System response curve correction method of runoff error for real-time flood forecast

Multiple factors including rainfall and underlying surface conditions make river basin real-time flood forecasting very challenging. It is often necessary to use real-time correction techniques to modify the forecasting results so that they reach satisfactory accuracy. There are many such techniques in use today; however, they tend to have weak physical conceptual basis, relatively short forecast periods, unsatisfactory correction effects, and other problems. The mechanism that affects real-time flood forecasting error is very complicated. The strongest influencing factors corresponding to this mechanism affect the runoff yield of the forecast model. This paper proposes a feedback correction algorithm that traces back to the source of information, namely, modifies the watershed runoff. The runoff yield error is investigated using the principle of least squares estimation. A unit hydrograph is introduced into the real-time flood forecast correction; a feedback correction model that traces back to the source of information. The model is established and verified by comparison with an ideal model. The correction effects of the runoff yield errors are also compared in different ranges. The proposed method shows stronger correction effect and enhanced prediction accuracy than the traditional method. It is also simple in structure and has a clear physical concept without requiring added parameters or forecast period truncation. It is readily applicable in actual river basin flood forecasting scenarios.

it is believed that there is a correlation between the previous residual series and the following residual series.
There is also a p-order autoregressive (AR) model available for error estimation. In a real-time flood forecasting scenario, however, errors occurring at different times do not meet such assumptions. For example, near the peak of the flood, there will be sudden changes in the flow and the errors are not related. If the AR model is used for correction, the result will not be ideal.
2. Truncated forecast period. Using the information from the errors between the measured and calculated flow at the exit section to establish an AR model (Todini & Jones ) and further correcting the flow calculation process severely shortens the forecast period. There is a demand for an effective error correction technique with physical meaning to support accurate real-time flood forecasting. Errors emerge in the real-time flood forecasting system for many reasons and the mechanism that affects these errors is complicated. Inductive analysis of these factors suggests that most of them affect the runoff yield of the forecast model. In view of this, a feedback correction method was established in this study that traces back to the source of information.
The concept of the response curve of the linear confluence model system was developed first. The resultant linear response model system response matrix feedback was then used to correct the outlet cross-section flow error. In other words, the runoff yield of the basin period was corrected and the corrected runoff was substituted into the model to recalculate the flow correction result of the river basin outlet section. It is important to emphasize that this method uses runoff as an input variable; each stage after runoff is a distinct system and the output is the streamflow of the outlet. The key to operating this method is to replace the correction of flow error of the outlet with the correction of the runoff process.

METHODOLOGY
Introduction to flow concentration system The river basin flow concentration can be regarded as a system. The input is the net rain process and the output is outlet section flow process (Figure 1).
In this system, the flow process at the catchment outlet can also be referred to as the response of the basin to the process of net rainfall, or simply, the 'river basin response'.
The relationship between river basin response Q(t) and net rainfall input I(t) can be expressed as follows: where φ is an operator denoting the operational relationship between the system input and response (Bao ).
Basin runoff is the first link in the formation of flow. It is a dynamic concept that presents spatial-temporal changes including the spatial development of runoff yield area at different times, as well as changes in runoff generation intensity over time as rain falls. The runoff is generated mainly on the slope of the basin; its role in the system may be very complicated (Zhao ). The proportions of the slope area are different in different basins as well. Various factors that influence the runoff generation on the slope, including vegetation, soil, slope ratio, land use, slope area, and location perform differently in basins of different sizes.
Therefore, there are many influencing factors in the runoff model simulation and the errors are complicated.
As shown in Figure 2, after runoff is generated in the river basin, the runoff yield R enters the confluence stage.
The confluence stage can be divided into two sub-stages: overland confluence and river network confluence. To facilitate the simplification of the confluence structure, before the overland confluence, the runoff yield R is divided into surface water, subsurface water, and groundwater portions each with different respective confluence characteristics; they participate in the calculation independently. Finally, they converge into the river basin outlet section at the river network confluence stage.
Runoff correction method based on the system response curve (RSRC) The confluence part of the hydrological forecast model is regarded as a system. The response function of the system can be expressed as follows: where X(t) represents various variables in the model (input   the basin, that is, there is a response process. The parameters of the model do not change within any certain period of time. Only the system response caused by the changes in the runoff yield R was considered in this study; no response caused by other variables is discussed here. Formula (2) can thus be rewritten as: where R ¼ [r 1 , r 2 , r 3 , Á Á Á , r n ] T indicates the runoff yield series. After performing total differential equation of the above system response functions, then: is the error value of the runoff yield, Q(R, θ, t) is the measured flow process Q O , and Q(R C , θ, t) is the calculated flow process Q C at the outlet section of the basin.
It is assumed that the length of the sample series is T and The matrix form of Formula (6) is: where ΔR is the magnitude of error of the runoff yield to be solved E ¼ [e 1 , e 2 , e 3 , Á Á Á , e T ] T is the error term and the white noise vector. The expression of the U matrix is: Each term in the U matrix of the above formula can be solved by the difference approximation of the following formula: where t ¼ 1…T, i ¼ 1…n. When i does not change and t changes from 1 to T, the T term difference value is a column in the U matrix. This column is exactly the system response series corresponding to the unit variation of runoff yield r i , which is referred to here as the system response curve corresponding to the runoff yield r i . This response curve is influenced by the runoff yield values in other periods. The runoff yield series changes in real time over time, so the system response curve corresponding to the runoff yield r i also changes dynamically.
The formula for calculating the correct amount of the runoff yield from Formula (6) is: and the corrected runoff yield series is: where R 0 C must satisfy R 0 C ! 0 and when PE > 0, R 0 C PE.
R 0 C in Equation (11) can be substituted into the calculation model for recalculation. The final correction result can then be obtained as follows: @Q @R in Equation (4) is the system response curve. The U matrix in Equation (8)

