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The upper and lower bounds on the level curve were estimated numerically by solving Equations (16) and (17), and assuming for simplicity (although other assumptions are possible) α1 = α2 = ɛ/2, with ɛ = 0.05. The upper and lower bounds are denoted as B1 and C1, respectively, in Figure 7. It is found that the bounds are close to the horizontal asymptote (i.e., w7 = 61.49 × 108 m3 for T= 1,000 and w7 = 50.23 × 108 m3 for T= 200) and vertical asymptote (i.e., qp = 22,800 m3/s for T= 1,000 and qp = 19,300 m3/s for T= 200) due to the small value assumed for the probability level ɛ. The upper and lower bounds were also calculated by the boundary identification method proposed by Volpi & Fiori (2012). The results are also presented in Table 5 and the derived bounds are denoted as B2 and C2, as shown in Figure 7. It is shown that the bounds estimated by the proposed method and that proposed by Volpi & Fiori (2012) are very similar.

Table 5

Comparison of the lower and upper bounds of the quantile curve

Boundary identification methodReturn periodLower bound
Upper bound
Qp (m3/s)W7 (108 m3)Qp (m3/s)W7 (108 m3)
Volpi & Fiori (2012)  1,000 22,930 65.84 26,080 61.54 
200 19,350 50.27 22,460 55.86 
Developed method 1,000 23,000 65.76 26,100 61.52 
200 19,400 54.49 22,500 50.26 
Boundary identification methodReturn periodLower bound
Upper bound
Qp (m3/s)W7 (108 m3)Qp (m3/s)W7 (108 m3)
Volpi & Fiori (2012)  1,000 22,930 65.84 26,080 61.54 
200 19,350 50.27 22,460 55.86 
Developed method 1,000 23,000 65.76 26,100 61.52 
200 19,400 54.49 22,500 50.26 

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