Computation of backwater effects in surface waters of tidal lowland catchments including control structures – An efficient and re-usable method implemented in the hydrological open source model Kalypso-

Backwater effects in surface water streams as well as on adjacent lowland areas caused by mostly complex drainage structures are not directly computed with hydrological approaches, yet. A solution of this weakness in hydrological modelling is presented in this article. The developed method enables to transfer discharges into water levels and to calculate backwater 10 volume routing along streams and adjacent lowland areas by balancing water level slopes. The implemented and evaluated method extends the application of hydrological models for rainfall-runoff simulations of backwater affected catchments with the advantages of (1) modelling complex drainage systems in tidal backwater affected lowlands, (2) less effort to parameterise river streams, (3) directly defined input factors of driving forces (climate change and urbanisation) and (4) runtime reduction of one to two orders of magnitude in comparison to coupled hydrodynamic models. The developed method is implemented in 15 the open source rainfall-runoff model Kalypso-NA (4.0). Evaluation results show the applicability of the model for modelling rainfall-runoff regimes and backwater effects in an exemplary lowland catchment (Hamburg, Germany) with a complex drainage system and where the drainage is influenced by a tidal range of about 4 m. The proposed method is applicable to answer a wide scope of hydrological and water management questions, e.g. water balances, flood forecasts and effectiveness of flood mitigation measures. It is re-usable to other hydrological numerical models, which apply conceptual hydrological 20 flood routing approaches (e.g. Muskingum-Cunge or Kalinin-Miljukov).


Introduction
Open demand exists in hydrological modelling of rainfall-runoff regimes in lowlands which are distinguished by complex flow routing in mostly intensively drained catchments by manifold control structures. The occurrence of backwater effects in such lowland river streams as well as on adjacent lowland areas pose an open research question in hydrological modelling. Adjacent 25 lowland areas in this article are distinguished by a low ground level and connection to rivers. The size of lowlands varies from narrow riparian areas, wetlands, shallow retention spaces, floodplains or vast partly urbanised marsh-or swamplands.
Hydrological models are applied to simulate processes of the (1) surface-atmosphere interaction, (2) the transition between soil-vegetation-atmosphere, (3) the processes in the vadose zone of the soil and (4) the flood routing in the receiving surface waters. In lowlands, the last two issues require more detailed considerations because of mostly high groundwater levels and 30 the drainage against fast changing water levels in tidal streams of complex drainage systems. For simulating the interaction between groundwater and surface water quite a few approaches are available (Brauer et al., 2014;Waseem et al., 2020;Sun et al., 2016). However, modelling backwater effects in tidal streams with fast changing water levels in complex drainage systems of lowland catchments directly with hydrological models is not implemented in most hydrological approaches up to now (Waseem et al., 2020). 35 The demand to solve this weakness in hydrological numerical models increases, since in low lying tidal catchments, the pressure on current storm water drainage systems raises due to combined impacts of enlarged urbanisation on the one hand and climate change induced sea level rise in combination with heavy storm events on the other hand (IPCC, 2013b(IPCC, , 2013a https://doi.org/10.5194/gmd-2021-140 Preprint. Discussion started: 31 May 2021 c Author(s) 2021. CC BY 4.0 License. UN DESA, 2018). Studies about the combined risk of high tides (storms) and stormwater events are given by (Lian et al., 2013;Nehlsen, 2017;Klijn et al., 2012;Zeeberg, 2009;Huong and Pathirana, 2013;Sweet et al., 2017). These selected 40 examples all show a conformity about the tendency that low lands will face higher pressures to mitigate flooding in the future.
A promising flood mitigation measure against the effects of (high) precipitation events in low lying catchments is the controlled temporary storage of water in retention areas. However, state-of-the-art hydrologic approaches reveal shortcomings in modelling the flood routing and retention volume in backwater affected lowland catchments.

Objectives 45
To resolve the shortcomings in hydrological approaches to model water depths and backwater effects, new concepts are required. Among others, the presented method in this article fulfils five objectives in hydrological modelling. The method is (1) applicable to model complex drainage systems in tidal backwater affected lowlands, (2) efficient by using short run-times for real-time operational model application, (3) open for further model developments, (4) re-useable for other hydrological model solutions and (5) parsimonious with regard to the complexity of input parameters. Reaching a balance between model 50 structure details (namely complexity) and data availability is an important issue to keep the model as parsimonious and efficient in runtime as possible, but complex enough to explain the heterogeneity in the areas and the dynamics in the hydrological processes. Most promising to accomplish the defined five objectives for a re-usable, open, efficient and parsimonious hydrological model, is the development of an extension approach for state-of-the-art flood routing methods (for instance Muskingum-Cunge or Kalinin-Miljukov), which can be transferred and implemented in different hydrological numerical 55 model approaches and on different model scales.

