bluuebunny/crossref_metadata_2025 huggingface dataset:
Implementation of the Kirchhoff integral for elastic waves in staggered‐grid modeling schemes
Authors: Mittet Rune
Published: October 2002

Implementation of boundary conditions in finite‐difference schemes is not straightforward for the elastic wave equation if a staggered grid formulation is used. Reverse time migration of VSP data requires a proper description of the recording surface so as not to excite false P‐ and S‐waves. Such contributions may cause artifacts in the imaging procedure. The boundary conditions for the elastic stress tensor can be implemented numerically in a staggered coarse grid modeling scheme by using band‐limited spatial delta‐functions and band‐limited first‐order derivatives of these spatial delta‐functions. A representation theorem for elastic waves is derived to test the implementation of the spatial part of the boundary condition. The implementation is tested in a 2-D numerical experiment for a closed, but curved, boundary S enclosing a volume V. The test condition is that within the volume V, the difference between the forward modeled field and the retropropagated field should be equal to zero. Both P‐ and S‐waves are properly recovered in a 2-D reverse time modeling example. The numerical artifacts related to the proposed spatial approximation of the boundary condition are found to be negligible.
Polarity convention for vibratory source/recording systems
Authors: Landrum Ralph A. , Brook Robert A. , Sallas John J.
Published: October 2002

This report extends the 1975 SEG Technical Standards Committee Recommendations for source/recorder system polarity to include P‐wave vibrators which are phase‐locked to the estimated ground force (weighted‐sum) or reaction mass signals, and to include horizontally polarized (S‐wave) vibrators, none of which were prevalent at the time of that first report.
Static delays in acoustic logging
Authors: Leveille Jacques P. , Nerf Richard B.
Published: October 2002

Borehole‐consistent analysis (analogous to surface‐consistent seismic refraction statics analysis) of compressional wave arrivals in full‐waveform acoustic logs reveal arrival‐time anomalies of tens of microseconds. These anomalies are correlated with lithology and are similar for logging sondes of different geometries. The magnitude of the anomalies makes it unlikely that they are caused by caliper errors or the presence of an altered annulus around the borehole. A possible explanation is given in terms of the Fourier phase spectrum of the first compressional arrivals. The theoretical delays depend on the formation elastic parameters, the borehole geometry and the properties of the borehole fluid; a sensitivity analysis of the relevant parameters is presented.
Electromagnetic ray‐trace tomography
Authors: Nekut A. G.
Published: October 2002

Electromagnetic (EM) tomography can be used to estimate the distribution of electrical rock properties between boreholes. For typical oil field well spacings and reservoir electrical characteristics, crosswell EM measurements are sensitive primarily to the electrical conductivity, not the permittivity. Ray‐tracing methods normally associated with undamped propagating waves can also be applied to the rapidly attenuated, diffusive EM waves encountered in the petroleum reservoir environment. Thermal noise considerations place an upper limit of approximately 20 skin depths on the separation between wells.
Wavefronts in dissipative anisotropic media
Authors: Carcione José M.
Published: October 2002

The purpose of this work is to draw attention to several differences between wave propagation in dis, sipative anisotropic media and purely elastic anisotropic media. In an elastic medium, the wavefront is defined as the envelope of the family of planes that makes the phase of the plane waves zero. It turns out that this definition coincides with the wavefronts obtained from the group and energy velocities, i.e., the three concepts are equivalent. However, for plane waves traveling in dissipative anisotropic media these concepts are different. Despite these differences, the velocity of the envelope of plane waves closely approximates the energy velocity, and therefore can represent the wavefront from a practical point of view. On the other hand, the group velocity describes the wavefront only when the attenuation is relatively low, i.e., for Q values higher than 100. The values of the different velocities and the shape of the wavefront are considerably influenced by the relative values of the attenuation along the principal axes of the anisotropic medium. This means that the anisotropic coefficients in attenuating anisotropic media may differ substantially from the corresponding elastic coefficients. Moteover, it is shown that the usual orthogonality properties between the slowness surface and energy velocity vector and the wavefront and wavenumber vector does not hold for dissipative anisotropic media.
Imaging of reflection seismic energy for mapping shallow fracture zones in crystalline rocks
Authors: Kim J. S. , Moon Wooil M. , Lodha Ganpat , Serzu Mulu , Soonawala Nash
Published: October 2002

