This study explores various aspects of catchment hydrology based on a mechanistic modeling of distributed watershed processes. A new physics-based, distributed-parameter hydrological model that uses an irregular spatial discretization is introduced. The model accounts, on a continuous basis, for the processes of rainfall interception, evapotranspiration, moisture dynamics in the unsaturated and saturated zones, and runoff routing. Simulations of several mid- to large-sized watersheds (∼103 km2) highlight various dynamic relationships between the vadose zone–groundwater processes and their dependence on the land surface characteristics. It is argued that the model inferences can be used for interpretation of distributed relationships in a catchment. By exploiting a multiple-resolution representation, the hydrologic features of the watershed terrain are captured with only 5–10% of the original grid nodes. This computational efficiency suggests the feasibility of the operational use of fully distributed, physics-based models for large watersheds.
 A central theme in distributed model development has been the premise that the explicit representation of spatially varying fields should lead to significant advances in the skill to simulate and forecast hydrologic response [e.g., Beven, 1989; Pessoa et al., 1993; Goodrich et al., 1995]. Distributed models can also serve to elucidate the complexity of hydrologic processes interacting in time and space. This paper uses an advanced distributed model to demonstrate the importance of process spatial and temporal interconnection in the basin response.
 The ability for detailed representation of spatial information (e.g., topography, soils, vegetation, and meteorological forcing) and hydrological dynamics comes at the expense of the computational burden. This common drawback significantly limits efficient distributed model calibration, real-time execution, and ensemble averaging of the model results. As a result, large-scale applications have traditionally used coarse model resolutions leading to the unavoidable loss of information in the detailed spatially distributed catchment data. The principal implication of model coarsening is the distortion of the simulated hydrological dynamics.
 This study provides new insights on spatially explicit basin dynamics and explores the potential for utilizing fully distributed approaches in studying catchment hydrology at large scales. The paper addresses the spatial aspects of hydrological mechanisms revealing the dynamic relationships between the unsaturated-saturated zone processes and their connection with the catchment land surface features. The work builds on a new modeling approach that represents the simulation domain via a triangulated irregular network (TIN). The flexible and multiscale computational structure, that can be tuned to the processes under study, thus allows to accurately capture the topography, drainage network, floodplain, and soil and landuse features without a significant loss in detail [Vivoni et al., 2004]. The following sections present the model and subsequently discuss the insights gained in simulating several mid- to large-scale watersheds located in Oklahoma.
2. Description of the Modeling Approach: General Overview of the Framework
2.1. Model Heritage: RIBS and CHILD
 The Real-time Integrated Basin Simulator (RIBS) of Garrote and Bras  implemented an event-based scheme for rainfall-runoff analysis. RIBS was the raster-based predecessor of the model detailed later: tRIBS (“t” stands for the “TIN-based”). tRIBS has inherited the functionality of RIBS while adding the hydrology necessary for continuous operation. The hydrologic framework has been fully integrated with the software and data platform of the TIN-based Channel-Hillslope Integrated Landscape Development model (CHILD) [Tucker et al., 2001a].
2.2. Continuous, Coupled Hydrologic Model
 The spatial variation of atmospheric forcing and land surface characteristics create complex time-varying interdependencies in catchment hydrology [Freeze and Witherspoon, 1968; Salvucci and Entekhabi, 1995; Kim et al., 1999]. In order to elucidate the principal physical mechanisms controlling the hydrologic response, the lateral and vertical interdependencies should be explicitly accounted for in a model. The hydrologic framework of tRIBS emphasizes the dynamic relationship between the vadose zone and the saturated zone by simulating the interaction of infiltration fronts with a variable groundwater surface through a simplified coupled system. The continuous mode of operation can simulate multiple storm events and handle moisture losses during interstorm periods. By modeling these processes, different runoff generation mechanisms are reproduced thus capturing the highly coupled hydrologic response in a basin.
2.3. Topographic Representation
 In a watershed model, topography can be represented utilizing a number of computational structures, including contour-based streamtubes, raster domains, and TINs. The primary motivation for the use of TINs is the multiple resolutions offered by the irregular domain. This translates to computational savings as the number of nodes can be significantly reduced. Another advantage is that the linear features can be precisely preserved in the mesh which allows mimicking of terrain breaklines, stream networks, and boundaries between heterogeneous regions. The utilized methodology for constructing a watershed TIN model from a raster-based digital elevation model (DEM) preserves critical hydrologic features in the terrain to properly account for the conceptual understanding of basin dynamics [Vivoni et al., 2004]. The methodology involves several steps, including sampling a high-resolution DEM, deriving a stream network, and embedding riparian or floodplain areas near the channel network into the TIN (Figure 1).
2.4. Geometry of the Computational Element
 A finite difference control volume approach is used to estimate the states of the dynamic variables. The computational framework relies on the basic geometry of the control volumes defined for the mesh nodes, Voronoi regions (same as Thiessen polygons (Figure 2a)), and the connectivity between elements [Tucker et al., 2001a, 2001b]. The reference system of a Voronoi cell is defined by the axes p and n, where p follows the direction parallel to the plane of the maximum slope and n follows the direction normal to that plane (Figure 2b). The state variables of the one-dimensional mass flow equations, when applied to a Voronoi cell, are a function of the direction n. The boundaries between Voronoi regions define the interfaces between adjoining cells. When a mass moves into a neighboring cell, the length of a given interface is used as the width in the flux computation.
2.5. Flow Pathways and Drainage Networks on a TIN
2.5.1. Overland Drainage Network
 A number of drainage algorithms have been developed for TIN-based models [Palacios-Velez and Cuevas-Renaud, 1986; Gandoy-Bernasconi and Palacios-Velez, 1990; Jones et al., 1990; Nelson et al., 1994; Braun and Sambridge, 1997]. Most methods assign flow pathways both across and between triangles using uniquely defined orientation and connectivity of the triangular elements. Besides their complexity, such schemes have difficulties in interpretation of the land surface parameters and hydrologic state variables. The assumed approach, following Tucker et al. [2001a], routes the surface flow along the steepest edge connected to a given node, i.e., along the direction of the p axis. The flow is thus constrained to follow triangle edges rather than the triangular surfaces. Tucker et al. [2001b] give additional details on the construction of the overland drainage network in the TIN model which includes resolving pathways from closed depressions and implementation of upstream-to-downstream computational order.
