Seasonal and inter-annual variability of water quality in the Uruguay River,Argentina

Abstract

Water quality of the Uruguay River was evaluated with multi-parametric (temperature, turbidity, conductivity, pH, dissolved oxygen) and sediment trap data (particle flux, total organic carbon and nitrogen contents) and correlated to precipitation, river discharge and El Niño Southern Oscillation (ENSO) indices for the period 2006–2011. Hydro-meteorological parameters averaged 24–85% variability with coincident precipitation (200–400 mm month^-1), discharge (7000–28 000 m³ s^-1) and turbidity peaks (50–80 NTU) in the austral spring, and absolute maxima during the El Niño 2009 episode. Spectral analysis of discharge and sea-surface temperature anomaly (SSTA) showed consistent variance maxima at approx. 3 and 1.5 years. Deseasonalized discharge was significantly correlated to SSTA. During river floods, pH decreased (from 7.5 to 6.6) and particle dynamics peaked (turbidity: 15–80 NTU; vertical fluxes: 20–200 g m^-2 d^-1; total solid load: <1000 to 100 000 t d^-1),whereas TOC remained stable (3.2 ± 0.8%) and C/N ratios increased (10–12) due to the higher contribution of terrestrial detritus.     

Numerical study of generation of the tidal shear front off the Yellow River mouth

Tidal shear front off the Yellow River mouth has been observed and modeled in the previous studies. However, a detailed investigation of the front generation has not been conducted. The aim of this paper is to use a three-dimensional tidal model coupled to a sediment transport module to examine the front formation. The model predicted a tidal shear front that propagated offshore and lasted 1–2 h at both flood and ebb phase off the Yellow River mouth. The sensitivity numerical experiments showed that the topography with a strong slope off the Yellow River mouth was a determining factor for the front generation, and a parallel orientation between the major axes of ellipses and co-tidal lines of maximum tidal current was a necessary condition. While the bottom friction and the river runoff had no effect on the front location but affected the front intensity, the front generation was not sensitive to the coastline variation. The study concluded that the bottom slope off the river mouth induces a strong variation in the bottom stress in a cross-shore direction, which produces both maximum phase gradient and sediment concentration variability across the tidal shear front. With the extending Yellow River delta, the tidal shear front under the new bathymetry of year 2003 has been strengthened and pushed further offshore due to an increased bottom slope.     

The application of two‐dimensional hydrodynamic models for underwater archaeological finds from the Vistula River in Warsaw,Poland

Low water levels occurring on the Warsaw section of the Vistula River during droughts in 2012 and 2015 stirred considerable interest by exposing parts of the riverbed. The river's low flow discharge created favourable conditions for archaeologists who, from the river's bottom, managed to salvage sculptures, architectural artefacts, and military accessories dating back to the middle of the 17th century. Literature shows that archaeological finds at the bottom of a major lowland river are unique and are attributed to very specific hydrological and geological conditions. Embankments and works meant to regulate the channel of the Vistula River in Warsaw narrowed the channel and accelerated the erosion of the river's bottom. A comparison of rating curves created between 1919 and 2015 shows that the river's bottom had lowered by 225 cm. The diameter of sediment particles that could be moved at maximum shear stress has been calculated using the Meyer–Peter and Muller formula. According to the calculated shear stress corresponding to the highest historical flood of 1844 particles smaller than 0.05 m were transported down the river, and it is true that artefacts of that size had not been collected at the site. Taking into consideration the very specific geological and hydrological conditions, we believe that finding so many 17th century artefacts with a well‐documented history at a site at the bottom of major lowland river with intensive channel processes is indeed unique. Additionally, it is the first time that results of hydrodynamic modelling have been successfully used for archaeological survey to define the critical shear stress and the smallest diameter of artefacts, which could remain stable on a river's bottom.     

