The Effects of Scaling and Model Complexity in Simulating the Transport of Inorganic Micropollutants in a Lowland River Reach

Karl-Erich Lindenschmidt,1* René Wodrich2 and Cornelia Hesse3

1GeoForschungsZentrum Potsdam, Section 5.4 – Engineering Hydrology, Telegrafenberg, 14473 Potsdam, Germany 2Applied University of Magdeburg-Stendal, Breitscheidstr. 42, 39114 Magdeburg, Germany 3Potsdam Institute for Climate Impact Research (PIK), Department of Global Change and Natural Systems, P.O. Box 60 12 03, 14412 Potsdam, Germany

A hypothesis stating that more complex descriptions of processes in models simulate reality better (less error) but with more unreliable predictability (more sensitivity) is tested using a river water quality model. This hypothesis was extended stating that applying the model on a domain of smaller scale requires greater complexity to capture the same accuracy as in large- scale model applications which, however, leads to increased model sensitivity. The sediment and pollutant transport model TOXI, a module in the WASP5 package, was applied to two case studies of different scale: a 90-km course of the 5th order (sensu Strahler 1952) lower Saale river, Germany (large scale), and the lock-and-weir system at Calbe (small scale) situated on the same river course. A sensitivity analysis of several parameters relating to the physical and chemical transport processes of suspended solids, chloride, arsenic, iron and zinc shows that the coefficient, which partitions the total heavy metal mass into its dissolved and sorbed fraction, is a very sensitive parameter. Hence, the complexity of the sorptive process was varied to test the hypotheses.

Key words: model complexity, error, sensitivity, scale, Saale river, TOXI, WASP5

Introduction Hence, this study should give new insight on scale com- parisons, shifting the focus from river catchments and This study combines two aspects in environmental model- their land surfaces (“area” scale) to transport mechanisms ling: scale issues and model complexity. Most studies on within the stream itself (“line” scale). scale issues in the water sciences, in which model results Many studies are also found in the literature which across different scales are compared, are found in the field explore the effect model complexity has on model uncer- of hydrology (see for example Kalma [1995], Sposito tainty. Elert et al. (1999) compiled the results of 13 dif- [1998] and Sivapalan et al. [2004]). These mostly consider ferent modelling exercises from seven research teams processes occurring on or in the land surfaces of catch- investigating the transport of surface contamination of a ment areas. The focus is mainly on water fluxes although pasture soil using three different radionuclides. No sim- some studies do include substance transport, for example ple relationship was found between model uncertainty erosion (Kandel et al. 2004), nitrate leaching (Hosang and complexity. 1998) and dissolved organic carbon (Aitkenhead et al. A hypothesis has been proposed by Snowling and 1999). Few studies consider scaling issues of processes of Kramer (2001) relating sensitivity and error, two charac- substance transport in the stream or river. Examples of teristics which constitute model uncertainty, with respect such cross-scale studies include the influence of the ripar- to model complexity (see Fig. 1). “Model sensitivity ian zone on stream water chemistry (Smart et al. 2001), increases with model complexity due to the larger num- the effect hydrological conditions have on stream water ber of degrees of freedom and the structure of the inter- acidity (Wade et al. 1999) and the relationship between actions between parameters and state variables. Model- soil organic carbon pools on dissolved organic carbon in ling error decreases with increasing model complexity as stream water (Aitkenhead et al. 1999). These studies have the more complex models are able to better simulate a focus on the stream but with the perspective from reality with more processes included and fewer simplify- processes occurring on or in the land surfaces connected ing assumptions” (Snowling and Kramer 2001). A utility to the stream or river network. The authors are unaware function can then be implemented which minimizes both of investigations focusing on instream processes on the error and sensitivity to help choose the “best” model. transport of substances in stream water at different scales. The function finds a compromise between models that can both simulate reality fairly closely (low error) and * Corresponding author; [email protected] still have high predictive power (low sensitivity).