2.5.2. Stream Network
 Surface overland flow reaches stream nodes, predefined in the input mesh, that correspond to the DEM-derived channel network. However, mesh undersampling and the presence of sinks in certain areas may result in drainage loops or disjoints in the channel network. An approach borrowed from graph theory [e.g., West, 1996] is used to resolve these difficulties. The general idea of the developed method consists of constructing globally optimized channel pathways that satisfy certain geometric considerations. In essence, each potential stream drainage link, i.e., each triangle edge that connects two stream nodes, is assigned a specific weight that represents a function of link length and slope. Starting from the basin outlet, the algorithm successively optimizes the choice of node connectivity to better fit the actual stream network.
 In order to apply a one-dimensional hydraulic routing model within the channel network, the basic computational domains, the stream reaches, have to be defined. This is done by identifying the confluence points after traversing the channel network from stream heads to the basin outlet.
3. Description of the Modeling Approach: Hydrologic Components
3.1. Rainfall Interception
 The canopy water balance model [Rutter et al., 1971, 1975] provides a complete method for modeling rainfall interception. It relates changes in the canopy storage C to the rainfall rate R, canopy drainage D, and potential evaporation rate Ep in the form
where parameters S and p are the canopy capacity and free throughfall coefficient. While the rainfall is prescribed from observations, the potential evaporation is computed based on meteorological and surface conditions. The canopy drainage, accounting for water losses from leaf dripping and stemflow, is modeled as [Shuttleworth, 1979]
where K and g are the drainage rate coefficient and exponential decay parameter.
3.2. Energy Balance and Surface Fluxes
 Current meteorological conditions and the antecedent soil moisture state define the amount of energy going into evaporation moisture loss. The surface energy balance describes the partitioning of net radiation Rn into sensible H, latent λE, and ground G heat fluxes at the soil surface:
Since each term can be expressed as a function of the soil surface temperature, an iterative scheme is used to solve each component for given meteorological conditions.
3.2.1. Short and Longwave Radiation
 The net radiation component Rn is composed of net incoming short wave radiation Rsi, incoming longwave radiation Rli, and outgoing longwave radiation Rlo: Rn = Rsi + Rli − Rlo. Parameterizations detailed in Bras [1990, pp. 31–47] are utilized for each radiative flux component. The incoming short wave radiation is a combination of various inputs resulting in its significant spatial variability:
where a is the albedo, Kt is the optical transmission coefficient, N is the cloud cover. Rdirs and Rdifs are the direct and diffuse solar radiation fluxes. These variables account for the geographic location, time of year, aspect of the element surface and its slope [Bras, 1990, pp. 21–42]. Rrefs is the radiation component which accounts for the reflected radiation from other sloping surfaces [Wilson and Gallant, 2000]. The incoming longwave radiation is modeled using gray body theory:
where σ is the Stefan-Boltzman constant, Kc is a function of cloud cover, Ea is the atmospheric thermal emissivity and Ta is the air temperature. Similarly, the outgoing longwave radiation is
where Es is the surface emissivity and Ts is the soil surface temperature.
3.2.2. Latent Heat Flux
 The Penman-Monteith approach [Penman, 1948; Monteith, 1965] combines the energy and mass transfer techniques as a tool for estimating the surface latent heat flux λE:
where Δ is the slope of Clausius-Clayperon relationship, γ is the psychometric constant, ρm is the moist air density, λν is the latent heat of vaporization, δqa is specific humidity deficit, ra is the aerodynamic resistance, and rs is the stomatal resistance. The state variables are calculated using standard meteorological relationships described in Rogers and Yau  and Bras . The resistance terms are computed utilizing methods described in Shuttleworth .
3.2.3. Sensible Heat Flux
 The sensible heat flux component H of the surface energy balance is computed from the gradient of the surface Ts and air temperature Ta using an aerodynamic surface resistance approach (Cp is the specific heat capacity of dry air):
3.2.4. Ground Heat Flux
 The ground heat flux G is determined using the force-restore model as described by Lin  and Hu and Islam . This method is based on solving the heat diffusion equation between a soil surface layer and a deeper soil profile. Both the surface Ts and the deep soil Td temperatures are obtained. The flux G is obtained from Lin  as
where Cs is soil heat capacity, ω is daily frequency of oscillation, d1 = is the soil heat wave damping depth, k = ks/Cs is the soil diffusivity, and ks is the soil heat conductivity. The parameter ξ is computed using the Hu and Islam  parameterization.
 Following Wigmosta et al. , three evaporation components are estimated: evaporation from wet canopy Ewc, canopy transpiration Edc, and bare soil evaporation Es. The latent heat flux, computed from the energy balance at the surface, provides an estimate for the actual evaporation Ea, while the potential evaporation rate Ep is obtained as [Wigmosta et al., 1994]
A vegetative fraction, v, for each computational element determines the proportion undergoing canopy and bare soil evaporation or transpiration. These processes are controlled by the amount of moisture available in the upper soil layer. The bare soil evaporation is [Deardorff, 1978]
where βe is determined from the saturation soil moisture θs and the soil moisture in the top 100 mm of the soil column θ100:
For the vegetation fraction, evaporation has two components: evaporation from the canopy Ewc and transpiration Edc which are related in the following way [Eltahir and Bras, 1993]:
Transpiration occurs at a rate
where βt accounts for the current soil moisture stress which limits the root water uptake. A simplified approach is used to parameterize this factor:
where θtop is the soil moisture content in the top meter of the soil column and θr is the residual moisture content [Brooks and Corey, 1964]. Of the three evapotranspiration components, only transpiration Edc and bare soil evaporation Es contribute to the depletion of the surface soil moisture. The wet canopy evaporation, Ewc, plays an important role in reducing canopy interception storage.
3.4. Infiltration Scheme
3.4.1. Basic Assumptions
 The infiltration model is based on the kinematic approximation for unsaturated flow of Cabral et al.  and Garrote and Bras  who analyzed moisture dynamics for a sloping heterogeneous anisotropic soil. Assumed soil heterogeneity is parameterized by the saturated conductivity exponentially decaying with normal depth [Beven, 1982, 1984]. Soil layering is represented by the dimensionless anisotropy coefficient defined as the ratio between the saturated conductivities in the parallel, p, and normal, n, to the soil surface directions. Soil hydraulic properties in the unsaturated phase are parameterized by using the Brooks and Corey  scheme. The reader is referred to Cabral et al.  for further details.