Comparison of hydrodynamic models of different complexities to model floods with emergency storage areas

A flood emergency storage area (polder) is used to reduce the flood peak in the main river and hence, protect downstream areas from being inundated. In this study, the effectiveness of a proposed flood emergency storage area at the middle Elbe River, Germany in reducing the flood peaks is investigated using hydrodynamic modelling. The flow to the polders is controlled by adjustable gates. The extreme flood event of August 2002 is used for the study. A fully hydrodynamic 1D model and a coupled 1D–2D model are applied to simulate the flooding and emptying processes in the polders and flow in the Elbe River. The results obtained from the 1D and 1D–2D models are compared with respect to the peak water level reductions in the Elbe River and flow processes in the polders during their filling and emptying. The computational time, storage space requirements and modelling effort for the two models are also compared. It is concluded that a 1D model may be used to study the water level and discharge reductions in the main river while a 1D-2D model may be used when the study of flow dynamics in the polder is of particular interest. Further, a detailed sensitivity analysis of the 1D and 1D–2D models is carried out with respect to Manning's n values, DEMs of different resolutions, number of cross-sections used and the gate opening time as well as gate opening/closing duration. Copyright © 2008 John Wiley & Sons, Ltd.     

Significance of particle interaction to the modelling of cohesive sediment transport in rivers

Aggregation processes of fine sediments have rarely been integrated in numerical simulations of cohesive sediment transport in riverine systems. These processes, however, can significantly alter the hydrodynamic characteristics of suspended particulate matter (SPM), modifying the particle settling velocity, which is one of the most important parameters in modelling suspended sediment dynamics. The present paper presents data from field measurements and an approach to integrate particle aggregation in a hydrodynamic sediment transport model. The aggregation term used represents the interaction of multiple sediment classes (fractions) with corresponding multiple deposition behaviour. The k–ε–turbulence model was used to calculate the coefficient of vertical turbulent mixing needed for the two‐dimensional vertical‐plane simulations. The model has been applied to transport and deposition of tracer particles and natural SPM in a lake‐outlet lowland river (Spree River, Germany). The results of simulations were evaluated by comparison with field data obtained for two levels of river discharge. Experimental data for both discharge levels showed that under the prevailing uniform hydraulic conditions along the river reach, the settling velocity distribution did not change significantly downstream, whereas the amount of SPM declined. It was also shown that higher flow velocities (higher fluid shear) resulted in higher proportions of fast settling SPM fractions. We conclude that in accordance with the respective prevailing turbulence structures, typical aggregation mechanisms occur that continuously generate similar distribution patterns, including particles that settle toward the river bed and thus mainly contribute to the observed decline in the total SPM concentration. In order to determine time‐scales of aggregation and related mass fluxes between the settling velocity fractions, results of model simulations were fitted to experimental data for total SPM concentration and of settling velocity frequency distributions. The comparison with simulations for the case of non‐interacting fractions clearly demonstrated the practical significance of particle interaction for a more realistic modelling of cohesive sediment and contaminant transport. Copyright © 2004 John Wiley & Sons, Ltd.     

Time‐varying suspended sediment‐discharge rating curves to estimate climate impacts on fluvial sediment transport

This study presents time‐varying suspended sediment‐discharge rating curves to model suspended‐sediment concentrations (SSCs) under alternative climate scenarios. The proposed models account for hysteresis at multiple time scales, with particular attention given to systematic shifts in sediment transport following large floods (long‐term hysteresis). A series of nested formulations are tested to evaluate the elements embedded in the proposed models in a case study watershed that supplies drinking water to New York City. To maximize available data for model development, a dynamic regression model is used to estimate SSC based on denser records of turbidity, where the parameters of this regression are allowed to vary over time to account for potential changes in the turbidity‐SSC relationship. After validating the proposed rating curves, we compare simulations of SSC among a subset of models in a climate change impact assessment using an ensemble of flow simulations generated using a stochastic weather generator and hydrologic model. We also examine SSC estimates under synthetic floods generated using a peaks‐over‐threshold model. Our results indicate that estimates of extreme SSC under new climate and hydrologic scenarios can vary widely depending on the selected model and may be significantly underestimated if long‐term hysteresis is ignored when simulating impacts under sequences of large storm event. Based on the climate change scenarios explored here, average annual maximum SSC could increase by as much as 2.45 times over historical values.     

