Anti-crosstalk

A regression to minimize crosstalk

Observing the geographically modeled data $\mathbf G\mathbf h$ correlating with the data drift $\mathbf u=\mathbf L\mathbf n$ we wish to articulate a regression that says they should not correlate. Since the drift $\mathbf u$ is a small correction to the data $\mathbf d$, in other words $\mathbf G\mathbf h\approx\mathbf d$, we can simplify the goal by asking that the dot product of $\mathbf d$ with $\mathbf u$ should vanish, vanish not necessarily over the entire data set; but that it should vanish under many triangular weighed windows.

Let us define $\mathbf D$ as a diagonal matrix with $\mathbf d$ on the diagonal. This may be a little unfamiliar. Often we see positive weighting functions on the diagonal. Here we see data (possibly with both polarities) on the diagonal. Additionally, let us define a matrix $\mathbf T$ of convolution with a triangle. Columns of $\mathbf T$ contain shifted triangle functions, likewise do rows. Take $\mathbf t'$ to be any row of $\mathbf T$. Then $\mathbf t'\mathbf D$ is a row vector of triangle weighted data. We want the regression $\mathbf 0 \approx \mathbf t'\mathbf D\mathbf u$ for all shifts of the triangle function. The way to express this is:

$$\mathbf 0 \approx (\mathbf T\mathbf D)\mathbf u$$ (2)

$$\mathbf 0 \approx (\mathbf T\mathbf D)\mathbf L \mathbf n$$ (3)

Hooray! Now we know what coding to do! But first, to better understand the regression (2) imagine instead that $\mathbf T$ is a square matrix of all ones, say $\mathbf 1$. That would be like super wide triangular windows. Then every component of the vector $\mathbf 1 \mathbf D \mathbf u$ contains the same dot product $\mathbf d \cdot \mathbf u$. Using $\mathbf T$ instead of $\mathbf 1$ gives us those dot products under a triangle weight, each final vector component having a shifted triangle.

What is a good name for $\mathbf T\mathbf D$? It measures the similarity of $\mathbf d$ and $\mathbf u$. It might be called the ``data similarity'' operator. What is a good name for its adjoint $\mathbf D\mathbf T$? Assuming whatever comes out of $\mathbf T$ is a smooth positive function, then $\mathbf D\mathbf T$ is a data gaining operator (its input being a gain function). Do we have any geophysical problems where the unknown is the gain?

Anti-crosstalk