Multioffset Vertical Seismic Profile Research Articles

Bandwidth-insensitive extended centroid frequency-shift method for near-surfaceQestimation

Seismic waves often are strongly attenuated in the near surface. The measurement and compensation for attenuation are crucial for high-resolution seismic imaging. The quality factor ([Formula: see text]), as a measure of attenuation, is usually estimated by frequency-based methods. We have extended the centroid frequency-shift (CFS) method for [Formula: see text] estimation to high loss media without the usual assumption of [Formula: see text]. The adaptability of this approach for an unknown source wavelet also is demonstrated. Skewness and kurtosis have been used for analyzing the spectral shape change during attenuated wave propagation. Synthetic data tests show a strong relationship between skewness and [Formula: see text] factor estimation accuracy for the conventional CFS method. Surprisingly, our extended CFS approach shows frequency band insensitivity and accuracy for strongly attenuating media. The relative insensitivity of the method to the selected or available frequency bandwidth is shown to be the theoretical basis for its good noise immunity. Finally, a layered medium [Formula: see text] inversion method is derived, which is appropriate to vertical seismic profile (VSP) surveying and is applied to a multioffset field VSP data set to obtain a reliable and stable internal [Formula: see text] value depth distribution.

Potential of full waveform inversion of vertical hard rock environment seismic profile data in

Complex structure of the subsurface in hard rock environment often complicates traditional processing and interpretation of seismic datasets based on the analysis of reflected waves. Full-waveform inversion (FWI), in turn, utilizes the whole wavefield, including the transmitted waves and reflections, to build the models of physical properties (such as P wave velocity and density), which is one of its main advantages over traditional methods of seismic imaging. We conduct a feasibility study of 2D full-waveform inversion (FWI) applied to vertical seismic profile (VSP) synthetic data computed in a model that is based on a real hard-rock survey site for the purpose of identification of heterogeneities. Using this complex model of the subsurface that contains steep interfaces, high seismic wave velocities and densities, we generate a synthetic VSP dataset with a finite-difference code. Multi-offset VSP geometry is considered due to the abundance of transmitted waves. We then apply FWI to this dataset. We use finite-difference modelling for the forward problem in FWI, optimization is conducted using the limited-memory BFGS method. FWI is applied in a multiscale manner, from 10 Hz to 190 Hz. Inversion results suggest that FWI of VSP data is a suitable tool for building the models of physical properties of the subsurface that are crucial for mineral exploration and mine planning.

Imaging the high-temperature geothermal field at Krafla using vertical seismic profiling

Geophysical exploration and in particular active-source seismic imaging of geothermal fields is important to assess and optimize the exploitation of natural heat sources for energy production and direct use. The first multi-offset (moving-source) vertical seismic profiling (VSP) experiment over the high-temperature geothermal field in Krafla (Iceland) was carried out in spring 2014 with the aim to test whether VSP is a suitable method to map volcanic stratigraphy, fractures, dykes, steam zones and magmatic bodies at this site and for volcanic environments in general. In this study, we present a workflow for processing the sparse Krafla VSP dataset recorded with receivers in either of two boreholes. The analysis involved first-arrival traveltime inversion and seismic reflection processing. The seismic velocity model obtained by traveltime tomography reveals structural information between the two boreholes and can be linked to an existing geological model, showing that the seismic velocities are mainly controlled by lithology. The zero-offset seismic reflection data were processed into two corridor stacks. Walk-away VSP reflection data were migrated with a novel multicomponent Kirchhoff migration algorithm that includes P- and S-wave isolation to obtain separate PP, PS and SS migrated images. The reflections imaged in the corridor stacks can be linked to the main lithological units known from borehole logging information. Migrated images from the walk-away data reveal reflectors below and to the sides of the two boreholes. Considering à priori information, such as hypocenter locations from earthquake seismology studies, the reflectors can be related to changes in lithology, fault zones, dykes and possibly the top of the Krafla magma chamber. We found that VSP is potentially a useful method to image the key lithological boundaries and volcanic stratigraphy in the complex magmatic environment at Krafla, but à priori information proved to be essential to constrain the processing and interpretation of the sparse array dataset.