Outline
The literature review in section 2 discusses current weaknesses in hydrological models to simulate backwater effects and subsequent flooding of adjacent lowland areas. The theoretical concept in section 3 and the developed method in section 4 explain the worked out solution. The implementation of the methodology is realised in the open source hydrological model 60 Kalypso-NA version 4.0 (section 5). The evaluation of the method is done using observed data of an exemplary tidal catchment study in Hamburg, Germany, where a complex drainage system and backwater affected streams have a significant impact on the results of the rainfall-runoff regime (section 6). The article closes in section 7 with a summary of the main findings and an outlook of follow-up research.

2
State-of-the-art in hydrological modelling to compute flood routing and backwater effects in lowlands 65 Flood routing describes the processes of translation and retention of a flood wave moving along a stream in downstream direction. To simulate the flood routing in rivers different approaches are applied: (1) pure black box (namely empirical, lumped), (2) hydrological conceptual or (3) hydrodynamic-numerical approaches. The applicable flood routing method needs to be chosen with respect to the modelling purpose and available data. Computation of water depths and backwater effects in rivers as well as on forelands by using hydrological approaches (1 and 2) is rarely done and up to now mostly linked with 70 comparatively high uncertainties. The missing applicability of hydrological approaches for simulating backwater effects is shown in a recent study within the North German lowlands (Waseem et al. 2020).
Up to now, simulating water depths and backwater effects in complex profiles as well as on floodplains and other adjacent lowlands demands for 1D, 2D or 3D hydrodynamic-numerical models with the numerical integration of the partial differential equations describing the flood routing processes. Hydrodynamic-numerical models show drawbacks in comparison to 75 hydrological models: (1) they require more effort to parameterise the river streams, (2) future impacts of climate change and urbanisation are not directly parameterised in the model approach and (3) simulation times are at least one to two orders of magnitudes longer. High resolution data describing the topography of the main channel and the natural flood plain in the case of bank overflow is necessary. Hence, the availability of suitable detailed profile data from measurements is significant for https://doi.org/10.5194/gmd-2021-140 Preprint. Discussion started: 31 May 2021 c Author(s) 2021. CC BY 4.0 License.
hydrodynamic-numerical modelling. The larger effort in data resources and runtime for hydrodynamic-numerical model 80 simulations is no limitation for answering special research questions. However, applying a coupled hydrologicalhydrodynamic model shows disadvantages in the application on meso to regional catchment scales (>100 km²) and for operational forecast applications. It is proposed in this article, that a stand alone hydrological approach is more suitable to enable parsimonious and efficient modelling of flood routing and backwater effects in lowlands.
Commonly applied conceptual hydrological approaches are the 'storage routing' by Puls (1928), 'Muskingum' or 85 'Muskingum-Cunge' routing described by McCarthy in (1938) or (Cunge, 1969, 'Kalinin and Miljukov routing' (1958) or 'linear reservoir and channel cascade routing' presented by Maddaus in (1969). The purpose of hydrological flood routing approaches is to compute the discharge hydrographs in the considered stream segments. For hydrological approaches, conceptual or empirical parameters are calibrated based on observed events like in the widely used Muskingum method. A compromise are hydrological methods using profile data of streams to model the flood routing, for example in the Muskingum-90 Cunge approach (Cunge, 1969) as well as the approach of Kalinin and Miljukov, 1957. These concepts use profile information in a conceptual way and require far less calculating effort for meso scale modelling (> 100 km²) than hydrodynamic numerical approaches.
Only few related studies are available with respect to model backwater effects in meso scale catchments with hydrological approaches, while none of the reviewed studies enabled the computation of backwater retention in lowland areas for mitigating 95 backwater induced flooding. Coupled hydrological-hydrodynamic computation models: like in MIKE SHE coupled with MIKE 11 (Waseem et al., 2020) or in the German Model NASIM coupled with a hydrodynamic computation model (Loch and Rothe, 2014;Dorp et al., 2017) are not part of this comparison, because of the afore described disadvantages in hydrodynamic approaches. A focus is set on direct or stand-alone hydrological model enhancements.
