Showing papers in "Geophysics in 1988"

Fourth-order finite-difference P-SV seismograms

Abstract: I describe the properties of a fourth-order accurate space, second-order accurate time two-dimensional P-Sk’ finite-difference scheme based on the MadariagaVirieux staggered-grid formulation. The numerical scheme is developed from the first-order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga-Virieux staggered-grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic-elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free-surface or within a layer and to satisfy free-surface boundary conditions. Benchmark comparisons of finite-difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite-difference and reflectivity solutions for elastic-elastic and acoustic-elastic layered models.

Static shift of magnetotelluric data and its removal in a sedimentary basin environment

Abstract: Previous modeling investigations of the static shift of magnetotelluric (MT) apparent resistivity curves have limited appeal in that the electric fields used were point measurements, whereas field observations are of voltage differences. Thus, inhomogeneities of dimension of the order of the electrode line length could not be investi­ gated. In this paper, by using a modeling algorithm that derives point voltages rather than point electric fields, I consider the effect on the MT responses of local near­ surface distorting structures, which are both outside of, and inside, the telluric electrode array. I show that static-shift effects are of larger spatial size but of less magnitude than would be expected from conventional modeling. Also, the field observation that static shift affects only the apparent resistivity curve but not the phase response can be replicated by the voltage differ­ ence modeling. If there exists within the earth a layer whose variation in electrical resistivity along the profile can be treated in a parametric fashion, then static shift of the apparent resistivity curves can be corrected. Deriving the modal value from a sufficient number of observations for the layer resistivity is the most useful approach.

Application of three‐dimensional interactive modeling in gravity and magnetics

Abstract: Three‐dimensional (3-D) interactive modeling permits integrated processing and interpretation of gravity and magnetic data, yielding an improved geologic interpretation. 3-D model bodies are constructed from polyhedra of suitable geometry and physical parameters (density and susceptibility), input on an interactive graphics terminal that is tied to a host computer. The method is especially designed for concurrent processing and interpretation in an interactive mode. The effect on gravity of a homogeneous polyhedron is calculated by transforming a volume integral into a sum of line integrals. Magnetic effects can be modeled by using either Poisson’s theorem or a slight modification of the formulas derived for gravity modeling. The interactive modeling program allows the user to change the geometry as well as the density and/or susceptibility of the elementary polyhedra and to observe results quickly during the course of processing. This capability enables the interpreter to decide immediately if and where ...

Correction for the static shift in magnetotellurics using transient electromagnetic soundings

Abstract: Shallow inhomogeneities can lead to severe problems in the interpretation of magnetotelluric (MT) data by shifting the MT apparent resistivity sounding curve by a scale factor, which is independent of frequency on the standard log-apparent-resistivity versus log-frequency display. The amount of parallel shift, commonly referred to as the MT static shift, can not be determined directly from conventionally recorded MT data at a single site. One method for measuring the static shift is a controlled-source measurement of the magnetic field. Unlike the electric field, the magnetic field is relatively unaffected by surface inhomogeneities. The controlled-source sounding (which may be a relatively shallow sounding made with lightweight equipment) can be combined with a deep MT sounding to obtain a complete, undistorted model of the earth. Inversions of the static shift-corrected MT data provide a much closer match to well-log resistivities than do inversions of the uncorrected data.The particular controlled-source magnetic-field sounding which we used was a central-induction Transient ElectroMagnetic (TEM) sounding. Correction for the static shift in the MT data was made by jointly inverting the MT data and the TEM data. A parameter which allowed vertical shifts in the MT apparent resistivity curves was included in the computer inversion to account for static shifts. A simple graphical comparison between the MT apparent resistivities and the TEM apparent resistivities produced essentially the same estimate of the static shift (within 0.1 decade) as the joint computer inversion.Central-induction TEM measurements were made adjacent to over 100 MT sites in central Oregon. The complete data base of over 100 sites showed an average static shift between 0 and 0.2 decade. However, in the rougher topography and more complex structure of the Cascade Mountain Range, the majority of the sites had static shifts of the order of 0.3 to 0.4 decade. The static shifts in this area are probably due to a combination of topography and surficial inhomogeneities. The TEM apparent resistivity (which is used to estimate the unshifted MT apparent resistivity) does not necessarily agree with either the transverse electric (TE) or the transverse magnetic (TM) MT polarization. TEM apparent resistivity may occur between the two, or may agree with one of the two polarizations, or may lie outside the MT polarizations.