NUMERICAL SIMULATION AND ANALYSIS Numerical simulation case
As discussed below, an ideal model with a unit hydrograph of confluence as the structure was used to verify the correctness, rationality, and effectiveness of the proposed runoff correction method. The 'ideal model' is one in which all terms of inputs, outputs, and errors are known (Bao et al. a, b). The structure of the ideal model is shown below.
As shown in Figure 3, with the system response curve matrix composed of the unit hydrograph, the runoff R is converted into an outlet flow process. This can be expressed as follows:  Q is the flow process of the outlet; U is a matrix composed of the unit hydrograph of the confluence curve and each column in the matrix is the confluence curve of the basin.
R is the runoff yield vector. U is an m × p-dimensional matrix composed of the unit hydrograph ordinates; the matrix is a lower triangular matrix.
The runoff depths of six periods were used as the model inputs. The single unit hydrograph and double unit hydrograph with the period rainfall as the variable were used as  (Table 1).
Four error models were set for the runoff errorsΔR, so four error models were used to validate the proposed method.
1. Uniform random distribution of error variation (À10% to When R 10 mm, unit hydrograph 1 was used; When R>10 mm, unit hydrograph 2 was used. Twelve years of historical data (1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999) including hourly precipitation, pan evaporation, and discharge were used in this study. Hydrological data were used for the case study including daily and hourly rainfall/runoff and daily evaporation rate. The runoff data were calculated based on the change in the water level of the reservoirs. The rainfall data were obtained from rain gauges near the dams. The daily evaporation data were obtained by using daily evaporation pan data from an evaporation station near the dams.
Hourly (30 flood event) data were used to test the performance of the flood forecasting correction method.

Evaluation criteria
In order to assess the accuracy of modeling results, several statistical indices were selected to judge the correction method performance including the Nash-Sutcliffe Efficiency (NSE)  NSE: Relative error of flood volume (ΔW ): Relative error of flood peak (ΔTP):  INS: where QC i and Q i are the calculated and measured flow, The relative flood volume error (ΔW ) and relative flood peak error (ΔTP) measure the bias of model performance.
The optimal value is 0.0, which means that the model has an unbiased flow simulation. An NSE value close to 1 means the model is of good quality and is highly reliable; NSE close to 0 means that the simulation results are close to the average value of the observed values, that is, the overall result is reliable, but the process simulation error is large. When NSE is far less than 0, the model is not credible.

NUMERICAL RESULTS AND DISCUSSION
Single system response curve case  Notes: W m is the measured flood volume; W o is the calculated flood volume without correction; W t is the calculated flood volume with correction; TP m refers to the measured flood peak; TP o refers to the calculated flood peak without correction; TP t refers to the calculated flood peak with correction; the relative error between the computed and measured flood volume without correction ΔW o ¼ (W o ÀW m )/W m * 100%; the relative error between the computed and measured flood volume with correction ΔW t ¼ (W t ÀW m )/W m * 100%; the relative error between the flood peak of the computed and the measured flow without correction ΔTPo ¼ (TPoÀTPm)/TPm * 100%; the relative error between the flood peak of the computed and the measured flow with correction ΔTP t ¼ (TP t ÀTP m )/TP m * 100%; NSE o is the deterministic coefficient without correction; NSE t is the deterministic coefficient with correction. The following tables are the same.

