L0+L1+L2 mixed optimization: a geometric approach to seismic imaging and inversion using concepts in topology and semigroup

摘要

The mathematical interpretation of L0, L1 and L2 is needed to understand howwe should use these norms for optimization problems. The L0 norm iscombinatorics which is counting certain properties of an object or an operator.This is the least amplitude dependent norm since it is counted regardless ofthe magnitude. The L1 norm could be interpreted as minimal geometricdescription. It is somewhat sensitive to amplitude information. In geophysics,it has been used to edit outliers like spikes in seismic data. This is a goodapplication of L1 norm. The L2 norm could be interpreted as the numericallysimplest solution to fitting data with a differential equation. It is verysensitive to amplitude information. Previous application includes least squaremigration. In this paper, we will show how to combine the usage of L0 and L1and L2. We will not be optimizing the 3 norms simultaneously but will go fromone norm to the next norm to optimize the data before the final migration.