Elastic full-waveform inversion of vertical seismic profile data acquired with distributed acoustic sensors

Distributed acoustic sensing (DAS) is a rapidly developing technology particularly useful for the acquisition of vertical seismic profile (VSP) surveys. DAS data are increasingly used for seismic imaging, but not for estimating rock properties. We have developed a workflow for estimating elastic properties of the subsurface using full-waveform inversion (FWI) of DAS VSP data. Whereas conventional borehole geophones usually measure three components of particle velocity, DAS measures a single quantity, which is an approximation of the strain or strain rate along the fiber. Standard FWI algorithms are developed for particle velocity data, and hence their application to DAS data requires conversion of these data to particle velocity along the fiber. This conversion can be accomplished by a specially designed filter. Field measurements show that the conversion result is close to vertical particle velocity as measured by geophones. Elastic time-domain FWI of a synthetic multioffset VSP data set for a vertical well shows that the inversion of the vertical component alone is sufficient to recover elastic properties of the subsurface. Application of the proposed workflow to a multioffset DAS data set acquired at the CO2CRC Otway Project site in Victoria, Australia, reveals salient subhorizontal layering consistent with the known geology of the site. The inverted [Formula: see text] model at the well location matches the upscaled [Formula: see text] log with a correlation coefficient of 0.85.

Near-surface seismic traveltime tomography using a direct-push source and surface-planted geophones

Information about seismic velocity distribution in heterogeneous near-surface sedimentary deposits is essential for a variety of environmental and engineering geophysical applications. We have evaluated the suitability of the minimally invasive direct-push technology for near-surface seismic traveltime tomography. Geophones placed at the surface and a seismic source installed temporarily in the subsurface by direct-push technology quickly acquire reversed multioffset vertical seismic profiles (VSPs). The first-arrival traveltimes of these data were used to reconstruct the 2D seismic velocity distribution tomographically. After testing this approach on synthetic data, we applied it to field data collected over alluvial deposits in a former river floodplain. The resulting velocity model contains information about high- and low-velocity anomalies and offers a significantly deeper penetration depth than conventional refraction tomography using surface-planted sources and receivers at the investigated site. A combination of refraction seismic and direct-push data increases resolution capabilities in the unsaturated zone and enables reliable reconstruction of velocity variations in near-surface unconsolidated sediments. The final velocity model structurally matches the results of cone-penetration tests and natural gamma-radiation data acquired along the profile. The suitability of multiple rapidly acquired reverse VSP surveys for 2D tomographic velocity imaging of near-surface unconsolidated sediments was explored.

Seismic cone penetration test and seismic tomography in permafrost

Two high-resolution multi-offset vertical seismic profile (VSP) surveys were carried out in a permafrost mound near Umiujaq in northern Quebec, Canada, while performing seismic cone penetration tests (SCPT) to study the cryostratigraphy and assess the body waves velocities and the dynamic properties of warm permafrost. Penetrometer-mounted triaxial accelerometers were used as the VSP receivers, and a swept impact seismic technique (SIST) source generating both compressional and shear waves was moved near the surface following a cross configuration of 40 seismic shot-point locations surrounding each of the two SCPTs. The inversion of travel times based on a simultaneous iterative reconstruction technique (SIRT) provided tomographic images of the distribution of seismic velocities in permafrost. The Young's and shear moduli at low strains were then calculated from the seismic velocities and the permafrost density measured on core samples. The combination of multi-offset VSP survey, SCPT, SIST, and SIRT for tomographic imaging led to new insights in the dynamic properties of permafrost at temperatures close to 0 °C. The P- and S-wave velocities in permafrost vary from 2400 to 3200 m/s and from 900 to 1750 m/s, respectively, for a temperature range between –0.2 and –2.0 °C. The Young's modulus varies from 2.15 to 13.65 GPa, and the shear modulus varies from 1.00 to 4.75 GPa over the same range of temperature.Key words: permafrost, seismic cone penetration test, vertical seismic profiling, seismic tomography, dynamic properties.

