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Keywords:

  • River Niger;
  • floodplains;
  • hydraulic modeling;
  • subgrid model

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method
  5. 3. Model Validation
  6. 4. Application—Niger Inland Delta
  7. 5. Conclusions
  8. Appendix A:  
  9. Acknowledgments
  10. References
  11. Supporting Information

[1] This paper presents a new computationally efficient hydraulic model for simulating the spatially distributed dynamics of water surface elevation, wave speed, and inundation extent over large data sparse domains. The numerical scheme is based on an extension of the hydraulic model LISFLOOD-FP to include a subgrid-scale representation of channelized flows, which allows river channels with any width below that of the grid resolution to be simulated. The scheme is shown to be numerically stable and scalable, before being applied to an 800 km reach of the river Niger in Mali. The Niger application focused on the performance of four different model structures: a model without channels (two-dimensional (2-D) model), a model without a floodplain (one-dimensional (1-D) model), a model of the main channels and floodplain (1-D/2-D model), and the subgrid approach developed here. Inclusion of both the channel network and the floodplain was shown to be essential, meaning that large scale models of this region, including routing models for land surface schemes, will require a floodplain component. Including subgrid-scale channels on the floodplain changed inundation dynamics over the delta significantly and increased simulation accuracy in terms of water level, wave propagation speed, and inundation extent. Furthermore, only the subgrid model showed a consistent parameterization when calibrated against either gauge or ICESat water level data, suggesting that connectivity provided by small channels is a strong control on the hydraulics of the floodplain, or, at the very least, that low resolution gridded hydraulic models require additional connectivity to represent the delta flow dynamics.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method
  5. 3. Model Validation
  6. 4. Application—Niger Inland Delta
  7. 5. Conclusions
  8. Appendix A:  
  9. Acknowledgments
  10. References
  11. Supporting Information

[2] Simulating the hydraulics of large river networks is becoming increasingly important as demands for regional scale hydrological risk assessment increase and interest in estimating river discharge from remote sensing becomes more widespread [Andreadis et al., 2007; Biancamaria et al., 2009, 2011; Durand et al., 2008; Neal et al., 2009b]. Methods for simulating river hydraulics have either concentrated on routing schemes that simulate wave propagation for global or regional hydrological and land surface models [Ngo-Duc et al., 2007; Oki and Sud, 1998; Pappenberger et al., 2010] or river and floodplain hydraulic models that simulate both wave propagation and water level [e.g., Bates and De Roo, 2000] but typically at model domain scales of <100 km.

[3] This paper presents a hydraulic model designed for large scale simulation of inundation extent, water level, and wave propagation in areas where limited or no ground based data are available. Specifically, this means it must be theoretically possible to build and evaluate the model using only remotely sensed data and that the time required to simulate flow for one year at the continental scale must be no more than a few hours. The model uses a computationally efficient finite difference numerical scheme adapted from the reach-scale inundation model ofBates et al. [2010]and utilizes gridded river network data. To run at a computationally feasible resolution at regional to continental scales (∼1 km on present-day computers) the model assumes that channel flows can be represented as a subgrid-scale process and that the same subgrid channel approach can be used to represent the flow connectivity necessary to simulate inundation dynamics on large predominantly natural floodplains at low resolution. The model utilizes only globally and freely available input data sets for topography and channel parameterization and was calibrated and evaluated on an 800 km stretch of the River Niger Inland Delta in Mali to investigate the performance of the model in terms of simulating water surface elevation, inundation, and wave propagation. The model was then used to assess the effect of the floodplain and complex network of floodplain channels on the flow dynamics of this delta. From these results, the impact of floodplain and tributary dynamics on wave propagation and inundation are discussed.Auxiliary material is also presented to validate the subgrid channel model against equivalent channel geometry at higher resolution, demonstrating the numerical stability and scalability of the subgrid approach.

1.1. Introduction to Flow Routing and Hydraulic Modeling at Large Scales

[4] Routing models are typically designed to convey runoff generated by global land surface or continental scale hydrological models to the sea, with the aim of accounting for this component of the water cycle. Most approaches apply a storage cell based routing scheme where the change in storage (V) over time (t) (dV/dt) is the sum of flows from upstream and runoff from the cell minus river flow to a downstream cell. The downstream cell is predetermined by a river network and flow direction map [Doll and Lehner, 2002; Fekete et al., 2001; Yamazaki et al., 2009], while the flow to it can be estimated using a wave velocity obtained via calibration to observations of discharge or a hydraulic equation relating river geometry to wave velocity (e.g., the Manning equation). Although the fundamental design of most routing schemes is the same, a wide range of variants on this theme with different river networks, resolutions, and process representations have been developed, including TRIP [Oki and Sud, 1998], TRIP2 [Ngo-Duc et al., 2007], HYDRA [Coe, 2000], WTM [Vorosmarty et al., 1989], and CaMa-Flood [Yamazaki et al., 2011].

[5] The simplicity of routing schemes allows ensemble and scenario based prediction of global runoff [Arnell, 1999, 2003] and Monte Carlo based approaches to parameter estimation [e.g., Pappenberger et al., 2010]. However, with simplicity comes the neglect of some important dynamic processes. Gong et al. [2009]point to the importance of convective delay when simulating wave propagation, where the inflow to a cell many kilometers in length would not be expected to instantaneously affect the outflow as in many routing model. Neglecting this effect can lead to an overprediction of wave speed or a scale-dependent wave speed when the model is calibrated. The method used to determine river network topology will also affect wave propagation and flow connectivity [Fekete et al., 2001; Oki and Sud, 1998; Vorosmarty et al., 2000]. For example, Yamazaki et al. [2009] illustrate how widely used steepest slope methods to determine flow direction using the eight neighboring cells (D8 algorithms) can lead to incorrect flow paths and accumulations, particularly when defined for low resolution grids (e.g., >10 km). A flow direction map will also be problematic in deltas, in wetlands, and during flood events, where flow networks can be dynamic due to the complex geometry and nonstationary hydraulic gradients [Alsdorf et al., 2007; Papa et al., 2008]. Flooded fraction, modeled as the inundated fraction associated with changing water surface elevation, has been included in routing models [e.g., Decharme et al., 2008] and can have a significant impact on simulated land–atmosphere water vapor flux [Dadson et al., 2010; Dawson et al., 2009; Pedinotti et al., 2012], although there can be distinct hysteresis in the wetting and drying of floodplains not captured by flooded fraction methods [Bates et al., 2006; Nicholas and Mitchell, 2003].

