In seismology, we use ray tracing to study subsurface geology and determine the structure of the earth. Rays are solutions of the Hamilton equations. In many cases, rays also can be obtained by following Fermat’s principle of stationary travel time.
According to Fermat's principle, ray travels along a curve that renders the travel time, minimum. So the ray path can be obtained by minimizing the travel time, so that the obtained path satisfies the ray equation. Hamilton equations can be solved by specifying boundary conditions. The most important case of boundary-value ray tracing is the two-point ray tracing.
The shooting and bending methods are two commonly used numerical approaches to solve boundary-value ray-tracing problems. The shooting method is based on solving ray equations as initial-value problems by specifying the takeoff angle. The takeoff angle is varied until the ray passes through the receiver position. The shooting method works well to find rays connecting sources and receivers in simple 2D media. It breaks down in areas where ray equations break down, such as shadow zones and in complicated media where a slight variation in the takeoff angle might result in a significantly different ray, thus causing difficulty in connecting sources to receivers. The bending method addresses intrinsically the problem of connecting sources to receivers. One begins by connecting source to receiver with an initial path. This initial path is bent according to a prescribed method based on minimizing travel time (Fermat’s principle) until the desired ray is obtained.
In this study, we minimized the travel time using simulated annealing to obtain global minima. Simulated annealing is based on the annealing process of solids in physics and is used in mathematics and physics to obtain an optimal solution to problems subjected led to constraints. Using this method, we found rays between fixed sources and receivers that render travel time globally minimal. By small change in the procedure, our algorithm can be modified to calculate rays of locally minimum travel time, such as reflected rays, by constraining the ray to pass through a set of points that are on the layers of the boundary. We formulate the concept of rays, which emerges from the Hamilton equations. Then, we show that these rays are solutions of the variation problem stated by Fermat’s principle.
The proposed method is applied to the three velocity models. In all of these cases, we found that the path renders global minimum travel time. We have also tested the method to determine the path for a reflected ray from a known reflector. The results are compared to that of a ray tracing method called "fast ray tracing algorithm.” For the first two models, the results were similar but for the third model, we had no response from fast ray tracing algorithm while we determined the correct ray path by the proposed method, which surpassed our expectations.
Our method overcomes two common shortcomings of other bending methods. First, solutions calculated using our algorithm is independent of the initial path. Second, our algorithm does not require using smoothly varying velocity models. We demonstrated the efficiency of our method by applying in to three different velocity models. The method is also generalized to a three-point problem, which is applicable for calculating rays constrained to pass through multiple interfaces.