Image gather reconstruction using StOMP |

(3) |

where and are the result of taking the norm of the first and second equations. The success of this approach relies on the accuracy of 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 can be transformed into through the linear operator 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 through,

where we are now estimating in the sense. We can then apply 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 |

2012-05-10