湖泊科学   2018, Vol. 30 Issue (2): 533-541.  DOI: 10.18307/2018.0224. 0

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SI Wei, BAO Weimin, QU Simin, SHI Peng. Real-time flood forecast updating method based on mean areal rainfall error correction. Journal of Lake Sciences, 2018, 30(2): 533-541. DOI: 10.18307/2018.0224.
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2017-03-31 收稿
2017-06-26 收修改稿

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(1: 河海大学商学院, 南京 210098)
(2: 河海大学水文及水资源学院, 南京 210098)

Real-time flood forecast updating method based on mean areal rainfall error correction
SI Wei 1,2, BAO Weimin 2, QU Simin 2, SHI Peng 2
(1: Business School of Hohai University, Nanjing 210098, P. R. China)
(2: College of Hydrology and Water Resources, Hohai University, Nanjing 210098, P. R. China)
Abstract: The accuracy of flood forecasts generated using spatially lumped hydrological models can be severely affected by errors in the estimates of mean areal rainfall. The quality of the latter depends both on the size and type of errors in point-based rainfall measurements, and on the density and spatial arrangement of rain gauges in the basin. Here, we use error feedback correction, based on the rainfall system response curve method, to compute updated estimates of the rainfall inputs. The capability of the method to improve the accuracy of real-time flood forecasts is demonstrated using the Xin'anjiang (XAJ) model applied to 16 flood events of the Wangjiaba Basin. The result shows that the forecast improvement is significant. For the Wangjiaba Basin, we also examinethe performance of the method for different rain gauge densities, and find that forecast improvement is more significant when gauge densities are lower. The method is relatively simple to apply and can improve the accuracy of real-time model predictions without increasing either model complexity and/or the number of model parameters.
Keywords: Real-time updating    mean areal rainfall error    system response curve    Xin'anjiang model    Wangjiaba Basin

1 修正方法 1.1 降雨系统响应曲线

 图 1 一般水文模型系统示意 Fig.1 A systems diagram of general hydrological model
 $Q\left( t \right) = f\left[ {P\left( t \right),E\left( t \right),X\left( t \right),\theta } \right]$ (1)

 $Q\left( P \right) = f\left( P \right)$ (2)

 ${\rm{d}}Q = \frac{{\partial Q\left( P \right)}}{{\partial P}}\left| {_{P = {P_0}}} \right.{\rm{d}}P$ (3)

 $Q\left( P \right) \approx Q\left( {{P_0}} \right) + \frac{{\partial Q\left( P \right)}}{{\partial {p_1}}}\left| {_{P = {P_0}}} \right.\Delta {p_1} + \frac{{\partial Q\left( P \right)}}{{\partial {p_2}}}\left| {_{P = {P_0}}} \right.\Delta {p_2} + L + \frac{{\partial Q\left( P \right)}}{{\partial {p_m}}}\left| {_{P = {P_0}}} \right.\Delta {p_m}$ (4)

 $\left\{ \begin{array}{l} {Q_1}\left( P \right) \approx {Q_1}\left( {{P_0}} \right) + \frac{{\partial {Q_1}\left( P \right)}}{{\partial {p_1}}}\left| {_{P = {P_0}}} \right.\Delta {p_1} + \frac{{\partial {Q_1}\left( P \right)}}{{\partial {p_2}}}\left| {_{P = {P_0}}} \right.\Delta {p_2} + L + \frac{{\partial {Q_1}\left( P \right)}}{{\partial {p_m}}}\left| {_{P = {P_0}}} \right.\Delta {p_m}\\ {Q_2}\left( P \right) \approx {Q_2}\left( {{P_0}} \right) + \frac{{\partial {Q_2}\left( P \right)}}{{\partial {p_1}}}\left| {_{P = {P_0}}} \right.\Delta {p_1} + \frac{{\partial {Q_2}\left( P \right)}}{{\partial {p_2}}}\left| {_{P = {P_0}}} \right.\Delta {p_2} + L + \frac{{\partial {Q_2}\left( P \right)}}{{\partial {p_m}}}\left| {_{P = {P_0}}} \right.\Delta {p_m}\\ {Q_L}\left( P \right) \approx {Q_L}\left( {{P_0}} \right) + \frac{{\partial {Q_L}\left( P \right)}}{{\partial {p_1}}}\left| {_{P = {P_0}}} \right.\Delta {p_1} + \frac{{\partial {Q_L}\left( P \right)}}{{\partial {p_2}}}\left| {_{P = {P_0}}} \right.\Delta {p_2} + L + \frac{{\partial {Q_L}\left( P \right)}}{{\partial {p_m}}}\left| {_{P = {P_0}}} \right.\Delta {p_m} \end{array} \right.$ (5)

 $Q\left( P \right) = Q\left( {{P_0}} \right) + S \cdot \Delta P + W$ (6)