Application of singular value decomposition to vertical seismic profiling

Abstract: An essential part of the interpretation of vertical seismic profiles (VSP) is the separation of the upgoing and downgoing waves. This paper presents a new approach which is based on the decomposition of time-shifted VSP sections into eigenimages, using singular value decomposition (SVD). The first few eigenimages of the time-shifted VSP section contain the contributions of the horizontally aligned downgoing waves. The last few eigenimages contain the contribution of uncorrelated noise components. The separated upgoing waves are recovered as a partial sum of the eigenimages. !mportant aspects of this approach are that regular sampling of the recording levels is not required, that the first-break times need not be measured with extreme accuracy, that noise rejection may be automatically included in the processing, and that eigenimages or sums of eigenimages which may be computed as part of the approach can provide important additional information.

Cellular‐automaton fluids: A model for flow in porous media

Abstract: Numerical models of fluid flow through porous media can be developed from either microscopic or macroscopic properties. The large‐scale viewpoint is perhaps the most prevalent. Darcy’s law relates the chief macroscopic parameters of interest—flow rate, permeability, viscosity, and pressure gradient—and may be invoked to solve for any of these parameters when the others are known. In practical situations, however, this solution may not be possible. Attention is then typically focused on the estimation of permeability, and numerous numerical methods based on knowledge of the microscopic pore‐space geometry have been proposed. Because the intrinsic inhomogeneity of porous media makes the application of proper boundary conditions difficult, microscopic flow calculations have typically been achieved with idealized arrays of geometrically simple pores, throats, and cracks. I propose here an attractive alternative which can freely and accurately model fluid flow in grossly irregular geometries. This new method s...

Porosity from seismic data: A geostatistical approach

Abstract: Using a geostatistical technique called cokriging, the areal distribution of porosity is estimated first in a numerically simulated reservoir model, then in an oilbearing channel-sand of Alberta, Canada. The cokriging method consistently integrates 3-D reflection seismic data with well measurements of the porosity and provides error-qualified, linear mean square estimates of this parameter. In contrast to traditional seismically assisted porosity mapping techniques that treat the data as spatially independent observations, the geostatistical approach uses spatial autocorrelation and crosscorrelation functions to model the lateral variations of the reservoir properties. In the simulated model, the experimental root-meansquare porosity error with cokriging is 50 percent smaller than the error in predictions relying on a leastsquares regression of porosity on seismically derived transit time in the reservoir interval. In the Alberta reservoir, a cross-validation study at the wells demonstrates that the cokriging procedure is 20 percent more accurate, in a mean square sense, than a standard regression method, which accounts only for local correlations between porosity and seismically derived impedances. In both cases, cokriging capitalizes on areally dense seismic measurements that are indirectly related to porosity. As a result, when compared to estimates obtained by interpolating the well data, this technique considerably improves the spatial description of porosity in areas of sparse well control.

Reflection seismology over azimuthally anisotropic media

Abstract: Recent surveys have shown that azimuthal anisotropy (due most plausibly to aligned fractures) has an important effect on seismic shear waves. Previous work had discussed these effects on VSP data; the same effects are seen in surface recording of reflections at small to moderate angles of incidence. The anisotropic effects one different polarization components of vertically traveling shear waves permit the recognition and estimation of very small degrees of azimuthal anisotropy (of order 2 1 percent), as in an interferometer. Anisotropic effects on traveltime yield estimates of anisotropy which are averages over large depth intervals. Often, raw field data must be corrected for these effects before the reflectors may be imaged; two variations of a rotational algorithm to determine the “principal time series” are derived. Anisotropic effects on moveout lead to abnormal moveout unless the survey line is parallel to the fractures. Anisotropic effects on reflection amplitude permit the recognition and estimation of anisotropy (hence fracture intensity) differences at the reflecting horizon, i.e., with high vertical resolution,

The pseudospectral method: Accurate representation of interfaces in elastic wave calculations

Abstract: When finite‐difference methods are used to solve the elastic wave equation in a discontinuous medium, the error has two dominant components. Dispersive errors lead to artificial wave trains. Errors from interfaces lead to circular wavefronts emanating from each location where the interface appears “jagged” to the rectangular grid. The pseudospectral method can be viewed as the limit of finite differences with infinite order of accuracy. With this method, dispersive errors are essentially eliminated. The mappings introduced in this paper also eliminate the other dominant error source. Test calculations confirm that these mappings significantly enhance the already highly competitive pseudospectral method with only a very small additional cost. Although the mapping method is described here in connection with the pseudospectral method, it can also be used with high‐order finite‐difference approximations.