Synergy of satellite remote sensing and numerical modeling for monitoring of suspended particulate matter

Monitoring and modeling of the distribution of suspended particulate matter (SPM) is an important task, especially in coastal environments. Several SPM models have been developed for the North Sea. However, due to waves in shallow water and strong tidal currents in the southern part of the North Sea, this is still a challenging task. In general there is a lack of measurements to determine initial distributions of SPM in the bottom sediment and essential model parameters, e.g., appropriate exchange coefficients. In many satellite-borne ocean color images of the North Sea a plume is visible, which is caused by the scattering of light at SPM in the upper ocean layer. The intensity and length of the plume depends on the wave and current climate. It is well known that the SPM plume is especially obvious shortly after strong storm events. In this paper a quasi-3-D and a 3-D SPM transport model are presented. Utilizing the synergy of satellite-borne ocean color data with numerical models, the vertical exchange coefficients due to currents and waves are derived. This results in models that for the first time are able to reproduce the temporal and spatial evolution of the plume intensity. The SPM models consist of several modules to compute ocean dynamics, the vertical and horizontal exchange of SPM in the water column, and exchange processes with the seabed such as erosion, sedimentation, and resuspension. In the bottom layer, bioturbation via benthos and diffusion processes is taken into account.Responsible Editor: Jörg-Olaf Wolff     

Numerical simulation of hydrodynamic characteristics and bedload transport in cross sections of two gravel-bed rivers based on one-dimensional lateral distribution method

Accurate evaluation and prediction of bedload transport are crucial in studies of fluvial hydrodynamic characteristics and river morphology.This paper presents a one-dimensional numerical model based on the one-dimensional lateral distribution method(1 D-LDM) and six classic bedload transport formulae that can be used to simulate hydrodynamic characteristics and bedload transport discharge in cross sections.Two gravel-bed rivers,i.e.the Danube River located approximately 70 km downstream from Bratislava in Slovakia and the Tolten River in south of Chile are used as examples.In the 1 D-LDM,gravity,bed shear stress,turbulent diffusion,and secondary flow are included to allow for accurate predictions of flow velocity and consequently bed shear stress in the cross sections.Six classic formulae were applied to evaluate the non-dimensional bedload transport rate,and the bedload transport discharge through a river cross section is obtained by integrating the bedload transport rate over the width of the cross section.The results show that the root mean square error(RMSE) and mean absolute error(MAE) of velocity and water discharge were less than 8% of the observed magnitude,while the correlation coefficient between model predictions and observations was close to unity.The formulae proposed by Ashida and Michiue(1972),in which particle collision with the bed is taken into account,and by Camenen and Larson(2005),which allows for yielding a non-zero bedload transport rate even when the bed shear stress is smaller than the critical bed shear stress value,appeared to be more appropriate for predicting the observed bedload transport rate in the studied cross sections of two gravel-bed rivers.If non-uniform sediment mixtures were considered,the bedload transport discharge through a cross-section could change considerably by up to 22.5% of the observed magnitude.The relations proposed by Ashida and Michiue(1972) and Egiazaroff(1965) for parameterizing the hiding factor yielded more realistic model predictions in comparison with observations for the measured data set collected for the Tolten River,while the one proposed by Wilcock and Crowe(2003) performs the best for the data set measured for the Danube River.     

Assessment of Water Quality in the Elbe River at Flood Water Conditions Based on Cluster Analysis,Principle Components Analysis,and Source Apportionment

An assessment of water quality measurements during a spring flood in the Elbe River is presented. Daily samples were taken at a site in the middle Elbe, which is part of the network of the International Commission for the Protection of the Elbe River (IKSE/MKOL). Cluster analysis (CA), principal components analysis (PCA), and source apportionment (APCS apportioning) were used to assess the flood‐dependent matter transport. As a result, three main components could be extracted as important to the matter transport in the Elbe River basin during flood events: (i) re‐suspended contaminated sediments, which led to temporarily increased concentrations of suspended matter and of most of the investigated heavy metals; (ii) water discharge related concentrations of pedogenic dissolved organic matter (DOM) as well as preliminary diluted concentrations of uranium and chloride, parameters with stable pollution background in the river basin; and (iii) abandoned mines, i.e., their dewatering systems, with particular influence on nickel, manganese, and zinc concentrations.     

