Frequency Analysis, Risk, and Uncertainty in Hydroclimatic Analysis

Abstract

Analysis of extreme events like severe storms, floods, droughts is an essential component of hydrology and hydroclimatology. The extreme events have catastrophic impact on the entire agro-socioeconomic sector of a country as well as of the whole world. It has aggravated in a changing climate. Thus, it has become really important to predict their occurrences or their frequency of occurrences. This chapter focuses on different methods to analyze these extreme events and forecast their possible future occurrences. At the very beginning of the chapter, the concept of return period has been discussed elaborately which is the building block of any frequency analysis. However, identification of the best-fit probability distribution for a sample data is essential for any frequency analysis. Concept of probability paper is important in this regard, and its construction is discussed along with graphical concept of frequency factor. Next, the concept of frequency analysis is discussed using different parametric probability distributions, such as normal distribution, lognormal distribution, log-Pearson type III distribution, Gumbel’s distribution. Basic concepts of risk, reliability, vulnerability, resiliency, and uncertainty are also explained which are inevitable in any kind of hydrologic design based on frequency analysis of extreme events.

Exercise

5.1

If the return period of a hurricane is 500 years, find out the probability that no such hurricane will occur in next 10 years. Consider occurrence of such hurricanes follows Poisson distribution. Ans:0.98

5.2

The annual rainfall magnitudes at a rain gauge station for a period of 20 years are given below in the table

Year	Annual rainfall (cm)	Year	Annual rainfall (cm)
1975	120	1985	100
1976	85	1986	108
1977	67	1987	105
1978	95	1988	113
1979	108	1989	98
1980	92	1990	93
1981	98	1991	76
1982	87	1992	83
1983	79	1993	91
1984	86	1994	87

Determine the following

• (a) The probability of occurrence of an annual rainfall more then 100 cm. Ans:0.286

• (b) Dependable (80%) rainfall at this rain gauge station. Ans:79.84 cm.

5.3

The records of peak annual flow in a river are available for 25 years. Plot the graph of return period versus annual peak flow, and estimate the magnitude of peak flow for (a) 50 year and (b) 100 year return period. Use Weibull plotting position formula. Ans: (a) 6991 cumec, (b) 7912 cumec.

Year	Annual peak flow (cumec)	Year	Annual peak flow (cumec)
1960	4780	1973	989
1961	2674	1974	1238
1962	4432	1975	1984
1963	1267	1976	2879
1964	3268	1977	2276
1965	3789	1978	3256
1966	2348	1979	3674
1967	2879	1980	4126
1968	3459	1981	4329
1969	4423	1982	2345
1970	5123	1983	1678
1971	4213	1984	1198
1972	3367

5.4

Use the annual peak flow data in Exercise 5.3, and find out the best-fits distribution for the data using probability paper among (a) normal distribution, (b) lognormal distribution, and (c) Gumbel’s distribution.

5.5

From analysis of flood peaks in a river, the following information is obtained

• (a) The flood peak data follows lognormal distribution.

• (b) Flood peak of 450 cumec has a return period of 50 year.

• (c) Flood peak of 600 cumec has a return period of 100 year.

Estimate the flood peak in the river with 1000-year return period. Ans:1347 cumec.

5.6

Repeat the Exercise 5.5 if the flood peak data follows Gumbel’s extreme value distribution. Ans:1096 cumec

5.7

Maximum annual flood at a river gauging station is used for frequency analysis using 30-year historical data. The frequency analysis performed by Gumbel’s method provides the following information.

Return period (years)	Max. annual flood (cumec)
50	1060
100	1200

• (a) Determine the mean and standard deviation of sample data used for frequency analysis. (Ans: mean $=$ 385 cumec, std. deviation $=$ 223 cumec)

  Sl. No.	Station	Sample size (years)	Mean annual flood (cumec)	Std. deviation of annual flood (cumec)
  1	A	92	6437	2951
  2	B	54	5627	3360

• (b) Estimate the magnitude of flood with return period 500- year (Ans: 1525 cumec).

