Discontinuity in the solution

The viewing angle (23 degrees off vertical) in Figure 2 might be such that the mountain blocks some of the landscape behind it. This leads to the interesting possibility that the phase function must have a discontinuity where our viewing angle jumps over the hidden terrain. It will be interesting to discover whether we can estimate functions with such discontinuities. I am not certain that the Vesuvius data really has such a shadow zone, so I prepared the synthetic data in Figure 6, which is noise free and definitely has one.

We notice the polarity of the synthetic data in 6 is opposite that of the Vesuvius data. This means that the radar altitude of Vesuvius is not measured from sea level but from the satellite level.

A reason I particularly like this Vesuvius exercise is that slight variations on the theme occur in various other fields. For example, in 3-D seismology we can take the cross-correlation of each seismogram with its neighbor and pick the time lag of the maximum correlation. Such time shifts from trace to trace can be organized as we have organized the $\Delta\phi$ values of Vesuvius. The discontinuity in phase along the skyline of our Vesuvius view is like the faults we find in the earth.

EXERCISES:

1. In differential equations, boundary conditions are often (1) a specified function value or (2) a specified derivative. These are associated with (1) transient convolution or (2) internal convolution. Gradient operator igrad2 is based on internal convolution with the filter (1,-1). Revise igrad2 to make a module called tgrad2 which has transient boundaries. Hint: Compare with icai2 and tcai2 .

 

synmod90
Figure 6 Synthetic mountain with hidden backside. For your estimation enjoyment.