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## Statics

An important example is the estimation of source and receiver time corrections. Here one has a set of observed traveltimes from the ith source to the jth receiver. After known systematic geometrical and velocity effects are removed, the time residual matrix tij remains. Then, near-source traveltimes si and near-receiver traveltimes rj are estimated from the tij by minimizing the error eij in

 eij = tij - si - rj. (2)

A trivial nonuniqueness is that an arbitrary constant added to all the si and subtracted from all the rj will give the same residuals. I was surprised to discover deeper nonuniqueness lurking in a simple example. Absolute error minimization reduced a 3-by-3 matrix of tij to the eij residual matrix
 (3)
As expected theoretically (by the solution method I used), there are 5 zeros representing the 5 independent unknowns of the 6 unknowns. Note that .Now modify source and receiver times by applying +12 to row 1 and -12 to column 1. We have
 (4)
still with .Now apply +12 to row 3 and -12 to column 3. We have
 (5)
Furthermore, we can generate an infinite set of eij (and hence source and receiver corrections) all with the same by taking residuals (3)-(5) and forming any convex combination (weighted combination where each weight is positive and the weights sum to one).

The existence of a sizeable nonuniqueness with absolute error minimization leaves us the uncomfortable feeling that the mathematical uniqueness of squared error is not genuine, i.e., that the uniqueness of results with squared error is not a realistic charactorization of our certainty.

Often, however, the this unfamiliar nonuniqueness does not arise. It depends on the data, not the mathematical structure of the problem. For example, I don't know any other minimum L1 solutions with the eij matrix:
 (6)
More details are found in Claerbout and Muir (1973) which is where I recovered this example.

Next: Curve through two points Up: EXAMPLES Previous: Best fitting straight line
Stanford Exploration Project
4/27/2000