Cooperative inversion of geophysical data

Abstract: Geophysical inversion by iterative modeling involves fitting observations by adjusting model parameters. Both seismic and potential-field model responses can be influenced by the adjustment of the parameters of the rock properties. The objective of this "cooperative in­ version" is to obtain a model which is consistent with all available surface and borehole geophysical data. Al­ though inversion of geophysical data is generally non­ unique and ambiguous, we can lessen the ambiguities by inverting all available surface and borehole data. This paper illustrates this concept with a case history in which surface seismic data, sonic logs, surface gravity data, and borehole gravity meter (BHGM) data are ad­ equately modeled by using least-squares inversion and a series offorward modeling steps. The traditional analysis of various geophysical data sets has been termed "integrated interpretation." Such analysis has traditionally involved much processing, modeling, and subjec­ tive interpretation in order to develop a geologic model which is consistent with various surface and borehole geophysical data. A method which automates some of the above pro­ cedures is least-squares inversion. This procedure perturbs an initial set of model parameters in order to fit the observed data. Least-squares inversion is described in many papers, in­ cluding those of Jackson (1972), Wiggins (1972), Jupp and Vozoff (1975), and Lines and Treitel (1984). This inversion technique could also be termed "automated iterative mod­ eling." In terms of modeling different geophysical data sets, there are many possible approaches which allow inversion of a given data set, and it is doubtful whether one single recipe will prove suitable for all problems. Nevertheless, by jointly invert­ ing various types of geophysical observations, we may lessen the extent of some of the ambiguities inherent in individual data sets. LaFehr (1984) gives a germane discussion of several reasons for this type of coordinated geophysical effort. In the present study, we define "cooperative inversion" as the esti­ mation of a subsurface model which is consistent with various independent geophysical data sets. In the case discussed in this paper, these data sets include surface and borehole observa­ tions of the seismic and gravity responses. In our definition, cooperative inversion may include either "joint inversion" or "sequential inversion."

Gravity inversion of an interface above which the density contrast varies exponentially with depth

Abstract: Mapping of an interface above which the density contrast varies exponentially with depth, as is common at the basement surface of sedimentary basins, is efficiently achieved by a theoretically precise gravity method which can be applied to either profile data or twodimensional data. The contrast in mass above the interface is modeled by an array of vertical rectangular prisms with density contrasts varying exponentially with depth. Gravity anomalies due to the prisms are calculated in the wavenumber domain and then converted to the space domain. The precision of the inverse numerical Fourier transform in this procedure is significantly increased by a shift‐sampling technique based on the discrete Fourier deviation equation. Depth to the interface is determined by iterative adjustment of the vertical extent of the prisms in accordance with observed gravity anomaly data. The basement surface of the Los Angeles basin, California, calculated by this method, closely duplicates the published configuration based...

Elastic wave‐field inversion of reflection and transmission data

Abstract: Elastic inversion of multioffset seismic data by wavefield fitting yields a maximum probability P-wave and S-wave velocity and density model of the Earth. Theoretically, the inversion accounts for all elastic waves including reflected and transmitted waves, mode conversions, shear waves, head waves, Rayleigh waves, etc. These different wave types tend to resolve different components of the Earth properties. By inverting two-component synthetic data, I show that reflection data mainly resolve high wavenumbers, while transmission data mainly resolve low wavenumbers of the P-wave and S-wave velocity model. The inversion of reflection data (shot gathers) yields a result that looks like a prestack elastic migration but the meaning of the inverted data is not simply reflectivity: it is the P-wave and S-wave velocity perturbation. The inversion of transmission data (VSPs) yields a solution that contains useful interval velocity information and is comparable to an elastic diffraction tomography result.

