HYDROLOGICAL PROCESSES Hydrol. Process. 13, 191±209 (1999)
Automated calibration applied to watershed-scale ¯ow simulations 1
Zhongbo Yu1* and Franklin W. Schwartz2 Earth System Science Center, The Pennsylvania State University, 248 Deike Building, University Park, PA 16802, USA 2 Department of Geological Sciences, The Ohio State University, 125 South Oval Mall, Columbus, OH 43210, USA
Abstract: Complexity in simulating the hydrological response in large watersheds over long times has prompted a signi®cant need for procedures for automatic calibration. Such a procedure is implemented in the basin-scale hydrological model (BSHM), a physically based distributed parameter watershed model. BSHM simulates the most important basin-scale hydrological processes, such as overland ¯ow, groundwater ¯ow and stream± aquifer interaction in watersheds. Here, the emphasis is on estimating the groundwater parameters with water levels in wells and groundwater base¯ows selected as the calibration targets. The best set of parameters is selected from within plausible ranges of parameters by adjusting the values of hydraulic conductivity, storativity, groundwater recharge and stream bed permeability. The base¯ow is determined from stream ¯ow hydrographs by using an empirical scheme validated using a chemical approach to hydrograph separation. Field studies determined that the speci®c conductance for components of the composite hydrograph were suciently unique to make the chemical approach feasible. The method was applied to the Big Darby Creek Watershed, Ohio. The parameter set selected for the groundwater system provides a good ®t with the estimated base¯ow and observed water well data. Copyright # 1999 John Wiley & Sons, Ltd. KEY WORDS
hydrological models; calibration; hydrograph separation; chemical mixing
INTRODUCTION Physically based hydrological modelling has emerged as an important tool for the study of watershed processes at a variety of scales. Beyond the diculties in actually constructing a model capable of representing complex hydrological processes are those of calibrating such a model. In watershed simulations, trial-and-error approaches are still common. These approaches involve a qualitative visual comparison between the observed and simulated hydrographs. For complex watersheds with long stream ¯ow records, this calibration procedure is inecient and often cannot make use of the limited information that may be available. This paper describes a strategy for calibrating the groundwater component of a hydrological model. The basin-scale hydrological model (BSHM) is capable of simulating hydrological processes in watersheds that include surface runo, groundwater base¯ow and stream±aquifer interaction (Yu and Schwartz, 1995, 1998). Hydraulic heads and groundwater base¯ows are used in our approach as calibration targets and constraints in the hydrological simulations. Furthermore, an indirect calibration approach is implemented to avoid the problems of ill-posedness, which is commonly associated with direct inverse procedures. The application of this approach is demonstrated in the Big Darby Creek Watershed, Ohio. The data for this exercise consist of * Correspondence to: Dr Zhongbo Yu, Earth System Science Center, Pennsylvania State University, 248 Deike Building, University Park, PA 16802, USA. Email addresses: Zhongbo Yu (
[email protected]); Franklin W. Schwartz (
[email protected]). CCC 0885±6087/99/020191±19$1750 Copyright # 1999 John Wiley & Sons, Ltd.
Received 22 June 1997 Revised 5 February 1998 Accepted 5 February 1998
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water level records for three wells and estimated base¯ows. An empirical/numerical method for hydrograph separation was used to estimate base¯ows. Chemically based approaches (for example, Hooper and Shoemaker, 1986; Kennedy et al., 1986; McDonnell et al., 1990; Robson et al., 1992; Harris et al., 1995) were used to validate the empirical procedure. METHODOLOGY Modules or submodels within the BSHM include the digital elevation generator (DEG), the versatile soil moisture budget (VB) model, the surface runo delay time model (RDTM) and the groundwater ¯ow model (GFM). The watershed is discretized into a set of small rectangular cells in BSHM. The BSHM operates on gridded data and performs a number of functions on each grid cell. On a cell by cell basis, precipitation is supplied to the model and the excess precipitation and evapotranspiration are estimated in the VB module. The excess precipitation is routed to the stream, and eventually to the watershed outlet. The calculated in®ltration is partitioned into soil zones and excess in®ltration becomes groundwater recharge. The iterative alternating direction implicit method (Prickett and Lonquist, 1971; Yu, 1997) is applied in the GFM to obtain a ®nite dierence solution of the second-order partial dierential equation of groundwater ¯ow. In each stream cell, base¯ow is calculated using Darcy equation based on the hydraulic gradient between the stream stage and groundwater hydraulic head. All of these submodels require parameters, which are commonly dicult to de®ne for complex watersheds. In this paper, we describe an automated approach for estimating the parameters required for the GFM. The outcome from the analysis is the set of groundwater