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 t_ij remains. Then, near-source traveltimes s_i and near-receiver traveltimes r_j are estimated from the t_ij by minimizing the error e_ij in

e_ij = t_ij - s_i - r_j.

A trivial nonuniqueness is that an arbitrary constant added to all the s_i and subtracted from all the r_j 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 t_ij to the e_ij residual matrix  

$$e_{ij} = \left[ \begin{array} {rrr} 0& -12 & 4 \\ 17& 0 & 0 \\ 0& 10 & 0 \end{array} \right].$$ (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 $\sum \vert e_{ij}\vert = 43$.Now modify source and receiver times by applying +12 to row 1 and -12 to column 1. We have  

$$\left[ \begin{array} {rrr} 0 & 0 & 16 \\ 5 & 0 & 0 \\ -12 & 10 & 0 \end{array} \right],$$ (4)

still with $\sum \vert e_{ij}\vert = 43$.Now apply +12 to row 3 and -12 to column 3. We have  

$$\left[ \begin{array} {rrr} 0 & 0 & 4 \\ 5 & 0 & -12 \\ 0 & 22 & 0 \end{array} \right].$$ (5)

Furthermore, we can generate an infinite set of e_ij (and hence source and receiver corrections) all with the same $\sum \vert e_{ij}\vert$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 e_ij matrix:

$$\left[ \begin{array} {rrr} 0 & 0 & 0 \\ 0 & 7 & -11 \\ 0 & -3 & 8\end{array} \right].$$ (6)

More details are found in Claerbout and Muir (1973) which is where I recovered this example.