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

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