Anisotropic Parameter Estimation Research Articles

Correction for the influence of velocity lenses on nonhyperbolic moveout inversion for VTI media

Nonhyperbolic moveout analysis plays an increasingly important role in velocity model building because it provides valuable information for anisotropic parameter estimation. However, lateral heterogeneity associated with stratigraphic lenses such as channels and reefs can significantly distort the moveout parameters, even when the structure is relatively simple. We analyze the influence of a low-velocity isotropic lens on nonhyperbolic moveout inversion for horizontally layered VTI (transversely isotropic with a vertical symmetry axis) models. Synthetic tests demonstrate that a lens can cause substantial, laterally varying errors in the normal-moveout velocity [Formula: see text] and the anellipticity parameter [Formula: see text]. The area influenced by the lens can be identified using the residual moveout after the nonhyperbolic moveout correction as well as the dependence of errors in [Formula: see text] and [Formula: see text] on spreadlength. To remove such errors in [Formula: see text] and [Formula: see text], we propose a correction algorithm designed for a lens embedded in a horizontally layered overburden. This algorithm involves estimation of the incidence angle of the ray passing through the lens for each recorded trace. With the assumption that lens-related perturbation of the raypath is negligible, the lens-induced traveltime shifts are computed from the corresponding zero-offset time distortion (i.e., from “pull-up” or “push-down” anomalies). Synthetic tests demonstrate that this algorithm substantially reduces the errors in the effective and interval parameters [Formula: see text] and [Formula: see text]. The corrected traces and reconstructed “background” values of [Formula: see text] and [Formula: see text] are suitable for anisotropic time imaging and producing a high-quality stack.

Crosshole seismic tomography including the anisotropy effect

Velocity anisotropy caused by fine layering has a strong effect on the travel paths at large vertical offsets in crosshole seismic data. This effect must be considered in seismic waveform tomography, so that the inversion procedure can match the waveforms in real data. In this paper, we propose to properly take into account this anisotropic effect in two stages. First, we invert for the anisotropy parameter simultaneously with the (horizontal) velocity in travel time tomography. Then we incorporate this estimated anisotropy parameter model in the waveform simulation during waveform tomography, to refine the velocity image. When considering this anisotropic effect, far-offset data are properly used in the inversion, then both travel time and waveform tomography may have a better ray coverage, producing a velocity image with a higher resolution.

Slowness surface construction and inversion from 3D VSP data

The emerging technology of three-dimensional vertical seismic profiles (3D VSPs) allows the extensive sampling of reflecting surfaces over a wide range of incident angles and azimuths. Such data can be used to provide additional insights into the effective rock properties associated with features that are below seismic resolution, such as layering or aligned systems of fractures which lead to elastic anisotropy. Here we analyse two 3D VSPs acquired in the Arabian Gulf and show how the data may be processed to extract slowness surfaces, how these surfaces can be used to estimate anisotropy parameters, and how the anisotropy relates to the geological characteristics of the formation and its stress history.

On the nonuniqueness of traveltime inversion in elliptically anisotropic media

Elliptical velocity dependence on the direction of wave propagation is the simplest type of anisotropy that might be encountered in the subsurface. When the symmetry axis is vertical, describing elliptical anisotropy (EA) requires only one parameter — the ellipticity coefficient — in addition to the conventional isotropic velocity. It is tempting, therefore, to use elliptical anisotropy as a testing ground for various anisotropic parameter-estimation techniques. This apparently straightforward thinking, however, fails to deliver satisfactory results because traveltime inversion in certain EA media is known to be nonunique. Here, I prove this nonuniqueness to be a general property of all EA media and explicitly show that a constant depth stretch generates a family of kinematically equivalent EA models for any given spatial distribution of the ellipticity coefficient and the velocity.

