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In order to understand the structure of the matrices and , we need to make some assumptions about the
relationship between the true model and the data
. A natural assumption is that if the model were known
exactly, the observed data would be related to it by a *forward
interpolation operator* as follows:

| |
(4) |

where is an additive observational noise. For simplicity,
we can assume that the noise is uncorrelated and normally distributed
around zero:
| |
(5) |

where is an identity matrix of the data size, and
is a scalar. Assuming that there is no linear correlation
between the noise and the model, we arrive at the following
expressions for the second moment matrices in
formula ():
| |
(6) |

| |
(7) |

Substituting equations () and () into
(), we finally obtain the following specialized form of
the Gauss-Markoff formula:
| |
(8) |

Assuming that is invertible, we can also rewrite
equation () in a mathematically equivalent form
| |
(9) |

The equivalence of formulas () and ()
follows from the simple matrix equality
| |
(10) |

It is important to note an important difference between
equations () and (): The inverted matrix
has data dimensions in the first case, and model dimensions in the
second case. I discuss the practical significance of this distinction
in Chapter .
In order to simplify the model estimation problem further, we can
introduce a local differential operator . A model complies with the operator if the residual after we apply
this operator is uncorrelated and
normally distributed. This means that

| |
(11) |

where the identity matrix has the model size. Furthermore,
assuming that is invertible, we can represent as follows:
| |
(12) |

Substituting formula () into () and
(), we can finally represent the model estimate in the
following equivalent forms:
| |
(13) |

| (14) |

where and .
The first simplification step has now been accomplished. By
introducing additional assumptions, we have approximated the
covariance matrices and with the forward
interpolation operator and the differential operator
. Both and act locally on the model.
Therefore, they are sparse, efficiently computed operators. Different
examples of operators , , and are
discussed later in this dissertation. In the next section, I proceed
to the second simplification step.

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** Up:** Fundamentals of data regularization
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Stanford Exploration Project

12/28/2000