Location identification of river bathymetric error based on the forward and reverse flow routing

River flow routing is a core issue in the flood control and water resource management. When using hydrodynamic flood routing models, the river bathymetric data is fundamental. Generally, the river bathymetric data derives from two sources (Luo et al. 2021): observations from field surveys, and integration data with postprocessing of bathymetric map and remote-sensing image as well as the digital elevation model (Liu et al. 2015). In practice, because of the high cost in time and labor and the difficulty in field surveys (Yu et al. 2014; Harada & Li 2018), the obtained bathymetric data are usually used over a long period. However, due to the measurement uncertainty and riverbed deformation, river bathymetric errors are inevitable. For example, the riverbed erosion and deposition at sharp river bends can significantly change the river morphology, and the sand mining usually deepens the water depth in a short time (Han et al. 2010). All these river morphological modifications induce large bathymetric errors to the flow routing. Recently, the above-mentioned issue has been widely studied, and a popular strategy to attenuate bathymetric errors is using data assimilation and data fusion with remote sensing images (Yoon et al. 2012; Wen et al. 2020), in which the variation of submerged topography cannot be identified. Therefore, it is of great significance to provide a method that can identify and then correct bathymetric data with large errors.

Research on the issue of river bathymetric errors at home and abroad is mainly concentrated on riverbed morphological modification (Byrnes et al. 2002; Zhang et al. 2002), the bathymetric data uncertainty assessment (Buhman et al. 2002; Harman et al. 2008), and the integration of multiple sources of topographic information (Shintani & Fonstad 2017; Luo et al. 2021), as well as the influence of bathymetric uncertainty on flood routing (McKean et al. 2014; Bures et al. 2019). Bailly et al. (2010) designed a methodology to assess the quality of LiDAR topographical data within rivers using a specific geostatistical method that conducts upscaling as well as interpolation of reference data. Cea & French (2012) estimated the influences of bathymetric error on the calibration and validation of estuarine data with a depth-averaged hydrodynamic model. Zhou et al. (2001) analyzed the abnormal water stage in some river sections of the Pearl River network, and found that river sand mining was mainly responsible for local bathymetric changes. McKean et al. (2014) compared flow model predictions using the lidar bathymetry with those made using a total station channel field survey and studied the effects of bathymetric lidar errors on flow properties. As the river topography directly determines the relationship between flow discharge and water stage, the disturbance of river topography can be reflected in the flow information. For example, Zhou et al. (2021) proposed a correction method of river bathymetry parameters based on stage–discharge rating curves as the rating curve represents the characteristics of the river section. Schaperow et al. (2019) used curve-fitting methods to predict unknown bathymetry with varying assumptions about height-width relationships. Nevertheless, the flow routing errors can only be calculated in those cross sections with observations, and there is still no feasible strategy to judge where the bathymetric error is located.

When the water flow is simulated at the cross section with large bathymetric errors, the difference between the flow information propagated from the upstream and reversely propagated from the downstream will be significant. This difference can be used to infer the bathymetric error. This study aims at proposing a method to identify the cross sections with large bathymetric errors, based on the forward and reverse flow routing. However, because the existing reverse flow routing models are mainly based on hydrological methods (Eli et al. 1974; D’ Oria & Tanda 2012), they share the same shortcomings in the feasibility of water stage simulation, as well as a deficiency in model stability and accuracy (Guan et al. 2006; Szymkiewicz 2008). To overcome the shortcomings, this study first developed stable models for both the forward and reverse flow routing, supporting the numerical simulation of water stage in different directions for the bathymetric error identification.

The location identification method of river bathymetric errors is introduced in the methodology section. After being tested and demonstrated in a hypothetical case, the proposed identification method is applied to the case of the Xunjiang River in a real-world case study. Following the two cases, the proposed method is briefly discussed. Finally, conclusions are given in the last section.

In this study, the governing equations were discretized using the finite difference method. The discretization of time t and space x formed a grid on which the dynamical model was solved (see Figure 1) using approximations of the partial derivatives. Here, the implicit four-point Preissmann scheme was adopted because it allows non-equidistant grids and computes the discharge and stage at the same point (Chau 1990), as well as for its stability and convergence characteristics (Lyn & Goodwin 1987). The labels n and represent the previous and present time steps, respectively. The labels j and represent the adjacent grid points in the x direction.

where represents the top width of cross section [m]. * denotes the last iterative value of Q and Z.

Finally, the linear Equations (7) and (8) were solved using the double-sweep method, which is an efficient method for solving a set of linear algebraic equations (Zhang 2005).

