A GIS-oriented hydrologic hybrid model (WBRM, Water Balance Raster Model) to bridge the gap between conceptual and physically based model structures MARCUS LEMPERT 1 MANFRED OSTROWSKI2 1
Kisters AG, Charlottenburger Allee 5, D 52068 Aachen,
[email protected] Fachgebiet Ingenieurhydrologie und Wasserbewirtschaftung, TU Darmstadt, Petersenstraße 13, D64297 Darmstadt,
[email protected] 2
Abstract The occurence of hydrological extremes all over the world proves the neccessity to further improve and apply reliable hydrological models. Although simplified linear models are still frequently used in hydrological science and practice, a clear trend is visible to change from conceptual linear to physical nonlinear models. The use of geographical information systems is now common to account for high spatial variability of basin characteristics, while the direct coupling of data base and models is still at an early stage. A model will be presented that can be applied using data organised in a GIS. This model replaces simplified linear differential equations by more physically based algorithms still making use of computationally efficient solution techniques. The model was developed by Lempert, 2000 in his PhD work and subsequently applied to several catchments at the mesoscale. This non commercial paper is intended to give an overview. Full references are given in the literature on the model. Key words Non linear hydrological, hybrid modelling, GIS INTRODUCTION Mathematical models play a key role in most sections of hydrology and water resources engineering, both in science and practice. They are the ultimate tools to describe, identify and solve existing complex problems. On the one hand, their continuous and increasing application in water resources aswell as environmental engineering and management during the last decades proves their usefulness in practice, on the other hand ongoing world wide research activities maintain the hypothesis that existing models are considered not to be adequately developed. Finally, new fields of application and more complex problems also require more advanced modelling techniques. Two examples are the coupling of models including hydrological ones in decision support systems for integrated river basin management and the prediction of flows in ungaged basins as defined by the PUB (Sivapalan et al, 2003) initiative. Attempting to classify existing deterministic models through a suitable set of criteria, which might also help to identify deficits and new directions in model development, Singh (1995) gave some directions. It can be concluded that a widespread agreement has been reached to differentiate between conceptual and physically based models as far as process description is concerned. Lumped and distributed models categorise spatial decomposition. Finally, continuous and event models with daily or shorter time steps in the range of minutes to hours are differentiated to account for temporal resolution. Since the early days of hydrological model development the question has been raised, whether there is a preference for a certain type of model or even for an overall optimum model structure. Due to the lack of other criteria, the most widely used approach is a set of evaluation criteria describing the ability of computed runoff data to fit measured values. It seemed to be accepted for more than a decade that all models fullfilling an adequate level of such criteria are equally acceptable (Equifinality as defined by
Beven,1995). This acceptance is avoiding potential claims of modelers to „own“ the best model. On the other hand from the scientific and practical point of view this conclusion decelerates the desirable and necessary search for better models or a temporally best model. For this search extended evaluation criteria beyond goodness of fit indices such as minimum uncertainty for application without measured runoff data must be defined. This could lead to a preference for physically based models as they might have advantages to optimally assume physically meaningful parameters without measured runoff. If this is confirmed, ways must be sought to move away from conceptual models towards more physically based models. The paper presents a model development which was initiated for this purpose, i.e. to close the gap between the conceptual and physically based models. CLASSIFICATION OF MODELS A recent classification of models and uncertainty assessment approaches was presented by Pechlivanidis et al, 2011. It covers the development of hydrological models from the beginning of the 90ies, but does not cover prior underlying models. According to his classification the model described is a distributed continuous strongly physically oriented but still conceptual nonlinear model. For the classification of process description, models can be differentiated into simplified linear models combining several subprocesses in the simplest case and into sophisticated models describing single subprocesses and their interactions by an empirically and/or physically defined set of coupled non linear differential equations in the most complex case. The set of equations used can be summarised as model structure. There is no completely sharp separation of different model types, transition is more or less continuous. Looking at deterministic models only, they can be also grouped into conceptual models based on empirical functions and physically based algorithms. While conceptual models have a long history in hydrology, e.g. the linear reservoir approach which goes back some eighty years (Zoch, 1934), sophisticated physically based models were first developed 50 years