Concepts¶

This section introduces the fundamental concepts behind rainfall-runoff modeling and explains the hydrological models, algorithms, and metrics implemented in HOLMES.

What is Rainfall-Runoff Modeling?¶

Rainfall-runoff modeling is the process of simulating how precipitation falling on a catchment transforms into streamflow at the outlet. This transformation involves complex physical processes: water infiltrates into the soil, evaporates back to the atmosphere, percolates to groundwater, and eventually reaches the stream through various pathways. A rainfall-runoff model attempts to represent these processes mathematically, allowing us to predict streamflow from meteorological inputs.

Understanding these models is essential for water resources management, flood forecasting, drought assessment, and infrastructure design. Rather than solving the full physics of water movement through soil and aquifers (which would require detailed spatial data rarely available), conceptual models use simplified representations that capture the essential behavior of catchment hydrology.

The Water Balance Concept¶

At its core, hydrological modeling relies on the water balance equation:

\[\frac{dS}{dt} = P - E - Q\]

where:

\(S\) is the water stored in the catchment (soil moisture, groundwater, snow)
\(P\) is precipitation (rain and snow)
\(E\) is evapotranspiration (water returning to the atmosphere)
\(Q\) is streamflow at the outlet

This simple equation states that changes in storage equal inputs minus outputs. All conceptual hydrological models are elaborations of this principle, adding reservoirs, pathways, and time delays to represent how water moves through the system.

The HOLMES Modeling Chain¶

HOLMES implements a complete modeling chain for rainfall-runoff simulation. Each step builds on the previous one:

1. Potential Evapotranspiration (PET)¶

Before running a hydrological model, we need to estimate the atmospheric demand for water. PET represents the maximum amount of water that would evaporate and transpire if water were unlimited. HOLMES uses the Oudin method, which estimates PET from temperature and solar radiation alone, making it practical when detailed meteorological data are unavailable.

2. Snow Accumulation and Melt¶

In catchments with significant snowfall, precipitation does not immediately contribute to runoff. Snow accumulates during cold periods and releases water during melt, fundamentally altering the timing of streamflow. The CemaNeige model tracks snowpack evolution using a degree-day approach, partitioning precipitation between rain and snow and calculating melt based on temperature.

3. Hydrological Transformation¶

The core of the modeling chain is the rainfall-runoff model that transforms effective precipitation (rainfall plus snowmelt) into streamflow. HOLMES implements several models:

Bucket model: A six-parameter model based on linear reservoir theory with explicit fast and slow flow paths. Offers more flexibility in flow partitioning and often captures recession behavior well.
CEQUEAU: A nine-parameter two-reservoir model (the "CEQU" simplification of the original CEQUEAU) that produces multiple threshold-based and continuous flow pathways, giving it considerable flexibility in hydrograph shape.
CREC: A six-parameter model featuring a sigmoid rainfall-splitting function that smoothly partitions precipitation between runoff and infiltration based on soil moisture. Uses nonlinear (quadratic) surface routing.
GARDENIA: A six-parameter BRGM model originally developed for rainfall → piezometric-level forecasting. Uses three reservoirs in series with a quadratic soil outflow law and a calibratable PET correction coefficient.
GR4J: A parsimonious four-parameter model widely used in research and operations. It represents the catchment as two stores (production and routing) connected by unit hydrographs.
HBV: A nine-parameter model following Bergström's HBV0 formulation from Perrin's thesis. Uses a non-linear soil production with five-sub-step integration, a two-outflow intermediate reservoir, capped percolation, and a triangular unit hydrograph.
IHACRES: A seven-parameter model from Jakeman et al. (1990) built around a dimensionless catchment moisture index with PET-modulated drying, midpoint-trapezoidal effective rainfall, and parallel fast/slow linear routing reservoirs sharing a multiplicative time-constant coupling.
HYMOD: A six-parameter model using a Pareto-distributed soil moisture store (variable-source-area runoff generation) combined with three linear reservoirs in cascade for fast flow and one linear groundwater reservoir for baseflow.
MARTINE: A seven-parameter BRGM model (Mazenc et al. 1984) with overflow-based surface production, a calibratable fast/slow distribution coefficient, quadratic direct routing, a dual-pathway intermediate reservoir (linear drainage + overflow), and linear groundwater recession.
MOHYSE: A seven-parameter minimalist model (Fortin & Turcotte 2007) with capacity-limited infiltration, dual soil/groundwater linear reservoirs, three-pathway linear drainage, and a gamma-shaped unit hydrograph for routing.
MORDOR: A six-parameter EDF model (Garçon 1999) with four cascading reservoirs (surface → intermediate → deep soil → groundwater), proportional rainfall partitioning, nonlinear cubic groundwater discharge, and three-component double-sided UH2 routing with exponent 2.5.
NAM: A ten-parameter port of HOOPLA's HM12 version of the Nielsen & Hansen (1973) Danish operational model. Seven reservoirs (surface, soil, two interflow cascade reservoirs, two overland-flow cascade reservoirs, and a groundwater deficit store) with three-branch evapotranspiration, capillary rise from the saturated zone, and a fractional-delay unit hydrograph.
PDM: An eight-parameter Probability-Distributed Model (Moore & Clarke 1981) with Pareto-distributed soil moisture capacity, threshold-gated drainage to a cubic groundwater store, two-stage linear cascade for fast routing, and a fractional-delay unit hydrograph.
SACRAMENTO: A nine-parameter variant of the Burnash et al. (1973) NWSRFS operational model following Perrin's simplification. Uses five reservoirs (interception, tension water, free water, lower-zone routing, direct routing) with a filling-feedback percolation scheme, interflow and hypodermic pathways, and an upward mass-balance correction between the lower-zone store and the free-water store.
SIMHYD: An eight-parameter Australian model (Chiew et al. 2002) with exponential infiltration capacity decaying with soil saturation, saturation-proportional interflow and groundwater recharge, and dual linear routing reservoirs (slow ground + fast routing) with fractional-delay routing.
SMAR: An eight-parameter model (O'Connell et al. 1970) with a 16-layer discretised soil column (25 mm per layer, 400 mm total), exponentially decaying ET with depth, moisture-dependent direct runoff, dual linear/quadratic routing reservoirs with calibratable flow partitioning, and a fractional-delay unit hydrograph.
TANK: A seven-parameter Perrin variant of the Sugawara (1979) model (HOOPLA HM17), organising the catchment as a vertical cascade of four linear reservoirs with dual-threshold side outlets on the surface store, geometric scaling of drain time constants, cascading top-down ET satisfaction, a calibratable PET correction, and a fractional-delay unit hydrograph routing the sum of five outflows.
TOPMODEL: A seven-parameter variant of the Beven & Kirkby (1979) topographic-index model following Perrin's simplification. An interception store, an unbounded groundwater deficit store with two sigmoid partition functions (recharge and evapotranspiration), a quadratic surface routing reservoir, and a fractional-delay unit hydrograph — pedagogically interesting because it replaces hard saturation thresholds with smooth probabilistic partitions.
WAGENINGEN: An eight-parameter conceptual model (Warmerdam et al. 1997, HOOPLA HM19) with a single soil-moisture threshold \(X_1\) that switches between percolation and capillary rise, cosine-damped ET below the threshold, flow dissociation via the \(T/X_5\) ratio, parallel fast/slow linear reservoirs with multiplicatively coupled time constants, and fractional-delay routing.
XINANJIANG: An eight-parameter variant of the Zhao et al. (1980) Chinese operational model. Uses two power-distributed saturation-excess reservoirs in series (soil + free-water) feeding a calibratable fast/slow routing split and a two-tap fractional-delay unit hydrograph.

4. Model Calibration¶

Hydrological models have parameters that cannot be measured directly and must be estimated by comparing model outputs to observed streamflow. This process, called calibration, searches for parameter values that minimize the difference between simulated and observed flows. HOLMES uses the SCE-UA algorithm, a global optimization method designed specifically for hydrological model calibration.

5. Performance Evaluation¶

After calibration, we need to assess how well the model performs. HOLMES provides several performance metrics that quantify different aspects of model accuracy:

RMSE measures average error magnitude
NSE measures skill relative to using the mean as a predictor
KGE decomposes performance into correlation, variability bias, and mean bias

Choosing the Right Model¶

The choice of model depends on your catchment characteristics and objectives. Each row below links to the full concept page for that model:

