Why take the derivative along the track?

In using a first derivative along the track, we assume that the tracks are shifted relative to each other by a constant value. If the satellite was traveling slow enough then tidal changes could be causing continuously changing shifts along the length of individual tracks. If this were the case, then a higher order derivative would be needed instead of the first derivative.

Observation of the convergence of the data-space residual revealed that the residual does shrink everywhere, which means that the model at least begins to match the data. However after enough iterations when the residual converges to a constant value, there still remain clumps of residual which move around in data-space with each additional iteration. Several possibilities could account for this. It could be the result of inaccuracies in the acquisition of the data or it could mean that the first derivative along the track is not the best function to use.

If the shifts between tracks were ramps rather than plateaus, then using the first derivative as described in equation (1) would create a model whose data-space derivative along track could not closely match that of the observed data. The difference between the derivative of the sampled model and the derivative along the observed data would not be shifted plateaus.

Jon Claerbout suggested calculating the 1D prediction-error filter on the rediduals to reveal the nature of the differences. If the differences were shifted plateaus, the PEF would be a first derivative. If the differences were shifted ramps, the PEF would be a second derivative. If the differences were curves, the PEF would be a higher order derivative, and so on.

To test this, I applied the above data fitting goal to get a model, m. I then calculated 1D prediction-error filters of different lengths along ${\bf r}$=${\bf W}$[${\bf L m}$ -${\bf d}$]. The results were:

PEF of length 2: 1.000 -.997

PEF of length 4: 1.000 -.997 -.001 .003

PEF of length 10: 1.000 -.998 .0001 .373 -.372 .005 .122 -.123 -.003 -.002

These PEFs are, to high accuracy, first derivatives, proving that the first derivative along the track is the desired function to use in the fitting goal (1).