[6] Kinematic wave models, where wave speed is determined by channel bed slope and friction, can efficiently simulate wave propagation and water levels from topographic gradients and inflow sources. Models which use such techniques include JULES G2G [Bell et al., 2007] and the LISFLOOD model used by the European Flood Alerts System [De Roo et al., 2000; Van Der Knijff et al., 2008]. However, kinematic models are unable to account for interaction effects with tributaries and may be difficult to use in areas with low relief due to the need for a monotonically decreasing river bed elevation and the presence of strongly diffusive flow regimes in such settings. Schemes based on diffusive wave models allow dynamic wave speeds and thus backwater effects from channels and floodplains to be simulated. Such models are more expensive but can still be applied at global scales. Yamazaki et al. [2011] applied a diffusive wave model with floodplain storage areas to simulate global inundation dynamics. The scheme was more accurate on most of the 30 major rivers basins evaluated than both a kinematic wave scheme and a scheme without the floodplain storage. However, the exchange of water between channel and floodplain was assumed to be instantaneous. Conveyance between floodplain units was also not considered in this study, but has subsequently been added to the model and tested in the Amazon basin by Getirana et al. [2012].

[7] Reach-scale hydraulic models solve the shallow water equations, or in some cases a simplification of them when the flow conditions are subcritical, and require distributed values for surface friction and ground elevations. They are designed to be utilized with detailed river cross sections and accurate floodplain digital elevation models (DEMs), which are not available globally or in many large catchments. They also require no prior processing of DEM data to determine flow direction and intrinsically represent wetting and drying hysteresis. Typically the spatial domain is discretized in either one or two spatial dimensions, with computational cost mainly depending on the number of dimensions, resolution of the model, and size of the model spatial domain. When used in two dimensions the models can simulate wave propagation across floodplains provided the flow controlling features are captured by the model topography. However, the computational cost and memory footprint will be too large for global applications at the <100 m resolutions typically used. As the resolution of the model decreases, flow controlling features and flow connectivity may be lost and simulation accuracy reduced. Methods have been developed to incorporate smaller scale features into models using porosity based techniques [Guinot and Soares-Fraza, 2006; McMillan and Brasington, 2007; Sanders et al., 2008; Yu and Lane, 2006] adaptive grids [Liang and Borthwick, 2009] or 1-D models for the channel component (e.g., coupled 1-D/2-D models), although these were typically designed with urban rather than global scale applications in mind [e.g.,Schubert and Sanders, 2012]. Nevertheless, the representation of flow physics in hydraulic models can be substantially more accurate than wave speed based routing models and kinematic wave models, especially in low relief or complex river networks.

[8] There are a number of examples of hydraulic models being implemented over large river catchments in data sparse areas including the Amazon [da Paz et al., 2011; Wilson et al., 2007], Congo [Jung et al., 2010], and Ob [Biancamaria et al., 2009] river basins. These highlight that the spatial scales at which routing and hydrodynamic models are used are converging, while routing models are now starting to include components of floodplain storage. However, despite the advances in modeling capability, it is still unclear to what extent the additional process representation available via the hydraulic modeling approach is necessary to simulate level and wave propagation over large river systems. Similarly, there is no consensus on how to represent floodplains in large scale hydraulic models where fine topographic controls cannot be represented explicitly, as would be the preferred approach at the reach scale. Simulating the hydraulics of large floodplains and wetlands and their effect on flow routing motivated the design of the subgrid hydraulic model that is presented in section 2. This is followed in section 3 by a summary of two synthetic test cases that were implemented to validate the numerical scheme. In section 4 the Niger Inland Delta is introduced as a test site. Finally, the performance of the subgrid model is assessed and compared with a number of alternative model structures in order to better understand the hydraulics of this globally important wetland.

2. Method

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method
  5. 3. Model Validation
  6. 4. Application—Niger Inland Delta
  7. 5. Conclusions
  8. Appendix A:  
  9. Acknowledgments
  10. References
  11. Supporting Information

[9] For the physical process representation of floodplains the existing two-dimensional hydraulic model ofBates et al. [2010] is used as a base model. This model simulates a shallow water wave without advection using an explicit finite difference scheme and was chosen because of its computational efficiency relative to both explicit diffusive and shallow water wave models. The Bates et al. [2010]model has been evaluated for the purpose of simulating reach-scale flood inundation dynamics byNeal et al. [2011] and Neal et al. [2012] and found to give similar results to diffusive and shallow water models for subcritical flows that vary gradually in time. Section 2.1briefly describes the conventional two-dimensional model LISFLOOD-FP which is used as the base model before the revisions to the model to include subgrid-scale channels are presented.

2.1. Two-Dimensional Base Model

[10] The two-dimensional base model LISFLOOD-FP uses an explicit forward difference scheme on a staggered grid as illustrated byFigure 1a. The model comprises two equations that handle the continuity of mass in each cell and continuity of momentum between cells. Although arranged in two dimensions, the momentum equation implemented across each face of each grid cell is in fact a one-dimensional calculation such that the fluxes through each cell face are decoupled from each other. For each cell the continuity equation over a time step inline image is

  • display math

where Q is the flow between cells, h is the water depth at the center of each cell, A is the water surface/cell area, and subscripts i and j are cell spatial indices in x and y directions, respectively. Because flows in the x and y directions are calculated using the same approach, the remainder of the method description will present only the calculation of flow in the x direction, dropping references to x and y directions from the notation. The momentum equation to calculate flow Q between two cells is described by the following equation:

  • display math

where inline image is the cell width, g is the acceleration due to gravity, qt is the is flow from the previous time step Qt divided by cell width inline image, S is the water surface slope between cells, n is Manning's roughness coefficient, and hflow is the depth between cells through which water can flow, defined from the water depths and cell elevations z as

  • display math
image

Figure 1. Conceptual diagram of (a) LISFLOOD-FP base model, (b) subgrid channels model, and (c) subgrid section.

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[11] The water surface slope S is therefore

  • display math

[12] To maintain stability the model time step must be limited to

  • display math

where inline image is a stability coefficient that ranges from 0.2 to 0.7 for most floodplains (0.7 was used for all simulations presented here) and inline imageis the maximum water depth in the model domain. This time-stepping equation is based on the Courant–Friedrichs–Lewy condition [Courant et al., 1928].

2.2. Subgrid Channel Two-Dimensional Model

[13] To extend the base model for large area application two key changes are made:

[14] 1. A subgrid-scale procedure for representing channel networks is introduced to allow any size of river channel below that of the grid resolution to be represented.

[15] 2. A means of estimating the unknown channel depth from observable variables such as channel width and bank elevation is incorporated into the model using hydraulic geometry theory.

