TOPMODEL Model¶

Overview¶

TOPMODEL (the TOPography-based hydrological MODEL) was introduced by Beven and Kirkby in 1979 as one of the first conceptual rainfall-runoff models to make explicit use of digital terrain information. Its central premise is that the spatial distribution of saturated areas in a catchment can be predicted from a single statistical descriptor of the topography — the topographic index \(\ln(a/\tan\beta)\), computed for every cell of a digital elevation model. This was a radical departure from the lumped reservoir models of its era, and TOPMODEL became the conceptual ancestor of an entire family of "semi-distributed" models that try to bridge the gap between fully distributed physically-based simulators and parsimonious lumped models.

The original Beven & Kirkby formulation has roughly ten degrees of freedom (capacities, transmissivities, plus the topographic-index distribution itself). Perrin (2000) showed that for daily lumped rainfall-runoff modelling without a digital elevation model in hand, the topographic-index distribution can be approximated by a logistic function of two parameters, collapsing the structure to seven calibratable parameters. This is the version implemented in HOLMES, following Perrin's Annex 1 fiche n°33 and HOOPLA's HM18 reference implementation.

Structurally, the simplified TOPMODEL represents a catchment as three reservoirs: an interception store \(S\), an unbounded groundwater deficit store \(T\), and a quadratic surface routing reservoir \(R\). What makes it distinctive — and what makes it pedagogically valuable — is that the partitioning of effective rainfall between recharge and surface runoff is not done by a hard threshold or a saturation curve, but by a sigmoid function that depends on the current state of \(T\). The same sigmoid mechanism is used a second time to partition residual evapotranspiration between the soil and the groundwater store. Students typically choose TOPMODEL when they want to study how a smooth probabilistic partition function behaves differently from the threshold-based partitions used in models like HBV or BUCKET.

Key Concepts¶

Topographic index: A spatial descriptor of the form \(\ln(a/\tan\beta)\), where \(a\) is the upslope contributing area per unit contour length and \(\tan\beta\) is the local slope. High values mark concave wet zones near the channel; low values mark dry hilltops. In Perrin's reduction this distribution is replaced by a two-parameter logistic curve.
Interception store (\(S\)): A small surface reservoir of capacity \(X_3\) that intercepts rainfall and consumes the first share of PET. Spill from \(S\) becomes the effective rainfall \(P_r\) that drives the rest of the model.
Groundwater deficit store (\(T\)): An unbounded reservoir that tracks the catchment-average groundwater state. Unlike the soil stores in GR4J or HBV, \(T\) has no upper or lower cap — it can drift positive (saturation) or negative (deep deficit). This unboundedness is structural to TOPMODEL and is part of why initialization matters.
Sigmoid recharge partition: The fraction of effective rainfall \(P_r\) that recharges \(T\) is given by the logistic function \(1/(1 + \exp(X_6 - T/X_5))\) — a smooth probabilistic alternative to the hard saturation thresholds used in other models. As \(T\) rises (catchment wetter), the sigmoid approaches 1 and almost everything recharges; as \(T\) drops, the sigmoid drops to zero and almost everything becomes overland flow.
Sigmoid groundwater ET: A second logistic function \(1/(1 + \exp(X_7 - T/X_5))\) partitions the residual PET between the soil and the groundwater store. Following Perrin's Fiche 33 — preserved literally from HOOPLA — this term is added to \(T\) rather than subtracted, an idiosyncrasy of the formulation that students often question.
Exponential baseflow: The groundwater discharge \(Q_t = X_2 \exp(T/X_2)\) is exponential in \(T\): it grows fast when the catchment saturates and decays slowly during dry periods. The fact that \(X_2\) appears as both the prefactor and the recession scale couples the baseflow magnitude to the recession length — an unusual choice with practical consequences for calibration.
Quadratic routing reservoir (\(R\)): A nonlinear surface routing store with discharge \(Q_r = R^2/(R + X_1)\). Compared to the linear stores used in BUCKET or HBV, this quadratic form delivers sharper flood peaks at high storage and gentler tails at low storage.
Fractional-delay routing: As in GR4J/GARDENIA/SACRAMENTO, the summed outflow \(Q_t + Q_r\) passes through a two-tap shift register of length \(\lceil X_4 \rceil + 1\), letting non-integer delays translate into a smooth time shift without per-step interpolation.

How It Works¶

The TOPMODEL model processes precipitation and evapotranspiration through the following steps:

Step 1: Interception store and effective rainfall. Precipitation \(P\) is added to the interception store \(S\), then PET consumes the first share \(E_s = \min(S, E)\). The residual PET, \(E' = E - E_s\), will drive groundwater evaporation downstream. If \(S\) exceeds its capacity \(X_3\), the spill \(P_r\) becomes the "effective rainfall" that enters the rest of the model.