The high‐resolution reflection seismic technique is being used increasingly to address geologic exploration and engineering problems. There are, however, a number of problems in applying reflection seismic techniques in a crystalline rock environment. The reflection seismic data collected over a fractured crystalline rock environment are often characterized by low signal‐to‐noise ratios (S/N) and inconsistent reflection events. Thus it is important to develop data processing strategies and correlation schemes for the imaging of fracture zones in crystalline rocks. Two sets of very low S/N, high‐resolution seismic data, previously collected by two different contractors in Pinawa, Canada, and the island of Äspö, Sweden, were reprocessed and analyzed, with special emphasis on the shallow reflection events occurring at depths as shallow as 60–100 m. The processing strategy included enhancing the signals hidden behind large‐amplitude noise, including clipped ground roll. The pre‐ and poststack processing includes shot f-k filtering, residual statics, careful muting after NMO correction, energy balance, and coherency filtering. The final processed seismic sections indicate that reflected energy in these data sets is closely related to rock quality in Äspö data and fracturing in Atomic Energy of Canada, Ltd. (AECL) data. The lithologic boundaries are not clearly mappable in these data. When thickness of the reflection zone is of the order of a wavelength, the top and bottom of the zone may be resolved. The major fracture zones in crystalline rocks correlate closely with the well‐log data and are usually characterized by very low velocity and produce low‐acoustic‐impedance contrasts compared to those of surrounding rocks. Because the incidence angles vary rapidly for shallow‐reflection geometries, segments of major fracture zones can effectively be analyzed in terms of reflectivity. Reflection images of each fracture zone were investigated in the common‐offset section, where each focused event was associated with a consistent incidence angle on the reflectivity map. The complex attributes of the data indicate that strong reflectors at shallow depth coincide with intensely fractured zones. These correlate well with instantaneous amplitude plots and instantaneous frequency plots. The instantaneous phase plot also identifies the major and minor fractures.
Obtaining smooth solutions to large, linear, inverse problems
Authors: VanDecar John C. , Snieder Roel
Published: October 2002

It is not uncommon now for geophysical inverse problems to be parameterized by [Formula: see text] to [Formula: see text] unknowns associated with upwards of [Formula: see text] to [Formula: see text] data constraints. The matrix problem defining the linearization of such a system (e.g., [Formula: see text]m = b) is usually solved with a least‐squares criterion [Formula: see text]. The size of the matrix, however, discourages the direct solution of the system and researchers often turn to iterative techniques such as the method of conjugate gradients to obtain an estimate of the least‐squares solution. These iterative methods take advantage of the sparseness of [Formula: see text], which often has as few as 2–3 percent of its elements nonzero, and do not require the calculation (or storage) of the matrix [Formula: see text]. Although there are usually many more data constraints than unknowns, these problems are, in general, underdetermined and therefore require some sort of regularization to obtain a solution. When the regularization is simple damping, the conjugate gradients method tends to converge in relatively few iterations. However, when derivative‐type regularization is applied (first derivative constraints to obtain the flattest model that fits the data; second derivative to obtain the smoothest), the convergence of parts of the solution may be drastically inhibited. In a series of 1-D examples and a synthetic 2-D crosshole tomography example, we demonstrate this problem and also suggest a method of accelerating the convergence through the preconditioning of the conjugate gradient search directions. We derive a 1-D preconditioning operator for the case of first derivative regularization using a WKBJ approximation. We have found that preconditioning can reduce the number of iterations necessary to obtain satisfactory convergence by up to an order of magnitude. The conclusions we present are also relevant to Bayesian inversion, where a smoothness constraint is imposed through an a priori covariance of the model.
The fields from a finite electrical dipole—A new computational approach
Authors: Sørensen Kurt I. , Christensen Niels B.
Published: October 2002

Controlled‐source, frequency‐domain, and time‐domain electromagnetic methods require accurate, fast, and reliable methods of computing the electric and magnetic fields from the source configurations used. Except for small magnetic dipole sources, all electric and magnetic sources are composed of lengths of straight wire, which may be grounded. If the source‐receiver separation is large enough, the composite electrical dipoles may be considered to be infinitely small, and in a 1-D earth model the fields are expressed as Hankel transforms of an input function, which depends only on the model parameters. The Hankel transforms can be evaluated using the digital filter theory of fast Hankel transforms. However, the approximation of the infinitely small dipole is not always valid, and fields from a finite electrical dipole must be calculated. Traditionally, this is done by numerical integration of the fields from an infinitesimal dipole, thus increasing computation time considerably. The fields from the finite electrical dipole are expressed as Hankel transforms and as integrals of Hankel transforms. The theory of fast Hankel transforms is extended to include integrals of Hankel transforms, and a method is devised for calculating the filter coefficients. Unlike the fast Hankel transform, the computation involved in the integrated Hankel transforms is not a true convolution, and so a set of filter coefficients must be calculated for each source‐receiver configuration. Furthermore, the method is extended to include the calculation of potential differences where one more integration is involved, which is what is actually measured in the field. The computation of filter coefficients is very fast, and for standard configurations, the coefficients need be computed only once. The method is as fast, accurate, and reliable as the fast Hankel transforms method, and is up to an order of magnitude faster than the usual numerical integration.
The Influence of clay content, salinity, stress, and wettability on the dielectric properties of brine‐saturated rocks: 10 Hz to 10 MHz
Authors: Garrouch Ali A. , Sharma Mukul M.
Published: October 2002