Rheological properties of dense natural cohesive sediments subject to shear loadings

Cohesive sediments exhibit complex rheological behaviors that are non-Newtonian and time-dependent when subjected to external loading. This paper presents the results of an investigation on the theological properties of three types of dense cohesive sediments, collected from the mouth of the Yangtze River, the shoal of the ttangzhou Bay, and the Yangcheng Lake in China. A set of rheological parameters （including viscosity, yield stress, etc.） was studied based on experiments that were conducted with a RheolabQC rheometer. Measurements of the flow curves, shear stress-time responses, and yield stresses were made. The solid-liquid transition of the dense cohesive sediments occurred both in the shear rate ramp tests and the shear stress ramp tests. This transition was not direct, but it was mediated by a transitional deformation regime or stress plateau. Both the Herschel-Bulkley model and Carreau model were able to describe the theological behavior of dense cohesive sediments, and the empirical expressions for calculating the parameters in these models were obtained by a dimensional and regression analysis. The yield stresses determined by the shear stress ramp test and by the vane method were compared and discussed. The influence of the water content on the rheological properties of dense cohesive sediments was considered.     

CHANNEL ADJUSTMENT OF AN UNSTABLE COARSE-GRAINED STREAM: OPPOSING TRENDS OF BOUNDARY AND CRITICAL SHEAR STRESS,AND THE APPLICABILITY OF EXTREMAL HYPOTHESES

Channel adjustments in the North Fork Toutle River and the Toutle River main stem were initiated by deposition of a 2.5 km³ debris avalanche and associated lahars that accompanied the catastrophic eruption of Mount St. Helens, Washington on 18 May 1980. Channel widening was the dominant process. In combination, adjustments caused average boundary shear stress to decrease non-linearly with time and critical shear stress to increase non-linearly with time. At the discharge that is equalled or exceeded 1 per cent of the time, these trends converged by 1991–1992 so that excess shear stress approached minimum values. Extremal hypotheses, such as minimization of unit stream power and minimization of the rate of energy dissipation (minimum stream power), are shown to be applicable to dynamic adjustments of the Toutle River system. Maximization of the Darcy–Weisbach friction factor did not occur, but increases in relative bed roughness, caused by the concomitant reduction in hydraulic depths and bed-material coarsening, were documented. Predictions of stable channel geometries using the minimum stream power approach were unsuccessful when compared to the 1991–1992 geometries and bed-material characteristics measured in the field. It is concluded that the predictions are not applicable because the study reaches are not truly stable and cannot become so until a new floodplain has been formed by renewed channel incision, retreat of stream-side hummocks, and establishment of riparian vegetation to limit the destabilizing effects of large floods. Further, prediction of energy slope (and consequently stream power) by the sediment transport equations is inaccurate because of the inability of the equations to account for significant contributions of finer grained (sand and gravel) bank materials (relative to the coarsened channel bed) from bank retreat and from upstream terrace erosion.     

Flow and sedimentary processes in the meandering river South Esk,Glen Clova,Scotland

Channel geometry, flow and sedimentation in a meander bend of the River South Esk were studied from bankfull stages (January–February) to low water stages (May) in 1974. Bed topography varied little over the study period, showing a typical pool and ripple geometry. Variation of mean depth and velocity with discharge differed from section to section around the bend, due primarily to locally varying flow resistance with stage. The flow pattern for all stages was dominated by a single spiral over the point bar, with a development zone at the bend entrance. Deviation of bed shear stress from the mean flow direction was in general accord with theory, especially for high stages. The use of a uniform longitudinal water surface slope in the calculation of bed shear stress is not justified because of a complicated water surface topography, also such calculated shear may not represent effective bed shear on grains, as it accounts also for energy losses associated with secondary flows. Dunes covered much of the bar at high stages, with increasing proportions of ripples, sand ribbons and lower phase plane beds at low stages. Local flow resistance generally decreases from dunes, diminished and ripple-backed dunes, ripples, sand ribbons to plane beds, and bed forms are predicted quite well by the stream power-grain size scheme. Mean size, sorting and skewness of sediment over the bed changes little with stage. In general, size decreases, sorting improves and skewness changes from positive to negative from the talweg to the inner bank, and in the downstream direction. Allen's (1970a, b) force balance equation for moving bed load particles is supported for bankfull stage, with some reservations, and textural characteristics are explained by progressive sorting in the direction of sediment transport. Large-scale trough cross stratification (with some flat bedding) formed at high stage by dunes (and lower phase plane beds) dominates the point bar sediments. Alternations of fine-medium sand (often cross-laminated) and vegetation-rich layers result from periodic deposition on the grassed upper bar surface. Fining upwards sequences produced by lateral channel migration are modified by a coarsening upward subsequence in the upstream bar region where spiral flow is developing from the bend upstream.     

