ML052060263

Analysis Performed by EA Engineering, Science, and Technology of Mississippi River Flow Data

ML052060263
Person / Time
Site:	Monticello
Issue date:	01/01/2004
From:	EA Engineering, Science, & Technology
To:	Office of Nuclear Reactor Regulation
Davis J A 415-6987
Shared Package
ML052060259	List: ML052060263
References
EA 2004a, EA 2004aU
Download: ML052060263 (21)

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Reference:

EA 2004a, uncited Analysis Performed by EA Engineering, Science & Technology of Mississippi River Flow Data EA Engineering Science and Technology (EA) performed an analysis of USGS Mississippi River data to provide flow statistics at the MNGP site. The central task of this analysis was to construct a historical record of flows at the MNGP site based upon drainage scaling from upstream and downstream USGS gaging stations. Because flow was available from different stations in different years, several drainage area scaling relationships were developed. The drainage area scaling relationships also accounted for the flows on two major tributaries. Elk River at Big Lake (USGS station 5275000) has a drainage area of 599 mi2. The confluence of Elk River with the Mississippi River is 2,500 ft upstream of the Mississippi River gage at Elk River. The Crow River at Rockford has a drainage area of 2,640 mi2 and enters the Mississippi River between the Elk River and Anoka gaging stations.

The daily flow data at Mississippi River gaging stations were obtained from the USGS NWIS web page. The historical periods available at each station are summarized in the following table.

River Drainage Area USGS Station Mile .2Available Record St. Cloud (5270700) 926.3 13,320 1988-2002 Elk River (5275500) 884.6 14,500 1915-1956 Anoka (5288500) 864.8 19,100 1931-2002 Elk River-trib (5275000) 599 1931-2002 Crow River (5280000) 2,640 1930-2002 The river flows downloaded from the USGS NWIS web pager are provided in the following files on the accompanying CD:

hlltp://nwis.waterdata.usgs.gov/mn/nwis/discharge.

St. Cloud MISS-CLD.txt Elk River MISS-ELK.txt Anoka MISS-ANK.txt Elk River at Big Lake ELK-BLAK.txt Crow River CROWRIV.txt The USGS flow data was extrapolated to the Monticello site using drainage area scaling to provide a continuous record of daily historical flows. The river flow at the MNGP site was based on the closer USGS gage in operation of that time. Prior to 1956, site flows were scaled from the gage at Elk River. Following 1988, site flows were scaled from the gage at St. Cloud. In the interval between 1956 and 1988, flows were scaled from the

-- a--

V 11 gage at Anoka. The drainage area scaling took into account two major tributaries; Elk River at Big Lake (5275000) and Crow River at Rockfort (5280000).

The MNGP site is located at river mile 900. Interpolating drainage area between St.

Cloud and Elk River by river mile, while accounting for the 599-mi2 area associated with the Elk River tributary, results in a drainage area for the MNGP site of 13,700 mi2 . Site flows were determined using the following relationships:

Prior to 1956 Qsite = 0.9855 (Flow Elk River - Elk River Tributary)

Qsite = 0.9448 (Flow Elk River) when missing data at Elk River tributary 1956-1988 Qsite = 0.8638 (Flow Anoka - Elk River Trib.- Crow River Trib.)

1988-2002 Qsite = 1.0285 (Flow St. Cloud)

The USGS flows at the 5 gaging stations were combined into a single file with a separate column for each gaging station (provided as file: USGS-HIS.prn). Using the above scaling relationships, a Fortran program was used to construct the historical Mississippi River flow file at the MNGP site (provided as file: QMONT.dat).

The historical Mississippi River flow data set was used to develop flow frequency distributions for periods prior and following MNGP operation. A flow frequency distribution by month for the 1940 to 1970 period is provided in Table 1 and a flow frequency distribution for the 1971 to 2001 period is provided in Table 2. Monthly average Mississippi River flows at the MNGP site are provided in Table 3 for the 1971 to 2001 period.

