Decomposition of model space

Weighted medians are equivalent to L₁ optimizations but medians themselves are just an idea that may or may not lead to successful results. In a convex optimization (such as L₁ or L₂) 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 L₁ 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.