Integrated prestack depth migration of vertical seismic profile and surface seismic data from the Rocky Mountain Foothills of southern Alberta, Canada

We have developed an anisotropic prestack depth migration code that can migrate either vertical seismic profile (VSP) or surface seismic data. We use this migration code in a new method for integrated VSP and surface seismic depth imaging. Instead of splicing the VSP image into the section derived from surface seismic data, we use the same migration algorithm and a single velocity model to migrate both data sets to a common output grid. We then scale and sum the two images to yield one integrated depth‐migrated section. After testing this method on synthetic surface seismic and VSP data, we applied it to field data from a 2D surface seismic line and a multioffset VSP from the Rocky Mountain Foothills of southern Alberta, Canada. Our results show that the resulting integrated image exhibits significant improvement over that obtained from (a) the migration of either data set alone or (b) the conventional splicing approach. The integrated image uses the broader frequency bandwidth of the VSP data to provide higher vertical resolution than the migration of the surface seismic data. The integrated image also shows enhanced structural detail, since no part of the surface seismic section is eliminated, and good event continuity through the use of a single migration–velocity model, obtained by an integrated interpretation of borehole and surface seismic data. This enhanced migrated image enabled us to perform a more robust interpretation with good well ties.

Vertical seismic profile at Pike's Peak, Saskatchewan, Canada: turning rays and velocity anisotropy

First-arrival traveltimes from a multi-offset vertical seismic profile (VSP) were used to estimate velocity anisotropy in the presence of a vertical velocity gradient. A numerical model consisting of two layers with vertical velocity gradients of 3.1 and 1.2 s−1, respectively, and global anisotropy parameters of ε=0.12±0.02 and δ=0.30±0.06 yielded first-arrival traveltimes that matched the observed traveltimes well. Shallow receivers were found to be crucial for constraining the vertical velocity field and for determining the parameters of anisotropy at depth.

A multioffset vertical seismic profiling experiment for anisotropy analysis and depth imaging

A multioffset vertical seismic profile (VSP) was carried out in the Rocky Mountain foothills of southern Alberta, Canada. The purpose of this experiment was to investigate whether the dipping shale strata exhibit P‐wave velocity anisotropy and, if so, to calculate the Thomsen anisotropy parameters for use in anisotropic depth migration. Traveltime inversion of first‐arrival data from the multioffset VSP revealed that the dipping Mesozoic clastics in the area exhibit seismic velocity anisotropy of about 10%. The anisotropy parameters derived from this experiment were then used in anisotropic prestack depth migration of data from a surface seismic line close to the VSP well. Comparison of the anisotropic migration with the corresponding isotropic prestack depth migration showed that the target was imaged incorrectly in the isotropic case; a lateral shift of 180 m in the updip direction of the overlying beds was observed. The image obtained with an anisotropic velocity model was also better focused than that obtained assuming isotropic velocities.

Laboratory results for the features of body‐wave propagation in a transversely isotropic media

Much of the earth’s crust appears to have some degree of elastic anisotropy (Crampin, 1981; Crampin and Lovell, 1991; Helbig, 1993). The phenomena of elastic wave propagation in anisotropic media are more complex than those in isotropic media. Shear‐wave propagation in an orthorhombic physical model is most complex when the direction of the wave is close to the neighborhood of the cusp on the group velocity surfaces (Brown et al., 1991). The first identification of singularities in wave propagation through sedimentary basins occurred in the examination of shear‐wave splitting in multioffset vertical seismic profiles (VSPs) at a borehole site in the Paris Basin (Bush and Crampin, 1991), where large variations in shear‐wave polarizations in propagation directions close to point singularities were observed. Computation of synthetic seismograms for layer sequences showed that the shear‐wave polarizations and amplitudes were irregular near point singularities (Crampin, 1991).