Minimum 7-Day Average Flow Statistics Minimum seven-day average Mississippi River flows for a range of recurrence intervals were calculated from the historical flows determined for the MNGP site based on drainage area scaling (QMONT.dat). The flow statistics were calculated using a Fortran program containing the standard Log-Pearson procedure (Linsley et al, 1082). Flow statistics were calculated for uniform 40-year periods both prior (1940-1970) and following (1971-2001) MNGP operation. The combined 1940-2001 period was also examined. A print-out from the program containing the minimum 7-day flow (cfs) for each year in the analysis and the calculated flows for a range of recurrence intervals is provided as Table 4.

River Flow Recorded at the MENGP Site by Xcel Energy The daily Mississippi River flows were recorded at the MNGP site by Xcel Energy from 1984 to 2003 (Holmes 2004). These daily river flows are provided in file MNGP-Q.xls on the accompanying CD. The monthly average flows from this data set are summarized in Table 5 for the 1988-2003 period. A 7Q10 analysis was performed on the 1990-2003 data, the period free from missing data, and the output listing is provided in Table 6. A

7Q10 analysis using the USGS flows scaled to the MNGP site for a similar 1990-2001 period is provided in Table 7.

References Holmes, Mark. 2004. Nuclear Management Company. Mississippi River Flow Measured at MNGP Site. Personal Communication with J. Yost (EA). 28 April 2004.

EA 2004aU Data files were obtained frofmTUSGS NWIS on the following website http://nwis.waterdata.usgs.gov/mn/nwis/discharge. Data for the following stations are contained in the identified files on the accompanying CD:

St. Cloud MISS-CLD.txt Elk River MISS-ELK.txt Anoka MISS-ANK.txt Elk River at Big Lake ELK-BLAK.txt Crow River CROWRIV.txt

,/

Table 1 Frequency Distribution of USGS Mississippi River Flows Scaled to the MNGP Site, 1940-1970 PERCENTILE FLOW (CFS)

(%) JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC TOTAL 0 840. 872. 907. 1867. 2046. 1664. 1049. 810. 866. 1084. 1088. 978. 810.

1 893. 926. 1202. 2438. 2381. 2227. 1256. 993. 1036. 1470. 1463. 1397. 1207.

5 1493. 1373. 1658. 3233. 3797. 3100. 1732. 1441. 1484. 1879. 1930. 1720. 1723.

10 1691. 1720. 2072. 4359. 5294. 3629. 2259. 1687. 1866. 2107. 2183. 1958. 2090.

15 1899. 1955. 2250. 5148. 5973. 4086. 2697. 2010. 2181. 2241. 2398. 2152. 2366.

20 2089. 2113. 2474. 5894. 6482. 4366. 3090. 2306. 2427. 2431. 2671. 2405. 2681.

25 2291. 2259. 2641. 6691. 7000. 4844. 3463. 2663. 2658. 2857. 2938. 2651. 2966.

30 2514. 2354. 2848. 7426. 7440. 5383. 3886. 2947. 2920. 3276. 3154. 2819. 3253.

35 2696. 2587. 3110. 8144. 8007. 6356. 4284. 3170. 3181. 3445. 3429. 2983. 3538.

40 2847. 2700. 3360. 9442. 8583. 6995. 4530. 3451. 3404. 3654. 3564. 3139. 3787.

45 2939. 2893. 3627. 10428. 9179. 7529. 4850. 3705. 3609. 3875. 375.3. 3272. 4034.

50 3095. 3036. 3762. 11541. 9879. 8200. 5193. 4043. 3845. 4072. 3972. 3430. 4301.

55 3400. 3292. 3885. 12518. 10454. 9007. 5673. 4297. 4087. 4295. 4189. 3541. 4605.

60 3637. 3637. 4067. 13616. 11277. 9500. 6305. 4565. 4689. 4577. 4375. 3665. 5071.