24 Scaling and Complexity in Water Quality Modelling 25

icant at the large scale need now be included in the small-scale model description to achieve better accuracy in model output. This increase in complexity also increases the overall model sensitivity since the inclusion of additional processes bring with it additional parameters and data input. The objective of this paper is to test these two hypotheses using a water quality model on two sections of the river Saale, Germany, representing two different scales. The transport of suspended solids and inorganic pollutants was simulated once on the large-scale model of a 90-km reach with a discretization of 500-m segments and a simulation time-step of one day, and again Fig. 1. Complexity related to uncertainty (sensitivity and on the small scale of a lock-and-weir system on the same error) (adapted from Snowling and Kramer [2001]). Maxi- mum model utility is achieved by minimizing both error and river using a segmentation of 100 m and a time-step of sensitivity. one hour. Care was taken that both temporal and spatial resolutions were made finer when downscaling (see Blöschl and Sivapalan 1995). The effect of locks and Snowling and Kramer (2001) tested their hypothesis weirs on the transport of substances at different scales is on a sorption model for radioactive zinc onto sediments also highlighted. in solution (LeBeuf 1992 cited in Snowling and Kramer 2001). The complexity was changed by varying the sorp- Methodology tion process (equilibrium or kinetic) and the number of solute and sorbed phases. Their hypothesis was verified Sensitivity in which increased complexity caused an increase in parameter sensitivity and a decrease in model error. The sensitivity, s, of the input parameter values, P, on They also tested the hypothesis on a second model which model output values, O, was calculated using: simulates the transport of a groundwater tracer plume. ∂O P The complexity was changed by varying the sorption s = · (1) ∂P O process (equilibrium or kinetic), degradation processes (zero or first order) and isotherms (linear, non-linear and First, a base run is simulated with the parameter set- Monod—in order of increasing complexity). The ting, Pbase, to give Obase. A parameter is then increased or hypothesis could only be verified in part: sensitivity does decreased by a certain fraction, x, designated as Px which increase with increasing complexity but no relation was gives the resulting Ox. The sensitivity then becomes: evident between error and complexity. ΔΟ P (O – O ) P Lindenschmidt (2006) tested and confirmed the s ≈ · = x base · base (2) ΔP O (P – P ) O hypothesis using a water quality model developed for the x base base Saale river, Germany. The model EUTRO, a module in the WASP5 simulation package (Ambrose et al. 1993) which describes the phytoplankton-nutrient-oxygen dynamics in a water body, was implemented. The complexity of the model can easily be increased by enabling more state variables in the simulation, which increases the number of processes interacting between the variables and the number of parameters required to control these processes. Both error and sensitivity followed the trends as expected by this hypothesis in relation to complexity. The authors propose a second hypothesis stating that there will be a shift in the error and sensitivity curves when implementing the same model for studies of different scale (see Fig. 2). For example, when reducing both the temporal and spatial scales, processes become more dynamic and short-lived (Blöschl and Sivapalan 1995) and increased model complexity is required to obtain the same reduction in model error. Additional Fig. 2. Proposed hypothesis in which the relationship of sen- processes which may be dampened or deemed less signif- sitivity and error to complexity shifts at different scales. 26 Lindenschmidt et al.

Since Pχ = (1 + x) · Pbase the equation reduces to: Table 1. Table 2 gives a synopsis of the weather and water conditions during the sampling campaigns. 1 O – O s = x base (3) xO( base ) Model Description and Complexity Levels where x was typically set to 0.1 (=10% difference). The transport of salts, suspended solids and heavy met- Error als was simulated using the computer model TOXI, which is a module of the WASP5 simulation package The error, ε, between model results and sampled data developed by the U.S. EPA (Ambrose et al. 1993). A was calculated using: mass balance equation is used accounting for all material entering and leaving the system by point and non-point ε = 1 – e(–σ) (4) loading, advective and dispersive transport and physical, which is an adaptation of a likelihood function from chemical and biological transformations: Beven (2001). σ is a normalized error variance between ∂C ∂ ∂ ∂ the measurements x and simulated x values normalized = – (UxC) – (UyC) – (UzC) m s ∂t ∂x ∂y ∂z to the average of the measured values x¯ m: ∂ ∂C ∂ ∂C ∂ ∂C 1 + ( Ex ) + ( Ey ) + ( Ez ) (7) σ = 2 ∂ ∂ ∂ ∂ ∂ ∂ √ ∑ (xm – xs) (5) x x y y z z x¯ m + S + S + S Taking the exponent of σ allows the error to lie in B K L the range between 1 (perfect fit) and 0 (no fit). where C is the substance concentration with ∂C/∂t representing its change with respect to time, t; Ex, Ey and Model Utility Ez are the longitudinal, vertical and lateral dispersion coefficients (only the first was implemented here), Both the sensitivity and the error values can be used to respectively; SB, SK and SL are the rates for boundary evaluate the “best” model for a particular application, in loading, kinetic transformations and loading from point terms of an index of utility, Um, for model, m (Snowling and non-point sources, respectively; and Ux, Uy and Uz and Kramer 2001): are the longitudinal, vertical and lateral advective veloci-