ago. Much credit has to be given to US-Americal environmental and water authorities together with the National Science Foundation who strongly supported the development of environmental. Hydrological models are only a small subsection of environmental models. The SWM was the first continuous semi-distributed physically oriented models working with sub-daily climate data. As frequently observed the development arrived in Europe about 10 years later, in the mid seventies R. Manley's HYSIM model was the first European model. In contrast to the US, where computer codes developed with public support remain open, commercial software codes are often inaccessible for research, i.e. they are rather useless. A new era of model development started with the arrival of GIS technology. All of a sudden the data was easily available which had to be derived manually from analogous maps before. It was expected that this extended availability of data would lead to new breakthrough in hydrological modelling. Indeed models based on digital elevation models and digital soil and land use maps initiated new modelling approaches; however, one important conclusion is the remaining uncertainty in parameter estimation. This has been in the centre of scientific hydrological research for some 20 years with interesting results, but without major positive impacts on applied hydrology. An intermediate step was the application of linear differential equations to a more detailed resolution of sub processes. i.e. the development of parallel linear reservoir cascade models. From the scientific point of view the highest level of model development is the representation of hydrological systems through coupled hydro- and thermodynamic mass and energy balance equations often coupled with linear and non linear biological differential equations. The choice of the system of differential equations will
directly influence the computational requirements, as most physically and biologically scientifically well based equations require numerical solutions, which normally also require a high spatio-temporal resolution. Basically, the degree of physicality and spatio-temporal resolution can be arbitrarily combined, i.e. we can use a linear reservoir on a five by five metre raster with a five minute time step or we can apply the Darcy-Richard equation on a 3 km2 sub catchment with a daily time step; yet it simply does not make sense from the scientific point of view. If we apply physically defined non linear equations, any transfer of the differential equation into a finite difference equation is connected with a loss of physical significance. Averaging of spatial information leads to the loss of information on the inhomogeneity which is equivalent to a loss of local extremes; any temporal averaging leads to a loss of extreme rainfall intensities, which in turn has to be compensated by parameter modifications. Parameters will partly loose their physical significance as they must be modified to account for averaging. A farspread spatial decomposition approach is the semi-distributed one, averaging basin characteristics within sub basins. Increasingly, more distributed models are based on rectangular, mostly quadratic raster or triangulated irregular networks (TIN). Raster elements and TINs allow to characterise processes on and within elements and their mutual dependencies in space and time in a physically defined way. Once these elements have been aggregated to form hydrologic similar units ( HSU), these interdependencies get lost. In between, sub basins and raster/TIN elements HSU are defined as space elements having the same geographic characteristics such as land use, soil, elevation, slope and precipitation. A basic assumption is the expectation that independent of their relative location in the basin HSU‘s will react in the same way to rainfall and thus can be aggregated. From these descriptions it becomes clear that physically based models require an adequately fine spatial and temporal decompositon. It does neither make sense to use physically based structures on semidistributed and lumped models nor does it make sense to use simple linear equations with highly distributed disaggregation. In the first case model parameters more or less completely loose their physical significance, in the second one it is useless to increase the computational effort only to feed a GIS to make results look good, even though they are much faster than non-linear models with numerical solutions. It can be concluded that hydrological modelling of land surfaces in contrast to e.g. hydrodynamic modelling of river bodies offers a multitude of possible and suitable approaches to solve a problem. In this respect hydrology is not a precise science but contains a considerable amount of subjective judgement, and it even has some slight aspects of art. What is considered a necessary requirement is the definition of a generic geographic representation of a basin and the set of combined differential equations for process simulation. A recent overview of the use of GIS for hydrological modelling is given by Fürst (2004). It can not be the only or dominant objective of science to reproduce measurements. It also matters, how this is achieved. In this respect physically based models are clearly superior to conceptual models. Consequently the question arises how to develop better physically based models. One approach might be to forget earlier approaches and to develop completely new models. The authors have chosen the alternative, to start from known modelling approaches and to contribute to building a bridge between conceptual and physically based models.