Normal moveout revisited; inhomogeneous media and curved interfaces

Abstract: The equation of normal moveout, t2=t02+x2/Va2, is valid for a reflection from the base of a single homogeneous and isotropic bed, but is only an approximation in the real world of multilayered, inhomogeneous media and curved interfaces. Using the theory of geometrical optics, we can find another second‐order equation which represents hyperbolas that are also symmetrical about the time axis. However, the centers of these hyperbolas do not coincide with the center of coordinates, but are shifted along the time axis. The equation describing this second type of hyperbola is (t+tp-t0)2=tp2+x2/V12, where tp is the time of focusing depth and V1, the velocity of the input medium. This equation is not only more accurate than the usual normal moveout, but its use is more economical on a vector computer because the traditional dynamic correction is a static correction in the tp analysis. This procedure makes it possible to compute velocities for all the samples of all the stacked traces and produces a velocity secti...

Attenuation of complex water-bottom multiples by wave-equation-based prediction and subtraction

Abstract: Multiple reflections that are generated by the water bottom in marine seismic data can be predicted by a combination of numerical wave extrapolation through the water layer and estimation of the water-bottom reflectivity. Attenuation of the multiples occurs when the predicted wave field is subtracted from the original record. I derive the expressions needed for prediction of the multiples, following the ideas of Morley, in a form that can be used to estimate the reflectivity of a water bottom that has a complex shape and has a reflectivity that varies with lateral position, frequency, and reflection angle. The specific form of the operations needed for prediction is implemented without assumptions about the simplicity or flatness of the water bottom. The derivation implies that the recorded wave field may be interpreted as both an upgoing and a downgoing wave. This interpretation is correct except for a simple surface ghost present in both interpretations.Application of the predictive method to data collected over a hard, complex water bottom demonstrates that it effectively attenuates water-bottom and peg-leg multiples even when the water bottom is complex and changes in character within the span of a shot gather. The same data were processed with various combinations of predictive attenuation, prestack automatic gain control, prestack moveout discrimination, and stacking. The multiple attenuation achieved with the predictive method alone was greater than that achieved by moveout discrimination alone; the net attenuation achieved by sequential application of all the methods was approximately additive.

Viscoacoustic wave propagation simulation in the earth

Abstract: Anelasticity of earth materials produces significant changes in the amplitude and phase spectra of seismic waves. The anelastic properties of real materials, particularly of porous rocks, are described using the theory of linear viscoelasticity based on Boltzmann's superposition principle. Wave-propagation simulation with this model requires implementing the convolutional relation in the equation of motion. The choice of a viscoacoustic constitutive relation based on a spectrum of relaxation mechanisms allows a realistic description of the anelastic effects, and the introduction of memory variables obviates storing the entire strain history required by the time convolution. A pseudospectral timeintegration technique is used to solve the equation of motion. Applications of viscoacoustic modeling suggest the need for considering the correct attenuation-dispersion effects for various fundamental seismic problems in anelastic earth models. Comparison of acoustic and viscoacoustic synthetic seismograms shows differences in the amplitudes and arrival times of the wave fields which are enhanced for particular combinations of anelastic and geometrical effects.

Magnetotelluric inversion for minimum structure

Abstract: Structure can be measured in terms of a norm of the derivative of a model with respect to a function of depth f(z), where the model m(z) is either the conductivity σ or log σ. An iterative linearized algorithm can find models that minimize norms of this form for chosen levels of chi‐squared misfit. The models found may very well be global minima of these norms, since they are not observed to depend on the starting model. Overfitting data causes extraneous structure. Some choices of the depth function result in systematic overfitting of high frequencies, a “blue” fit, and extraneous shallow structure. Others result in systematic overfitting of low frequencies, a “red” fit, and extraneous deep structure. A robust statistic is used to test for whiteness; the fit can be made acceptably white by varying the depth function f(z) which defines the norm. An optimum norm produces an inversion which does not introduce false structure and which approaches the true structure in a reasonable way as data errors decrease...