2 2 ties (only the first is required for our one-dimensional ws · ˆs total,m + wE · ˆε total,m (6) Um = 1– case), respectively. The velocities are provided by the √ w + w s E hydrodynamic simulations using DYNHYD, which is 2 2 where sˆ total,m and ˆε total,m are the sensitivity and error also a module in the WASP5 package. of each model normalized to 1; wS and wE are weighting The substances transported are any combination of factors for sensitivity and error , respectively, and both three dissolved and three particulate substances. Most equal 1 for no preference. Increasing one factor empha- salts can be modelled as conservative substances; hence, sizes that particular characteristic. The aim is to maxi- no reaction terms, SK, are required. The transport of sus- mize Um by decreasing both sensitivity and error. pended solids, SS, requires additional sink and source terms to describe the movement of particles to and from Calibration and Validation the bottom sediments. Settling, deposition and resuspension rates are described by velocities and surface areas. Since the number of parameters is not large (up to six), a The sedimentation rate, vsed, is set within the range of manual fitting of the simulation results to the sampling Stoke’s velocities corresponding to the suspended particle points by inspection was sufficient to calibrate the mod- size distribution. This rate is multiplied by a probability of els. The models were then validated using independent deposition to obtain the deposition rate. The probability data sets from different time periods, indicated in of deposition depends upon the shear stress on the benthic

TABLE 1. Characteristics of the modelling resolutions and simulation time periods at the different scales.

Characteristic Saale (large scale) Calbe (small scale) Spatial resolution (longitudinal) 500 m 100 m Time resolution 1 d 1 h Calibration 8–10 September 2003 19–21 June 2002 10–11 June 2001 Validation 5–18 June 2001 23–24 June 2002 Scaling and Complexity in Water Quality Modelling 27

TABLE 2. Water and weather conditions during the sampling campaigns

5–18 June 2001 (14 d) 19–24 June 2002 (6 d) 8–10 September 2003(3 d) Mean flow (m3/s)a 54.7 76.6 36.0 Mean water temperature (ºC)a 17.0 22.5 17.6 Secchi depth (m)a 0.9 0.7 0.6 Mean daytime air temperature (ºC)b 14.3 19.5 15.0 Mean global rad. (J/cm2/d)b 1574 1606 700 Total precipitation (mm)b 11.4 17.4 8.4 aMeasured at Calbe. bMeasured at Bad Lauchstedt (see Fig. 3).

surface and the suspended sediment size and cohesiveness. where KOC is a constant and represents the organic Likewise, the resuspension rate, vres, depends upon the carbon partition coefficient and is calibrated for each shear stress, the bed sediment size and cohesiveness, and heavy metal separately. the state of consolidation of surficial benthic deposits (Ambrose et al. 1993). Diffusion of dissolved substances Complexity 3: Equilibrium Sorption from the bottom sediments into the water column is dri- ven by the gradient of the substance concentration in the In this complexity, sorption reactions are fast relative to sediments, csed, and the overlying water. The rate is con- other reactive terms and are assumed to be in equilibrium trolled by the diffusion coefficient, Dy. in which the transfer rates of metals from the dissolved to Sorption processes must also be included in the reac- the solid phase and vice versa are equal. The partition tion term when the transport of heavy metals is simu- coefficient, KD, is a constant and relates the concentra- lated. Sorption is the bonding of dissolved chemicals to tions of the metal phases and the suspended solids as: the particulate solid material in suspension or in the sed- Cpart iments. The process is described using a partition coeffi- KD = (9) Cdis · SS cient, KD, which represents the fraction of dissolved and particulate fractions of the heavy metals in relation to where Cdis and Cpart are the dissolved and particulate the concentration of suspended solids. Sorption is the fractions of the heavy metal, respectively, and SS is the most sensitive parameter (details upcoming in the concentration of suspended solids. Results section), hence its complexity was varied as fol- lows and summarized in Table 3. Complexity 4: Dynamic Sorption

