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An alternative method to combine the vector field
of expression (15)
to a scalar field is the combination of the forward and adjoint
of the prediction-error operator:

The space of the output is equal the space of the input *g*.
Unfortunately,
I do not know a insightful interpretation of the procedure.
If prediction-error filter is similar to a derivative operator,
the combination of forward and adjoint corresponds to a sequence
of two derivative operations.

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Stanford Exploration Project

3/8/1999