Sediment yield error correction by dynamic system response curve method in real-time flood forecasting

The loess region of the Yellow River is famous for its characteristics of high sediment loads and transport. Before-storm and raindrop splash soil erosions are the original sources for suspended sediment yield. These two processes usually provide enough loose soil at most locations due to the distinctive soil properties (Xian et al. 1983; Fang et al. 2008). Runoff, which is in response to rainfall events, acts directly on the sediment transportation (Hairsine et al. 1999; Jiao et al. 2002; Nearing et al. 2005; Rouhipour et al. 2006; Wu et al. 2018). To give better understanding of the physical law of flow and sediment transport in the loess region, a streamflow-sediment coupled (SSC) model with clear physical meaning, simple structure and parameters was established by Bao (1995). Numerous attempts have been made to illustrate the proposed SSC model is rational and applicable in forecasting streamflow-sediment processes within the loess region (Bao et al. 2019; Si et al. 2017). However, flow simulation always shows relatively higher accuracy than the simulation of sediment transport processes at both event and annual temporal scales.

Sediment generation is a complex function of climate, hydrology, basin topography and land cover (Bingner et al. 1997; Merritt et al. 2003; De Vente & Poesen 2005; De Vente et al. 2010; Rompaey et al. 2010; Si et al. 2016). In-channel proximal deposits of sediment make significant contributions to the total sediment load. Sediment distal supply and proximal sediment availability are strongly related with the time-varying streamflow and erosion/deposition processes (Si et al. 2017; Juez et al. 2018a, 2018b). Many factors influence the forecasting result, among which the sediment yield error dominates the volume error which directly affects the accuracy of sediment transport estimates. A widely used approach to reduce the size of forecast errors is real-time error correction (Babovic et al. 2001; De Chant & Moradkhani 2012; Bao et al. 2014; Chen et al. 2015; Si et al. 2019). According to different corrected variables, error correction is classified into four categories (WMO 1992): input, state, parameters, and output correction. Different methods have been proposed for the correction of different variables. For example, the AR model (Pegram & James 1972; Babovic et al. 2001; Komma et al. 2008; Ju et al. 2009) is commonly used for output correction. However, mutation occurs around the flood peak and the errors there are not correlated, which leads to unreasonable results when using the AR model (Bao 2006). Besides, either an Ensemble Kalman Filter (EnKF) or Particle Filter (PF) can be used to recursively update the model states and parameters via data assimilation (Vrugt et al. 2006; Weerts et al. 2006; Neal et al. 2009; De Chant & Moradkhani 2012; Moradkhani et al. 2012). The results show that the EnKF is consistently overconfident in predicting streamflow.

In view of this, a dynamic system response curve (DSRC) method is proposed to update estimates of the inputs, state variables and outputs. Much subsequent experience supports its validity for updating flow at the outlet in hydrological models. For instance, the DSRC method is compared with the AR model in correcting both input and state errors, including free-water storage (Bao et al. 2015), soil water content, rainfall (Si et al. 2013a, 2015, 2016, 2019; Yang et al. 2015) and runoff (Si et al. 2013b; Bao et al. 2014; Zhang et al. 2014; Liu et al. 2015). The DSRC method always presents better performance without requiring modifications to model structure (Gupta et al. 2012) or parameters calibration (Gupta et al. 2008). Thus, updating real-time flood forecasts based on system differential response theory is of increasingly high value for real-time error correction studies. The DSRC method has the dominant features of clear physical concept (Bao et al. 2014), simple structure and stable performance (Si et al. 2015, 2019). It can provide an effective and stable correction for the outputs without losing the lead time. Existing studies have paid more attention to updating the real-time flood forecasting by correcting streamflow phase errors, while little attention has been given to the sediment transport phase, of which the accurate predictions are critically informative for efficient reservoir operation, river management, flood control, and warning in the Yellow River basin (Zhang et al. 2007; Tamene et al. 2010; Palazon et al. 2014). Therefore, investigation of those errors resulting from uncertainties in the sediment transport simulation is needed.

