## 1. Introduction

[2] In this paper we address the problem of process integration for hydrologic prediction in watersheds and river basins. Simulation is now widely utilized as a complementary research methodology to theory and experiment [*Post and Votta*, 2005]. However, the grid resolution, scale of the model, and range of hydrologic processes operating in watersheds and river basins offer the dilemma of what is necessary to predict hydrologic response or to simulate certain behaviors of the coupled system. In this paper we formulate a multiscale strategy that incorporates constitutive relationships representing volume-average state variables. For small watersheds and fine numerical grids, local continuum relationships (e.g., Darcy's law) lead to a fully coupled, physics-based, distributed model. At larger scales and coarse grids, empirical relationships with large-scale volume averages are applied, and the model becomes a semidistributed model. A brief review of hydrologic modeling strategies demonstrates the issues involved with integration and coupling of multiple processes and clarifies the purpose of this paper.

[3] Current hydrologic models may be described from two perspectives: physically based, spatially distributed models, and lumped conceptual models. *Freeze and Harlan* [1969] developed the first blueprint for numerical solutions to physically based, distributed watershed models starting from a continuum perspective (i.e., Richards' equations for subsurface flow, Saint Venant equations for surface flow and channel routing). It was some years before the SHE model [*Abbott et al.*, 1986a, 1986b] and its variants produced a second generation where the coupled physical equations are actually solved on a regular grid, with coupling handled through a sophisticated control algorithm that passes information between processes (e.g., surface water–groundwater exchange).

[4] The approach of coupling multiple processes through time-lagging and iterative coupling through boundary conditions is generally considered a weak form of coupling, in that it may lead to significant instability and errors [*LaBolle et al.*, 2003]. The approach also requires considerable reprogramming if changes are made to the physical equations for a specific application. More recently, *Panday and Huyakorn* [2004] have developed an approach where all equations in the model are of the diffusive type, which are solved in a single system on a regular grid (e.g., Richard's equation and diffusive wave equation), while equations for other processes (vegetation, energy, snow) are dealt with separately (iteratively). *Yeh et al.* [1998] have used a similar approach but with finite elements. As will be described later, our approach couples all dynamical equations within the same prismatic volume (a prism is defined by a triangle projected from the canopy, through the land surface to the lower boundary of groundwater flow); and all equations are solved simultaneously, eliminating the need for a controller, delayed, or off-line process equations.

[5] Lumped or spatially integrated models are widely used today, where the goal of the prediction is outflow from forcing (e.g., rainfall-runoff, recharge-baseflow, precipitation-infiltration). Lumped systems are low-dimensional and conveniently solved, but still require an empirical relationship for flux discharge that is generally assumed to be linear or weakly nonlinear and fitted or calibrated to the data. The reduced parameter set of this approach can resolve the overall mass balance but cannot by definition inform the internal space-time variation of physical processes. The Stanford watershed model is an early example of the lumped model that includes watershed processes [*Crawford and Linsley*, 1966]. There have been efforts to try to bridge these two approaches. *Duffy* [1996] describes a two-state model by integrating Richards' equation over a hillslope into saturated and unsaturated states, and later extended this approach to the problem of mountain-front recharge using hypsometry to partition the upland, transition, and flood plain zones into a intermediate-dimensional system [*Duffy*, 2004]. *Reggiani et al.* [1998, 1999] proposed a comprehensive semidistributed framework in which integrated conservation equations of mass, momentum, and energy are solved over a representative elementary watershed (REW). They discuss the issues involved in parameterizing the integral flux-storage relation at the REW scale, and refer to this as hydrologic closure.

[6] The decision of using a lumped, distributed, or semidistributed approach to model watershed systems ultimately depends on the purpose of the model, and each has its advantages and disadvantages. For the distributed case, the governing equations are derived from local constitutive relationships. For instance, the Darcy equation is applicable at the plot or perhaps hillslope scale, but it is not clear what should be the effective relation of flux-to-state variable when integrated over larger scales where semidistributed or lumped models are used (e.g., the hydrologic closure problem discussed by *Beven* [2006]). At present there is considerable discussion in the literature about the relation of data needs and predictive models, including the issues of model type (lumped, semidistributed, distributed), uniqueness, and the appropriate scales of integration [*Sivapalan et al.*, 2002].

[7] In the present paper a new strategy for integrated hydrologic modeling is proposed that naturally handles physical processes of mixed partial differential equations (PDEs) and ordinary differential equations (ODEs) as a fully coupled system. The model formulates the local physical equations via the finite volume method, using geographic information systems (GIS) tools to decompose the model domain on an unstructured grid, as well s distributing a priori parameter estimates to each grid cell. In the limit of small-scale numerical grids, the finite volume method implements classical (e.g., contiuum) constitutive relationships. For larger grid scales the method reflects the assumptions of the semidistributed approach described above, but with full coupling of all elements. The process of altering the physical model to accommodate effective parameterizations or new equations is a relatively simple process, since all equations reside in the same location in the code (i.e., the kernel). In this approach, the interactions are assembled on the right-hand side of the global ODE system, which is then solved with a state-of-the-art solver designed for stiff, nonlinear systems. The approach utilizes a triangular irregular grid that covers the domain with the fewest number of triangles [*Palacios-Velez and Duevas-Renaud*, 1986; *Polis and McKeown*, 1993] subject to constraints as defined by the particular problem.