[16] Introducing a subgrid channel is relatively simple because each cell face can be treated as a one-dimensional model and therefore implemented as a series of separate flow calculations with different widths and bed elevations. Two variables are added to the base model (seeFigure 1b) to represent the channel bed elevation inline image (the estimation of which from bank elevation inline image will be described later) and channel width w. Friction parameters can also be specified separately for channels and floodplain if desired. In most cells there will be no channel and the model remains identical to the base model. However, if a subgrid-scale channel flows between two cells, in either direction, the following scheme is implemented starting with the calculation of the flux in the channel, where the water surface slope is found by

  • display math

where inline image is the length of the channel, which in this case is set as inline image. The depth of flow inline image in the channel is then calculated in the same way as the base model but using the channel bed elevations

  • display math

[17] The channel slope inline image and depth of flow inline image are then used to calculate the channel flow inline image using a reformulation of the dynamic wave equation from the base flow model that uses channel hydraulic radius and area instead of flow depth and cell width inline image. In previous versions of LISFLOOD-FP, channels were assumed to be rectangular and the hydraulic radius was approximated by the flow depth [Bates and De Roo, 2000; Trigg et al., 2009]. However, since the subgrid model could be required to simulate relatively small, narrow, and deep channels the model formulation was derived for the more general case where the hydraulic radius of the channel is defined as the area of flow between cells inline image divided by the wetter perimeter as defined by the channel width inline image and water depth inline image. This formulation presents a potential stability problem where there is flow from a large to small channel because a large discharge into a small channel would cause water surface elevation instability. Since channel geometry is defined for cells rather than cell edges, the flow area and hydraulic radius from the cell with the smallest width is used, while a rectangular single thread channel with a length of inline image was assumed here for simplicity. Thus, the width of flow at each edge is constant and can be calculated by a preprocessor to save computation time such that

  • display math

[18] Width of flow can then be used to calculate the channel flow area based on the depth of flow

  • display math

with channel hydraulic radius defined by

  • display math

[19] Deriving the momentum equation for a channel with any hydraulic radius (see Appendix A for derivation), using the same approach as Bates et al. [2010] to stabilize the scheme, yields the flow equation:

  • display math

[20] In this case, the channel is assumed to be rectangular; however, any geometry could be used to calculate hydraulic radius and area for this equation. Furthermore, we assume that the volume of channel storage in a cell is the width multiplied by the channel depth and channel length without additional storage for tributaries that may enter a cell. To calculate floodplain flow inline image the water depth inline image is reduced to account for the depth of the channel which does not flow on the floodplain

  • display math

[21] If inline image or inline image is greater than zero the scheme proceeds to calculate a floodplain flow inline image in addition to the channel flow already calculated. To do this the floodplain water surface slope inline image and depth of floodplain flow inline image are recalculated using floodplain water depths inline image and elevations inline image. Floodplain flow inline image is then calculated using the dynamic wave equation (2) by treating the flow width as the cell size as Δx minus the channel width inline image:

  • display math

[22] Floodplain and channel flows are then combined to obtain the cell face discharge that is needed by the continuity equation

  • display math

[23] The continuity equation is similar to that of the base model but must account for the different surface areas of the channel and floodplain. Therefore, the water surface area Ai,j in the continuity equation is either equal to inline image if the water is above bankfull or inline image if water is in channel. Since there is likely to be a significant change in water surface area as the river transitions between in and out of bank flow a correction is necessary when this happens in order to conserve mass. If inline image is the depth before the update and inline image is the depth after the update then the correction to inline image when the depth crosses from below to above bankfull is

  • display math

and therefore

  • display math

when depth returns within bank. The model time-stepping criterion is identical to that of the base model and the scheme was parallelized for multiprocessor computation using the OpenMP based approach outlined byNeal et al. [2009a].

[24] For the Niger test case in section 4 a method of simulating evaporation from open water was implemented for all cells where water was present. This was done by subtracting an evaporation rate inline image in m s−1 multiplied by the model time step inline image from the simulated depth after the implementation of the continuity equations (1), (15), and (16). Hence,

  • display math

[25] This reduces the volume of water in the domain at the next time step

2.3. Calculation of Channel Bed Elevation

[26] For the model to operate in data sparse areas where channel bed data may not be available an approximation of the bed elevation is necessary. Whatever approximation is used here may well introduce greater errors to the model structure than using ground survey data of typical accuracy, but such information is not available everywhere. Data sets that can, at least in theory at certain scales, be observed remotely are the channel width and height of the channel banks. Therefore the channel bed is defined as a depth from the top of the channel bank using a power law relationship between channel width and depth, which can be approximated from empirical data then refined through model calibration or data assimilation. Therefore, the channel bed at a cell is found by a preprocessor:

  • display math

where r and p are coefficients that need to be estimated. As stated earlier the intention is to estimate these coefficients by calibrating to observations; however, prior estimates can be obtained from classic downstream hydraulic geometry descriptions of channel form [Leopold and Maddock, 1953]. Hydraulic geometry relationships are a series of power laws that relate river width w, depth d, and velocity v to a measure of discharge such as bankfull discharge inline image.

  • display math
  • display math
  • display math

where inline image and inline image. To find r and p we ignore the velocity equation and, rearranging the equations for channel width and depth to equal inline image, we obtain:

  • display math

[27] By combining the equations for w and d the power law relationship between depth and observable width can then be found:

  • display math

where the terms in parentheses to the left of w become r and the fractional exponent to the right of w becomes p. For example, Table 1lists the average hydraulic geometry relationships of British gravel-bed rivers fromHey and Thorne [1986] along with the derived coefficient and exponent for the subgrid channel model. For any number of cells where channel bed elevation has been observed, the estimates from the preprocessor can simply be replaced with the observations.

Table 1. Example Empirical Values of Hydraulic Geometry Coefficients and Exponents for Gravel-Bed Rivers in Britain FromHey and Thorne [1986]
CoefficientsExponents
a, 3.67b, 0.45
c, 0.33f, 0.35
r, 0.12p, 0.78

3. Model Validation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method
  5. 3. Model Validation
  6. 4. Application—Niger Inland Delta
  7. 5. Conclusions
  8. Appendix A:  
  9. Acknowledgments
  10. References
  11. Supporting Information

[28] The numerical scheme for the base model was evaluated previously by Bates et al. [2010] using simple analytical tests that required the model to simulate wave propagation, while Neal et al. [2012]compared the scheme to industry hydraulic models using 2-D model benchmarking test cases developed by the Environment Agency of England and Wales [Néelz and Pender, 2010]. In both cases the model was found to adequately simulate flow dynamics for subcritical flows that vary gradually in space and time. As the subgrid channel component of the model is new, two design experiments were conducted to confirm whether the model behaved as expected and in a satisfactory manner, given equivalent models at higher resolution. These experiments are included in the supporting online material (DS01) and compare a high resolution model of a straight one cell wide river channel, which has a wide floodplain that is flat orthogonal to the channel, with a 1:10 resolution model with and without subgrid channels. The following dynamics were simulated:

[29] 1. Constant within bank inflow over an initially dry channel.

