bluuebunny/crossref_metadata_2025 huggingface dataset:
Surface‐consistent deconvolution
Authors: Levin Stewart A.
Published: October 2002

Using first principles, I formulate surface‐consistent deconvolution as a problem in optimization, to which I apply methods for nonlinear least‐squares. To minimize processing artifacts, I avoid data transformations such as Fourier transforms or slant stacks. My deconvolution filters are computed as approximate solutions to a large, sparse, least‐squares system. Useful solutions are obtained in a few iterations, with each iteration equivalent in cost to performing a single‐trace deconvolution of the data. The use of optimization removes a great deal of uncertainty about the results of surface‐consistent deconvolution. If, with a given choice of filter length, etc., a good deconvolution can be obtained, it is obtained. I illustrate the method and its limitations with three instructive field examples. The first underlines the issues of uncertainty and reliability. The other two show how surface‐consistent deconvolution can degrade as seismic data depart from the surface‐consistent model by small and large amounts, respectively.
Incorporation of topography into two‐dimensional resistivity inversion
Authors: Tong Lun‐Tao , Yang Chieh‐Hou
Published: October 2002

Direct current resistivity data acquired on rough terrain can be interpreted by using an appropriate inversion technique after a topographic correction. In order to avoid the influence of a possibly incomplete topographic correction, an improved two‐dimensional resistivity inversion algorithm has been developed to estimate the subsurface resistivity distribution in the presence of topography. In this paper, fully discretized modeling is based on the finite‐element method, and the iterative inversion scheme is derived from the second‐order Marquardt damped least‐squares method. The algorithm has been tested on both synthetic and field resistivity data with topography incorporated explicitly into the inversion model. Both theoretical and field studies indicate that the technique is computationally efficient and provides an improved geologic interpretation for complex subsurface structures. A four‐electrode configuration is used in the algorithm so that the inversion can represent most field measurements.
Ray‐tracing‐based prediction and subtraction of water‐layer multiples
Authors: Calvert Andrew J.
Published: October 2002

Many methods of multiple suppression break down when the structure that produces the reverberation possesses significant lateral variation; a common example of this situation occurs in marine data with the multiple reflections that are generated by seafloor topography. Such multiples may be suppressed by techniques based upon wave‐equation extrapolation; the recorded seismic data are mathematically propagated through a simulated water layer to generate a set of multiple arrivals which may, after data matching, be subtracted. However, the computational effort required to propagate prestack data to a laterally varying datum is very large. In this paper, a method of suppressing selected multiples with arbitrary moveout is presented. In order to reduce the computational cost, prediction of the multiple arrival times is performed by ray tracing through a model of the laterally varying water layer and, possibly, the subsurface. An estimate of the multiple waveform on each trace is obtained by stacking a window of data about the calculated arrival times. The multiple arrival can then be attenuated by subtracting this wavelet from each trace in the prestack gather from which the estimate is derived. In practice, calculations of the variation in multiple amplitude and of any errors in the moveout correction require the multiple reflections to be of comparable, or higher, amplitude than contemporary primary events, a situation that is often the case where multiple contamination is a problem.
Offset‐dependent geometrical spreading in a layered medium
Authors: Ursin Bjørn
Published: October 2002

The geometrical spreading for a point source in a horizontally layered medium has been computed by Ursin (1978) and Hubral (1978) as a Taylor series in the offset coordinate. The coefficients in the Taylor series depend on the thicknesses and the velocities of the layers. Here, I start with the exact expression for geometrical spreading and show that it can be expressed as a function of the velocity in the first layer, the offset, and the first‐ and second‐order traveltime derivatives. A shifted hyperbolic traveltime approximation (Castle, 1988) and the usual hyperbolic traveltime approximation are used to derive approximate expressions for geometrical spreading. These expressions can also be derived from a truncated Taylor series by making additional approximations, but this procedure is not so obvious.
A new concept in modeling the dielectric response of sandstones: Defining a wetted rock and bulk water system
Authors: Knight Rosemary , Endres Anthony
Published: October 2002

Experimental data for the real part of the dielectric constant (K′) of three sandstone samples are considered as a function of the level of water saturation [Formula: see text] in the frequency range 60 kHz to 4 MHz. Existing theoretical models have previously shown poor agreement with K′ versus [Formula: see text] data for rock samples, undoubtedly due to the complexity involved in adequately accounting for geometrical and electrochemical effects. In analyzing the data presented here, we find a pronounced increase in K′ in the low saturation region which in all cases can be attributed to the establishment of geometrical and surface effects associated with the rock‐water interface. When this increase in K′ is accounted for by defining wetted matrix parameters, the data show excellent agreement with existing theoretical models.
A shallow seismic reflection survey in basalts of the Snake River Plain, Idaho
Authors: Miller Richard D. , Steeples Don W.
Published: October 2002

