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Weighted medians are equivalent
to *L*_{1} optimizations
but medians themselves are just an idea
that may or may not lead to successful results.
In a convex optimization (such as *L*_{1} or *L*_{2})
in principle
(and in fact when the dimensionality is low)
we can find the optimum
by blundering about in random directions,
as long as we always move to the minimum in each direction.
Unweighted medians are not related to convex optimization
so they may require the ``right'' basis.
In the migration problem we are solving an *L*_{1} optimization.
We should be able to use random directions,
but we could use the gradient.
If chosing the ``right'' direction were important,
I would reason that the earth is a sedimentary section
and local planes are the proper model space
and that we should median-search the data space
for the effect of each local plane in model space.

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Stanford Exploration Project

11/12/1997