[30] 2. Unsteady flow, initially within bank, with transitions from within- to out-of-bank flow.

[31] The tests demonstrated that the channel model was stable, had insignificant mass errors, and that without the subgrid channel the low resolution model could not simulate the dynamics of the higher resolution model benchmark solution. In section 4 the model will be subjected to a more complex real world test case.

4. Application—Niger Inland Delta

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method
  5. 3. Model Validation
  6. 4. Application—Niger Inland Delta
  7. 5. Conclusions
  8. Appendix A:  
  9. Acknowledgments
  10. References
  11. Supporting Information

[32] To evaluate the model formulation, an 800 km stretch of the river Niger in Mali, including the extensive seasonal wetlands of the Niger Inland Delta, was modeled. The river has a low gradient, descending ∼20 m over the test site, and bifurcates into an inland delta below the confluence with the Bani, before returning to a single braided channel near Timbuktu (see Figure 2). As the Niger Inland Delta is in a semiarid region the majority of the discharge in the river derives from upstream of the study site, while around 30%–50% of the water entering the delta can evaporate before Timbuktu depending on the inundation extent [Dadson et al., 2010; Zwarts et al., 2005]. In contrast, infiltration is believed to be negligible as borehole surveys have found much of the delta to have a high water table throughout the year and to be underlain by an impermeable clay layer [UNICEF, 2010]. Although a groundwater aquifer is thought to maintain flow during the dry season [Pedinotti et al., 2012] this can be <1% of the annual peak flow. The surface of the floodplain is covered with a network of small channels. The effect these have on floodplain hydraulics will be investigated with the subgrid model. For a comprehensive overview of the delta's physical characteristics see Zwarts et al. [2005]. There were essentially two aims of this test.

image

Figure 2. DEM of River Niger test site using Shuttle Radar Topography Mission (SRTM) data.

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[33] 1. To assess how accurately the model could simultaneously simulate wave propagation, water surface elevation dynamics, and inundation extent over the inland delta, while characterizing the simulation sensitivity to the hydraulic geometry characteristics and friction coefficient. To do this we simply run an ensemble of model simulations with channel geometry and friction parameters sampled from some predefined distributions and compare the model output with water surface elevations measured using the ICESat laser altimeter.

[34] 2. To assess the role of floodplain and channel hydraulics on the attenuation of the seasonal flood wave across the delta. For this aim the four model structures listed in Table 2 were implemented, each with a different component of the system omitted.

Table 2. List of Model Structures
Model StructureComponents of Models
(A) Subgrid ModelThe full model including subgrid representation of the main rivers and floodplain channels.
(B) Main Channels Only (1D/2D Model)Subgrid channels for the rivers Niger and Bani tributary only, floodplain flow simulated using 2D floodplain model. This is an analog of a coupled 1D/2D model of river and floodplain where only the main channel is simulated by a 1D river network model and the remainder of the domain (including any floodplain channels) is handled in 2D.
(C) No Floodplain (1D Model)The subgrid channels without floodplain storage. This is an analog of a simple 1D model.
(D) No Channels (2D Model)A 2D floodplain model without the subgrid channels (e.g., flow over a DEM). This is an analog to a full 2D model, although at coarse resolutions channels will be poorly represented.

[35] A further test, especially of the analog to a 1-D/2-D model (model B), would have been to compare results with the existing 1-D/2-D model in LISFLOOD-FP [Trigg et al., 2009], however this was not possible because the Trigg et al. 1-D model is incapable of simulating wetting and drying (e.g., ephemeral rivers) or bifurcations.Section 4.1 describes the setup of the hydraulic models from source data.

4.1. Model Setup

[36] The model requires five inputs: floodplain topography, river channel widths, banks heights, model parameters, and hydrology. Channels were assumed to be the cell width Δx in length, meaning the map of river widths also represents the river centerline. The process of obtaining each of these data sets is described below.

4.1.1. Floodplain Topography

[37] Digital elevation models of floodplain topography with submeter vertical accuracy are increasingly available for many regions of the world. However, they may be difficult to obtain and are not always freely available; therefore, for large scale applications the DEM from the Shuttle Radar Topography Mission (SRTM) remains the most complete. Conveniently, the SRTM data were obtained in February at the start of the Niger Delta dry season, meaning that lake levels were below average and the floodplain was not inundated at the time. The limited vegetation cover in this semiarid region also avoids the problem of tree canopies increasing the floodplain elevation, as seen in hydraulic model applications to tropical rivers [e.g., Wilson et al., 2007; Jung et al., 2010]. The SRTM DEM has a ground resolution of approximately 90 m over the test site, while elevation errors were found to have a standard deviation σ of 4.68 m over Africa by Rodriguez et al. [2006]. The magnitude of these errors, although typically lower over flat terrain, can be a problem for inundation models where floodplain depths are typically in the order of a few meters as demonstrated by Sanders [2007]. However, computational constraints dictate that the model will need to run at a resolution coarser than 90 m anyway, while the near random spatial structure of the SRTM errors between 90 m and 220 km [Rodriguez et al., 2006] means that the SRTM error can be reduced by averaging. Assuming errors are normally distributed and the terrain is flat, the sampling error in the cell elevation will decrease proportionally to inline image, where N is the number of samples. Therefore, the standard deviation of errors reduces to 1.48 m when averaged over 10 cells, 0.468 m over 100 cells, and 0.148 m over 1000 cells. Given these error assumptions, to have a 95% chance of representing floodplain connectivity between cells over flat terrain for a flow depth of 1 m, the model cell elevation will need to be an average of ∼100 SRTM cell elevations. This resulted in a DEM of 905 m resolution where the cell elevation is an average of 100 SRTM DEM elevations of ∼90 m. The computational cost of the hydraulic model increases by an order of magnitude every time the resolution is halved, due to the combined effect of more cells and shorter time steps. There are also additional I/O and storage requirements at high resolution. Therefore, the 905 m resolution model is expected to be 3–4 orders of magnitude faster than a model built at the SRTM native resolution as well as being significantly less noisy. The digital elevation model is displayed in Figure 2.

4.1.2. Channel Widths

[38] Channel widths were estimated from Landsat Enhanced Thematic Mapper (EPM) imagery with a pixel resolution of 30 m. During periods of high discharge it can be difficult to identify the channels due to overbank inundation, while to the south of the inland delta the Niger develops a meandering thalweg and narrow width during the dry season. Therefore only Landsat ETM images where the Niger was within bank and did not show a meandering thalweg were used for width extraction. The images used are listed in Table 3, which shows most were acquired in July and August at the end of the dry season, although image L71198050 at the southwest end of the model domain was acquired during high water in an area not prone to seasonal flooding. Note that it was not possible to obtain cloud-free images for the whole domain from the same year in these months, and images acquired by Landsat TM5 from 2006 to 2007 were deliberately withheld to evaluate inundation extent simulation (seesection 4.3.2).

