SOME EXAMPLES

Figure  shows the modeling results for a v(z) medium with a strong velocity gradient (10 ms^-1/m). In this medium the wavefronts overturn. The rectangular coordinate modeling code does not model these waves because the wavefield is always extrapolated downwards and any energy in the evanescent zone (with a dip greater than $90^\circ$ ) is removed at each downwards extrapolation step. However it is surprising to see energy in the rectangular coordinate frame that is apparently upgoing. This is a byproduct of the high velocity gradient, the energy has a phase slowness that points horizontally but a group slowness that points upwards. There is less energy in the ``overturned'' region that there should be, because the true overturned energy has not been modeled. The two polar coordinate frames show the correct amount of overturned energy. In this particular example the $45^\circ$ result may be better that the PSPI result in polar coordinates. This is because the very strong gradient is not well handled by the PSPI algorithm. The gradient affects the polar coordinate PSPI code because there is a strong gradient in $\theta$,even though there in no gradient in x.

Figure  shows another result with upgoing energy. In this case the energy may not be desirable. The model is a two layer medium with a step function in velocity at 300m. The rectangular coordinate frame phase shift result shows only downgoing waves. In contrast the two polar coordinate results display a small piece of reflected energy. This is energy that is upgoing, but still outgoing! Once the angle of incidence exceeds $45^\circ$ to the vertical the reflected energy is propagated outwards in the polar coordinate frame. There is also some dispersed energy in the $45^\circ$ result. This is probably due to reflected energy that travels at more than $45^\circ$to the polar grid and is thus poorly modeled.

Figure  shows the result of modeling in a medium with a small low velocity anomaly. The anomaly is a Gaussian blob with a half width of 100m. Apart from the loss of high dips in the rectangular coordinate frame, all three methods produce similar results. The main difference is that the $45^\circ$ $\omega-x$ method was much cheaper to produce.