In (Waseem et al., 2020), a recent comparison of models to simulate decisive hydrological processes in coastal lowlands shows 100 weaknesses in the model SWIM (soil and water integrated model) and HSPF (hydrological simulation program-FORTRAN).
The approaches in SWAT (soil and water assessment tool) und MIKE SHE show a good conformity to model processes in lowlands while both are not applicable to model backwater effects in the river, on floodplains or other adjacent lowlands and backwater effects caused by control structures (sluices, pumping stations and tide gates). An enhanced approach in SWAT for riparian wetlands (SWATrw) is presented in (Rahman et al., 2016) to compute the surface water interaction between river 105 streams and explicitly defined wetlands, while backwater effects in streams are unconsidered. The modified SWAT-Landscape Unit (SWAT-LU) model enables to compute horizontal hydraulic interactions between a river and the aquifer beneath the adjacent floodplain (Sun et al., 2016). Similarly, in the Rainfall-Runoff Modell WALRUS (Wageningen Lowland Runoff Simulator) a lumped approach is realized to model the following processes: (1) groundwater-unsaturated zone coupling, (2) groundwater-surface water feedbacks and (3) seepage and surface water supply (Brauer et al., 2014). These are important 110 model features to model the runoff regime in lowlands, but neither of the approaches enable to compute backwater effects on the land surface or along the streams and control structures in the receiving streams.
More national specific studies to model backwater effects in streams are done with the German model ArcEGMO (by the 'Büro für Angewandte Hydrologie', Berlin). The hydrological model 'ArcEGMO' takes into account backwater effects by hindering the downstream flood routing when the water level at the downstream segment is higher than the upstream one 115 (Pfützner, 2018). This method calculates a retained flood rooting, but neither computes backwater volume being routed into In a study by (Messal, 2000), backwater effects among river streams and the subsurface flow in river banks are modelled exemplarily for the catchment Stör (1157 km²) in Schleswig-Holstein. Messal applies a proportional relationship between upstream and downstream elements for calibration purposes. The model serves well for the catchment study Stör, but the parameter values are non-transferable to other catchments because of a lack in physical descriptions. 125 Another approach is presented by (Riedel, 2004) to model the backwater effects among river streams in German lowlands on the example of two tidal tributaries of the Weser river. The approach uses the reservoir cascade theory including the input parameters of the roughness coefficient by Manning-Strickler and geometric descriptions of the profiles for the flood routing computation. The river is modelled as a cascade of reservoirs (namely a NASH-cascade), while the water level from the previous time step of the downstream segments are taken into account to compute the flood routing. A time step shift in the 130 computational approach is accepted by (Riedel, 2004) because he reduced the simulation time step size to one minute. The model computes a reservoir cascade on the basis of a defined boundary condition at the downstream segment. However, the explicit simulation of backwater induced flooding of flood prone areas or adjacent lowland areas is not included.
These reviewed hydrological methods compute backwater effects in a more or less conceptual way with the described weaknesses and limitations. None of these studies analysed the backwater induced flooding of lowland areas or in this specific 135 case, retention areas. Consequently, none of the studies accomplish to simulate a controlled retention of backwater volume in such areas, a subsequent drainage and neither the computation of hydrological processes influenced by backwater induced flooding. Further on, most studies do not apply physical-based parameters to transfer validated values and knowledge from one catchment to other studies. A methodology to solve these shortcomings is proposed in this article.

3
Theoretical approach to enhance a hydrologic conceptual flood routing method to compute backwater effects 140 To reach the described objectives, a state-of-the art conceptual hydrological method is extended to be applicable for the computation of backwater effects in streams and adjacent lowland areas (incl. retention areas). This section describes the theory of the conventional hydrological approaches to compute the flood routing (3.1), the concept of modelling control structures in tidal lowlands (3.2) and the approach to compute backwater effects with a conceptual hydrological approach in streams and adjacent lowland areas (3.3). 145

