Tools and Algorithms to Link Horizontal Hydrologic and Vertical Hydrodynamic Models and Provide a Stochastic Modeling Framework

Abstract

[1] We present algorithms and tools we developed to automatically link an overland flow model to a hydrodynamic water quality model with different spatial and temporal discretizations. These tools run the linked models which provide a stochastic simulation frame. We also briefly present the tools and algorithms we developed to facilitate and analyze stochastic simulations of the linked models. We demonstrate the algorithms by linking the Gridded Surface Subsurface Hydrologic Analysis (GSSHA) model for overland flow with the CE-QUAL-W2 model for water quality and reservoir hydrodynamics. GSSHA uses a two-dimensional horizontal grid while CE-QUAL-W2 uses a two-dimensional vertical grid. We implemented the algorithms and tools in the Watershed Modeling System (WMS) which allows modelers to easily create and use models. The algorithms are general and could be used for other models. Our tools create and analyze stochastic simulations to help understand uncertainty in the model application. While a number of examples of linked models exist, the ability to perform automatic, unassisted linking is a step forward and provides the framework to easily implement stochastic modeling studies.

1. Introduction

[2] Hydrologic modeling has two distinct flow regimes: overland flow/routing and hydrodynamic reservoir/channel simulation [McCuen, 2005]. Computational codes are generally simplified and optimized for one of these regimes: two-dimensional (2D) horizontal for overland flow and 2D-vertical or three-dimensional (3D) for reservoir mixing. In general, overland flow models are optimized to simulate area processes (surface and subsurface flow) implementing 2D horizontal simplifications. While hydrodynamic water body mixing models are optimized to simulate vertical processes such as thermoclines, reservoir mixing, and other processes, often implemented with 2D vertical simplifications. A model optimized to simulate one regime generally uses simplifying assumptions (for the other regime) to reduce model setup, complexity, and run time. These simplifications may not be optimal for the other regime and linking the two models is not a trivial task [Beaujouan et al., 2001; Flipo et al., 2007; Flipo et al., 2004].

[3] Models simplified for overland flow modeling typically use a 2D horizontal or area-based approach (e.g., horizontal grids or sub-basins). Vertical components, such as infiltration are typically handled with approximations such as layers or one-dimensional boundary models, rather than implementing a full 3D model. Examples include the Gridded Surface Subsurface Hydrological Analysis (GSSHA), Soil and Water Assessment Tool (SWAT), and Hydrologic Simulation Program-Fortran (HSPF) models which all use two-dimensional or area approaches [Downer and Ogden, 2006; US-EPA, 2007]. This class of models usually include routines for flow routing but do not implement hydrodynamic vertical mixing or process simulation capabilities.

[4] Reservoir models that simulate vertical mixing (for water quality or thermal modeling) use vertical discretization schemes [Salah, 2009] and can be either 2- or 3D, with the decritization designed to simulate mixing and other reservoir processes which have significant vertical components. For these models, overland flow is generally simplified as either inflow or boundary conditions for the model. One example of this type of model is CE-QUAL-W2 (W2) which has complex hydrodynamic capabilities for water quantity and quality modeling and includes chemical, thermal, biological, sediment, and other reservoir process modeling capabilities [Cole and Wells, 2007].

[5] In addition to different spatial discretization (horizontal versus vertical), temporal discretization (or time stepping) can also be different between models, even when modeling the same basin and time frame. This is to optimize computing resources as time scales or periods of interest for reservoir mixing processes are typically different than those for overland flow processes, though there is considerable overlap.

[6] In some cases, for example, in total maximum daily load (TMDL) studies, modelers need to use both advanced overland flow and advanced reservoir quality/mixing models to evaluate impacts and make management decisions [Beaujouan et al., 2001]. This can be done by using an overland flow model to provide accurate boundary or input conditions for a reservoir model. Traditionally, modelers have manually linked the output of the overland flow models to the input of the reservoir mixing models [e.g., Beaujouan et al., 2001; Flipo et al., 2007; Flipo et al., 2004Melching and Bauwens, 2001; US-EPA, 2007]. This effort is difficult, time consuming, and is generally specific to the watershed being modeled and the models used for simulation. Semi-automated approaches have been developed, such as the Environmental Protection Agency (EPA) Better Assessment Science Integrating point & Non-point Sources (BASINS) modeling system [US-EPA, 2007], that implement both simplified algorithmic, and manual steps for linking the models for the different flow regimes. We present a fully automated approach that links and runs the two model regimes. This provides a framework that easily supports stochastic simulations. This is an extension of previous work, developing algorithms to link models without human intervention and run the resulting simulation system.