Compared with forward flow routing, the reverse routing can be carried out in three different ways, namely, only reverse in space, only reverse in time, and reverse in both space and time (Eli et al. 1974), as shown in Figure 2. The forward model predicts the future states along the flow direction, while the reverse model traces the historical trajectory from downstream to upstream. In a numerical simulation, the reverse iteration to solve the Saint-Venant equations is usually unstable because of attenuation along the flow direction (sharpness in reverse). Therefore, most previous research on reverse flow routing shares the same instability issue, particularly for the way of reverse in both space and time (Figure 2(b)). In this study, the way of reverse in time but forward in space was also not considered because the upstream information is usually unknown in reverse flow routing (Figure 2(d)).

To improve model stability, an anticlockwise rotation transformation to the iteration direction of forward model was implemented to use the way of only reverse in space (see Figure 2(c), Figure 3).

The linear Equations (11) and (12) were solved using the double-sweep method, after the boundary and initial conditions respectively obtained from the initial and boundary conditions of the forward flow routing model. The difference between the forward and reverse flow routing models are compared in Table 1.

According to Equation (13), the at different locations can be calculated. Then, the cross section with large bathymetric errors can be identified, since at this location is much larger than its adjacent cross sections with no or small bathymetrical errors.

In this study, the proposed method was firstly examined with it applied to a prismatic trapezoidal river, whose detailed information is as follows: a bottom width of 30 m, a side slope of 2.5, a bottom slope of 0.0001, and a roughness coefficient of 0.017. The river length is 20 km, and the gauging station is located at 16 km away from the river inlet. The study explores the scenarios in which there are bathymetric errors in the riverbed elevation, side slope and bottom width, at a single cross section (5 km and 10 km away from the river inlet, respectively) or at multiple cross sections. In the reverse flow routing, the gauging cross section was taken as the starting calculation section from downstream to upstream, and the flow and water stage hydrograph at this location is shown in Figure 4.

At first, the water stage simulated respectively by the forward and reverse routing models were compared in Figure 5, without any bathymetric error considered. In Figure 5, the time before the flow peak (t = 24 h), around flow peak (t = 36 h), and after flow peak (t = 48 h) were presented. According to the comparisons, the simulated water stages from two routing models are very close to each other. However, due to the model uncertainty, there still exists some differences between the water stages from two models, with the maximum value about 0.03 m, although no bathymetric error considered.

Natural rivers are frequently modified by human activities. For example, the sand mining can deepen the riverbed elevation. Therefore, this study considered several scenarios with bathymetric errors including the decrease of the riverbed elevation, the decrease of the side slope, and the increase of the bottom width. To distinguish with the influence of model uncertainty, this study compared the variation of in the case with no bathymetric errors. The test results are shown in Figures 6 and 7.

In Figures 6(a) and 7(a), it can be found that the model uncertainty induces obvious water stage errors, and a maximum value about 0.005 m of can be observed when no bathymetric error is considered. However, different from this kind of continuous and smooth variations by the model itself, the variation of in the case with bathymetric errors shows an abrupt change. At the location of 5 km or 10 km with bathymetric errors, the corresponding is significantly abnormal, and this cross section can be identified according to the anomalies of abrupt change.

Due to frequent human activities, it's likely that multiple cross sections of natural rivers undergo significant bathymetric modifications at the same time (Zang et al. 2020). To further examine the practicability of the identification method, the variation of in the cases with bathymetrical errors at two or three cross sections was analyzed, as shown in Figures 8 and 9.

The results in Figures 8 and 9 also verify the model performance, as it can also identify multiple locations with large bathymetric errors. In addition, if the large bathymetric error is located at the gauging cross section, it can be directly judged by comparing the simulations from the forward model and the observations at this location.

After being tested in the hypothetical case, the proposed location identification method was then applied to the Xunjiang River case to further demonstrate its feasibility. The main reach of the Xunjiang River from the Dahuangjiangkou gauging station to the Tengxian gauging station (96.19 km away from the Dahuangjiangkou station) was selected for the case study. The bathymetric data used for identification came from 27 cross sections measured in 2011 (Table 2). The measured data were spatially interpolated and input into the river flow calculation models with the uniform space step 500 m. The river location and its gauging stations are shown in Figure 10.

To analyze the influence of flow patterns, three different flow hydrographs since 2016 (Figure 11) were selected for comparison. The water stage observations were available at Pingnan, the other gauging station along this study river, and were used to verify the simulations from the two models, as shown in Figure 12.