Complex impedance measurements have been performed on 14 shaly sand samples, Berea sandstone, and Ottawa sand‐bentonite packs in a frequency range of 10 Hz to 10 MHz, using both the two‐ and four‐electrode techniques. Measurements have been conducted at an effective radial stress varying from ambient pressure to 4000 psi for brine‐saturated oil‐wet and water‐wet samples. The dielectric permittivity is found to correlate with the clay volume fraction, the cation exchange capacity, and electrochemical potential of the rock samples and to depend strongly on the salinity of the brine used. Stress and wettability are shown to have a small influence on the dielectric constant of fully brine‐saturated rocks. A lower critical frequency is found to characterize the geometry of the pore space. Empirical correlations between the dielectric constant, frequency, permeability, cation exchange capacity, and porosity are presented for the shaly sands used in this study. These correlations provide a means of estimating important petrophysical parameters such as the permeability and the clay content from a nondestructive complex impedance sweep of shaly sands fully saturated with brine.
Interpretation of the magnetic anomaly over the Omaha Oil Field, Gallatin County, Illinois
Authors: Sparlin Mark A. , Lewis Richard D.
Published: October 2002

A 40 nanoTesla (nT) magnetic anomaly identified in an aeromagnetic survey over southern Illinois contours as a localized magnetic high on the west flank of a regional magnetic low. This magnetic anomaly is generally coincident with the Omaha Oil Field in northwest Gallatin County, Illinois. It was initially assumed that cultural sources of steel associated with this oil field were the primary source of the magnetic feature; however, similar oil fields overflown by the survey do not exhibit magnetic anomalies in the data set. The Luther Rister et ux ♯1 well, drilled near the apex of the Omaha structural dome, encountered two zones of ultramafic intrusive rock containing 9.0 percent by volume magnetite. These intrusives were identified to be alnöites which are a class of mantle‐derived ultramafic rock that can be associated with the incipient stages of crustal rifting. A ground magnetic survey verified the presence of the anomaly, and provided detailed data for 3-D modeling of the source. Petrophysical evaluations, magnetic susceptibility measurements and thin section modal analysis were made on drill cuttings from the ultramafic intrusives encountered in the Luther Rister ♯1 well. These measurements were made to constrain the 3-D magnetic modeling by the petrophysical characteristics of the source. After removal of the regional magnetic field, the resulting 140 nT residual magnetic anomaly was successfully modeled using two ultramafic sills with an igneous feeder plug. The two igneous sills adequately account for the structural closure exhibited in the Omaha Oil Field and raise the interesting possibility of other hydrocarbon trapping structures generated by intrusives emplaced into the sedimentary section.
Looking ahead with vertical seismic profiles
Authors: Payne Michael A.
Published: October 2002

Several operations enhance our ability to predict the subsurface below the bottom total depth (TD) of the well when applied to zero‐offset vertical seismic profiling (VSP) data. Other key issues regarding the use of VSP data in this fashion are resolution and look‐ahead distance. An impedance log is the most useful form for presenting VSP data to look ahead of the drill bit. The VSP composite trace must first tie reliably to the surface seismic section and to the well log synthetic seismogram. The impedance log is obtained by inverting this VSP composite trace. However, before performing the inversion, we need to (1) correct the composite trace for attenuation effects below TD and (2) input velocities to provide low‐frequency information. An exponential gain function applied to the VSP data below TD adequately compensates for the loss of amplitude caused by attenuation. A calibration of the seismically derived velocities with VSP velocities yields the necessary low‐frequency information. These concepts are illustrated using a field data set and its subset truncated above TD. The output of these operations on the VSP data are compared to well log data. The question of resolution with these data was determined with a model VSP data set based on the well log data. The investigations indicate that the resolution attainable from look‐ahead data is on the order of 50–75 ft (15–23 m). This is one‐quarter seismic wavelength for the frequencies present in these data. In addition, the maximum look‐ahead distance for these data is shown to be easily 2000 ft (600) m and, perhaps, 4000 ft (1200 m5). By way of illustration, the techniques described and investigated 6were applied to an offshore VSP data set to yield an impedance log. After calibrating this curve with the well log data, the base of the target sand was correctly identified below TD. This prediction successfully yielded the thickness of the sand. Individual zones within the sand unit were identified with less confidence.
THE EFFECT OF DENSITY ON SEISMIC REFLECTIONS
Authors: West S. S.
Published: October 2002

