Equivalence of modeling equations

First I show that equation (5) and (4) are numerically equivalent and produce identical results, as predicted analytically. Figure illustrates a wave field propagating in the model that shows a step discontinuity but is otherwise homogeneous. The elastic wave propagates across a discontinuity in v_p and v_s. Figure shows a comparison of two modeling responses. One result was calculated using equation (5) and one result was obtained using equation (4). For comparison I extract a single trace from the recorded wave fields, as shown in Figure . There is no noticeable difference between the two curves for each of the two components. For both components the resulting responses overlie each other perfectly and indicate that both modeling equations are also numerically equivalent.

 

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Figure 1 The x component of an elastic wave field in a homogeneous medium with a step discontinuity has propagated ca. 240 timesteps.

 

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Figure 2 Trace 90 is taken from two wave fields. The results from the two different modeling equations are overlain and confirm that they are not only analytically but also numerically equivalent. The modeling was carried out on a staggered grid. The source is located on top of the step and the receivers are located below.