3.4.2. Ponded Infiltration
 The standard Green-Ampt model of ponded infiltration [Green and Ampt, 1911; Morel-Seytoux and Khanji, 1974; Neuman, 1976] follows from assuming that for a moisture wave infiltrating into a semi-infinite soil with a uniform initial volumetric water content, there exists a sharply defined wetting front for which the water pressure head hf remains constant with time and position. A modified formulation of the model presented by Childs and Bybordi  and Beven  for layered soils is used:
where qn(Nf) is the normal component of the flow vector, Nf is the wetting front depth, and Keff is the harmonic mean of conductivities over the saturated depth. For the surface saturated conductivity K0n exponentially decaying with depth at the rate f, Keff can be expressed as
The effective wetting front capillary pressure which explicitly accounts for changes in the soil moisture and conductivity with depth is parameterized as [Ivanov, 2002]
where Sei(Nf) = (θi(Nf) − θr)/(θs − θr) and λ(Nf) = ψb is the air entry bubbling pressure, λ0 is the pore-size distribution index, and θi (Nf) is the moisture content at the depth Nf of the initial moisture profile. Taking into account the change in the gravity gradient with slope of the soil element, equation (16) can be rewritten in the following form:
where α is the slope of the soil column and the term Ψis(Nf) = −Keff(hf(Nf)/Nf) represents the flux rate due to capillary forces in the soil. The index “is” denotes the soil moisture range θi(Nf) to θs for which the term is evaluated. Expression (19) constitutes the basis for modeling saturated infiltration when the rainfall rate is higher or equal to the qn.
3.4.3. Infiltration Under Unsaturated Conditions
 At the onset of an infiltration event, if the rainfall rate is lower than the infiltration capacity of the soil, the movement of water in the soil occurs under unsaturated conditions. This phase includes the development of a wetted unsaturated wedge and, if the rainfall intensity is sufficiently high, the formation of a perched zone may follow. A schematic soil moisture profile in Figure 3a, simulated in the tRIBS model, depicts the wetting and the top front. The wetting front separates the infiltrated rainfall from the initial soil moisture profile in a discontinuous fashion. The top front represents the ascent of the shock wave caused by the formation of the perched saturation zone. The normal depths to the wetting and the top front, Nf and Nt correspondingly, coincide if there is no perched layer (Figure 3b).
126.96.36.199. Wetted Wedge Dynamics: Unsaturated Phase
 One of the key assumptions made in the infiltration model is that while recognizing the importance of the capillary forces, gravity is considered to be the dominant component in the infiltration process [Cabral et al., 1992]. Capillarity effects are accounted in a simple way by an analogy with (19). The redistribution flux in the normal to the surface direction is formulated for the unsaturated wetted wedge as
where Re is an “equivalent” rainfall rate defined as the value that leads to the same moisture content above the wetting front as from a constant rainfall at rate Re under equilibrium conditions [Garrote and Bras, 1995] and Ψie(Nf) is the capillary drive across the wetting front in the unsaturated conditions. For the discontinuous profile as in Figure 3b, Ψie(Nf) is evaluated for the range of values [θe (Re, Nf), θi(Nf)], where θe(Re, Nf) is the maximum moisture value in the wedge (Figure 3b). For an unsaturated form of Darcy's law [Smith et al., 1993], Ψie (Nf) is
where KSn(Nf) is the saturated conductivity at the depth Nf and hf(Nf, θi, θe) is the effective unsaturated capillary pressure evaluated for an arbitrary moisture range in soils with decaying saturated conductivity. This term is approximated by generalizing (18) (similar to Smith et al. ):
where See(Nf) = (θe(Re, Nf) − θr)/(θs − θr).
188.8.131.52. Wetted Wedge Dynamics: Perched Zone Formation
 Given that the saturated conductivity decreases with normal depth, the saturation may develop at some depth N*. If the moisture influx above the wetting front is high enough, water accumulates above N* and perched saturation develops (Figure 3a). An analogous expression to (19) for the normal flux can be written as
where Keff is as previously the harmonic mean of the conductivities over the saturated thickness:
3.4.4. Evolution of Fronts
 The formulation of the wetting and top front evolution for unsaturated and surface saturated state (Figures 4b and 4d) is similar to the one described by Cabral et al.  and Garrote and Bras . For the perched saturated state (Figure 4c), it is assumed that at any instantaneous time the top front depth represents the saturation level of the steady state profile corresponding to some Re, determined by the moisture content above the wetting front.
3.4.5. Basic Soil Moisture States and Runoff Generation Potentials
 Five distinct cell states are considered, each defined by the dynamic variables describing the moisture state of the soil column (Figure 4). These states have different potentials for runoff generation among the four mechanisms considered by the rainfall-runoff scheme.
 For the first three states: initial, unsaturated, and perched saturated state (Figures 4a–4c), the soil infiltration capacity is not constrained by the surface conductivity unless the top of the soil column reaches immediate saturation, i.e., transits to a surface saturated state. In the surface saturated state (Figure 4d), the infiltration capacity is constrained by conductivity at the bottom of the saturated profile. Depending on soil parameters and rainfall intensity, infiltration excess runoff may be produced. Within the model, runoff is considered to be of infiltration excess type when the redistribution rate of the top saturated layer is lower than the rainfall intensity, irrespective of the preceding infiltration history. In the context of literature definitions [e.g., Freeze, 1974; Bras, 1990; Beven, 2000], this is a mixture of “Hortonian” runoff type, occurring when soil becomes saturated from above by a high-intensity rainfall, and saturation from below runoff occurring due to the development of perched groundwater (e.g., Beven  also refers to the latter mechanism as of infiltration excess type). Perched subsurface stormflow may also occur at an element in this state if the outflux from the vadose zone of an upslope cell discharges onto the surface of the element. The fully saturated state implies that the wetting front has reached the water table and the top front is at the soil surface (Figure 4e). The cell infiltration capacity in this state is zero and if rainfall persists, the element produces saturation excess runoff. Lateral redistribution fluxes in the phreatic aquifer may result in groundwater runoff production. Perched subsurface stormflow may also be produced at an element in this state.