By means of existing data on the density of rocks, it is shown that reflection coefficients of seismic waves, calculated from velocities alone, often differ greatly from the actual values. Data on a large number of density discontinuities have been tabulated, and the true reflection coefficients calculated for six cases in which velocities are also known.
THE ATTENUATION CONSTANT OF EARTH MATERIALS
Authors: Born W. T.
Published: October 2002

This paper discusses briefly the nature of viscous losses and solid friction losses, both of which may cause sound waves to be attenuated as they travel through a physical medium. A simple experimental technique for determining the nature and magnitude of the loss factor in small rock samples is described, and data are given which indicate that solid friction losses are primarily responsible for the observed attenuation of the seismic waves employed in the seismic reflection method. A method of estimating the attenuation factor of earth materials from seismic reflection records is outlined and it is shown that the values so obtained are not inconsistent with the laboratory data. Frequency characteristic curves of seismic wave paths are derived on the basis of the experimental data.
GRAVITY METER SURVEY OF THE WELLINGTON FIELD, LARIMER COUNTY, COLORADO
Authors: Wilson John H.
Published: October 2002

The results of a gravity meter survey across the Wellington Field, Larimer County, Colorado, both before and after elimination of the regional effect is presented in relation to the subsurface structure of the field. The method of eliminating the regional effect is shown. This information is presented as a “case book” type, illustrating an anomaly related to local structure that is almost obscured by strong regional gradients.
Experimental observation of surface wave propagation for a transversely isotropic medium
Authors: Chang Chih‐Hsiung , Gardner Gerald H. F. , McDonald John A.
Published: October 2002

Velocity anisotropy of surface‐wave propagation in a transversely isotropic solid has been observed in a laboratory study. In this study, Phenolite™, an electrical insulation material, was used as the transversely isotropic media (TIM), and a vertical seismic profiling (VSP) geometry was used to record seismic arrivals and to separate surface waves from shear waves. Results show that surface waves that propagate with different velocities exist at certain directions.
General solutions and causality for a Voigt medium
Authors: Duren Richard E. , Heestand Richard L.
Published: October 2002

A 1-D wave equation solution for a propagating seismic pulse in a Voigt medium can be obtained by using a separation of variables to find time harmonic particular solutions and then superimposing the particular solutions. This superposition is a time convolution of the boundary condition (or incident pulse) and the medium’s impulse response. Even though causality is not introduced during the solution of the wave equation, the general solution is causal since the boundary condition is causal and the medium’s impulse response can be shown to be causal. The relationship between attenuation and phase velocity as well as their dependence on frequency arise from the form chosen for the particular solutions. The arbitrary constants associated with the particular solutions are determined by the boundary condition, and the initial condition is also dependent on the boundary condition; however, the initial condition is properly determined and does not depend on times after the initial time (thereby satisfying causality). The convolutional nature of the general solution allows it to also be expressed as a time convolution of the boundary condition’s time derivative and the medium’s step function response. This expression can be viewed as a superposition of step function responses where the step function response is a particular solution to the wave equation obtained using an approach that is similar to one recently developed for propagating electric pulses. This new solution is obtained with the initial and boundary conditions being independently introduced during the solution of the wave equation. There is no frequency dependence in this solution, and the general solution is causal since it is a superposition of causal step function responses.
Normal moveout from dipping reflectors in anisotropic media
Authors: Tsvankin Ilya
Published: October 2002

Description of reflection moveout from dipping interfaces is important in developing seismic processing methods for anisotropic media, as well as in the inversion of reflection data. Here, I present a concise analytic expression for normal‐moveout (NMO) velocities valid for a wide range of homogeneous anisotropic models including transverse isotropy with a tilted in‐plane symmetry axis and symmetry planes in orthorhombic media. In transversely isotropic media, NMO velocity for quasi‐P‐waves may deviate substantially from the isotropic cosine‐of‐dip dependence used in conventional constant‐velocity dip‐moveout (DMO) algorithms. However, numerical studies of NMO velocities have revealed no apparent correlation between the conventional measures of anisotropy and errors in the cosine‐of‐dip DMO correction (“DMO errors”). The analytic treatment developed here shows that for transverse isotropy with a vertical symmetry axis, the magnitude of DMO errors is dependent primarily on the difference between Thomsen parameters ε and δ. For the most common case, ε − δ > 0, the cosine‐of‐dip–corrected moveout velocity remains significantly larger than the moveout velocity for a horizontal reflector. DMO errors at a dip of 45 degrees may exceed 20–25 percent, even for weak anisotropy. By comparing analytically derived NMO velocities with moveout velocities calculated on finite spreads, I analyze anisotropy‐induced deviations from hyperbolic moveout for dipping reflectors. For transversely isotropic media with a vertical velocity gradient and typical (positive) values of the difference ε − δ, inhomogeneity tends to reduce (sometimes significantly) the influence of anisotropy on the dip dependence of moveout velocity.
Squirt flow in fully saturated rocks
Authors: Dvorkin Jack , Mavko Gary , Nur Amos
Published: October 2002