Step 2: Sigmoid recharge to the groundwater store (\(P_r \to T\)). The effective rainfall \(P_r\) is split into a recharge component \(P_s\) (going to \(T\)) and a quickflow component \(P_r - P_s\) (going to the surface routing store \(R\)). The split is governed by the logistic function \(P_s = P_r / (1 + \exp(X_6 - T/X_5))\) — when \(T\) is high (wet catchment), most water recharges \(T\); when \(T\) is low (dry catchment), most water becomes quickflow. The groundwater state \(T\) is then updated by \(T \leftarrow T + P_r - P_s\) — i.e., \(T\) receives only the non-recharged fraction, because the recharged \(P_s\) is treated as having flowed straight through to deeper drainage.

Step 3: Sigmoid groundwater evapotranspiration. The residual PET \(E'\) is partitioned by a second sigmoid: \(E_t = E' / (1 + \exp(X_7 - T/X_5))\). Following Perrin's Annex 1 fiche n°33 and HOOPLA HM18, this term is added to \(T\) rather than subtracted: \(T \leftarrow T + E_t\). This sign convention is preserved literally from the source even though it conflicts with the obvious physical reading; it is part of the formulation and removing it would change the calibrated behavior.

Step 4: Surface routing reservoir (\(R\), quadratic discharge). The quickflow \(P_s\) (interpreted as the saturation-excess fraction not absorbed by \(T\)) is added to the surface routing reservoir \(R\). The reservoir then drains nonlinearly as \(Q_r = R^2 / (R + X_1)\), a quadratic discharge function that produces sharper peaks at high storage and softer tails at low storage than a linear store would.

Step 5: Exponential baseflow from \(T\). The groundwater store \(T\) discharges as \(Q_t = X_2 \exp(T/X_2)\), then \(T\) is updated by \(T \leftarrow T - Q_t\). Because \(X_2\) appears as both the prefactor and the recession scale, doubling \(X_2\) both doubles the baseflow magnitude and halves the recession decay rate of \(T\) — the two are not independently calibratable, which is a known limitation of the seven-parameter reduction.

Step 6: Fractional-delay routing of the total outflow. The two outflows \(Q_t\) and \(Q_r\) are summed and pushed through a shift-and-add register \(\{HY_k\}\) of length \(n = \lceil X_4 \rceil + 1\), with two non-zero weights at \(DL_{n-2}\) and \(DL_{n-1}\) that encode the non-integer delay \(X_4\). The first element of the register, clamped at zero, is returned as the simulated streamflow for the current time step.

Parameters¶

The TOPMODEL has seven calibratable parameters.

Parameter	Description	Range	Units	Physical Interpretation
\(X_1\)	Quadratic routing reservoir capacity	1–1000	mm	Storage scale of the surface routing reservoir \(R\). Larger values smooth the hydrograph by spreading the surface response over more time steps.
\(X_2\)	Exponential groundwater drainage parameter	0.1–50	mm	Sets both the baseflow magnitude at saturation (\(Q_t = X_2\) when \(T = 0\)) and the recession decay length. The two roles are coupled.
\(X_3\)	Interception reservoir capacity	0.1–100	mm	Capacity of the canopy/depression store \(S\). Above \(X_3\) the interception spill becomes effective rainfall. Often weakly identifiable.
\(X_4\)	Routing delay	0.5–10	days	Fractional delay of the unit hydrograph register. Affects timing only, not shape.
\(X_5\)	Topographic-index distribution scale	1–200	mm	Sets the steepness of both sigmoid partition functions through the term \(T/X_5\). Smaller values make the sigmoids switch between the wet and dry regimes more abruptly.
\(X_6\)	Topographic-index sigmoid offset	-10 – 10	-	Shifts the recharge sigmoid horizontally. Negative values bias toward more recharge; positive values bias toward more quickflow.
\(X_7\)	Groundwater PET sigmoid offset	-10 – 10	-	Shifts the groundwater-ET sigmoid horizontally. Controls how much of the residual PET ends up modulating \(T\).

Understanding the parameters:

\(X_2\) is dual-roled and the most sensitive parameter: it sets both the baseflow rate at saturation and the exponential recession length. Doubling \(X_2\) doubles the baseflow magnitude and doubles the recession length scale, so an under-calibrated catchment can easily lead to runaway baseflow magnitudes if the bounds are too generous. Stay in the 1–30 mm range for most catchments unless you have a strong reason otherwise.
\(X_5\) controls the sigmoid sharpness, while \(X_6\) and \(X_7\) control the sigmoid offsets. These three parameters together fully determine the partitioning behavior of TOPMODEL. A common calibration trap is to fix \(X_5\) at a moderate value (say 50 mm) and explore \(X_6\) and \(X_7\) first — the model is much less identifiable when all three are free.
\(X_1\) shapes the flood response without affecting the long-term water balance. Small \(X_1\) produces flashy peaks; large \(X_1\) smooths the hydrograph. It is the easiest parameter to tune by visual inspection.
\(X_3\) is often weakly identifiable — interception capacities below ~5 mm produce nearly indistinguishable hydrographs in most temperate catchments. Perrin's discussion in Fiche 33 explicitly notes that \(X_3\) "may be fixed" without much loss.
\(X_4\) is a pure translation parameter: it shifts the hydrograph in time without changing its shape. Calibrate it last, after the others have settled into a reasonable shape.