Table 3. Landsat Images Used to Derive the Wet Mask
Image NameDate
L7119805016 Oct 2002
L7119705017 Aug 2000
L7119704917 Aug 2000
L7119604828 Jul 2001

[39] From the Landsat imagery, a binary water mask was created using the normalized difference water index (NDWI) where any negative value was classified as water [McFeeters, 1996]. NDWI for Landsat data is defined by

  • display math

where Band4 is the radiance in near-infrared wavelengths (0.76–0.90 μm) and Band2 is radiance at green wavelengths (0.52–0.60 μm). Modifyingequation (24)to use the middle-infrared band instead of the near-infrared band can sometimes yield better results [Xu, 2006]; however, this was found to have very little effect on the water mask at this site. No ground data were available to assess the accuracy of the water mask. Once the water mask was obtained, the RiverWidth program [Pavelsky and Smith, 2008] was used to extract river width and centerline estimates at 30 m intervals for the Niger and Bani rivers (main channels) and for all other channels where width was consistently above 120 m (e.g., 4 Landsat TM pixels). The centerline for channels below 120 m were drawn manually because occasional breaks in connectivity along the channels, due to classification errors and the resolvable width in Landsat imagery, made implementing the automated approach difficult. Manually drawn channels were assigned widths of 30, 60, or 90 m. To upscale the channel widths to 905 m resolution, river widths from the main channel, from channels above 120 m wide, and from other channels (e.g., those drawn manually) within a 905 m DEM cell were averaged separately; only the maximum width from these three classes was used by the model (e.g., we retain the larger channels in the model but do not add the smaller channel widths to the main channel width due to the nonlinear nature of the width–depth relationship).

[40] Mean river width for the Niger was 742 m, while the 5th and 95th percentiles of the width distribution were 523 and 942 m, respectively. Local variations in river channel depth and width may actually show an inverse correlation as the river adjusts to maintain a consistent conveyance capacity. Therefore, the averaging of river widths over short reaches, 905 m in this case, is essential such that the hydraulic geometry estimates of depth also reach average values. Averaging also overcomes the problem of the 1 pixel (30 m) precision on the width values at any single location. The final width map is plotted in Figure 3 along with a histogram of observed widths from the River Niger before and after upscaling to 905 m resolution. Note that the narrowest main river channels are in the center of the domain as the river bifurcates over the delta, while most of the widths above 1 km are due to connected lakes.

image

Figure 3. Map of river widths, ICESat observation locations, gauging stations, and locations used for level time series plots in Figure 5.

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4.1.3. Bank Heights

[41] A fundamental difficulty with modeling river hydraulics from remote sensing is the lack of bathymetric data for the river bed, which must therefore be estimated from something that is observable. Here the elevation of the floodplain adjacent to the channel was used as a proxy for the height of the banks, from which the elevation of the channel bed was calculated using hydraulic geometry theory. The water mask, originally created to calculate the channel widths, was used to find the SRTM elevations adjacent to the outer edge of the water mask. These data will be subject to instrument noise and must therefore be smoothed to reduce the errors in the bank heights used by the model. Averaging the bank heights for a straight channel that is orthogonal to a 905 m cell would provide a sample of 20 elevations, which would reduce the standard deviation of the mean errors inline image to 1.04 m. Assuming this is too large we calculate the number of samples needed such that 95% of errors are within 0.5 m (note that in this case the 0.5 m threshold was chosen arbitrarily and a later paper will look at optimizing this value). The number of samples required works out to be 350, or 15.75 km of river. Therefore, to ensure at least 350 samples are used, the bank elevation in a 905 m cell was assumed to be the average of all bank elevations within 7.9 km of the cell. The banks will obviously not be flat over this distance; however the average slope of the Niger at this site is typically <2 × 10−5 m m−1, which means the elevation is expected to change by <1.6 × 10−4 m over the search radius. As the bank height errors are orders of magnitude greater than the elevation change over the smoothing window, the analysis of slope actually suggests that a larger smoothing window could be used to further reduce the bank elevation errors. However, it is not clear at what point the smoothing would start to remove hydraulically important topography or that this is necessary, given that the hydraulic model will simulate smooth transitions in slope and level due to the significance of diffusion in low gradient rivers anyway. Note that bank heights for the main rivers, rivers over 100 m wide, and rivers below 100 m were processed separately to avoid possibly steeper tributaries with smaller floodplains influencing bank heights on larger rivers. Segmenting the processing by stream order would achieve a similar outcome.

4.1.4. Model Parameters

[42] The model has three parameters which to some extent can all be estimated (not necessarily very accurately) prior to implementing the model; they are the friction, and the exponent, and coefficient of the hydraulic geometry. The friction parameter for hydraulic models has received much attention in the literature [Mason et al., 2003; Werner et al., 2005] and it is widely acknowledged that friction varies in space and that simulation accuracy can be increased via calibrating and the use of effective friction coefficients [Aronica et al., 1998; Pappenberger et al., 2005]. In this case, two additional parameters have been included that control the capacity of the river channel that in more data-rich areas would be provided by field survey. The two channel parameters are interconnected in that both affect the area and hydraulic radius of the channel cross section, so a degree of equifinality between them is expected. Models were run using parameter values defined by the ranges and sampling intervals presented inTable 4, the results of which are discussed at the start of section 4.3. The range of friction values was set based on user experience of the values typically found when calibrating previous versions of the LISFLOOD-FP model [Bates and De Roo, 2000; Bates et al., 2010; Hunter et al., 2005; Neal et al., 2009c], while the channel parameters were restricted to prevent the main channel depth being physically unrealistic (e.g., no more than a few decameters deep and no less than a meter).

Table 4. Ranges of Model Parameters and Calibration Sampling Intervals
 MinimumMaximumSampling Interval
Coefficient r0.0250.1750.025
Exponent p0.690.820.01
Channel Friction n0.0250.050.005
4.1.5. Hydrology

[43] Inflows to the model domain were represented using daily gauge data from Kirango Aval [Niger near Markala: Global Runoff Data Centre (GRDC) station 1134250] and Beneny Kegny (Bani: GRDC station 1134450). The Niger gauge is at the edge of the model domain, while the Bani gauge is 13 km upstream of the model domain, thus no adjustments were made to the gauged flow timings. Local hydrological inputs due to rainfall, typically between 600 and 200 mm per year in this region, were ignored in the model setup because of their relatively small impact on discharge and because of the focus of this study on the river hydraulics. However, observations of evaporation at Mopti (see Figure 3) provided by UNESCO-IHE were included due to the high evaporation rate [Dadson et al., 2010].

