Anti-crosstalk

Crosstalk in a more general context

We seemed to escape nonlinearity in the lake depth sounding example above, but that was a lucky accident. Since the data there was mostly explained by geography with a small perturbation by rain and drain, the crosstalk while fundamentally nonlinear was practically linear. More generally anti-crosstalk strategies seem little (if ever!) developed because they lead us directly into nonlinear regression. Let us work through the general case, the nonlinear theory.

Consider data $\mathbf d$ a shot gather or CMP gather. We might choose to model it as reflections (hyperbolas) $\mathbf d_1$ plus linear events $\mathbf d_2$ (noises or head waves). We might thus set up the regression

$$\mathbf 0 \approx \mathbf d_1 + \mathbf d_2-\mathbf d$$ (4)

$$\mathbf 0 \approx \mathbf F_1 \mathbf m_1 + \mathbf F_2 \mathbf m_2 -\mathbf d$$ (5)

Of course we need some damping regularization on $\mathbf m_1$ and $\mathbf m_2$ which for simplicity of exposition I will take to be $\mathbf 0 \approx \mathbf m_1$ and $\mathbf 0 \approx \mathbf m_2$. Is that all there is to this problem? Not necessarily. We'll be annoyed if we discover a lot of cross talk between $\mathbf d_1$ and $\mathbf d_2$. Two different physical mechanisms are supposed to have created our data. We'll be annoyed to discover they both make the same contribution or that they make opposite contributions. The regularization should reduce (or prevent) the contributions from coming out opposite. If they are opposites, it could represent our lack of analytic skills in formulating the regularization, or it could represent our need for the anti-crosstalk methodology being proposed here.

We'd like that the modeled data parts $\mathbf d_1$ and $\mathbf d_2$ do not "look like" each other. It's not enough that the dot product $\mathbf d_1 \cdot \mathbf d_2$ vanish. That dot product should be small under all shifted (say triangular) weighting windows. Since $\mathbf d_1$ is a linear function of the model $\mathbf m_1$ and likewise for $\mathbf d_2$, the orthogonality we seek involves the product of $\mathbf m_1$ with $\mathbf m_2$ so our goals are a non-linear function of our unknowns. Never fear. We have done non-linear problems before. They don't turn out badly when we are able to define a good starting location (which we do by solving the linearized non-linear problem first).

Anti-crosstalk