65 3799. 3769. 4274. 14706. 12229. 10270. 6949. 4804. 5085. 5009. 4555. 3848. 5618.

70 3952. 3877. 4471. 16255. 13737. 11540. 7581. 5129. 5464. 5344. 4819. 3997. 6354.

75 4060. 4031. 5014. 17520. 14783. 12827. 8222. 5635. 6099. 5710. 5206. 4228. 7352.

80 4287. 4238. 5716. 19439. 16291. 14269. 9073. 6331. 6648. 6014. 5575. 4346. 8533.

85 4492. 4779. 6150. 21587. 17411. 16167. 9938. 7425. 7481. 6800. 6289. 4514. 10163.

90 4681. 5150. 7769. 24084. 19944. 19583. 11486. 9055. 8577. 8096. 7019. 4790. 13052.

95 5627. 5445. 12542. 29431. 22497. 23322. 16273. 13140. 9482. 11225. 8075. 6526. 17533.

99 7530. 9769. 25250. 45744. 33010. 27962. 22794. 21879. 11452. 16265. 10982. 8973. 26945.

MEAN 3274. 3287. 4792. 13307. 11436. 9768. 6467. 4899. 4570. 4791. 4332. 3561. 6209.

MAX 8291. 10987. 32080. 54045. 35755. 31112. 26433. 23956. 13194. 17981. 14359. 12170. 54045.

OBS 961 876 961 930 961 930 961 961 930 961 930 961 11323

Table 2 Frequency Distribution of USGS Mississippi River Flows Scaled to the MNGP Site, 1971-2001 PERCENTILE FLOW (CFS)

(%) JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC TOTAL 0 819. 986. 1076. 2459. 1696. 911. 682. 707. 586. 815. 860. 624. 586.

1 970. 1121. 1506. 3270. 1985. 1178. 854. 930. 828. 874. 1018. 947. 946.

5 1810. 1807. 2593. 4832. 2935. 2223. 1779. 1183. 2020. 2489. 2674. 2018. 1954.

10 2078. 2006. 3154. 5871. 3899. 2787. 2288. 1677. 2273. 2962. 3178. 2499. 2595.

15 2564. 2737. 3468. 6552. 4474. 3958. 2886. 2119. 2499. 3137. 3625. 2859. 3044.

20 2896. 3086. 3651. 7087. 5688. 4856. 3826. 2445. 2726. 3276. 3950. 3023. 3429.

25 3343. 3271. 3877. 7765. 6518. 5297. 4297. 2708. 2983. 3415. 4515. 3343. 3791.

30 3536. 3462. 4280. 8478. 7354. 5657. 4834. 2996. 3252. 3575. 5040. 3913. 4166.

35 3857. 3703. 4577. 9247. 8197. 6058. 5151. 3316. 3600. 3864. 5321. 4270. 4526.

40 4176. 3806. 4994. 10279. 9041. 6552. 5667. 3754. 3867. 4109. 5734. 4670. 4847.

45 4320. 3928. 5266. 11108. 9802. 7210. 6167. 4279. 4154. 4486. 6134. 4865. 5194.

50 4474. 4114. 5503. 11943. 10812. 7724. 6658. 4587. 4398. 5215. 6541. 5105. 5575.

55 4628. 4268. 5832. 12754. 11931. 8444. 7056. 4893. 4598. 5766. 6866. 5374. 5966.

60 4769. 4432. 6150. 13613. 12987. 9255. 7704. 5165. 4827. 6058. 7179. 5723. 6419.

65 4937. 4559. 6468. 14811. 14181. 9936. 8359. 5494. 5030. 6430. 7487. 5873. 6973.

70 5143. 4625. 6788. 16045. 15499. 10726. 9154. 5960. 5190. 7037. 7746. 6120. 7726.

75 5279. 4988. 7570. 18514. 16868. 11828. 10210. 6418. 5852. 7776. 8218. 6383. 8720.

