Problem

The following annual total rainfall data for Houston Intercontinental Airport were collect...

The following annual total rainfall data for Houston Intercontinental Airport were collected over a 21-yr period.

Year	Rainfall (in.)	Year	Rainfall (in.)	Year	Rainfall (in.)
1970	48.19	1977	34.94	1984	48.19
1971	37.83	1978	44.93	1985	49.14
1972	50.80	1979	58.97	1986	44.93
1973	70.16	1980	38.99	1987	40.60
1974	49.29	1981	55.98	1988	22.93
1975	50.97	1982	42.87	1989	52.73
1976	54.62	1983	53.21	1990	40.37

(a) Compute the mean, variance, and the skewness coefficient (Cs).
(b) Plot a histogram using 5-in. intervals.
(c) Fit the data with the normal distribution. Sketch the normal PDF on the histogram of part (b), scaling such that the areas under the histogram and under the PDF are the same (e.g., see Figure).
(d) Find the value of the 10-yr annual rainfall total.
(e) Which years most closely represent the mean annual and 10-yr rainfalls for Houston?

Figure

Four PDFs fit to data for the Siletz River. Fit is by the method of moments, as shown in the text,

with moments given in Example.

Example

MOMENTS OF AN ANNUAL MAXIMUM SERIES

The series of 75 annual maximum flows for the Siletz River is shown in Table 1. Evaluate the mean and standard deviation of the original data and of the logs (base 10) of the data using Equations 1 and 2. Compare the various skewness estimates.

SOLUTION

This exercise is easily performed in a spreadsheet. For example, Excel functions to perform the moment calculations for the column of data in Table 1 are shown below. Moments for log10 DATA are performed on the log10 transformation of the column of flows. Note, for instance, that the log of the mean flow is not equal to the mean of the logs; that is, log (20452) = 4.3107 ≠ 4.2921. (Ample significant figures should be carried when working with logarithms.) The results are presented in Table 2.

A coefficient of variation of the flow data of 30% indicates wide variability of the flows, as is evident in Figure 1. Considering the logi0 values, using the regional data from Figure 4 gives Cm = 0.0 for the north-central coastline of Oregon. A weighted average using Equation 4 gives an alternative estimate for the skewness of the logs of −0.1242, somewhat less in magnitude than the station value given by Equation 3. Which value is more nearly correct could be determined from a study of other stations in the region; the practical effect of the small difference in this case is minor. For purposes of examples in this chapter, the weighted value of −0.1242 will be used [Equation 4]. The weighted average skewness is computed as follows:

For the Siletz River data for Oregon, using Equation 4, Cm = 0.0 and | Cs | = 0.1565. Therefore,

A = − 0.33 + 0.08(0.1565) = − 0.31748

B = 0.94 -0.26(0.1565) = 0.899315

and V(Cm) = 0.302 for the map.

Finally,

and 1 − W = 0.207

and

Cw = 0.793(−0.1565) + (0.207)(0.0) = − 0.1242.

Equation 1

Equation 2

Equation 3

Equation 4

Table 1

Data and Freauency Analysis Computations for the Siletz River, at Siletz, Oreaon

Wate Year	Peak Flow (cfs)	Water Year	Peak Flow (cfs)	Water Year	Peak Flow (cfs)
1925	18,800	1950	16,400	1975	21,500
1926	16,800	1951	16,600	1976	23,600
1927	19,500	1952	19,400	1977	8630
1928	30,700	1953	24,600	1978	23,100
1929	11,200	1954	21,900	1979	16,600
1930	11,500	1955	21,200	1980	14,500
1931	34,100	1956	22,700	1981	26,500
1932	21,800	1957	20,900	1982	21,400
1933	19,800	1958	22,200	1983	18,300
1934	28,700	1959	14,200	1984	11,300
1935	15,000	1960	14,200	1985	13,600
1936	19,600	1961	24,400	1986	17,100
1937	16,100	1962	20,900	1987	20,000
1938	30,100	1963	26,300	1988	19,200
1939	17,800	1964	19,700	1989	15,400
1940	21,400	1965	32,200	1990	17,200
1941	13,200	1966	19,500	1991	20,500
1942	25,400	1967	19,100	1992	10,800
1943	26,500	1968	18,600	1993	12,000
1944	12,800	1969	14,500	1994	18,300
1945	22,400	1970	17,200	1995	18,800
1946	21,600	1971	18,100	1996	34,700
1947	28,000	1972	31,800	1997	22,700
1948	21,900	1973	19,700	1998	16,800
1949	29,000	1974	20,900	1999	40,500

Data were downloaded for USGS Gage Number 14305500 from the USGS website, http://water. usgs.gov/nwis/. The continuous record from 1925 has been used.

Table 2

Computed Moments for Annual Maximum Floods for the Siletz River, Near Siletz, Oregon, Water Years 1925-1999.

	Excel Function	Original Data (cfs)	login Data (log cfs)
Number of data points	COUNT	75	75
Mean [Equation 1]	AVERAGE	20,452 cfs	4.2921
Variance [Equation 2)]	VAR	37,079,690 cfs2	0.01665
Standard deviation	STDEV	6089 cfs	0.12905
Coefficient of variation		0.298	0.030
Skewness [Equation 3]	SKEW	0.7889	-0.1565
Weighted skewness [Equation (4)]			-0.1242

Figure 1

Time series of annual maximum peak flows for the Siletz River, near Siletz, Oregon. Also shown is the 5-yr running mean, from which longer-term trends can sometimes be discerned. Only quantitative methods of time-series analysis can determine for sure whether or not there are periodicities or nonstationary components in the data, but none are obvious visually.