THE MODELLING SYSTEM WBRM
macropore
The modelling system WBRM is based on the concept of elementary units in the form of vertically layered quadratic raster elements as given in figure 1.
infiltration- layer root - layer
transport layer groundwater
Figure 1: Structure of WBrM (Lempert(1999)) Three of the five layers defined account for the interaction between atmosphere, vegetation and soil water dynamics. Vegetation can partly cover an element, canopy density is a function of vegetation type and season. No direct coupling of vegetation growth and soil water dynamics is included, although desirable on a longterm perspective. Seasonal variation of vegetation parameters, however, is included. An element can be partly impervious due to urbanisation. In case of vegetation cover precipitation is intercepted on the biomass and evaporated during and after the end of the storm. In case of missing coverage it will reach the soil or impervois area. Runoff from impervious areas is simulated separately. The soil column itself is separated into three layers. The first layer accounts for infiltration and evaporation from the soil matrix, the second one represents the root zone ending at maximum rooting depth being a monthly function. The third layer is the transient or transport zone which transforms percolation into groundwater renewal. Soil matrix flow is both vertical and lateral in the two upper layers with the simplifying assumption to occur parallel to hillslope. In Figure 2 the structure of the submodel for the infiltration layer is given. The dotted arrows indicate the ability of the model to account for backwater effects, both in vertical and lateral upward directions. Parallel to these three layers with vertical transport a separate macropore system is defined for the first and second layer. Precipitation reaching the soil surface can enter the soil layer up to the maximum infiltration rate into the macro pores being possible at the given soil moisture. When this is exceeded the macropore system is assumed filled. Macropore flow can be both vertical and lateral, feeding water into the soil matrix. Once both soil and macropore infiltration are exceeded surface runoff occurs which will be stored on the surface and transported laterally. Subsequent infiltration from the surface is possible reducing surface runoff. Water in the infiltration layer is evaporated as a function of actual soil moisture and potential evapotranspiration or transported in lateral or vertical direction towards the second layer, where water is absorbed by plants through transpiration. In case of high groundwater
levels capillary rise is possible into the root zone.
flows 1 infiltration 2 deep infiltration 3 percolation 4 groundwater renewal 5 overland flow 6 upper interflow 7 lower interflow
backwater effects
1
51 6
2 3
Infiltration Zone
72
Root Zone
3
Transition Zone 4
4
Groundwater Zone
Figure 2 Model structure of soil moisture in the unsaturated soil zone Table 1 gives an overall overview of the algorithms used in the model. Hydrological Processes
Simulation approach
Interception Evapotranspiration (pot)
Rutter approach / Linear storage (Rutter, 1975) Haude approach combined with vegetation index (Haude, 1958)
Snow hydrology
Temperature index coupled with snow compaction (Bertle, 1966)
Surface runoff
Kinematic wave (Henderson, Wooding, 1964)
Macropore flow
Kinematic wave (Beven & Germann, 1982)
Infiltration
Modified Holtan approach (Holtan & Lopez, 1971)]
Perkolation
Modified Brooks-Corey approach (Brooks & Corey, 1964)
Evapotranspiration (act) Nonlinear function of soil moisture (Albert, 1989) To compute river flows, local processes occuring on and in elementary units have to be aggregated along their flow lines within the coupled system. The sequence of aggregation results from the relative position of elements and related driving forces, which are with few exceptions gravitational forces. Spatial aggregation follows gravity according to steepest descend. This sequence is assumed to be stationary and thus can be pre-determined before process simulation occurs. Through a search algorithm following closely the approach developed by Martz and Garbrecht, 1999 a priori all elements are identified along the watershed which have no uphill predecessor. If no external disturbance occurs e.g. through human impact flows occur into the direction of steepest slope. Lateral flow components following these pathes are surface runoff, macropore flow and interflow. As simulation of groundwater flow is given less weight in the model, all element renewal rates flow into one groundwater compartment modelled as a linear reservoir under the raster element with no feedback to upper layers. All other processes are coupled with each other in each element and can thus locally exchange mass. However, in contrast to fully physically defined models one major limitation of the model is the assumption that all lateral components follow surface slopes. When surface storage occurs, excess water which cannot be stored is transported downhill. Stored water is still available for infiltration, i.e. it is available for macropore flow and subsequently increase of soil moisture. In case of partially impervious elements water can not infiltrate and is moved downstream to the next raster where it might infiltrate. Finally, only water which could not be stored or