The objective of this paper is to employ the DSRC method coupled with the SSC model for improving the accuracy of sediment prediction results through correcting the sediment yield time series in the loess region. The physical basis of the system response curve is the sediment concentration of the SSC model. The theoretical basis for using DSRC – to correct the error of slope sediment yield – is the differential of the system response function of the slope sediment yield time series (Si et al. 2015, 2019). This proposed method is tested in three steps: (1) an ideal case is selected to demonstrate the applicability of the DSRC method using the SSC model; (2) three real basins with different scales in the Yellow River are used to further test the method performance and provide insights into correction effectiveness; (3) the initial calculated and the corrected forecasts are compared by further contrastive analyses using certain performance indices. Results show that the new correction approach is reasonable and effective for forecast-updating, and can result in improved prediction accuracy.

Sediment in the Yellow River mainly comes from the mid-streams which are rich in sediment. Therefore, three different scale basins in the middle reach of the Yellow River basin with 101 historical floods are selected to further test the proposed error feedback correction method based on the DSRC. Jiuyuangou basin (the area is 70.7 km²), which is a tributary in the middle reaches of Wuding River with 11 rainfall stations, is located between latitudes 37°33N and 37°38N, longtitudes 110°16E and 110°26E. The annual mean precipitation is about 500 mm, of which 70% is concentrated in the flooding season (from June to September). The mean modulus of soil erosion varied from 18,000 to 19,900 during the natural period (Wang et al. 2016a). Since the year 1953, considerable soil and water conservation management has been carried out across the basin, and more strikingly, the dam system emerged (Wang et al. 2016b). It is recorded that during the period of the 1970 and 1980s, check-dams rapidly developed up to over 250, whose total storage capacity reached 0.43 million m³.

Chabagou basin is a larger tributary of the Dali River, which belongs to the first sub-area of hilly and gully in loess plateau, with a drainage area of 187 km² and 46 rainfall stations. Caoping hydrological station is an important controller at the basin outlet. The average annual precipitation in the basin is about 447 mm, 80% of the total precipitation, which is more concentrative spatially and temporally from June to September with high intensity. High sediment loads of flood are linked to obvious changes to the annual runoff and sediment transport (Qi et al. 2011), with an erosion modulus of almost 222,000 ⁠. Check-dams have been constructed in the Chabagou watershed and boomed in the 1970s before stagnating after the 1980s (Zhang et al. 2010), which depressed the sediment yield by reducing the frequency and intensity of hyper-concentration streamflow and soil erosion in gullies.

Dali River basin is the largest tributary of the Wuding River, with a drainage area of 3,906 km² and 67 rainfall stations. Sparse vegetation coverage, fragmented terrain and the dry climate cause serious water and soil loss in the loess plateau and further contribute to the majority of sediment in the Yellow River in the form of suspended sediment. In the late 1980s, more than 3,100 dams were built for water and soil conservation within the basin. The annual mean precipitation in this basin is 439.5 mm in the form of torrential rain. Because of the weak erosion resistance resulting from the low-level management before the 1970s, the average sediment transport modulus was 16,000 (Qi et al. 2011). Suide hydrological station is the major control station at the outlet of Dali River basin.

Precipitation across all study basins is characterized by small scope, short duration and high intensity. Severe soil erosion, therefore, is a result of rainstorm in the loess region. Moreover, the annual distribution of precipitation is extremely uneven, accompanied by significant inter-annual variation. Precipitation in the flooding season (from May to September) accounts for more than 80% of the total annual precipitation. Sediment loads caused by rainstorms are much higher than those caused by common rainfall and are more concentrated in the flooding season than the dry season. The locations of rainfall and hydrological stations in these study basins are shown in Figure 1(a)–1(c).

The proposed sediment yield model is coupled with a conceptual hydrologic model. It has a simple structure and specific physical meaning. The model has been successfully applied in the watersheds with similar soil texture to that of the loess plateau region (Bao et al. 2019). Therefore, the SSC model is chosen to study the streamflow-sediment simulation and forecasting in the loess region in the middle reach of the Yellow River.

