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A problem arises with partitioned operators.
Here we are fitting observed data to theoretical data
where there are two classes of model parameters
and .We seek to minimize
the norm of the residual defined by
| |
(9) |

where and are arbitrary scaling constants.
The residual is independent of and ,but when we ``solve" this system using (as we must)
the idea that ,we see the result depends on and ,namely it contains and
| |
(10) |

Let us find the best and .Inserting the image (10)
into the residual (9)
we get
| |
(11) |

| (12) |

| (13) |

which defines two vectors and .
We find the best scaling factors by
setting to zero the derivative of with respect to and

| |
(14) |

solving gives
| |
(15) |

I believe it can be shown that the values
and are positive.
Recalling that
and
,let us now define
and
so
and
.

In imaging applications we customarily ignore the scaling factor
which is the common part of and ,namely, the denominator determinant in (15).
We have the proportions

| |
(16) |

| (17) |

| (18) |

A deeper problem of interest
arises when we seek the best diagonal matrix scaling.
Then we replace with two diagonal matrices, one before and one after .Likewise with and .We need help finding those four diagonal matrices.

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Stanford Exploration Project

11/11/1997