Complexity 1: Conservative Transport In this complexity, the partition coefficient, KD, is (No Sorption) allowed to increase or decrease in the flow direction at a particular rate. This spatial (and temporal) dependency In the simplest complexity, all substances, both particu- of the phase partitioning is important when sorption late and dissolved fractions, are transported conserva- does not occur quicker than other reaction processes. tively without any sorption reactions between the phases This is particularly the case when large loads of dis- or with different substances. solved heavy metals are emitted into a river causing a large increase in the metal concentrations in the river. Complexity 2: Dependency on Organic Carbon This is the case for the tributary Schlenze which drains large amounts of copper, lead and zinc from a large The complexity of heavy metal transport is increased by abandoned underground mine (details in next section). considering the organic carbon content in the sorption kinetics. Many metals have an affinity to sorb either to the fraction of organic carbon, fOC, of the particulate matter or to bind with the dissolved organic carbon TABLE 3. Partition coefficient functions of various complexities (DOC) fraction to form colloids. DOC remained fairly constant in the flow direction whereas a large variability Complexity Sorption process in fOC was observed. Hence, the dependence of the parti- 1 Conservative transport tion coefficient KD on fOC was explored: 2KD = f(fOC) 3KD = f(Cdis, Cpart, SS) KD = fOC · KOC (8) 4KD = f(Cdis, Cpart, SS, time) 28 Lindenschmidt et al.

Description of Study Sites and their Model Setup

The Saale river is an important tributary of the Elbe river in Germany. Its course is 413 km in length but only the lower 90-km stretch will be considered in this study (see Fig. 3). This course is a national waterway and is regulated with locks and channelized to aid ship naviga- tion. The river’s water quality is strongly influenced by mining activities and abandoned mines in the river basin. Presently, potash, salt, uranium, copper and lignite are the most important materials mined. The Saale’s most downstream lock-and-weir system is situated at Calbe and stretches between 19 and 21 km upstream from the Saale confluence. The system divides the flow between the lock, weir and diversion reaches (see Fig. 4) with the majority of the flow discharging over the weir.

Data Sampling

For the calibration and validation of the models three data sets were available which were sampled along the lower Saale and its tributaries by UFZ – Centre for Envi- Fig. 4. Calbe lock-and-weir system with sampling stations: km ronmental Research, Magdeburg, Germany (Baborowski 20.8, lock entrance, km 18.5, upper weir, ferry and diversion. et al., Unpublished data). The sampling periods are: (i) 5–18 June 2001, (ii) 19–21 June 2002 and (iii) 8–10 Sep- and Bode were sampled at their confluences. The sam- tember 2003 (see Table 1). Mean climatic and hydrolog- pling stations for the first sampling period include Halle- ical conditions for the various periods are given in Trotha, Wettin, Bernburg, Calbe and Groß Rosenburg. Table 2. The tributaries Salza, Schlenze, Wipper, Fuhne Nienburg was included as a station in the second sampling campaign. The sampling resolution was increased to five additional stations for the third sampling campaign with the addition of Brachwitz, Döblitz, Könnern, Alsleben, Gröna and the Saale confluence. An Eulerian sampling strategy was carried out for the first campaign in which each station was sampled once daily or every second day over the two-week period. A Lagrangian approach was taken for the campaigns in 2002 and 2003 in which the sampled water parcel at the most upstream station (Halle-Trotha) was tracked and sampled along the course of the river. A 24-hour diel study at the Calbe lock-and-weir com- plemented the first two sampling campaigns on 10–11 June 2001 and 23–24 June 2002. Only the most upstream and downstream stations were sampled. Trends of the data at the other stations, lock entrance, upper weir, ferry and diversion, were interpolated from a weekly sampling program of all stations during the first half of 2002 (Eck- ert 2002). An additional long-term validation period was simulated from 14 May to 31 July 2002 (79 days) using data from LAU – Saxony-Anhalt Bureau for Environmen- tal Protection (Approval # LAU/3.3/01/2004). The substances sampled were suspended solids, salts (calcium, chloride, magnesium, potassium, sodium, sul- fate) and both dissolved and particulate fractions of Fig. 3. The course of the lower Saale river (adapted from heavy metals (copper, chromium, iron, lead, manganese, Lindenschmidt [2006]). nickel, zinc) and arsenic. Scaling and Complexity in Water Quality Modelling 29