 $S = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial {Q_1}\left( P \right)}}{{\partial {p_1}}}}& \cdots &{\frac{{\partial {Q_1}\left( P \right)}}{{\partial {p_m}}}}\\ {\frac{{\partial {Q_2}\left( P \right)}}{{\partial {p_1}}}}& \cdots &{\frac{{\partial {Q_2}\left( P \right)}}{{\partial {p_m}}}}\\ {}& \vdots &{}\\ {\frac{{\partial {Q_L}\left( P \right)}}{{\partial {p_1}}}}& \cdots &{\frac{{\partial {Q_L}\left( P \right)}}{{\partial {p_m}}}} \end{array}} \right]$ (7)

 ${\rm{d}}Q = S \cdot {\rm{d}}P$ (8)

 $\frac{{\partial {Q_j}\left( P \right)}}{{\partial {P_i}}} = \frac{{{Q_j}\left( {{p_1}, \cdots ,{p_i} + \Delta {p_i}, \cdots ,{p_m}} \right) - {Q_j}\left( {{p_1}, \cdots ,{p_i}, \cdots ,{p_m}} \right)}}{{\Delta {p_i}}}\left( {i = 1、\cdots 、m,j = 1、\cdots 、L} \right)$ (9)

 图 2 降雨系统响应曲线示意图 Fig.2 The schematic diagram of rainfall system response curve
1.2 降雨误差估计

 $\left\{ \begin{array}{l} {e_{{Q_1}}} = {S_{11}} \cdot {e_{{P_1}}} + {S_{12}} \cdot {e_{{P_2}}} + \cdots + {S_{1m}} \cdot {e_{{P_m}}}\\ {e_{{Q_2}}} = {S_{21}} \cdot {e_{{P_1}}} + {S_{22}} \cdot {e_{{P_2}}} + \cdots + {S_{2m}} \cdot {e_{{P_m}}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \\ {e_{{Q_L}}} = {S_{L1}} \cdot {e_{{P_1}}} + {S_{L2}} \cdot {e_{{P_2}}} + \cdots + {S_{Lm}} \cdot {e_{{P_m}}} \end{array} \right.$ (10)

 ${E_Q} = S \cdot {E_P}$ (11)

 ${E_Q} = S \cdot {E_P} + \zeta$ (12)

 ${{E'}_P} = {\left( {{S^T} \cdot S} \right)^{ - 1}} \cdot {S^T} \cdot {E_Q}$ (13)

2 应用检验

(1) NS(Nash-Sutcliffe)系数：

 $NS = 1 - \sum\limits_{j = 1}^L {{{\left( {{M_{\rm{C}}}\left( j \right) - {M_{\rm{O}}}\left( j \right)} \right)}^2}} /\sum\limits_{j = 1}^L {{{\left( {{M_{\rm{O}}}\left( j \right) - \overline {{M_{\rm{O}}}} } \right)}^2}}$ (14)

(2) 相对误差：

 $\Delta M\left( \% \right) = \left[ {\left( {{M_{\rm{C}}} - {M_{\rm{O}}}} \right)/{M_{\rm{O}}}} \right] \times 100\%$ (15)

(3) 修正后相对误差提高幅度：

 $IRM = \left| {R{M_{{\rm{bu}}}} - R{M_{{\rm{au}}}}} \right|/\left| {R{M_{{\rm{bu}}}}} \right| \times 100\%$ (16)

(4) NS系数提高幅度：

 $INS = \left( {N{S_{{\rm{bu}}}} - N{S_{{\rm{au}}}}} \right)/\left( {1 - N{S_{{\rm{bu}}}}} \right) \times 100\%$ (17)

2.1 理想案例应用检验

 图 3 给定雨量误差与估计雨量误差对比 Fig.3 Comparison between the given rainfall errors and the estimated rainfall errors

 图 4 理想案例流量过程对比 Fig.4 The hydrographs comparison of ideal case
 ${Q_{\rm{o}}} = Q\left( {P + \Delta P} \right) + \varepsilon$ (18)

2.2 实际流域应用

 图 5 王家坝流域3种不同站点密度的水系 Fig.5 Three different rain gauge distributions of Wangjiaba Basin

 图 6 王家坝流域16场洪水的IRRD和IRPF效果对比 Fig.6 The comparison between IRRD and IRPF of 16 flood events in Wangjiaba Basin

 图 7 王家坝流域不同雨量站密度下修正效果提高幅度对比 Fig.7 The improvement comparisons of different rain gauge densities in Wangjiaba Basin

3种雨量站密度情况下，本修正方法对于径流深修正效果的平均提高幅度均比洪峰流量修正效果的提高幅度大，进一步说明此方法能够更好的修正洪水预报过程中水量平衡误差(图 8).

 图 8 王家坝流域不同雨量站密度下IRRD与IRPF对比结果 Fig.8 The comparison of IRRD and IRPF of different rain gauge densities in Wangjiaba Basin

3 结论

4 参考文献