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The classical approach of solvability using group theory is well known andone original motivation is to solve polynomials by radicals. Radicals aresquare, cube, square root, cube root etc of the original coefficients for thepolynomial. A polynomial is solvable by radicals if the permutation group issolvable. This is exact solvability via group theory. With modern computers, wemight need to relax our definition of exact solvability and move towardspractical solvability. We will address seismic imaging as an example ofpractical solvability by semigroup theory. The difference between semigroup andgroup is that the semigroup operators do not have to be invertible as in groupoperators. Using the metaphor of complex decomposition, we will decompose anoperator into simple part and complex part. The simple part of the operator issolvable by numerical methods. The complex part of the operator isinterpretable but not numerically solvable. It is sometimes called theevanescent energy in geophysics.

Solvability by semigroup : Application to seismic imaging with complex decomposition of wave equations and migration operators with idempotents