applying, in addition to NMO, a prestack process known variously as DEVILISH, prestack partial migration, and dip‐moveout. As the latter term implies, this process is a dip‐dependent moveout correction that enables reflections from both horizontal and dipping reflectors to be stacked with the same NMO velocity. Stated another way, NMO velocities estimated from dip‐moveout‐corrected seismograms are independent of the dips o...

Kirchhoff integral extrapolation and migration of nonplanar data

Abstract: The form of the Kirchhoff integral commonly used for migration of seismic data assumes that the observation surface is a flat, horizontal plane. This restriction is not necessary in theory or practice. A useful integral expression can be obtained for an arbitrary observation surface by use of the Kirchhoff approximation for relating the field on the observation surface to its normal derivative. Kirchhoff migration is usually carried out by an integral that represents summation over receivers for a given shot. Summation over shots is not explicitly handled. By invoking the principle of reciprocity and applying the Kirchhoff integral twice, one arrives at an integral expression that explicitly represents shot and receiver summation, contains the factors needed to compensate for nonplanar survey geometry, and can be used for wave extrapolation as well as migration. Adaptation of this integral to handling 2-D or line survey data is straightforward and, with the usual approximations concerning spherical wave s...

Diffusion of electromagnetic fields into a two-dimensional earth; a finite-difference approach

Abstract: We describe a numerical method for time‐stepping Maxwell’s equations in the two‐dimensional (2-D) TE‐mode, which in a conductive earth reduces to the diffusion equation. The method is based on the classical DuFort‐Frankel finite‐difference scheme, which is both explicit and stable for any size of the time step. With this method, small time steps can be used at early times to track the rapid variations of the field, and large steps can be used at late times, when the field becomes smooth and its rates of diffusion and decay slow down. The boundary condition at the earth‐air interface is handled explicitly by calculating the field in the air from its values at the earth’s surface with an upward continuation based on Laplace’s equation. Boundary conditions in the earth are imposed by using a large, graded grid and setting the values at the sides and bottom to those for a haft‐space. We use the 2-D model to simulate transient electromagnetic (TE) surveys over a thin vertical conductor embedded in a half‐space...

Array estimators and the use of microseisms for reconnaissance of sedimentary basins.

Abstract: A “natural field” seismic technique is possible to attain by observing microseisms with a suitably designed array and by digitally processing the data to obtain estimates of the phase velocities of Rayleigh waves. Wavelengths of interest in detecting depth to the basement of sedimentary basins are in the range 2 to 20 km, and correspond to wave periods from 1 to 7 s. An array of five or seven seismometers deployed as an expanding cross configuration simplifies field procedures and is adequate for phase velocity measurements of Rayleigh waves in the required wavelength range, provided high‐resolution frequency‐wavenumber spectral analysis is used. This analysis can be implemented on a minicomputer in the field. Results obtained from observation in a sedimentary basin of known structure show predominantly fundamental‐mode Rayleigh wave propagation. The scatter of velocity estimates is small enough to allow inversion by curve matching, and depth to the basement can be computed to an accuracy of ±30 percent w...

Two-dimensional resistivity inversion

Abstract: Resistivity data on a profile often must be interpreted in terms of a complex two‐dimensional (2-D) model. However, trial‐and‐error modeling for such a case can be very difficult and frustrating. To make interpretation easier and more objective, we have developed a nonlinear inversion technique that estimates the resistivities of cells in a 2-D model of predetermined geometry, based on dipole‐dipole resistivity data. Our numerical solution for the forward problem is based on the transmission‐surface analogy. The partial derivatives of apparent resistivity with respect to model resistivities are equal to a simple function of the currents excited in the transmission surface by transmitters placed at receiver and transmitter sites. Thus, for the dipole‐dipole array the inversion requires only one forward problem per iteration. We use the Box‐Kanemasu method to stabilize the parameter step at each iteration. We have tested our inversion technique on synthetic and field data. In both cases, convergence is rapi...

Gravity and magnetic inversion with minimization of a specific functional

Abstract: The nonuniqueness of gravity or magnetic data inversion is well known. In order to remove ambiguity, some authors have sought solutions minimizing a functional describing geometrical or physical properties. Last and Kubik (1983), in particular, developed a method explaining the observed anomaly by structures of minimum volume. In this method the domain where anomalous sources are searched is divided into elementary prisms of a constant density or susceptibility contrast. Each elementary contrast is allowed to vary individually. Thus a contrast distribution is computed. The search for this kind of solution leads in general to geologically more appropriate bodies, but exceptions do occur. In this paper, the technique is broadened to include the search for solutions minimizing the moment of inertia with respect to the center of gravity or with respect to a given dip line passing through it. The resulting structures are both deeper and more compact, precisely as is required in specific cases. Theoretical and ...

Complex seismic trace analysis of thin beds

Abstract: Displays of complex trace attributes can help to define thin beds in seismic sections. If the wavelet in a section is zero phase, low impedance strata whose thicknesses are of the order of half the peak‐to‐peak period of the dominant seismic energy show up as anomalously high‐amplitude zones on instantaneous amplitude sections. These anomalies result from the well‐known amplitude tuning effect which occurs when reflection coefficients of opposite polarity a half period apart are convolved with a seismic wavelet. As the layers thin to a quarter period of the dominant seismic energy, thinning is revealed by an anomalous increase in instantaneous frequency. This behavior results from the less well‐known but equally important phenomenon of frequency tuning by beds which thin laterally. Instantaneous frequency reaches an anomalously high value when bed thickness is about a quarter period and remains high as the bed continues to thin. In this paper, complex trace analysis is applied to a synthetic model of a we...