Seismic velocity and Q model for the shallow structure of the Arabian Shield from short-period Rayleigh waves

Abstract: The shear velocity and Q (sub beta ) structure for the upper 1 km in different tectonic regions of the Arabian shield have been investigated using high-frequency vertical component records of Rayleigh waves (1-20 Hz), which were recorded at source-to-receiver separations 55-80 km during a deep seismic refraction survey. Group and phase velocities of the fundamental and first higher modes were inverted for the shear-wave velocity structure; Rayleigh-wave attenuation coefficients were determined from the decay of the amplitude spectrum of the fundamental mode and used to invert for the Q (sub beta ) structure. Models derived from the data were tested by calculating synthetic seismograms for the fundamental and the first higher modes from surface-wave theory with a center of compression used to represent the source; both band-pass filtered step and Dirac delta source time functions were tested. Modeling indicates that the shear-wave velocity of the shield increases from 2.6 km/s to 3.4 km/s in the upper 400 m of the crust. Q (sub beta ) increases from 30 in the upper 50 m to 150 at 500 m depth. The underlying material has a Q (sub beta ) of 400-500 for the outcropping igneous rocks such as granite and may reach values higher than 700 for the metamorphic green schist rock. A band-pass filtered Dirac delta source time function produces the synthetic that is the best fit with observations.

Three-dimensional elastic modeling by the Fourier method

Abstract: Earlier work on three-dimensional forward modeling is extended to elastic waves using the equations of con­ servation of momentum and the stress-strain relations for an isotropic elastic medium undergoing infinitesimal deformation. In addition to arbitrary compressional (or P-wave) velocity and density variation in lateral and vertical directions, elastic modeling permits shear (or S-wave) velocity variation as well. The elastic wave equation is solved using a generalization of the method for the acoustic case. Computation of each time step begins by computing six strain components by per­ forming nine spatial partial differentiation operations on the three displacement components from the pre­ vious time step. The six strains and two Lame constants are linearly combined to yield six stress components.

The application of diffraction tomography to cross-hole seismic data

Abstract: Previously published equations for diffraction tomography do not solve the “two and one‐half dimensional problem” (point source illumination of two‐dimensional geology) if sources and receivers are confined to linear arrays. In spite of this lack of a formal solution, useful images can be formed by the application of two‐dimensional formulas to such problems. The estimation of difference fields, of crucial importance in diffraction tomography, reduces to the problem of estimating the source function. Using assumptions about the consistency of the source behavior, we extract the source function in a statistical fashion from cross‐hole data. Using this technique, the difference fields are computed directly from the recorded wave fields for two experiments and diffraction tomographic images are obtained. In the first experiment, the data are generated using a two‐dimensional finite‐difference modeling algorithm. In the second, a physical scale model of a crosshole experiment is performed in an ultrasonic mod...

Imaging discontinuities on seismic sections

Abstract: In routine seismic processing, normal moveout (NMO) corrections are performed to enhance the reflected signals on common-depth-point or common-midpoint stacked sections. However, when faults are present, reflection interference from the two blocks and the diffractions from their edges hinder fault location determination. Destruction of diffraction patterns by poststack migration further inhibits proper imaging of diffracting centers.This paper presents a new technique which helps in the interpretation of diffracting edges by concentrating the signal amplitudes from discontinuous diffracting points on seismic sections. It involves application to the data of moveout and amplitude corrections appropriate to an assumed diffractor location. The maximum diffraction amplitude occurs at the location of the receiver for which the diffracting discontinuity is beneath the source-receiver midpoint. Since the amplitudes of these diffracted signals drop very rapidly on either side of the midpoint, an appropriate amplitude correction must be applied. Also, because the diffracted signals are present on all traces, one can use all of them to obtain a stacked trace for one possible diffractor location. Repetition of this procedure for diffractors assumed to be located beneath each surface point results in the common-fault-point (CFP) stacked section, which shows diffractor locations by high amplitudes.The method was tested for synthetic data with and without noise. It proves to be quite effective, but is sensitive to the velocity model used for moveout corrections. Therefore, the velocity model obtained from NMO stacking is generally used for enhancing diffractor locations by stacking. Finally, the technique was applied to a field reflection data set from an area south of Princess well in Alberta.