We estimate velocity/frequency dispersion and attenuation in fully saturated rocks by employing the squirt‐flow mechanism of solid/fluid interaction. In this model, pore fluid is squeezed from thin soft cracks into the surrounding large pores. Information about the compliance of these soft cracks at low confining pressures is extracted from high‐pressure velocity data. The frequency dependence of squirt‐induced pressure in the soft cracks is linked with the porosity and permeability of the soft pore space, and the characteristic squirt‐flow length. These unknown parameters are combined into one expression that is assumed to be a fundamental rock property that does not depend on frequency. The appropriate value of this expression for a given rock can be found by matching our theoretical predictions with the experimental measurements of attenuation or velocity. The low‐frequency velocity limits, as given by our model, are identical to those predicted by Gassmann’s formula. The high‐frequency limits may significantly exceed those given by the Biot theory: the high‐frequency frame bulk modulus is close to that measured at high confining pressure. We have applied our model to D’Euville Limestone, Navajo Sandstone, and Westerly Granite. The model realistically predicts the observed velocity/frequency dispersion, and attenuation.
Integrated reservoir characterization: Beyond tomography
Authors: Lines Larry , Tan Henry , Treitel Sven , Beck John , Chambers Richard , Eager John , Savage Charles , Queen John , Rizer William , Buller Paul , Cox Dale , Sinton John , Ballard James , Kokkoros George , Track Antoine , Guerendel Philippe , Harris Jerry , Lazaratos Spyros , Van Schaak Mark
Published: October 2002

In 1992, there was a collaborative effort in reservoir geophysics involving Amoco, Conoco, Schlumberger, and Stanford University in an attempt to delineate variations in reservoir properties of the Grayburg unit in a West Texas [Formula: see text] pilot at North Cowden Field. Our objective was to go beyond traveltime tomography in characterizing reservoir heterogeneity and flow anisotropy. This effort involved a comprehensive set of measurements to do traveltime tomography, to image reflectors, to analyze channel waves for reservoir continuity, to study shear‐wave splitting for borehole stress‐pattern estimation, and to do seismic anisotropy analysis. All these studies were combined with 3-D surface seismic data and with sonic log interpretation. The results are to be validated in the future with cores and engineering data by history matching of primary, water, and [Formula: see text] injection performance. The implementation of these procedures should provide critical information on reservoir heterogeneities and preferential flow direction. Geophysical methods generally indicated a continuous reservoir zone between wells.
Radio frequency electromagnetic tunnel detection and delineation at the Otay Mesa site
Authors: Mahrer Kenneth D. , List David F.
Published: October 2002

Using three radio frequency electromagnetic field procedures: surface‐to‐surface, borehole‐to‐surface, and borehole‐to‐borehole, we detected and delineated a horizontal, 2 1/2 × 5-ft (3/4 × 1 1/2-m), and 45-ft (14-m) deep tunnel crossing the US‐Mexico border near Otay Mesa east of San Diego, CA. We used continuous monochromatic signals to delineate the tunnel by (1) locating and identifying signature variations in signal transmissivity from the tunnel void in the local rock (sandstone) and (2) inducing and mapping the (secondary) radiation from electrical cabling within and running the length of the tunnel. In the surface‐to‐surface study we evaluated both a single magnetic dipole and a dipole gradiometer receiver and both a magnetic dipole and a grounded long‐wire transmitter operating between 22 kHz to approximately 300 kHz to isolate the induced radiation. We found that both sources operating at approximately 100 kHz gave the best transmitter‐cable coupling, and the nulling effect of the gradiometer was necessary to eliminate the primary field. In the surface‐to‐borehole study we placed a dipole receiver in boreholes and a dipole transmitter on the surface and operated at approximately 100 kHz. Horizontal traverses of the transmitter found peaks in the signal amplitude when the transmitter was directly above the tunnel. Vertical traverses of the receiver found (secondary) radiation peaks in the signal amplitude at depths corresponding to the tunnel depth. In the borehole‐to‐borehole studies, using zero‐offset (i.e., transmitter and receiver at the same depth) traverses and 1 MHz and 15 MHz signals, we found peaks in the signal amplitude at depths corresponding to direct ray transmission through the tunnel.
Viscoacoustic wave propagation in 2-D random media and separation of absorption and scattering attenuation
Authors: Kneib Guido , Shapiro Serge A.
Published: October 2002