Conceptual hydrological flood routing approach
State-of-the-art hydrological flood routing theory in free flow conditions describes the flood wave propagation in streams which are not affected by downstream conditions. This means that an afflux in front of obstacles downstream of the considered stream segment is assumed to have no impact on the upstream segments. With this assumption, backwater effects are not considered.
Flood routing processes depend on the characteristics of the drainage network comprising the geometry of profiles, gradients 150 and roughness of the streams. Linear or non-linear Muskingum approaches have no physically based parameterisation and require input parameters, which are based on observed data in upstream and downstream segments of rivers. Therefore, these hydrological approaches are not suitable for the simulation with changed geometries or changed flow conditions in streams where no observed data is available. This lack is solved in two approaches, which are based on physical characteristics such as river geometry, stream length, roughness coefficient and river bed slope. On the one hand, the Muskingum-Cunge (often 155 used in the United States) and on the other hand, the Kalinin-Miljukov (KM) flood routing approach are applicable. For this work, the approach of Kalinin-Miljukov is chosen, since this approach is widely applied in Germany and Eastern Europe.
The approach of (Kalinin and Miljukov, 1957) (KM-approach) divides a stream into a number of characteristic lengths. Each length is considered to be short enough for assuming a quasi-stationary relationship on the basis of a hysteresis curve. Different With such conceptual hydrological flood routing approaches the magnitude and time of flow along a stream on the basis of stream characteristics is determined. It describes the (free flow) propagation of discharge through streams, whereby translation and retention processes along the stream changes the shape of the hydrograph from an upstream to a downstream point. The explicit direction of computation from upstream to downstream in flood routing approaches limits to include effects derived 165 from downstream obstacles. Backwater effects caused by an afflux are not implemented in these conceptual hydrological approaches yet and an extension is therefore developed in this article (section 3.3).

Concept to model control structures in lowland catchments
A backwater effect in a catchment is often caused at obstacles like weirs, (tide) gates, retention or detention reservoirs, which also function as control structures in streams. It is required to model these structures in hydrological models since such control 170 structures are regularly used to control the flow in catchments. In this article, we focus on control structures frequently applied in tidal lowland drainage areas. Operation rules of control structures are mostly pre-defined depending on operative criteria.
The criteria are normally based on thresholds of water level, discharge or precipitation intensity within hindcasted or forecasted data (see Fig. 1). Since the data time series influence the status of control structures, they are defined in this article as drivers.
There is a difference between pre-set and on-the-fly processed driver data. Pre-set data time series are imported such as 175 observed water level or precipitation data. Additionally, data series which are computed during runtime (e.g. discharge) can serve likewise as drivers and are processed on-the-fly.
When a threshold of an operative criteria is reached during the runtime of the model, the status of the system is changed (e.g. opening or closing a gate). The change of the status based on reached thresholds is described in control functions, which are checked per time step. In a control structure the retained water can cause backwater effects in upstream direction if an afflux 180 of water occurs. Control structures are one component type within a hydrological network. Other component types are streams (linear data structures), areas (spatial data structures) and nodes (point data structures). An explanation of these components of a hydrological network is given in the supplement (suppl. section 3).

Concept of the flood routing enhancement to compute backwater effects
The afore described hydrological conceptual approach (here, of Kalinin and Miljukov) is enhanced by using the resulting water 185 level, volume and discharge (WVQ) relation to compute backwater effects per stream element. The concept enables to compute a backwater volume routing according to the water level slope. This is illustrated in a scheme in Fig. 2 for a river longitudinal segment which is separated in several strands. At the downstream segment a tide gate is located. In stage (1) the free flood routing in downstream direction is computed. When the barrier (e.g. a tide gate) is closed by control functions (stage 2), an afflux of water is generated (stage 3). The afflux initiates a 'backwater volume routing' (stage 4), meaning that the water 190 volume is routed in upstream direction to equalise the surplus water level of the afflux. When the barrier is opened, the backed up water volume is routed downstream (stage 5). These five stages are computed according to the water level slope in each time step. The methodology to realise the coding of this theoretical concept into a numerical hydrological model is explained in the following chapter 4.

Methodology to compute backwater effects in rivers and adjacent lowland areas with complex drainage systems 195
The methodology to calculate backwater effects with a hydrological conceptual approach, consists of three main algorithms: a transfer of discharges to water levels and volumes per stream segment and time step (section 4.1), the calculation of (inter-) active control structures (section 4.2) and a backwater volume routing according to the water level slope along stream segments and adjacent lowland areas (section 4.3).

