Lúcio T. Santos

2.5-D true-amplitude Kirchhoff migration to zero offset in laterally inhomogeneous media

The proposed new Kirchhoff-type true-amplitude migration to zero offset (MZO) for 2.5-D common-offset reflections in 2-D laterally inhomogeneous layered isotropic earth models does not depend on the reflector curvature. It provides a transformation of a common-offset seismic section to a simulated zero-offset section in which both the kinematic and main dynamic effects are accounted for correctly. The process transforms primary common-offset reflections from arbitrary curved interfaces into their corresponding zero-offset reflections automatically replacing the geometrical-spreading factor. In analogy to a weighted Kirchhoff migration scheme, the stacking curve and weight function can be computed by dynamic ray tracing in the macro-velocity model that is supposed to be available. In addition, we show that an MZO stretches the seismic source pulse by the cosine of the reflection angle of the original offset reflections. The proposed approach quantitatively extends the previous MZO or dip moveout (DMO) schemes to the 2.5-D situation.

Restricted optimization: a clue to a fast and accurate implementation of the Common Reflection Surface Stack method

For a fixed, central ray in an isotropic elastic or acoustic media, traveltime moveouts of rays in its vicinity can be described in terms of a certain number of parameters that refer to the central ray only. The determination of these parameters out of multi-coverage data leads to very powerful algorithms that can be used for several imaging and inversion processes. Assuming two-dimensional propagation, the traveltime expressions depend on three parameters directly related to the geometry of the unknown model in the vicinity of the central ray. We present a new method to extract these parameters out of coherency analysis applied directly to the data. It uses (a) fast one-parameter searches on different sections extracted from the multi-coverage data to derive initial values of the sections parameters, and (b) the application of a recently introduced Spectral Projected Gradient (SPG) optimization algorithm for the final parameter estimation. Application of the method on a synthetic example shows an excellent performance of the algorithm both in accuracy and efficiency. The results obtained so far indicate that the algorithm may be a feasible option to solve the corresponding, harder, full three-dimensional problem, in which eight parameters, instead of three, are required.

Impedance‐type approximations of the P–P elastic reflection coefficient: Modeling and AVO inversion

The normal-incidence elastic compressional reflection coefficient admits an exact, simple expression in terms of the acoustic impedance, namely the product of the P-wave velocity and density, at both sides of an interface. With slight modifications a similar expression can, also exactly, express the oblique-incidence acoustic reflection coefficient. A severe limitation on the use of these two reflection coefficients in analyzing seismic reflection data is that they provide no information on shear-wave velocities that refer to the interface. We address the natural question of whether a suitable impedance concept can be introduced for which arbitrary P–P reflection coefficients can be expressed in a form analogous to their acoustic counterparts. Although no closed-form exact solution exists, our analysis provides a general framework for which, under suitable restrictions of the medium parameters, possible impedance functions can be derived. In particular, the well-established concept of elastic impedance and the recently introduced concept of reflection impedance can be better understood. Concerning these two impedances, we examine their potential for modeling and for estimating the AVO indicators of intercept and gradient. For typical synthetic examples, we show that the reflection impedance formulation provides consistently better results than those obtained using the elastic impedance.

Seismic modeling by demigration

Kirchhoff-type, isochron-stack demigration is the natural asymptotic inverse to classical Kirchhoff or diffraction-stack migration. Both stacking operations can be performed in true amplitude by an appropriate selection of weight functions. Isochron-stack demigration is closely related to seismic modeling with the Kirchhoff integral. The principal objective of this paper is to show how demigration can be used to compute synthetic seismograms. The idea is to attach to each reflector in the model an appropriately stretched (i.e., frequency-shifted) spatial wavelet. Its amplitude is proportional to the reflection coefficient, transforming the original reflector model into an artificially constructed true-amplitude, depth-migrated section. The seismic modeling is then realized by a true-amplitude demigration operation applied to this artificially constructed migrated section. A simple but typical synthetic data example indicates that modeling by demigration yields results superior to conventional zero-order ray theory or classical Kirchhoff modeling.

