Improved generalized moveout approximation with a novel parameterization

Summary Numerical and physical modelling studies have been used to investigate the accuracy and limitations of the small-offset P-wave NMO velocity estimation in transversely isotropic media. An expression for the P-wave small-offset NMO velocity for a single horizontal reflector in a vertically transversely isotropic layer has been previously obtained: V nmo =+ αδ 0 12 where αο and δ respectively represent the vertical P-wave velocity and the P-wave critical anisotropy at oblique angles. The above equation is presented as being exact and valid for any degree of anisotropy and has been used in estimating the small-off set NMO velocity in the offshore Zaire seismic data acquired by Unocal in West Africa. The accuracy of this expression is subject t o practical testing in this paper. A one-layer horizontal model was used to carry out the numerical modelling studies. A computer program which calculates Pwave NMO velocities in transversely isotropic media was used. This program uses the exact equations for the phase and ray velocities in generating velocity functions to be used in ray tracing; it is known to give accurate results. Using the publishe d elastic parameters for anisotropic rocks, the small-offset NMO velocities are computed and results are compared with values obtained using the above expression. The degree of anisotropy varied from very weak to strong and contrasting velocity function s (positive and negative δ) were used in numerical modelling simulations. Phenolite materials with contrasting velocity functions were used to simulate transversely isotropic media with vertical axis of symmetry in physical modelling tests. Experimental reflection data were collected and velocity analysis was conducted for small-offset tr aces, and the NMO velocity determined by the maximum stack response. The elastic parameters αο and δ of the Phenolite materials were recovered by an anisotropic inversion technique. These parameters were used to estimate the small-offset P-wave NMO velocity in Phenolite with the above expression and also with our computer program. The NMO velocities obtained from these two methods were compared to that obtained from velocity analysis of the experimental seismic data. These comparisons enabled the validity of the above expressio n to yield accurate P-wave NMO velocity, in the limit of small offset, to be tested. Numerical and physical modelling results indicate that the accuracy of the small-offset P-wave NMO equation depends on the nature and degree of the anisotropy encountered in a given area, and is not valid for all degrees of P-wave anisotropy. The sma lloffset NMO (stacking) velocity from horizontal reflectors is known to influence the effects of vertical transverse isotropy on normal moveout (NMO), dip moveout (DMO) corrections and time migration. Determination of the correct small-offset P-wave NMO velocity is essential for obtaining accurate seismic images, and subsequently correct interpretations.

Anisotropic P-wave velocity analysis and seismic imaging in onshore Kutch sedimentary basin of India

Summary Transversely isotropic with vertical symmetry axis (VTI) is a practical assumption for wave propagation environment in many 2D seismic data, due to its acceptable accuracy and limited number of input parameters. Explicit VTI traveltime approximations are commonly used in many seismic data processing and inversion algorithms. In this study we propose a new traveltime approximation for P-wave propagation in VTI media that is more accurate than the previously reported most accurate approximations. We compare its accuracy, and show that it can outperform other three-parameter approximations in velocity analysis and normal moveout correction.

The anisotropic phenomenon in wave propagation has been widely recognized on various scales. Knowledge of anisotropic effects is essential for interpretation and processing of seismic data. The existence of anisotropy leads to the moveout of nonhyperbolic even in a homogenous layer. A transversely isotropic medium with a vertical symmetry axis (VTI) is a reasonable approximation of horizontally layered anisotropic medium. The approximation of travel time is important for reducing the residual normal moveout of layered VTI media. The aim of this study is to develop globally optimized generalized moveout approximation in reducing residual normal moveout in VTI media. A comparative analysis was carried out to recent method for given an ellipticity parameter (0 ≤ η ≥ 0.5 ) and wide offset depth to ratio (0 to 4). The result shows that the globally optimized generalized moveout approximation is better in reducing residual moveout at large offsets with a stronger anellipticity parameter than existing methods. This is essential for reducing the accumulation of error especially for deeper substructures.

