Ellen N. S. Gomes

Resolution analysis for stereotomography in media with elliptic and anelliptic anisotropy

Stereotomography is a tomographic method that uses slowness-vector components to improve traveltime inversion. We extend it to general anisotropic media and implement it for media with elliptic and anelliptic anisotropy. Such media depend on only three elastic parameters, which makes stereotomography less sensitive to ambiguity resulting from limited coverage of surface seismic experiments than it would be in transversely isotropic or orthorhombic media. The corresponding approximations of the slowness surface restrict the validity of the present approach to qP events and mild anisotropy. Our numerical analysis demonstrates that stereotomography can be used to estimate macrovelocity models in the presence of anisotropy. For elliptic anisotropy, the parameter that is most difficult to estimate is the orientation of the symmetry axis. For anelliptic anisotropy, estimation of stereotomography parameters is possible in principle; however, it will be mostly ill conditioned in practice, even for low noise levels. Better results are achieved, if transmission events from multiple-offset vertical seismic profiling experiments are included in the inversion.

Sensitivity Analysis for Stereotomography in Elliptic and Anelliptic Media

Stereotomography is extended to general anisotropic models and implemented for elliptical and anelliptical anisotropy. Elliptical and anelliptical models depend on three parameters only, which makes them less sensitive to ambiguity due to limited coverage of surface seismic experiments than transversely isotropic or orthorhombic models. The corresponding approximations of the slowness surface restrict the validity of the present approach to qP events and mild anisotropy. Numerical experiments show the potential and the limitations of stereotomography for estimating macrovelocity models in the presence of anisotropy as well as the importance of transmission events from multiple-offset VSP experiments for the success of the approach. INTRODUCTION The determination of a macrovelocity model is essential for time and depth imaging of seismic reflectors in the earth. Among the many methods that try to to achieve this aim are so-called tomographic methods that are based on the inversion of traveltimes of seismic reflection events. One of these is stereotomography, which uses slowness vector components to improve and stabilize the traveltime inversion. Stereotomography was initially proposed by Billette and Lambaré (1998) as a robust tomographic method for estimating velocity macro models from seismic reflection data. They had recognized the potential efficiency of traveltime tomography (Bishop et al., 1985; Farra and Madariaga, 1988) but also the difficulties associated with a highly interpretative picking. The selected events have to be tracked over a large extent of the pre-stack data cube, which is quite difficult for noisy or complex data. The idea is to use locally coherent events characterized by their slopes in the pre-stack data-volume. Such events can be interpreted as pairs of ray segments and provide independent information about the velocity model. Recently, Billette et al. (2003) demonstrated the successful use of stereotomography to recover isotropic background media. According to Gosselet et al. (2005), the use of reflection events only is insufficient to recover anisotropic models. Here, we study the limitations of stereotomography applied in anisotropic media, using reflected and transmitted events. For this purpose, we use approximations for weak elliptic and anelliptic anisotropy that are valid for qP waves. Any tomographic method is based on ray theory. Our approach follows the lines of Farra and Madariaga (1987) who applied perturbation theory to the Hamiltonian systems that describe the rays in media with arbitrary anisotropy (Goldstein, 1980). Perturbation theory allows to calculate linear approximations to the observed data, the so-called Fréchet derivatives. Here, we extend the work of Farra and Madariaga (1987) to arbitrary anisotropy and restrict it later on for application purposes to elliptic and anelliptic media. STEREOTOMOGRAPHY IN ANISOTROPIC MEDIA Stereotomography differs from conventional reflection tomography by the data that are used for the inversion (Billette et al., 2003). Firstly, the traveltimes are picked from locally coherent events that are 210 Annual WIT report 2006 interpreted as primary reflections or diffractions. Secondly, in-line slowness vectors components of these events, detected in common-shot or common-receiver gathers, are used in addition to positions and traveltimes of sources and