Transfer of discharges to water levels and volumes 200
The flood routing in stream segments of the hydrological network is computed with conceptual hydrological approaches like Kalinin-Miljukov

Calculating (interactive) control functions of drainage systems
A control structure of a linear stream segment is defined with unsteady WVQ-relations and the flood routing is modelled with a storage indication method. In this work the modified Puls method is applied. Operative criteria of control structures are defined for three types of driver time series which are precipitation intensity, water level stages and discharge values. 220 Hydrographs of water level stages and discharges are results given at junction nodes, while precipitation time series are part of spatial structures (namely subcatchments). The status of control structures is checked per time step during the execution of the numerical model. A differentiation between three functions of control structures is done according to their operative criteria depending on pre-set (external pre-processed) or on-the-fly (internal processed) driver time series. The three functions of control structures and operative criteria are listed in Fig. 4 (left). Control function type (1) depend on observed or externally 225 forecasted driver time series for instance, precipitation or water level gauge data. These control functions are computed in the pre-processing phase of the simulation run to set the status of a control structure. With forecasted data a time duration can be set to change the status of control functions (closing or opening a gate) with a specific leadtime before the threshold (operative criteria) is reached. In the control functions type (2), criteria depend on the output of computed parameters of the hydrological network, namely water level or discharge. The functions are computed during the simulation run "on-the-fly". This procedure 230 depends on the condition that the driver elements are located upstream of the control structure and are not influenced by backwater. If the criteria of a control structure depend on downstream or backwater affected conditions in an interactive system, a recursive calculation routine is started to compute the control function type (3). The recursive calculation routine is explained in the following section 4.3.

4.3
Calculating backwater effects along river streams and adjacent lowland areas 235 An afflux due to natural or artificial obstructions (for instance gates or weirs) leads to a rise of water level in upstream segments.
To simulate the resulting backwater effects, the downstream directed surplus water volume is reversed as backwater, when the downstream water level is higher than upstream. This concept is illustrated in the theoretical approach in section 3. Additionally, per backwater system (j) and per time step (t) a query checks if an interactive backwater system with a control function type (3) is defined. An interactive system depends on both, downstream and upstream conditions. In case of an interactive system, the flag for a 'recalculation' loop is activated. The final-balanced stage is reached when in a backwater affected system the downstream water levels are not higher than the upstream water levels within a range of a minimum 250 'tolerated' water level difference. The method demands to define a minimum difference (ΔWmin)  In the calculation routine a (Fig. 5), the initialisation of formal parameters of each linear and spatial data structure for the backwater effect computation is performed. This includes an initialisation of the water level, volume and discharge per time step. Discharges are computed with the flood routing approaches described in section 3.1. The corresponding water levels and retained water volumes are derived from the calculated WVQ-relations per stream segment (see section 4.1). The algorithm and equations are given in the suppl. section 5. 265 In the calculation routine b (Fig. 5), the backwater effect computational loop in upstream direction is activated, while afflux conditions are present in the backwater system. The calculation is done per stream segment in a computational loop starting at the downstream element ( = ). If the difference in water levels between the actual and the upstream segment is larger than the defined tolerated water level difference ! , an algorithm to compute the backwater effect is activated. The backwater quantity derived from an afflux at the downstream segment, is routed to the upstream segments. Along the streams, 270 spatial structures (like lowland catchments) are linked, where the water is retained or causes backwater flooding. This developed concept is illustrated in the scheme in Fig. 6, where the backwater effect computation between stream segments with linked spatial structures (retention areas) is shown. The formal parameters of the WVQ-relations of the current (i) and the upstream (i-1) segment are processed. The computation is done in three sub-calculation routines (namely A, B and C) to compute the water level and volume stages. structure as illustrated in Fig. 6 (case C), the balancing of water level and volume is done respectively to the procedure in (A).
As long as a backwater effect is present in any river segment or adjacent lowland area, the calculation is repeated (till = 10 000). The algorithm and explanations to calculate the revised flow regimes in the stages A, B and C are given in the suppl. 285 section 6.
In the calculation routine c (Fig. 5), the backwater volume is routed downstream, if the afflux conditions at the downstream segment of the backwater system is not present anymore, for instance by opening a gate or starting additional pumping. The water level and storage volume in the stream segments are reduced per time step until free flow conditions are reached. In the developed calculation routine the drainage process of the backed up water volume is calculated. The stream 290 segments are computed in the order from upstream (i = 1) to downstream (i = n). The algorithm for the computation of the subsequently drained backwater in downstream direction is done step wise with the current (i) and the downstream (i+1) data structures using the sub-calculation routines (C) to (A) in reversed order (see Fig. 6).
In calculation routine d (Fig. 5) interactive systems are computed. When a control structure depends on criteria of a downstream backwater affected system, an interactive computational loop is activated. In this case a 'recalculation' loop is 295 started and revises control structure settings if the results of the interactive backwater system are available. Then the recalculation loop restarts the computation of the calculation routines (a) to (c) (Fig. 5). The results of this developed algorithm to compute backwater effects are the time series of water levels (m a.s.l), discharges (m 3 /s) and volumes (m 3 ) for stream segments and linked spatial data structures (e.g. lowland catchments). Additionally, the activated control functions per control structure are given as time series for verification purposes. is applied in the extended algorithm to model processes in sub-catchments like the soil water balance and the downstream directed flood routing. This algorithm is explained in more details in the journal paper (Hellmers and Fröhle, 2017). 325 Additionally, an algorithm is implemented where spatial calculation routines are nested in time loops. This secondary algorithm provides the overall results of a backwater affected system per time step before calculating the next time step. The time loop is additionally nested in a backwater system loop. In that calculation routine the backwater effects in streams and adjacent lowland areas as well as the evaporation from submerged water surfaces are computed. This implementation is labelled as space-before-time algorithm and is illustrated in (Fig. 5). The implemented hydrological model approach is 330 applicable to other catchment studies, while using physical-based input parameters. The input and output parameters are listed in the suppl. section 2 and 7. The compiled code is freely available at http://kalypso.wb.tu-harburg.de/downloads/KalypsoNA/ and the source code of the modified part of the model presented in this paper can be provided upon request to the corresponding author.

