## 2 Materials and Methods  
### Formulation of the Model  
This work takes the entire catchment as a single unit where the spatial variability is disregarded. In such a modeling approach, one tries to relate the system output to the forcing without going into the details of what is actually happening inside the system. That is, the present study assesses the catchment response simply at the outlet without counting for individual sub-basin responses. Furthermore, this work considers a catchment with a variable input, \(Ci(t)\), and an output, \(C(t)\), as a linear time-invariant system. If the system represents a convolution, \(C(t)\) and \(Ci(t)\) may then be connected by the integral form of [26; 27; 28]  
\[C(t)=\int\limits_{0}^{t}g(t-s)C_{i}(s)ds \tag{1}\]  
where \(g(t)\) is the transfer function (travel time), \(t\)-\(s\) is interpreted as the time lag between the input and the output time and \(s\) is used as a dummy variable of integration, which disappears after the integral is evaluated. The convolution integral operation may be thought of as a filtering operation on the input, where the transfer function is acting as the filter. Equation (1) treats the input flow as uniform over the catchment and ignores the internal spatial variation of catchment flow. The variables \(Ci(t)\) and \(C(t)\), as considered here, are represented as temporal fluctuations in inflow and outflow concentrations of an inert solute, respectively, and regarded as stochastic (random) processes in time to account for their irregular temporal variation and the uncertainty of their distribution. This study uses the variance as an index of variability of a random variable \(C(t)\) in time. The transfer function \(g(t)\) in Equation (1) is a physical measure that integrates flow path heterogeneity and is directly related to internal catchment processes [29]. For simplicity, its temporal distribution is treated as deterministic. That is, it represents an ensemble mean (an average). Following the parameterization proposed by [PERSON] et al. [7; 8] and [PERSON] et al. [30], the present study uses the gamma distribution to represent the temporal behavior of \(g(t)\). \[g(t)=(\frac{\alpha}{\tau_{m}})^{\alpha}\frac{t^{\alpha-1}}{\Gamma(\alpha)}\exp [-\alpha t/\tau_{m}] \tag{2}\]  
where \(\alpha\) is a shape parameter, \(\tau_{m}\) is the catchment mean travel time constant, and \(\Gamma(\alpha)\) denotes the incomplete gamma function of argument \(\alpha\). Note that the time required for water to travel through a catchment is a physical measure that integrates flow path heterogeneity. It has become apparent from previous studies [5; 31; 32] that the convolution integral model based on travel time distributions of solute migration may provide an alternative way of representing the large-scale transport process. As mentioned by [PERSON] et al. [7], the gamma travel time distribution with two parameters is both conceptually and mathematically a more suitable representation of the resulting solute mixing processes as its higher degree of freedom offers greater flexibility in accounting for different flow path distributions. The \(\alpha\) in Equation (2) reflects the heterogeneity in the flow path lengths and velocities of catchments [30]. The smaller the value of \(\alpha\), the greater the variability in the catchment flow path length and velocity compared to the mean. For \(\alpha>1\), the gamma distribution rises to a peak and then falls off, similar to a typical storm hydrograph, which is often used to model rainfall-runoff relationships. For \(\alpha=1\), the gamma distribution is a close approximation of the exponential distribution, implying that the mixing of a solute plume only occurs at the sampling point; thus, there is no exchange between flow lines. For \(\alpha<1\), however, the gamma distribution has a completely different shape, having maximum weight at lags near zero and a relatively long tail.