Assessment of Water Quality in the Elbe River at Low Water Conditions Based on Factor Analysis

An assessment of water quality measurements during a long‐lasting low water period in the Elbe River is presented. Weekly samples were taken from May to December 2003 at a sampling site in the middle part of the Elbe River. For multivariate data analysis, 34 parameters of 46 samplings were considered. As a result of this analysis, 78% of the variance of the data set is explained by five factors. They can be assigned to the following latent variables: season (37.5%) > tributaries (12.7%) > re‐suspension (10.4%) > discharge (9.4%) > complexation (8.5%). For the investigated sampling site, two key processes were identified as dominating factors on the water quality during low water conditions. First, seasonal phytoplankton development caused changes in redox conditions with consequences for re‐solution of pollutants from sediments. Second, tributaries have a higher impact on the main stream, due to changes in mixing processes. Therefore, in addition to flood investigations, monitoring strategies, and management plans should be developed in order to survey changes in water quality during low water conditions.     

River ice cover influence on sediment transportation at present and under projected hydroclimatic conditions

A large number of rivers are frozen annually, and the river ice cover has an influence on the geomorphological processes. These processes in cohesive sediment rivers are not fully understood. Therefore, this paper demonstrates the impact of river ice cover on sediment transport, i.e. turbidity, suspended sediment loads and erosion potential, compared with a river with ice‐free flow conditions. The present sediment transportation conditions during the annual cycle are analysed, and the implications of climate change on wintertime geomorphological processes are estimated. A one‐dimensional hydrodynamic model has been applied to the Kokemäenjoki River in Southwest Finland. The shear stress forces directed to the river bed are simulated with present and projected hydroclimatic conditions. The results of shear stress simulations indicate that a thermally formed smooth ice cover diminishes river bed erosion, compared with an ice‐free river with similar discharges. Based on long‐term field data, the river ice cover reduces turbidity statistically significantly. Furthermore, suspended sediment concentrations measured in ice‐free and ice‐covered river water reveal a diminishing effect of ice cover on riverine sediment load. The hydrodynamic simulations suggest that the influence of rippled ice cover on shear stress is varying. Climate change is projected to increase the winter discharges by 27–77% on average by 2070–2099. Thus, the increasing winter discharges and possible diminishing ice cover periods both increase the erosion potential of the river bed. Hence, the wintertime sediment load of the river is expected to become larger in the future. Copyright © 2015 John Wiley & Sons, Ltd.     

Properties of suspended sediment in the estuarine turbidity maximum of the highly turbid Humber Estuary system, UK

Measurements are presented of the properties of suspended particulate matter (SPM) in the estuarine turbidity maximum (ETM) of the upper Humber and Ouse estuaries during transient, relatively low freshwater inflow conditions of September 1995. Very high concentrations of near-bed SPM (more than 100 g l⁻¹) were observed in the low-salinity (less than 1), upper reaches. SPM within the ETM consisted largely of fine sediment (silt and clay) that existed as microfloc and macrofloc aggregates and individual particles. Primary sediment particles were very fine grained, and typically, about 20–30% was clay-sized at high water. The clay mineralogy was dominated by chlorite and illite. There was a pronounced increase in particle size in the tidal river, up-estuary of the ETM. The mean specific surface area (SSA) of near-bed SPM within the ETM was 22 m² g⁻¹ on a spring tide and 24 m² g⁻¹ on a neap tide. A tidal cycle of measurements within a near-bed, high concentration SPM layer during a very small neap tide gave a mean SSA of 26 m² g⁻¹. The percentage of silt and clay in surficial bed sediments along the main channel of the estuary varied strongly. The relatively low silt and clay percentage of surficial bed sediments (about 10–35%) within the ETM’s region of highest near-bed SPM concentrations and their low SSA values were in marked contrast to the overlying SPM. The loss on ignition (LOI) of near-bed SPM in the turbid reaches of the estuary was about 10%, compared with about 12% for surface SPM and more than 40% in the very low turbidity waters up-estuary of the ETM. Settling velocities of Humber–Ouse SPM, sampled in situ and measured using a settling column, maximized at 1.5 mm s⁻¹ and exhibited hindered settling at higher SPM concentrations.     

