Spatial runoff updating based on the hydrologic system differential response for flood forecasting

Hydrological processes are highly nonlinear, time-dependent and spatially varying due to spatial–temporal variability of basin characteristics, precipitation and evaporation (Beven 2002; McDonnell 2003; McDonnell et al. 2007; Chang et al. 2019). The processes of rainfall–runoff and runoff–riverflow are complex. Reliable hydrological modeling remains an ongoing challenge; there exist inevitably various errors in flood forecasting (e.g. Wagener & Gupta 2005; Pebesma et al. 2007; Kavetski & Clark 2011; Montanari & Baldassarre 2013; Pianosi & Wagener 2016; Seibert et al. 2018; Brunnera & Sikorska-Senoner 2019). An error updating procedure is essential for a reliable and accurate real-time flood forecasting system (WMO 1992).

Currently, many techniques are available to correct model predictions, such as input variables updating (e.g. Si et al. 2015), state variable updating (e.g. Spaaks & Bouten 2013), parameter updating (e.g. Moradkhani et al. 2005) and output updating (e.g. Valipour et al. 2013). Moreover, from the perspective of information utilization, some effective updating strategies are developed (e.g. Hsu et al. 2003; Bao et al. 2011; Tiedeman & Green 2013; Zhang & Bao 2013; Li et al. 2016). The hydrologic system differential response (HSDR) between errors of model outputs (dependent variables) and related factors (independent variables) is useful for correcting independent variables (Kirchner 2009). The dynamic system response curve method based on the HSDR has been proposed to update areal mean variables in the whole basin, including areal mean runoff (AMR; Bao et al. 2014; Zhang et al. 2014), areal mean rainfall (Si et al. 2015), areal mean state variables (Sun et al. 2018a) and areal mean initial conditions (Sun et al. 2018b). In previous works, little research has been conducted on the spatial variability of updated variables. This prompts us to examine the performance of the HSDR method on updating spatial distributed variables in flood forecasting.

The calculation of runoff generation controls water volume in rainfall–runoff models; it plays an especially important role in flood forecasting. The spatial–temporal variabilities in basin underlying, precipitation, evaporation and runoff production significantly affect runoff calculated by hydrological models (Singh 1997; Smith et al. 2004; Huza et al. 2014; Fraga et al. 2019; Guse et al. 2019; Visser-Quinn et al. 2019). The effects of all error sources are compressed into model output; it is difficult to distinguish and identify error sources from the residuals between observations and calculations (Renard et al. 2011; Li et al. 2017; Zhang et al. 2019). Extracting the spatial–temporal properties of runoff errors from discharge residuals is important for flow updating. How to consider the spatial–temporal variability in runoff updating by using the HSDR method is an interesting problem that needs to be investigated.

The main objective of the present study is to extend the HSDR method on spatial error updating in flood forecasting. The HSDR with stepwise approximation is proposed to gradually update spatial distributed runoff (SDR) and AMR, referred to as HSDR_SDR and HSDR_AMR, respectively. Furthermore, the performance of the HSDR and the autoregressive (AR) technique with increasing lead time is compared. This study can provide an investigation of the HSDR on spatial runoff updating; it could help improve real-time flood forecasting and extend our understanding of error feedback in the hydrological process.

The Xinanjiang (XAJ) model developed by Zhao (1992) is a simple conceptual rainfall–runoff model based on the concept of runoff formation upon the repletion of storage. It has been extensively applied for flood forecasting in humid and semi-humid regions in China (e.g. Jayawardena & Zhou 2000; Li et al. 2009; Liu et al. 2009; Lü et al. 2013; Yao et al. 2014). The model structure consists of evaporation, runoff production, runoff component separation, flow concentration and flood routing. The evaporation is represented by a three-layer model. The runoff production occurs on the repletion of storage. The total runoff is separated into surface runoff, subsurface runoff and groundwater components. The flow concentration for each runoff component to the outflow of each sub-basin is routed by using the linear reservoir technique. The outflow is finally routed to the basin outlet by using the Muskingum method.

The flowchart of the XAJ model is given in Figure 1. All symbols inside the box are variables including inputs, outputs and state variables. The model inputs are rainfall P and pan evaporation EM; and the outputs are actual evaporation E and discharge Q_T at the basin outlet. The variables outside the box are parameters, including evapotranspiration parameters (K, WUM, WLM, C), runoff production parameters (WM, B, IM), runoff separation parameters (SM, EX, KI, KG), runoff concentration parameters (CS, CI, CG) and Muskingum parameters (KE, XE). More details of the symbols can be found in the literature (Zhao 1992).