Discretization significant with regard to water volume. No additional inflows into the Saale occur in Calbe. In previous studies The course of the lower Saale was discretized using 182 flows were simulated dynamically for the Saale (von segments, each approximately 500 m in length. Cross- Saleski et al. 2004) and the Calbe lock-and-weir system sectional profiles every 100 m along the river were avail- (Lindenschmidt et al. 2004a; Wodrich et al. 2004) using able from which initial hydraulic radii and segment water the hydrodynamic model DYNHYD, which is also pro- volumes (calculated from mean water levels) of each seg- vided with the WASP5 package. Discharges, flow veloci- ment were determined. Simulation results are output on a ties and depths are simulated for each time step and stored daily time step. The Calbe lock-and-weir system was dis- in a hydrodynamic file which is then read by TOXI. cretized using 63 segments each approximately 100 m in length (see Fig. 5). A detailed description of the river Exchanges morphology was derived from sonar graphs provided by the Water and Shipping Authority (WSA), ABZ Bern- Dispersion of substances is modelled as an exchange burg, Germany. Each segment is divided into a water col- between adjacent water column compartments. Diffu- umn and a sediment compartment. sion of dissolved substances occurs between each water column compartment and their underlying sediment Flows compartments.

The mean discharge of this lower river course varies between 99 m3/s at Trotha and 115 m3/s at Calbe and is Loads and Boundary Conditions regulated by six lock-and-weir systems at Trotha, Wettin, The most important source of substances into the lower Rothenburg, Alsleben, Bernburg and Calbe. Five tribu- Saale river is the soda production plant at Bernburg in taries, Salza, Schlenze, Wipper, Fuhne and Bode, (mean which soda ash (Na CO ) is produced using salt (NaCl discharges = 1.0, 0.57, 2.5, 1.1, 14.3 m3/s, respectively) 2 3 and CaCO ) causing large loads of chloride, calcium and drain into the lower Saale of which the Bode is the most 3 sodium into the Saale. Data on the loads were not available but could be derived from general load values pub- lished by the German Environmental Agency (UBA 2004) and from data produced by the Staßfurt soda plant located upon the Bode river (Sodawerk Staßfurt GmbH & Co. KG 2002). Boundaries to the Saale model are the five tributaries and the most upstream segment at Halle- Trotha. Evident is the high input from the Schlenze which drains a large abandoned underground mine (named “Schlüsselstollen”), despite the tributary’s low discharge.

Parameters

Up to six parameters were used for the calibration of the model:

i) Longitudinal dispersion, Ex: can be calibrated alone by simulating the transport of a conservative substance such as chloride. This parameter plays an important role for model setups in which mean velocities vary substantially between discretized units, which is the case for the lock-and-weir system. ii) Vertical diffusion, Dy: describes the movement of substances between the bottom sediments and the water column. This parameter is important for the transport of inorganic substances which for the case of heavy metals tend to have higher concentrations in the bottom sediments than in the overlying water. iii) Sedimentation rate, vsed: serves as a sink of suspended, particulate matter through vertical transport from the Fig. 5. Discretization of the Calbe lock-and-weir system water column to the sediments. Its rate is assumed to (adapted from Wodrich et al. [2004]). increase immediately upstream from weirs. 30 Lindenschmidt et al.

iv) Resuspension rate, vres: describes the transport of concentrations for all the discretized segments were lin- particulate matter from the bottom sediments into early interpolated between the sampling stations. the water column. Its rate can increase locally immediately downstream from weirs. Results v) Substance concentration in the bottom sediment, csed: is particularly important for substances with Saale—Calibration and Validation strong concentration gradients between bottom sediments and water, which is the case for heavy metal There is good agreement between the simulation and transport. measured values of suspended solid concentrations, vi) Partition coefficient for sorption of dissolved sub- which remained fairly steady along the course of the stances onto suspended matter, KD: has different lower Saale river (see Fig. 6). There is higher uncertainty descriptions depending on the model complexity in some sampled values: at Döblitz and Wettin due to used (described above). unavoidable excessive resuspension of bottom sediments during sampling at the shore and at Nienburg due to Initial Conditions mixing effects by the Bode inflow. Increased sedimentation at the lock-and-weir systems is not evident. A good A longitudinal profile of all the state variables is required fit of the simulation to the data was also obtained for at the commencement of each model simulation. Initial the chloride concentrations (see Fig. 6). The industrial