Integrated geological and geophysical studies in the SG4 borehole area, Tagil Volcanic Arc, Middle Urals: Location of seismic reflectors and source of the reflectivity

Near‐vertical incidence reflection seismic data acquired in the Tagil Volcanic Arc (Middle Urals) show the upper crust to be highly reflective. Two intersecting seismic lines located near the ongoing ∼5400 m deep SG4 borehole show that the main reflectivity strikes approximately N‐S and dips ∼35°–55° to the east. Prominent reflections intercept the borehole at ∼1000, ∼1500, 2800–2900, ∼3400, and between ∼4000 and 5400 m, which correspond to intervals of low velocity/low density/low resistivity. The surface projections of these reflections lie parallel to the strike of magnetic anomaly trends. Multioffset vertical seismic profile (VSP) data acquired in the SG4 borehole show a seismic response dominated by P to S reflected converted waves from the moderately east dipping reflectivity and from a set of very steep east dipping reflectors not imaged by the surface data. Modeling of the VSP data constrains the depth at which reflectors intercept the borehole and suggests that the P to S conversions are best explained by low‐velocity porous intervals rather than higher‐velocity mafic material. The most prominent east dipping reflection on the surface seismic data is only imaged on VSP shots that sample the crust closer to the E‐W seismic line. This discrepancy between the VSP and the surface seismic data is attributed to rapid lateral changes in the physical properties of the reflector. Surface and borehole data suggest that the low‐velocity/low‐density/low‐resistivity intervals are the most important source of reflectivity in the SG4 borehole area, although lithological contrasts may also play a role. Drill cores from the these zones contain hydrothermal alteration minerals indicating interaction with fluids. Tectonic criteria suggest that they might represent imbricated fracture zones often bounding different lithologies and/or intrusions. Some of them might also represent high‐porosity lava flows or pyroclastic units, common in island arc environments.

A Multi-Offset Vertical Seismic Profiling (VSP) Experiment for Anisotropy Analysis and Imaging

Editors Note Editor's note: The Canadian International Petroleum Conference held June 4 - 8, 2000, featured a Graduate Student Presentation Competition as part of the Petroleum Society's tradition at the Annual Technical Meeting. Graduate students from respected universities across North America competed in these sessions. The objective of the competition was to give participants the opportunity to present highlights of their work to petroleum industry experts from around the world. The presentations were judged on technical content, applicability, and overall presentation quality. Judges included Dr. Nick Mungan, Mungan Petroleum Consultants Ltd.; Dr. Don Towson, IRAP; and Dr. Norman Freitag, Saskatchewan Research Council. A "Best Graduate Student Presentation Award" and two runner-up awards were presented by Don Mallory, chair of the Student Competition, shown here with the 1st Place presentation author, Graziella Grech of the University of Calgary. On behalf of the Petroleum Society, the JCPT is pleased to present Ms. Grech's award winning presentation. Abstract A multi-offset Vertical Seismic Profiling (VSP) experiment was carried out in the Rocky Mountain Foothills of Southern Alberta, Canada. The purpose of this survey was to determine whether the dipping shale strata in the study area exhibit seismic velocity anisotropy, and to obtain an estimate of the error in the imaged location of the target underneath these shales if anisotropy is not taken into account during seismic data processing. Traveltime inversion of the first arrival data from the multioffset VSP has revealed that the shales exhibit velocity anisotropy of about 10%. For a target depth of about 3,000 m and moderate dips of 30 ° to 50 ° in the anisotropic overburden, this may lead to a lateral shift in the imaged location of the target of up to 300 m in the up-dip direction of overlying bedding. The anisotropic parameters derived from this study will also be used in the anisotropic prestack depth migration of the VSP and surface seismic data. FIGURE 1: Interbedded thin layers of sandstone and shale found in the Western Canada Sedimentary Basin. Due to anisotropy, the velocity parallel to bedding (V90) is higher than the velocity perpendicular to bedding (V0). This class of anisotropy is termed Transverse Isotropy. (Photo courtesy D. Spratt) (Available in full paper) Introduction Vertical Seismic Profiling (VSP) and surface seismic data are used to image and locate hydrocarbon targets in the subsurface. It is well established that these depth images depend on the accuracy of the velocity model(1,2). If rocks overlying the target are anisotropic, and if this property is not accounted for during velocity model building and depth imaging, then the final image will be incorrect, hence increasing the risk of dry holes. It is therefore important to determine which formations exhibit seismic velocity anisotropy and quantify their parameters for use during seismic imaging. Seismic Velocity Anisotropy Seismic velocity anisotropy is the change in velocity with the direction in which it is measured. In rocks, it can be caused by the alignment of mineral grains, cracks and crystals in a preferred direction, and by the consecutive layering of thin beds.