parameters that provide the best overall ®t of simulated variables with limited quantities of observation data. While the parameters in soil pro®le, surface water and groundwater could aect the model calibration, the calibration procedure in this study involves four parameters Ð groundwater recharge (R), hydraulic conductivity (K), storativity (S) and stream bed permeability (C). The other model parameters and model structure are assumed to be ®xed from other calibration exercises or existing data (Yu and Schwartz, 1998). The values for those four parameters (R, K, S, C) are assumed to fall within plausible ranges. Values of each parameter can be varied spatially within a limited number of areas, although here K, S, and C are assumed to be single-valued parameters for the watershed. Calibration targets for the inverse procedure are records of water levels from various wells, and the estimated base¯ows from the watershed. The former data are generally available in most watersheds with wells. The latter information is not available, but must be determined through observations or calculations. We use the numerical method for hydrograph separation outlined in the following section, where the empirical scheme for hydrograph separation is validated using the geochemical response of the stream to storm ¯ow events. Estimating daily groundwater discharge This section presents an approach to estimating base¯ow at the watershed outlet. During storm ¯ow events, the hydrograph includes contributions from both surface runo and base¯ow components, while inter¯ow is considered to be insigni®cant. Historically, a variety of graphical and chemical methods have been used for the hydrograph separation. These approaches vary in their rigour and characteristically yield uncertain estimates. The graphical methods are generally empirical and should be considered as a crude estimate. The geochemical methods are dierent because they utilize actual data collected for the stream and the various components of the composite hydrograph. In many watersheds, however, long-term monitoring records for geochemical parameters are likely not to be available. Our approach involves collecting a limited suite of geochemical data to verify the accuracy of an empirical model for hydrograph separation. A procedure developed by Rutledge (1993) was found to yield separations comparable with those obtained using chemical techniques. His procedure for base¯ow estimation is similar to other stream ¯ow partitioning methods (Shirmohammadi et al., 1984, 1987). In this method, the separation of base¯ow is accomplished by joining the beginning of surface runo to the end of direct runo with a straight line (Figure 1). A linear Copyright # 1999 John Wiley & Sons, Ltd.
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Figure 1. Hydrograph separation of surface runo and base¯ow
interpolation scheme is used with daily discharge estimates to separate the composite hydrograph. This method is based on an antecedent stream ¯ow recession. Following a storm, a return to base¯ow is determined for the day when ¯ow ®rst declines by less than 0.1 log cycle: a daily decline of more than 0.1 log cycle indicates surface ¯ow (note that a user can specify a dierent value) (Barnes, 1939; Rutledge, 1993). The base¯ow through the period of storm runo is determined by linear interpolation through the period of storm ¯ow. A linear interpolation scheme may be adequate in some cases, but it may not represent the real response of groundwater base¯ow where the subsurface geology is complex. To account for dierent responses of groundwater base¯ow, this method was modi®ed to allow input about the base¯ow hydrograph from the user's knowledge during the storm events (e.g. the peak timing and value). Two techniques (three-point linear and polynomial function interpolations) are selected and coded to describe the base¯ow hydrograph for the period of the beginning of base¯ow increase to the end of surface runo during a storm event (Figure 1). Chemical hydrograph separation In this study, a chemical approach to hydrograph separation for a 68-day record is used to verify the empirical method described above. The chemical approaches for hydrograph separation are based on chemical mixing theory (Pinder and Jones, 1969; Pilgrim et al., 1979; Hooper and Shoemaker, 1986; Harris et al., 1995). The three main in¯ow components include: (1) direct precipitation on stream channels; (2) surface runo; and (3) groundwater. The mixing theory is based on the conservation of water mass and tracer for a continuous and open system with a stream chemistry as a result of mixing from the three sources. The chemical mixing method is not generally applicable or appropriate to problems in which continuous mixing and changes in storage are important (Harris et al., 1995). The calculation of base¯ow using this method requires several assumptions. (1) The precipitation and its chemical character do not vary spatially over the watershed, but may vary with time. (2) The chemistry of surface runo remains the same throughout the event. (3) The contribution to the stream from under¯ow and surface storage are insigni®cant. A mixing equation for a conservative tracer at each time-step is written as follows Cs Qs Cg Qg Cp Qp Ct Qt Copyright # 1999 John Wiley & Sons, Ltd.
