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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,
nearsource traveltimes s_{i} and nearreceiver traveltimes r_{j}
are estimated from the t_{ij} by minimizing the error e_{ij}
in

e_{ij} = t_{ij}  s_{i}  r_{j}.

(2) 
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 3by3 matrix
of t_{ij} to the e_{ij} 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 e_{ij}
(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 L_{1} solutions with the e_{ij} 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