## 1 Introduction  
Agricultural chemicals are frequently mixed into shallow soil layers. Runoff through agricultural land may pick up soil chemicals and introduce them into surface waters. During the periods of heavy rainfall, the soil chemicals in runoff water can pose a potential threat to the quality of the environment and water supplies [1; 2; 3]. Modeling of the transfer of agricultural chemicals into surface runoff is therefore needed in order to predict the development of the contaminant plume. The field evidence indicates very irregular patterns of rainfall in time and, in turn, very complex runoff production in a catchment during a hydrologic event [4]. It therefore requires a large quantity of measurements for accurate predictions of the hydrological response of the catchment. This situation is, however, far from that encountered in practice. The amount of information on all coefficients and parameters appearing in the model for accurate predictions is never enough, so one has to accept a level of uncertainty. As such, the runoff and associated solute transport are often formulated as stochastic processes in order to reflect the uncertainty of our knowledge [5; 6; 7; 8; 9; 10; 11]. Since the hydrological variables are regarded as random, predictions are therefore made in terms of probabilities (or moments) rather than in the traditional deterministic framework [12; 13; 14; 15]. Many contamination incidents are geographically isolated and their sources can be considered as point sources of pollution. Although locally important, contamination from these sources may not be significant throughout the catchment area. In contrast, nitrates and pesticides from agricultural areas, for example, which come from distributed, or non-point sources of pollution can have a significant impact on the water quality of the catchment area. The main interest is therefore in long-term basin-wide changes in water quality due to distributed contaminant sources, ignoring the special variations and focusing on the temporal variation of the mean contaminant concentration in the catchment area. This type of model, which does not require specification of heterogeneity on a small spatial scale, is called a lumped-parameter model [16; 17]. In many regional groundwater studies, a lumped model approach is often used for long-term basin-wide predictions due to the uniformly distributed recharge sources [18; 19; 20]. Since the temporal variations of the runoff are usually observed at the outflows of the catchment, the lumped models require to specify the nature of the input to the model and the model itself, which relates inputs, outputs, and system states. They use simpler representation of physical phenomena to convert input (recharge) into output (discharge), reflecting the overall hydrogeological configuration of the catchment. Consequently, a lumped model provides basic information on the hydraulic behavior of the output response to the input into the catchment system. A clear understanding of the relation between the output and input of the catchment is essential to predict the impact of changes in the concentration of the input solutes on the variability of the discharge concentration at the outlet of the catchment, which can improve our ability to deal with the problem of surface water quality. The goal of this work is, therefore, to use a lumped model approach for quantification of temporal variability in the catchment response to a non-stationary random input concentration of an inert solute, which so far has attracted only limited attention in the hydrology literature. This work treats a catchment with a random variable input and an output as a linear time-invariant system [21; 22]. This means that the transfer from rainfall to recharge is assumed to be linear. It is considered a black box process at any point in space and the classical theory of linear systems [23], also known as the theory of unit hydrograph in surface water hydrology, can be used.