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# Representing covariance matrices by sparse operators

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.

Next: Data regularization as an Up: Fundamentals of data regularization Previous: Statistical estimation
Stanford Exploration Project
12/28/2000