*3.2. Auto-Correlation and Cross-Correlation Analyses*

Figure 4a shows that the auto-correlation coefficients of rainfall decay quickly close to zero, and all of the correlograms become null within 3 months, implying that the rainfall is relatively random. By comparison, GWLs present a long memory effect relative to rainfall (Figure 4b). The auto-correlation functions of GWLs show that the order of increasing inertia ranks as follows: W2→W1→W3→W4. For example, the auto-correlation slope (the slope of the auto-correlation coefficient before the curve becomes flat) increases from −12.7 × <sup>10</sup>−<sup>2</sup> month−<sup>1</sup> at W2 to −5.0 × <sup>10</sup>−<sup>2</sup> month−<sup>1</sup> at W4 (Table 3), and the time lag required for auto-correlation coefficients to reach 0.2 (*k*0.2 values) also rises from W2 (5.7 months) to W4 (12.3 months). Note that without considering W2, there is an upward trend in persistence from upstream to downstream, which has also been identified by Duvert et al. [3] in a subtropical agricultural catchment dominated by alluvial aquifers in southeast Queensland, Australia.

**Figure 4.** Auto-correlation functions for (**a**) rainfall, and (**b**) GWLs.

**Table 3.** Parameters of the auto-correlation functions.


Figure 5 shows that the peak value of *rxy* between precipitation and GWLs is the maximum of 0.52 at W2, followed by 0.45 at W1, 0.41 at W3, and 0.40 at W4. It is interesting that this order is consistent with the above ranking result from the auto-correlation functions of GWLs. That is, the shorter the memory time, the greater the correlation coefficient. The time lags corresponding to the peak values are 0.67 months at W2and W3, and 1.33 months at W1 and W4.

**Figure 5.** Cross-correlation diagrams between rainfall and GWLs.

#### *3.3. Continuous Wavelet Spectra*

Wavelet power spectra for rainfall and GWLs were plotted in Figure 6. Warmer colors denote higher power. It is statistically significant that the rainfall spectrum has a clear annual periodicity throughout the study period, which is mainly caused by the annual wet/dry cycle. For groundwater, this annual periodicity was identified during 2009–2014 and 2015–2017 for W1, 2008–2015 for W2, 2006–2011 and 2014–2016 for W3, and 2014–2016 for W4.

It can be seen that high-power frequencies in the rainfall spectra are absorbed and filtered by the aquifer to produce the groundwater signals. Therefore, aquifers serve as lowpass filters, which is consistent with the research of Imagawa et al. [5] and Duvert et al. [3]. It is interesting that the period when the maximum value in the global wavelet spectrum is achieved increases gradually from 2.1 years at W1 (upstream) to 3.7 years at W4 (downstream). Gómez et al. [39] also identified "longer aquifer regulation times in larger basins". The increasing time period from upstream to downstream we observed here further demonstrates the impacts of regional water circulation.

**Figure 6.** Continuous wavelet spectra for both precipitation and GWLs at (**a**) W1, (**b**) W2, (**c**) W3, and (**d**) W4, with the global wavelet spectrum right side of each subplot. Zones surrounded by black lines have significant wavelet power at the 95% confidence level. White lines denote the cone of influence.