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where C is the concentration of each water source, Q is the ¯ux and the subscripts s, g, p and t refer to the surface runo, groundwater base¯ow, direct precipitation on the stream and total ¯ux at the watershed outlet, respectively. One additional constraint is Qs Qg Qp Qt
2
If the quantity of direct precipitation on stream channels is insigni®cant compared with the total area of the watershed, then the proportion of ¯ux that arises from groundwater may be written as X
Ct ÿ Cs Cg ÿ Cs
3
These equations form the basis for the chemical method for hydrograph separation. Inverse procedure for calibration While trial-and-error calibration is still the most common method for calibrating models of all kinds, many direct or indirect techniques are available to solve an inverse problem in a groundwater system (Neuman, 1973, 1982; Yeh, 1988; Sun and Yeh, 1992; Brooks et al., 1994; RamaRao et al., 1995; Sun et al., 1995). The direct approaches use an inverse operator to solve the problems directly, while the indirect approaches use an iterative scheme. Our procedure is based on this latter, indirect, approach. Operationally, we use a process like latin hypercube sampling to cover the range of adjustable variables systematically. In eect, the simulation problem is solved many times over with coverage of the potential solution space. Ultimately, the process provides a set of parameters yielding the best ®t with observed data. Water levels in wells provide the initial calibration target. Let ho be the observed groundwater level and hc be the simulated groundwater level. Then, according to Taylor's series expansion, the equation can be written as m X @hc i
j 1
@pj
o
c
dpj hi
p dp ÿ hi
p i 1; . . . ; n
4
In matrix form, the above equation is written as Jdp r
5
where J is a rectangular Jacobian matrix, p is the parameter vector to be inverted, dp is the parameter correction vector and r is the residual vector between the measured and simulated hydraulic heads. In order to solve Equation (5), the transpose of matrix J multiplies both sides of the equation. The result is a system of equations with m unknowns. A problem occurs because the solution of the matrix system is unstable. Marquardt's approach (Marquardt, 1963; Press et al., 1989), which involves adding terms to the diagonals of the matrix JTJ, can overcome this problem. The Levenberg±Marquardt method or Marquardt method (iterative procedures) can be used to solve this system. The algorithm is equivalent to the minimization of the sum of the squares of the residuals (also called objective function or ®tness measure). The general form for a steady state is given by Obj
n X i1
o
c 2
wi si ÿ si
6
where wi is a weighted factor, soi is the observed data, sci is a simulated result, i is the index for location points and n is the number of location points. Copyright # 1999 John Wiley & Sons, Ltd.
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The method used in our study is similar to that of Brooks et al. (1994). Factors, f, for each parameter (R, K, S, C) are applied to each grid node (e.g. f can have an increment of 0.1 from the low end of a plausible range or ÿ01 starting from the high end). It is feasible to zone spatially, although execution times will increase signi®cantly as the degrees of freedom in the parameter values increase. Dierent combinations of the four factors were selected for each simulation run. Two objective functions, based on a weighted (e.g. 0.33 for each of three wells) least-square error criterion, are used to drive the calibration module. The ®rst objective is to minimize the dierence between measured and simulated hydraulic heads over time. The second objective is to minimize the dierence between the simulated base¯ow hydrograph and the base¯ow hydrograph determined from hydrograph separation. The objective function for a transient simulation is then formed as Obj
m X n X j 1 i1
o
c 2
wij sij ÿ sij
7
where m is the number of time steps, j is the index for time-steps and the other variables are as de®ned previously. BIG DARBY CREEK WATERSHED The Big Darby Creek Watershed is located approximately 50 km west of Columbus, Ohio (Figure 2). Big Darby and Little Darby creeks are the major named streams in the watershed. The watershed covers 1437 km2 with approximately 395 km of streams and tributaries. Slopes in the basin are gentle with elevations ranging from 410 m above sea level in the upland areas of the end of the watershed to 210 m above sea level at the watershed outlet. The major land use is corn and soybean farming, with 2% forest. Details of the geology and hydrogeology are given by Yu and Schwartz (1998) and will only be summarized here. The uppermost bedrock units are the Silurian±Devonian limestones and dolomites. They are represented as a single hydrostratigraphic unit, the so-called Carbonate Aquifer. Transmissivity values for this unit range from 190 to 400 m2/day. Storage coecients vary from 10 10ÿ3 to 10 10ÿ5 . Sur®cial deposits include ground moraine, outwash and alluvium. Ground moraine in the form of till covers 85% of the watershed to depths ranging from 1 to 30 m. Glacial outwash is commonly found along the stream valleys. In addition, sand and gravel bodies occur frequently in the glacial till and are of sucient size to provide a water supply to farmers. The hydraulic conductivity of the various sand aquifers ranges from 40 to 120 m/day. The vertical permeability of the till falls in a range from 10 10ÿ3 to 30 10ÿ5 m/day. Water chemistry The application of chemical methods for hydrograph separation requires knowledge of the chemistry of the in¯ow components, especially constituents that are used in the balance calculations. Although some chemical analyses are available for the stream and groundwater, they are limited in number and distribution. As shown in Table I, the surface waters are of calcium±magnesium bicarbonate type. At base¯ow, the stream has a composition similar to groundwater. However, during storm ¯ows, the chemistry changes with speci®c conductance, and Ca2 , Na , and Cl ÿ concentrations decrease as the discharge increases. pH varies from 7.6 to 8.6. The Ohio Geological Survey also has chemical analyses available for water samples collected at low ¯ows. Furthermore, chemical data are available for wells and piezometers completed in the bedrock and glacial deposits. Representative chemistry for waters from these units is also summarized in Table I. While existing data are reasonably abundant for the watershed, the discontinuous record is inadequate to construct a chemical hydrograph at the Darbyville Station (close to the watershed outlet). In addition, there were no data available for rain and overland ¯ow components. A limited ®eld study was undertaken to provide additional chemical data. Based on an examination of other work, and the preliminary data available for the Big Darby Creek, speci®c conductance was selected as the chemical variable for hydrograph separations. In all, 165 samples (stream: 90, surface runo: 45, small Copyright # 1999 John Wiley & Sons, Ltd.