Surface reflection tomography is an inversion method that attempts to determine simultaneously the subsurface elastic parameters and reflector depths from pre-stack seismic data. To date, this technique has been applied to acoustic and isotropic elastic media. In this paper, we study surface reflection tomography in the anisotropic elastic case. Anisotropic media are described by many more elastic parameters than are needed to describe isotropic media; these additional parameters are useful for obtaining rock properties such as crack density, pore shape and fracture strike that cannot be found with isotropic methods. Similar to isotropic surface reflection tomography, many features of an anisotropic model are poorly resolved due to limited ray path coverage; the vertical smearing of isotropic tomography occurs in each elastic parameter in the anisotropic problem. Unlike the isotropic problem, however, there is additional indeterminacy in the solution of the anisotropic tomography problem because of ambiguity amongst the several elastic parameters needed to describe anisotropic media. Also unlike the isotropic problem, the reflector depths participate very strongly in this ambiguity. We investigate the nature of this indeterminacy by studying the null space for linearized tomography, that is, the class of model perturbations of a background medium which, to first order, cause no perturbation at all in the surface reflection traveltimes. Such model perturbations cannot be determined from the traveltime perturbations; describing these null space model perturbations gives insight into the indeterminacy in the anisotropic problem. Complementary to computational approaches towards identifying the null space for discrete formulations of tomography, we study a continuum formulation. Our first set of results concerns the ‘elastic’ null space: perturbations of the elastic parameters (with zero depth perturbation) which cause no perturbation in the traveltimes. The results here are very similar to previous results for cross-well transmission tomography. As expected, the ‘elastic’ null space is larger than the isotropic null space due to the ambiguity amongst the elastic parameters. We identify three categories of model perturbations in the ‘elastic’ null space. The first category consists of model perturbations for which the perturbation in each of the individual elastic parameters is itself either in the isotropic null space, or in a closely related set of model perturbations which we call the odd isotropic null space. Elements in the second category are anisotropic versions of the most well-known isotropic null space elements: perturbations which are polynomials in the horizontal variable with coefficients which are functions of the depth variable satisfying certain linear integral constraints; unlike the isotropic problem, the integral constraints in the anisotropic problem couple together the several elastic parameters. The third category consists of model perturbations satisfying zero boundary conditions at the surface and at the reflector for which a specific linear combination of integrals and derivatives of the several elastic parameters is in the isotropic null space. In particular, there are model perturbations in this third category which represent anomalies that are completely contained in the interior of the model and yet are in the ‘elastic’ null space; this behaviour is different from the isotropic problem. These categories are sufficient to describe the ‘elastic’ null space completely. We demonstrate that every model perturbation in the ‘elastic’ null space is the sum of an element in the first category (indicating an indeterminacy of the same nature as in the isotropic problem in each of the elastic parameters separately) and an element in the third category (indicating an ambiguity amongst the parameters). The second category gives a rich family of examples of sums of null space elements in the first and third categories, and thereby gives a sense of just how large the ‘elastic’ null space is. Moreover, we show that the traveltime perturbations caused by an elastic perturbation determine only a small number of features of the elastic perturbation which distinguish between the several elastic parameters. We identify these features pre-cisely: they are functions of the horizontal variable representing vertical averages of combinations of the elastic parameters and their derivatives. Elastic parameters influencing the vertical velocity appear more prominently in these features than those influencing the horizontal velocity, and these features are closely related to the zero-offset traveltimes and the normal moveout velocities. Our second set of results concerns the so-called velocity–depth ambiguity, more precisely here the elastic parameter versus the depth parameter ambiguity. We demonstrate that for any smooth depth perturbation, there is a perturbation in the elastic parameters for which the combined elastic and depth perturbations cause zero traveltime perturbations; moreover, this can be accomplished by an elliptically anisotropic elastic perturbation with a vertical axis of symmetry, perturbing the vertical velocity alone. This implies that the velocity–depth ambiguity cannot be resolved here as it can be in the isotropic case. We show an example of an anticlinal structure embedded in a homogeneous background whose reflection traveltimes exactly match those from a model with elliptically anisotropic elastic perturbations overlying a flat reflector. We also draw a number of conclusions from these results for the special case of transversely isotropic media with a vertical symmetry axis.