infiltrated will reach the drainage system as direct runoff, which in simulation and reality occurs with a low probability. Despite the unidirectional transport assumed the lateral sub soil movement of water in WBrM offers the opportunity to simultaneously combine Hortonian flow with saturation area concepts, of which the basic assumption is the occurence of parallel lateral surface and sub surface flow paths. Basically, this assumption is similar to the one used in the TOPMODEL (BEVEN u.a. 1995). Before surface flow is created, the macropore system is activated. This system can only transport water laterally, if it is not quickly percolated to deeper soil layers where it infiltrates into the soil matrix. If macropore flow is created, it is transported to the next downstream element. In case this does not have a macropore system, this flow reappears on the surface. Lateral transport in the soil matrix can occur in all three soil layers. Flows in the infiltration layer are transported downhill increasing soil mositure. If the maximum capacity is exceeded excess water becomes return flow adding to surface flow. All other inputs from other neighbouring elements and from rainfall become flow from this saturated area. In the two lower layers backwater effects have to be considered. If not all water can be absorbed by the downhill element, inflow is limited and excess water is distributed to uphill elements adding to soil moisture. This is acchieved by stepwise reducing the conductivity of uphill elements, until excess water is distributed. Backwater phenomena occur in case of dominantly horizontal transport processes, when less permeable geologic formations or B-horizons are present. In these cases backwater effects occur also in vertical direction, which can also be modelled through redistribution of excess water within upper layers. The procedure was developed earlier and is described in LWA (1995). Although the procedure described is still a simplified description of physical processes, it leads to plausible results. Figure 3 shows the development of surface storage in a small forested watershed (Solchbach, Eifel,). The grid size is 25 m, the time stept is 15 minutes. According to natural conditions areas behind elements with limited permeability become increasingly saturated also into uphill regions. The approach allows to efficiently model 3D effects with a 1-D modelling approach.
Figure 3 Wettening of a small catchment during a storm
Use of a quasi non-linear multi input-output storage module The linear storage approach was a milestone for the development of hydrological models and has been the most important module for more complex modelling systems over many decades. However, it can per definition model natural non linear effects with limited accuracy only. Consequently it is desirable to replace the linear storage concept with a non linear storage approach without giving up the advantages of linear first order homogeneous or inhomogeneous differential equations having stable and computionally highly efficient analytical solutions. For WBrM a generalised module was used for replacing multiple input/output storage functions through piecewise linearised non linear functions. This module was first developed by Ostrowski (1992) and the further modified and improved through a group of researchers. An overview of the present state of development is given by Ostrowski (2002). The basic structure of the generic module is given in Figure 4. The linearisation is demonstrated by Figure 5. V(t)
Q2(t)
Q1(t) I1(t)
Q3(t)
V(t)
Q1(t)
Q2(t)
I2(t)
Q3(t)
I3(t)
Qj(t)
Figure 4: Concept of a generic multi input/output non lonear storage module
V Vi+1
Vi V(t) ki
Vi-1
Q i-1
Q (t)
Qi
Q i+1
Q
Figure 5: Linearisation of non linear function Q(t) = f (V(t)) It is evident that the number of linear reaches defined determines the accuracy with which the non linear function is approximated. Each linear reach behaves like a single linear storage with the known anlytical solution. The algorithm procedes stepwise to the next lower (emptying) of upper segment until the boundary of the timestep used is reached. Although numerical solutions of the original non linear differential equations become continuously faster and more stable, the approach still offers considerable
advantages concerning computational efficiency, stability and correctness of mass balance. The module is not only used for most non linear sub processes in the WBrM system, but has also been most successfully applied for the simulation and operation of water resources systems. MODEL DEVELOPMENT – STRATEGY AND TEST The most important modules soil moisture and land surface flow applied for process simulation were first tested on different spatial scales before they were assembled in the WBRM system which was then applied to the hydrological research station Weiherbach in Southwest Germany. Surface runoff module The kinematic wave concept is basically known to simulate surface runoff well. The approach was tested with experimental data provided by Izzard, 1946. Besides proving the applicability of the approach the procedure of the non linear storage module was tested concerning its numerical accuracy. Figure 6 shows the results of the analysis after optimum piecewise approximation and surface roughness estimation.
