Applied Hydrology 622 U2600

Department of Bioenvironmental Systems Engineering, NationalTaiwanUniversity

Homework 1–Frequency Analysis (I) [Due 03/24/2016]

Annual maximum rainfall depths of various design durations at a raingauge site are shown in Table 1. It is assumed that all annual maximum series can be characterized by the Pearson Type III distribution.

  1. Conduct the chi-square GOF test (using five equiprobable categories) to see whether the assumption is reasonable.
  2. Establish the intensity-duration-frequency relationships for the raingauge site. Design storm depths of 5, 10, 25, 50, and 100-yr return periods should be calculated and the Horton’s equation with b=0 be used to fit the IDF curves.
  3. A random variable X with its cumulative distribution F(X) is shown in Figure 1. Random variable Y is a monotonic transformation of X by a function Y=g(X). Sketch in figure 1 the transformation between X and Y.
  4. Use the probability plotting GOF test to see whether the data in Table 2 is originated from a Gumbel distribution. (Use the Weibull plotting position formula for calculation of sample quantiles.)

Table 1. Data of annual maximum rainfall depths (mm).

Duration
Year / 1hr / 2 hrs / 6 hrs / 12 hrs / 18 hrs / 24 hrs
54 / 535 / 662 / 1082 / 1315 / 1486 / 1662
55 / 455 / 653 / 1290 / 1744 / 1775 / 1775
56 / 682 / 915 / 1410 / 1558 / 1558 / 1558
57 / 589 / 934 / 2064 / 3034 / 3369 / 3924
58 / 471 / 898 / 1787 / 2427 / 3017 / 3411
59 / 418 / 538 / 794 / 963 / 1032 / 1033
60 / 529 / 842 / 1936 / 2599 / 2760 / 2817
61 / 185 / 303 / 701 / 1313 / 1741 / 2042
62 / 908 / 1486 / 2892 / 3412 / 4398 / 4424
63 / 386 / 561 / 1144 / 1975 / 2646 / 2965
64 / 502 / 902 / 1199 / 1205 / 1205 / 1205
65 / 359 / 526 / 659 / 1028 / 1218 / 1287
66 / 662 / 1020 / 1748 / 2152 / 2943 / 3156
67 / 432 / 740 / 1690 / 2720 / 3570 / 4370
68 / 660 / 950 / 1370 / 2160 / 2510 / 2560
69 / 320 / 560 / 1150 / 2030 / 2030 / 2030
70 / 670 / 820 / 1200 / 1930 / 2500 / 2620
71 / 390 / 600 / 1010 / 1590 / 1720 / 2270
72 / 290 / 510 / 920 / 1270 / 1520 / 1680
73 / 600 / 870 / 1730 / 1820 / 1820 / 1860
74 / 490 / 690 / 1020 / 1760 / 1980 / 2100
75 / 610 / 990 / 1940 / 2270 / 2510 / 2900
76 / 790 / 1170 / 3060 / 5730 / 7490 / 9240
77 / 860 / 1200 / 2260 / 2470 / 2470 / 2470
78 / 370 / 580 / 1510 / 2590 / 3090 / 3110
79 / 570 / 910 / 1320 / 1860 / 2000 / 2140
80 / 850 / 1190 / 1780 / 1830 / 1830 / 1830
81 / 310 / 480 / 1000 / 1390 / 1390 / 1390
82 / 590 / 1050 / 1270 / 1500 / 1660 / 1660
83 / 680 / 890 / 1490 / 2180 / 2430 / 2800

Figure 1.

Table 2.

35.44 / 62.02 / 65.63 / 87.12 / 59.94 / 99.30 / 94.41 / 54.91 / 5.58 / 21.40
43.76 / 23.43 / 10.92 / 48.16 / 38.64 / 69.36 / 23.29 / 59.49 / 42.03 / 47.26
58.41 / 80.71 / 45.33 / 21.52 / 37.78 / 43.23 / 20.25 / 72.35 / 58.53 / 59.47
117.41 / 26.50 / 53.43 / 15.13 / 130.77 / 50.01 / 59.96 / 37.64 / 165.26 / 3.77
4.64 / 21.08 / 0.24 / 26.44 / 39.44 / 52.90 / 97.43 / 35.05 / 60.45 / 53.21
  1. Goodness-of-fit tests (I)

(1)Generate a random sample of size n=60 from a Gaussian (normal distribution with .

(2)Conduct goodness-of-fit test on the generated data with Gaussian distribution as the null hypothesis using chi-squared test, K-S test and LMRD-based test.

  1. Goodness-of-fit tests (II)

(1)Generate a random sample of size n=1000 from a Gaussian (normal distribution with .

(2)Conduct goodness-of-fit test on the generated data with Gaussian distribution as the null hypothesis using chi-squared test, K-S test and LMRD-based test.