Paraxial ray Kirchhoff migration

Abstract: A rapid nonrecursive prestack Kirchhoff migration is implemented (for 2-D or 2.5-D media) by computing the Green’s functions (both traveltimes and amplitudes) in variable velocity media with the paraxial ray method. Since the paraxial ray method allows the Green’s functions to be determined at points which do not lie on the ray, two‐point ray tracing is not required. The Green’s functions between a source or receiver location and a dense grid of thousands of image points can be estimated to a desired accuracy by shooting a sufficiently dense fan of rays. For a given grid of image points, the paraxial ray method reduces computation time by one order of magnitude compared with interpolation schemes. The method is illustrated using synthetic data generated by acoustic ray tracing. Application to VSP data collected in a borehole adjacent to a reef in Michigan produces an image that clearly shows the location of the reef.

Multivariate statistical analyses applied to seismic facies recognition

Abstract: One of the most important goals of seismic stratigraphy is to recognize and analyze seismic facies with regard to the geologic environment. The first problem is to determine which seismic parameters are discriminant for characterizing the facies, then to take into account all those parameters simultaneously. The second problem is to be sure that there is a link between the seismic parameters and the geologic facies we are investigating. This paper presents a methodology for automatic facies recognition based upon two steps. The first, or learning step, begins with the definition of learning seismic traces for each facies we wish to recognize. The choice of learning traces is based upon either well data or a seismic stratigraphic interpretation. A large number of seismic parameters are then computed from the learning traces; multidimensional analyses are carried out in order to validate the choice of learning traces and to select, among all the available parameters, those that discriminate best. At this st...

Full-wave synthetic acoustic logs in radially semiinfinite saturated porous media

Abstract: The wave field generated by a point source in an axisymmetric fluid‐filled borehole embedded in a saturated porous formation is studied in both the spectral domain and time domain. The formation is modeled following Biot theory modified in accordance with homogenization theory. When the borehole wall is permeable, guided waves can be significantly affected by the permeability of the formation. Whatever the formation, fast or slow, Stoneley‐wave phase velocity and energy decrease and attenuation (in the sense of Q-1) increases with increasing permeability. These effects are more important in the very low‐frequency range, where Darcy’s law governs the fluid motion and the wave energy at the interface is maximum, than at higher frequencies. The effects increase and persist over a larger frequency range with decreasing viscosity and increasing compressibility of the saturant fluid, with increasing pore‐fluid volume, and with decreasing borehole radius. In contrast, the effects decrease with decreasing stiffne...

Low-frequency tube waves in permeable rocks

Abstract: Low‐frequency tube waves in permeable boreholes are of interest because their propagation depends upon in‐situ rock permeability. In this paper, we use low‐ frequency asymptotics to unify two approaches that both describe tube waves. The first approach is due to White, in which the low‐frequency tube‐wave velocity and attenuation are expressed explicitly as functions of frequency. The expressions include contributions from three factors: the borehole fluid compressibility, the wall rigidity, and the wall impedance due to the permeability of the solid. The second approach is the full solution of tube‐wave properties based on Biot theory and the appropriate boundary conditions between the fluid and the porous solid. By taking the low‐frequency approximations to the full solution, we derive an explicit expression for the tube‐wave properties, similar to White’s expression. White’s formulation is hereby corrected to include the elasticity of the solid frame around the borehole. For a 20 cm diameter borehole, ...

Absorbing boundary condition for the elastic wave equation

Abstract: Extant absorbing boundary conditions for the elastic wave equation are generally effective only for waves nearly normally incident upon the boundary. High reflectivity is exhibited for waves traveling obliquely to the boundary. In this paper, a new and efficient absorbing boundary condition for two‐dimensional and three‐dimensional finite‐difference calculations of elastic wave propagation is presented. Compressional and shear components of the incident vector displacement fields are separated by calculating intermediary scalar potentials, allowing the use of Lindman’s boundary condition for scalar fields, which is highly absorbing for waves incident at any angle. The elastic medium is assumed to be homogeneous in the region immediately adjacent to the boundary. The reflectivity matrix of the resulting absorbing boundary for elastic waves is calculated, including the effects of finite‐difference truncation error. For effectively all angles of incidence, reflectivities are much smaller than those of the co...