Stormflow dynamics and loads of Escherichia coli in a large mixed land use catchment

Storm events are major transporters of faecal microbial contaminants, but few studies have reported storm loads or concentration dynamics in relation to discharge or other pollutants, notably fine sediment. Episodically, high loads of faecal contamination during storm flows impact downstream uses of water bodies, particularly contact recreation and shellfish harvesting. We examined the storm dynamics of Escherichia coli, turbidity and discharge in the mixed land use Motueka catchment (2047 km²; 60% forest and 19% pasture) to gain insights into E. coli sources and transport. We also explored different approaches for calculating E. coli loads. Discharge and field turbidity were recorded continuously, and E. coli concentrations were sampled during events, over a 13‐month period near the mouth of the Motueka River. E. coli loads were estimated by interpolation, averaging estimators and by using linear regression with smearing correction of the log‐transformed variables: discharge, turbidity, and both turbidity and discharge. The annual E. coli load was dominated (～98%) by export during events. Comparison of monthly monitoring with the intensive storm monitoring campaign suggests that simple stratification of the sampling into storm and baseflow would greatly improve export estimates. E. coli peak concentrations always preceded discharge and turbidity peaks (which had similar timing). Turbidity can be a useful surrogate for faecal microbes in smaller catchments, but in the Motueka turbidity was no better for predicting E. coli concentration than discharge. Runoff from grazed pasture and direct deposition from livestock are probably the ultimate E. coli sources in the Motueka catchment. However, in‐channel stores seem to dominate E. coli dynamics during events and account for the typical feature of bacterial concentrations peaking ahead of discharge and turbidity. This study demonstrates the importance of storm events to faecal microbial loads and shows that E. coli concentration dynamics may contrast with those of turbidity. Copyright © 2009 John Wiley & Sons, Ltd.     

Assessment of Primary Production by Statistical Analysis of Water‐quality Data

Time series of weekly water‐quality data at Schnackenburg on the Elbe River (1985—2000) were subjected to principal component analysis (PCA). Considering the amplitudes of composite patterns of variables is a step towards a process‐oriented interpretation of waterquality data. One specific objective was to investigate the impact of improved water quality after the German reunification in 1990 on primary production and the oxygen budget. To discriminate anthropogenic signals from natural fluctuations a separation of the impact of discharge was attempted based on a linear regression approach. A dominant pattern of co‐variation in the residual data could be attributed to biological activity (primary production). The most relevant variables of this 'biomode' are oxygen saturation, pH, and orthophosphate. We conclude that multivariate statistical analysis of water‐quality data can help to estimate primary production when direct observations of algal concentrations are missing. In the years from 1998—2000 the trend of the ‘biomode’ indicates an increased load of oxygen consuming biomass caused by enhanced primary production in the middle stretches of the Elbe River which corresponds with the observation of more severe oxygen deficits in the tidal section of the river.     

Grain size characteristics of surface sediment and its response to the dynamic sedimentary environment in Qiantang Estuary,China

The composition and grain size characteristics of sediment in estuarine and coastal environments provide important information on the material source, hydrodynamic environment, environmental events,etc. However, few studies have focused on the sediment characteristics of the entire Qiantang Estuary,particularly, on the correlation between hydrodynamics and sediment properties. Through systematic sampling of the surface sediment in a large area of the Qiantang Estuary, the spatial distribution ch...     

Reduced‐complexity flow routing models for sinuous single‐thread channels: intercomparison with a physically‐based shallow‐water equation model

Reduced‐complexity models of fluvial processes use simple rules that neglect much of the underlying governing physics. This approach is justified by the potential to use these models to investigate long‐term and/or fundamental river behaviour. However, little attention has been given to the validity or realism of reduced‐complexity process parameterizations, despite the fact that the assumptions inherent in these approaches may limit the potential for elucidating the behaviour of natural rivers. This study presents two new reduced‐complexity flow routing schemes developed specifically for application in single‐thread rivers. Output from both schemes is compared with that from a more sophisticated model that solves the depth‐averaged shallow water equations. This comparison provides the first demonstration of the potential for deriving realistic predictions of in‐channel flow depth, unit discharge, energy slope and unit stream power using simple flow routing schemes. It also highlights the inadequacy of modelling unit stream power, shear stress or sediment transport capacity as a function of local bed slope, as has been common practice in a number of previous reduced‐complexity models. Copyright © 2009 John Wiley & Sons, Ltd.