Wave theoretical analysis of scalar, time‐harmonic waves propagating in a constant density medium with isotropic, random velocity fluctuations and being scattered mainly in the forward direction yields a simple and robust procedure that combines the logarithm of the mean wave amplitude with the mean logarithm of the wave amplitude to perform a separation of scattering attenuation and absorption effects. Finite‐difference simulations of wave propagation in 2-D random media with a Voigt‐body rheology illustrate the evolution of wave field fluctuations and demonstrate that the separation procedure works for a wide range of seismic albedos. In the case of no absorption, the logarithms of seismic amplitudes will have a nonlinear dependence on the travel distance if the wavefield fluctuations are small compared to the amplitude of the coherent field. If these fluctuations are large, the logarithms of seismic amplitudes will tend to constant levels independent of the travel distance. In the case of random viscoacoustic media and at propagation distances larger than the inverse of the scattering coefficient of the coherent field, and apart from geometrical spreading, the overall amplitude decrease will be predominated by absorption, even if the absorption coefficient is one order smaller than the scattering coefficient of the coherent field.
Determination of background velocities by multiple migration fitting
Authors: Chavent Guy , Jacewitz Chester A.
Published: October 2002

We present an approach called multiple migration fitting (MMF) designed to automatically determine 2-D background velocities from prestack seismic data. In this approach, we maximize a scalar similarity index (SI) for a collection of migrated sections obtained by various illuminations of the same earth. Numerical investigation shows that this index is a rather smooth, nonoscillatory function of velocity that tends to be a maximum for good velocity profiles, and hence is amenable to maximization by local gradient techniques. This maximization will be practically feasible, as we prove that the exact gradient of SI can be computed at an additional cost of only twice that required for the computation of the collection of migrated sections, independently of the number of velocity unknowns. Application to synthetic data shows that MMF leads to enhanced background velocities and stacked migrated sections.
An analysis of 3-D analytic signal
Authors: Mohan N. L. , Babu L. Anand
Published: October 2002

The mathematical and physical basis of defining the 3-D analytic function and the corresponding analytic signal is critically examined, and it is proved that 3-D analytic signals based on (1) scalar and (2) vector additions of the horizontal derivatives of the total magnetic fields are completely identical. Two sets of simulated gridded data are considered, and 3-D analytic signals are computed using both scalar and vector additions and are found to be identical. The equality of scalar and vector additions of 3-D analytic signals is further demonstrated with the help of gridded‐surface vertical magnetic field data from the Krishna‐Godavari Basin, Andhra Pradesh, India.
Decomposition of multicomponent sea‐floor data into upgoing and downgoing P‐ and S‐waves
Authors: Amundsen Lasse , Reitan Arne
Published: October 2002

A method for decomposing multicomponent sea‐floor measurements into upgoing and downgoing P‐ and S‐waves is presented. We assume that a marine survey employing a marine source in the water layer is conducted over a plane‐layered medium. From recordings of the pressure just above the sea floor and the particle velocity vector just below the sea floor, decomposition filters can be determined by plane‐wave analysis. The decomposition filter coefficients depend on the P‐ and S‐wave velocities and the density at the sea bottom. We show how to decompose the multicomponent measurements into upgoing and downgoing P‐ and S‐vertical traction components, vertical‐particle velocity components, and horizontal particle velocity components. The decomposition filters are applied with good results to synthetic data modeled in a plane‐layered medium.
A critical review of acoustic wave modeling procedures in 2.5 dimensions
Authors: Williamson Paul R. , Pratt R. Gerhard
Published: October 2002

In a short note Liner (1991) postulated a wave equation that would allow efficient computation of 3-D wavefields in c(x, z) media when the sources and receivers lie in a common, y = 0 plane. Liner and his co‐researchers continued to investigate this wave equation in two further papers (Liner and Stockwell, 1993; Bording and Liner, 1993). This problem, known as the 2.5-D problem (after Bleistein, 1986), is of great importance because of the prevalence of 2-D surveys in seismic exploration and the cost of full 3-D modeling. We have discovered that a solution to the 2.5-D problem based on previously published information about filters is more accurate than Liner’s method.
Borehole seismic‐source radiation in layered isotropic and anisotropic media: Boundary element modeling
Authors: Dong Wenjie , Bouchon Michel , Toksöz M. Nafi
Published: October 2002

In modeling waves radiated from a borehole seismic source in layered isotropic or anisotropic media, the commonly used numerical methods (e.g., finite difference and finite element) encounter difficulties because of the large scale difference between the borehole diameter and the formation extent. To get around this problem, we apply the indirect boundary element method to establish a general algorithm for modeling source radiation from open and cased boreholes in layered transversely isotropic (TI) media. The essence of the algorithm is to use discrete secondary sources (unknowns) on both sides of the borehole wall (formation/cement interface) to represent the influence of the interface on wave scattering, so that wave propagation inside and outside the borehole can be carried out by Green’s functions. The discrete distribution of the secondary sources is determined by matching boundary conditions on the borehole wall. Comparison with the discrete wavenumber method validates the implementation. Applications to fluid‐filled open and cased boreholes in three‐layer media demonstrate the creation of guided modes in low velocity layers. Presence of anisotropy complicates the guided modes as a result of dispersion and P‐ and SV‐waves coupling in homogeneous TI media. Presence of casing and cement enhances the visibility of the guided modes.
Pitfalls in crosshole seismic interpretation as a result of 3-D effects
Authors: Mufti Irshad R.
Published: October 2002