receivers. Thus, the data space is given by d = [(x,x, s, s, T sr)n] (n = 1, . . . , N) . (1) where x and x are the source and receiver positions, T sr are the traveltimes, and s and s are the slowness-vector projections into the receiver line. Moreover, N is the number of selected events. Stereotomography also uses a different model parameterization than conventional reflection tomography. In 2D, the model to be estimated includes: the parameters describing the velocity model, p, the scattering-point coordinates, X, the emergence angles, θ and θ, and the ray traveltimes, τ s e τ . In other words, the model vector is m = {p, (X, θ, θ, τ , τ )n} (n = 1, . . . , N) . (2) To solve the inverse problem using linear iterations, an initial reference model must be given. In this model m0, ray tracing is performed to calculate the synthetic data, equation (1), denoted as d. The difference between the observed and calculated data, d − d, defines the deviation δd. This deviation is modeled in linear approximation as δd = DF(m0)δm , (3) where DF denotes the approximate operator describing the direct problem under variation of the reference model m0. The operatorDF(m0) is known as the Fréchet derivative (see, e.g., Menke, 1989). The solution of the linear system in equation (3) determines a new reference model m 0 = m0 + δm. (4) The process continues iteratively until the norm of the deviation ‖δd‖ is smaller than a given tolerance value (in case of convergence) or until a maximum number of steps. In this work, we use the standard L2 norm (Menke, 1989). Rays in anisotropic media The ray tracing system in generally anisotropic media can be represented as (Červený, 2001) dx dτ = ∇sH , ds dτ = −∇xH , (5) where ∇x and ∇s represent the gradients with respect to the position and slowness vectors, x and s, respectively, and where τ is the traveltime along the ray. Moreover, H(x, s;p) = 0 along the ray. In tomographic applications, this system (5) is solved numerically. Upon perturbation of the medium parameters p, the position and slowness vectors of a ray get perturbed. Retaining only first-order effects in these perturbations δp, δx and δs, the system becomes d dτ  δx δs  =  ∇s∇xH ∇s∇s H −∇x∇xH −∇x∇s H  δx δs  +  ∇s∇pHδp −∇x(∇pHδp)  . (6) Initial conditions Initial conditions for δx e δs can be established upon requiring that the first-order perturbations of the Hamiltonian at the starting point must be zero. This condition guarantees that the paraxial rays satisfy, to the first order, the Hamiltonian equations. For stereotomography, it is necessary to integrate the above Annual WIT report 2006 211 system (6) for one initial condition for each possible perturbations. Therefore, the determination can be reduced to three cases: (1) Perturbation of the slowness direction: δs = s ( I− n∇ T s H ∇s Hn ) dn dθ δθ . (7) (2) Perturbation of the diffraction point position: δs = − ∇sH ‖∇sH‖ ∇xHδX ‖∇sH‖ , (8) (3) Perturbation of the elastic parameters: δs = − ∇sH ‖∇sH‖ ∇pHδp/‖∇sH‖. (9) With these initial conditions, the system in equation (6) can be integrated along a ray in the reference medium. In this way, the Fréchet derivatives with respect to the perturbations of the initial position, initial angle, and elastic parameters can be numerically evaluated. System (6) can be efficiently solved for each choice of initial conditions by means of the propagator method (Červený, 2001). With the central ray, i.e., x(τ) and s(τ), supposed to be known, system (6) takes the form dy dτ = A(τ)y + f(τ) , (10) where A(τ) =  ∇s∇xH ∇s∇s H −∇x∇xH −∇x∇s H  , y = [ δx δs ] , and f = [ ∇s(∇pHδp) −∇x(∇pHδp) ] . (11) Equation (10) is a system of linear ordinary differential equations. The propagator methods allow to represent the solution to this system in an interval (τ0, τ), satisfying the initial condition y(τ0) = y0, as y(τ) = P(τ, τ0)y0 + ∫ τ τ0 P(τ, ξ)f(ξ)dξ . (12) The initial condition for the propagator matrix P(τ, τ0) is then P(τ0, τ0) = I, where I is the identity matrix. Numerically, P(τ, τ0) can be determined using Runge-Kutta schemes. For stereotomography, this approach has the advantage that the propagator matrix P(τ, τ0) can be determined independently of y0, which means that it needs to be calculated only once. The integration of system (6) is based on the assumption that the elastic parameters vary smoothly. Actually, they need to be second-order differentiable. Moreover, the model needs to be specified by a finite number of parameters. To satisfy these conditions, the parameters must be interpolated. In our implementation, each medium parameter is represented using the tensor product of third-order B-splines as

Processing of large offset data: experimental seismic line from Tenerife Field, Colombia