[44] The approach taken here remains reliant on ground data. However, the uncertainty introduced into the modeling by using discharges from a hydrological model for the upstream boundary conditions would have made it more difficult to assess the different hydraulic model structures tested in this paper. Future work will need to examine the effect of simulated boundary conditions on model accuracy, but this is beyond the scope of this paper.

4.2. Data for Model Calibration and Evaluation

[45] Three data sets were used to assess the hydraulic models.

[46] 1. Water surface elevations from the ice, cloud, and land elevation satellite (ICESat) laser altimeter (2003–2009)

[47] 2. Discharge observations from the Diré gauge station (Figure 3: GRDC station 1134700) (2002–2009).

[48] 3. Observations of inundation extent from 24 Landsat images (2006–2007).

[49] ICESat data were obtained from the GLA14 Land Elevation Product, Release 31. A total of 127 observations of water surface elevation at the 18 locations marked on Figure 3 were used for model evaluation. Each observation can be a composite of multiple ICESat observations of the river water surface at a particular location and time, the number of which depended on the river width and track orientation relative to the water surface. The study by Hall et al. [2012] recovered river water surface elevations with a mean absolute error of 0.19 m using ICESat, an error that is expected to be well below that of the hydraulic model, given the terrain data errors discussed previously. The satellite footprint of ∼70 m means that relatively few interactions with topography surrounding the river were expected. ICESat overpass times were not uniformly spread across the year. The satellite operated during the months of September or October every year from 2003 to 2009 which coincided with high flow periods. The intermediate to low flow period of February to March was also sampled each year from 2003 to 2008. Low flows during May were sampled only in 2004–2006.

4.3. Results of Model Simulations

[50] A total of 588 simulations were run for each of the four model structures using the parameter ranges defined in section 4.1.4. The models had 256,878 cells and covered an area of 210,389 km2, while the subgrid model included over 9500 km of river network. The optimal subgrid model took 106 min on a quad core 2.8 GHz Intel E5462 processor. Model simulations were run from 1 January 2002 to 31 December 2009. This allowed a one year warm-up period, which is more than adequate to provide initial conditions for the next wet season, before simulation results were compared to the 2003–2009 ICESat water surface elevation and Diré gauge discharge record. A short animated video clip of the subgrid model inundation simulation is available with the supporting online material (MS01).Table 5 shows a cross comparison of the performance of the models in terms of root mean square error (RMSE) to ICESat water surface elevations and Nash–Sutcliffe efficiency (NS) [Nash and Sutcliffe, 1970], given observed discharge at the Diré gauge. The channel parameters found to be optimal given either the RMSE or NS efficiency are also shown, along with the mean water surface elevation error (ME) and channel depth (h) for the average width of the Niger. The subgrid model (model A) was the most accurate in terms of water surface elevation. Furthermore, when calibrated using water surface elevation, the model simulated discharge with only a 0.002% loss in efficiency compared to calibrating with gauged discharge. This meant that the model could be calibrated using only the remotely sensed information and still simulate downstream discharge. Optimal channel parameters were also broadly consistent when calibrated with either performance measure. For example, given the average channel width of the Niger (742 m) the channel would be 11.9 m deep for the RMSE calibration and 10.9 m deep for the NS calibration.

Table 5. Performance Statistics and Optimal Parameters for the Four Model Structures
Model StructureCalibration MetricaModel PerformanceChannel ParametersWidth h (m) at Depth of 724 m
RMSENSMErpn (m1/3 s−1)
  • a

    ME is the mean level error at ICESat overpass locations in meters. RMSE is the root mean squared level error at ICESat overpass locations in meters. NS is Nash–Sutcliffe efficiency of gauged discharge at Diré, where 1 is perfect efficiency.

(A) Subgrid ModelRMSE1.210.9110.240.1250.690.04511.9
NS1.260.9130.550.10.710.04010.9
(B) Only Main ChannelsRMSE1.550.760.650.0250.780.0404.33
NS1.720.930.870.0250.740.0453.32
(C) No Floodplain (1-D Model)RMSE1.65−0.83−0.030.1250.770.03020.3
NS9.450.538.490.050.770.0258.1
(D) No Channels (2-D Model)RMSE3.34−1.322.560.050
NS3.51−1.132.790.025

[51] The model without the floodplain channels (model B) was less accurate in terms of RMSE, although more importantly the optimal channel geometry for water surface elevation simulation resulted in NS efficiency 18% lower than obtained when calibrating to the gauge data. Model B obtained the highest NS efficiency when calibrated to the gauged discharge. However, in obtaining this efficiency the level RMSE increased by 0.17 m (11%) relative to the RMSE calibrated parameter set and by 0.46 m (37%) compared to the NS calibrated full subgrid model. Given the optimal parameterizations for the two performance measures, a 742 m wide channel would have a depth of 4.33 m when calibrated using water level RMSE and 3.32 m when calibrated using the efficiency of gauged discharge simulation. It appears somewhat counterintuitive that a model with fewer channels for flow conveyance than the full subgrid model would need a shallower main channel. However, this can be explained by the channel's interaction with the floodplain. Without the connectivity to the floodplain provided by the subgrid channels, a higher channel water surface elevation is needed to inundate the floodplain, which in turn is necessary to obtain a reasonable simulation of wave dynamics. This explains why ME increased relative to the full subgrid model when the floodplain channels were removed.

[52] The model without the floodplain (model C) was unable to simulate wave propagation when calibrated with levels, the negative NS efficiency after calibration using RMSE indicating that this approach is worse than assuming a constant mean annual discharge. To obtain the optimal NS efficiency with this model it was necessary to substantially overpredict water surface elevations, which allowed the model to increase the number of small subgrid channels inundated and obtain additional friction effect from the channel banks. The model without a channel network (model D) was the least accurate model for water surface elevation simulation with RMSE always over 3 m. Furthermore, the errors in simulated flow paths are so significant (see later analysis of inundation extent in section 4.3.2) that the model could not simulate wave propagation with a positive NS efficiency given realistic friction coefficients between 0.025 and 0.050. In fact, for this model the optimal friction parameterizations for simulating water surface elevation and wave propagation have diverged to opposite ends of the specified range, suggesting this model would not improve even with physically meaningless friction coefficients.

