Preconditioning |

IID random variables have uniform variance in both physical space and Fourier space. |

In a geophysical project, it is important the residual between observed data and modeled data is not far from IID. To raw residuals, we should apply weights and filters to get IID residuals. We minimize sums of squares of residuals. If any residuals are small, the squares are tiny, so such regression equations are effectively ignored. We would hardly ever want residuals ignored. Echo seismograms get weak at late time. So, even with a bad fit, the difference between real and theoretical seismograms is necessarily weak at late times. We do not want the data at late times to be ignored. So, we boost up the residual there. We choose to be a diagonal matrix that boosts late times in the regression .

An example with too much low (spatial) frequency in a residual might arise in a topographic study. It is not unusual for the topographic wavelength to exceed the survey size. Here, we should choose to be a filter to boost up the higher frequencies. Perhaps, should contain a derivative or a Laplacian. If you set up and solve a data-modeling problem and then find is not IID, you should consider changing your . Chapter provides a systematic approach to whitening residuals.

Now, let us include regularization
and a preconditioning variable
.
We have our data-fitting goal and our model-styling goal;
the first with a residual
in data space, the second with a residual
in model space.
We have had to choose a regularization operator
and a scaling factor
.

This system of two regressions could be packed into one; the two residual vectors stacked on top of each other, likewise the operators and . The IID notion seems to apply to this unified system which gives us a clue as to how we should have chosen the regularization operator . Not only should be IID, but also should --within a scale , . Thus, the preconditioning variable is not simply something to speed computational convergence. It is a variable that should be IID. If it is not coming out that way, we should consider changing . Chapter addresses the task of choosing an , so comes out IID.

We should choose a weighting function (and/or operator) , so data residuals are IID. We should also choose our regularization operator so the preconditioning variable comes out IID. |

Preconditioning |

2015-05-07