Michael A. Slawinski

Seismic rays as Finsler geodesics

We prove that, in general, for anisotropic nonuniform continua, seismic rays are geodesics in Finsler geometry. In particular, for separable velocity functions, the geometry is Wagnerian. We provide concrete examples with theoretical discussions and introduce the seismic Finsler metric.

Ray tracing by simulated annealing: Bending method

We propose a new ray-tracing method based on the concept of simulated annealing. Using this method, we find rays between fixed sources and receivers that render traveltime globally minimal. With our method, we are able to construct rays and their associated traveltimes to satisfactory precision in complex media. Furthermore, our algorithm can be modified to calculate rays of locally minimum traveltime, such as reflected rays, by constraining the ray to pass through a set of points that we are free to specify.

Invariant Properties for Finding Distance in Space of Elasticity Tensors

Using orthogonal projections, we investigate distance of a given elasticity tensor to classes of elasticity tensors exhibiting particular material symmetries. These projections depend on the orientation of the elasticity tensor; hence the distance is obtained as the minimization of corresponding expressions with respect to the action of the orthogonal group. These expressions are stated in terms of the eigenvalues of both the given tensor and the projected one. The process of minimization is facilitated by the fact that, as we prove, the traces of the corresponding Voigt and dilatation tensors are invariant under these orthogonal projections. For isotropy, cubic symmetry and transverse isotropy, we formulate algorithms to find both the orientation and the eigenvalues of the elasticity tensor endowed with a particular symmetry and closest to the given elasticity tensor.

Estimating effective elasticity tensors from Christoffel equations

We consider the problem of obtaining the orientation and elasticity parameters of an effective tensor of particular symmetry that corresponds to measurable traveltime and polarization quantities. These quantities — the wavefront-slowness and polarization vectors — are used in the Christoffel equation, a characteristic equation of the elastodynamic equation that brings seismic concepts to our formulation and relates experimental data to the elasticity tensor. To obtain an effective tensor of particular symmetry, we do not assume its orientation; thus, the regression using the residuals of the Christoffel equation results in a nonlinear optimization problem. We find the absolute extremum and, to avoid numerical instability of a global search, obtain an accurate initial guess using the tensor of given symmetry closest to the generally anisotropic tensor obtained from data by linear regression. The issue is twofold. First, finding the closest tensor of particular symmetry without assuming its orientation is challenging. Second, the closest tensor is not the effective tensor in the sense of regression because the process of finding it carries neither seismic concepts nor statistical information; rather, it relies on an abstract norm in the space of elasticity tensors. To include seismic concepts and statistical information, we distinguish between the closest tensor of particular symmetry and the effective one; the former is the initial guess to search for the latter.

Analytic solution of ray-tracing equations for a linearly inhomogeneous and elliptically anisotropic velocity model

We study wave propagation in anisotropic inhomogeneous media. Specifically, we formulate and analytically solve the ray-tracing equations for the factorized model with wavefront velocity increasing linearly with depth and depending elliptically on direction. We obtain explicit expressions for traveltime, wavefront (phase) angle, and ray (group) velocity and angle, and study these seismological quantities for a model that successfully describes field measurements in the Western Canada Basin. By considering numerical examples, we also show that the difference between the wavefront and ray velocities depends only slightly on the anisotropy parameter, whereas the difference between the wavefront and ray angles is, in a first-order approximation, linear in the anisotropy parameter.

VSP traveltime inversion for linear inhomogeneity and elliptical anisotropy

To account for measured vertical seismic profiling (VSP) traveltimes, we study a velocity model described by three parameters. We assume that the velocity increases linearly with depth and is given in terms of parameters a and b, whereas the anisotropy is the result of elliptical velocity dependence and is given in terms of parameter χ. Using this model, we formulate an analytical expression for traveltime between a given source and a given receiver. This traveltime expression contains the three parameters that are present in the velocity model. To obtain the values of a, b, and χ, we use least‐squares fitting of this traveltime expression, with respect to measured traveltimes. This process of obtaining the parameters is exemplified by a study of traveltime data acquired with a two‐offset VSP in the Western Canada Basin. Having obtained a, b, and χ, we perform a statistical analysis, which shows good agreement between the field data and the modeled data. Furthermore, it shows that the elliptical velocity ...

On Backus Average for Generally Anisotropic Layers

In this paper, following the Backus (in J. Geophys. Res. 67(11):4427–4440, 1962) approach, we examine expressions for elasticity parameters of a homogeneous generally anisotropic medium that is long-wave-equivalent to a stack of thin generally anisotropic layers. These expressions reduce to the results of Backus (1962) for the case of isotropic and transversely isotropic layers.In the over half-a-century since the publications of Backus (1962) there have been numerous publications applying and extending that formulation. However, neither George Backus nor the authors of the present paper are aware of further examinations of the mathematical underpinnings of the original formulation; hence this paper.We prove that—within the long-wave approximation—if the thin layers obey stability conditions, then so does the equivalent medium. We examine—within the Backus-average context—the approximation of the average of a product as the product of averages, which underlies the averaging process.In the presented examination we use the expression of Hooke’s law as a tensor equation; in other words, we use Kelvin’s—as opposed to Voigt’s—notation. In general, the tensorial notation allows us to conveniently examine effects due to rotations of coordinate systems.

Backus Average Under Random Perturbations of Layered Media

We examine the Backus averaging method---which, in general, allows one to represent a series of parallel layers by a transversely isotropic medium---using a repetitive shale-sandstone model. To examine this method in the context of experimental data, we perturb the model with random errors, in particular, the values of layer thicknesses and elasticity parameters. We analyze the effect of perturbations on the parameters of the transversely isotropic medium. Also, we analyze their effect on the relation between layer thicknesses and wavelengths. To gain insight into the strength of anisotropy of that medium, we invoke the Thomsen parameters.

On Choosing Effective Elasticity Tensors Using a Monte-Carlo Method

A generally anisotropic elasticity tensor can be related to its closest counterparts in various symmetry classes. We refer to these counterparts as effective tensors in these classes. In finding effective tensors, we do not assume a priori orientations of their symmetry planes and axes. Knowledge of orientations of Hookean solids allows us to infer properties of materials represented by these solids. Obtaining orientations and parameter values of effective tensors is a highly nonlinear process involving finding absolute minima for orthogonal projections under all three-dimensional rotations. Given the standard deviations of the components of a generally anisotropic tensor, we examine the influence of measurement errors on the properties of effective tensors. We use a global optimization method to generate thousands of realizations of a generally anisotropic tensor, subject to errors. Using this optimization, we perform a Monte Carlo analysis of distances between that tensor and its counterparts in different symmetry classes, as well as of their orientations and elasticity parameters.

Minimizing blossoms under symmetric linear constraints

In this paper, we show that there exists a close dependence between the control polygon of a polynomial and the minimum of its blossom under symmetric linear constraints. We consider a given minimization problem P, for which a unique solution will be a point γ on the Bezier curve. For the minimization function f, two sufficient conditions exist that ensure the uniqueness of the solution, namely, the concavity of the control polygon of the polynomial and the characteristics of the Polya frequency-control polygon where the minimum coincides with a critical point of the polynomial. The use of the blossoming theory provides us with a useful geometrical interpretation of the minimization problem. In addition, this minimization approach leads us to a new method of discovering inequalities about the elementary symmetric polynomials.