The gamma distribution may be written in many different (but mathematically equivalent) fo...

The gamma distribution may be written in many different (but mathematically equivalent) forms. Excel uses the following form for the two-parameter gamma distribution in its functions GAMMADIST(x1) = F(x1) and GAMMAINV(F) = x1:

The three-parameter gamma distribution with lower bound x0 is easily evaluated by replacing the lower and upper limits by x0 and x1 − x0, respectively. Method-of-moments estimates for the three parameters are

If CS< 0, b will be negative (not allowed in Excel). In this case, | β | is used in the Excel functions, and GAMMADIST returns 1 − F, and the argument of GAMMAINV is 1 − F. Method-of-moments estimates for the two-parameter gamma distribution are:

The skewness of the two-parameter gamma distribution is

which provides a way of computing frequency factors, as described below.

Of course, moments are computed using logarithms of the data for the LP3 distribution. For given values of α and β, Excel function GAMMADIST returns F(x1) given x1, and GAMMAINV returns x1 given F. Letting x1 =  + K • Sx, where K(F, Cs) is the frequency factor, and using the parameter estimates given above, the upper limit of the integral can be manipulated to give

where Ginv = GAMMAINV(F, α, β). To evaluate K, set β = 1 and  Hence, frequency factors can be computed only as a function of the CDF and skewness for any values of either parameter, and Table can be avoided. For negative skewness, the symmetry of the distribution is exploited, and the same relationship holds for K, but with 1 − F as the argument of GAMMAINV, instead of F. (The negative sign of Cs is retained in the equation.) For the special case of Cs = 0, K = z = standard normal variate.

(a) Compute frequency factors for T = 2 yr and 100 yr and for Cs = + 0.5 and − 0.5. Compare the four values with the values given in Table.

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(b) Repeat Problems 1, 2, and 3 using Excel functions GAMMADIST and GAMMAINV.

Table

Frequency Factors K for Gamma and Log Pearson Type 3 Distributions

Source: IACWD (1982).

Problem 1

Refer to the updated 1925-2010 Siletz River data developed in Problems 1 and 2. For consistency, assume that the following moments are valid for the period water years 1925-2010:

Problem

Assume that the Siletz River data may be fit by a three-parameter Gamma (Pearson 3) distribution. Find the following.

(a) Peak flow of the 100-yr flood

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(b) Peak flow of the 50-year flood

Problem 1

This chapter employs Siletz River data for the continuous period 1925-1999. As of this writing, peak flow data through water year 2010 are available on the USGS Oregon surface water data webpage http://waterdata.usgs.gov/or/nwis/sw, including for Gage Number 14305500, which is the Siletz River. You will be asked in different problems in this chapter to update the Chapter 3 examples by using the 86-yr record, 1925-2010.

(a) Download the peak flow data for the Siletz River, and enter the data into a spreadsheet.

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(b) Plot the 86 years of peak flows (1925-2010) and a 5-yr running mean vs. their water year, to update Figure 1. Comment in general about the appearance of this time series, in the manner of the discussion of Figure 1.

---

(c) Using the updated data, develop new relative frequency and cumulative frequency histograms; that is, update Figures 2 and 3.

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.

Figure 2

Relative frequencies (probabilities) for the Siletz River, plotted vs. their class mark.

Figure 3

Cumulative frequency histogram for the Siletz River, plotted vs. class intervals.

Problem 2

(a) Use the data found in Problem 3 to calculate the mean, standard deviation, and skew coefficient (Equation 1, 2, and 3) of the updated Siletz River data (1925-2010).

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(b) Repeat part (a) using the logs (base 10) of the Siletz River data.

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(c) Develop the weighted skewness of the logs according to the Bulletin 17B protocol. That is, redo Example. At the conclusion of this problem, you should have an updated version of Table 2.

Problem 3

This chapter employs Siletz River data for the continuous period 1925-1999. As of this writing, peak flow data through water year 2010 are available on the USGS Oregon surface water data webpage http://waterdata.usgs.gov/or/nwis/sw, including for Gage Number 14305500, which is the Siletz River. You will be asked in different problems in this chapter to update the Chapter 3 examples by using the 86-yr record, 1925-2010.

(a) Download the peak flow data for the Siletz River, and enter the data into a spreadsheet.

---

(b) Plot the 86 years of peak flows (1925-2010) and a 5-yr running mean vs. their water year, to update Figure 1. Comment in general about the appearance of this time series, in the manner of the discussion of Figure 1.