Seismic critical-angle anisotropy analysis in the τ -p domain

Seismic critical-angle reflectometry is a relatively new field for estimating seismic anisotropy parameters. The theory relates changes in the critical angle with azimuth of the seismic line to the principal axis and anisotropy parameters. Current implementation of the critical-angle reflectometry process has certain shortcomings in that the critical angle is determined from critical offset and the process is vulnerable to different approximation errors. Seismic critical-angle analysis in the plane-wave [Formula: see text] domain can handle these issues and has the potential to become an independent tool for estimating anisotropy parameters. The theory of seismic critical-angle reflectometry is modified to make it suitable for [Formula: see text] domain analysis. Then using full-wave synthetic seismograms at three different azimuths for a transversely isotropic medium with a horizontal axis of symmetry (HTI), the effectiveness of anisotropy parameter estimation is demonstrated.

Revealing the Anisotropy in MgB2 Superconductor and Investigation of Critical Current near Critical Temperature by Using Hall Probe Susceptibility Technique

We report the results of an investigation on critical current measurements in bulk MgB2 superconductor by using Hall probe ac susceptibility. The temperature versus ac susceptibility has been measured as a function of external ac magnetic field with no dc part and magnitude in the range 240–1200 A/m by using the Hall probe method. The real part of the Hall probe susceptibility showed two-step transitions near critical temperature. We attributed this behavior to the anisotropic nature of the MgB2 compound. Close to the critical temperature the estimated anisotropy parameter is γ≈2.5, a value that is in fair agreement to the ones reported in the literature for polycrystalline and single crystals MgB2. We have also analyzed the susceptibility data within the framework of a critical state model (using the real part of the susceptibility) and an anisotropic nature of MgB2 compound. The inter-grain critical current is found to be 4×106 A/m2 at 39.59 K by using the critical state model. The temperature dependence of the critical current density was found to be in the form of J c (T)∝(1−T/T c ) β , where β was calculated as 2.30 by using critical state model, respectively. Scanning electron microscopy was used for grain size estimation.

Azimuthal τ-panalysis in anisotropic media

SUMMARY For the purpose of transversely isotropic (TI) normal moveout (NMO) correction, we propose analysis of plane wave transformed azimuthal gathers, interactively using a single azimuth data at a time and a new delay time equation (developed in this paper), which is a function of two parameters at each azimuth. Results from independently estimated multi-azimuth gathers, then, can be combined to estimate stiffness or Thomsen coefficients. Azimuthal τ–p analysis also avoids numerical ray tracing, resulting in a rapid algorithm. We demonstrate the applicability of our method using a set of P-wave synthetic seismograms from a multilayered medium, consisting of isotropic and HTI layers. Azimuth-dependent anisotropy parameters are derived by delay time fitting and NMO correction. The reflections from the bottom interface of an isotropic layer with an anisotropic overburden show apparent anisotropic traveltime behaviour, which is easily accounted for by our layer-stripping based azimuthal NMO analysis. Unlike the previous approximate HTI NMO correction equation, this equation performs better NMO correction for the HTI medium and is also applicable to the VTI medium. Presence of only two reduced parameters in the equation helps the anisotropic parameter estimation become less ambiguous.

Rational interpolation of qP-traveltimes for semblance-based anisotropy estimation in layered VTI media

The [Formula: see text] domain is the natural domain for anisotropy parameter estimation in horizontally layered media. The need to transform the data to the [Formula: see text] domain or to pick traveltimes in the [Formula: see text] domain is, however, a practical disadvantage. To overcome this, we combine [Formula: see text]-derived traveltimes and offsets in horizontally layered transversely isotropic media with a vertical symmetry axis (VTI) with a rational interpolation procedure applied in the [Formula: see text] domain. This combination results in an accurate and efficient [Formula: see text]-based semblance analysis for anisotropy parameter estimation from the moveout of qP-waves in horizontally layered VTI media. The semblance analysis is applied to the moveout to search directly for the interval values of the relevant parameters. To achieve this, the method is applied in a layer-stripping fashion. We demonstrate the method using synthetic data examples and show that it is robust in the presence of random noise and moderate statics.