Exemplary model application and evaluation 335
Objective of the model evaluation is to determine the reliability of the numerical model results to be in a sufficient range of accuracy for the designated field of application (Law, 2008;Oberkampf and Roy, 2010;Refsgaard and Henriksen, 2004;Sargent, 2014). An evaluation of the extended model Kalypso-NA (4.0) is performed by comparing the results of the numerical model with observed data of gauging stations in the mesoscale catchment 'Dove-Elbe'. This exemplary catchment comprises a tide gate as well as several sluices, weirs and low lying catchments drained by pumping stations. The drainage through the 340 tide gate depends on low tide conditions. At high tide, the gate is closed causing backwater effects in the streams.

Description of the backwater affected lowland catchment 'Dove-Elbe'
The mesoscale catchment area 'Vier-und Marschlande" has a size of 175 km² and is located in the South-East of Hamburg, Germany (see Fig. 8). The downstream river segment Dove-Elbe is a stream of 18 km in length and is a tributary of the tidal influenced Elbe River. Further tributary streams which drain into this main river segment are the Gose Elbe, Schleusengraben, 345 Brookwetterung and a downstream segment of the Bille. These streams are part of the analysed mesoscale catchment. The soil is mainly peat and clay with a varying spatial distribution and thickness. Another regional scale catchment (namely of the river Bille) with a size of about 337 km² drains into the study area 'Vier-und Marschlande'. Thus, an overall catchment area of about 512 km² is drained through the tide gate 'Tatenberger Deichsiel'. The downstream situated water level in front of the tide gate is affected by a mean tidal range of about 3.7 m (Nehlsen, 2017) The locations of gauging stations and control structures are indicated in Fig. 8.
The results at the downstream gauging station ("Allermöher Deich") are illustrated in Fig. 9 for the opening and closing 370 function of the tide gate (in red) according to water levels at the downstream gauging station 'Schöpfstelle' in the Elbe River water level increased due to backwater effects caused by high flood discharge from upstream catchments. Here, a difference of less than 0.01 m is shown between observed and simulated peak water levels. The scatter plot, the R² and the RMSE for the flood event analysis on the 07.02.2011 to 08.02.2011 show a good result. An interactive backwater system is present for the control structures 'Reitschleuse" (blue, Fig. 9) and 'Dove-Elbe Schleuse" (green, Fig. 9) which depend on thresholds of the downstream water levels in the Dove-Elbe stream segments (black, Fig. 9). In this case, the method to model interactive control 385 systems is applied and evaluated. Additionally to the presented evaluation studies, a flood peak reduction measure is analysed in the research project 395 StucK. By excavating three retention areas with a total size of 330.000 m² from +2 m a.s.l. to +1 m a.s.l., an additional retention volume of 330.000 m³ is created when the water level exceeds the river banks at +1 m a.s.l. The location of retention areas is indicated in Fig. 8. With the additional retention volume, the peak water level can be reduced by 0.08 m. For the event 2011 the result is shown in the suppl. section 8. More results of the model application for the research project StucK are published in (Fröhle and Hellmers, 2020). 400

