Young water fraction
The young water fraction (Fyw) is the fraction of water with transit times between zero and a young water threshold (ty ) is described by the following:
\(F_{\text{yw}}=\int_{0}^{t_{y}}{h(\tau)d\tau}\) (1)
where \(h(\tau)\) is the flow model of the hydrological system based on the distribution of water fluxes in the catchment.
The sine-wave approach for the young water fraction expresses the seasonal cycles of stream water and precipitation isotopes developed by Kirchner (2016a). The seasonal isotope cycles in precipitation and stream water can be described by the following:
\(C_{p}\left(t\right)=A_{p}\sin\left(2\pi ft-\varphi_{p}\right)+k_{p}\)(2)
\(C_{s}\left(t\right)=A_{s}\sin\left(2\pi ft-\varphi_{s}\right)+k_{s}\)(3)
where A is the amplitude (‰), \(\varphi\) is the phase of the seasonal cycle (in radians, with \(2\pi\) rad equalling 1 year),t is the time (decimal years), f is the frequency (yr-1) and k (‰) is a constant describing the vertical offset of the isotope signal. The subscripts p ands represent precipitation and stream water, respectively. The amplitudes As and Ap of the seasonal isotope cycles in Eqs. (2) and (3) can be estimated by using multiple linear regression to obtain the coefficients a andb in the following:
\(C_{p}\left(t\right)=a_{p}\cos\left(2\pi ft\right)+b_{p}\sin\left(2\pi ft\right)+k_{p}\)(4)
\(C_{s}\left(t\right)=a_{s}\cos\left(2\pi ft\right)+b_{s}\sin\left(2\pi ft\right)+k_{s}\)(5)
The amplitudes AS and APare then determined by the following:
\(A_{p}=\sqrt{a_{p}^{2}+b_{p}^{2}}\) (6)
\(A_{s}=\sqrt{a_{s}^{2}+b_{s}^{2}}\) (7)
Following Kirchner (2016a), we calculate Fyw as the amplitude ratio As/Ap , for the hillslope unit and catchment. And the threshold age for Fyw (\(t_{y}\) in Eq. (1)) is 0.189 years (69 days), which equals the threshold age for seasonal cycles convolved with an exponential TTD. This method of quantifying the Fyw can be much more reliable than using the MTT, particularly in heterogeneous and nonstationary catchments (Kirchner, 2016a, b).
According to the method, we fitted sine waves to the daily precipitation and streamflow isotope data, from November 2016 to October 2017, at a hillslope spring and catchment outlet (in Fig.1) using iteratively reweighted least squares (IRLS), with rainfall/discharge-weighting. The statistical analysis was completed using the R language (R Core Team, 2017), and the significance level of the hypothesis test was 0.05.