A flexible functional form can provide a second-order approximation to an arbitrary unknown function at a single point. Except in special cases, the parameters of flexible forms will vary from one point of approximation to another. I use this property to show that, in general, if an unknown function is homogeneous then i) Euler's Theorem gives rise to linear equality constraints involving both the data and a set of observation-varying flexible form parameters, ii) the common practice of imposing homogeneity on flexible functional forms is unnecessarily restrictive, and iii) it is possible to obtain estimates of the observation-varying parameters of approximating flexible forms using a Singular Value Decomposition (SVD) estimator. Two illustrations are provided: artificially-generated data is used to estimate the characteristics of a generalised linear production function; and Canadian data is used to estimate the characteristics of a consumer demand system.

ABSTRACTComputation of complex‐valued traveltimes provides an efficient approach to describe the seismic wave attenuation for applications like attenuation tomography, inverse Q filtering and Kirchhoff migration with absorption compensation. Attenuating acoustic transverse isotropy can be used to describe the directional variation of velocity and attenuation of P‐waves in thin‐bedding geological structures. We present an approximate method to solve the acoustic eikonal equation for an attenuating transversely isotropic medium with a vertical symmetry axis. We take into account two similar parameterizations of an attenuating vertical symmetry axis medium. The first parameterization uses the normal moveout velocity, whereas the second parameterization uses the horizontal velocity. For each parameterization, we combine perturbation theory and the Shanks transform in different ways to derive analytic solutions. Numerical examples show that the analytic solutions derived from the second parameterization yield better accuracy. The Shanks transform solution with respect to only the anellipticity parameter from the second parameterization is demonstrated numerically to be the most accurate among all the analytic solutions.

ABSTRACTWe study the azimuthally dependent hyperbolic moveout approximation for small angles (or offsets) for quasi‐compressional, quasi‐shear, and converted waves in one‐dimensional multi‐layer orthorhombic media. The vertical orthorhombic axis is the same for all layers, but the azimuthal orientation of the horizontal orthorhombic axes at each layer may be different. By starting with the known equation for normal moveout velocity with respect to the surface‐offset azimuth and applying our derived relationship between the surface‐offset azimuth and phase‐velocity azimuth, we obtain the normal moveout velocity versus the phase‐velocity azimuth. As the surface offset/azimuth moveout dependence is required for analysing azimuthally dependent moveout parameters directly from time‐domain rich azimuth gathers, our phase angle/azimuth formulas are required for analysing azimuthally dependent residual moveout along the migrated local‐angle‐domain common image gathers. The angle and azimuth parameters of the local‐angle‐domain gathers represent the opening angle between the incidence and reflection slowness vectors and the azimuth of the phase velocity ψphs at the image points in the specular direction. Our derivation of the effective velocity parameters for a multi‐layer structure is based on the fact that, for a one‐dimensional model assumption, the horizontal slowness and the azimuth of the phase velocity ψphs remain constant along the entire ray (wave) path. We introduce a special set of auxiliary parameters that allow us to establish equivalent effective model parameters in a simple summation manner. We then transform this set of parameters into three widely used effective parameters: fast and slow normal moveout velocities and azimuth of the slow one. For completeness, we show that these three effective normal moveout velocity parameters can be equivalently obtained in both surface‐offset azimuth and phase‐velocity azimuth domains.