Fig. 6. Calibrated longitudinal profiles of suspended solids and chloride along the Saale. Scaling and Complexity in Water Quality Modelling 31 effluent at Bernburg proved to be an important point stream from the Schlenze confluence (see Fig. 7). The load for the overall substance balance downstream from simulations of the total zinc concentrations were initially Bernburg. The Schlenze tributary is an important source overestimated and could only be corrected by assuming of heavy metals to the Saale, evident in the drastic increased sedimentation of zinc at the lock-and-weirs. increase of total zinc concentrations immediately down- Zinc in the Schlenze is predominately present in dis-

Fig. 7. Calibrated longitudinal profiles of total zinc and its particulate and dissolved fractions along the Saale. 32 Lindenschmidt et al.

solved form and requires approximately 30 h for the dis- weir system. The concentrations in the lock reach are solved and particulate fractions to reach an equilibrium less due to the low flow through the lock and the long sorption state (in this simulation the time corresponds to duration within this reach for solids to settle out. The a flow distance between the Schlenze confluence and sedimentation rate above and below the weir were, Bernburg). By linearly increasing KD from 10,000 L/kg respectively, increased and decreased so as to better fit at the Schlenze confluence to 40,000 L/kg at Bernburg, the simulations to the sampled data. This trend occurred this dynamic sorption process was accurately simulated. for most heavy metals as well and was replicated in the Simulation results and sampling data for suspended validation. solids, chloride and total and dissolved zinc at Groß Rosenburg are shown in Fig. 8 for the validation time Calibrated Parameter Values period. Generally, the model was able to predict the substance concentrations well. The simulations for the total The calibrated parameter values of the model for each zinc fractions were somewhat overestimated for the last scale are given in Table 4. The bottom sediments play a three days of the time period due to high sampled sus- vital role for substance transport on the smaller scale for pended solids measurements. which the parameters Dy and csed are more variable depending on the substance being transported. The cali- Calbe—Calibration and Validation brated values increase as the particulate fractions of the substances increase. This same trend is also evident for Figure 9 gives a snapshot at 6:00 p.m. of the concentra- KD which has lower values on the smaller scale than for tions of suspended solids in all areas of the lock-and- the corresponding substance on the large scale. The sedimentation rates are greater and the resuspension rates are less for the lock-and-weir system compared to those values calibrated for the entire lower Saale reach. Ex is also more variable on the small scale but this parameter is quite insensitive to substance transport in both scales (see also Fig. 10 and the following).

Sensitivity

Figure 10 shows the sensitivities of the parameters and discharge boundary condition on the concentrations of suspended solids and chloride (top diagrams) and dissolved (middle diagrams) and particulate (sorbed) (bottom diagrams) fractions of iron, zinc and arsenic. The diagrams are juxtaposed for the large-scale (left side) and small-scale (right side) modelling exercises. A posi-

Fig. 8. Time series of model validation (5–18 June 2001) for Fig. 9. Calibration and validation of suspended solids for the suspended solids, chloride and total and dissolved zinc at corresponding reaches of the Calbe lock-and-weir system at Groß Rosenburg on the Saale. 6:00 p.m. (adapted from Wodrich et al. [2004]). Scaling and Complexity in Water Quality Modelling 33

TABLE 4. Values of calibration parameters for each scale

Saale (large scale) Calbe (small scale)

Parameter Units SS Fe Zn As SS Fe Zn As Dissolved % 0 10 68 87 0 8 63 90 2 Ex m /s 0.2 0.2 0.2 0.2 1 0.01 0.01 10 2 -4 -4 -4 -4 -5 -6 -6 -6 Dy m /s 10 10 10 10 10 8 × 10 2 × 10 1.2 × 10 vsed m/d0.20.20.20.22222 -5 -5 -5 -5 -6 -6 -6 -6 vres m/d 10 10 10 10 10 10 10 10 6 5 csed mg/L 10 50 50 50 5 × 10 27,800 850 20 6 KD L/mg 10 40,000 10,000 77,000 27,000 4900 tive sensitivity means that an increase (or decrease) in tions on the total concentrations: Fe ≈ 10%, Zn ≈ 65% the parameter or boundary value will increase (or and As ≈ 90% (see also Table 4). The concentrations are decrease) the variable concentrations; a negative sensitiv- much more sensitive to changes in parameters and flow ity means that the variable value will increase or boundaries on the small scale. For both scales, particulate decrease contrary to the change in the parameter setting. and dissolved fractions are most sensitive to KD when The figure indicates that chloride is affected only on their respective percentages to the total concentration are the large scale by the boundary discharges and none of low. On the small scale the diffusion coefficient, sedimen- the parameters, which is to be expected for conservative tation rate and boundary discharges are relatively sensitive transport of dissolved substances. At both scales sus- to these substances. For KD there is a distinct trend of sen- pended solids concentrations are negatively sensitive to sitivity with dissolved to particulate fraction ratios. the sedimentation rate. However, both the solids concen- The results indicate that the chemical processes tration in the sediments and the resuspension rate play a between metal phases, such as sorption of dissolved met- dominant role in the transport of suspended particulate als onto solids and diffusion of dissolved metals from the matter on the large scale but not on the small scale. bottom sediments into the water column play a vital role For the sensitivity analysis iron, zinc and arsenic were in the transport of metals on the smaller scale, more so selected to reflect different percentages of dissolved frac- than physical processes such as advection and net sedimentation. As the scale becomes large the influence of chemical processes on metal transport dampens. For suspended solids the transport both in the longitudinal (advection) and vertical (sedimentation and resuspension) directions are important transport components on the large scale, not so on the small scale. It appears that chemical processes increasingly override the physical processes as the scale becomes smaller.

