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

TL;DR

This paper proposes a practical approach to solvability in seismic imaging by decomposing data into a solvable simple part and an interpretable complex residual, emphasizing interpretation over absolute solvability.

Contribution

It introduces a complex decomposition method for seismic data, highlighting the importance of both simple and complex parts for practical solvability and interpretation.

Findings

Decomposition separates data into solvable and interpretable parts.

Complex residuals are valuable for geological interpretation.

Practical solvability emphasizes data interpretation over mathematical solvability.

Abstract

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.