Stochastic numerical optimization with applications to seismic exploration
Description
Deterministic Numerical Optimization methods that are used in Science and Engineering have several short-comings that limit their applicability. If the phenomena that cause a deterministic Numerical Optimization method to fail are known, then it may be possible to modify that method to overcome the specific effects that cause it to fail Our method of Stochastic Numerical Optimization attempts to overcome the limitations of deterministic Numerical Optimization methods, even when the specific cause of the limitation is unknown, by deliberately introducing noise into each step of the deterministic method. We use the Newton-Raphson (N-R) method of root finding as a paradigm for the class of Numerical Optimization methods; we present the details of the development of the Stochastic Newton-Raphson (SN-R) method. We show that the SN-R method converges to a 'best' solution of G(x) = 0, G: $\IR\sp{\rm m}$ $\to$ $\IR\sp{\rm n}$ for a larger class of functions than that for which the N-R method converges The precursor to our SN-R method is the method of Stochastic Approximations developed by Robbins and Munro in 1951. Many similarities and differences between SN-R and Stochastic Approximations are pointed out in the course of the work Finally, we demonstrate the practical application of the SN-R method to a problem in seismic exploration. We use SN-R to determine the thickness and the characteristic seismic velocity of subsurface strata from reflection seismic data