2. Objectives

[7] We have two main objectives in this research:

[8] Develop general automated algorithms to spatially and temporally link and run overland flow and reservoir mixing models.

[9] Develop tools for easily implementing, executing, analyzing and presenting stochastic simulations.

[10] Both these areas are currently addressed with various manual and semi-automated tools [e.g., US-EPA, 2007]. We extend this field by developing, implementing, and demonstrating automated tools to simplify linked and stochastic modeling.

3. Methodology Overview

[11] Linking an overland flow and reservoir mixing models is challenging because the two models use different discretization schemes (i.e., 2D horizontal versus 2D vertical) making the determination of interface points non trivial; the models have different grid sizes and time steps making it difficult to determine how to partition fluxes from an overland grid cell to a reservoir grid cell. Model linking using semi-automated and manual approaches is not new [Beaujouan et al., 2001; US-EPA, 2007; Flipo et al., 2007; Flipo et al., 2004]. Our approach is fully automated and provides high-resolution links and a framework that supports stochastic simulation.

[12] We implemented the algorithms as tools in the Watershed Modeling System (WMS) to link and run GSSHA and W2 [Ogden, 2006; Cole and Wells, 2007; Downer and Nelson, 2010]. The tools create input files, run the models and this provides a framework to easily implement stochastic simulations to explore parameter uncertainty. Using the tools, a modeler can assign statistical distributions to model parameters, perform a number of runs using parameters selected from the chosen distributions with three different methods, and analyze the results using statistical and presentation tools.

[13] Our algorithms identify interface cells, distribute fluxes among cells, configure and run GSSHA repeatedly using the model input and stochastically selected parameters. The tools then create the W2 input based on the interface cells, flux distribution, and output from the algorithms and the GSSHA runs. The tools then run the W2 model repeatedly using different GSSHA results. The tools aggregate and present the results of stochastic runs.

[14] These algorithms automatically create detailed links at a finer resolution than would commonly be done by hand. They support model development and model execution, both as single runs and as stochastic simulations that use parameter distributions rather than single values. Our tools reduce the time and resources required to develop and use linked models and perform linked stochastic simulations. These tool provides repeatable and defensible results, an essential criteria for stochastic simulations [Salah, 2009].

4. Models and Tools

[15] GSSHA and W2 are both developed by the U.S. Army Corps of Engineers (USACE). GSSHA is 2D, horizontal gridded, finite difference, distributed parameter, hydrologic model. GSSHA simulates the hydrologic response of a watershed including surface and ground water hydrology, erosion and sediment transport [Downer and Ogden, 2006]. W2 is a laterally averaged 2D longitudinal-vertical hydrodynamic and transport model. It can perform long-term, time-varying water quality simulations of networks of rivers, lakes, reservoirs, and estuaries. W2 accurately reproduces vertical and longitudinal water quality gradients [Cole and Wells, 2007].

[16] GSSHA uses a regular square grid with an underlying digital elevation model. GSSHA is capable of long-term simulations of multiple events and requires spatially and temporally distributed meteorological variables and surface energy-balance parameters [Downer; Ogden, 2006; Downer et al., 2004]. Features of GSSHA include 2D overland flow, overland erosion and sediment transport, 1D stream flow, evapo-transpiration, 1D infiltration, 2D groundwater, and full coupling between the groundwater, vadoze zone, streams, and overland flow. GSSHA has advanced channel routing routines allowing flow in any direction and meander independent of grid resolution. Routing includes routines to calculate the impact of weirs, culverts, lakes, wetlands, rating curves, and scheduled releases. Overland and channel flow modeling is based on an explicit, finite-volume, diffusive scheme with dynamic time stepping to improve model stability and decrease simulation times. Groundwater is coupled to streams in GSSHA using a two-dimensional finite-difference groundwater model with a stream bed conductance layer [Downer; Ogden, 2006; Downer et al., 2004].

[17] W2 solves the unsteady hydrodynamic and advective-diffusion equations on a 2D vertical grid assuming laterally averaged equations of momentum, continuity, and transport. Density changes from constituents such as temperature and salinity are included through an equation of state. The formulations include surface wind stress and the bottom stress due to friction. W2 computes water quality for many nutrient and nutrient-related parameters as well as suspended solids, coliforms, total dissolved solids, and numerical tracers [Cole and Wells, 2007].