The characteristic feature of the SSC model is the concept of hydraulic erosion capacity and soil erosion resistance capacity, and the distribution curve of soil erosion resistance capacity. Detachment and deposition along the flow path are integrated results determined by time-varying hydraulic factors, as well as sediment distal supply and proximal sediment availability (Si et al. 2017; Juez et al. 2018a, 2018b). Thus, an event responsive watershed sediment generation structure and conceptual structure of sediment movement in gully and stream are established. The SSC model is divided into the following five parts:

• (1)

  Runoff generation: The vertically-mixed runoff generation method combines the spatial distribution curve of infiltration capacity with the water storage capacity curve (Bao & Wang 1997). Runoff (R) is generated when rainfall (P) hits the ground, and in general is initially separated into surface runoff and infiltration streamflow through the spatial distribution curve of infiltration capacity. The infiltration streamflow (RR), including subsurface and groundwater runoff, replenishes soil moisture deficiency generated by the evapotranspiration. Underground runoff is generated where the soil moisture content of the aeration zone reaches field capacity. Surface runoff formation in the excess of infiltration is determined by rainfall intensity and infiltration; and, on the other hand, subsurface and groundwater runoff generation upon repletion of storage are dependent on early soil water shortage and the volume of infiltration.

• (2)

  Separation of runoff components: The free water reservoir division method (Bao et al. 2019) is applied to divide the previously determined total runoff into three components: surface runoff (RS), subsurface runoff (RI), and groundwater runoff (RG).

• (3)

  Streamflow concentration: The three runoff components are further transferred into QS, QI, and QG, and together form the total inflow to the channel network of the sub-basin. The outflow of the sub-basin is Q. By applying the Muskingum method to successive sub-reaches (Zhao 1993), flood routing from sub-basin outlets to total basin outlets is achieved. Eventually, the discharge (TQ) is obtained.

• (4)

  Sediment generation: According to the proposed spatial distribution curve of soil erosion resistance capacity (Si et al. 2017; Bao et al. 2019), slope sediment yield (SS) could be acquired by the integral of the difference between sediment carrying capacity and soil erosion resistance capacity. Gully sediment generation (SG), which varies with the hydraulic condition of the channel, is calculated by the suspended sediment formula of Bagnold's River channel (Bagnold 1960). Due to the parameter CGM (channel sediment concentration on average) based on the physical meaning, the relationship between hydraulic factors and the sediment yield rate of gully erosion is well explained (Bao et al. 2019). Slope and gully sediment generation together form the total suspended sediment load within the stream section.

• (5)

  Sediment transport: The transport capacity is the threshold for erosion and deposition (Si et al. 2017). If the actual sediment concentration is less than its capacity, erosion occurs; otherwise, deposition occurs. The corresponding coefficient is introduced to reflect the erosion and deposition relationship. By referring to the similarity of streamflow concentration, sediment transportation in the stream section is calculated by sediment continuity and motion equation. The slope and gully sediment yield are further collected into TSS and TSG. Finally, the sediment process at the total basin outlet (TS) is obtained by continuous routing.

The structural diagram of the SSC model is shown in Figure 2. The symbols inside the blocks are variables including inputs, outputs, state variables, and internal variables while those outside are parameters. The inputs of the model are rainfall (P) and measured pan evaporation (EM). The outputs are the discharge (TQ), the sediment transport rate (TS) from the whole basin and the actual evapotranspiration (E), which includes three components, EU, EL, and ED. The state variables are the area mean tension water storage (W) and the area mean free water storage (SF). The area mean tension water (W) had three components (WU, WL, and WD) in the upper, lower, and deep layer. FR is the runoff contributing area factor that is related to W. The rest of the symbols inside the blocks are all internal variables. The functions, methods, parameters meaning and corresponding value ranges of streamflow and sediment part are shown in Tables 1 and 2 respectively. More detailed calculation formula and the physical explanation of each parameter are illustrated in Bao et al. (2019).

To apply the DSRC method, the SSC model is systematically generalized into a nonlinear response system except for the parts before sediment generation (Figure 3). The dashed box is the system used for slope sediment yield correction in this study. Because this correction is affected by the values of sediment yield at other times, it is time-varying, and the system response curve is dynamic.

The additional term could control the range of change ⁠, which helps to effectively alleviate or avoid the ‘oscillation’ phenomenon, and further enhance the stability of this correction method.

