Mrinal K. Sen

Geophysical applications of global optimization methods

Model based inversion methods as applied to geophysical problems involve finding the global minimum of an objective function (given by a suitably chosen norm) that corresponds to an earth model which best explains the observed geophysical data. It has long been realized that geophysical inverse problems are nonlinear and the objective functions are multimodal. Traditionally such problems have been solved using iterative linear methods which require a good starting solution.

Bayesian inference, Gibbs sampler and uncertainty estimation in nonlinear geophysical inversion

The Bayes or the Tarantola-Vallette formulation of the geophysical inverse problem describes the solution of the inverse problem as the a posteriori probability density (PPD) function in model space. Since the complete description of the PPD is impossible in the highly multidimensional model space of geophysical applications, several measures such as the highest posterior density regions, marginal PPD and several orders of moments are often used to describe the solutions. Calculation of such quantities requires evaluation of multi-dimensional integrale. A faster alternative to enumeration and blind Monte-Carlo integration is importance sampling which may be useful in several applications. Importance sampling can be carried out most efficiently by a Gibbs sampler (Geman and Geman, 1984). We introduce here a new method called the parallel Gibbs sampler (PGS) based on genetic algorithms and show that the results from the two samplers are nearly identical.

Plane‐wave Gaussian‐beam prestack depth migration

Plane-wave migration methods use data transformed to the tau-p domain (Stoffa et al., 1981; Akbar et al., 1996; Xu and Lambare, 1998; Sun and Schuster, 1999; De Hoop and Brandsberg-Dahl, 2000; Albertin et al., 2001); an appropriate extrapolation operator is then designed to continue the wavefield (characterized by its surface ray parameter) downward in depth. Stoffa et al. (2006) considered depth migration methods based on one or more plane wave decompositions of the original seismic data. Based on an ART approximation, they developed a new depth migration method using both source and receiver plane wave decompositions of the seismic data. Their method can be improved using a multi-valued ray tracing method. However, it still only deals with the caustics problem one by one based on Maslov solutions (Foster et al., 2002).

On two approaches to wave equation based multiple attenuation

Free surface multiple elimination remains as one of the most challenging problems in seismic data processing. Recently wave equation based multiple elimination methods have become quite popular. The formulation of the problem is generally based on invariant embedding assuming a ID earth model. In higher dimensions it is derived from inverse quantum scattering, Kirchhoff Helmholtz integral, reciprocity theorem or invariant embedding in laterally varying media. The multiple elimination problem is usually cast as an optimization problem in which a seismic trace with minimum energy (multiple-free) is sought. Essentially there exist two approaches: one approach involves estimation of the subsurface reflectivity (or an earth model) and the other involves estimation of an effective source function in an iterative manner such that we obtain minimum energy seismograms.

3-D Traveltime Calculations

This equation is the basis for both the 2-D and 3-D traveltime code. Figure 1 shows that the update time t is the time of the minimum raypath, t0 for example, added to the time from z0 to the update point at (x,z2). In other words t = t0 + s[(z2 – z0) + Δx2]1/2 To find the minimum raypath location z0, this equation is differentiated w.r.t. z0 giving dt/dz0 = z0w/t0 – s(z2 – z0)[(z2 – z0) + Δx2]-1/2