I am a geophysicist learning about geophysical inverse problems. In many papers, the authors discuss the "covariance matrix" as it applies to the inverse problem.

In most geophysical applications, you have some data vector, $\mathbf{d}$, with a length $N$. You also have a vector of "data errors", $\mathbf{\Delta d}$. You then create some model which produces predicted data values, $\mathbf{F}$. You can compare the two by computing the misfit (e.g. normalized root mean square or chi-square):

$$\chi^2 = \sum_k^N\left ( \frac{d_k - F_k}{\Delta d_k} \right )^2$$

Many authors write this as:

$$\chi^2 = (\mathbf{d}-\mathbf{F})^T \mathbf{C_d}^{-1} (\mathbf{d} - \mathbf{F})$$

where $\mathbf{C_d}$ is the "data covariance matrix".

My understanding is that

$$\mathbf{C_d} = \begin{bmatrix} \Delta d_1 & & & &\\ & \Delta d_2 & & \\ & & ... & \\ & & & \Delta d_N \end{bmatrix} $$

Is this correct?

Futhermore, can someone elaborate on why this is called the "data covariance matrix"? When I look at the Wikipedia entry on covariance or covariance matrices, I do not see how that relates to data errors in this application. It also seems that the covariance matrix is usually a full matrix according to Wikipedia whereas in my application, my understanding is that it is a diagonal matrix.


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