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Figure 2. The location of the Big Darby Creek Watershed
creek: 10, rainfall: 20) were collected from 8 March to 15 May 1996. The sampling locations are shown in Figure 2. One stream water sample was collected daily and three during the storm ¯ow events. Three to seven samples were collected from surface runo during the rainfall events. Samples of both rain and snow were collected as well. The speci®c conductance of groundwater was determined from groundwater sample results placed on ®le with the Ohio Department of Natural Resources. The mean speci®c conductance for groundwater was 843 mS/m. Figure 3 summarizes the precipitation, discharge and speci®c conductance during this measurement period. As is apparent from the ®gure, there were several major storm events with a maximum daily discharge of 245 m3/s. During this period, the speci®c conductance of stream water averaged 489 mS/m with a range of 230 to 619 mS/m (Figure 3). The speci®c conductance of surface runo ranged from 200 to 332 mS/ m, with an average of 279 mS/m. The rain has a very low speci®c conductance (19±34 mS/m) compared with stream water, surface runo and groundwater. Copyright # 1999 John Wiley & Sons, Ltd.
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Table I. Summary of chemistry of surface water and groundwater in Big Darby Creek Speci®c conductance (ms)
Ionic concentration (mg/l) Ca2
Mg2
Na
K
HCOÿ 3
Surface water Average Maximum Minimum
686 864 492
72.5 81.0 60.0
33.3 41.0 25.0
17.3 23.0 12.0
2.5 3.3 2.1
278.3 340.0 190.0
Groundwater Average Maximum Minimum
843 1110 579
94.5 143.0 54.0
43.2 53.0 32.0
32.3 63.0 19.0
1.8 3.6 1.1
440.7 524.0 356.0
SO2ÿ 4 0.09 0.25 0.01 98.4 258.0 34.0
Cl ÿ 72.9 120.0 43.0 9.4 36.0 0.4
Figure 3. The distribution of speci®c conductances of rainfall, surface runo and stream ¯ow along with oberved rainfall and stream ¯ow Copyright # 1999 John Wiley & Sons, Ltd.
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Figure 4. Estimated daily base¯ows by the empirical scheme and chemical approach versus time from 8 March to 17 May, 1996
The daily groundwater base¯ows, estimated by the empirical scheme and the chemical method for hydrograph separation from observed stream ¯ow hydrograph, are plotted in Figure 4. These results show that the rainfall signals on stream ¯ows are more sensitive in the chemical approach than those in the empirical approach. The general trends of base¯ow hydrographs compared well for the two approaches. The major dierence is seen at a later time characterized by a series of frequent rainfall events, where the base¯ows estimated by the chemical technique are more `up-and-down' and recede faster than those estimated by the empirical scheme. This may be partially attributed to two eects. One is that the use of a constant composition for the groundwater in the chemical approach is an oversimpli®cation of this situation. In reality, the speci®c conductance can decrease in the shallow groundwater after the rainfall events because the less saline water can recharge the shallow groundwater and cause a dilution in the chemistry. Also, the daily speci®c conductance data may not be constant for the entire day because only two or three samples were collected during the rainfall events. The estimated daily surface runo and base¯ow from September 1992 to August 1993 are plotted in Figure 5. The average of discharge is 20.30 m3/s with a maximum stream discharge of 241.54 m3/s in July
Figure 5. The base¯ow and stream ¯ow hydrograph for the 1992±1993 water year Copyright # 1999 John Wiley & Sons, Ltd.
