Image gather reconstruction using StOMP

Compressive sensing

Compressive sensing is a statistical technique attributed to Donoho (2006), but whose start could be placed as early as the basic pursuit work of Mallat and Zhang (1993). A compressive sensing problem at its heart is a special case of a missing data problem. In geophysics, we often think of a missing data problem as solving for a model $\bf m$ given some data $\bf d$ which exist in the same vector space. We have a masking operator $\bf R$ (1 where the data is known, 0 elsewhere). We add in some knowledge of the covariance of the model through a regularization operator $\bf A$ . We then estimate the best model from the following system of equations in a $\ell_2$ sense,

$$\bf0 \approx \bf r_d = \ell_2(\bf d - \bf R \bf m)$$

$$\bf0 \approx \bf r_m = \ell_2(\bf A \bf m ),$$ (3)

where $\bf r_d$ and $\bf r_m$ are the result of taking the $\ell_2$ norm of the first and second equations. The success of this approach relies on the accuracy of $\bf A$ to describe the covariance of the model.

Compressive sensing approaches the problem from a different perspective. It starts from the notion that there exists a basis function that $\bf d$ can be transformed into through the linear operator $\bf L^T$ in which very few non-zero elements are needed to represent the signal. The compressive sensing approach is then to set up the missing data problem in two phases. First, estimate the elements of the sparse basis function $\bf m$ through,

$$\bf0 \approx \bf r = \ell_1(\bf d- \bf R \bf L \bf m),$$ (4)

where we are now estimating $\bf m$ in the $\bf\ell_1$ sense. We can then apply $\bf L$ to recover the full model. The magic of compressive sensing is that you only need to collect a small multiple, typically 4-5, more data points than the number of non-zero basis elements. In the case of correlation gather compression this would indicate collecting in the range of 5% of the correlations should be sufficient to recover the entire model, much smaller than what the Nyquist-Shannon (Nyquist, 1928) criteria would suggest.

Image gather reconstruction using StOMP