Seismic anisotropy in the upper 500 m of the Southern Sydney Basin

An analysis of seismic anisotropy at a BHP mining site in the Southern Sydney Basin by combined use of crosshole and vertical seismic profiling (VSP) data is presented. The upper 250 m in this area is highly heterogeneous and has a major impact on the analysis of P-wave traveltimes. It is shown that using P-wave information solely would not, at least in this case, lead to any reasonable estimate of the elastic constants, in particular C13, even if the measurements contained a full range of incident angles. However, if the measurements of SV-waves are available, even over a small range of incident angles, then C13 is determined more accurately. P-wave velocities measured in the vertical and horizontal directions show that anisotropy is present in the area. Additional measurements, along different incident angles, indicate that the rock down to 500 m depth is predominantly transversely isotropic (TI) with a vertical axis of symmetry. The P-wave anisotropy can be approximated as elliptical. Using the elastic constants estimated from the data analyses, synthetic seismograms for heterogeneous TI media were generated. Comparison of the seismic modeling with real crosshole data shows that it is necessary to include both fault zones and gas accumulations in the model to qualitatively match the real data. By using SV-waves in the multioffset VSP data, reflectors are mapped more accurately than by using P-waves, even under the assumption of isotropy and in the presence of heterogeneity. Mapping of converted P-SV waves by a straight ray approach also produced better results than the corresponding isotropic P-wave mapping. Inclusion of elliptical anisotropy into Kirchhoff migration resulted in better P-wave images than using an isotopic migration code. We conclude that both P-wave VSP multioffset mapping and tomographic inversion methods need to account for anisotropy to be accurate in this area, while SV-waves may be handled using isotropic codes. The same is true for crosshole and surface seismic data.

Analysis of multiazimuthal VSP data for anisotropy and AVO

Multioffset vertical seismic profile (VSP) experiments, commonly referred to as walkaways, enable anisotropy to be measured reliably in the field. The results can be fed into modeling programs to study the impact of anisotropy on velocity analysis, migration, and amplitude versus offset (AVO). Properly designed multioffset VSPs can also provide the target AVO response measured under optimum conditions, since the wavelet is recorded just above the reflectors of interest with minimal reflection point dispersal. In this paper, the multioffset VSP technique is extended to include multioffset azimuths, and a multiazimuthal multiple VSP data set acquired over a carbonate reservoir is analyzed for P-wave anisotropy and AVO. Direct arrival times down to the overlying shale and reflection times and amplitudes from the carbonate are analyzed. Data analysis involves a three‐term fit to account for nonhyperbolic moveout, dip, and azimuthal anisotropy. Results indicate that the overlying shale is transversely isotropic with a vertical axis of symmetry (VTI), while the carbonate shows 4–5% azimuthal anisotropy in traveltimes. The fast direction is consistent with the maximum horizontal stress orientation determined from break‐out logs and is also consistent with the strike of major faults. AVO analysis of the reflection from the top of the carbonate layer shows a critical angle reduction in the fast direction and maximum gradient in the slow direction. This agrees with modeling and indicates a greater amplitude sensitivity in the slow direction—the direction perpendicular to fracture strike. In principle, 3-D surveys should have wide azimuthal coverage to characterize fractured reservoirs. If this is not possible, it is important to have azimuthal line coverage in the minimum horizontal stress direction to optimize the use of AVO for fractured reservoir characterization. This direction can be obtained from multiazimuthal walkaways using the azimuthal P-wave analysis techniques presented.

