Blocky models via the L1/L2 hybrid norm

UNKNOWN SHOT WAVEFORM

A one-dimensional seismogram $d(t)$ is unknown reflectivity $c(t)$ convolved with unknown source waveform $s(t)$. The number of data points ND$\approx$NC is less than the number of unknowns NC+NS. Clearly we need a "smart" regularization. Let us see how this problem can be set up so reflectivity $c(t)$ comes out with sparse spikes so the integral of $c(t)$ is blocky.

This is a nonlinear problem because the convolution of the unknowns is made of their product. Nonlinear problems elicit well-warranted fear of multiple solutions leading to us getting stuck in the wrong one. The key to avoiding this pitfall is starting ``close enough'' to the correct solution. The way to get close enough (besides luck and a good starting guess) is to define a linear problem that takes us to the neighborhood where a nonlinear solver can be trusted. We will do that first.

Blocky models via the L1/L2 hybrid norm