A Stochastic Numerical Method for Estimating the Thickness of Subsurface Strata
Authors: Joseph G.J. , Levine A. , Liukkonen J.R. 
Published: October 2008

ABSTRACT
A new method for determining the seismic velocity and the height of subsurface layers directly from CDP gathers is presented here. This method is used sequentially from shallow layers downward. In each layer we establish a criterion function G(V, h) of the velocity V and the height h which has the property that G(V, h) = 0 at the true values of V and h associated with the layer and use a numerical process to estimate these values of V and h. The characteristics of the criterion function as determined by the physical nature of the problem are such that deterministic algorithms do not yield acceptable results. We overcome this difficulty by introducing noise into the deterministic algorithm, The stochastic algorithm thus created is shown to converge to the true solution even when the deterministic algorithm fails. This method is free from the restrictive conditions required by the current procedure that is based on the work of Dix. Furthermore, this new method does not require the very expensive data processing needed to "present the data" nor does it use an interpretive velocity analysis stage.
Offshore exploration relies upon subsurface maps produced from information obtained through the seismic method. A significant portion of the cost associated with this mapping effort may be avoided by using this direct method of estimating the required parameters.
We use a two layer model and synthetic data to demonstrate this method. The estimates obtained are accurate to within 0.570 in this case. Details of the steps taken to obtain estimates of V and h from the data are included herein.
This stochastic method can be adapted to online operation. The methodology thereby has important implications for offshore exploration in that it can significantly reduce both costs and time of mapping and search procedures.

INTRODUCTION
Mapping subsurface strata using the reflection seismic method is the basis of much of the modern exploration for reservoir of hydrocarbons and for underground reservoirs of water (Geisler (1989)). It is through these mappings that the structure of subsurface layers at various depths are determined. The accuracy of these mappings depends upon the accuracy of estimates of the local velocity propagation of seismic waves through the subsurface strata. Consequently, "velocity analysis" is a significant part of the seismic data processing effort.
The method of velocity analysis that is now widely used is based on the works of Dix (1955) who developed an approximation for the relationship between the travel time from the shot to a receiver and the distance between the shot and the receiver. The goal of the velocity analysis is to identify interfaces (seismic horizons) between strata and to determine the average velocity of propagation of the seismic wave, as defined by Dix, to these horizons. Interval velocities are then calculated from this information (see Hubrel and Krey (1980)). The procedure for velocity analysis requires an interpretive step which is based on an assumed velocity model.