## 3 Results and Discussion
Figure 1: The dependence of dimensionless variance of outflow concentration (Equation (24)) upon the dimensionless inflow response time at various dimensionless time scales. Figure 2: The dependence of dimensionless variance of outflow concentration (Equation (24)) upon the dimensionless catchment mean travel time for various values of the dimensionless correlation scales. For \(\alpha=2\), the transfer function \(g(t)\) in Equation (2) has the form  
\[g(t)=4\frac{t}{\tau_{m}}\eta_{3}^{2} \tag{25}\]  
which leads Equation (14) eventually to  
\[S_{CC}(t;\omega)=\frac{16}{\pi}\sigma_{\xi}^{2}\frac{\lambda\tau_{C}^{2}\eta_{3} ^{4}}{\mu_{1}\mu_{2}}[\Gamma_{1}^{2}+\frac{1}{\mu_{4}^{2}}[\Gamma_{2}+\Gamma_ {3}\rho^{2}+(\Gamma_{4}+\Gamma_{5}\rho^{2})\cos(\rho)+\Gamma_{6}\rho\sin(\rho )]] \tag{26}\]  
where \(\mu_{4}=4+\tau_{m}^{2}\omega_{2}\) and expressions for \(\Gamma_{1}-\Gamma_{6}\) are given in the Appendix B. It follows from Equation (25) that the associated variance of outflow concentration is obtained as  
\[\sigma_{C}^{2}(t)=16\sigma_{\xi}^{2}\lambda\tau_{C}\eta_{3}^{4}[\frac{\Gamma_ {1}^{2}}{\Gamma_{1}+\gamma_{1}}+\frac{1}{1-\gamma_{1}^{2}}[\frac{1}{16}\Phi_{ 1}\Gamma_{2}+\frac{1}{4}\epsilon_{2}^{2}\gamma_{2}\Phi_{2}\Gamma_{3}+\Gamma_ {4}(\Phi_{3}-\Phi_{4})-2\frac{\gamma_{2}}{\eta_{3}^{2}}(\Phi_{5}-\Phi_{6}) \Gamma_{1}+2\frac{\gamma_{2}}{\eta_{3}^{2}}(\Phi_{7}-\Phi_{8})(4\Gamma_{1}- \epsilon_{2})]] \tag{27}\]  
where expressions for \(\Phi_{1}-\Phi_{8}\) are given in the Appendix C.  
Figure 3 shows a comparison between the outflow concentration evolutionary power spectra given by Equations (23) and (26) at a given time. Evolutionary power spectra have the interpretation of decomposition of total energy (or the strength of the variations) over frequency at time \(t\)[40]. It is apparent from Figure 3 that high-frequency variations in the temporal input series are attenuated. A similar behavior is shown in [PERSON] et al. [7, 8] for the case of stationary outflow concentration time series. The figure also implies that the persistence of the concentration field is enhanced by a smaller value of\(\alpha\). A smaller \(\alpha\) will introduce greater variability in outflow concentration. The parameter \(\alpha\) in Equation (2) could be a fractional number. In this case, the integral in Equations (11) and (13) may not be evaluated analytically. Since the transfer function is acting as a filter, it is expected that the outflow concentration variability behavior for the case of \(\alpha\) being an integer will be qualitatively similar to that for the case of \(\alpha\) being a fractional number, but not quantitatively. The determination of the transfer function in Equation (1) is of great importance in an application of the convolution integral model for modelling of hydrological processes. A brief introduction of  
Figure 3: Comparison between the dimensionless evolutionary power spectra for the outflow concentration processes in Equations (23) and (26) at a given dimensionless time. using the cross-correlation technique in determining the transfer function is given as follows. Using Equations (6) and (7), it follows that  
\[R_{CC_{i}}(t)=E[C_{i}(t)C^{*}(t)]=\int\limits_{-\infty}^{\infty}\Lambda_{C_{i}}(t; \omega)\Lambda_{C}(t;-\omega)S_{\xi\xi}(\omega)d\omega \tag{28}\]  
where \(R_{CC_{i}}\) the cross-correlation between \(C_{i}(t)\) and \(C(t)\) processes. Following [PERSON] [40], \(R_{CC_{i}}\) has a Fourier-Stieltjes representation of the form  
\[S_{CC}(t;\omega)=\sigma_{\xi}^{2}\frac{\lambda\tau_{C}^{2}}{\pi(1-\gamma_{2}) \mu_{1}\mu_{2}\mu_{3}}\{\phi_{1}+2[\phi_{2}+\phi_{3}\cos(\omega t)+\phi_{4} \rho\sin(\omega t)]\} \tag{23}\]0  
where \(S_{CC_{i}}\) is the cross-spectrum between \(C_{i}(t)\) and \(C(t)\) processes.