In the application of the XAJ model, the study basin is commonly divided into a set of sub-basins to represent spatial characteristics. The basic inputs are the areal mean rainfall and pan evaporation in each sub-basin. The areal mean rainfall of each sub-basin is represented by the point measurements obtained from the rain gauges in the corresponding sub-basin. The rainfall–runoff modeling is conducted in each sub-basin. The outflow of each sub-basin is routed down to the outlet of the whole basin.

The schematic of the HSDR for runoff updating in the XAJ model is displayed in Figure 2. Every process in the XAJ model can be generalized as a system, and several related processes also can be regarded as a bigger system. The procedure of evaporation and runoff production is referred as the System A, and that including runoff component separation, flow concentration and flood routing is referred as the System B. In flood forecasting, we can use real-time residuals between calculations and observations to update variables through the basic mathematical foundation of the hydrologic system. The streamflow prediction can be improved by re-running the hydrologic model with updated variables. We adopt a stepwise approximation for the HSDR to reduce the effect caused by the linearization of the nonlinear system, which is different from the previous studies without iteration (Bao et al. 2014; Zhang et al. 2014; Si et al. 2015; Sun et al. 2018a, 2018b). We employ the root mean square error RMSE as the objective function for iteration termination. The error updating is operated when the newly corrected performance is better than the original; it can assure that there is no deterioration in the model performance when using the HSDR method.

Equation (7) determines how basin outlet discharge responds to spatial runoff. It also can be used to understand how error in runoff accumulates as it propagates through the System B. Figure 3 displays two strategies for runoff updating by using the HSDR, namely the AMR updating in the whole basin (referred as HSDR_AMR) shown in Figure 3(a) and the SDR updating that gradually updates the AMR in each sub-basin (referred as HSDR_SDR) shown in Figure 3(b). Figure 3 is the key part of the HSDR method shown in the right blue dash line box in Figure 2. The major differences between the HSDR_SDR and HSDR_AMR are the consideration of differential response matrix U for each sub-basin or the whole basin (blue dash line boxes in Figure 3) and the assignment of the estimated incremental runoff for each sub-basin (black dash line boxes in Figure 3).

As the HSDR_AMR shown in Figure 3(a), the differential response matrix U for the whole basin is calculated from the change of outlet discharge to the perturbations of AMR in the whole basin. The procedure of the HSDR_AMR mainly includes:

Step 1: Calculate runoff R_i (i = 1 to b) at the ‘j’ time in each sub-basin via the System A;

Step 2: Generate a perturbed runoff series by increasing the R_i (i = 1 to b) at the ‘j’ time by one unit in each sub-basin, while the runoff at other time remains unchanged;

Step 3: Calculate the difference between the flow series via the System B using the R_i and ⁠;

As the HSDR_SDR shown in Figure 3(b), the differential response matrix U for each sub-basin is calculated according to the response of outlet discharge to runoff change in the corresponding sub-basin. The main procedure of the HSDR_SDR is shown as follows:

Step 1: Calculate runoff R_i (i = 1 to b) at the ‘j’ time in each sub-basin via the System A;

Step 2: Generate a perturbed runoff series by increasing the runoff in the ‘i’ sub-basin at the ‘j’ time by one unit while keeping the runoff at other times in the ‘i’ sub-basin and all runoff in other sub-basins unchanged;

Step 3: Compute the difference between the flow series via the System B using the R_i and ⁠;

Step 4: Repeat Steps 1–3 at different times (j = 1 to n), all the columns of matrix U for the ‘i’ sub-basin are obtained. By using the U for the ‘i’ sub-basin, we can estimate in the whole basin. The estimate of for the ‘i’ sub-basin is assigned according to the area proportion of the ‘i’ sub-basin to the whole basin.

Step 5: Repeat Step 4 for each sub-basin (i = 1 to b), we can obtain the estimated error of AMR in each sub-basin.

In previous works, the HSDR_AMR without stepwise approximation was used (Bao et al. 2014; Si et al. 2015). In the HSDR_SDR, the information of residuals between observations and calculations can be decomposed by using the differential response due to SDR in each sub-basin. Compared with the HSDR_AMR, the HSDR_SDR can account for spatial variability of runoff, especially in basins with various spatial characteristics.

