Abandoned strategy for attenuating tracks

An earlier strategy to remove the ship tracks is to filter the residual as follows:  

$$\begin{array} {lllll} \bold 0 &\approx& \bold r_d &=& \bold... ... \bold 0 &\approx& \bold r_p &=& \epsilon \bold p, \end{array}$$ (25)

where $\frac{d}{ds}$ is the derivative along the track. The derivative removes the drift from the field data (and the modeled data). An unfortunate consequence of the track derivative is that it creates more glitches and spiky noise at the track ends and at the bad data points. Several students struggled with this idea without good results.

One explanation (of unknown validity) given for the poor results is that perhaps the numerical conditioning of the algebraic problem is worsened by the operators $\bold W$, $\frac{d}{ds}$, $\bold B$, and $\bold {H^{-1}}$,drastically slowing the convergence. Another explanation is that the operator $d\over ds$ is too simple. Perhaps we should have a five or ten point low-cut filter--or maybe a PEF. A PEF could be estimated from the residual itself. Unfortunately, such a longer filter would smear the bad effect of noise glitches onto more residuals, effectively spoiling more measurements.

We concluded that the data is bad only in a very low frequency sense. Perhaps the lake is evaporating, or it is raining, or the load in the boat has been changed or shifted. It's a fact that any very low-frequency reject filter is necessarily a long filter, and that means that it must catch many noise spikes. Thus we should not attempt to filter out the drift from the residual. Instead we should model the drift.

In the presence of both noise bursts and noise with a sensible spectrum (systematic noise), the systematic noise should be modeled while the noise bursts should be handled with $\ell^1$.