3.4.6. Subsurface Flow Exchange in the Vadose Zone
 A simplified scheme is adopted to account for the moisture transfers in the vadose zone between contiguous elements based on the formulation provided by Cabral et al. . The spatial orientation of flows entering a cell is assumed to be parallel to the line of maximum terrain slope, the direction p, irrespective of the orientation of triangular facets of the TIN that compose the surface of the Voronoi cell (Figure 2b) as well as discontinuities at the cell boundaries associated with slopes in adjacent elements.
3.4.7. Soil Water Redistribution During Interstorm Periods
 Adequate representation of the soil water redistribution and dynamic adjustments of the moisture profile during evapotranspiration are required for continuous model operation. Interstorm conditions are modeled using various state transitions detailed in Ivanov  and are briefly outlined below.
 The first transition describes the water table drop from the soil surface as a result of evaporation conditions applied to the computational element. A hydraulic equilibrium profile (section 4.1) is assumed to be attained in the unsaturated zone and a mass conservative scheme is used to define the new groundwater level. The second transition deals with evaporation conditions applied to elements having water table located at some depth below the land surface. The implemented scheme subtracts moisture from the whole unsaturated profile resulting in the drop of the groundwater level and reinitialization of the soil water profile. Adjustments of the moisture profile are also made during simulation of rainfall of decreasing intensity, rainfall hiatus, or interstorm periods when a wedge of infiltrated water exists in the soil. The moisture wedge may transit to various states depending on the prior cell state, the soil parameters and the intensity of evaporative demand. The principal transitions are illustrated in Figure 5.
 It is assumed that after a given time with no rainfall the interstorm period begins. This marks the time at which redistribution of moisture excess in the unsaturated zone occurs and the resulting soil moisture will be the initial state for any subsequent storm. The characteristic time that marks the beginning of an interstorm period is assumed to be climatically dictated and varies for different regions, commonly to be in the range 1–3 days [Restrepo-Posada and Eagleson, 1982].
3.5. Groundwater Model
 A quasi three-dimensional “cascade” groundwater model is utilized. An explicit cell to cell approach is used to route the lateral saturated subsurface flow. For each direction of a TIN edge j, the total groundwater flow from a Voronoi cell is
where QSout is the outflux from a saturated layer of width W along the negative hydraulic gradient, approximated as a local gradient of the water table, tan(β), where β is the local slope of the groundwater level. The index j refers to values of the width and hydraulic gradient defined in the jth direction. The aquifer transmissivity, T, nonlinearly depends on the groundwater depth Nwt and bedrock depth η′ due to the exponential decay of the saturated conductivity:
where ar is the soil anisotropy ratio. The total influx for any given Voronoi cell is obtained by summing the outfluxes from elements that contribute to that cell. The corresponding changes in depth to the water table are modeled as
where A is the Voronoi cell area. From a computational standpoint, the treatment of the specific yield Sy in equation (27) is inconvenient, especially when moisture fronts are present in the vadose zone. An approach with an “implicit” computation of the specific yield based on mass conservation in the element is used instead [Ivanov, 2002]. Various possible transitions allow simulation of the subsurface saturated exchanges and couple the subtraction/addition of water from/to the groundwater with the corresponding adjustments of the soil moisture profile in the vadoze zone.
3.6. Runoff Generation Scheme
 The actual infiltration I, rainfall R, infiltration capacity fc, and runoff Rf can be related as
Return flow is the result of lateral subsurface exchange and is produced under similar conditions:
where ΣQU is the net sum of subsurface lateral inflows and outflows in the unsaturated zone. The runoff generation due to groundwater corresponds to the following conditional expression:
where ΣQS is the net positive sum of fluxes in the saturated zone. The total surface flow generated in the element is the sum of all runoff types produced under the described conditions. A reinfiltration scheme is not considered and runoff produced in a cell is assumed to contribute to streamflow at the catchment outlet.
3.7. Runoff Routing
 The chosen methodology for routing runoff represents a trade-off between exploiting the efficient TIN structure and the complexity of the routing problem over the TIN surface. Runoff is assumed to follow TIN edges in accordance with the consecutive drainage directions. The total runoff travel path lt consists of a hillslope fraction lh and a stream fraction ls: lt = lh + ls. Two different routing models are applied for these path fractions.
3.7.1. Hydrologic Routing
 For every hillslope path, the bulk transport of water is assumed to be the dominant factor in runoff routing. The effect of dispersion is introduced in a simplified manner that keeps the model parameters to a minimum and leads to high computational efficiency. Each hillslope node is assigned an “outlet” stream node to which it contributes flow via consecutive drainage paths of the TIN (Figure 6). The travel time tτ of runoff between a hillslope node and its “outlet” can be defined as: tτ = lh/υh(τ), where υh(τ) is the hillslope velocity at time τ. This velocity is allowed to vary in space and time as
where Q(τ) is the discharge at the “outlet” stream node at time τ, Ac is the surface contributing area of the “outlet” node, and cν and r are uniform parameters for a given basin. Assuming steady state flow conditions, the relationship Q(τ)/Ac for any given point in the stream network is constant [Leopold and Maddock, 1953; Rodriguez-Iturbe and Valdes, 1979] and therefore the latter parameters can be roughly approximated through an analysis of various flow regimes.
 A velocity approximation in the form (28) allows for a simple computation of the hillslope hydrograph at each hillslope “outlet” node. The total hillslope response at time T is obtained by adding the incremental responses from the hillslope nodes qτ(t) since the beginning of the storm:
3.7.2. Hydraulic Routing
 Hillslope hydrographs, from equation (29), in the stream nodes represent lateral inflow into the channel network of the basin. A kinematic wave routing model is used to simulate transport of water in the channel network [e.g., Goodrich et al., 1991; Singh, 1996]. The one-dimensional continuity equation for unsteady free surface flow is
where F is the cross-sectional area, Q is the discharge along the x axis, Rb is the lateral influx of water into the channel per unit length. If the channel cross section is approximated by a rectangle and Manning's equation is used to parameterize the unsteady flow velocity:
where H is the depth, i0 is the channel slope, ne is the channel roughness, and b is the channel width. The channel width b is either obtained from measurements or approximated using regional geomorphological relationships of the form: b = f(Ac) [e.g., Orlandini and Rosso, 1998].
