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I have a linear ill posed problem with the form: $$Ax=b.$$ One approach to this problem is Tikhonov regularization, replacing it with $$\min_x( \|Ax-b\|^2+\alpha\|x\|^2 ).$$

In https://en.wikipedia.org/wiki/Tikhonov_regularization said that you can use covariance matrix of dataset to find optimal regularisation parameter for Tikhonov regularization. I also heard in a conference that a resolution matrix can be very useful in this task.

What is the resolution matrix of a linear problem, what does it mean? and more important: How can I use standard deviation of my data to determine Tikhonov regularization parameter?

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It seems to be measurement concatenated with inversion.

$$R^M = (\hat S ^\dagger \hat S)$$

where I presume $^\dagger$ is some pseudo-inverse.

according to this source http://www.resistivity.net/dc2dinvres/invres.pdf

In other words what something looks like according to the model. The error would be $I-R^M$ real data minus what the model can catpure.

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