Moveout-based geometrical-spreading correction for PS-waves in layered anisotropic media

This paper is devoted to pre-stack amplitude analysis of reflection seismic data from anisotropic (e.g., fractured) media. Geometrical-spreading correction is an important component of amplitude-variation-with-offset (AVO) analysis, which provides high-resolution information for anisotropic parameter estimation and fracture characterization. Here, we extend the algorithm of moveout-based anisotropic spreading correction (MASC) to mode-converted PSV-waves in VTI (transversely isotropic with a vertical symmetry axis) media and symmetry planes of orthorhombic media. While the geometrical-spreading equation in terms of reflection traveltime has the same form for all wave modes in laterally homogeneous media, reflection moveout of PS-waves is more complicated than that of P-waves (e.g., it can become asymmetric in common-midpoint geometry). Still, for models with a horizontal symmetry plane, long-spread reflection traveltimes of PS waves can be well approximated by the Tsvankin–Thomsen and Alkhalifah–Tsvankin moveout equations, which are widely used for P-waves. Although the accuracy of the Alkhalifah–Tsvankin equation is somewhat lower, it includes fewer moveout parameters and helps to maintain the uniformity of the MASC algorithm for P- and PS-waves. The parameters of both moveout equations are obtained by least-squares traveltime fitting or semblance analysis and are different from those for P-waves. Testing on full-waveform synthetic data generated by the reflectivity method for layered VTI media confirms that MASC accurately reconstructs the plane-wave conversion coefficient from conventional-spread PS data. Errors in the estimated conversion coefficient, which become noticeable at moderate and large offsets, are mostly caused by the offset-dependent transmission loss of PS-waves.

Depth migration anisotropy analysis in the time domain

ABSTRACTIn areas of complex geology such as the Canadian Foothills, the effects of anisotropy are apparent in seismic data and estimation of anisotropic parameters for use in seismic imaging is not a trivial task. Here we explore the applicability of common‐focus point (CFP)‐based velocity analysis to estimate anisotropic parameters for the variably tilted shale thrust sheet in the Canadian Foothills model. To avoid the inherent velocity‐depth ambiguity, we assume that the elastic properties of thrust‐sheet with respect to transverse isotropy symmetry axis are homogeneous, the reflector below the thrust‐sheet is flat, and that the anisotropy is weak. In our CFP approach to velocity analysis, for a poorly imaged reflection point, a traveltime residual is obtained as the time difference between the focusing operator for an assumed subsurface velocity model and the corresponding CFP response obtained from the reflection data. We assume that this residual is due to unknown values for anisotropy, and we perform an iterative linear inversion to obtain new model parameters that minimize the residuals. Migration of the data using parameters obtained from our inversion results in a correctly positioned and better focused reflector below the thrust sheet. For traveltime computation we use a brute force mapping scheme that takes into account weakly tilted transverse isotropy media. For inversion, the problem is set up as a generalized Newton's equation where traveltime error (differential time shift) is linearly dependent on the parameter updates. The iterative updates of parameters are obtained by a least‐squares solution of Newton's equations. The significance of this work lies in its applicability to areas where transverse isotropy layers are heterogeneous laterally, and where transverse isotropy layers are overlain by complex structures that preclude a moveout curve fitting.

Seismic critical-angle reflectometry: A method to characterize azimuthal anisotropy?

