Practical approach to solvability: Geophysical application using complex decomposition into simple part (solvable) and complex part (interpretable) for seismic imaging

The classical approach to solvability of a mathematical problem is to define a method which includes certain rules of operation or algorithms. Then using the defined method, one can show that some problems are solvable or not solvable or undecidable depending on the particular method. With numerical solutions implemented in a computer, it might be more practical to define solvability of a mathematical problem as a complex decomposition problem. The decomposition breaks the data into a simple part and a complex part. The simple part is the solvable part by the method prescribed in the problem definition. The complex part is the leftover of the simple part. Complex part can be viewed as the "residual" of data or operator. It should be interpreted and not to be discarded as useless. We will give different examples to illustrate the more practical definition of solvability. The complex part is not noise and should not be viewed as useless part of the data. It has its own merit in terms of topological or geological interpretation. We have de-emphasized absolute solvability and have emphasized the practical solvability where the simple and complex parts both play important roles.