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Table II. The plausible ranges of hydraulic parameters for model automated calibration Hydraulic parameters Transmissivity (m2/s) Hydraulic conductivity (m/s) Storativity (dimensionless) Stream bed permeability (m/s) Recharge (cm)
Plausible ranges
Trial-and-error values*
1.0 10 ÿ6 ±1.0 10 ÿ1 1.0 10 ÿ7 ±1.0 10 ÿ2 1.0 10 ÿ6 ±1.0 10 ÿ1 1.0 10 ÿ10 ±1.0 10 ÿ5 23.3±60.6
2.9 10 ÿ3 1.4 10 ÿ4 4.8 10 ÿ3 7.2 10 ÿ7 47.9
* From Yu and Schwartz (1997).
1993 and a minimum of 0.25 m3/s in October 1992. Groundwater is the major stream ¯ow component during winter and spring. The annual surface runo is 24.07 cm while the annual base¯ow is 20.62 cm. The estimated base¯ow hydrograph was used as one calibration target along with water level hydrographs from three wells. Geographically, two of the wells are located in the upper end of the watershed and one is located close to the outlet of the watershed (Figure 2). Watershed parameters The hydrological model uses a variety of information about the watershed. The overall approach to providing the necessary information on climate, soils, land use, topography, and groundwater are discussed in a previous paper (Yu and Schwartz, 1998). The `best-®t' set of parameters as determined by trial-and-error calibration are summarized in Table II. Details on how these values were established are presented by Yu and Schwartz (1998). This calibrated idealization of the watershed is taken as the starting point for the parameter-estimation exercise as applied to the groundwater system. RESULTS To accommodate routing of ¯ows in the stream network, Big Darby Creek Watershed is subdivided into seven sub-basins, ws1, ws2, ws3, ws4, ws5, ws6 and ws7 (Figure 2). The automatic calibration procedure is carried out with the watershed discretized by a 33 82 grid. One water year-long record (from September 1992 to August 1993) was utilized for the simulation. The rainfall data for the 1992±1993 water year are shown in Figure 6. The reason for the relatively coarse grid and short simulation period is that the automated calibration requires a large number of iterations in searching for the best ®ts between the observed and simulated hydraulic heads, and between the estimated and simulated daily base¯ows. With these grid and
Figure 6. The rainfall data from 6 September 1992 to 31 August 31 1993 Copyright # 1999 John Wiley & Sons, Ltd.
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time-scales, one simulation trial requires about 90 seconds on a Silicon Graphics workstation (Indigo 2 Extreme). The complete automated calibration procedure involves 20 individual model trials. It would be inappropriate simply to run the model for the simulation period of interest (i.e. 1992±1993), because the initial conditions cannot be de®ned accurately. The process of developing a set of reasonable initial conditions involves running the model in three stages. We exercise the model using an initial one-year period with a constant recharge rate of 0.09 cm/day (32.85 cm annually) to bring the model to a steady state. A second year is simulated using actual climatic data providing a further period of adjustment. The automated comparisons between simulated results and observed data are performed in the third period. Automated calibration The automated calibration procedure ®nds the best ®t between the estimated and simulated groundwater base¯ows, while the calibrated hydraulic heads are considered as constraints in simulating the variation in the water table con®guration. The other model parameters, governing runo, in®ltration and evapotranspiration, were ®xed, having been calibrated previously through a manual trial-and-error procedure (Yu and Schwartz, 1998). The automated calibration is performed in two stages. In the ®rst stage, the model is run to ®nd the best-®t between the observed and simulated hydraulic heads without considering the ®t between the estimated and simulated groundwater base¯ows. This stage was accomplished by adjusting the values of R, K and S. In the second stage, the model was run to ®nd the best ®t between the estimated and simulated daily groundwater base¯ows by adjusting the value of C based on the calibrated values of R, K and S from the ®rst stage. This two-stage approach signi®cantly reduces computer processing times. The plausible range of each unknown parameters (R, K, S and C) was initially divided into 10 subdivisions Ð the so-called `coarse subdivision'. This approach is shown schematically in Figure 7. The calibration requires 1010 individual model trials to cover the spectrum of possible solutions. This step enables us to reduce the plausible range of distributions and to re®ne estimates through a ®ne-scale step. The searches were able to ®nd minimum function values, although one cannot guarantee that the global optimum has been found without the presence of Lipschitz conditions (Torn and Zilinskas, 1989; Brooks et al., 1994). The ®ne-subdivision searches were repeated until the ®tness was acceptable, or the relative error dierence between consecutive searches was less than 5%. The results of coarse- and ®ne-subdivision searches are presented in Table III.
Figure 7. Schematic digram of best-®t searches Copyright # 1999 John Wiley & Sons, Ltd.