 A union of connected segments without tributaries constitutes a stream reach which serves as a basic one-dimensional finite element domain for the routing model. Using piecewise polynomial basis functions, F(x, t) and Q(x, t) can be approximated continuously within each channel reach [Kuchment et al., 1983]. The Galerkin method is used to minimize the errors of the approximation. An implicit numerical scheme results in a system of nonlinear equations which is solved using Newton-Raphson iteration method combined with line searches and backtracking [Press et al., 1999].
4. Model Initialization and Data Mapping
4.1. Initialization of the Water Table Depth and Soil Moisture Distribution
 Catchment initial conditions, required for modeling the rainfall-runoff response, involve two distinct but interconnected aspects: the initial water table depth and the initial moisture profile in the vadose zone. The former represents a measure of the storage capacity of a basin while the latter determines the moisture deficit and the infiltration characteristics. Spatial information for both states is not usually available and, consequently, reasonable approximations have to be made. It is important to point out, however, that the effect of initial conditions has less influence on the simulation results in the case of continuous modeling. The influence of the initial state diminishes with the simulation time as storm and interstorm periods redistribute soil moisture in the watershed in vertical and lateral directions. The long-term continuous modeling is therefore suitable for elucidating the inherent catchment dynamics.
 To initialize the groundwater state, an assumption of quasi steady state conditions in the groundwater system is used [Sivapalan et al., 1987]. If a spatially pseudo-uniform soil is assumed with the “effective” basin scale parameters K0n, f, and ar, then the distribution of the depth to water table in the formulation presented by Sivapalan et al.  is controlled only by catchment topography. Sivapalan et al.  and Troch et al.  discuss estimation of the mean depth to the water table and aquifer hydraulic properties based on recession flow analysis.
 Specifying the initial moisture state of the catchment is a common problem in rainfall-runoff modeling [Salvucci, 1993; Salvucci and Entekhabi, 1994]. It is assumed that the depth to the water table significantly controls wetness conditions in the basin and therefore defines the soil's initial infiltration capacity. The implemented approach assumes hydrostatic equilibrium for the vertical distribution of pressure head which corresponds to zero initial flux in the unsaturated zone: ∂ψ/∂n = 1 [Sivapalan et al., 1987; Famiglietti and Wood, 1991; Troch et al., 1993; Coles et al., 1997]. The suction head at any depth N < (Nwt + ψb) is given as ψ(n) = Nwt − n. Using the Brooks-Corey parameterization and Miller scaling [Miller and Miller, 1956], the soil moisture profile can be expressed as
where λ(n) accounts for changes in pore size with depth of the form: λ(n) = λ0e−fn/2 [Selker et al., 1999]. However, a computationally convenient assumption is to use (32) with the exponent approximated by the surface pore-size distribution λ(n) ≈ λ0 [Ivanov, 2002].
4.2. Land Surface and Hydrometeorological Data
 Information about the land surface characteristics of a river basin is readily available from various soils and landuse databases such as the NRCS State Soil Geographic Databases (STATSGO, SSURGO), the USGS Land Use and Land Cover (LULC) database, and other databanks supported by scientific communities. The land surface descriptors are used to derive soil and landuse types. The corresponding parameters are assigned their initial values based on information provided in the literature [e.g., McCuen et al., 1981; Rawls et al., 1982, 1983; Shuttleworth, 1992].
 One of the driving reasons for using spatially distributed approaches is to take advantage of the rainfall estimates provided by weather radars [e.g., Wyss et al., 1990; Pessoa et al., 1993]. Estimates from the NWS Next-Generation Weather Radar system (NEXRAD) are used as the model input. Measurements from surface weather stations (e.g., air temperature, humidity, wind, and rain gauge data) are utilized to compute the surface energy fluxes and evaporation potential as well as to assess the quality of weather radar data. Access to the data is typically available through surface observation networks, e.g., the National Climatic Data Center (NCDC).
4.3. Data Mapping to Voronoi Regions
 Both gridded and point data can be inputted to assign the data values to Voronoi cells. Depending on form of the input, either a unique or weighted average value is computed from the input data layer. Unique values are assigned in the case of point station or gridded data when continuous fields are represented by discrete types, e.g., soil and landuse type indices. The data values are assigned to the Voronoi nodes that have their centers of mass inside of a corresponding Thiessen polygon/grid cell. When gridded input data represent continuous fields of a hydrometeorological variable (e.g., rainfall intensity or depth to the groundwater), a weighted average value is derived based on the location of a Voronoi region with respect to the cells of the input grid.
5. Space-Time Aspects of Catchment Hydrology
 In order to fully demonstrate the potential of the spatially distributed approach in studying the causal connections in a catchment, the hydrologic dynamics and states are illustrated in this section at different spatial and temporal scales. The physical mechanisms of hydrologic response and their interdependent feedbacks are elucidated from various perspectives illustrating the integral catchment behavior. The following discusses the results of the model application to several mid- to large-size basins in the Arkansas–Red River basin.
5.1. Calibration and Validation
 An essential component in evaluating the reliability of hydrological estimates is validation against the observed data. Historically, the most often used measure is the discharge at the catchment outlet. The most promising advantage of spatially distributed approaches is their representation of internal catchment dynamics which allows one to obtain fields of the state variables of a system. Streamflow from multiple nested gaged locations, spatially explicit maps of surface layer soil moisture, soil temperature, and water table depth are the type of data that can be anticipated to become available in the future to help test and calibrate physically based models [Grayson and Bloschl, 2000; Refsgaard, 2000; Grayson et al., 2002]. To date, however, there have been only a few examples of model validation based on the interior catchment information [e.g., Houser et al., 1998; Refsgaard, 2000; Senarath et al., 2000]. No spatial data currently exist to fully test distributed model simulations. However, the “essential” physical realism [Dietrich et al., 2003] can be achieved via a consistent physical approximation of principal mechanisms and processes provided they are corroborated by the reproduced dynamics and states.
