In investigating hydrological quantities, one interesting issue is to understand if two time series are correlated and especially if the correlation comes with a lag time, and, in case which is this lag time. This is nothing different than in many other analysis and, in fact the tools developed are ubiquitous in science. Looking for the how to correlate rainfall and discharges I stumbled in this ready-made post, “Four ways to quantify synchrony between time series data” by Jin Hyun Cheong, PhD.
The added value of this post is that the tools described are also available as open source Python scripts embedded in Jupyter Notebook and therefore anybody can re-execute them easily and learn as they work. I believe that when you go to apply the notebook to your data set it will not be hassle-free. However it is a good starting point. Certainly also you’ll have to dig a little in literature to get the sense of what you were doing but this is a great starting point for those who needs to cope with this type of analyses. Jin Hyun material is available on OSF. Please, if you use it, cite it.
A second way to see their relation is to use the Kullback-Leibler mutual information, a concept derived form Information Theory (see also here) that you can find a little illustrated in the Veyrat-Charvillon and Standaert (2009) paper cited below. Here a notebook that teaches how to estimate it in Python using pyTOrch. Here a bottom-up calculation with standard Python.
The above time series analysis performed are quite interesting because they can also suggest new type of comparison between modelled and simulated time series if you start to get bored by the standard indicators of goodness of fit, like Kling-Gupta-Efficiency and Nash-Shutcliffe.
If your main focus is the rainfall-runoff times series relationships, a recent paper to mention, is the one by Giani et al, below in References. But also the work of Serinaldi and Kilsby (2013) that seems quite complicate (boring or interesting? I still do not have read it) contains information.
Giani, G., M. A. Rico‐Ramirez, and R. A. Woods. 2021. “A Practical, Objective, and Robust Technique to Directly Estimate Catchment Response Time.” Water Resources Research 57 (2). https://doi.org/10.1029/2020wr028201.
Veyrat-Charvillon, Nicolas, and François-Xavier Standaert. 2009. “Mutual Information Analysis: How, When and Why?” In Cryptographic Hardware and Embedded Systems – CHES 2009, 429–43. Springer Berlin Heidelberg.
Serinaldi, Francesco, and Chris G. Kilsby. 2013. “The Intrinsic Dependence Structure of Peak, Volume, Duration, and Average Intensity of Hyetographs and Hydrographs.” Water Resources Research 49 (6): 3423–42.