The proposed method is evaluated in the Baishuikeng (BSK) reservoir basin and the Shaowu (SW) basin in China (Figure 4). The characteristics of the two basins are summarized in Table 1. The BSK is a reservoir basin. The ‘observed’ reservoir inflows in the BSK basin are calculated according to the reservoir water level and outflow discharge; the ‘observed’ hydrographs are serrated displaying fluctuation. Most hydrographs at the outlet of the SW basin have multi-peaks. According to the collected history of floods, the average duration time from the beginning of the flood to the occurrence of flood peak is 34.6 h in the BSK basin and 61.1 h in the SW basin, which can affect the length of observations used for updating with different lead times.

The BSK and SW basins are divided into seven sub-basins and nine sub-basins, respectively. The rainfall, pan evaporation and discharge data are provided by the local hydrological bureau. The XAJ model parameters are calibrated by using historical daily data and hourly data during flood events. Twenty of 27 flood events in the BSK basin and 25 of 34 flood events in the SW basin are used for parameter calibration and the remainder of the flood events for validation. The data for all flood events are at an hourly time step. The calibrated model parameters are listed in Table 2.

The performance of the HSDR on runoff updating is examined to correct the flood predictions by the XAJ model in two basins with different rain gauge density and different flood characteristics. The comparisons of the HSDR updating with and without stepwise approximation are presented. The performances of the HSDR_AMR and HSDR_SDR are compared. Moreover, the performance of the HSDR and the second-order AR technique with different lead times is demonstrated. The AR technique based on residuals between observations and calculations is a traditional and widely used correction method. The lead time in this study represents the time ahead of flood peak timing. We set different lead times by using different numbers of observations for updating. Shorter lead time means more observations used for updating. Note that the parameter values remain unchanged during both the prediction and updating processes.

The results by using the XAJ model without correction are shown in Table 3. The mean results for the HSDR with and without stepwise approximation are shown in Figure 5 for the BSK basin and Figure 8 for the SW basin. The mean evaluation measures by the HSDR with stepwise approximation are labeled as ‘AMR’ and ‘SDR’, respectively, and those without it are represented as ‘’ and ‘’, respectively. The mean results for the HSDR_SDR, HSDR_AMR and AR with different lead times are compared in Figure 6 for the BSK basin and Figure 9 for the SW basin. The ‘XAJ’ represents the original results calculated by the XAJ model without correction. The ‘IN’ represents the number of average iterations. Some hydrographs with different lead times are displayed in Figures 7 and 10. The ‘Q_obs’ represents the observed discharge at the basin outlet.

As expected, the results for the HSDR_SDR and HSDR_AMR with stepwise approximation are improved compared with those without it. Both in the BSK and SW basins shown in Figures 5 and Figure 8, the RMSE and ARPE values are reduced and the NSE values are increased. It indicates that the HSDR with iteration can reduce the effect of the linearization of nonlinear system in error updating.

From Figures 6 and 9, in comparison with the HSDR_AMR, the values of RMSE for the HSDR_SDR are all smaller and the values of NSE for the HSDR_SDR are all greater. It shows that the HSDR_SDR obviously performs better than the HSDR_AMR with an increasing lead time. The iterations for both the HSDR_AMR and HSDR_SDR are all fewer than 10. Moreover, when increasing lead time, the performance of the AR technique is considerably decreased as compared with the HSDR method. In the BSK basin, the corrected results by the AR are even poorer than the original XAJ results when the lead time is larger than 6 h. When the lead time is greater than 1 h in the BSK basin and 3 h in the SW basin, the HSDR_SDR performs better than the AR.

As Figures 7 and 10 exhibited, the hydrographs for the HSDR_SDR and AR agree well with the observations when lead time is 1 h. When increasing lead time, the simulations by the HSDR_SDR are generally better matched to the observations and especially better capture flood peaks. It is found that there exist large errors around flood peak for some floods by using the AR, such as Figure 10 with the lead time of 6 h, which may be due to poor correlation between the discharge residuals around flood peak with increasing lead time.

For all the flood events in the study basins, the model performances are improved using the proposed method as compared with the original predictions using the XAJ. Overall, both in the large basin SW with sparse-density rain gauge and small reservoir basin BSK with high-density rain gauge, the HSDR_SDR and HSDR_AMR with iteration outperform those without it; the performances of the HSDR_SDR are generally superior to the HSDR_AMR; and the HSDR for runoff updating exhibits more stable performance with increasing lead time as compared with the AR technique.