Complexity and Scale

The complexities versus error and sensitivity curves are shown in Fig. 11 for total zinc (ZnT) and its particulate (ZnP) and dissolved (ZnD) fractions. The curves are shown for both scales. In general, the trends for error decrease and for sensitivity increase with increasing complexity. An exception is the error for ZnD at the small scale where no particular trend is noticeable. The errors are generally higher for the model complexities of Calbe than for the Saale river. It appears as if complexity must be increased more for an extrapolated Fig. 10. Sensitivities of the calibration parameters and error curve to reach values comparable to values attained boundary discharges on suspended solids (SS), chloride (Cl) by the large scale. The sensitivities are also larger for the and the dissolved and particulate fractions of iron (Fe), zinc model applied on the small scale compared to that of the (Zn) and arsenic (As) for the Saale (large scale) and Calbe lock-and-weir system (small scale) modelling exercises (data large scale. However, there is a levelling off of the sensi- compiled from Lindenschmidt et al. [2004b] and Wodrich et tivities at higher complexities suggesting that there is an al. [2004]). upper bound of model sensitivity. Both error and sensitiv- 34 Lindenschmidt et al.

large-scale model is higher than for the small-scale model. Hence, TOXI is best implemented for modelling exercises of large river reaches.

Discussion

Uncertainty, Complexity and Scale

The modelling exercises confirm the hypothesis by Snowling and Kramer (2001) for both scales. Greater complexity increased model sensitivity and decreased the error in the output simulations. In a more theoreti- cal framework Cox (1999) also shows that for models used in risk assessment, greater complexity leads to more certainty in risk estimates. He does state, though, that the additional complexity included in the model must allow additional relevant observations to be incor- porated. This is the case in our models in which added process complexity is accompanied by substance concentrations that also act as state variables in the model (e.g., the addition of suspended solids and metals concentration in Complexity 3). The error and sensitivities tend to shift in relation to complexity at different scales. Models of smaller scale require a more detailed description of the processes to accurately simulate the state of the modelled area for a given time frame. Sivapalan (2003) mentions that increased complexity is required to capture the hydrological response at the hillslope scale compared to the catchment scale. For example, Butts et al. (2004) and Perrin et al. (2001) found that hydrological variables modelled for large river basins were more accurately Fig. 11. Complexity versus error and sensitivity for zinc simulated using simpler process descriptions. van der derived from the Saale (large scale) and Calbe lock-and-weir system (small scale) modelling exercises. Linden and Woo (2003), who applied models with increasing complexity to simulate hydrological conditions in subarctic catchments, also found that with ity are generally lower for ZnT than for the particulate decreasing temporal and spatial scale, process represen- and dissolved fractions. This difference is more pro- tation needs to be more complex. nounced at the larger scale. For the utility calculations, TOXI is more suitable The modelling exercises confirm the hypothesis by for applications at larger scales (long river reaches) such Snowling and Kramer (2001) for both scales. The error and sensitivities tend to shift in relation to complexity at different scales.