In surface seismics, much of the reflection data originate from more or less stratified structures with large lateral extensions. Consequently, the data can often be interpreted without introducing serious errors by treating the earth as a 2-D medium. In crosshole reservoir tomography, however, the target of interest is a finite object with geometric features comparable in size to the dominant wavelength of the source signal. Under these conditions, a significant amount of energy can bypass the object under investigation as a result of diffraction. When the velocity of the anomalous body is less than that of the surrounding medium, the bypassed energy can appear as the first arrival in the receiver borehole in accordance with Fermat’s principle. Crosshole seismic response of a number of simple 3-D reservoirs is investigated by finite‐difference models based on the acoustic wave equation. The nature of the signals computed along the receiver borehole confirms the presence of both the transmitted and the bypassed energy. Our observations indicate that unless the two surfaces at which the energy enters and leaves the reservoir are parallel, a 2-D inversion algorithm will lead to incorrect results. Moreover, a velocity anomaly as a result of a steam flood may have negligible effect on the first arrivals because most of the energy bypasses the anomaly in the third dimension. All these conditions point to potential pitfalls in standard 2-D tomographic inversion which considers the first arrivals only. More reliable results can be obtained by simultaneous recording of data along more than one borehole, identifying the bypassed signals, utilizing as many events as possible in the recorded data, and performing model‐driven interpretation.
Crosswell electromagnetic tomography in steel‐cased wells
Authors: Nekut Anthony G.
Published: October 2002

The applicability of crosswell electromagnetic (EM) tomography is limited when the boreholes are steel cased. Steel casing is typically eight orders of magnitude more conductive than the formations around the borehole. Consequently, the high frequency signals which are needed to resolve the formation conductivity distribution in the interwell region cannot be radiated or received by antennas inside the casing. Introduction of small insulating gaps in the casing eliminates these problems by creating ports where signals can exit and enter the casing, making the casing an electric bipole antenna. The gapped casing can be operated in a shorted‐port or open‐port mode. In the shorted‐port mode, the voltage is specified at a source port, and the current is measured at all the remaining shorted ports. In the open‐port mode, the current is specified at a source port, and the voltage is measured at all the remaining open ports. For high‐frequency shorted‐port operation, the effective length of the casing bipole antenna is determined by the skin depth in the formations surrounding the casing. For high‐frequency open‐port operation, the effective length of the casing bipole antenna is determined by the casing gap interval, provided that it is less than the formation skin depth. The open‐port mode is preferred. The interval between casing gaps is determined mainly by the well spacing. If the effective bipole length is proportional to the well spacing, the crosswell signal‐to‐noise ratio is approximately independent of the well spacing and the interwell resolution is approximately proportional to the well spacing at constant source input power. For large well spacings, gapped casing antennas outperform open‐hole tools. Steel‐cased wells completed or retrofitted with insulating gaps offer a stable, low‐cost, permanent set of electrodes to implement long‐term monitoring of petroleum reservoir fluid movements. Gapped casings offer a practical way to achieve high‐frequency, wide‐aperture EM tomographic datasets at large well spacings. Gapped casings can also be used for deep EM probing of the formations around a single well.
Anisotropic wave propagation through finite‐difference grids
Authors: Igel Heiner , Mora Peter , Riollet Bruno
Published: October 2002

An algorithm is presented to solve the elastic‐wave equation by replacing the partial differentials with finite differences. It enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models. The space derivatives are calculated using discrete convolution sums, while the time derivatives are replaced by a truncated Taylor expansion. A centered finite difference scheme in Cartesian coordinates is used for the space derivatives leading to staggered grids. The use of finite difference approximations to the partial derivatives results in a frequency‐dependent error in the group and phase velocities of waves. For anisotropic media, the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the Hook sum. This interpolation induces an additional error in the wave properties. The overall error depends on the precision of the derivative and interpolation operators, the anisotropic symmetry system, its orientation and the degree of anisotropy. The dispersion relation for the homogeneous case was derived for the proposed scheme. Since we use a general description of convolution sums to describe the finite difference operators, the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model. In particular, phase and group velocities of the three wave types can be determined in any direction. We demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed.
Uneven effect of random noise in magnetotelluric apparent resistivity definitions
Authors: Gómez‐Treviño E. , Mondragón Miguel
Published: October 2002