Existing anisotropic parameter-estimation algorithms that operate with long-offset data are based on the inversion of either nonhyperbolic moveout or wide-angle amplitude-variation-with-offset (AVO) response. We show that valuable information about anisotropic reservoirs can also be obtained from the critical angle of reflected waves. To explain the behavior of the critical angle, we develop weak-anisotropy approximations for vertical transverse isotropy and then use Tsvankin’s notation to extend them to azimuthally anisotropic models of orthorhombic symmetry. The P-wave critical-angle reflection in orthorhombic media is particularly sensitive to the parameters [Formula: see text] and [Formula: see text] responsible for the symmetry-plane horizontal velocity in the high-velocity layer. The azimuthal variation of the critical angle for typical orthorhombic models can reach [Formula: see text], which translates into substantial changes in the critical offset of the reflected P-wave. The main diagnostic features of the critical-angle reflection employed in our method include the rapid amplitude increase at the critical angle and the subsequent separation of the head wave. Analysis of exact synthetic seismograms, generated with the reflectivity method, confirms that the azimuthal variation of the critical offset is detectable on wide-azimuth, long-spread data and can be qualitatively described by our linearized equations. Estimation of the critical offset from the amplitude curve of the reflected wave, however, is not straightforward. Additional complications may be caused by the overburden noise train and by the influence of errors in the overburden velocity model on the computation of the critical angle. Still, critical-angle reflectometry should help to constrain the dominant fracture directions and can be combined with other methods to reduce the uncertainty in the estimated anisotropy parameters.

Anisotropic velocity updating for converted-wave prestack time migration

The conventional method used to estimate velocities for converted-wave (C-wave) prestack time migration is awkward because the P-wave velocity [Formula: see text] comes from P-wave processing, the velocity ratio gamma [Formula: see text] is estimated from C-wave data, and the S-wave velocity [Formula: see text] is then derived from [Formula: see text] and gamma. Instead, by using the C-wave velocity [Formula: see text], effective gamma [Formula: see text], and anisotropy parameter [Formula: see text], velocity updating becomes straightforward and more reliable. To update [Formula: see text] for converted-wave time migration, one can carry out hyperbolic moveout analysis on the hyperbolic-moveout-migrated-common-midpoint (HMO-MCMP) gathers. However, the errors in initial [Formula: see text] and anisotropy parameter [Formula: see text] can only be corrected by trial and error. In this article, we propose to remove the effects of initial [Formula: see text] and [Formula: see text] in the HMO-MCMP gathers by inverting the moveout related to the initial [Formula: see text] and [Formula: see text]. This enables a full nonhyperbolic velocity analysis to update not only [Formula: see text] but also [Formula: see text] and [Formula: see text]. To obtain reliable [Formula: see text], we also develop a simultaneous PP/PS anisotropic-parameter estimation method so the [Formula: see text] estimated from P-wave data is compared immediately with the [Formula: see text] derived from [Formula: see text] by using C-wave data. This provides a better constraint for estimating anisotropy parameters. The method has been tested and shows consistent improvement in converted-wave prestack time-migration velocity estimations.

VTI anisotropic corrections and effective parameter estimation after isotropic prestack depth migration

In the method for applying anisotropic corrections after isotropic prestack depth migration (PSDM), the correction, which is calculated and implemented in the depth domain, is defined as a time difference between isotropic and anisotropic traveltimes, under the assumption that the vertical velocity is known. The definition of this correction uses a special postmigration common-image-gather (CIG) ordering, which collects the migrated data according to the input-trace’s source and receiver distance from the surface CIG location. In this postmigration domain, the dip of the events can be directly related to their horizontal position in the CIG, called the imaging offset, and the separation of flat and dipping reflectors becomes easy to perform. The dependency of the seismic anisotropic effect on the subsurface dip angle is well pronounced in these CIGs. After application of an isotropic PSDM, effective anisotropic-parameter estimation is performed at selected CIG locations by using a simple two-parameter scan procedure. The optimal anisotropic parameters can be used to perform a final anisotropic PSDM or to apply a residual correction to the isotropically migrated data. We demonstrate the method for P-wave data in 2D media with vertical transverse isotropy (VTI) symmetry by using both synthetic and real data. We also present a strategy for handling the ambiguity between the vertical velocity and the anisotropic parameters.

Spin-triplet excitons in theS=12gapped antiferromagnetBaCuSi2O6: Electron paramagnetic resonance studies

BaCuSi$_2$O$_6$, a $S=1/2$ quantum antiferromagnet with a double-layer structure of Cu$^{2+}$ ions in a distorted planar-rectangular coordination and with a dimerized spin singlet ground state, is studied by means of the electron paramagnetic resonance technique. It is argued that multiple absorptions observed at low temperatures are intimately related to a thermally-activated spin-triplet exciton superstructure. Analysis of the angular dependence of exciton modes in BaCuSi$_2$O$_6$ allows us to accurately estimate anisotropy parameters. In addition, the temperature dependence of EPR intensity and linewidth is discussed.