Summary and outlook
Numerical models are required in forecast simulations and to assess the consequences by future impacts like changes in magnitude as well as probability of stormwater events, changes in urbanisation and predicted mean sea level rise on the runoff regime in catchments. Especially in coastal lowlands, the pressure on stormwater drainage systems raises due to a combination of all three impacts. The literature review shows weaknesses in modelling water depths and backwater effects in streams and 405 lowland areas using hydrological numerical models. A method to resolve these weaknesses is presented in this article. The developed numerical method is: (1) applicable to model complex drainage systems in tidal backwater affected lowlands, (2) efficient by using short runtimes for real-time operational model application, (3) open for further model developments, 410 (4) re-useable for other hydrological model solutions and (5) parsimonious with respect to the complexity of input parameters.
The developed, implemented and evaluated method for modelling backwater effects transfers discharges into water levels using a conceptual approach. Backwater volume routing is calculated by taking into account the water level slope along streams and adjacent lowland areas. Using physical-based input parameters enables to apply the presented hydrological model for other 415 catchment studies. The input parameters comprise data of the stream profiles, gradients and roughness along the flow path.
The implementation of the method is realised in the open source rainfall-runoff model Kalypso-NA 4.0 (published on 29.01.2021). The evaluation results in the application study of the complex and tidal influenced lowland catchment 'Vier-und Marschlande' illustrate good conformance in the simulated control functions of tide gates and sluices. Water level differences are determined by comparing observed gauge station measurements with numerical model results. The differences in peak 420 water levels are in the range of 0.01 m to 0.10 m. This corresponds to a variation of 1 to 10 % in the streams with a backwater affected water level variation larger than 1 m. The RMSE ( < 0.12 m) and R² ( > 0.9) of the flood event analysis confirm the good result evaluation. In the presented application studies a standard desktop computer with i7-5600U CPU processor and 2.6 GHz is applied. The computation time is in the range of max 3 minutes even for large catchments (here 175 km²) using a time step size of 15 minutes for a 14 days simulation. With respect to a runtime of a few minutes, the model is more applicable 425 for real-time operational simulations in flood forecasting, than hydrodynamic numerical models with simulation times of at least one to two orders of magnitudes longer.
Additionally to the findings in this article, the published outcomes in (Hellmers, 2020;Fröhle and Hellmers, 2020) show the reliability of the numerical model results to be in a sufficient range of accuracy for the designated field of application to answer a wide range of hydrological and water management questions. The numerical model is suitable for operational flood 430 forecasting, real-time control, risk analyses, scenario analyses and time series gap filling in micro to regional scale catchments.
The presented method is re-useable for other hydrological numerical models which apply conceptual hydrological flood routing approaches (e.g. Muskingum-Cunge or Kalinin-Miljukov).

Outlook 435
The presented method in the model Kalypso-NA (4.0) to compute backwater affected flood routing will be adapted to model hydrological processes in local scale drainage measures (aka SUDS, GI, BMP as parts of nature based solutions). Preliminary research study results of local scale drainage measures are published in (Hellmers and Fröhle, 2017) and in (Hellmers, 2020).
The integration of Kalypso-NA in flood forecasting systems (e.g. Delft-FEWS) is in progress.

8
Acknowledgement 440 The model development and evaluation study is part of the research project StucK (Long term drainage management of tide-

Author contribution
The lead author of this article, SH, formulated the research topic. She placed the topic in the current state of research and defined the purpose of the work. The presented approaches, methods, implementations and evaluation results have been worked out by SH and were discussed with PF. The conceptualization of the paper was a joint effort from SH and PF, as were 465 the discussion and refinement of the methods presented.

Competing interests:
The authors declare that they have no conflict of interest.

Review statement
(Will be added after the reviews are available.)