Long-term data-driven studies have become indispensable in many areas of science. Often, the data formats, structures and semantics of data change over time, the data sets evolve. Therefore, studies over several decades in particular have to consider changing database schemas. The evolution of these databases lead at some point to a large number of schemas, which have to be stored and managed, costly and time-consuming. However, in the sense of reproducibility of research data each database version must be reconstructable with little effort. So a previously published result can be validated and reproduced at any time. Nevertheless, in many cases, such an evolution can not be fully reconstructed. This article classifies the 15 most frequently used schema modification operators and defines the associated inverses for each operation. For avoiding an information loss, it furthermore defines which additional provenance information have to be stored. We define four classes dealing with dangling tuples, duplicates and provenance-invariant operators. Each class will be presented by one representative. By using and extending the theory of schema mappings and their inverses for queries, data analysis, why-provenance, and schema evolution, we are able to combine data analysis applications with provenance under evolving database structures, in order to enable the reproducibility of scientific results over longer periods of time. While most of the inverses of schema mappings used for analysis or evolution are not exact, but only quasi-inverses, adding provenance information enables us to reconstruct a sub-database of research data that is sufficient to guarantee reproducibility.