WAGENINGEN Model¶

Overview¶

WAGENINGEN is an eight-parameter daily lumped rainfall-runoff model developed by Warmerdam, Kole, and Chormanski (1997) at Wageningen Agricultural University in the Netherlands. It was designed as a parsimonious conceptual model suited to humid-temperate catchments and operational forecasting.

The variant implemented in HOLMES is the HOOPLA HM19 port described in Annex 1 of Perrin (2000). The model represents the catchment as three reservoirs in a production–routing chain: a soil reservoir \(S\) that controls evapotranspiration and percolation, a slow sub-intermediate reservoir \(T\) that can feed water back up to \(S\) by capillary rise, and a fast surface reservoir \(R\) — both routing stores drain through a common fractional-delay unit hydrograph.

The defining feature of WAGENINGEN is the threshold-based split at soil-moisture level \(X_1\). When \(S \geq X_1\), the soil is wet enough to drain to the routing system and evapotranspires at the full atmospheric demand; when \(S < X_1\), percolation stops, capillary rise transfers water upward from \(T\), and evapotranspiration follows a cosine envelope that tapers smoothly toward zero as the soil dries. This gives a pedagogically useful illustration of how a single wetness threshold can generate four coupled process regimes (drainage, capillary rise, full ET, suppressed ET) without introducing additional calibration parameters.

Key Concepts¶

Soil-moisture threshold \(X_1\): The single wetness level that switches the model between a wet regime (percolation active, ET unrestricted) and a dry regime (percolation off, capillary rise from \(T\) into \(S\) active, ET damped).
Cosine-envelope evapotranspiration: Below the threshold, actual ET is scaled by \(\cos\!\bigl((\pi/2)\cdot(X_1 - S)/X_1\bigr)\). This produces a smooth, concave taper from full atmospheric demand at \(S = X_1\) down to zero at \(S = 0\) — gentler than a linear reduction and numerically stable.
Capillary rise \(I_t\): When \(S < X_1\), water moves upward from the slow reservoir \(T\) into \(S\) at a rate proportional to both the storage in \(T\) and the soil-moisture deficit \((X_1 - S)\). This is the only HOLMES model that represents capillary rise explicitly.
Flow dissociation via \(\mathrm{DIV} = \min(1, T/X_5)\): The ratio of storage in \(T\) to the dissociation threshold \(X_5\) decides the split of percolation \(I_s\) between fast and slow routing — when \(T\) is full the slow path saturates and everything goes fast; when \(T\) is empty all percolation goes slow to refill it.
Fast reservoir \(R\): Linear routing store with time constant \(X_6\), producing storm-event peaks.
Slow reservoir \(T\): Linear routing store with time constant \(X_6 \cdot X_7\) (with \(X_7 \geq 1\)), producing sustained baseflow. The same store supplies capillary rise into \(S\).
Fractional-delay routing: A pure channel delay \(X_8\) (in days) shifts the combined hydrograph, interpolating between the two adjacent integer delays.

How It Works¶

Step 1: Soil-moisture update. Precipitation \(P\) is added to the soil reservoir: \(S \leftarrow S + P\). The current value of \(S\) relative to the threshold \(X_1\) determines which of the two regimes applies for the rest of the step.

Step 2: Percolation or capillary rise. If \(S \geq X_1\), percolation drains the soil: \(I_s = (S/X_2) \cdot (S - X_1)/X_3\), and no capillary rise occurs. If \(S < X_1\), percolation stops and capillary rise lifts water from \(T\) back to \(S\): \(I_t = (T/X_4) \cdot (X_1 - S)\). The soil reservoir is then updated: \(S \leftarrow S + I_t - I_s\).

Step 3: Evapotranspiration. Above the threshold, actual ET equals the demand: \(E_s = E\). Below the threshold, ET is damped by a cosine envelope: \(E_s = E \cdot \cos\!\bigl((\pi/2)\cdot(X_1 - S)/X_1\bigr)\). The soil storage is updated and clamped at zero: \(S \leftarrow \max(0, S - E_s)\).

Step 4: Flow dissociation. The percolated water \(I_s\) is split between the two routing reservoirs by the dissociation ratio \(\mathrm{DIV} = \min(1, T/X_5)\). A fraction \(\mathrm{DIV} \cdot I_s\) goes to the fast reservoir \(R\) and the remainder \((1 - \mathrm{DIV}) \cdot I_s\) goes to the slow reservoir \(T\). When \(T\) is near empty, the slow path absorbs most flux; when \(T\) is well-filled, the fast path dominates.

Step 5: Linear routing. Both routing reservoirs drain with linear-reservoir laws. The fast reservoir empties with time constant \(X_6\): \(Q_r = R/X_6\). The slow reservoir empties with time constant \(X_6 \cdot X_7\): \(Q_t = T/(X_6 \cdot X_7)\). Since \(X_7 \geq 1\), the slow reservoir is always at least as slow as the fast one.

Step 6: Delay routing. The combined outflow \(Q_r + Q_t\) is injected into the fractional-delay register of size \(\lceil X_8 \rceil + 1\). The register shifts by one step each day, and the first element, clamped at zero, is returned as simulated streamflow.