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Table III. The calibrated factors of hydraulic parameters for coarse- and ®ne-subdivision searches Hydraulic parameters Recharge factor Hydraulic conductivity factor Storativity factor Stream bed permeability factor Objective function 1
Coarse grid
Fine grid
1.30 1.01 10.01 2.41 193.19
1.26 0.75 4.50 3.01 156.25
Hydraulic head data consisted of measurements at three wells during the 1992±1993 calibration year. In well 2, 365 continuous daily hydraulic head measurements were available. Less complete hydraulic head data were available for wells 1 and 3. In all, 930 hydraulic head measurements were available for the model calibration. Results from the calibration for hydraulic heads are quite good, both in terms of the match between the simulated and measured hydraulic heads (Figure 8), and in terms of the calibrated parameters, which fell within the expected ranges. The comparison between the simulated and measured hydraulic heads in wells 1 and 3 is not as good as that for well 2, in terms of daily variations. The problem with wells 1 and 3 could be partially caused by local heterogeneity not represented in the model, the simpli®cation that is made in representing the system as two-dimensional or the coarse model grid, which makes it impossible to match the nodes exactly with the well locations. Such problems of ®t are typical in ¯ow models and make it unlikely that any choice of parameters will produce simulated hydraulic heads that closely ®t the observed heads (Brooks et al., 1994). In all three wells, the simulated general trends of groundwater ¯uctuation are in good agreement with the observations. Unlike many hydrographs, those from wells 1 and 3 are characterized by frequent small changes in water levels. This sinuosity in the measured water level hydrographs may be attributed to local eects such as nearby pumping. It is possible that a dierent conceptual model, more resolved discretization of parameters or temporal changes in the parameters (such as coecients in VB and RDTM) would improve the ®ts. However, too small a grid can lead to over®tting and the temporal adjustment of parameters would require prohibitively large execution times. Overall, 365 base¯ow data points were derived from the observed daily stream ¯ow record for the 1992± 1993 water year using a numerical method for hydrograph separation. The results of the calibration run are plotted along with the estimated base¯ow hydrograph in Figure 9. The simulated base¯ows are in reasonable agreement with the estimated base¯ows. In general, the simulated base¯ow peaks occur somewhat earlier than the estimated base¯ow peaks after rainfall events. This dierence may be partially due to the direct treatment of rainfall excess in the VB model as an instantaneous recharge to the groundwater in the BSHM. Eect of changing hydraulic parameters on simulations The objective of this analysis is to evaluate the groundwater system response to various parameter disturbances based on the calibrated hydraulic parameters (R, K, S) and how combined constraints (hydraulic heads and estimated base¯ows) can improve the calibration process in the groundwater ¯ow system. This analysis involves perturbing the values of the hydraulic parameters with respect to the best estimates and examining how the simulated daily hydraulic heads and base¯ow discharges change. There are many approaches to determine the sensitivity (Yeh, 1986; Carrera, 1987). The sensitivity analysis of our calibrated model involved varying the hydraulic parameter values to look at the daily changes in hydraulic heads and base¯ow discharges according to the calibrated values from the previous section. For comparison purposes, the factors for the calibrated parameter values were normalized to 1.0. In the ®rst experiment, the hydraulic parameters conductivity (K), storativity (S) and groundwater recharge (R) were evaluated in this study by conducting two simulation trials for each parameter. One trial used a larger parameter value than Copyright # 1999 John Wiley & Sons, Ltd.
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Figure 8. Measured (cross) and simulated (line) hydraulic heads versus time in three wells (elevation: metres above sea level)
the calibrated value (1.0X) while the other used a smaller value. All other parameters were kept constant while a given parameter was being evaluated. In the second experiment, all three parameters are varied simultaneously to examine the behaviour of the groundwater ¯ow system under dierent situations. The non-uniqueness of solution of the parameter estimations for the groundwater ¯ow system can be evaluated. Copyright # 1999 John Wiley & Sons, Ltd.