MULTIPARAMETRIC TRAVELTIME INVERSION

In conventional seismic processing, the classical algorithm of Hubral and Krey is routinely applied to extract an initial macrovelocity model that consists of a stack of homogeneous layers bounded by curved interfaces. Input for the algorithm are identified primary reflections together with normal moveout (NMO) velocities, as derived from a previous velocity analysis conducted on common midpoint (CMP) data. This work presents a modified version of the Hubral and Krey algorithm that is designed to extend the original version in two ways, namely (a) it makes an advantageous use of previously obtained common-reflection-surface (CRS) attributes as its input and (b) it also allows for gradient layer velocities in depth. A new strategy to recover interfaces as optimized cubic splines is also proposed. Some synthetic examples are provided to illustrate and explain the implementation of the method.

The probability density function to the random linear transport equation

We present a formula to calculate the probability density function of the solution of the random linear transport equation in terms of the density functions of the velocity and the initial condition. We also present an expression for the joint probability density function of the solution in two different points. Our results have shown good agreement with Monte Carlo simulations.

Multifocus moveout revisited: derivations and alternative expressions

The multifocus moveout of Gelchinsky et al. [Gelchinsky, B., Berkovitch, A., Keydar, S., 1997. Multifocusing homeomorphic imaging: Parts I and II: Course Notes, Special Course on Homeomorphic Imaging. Seeheim, Germany] is a powerful tool for stacking multicoverage data in arbitrary configurations. Based on general ray theoretical assumptions and on attractively simple geometrical considerations, the multifocus moveout is designed to express the traveltimes of neighbouring rays arbitrarily located around a fixed central, primary reflected or even diffracted, ray. In this work, the basic derivations and results concerning the multifocus approach are reviewed. A higher-order multifocus moveout expression that generalizes the corresponding one of Gelchinsky is obtained from slight modifications of the original derivation. An alternative form of the obtained multifocus expression that is best suited for numerical implementation is also provided. By means of a simple numerical experiment, we also comment on the accuracy of the multifocus traveltime approximations.

Automatic smoothing by optimal splines

We propose a method that is capable to filter out noise as well as suppress outliers of sampled real functions under fairly general conditions. From an a priori selection of the number of knots that define the adjusting spline, but not their location in that curve, the method automatically determines the adjusting cubic spline in a least-squares optimal sense. The method is fast and easily allows for selection of various possible number of knots, adding a desirable flexibility to the procedure. As an illustration, we apply the method to some typical situations found in geophysical problems.

Evaluation of subsurface contact stresses in railroad wheels using an elastic half-space model

Railroad wheels fail in two main modes: rolling surface defects like spalling, shelling and wear, and internal defects including cracks propagating after a change takes place in the original stress pattern. Although the effects of the latter are almost always catastrophic, the former is more usual. The onset of rolling surface defects depends on the strength of the surface and the applied loads. The strength is related to surface hardness and can be determined through experimental evaluation under controlled conditions. Evaluating the loads is one of the challenges for contact researchers. This paper presents the evaluation of the stress field inside elastic rolling bodies with an elliptic area of contact. This kind of model can be applied to wheel-rail contact phenomena. Typical high freight transportation loads are used in conjunction with regular recommended wheel and rail sizes. The results have shown that shear stress reaches the maximum magnitude below the surface of contact, and this explains the presence of shelling defects in service. They have also shown that a new model including plasticity is required, because the range of the stresses reached surpasses, by far, the elastic limit

New theoretical results on recursive quadratic programming algorithms

Recursive quadratic programming is a family of techniques developed by Bartholomew-Biggs and other authors for solving nonlinear programming problems. The first-order optimality conditions for a local minimizer of the augmented Lagrangian are transformed into a nonlinear system where both primal and dual variables appear explicitly. The inner iteration of the algorithm is a Newton-like procedure that updates simultaneously primal variables and Lagrange multipliers. In this way, as observed by Gould, the implementation of the Newton method becomes stable, in spite of the possibility of having large penalization parameters. In this paper, the inner iteration is analyzed from a different point of view. Namely, the size of the convergence region and the speed of convergence of the inner process are considered and it is shown that, in some sense, both are independent of the penalization parameter when an adequate version of the Newton method is used. In other words, classical Newton-like iterations are improved, not only in relation to stability of the linear algebra involved, but also with regard to the ovearll convergence of the nonlinear process. Some numerical experiments suggset that, in fact, practical efficiency of the methods is related to these theoretical results.