Parameters¶

The WAGENINGEN model has eight calibratable parameters:

Parameter	Description	Range	Units	Physical Interpretation
\(X_1\)	Percolation/ET threshold	1–500	mm	Soil-moisture level above which percolation activates and ET runs unrestricted. Acts as the "field capacity" of the model — the higher it is, the larger the storage the catchment must build before generating runoff.
\(X_2\)	Soil reservoir characteristic capacity	10–2000	mm	Appears in the denominator of the percolation rate. Larger values slow the drainage of the soil store and favor sustained moisture.
\(X_3\)	Infiltration emptying constant	0.1–1000	mm	Second denominator in the percolation rate. Together with \(X_2\) it controls how rapidly the soil drains above threshold.
\(X_4\)	Capillary rise constant	1–1000	d	Time-constant analogue for the upward flux from \(T\) to \(S\). Smaller values produce faster capillary recovery from drought.
\(X_5\)	Flow dissociation threshold	0.1–500	mm	Storage level in \(T\) at which the slow path saturates and all percolation goes fast. Small values make the model flash-prone; large values favor groundwater storage.
\(X_6\)	Fast-reservoir time constant	0.5–50	d	Residence time of \(R\). Controls storm recession speed.
\(X_7\)	Slow/fast time-constant ratio	1–50	-	Multiplicative factor: slow-reservoir time constant is \(X_6 \cdot X_7\). Constrained \(\geq 1\) so the slow path is genuinely slower than the fast path.
\(X_8\)	Routing delay	0.5–5	d	Pure channel travel-time shift applied at the outlet.

Understanding the parameters:

\(X_1\) is the single most sensitive parameter. It sets the wet/dry regime boundary: too low and the model almost always percolates and evaporates at full rate (linear-reservoir behavior); too high and the soil never drains (all precipitation lost to ET). Expect calibrated values between 50 and 200 mm on humid catchments.
\(X_2\) and \(X_3\) jointly scale the percolation rate via \((S/X_2) \cdot (S - X_1)/X_3\). Because they appear multiplicatively in a denominator, they are strongly equifinal — increasing one and decreasing the other gives the same drainage, so calibration typically fixes their ratio rather than each individually.
\(X_5\) controls flashiness. If \(T\) oscillates around \(X_5\), small changes in \(T\) can swing \(\mathrm{DIV}\) from 0 to 1 and cause the model to suddenly dump percolation into the fast reservoir. Values well above the typical \(T\) storage produce a mostly-slow response; values far below give a mostly-fast response.
\(X_6\) and \(X_7\) are the routing knobs. \(X_6\) sets the fast recession timescale (typical catchment storm recessions are 2–10 days) and \(X_7\) lengthens the slow path (baseflow recession over weeks to months implies \(X_7 \approx 10\)–\(30\)).
Capillary rise \(X_4\) is only active in the dry regime. On catchments with a persistently wet soil (\(S \geq X_1\) most of the time), \(X_4\) barely moves the objective function — expect weak identifiability.

Mathematical Formulation¶

Initialization¶

Fixed reservoir states at \(t = 0\) (from HOOPLA ini_HydroMod19.m):

\[S_0 = 30 \text{ mm}, \quad R_0 = 0 \text{ mm}, \quad T_0 = 200 \text{ mm}\]

The fractional-delay register \(\{H_k\}\) has \(n = \lceil X_8 \rceil + 1\) elements, initialized to zero, with weights:

\[d_{n-2} = \frac{1}{X_8 - n + 3}, \quad d_{n-1} = 1 - d_{n-2}\]

Soil Moisture Update¶

\[S \leftarrow S + P\]

Percolation and Capillary Rise¶

Mutually exclusive fluxes depending on the threshold \(X_1\):

\[ I_s = \begin{cases} \dfrac{S}{X_2} \cdot \dfrac{S - X_1}{X_3} & \text{if } S \geq X_1 \\ 0 & \text{if } S < X_1 \end{cases} \]

\[ I_t = \begin{cases} 0 & \text{if } S \geq X_1 \\ \dfrac{T}{X_4} \cdot (X_1 - S) & \text{if } S < X_1 \end{cases} \]

\[S \leftarrow S + I_t - I_s\]

Evapotranspiration¶

\[ E_s = \begin{cases} E & \text{if } S \geq X_1 \\ E \cdot \cos\!\left(\dfrac{\pi}{2} \cdot \dfrac{X_1 - S}{X_1}\right) & \text{if } S < X_1 \end{cases} \]

\[S \leftarrow \max(0,\; S - E_s)\]

Flow Dissociation¶

\[\mathrm{DIV} = \min\!\left(1,\; \frac{T}{X_5}\right)\]

\[T \leftarrow T + (1 - \mathrm{DIV}) \cdot I_s\]

\[R \leftarrow R + \mathrm{DIV} \cdot I_s\]

Linear Routing¶

\[Q_r = \frac{R}{X_6}, \quad R \leftarrow R - Q_r\]

\[Q_t = \frac{T}{X_6 \cdot X_7}, \quad T \leftarrow T - Q_t\]

Delay Routing¶

\[Q = Q_r + Q_t\]

\[H_k \leftarrow H_{k+1} + d_k \cdot Q \quad \text{for } k = 0, 1, \ldots, n - 2\]

\[H_{n-1} \leftarrow d_{n-1} \cdot Q\]

\[Q_{\text{sim}} = \max(0,\; H_0)\]

References¶

Warmerdam, P. M. M., Kole, J., & Chormanski, J. (1997). Modelling rainfall-runoff processes in the Hupselse Beek research basin. IHP-V Technical Documents in Hydrology, 14, 155–160. UNESCO, Paris.

Perrin, C. (2000). Vers une amélioration d'un modèle global pluie-débit au travers d'une approche comparative. PhD dissertation, Institut National Polytechnique de Grenoble, France. Annex 1, Fiche n°34 (WAGE), pp. 461–464.

Thiboult, A., Seiller, G., Poncelet, C., & Anctil, F. (2020). The HOOPLA toolbox: a HydrOlOgical Prediction LAboratory. Hydrology and Earth System Sciences Discussions.