SIMHYD Model¶

Overview¶

SIMHYD (SIMple HYDrological model) is an eight-parameter conceptual rainfall-runoff model developed in Australia by Chiew, Peel, and Western (2002). It belongs to the family of threshold-based models where infiltration capacity decays exponentially with soil saturation, a common simplification in Australian water-resources practice.

The HOLMES implementation follows the modified version from the HOOPLA toolbox (Thiboult et al., 2020, HM15), which adds two routing reservoirs and a fractional-delay parameter to the original formulation. The model represents a catchment as an interception store, a soil moisture store, and two linear routing reservoirs (ground and main) connected by a time-delay register.

SIMHYD is a good choice when a simple model with explicit interflow and groundwater-recharge pathways is desired. Its eight parameters are easy to interpret physically, making it well-suited for teaching and for benchmarking against more complex models.

Key Concepts¶

Interception store: a conceptual reservoir of capacity \(X_1\) that captures precipitation before it reaches the soil, limited by both rainfall and available PET.
Soil moisture store: a bucket of capacity \(X_2\) that receives infiltrated water and loses water to evapotranspiration, interflow, groundwater recharge, and overflow.
Exponential infiltration: the maximum infiltration rate decays as \(X_8 \cdot e^{-2\,S/X_2}\) — a saturated soil admits very little additional water.
Interflow: lateral subsurface flow proportional to soil saturation and inversely proportional to \(X_6\).
Groundwater recharge: deep percolation proportional to soil saturation and inversely proportional to \(X_7\).
Ground reservoir: a slow linear reservoir (emptying constant \(X_3 \cdot X_5\)) that receives soil excess and recharge.
Routing reservoir: a fast linear reservoir (emptying constant \(X_5\)) that collects all flow components before delay routing.
Fractional delay: a two-weight unit hydrograph of base \(X_4\) days that shifts the routed flow in time.

How It Works¶

Step 1: Interception. Incoming precipitation \(P\) is first intercepted up to a limit set by both the interception capacity \(X_1\) and the available PET \(E\). The intercepted amount is \(\text{CAP} = \min(P,\, X_1,\, E)\). If precipitation exceeds this amount, the excess \(\text{EXC} = P - \text{CAP}\) proceeds to infiltration; otherwise all precipitation is intercepted and no excess remains.

Step 2: Infiltration. The excess rainfall is partitioned between surface runoff and infiltration based on the soil's ability to absorb water. The infiltration capacity is \(\text{RINF} = X_8 \cdot e^{-2\,S/X_2}\), which decays exponentially as the soil saturates. If the excess exceeds this capacity, the surplus becomes surface runoff \(Q_\text{srun}\) and only \(\text{RINF}\) infiltrates.

Step 3: Interflow and groundwater recharge. From the infiltrated water, a fraction proportional to soil saturation \(S/X_2\) is diverted as interflow (\(\text{SINT}\), governed by \(X_6\)) and groundwater recharge (\(\text{REC}\), governed by \(X_7\)). Both quantities increase as the soil approaches saturation, representing the idea that wetter soils lose water faster through lateral and vertical pathways.

Step 4: Soil moisture accounting. The soil store \(S\) receives the infiltrated water minus what was lost to interflow and recharge. If \(S\) exceeds capacity \(X_2\), the excess \(\text{EX}_2\) overflows to the ground reservoir. Remaining PET (after interception) drives actual evapotranspiration from the soil, limited to \(10 \cdot S / X_2\) mm, and the soil level is clamped to zero.

Step 5: Routing. Soil excess and recharge feed the ground reservoir \(R\) (slow pathway, emptying at rate \(1/(X_3 \cdot X_5)\)). Surface runoff, interflow, and the ground reservoir outflow feed the routing reservoir \(T\) (fast pathway, emptying at rate \(1/X_5\)). This two-reservoir arrangement lets the model separate fast response from slow baseflow.

Step 6: Delay and streamflow. The routing reservoir output \(Q_t\) is convolved with a two-weight fractional-delay function of base \(X_4\) days, producing the final simulated streamflow \(Q\). The delay shifts the hydrograph peak in time without changing its volume, representing travel time through the channel network.