Figure 4 provides a further explanation for performing the dynamic correction process. Observed rainfall ⁠, runoff ⁠, discharge and sediment transport rate processes are shown as graphical information for the case contained in Figure 4. By using the DSRC method, corresponding corrected processes such as ⁠, are generated by each original rainfall inputs series t, respectively. The ultimate updating process (S_t+1(t) in Figure 4 is, therefore, dynamically corrected against an observed time series of input data.

The detailed steps for dynamic correction of sediment transport rate processes using the DSRC method are presented as follows: 1. Slope sediment yield are first calculated by using the observed data of streamflow and sediment before the initial time t of rainfall (including current time), and therefore the calculated sediment transport rate series is acquired; 2. Calculated series from the former computation are fixed to the constant values, while adding slope sediment yield by one step ⁠. A new series of slope sediment yield is then obtained; 3. The new series is applied to recalculate the sediment transport rate series ⁠; 4. The system response curves of are then computed from the differences between and divided by the step ⁠. Each column in the matrix is obtained in the same way; 5. Based on the error correction method through DSRC mentioned above (Equations (5)–(14)), the new series of sediment transport rate is acquired. Moreover, only the value in the next moment of the original series calculated by the SSC model is substituted by the new computed one at the corresponding time. Therefore, the corrected (not the final) sediment transport rate process is obtained; 6. Steps 1–5 are repeated in the next correction period and terminated in the last period which is the end of rainfall in this flood. Eventually, the final corrected sediment transport rate process is obtained by continuous replacement over the range of correction spanned by the rainfall period.

In order to provide a statistical analysis of each correction in different flood events, it is necessary to identify multiple performance measures to evaluate the goodness-of-fit (Song et al. 2011). Three quantitative measures, including the Nash–Sutcliffe (NS) coefficient, the relative error of sediment peak and relative error of sediment yield are used to check the correction performance of the DSRC method. The level of effective improvement, in comparison to the results before correction, is measured by two quantitative indices with respect to sediment peak and NS improvement in both the ideal case and real-data simulation. Model performance is evaluated according to the following metrics:

• 1.
  Nash–Sutcliffe (NS) (Yapo et al. 1996):

  

  (15)
• 2.
  The relative error of sediment peak (Bao et al. 2012):

  

  (16)
• 3.
  The relative error of sediment yield:

  

  (17)
• 4.
  The effective improvement of sediment peak:

  

  (18)
• 5.
  The effective improvement of NS:

  

  (19)

  where ⁠, are computed and observed sediment transport rate (t/s); is the mean value of observed sediment transport rate (t/s); N is the number of time periods in a flood event; ⁠, ⁠, are calculated, observed and corrected sediment peak (t/s); ⁠, represent calculated and observed sediment yield respectively (t); ⁠, represent computed and observed occurrence time of sediment peak (t/s); denotes the corrected sediment transport rate calculated by the updated slope sediment yield (t/s); i is the number of time period. Terms ⁠, are used to evaluate the relative effectiveness of the correction method. Note that larger values for ⁠, mean that better improvements are achieved by updating and a maximum value is 1.0.

Twenty-two parameters of both streamflow and sediment parts in the SSC model are included. Floods in each basin used for calibration are event-based for which hourly streamflow and sediment datasets are available. To calibrate the model parameters, an optimization algorithm which merges the strengths of the auto-estimated method based on parameter function surface (Bao et al. 2013) is used. Three performance measures, the Nash–Sutcliffe efficiency (NS), relative error of sediment yield and relative error of sediment peak ⁠, are taken as the objective functions (Bardossy & Singh 2008; Wagener et al. 2009) to determine the optimal model parameters. The final calibrated parameter sets of streamflow and sediment are listed in Tables 3 and 4 and, once calibrated, generally do not change during both calculation and correction periods.

The calculated sediment transport rate is obtained by the SSC model using the series. Assigned error and estimated error (computed by the DSRC method) of slope sediment yield, along with their inter-correlation among the time period, is shown in Figures 5 and 6 respectively. T is the number of time period. Though the two series are slightly different and varied with the corresponding slope sediment yield, the error values estimated by the DSRC method are very close to the actual (assigned) values. The correlation coefficient is 0.978 (Figure 6). Furthermore, the sum of slope sediment yield errors is 39.4 t, while that of the estimated series corrected by DSRC method reached 38.7 t – a relative error of only 1.9%.

