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The classical approach to solvability of a mathematical problem is to definea method which includes certain rules of operation or algorithms. Then usingthe defined method, one can show that some problems are solvable or notsolvable or undecidable depending on the particular method. With numericalsolutions implemented in a computer, it might be more practical to definesolvability of a mathematical problem as a complex decomposition problem. Thedecomposition breaks the data into a simple part and a complex part. The simplepart 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 viewedas the "residual" of data or operator. It should be interpreted and not to bediscarded as useless. We will give different examples to illustrate the morepractical definition of solvability. The complex part is not noise and shouldnot be viewed as useless part of the data. It has its own merit in terms oftopological or geological interpretation. We have de-emphasized absolutesolvability and have emphasized the practical solvability where the simple andcomplex parts both play important roles.

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