mentioned above, the effect coefficient INS indicates
whether the correction effect is good or poor, that is, the effect relative to the original model error.
From the perspective of model calculations, Tables 2-5 show that after the four distribution functions produce errors. The model calculation results have an average  and À11% (Tables 4 and 5, ΔW o ), respectively. The flood peak forecast errors at ±10% uniform distribution and that subject to the N (0, 0.05 2 ) normal distribution reach À10 and À9.5% (Tables 2 and 3, ΔTP o ), respectively, while those of flood volume reach À4 and À4% (Tables 2 and 3, ΔW o ).
In effect, the unit hydrograph forecasting model is applicable under the given assumptions for the basin but must be further corrected in real time during the forecasting process. normal distribution reach À1.82 and À1.45% (Tables 2 and   3, ΔTP t ) while those of the flood volume reach À0.66 and À1.24% (Tables 2 and 3  or later in time and the resulting peak shapes also fluctuate.
The hydrograph shown in Figure 8 also indicates that the corrected streamflow obtained using the proposed correction method is closer to the observed streamflow than that without correction.
Based on the above results, the proposed correction method, which seeks the feedback corrected runoff based on the model structure from the perspective of a single system response curve, is reasonable and effective with relatively high calculation accuracy under different variation ranges of the same distribution or different characteristic value distributions. The results also suggest that the correction effect improves as the random error increases.
However, after the runoff yield correction, the simulated uniform distribution subject to (À20%, 20%) and the normal distribution subject to N (0, 0.1 2 ) improved substantially.
The flood peak errors (À20%, 20%) of uniform distribution are À3.52 and À3.54%. The errors of flood volume are À1.51 and À1.49%. The flood peak errors of normal     As shown in Figure 9, the errors plus the error simulation effects of two different distributions were ultimately improved by the correction method based on the system response curve feedback to correct the runoff yield, especially in the vicinity of the flood peak. The proposed method, based on the feedback correction of the system response curve, remains feasible and effective as the system response curve changes.
Numerical results Figure 10 shows the relative flood volume error (ΔW ) of four error models with correction compared with without correction in a single system response curve (RSRC-S).
The relative flood peak error (ΔTP) is shown in Figure 11. Comparison between the correction result and the ΔTP t value reveals a small range from 0.59 to 1.3%, while relative flood peak error (ΔTP) decreases by 79.2% on average compared with the uncorrected value.   and a 10% improvement with N (0, 0.1 2 ). The double system response curve correction method provides better results for real-time correction. This is mainly because this approach uses more information about the unit hydrograph.
Under the same error distribution, it can obtain better results than a method with a single curve.

Real study results
Detailed results obtained by the application of the RSRC in an actual basin are provided in Table 8  After correction, ΔW t was only 3.16% marking a decrease of 10.29%. The relative error of flood volume did not decrease in certain floods but rather may have increased (e.g., 970,702, 960,328, 950,625, 940,425, 920,704). The relative error of flood volume calculated by the original model was small in these cases but the certainty coefficient was below 0.900, which suggests that the model simulation was relatively balanced in terms of the total amount but it was unsatisfactory from the start of the process. After correction, the relative error of the flood volume did not significantly increase. The revised flood volume was still below 3% and the certainty coefficient significantly improved, indicating that the overall process simulation improved compared to that before the correction.
The proposed method produced marked effects in the actual data for Qilijie Station. Under the model response theory, the proposed runoff correction is reasonable and effective in terms of the physical cause mechanism and yields highly accurate calculations in terms of actual watershed inspection precision.  context. It is necessary to introduce an error correction technique to the real-time flood forecasting system.
The concept of basin linear confluence system was utilized in this study to construct a conceptual runoff error correction model. A model system response curve correction method was established accordingly which corrects the runoff by the model system response curve feedback based on the errors between the calculated and measured flow results. An error correction theory was also developed based on the structure itself, which has a physical basis. The proposed method enhances forecasting accuracy without truncating the forecast period or requiring additional parameters.
For verification based on the ideal model, the linear system ideal model was first divided into single system was 11%. To this effect, the proposed method is valid. The method presented in this paper is one of the few error correction models that has a physical formation mechanism and requires no additional model parameters to operate. The method is readily applicable. However, the applicability of the correction method to areas with significant nonlinearity in the basin confluence system still needs further research.