Heat flow in the Uinta Basin determined from bottom hole temperature (BHT) data

Abstract: The thermal resistance (or Bullard) method is used to judge the utility of petroleum well bottom‐hole temperature data in determining surface heat flow and subsurface temperature patterns in a sedimentary basin. Thermal resistance, defined as the quotient of a depth parameter Δz and thermal conductivity k, governs subsurface temperatures as follows: TB=T0+q0∑z=0BΔzk, where TB is the temperature at depth z=B, T0 is the surface temperature, q0 is surface heat flow, and the thermal resistance (Δz/k) is summed for all rock units between the surface and depth B. In practice, bottom‐hole and surface temperatures are combined with a measured or estimated thermal conductivity profile to determine the surface heat flow q0 which, in turn, is used for all consequent subsurface temperature computations. The method has been applied to the Tertiary Uinta Basin, northeastern Utah, a basin of intermediate geologic complexity—simple structure but complex facies relationships—where considerable well data are available. Bot...

Seismic vibrator control and the downgoing P-wave

Abstract: While the need for phase compensation is well established, the best method to measure the seismic vibrator output is not. Phase control of the force exerted by a seismic vibrator upon the earth’s surface (ground force) is shown to be useful in producing consistent downhole P-wave signatures. Experimental results are presented which compare the downhole correlation wavelets produced by phase‐locking to ground force; reaction mass acceleration and baseplate acceleration as changes in vibrator type, sweep bandwidth, drive level, and coupling medium are made. The empirical results support earlier theoretical work which predicts with suitable assumptions that ground force and far‐field particle displacement are in phase except for a time delay.

Noise processing techniques for time‐domain EM systems

Abstract: A variety of signal processing techniques can be used to minimize the effects of noise on linear, wideband, electromagnetic (EM) systems operating in the time‐domain. All systems use repetitive waveforms with polarity reversal in alternate half‐cycles. Exponential averaging or digital integration (stacking) is employed to increase signal‐to‐noise (S/N) ratios by limiting the noise acceptance to narrow frequency bands centered on odd harmonics of the repetition frequency, the width of the acceptance bands being inversely proportional to stacking time. For certain types of nonstationary noise (e.g., occasional transients) or coherent noise (e.g., powerlines) it is possible to increase S/N ratios above those obtained by simple stacking for an equal time by use of techniques such as pruning, tapered stacking or randomized stacking. With some system waveforms and when the noise spectrum is not “white”, use of preemphasis filtering in the transmitter and a corresponding de‐emphasis filter in the receiver may si...

A two-way nonreflecting wave equation

Abstract: Highly reduced reflection coefficients for transmission across material boundaries. For homogeneous regions of space, however, this wave equation becomes identical to the full acoustic wave equation. Possible applications of this wave equation for forward modeling and for migration are illustrated with simple models.--Modified journal abstract.

Diffraction of axisymmetric waves in a borehole by bed boundary discontinuities

Abstract: This paper presents the calculation of the diffraction of axisymmetric borehole waves by bed boundary discontinuities. The bed boundary is assumed to be horizontal and the inhomogeneities to be axi...

Geophone ground coupling

Abstract: For frequencies much lower than the coupling resonant frequency, the geophone accurately follows the ground motion, but for higher frequencies the coupling can alter both the amplitude and phase of the seismic signal. The normal planting of vertical geophones in the field results in coupling adequate for conventional recording that uses frequencies less than 100 Hz. However, for very loose soils or for high-frequency seismic recording, I recommend that the geophones be buried to place the geophones in firmer soil. The coupling resonant frequency for vertical geophones is determined by the firmness of the soil, and I have measured resonant frequency ranging from 100 to 500 Hz at different locations. Because the firmness of the soil increases with depth, the coupling resonant frequency can be increased by burial of the geophones or by the use of longer spikes. The rocking of horizontal geophones causes a low-frequency coupling resonance. It is crucial that horizontal geophones be planted with their bases firmly contacting the soil. Geophones so planted have a resonance around 130 Hz, whereas those 1 cm off the ground can have a resonance of 30 Hz or lower. Soil conditions have little effect on the resonant frequency. Horizontal geophones with 1-inch spikes are as well coupled as those with longer spikes, but the best coupling is achieved by burial of the geophones.--Modified journal abstract.

Signal/noise separation and velocity estimation

Abstract: A signal/noise separation must recognize the lateral coherence of geologic events and their statistical predictability before extracting those components most useful for a particular process, such as velocity analysis Events with recognizable coherence we call signal; the rest we term noise Let us define “focusing” as increasing the statistical independence of samples with some invertible, linear transform L By the central limit theorem, focused signal must become more non‐Gaussian; the same L must defocus noise and make it more Gaussian A measure F defined from cross entropy measures non‐Gaussianity from local histograms of an array, and thereby measures focusing Local histograms of the transformed data and of transformed, artificially incoherent data provide enough information to estimate the amplitude distributions of transformed signal and noise; errors only increase the estimate of noise These distributions allow the recognition and extraction of samples containing the highest percentage of sig