SIMPLE TEST

To demonstrate how the tau back projection operator is less affected by our initial slowness model, we constructed a simple 1-D synthetic. The model, Figure 2, is composed of two 2.3 km/s zones in a constant 2 km/s background. For this test we assumed that our slowness model had correctly resolved the bottom anomaly in vertical travel time. Our choice of vertical travel time is quite important, as when doing velocity estimation, we must always preserve zero-offset travel time. In this simple 1-D synthetic, that means that the vertical travel-time to the layer boundaries and to the reflector must be kept constant. Therefore, in depth, we will misplace the location of of the bottom high velocity zone but preserve the correct vertical travel times to the layer top and bottom. After constructing the model we found the ray that hit the reflector at 2 km depth, 2 km away from source in both ($\tau,x$) and (z,x) space (Figure 3.) Following the method outlined in Clapp and Biondi 1998, we built the tomography operator for both tau (${\bf T^{'}_{0,\tau}}$) and depth (${\bf T^{'}_{0,z}}$), Figure 4.

 

cor Figure 2 Synthetic 1-D velocity function in $\tau$.

 

vel0
Figure 3 Initial guess at the velocity function overlaid by ray hitting reflector at 4 km with a half-offset of 2 km. Left panel is in depth, right panel is in tau.

For comparison, we ray traced through the `correct' velocity model in both spaces (Figure 5) and calculated the corresponding operators. By comparing the correct and initial operator for tau and depth, or by looking at the difference between the two operators (Figure 7), we can clearly see that our initial guess for our tau operator is overall better than our initial guess for our depth operator. In the upper layer, we see marginally more change in the tau operator but at the lower reflector boundaries (which move in the case of depth but remain constant in tau) we see significantly more error in depth. In addition, the change in reflector position has caused a spike in the difference panel for the depth case. Finally, the change in the tau operator is smooth, while the change in the depth operator shows dramatic jumps. Our successive relinearizing have an underlying assumption that we are smoothly converging to the correct operator. In tau space, this assumption seems to be more valid. With a more complicated model our positioning of layer boundaries, will be subject to more change, making the tau compared to depth difference even more dramatic.

 

operator0
Figure 4 The operator calculated from our initial guess at velocity and the resulting ray paths in depth (left) and tau (right).

 

vel1
Figure 5 ``Correct'' velocity function overlaid by ray hitting reflector at 4 km with a half-offset of 2 km. Left Panel is in depth, right panel is in tau.

 

operator1
Figure 6 The operator calculated from the ``correct'' velocity and the resulting ray paths in depth (left) and tau (right).

 

diff
Figure 7 The difference between the operators calculated from the correct and our initial guess at velocity, for depth (left) and tau (right. Note the significant spikes at the reflector and at the lower layer boundary in the depth case.