80 5496. 5262. 9021. 20117. 18102. 12843. 11276. 7210. 6953. 9072. 8660. 6702. 10008.

85 5760. 5554. 10153. 22276. 19542. 13658. 12240. 8413. 8570. 11054. 9180. 7080. 11622.

90 6043. 5863. 11879. 25508. 22230. 14908. 13902. 9822. 9915. 12596. 10183. 7548. 14019.

95 6487. 6038. 16835. 33736. 27154. 16990. 16265. 11108. 11442. 15608. 12034. 8403. 18308.

99 7110. 9743. 22104. 43198. 34870. 29204. 21274. 17586. 22544. 21393. 18738. 10203. 29670.

MEAN 4284. 4135. 6674. 14140. 12304. 8759. 7622. 5155. 5224. 6455. 6715. 5055. 7217.

MAX 7649. 11315. 25403. 46387. 41167. 31370. 35172. 26655. 28711. 25766. 22151. 11747. 46387.

OBS 961 876 961 930 961 930 961 961 930 961 930 961 11323

Table 3 Monthly Average USGS Mississippi River Flows Scaled to the MNGP Site, 1971-2001 FLOW (CFS)

YEAR JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC ANNUAL 1971 3399.5 3329.7 4596.7 19187.5 11825.0 7327.8 5035.1 2548.6 2485.2 5507.5 17355.9 7965.0 7537.3 1972 6640.6 5664.6 11526.4 18381.1 16993.3 8575.3 12492.3 16522.9 8971.0 7354.5 9431.5 5306.2 10688.5 1973 5317.1 4345.7 13093.5. 8704.1 8979.5 6471.8 3627.6 5922.8 4348.0 12670.6 9159.6 5787.3 7396.0 1974 4724. 6 4447.0 5666.6 14109.2 16561.1 14286.4 5634.0 5077.5 2781.5 2820.1 5548.3 3961.8 7134.5 1975 3944.9 4467.8 4653.8 17875.4 25073.9 11588.0 14179.8 5259.7 4943.2 4512.4 5062.1 4905.8 8897.5 1976 4339.3 4683.9 7533.4 13307.8 4081.2- 2353.3 2030.8 1111.4 852.7 919.9 1113.1 1006.4 3593.6 1977 1029.1 1194.6 2887.5 3630.9 2342.7 2670.8 2538.8 1426.7 4572.3 6949.1 7799.9 7099.8 3688.1 1978 4546.5 3455.5 4801.8 16632.4 8088.9 9320.9 12391.5 7403.2 9213.1 6056.9 4654.7 3654.0 7525.0 1979 3252.6 3154.6 4512.5 21180.2 21348.4 11882.5 11001.9 5413.6 4463.4 3445.8 7998.1 4453.1 8521.4 1980 3855.7 3884.7 4612.5 10879.4 4212.7 4544.2 2279.3 2422.8 4674.4 3574.7 3550.8 2547.4 4238.1 1981 2134.9 2017.7 2701.0 5442.6 6785.5 7396.8 5793.6 5078.9 3848.5 7903.4 7135.1 4126.8 5045.0 1982 3744. 9 3805.7 4037.4 23050.8 18790.9 9248.1 5982.9 3223.9 3685.5 10592.0 7978.5 7407.2 8472.9 1983 6143.0 5160.5 12220.0 10689.6 6701.4 8752.1 10661.4 5225.1 4423.8 5812.7 6279.5 6005.3 7355.3 1984 5527.5 6798.0 8279.3 11777.5 11545.2 17816.6 7304.5 3095.3 2740.3 9949.3 9075.2 5734.2 8293.5 1985 4904.2 4051.0 9540.4 12048.0 17194.1 13070.1 12418.6 8832.1 11156.7 11716.9 7257.6 7038.9 9973.7 1986 6507.4 5754.9 6727.1 25916.4 27771.7 