In Figure 7, ⁠, ⁠, ⁠, represent runoff depth, calculated, corrected, and observed sediment transport rate process respectively. As shown in Figure 7, the DSRC method significantly improves the forecast in sediment transport process compared to the original calculation. decreases from –11.60 to 2.56%, decreases from –3.33 to 0.04%, while coefficient increases from 0.978 to 0.998. Even when the sediment yield errors vary with time, the DSRC method could still accurately estimate the error series by extracting useful information from the observed data. The results reflect the fact that through accurate time-varying slope sediment yield error estimates, forecast accuracy in the sediment transport rate process at the basin outlet tends to be effectively improved using the DSRC method.

The DSRC method is further tested in three real basins with different spatial scales and rain gauge densities. The flood data and more detailed basin information used for each of the three cases are summarized in Table 5. In accordance with the national criteria for flood forecasting in China (MWR 2006), the results are acceptable (also known as qualified simulation) when the absolute errors between observed and corrected sediment yield in flood events are equal to or less than 30%. A value of is acceptable when it shows a desirable limit within 30%, with ‘√’ in the corresponding table; while ‘ × ’ indicates too much lower accuracy in simulation. The sediment yield and peak dominate the process characteristics and are thus more important for simulation, which is a better measure of how well the model corrects the overall sediment process. The observed, computed, and updated results of each basin are listed in the supporting information, Tables S1–S3.

Over the 11 floods in supporting information Table S1, it can be noted that the DSRC method produces a higher NS value and a lower relative error of both sediment yield and peak than the initial calculation in nearly every correction. In this case, the average improvement in NS value has been increased from 0.500 to 0.849, with an average effective improvement in NS (INS) being 0.686. As shown in Figure 8(a), the NS performance metric values for the DSRC method are significantly larger than the NS of the SSC model without updating. That is, the DSRC updating method is effective at improving the forecast accuracy. The better NS performance is especially reflected in the decrease of relative error of sediment peak (from 49.69 to 8.70%) and in the decrease of relative error of sediment yield (from 23.36 to 13.48%). The improved qualified rate (the proportion of qualified floods) hence results from significantly better model forecasts (36.4% improvement).

Table S2 shows the comparison results concerning the sediment process focused precision indices. The accuracy has significantly improved when using the DSRC method over the period, whether human activities intervene or not. Specifically, the qualified rate has been increased from 84.4 to 97.8%, with an increment of 13.4%. Note that the average NS value of 45 historical floods improved from 0.442 to 0.929, and the value greater than 0.9 in flood events is 0 and 42 before and after correction, respectively. The NS values (before and after updating) for every flood event are shown in Figure 8(b). In all flood events, the NS performance is improved by updating, showing the significant updating effect of the DSRC method. For some floods like No. 19610926, sediment yield estimates seem to be overcorrected with the increasing relative error. On average, however, the relative error decreases from 23.74 to 11.56%. The mean relative error of sediment peak decreases from 56.87 to 9.99%, showing an effective improvement, as much as 0.719 on average.

The corrected outcomes of Dali River basin also show consistency with that of other basins, however, relatively poor performance in sediment peak correction is obtained. Table S3 shows a direct comparison of the results. Note that the qualified rate is improved from 55.6 to 95.6% (40% improvement) and the average NS value is improved from 0.301 to 0.816 (0.515 improvement). NS values (before and after updating) for every flood event are compared in Figure 8(c). As shown in Figure 8(c), the accuracies of flood estimates for the 45 events in the Dali River basin have, again, been significantly improved by the DSRC updating method. The number of floods with NS greater than 0.9 increased from 0 to 7. Though the percentage of absolute error of corrected sediment peak falling within the qualified range of <30% decreased from 33.46 to 19.99%, the negative percentage indicates the corrected mean value exceeds the observed one. The mean effective improvement in NS value reaches 0.696, whereas the improvement of sediment peak is less effective due to overcorrection. Overcorrection also remains in some floods, while the mean relative error of sediment yield still decreases from 34.73 to 19.01%.