Explicit expressions for prestack map time migration in isotropic and VTI media and the applicability of map depth migration in heterogeneous anisotropic media

We present 3D prestack map time migration in closed form for qP-, qSV-, and mode-converted waves in homogeneous transversely isotropic media with a vertical symmetry axis (VTI). As far as prestack time demigration is concerned, we present closed-form expressions for mapping in homogeneous isotropic media, while for homogeneous VTI media we present a system of four nonlinear equations with four unknowns to solve numerically. The expressions for prestack map time migration in VTI homogeneous media are directly applicable to the problem of anisotropic parameter estimation (i.e., the anellipticity parameter η) in the context of time-migration velocity analysis. In addition, we present closed-form expressions for both prestack map time migration and demigration in the common-offset domain for pure-mode (P-P or S-S) waves in homogeneous isotropic media that use only the slope in the common-offset domain as opposed to slopes in both the common-shot and common-receiver (or equivalently the common-offset and common-midpoint) domains. All time-migration and demigration equations presented can be used in media with mild lateral and vertical velocity variations, provided the velocity is replaced with the local rms velocity. Finally, we discuss the condition for applicability of prestack map depth migration and demigration in heterogeneous anisotropic media that allows the formation of caustics and explain that this condition is satisfied if, given a velocity model and acquisition geometry, one can map-depth-migrate without ambiguity in either the migrated location or the migrated orientation of reflectors in the image.

The effects of near‐surface conditions on anisotropy parameter estimations from 4C seismic data

ABSTRACTWe present a study of anisotropic parameter estimation in the near‐surface layers for P‐wave and converted‐wave (C‐wave) data. Near‐surface data is affected by apparent anisotropy due to a vertical velocity compaction gradient. We have carried out a modelling study, which showed that a velocity gradient introduces apparent anisotropy into an isotropic medium. Thus, parameter estimation will give anomalous values that affect the imaging of the target area.The parameter estimation technique is also influenced by phase reversals with diminishing amplitude, leading to erroneous parameters. In a modelling study using a near‐surface model, we have observed phase reversals in near‐surface PP reflections. The values of the P‐wave anisotropy parameter η estimated from these events are about an order of magnitude larger than the model values. Next, we use C‐wave data to estimate the effect of anisotropy (χ) and compute η from these values. These calculated η‐values are closer to the model values, and NMO correction with both η‐values shows a better correction for the calculated value. Hence, we believe that calculating η from χ gives a better representation of the anisotropy than picked η from the P‐wave.Finally, we extract the anisotropy parameters η and χ from real data from the Alba Field in the North Sea. Comparing the results with reference values from a model built according to well‐log, VSP and surface data, we find that the parameters show differences of up to an order of magnitude. The η‐values calculated from the C‐wave anisotropy parameter χ fit the reference values much better and show values of the same order of magnitude.

Integrated velocity model estimation for improved positioning with anisotropic PSDM

There are many geologic settings where anisotropic migration is necessary to obtain accurate seismic images. While this is well known, stable anisotropic parameter estimation has posed a serious challenge. Seismic data, though extensive in coverage, cannot constrain the anisotropy parameters alone (Tsvankin and Thomsen, 1995). The set of parameters is better constrained by integrating the seismic information with certain types of well data. However, the well data are generally sparse, so the parameters are only constrained at a few locations. Nonuniqueness is obviously a fundamental issue in our estimation problem.

