Transformation Semigroup and Complex Topology: a study of inversion with increasing complexity

TL;DR

This paper classifies the complexity of inversion in systems based on topology and algebraic structure, illustrating the classes with synthetic models and analyzing implications for complex topological systems like Cantor layers.

Contribution

It introduces a three-class classification of inversion complexity using synthetic models, extending previous work on complex topology and invertibility.

Findings

Systems with symmetry have clean inversion.

Semigroup systems lack guaranteed global invertibility.

Complex topological systems require infinite constructions for inversion analysis.

Abstract

This paper is a continuation of our 2005 paper on complex topology and its implication on invertibility (or non-invertibility). In this paper, we will try to classify the complexity of inversion into 3 different classes. We will use synthetic models based on well control to illustrate the different classes. The first class is systems which have a group of symmetry. This class has clean inversion. We will use two examples which include 2-term AVO on 1-D layers as an example. The second class does not have a group support. It is in general described by a semigroup which is a set of operators with or without inverses. There is no guarantee in general of invertibility in a global sense. Even though this class does not have invertibility in general, there could still be local weak invertibility embedded in the semigroup. The last class is system with complex topology where the underlying topology/geometry requires infinite construction. In our previous 2005 paper, we gave the 1-D Cantor layers as the forerunner of all complex topological interface. We will re-examine this last class in detail. The idea of constructing the Cantor layers as inverse limit was introduced in the previous paper. We will examine the finite approximation of the Cantor layers and its implication on inversion.