The objective of this feasibility study was to determine if the seismic reflection method could help to optimize placement of water‐quality monitoring wells near a radioactive storage facility. Seismic reflections from depths less than 30 m were recorded along a 500 m long line over a basalt, rhyolite, and sedimentary sequence in the Snake River Plain. Some shallow reflections at 40 to 50 ms on the field files are of exceptional quality with frequency exceeding 150 Hz. Reflections and refractions from selected seismograms along the line possess vastly different normal‐moveout (NMO) and apparent velocities as well as wavelet characteristics. Extreme variations in quality, seismic character, and reflector geometries observed on seismograms give the appearance of varying geologic settings and are uncommon for such short distances. Severe surgical muting was necessary for accurate velocity and statics analyses. The seismic reflection data show apparent structural lows in a sedimentary layer sandwiched between basalt flows. Interpreted structural lows must be verified by drilling before a monitoring plan can be fully developed. Similar shallow reflection surveys could also be used to improve deeper conventional seismic data in this and other basaltic terrain.
The application of Walsh transforms to interpret gravity anomalies due to some simple geometrically shaped causative sources: A feasibility study
Authors: Shaw R. K. , Agarwal B. N. P.
Published: October 2002

Walsh functions are a set of complete and orthonormal functions of nonsinusoidal waveform. In contrast to sinusoidal waveforms whose amplitudes may assume any value between −1 to +1, Walsh functions assume only discrete amplitudes of ±1 which form the kernel function of the Walsh transform. Because of this special nature of the kernel, computation of the Walsh transform of a given signal is simpler and faster than that of the Fourier transform. The properties of the Fourier transform in linear time are similar to those of the Walsh transform in dyadic time. The Fourier transform has been widely used in interpretation of geophysical problems. Considering various aspects of the Walsh transform, an attempt has been made to apply it to some gravity data. A procedure has been developed for automated interpretation of gravity anomalies due to simple geometrical causative sources, viz., a sphere, a horizontal cylinder, and a 2-D vertical prism of large depth extent. The technique has been applied to data from the published literature to evaluate its applicability, and the results are in good agreement with the more conventional ones.
Digital image processing of airborne geophysical data for uranium‐mineralized breccia pipes exploration in northwestern Arizona
Authors: Kwarteng A. Y. , Chavez P. S.
Published: October 2002

Digital image processing and integration of data sets have been used to develop exploration models from airborne electromagnetics (EM), magnetics, and very‐low‐frequency electromagnetics (VLF-EM) data collected over an area in northwestern Arizona. The area has potential for the occurrence of uranium‐mineralized breccia pipes. Apparent resistivity and overburden thickness were derived from the EM measurements using half‐space models. Digital image processing techniques applied to the geophysical data sets included: (1) conversion of the data into gridded‐scale images, (2) spatial filtering for noise removal, (3) integration and analysis of the data sets, and (4) modeling using various parameter combinations. The general relationships between the geophysical variables/parameters and their ability to detect metallic deposits were used as guides in selecting digital number ranges that were used as input into various models. One of the best models incorporated apparent resistivity and total‐field magnetics; the results of this model outlined 13 anomalous combinations in the survey area. Field checking confirmed that two of the anomalies were previously known orebodies, and most of the other anomalies corresponded to suspected pipes that were under evaluation by the group that is exploring the property.
Relationships among compressional wave attenuation, porosity, clay content, and permeability in sandstones
Authors: Klimentos T. , McCann C.
Published: October 2002

Anelastic attenuation is the process by which rocks convert compressional waves into heat and thereby modify the amplitude and phase of the waves. Understanding the causes of compressional wave attenuation is important in the acquisition, processing, and interpretation of high‐resolution seismic data, and in deducing the physical properties of rocks from seismic data. We have measured the attenuation coefficients of compressional waves in 42 sandstones at a confining pressure of 40 MPa (equivalent to a depth of burial of about 1.5 km) in a frequency range from 0.5 to 1.5 MHz. The compressional wave measurements were made using a pulse‐echo method in which the sample (5 cm diameter, 1.8 cm to 3.5 cm long) was sandwiched between perspex (lucite) buffer rods inside the high‐pressure rig. The attenuation of the sample was estimated from the logarithmic spectral ratio of the signals (corrected for beam spreading) reflected from the top and base of the sample. The results show that for these samples, compressional wave attenuation (α, dB/cm) at 1 MHz and 40 MPa is related to clay content (C, percent) and porosity (ϕ, percent) by α=0.0315ϕ+0.241C−0.132 with a correlation coefficient of 0.88. The relationship between attenuation and permeability is less well defined: Those samples with permeabilities less than 50 md have high attenuation coefficients (generally greater than 1 dB/cm) while those with permeabilities greater than 50 md have low attenuation coefficients (generally less than 1 dB/cm) at 1 MHz at 40 MPa. These experimental data can be accounted for by modifications of the Biot theory and by consideration of the Sewell/Urick theory of compressional wave attenuation in porous, fluid‐saturated media.
Localization and backscattering spectrum of seismic waves in stratified lithology
Authors: White Benjamin , Sheng Ping , Nair Balan
Published: October 2002