Why the groundwater store is unbounded: TOPMODEL's \(T\) represents a deficit relative to a notional saturation reference, not a stock of water. There is no physical reason for it to be capped — a catchment can be arbitrarily wet (positive \(T\)) or arbitrarily dry (negative \(T\)). The exponential discharge function \(Q_t = X_2 \exp(T/X_2)\) makes the model self-regulating: when \(T\) grows, baseflow grows exponentially and pulls \(T\) back down. This is structurally different from the bounded soil stores in HBV/SACRAMENTO/GR4J.

Mathematical Formulation¶

Initialization¶

Initial reservoir states (from HOOPLA's ini_HydroMod18.m, matching Perrin's fiche):

\[S_0 = 10 \ \text{mm}, \quad T_0 = -50 \ \text{mm}, \quad R_0 = 0.2 \cdot X_1\]

The deeply negative initial \(T\) is intentional: it suppresses the exponential baseflow \(Q_t = X_2 \exp(T/X_2)\) during spin-up, letting recharge fill \(T\) toward its steady-state equilibrium without producing a startup discharge spike.

The fractional-delay routing array \(\{DL_k\}\) of length \(n = \lceil X_4 \rceil + 1\) is built so that only the last two elements are non-zero:

\[DL_{n-2} = \frac{1}{X_4 - n + 3}, \quad DL_{n-1} = 1 - DL_{n-2}\]

This is the same two-tap stencil used by GR4J, GARDENIA, and SACRAMENTO.

Surface Phase (interception store)¶

\[S \leftarrow S + P\]

\[E_s = \min(S, E), \quad S \leftarrow S - E_s, \quad E' = E - E_s\]

\[P_r = \max(0, \ S - X_3), \quad S \leftarrow S - P_r\]

Sigmoid Recharge to the Groundwater Store¶

The effective rainfall \(P_r\) is partitioned by a logistic function depending on \(T\):

\[P_s = \frac{P_r}{1 + \exp\left(X_6 - \dfrac{T}{X_5}\right)}\]

\[T \leftarrow T + (P_r - P_s)\]

Sigmoid Groundwater Evapotranspiration¶

\[E_t = \frac{E'}{1 + \exp\left(X_7 - \dfrac{T}{X_5}\right)}\]

\[T \leftarrow T + E_t\]

(The sign matches Perrin's Annex 1 fiche n°33 and HOOPLA HM18 literally — see the implementation note in the source.)

Surface Routing Reservoir¶

The quickflow \(P_s\) enters the quadratic routing reservoir \(R\):

\[R \leftarrow R + P_s\]

\[Q_r = \frac{R^2}{R + X_1}, \quad R \leftarrow R - Q_r\]

Exponential Baseflow from \(T\)¶

\[Q_t = X_2 \exp\left(\frac{T}{X_2}\right), \quad T \leftarrow T - Q_t\]

Total Streamflow and Fractional Delay¶

The two outflows are summed and pushed through the shift-and-add delay register \(\{HY_k\}\) of length \(n = \lceil X_4 \rceil + 1\):

\[Q = Q_t + Q_r\]

\[HY_k \leftarrow HY_{k+1} + DL_k \cdot Q \quad \text{for } k = 0, 1, \ldots, n - 2\]

\[HY_{n-1} \leftarrow DL_{n-1} \cdot Q\]

\[Q_{\text{sim}} = \max(0, \ HY_0)\]

The first element of the register is returned as the simulated streamflow; the register then advances by one position, ready for the next step.

References¶

Beven, K. J., & Kirkby, M. J. (1979). A physically based, variable contributing area model of basin hydrology. Hydrological Sciences Bulletin, 24(1), 43–69. https://doi.org/10.1080/02626667909491834

Beven, K. (1997). TOPMODEL: a critique. Hydrological Processes, 11(9), 1069–1085.

Franchini, M., Wendling, J., Obled, C., & Todini, E. (1996). Physical interpretation and sensitivity analysis of the TOPMODEL. Journal of Hydrology, 175, 293–338.

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°33 (TOPMODEL), pp. 453–458.

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