[18] W2 can simulate routing, mixing, and eutrophication processes. It models relationships among temperature, nutrients, algae, dissolved oxygen, organic matter, and sediments. W2 includes water flow, contaminant transport, chemical reactions, and biological reactions with vertical mixing for interconnected rivers, reservoirs and estuaries [Cole and Wells, 2007]. Since it is laterally averaged, W2 divides water bodies, such as reservoirs, into branches, with each branch being modeled as a 2D vertical grid [Cole and Wells, 2007]. W2 further breaks down each branch into segments or longitudinal grid cells. Boundary conditions include fluxes at the open end of branches and distributed inflow to other segments.

[19] GSSHA and W2 are state-of-the-art models for overland flow and routing (GSSHA) and reservoir and river water quality (W2). We implemented tools to link these two models as a specific case study of the generic algorithms we developed to link overland flow and hydrodynamic models [Salah, 2009].

[20] We implemented our algorithms in the WMS model development and analysis environment to take advantage of its GIS tools and other pre- and post-processing capabilities. Our implementation included the algorithms, tools and a user interface to link the models, select stochastic parameters and assign distribution (for stochastic runs), edit the selections, run the models, and analyze the results. WMS is a commercial comprehensive graphical user environment to develop and analyze models for watershed hydrology and hydraulics [Aquaveo, 2010]. WMS supports a number of hydrologic modeling packages including GSSHA and W2 [Nelson, 2010].

5. Spatial and Temporal Linking

[21] In this section we discuss the algorithms developed for spatial and temporal linking. We present the tools we developed to configure and analyze stochastic runs. We, then, summarize the modeling environment and how the linkage is implemented and present a modeling case study to demonstrate the tools and techniques.

[22] Figure 1 is a schematic diagram illustrating the spatial linkage concept. Output from the GSSHA horizontal grid is routed to the vertical grid of W2. This includes water quality and sediment information from GSSHA, in addition to flow. Figure 1 also indicates the surface-subsurface interaction within GSSHA. W2 uses the GSSHA input as boundary conditions for detailed reservoir mixing analysis. This figure highlights the integrated watershed concept where the overland portion of the watershed is modeled by GSSHA (horizontal grid) and the hydrodynamic, thermal, chemical, and biological reservoir processes are modeled by W2 (vertical grid).

Figure 1. Linking Two Models with Different Domains; 2D Horizontal to 2D Vertical. While the figure shows the horizontal grid as three pieces, the grid is a single grid that extends over the vertical grid.

5.1. Spatial Model Linking

[23] To link these two dissimilar domain models (Figure 1) the algorithms need to identify the GSSHA cells that bound water bodies and link the outflow from these cells as input to the W2 model.

[24] We evaluated a number of approaches to spatially link the two models. First we tried using the internal GSSHA routing packages coupled with a GSSHA lake that matched the spatial extent of the W2 reservoir to identify boundary cells. We found that this approach required bi-directional model links because changing water levels alters the location of the lake and overland flow cells, so this approach was not used. We, then, evaluated algorithms that tested cells using a spatial Land-Water Interface (LWI) approach. We developed algorithms to link the models and address situations where the number of cells and complicated boundaries resulted in incorrect mass balances. While this approach was viable, we created another approach called the Segment ID Index Map (SIIM), that solved these problems in a straight forward manner, consistent with the GIS-based conceptualization of WMS and the index map of GSSHA. We fully implemented SIIM to compute the spatial links between the two models. The appendix, Section 11.1 presents a brief overview of the approaches that were tried and rejected.

5.2. Segment ID Index Map (SIIM)

[25] The SIIM approach links the two models using a segment ID index map. SIIM relies on the fact that each water body in a typical W2 model is divided into branches and each branch is divided into segments (Figure 2). A segment is similar to a grid cell in a standard finite difference mesh. In this approach, we first generate an index map with a SIIM cell for each GSSHA cell (Figure 3). The SIIM cells (index map) are overlain on the W2 segment polygons using WMS spatial intersection routines. The SIIM cells are assigned index values as follows:

Figure 2. An example W2 Model with one branch and six Segments (segment ID displayed). While this figure shows the width of the reservoir, notice the grid is 2D only in the vertical direction.

Figure 3. Spatial Linkage, Segment ID Index Map for the Same Example in Figure 2.

[26] Zero: for GSSHA cells that do not overlap the water body (Figure 3) and

[27] Non-zero: for GSSHA cells that overlap the W2 water body. The non-zero value is the corresponding W2 segment number (Figure 3).

[28] Figure 2 shows an example W2 grid representing a water body with only one branch and six segments (comprising this branch) with the segment IDs displayed from 2 to 7. Figure 3 shows the SIIM that results from the GSSHA grid and these six segments. The grid cell size plays a major role in the conversion process and how accurately each segment is represented by the SIIM. Border cells are crucial since they represent the interface between the overland flow and reservoir models. The SIIM algorithm does not consider inner reservoir cells while identifying the links. The water body boundary is the most important aspect when linking the two models as this determines the resolution and accuracy of the flux partitioning.