According to the results obtained in different basins (listed in the supporting information), the DSRC method has been proven as a useful tool for improving the accuracy of real-time flood predictions. This method has a significant correction effect for both sediment peak and total sediment yield. This is the case for both smaller and larger basin scales. Besides, the improvements in the three study basins vary with basin size and rain gauge density. A larger relative (percentage) improvement is obtained for larger basin scales, and the mean NS value gradually increases with increasing rain gauge density. Though the DSRC method is effective and stable at improving the forecast accuracy for the sediment transport process, the corrected relative error of sediment yield of some floods is slightly larger than that without updating.

Chabagou basin is chosen as the target basin for further statistical analysis. Unqualified flood No. 19950904 is considered for its poor performance in the relative error of sediment yield over 30% without proper correction to sediment peak (listed in the supporting information Table S2). The initial error of calculated sediment peak can be partially linked to the stationary assumption of parameters like ⁠, and in the sediment transport phase. Moreover, the uncorrected peak value indicates that the short rainfall duration leads to a short correction period before peak time. The short correction period has restricted the use of effective observed information for tracking errors by the DSRC method, and therefore incurs a much lower NS value (only 0.313), but still 0.097 higher than that before correction.

Four typical floods in Chabagou basin (No. 19680813, No. 19730630, No. 19890721, and No. 20060730, including human intervention periods) are presented in Figure 9 to reveal the corrective effectiveness of the DSRC method. Except for No. 20060730, the relative error of sediment yield is reduced among the above four floods. As shown in Figure 9, the estimates of the sediment transport rate process at the basin outlet have been effectively corrected through a process of estimating the time-varying size of the slope sediment yield error. For the selected four floods, peak values are obviously corrected without time difference. The NS value is increased from 0.487 to 0.957, from 0.008 to 0.962, from 0.249 to 0.929 and from 0.601 to 0.978, respectively, while the significant improvement in sediment peak brings about the increased relative error of sediment yield in No. 20060730. It is worth noting here that a better simulation of rising and falling stages is achieved, which supports the view that the DSRC method can obtain sufficient effective information for identifying errors.

With respect to the comparison of the correction effect of each basin, performance statistics for non-updated and updated sediment processes are shown in Table 6 (ranked in order of ascending size of drainage area). Overall, in all real cases, better effectiveness by using the DSRC method for updating in both quantity and processes is achieved. Qualified rates of the DSRC method are all more than 90% with the highest improvement in Dali River basin (represents 41.8%). The average NS values after updating are all greater than 0.8 meaning a good-of-fitness in process forecasting, while the least increment of 0.349 is in Jiuyuangou basin and the greatest increment of 0.515 is in Dali River basin. Meanwhile, sediment yield estimates of some floods within the study basins seem to be overcorrected. This is primarily explained by the fact that the sediment peak is significantly improved. Such results also provide an indication of the limited use of effective information for forecast correction. However, the overall trend is the decreasing error in simulation, indicating that improved accuracy is acquired in real-time flood forecasting when using the DSRC method. As shown in Table 6, the decrement of the relative error of sediment yield increases with increasing basin area; 9.88% in Jiuyuangou basin, 12.18% in Chabagou basin, and 15.72% in Dali River basin. Furthermore, it is worth mentioning that the correction effect of sediment yield tends to be less stable compared to the improvement on the peak, the choice of objective function could be a decisive factor.