5.1.1. Test Basins
 Three basins are used in the following examples: Baron Fork at Eldon, Blue River at Blue, and Illinois River near Watts (Figure 7). Basic topographic and hydrologic characteristics for these basins are summarized in Table 1 (based on Slack et al. , USGS streamflow data, and STATSGO and LULC data). The topography of Baron Fork watershed is characterized by gently rolling relief at the basin headwaters (east) and rugged terrain in its lower (west) area. The watershed vegetation cover (about 55% of the area) is dominated by deciduous forest. The soil has silt loam as the dominant type. The catchment has a nested basin with measured streamflow: Peacheater Creek (64 km2). Illinois River near Watts basin is located just north from the Baron Fork watershed and has similar topography characteristics: flatter areas in the upstream region with more complex terrain in the center of the watershed and it is downstream regions. Catchment landuse is characterized by significant areas occupied by croplands and pastures 65%, about 28% is occupied by forests of predominantly deciduous type. The dominant soil type is loam. The catchment has an interior gauge near Savoy (Figure 7) with measured streamflow (432 km2). Blue River basin topography is characterized by low relief along the course of the river. Woody savanna represents the dominant type of vegetation (about 77%), deciduous forests occupy about 17% of the catchment area. There were defined seven dominant soil types varying from fine sand to clay with about 53% of the area occupied by loam and silty loam soil type, 16% by fine sandy loam, and 14% by clay.
Table 1. Basic Topographic and Hydrologic Characteristics of the Test Basins
Basin (USGS Gauge Number)
Basin Area, km2
Number of Elements Used to Represent Basin
Mean Elevation, m above NGVD29
Length of Main Stream, km
Average Stream Slope, m/km
Mean Annual Precipitation, mm
Mean Annual Flow, mm/cms
Baron Fork at Eldon (USGS 07197000)
Illinois River at Watts (USGS 07195500)
Blue River at Blue (USGS 07332500)
5.1.2. Streamflow Calibration and Validation
 Only a few calibration and validation results are illustrated for Baron Fork and Blue River basins, two systems of different topographic, soil, and landuse characteristics. A more comprehensive study that concerns streamflow simulation skills of the model is presented by Reed et al. .
 Rainfall forcing data in the form of hourly NEXRAD 4-km gridded estimates were used to obtain spatial distribution of precipitation fields. Hourly 1/8 degree gridded meteorological data developed by the University of Washington [Maurer et al., 2002]: air and dew point temperature, cloudiness, wind speed, and atmospheric pressure were utilized to compute the surface energy fluxes and evaporation potential. For each basin, a location associated with the catchment center was chosen to obtain the corresponding grid values of the hydrometeorological variables that were considered to be spatially uniform throughout the basin. The latter assumption is not restrictive provided that the spatially distributed information is available (see section 4.3).
Figure 8 shows the results of manual calibration for Baron Fork covering period 1–25 November 1995. A preceding calibration for the nested basin, Peacheater Creek, based on several events had provided parameter sets which guided the choice of the best values for the presented case (e.g., parameters K0n, f, ar, cυ, r, and ne were involved in calibration). Table 2 lists all of the tRIBS model parameters and their respective ranges for the soil and landuse types used in the discussed example. Figure 8b compares the observed and simulated series of discharge at the basin outlet as a response to the sequence of rainfall events shown in Figure 8a. Hourly streamflow records at this outlet were the only observed data used in the calibration procedure whose objective was to minimize the difference between the simulated and observed discharge values. As it can be seen in Figure 8b, there is a generally good agreement between the series. Partitioning of the simulated hydrograph into the simulated runoff types is provided in Figure 8c. The illustration is consistent with the theoretical assumptions one could make based on the observed series of discharge. Basin response during rainfall events of higher intensity is a mixture of infiltration excess and saturation excess runoff types. Runoff production during events of lower rainfall intensity is dominated by the saturation excess mechanism. Interflow constitutes a significant proportion of the hydrograph recession phase and the groundwater runoff is the major background streamflow component. Figure 8d shows the observed and simulated streamflow for Peacheater Creek basin outlet considered as an interior point of the Baron Fork drainage network. Calibration for this basin was not performed for that particular event. At hour 300, the observed streamflow series for Peacheater Creek does not have a peak. The discharge records for Baron Fork, however, do show a streamflow event for that time which was not reproduced by the model. A variety of reasons that may involve erroneous model structure, assumptions, and issues of initialization could lead to such a behavior. However, analysis of the ground-observed precipitation data (not presented) implies to a credible degree that the actual rainfall rates had significant spatial variability which was not captured by the radar data and, therefore, was not properly simulated.
Table 2. The tRIBS Model Parameters and Their Respective Ranges Used to Simulate the Hydrologic Response of the Baron Fork Basina
The column “Source” indicates “Calibration” for the parameters whose initial values obtained from literature were modified through the model calibration. The actual calibration effort was different for different parameters.
free throughfall coefficient
canopy drainage rate coefficient
canopy drainage exponent
optical transmission coefficient
average canopy stomatal resistance
Soil Hydraulic and Thermal Properties
saturated hydraulic conductivity
saturation soil moisture content
residual soil moisture content
pore distribution index
air entry bubbling pressure
−0.4 to −0.1
conductivity decay parameter
volumetric heat conductivity
J/m s K
soil heat capacity
Channel and Hillslope Routing Parameters
Manning's channel roughness
channel width–area coefficient
channel width–area exponent
hillslope velocity coefficient
hillslope velocity exponent
Figure 9 illustrates validation results for the Blue River basin for an 8-month period of continuous simulation based on previous calibration limited to several events not included in the considered time period. The observed and simulated streamflow series agree well (Figure 9b). While the basin response is dominated by the saturation excess runoff type (Figure 9c), a substantial amount of infiltration excess runoff can be generated during periods of high rainfall rates over the basin area with clayey soils (southern part of the basin). The groundwater response to the wetting and drying phases of hydrometeorological cycle are also clearly seen in Figure 9c.
 An example of simulation results obtained in a blind validation test based on the transfer of the hydrological parameters is given in Figure 10. The Illinois River basin near Watts is similar to the Baron Fork watershed in terms of soil characteristics, vegetation cover, and topography. Using the same classification scheme of land surface features, the corresponding groups were assigned the calibrated parameter values obtained for Baron Fork basin. As it can be seen in Figure 10, the results of the test are satisfactory, particularly since no calibration has been done for this basin. The results support the use of physically based distributed approaches for representing ungauged watersheds [Westrick et al., 2002].