4.3.1. Simulation of Level and Wave Propagation

[53] This section focuses on results from the four model structures after calibration using the water surface elevation data (RMSE). These models were selected for further analysis because it is desirable to have hydraulic models that can be calibrated using water levels as they are remotely observable. Hence, these are the most relevant results for applications such as regional flood risk analysis and discharge estimation [Andreadis et al., 2007]. Simulations and observations of discharge at the Diré gauge are plotted in Figure 4, the total inflow to the model domain was also plotted to indicate the wave travel time and attenuation. Results in Table 5 demonstrated that the full subgrid model (model A) was the most accurate simulator of gauged discharge based on the calibration to RMSE at this location. However, the discharge time series also shows that the subgrid model was the most accurate simulator of peak discharge, wave speed, and attenuation. The root mean square errors between the simulated and gauged discharge at Diré were also smallest for the subgrid model, with values of 220 (model A), 357 (model B), 999 (model C), and 1077 m3 s−1 (model D). For the subgrid model the errors in low flow simulations were the greatest; however, this was not surprising given the lack of local hydrological inputs to the model. For example, the wet season in Mali runs from June to October, meaning the underprediction of discharge early on the rising limb of each flood season is likely to be due to the absence of local runoff inputs to the model. Furthermore, the river width decreases substantially at low flow and exhibits a meandering thalweg, which will not be represented by the rectangular geometry assumed here. Pedinotti et al. [2012] found that including a linear aquifer model in the TRIP routing scheme (with floodplain storage cells) could improve low flow simulation and flood wave recession in this region. The relatively poor low flow simulation by the subgrid model supports that conclusion; however, the flood recession is among the most accurate aspects of the subgrid model simulation, suggesting the more detailed floodplain and inflow representation in the subgrid model could be important for this aspect of model response. Without the floodplain (model C), the wave speed was too fast, resulting in early arrivals of the flood peak for the no floodplain model. Although wave speed could be reduced by increasing the friction, this further decreased the accuracy of the water surface elevation simulation.

image

Figure 4. Observed discharge at Diré with simulated discharge from the four model structures and model upstream inflows.

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[54] Water surface elevation simulations by the four model structures are presented in Figure 5 at three ICESat overpass locations where there are multiple water surface elevation observations over time. These plots indicate that water surface elevations vary by >5 m throughout the year. The spatial locations of these time series are indicated in Figure 6, along with the mean error in simulated water surface elevations from the subgrid model with the smallest RMSE (see Table 5). At location A, an ICESat overpass conveniently coincided with the expected flood peak each year, both the subgrid and main channels-only model (models A and B) underpredicted the water level by between 2.3 and 2.4 m at this location, while both of these models and the no floodplain model (model C) underpredicted water surface elevation at this site at all times. The most accurate simulator of water surface elevation at this location was the no-channels model (model D). Therefore, it would appear that the channel parameters found to be optimal over the whole domain are simulating a channel bed which is too low for this particular area of the domain. At locations B and C the subgrid model (model A) was the most accurate simulator of water surface elevation with a maximum error of 2.4 m and mean errors of 0.58 (B) and 0.62 m (C). The no-channels model (model D) made progressively worse predictions of water surface elevation with increasing distance downstream and a number of the elevations for this model are in fact dry DEM elevations (this was not the case for the other models with channels). With the exception of model D (no channels), simulations were most accurate around mean annual flow. River widths were observed at flows close to annual mean, which may contribute to the elevated accuracy.

image

Figure 5. Water surface elevation simulations from the four model structures and observations at the three ICESat overpass locations marked on Figure 3. Each model was calibrated using the RMSE to all ICEsat elevations over the domain.

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image

Figure 6. Spatial distribution of water surface elevation errors.

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[55] Examining the mean water surface elevation errors for the subgrid model (Figure 6) shows that the model systematically underpredicted levels south of Lake Debo and systematically overestimated levels to the north of the lake, with the exception of one low flow observation just upstream of the Bani-Niger confluence at Mopti. Assuming the gauged discharges are accurate, this indicates that the model should benefit from separate channel parameterizations above and below the delta. This was not unexpected given that the geomorphology of the river changes from a fluvial fan south of the lake to a branched network confined by sand dunes to the north. In fact, calibrating the model to observations only above or only below the lake reduces RMSE by almost half, to 0.7 m. Future work will look at optimizing the channel parameters for separate geomorphic units of the river; however, this paper will continue to focus on the model structure and retain a simple global parameterization.

4.3.2. Simulation of Inundation Extent

[56] This final section of results looks briefly at the simulations of inundation extent. In total a series of 24 Landsat TM5 images, acquired on 12 occasions between November 2006 and March 2007, were analyzed (e.g., from the wet to dry season). Image pixels were classified as “open water” if NDWI was greater than zero, “recently or currently inundated” if the normalized difference vegetation index (NDVI) was greater than 0.5, or “dry” in all other cases. Here only two example times are presented for brevity and these are shown in Figure 7 as false color images (TM5 bands 2, 3, 4). The image on 7 November 2006 is indicative of a flood extent during high water, while the 27 February 2007 image is indicative of low flow inundation conditions and is also the period of time when the topography was measured by the Shuttle Radar Topography Mission. Water depths simulated by the three models with floodplain components at these times are plotted in Figure 8, along with the classified images of inundation. Quantitatively comparing these data proved rather difficult because of the differing resolutions of the images data (30 m) and model outputs (905 m), along with the uncertainty over how to handle the “wet or recently wet category.” For example, using the spatial performance measure proposed by Aronica et al. [2002]could result in model performances between 0.4 and 0.7 (where 1 is perfect agreement and zero is no skill) for the subgrid model for the 7 November 2006, depending on how the observations were resampled to 905 m and on the treatment of the wet or recently wet class. Nevertheless, a number of qualitative conclusions can be made from the data. First, the no channels model (model D) inundates a section of the domain toward the upstream end of the Niger that should not be inundated. Second, both the no channels and main channels only models (model D&B) fail to dewater the inland delta during the dry season leading to an overestimation of inundation at this time. Third, only the subgrid model (model A) inundates the south western segment of the inland delta and the wetlands region between the Bani and Niger. Finally, and perhaps of most significance to modeling at other sites, is that striping with a 30–100 km wavelength (running north west to south east) is evident in the DEM topography (note that the area to the north of Lake Debo is a dune field rather than SRTM striping and is obvious as a set of linear features aligned east-west inFigure 2). These stripes, with amplitudes of a few meters, had a greater impact on the model's ability to simulate inundation extent than the 4.68 m vertical noise error for each SRTM cell, which was largely removed by averaging to 905 m. For example, the under prediction of open water extent on the western edge of the delta by the subgrid model was largely due to these stripes because the subgrid channels transport water to this region along the western edge of the delta from the Niger above its confluence with the Bani. Actually, the flow paths along which the subgrid model and main channels only model (models A&B) inundate the delta are quite different. The main channels only model (model B) inundated the delta via backwatering from Lake Debo and nearby main channels, while the subgrid model (model A) also transported water from upstream of the Bani confluence onto the delta. The subgrid model increased the connectivity between the main channels and the floodplain, which in this case improved the simulation of inundation. To further improve the inundation simulation it will be necessary to correct for the SRTM striping.

image

Figure 7. Landsat false color images from 7 November 2006 and 27 February 2007 (TM5 bands 2, 3, 4).