Vertical seismic profile results from the Kola Superdeep Borehole, Russia

Multi-offset vertical seismic profiles (VSPs) from the Kola Superdeep Borehole (SG-3), as part of a larger seismic study of the Kola region conducted during the spring of 1992, sample the dipping Pechenga complex from 2175 m to 6000 m and contribute to the understanding of reflectivity in crystalline and Precambrian environments. From the surface to 6000 m, the SG-3 borehole penetrates interlayered Proterozoic metavolcanic and metasedimentary units and a mylonitic shear zone ranging from greenschist to amphibolite metamorphic grade, respectively. The Kola VSPs display a 6% velocity decrease which coincides to a mylonitic shear zone located between 4500 m and 5100 m within the SG-3 borehole. Seismic interfaces are identified by mode-converted energy (PS, and SP transmissions and reflections) in addition to primary seismic phases. The VSP shear wave energy is generated at or near the source by vertical vibrators. P-wave and S-wave reflections are generally detected from the same reflecting horizons, but increases in relative S-wave and SP reflection amplitudes originate at 1900 m, 3800 m, 4500 m, and 5100 m depths. These depths coincide with zones of elevated V p V s and may support the presence of free pore fluid which is reported from initial drilling. For the Proterozoic lithologies sampled by the VSP, reflection events result from five mylonitic shear zones and three lithologic contrasts.

Estimation of anisotropy and anisotropic 3-D prestack depth migration, offshore Zaire

Chevron (operator), Teikoku Oil, and Unocal hold the offshore Zaire concession, West Africa. Recently, Unocal performed an anisotropic 3-D prestack depth migration in an attempt to optimally image tilted fault blocks in the presalt section. Depth migration was required because large‐displacement faults juxtaposed a high‐velocity massive carbonate and a low‐velocity marl. Large compressional‐wave anisotropy caused by thin layering was inferred for most of the depth section. Evidence on the nature and magnitude of the anisotropy came from comparing stacking velocities with well velocities, analyzing a multioffset vertical seismic profile (VSP), thin‐layer modeling in the massive carbonate using well log data, and rock physics laboratory velocity measurements on core from the marl. The subsurface was characterized as being locally transversely isotropic and the Thomsen parameters were used to describe the anisotropy. This required that five parameters be specified at each subsurface location to represent the compressional‐wave 3-D velocity field. The observed anisotropy deviated far from the elliptical condition, and modeling indicated that isotropic migration would exhibit misfocusing and mispositioning of events, and its quality would be dip‐dependent. The anisotropic migration was performed using Kirchhoff summation for which the traveltimes were determined using a finite‐difference scheme. Interpreted horizons in the migrated depth section accurately tied well depths.