For the DSRC method, the updating effect is more significant for larger basins and for basins with lower rain gauge densities. According to the indices and ⁠, in contrast to the performance of Chabagou basin, Jiuyuangou and Dali River basins perform less well in terms of NS performance and sediment peak correction. An important reason being that the reducing rainfall station density brings about decreasing quality of mean areal precipitation input data. For basins with lower rain gauge densities, the location of the storm center and the distribution of rainfall intensities are more difficult to monitor, resulting in a tendency to misrepresent the amount of actual rainfall falling over the basin. The information for each basin is listed in Table 5; note that the largest effective rainfall gauge area (smallest density of rainfall stations) is in the Dali River basin (58.30 km²/Gauge) while the least (largest density) is in Chabagou basin (4.07 km²/Gauge). Thus, less satisfying corrected results for sediment simulation of Dali River basin are received. However, the Dali River basin results in much larger improvements to the corrected performance, due to the lower accuracies of the original flood estimates. This view could also be further supported by more study results in Si et al. (2015). Correction in much smaller basin areas (Jiuyuangou basin in this study) where sediment concentration time is short seems to get less satisfying performance than larger ones due to its lack of using effective information. In summary, the error feedback correction based on DSRC provides ways to track errors by using observed and calculated information. For real cases with different basin scales, improved sediment yield estimates result in improved forecast accuracy that then correct forwards in time to reduce future sediment transportation errors. Moreover, the sensitivity of results to rainfall station density would become a good channel to further investigate the correction effect for sediment yield of the DSRC method.

The streamflow-sediment coupled model is simple, the model parameters are easily calibrated, and the model structure is physics-based. Compared with mathematical models (De Roo & Jetten 1999; Ramsankaran et al. 2013; Juez et al. 2018a, 2018b), the SSC model is not restricted to study watersheds where there has been considerable work undertaken for the description of the watershed and necessary inputs. The physical model structure can be simplified to reflect the key processes by considering the time-varying factors in structure and reflecting the time-invariant factors by parameters (Bao 1995; Bao & Wang 1997). It has been successfully used to predict streamflow and sediment transport processes in the loess region, whereas the results in the sediment simulation are more or less unsatisfactory. A real-time error correction method is needed to decrease the error of forecasting. The DSRC method, therefore, been developed and incorporated into the SSC model to correct the slope sediment yield error for improving sediment transport forecasts. The theoretical basis for the approach is the local (i.e. at current conditions) sensitivity of the system response to perturbations in the sediment yield time series. The DSRC method has the following advantages: it is simple and the performance indices will not deteriorate as the forecasting period increases. Furthermore, it is based on physical mechanisms and will not increase either the number of model parameters or model complexity.

Ideal and three real-data cases, over the range spanned by different scales of basin areas, are used to verify the effectiveness of the proposed DSRC method. Results of all cases show the accuracies of flood forecasts are greatly improved by the DSRC method. The analysis of improvement further confirms that the SSC model performs with a sound physical structure of sediment transport. This method has a significant correction effect for both sediment peak and total sediment yield. The limitation of effective information for error estimates use is the direct factor which leads to the poor performance. Meanwhile, the improvements vary with basin size and rain gauge density (as reported also by Si et al. (2015)), which is mainly attributed to the larger uncertainties in estimates of sediment yield error resulting from larger basins and lower rainfall station densities. Moreover, the gain decreases with the increase in rainfall station densities. Over the human intervention period when there are many water conservancy projects and measures involved within the basins, the DSRC method also presented a significant corrective effect to the flood forecasts.

Overall, the DSRC method merges the strengths of error tracing and feedback with the utilization of effective information. It can be directly applied to both linear and nonlinear hydrological model structures. DSRC is a comprehensive correction method, which means that it can be used to input, state, and output correction in real-time updating. The improvement of sediment prediction accuracy has direct relevance for guiding streamflow-sediment forecasting, river management and for reservoirs operation in the loess plateau area of the Yellow River. Therefore, the newly developed method is a reasonable and effective sediment yield correction method for real-time flood forecasting. While the DSRC method provides a definite improvement in performance, some forecasts are slightly overcorrected. The correction effect varies greatly with the objective function used, and the sediment balance information in the choice of objective function would require further consideration. Further ways can be explored to optimize model parameters calibration based on the DSRC method. More extensive work will continue to further evaluate the effectiveness of the DSRC method for real-time streamflow-sediment forecast updating widely, and to specifically quantify the uncertainty of sediment yield in the future.

This work was supported by the Fundamental Research Funds for Central Universities (B200201027), the National Natural Science Foundation of China (No.51709077), the Open Research Fund of Yellow River Sediment Key Laboratory (201804), Special Foundation for National Program on Key Basic Research Project (2016YFC0402703), and the Management of Central Public-Interest Scientific Institution Basal Research Fund (HKY-JBYW-2017-12).

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