Manuscript ID: 20170072. Received on: 05/21/2017. Approved on: 11/13/2017. ABSTRACT: Exploration seismology provides the main source of information about the Earth’s subsurface, which in many cases can be presented as a simple model of horizontal or near-horizontal layers. After the seismic acquisition step, conventional seismic processing of reflection data provides an image of the subsurface by using information about the reflections of these layers. The traveltime from a source to different receivers is adjusted using a hyperbolic function. This expression is used in the case involving an isotropic medium, which is a simplification of nature, whereas geologically complex media are generally anisotropic. A subsurface model that more closely resembles reality is the vertical transverse isotropy, which defines two parameters that are required to correct the traveltimes: the NMO velocity and the anellipticity parameter. In this paper, we reviewed the literature and methodology for velocity analysis of seismic data acquired from anisotropic media. A model with horizontal layers and anisotropic behavior was developed and evaluated. The anisotropic velocity was compared to the isotropic velocity, and the results were analyzed. Finally, the methodology was applied to real seismic data, i.e. an experimental landline from Tenerife Field, Colombia. The results show the importance of the anellipticity parameter in models with anisotropic layers.

Improvement of local anisotropy estimation from VSP data through experimental design

We investigate a linear inversion scheme to determine the local anisotropy around a well using multi-azimuthal walkaway VSP data. The input data consist of the vertical components of the slowness vector and the polarization vector of direct and reflected P-wave measurements. The inversion assumes that weak anisotropy can be modelled by first-order perturbations around an isotropic reference medium. This inversion scheme is not restricted to symmetrical classes of anisotropy and the well local orientation. Based on numerical simulations, we show that the inversion is sensitive to the number and orientation of the acquired azimuths. Using conservative assumptions regarding the noise level, phase velocity can be well estimated for a limited angular opening around the well local orientation.

Modelagem Sísmica de Alvo Exploratório em Bacia Paleozóica

O imageamento sismico abaixo das soleiras de diabasio e um grande desafio para a sismica de reflexao em bacias paleozoicas, ocasionando o aumento do risco exploratorio na regiao. Varios fatores contribuem para essa dificuldade como a alta refletividade na interface entre o topo do diabasio e o pacote sedimentar, multiplas de curto periodo devido a heterogeneidade dentro do diabasio e o espalhamento de energia devido a rugosidade na topografia do diabasio. Para se obter um melhor entendimento acerca da propagacao de ondas sismicas neste tipo de ambiente, construimos um modelo acustico 3D representativo de alvo exploratorio de uma bacia paleozoica. Em seguida inserimos propriedades fisicas no modelo atraves de dados de pocos, simulamos uma aquisicao 3D e 2D usando diferencas finitas e apresentamos o resultado para uma secao 2D na regiao de maior espessamento do diabasio. Descrevemos os procedimentos utilizados para a construcao do modelo 3D analisando as caracteristicas dos dados obtidos pela modelagem sismica 3D e o resultado do imageamento sismico em tempo e em profundidade de uma secao 2D. Os eventos de pull-up sao observados abaixo da soleira de diabasio como resultado do imageamento.

Estimation of fractures orientation from qP reflectivity using multiazimuthal AVO analysis

Investigamos a estimativa da orientacao de fratura; mergulho e direcao, atraves da analise de AVO multiazimutal de ondas qP e de suas convertidas qS1 e qS2. Assumimos fraco contraste de impedância, fraca anisotropia e ainda que, o meio fraturado comporta-se efetivamente como um meio transversalmente isotropico (TI). Dentro destas hipoteses, a estimativa da orientacao de fratura reduz-se a estimativa da orientacao do eixo de simetria a partir de dados de refletividade de ondas qP. Aproximacoes lineares da refletividade da onda qP sao utilizadas na inversao.

Linearização dos coeficientes de reflexão de ondas qP em meios anisotrópicos

The reflection coefficients at a planar interface separating two anisotropic media have a nonlinear dependence on the elastic parameters and densities of both media. Linear approximations on the elastic parameters for the qP wave reflectivity are more convenient for AVO/AVD analysis. We present the solution of the Zoeppritz equations in terms of impedance and polarization matrices. Using this approach and assuming weak impedance contrast and weak anisotropy, a simple derivation of linearized approximations for qP the reflectivity is presented for general anisotropy. The linear approximations of reflection coefficients, qP and converted waves, for qP incidence are very close to the exact results for incidence angles up to 30 degrees considering moderate impedance contrast and anisotropy.