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## An insight into the Engquist-Osher scheme applied to Van Trier's algorithm

The practical implementation of the above formulation is centered around the problem of advancing the computational front. The Engquist-Osher scheme provides a way to compute by imposing a time minimization condition along three points of the finite-difference stencil. The scheme calculates to approximate the partial derivative (with respect to ), of the function , by using the values of in points of minimum traveltime; varies only in along the constant radius computational front. Given three consecutive points on the computational front with constant radius, the values of the functions and vary only in the variable .Each function will have the values uj-1, uj, uj+1 and vj-1, vj and vj+1 respectively, in the three points of the stencil. From equation (6) we can write as a function of :

The Engquist-Osher scheme computes as
 (7)
where , at the point where . For this case the value of from equation (6). Because the values of uj-1, uj and uj+1 are compared against zero (), the scheme needs only the sign of the function u in the three points of the stencil. Equation (7) allows for eight cases as a function of the positive or negative values of uj-1, uj and uj+1 in the three points of the stencil. The eight cases are shown in Figure 1.

The calculation of is done for locations where the value of the traveltime function , is minimum. From Figure 1 one can see that only cases 1 and 5 will actually give a first order correct value. In both cases the values for the function v are chosen from the three points of the stencil where the value of the time is minimum. While for cases 1 and 5 the accuracy of the scheme is unquestionable, for the rest of the cases some approximations are introduced. The other six cases also calculate the values of using the points where the value of the time is minimum. However, the value of is divided by a constant ,even though the function is estimated over a different interval .A potentially more accurate algorithm would calculate the exact value of for each intermediate case. The algorithm can be designed to calculate the locations of the minimum travel time in the three point stencil interval and the span over the axis, necessary to divide the value .

Next: 3-D ALGORITHM Up: THE BASIC FINITE-DIFFERENCE SCHEME Previous: THE BASIC FINITE-DIFFERENCE SCHEME
Stanford Exploration Project
12/18/1997