precipitation runoff measured runoff calculated
Figure 6 Computed and measured flow from impervious surfaces SVAT module The most important module of any physically based hydrologic model is the soil vegetation atmosphere transfer module (SVAT-module), which accounts for the local separation of precipitation into the water balance compartments. Before continuing with lateral flow components it must be shown that the SVAT module describes vertical flows adequately. This is frequently achieved by modelling lysimeter measurements. In this case the lysimeter unit Senne close to Bielefeld in Northwest Germany has been used. The Senne site has three lysimeter units, each 2 m deep with a circular cross section of 1 m2. The lysimeters were filled with different soils each covered with grass. Further information on the lysimeter was given by Schroeder (1991). Before applying the SVAT module, soil parameters were pre-estimated from laboratory analyses and from measurements. The most important characteristic used in the
Percolation Sickerwasser [mm/d] [mm/d]
module is the unsaturated hydraulic conductivity as a function of soil moisture according to figure 7. The pF curves for the lysimeters were known from laboratory tests for three levels in each lysimeters. In addition the scatter diagrams according to figure 5 supplied more information about the relationship between soil moisture and percolation. 16
FK
WP
14
1984 1985 1975
12 10 8 6 4 2 0
20
40
60 80 100 soilmoisture [mm] Bodenfeuchte [mm] LysimeterSenne Senne11 lysimeter
120
140
Figure 7 Derivation of unsaturated conductivity from lysimeter measurements Finally the measured percolation was compared with the computed one. The approach proved to be adequate to simulate local vertival soil moisture storage and related processes. A typical result is given in Figure 8. Percolation measured Percolation calculated
Change of soilmoisture measured Change of soilmoisture calculated
Figure 8 Measured and computed soil moisture and percolation rates
In a further analysis the ability of the soil moisture model was tested to model two dimensional subsurface flow. The test site was a hillslope of a solid waste dump near Frankfurt Main, Germany covered with a stratified so called capillary barrier. The concept of such a barrier is the diversion of vertical percolation into interflow due to soil texture characteristics. A coarse sand layer is covered with a fine sand layer with a textile in between to avoid wash out. Residual soil moisture of the fine sand is some percent above that of coarse sand leading to lateral flow in the fine sand layer while the coarse material can not yet transport water. It could be shown that the model is able to simulate this interesting phenomenon adequately according to figure 9. Finally, to assess the model‘s ability to simulate water balances of river basins, the experimental basin Weiherbach was chosen, which had been installed and run earlier by (PLATE ET AL, 1992) from 1989 to 1997 involving several reasearch teams from different disciplines. Part of the initiative was the collection of hydrologic, meteorological, geologic, pedologic und biologic data, many of them directly usable for model application. The model WBrM was applied to a sub basin up to gage Menzingen, both for modelling continuous hydrologic balances for the total observation period as well as modelling selected flood events in higher temporal detail. Continuous modelling was based on daily timesteps. The dominant objective was to achieve a realistic simulation of the basin‘s water balance. The criteria used to assess the plausibility of results were measured flow at the gage as well as spatial distribution of processes and their integrals over time. Figure 10 shows measured and computed flows for the hydrologic year 1994 (Nov 93-Oct 94). It can be concluded that both dry and wet periods could be modelled in a highly realistic way. In general also the areal distribution of yearly percolation and evapotranspiration depth corresponds well to soil and land use distribution in the basin. Special weight was given to soil moisture dynamics. This could be assessed comparing measured soil moisture changes with computed ones as several continuous soil moisture measurements were available. [mm/d]
flow capillarlayer calculated
4.0 3.5 3.0 2.5 2.0 1.5 1.0
flow capillarlayer measured
0.5
[mm/d]
1.Jan 1995
1.Jul
2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25
1.Jan 1996
1.Jul
1.Jan 1997
flow capillarbarrier calculated flow capillarbarrier measured
1.Jan 1995
1.Jul
1.Jan 1996
1.Jul
1.Jan 1997
Figure 9 Computed and measured flows in the fine sand layer (up) and gravel layer (down)
3
Figure. 10
[m /s] 0.1
0.387 (meas.) / 0.391(cal.)