In the magnetotelluric method, measurements of electromagnetic fields on the surface of the earth are used to investigate the subsurface distribution of electrical resistivity from depths of a few meters to hundreds of kilometers. In the initial stage of interpretation, the measurements are conveniently normalized and expressed in terms of apparent resistivity. The formula to normalize the measurements was derived by Cagniard (1953) in the classical paper where he also established the principles of the method.
Model‐based inversion of amplitude‐variations‐with‐offset data using a genetic algorithm
Authors: Mallick Subhashis
Published: October 2002

I cast the inversion of amplitude‐variation‐with‐offset (AVO) data into the framework of Bayesian statistics. Under such a framework, the model parameters and the physics of the forward problem are used to generate synthetic data. These synthetic data are then matched with the observed data to obtain an a‐posteriori probability density (PPD) function in the model space. The genetic algorithm (GA) uses a directed random search technique to estimate the shape of the PPD. Unlike the classical inversion methods, GA does not depend upon the choice of an initial model and is well suited for the AVO inversion. For the single‐layer AVO inversion where the amplitudes from a single reflection event are inverted, GA estimates the normal incidence reflection coefficient [Formula: see text] and the contrast of the Poisson’s ratio (Δσ) to reasonable accuracy, even when the signal‐to‐noise ratio is poor. Comparisons of single‐layer amplitude inversion using synthetic data demonstrate that GA inversion obtains more accurate results than does the least‐squares fit to the approximate reflection coefficients as is usually practiced in the industry. In the multilayer AVO waveform inversion, all or a part of the prestack data are inverted. Inversion of this type is nonunique for the estimation of the absolute values of velocities, Poisson’s ratios, and densities. However, by applying simplified approximations to the P‐wave reflection coefficient, I verify that [Formula: see text], the contrast in the acoustic impedance (ΔA), and the gradient in the reflection coefficient (G), can be estimated from such an inversion. From the GA estimated values of [Formula: see text], ΔA, and G, and from reliable estimates of velocity and Poisson’s ratio at the start time of the input data, an inverted model can be generated. I apply this procedure to marine data and demonstrate that the the synthetics computed from such an inverted model match the input data to reasonable accuracy. Comparison of the log data from a nearby well shows that the GA inversion obtains both the low‐ and the high‐frequency trends (within the bandwidth of seismic resolution) of the P‐wave acoustic impedance. In addition to P‐wave acoustic impedance, GA also obtains an estimate of the Poisson’s ratio, an extremely important parameter for the direct detection of hydrocarbons from seismic data.
Transforming walk‐away VSP data into reverse VSP data
Authors: Mittet Rune , Hokstad Ketil
Published: October 2002

Marine walk‐away vertical seismic profiling (VSP) data can be transformed into reverse VSP data using an elastic reciprocity transformation. A reciprocity transform is derived and tested using data generated with a 2-D high‐order, finite‐difference modeling scheme in a complex elastic model. First, 201 shots are generated with a walk‐away VSP experimental configuration. Both the x‐component and the z‐component of the displacement are measured. These data are collected in two common receiver data sets. Then two shots are generated in a reverse VSP configuration. We demonstrate that subtraction of the reverse VSP data from the walk‐away VSP data gives very small residuals. The transformation of walk‐away data into reverse VSP data makes prestack shot‐domain migration feasible for walk‐away data. Synthetic data from a multishot walk‐away experiment can be obtained from one or a few modeling operations with a RVSP experimental configuration. The required computer time is reduced by two orders of magnitude.
Linear inversion of gravity data for 3-D density distributions
Authors: Bear Glenn W. , Al‐Shukri Haydar J. , Rudman Albert J.
Published: October 2002

We have developed an improved Levenburg‐Marquart technique to rapidly invert Bouguer gravity data for a 3-D density distribution as a source of the observed field. This technique is designed to replace tedious forward modeling with an automatic solver that determines density models constrained by geologic information supplied by the user. Where such information is not available, objective models are generated. The technique estimates the density distribution within the source volume using a least‐squares inverse solution that is obtained iteratively by singular value decomposition using orthogonal decomposition of matrices with sequential Householder transformations. The source volume is subdivided into a series of right rectangular prisms of specified size but of unknown density. This discretization allows the construction of a system of linear equations relating the observed gravity field to the unknown density distribution. Convergence of the solution to the system is tightly controlled by a damping parameter which may be varied at each iteration. The associated algorithm generates statistical measures of solution quality not available with most forward methods. Along with the ability to handle large data sets within reasonable time constraints, the advantages of this approach are: (1) the ease with which pre‐existing geological information can be included to constrain the solution, (2) its minimization of subjective user input, (3) the avoidance of difficulties encountered during wavenumber domain transformations, and (4) the objective nature of the solution. Application to a gravity data set from Hamilton County, Indiana, has yielded a geologically reasonable result that agrees with published models derived from interpretation of gravity, magnetic, seismic, and drilling data.