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Figure 9. Estimated (cross) and simulated daily base¯ows versus time
Table IV. The water level variations in three wells owing to changing hydraulic parameters Parameters 1.0K, 1.0S, 1.0R 0.1K, 1.0S, 1.0R 10.0K, 1.0S, 1.0R 1.0K, 1.0S, 1.0R 1.0K, 10.0S, 1.0R 1.0K, 1.0S, 0.7R 1.0K, 1.0S, 1.3R
Water level (m)
Well 1
Well 2
Well 3
Average Minimum Maximum Average Minimum Maximum Average Minimum Maximum Average Minimum Maximum Average Minimum Maximum Average Minimum Maximum Average Minimum Maximum
216.29 212.67 226.81 232.85 221.50 243.15 212.73 212.23 221.12 215.40 212.24 247.14 219.63 216.11 228.41 215.39 212.57 226.81 217.11 212.74 226.82
284.67 275.98 306.85 317.55 293.00 337.73 271.16 268.56 299.32 282.36 268.59 370.13 292.53 288.44 309.11 281.90 274.63 306.78 287.18 277.08 306.91
293.56 283.75 318.55 331.32 302.46 355.24 275.59 271.65 310.71 290.55 271.74 409.47 303.44 300.22 321.42 290.14 281.40 318.44 296.64 284.97 318.64
The variation in the groundwater levels at the three wells from the model calibration is summarized in Table IV. Two simulation trials were carried out to assess the role of hydraulic conductivity in in¯uencing the variation in groundwater levels and base¯ow hydrographs. In the two simulation trials, two hydraulic conductivity values were used: one with ten times larger than the calibrated conductivity (10.0K) and one with 10 times smaller (0.1K). The results of the two simulation trials show signi®cant dierences in Copyright # 1999 John Wiley & Sons, Ltd.
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the groundwater levels and base¯ow hydrographs compared with the simulated results in the model calibration (Table IV and Figure 10a). In general, decreasing the conductivity value results in an increase in the groundwater level and the range in the ¯uctuation in the groundwater levels. Increasing the hydraulic conductivity value produced the opposite behaviour.
Figure 10. Simulated base¯ow hydrographs resulting from the change in the hydraulic parameters. (a) Change in conductivity; (b) change in storativity; (c) change in groundwater recharge Copyright # 1999 John Wiley & Sons, Ltd.
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The same trend was observed in the base¯ow hydrographs. Increasing the conductivity value tends to reduce the daily base¯ows during the low ¯ow periods and to increase the daily base¯ows during the rainfall events. In some low ¯ow periods, a negative daily base¯ow was recorded, indicating stream recharge to the groundwater. The trends are opposite in the case of a decrease in the conductivity value. The results show the non-linear relationship between the base¯ow hydrographs and the hydraulic conductivity. Two simulation trials for two storativity values (0.1S and 10.0S) were conducted to examine how changing the storativity value aects the groundwater level and base¯ow hydrograph shape. The variations in the daily groundwater levels and base¯ow hydrographs resulting from changing the storativity value are more extreme than those from changing the hydraulic conductivity value (Table IV and Fig. 10a and b). In general, a decrease in storativity results in more widespread declines in the groundwater level and an increase in the ranges of the groundwater levels compared with the simulated results in the model calibration. It also increases the quantity of groundwater released to the stream just after rainfall events. Decreasing the storativity tends to reduce the daily base¯ows during low ¯ow periods, to increase the daily base¯ows during rainfall events and to increase the overall range in the base¯ow. The results also show the non-linear relationship between the base¯ow hydrographs and storativity. Figure 10c and Table IV show the simulated groundwater levels and base¯ows for two simulation trials with two dierent groundwater recharge conditions (0.7R and 1.3R). In general, the base¯ow hydrographs with dierent conditions of groundwater recharge mimic each other (hydrographs parallel each other). The behaviour in terms of the peak evolution and timing is dierent from the cases when values of hydraulic conductivity and storativity are changed. In the second experiment, eight runs with various combinations of dierent hydraulic conductivity (0.1K and 10.0K), storativity (0.1S and 10.0S), and recharge values (0.7R and 1.3R) were performed. The resulting hydrographs, with six dierent combinations, are shown in Figure 11, while two runs with the combinations 10.0K, 0.1S, 0.7R and 10.0K, 0.1S, 1.3R produced no solution (¯oating point error and non-converge). The simulated hydrographs with the combinations 0.1K, 10.0S, 0.7R and 0.1K, 10.0S, 1.3R are ¯at without noticeable recharge signals (Figure 11a). In Figure 11b, the simulated hydrograph with the combination 0.1K, 0.1S, 0.7R compares well with the calibrated hydrograph while the simulation with the combination 0.1K, 0.1S, 1.3R provides much higher base¯ows than the calibration run during the rainfall event. This result suggests that dierent combinations of three parameters could provide other good solutions (good ®ts between the simulated results and observed data). The simulations with the combinations 10.0K, 10.0S, 0.7R and 10.0K, 10.0S, 1.3R in Figure 11c further demonstrate that simulation with dierent combinations of these three hydraulic parameters produce good ®ts. Three additional experiments were conducted to examine the eect of various combinations of hydraulic parameters (K, S, R) on the model solution. Each experiment consisted of 441 combinations of two hydraulic parameters when the third parameter was ®xed at the calibrated value (1.0X). According to the values of objective function, the ®tness of each run was ranked from the best (solid circle: an average daily 1.18 m3/s value of objective function) to the worst or non-solution (the smallest cycle) (Figure 12). Extreme high or low values of any hydraulic parameter combined with any value of the other hydraulic parameter produced a solution with very high values of the objective function or non-solution. In Figure 12a, the solution with the best-®t (solid circle) is surrounded by a cluster of solutions with good ®ts (the values of the objective function within the same order of magnitude as the best-®t). The goodness-of-®ts decrease as the combinations of K and S distance from the best-®t. The values of the objective function for the same R value increases as the values of K and S increase or decrease from the values (1.0K, 1.0S) at the best-®t (Figure 12b and c). The values of the objective function increase as the R value increases or decreases from the best-®t value (1.0R). Because the values of the objective function vary over many orders of magnitude, the variation in the values of the objective function due to changing R value cannot be displayed in Figures 12b and c. This analysis indicates that all four parameters in¯uence the transient variation in groundwater level and the base¯ow hydrograph. The combined variation of hydraulic conductivity and storativity plays a Copyright # 1999 John Wiley & Sons, Ltd.