Anisotropic waveform tomography (AWT) uses anisotropic traveltime tomography followed by anisotropic full-waveform inversion (FWI). Such an approach is required for FWI in cases in which the geology is likely to exhibit anisotropy. An important anisotropy class is that of transverse isotropy (TI), and the special case of TI media with a vertical symmetry axis (VTI) media is often used to represent elasticity in undeformed sedimentary layering. We have developed an approach for AWT that uses an acoustic approximation to simulate waves in VTI media, and we apply this approach to crosshole data. In our approach, the best-fitting models of seismic velocity and Thomsen VTI anisotropy parameters are initially obtained using anisotropic traveltime tomography, and they are then used as the starting models for VTI FWI within the acoustic approximation. One common problem with the acoustic approach to TI media is the generation of late-arriving (spurious) S-waves as a by-product of the equation system. We used a Laplace-Fourier approach that effectively damps the spurious S-waves to suppress artifacts that might otherwise corrupt the final inversion results. The results of applying AWT to synthetic data illustrate the trade-offs in resolution between the two parameter classes of velocity and anisotropy, and they also verify anisotropic traveltime tomography as a valid method for generating starting models for FWI. The synthetic study further indicates the importance of smoothing the anisotropy parameters before proceeding to FWI inversions of the velocity parameter. The AWT technique is applied to real crosshole field gathers from a sedimentary environment in Western Canada, and the results are compared with the results from a simpler (elliptical) anisotropy model. The transversely isotropic approach yields an FWI image of the vertical velocity that (1) exhibits a superior resolution and (2) better predicts the field data than does the elliptical approach.

The main difficulty in extending seismic processing to anisotropic media is the recovery of anisotropic velocity fields from surface reflection data. We suggest carrying out velocity analysis for transversely isotropic (TI) media by inverting the dependence of P-wave moveout velocities on the ray parameter. The inversion technique is based on the exact analytic equation for the normal-moveout (NMO) velocity for dipping reflectors in anisotropic media.We show that P-wave NMO velocity for dipping reflectors in homogeneous TI media with a vertical symmetry axis depends just on the zero-dip value V nmo (0) and a new effective parameter eta that reduces to the difference between Thomsen parameters epsilon and delta in the limit of weak anisotropy. Our inversion procedure makes it possible to obtain eta and reconstruct the NMO velocity as a function of ray parameter using moveout velocities for two different dips. Moreover, V nmo (0) and eta determine not only the NMO velocity, but also long-spread (nonhyperbolic) P-wave moveout for horizontal reflectors and the time-migration impulse response. This means that inversion of dip-moveout information allows one to perform all time-processing steps in TI media using only surface P-wave data. For elliptical anisotropy (epsilon = delta ), isotropic time-processing methods remain entirely valid. We show the performance of our velocity-analysis method not only on synthetic, but also on field data from offshore Africa.Accurate time-to-depth conversion, however, requires that the vertical velocity V P0 be resolved independently. Unfortunately, it cannot be done using P-wave surface moveout data alone, no matter how many dips are available. In some cases V P0 is known (e.g., from check shots or well logs); then the anisotropy parameters epsilon and delta can be found by inverting two P-wave NMO velocities corresponding to a horizontal and a dipping reflector. If no well information is available, all three parameters (V P0 , epsilon , and delta ) can be obtained by combining our inversion results with shear-wave information, such as the P-SV or SV-SV wave NMO velocities for a horizontal reflector.Generalization of the single-layer NMO equation to layered anisotropic media with a dipping reflector provides a basis for extending anisotropic velocity analysis to vertically inhomogeneous media. We demonstrate how the influence of a stratified anisotropic overburden on moveout velocity can be stripped through a Dix-type differentiation procedure.

Econometric methods can be helpful when one is trying to understand the nature of competition between products in differentiated products industries, an issue that arises in many competition policy contexts. However, misspecification of the consumer demand system can result in biased econometric results and misleading conclusions. We discuss some considerations that go into choosing a demand system specification and suggest that a ‘flexible functional form’ be used. This approach avoids placing restrictions on the demand elasticities. We identify and discuss other issues that arise in the estimation of demand systems. We conclude with an empirical example concerning the calculation of lost profits damages in a patent infringement case. This example demonstrates how using a flexible functional form leads to a substantially more reliable damages calculation than using the particular restrictive functional form that is commonly used to calculate lost profits damages in patent litigation.