Using four sonic well logs from diverse geological environments, we analyze the statistics of lithological layers relevant to seismic wave propagation. The autocorrelation functions are found to be well approximated by exponentials with correlation lengths generally in the 1.5 to 3 m range. We use localization theory to calculate the apparent attenuation caused by random scattering as a function of frequency. This attenuation has a peak at 50 to 150 Hz with a corresponding localization length of 1.6 to 4.8 km (1 to 3 mi) and an apparent Q of 120 to 450. At seismic frequencies (20 Hz) the attenuation is smaller, with localization lengths in the 16 to 32 km (10 to 20 mi) range and an apparent Q of 300 to 700. These values of apparent Q are consistent with observations of previous authors who used well‐log calculations. Using finite differences we compute synthetic traces for a 20-Hz pulse with multiple backscattering from the lithology given by each of the well logs. The power spectral density of the trace is computed for different time gates and compared with theoretical predictions. The nonwhite, [Formula: see text] dependence of the spectrum as predicted by theory is confirmed. Furthermore, when plotted as a function of the scaled variable inversely proportional to the localization length, the average of the four power spectra exhibits excellent agreement with the dimensionless scaling function derived using localization theory.
Split‐step Fourier shot‐record migration with deconvolution imaging
Authors: Lee Delman , Mason Iain M. , Jackson Geoffrey M.
Published: October 2002

The split‐step Fourier method is used to prestack migrate two synthetic borehole‐to‐surface shot gathers. Model structures in the zone of specular illumination beneath the shot are reconstructed by using the split‐step Fourier method both to back‐propagate the recorded wavefield and to forward propagate the source wavelet. The overburden is vertically and laterally inhomogeneous. Each depth interval is treated as a homogeneous strip with the mean velocity plus an inhomogeneity correction term. The inhomogeneity correction term is split and spatially multiplied with each spectral component of the wavefield on its entry to and upon its exit from each strip. Propagation through each strip is effected by multiplication in the spatial frequency domain. The split‐step Fourier method offers a valuable alternative to finite‐difference migration for machines with limited memory. Three imaging methods are compared for two signal‐to‐noise ratios. They are: image extraction by traveltime, crosscorrelation with source wavelet, and deconvolution with source wavelet. At high signal‐to‐noise level, the image formed by deconvolution offers better spatial resolution than images formed by crosscorrelation with the source wavelet or by extraction using traveltime. If the signal‐to‐noise level is low, traveltime imaging deteriorates rapidly, while deconvolution images degrade towards those created by crosscorrelation imaging.
Nonlinear multiparameter optimization using genetic algorithms: Inversion of plane‐wave seismograms
Authors: Stoffa Paul L. , Sen Mrinal K.
Published: October 2002

Seismic waveform inversion is one of many geophysical problems which can be identified as a nonlinear multiparameter optimization problem. Methods based on local linearization fail if the starting model is too far from the true model. We have investigated the applicability of “Genetic Algorithms” (GA) to the inversion of plane‐wave seismograms. Like simulated annealing, genetic algorithms use a random walk in model space and a transition probability rule to help guide their search. However, unlike a single simulated annealing run, the genetic algorithms search from a randomly chosen population of models (strings) and work with a binary coding of the model parameter set. Unlike a pure random search, such as in a “Monte Carlo” method, the search used in genetic algorithms is not directionless. Genetic algorithms essentially consist of three operations, selection, crossover, and mutation, which involve random number generation, string copies, and some partial string exchanges. The choice of the initial population, the probabilities of crossover and mutation are crucial for the practical implementation of the algorithm. We investigated the effects of these parameters in the inversion of plane‐wave seismograms in which a normalized crosscorrelation function was used as the objective or fitness function (E). We also introduce the concept of “update” probability to control the influence of past generations. The combination of a low value of mutation probability (∼0.01), a moderate value of the crossover probability (∼0.6) and a high value of update probability (∼0.9) are found to be optimal for the convergence of the algorithm. Further, we show that concepts from simulated annealing can be used effectively for the stretching of the fitness function which helps in the convergence of the algorithm. Thus, we propose to use exp (E/T) rather than E as the fitness function, where T (analogous to temperature in simulated annealing) is a properly chosen parameter which can change slowly with each generation. Also, by repeating the GA optimization procedure several times with different randomly chosen initial model populations, we derive “a very good subset” of models from the entire model space and calculate the a posteriori probability density σ(m) ∝ exp (E(m)/T). The σ(m) ’s are then used to calculate a “mean” model, which is found to be close to the true model.
Comparison of the least‐squares criterion and the Cauchy criterion in frequency‐wavenumber inversion
Authors: Amundsen Lasse
Published: October 2002