Model	Params	Soil store	Flow partitioning	Routing	GW exchange	Equifinality	Best for
Bucket	6	Single bucket	Calibratable (\(\alpha\), \(\beta\))	Linear reservoirs	No	Higher	Catchments with distinct recession components
CEQUEAU	9	Two-reservoir (surface + groundwater)	Threshold + continuous pathways	Pure time delay	No	Higher	Flexible hydrograph shapes, threshold-driven response
CREC	6	Single bucket + sigmoid split	Sigmoid (moisture-dependent)	Quadratic + linear stores	No	Moderate	Catchments with moisture-dependent runoff generation
GARDENIA	6	Surface + soil + groundwater in series	Overflow at surface + quadratic soil outflow	Fractional delay	No	Moderate	Catchments with a strong aquifer component; rainfall → piezometric-level use cases
GR4J	4	Single reservoir	Fixed 90% / 10%	Unit hydrographs + nonlinear store	Yes (\(X_2\))	Lower	Humid temperate catchments, benchmarking
HBV	9	Non-linear soil (five sub-steps)	Threshold upper + linear lower intermediate reservoir	Triangular unit hydrograph	No	Higher	Nordic / temperate catchments, operational forecasting
IHACRES	7	Dimensionless moisture index (unbounded, PET-modulated decay)	Calibratable fast/slow fraction (\(X_2\))	Parallel linear reservoirs (\(X_3\) / \(X_3 \cdot X_4\)) + fractional delay	No	Moderate	Catchments where recession analysis drives calibration; teaching contrast to soil-bucket models
HYMOD	6	Pareto-distributed (variable source area)	Saturation excess + calibratable \(\alpha\)	Three linear reservoirs cascade + one slow reservoir	No	Moderate	Catchments where saturated-area runoff dominates
MARTINE	7	Single bucket (overflow)	Calibratable fast/slow fraction (\(X_5\))	Quadratic direct store + dual-pathway intermediate + linear GW + fractional delay	No	Moderate	Regionalization studies; catchments with distinct interflow and baseflow components
MOHYSE	7	Capacity-limited single bucket	Linear vadose drainage to river + GW	Gamma unit hydrograph (80-step memory)	No	Moderate	Benchmarking; minimal-complexity baseline; teaching production-routing-convolution chain
MORDOR	6	Four cascading reservoirs (U → L → Z → N)	Proportional to U filling + Z-ratio partitioning of L drainage	Three-component double-sided UH2 (exponent 2.5)	No	Moderate	Catchments with significant baseflow; teaching cascading reservoir chains
NAM	10	Surface store + soil store with capillary rise	Three-branch ET + soil-ratio-driven overland/interflow/percolation split	Two parallel two-reservoir cascades + fractional-delay UH	Yes (deficit-based, threshold \(X_1\))	Higher	Catchments where overland flow and interflow must be modelled separately; Scandinavian operational use cases
PDM	8	Pareto-distributed (variable source area)	Saturation excess + infiltration excess + threshold drainage	Two linear reservoirs cascade + cubic GW store + fractional delay	No	Moderate	Catchments with variable-source-area runoff and nonlinear baseflow recession; British operational use cases
SACRAMENTO	9	Interception + tension + free water (three-store cascade)	Percolation with filling-feedback + threshold overflow	Direct routing store + fractional-delay register	Yes (via \(X_8\) deep percolation)	Higher	Catchments with clear baseflow separation; operational NWS-style use cases
SIMHYD	8	Interception + soil bucket (exponential infiltration)	Saturation-proportional interflow + GW recharge	Ground (slow) + routing (fast) linear reservoirs + fractional delay	No	Moderate	Australian catchments; benchmarking threshold-based infiltration models
SMAR	8	16-layer discretised soil column (400 mm)	Moisture-dependent direct runoff + exponential infiltration	Linear GW + quadratic surface reservoirs + fractional delay	No	Moderate	Teaching vertical soil discretisation; catchments where ET depth-profile matters
TANK	7	Four linear reservoirs in vertical cascade (S → R → T → L)	Dual-threshold side outlets on \(S\) + single side outlets on \(R\), \(T\)	Geometric drain-time scaling (\(x_3\), \(x_3 x_4\), \(x_3 x_4 x_7\), \(x_3 x_4 x_7^2\)) + fractional delay	No	Moderate	Catchments with emergent multi-timescale recessions; teaching storage-driven flow separation
TOPMODEL	7	Interception + unbounded groundwater deficit	Sigmoid recharge + sigmoid groundwater ET (logistic, no thresholds)	Quadratic surface store + exponential baseflow + fractional delay	No	Moderate	Catchments where smooth saturation-area dynamics matter; pedagogical contrast with threshold-based models
WAGENINGEN	8	Single soil reservoir with threshold \(X_1\) and capillary rise from \(T\)	Flow dissociation via \(\mathrm{DIV} = \min(1, T/X_5)\)	Parallel fast (\(X_6\)) + slow (\(X_6 \cdot X_7\)) linear reservoirs + fractional delay	Upward (capillary rise \(T \to S\))	Moderate	Humid-temperate catchments with clear wet/dry regime switching; teaching threshold-driven process coupling
XINANJIANG	8	Two power-distributed reservoirs (soil + free-water)	Saturation excess (fixed \(B = 0.25\), calibratable \(X_8\))	Fast/slow linear reservoirs + two-tap unit hydrograph	No	Moderate	Catchments with strong spatial variability of storage capacity; Chinese / monsoon operational use cases

For catchments with significant snow, enable CemaNeige regardless of which hydrological model you choose.