Topological data analysis provides compact representations of the characteristic structures of scientific data, notably through persistence diagrams. When these data evolve over time, their comparison must account for the fact that similar phenomena may occur at different times or evolve at different rates.
This thesis introduces the Continuous Edit Distance (CED), an elastic metric for comparing time-varying persistence diagrams. It combines diagram comparison through the Wasserstein distance with temporal alignment, insertion, and deletion operations. We establish its metric properties and provide an explicit construction of geodesics, together with methods for computing barycenters that represent an average evolution. These tools are then used for temporal pattern search and clustering similar evolutions. A C++ implementation within the Topology ToolKit (TTK) enables the evaluation of these methods on scientific datasets, including simulation data, and the study of their robustness to perturbations and temporal shifts.