One alternative to the least‐squares inversion technique is the use of a Cauchy error criterion. We show how inversion algorithms of the Gauss‐Newton type based on the least‐squares method can be modified to handle the Cauchy norm. A criterion for the lower bound of the scale parameter in the Cauchy norm is given. We compare the least‐squares and Cauchy error criteria by inverting synthetic data corrupted by random noise and weather noise. The data are transformed to the frequency‐wavenumber domain before the inversion starts. The numerical examples show that the algorithm based on the Cauchy criterion is more robust in the presence of the noise tested here. Per iteration, the computer costs of the two algorithms are approximately the same.
Solving elastic wave equations with the Hartley method
Authors: Saatcilar R. , Ergintav S.
Published: October 2002

Elastic modeling is one of the basic approaches to determining the seismic response of subsurface structures. There are different numerical techniques for computing the solution of the elastic wave equation, such as the finite‐difference, the finite‐element, and the Fourier methods.
Stoneley‐wave propagation in a fluid‐filled borehole with a vertical fracture
Authors: Tang X. M. , Cheng C. H. , Toksöz M. N.
Published: October 2002

The propagation of Stoneley waves in a fluid‐filled borehole with a vertical fracture is investigated both theoretically and experimentally. The borehole propagation excites fluid motion in the fracture and the resulting fluid flow at the fracture opening perturbs the fluid‐solid interface boundary condition at the borehole wall. By developing a boundary condition perturbation technique for the borehole situation, we studied the effect of this change in the boundary condition on the Stoneley propagation. Cases of both hard and soft formations have been investigated. The fracture has minimal effects on the Stoneley velocity, except in the very low frequency range in which the Stoneley velocity drastically decreases with decreasing frequency. Significant Stoneley‐wave attenuation is produced because of the energy dissipation into the fracture. The quantitative behavior of these effects depends not only on fracture aperture and borehole radius, but also on the acoustic properties of the formation and fluid. Ultrasonic experiments were performed to measure Stoneley propagation in laboratory fracture borehole models. Aluminum and lucite were used to simulate a hard and a soft formation, respectively. Array data for wave propagation were obtained and were processed using Prony’s method to give velocity and attenuation of Stoneley waves as a function of frequency. In both hard and soft formation cases, the experimental results agreed with the theoretical predictions. The important result of this study is that it presents a quantitative relationship between the Stoneley propagation and the fracture character in conjunction with formation and fluid properties. This relationship provides a method for estimating the characteristics of a vertical fracture by means of Stoneley wave measurements.
Correction of topographic distortion in gravity data
Authors: Xia J. , Sprowl D. R.
Published: October 2002

The numerical equivalent source is used to correct distortions in gravity anomalies caused by topographic relief. The error involved in the topographic correction is a function of the depth of the equivalent source. The optimum depth of the equivalent source is that which maximizes the smoothness of the calculated anomaly between control points. The technique is straightforward and can be incorporated easily into a standard data processing sequence.
Tomographic traveltime inversion using natural pixels
Authors: Michelena R. J. , Harris J. M.
Published: October 2002

Traditionally in the problem of tomographic traveltime inversion, the model is divided into a number of rectangular cells of constant slowness. Inversion consists of finding these constant values using the measured traveltimes. The inversion process can demand a large computational effort if a high‐resolution result is desired. We show how to use a different kind of parameterization of the model based on beam propagation paths. This parameterization is obtained within the framework of reconstruction in Hilbert spaces by minimizing the error between the true model and the estimated model. The traveltimes are interpreted as the projections of the slowness along the beampaths. Although the actual beampaths are described by complicated spatial functions, we simplify the computations by approximating these functions with functions of constant width and height, i.e., “fat” rays, which collectively form a basis set of natural pixels. With a simple numerical example we demonstrate that the main advantage of this parameterization, compared with the traditional decomposition of the model in rectangular pixels, is that 2-D reconstructed images of similar quality can be obtained with considerably less computational effort. This result suggests that the natural pixels can provide considerable computational advantage for 3-D problems.
Wave‐equation tomography
Authors: Woodward Marta Jo
Published: October 2002