Flow at a basin outlet is contributed by all related spatial–temporal information. Qualifying the relative contribution of error sources to model outputs is a challenging task. Runoff distribution is spatially and temporally uneven in a basin. The HSDR_SDR can account for the influence of spatial variability in runoff updating on the hydrologic response through the differential response matrix for each sub-basin. To some extent, the HSDR_SDR can apportion the total response error to contributing factors for runoff updating in each sub-basin. It is simple to operate with no more parameters nor change in the hydrological model. Theoretically and practically, compared with the HSDR_AMR that uses the differential response matrix for the whole basin, the HSDR_SDR can better improve flood predictions due to the consideration of spatial variability in runoff updating on the hydrological response. Operational flood forecasting requires accurate forecasts with a suitable lead time for the issuance of a flood warning. We suggest the use of the HSDR_SDR with stepwise approximation for long lead time predictions.

The first-order partial derivative is adopted to reflect the hydrological response to the change of variables in a nonlinear hydrologic system, and a stepwise approximation approach is adopted to reduce the error caused by the linearization of the nonlinear system in model solution. The case studies demonstrate that the proposed method based on the HSDR can produce satisfactory discharge correction by updating runoff. Even in a reservoir basin such as the BSK, the HSDR_SDR can obtain satisfactory correction for reservoir inflow. The HSDR performance decreases slowly when increasing lead time; it indicates that the HSDR for runoff updating is effective with strong convergence and stable performance. The more observed data used for updating the better the HSDR performs. Moreover, the HSDR approach is more robust than the AR technique on streamflow correction with increasing lead time.

The residuals between observed and calculated discharges at the basin outlet are comprehensively caused by various errors such as model structure, parameters and states. We use the HSDR to correct flood predictions by adjusting runoff based on the information of the outlet discharge residuals. Unfortunately, there exist various errors in hydrological modeling that we cannot quantify. It may be improper that all errors of discharge at the basin outlet is primarily attributed to runoff, which is the possible reason for why there is little improvement in the model performance for several of the flood events in the study basins. The performance of the proposed approach depends on whether the corrected variables are the main error sources of the discrepancy between observed and calculated streamflow. From the RMSE and NSE in Figure 9, the AR performs better for the 1-, 2-, and 3-h predictions in the SW basin. It is possible due to the fact that the AR technique takes all errors into consideration from a black-box perspective. Runoff controls the water volume in the rainfall–runoff process; it is also the dominant error source in operational forecasting. Future research should identify and examine the principle sources of error.

In the study case, the RMSE is set as the objective function for iteration termination. We find that the RMSE and NSE for the corrected results are all improved by using the HSDR, but the ARPE for several of the flood events are not reduced. The multi-objective functions that accout for both flood peak and RMSE will be investigated in the application of the proposed method (e.g. Engeland et al. 2006; Fowler et al. 2018; Her & Seong 2018).

Generally, the proposed method for spatial runoff updating has the capacity to improve real-time flood forecasting without increasing model complexity and model parameters, which make it easy to operate in real-time forecasting. It is worth pointing out that the HSDR method is general rather than a specific one and can be extended for other variables updating (such as rainfall, soil moisture, parameter and state) in other models to produce reasonable flood forecasting results.

The HSDR can represent the relation between model outputs and influencing variables. We present an enhanced HSDR approach with stepwise approximation for SDR updating in flood forecasting. The study area is divided into a number of sub-basins in the application of the XAJ model. The HSDR method is used to gradually update area mean runoff in each sub-basin (SDR) and AMR in the whole basin, respectively. The results are as expected. The HSDR_SDR outperforms the HSDR_AMR considering the spatial variability of runoff. Furthermore, compared with the AR technique, the HSDR method exhibits less deterioration on the improvement of model performance when increasing lead time.

It is highlighted that the HSDR for error updating is based on the hydrologic response to the updated variable through the adopted hydrological model; the HSDR for runoff updating is specifically based on the runoff–riverflow process expressed in the hydrological model. Obviously, the HSDR method has good theoretical foundations and strong potential for improving real-time flood forecasting. It can provide an insight of how basin outflow responds to variable changes through the physical mechanism described by hydrological models. The proposed approach based on the HSDR in this study can identify the total response error to the contributing factors; it provides an effective approach for error decomposition and correction. The performance of the HSDR on other spatial variables updating will be explored; it would help evaluate the proposed method and ultimately extend our understanding of error feedback in the hydrologic process. Future work will continue to further examine the proposed method through physically based distributed hydrologic models and quantify the uncertainty of updated variables through uncertainty estimation techniques.

The study was supported by the National Key R&D Program of China (2019YFC0409003), the National Natural Science Foundation of China (51709076) and Fundamental Research Funds for the Central Universities (B200202026). Finally, the authors would like to thank the editor and anonymous reviewers for their helpful comments and suggestions.

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