To reduce the time needed for analysis and printing, I reduced the data size two different ways, by decimation and by local averaging, as shown in Figure . The decimation was to about 1 part in 9 on each axis, and the local averaging was done in windows giving the same spatial resolution in each case. The local averaging was done independently in the plane of the real part and the plane of the imaginary part. Putting them back together again showed that the phase angle of the averaged data behaves much more consistently. This adds evidence that the data is not troubled by noise bursts.
From Figures and we see that contours of constant phase appear to be contours of constant altitude; this conclusion leads us to suppose that a study of radar theory would lead us to a relation like Z=e^{ih} where h is altitude (in units unknown to us nonspecialists). Because the flat land away from the mountain is all at the same phase (as is the altitude), the distance as revealed by the phase does not represent the distance from the ground to the satellite viewer. We are accustomed to measuring altitude along a vertical line to a datum, but here the distance seems to be measured from the ground along a angle from the vertical to a datum at the satellite height.
Phase is a troublesome measurement because we generally see it modulo .Marching up the mountain we see the phase getting lighter and lighter until it suddenly jumps to black which then continues to lighten as we continue up the mountain to the next jump. Let us undertake to compute the phase including all of its jumps of .Begin with a complex number Z representing the complex-valued image at any location in the (x,y)-plane.
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We will be handling the differential equation as a difference equation using an exact representation on the data mesh. By working with the phase difference of neighboring data values, we do not have to worry about phases greater than (except where phase jumps that much between mesh points). Thus we solve (73) with finite differences instead of differentials. Module igrad2 is a linear operator for the difference representation of the operator representing the gradient of a potential field. Its adjoint is known as the divergence of a vector field. igrad2gradient in 2-D To do the least-squares fitting (73) we pass the igrad2 module to the Krylov subspace solver. (Other people might prepare a matrix and give it to Matlab.)
The difference equation representation of the fitting goal (73) is:
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