The relation between ray‐trace and diffraction tomography is usually obscured by formulation of the two methods in different domains: the former in space, the latter in wavenumber. Here diffraction tomography is reformulated in the space domain, under the title of wave‐equation tomography. With this transformation, wave‐equation tomography projects monochromatic, scattered wavefields back over source‐receiver wavepaths, just as ray‐trace tomography projects traveltime delays back over source‐receiver raypaths. Derived under the Born approximation, these wavepaths are wave‐theoretic back‐projection patterns for reflected energy; derived under the Rytov approximation, they are wave‐theoretic back‐projection patterns for transmitted energy. Differences between ray‐trace and wave‐equation tomography are examined through comparison of wavepaths and raypaths, followed by their application to a transmission‐geometry, synthetic data set. Rytov wave‐equation tomography proves superior to ray‐trace tomography in dealing with geometrical frequency dispersion and finite‐aperture data, but inferior in robustness. Where ray‐trace tomography assumes linear phase delay and inverts the arrival time of one well‐understood event, wave‐equation tomography accommodates scattering and inverts all of the signal and noise on an infinite trace simultaneously. Interpreted through the uncertainty relation, these differences lead to a redefinition of Rytov wavepaths as monochromatic raypaths, and of raypaths as infinite‐bandwidth wavepaths (Rytov wavepaths averaged over an infinite bandwidth). The infinite‐bandwidth and infinite‐time assumptions of ray‐trace and Rytov, wave‐equation tomography are reconciled through the introduction of bandlimited raypaths (Rytov wavepaths averaged over a finite bandwidth). A compromise between rays and waves, bandlimited raypaths are broad back‐projection patterns that account for the uncertainty inherent in picking traveltimes from bandlimited data.
Epigenetic versus syngenetic magnetite as a cause of magnetic anomalies
Authors: Gay S. Parker
Published: October 2002

Studies carried out a decade ago at Cement Field, Oklahoma, suggesting evidence of epigenetic (diagenetic) magnetite are shown to have been flawed for two reasons. Aeromagnetic surveying over the field could not have yielded the type of broad, low‐amplitude anomalies expected from hypothesized diagenetic magnetite sources because of the occurrence of numerous high‐amplitude magnetic spikes over cultural sources. Applied Geophysics, Inc., flew a low level (approximately 30 m above ground) aeromagnetic profile along the long axis of the field in 1983 and encountered nearly 40 such spikes due to culture. Additionally, studies of drill cuttings from 23 wells in the field showed them all to be contaminated by rust, scale, and bit shavings, in approximately the same percentages as had been postulated for diagenetic magnetite in 5 wells examined in the prior study. Subsequent studies in support of the diagenetic magnetite hypothesis in other areas have relied on the occurrence of “high‐wavenumber” (i.e., high‐frequency) aeromagnetic or ground magnetic anomalies as indicators of such magnetite. These studies do not take into account the possibility of detrital magnetite as the cause of the high‐frequency anomalies. Detrital magnetite is of near‐universal occurrence in small quantities in both unconsolidated sediments and sedimentary rocks and is easily concentrated by fluvial processes which would affect ground magnetic surveys in particular, but airborne surveys as well. In rocks of Mesozoic age or younger, detrital magnetite occurs in sufficient quantity in some areas to give quite detectable airborne magnetic anomalies. Four such localities have recently been documented, one in the Utah‐Idaho‐Wyoming USA thrust belt; and two in the western United States and one in Central America. However, recent measurements of soil magnetic susceptibility over oil fields do present intriguing evidence for anomalous near‐surface magnetization associated with hydrocarbon leakage plumes. Based on previous experience with sulfide mineral deposits, I postulate that small quantities of pyrite, and perhaps lesser quantities of other sulfide minerals that are sometimes precipitated by rising hydrocarbons, are oxidized above the water table in the vadose zone to secondary magnetic iron oxides. This thin layer of weak anomalous magnetization, when properly distinguished from magnetization due to detrital magnetite, would be recognizable with shallow magnetic susceptibility measurements, and possibly ground magnetic surveys, but probably not with airborne surveys, except under exceptional circumstances.
2-D and 3-D resistivity image reconstruction using crosshole data
Authors: Shima Hiromasa
Published: October 2002

Theoretical changes in the distribution of electrical potential near subsurface resistivity anomalies have been studied using two resistivity models. The results suggest that the greatest response from such anomalies can be observed with buried electrodes, and that the resistivity model of a volume between boreholes can be accurately reconstructed by using crosshole data. The distributive properties of crosshole electrical potential data obtained by the pole‐pole array method have also been examined using the calculated partial derivative of the observed apparent resistivity with respect to a small cell within a given volume. The results show that for optimum two‐dimensional (2-D) and three‐dimensional (3-D) target imaging, in‐line data and crossline data should be combined, and an area outside the zone of exploration should be included in the analysis. In this paper, the 2-D and 3-D resistivity images presented are reconstructed from crosshole data by the combination of two inversion algorithms. The first algorithm uses the alpha center method for forward modeling and reconstructs a resistivity model by a nonlinear least‐squares inversion. Alpha centers express a continuously varying resistivity model, and the distribution of the electrical potential from the model can be calculated quickly. An initial general model is determined by the resistivity backprojection technique (RBPT) prior to the first inversion step. The second process uses finite elements and a linear inversion algorithm to improve the resolution of the resistivity model created by the first step. Simple 2-D and 3-D numerical models are discussed to illustrate the inversion method used in processing. Data from several field studies are also presented to demonstrate the capabilities of using crosshole resistivity exploration techniques. The numerical experiments show that by using the combined reconstruction algorithm, thin conductive layers can be imaged with good resolution for 2-D and 3-D cases. The integration of finite‐element computations is shown to improve the image obtained by the alpha center inversion process for 3-D applications. The first field test uses horizontal galleries to evaluate complex 2-D features of a zinc mine. The second field test illustrates the use of three boreholes at a dam site to investigate base rock features and define the distribution of an altered zone in three dimensions.
Plane‐wave reflection and transmission coefficients for a transversely isotropic solid
Authors: Graebner Mark
Published: October 2002