---

(c) Using the updated data, develop new relative frequency and cumulative frequency histograms; that is, update Figures 2 and 3.

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.

Figure 2

Relative frequencies (probabilities) for the Siletz River, plotted vs. their class mark.

Figure 3

Cumulative frequency histogram for the Siletz River, plotted vs. class intervals.

Equation 1

Equation 2

Equation 3

Equation 4

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.

Table 2

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

Table 1

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

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.

Figure 4

Generalized skew coefficients of logarithms of annual maximum streamflow. (From IACWD, 1982.)

Problem 2

Refer to the updated 1925-2010 Siletz River data developed in Problems 1 and 2. For consistency, assume that the following moments are valid for the period water years 1925-2010:

Problem

Assume that the Siletz River data may be fit by a log-Pearson 3 (LP3) distribution. Find the following.

(a) Peak flow of the 100-yr flood

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(b) Peak flow of the 50-yr flood

Problem 1

This chapter employs Siletz River data for the continuous period 1925-1999. As of this writing, peak flow data through water year 2010 are available on the USGS Oregon surface water data webpage http://waterdata.usgs.gov/or/nwis/sw, including for Gage Number 14305500, which is the Siletz River. You will be asked in different problems in this chapter to update the Chapter 3 examples by using the 86-yr record, 1925-2010.

(a) Download the peak flow data for the Siletz River, and enter the data into a spreadsheet.

---

(b) Plot the 86 years of peak flows (1925-2010) and a 5-yr running mean vs. their water year, to update Figure 1. Comment in general about the appearance of this time series, in the manner of the discussion of Figure 1.

---

(c) Using the updated data, develop new relative frequency and cumulative frequency histograms; that is, update Figures 2 and 3.

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.

Figure 2

Relative frequencies (probabilities) for the Siletz River, plotted vs. their class mark.

Figure 3

Cumulative frequency histogram for the Siletz River, plotted vs. class intervals.

Problem 2

(a) Use the data found in Problem 3 to calculate the mean, standard deviation, and skew coefficient (Equation 1, 2, and 3) of the updated Siletz River data (1925-2010).

---

(b) Repeat part (a) using the logs (base 10) of the Siletz River data.

---

(c) Develop the weighted skewness of the logs according to the Bulletin 17B protocol. That is, redo Example. At the conclusion of this problem, you should have an updated version of Table 2.

Problem 3

This chapter employs Siletz River data for the continuous period 1925-1999. As of this writing, peak flow data through water year 2010 are available on the USGS Oregon surface water data webpage http://waterdata.usgs.gov/or/nwis/sw, including for Gage Number 14305500, which is the Siletz River. You will be asked in different problems in this chapter to update the Chapter 3 examples by using the 86-yr record, 1925-2010.

(a) Download the peak flow data for the Siletz River, and enter the data into a spreadsheet.

---

(b) Plot the 86 years of peak flows (1925-2010) and a 5-yr running mean vs. their water year, to update Figure 1. Comment in general about the appearance of this time series, in the manner of the discussion of Figure 1.

---

(c) Using the updated data, develop new relative frequency and cumulative frequency histograms; that is, update Figures 2 and 3.

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.

Figure 2

Relative frequencies (probabilities) for the Siletz River, plotted vs. their class mark.

Figure 3

Cumulative frequency histogram for the Siletz River, plotted vs. class intervals.

Equation 1

Equation 2

Equation 3

Equation 4

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.

Table 2

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

Table 1

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

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.

Figure 4

Generalized skew coefficients of logarithms of annual maximum streamflow. (From IACWD, 1982.)

Problem 3

Using graphs from Problem and additional computations as appropriate, estimate the return period and nonexceedance probability, F(Q), of a flood of magnitude 30,000 cfs for the Siletz River, 1925-2010.

Problem

Generate a new flood frequency plot for the updated Siletz River data, 1925-2010. That is, generate an updated version of Figure, but only plot the Gamma-3 and LP3 fits. You may omit some data points in the middle of the ordered series to ease crowding. If you are unable to obtain lognormal probability paper, a plot of magnitude vs. log(T) or magnitude vs. log(l-F) can serve as a substitute. That is, plot magnitude on the arithmetic scale vs. T or 1-F on the log scale on semilog paper or in Excel.

Figure

Comparison of four fitted CDFs for Siletz River flows, 1925–1999.