2.4.2. Delayed Response Time and Correlation

The response of the groundwater level to the rise and fall of the river stage was analyzed using the cross-correlation analysis method. Through cross-correlation analysis, the response relationship between the two timeseries could be estimated, and the time shift could be quantified. For two discrete timeseries signals *Y*1(*m*) and *Y*2(*m*), their crosscorrelation function *R*(*τ*) is expressed in Equation (2) [40].

$$R(\tau) = \sum\_{m=0}^{M} Y\_1(m) \cdot Y\_2(m+\tau),\tag{2}$$

where *m* is the moment, *M* is the length of the timeseries, *τ* is the delayed time, and *Y*1(*m*) and *Y*2(*m*) are the timeseries of the river water level and the groundwater level, respectively. A larger *R*(*τ*) value indicates a greater correlation between the two sequences. If *Y*1(*m*) is equal to *Y*2(*m*), then *R*(*τ*) represents the self-correlation function value of the two sequences at *τ*. Normalization is usually performed on *R*(*τ*), so that the self-correlation is equal to one at the delayed time of zero. The delayed response time of groundwater level to the rise and fall of the river stage corresponded to the maximum *R*(*τ*).