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image

Figure 8. Plots of simulated inundation extent and classified images of inundation at two times.

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5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method
  5. 3. Model Validation
  6. 4. Application—Niger Inland Delta
  7. 5. Conclusions
  8. Appendix A:  
  9. Acknowledgments
  10. References
  11. Supporting Information

[57] This paper has presented a new subgrid hydraulic model designed for large scale application in data sparse areas where detailed channel information is not available. The scheme was shown to be numerically stable and scalable, with the aid of some simple test cases, before it was applied to an 800 km reach of the river Niger in Mali. The application on the Niger focused on the performance of four model structures: a model without channels (2-D model), a model without a floodplain (1-D model), a model of the main channels and floodplain (1-D/2-D model) and the subgrid approach developed on this paper. Inclusion of both the channel network and the floodplain was shown to be essential to the extent that models without either of these components lacked any predictive skill (e.g., they were worse than assuming average conditions all the time). Thus large scale models that simulate the hydraulics of this region, including routing models for land surface schemes, will require a floodplain component to provide accurate simulations.

[58] Including subgrid channels on the floodplain changed inundation patterns over the delta and resulted in increased model accuracy in terms of water level simulation, wave propagation speed, and inundation extent. Perhaps most significantly, only the subgrid model showed a consistent parameterization when calibrated against gauge and ICESat water level data. This suggests that connectivity provided by small channels has a significant effect on the hydraulics of the floodplain, or at the very least that at ∼1 km resolution a gridded hydraulic model requires additional connectivity to represent the flow dynamics at the test site. The implication of adding floodplain connectivity to improve simulation accuracy is that small channels are a more important control on the inundation dynamics than previously thought. Recent studies suggest similar conclusions in many floodplain systems at a range of scales and climatic settings: rural temperate [Nicholas and Mitchell, 2003], urban temperate [Neal et al., 2011], large scale tropical [Trigg et al., 2012], and large scale semiarid (this paper).

[59] A major inaccuracy in the subgrid model's simulation of level was due to the use of a global channel parameterization, while at low flow the lack of local hydrology and assumption of rectangular channel geometry may become important. This paper has focused on surface water routing to avoid a detailed analysis of other hydrological processes such as infiltration or runoff. However, it is likely that the model would ultimately need to be used in concert with other models to achieve a more complete process description. Fortunately, it is relatively simple to treat sources and sinks of water from other models as lateral inflows to the continuity equation (equation (1)), in a similar manner to the treatment of evaporation (equation (17)). The main barrier to improved inundation simulation appears to be long range stripping (30–100 km) in the SRTM data used for the DEM in this model, which are a similar magnitude to the subgrid model water level RMSE. Given that the spatial scale of this effect is much greater than those of interest for flood hazard assessment the current globally available DEM, based on SRTM data, will be a significant source of error for any regional flood risk assessment unless these effects can be corrected.

Appendix A:  

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method
  5. 3. Model Validation
  6. 4. Application—Niger Inland Delta
  7. 5. Conclusions
  8. Appendix A:  
  9. Acknowledgments
  10. References
  11. Supporting Information

[60] This section describes the derivation of the subgrid channel model for simulating discharge between neighboring cells. Start with a one-dimensional shallow water equation with convective acceleration terms (inertia) omitted [Cunge et al., 1980]

  • display math

[61] Rearrange to find Q at the next time step

  • display math

[62] Make the Q2terms on the right-hand side semi-implicit to improve scheme stability [Bates et al., 2010; Liang et al., 2006]

  • display math

[63] Rearrange so Qat the next time step is on the left-hand side to give an explicit model where hydraulic radius is the channel cross-sectional area divided by the wetted perimeter

  • display math

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method
  5. 3. Model Validation
  6. 4. Application—Niger Inland Delta
  7. 5. Conclusions
  8. Appendix A:  
  9. Acknowledgments
  10. References
  11. Supporting Information

[64] Jeffrey Neal was funded by the Leverhulme Trust Early Career Fellowship scheme, while Paul Bates was supported by the Willis Research Network and received additional funding from the European Union via grant FP7-ENV-2010-265280. The authors would like to thank Brett Sanders, Tamlin Pavelsky, and an anonymous reviewer for their constructive comments and suggestions, which have significantly improved the manuscript.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method
  5. 3. Model Validation
  6. 4. Application—Niger Inland Delta
  7. 5. Conclusions
  8. Appendix A:  
  9. Acknowledgments
  10. References
  11. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Method
  5. 3. Model Validation
  6. 4. Application—Niger Inland Delta
  7. 5. Conclusions
  8. Appendix A:  
  9. Acknowledgments
  10. References
  11. Supporting Information

This auxiliary material submission contais results from two design experiments and an animation of a sub-grid model depth simulation.

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Additional file information is provided in the readme.txt.

FilenameFormatSizeDescription
wrcr13623-sup-0001-readme.txtplain text document2Kreadme.txt
wrcr13623-sup-0002-S01.epsPS document590KFig. S1. Digital elevation model for high resolution test case. The inflow point is at 0, 250 m (top of channel). Dots show control points 1-3.
wrcr13623-sup-0003-S02.epsPS document1020KFig. S2. Wave propagation through test case over time, plotting results from the high resolution model, a model at 1/5th of the resolution without the channel and a subgrid channel model.
wrcr13623-sup-0004-S03.epsPS document730KFig. S3. (a) Plot of inflows to model domain and outflows from the domain for the three models. (b) Plot of residuals between outflow discharges from the high resolution model and the low resolution model with and without subgrid channels.
wrcr13623-sup-0005-S04.epsPS document1044KFig. S4. (a, c, e) Plots of simulated levels from the three models at the three control points shown on Figure S1. (b, d, f) Plots of residuals between high resolution model and subgrid model, results from the low resolution model are not shown because of the large magnitude of the residuals relative to the sub-grid model.
wrcr13623-sup-0006-s00.mpgMPEG video10601KAnimation S1.  
wrcr13623-sup-0007-t01.txtplain text document0KTab-delimited Table 1.
wrcr13623-sup-0008-t02.txtplain text document1KTab-delimited Table 2.
wrcr13623-sup-0009-t03.txtplain text document0KTab-delimited Table 3.
wrcr13623-sup-0010-t04.txtplain text document0KTab-delimited Table 4.
wrcr13623-sup-0011-t05.txtplain text document1KTab-delimited Table 5.

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