Numerous investigators have studied the P-SV reflection and transmission coefficients of an isotropic solid (Zoeppritz, 1919; Nafe, 1957; Frasier, 1970; Young and Braile, 1976; Kind, 1976; Aki and Richards, 1980).
Ray tracing using reciprocity
Authors: Matsuoka Toshifumi , Ezaka Teruya
Published: October 2002

Conventional ray‐tracing techniques like the shooting method have many difficulties when applied to tomography analysis. For example, (1) head waves are generally not included, (2) a single raypath is generally assumed for each pair of source and receiver, (3) large computation time is required in a cell structure model for many source‐receiver pairs, and (4) it is difficult to find a raypath in a complicated velocity structure. A new ray‐tracing technique can overcome these difficulties. This technique is based on the two well‐known principles: the reciprocity principle and Fermat’s principle. The algorithm is divided into two steps: (1) calculating the traveltime of first break at each grid point, and (2) estimating raypaths using calculated traveltime data. In this second step, we use total traveltime which is the sum of traveltime from a source point and that from a receiver point. A minimum of the total traveltime represents first break raypaths. Therefore, the method can treat not only a single first‐break raypath, but also multipaths very easily. Also the method can be applied to ray tracing for reflected waves since stationary points of the total traveltime along the reflection boundary become reflection points. Examples using simple models show that this method is an appropriate ray‐tracing approach for a cell model.
Velocity estimation by beam stack
Authors: Biondi Biondo
Published: October 2002

Imaging seismic data requires detailed knowledge of the propagation velocity of compressional waves in the subsurface. In conventional seismic processing, the interval velocity model is usually derived from stacking velocities. Stacking velocities are determined by measuring the coherency of the reflections along hyperbolic moveout trajectories in offset. This conventional method becomes inaccurate in geologically complex areas because the conversion of stacking velocities to interval velocities assumes a horizontally stratified medium and mild lateral variations in velocity. The tomographic velocity estimation proposed in this paper can be applied when there are dipping reflectors and strong lateral variations. The method is based on the measurements of moveouts by beam stacks. A beam stack measures local coherency of reflections along hyperbolic trajectories. Because it is a local operator, the beam stack can provide information on nonhyperbolic moveouts in the data. This information is more reliable than traveltimes of reflections picked directly from the data because many seismic traces are used for computing beam stacks. To estimate interval velocity, I iteratively search for the velocity model that best predicts the events in beam‐stacked data. My estimation method does not require a preliminary picking of the data because it directly maximizes the beam‐stack’s energy at the traveltimes and surface locations predicted by ray tracing. The advantage of this formulation is that detection of the events in the beam‐stacked data can be guided by the imposition of smoothness constraints on the velocity model. The optimization problem of maximizing beam‐stack energy is solved by a gradient algorithm. To compute the derivatives of the objective function with respect to the velocity model, I derive a linear operator that relates perturbations in velocity to the observed changes in the beam‐stack kinematics. The method has been successfully applied to a marine survey for estimating a low‐velocity anomaly. The estimated velocity function correctly predicts the nonhyperbolic moveouts in the data caused by the velocity anomaly.
Similarity and the inverse Q filter: The Pareto‐Levy stretch
Authors: Bickel Samuel H.
Published: October 2002

The seismic impulse response of a constant Q attenuating material has the same mathematical form as the density function for the Pareto‐Levy probability law. Since this law is stable it follows that the seismic pulse is closed under convolution. This means that if the distance from the source x is the sum of two distances [Formula: see text] then the pulse at x is the convolution of the pulse at [Formula: see text] with the one at [Formula: see text]. Its amplitude is proportional to [Formula: see text] where the parameter [Formula: see text] depends only on Q with α = 1 corresponding to a loss‐free material (Q = ∞). As the pulse loses amplitude it stretches along the time axis so that its area is preserved. This scaling between time and distance is illustrated by the stationary phase approximation for the constant Q pulse. The approximation becomes increasingly more accurate for small values of Q until [Formula: see text] where it is exact; it becomes the solution to the heat equation. Log‐stretch inverse Q filtering of seismic data is performed by logarithmically stretching the time axis of the data about a reference time so that all broad‐band pulses have the same shape. A time‐stationary inverse filter is then applied in the stretch domain. However the stretching operation is nonlinear and so band‐limited signals will not have the same shape in the stretch domain—the bandwidths of the stretched wavelets will depend upon wavelet position. To avoid energy loss the bandwidth of the inverse filter must be broader than that of the reference wavelet. For events arriving ahead of the reference wavelet the low‐frequency limit must be lowered, while the high‐frequency limit must be increased for later arrivals. Furthermore, Pareto‐Levy scaling suggests [Formula: see text] stretching corrects the residual traveltime errors that accompany the log‐stretch method of inverse Q filtering.
Four‐component surface‐consistent deconvolution
Authors: Cary Peter W. , Lorentz Gary A.
Published: October 2002

