Preconditioning |

- is a data vector in which components range over the vertical traveltime depth . Its component values contain the scaled RMS velocity squared , where is the index on the time axis.
- is a diagonal matrix along which we lay the given measure of data quality. We use it as a weighting function.
- is the matrix of causal integration, a lower triangular matrix of ones.
- is the matrix of causal differentiation, namely, .
- is a vector containing the interval velocity squared ranging over the vertical traveltime depth .

(31) |

In other words, any component of measures the integral of a material property from the Earth surface to the depth of . We wish to find the material property everywhere, which is . If we integrate it from the surface downward with causal integration , we should get the measurements .

With imperfect data, our data fitting goal is to minimize the residual:

(32) |

where is some weighting function, we need to choose. To find the interval velocity where there is no data (where the stack power theoretically vanishes), we have the ``model damping'' goal to minimize the wiggliness of the squared interval velocity .

(33) |

We precondition these two goals
by changing the optimization variable from
interval velocity squared
to its wiggliness
.
Substituting
gives the two goals
expressed as a function of wiggliness
.

Preconditioning |

2015-05-07