0.09
runoff Abflußcalculated berechnet runoff Abflußmeasured gemessen
0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 1.11.93
1.12.93 31.12.93 30.1.94
1.3.94
31.3.94
30.4.94
30.5.94
29.6.94
29.7.94
28.8.94
27.9.94 27.10.94
1994
Measured and computed flows at gage Menzingen hydrologic year 1994 In Figure 11 measurements and computed values of soil moisture are presented for 3 years from 1992 to 1994 for one point measurement showing again realistic simulations of dynamics although some lartge deviations also occurred, which indicates that the model structure chosen represents well the major processes. It should be mentioned that these measurements have not been used for calibrating flows at the gage. Vol % 40 soilmoisture at Meßpunkt point 8 Bodenfeuchte 8
35 30 25 20 15 10 5
Bodenfeuchte gemessen soilmoisture measured soilmoisture calculatetd Bodenfeuchte berechnet
0 N-91 D-91 F-92 A-92 J-92 A-92 O-92 D-92 F-93 A-93 J-93 A-93 O-93 D-93 F-94 A-94 J-94 A-94 O-94 1992 - 1994
Abb. 11 Measured and computed soil moisture in the Weiherbach basin Simulation of single flood periods confirm these longterm results. Seven events of varying magnitude and type of generating precipitation were analysed, leading to the conclusion that the model can represent all type of potential floods.
Summary The contribution reports about the newly developed WBRM model being highly distributed in space and time using non linear physically oriented, but still conceptual algorithms to simulate flows and storage contents in several compartments. The basic objective was to bridge the gap between linear conceptual and non linear physically based models. Non linear processes are approximated by piecewise linear differential equations with analytical solutions. The result is a realistic description of processes and high computational efficiency at the same time. Modules describing sub processes for simulation of surface flow and soil moisture for example have been tested and verified seperately. Continuous simulation and single storm event simulation in the Weiherbach catchment resulted in good agreement of measured and computed observation values of both flows and soil moisture. In addition the spatial and integral values of other variables without measured values such as real evapotranspiration and groundwater renewal proved to be within plausibility limits. This proves that WBRM cannot only reproduce flows but also other components of the water balance realistically. Thus, it seems possible to apply the model also in ungaged catchments.
References Albert, W. (1989): Die Gebietsverdunstung von Waldstandorten aus der Simulation von Grundwasserganglinien mit klimatischem Bodenwasserhaushaltsmodell. Mitteilungen des Instituts für Wasserbau und Wasserwirtschaft, Nr. 30, Technische Universität Darmstadt, Darmstadt. Bertle, F.A. (1966) Effect of snow compaction on runoff from rain on snow, Engineering monograph No 35 Beven, K.J., Germann, P. (1982) Macropores and water flow in soils, Water resources Research, No. 18, p. 1311-1325 Beven, K.J., Lamb R., Quinn P.F., Romanowicz R., Freer J. (1995): Topmodel. in: Singh, V. P. (ed.): Computer models of watershed hydrology. 1. Aufl., Water Resources Publication, Colorado, S. 627 - 668. Brooks, R.H., Corey, A.T. (1964) Hydraulic properties of porous media, Hydrology papaer No. 3, Colorado State University, Fort Collins, Colorado Crawford, N.H. and R.K. Linsley, 1966. Digital Simulation in Hydrology: Stanford Watershed Model IV. Technical Report No. 39, Department of Civil Engineering, Stanford University, p. 210. Fürst, J. (2004):GIS in Hydrologie und Wasserwirtschaft, Herbert Wichmann Verlag, Hüthig GmbH & Co. KG, Heidelberg, ISBN 3-87907-413-5 Haude, W. (1958) Über die Verwendung verschiedener Klimafaktoren zur Berechnung der potentiellen Evaporation und Evapotranspiration, Meteorologische Rundschau, No. 11, P. 96-99 Henderson, F.M., Wooding, R.A. (1964) Overland flow and groundwater flow from a steady rainfall of finite duration, Journal of geophysical research, No. 69, p. 1531-1540 Holtan, H.N., Lopez, N.C. (1971) USDAHL-70 Model of watzershed hydrology, US Department of Agriculture, Technical Bulletin 1435 Izzard, C.F. (1946): Hydraulics of runoff from developed surfaces. Proc. of the annual meeting of the highway research board, Nr. 26, S. 129. Lempert, M. (2000): Ein GIS gekoppeltes rasterbasiertes Modell zur Berechnung des Wasserhaushaltes kleiner Einzugsgebiet, Dissertation, Mitteilungen des Instituts für Wasserbau und Wasserwirtschaft, Nr. 110, TU-Darmstadt, Darmstadt. Lohr, H. (2001) : Simulation, Bewertung und Optimierung von Betriebsregeln für wasserwirtschaftliche Speichersysteme, Dissertation, Mitteilungen des Instituts für Wasserbau und Wasserwirtschaft, Nr. 118, TU-Darmstadt, Darmstadt. LWA (1995): Modellbaustein zur Simulation von Bodenfeuchteprozessen. Forschungsbericht, Landesamt für Wasser und Abfall Nordrhein-Westfahlen, Düsseldorf, unveröffentlicht. Martz, L.W. and Garbrecht, J. (1999). An outlet breaching algorithm for the treatment ofclosed depressions in a raster DEM. Computers and Geosciences 25, 835-844.