When performing four‐component surface‐consistent deconvolution, it is assumed that the decomposition of amplitude spectra into source, receiver, offset, and common‐depth‐point components enables accurate deconvolution filters to be derived. However, relatively little effort has been put into the verification of this assumption. Some verification of the assumption is available by analyzing the results of the surface‐consistent decomposition of real seismic data. The surface‐consistent log‐amplitude spectra of land seismic data are able to provide convincing evidence that the source component collects effects of the source signature and near‐source structural effects, and that the receiver component collects receiver characteristics and near‐receiver structural effects. In addition, the offset component collects effects due to ground roll and average reflectivity, and the CDP component collects mostly random noise unless it is constrained to be smooth. Based on the results of this analysis, deconvolution filters should be constructed from the source and receiver components, while the offset and CDP components are discarded. The four‐component surface‐consistent decomposition can be performed efficiently by making use of a simple rearrangement of the Gauss‐Seidel matrix inversion equations. The algorithm requires just two passes through the prestack data volume, regardless of the sorted order of the data, so it is useful for both two‐dimensional and three‐dimensional (2-D and 3-D) data volumes.
Dynamic poroelasticity: A unified model with the squirt and the Biot mechanisms
Authors: Dvorkin Jack , Nur Amos
Published: October 2002

The velocities and attenuation of seismic and acoustic waves in rocks with fluids are affected by the two most important modes of fluid/solid interaction: (1) the Biot mechanism where the fluid is forced to participate in the solid’s motion by viscous friction and inertial coupling, and (2) the squirt‐flow mechanism where the fluid is squeezed out of thin pores deformed by a passing wave. Traditionally, both modes have been modeled separately, with the Biot mechanism treated in a macroscopic framework, and the squirt flow examined at the individual pore level. We offer a model which treats both mechanisms as coupled processes and relates P‐velocity and attenuation to macroscopic parameters: the Biot poroelastic constants, porosity, permeability, fluid compressibility and viscosity, and a newly introduced microscale parameter—a characteristic squirt‐flow length. The latter is referred to as a fundamental rock property that can be determined experimentally. We show that the squirt‐flow mechanism dominates the Biot mechanism and is responsible for measured large velocity dispersion and attenuation values. The model directly relates P‐velocity and attenuation to measurable rock and fluid properties. Therefore, it allows one to realistically interpret velocity dispersion and/or attenuation in terms of fluid properties changes [e.g., viscosity during thermal enhanced oil recovery (EOR)], or to link seismic measurements to reservoir properties. As an example of the latter transformation, we relate permeability to attenuation and achieve good qualitative correlation with experimental data.
3-D true‐amplitude finite‐offset migration
Authors: Schleicher Jorg , Tygel Martin , Hubral Peter
Published: October 2002

Compressional primary nonzero offset reflections can be imaged into three‐dimensional (3-D) time or depth‐migrated reflections so that the migrated wavefield amplitudes are a measure of angle‐dependent reflection coefficients. Various migration/inversion algorithms involving weighted diffraction stacks recently proposed are based on Born or Kirchhoff approximations. Here a 3-D Kirchhoff‐type prestack migration approach is proposed where the primary reflections of the wavefields to be imaged are a priori described by the zero‐order ray approximation. As a result, the principal issue in the attempt to recover angle‐dependent reflection coefficients becomes the removal of the geometrical spreading factor of the primary reflections. The weight function that achieves this aim is independent of the unknown reflector and correctly accounts for the recovery of the source pulse in the migrated image irrespective of the source‐receiver configurations employed and the caustics occurring in the wavefield. Our weight function, which is computed using paraxial ray theory, is compared with the one of the inversion integral based on the Beylkin determinant. It differs by a factor that can be easily explained.
Boundary element modeling of nondissipative and dissipative waves
Authors: Chen Genmeng , Zhou Hua‐wei
Published: October 2002

A boundary element (BE) algorithm is developed to compute acoustic or SH‐waves in models consisting of limited or unlimited volumes and irregular interfaces. We solve the BE system in the frequency domain so that anelasticity can be easily represented by different viscoelastic models, such as the Kelvin‐Voigt type. Three illustrative computations are shown. The waveform given by the BE method for a circular inclusion model agrees well with that given by the finite‐difference (FD) method. Dissipation of waves at high frequency caused by the presence of multi‐cracks in an elastic medium resembles the masking effect of anelasticity. The waveforms for nondissipative and dissipative models containing hexagonal inclusions illustrate some interesting characteristics of the composite media.
Seismic detection of a hydraulic fracture from shear‐wave VSP data at Lost Hills Field, California
Authors: Meadows Mark A. , Winterstein Don F.
Published: October 2002