A stable method for linearized inversion of elastic parameters

An inversion method that stabilizes the multiparameter inverse problem for a constant background elastic isotropic medium is developed. The inverse problem is formulated in the wavenumber domain and the operators acting on the individual elastic parameters are displayed and analysed. From here it is noticed that for certain scattering angles some parameters produce no scattering. This scattering angle dependence is combined with the frequency dependence of the parameters to reduce the multiparameter inverse problem to a single parameter inversion problem, which is known to yield stable results. The method developed here is an extension of an acoustical medical imaging method by Norton (1983) to the elastic seismic imaging problem which inherits limited view constraints. This method is compared to the elastic extension of a multiparameter acoustic inversion method developed by Devaney (1985) and the resulting improvements in stability are demonstrated on synthetic examples. From the theory it is observed that a fixed 90° scattering angle can be used with multiple frequencies to achieve highly stable inversion results of all elastic parameters provided specific elastic parameters are extracted from specific scattering modes. The attempt to extract all elastic parameters from the P-to-P scattering mode using multiple scattering angles is seen to be very ill-conditioned and not recommended. For the 90° scattering angle we find that surface reflection profiling (SRP) and cross-hole geometries have the same wavenumber domain coverage and the multi-offset vertical seismic profiling (MVSP) geometries give a better coverage than only a SRP or cross-hole geometry.

The effect of a changing aspect ratio of aligned cracks on shear wave vertical seismic profiles: A theoretical study

Rocks containing parallel cracks or preferentially oriented pores behave at low frequencies as transversely isotropic solids. A change in the aspect ratio of these inclusions affects the resultant anisotropy. Curves showing the velocity dependence of the two shear waves on the direction of wave propagation in such rocks intersect in a direction of propagation which depends on the aspect ratio. The angle this direction makes with the symmetry axis of the anisotropic rock decreases for increasing aspect ratio. Characteristic of wave propagation around this angle is a change in the polarization of the leading shear wave. To demonstrate how this affects vertical seismic profile (VSP) measurements, multioffset VSPs at different azimuths are modeled with synthetic seismograms for rocks containing inclusions with a range of aspect ratios. For inclusions with small aspect ratios, the polarization of the initial shear wave may change abruptly as the direction of propagation changes, whereas for large aspect ratios there is no change or the change may be more gradual and spread out over a wider range of directions. This feature may be of interest in assessing the physical character of fluid reservoirs. The observations of changes of polarization and variation of delays between the split shear waves may also be used to monitor changes in stresses in the crust and in monitoring changes during hydrocarbon extraction.

Computation of synthetic seismograms for stratified azimuthally anisotropic media

We outline a method of computing synthetic seismograms for stratified, azimuthally anisotropic, viscoelastic earth models. This method is an extended form of the Kennett algorithm that is efficient for multioffset vertical seismic profiling. The model consists of a stack of homogeneous plane layers, and the response is computed iteratively by successive inclusion of deeper layers. In each layer, the 6×6 system matrix A is diagonalized numerically; this permits treatment of triclinic materials, i.e., those with the lowest possible symmetry. Jacobi iteration is an efficient way to diagonalize A because the entries of A change little from one wavenumber to the next. When the material properties are frequency dependent, the wavenumber loops are inside the frequency loop, and the computation is slow even on a supercomputer. When the material parameters are frequency independent, it is better to make frequency the deepest loop, with diagonalization of A outside the loop, in which case vectorization gives a relatively rapid computation. Temporal wraparound is avoided by making use of complex frequencies, and spatial aliasing is avoided by using a generalized Filon's method to evaluate both the wavenumber integrals. Various methods of generating anisotropic elastic constants from microlayers, cracks, and fractures and joints are discussed. Example computations are given for azimuthally isotropic and azimuthally anisotropic (AA) earth models. Comparison of computations using single and double wavenumber integrations for a realistic AA model shows that single wave‐number integration often gives incorrect answers especially at near offsets. Errors due to use of a single wavenumber integration are explained heuristically by use of wave front diagrams for point and line sources.

Application of multioffset vertical seismic profiling in fracture mapping : Carswell, A; Moon, W M Geophysics V54, N6, June 1989, P737–746

Application of multioffset vertical seismic profiling in fracture mapping : Carswell, A; Moon, W M Geophysics V54, N6, June 1989, P737–746