Manley, R.(1977) The soil moisture component of mathematical models, Journal of Hydrology,35, p. 341-356 Manley, R. (1978) Simulation of flows in ungauged basins, Hydrological Sciences Bulletin, 23, p. 85-101 Maurer, Th. (1997): Physikalisch begründete, zeitkontinuierliche Modellierung des Wassertransports in kleinen ländlichen Einzugsgebieten. Mitteilungen des IHW, Nr. 61, Institut für Hydrologie und Wasserwirtschaft, Universität Karlsruhe, Karlsruhe. Mehler, R. (2000) : Mischwasserbehandlung - Verfahren und Modellierung, Dissertation, Mitteilungen
des Instituts für Wasserbau und Wasserwirtschaft, Nr. 113, TU-Darmstadt, Darmstadt. Merz, B. (1996): Modellierung des Niederschlag-Abfluss-Vorgangs in kleinen Einzugsgebieten unter Berücksichtigung der natürlichen Variabilität. Mitteilungen des IHW, Nr. 56, Institut für Hydrologie und Wasserwirtschaft, Universität Karlsruhe, Karlsruhe. I.G. Pechnlivanidis, I.G., B.M. Jackson. B.M., McIntyre, N.R., Wheater, H.S.:Catchment scale hydrological modelling: A review of model types, calibration approaches and uncertainty analysis methods in the context of recent developments in technology and applications, Global NEST Journal, Vol 13, No 3, pp 193-214, 2011 Ostrowski, M. (1992): Ein universeller Baustein zur Simulation hydrologischer Prozesse. Wasser und Boden, Nr. 11, S. 755 - 760. Ostrowski, M. (2002): Ein allgemeiner Ansatz für die nichtlineare hydrologische Modellierung – Theorie und Beispiele, Tag der Hydrologie 2002, Forum für Hydrologie und Wasserbewirtschaftung, Heft 1 Plate, E. (1992): Weiherbachprojekt. "Prognosemodell für die Gewässerbelastung durch Stofftransport aus einem kleinen ländlichen Einzugsgebiet", Schlußbericht zur 1. Phase des BMFT-Verbundprojektes. Mitteilungen des IHW, Nr. 41. Refsgaard, J.C., Knudsen, J. (1996): Operational validation and intercomparison of different types of hydrological models. Water Resources Research, Nr. 32, S. 2189 - 2202. Rutter, A.J., Morton, A.J. (1975) A predictive model of rainfall interception in forests, 2. Generalisation of the model and comparison with observations in some coniferous and hardwood stands, Journal of applied ecology, No. 12, p. 267-380 Schroeder, M. (1991) Meßergebnisse der Jahre 1979 bis 1985 an der wägbaren Lysimeteranlage Senne. LWA Schriftenreihe, Nr. 48, Landesamt für Wasser und Abfall Nordrhein-Westfahlen, Düsseldorf. Singh,V.P. (ed.) 1995): computer models of watershed hydrology. 1. aufl., water resources publication, colorado. Sivapalan, M., Schaake, J. (2003): IAHS decade on predictions in ungauged basins (pub): 2003-2012, PDF on URL http://www.cig.ensmp.fr/~iahs/, September 2003 Zoch, R. T. (1934,1936,1937): On the relation between rainfall and stream flow. Monthly Weather Review, 62(9)/64(4)/65(4), 315-322/105-121/135-147.