A shear‐wave (S‐wave) VSP experiment was performed at Lost Hills Field, California, in an attempt to detect hydraulic fractures induced in a nearby well. The hydrofrac well was located between an impulsive, S‐wave source on the surface and a receiver well containing a clamped, three‐component geophone. Both direct and scattered waves were detected immediately after shut‐in, when the hydraulic pumps were shut off and recording started. The scattered energy disappeared within about an hour, which is consistent with other measurements that indicate some degree of fracture closure and leak‐off within that period. Although S‐wave splitting was evident, no change was detected in the fast wave (polarized parallel to the fracture). However, the slow wave (polarized perpendicular to the fracture) did change over a period of about an hour, after which the prehydrofrac wavelet shape was recovered. The fact that only the wave polarized perpendicular to the fracture was affected is a dramatic confirmation of both theoretical predictions and laboratory observations of S‐wave behavior in a fractured medium. Subtracting the prehydrofrac wavelet from the wavelets recorded within the first hour after shut‐in revealed scattered wavelets that were diminished and phase‐rotated versions of the incident (prehydrofrac) wavelet. Arrival times of the direct and scattered waves were matched by ray tracing. We accounted for the scattered‐wave amplitudes by using numerical solutions of S‐wave diffractions off of ribbon‐shaped fractures. Amplitudes derived from full‐wavefield Born scattering, however, did not match recorded amplitudes. The phase of the scattered wavelets was matched very well by Born scattering when the incident wavelet was input, but only for fracture lengths no larger than half those predicted from fracture‐simulator models. These results show that a carefully controlled experiment, combined with accurate modeling, can provide important information about the geometry of induced fractures.
Pulse distortion in depth migration
Authors: Tygel Martin , Schleicher Jörg , Hubral Peter
Published: October 2002

When migrating seismic primary reflections obtained from arbitrary source‐receiver configurations (e.g., common shot or constant offset) into depth, a pulse distortion occurs along the reflector. This distortion exists even if the migration was performed using the correct velocity model. Regardless of the migration algorithm, this distortion is a consequence of varying reflection angle, reflector dip, and/or velocity variation. The relationship between the original time pulse and the depth pulse after migration can be explained and quantified in terms of a prestack, Kirchhoff‐type, diffraction‐stack migration theory.
Anisotropic effective‐medium modeling of the elastic properties of shales
Authors: Hornby Brian E. , Schwartz Larry M. , Hudson John A.
Published: October 2002

Shales are complex porous materials, normally consisting of percolating and interpenetrating fluid and solid phases. The solid phase is generally comprised of several mineral components and forms an intricate and anisotropic microstructure. The shape, orientation, and connection of the two phases control the anisotropic elastic properties of the composite solid. We develop a theoretical framework that allows us to predict the effective elastic properties of shales. Its usefulness is demonstrated with numerical modeling and by comparison with established ultrasonic laboratory experiments. The theory is based on a combination of anisotropic formulations of the self‐consistent (SCA) and differential effective‐medium (DEM) approximations. This combination guarantees that both the fluid and solid phases percolate at all porosities. Our modeling of the elastic properties of shales proceeds in four steps. First, we consider the case of an aligned biconnected clay‐fluid composite composed of ellipsoidal inclusions. Anisotropic elastic constants are estimated for a clay‐fluid composite as a function of the fluid‐filled porosity and the aspect ratio of the inclusions. Second, a new processing technique is developed to estimate the distribution of clay platelet orientations from digitized scanning electron microphotographs (SEM). Third, the derived clay platelet distribution is employed to estimate the effective elastic parameters of a solid comprising clay‐fluid composites oriented at different angles. Finally, silt minerals are included in the calculations as isolated spherical inclusions.
Multipole logging in a formation with stress‐relief‐induced anisotropy
Authors: Renlie Lasse
Published: October 2002

The stress relief associated with the drilling of a borehole may induce a mechanically damaged zone with radial transverse isotropy (RTI), where the properties in the radial direction differ from those in the axial and tangential directions. The effect of such a zone on multipole acoustic full‐waveform logging is investigated using a numerical model based on the frequency‐axial‐wavenumber method. Calculations of the spectral behavior show that the fundamental mode associated with the multipole source behaves the same way in an RTI zone as it does in a damaged zone with isotropic properties. In a slow virgin formation, calculations of full waveforms show that the presence of a damaged zone with RTI is more difficult to detect than a damaged zone with isotropic elasticity because the refracted P‐wave encounters an isotropic zone but not an RTI zone. The results indicate that a damaged zone with RTI, which is a precursor to destructive events such as borehole instability and sand production, can be detected only by analyzing the spectral behavior of the fundamental mode.