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The surface run-off generated from the landscape is routed (attenuation and lag) to the catchment outlet using the linear cascade model of Nash (1960). The model was obtained as a general solution relating a given input of unit volume to a given output as in equation 1.
| Equation 1 |
|---|
where:
t = simulation time-step (d);
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The generated surface runoff (rs mm·d-1) and the routed runoff (QrT mm·d-1) can be time averaged, as in equations (2) and (3), to represent the daily values.
| Equation 2 |
|---|
| Equation 3 |
|---|
The linear model described by equation 4 (below) is the simplest representation of a causal, time invariant, relationship between an input function of time (generated runoff) and the corresponding output function (routed runoff). It is used in conceptual modelling, as a component, representing the routing or diffusion, effects of the catchment on those components of the rainfall hyetograph contributing to the outflow.
| Equation 4 |
|---|
where:
m = memory of the pulse response function (d).
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The mass balance equation for the groundwater system can be written as in equation 5:
| Equation 5 |
|---|
where:
QTrech = recharge to the groundwater system (mm.s-1).
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The pulse-response function for the groundwater component can be obtained in a manner analogous to equation 1 as in equation 6 (i.e. equation 1 with n and Γ(n) equal to 1; Vaze et al., 2004).
| Equation 6 |
|---|
The recharge QTrech and the discharge QTg can be time averaged to mm·d-1 in an analogous manner to the generated surface runoff (rs) and the routed runoff (QrT), as in equations 2 and 3.
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Table 1. Parameters in SMARG and their default values
| Parameter | Description | Units | Default | Range |
|---|---|---|---|---|
| C | Evaporation coefficient | none | 0 | 0-1 |
| G | used to estimate the proportion of moisture in excess of soil moisture storage capacity recharging groundwater (and also discharged to the stream) | none | 0 | 0-1 |
| H | used to estimate the proportion of rainfall excess contributing to the generated runoff as saturation excess runoff or the Dunne runoff | none | 0 | 0-1 |
| Kg | Time lag parameter for groundwater routing | none | 0.01 | 0.01-200 |
| n | Surface runoff hydrograph ‘shape’ parameter (i.e. number of linear reservoirs) | none | 1 | 1-10 |
| nK | Surface runoff hydrograph ‘scale’ parameter (i.e. time lag parameter in Nash cascade model) | none | 1 | 1-10 |
| T | Ratio of potential evapotranspiration to pan evaporation | none | 0 | 0-1 |
| Y | Infiltration capacity of the soil | mm.d-1 | 0 | 0-100 |
| Z | Effective moisture storage capacity of the soil contributing to the runoff generation mechanisms | mm | 0 | 0-125 |
| Info | ||
|---|---|---|
| ||
| Note: the number of soil layers is determined from "Z" (Soil Moisture Storage Capacity) and a constant in the code which is 25 mm (the depth of each of the "groundwater"/soil layers in mm of water). |
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Table 2. Recorded variables
| Variable | Parameter | Frequency | Notes |
|---|---|---|---|
| PET | Potential evapotranspiration | time-step | |
| x | Excess rainfall | time-step | see Figure 1 |
| INF | Infiltration | time-step | Estimated from (1-H’)x (see Figure 1) |
| r1 | Direct runoff | time-step | see Figure 1 |
| r2 | Rainfall in excess of infiltration capacity (Hortonian runoff) | time-step | see Figure 1 |
| r3 | Moisture in excess of soil moisture capacity discharged to stream | time-step | see Figure 1 |
| r9 | Moisture in excess of soil moisture capacity recharging (percolating to) groundwater | time-step | see Figure 1 |
| rs | Generated surface runoff | time-step | see Figure 1 |
| QOUTsurf | Routed surface runoff (from gamma function) | time-step | see Figure 1 |
| QOUTgw | Routed groundwater runoff | time-step | see Figure 1 |
| SMStot | Soil moisture store contents (total of all layers) | time-step | see Figure 1 |
| Layers 1...etc | List of soil moisture store layers | one-off item | see Figure 1 |
Reference list
Kachroo, R.K. (1992). River flow forecasting. Part 5. Applications of a conceptual model, Journal of Hydrology, 133: 141–178.
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