Supplementary material to "Detecting hydrological connectivity using causal inference from time-series: synthetic and real karstic study cases"

The ρ coefficient is a standardized measure of linear dependencies that can be interpreted as the slope of a linear regression between the two standardized variables (i.e., zero mean and unit variance). Accordingly, ρ is ranging between -1 and 1, meaning respectively perfectly anti-correlated or correlated. A ρ of zero indicates the absence of linear dependencies. The significance of the hypothesis that ρ is different from zero is usually assessed analytically through a Student’s-t test reporting a p-value. The p-value estimates the probability that the correlation between the two time-series is the output of an uncorrelated process. The p-value is sensitive to the number of overlapping samples such that more samples are required to have a significant p-value if |ρ| is low. For a significance level α, significant relationships are considered when the p-value is lower than α. In the main manuscript, significant correlations output by the CCF method are considered as a way to reveal potentially causal links between two time-series at a causal delay d, in virtue of the principle of priority. The case of d = 0 refers to a contemporaneous dependency and does not allow to infer a direction for the causal relationship. Correlation and the significance test were performed using Python and the Scipy library (Virtanen et al., 2020).