Model Utility

The utility of each complexity is exemplified in Fig. 12 for the simulation of the total concentration of zinc. The trend of the error with complexity curve was taken for the utility calculation and both error and sensitivity have equal weighting. For both scales, Complexity 2 is the “best” model, for which sorption is a function of the fraction of particulate matter consisting of organic carbon. Although the more complex models have a larger reduction in simulation errors, predictive ability is dimin- Fig. 12. Utility of each model complexity (see Table 3) at the ished. The utility of Complexities 3 and 4 decreases more two different scales for the simulation of the total concentra- for the small-scale model and overall the utility for the tion of zinc. Scaling and Complexity in Water Quality Modelling 35 as developing computer-aided decision support systems for river basin management. On the smaller scale, more processes need to be implemented to acquire the accuracy attained on the larger scale. Moreover, on the small scale, the bottom sediments play a crucial role in the transport of inorganic substances. Hence, more dynamics in substance turnover should be included in the sediment layers differentiating between aerobic and anaerobic zones. For future work on small-scale applications it may be worth- while to incorporate a geochemical model such as PHREEQC (http://wwwbrr.cr.usgs.gov/projects/ GWC_coupled/phreeqc/) into TOXI, which would enable a more differentiated and detailed simulation of the bottom sediment. TOXI could still provide the advective transport of the substances in the overlying water.

Influence of Locks and Weirs at Different Scales Fig. 13. Particulate organic carbon (POC) and chlorophyll a (Chl-a) increase with flow direction along the Saale river Lindenschmidt et al. (2004a) found that locally on the (mean values for 5–18 June 2001). small scale locks and weirs influence the transport of both suspended solids and the total concentrations of most heavy metals substantially. This is particularly Conclusions due to the large differences in the mean velocities between the various areas in the lock-and-weir system. Important conclusions from this study are: Increased sedimentation immediately upstream from • The hypothesis proposed by Snowling and Kramer the weir and higher resuspension rates downstream (2001) could be confirmed for river water quality from the weir were modelled. Also, for this particular modelling exercises at both large and small scales. lock the low-flow conditions led to more sedimentation • The hypothesis proposed by the authors that the of solids than elsewhere in the system. On the larger relation of model error and sensitivity with com- scale for the lower Saale river and for similar low-flow plexity, as suggested by Snowling and Kramer conditions the lock-and-weir systems have little effect (2001), shifts for different scales was confirmed. on the suspended solid concentration. The simulations • Increased sedimentation rates occurred on the small agree well with the measurements without having to scale in the lock reach and immediately upstream include higher sedimentation rates in the upstream from the weir. Resuspension was enhanced immedi- areas of the weirs. This is contrary to the total zinc concentration for which higher sedimentation rates need to be included for the simulations to coincide with sampled values. We hypothesize that two processes exist which counteract each other: i) large particulate zinc fraction (mostly inorganic) that is formed immediately downstream from the large emission of dissolved zinc from the Schlenze tributary settles out and causes a loss of suspended solids, and ii) inorganic solids that are settled out reducing the water’s turbidity allowing an increase in phytoplankton growth which replaces the settled inorganic fraction of the suspended solids. Figure 13 shows that chlorophyll a and particulate organic carbon (POC) both increase in the flow direction in the Saale. The increase of the ratio of organic to inorganic material along the river’s course is also confirmed in Fig. 14, in which both the weight fraction of Fig. 14. Both the weight fraction of the total carbon in the solid material (fOC) increases and the loss-on-ignition have the total carbon in the solid material (fOC) and the loss- increasing trends in the flow direction along the Saale from on-ignition increase in the flow direction along the Saale Wettin to the Saale confluence (mean values for 5–18 June from Wettin to the Saale confluence. 2001). 36 Lindenschmidt et al.

ately downstream from the weir. For suspended parameter sensitivity and model uncertainty in river solids these effects occurred only locally and were water quality modelling. Ecol. Model. 190(1–2):72–86. averaged out in the large-scale modelling exercise. Lindenschmidt K-E, Eckhardt S, Wodrich R, Eckert U, • Physical processes are more dominant on the large Baborowski M, Guhr H. 2004a. Water quality model- scale whereas chemical processes become more ling of a lock-and-weir system of the lower Saale river important on the smaller scale. (in German). Gas- und Wasserfach: Wasser-Abwasser • TOXI is a model suited for the transport of inor- 145(9):612–621. ganic substances on the large scale. For small-scale Lindenschmidt K-E, Hesse C, Guhr H, Baborowski M. applications additional processes must be incorpo- 2004b. Modellierung anorganischer Schadstoffe in der rated which describe the complexity of the transport Unteren Saale. 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