Anti-crosstalk

Lake example

A depth sounding survey was made of a lake. A boat with a depth sounder sailed a gridwork of passes on the lake. Upon analysis the final image contained obvious evidence of the survey grid. Oops! We should always hide our data acquisition footprint. The footprint is not the geography or geology we wish to show. How did this happen? We guessed the water level changed during the survey. Perhaps it rained or perhaps the water was used for agriculture. Perhaps the wind caused the lake to ``pendulum" (seiche). Perhaps the operator sat in the front of the boat, or sometimes its back, or ran it at various speeds giving the depth measurement a different bias. Our data measured the difference between the top and the bottom of the lake; yet we had no idea how to model the top. Eventually we modeled the top as an ``arbitrary low frequency function'' in data space (a one dimensional function following the boat). This got rid of the tracks in the model space (map space) but led to much embarrassment. It was embarrassing to discover that the geography (as seen in data space) was correlated with the lake surface (rain and drain).

Let us express these ideas mathematically: $\mathbf d$ is the data, depth along the survey coordinate $d(s)$. Here $s$ is a parameter like time. It increases steadily whether the boat is sailing north-south or east-west or turning inbetween. The model space is the depth $h(x,y)$. There will be a regularization on the depth, perhaps $0\approx \nabla h(x,y)$.

For the top of the lake with the ship we need some slowly variable function of location $s$. It's embarrassing for us to need to specify it because we have no good model for it. So, we specify a slowly variable function $\mathbf u = u(s)$ by asking a random noise function $\mathbf n=n(s)$ to run thru a low frequency filter, say $\mathbf L$. We are not comfortable also about needing to choose $\mathbf L$. We call the function $\mathbf u=\mathbf L\mathbf n$ the rain and drain function. We take the regularization for the unknown $\mathbf n$ to be $\mathbf 0\approx\mathbf n$. (Least squares will tend to drive components of $\mathbf n$ to similar values, and under some conditions likewise the spectrum of $\mathbf n$ will tend to white, so we expect (and often find) the spectrum of $\mathbf u$ comes out that of $\mathbf L$.)

The operator we do understand very clearly is the geography operator $\mathbf G$. Given we wish to make a theoretical data point (water depth), the geography operator $\mathbf G$ tells us where to go on the map to get it. Of course each of the two regularizations $0\approx \nabla h(x,y)$ and $\mathbf 0\approx\mathbf n$ has its own epsilon which is annoying because we need to specify those too. With all these definitions our unknowns are the geography $\mathbf h$ and the noise $\mathbf n$ that builds us a drift function. Our data fitting goal says the data should be the separation of the top and bottom of the lake.

$$\mathbf 0 \quad \approx \quad \mathbf G \mathbf h + \mathbf L \mathbf n -\mathbf d$$ (1)

In my free on-line textbook GEE all this seemed rather conventional and rather fine. Our embarrassment came when we compared the geographically modeled part of the data $\mathbf G\mathbf h$ to the drift (rain and drain) modeled part of the data $\mathbf L\mathbf n$. They were visibly correlated. This is crazy! The boat being in deep water should not correlate with rain (or drain). We needed to add an ingredient to the formulation saying $\mathbf u=\mathbf L\mathbf n$ should be orthogonal to $\mathbf G\mathbf h$ (which is practically the same as $\mathbf d$) in some generalized sense. Let us see how this might be done.

Anti-crosstalk