[29] GSSHA output fluxes are aggregated for input to the W2 model based on the SIIM values. The net flux (flow, constituents, sediments, etc) to a W2 segment is aggregated only at the interface from a zero to a non-zero cell. This approach solves many of the issues related to complicated boundaries (discussed in the appendix) in an easily implemented manner.

[30] GSSHA streams are routed to the appropriate W2 segments by identifying GSSHA stream nodes that intersected W2 segments. At these locations, we used the GSSHA output hydrograph to calculate the flow and constituent input to W2 as a boundary condition.

5.3. Temporal Linking

[31] The duration, time step, start time, and end time of the two models need to be linked to allow inter-model routing. Salah, 2009] suggests three primary ways to temporally link models together:

[32] Identical time step: using the same model time step,

[33] Interpolated time step: using a different time step for the two models and interpolating the results. This is useful in two-way model linkage where input/output data is transferred between the two models at run time and

[34] Independent time step: where the two models can have two different time steps and run independently.

[35] We used identical time step approach while allowing each model to be optimal for its task and simplifying issues associated with time step interpolation. In our approach, the time steps of the two models are the same, but the model durations can be different. We set the W2 time step equal to the hydrograph output frequency in GSSHA. In practice we found that using identical time-steps eliminates a number of potential issues and with modern computers there is little penalty for the finer temporal resolution [Downer et al., 2006; Cole et al., 2007]. Identical time steps allow aggregated fluxes from GSSHA to be used as direct inputs for the W2 segment inflow file. Interpolated time steps would require additional effort and create potential errors from the interpolation plus the selection of the interpolation technique becomes a concern.

[36] The GSSHA run duration does not need to match the W2 run duration. With the same start time there are three cases:

[37] GSSHA run duration is longer than W2 run duration. In this case, the algorithm trims the flux files generated by GSSHA to match the total run duration of W2, specified in the control file (Main W2 input).

[38] GSSHA run duration is shorter than W2 run duration. In this case, the algorithm extends the flux files generated by GSSHA using a dummy record with a flow of zero and Julian date matching the end of the W2 model. This is analogous to a case where overland flow ceases after a precipitation event.

[39] GSSHA run duration is the same as the W2 run duration. In this case, the algorithm evaluates and adjust, if necessary, the last time step in the GSSHA-generated flux files to exactly match W2.

6. Stochastic Modeling

6.1. Simulations

[40] To demonstrate stochastic modeling, we selected and implemented six GSSHA parameters to treat stochastically, provided ways to assign statistical distributions to these parameters, implemented several methods to select values from these distributions for simulations, and provided a way to save and edit the resulting parameter sets.

[41] There is often uncertainty associated with model input parameters and in some cases, we have quantitative information for, or can estimate the parameter's statistical distribution and uncertainty. The stochastic approach propagates this statistical knowledge about the input parameters to provide an estimate of the uncertainly in the model results. Our tools present the consequent model results as an expected range along with the best estimate.

[42] The tools we developed for linked stochastic modeling Include 1) a way to specify the number of runs to execute, 2) a selection list of GSSHA parameters that can be treated stochastically, 3) a way to assign a statistical distribution to the selected parameters, 4) algorithms to select values for a given realization, 5) a mean to save and edit the parameters sets and values, and 6) tools to aggregate, analyze, and communicate the stochastic results.

[43] The six GSSHA parameters for stochastic treatment are capillary head, hydraulic conductivity, initial moisture, Manning's n, porosity, and precipitation as suggested by Salah, 2009]. Modelers can select any number of, or all, parameters for stochastic simulation. The parameters can be assigned a uniform, normal, log-normal, beta, or user defined distribution. The Beta distribution is useful because it can approximate both the normal and log-normal distributions, ranges between true minimum and maximum. It does not require truncation, as the normal and log-normal do, to match physical constraints [Bury, 1999, Ashkar and Arsenault, 1998].

[44] We implemented three selection methods to generate parameter sets for the stochastic runs: Monte Carlo (random), Latin hypercube (weighted), and a user assigned method [Landau and Binder, 2005]. The user selects the number of runs. Our tools select parameter values from the assigned distribution using the chosen method and selected number of runs. Before starting the simulation, these values are saved and presented for editing so the given set of realizations can be re-run with other model values.