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Figure 11. Simulated base¯ow hydrographs resulting from the combined change in hydraulic parameters
signi®cant role in shaping the base¯ow hydrograph, while the recharge factor and the stream bed permeability control the magnitude of the peaks of the base¯ow hydrograph. It is likely that various combination of these parameters could produce similar groundwater level and base¯ow hydrographs. However, the approach of combined transient constraints limits the number of possible solutions for the groundwater ¯ow system and is able to ®nd the best-®t given the assumptions and model structure. Copyright # 1999 John Wiley & Sons, Ltd.
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Figure 12. Solutions of changing hydraulic parameters. (a) Varying hydraulic conductivity and storativity; (b) varying recharge and hydraulic conductivity; (c) varying recharge and storativity. Note: solid circle representing the best-®t (1.18 m3/s); the largest circle to the smallest representing, 1.18±3.88 m3/s, 3.88±4.66 m3/s, 4.66±7.76 m3/s, and 7.76 m3/s non-solution, respectively Copyright # 1999 John Wiley & Sons, Ltd.
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SUMMARY AND CONCLUSION A methodology for simulating the hydrological response of the watershed has been presented, along with a procedure for automated model calibration. This calibration approach uses observed daily hydraulic heads and estimated daily base¯ows derived from a scheme for hydrograph separation. The calibration model was able to ®nd the best-®ts between the simulated and observed hydraulic heads, and between the simulated and estimated base¯ows. Although the approach is computationally intensive, the method converges on a best-®t more rapidly and more systematically than a manual trial-and-error method alone. The groundwater ¯ow system is better constrained because of the inclusion of both groundwater level and stream ¯ow data. The ¯exibility and eciency of the BSHM applied to regional surface water and groundwater problems in large watersheds are enhanced by this automated model calibration. The general trend of the base¯ow estimated by an empirical scheme compared well with a chemical approach for hydrograph separation. We consider that this is a reasonable validation test of the scheme for hydrograph separation. The results have suggested that the base¯ow estimated by the two techniques might not be valid during a series of frequent rainfall events. These analyses oversimplify the base¯ow hydrograph (empirical) or overamplify the rainfall signal in the base¯ow hydrograph (chemical). The eect of changing hydraulic parameters (R, K, and S) on the simulations points to a non-linear relationship between the base¯ow (or hydraulic head) and hydraulic parameters. An increase in the hydraulic conductivity and decrease in the storativity were manifested by an increase in the range of hydraulic heads and base¯ows, a decrease in average hydraulic heads and an increase in the amplitude of the hydrograph storm peaks. Decreasing the conductivity value and increasing the storativity value tended to smooth the pulse signal in the base¯ow hydrograph during rainfall events. The base¯ow hydrographs resulting from varying the groundwater recharge were very similar to each other. Experiments where the hydraulic conductivity, storativity and groundwater recharge were varied simultaneously were also conducted to further examine the model behaviour. Analysis of these parameters implies that the combination of changing these parameters could provide more than one good solution for the entire groundwater system. Two objective functions (one for hydraulic head and another for groundwater base¯ow) were able to constrain the groundwater ¯ow system better than just one. Using additional stream ¯ow data in the calibration will probably reduce the number of possible solutions with the combination of various parameters. In conclusion, the physically based BSHM provides results that compare well with other approaches. The hydrological components identi®ed by the short chemical analysis exercise compared well with the BSHM simulation results. ACKNOWLEDGEMENTS
This research was supported by a research grant from Ohio State University. We would like to acknowledge support from the Ohio State Supercomputer Center for use of the Cray Y-MP and Cray T3D and support from the Earth System Science Center at Pennsylvania State University. Special thanks go to Dr Chris Yoder at Ohio Environmental Protection Agency for his help in water sampling. REFERENCES
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