The results in Figure 12 indicate that the simulated water stages from two models were very close to the observations with small errors. It can be found that the simulations from the forward model were slightly better than that from the reverse model. Nevertheless, when only a single model (forward routing model or reverse routing model) was used, it could be difficult to judge the location of bathymetric errors. For the cross section with flow routing results compared to the observations, the time-variation of simulation errors was almost in accordance with the observed water stage while the information of bathymetric error was not reflected.

After model verification, the location identification method was then applied to the three selected flows, and the results are shown in Figure 13.

In Figure 13, there are significant anomalies in the spatial variation of around Mengjiangkou (ID 20, 73.89 km). The identification results indicate that there should be a large bathymetric error around the Mengjiangkou cross section, which was likely to be caused by the scour and deposition of its tributary, Mengjiang River. In addition, comparing the results of three flows in Figure 13, it can be found that the abrupt change of is closely related to the flow magnitude. Generally, the spatial variation of is more significant in the larger flow case. Similarly to the hypothetical river case, the at the upper river reach is larger than that at the lower reach, and the tends to decrease along the river, which might be attributed to the effects of boundary conditions.

Due to the irregular hydraulic geometry of this natural river, the variation of were much more dramatic, resulting in that the bathymetric error identification was worse than the hypothetical case, especially for the case with small flow magnitude. Therefore, when using the proposed identification method, a flow with a larger magnitude is suggested to be selected. Overall, the proposed method in this study is feasible for the location identification of bathymetric errors in a natural river.

The proposed location identification method was successfully applied to a hypothetical and a real-word river case, and the cross sections with large bathymetric errors could be picked out according to the spatial variation of along the river. Nonetheless, several points should be discussed to better understand the identification method.

• (1)

  In Figures 6–9 and 13, the maximum respectively caused by model uncertainty and bathymetric error were close to each other. However, the spatial variations were significantly different. Different from the continuous and smooth variations by model itself, the variation of by bathymetric error presented an abrupt change. Thus, the identification was mainly determined by the spatial variation rule of ⁠, although its performance was affected by flow magnitude, which was quite different in the two cases.

• (2)

  In the hypothetical case, the abrupt change of caused by bottom width, bottom elevation and slope deformation could not be distinguished in Figures 6–9. The identification method itself was also not able to determine the detailed reason for bathymetric error. In the real-world case, after the location identification, the possible reasons for bathymetric error can be further analyzed by integration with field surveys, as well as the remote sensing images. After identification, the subsequent work to improve the bathymetric data as well as the flow simulation is also important. The idea of data assimilation using EnKF (Wang et al. 2019) associated with field surveys will be tried in our following studies.

• (3)

  Compared with previous works on this topic, the main contribution of this study is to put forward an identification indicator ⁠, whose spatial variation along the river can partition the effects respectively from model uncertainty and bathymetric error. And the developed flow routing models that can well simulate the water stage made the calculation of feasible. Although the adopted numerical schemes are closely related to the calculation of ⁠, the identification results are not affected by the selected simulation methods, because the abrupt change of attributed to bathymetric error can be judged as long as the models can well simulate the water stage in both forward and reverse directions.

In this research, a method for identifying the river bathymetric error location was proposed, based on the forward and reverse flow routing. This study first built the forward and reverse flow routing models, and then quantified the influence of the river bathymetric errors on water flow simulation according to the variation characteristics of simulated water stage. The identification indicator was calculated, using the water stage respectively obtained from the forward and the reverse flow routing models, and the location of bathymetric errors was then determined.

The proposed method was examined by a hypothetical river case and then applied to a real-world river case. The main conclusions are given as follows: (1) For the hypothetical river, the method effectively identified single and multiple locations with bathymetric errors in the riverbed elevation, bottom width and side slope. (2) In the case of the natural river, the identification results were affected by the flow magnitude. Generally, the spatial variation of was more significant in the larger flow case. Nonetheless, the proposed method in this study was feasible for the location identification of bathymetric error in a natural river. (3) When using the proposed identification method, a flow with larger magnitude is suggested to be selected. (4) The proposed method can effectively identify the location of river bathymetric errors, and the research results are helpful for improving the bathymetric data and the accuracy of water flow simulation.

This work was supported by the Guangdong Basic and Applied Basic Research Foundation (2020A1515110906), the National Natural Science Foundation of China (52109047), and the Joint Open Research Fund Program of State Key Laboratory of Hydroscience and Engineering and Tsinghua – Ningxia Yinchuan Joint Institute of Internet of Waters on Digital Water Governance (sklhse-2021-Iow01).

We would like to offer thanks for the review work by the editors and the anonymous reviewers.

The authors declare no conflicts of interest.

All relevant data are included in the paper or its Supplementary Information.