5.2. Coupled Hydrological Systems at the Scale of Basic Computational Element
 Hydrologic dynamics at the scale of the basic computational element are presented in the following to fully demonstrate the coupling of various mechanisms. The illustrated Voronoi cell is located in the Baron Fork basin. The considered soil column is bounded from below by the dynamic water table and confined laterally by boundaries of the Voronoi region (e.g., Figure 2b). Evolution of the cell's state variables is considered in one dimension. The presented scenario case corresponds to a clayey loam soil covered with deciduous forest (v = 0.60). The considered Voronoi element (2,300 m2) is located in a downslope region where there is a significant contribution of the subsurface flow from upstream cells. The interception and evapotranspiration parameters are obtained from published literature values and correspond to: p = 0.35, S = 1.0 mm, K = 0.18 mm/h, g = 3.9 mm−1, a = 0.16, Kt = 0.60, rs = 100 s/m (scaled in the daily cycle according to Shuttleworth ). The soil hydraulic parameter set used in the example is the following: K0n = 3.0 mm/h, ar = 400, f = 0.0007 mm−1, λ0 = 0.23, ψb = −200 mm, θs = 0.4, and θr = 0.05. The initial water table position in a cell is Nwt = 4.1 m with the bedrock located at 10 m deep. The discussed below results correspond to a simulation that starts on 18 May 2000 and spans 1100 hours.
 The Rutter model relates the change in the canopy storage to both drainage and evaporation demand. The latter component is considered to be a dependent variable of meteorological conditions. Figure 11a shows a time series of net precipitation that accounts for the interception (Figure 11b) and drainage. Figure 11c shows the dynamics of the canopy storage modeled by (1). As it can be seen, when the storage is higher than the canopy capacity, the drainage is high. For lower values of the storage, the water is primarily removed by evaporation.
5.2.2. Energy Balance and Evapotranspiration
 The surface energy balance is used to compute evaporative forcing through the surface flux components accounting for landuse properties of an element and its spatial location. Evaporative fraction, defined as the ratio between latent heat flux and available energy, is a representative indicator of the relative wetness conditions of a site since evaporation explicitly depends on the amount of moisture contained in the soil layer. Figure 12a shows the series of rainfall that supplies water to the soil. Depending on the soil state that controls energy partitioning by (12) and (15), the corresponding series of daily evaporative fraction reflects higher or lower magnitude of the latent heat flux (Figure 12b). Periods with high latent and low sensible heat fluxes correspond to the wetter states of the soil surface and result in higher values of evaporative fraction (e.g., hours 100–200, 720–850). When the magnitudes of the sensible and latent heat flux are comparable implies drier soil conditions and therefore lower values of evaporative fraction (e.g., hours 350–460).
 The latent heat flux, estimated as a bulk quantity for both the bare soil and vegetated portions of a given element, is partitioned into various components by (11), (13), and (14). Figure 12c shows the composition of the mean diurnal cycle of total evapotranspiration over the considered period. Evaporation from the canopy storage, transpiration, and evaporation from bare soil exhibit different relative magnitudes defined by the time-invariant (i.e., vegetative fraction) and time-dependent state variables (rainfall and soil moisture).
5.2.3. Unsaturated Zone and Groundwater Dynamics
 The difference between the net precipitation and evapotranspiration is applied to the soil column as the upper boundary condition (Figure 13a). The subsurface moisture fluxes also participate in the water balance of the wetted wedge that goes through the unsaturated and saturated phases (Figure 13b). During the rainfall events, there is almost constant positive balance of water above the wetting front as long as the subsurface influx exceeds the outflux (Figure 13c), an implication of high soil anisotropy and the element location in the convergent topographic area. As a result of moisture accumulation above the wetting front and small redistribution rates, perched saturation develops in the element (Figure 13b) (hours 131–135, 499–501, 635–643, 726–745). In certain cases the element reaches the surface saturation and generates runoff (Figure 13c). The relative soil moisture in the top 100 mm, θ100(t)/θs, is shown in Figure 13d (θ100(t) is the moisture value in the top 100 mm). Sudden drop of the moisture content (hour 277) correspond to the time when the moisture excess in the unsaturated zone is redistributed as an instantaneous groundwater recharge. According to section 3.4.7, this marks the beginning of an interstorm period during which the element is in the initial state for any subsequent storm. The same also applies to the sudden change in the groundwater exchange flux, computed as the numerator in the right hand part of (27). Positive values of the groundwater exchange flux imply a drop in the water table with time (Figure 13e). Because the water table position dictates the moisture distribution in the unsaturated zone, a longer simulation period could reveal surface soil moisture dependence on the dynamics of the groundwater. An impact would be observed in the drier or wetter initial states of the element.
5.3. Spatial Heterogeneity of Soil Water States and Fluxes
 Catchment topographic features and soil properties define the magnitude of the lateral subsurface moisture redistribution and the local infiltration characteristics. Adequate terrain representation and simulation of the relevant processes should result in a realistic representation of the essential catchment dynamics. For instance, the temporal variation of the soil moisture and groundwater spatial patterns is an important indicator of distinctive basin properties. While the instantaneous state of a basin describes its immediate response, time-integrated state variables provide a clearer representation of the catchment interior features. Figure 14a shows the spatial distribution of the frequency of surface saturation, at the level of detail dictated by the TIN resolution, as a percentage of the total run time over 85 months of simulation for Baron Fork. It is necessary to note that the surface saturation implies that either the soil column has reached complete saturation or the surface layer is saturated. The latter case occurs during formation of the perched layer in conditions of low rates of lateral soil moisture redistribution. Figure 14a illustrates both situations: the areas of low relief (e.g., the northern part of the basin) exhibit a combination of both cases due to the negligible lateral drainage while near stream elements are almost constantly saturated due to contribution from the upstream areas. The cells located in the floodplain depict areas of gradually lower saturation frequency reflecting expansion and contraction of the variable source areas [Hewlett and Nutter, 1970].