Imaging the deep: a practical approach to handling anisotropy in the context of prestack migration velocity analysis

This article suggests a novel prestack migration model to represent the Earth subsurface, in which the shallow layers are smooth and anisotropic and the deep layers are complex and isotropic. This model circumvents the theoretical limitations associated with anisotropic parameter estimation from surface seismic P-wave data in a fully laterally inhomogeneous media. It also fits well with our understanding of the behaviour of the Earth's subsurface in many places, like offshore West Africa and South America. It is a clear improvement over ignoring anisotropy altogether. Thus, we develop an imaging approach based on resolving the anisotropy in the shallow smooth layers and using it to apply prestack migration velocity analysis on the complex deep. We, specifically, use prestack migration velocity analysis in the (x-t) domain to image and resolve the complex deep, taking the anisotropy up shallow into account. Nevertheless, only the interval velocity of the medium is updated from one iteration of prestack TAU migration velocity analysis to another. Using this simplified model along with prestack TAU migration velocity analysis, we managed to better image data from offshore Trinidad.

Migration velocity analysis in factorized VTI media

One of the main challenges in anisotropic velocity analysis and imaging is simultaneous estimation of velocity gradients and anisotropic parameters from reflection data. Approximating the subsurface by a factorized VTI (transversely isotropic with a vertical symmetry axis) medium provides a convenient way of building vertically and laterally heterogeneous anisotropic models for prestack depthmigration.The algorithm for P‐wave migration velocity analysis (MVA) introduced here is designed for models composed of factorized VTI layers or blocks with constant vertical and lateral gradients in the vertical velocity VP0. The anisotropic MVA method is implemented as an iterative two‐step procedure that includes prestack depth migration (imaging step) followed by an update of the medium parameters (velocity‐analysis step). The residual moveout of the migrated events, which is minimized during the parameter updates, is described by a nonhyperbolic equation whose coefficients are determined by 2D semblance scanning.For piecewise‐factorized VTI media without significant dips in the overburden, the residual moveout of P‐wave events in image gathers is governed by four effective quantities in each block: (1) the normal‐moveout velocity Vnmoat a certain point within the block, (2) the vertical velocity gradient kz, (3) the combination kx[Formula: see text] of the lateral velocity gradient kxand the anisotropic parameter δ, and (4) the anellipticity parameter η. We show that all four parameters can be estimated from the residual moveout for at least two reflectors within a block sufficiently separated in depth. Inversion for the parameter η also requires using either long‐spread data (with the maximum offset‐to‐depth ratio no less than two) from horizontal interfaces or reflections from dipping interfaces.To find the depth scale of the section and build a model for prestack depth migration using the MVA results, the vertical velocity VP0needs to be specified for at least a single point in each block. When no borehole information about VP0is available, a well‐focused image can often be obtained by assuming that the vertical‐velocity field is continuous across layer boundaries. A synthetic test for a three‐layer model with a syncline structure confirms the accuracy of our MVA algorithm in estimating the interval parameters Vnmo, kz, kx, and η and illustrates the influence of errors in the vertical velocity on the image quality.

Anisotropic ambiguities in TI media

Introduction In recent years we have seen processing and imaging algorithms re-written to handle anisotropic effects. The most common type of anisotropy that one deals with in seismic data is polar anisotropy (transverse isotropy). Media with vertical (VTI), tilted (TTI), and horizontal (HTI) axes have been shown to exist as a result of sedimentary deposition or fracturing. The consequences of ignoring polar anisotropy vary depending on the degree and type of anisotropy. Typically, using isotropic imaging in an anisotropic medium results in mis-ties between the preSDM and the well depths. Such mis-ties, in some extreme cases, can exceed 10% of the true depth (for example, in the Franklin-Elgin field operated by TFE in the North Sea there is a 600m mis-tie at a depth of 5km). In addition to the vertical depth error, there is also a lateral shift, most pronounced for the steepest dips, and noticeable in fault surface reflections. Experience has shown that one cannot image simultaneously flat and steep dips with an isotropic velocity field. The difficulty in addressing anisotropy lies in the estimation of reliable parameters to be used with the processing or imaging algorithms. In this work, we assess the effects of errors in anisotropic parameter estimation for the case of polar anisotropy, and attempt to quantify the consequences of these errors with some specific synthetic examples.