A simple example of an adjoint with rough output is normal moveout. Performing normal moveout on a trace means looping over values of vertical travel time and solving for the appropriate travel time t. The trace, collected at some offset, is stretched upward to simulate a zero offset trace. Each output bin collects one value from the input space. Performing the adjoint means pushing each input value into an output bin. Multiple inputs will end up in some output bins. The roughness with which the output space samples the input space is dictated by the velocity.
Here is a simple inversion, following an example from TDF Claerbout (1994). Begin with a single seismic trace, which we label the model, m. We also have a solution trace, labeled x, which we wish to construct from a synthetic gather d = Mm, where M is a normal moveout 0 operator, and M' its adjoint stacking operator. Thus our regression is:
We have some freedom in the way we define M and M'. M is our 0 operator, but we can choose it to sample the input trace or to sample the output gather, and the same is true of the stacking operator M'. In order to be precisely adjoint to each other, both M and M' must sample the same space. Further, the mathematics of normal moveout recommend always sampling the trace, because of the explicit relationship between vertical travel-time and velocity. So initially we choose our NMO operator to model by pushing values from an input trace into the output gather, and to stack by pulling values into the trace. This gives an exact adjoint.
I use this operator pair to solve the above regression and get results as seen in Figure . We see that the original waveforms are reconstructed almost exactly after just four or five iterations.
Figure 1 Iterations of the inversion . Below is shown the original trace. Note that the original is duplicated almost exactly after just a few iterations.
Now we redefine the operator. M' stays unchanged, but M is replaced. Maintaining similar naming conventions, we can define a stacking operator S which samples an input gather and pushes values into a stacked trace output. S' then must model by pulling values into the gather. We replace M by S' and obtain the regression:
Using this new operator pair in the inversion yields the results seen in Figure . Amplitudes for most events are reconstructed more quickly, after a single iteration.
Figure 2 Iterations of the inversion . Below is shown the original trace. Note that amplitudes on most events are recovered more quickly than with the exact adjoint.
To quantify the results, Figure displays the misfit of the model at each iteration with the original trace: .This misfit is the model residual, normally an unknown quantity, but a useful measure of success in synthetic problems where the exact answer is known. The inversion which uses exclusively pull operators converges more quickly than that which uses exact adjoints. For this simple example, instant convergence in the model residual would mean unitarity; accelerated convergence suggests that the pull adjoint gives something closer to a unitary operator than the exact adjoint.
Figure 3 Residuals for the two inversions as a function of iteration. Dashed line denotes inversion with the exact adjoint, continuous line with the pull adjoint.
To satisfy curiosity, I measured data residuals (that quantity minimized by the inversion) after several iterations at various slownesses. The results are shown in Figure . The smooth curve corresponds to the pull adjoint and the rough one to the exact adjoint. At all velocities, there is a noticeable advantage to using pull adjoint NMO. It is surprising that there is so little velocity dependence, because it seems the aliasing at high dips (or low velocities) should be a problem for push operators more than for pull operators. Further, the magnitude by which the operator pair which includes the pull adjoint fails the dot product test is not systematically dependent on velocity.
Figure 4 Comparison of residual after one iteration for the pull and exact adjoint. The smooth curve denotes the pull adjoint. Click to see movie of residuals at higher numbers of iterations.
Figure 5 Error in the dot product test is independent of velocity.