6.2. Stochastic Analysis Tools

[45] WMS aggregates the output (hundreds of runs) and presents the results over time with the associated uncertainty and provides statistical tools for further analysis. The results are a set of W2 output time series for selected observation locations and output parameters. These can be single observation points, or a full 2D grid representing the changes in a reservoir over time. Each observation point and chosen constituent(s) has a number of associated time series, equal to the number of stochastic runs. WMS computes the mean value for each point in a time series set and the associated credible interval. The user can select the uncertainty associated with the credible interval, with values generally in the 60% to the high 90% range.

6.3. Credible Intervals

[46] Good and Hardin [2006] state that the use of the best estimate plus or minus a multiple of the standard error is an incorrect way to evaluate and communicate uncertainty because this method is very sensitive to outliers and for stochastic modeling, it may be misleading. Ramsey and Schafer [2002] suggest there are three ways to better address uncertainty: confidence, tolerance, and credible intervals. Confidence intervals are ranges in which the parameter may lie most of the time. Tolerance intervals are ranges covering a stated proportion of the population most of the time. Credible intervals are ranges in which the parameter probably lies [Hays and Winkler, 1970]. We implemented credible intervals to present and analyze uncertainty associated with stochastic results.

[47] The Credible Interval half width is equal to:

• 

Where: CI_w is the credible interval half width, s is the sample standard deviation, n is the sample size, and k is an adjustment factor which, in this case, is Z-score. This relies on a number of assumptions (for example: Wu et al., 2006; Ramsey and Schafer [2002; Good and Hardin [2006; or Melching and Bauwens, 2001]. The main assumption is the variation in the results is normally distributed for each point in the time series. Based on the central limit theorem, this holds true as the variation is a result of combining various parameter distributions. This does not mean the results are normally distributed, only that variation or spread in the results is normally distributed for each point in time. Also, Z-scores are robust and can be used if the normality constraint is slightly violated [Ramsey and Schafer [2002].

[48] The Z-score is based on the credible level desired and is obtained by lookup from most statistical text books [e.g., Hays and Winkler, 1970; or Ramsey and Schafer [2002]. Table 1 presents various credible levels and the associated Z-scores.

Table 1. Z Scores for Various Confidence Levels

Credible Level	α	α/2	Z-Score
50.0%	0.500	0.2500	±0.675
80.0%	0.200	0.1000	±1.282
90.0%	0.100	0.0500	±1.645
95.0%	0.050	0.0250	±1.960
98.0%	0.020	0.0100	±2.326
99.0%	0.010	0.0050	±2.576
99.9%	0.001	0.0005	±3.291

[49] Our tools allow users to select a credible level. WMS then computes the associated CI_w and mean result for each time step. The results are presented as the mean (or best estimate) ± CI_w in a time series plot as shown in Figure 4. Figure 4 shows a part of time series plot with the mean value and upper and lower bounds of the 99% interval.

Figure 4. Stochastic time series plot in a W2 cell. The plot shows the mean value at each time step (not the results of the median run) and the associated 99% credible interval at each time step. This plot is based on 50 runs.

7. Modeling Environment

7.1. Overview and Implementation

[50] To use our tools, a modeler first develops the GSSHA and the W2 models using (or loading them into) WMS. This includes creating the grids, assigning parameters, choosing output, etc. In WMS the modeler chooses whether to use the stochastic capabilities or not. If so, then the modeler selects the number of runs and the parameters that will be described by statistical distributions. The modeler assigns distributions to each parameter and chooses the selection method that will be used to generate the data sets. The GSSHA and W2 models should be individually validated and calibrated to the extent possible before linking.

[51] After the modeler has created and calibrated the models, WMS links the two models in the following sequence, WMS; 1) verifies that both W2 branch and segment coverages exist, 2) verifies that a GSSHA grid exists and covers at least the bounding box of all segments in the W2 model, 3) identifies the W2 segment ID range; i.e. minimum and maximum segment ID (minimum segment ID should always be 2, as W2 adds a dummy segment upstream of the first segment and assigns ID of 1 to this dummy segment), 4) scans GSSHA grid cells and determines the cell centroids, 5) finds the segment polygon (polygon used by WMS to define the W2 segments) that encloses the cell, 6) assigns the W2 segment ID to the GSSHA grid cell, 7) creates the Segment ID Index Map file, 8) creates GSSHA input files and runs the GSSHA model, 9) uses the GSSHA output and the Segment ID Index map to create W2 input files (using the appropriate GSSHA outputs and W2 inputs), and 10) runs the W2 model and stores the results.

[52] For stochastic runs, values for the stochastic parameters are selected using the designated statistical distributions, and are used to create the GSSHA input file in Step 8. The algorithm loops from Step 8 to 10 until the specified number of runs have been completed.