Flexible functional forms and tests of homogeneous separability

Moveout analysis of long‐spread P‐wave data is widely used to estimate the key time‐processing parameter η in layered VTI (transversely isotropic with a vertical symmetry axis) media. Inversion for interval η values, however, suffers from instability caused by the tradeoff between the effective moveout parameters and by the subsequent error amplification during Dix‐type layer stripping. Here, we propose an alternative approach to interval parameter estimation based on the velocity‐independent layer‐stripping (VILS) method of Dewangan and Tsvankin (2006). This method makes it possible to compute the interval moveout in a layer of interest without knowledge of the medium parameters in the overburden. Thus, the layer stripping of moveout parameters in the conventional method is replaced by the much more stable stripping of reflection traveltimes. We also develop the 3D version of the layer‐stripping method and apply it to interval parameter estimation for orthorhombic media using wide‐azimuth P‐wave data. The superior accuracy and stability of our algorithm is illustrated on noise‐contaminated ray‐traced synthetic data for typical layered VTI and orthorhombic models. The algorithm is also tested on wide‐azimuth P‐wave reflection data recorded above a fractured reservoir at Rulison field in Colorado. The interval η values estimated by VILS are more plausible and stable than those obtained by the Dix‐type method.

It is necessary to account for anisotropy in full waveform inversion (FWI) of wide-azimuth and wide-aperture seismic data in most geologic environments, for correct depth positioning of reflectors, and for reliable estimations of wave speeds as a function of the direction of propagation. In this framework, choosing a suitable anisotropic subsurface parameterization is a central issue in monoparameter and multiparameter FWI. This is because this parameterization defines the influence of each physical parameter class on the data as a function of the scattering angle, and hence the resolution of the parameter reconstruction, and on the potential trade-off between different parameter classes. We apply monoparameter and multiparameter frequency-domain acoustic vertical transverse isotropic FWI to synthetic and real wide-aperture data, representative of the Valhall oil field. We first show that reliable monoparameter FWI can be performed to build a high-resolution velocity model (for the vertical, the horizontal, or normal move-out velocity), provided that the background models of two Thomsen parameters describe the long wavelengths of the subsurface sufficiently accurately. Alternatively, we show the feasibility of the joint reconstruction of two wave speeds (e.g., the vertical and horizontal wave speeds) with limited trade-off effects, while Thomsen parameter [Formula: see text] is kept fixed during the inversion. The influence of the wave speeds on the data for a limited range of scattering angles when combined each other can, however, significantly hamper the resolution with which the two wave speeds are imaged. These conclusions inferred from the application to the real data are fully consistent with those inferred from the theoretical parameterization analysis of acoustic vertical transverse isotropic FWI performed in the companion report.

We present a new inverse method for retrieving the mantle structure, using the fundamental and higher modes surface waves. The fundamental mode is only sensitive to structures down to 400–500 km but higher modes have a depth resolution down to 1000–1500 km. So, they are well suited to the study of the transition zone between 400 and 1000 km depth. A data set of seismograms corresponding to events all located in a small area and recorded at a single station is selected. The inversion of the seismograms is divided into two steps. First, the difference between the real spectra and the sum of the higher modes synthetic spectra is inverted for retrieving the phase velocities of the first higher modes. In a second step, these phase velocities are inverted at depth in order to retrieve the average mantle structure between the receiver and the epicentral area. This structure is modelled as a transversely isotropic medium with a vertical symmetry axis. Because of the non-linearity of the problem, it is necessary to iterate the procedure and to recalculate the Earth eigenfunctions and the new synthetic spectra at each iteration. Numerical tests on synthetic data have been performed and they demonstrate that we have a good resolution at depth down to 1500 km. Some results obtained on real data recorded on GEOSCOPE network are also presented. This method can now be applied systematically to real data.