Parameters¶

The SIMHYD model has 8 calibratable parameters:

Parameter	Description	Range	Units	Physical Interpretation
\(X_1\)	Interception store capacity	0.5 – 10.0	mm	Maximum rainfall that can be held by vegetation canopy before reaching the soil
\(X_2\)	Soil moisture store capacity	1.0 – 500.0	mm	Total amount of water the soil can hold before overflowing
\(X_3\)	Ground reservoir emptying constant	1.0 – 1000.0	-	Multiplier that slows the ground reservoir relative to the routing reservoir
\(X_4\)	Delay	0.5 – 5.0	d	Fractional time shift applied to routed streamflow
\(X_5\)	Routing reservoir emptying constant	1.0 – 500.0	d	Controls how quickly the main routing reservoir drains
\(X_6\)	Interflow constant	1.0 – 1000.0	-	Larger values reduce interflow; inversely proportional to the standard SIMHYD SUB parameter
\(X_7\)	Groundwater recharge constant	1.0 – 1000.0	-	Larger values reduce deep percolation; inversely proportional to the standard SIMHYD CRAK parameter
\(X_8\)	Maximum infiltration capacity	1.0 – 500.0	mm	Infiltration rate when the soil is completely dry; decays exponentially with saturation

Understanding the parameters:

\(X_1\) is small (typically 1–5 mm) because canopy interception is a minor component of the water balance. It mainly affects peak timing and ET partitioning.
\(X_2\) controls the overall wetness of the catchment — larger values mean more buffering before soil excess triggers overflow to the ground reservoir.
\(X_3\) and \(X_5\) together control the slow/fast flow separation: the routing reservoir empties at rate \(1/X_5\) while the ground reservoir empties at rate \(1/(X_3 \cdot X_5)\). Increasing \(X_3\) makes baseflow more persistent.
\(X_6\) and \(X_7\) compete for infiltrated water — when both are small, the soil loses water quickly through lateral and vertical pathways; when both are large, water stays in the soil longer and is eventually lost to ET or overflow.
\(X_8\) sets the infiltration ceiling. On dry soils the full \(X_8\) mm is available, but as \(S \to X_2\) the effective infiltration drops to about \(14\%\) of \(X_8\) (because \(e^{-2} \approx 0.135\)).

Mathematical Formulation¶

Initialization¶

\[S_0 = \frac{X_2}{2}, \quad R_0 = 80, \quad T_0 = 1\]

Interception¶

\[\text{CAP} = \min(P,\, X_1,\, E)\]

\[\text{EXC} = \max(0,\; P - \text{CAP}), \quad E_1 = \begin{cases} \text{CAP} & \text{if } P > \text{CAP} \\ P & \text{otherwise} \end{cases}\]

Infiltration¶

\[\text{RINF} = X_8 \cdot \exp\!\left(-2\,\frac{S}{X_2}\right)\]

\[Q_\text{srun} = \max(0,\; \text{EXC} - \text{RINF}), \quad \text{FILT} = \min(\text{EXC},\; \text{RINF})\]

Interflow and Groundwater Recharge¶

\[\text{SINT} = \frac{S}{X_2} \cdot \frac{\text{FILT}}{X_6}\]

\[\text{REC} = \max\!\left(0,\; \frac{S}{X_2} \cdot \frac{\text{FILT} - \text{SINT}}{X_7}\right)\]

Soil Moisture Accounting¶

\[S \leftarrow S + \text{FILT} - \text{SINT} - \text{REC}\]

\[\text{EX}_2 = \max(0,\; S - X_2), \quad S \leftarrow \min(S,\; X_2)\]

\[\text{ET} = \min\!\left(E - E_1,\; 10 \cdot \frac{S}{X_2}\right), \quad S \leftarrow \max(0,\; S - \text{ET})\]

Ground Reservoir (Slow)¶

\[R \leftarrow R + \text{EX}_2 + \text{REC}\]

\[Q_r = \frac{R}{X_3 \cdot X_5}, \quad R \leftarrow R - Q_r\]

Routing Reservoir (Fast)¶

\[T \leftarrow T + \text{SINT} + Q_\text{srun} + Q_r\]

\[Q_t = \frac{T}{X_5}, \quad T \leftarrow T - Q_t\]

Total Streamflow¶

\[Q = \text{delay}(Q_t,\; X_4)\]

where \(\text{delay}(\cdot, X_4)\) is a two-weight fractional-delay convolution of base \(\lceil X_4 \rceil + 1\) time steps.

References¶

Chiew, F.H.S., Peel, M.C., & Western, A.W. (2002). Application and testing of the simple rainfall-runoff model SIMHYD. In V.P. Singh, D.K. Frevert (Eds.), Mathematical Models of Small Watershed Hydrology and Applications (pp. 335–367). Water Resources Publications.
Thiboult, A., Seiller, G., Poncelet, C., & Anctil, F. (2020). The HOOPLA toolbox: a HydrOlOgical Prediction LAboratory. Hydrology and Earth System Sciences Discussions.
Perrin, C. (2000). Vers une amélioration d'un modèle global pluie-débit au travers d'une approche comparative (PhD thesis). INPG, Grenoble.