The traditional formulation of geophysical inverse problems requires a noise covariance matrix and a model covariance matrix. In my experience these are hard to define. I have recently made considerable progress defining the model covariance matrix from a model or a model estimate. (See my companion paper: Information from smiles, monoplane-annihilator weighted regression.)

For starters, I reorganize inverse theory to define the image as the migration operator applied to the complete data set and I take the unknowns to be the missing data. This formulation enables me to work with data and model covariances instead of model and noise covariances. If it shows promise we can always come back and look for the noise covariance.

Start with the definitions:

k |
known data |

u |
unknown values in data space |

d=k+u |
the complete data space. |

d=_{0}k+g |
raw data with gaps filled by gapfill program |

K |
Kirchhoff migration operator (for example) |

M |
matrix diagonal with 1 where data missing, 0 otherwise |

m=Kd |
model or image, migration of the complete data space. |

A_{d} |
array of 2-D prediction-error filters in data space |

A_{m} |
array of 2-D prediction-error filters in model space |

gain control function in model space | |

r_{d}=A_{d} d |
data residual |

model residual | |

perturbation of the unknown data |

A regression I might try for the unknown *u* is

(1) | ||

(2) |

11/18/1997