 The mean frequency of surface saturation shown in Figure 14a can be related to the basin topographic attributes. Figure 14b illustrates this variable as a function of the topographic index. Vertical bars from points in the plot represent standard deviations of the values computed for the corresponding bins. As the figure shows, there is a general increase in occurrence of surface saturation for both high and small values of the topographic index. In the former case, it can likely be attributed to the more permanent soil saturation by the groundwater in convergent topographic forms (e.g., the floodplain or hollows of the rugged terrain in the western part of the basin), while in the latter to the negligible rates of lateral moisture redistribution in areas of low relief (e.g., the northern part of the basin near the watershed divide). On the other hand, steeply sloped sites with transitional groundwater and unsaturated zone dynamics can efficiently redistribute soil moisture laterally thus leading to rare occurrence of surface saturation (Figure 14b).
 Groundwater spatial patterns can be integrated in time to evaluate consistency of their dynamics in relation to the catchment topography. The net long-term flux across the water table can be analyzed to discern regions where it has tendency to be positive, negative, or close to zero [Freeze and Cherry, 1979, pp. 193–195]. The hillslope areas can be correspondingly classified into the groundwater recharge, discharge and transitional regions. Figure 15a illustrates the relation of the groundwater to terrain topography for an interior area of the Baron Fork basin at the end of 85 month simulation period (bedrock was assumed to be spatially uniform at 10 m deep). Figure 15b categorizes the area into regions of substantially different groundwater dynamics. Notwithstanding the short simulation period and unavoidable errors in the initialization, the basic features of the groundwater dynamics are captured with the hydrological model. Convergent topographic areas are classified as discharge regions while flat areas of topographic highs are categorized as recharge zones which is in good agreement with reported observations and theoretical arguments. Figure 16a, for instance, shows the mean values of the topographic index and its variability for the identified groundwater dynamics regions. As it can be seen, there is an increase in topographic index as one moves from the groundwater recharge to discharge regions. Figure 16b shows the mean soil moisture content in the top 1 m of soil for the defined regions, a characteristic which is also an indicator of the connection between the groundwater and surface processes. Recharge regions primarily supply moisture to the groundwater system and are little affected by its dynamics. These regions, therefore, exhibit lower values of the mean moisture content and its variability. As one moves into the midline and discharge regions, the effect of periodic wetting and drying of the groundwater on soil moisture becomes higher. This is reflected in higher magnitudes and variability of the soil moisture. As it can also be seen in Figure 16b, land surface characteristics have a substantial impact on magnitude and variability of the root zone soil moisture.
5.4. Spatial Organization of Runoff Types
 Four different runoff production mechanisms are considered in the model, their general schemes of simulation are discussed in section 3.4.5. Figure 17 presents spatial distribution of relative frequencies of infiltration and saturation excess runoff types as a percentage of the total run time over 33 months of simulation for Baron Fork watershed (OK). The infiltration excess runoff is consistently simulated either in the hillslope bottom areas or in the topographic highs that have low relief. In the former case, it occurs as the result of moisture convergence due to subsurface fluxes in the vadose zone. In the latter case, infiltration excess is caused by water accumulation in the unsaturated zone in conditions of extremely low lateral subsurface redistribution rates. It is also worth noting that steep terrain within the watershed is covered with deciduous forest vegetation. The corresponding soil type is parameterized to have very high conductivity anisotropy in the top soil layer that creates significant interflow in the system. Flat areas are mostly occupied by grasslands, pastures, and croplands and have lower hydraulic conductivity and anisotropy values. The patchy areas of the highest frequency of infiltration excess runoff production are urbanized areas that have low infiltration potential.
 Locations of the frequent saturation excess runoff (Figure 17b) reflect the perennial stream network and its floodplain. The flat areas in the northern part of the watershed also exhibit somewhat higher frequency because of the small lateral redistribution rates. The lower-frequency locations can be attributed to the zeroth order hollows or ephemeral tributaries that exist only during rainfall events having specific characteristics. The TIN representation of the terrain allows to consistently capture these features. Perched subsurface stormflow and groundwater runoff occurrence agrees well with Figure 17b since these runoff types can also be attributed to the channelized forms of topography and the floodplain.
 The mechanistic inferences that can be derived from detailed modeling the principal processes of the hydrological cycle, i.e., rainfall interception, evapotranspiration, infiltration with continuous soil moisture accounting, lateral moisture transfer in the unsaturated and saturated zones, and overland and channel runoff routing, allow for a physically significant interpretation of distributed relationships in a catchment. This paper discusses various new aspects of studying catchment behavior with a TIN-based, physically oriented, distributed-parameter model. The model was applied to several catchments varying in size and land surface characteristics. The case study examples illustrate the potential for utilizing the physically based approaches in providing insights on states and dynamics integrated over various spatial and temporal scales. They yield modeling evidence about existing links between the catchment hydrological processes and land surface characteristics. They also confirm the strongly coupled nature of surface water–groundwater dynamics. No spatial data about catchment states exist yet at such scales in the amount and detail sufficient for thorough validation of the simulated distributed dynamics. Only partial verification is possible. However, the essential physical realism of basin hydrological function can be nonetheless reproduced allowing to elucidate links and mechanisms not detectable with traditional approaches.
 The TIN structure is capable of representing the simulation domain with multiple resolution without a significant loss in topographic detail. Through a proper TIN implementation, the quantity of computational elements is significantly reduced thus making the described approach suitable for large-scale applications. For instance, compared to a high-resolution DEM, the amount of computational elements for the considered basins was in the order of 5–10% of the original number of grid cells. While fine computational time steps were used in the provided examples (4 min. for unsaturated zone dynamics and kinematic routing, 30 min. for the groundwater model, and 1 hour for computing the interception and radiation budget), the reduction in the number of computational elements and the physical realism captured by the TIN-based model provide evidence that fully distributed hydrologic simulations of large watersheds are feasible in operational settings.
 The distributed hydrologic model, tRIBS, discussed in the first part of this paper, is currently available through the MIT Licensing Office. The model user manual, associated publications, and relevant input data can be accessed at http://hydrology.mit.edu/tRIBS.
 This work was supported by the National Aeronautics and Space Administration (contract NAG57475) and the National Oceanic and Atmospheric Administration (contract NA97WH0033). The authors thank anonymous reviewers for the constructive suggestions.