[53] This framework allows other modeling codes to be used with minimal work, which would not be possible with a monolithic modeling system. Details on how these links were implemented and the associated files and code modifications are provided in the Appendix, Section 11.2.

7.2. Stochastic Input Tools

[54] Figure 5 shows the stochastic simulation interface which provides a list of each potential stochastic parameter. The modeler selects the stochastic parameters and assigns a distribution with its associated parameters. The modeler then chooses which selection method will be used to generate values for each realization (or run): Monte Carlo, Latin hypercube, or user defined (Figure 5).

Figure 5. Stochastic GSSHA dialog showing the parameters selected for stochastic treatment (upper right to center), the number of runs (upper right), parameter selection method (near upper right), the distribution for a parameter (center left), associated distribution parameters (center left), and the values selected for each run (lower left).

[55] The routines then generate and present the parameter sets so they can be edited and saved before the models are executed. This allows the modeler to review the parameter picks, and also to re-run a set of stochastic runs using the same values. This can be beneficial if modelers need to explore the impacts of other modeling scenarios or parameters, but want to compare between runs.

8. Case Study Example

[56] We tested and demonstrated the algorithms and tools we developed with two case studies, Lake Zapotlan, Mexico and Eau Galle Reservoir, Wisconsin, USA [Salah, 2009]. We briefly present some of the findings for Eau Galle Reservoir here to demonstrate these tools. The case studies are discussed in detail in Salah, 2009].

[57] We simulated the Eau Galle Reservoir Watershed using a GSSHA model with a grid cell size of 100 m with 468 rows and 305 columns to model an area of approximately 140 hectares. The model simulation was 1000 minutes (approximately 17 hours) with a computational time step of 5 seconds. The Green and Ampt method was used for infiltration and the diffusive wave equation was used to compute overland flow and channel routing for Eau Galle River. The 17 hour period represents a specific storm event that was studied.

[58] Eau Galle Reservoir is a 60 hectare impoundment located just north of Spring Valley, Wisconsin and 80 kilometers east of the Twin Cities (US-ACE, 2007). The Eau Galle River is located in western Wisconsin in the United States and is about 55 kilometer long. It is part of the Mississippi River watershed. Eau Galle Reservoir was simulated using a W2 model with one branch and seven segments. The upstream and downstream segments of the reservoir were discretized to 2 and 9 W2 layers, respectively. Intermediate segments were modeled which gradually increased from 2 to 9 layers down the reservoir. The layers were 3 meters thick and they simulate a reservoir depth of up to 27 meters. The W2 model duration was two years.

[59] The W2 distributed reservoir input is from the GSSHA computed overland flow, with the W2 river input from the GSSHA computed flow in the Eau Galle River using GSSHA routing modules. The linked cells were selected using the SIIM algorithm and the GSSHA and W2 model grids. We chose 50 runs to demonstrate a stochastic simulation.

[60] To investigate the various aspects of the stochastically linked GSSHA to W2 models, we evaluated 4 different scenarios:

[61] Deterministic run: This represented a single GSSHA run linked to a single W2 run. The GSSHA model start date was identical to W2 model start date. However, the duration of GSSHA was considerably less than the duration of the W2 model (17 hours for GSSHA and 2 years long term simulation for W2). This was considered the base deterministic model.

[62] Base stochastic run: This included 50 stochastic linked GSSHA - W2 runs. Similar to the base deterministic model, the GSSHA model start Julian date coincides with the W2 model start date on all 50 runs. This was considered the base stochastic model. We treated capillary head, hydraulic conductivity, initial moisture, and porosity stochastically (Figure 5).

[63] Delayed stochastic run: This is identical to the base stochastic model with GSSHA input delayed to Julian date 381. In this model, the first 380 days have essentially the same input and therefore, the credible interval width is zero up until the stochastic GSSHA input comes into effect (Figure 6).

Figure 6. W2 Output: water depth for delayed stochastic runs 1, 2, 6 and 13. Runs 6 and 13 are the minimum and maximum runs, respectively. Runs 1 and 2 are results in the middle of the distribution.

[64] Stochastic temperature run: This is identical to the base stochastic model with one exception; i.e. the tributary temperature were modeled stochastically. The tributary input temperatures were assigned a Normal distribution.