Datum correction by wave-equation extrapolation

Abstract: Seismic sections are usually datum corrected by static shifting. For small differences in elevation and slow velocity variations between the input datum and the output datum, static shifting is a sufficiently accurate datum correction procedure. However, for significant differences in elevations and a more complicated velocity model, the accuracy of the static solution may prove to be insufficient; and a more exact method should be used.In this paper, we study the limitations of the static method of datum correction and develop simple and effective extrapolation schemes based on the wave equation, schemes which lead to more accurate datum correction. The distortions of seismic events caused by static correction are illustrated by a number of simple examples. To reduce the distortions, we propose a number of extrapolation schemes based on the asymptotics of the Kirchhoff integral solution of the 2-D scalar wave equation. Application of the extrapolation algorithms to synthetic data shows that they provide accurate datum corrections even for a nonplanar input datum and vertical and lateral velocity variations. The algorithms have been successfully applied to real data.

Ultrasonic laboratory tests of geophysical tomographic reconstruction

Abstract: In this study, we test geophysical ray tomography and geophysical diffraction tomography by scaled model ultrasonics experiments. First, we compare the performance of these two methods under limited view‐angle conditions. Second, we compare the adaptabilities of these two methods to objects of various sizes and acoustic properties. Finally, for diffraction tomography, we compare the Born and Rytov approximations based on the induced image distortion by using these two approximation methods. Our experimental results indicate the following: (1) When the scattered field can be obtained, geophysical diffraction tomography is in general superior to ray tomography because diffraction tomography is less sensitive to the limited view‐angle problem and can image small objects of size comparable to a wavelength. (2) The advantage of using ray tomography is that reconstruction can be done using the first arrivals only, the most easily measurable quantity; and there is no restriction on the properties of the object b...

Reverse-time wave-field extrapolation, imaging, and inversion

Abstract: The scattered wave field propagated backward in time into an arbitrary background medium is related via a volume integral to perturbations in velocity about the background, which are expressed as a scattering potential. In general, there is no closed-form expression for the kernel of this integral representation, although it can be expressed asymptotically as a superposition of plane waves backpropagated from the receiver array. When the receiver array completely surrounds the scatterer, the kernel reduces to the imaginary part of the Green's function for the background medium.This integral representation is used to relate the images obtained by imaging algorithms to the actual scattering potential. Two such relations are given: (1) for the migrated image, obtained by deconvolving the extrapolated field with the incident field; and (2) for the reconstructed image, obtained by applying a one-way wave operator to the extrapolated field and then deconvolving by the incident field. The migrated image high-lights rapid changes in the scattering potential (interfaces), whereas the reconstructed image can, under ideal conditions, be a perfect reconstruction of the scattering potential. 'Ideal' conditions correspond to (1) weak scattering about a smoothly varying background medium, (2) a receiver array with full angular aperture, and (3) data of infinite bandwidth.Images obtained from a multioffset vertical seismic profile (VSP) illustrate some of the practical differences between the two imaging algorithms. The reconstructed image shows a much clearer picture of the target (a reef structure), in part because the one-way imaging operator eliminates artifacts caused by the limited aperture of the receiver array.

Fracture detection using P-wave and S-wave vertical seismic profiling at The Geysers

Abstract: In a pilot vertical seismic profiling study, P-wave and cross-polarized S-wave vibrators were used to investigate the potential utility of shear-wave anisotropy measurements in characterizing a fractured rock mass. The caprock at The Geysers geothermal field was found to exhibit about an 11 percent velocity variation between SH-waves and SV-waves generated by rotating the S-wave vibrator orientation to two orthogonal polarizations for each survey level in the well. The effect is generally consistent with the equivalent anisotropy expected from the known fracture geometry.

Calculation of the magnetic gradient tensor from total field gradient measurements and its application to geophysical interpretation

Abstract: The very low inherent noise levels of superconducting quantum interference device (SQUID) sensors have led to proposals for the use of airborne SQUID magnetic gradiometers as geophysical interpretation tools. The quantity measured by such systems will be the gradient tensor, the spatial rate of change of the vector components of the magnetic field. By contrast, existing airborne gradiometers measure the spatial rate of change of the magnitude of the total field. This work describes a technique whereby the gradient tensor can be calculated from measurements of either the vertical or horizontal total field gradients throughout a plane. The signal‐to‐noise ratio of the calculated tensor components is essentially the signal‐to‐noise ratio of the original total field gradient measurements. The resulting tensor components may be upward or downward continued with standard techniques. Two advantages of using the tensor gradients instead of the total field gradients have been determined. Because the tensor compone...