5.8

Consider the following annual flood data at two river gauging stations

• (a) Estimate the 100- and 1000-year floods for both the stations. Use the Gumbel method.

• (b) Determine the 95% confidence interval for the predicted value.

Ans: (a) $Q_{100} =16359\pm 2554\,$cumec and $Q_{1000} =22023\pm 3744\,$cumec and (b) $Q_{100} =17298\pm 3885\,$cumec and $Q_{1000} =23935\pm 5721\,$ cumec.

5.9

A structure is proposed to be built within the 50-year flood plain of the river. If the life of the industry is 25 years, what is the reliability that the structure will never face flood. (Ans: 0.603)

5.10

A bridge with 25 years expected life is designed for a flood magnitude of 100 years. (a) What is the risk involved in the design? (b) If only 10% risk is acceptable in the design, what return period should be adopted in the design? (Ans: (a) 0.222 (b) 240 years).

5.11

Frequency analysis of flood data at a river gauging station is performed by log-Pearson type III distribution which yields the following information

Coefficient of skewness $=$ 0.4

Return period (years)	Max. annual flood (cumec)
50	10600
100	13000

Estimate the magnitude of flood with return period of 1000 years (Ans:23875 cumec).

5.12

The following table gives annual peak flood magnitudes in a river. Estimate the flood peaks with return period 10, 100, and 500 years using (a) Gumbel’s extreme value distribution, (b) log-Pearson type III distribution, and (c) lognormal distribution

Year	Q (cumec)	Year	Q (cumec)	Year	Q (cumec)	Year	Q (cumec)
1950	1982	1965	1246	1980	2291	1995	1252
1951	1705	1966	2469	1981	3143	1996	983
1952	2277	1967	3256	1982	2619	1997	1339
1953	1331	1968	1860	1983	2268	1998	2721
1954	915	1969	1945	1984	2064	1999	2653
1955	1557	1970	2078	1985	1877	2000	2407
1956	1430	1971	2243	1986	1303	2001	2591
1957	583	1972	3171	1987	1141	2002	2347
1958	1325	1973	2381	1988	1642	2003	2512
1959	2200	1974	2670	1989	2016	2004	2005
1960	1736	1975	1894	1990	2265	2005	1920
1961	804	1976	1518	1991	2806	2006	1773
1962	2180	1977	1218	1992	2532	2007	1274
1963	1515	1978	966	1993	1996	2008	2466
1964	1903	1979	1484	1994	1540	2009	2387

(Ans: (a) $Q_{10}= 2829$ cumec, $Q_{100}= 4066$ cumec, $Q_{500}= 4672$ cumec, (b) $Q_{10}= 2762$ cumec, $Q_{100}= 3351$ cumec, $Q_{500}= 3553$ cumec, (c) $Q_{10}= 2851$ cumec, $Q_{100}= 4142$ cumec, $Q_{500}= 5045$ cumec)

5.13

The following table gives soil moisture (SM) data at a particular location. Consider PWP as 0.12 and evaluate reliability, resilience, and vulnerability of the data.

Day	SM	Day	SM	Day	SM	Day	SM
1	0.0816	11	0.4080	21	0.0717	31	0.2834
2	0.2253	12	0.3745	22	0.2253	32	0.2953
3	0.1944	13	0.1647	23	0.4149	33	0.1647
4	0.3370	14	0.2654	24	0.3370	34	0.1190
5	0.1208	15	0.1300	25	0.2500	35	0.0655
6	0.0954	16	0.2703	26	0.1423	36	0.0532
7	0.0562	17	0.3837	27	0.1258	37	0.0296
8	0.2382	18	0.3152	28	0.1228	38	0.2145
9	0.1949	19	0.1448	29	0.2948	39	0.1526
10	0.3500	20	0.1152	30	0.4024	40	0.1210

(Ans: Reliability $=$ 0.775, resilience $=$ 0.889, vulnerability $=$ 0.044).