[65] The complete results for these four scenarios are beyond the scope of this paper. We present the computed reservoir water depth for the delayed stochastic run as an example. Figure 6 presents reservoir water depth for four runs (runs 1, 2, 6 and 13) over the 2-year simulation period. The runs were selected to show the range of the stochastic simulations. Run 6 represents the minimum depth values simulated while run 13 results represents the highest depth values simulated. Runs 1 and 2 represent middle values. Each run is a single realization of the possible parameter range starting day 381 (1^st Jan 2007). Prior to the GSSHA-modeled storm input (i.e. before Julian day 381), there are no variations in the water surface elevations because all the runs – up to- day 380 are the same; GSSHA stochastic variations begin at day 381.

[66] Figure 6 presents these four different results that show the range of impact of the storm runoff on the lake resulting from 50 runs and the 4 varying parameters: capillary head, hydraulic conductivity, initial moisture, and porosity stochastically.

[67] Figure 6 shows the water depth in the reservoir through the entire W2 simulation period of 2 years for the 50 W2 runs. The GSSHA model was stochastically run for approximately one day at the beginning of January 2007 (the 17 hour storm period). The Figure shows that prior to the GSSHA stochastic input the four runs, 1, 2, 6 and 13 produced the same reservoir depth. This makes sense because there are no changes in the boundary conditions. During the period where GSSHA provides stochastic input (changing boundary conditions between runs) we start to see difference in the water depths between these respective runs. As noted, these runs were selected to show the range of W2 stochastic results, with the minimum, maximum, and two middle value runs.

[68] Figure 7 shows an temperature variation output from the stochastic W2 model, in particular, the interval half width (CI_w) for temperature in a specific cell (i.e. layer and segment) in the reservoir. We can see that the interval width is zero until 1^st Jan 07 when the GSSHA stochastic variation starts. The negative values on the left axis in Figure 7 indicates a reduction in the temperature. Each point on this figure represents the difference between the highest and lowest temperature, as stochastically modeled, for the specific cell for the respective credible interval.

Figure 7. Temperature variation for Delayed Stochastic Run with three separate credible interval widths. The temperature variation is symmetrical for the stochastic simulation, this plot shows only the lower half (temperature decrease) for space reasons.

9. Summary and Conclusions

[69] We developed general algorithms and tools to spatially and temporally link an overland flow model with a stream and reservoir routing model including tools to treat overland flow parameters statistically, using stochastic simulation to provide a way to incorporate parameter uncertainty into the simulation results. We implemented specific algorithms to llink GSSHA and W2 using WMS as a pre- and post-processer to setup the models, perform the runs and present and analyze the results.

[70] We evaluated three ways to implement the spatial links between the two models; i.e. GSSHA Lake, LWI, and SIIM. Based on implementing these three methods, we determined that SIIM is the most appropriate approach. We evaluated using various techniques to link the models temporally. We found that identical time steps in GSSHA and W2 are not required but are expedient. In our tools, the start date, end date, and duration of both models do not need to be identical provided that the start date of the GSSHA model is between the start and end dates of the W2 model. We demonstrated our tools for the Eau Galle Reservoir where the W2 model runs for a significantly longer period than the GSSHA model. This example shows the impact of a single storm on reservoir processes.

[71] We developed tools to calculate and present the stochastic results credible intervals. We recommend that the modeler carefully evaluates the number of runs used for a simulation to balance the computer time required with having enough data to support the statistical analysis. If more parameters are chosen to evaluate statistically, more runs are required to develop a reasonable estimates for the credible intervals.

[72] This work provides a comprehensive integrated water resources management tool to answer hydrologic, hydrodynamic and water quality questions on a watershed scale in a stochastic context. We show how individual models with different domains and geometric aspects can be linked, and how to implement stochastic tools for these modeling applications.

[73] The approach and algorithms we developed can be generalized to other models. However, it requires a thorough understanding of the model domains to determine how to match material flows between dissimilar descretizations.

Appendix A

[74] This appendix provides additional information on two separate topics. The first are approaches we explored for linking the models, but ultimately rejected. Even though these rejected approaches were not the best for our particular implemetnation, we feel that these approaches will be useful in other applications. The second is more detail on the mechanics of how the models were linked. This portion of the appendix (i.e. 11.2) describes how input files were modified and the various approaches accomplished.

A1. Rejected Approaches

[75] We initially tried a number of approaches to link the two models before selecting the SIIM approach. These approaches were rejected for various reasons. We briefly present the two rejected approaches to provide researchers with the insight we gained from these investigations.

A2. GSSHA Lake

[76] A “lake” in GSSHA is a set of grid cells representing the initial, minimum and maximum water surface [Byrd et al., 2005]. We evaluated an algorithm that used the GSSHA lake boundaries as routing points for W2 reservoirs. This required extending the GSSHA stream network to the minimum water surface elevation cells, not the maximum or initial cells. If the GSSHA stream segments were not extended, the stream outflow (i.e., lake input) was not properly routed to the appropriate W2 segment (in fact it was not routed to any segment, and this resulted in mass balance errors). In addition, the computed surface elevations in GSSHA and W2 lakes/reservoirs were not consistent. To address this inconsistency, a bi-directional link was necessary, requiring a significant programming. This approach was not pursued after the initial implementation.

A2.1. The Land Water Interface (LWI)

[77] The LWI approach uses a modeler-assigned border that separates water bodies from overland. While simple in principle, the distinction is not always clear when developing boundaries in dissimilar model domains. The LWI algorithm spatially tests each GSSHA cell to determine if it is an interface cell (bounds a water body), then the algorithm tests the surrounding cells to determine if they are interface cells. If they are interface cells, the algorithm determines how flow should be routed between the cells, and across the interface. There are multiple tests to address the various potential shore configurations, some of which were algorithmically difficult. The difficult cases mostly arose when the W2 lake shore resolution was similar to, or at a finer resolution than the GSSHA lake shore. In these cases it is difficult to determine how to route flow from a large GSSHA cell, to several smaller W2 segments.

[78] The LWI approach evaluates each cell in the GSSHA model against the listed criteria. We developed tests for foreseeable conditions, but in practice, we found that implementation was difficult, and that criteria resulted in problems with complex boundaries. These issues are compounded if the W2 and GSSHA grid discretization is on the same scale; the optimal scale for the cells. Because of these issues, the LWI approach, after implementation and evaluation, was not used.

A2.2. Mechanics of Implementation

[79] In this section we present the details of how we modified the programs and implemented the algorithmic links. Both programs, W2 and GSSHA, use text input files. While there were some minor code modifications, in general, the tools we developed to link the models create and modify these input files, run the models, and use the resulting output to create and modify other input files.

[80] Figure 8 shows an example GSSHA project file to illustrate the modeling process and generated WMS files. This file is divided into four main sections: 1) Introduction: including title and additional remarks, 2) Batch Mode: indentifies the parameter and value files (Figure 8-A) for stochastic runs, 3) Regular Input: includes non-stochastic parameters, and 4) GSSHA to W2: indentifies the flux file, Segment ID index map and the stream nodes file (Figure 8-B).

Figure 8. Sample GSSHA Project File: (A) Batch Mode Section, (B) GSSHA to W2 Section.

[81] The flux file within GSSHA project file includes GSSHA directives that route output to each W2 segment for each time step. It identifies each GSSHA cell that routes to an individual W2 segment. This section of the file also directs WMS to create W2 input files for each segment that will receive input from GSSHA. GSSHA stream nodes are handled in a similar manner.

[82] W2 input accepts GSSHA output using a flow tributary inflow file. WMS creates a separate W2 flow tributary input file for each segment to include GSSHA output cell or stream index node. This is shown in Figure 9.

Figure 9. Flux File Break Down for the Same Example in Figure 2. The Main File is Broken Into Segments.

[83] In this algorithm, each GSSHA stochastic parameter is defined with a unique key value that is a negative integer. This key is written to the GSSHA input file, which is repeatedly replaced with an actual value picked from the selected statistical distribution (parameter and value files as outlined in Figure 10) during the stochastic simulations. WMS generates two files, the parameter and value files (Figure 10), to implement stochastic simulations. The parameter file lists all the parameters to be stochastically modeled and the value file contains the associated values.

Figure 10. Value and Parameter Files Example Showing 5 Parameters.

[84] We modified the FORTRAN code of W2 generic version (Microsoft Windows platform) to take arguments specifying the path of the control file name (i.e. main W2 input file). This was necessary to automatically run W2 for stochastic runs. The tools were developed to run within WMS in Microsoft Windows.

[85] Figure 11 presents the overall phases required to run the linked models. This includes: Phase I where the files necessary to run stochastic GSSHA runs are developed; Phase II where files to perform the link between the GSSHA and W2 are generated; Phase III where W2 is run using the stochastic input from GSSHA, and Phase IV where the stochastic output at selected monitoring points in the W2 model are aggregated and presented to evaluate and communicate results and associated uncertainty.

Figure 11. Overall Conceptual Framework of the GSSHA/W2 Linkage.

Acknowledgments

[86] The authors are greatly indebted to Brigham Young University, department of Civil and Environmental Engineering for providing funds and resources for this research. The authors would like to thank Dr. Christopher Smemoe with Aquaveo for providing support with algorithm implementation in WMS. Special thanks go to the US-Army Corps of Engineers, ERDC in Vicksburg, Mississippi for providing part of the research fund.