12834.1 9940.1 9733.5 17057.8 14705.9 8448.3 7049.5 12724.2 1987 5682.9 5442.2 6840.6 6538.2 5539.7 5912.9 2996.2 3869.5 2935.4 3417.5 3559.6 3031.6 4639.7 1988 2419.7 2642.3 4413.4 7154.0 3742.3 1437.4 880.3 2444.7 7570.3 3782.0 3036.9 2899.8 3528.7 1989 2867.3 3034.2 3969.8 16213.0 10893.1 6007.6 4041.8 1578.3 4455.6 3694.4 3235.4 2452.5 5196.5 1 990 2048.1 1867.1 7773.0 6764.0 9152.6 10384.0 4907.1 2160.6 2362.5 4295.6 4059.6 2376.2 4858.9 1991 1981.7 1985.4 4147.9 9356.9 12833.1 6980.3 7411.7 3465.2 4516.3 3242.8 5307.9 5342.7 5566.1 1992 4756.1 3665.1 6947.5 8759.6 6387.2 3850.1 5713.3 2571.6 4201.5 3194.7 3672.2 2947.9 4726.6 1993 3447.2 3298.6 4013.6 11081.4 11061.3 13098.0 17305.8 9338.0 7421.5 6043.1 6415.6 5511.6 8195.1 1994 5259.4 5063.6 9102.2 16099.9 14240.8 7822.0 10348.7 5588.2 5124.8 8058.7 7027.3 5404.4 8279.7 1995 4696.1 3981.9 10900.7 12359.8 12562.0 6013.1 7158.2 6610.8 5955.2 16124.0 9951.4 6337.1 8591.6 1996 5460.5 5691.0 8435.9 19434.7 17743.8 8115.1 6973.4 4784.0 2673.1 4488.7 8341.4 7645.9 8322.7 1997 5776.4 5961.8 6358.3 30444.4 12709.0 6012.4 12687.4 7678.8 5092.2 6233.9 6010.7 5630.0 9213.7 1998 4311.2 5782.5 7042.4 9650.0 5813.8 9673.3 9113.1 3625.7 2371.8 5771.7 7832.9 6906.1 6487.3 1999 4397.8 4270.2 6207.7 14796.8 20154.5 11240.8 10133.7 9963.8 10041.5 8401.8 7074.9 5261.4 9354.3 2000 4678.1 4563.1 8362.9 6309.0 7636.7 5958.6 6232.6 3296.6 3083.9 3908.1 9574.2 5374.9 5752.7 2001 4996.0 4568.9 4983.7 30560.7 22650.8 20892.8 7056.7 4518.2 3911.8 4948.6 5231.8 5530.8 9975.5 2002 4621.1 3817.7 3758.8 10646.0 9661.2 5635.3 13149.6 7163.5 5810.5 5366.1

Table 4 Output Listing from Log-Pearson Program for Flow Analysis at MINGP Site Using Mississippi River Data Scaled from USGS Stations Analysis of 1940-1970 Data Set COL= 1 YEAR LISTING 1940 1310.5 Flow (cfs) 1941 2592.2 1942 3089.0 1943 3074.9 1944 3359.3 1945 2842.6 1946 2800.3 1947 2646.9 1948 1753.6 1949 2425.1 1950 2302.2 1951 4385.4 1952 3139.7 1953 4229.9 1954 2720.9 1955 2118.0 1956 1790.4 1957 2865.8 1958 1329.7 1959 1708.5 1960 1382.0 1961 913.2 1962 1955.8 1963 1758.5 1964 1762.1 1965 2951.8 1966 2487.1 1967 1561.5 1968 2222.4 1969 1908.0 1970 1362.4 RETURN FLOW (CFS) 2 2250.59 5 1635.53 10 1366.22 25 1116.55 50 975.58 100 862.33 Analysis of 1971-2001 Data Set COL= 1 YEAR LISTING 1971 2139.8 1972 3582.1 1973 2737.5 1974 2002.7 1975 3986.0 1976 787.1

1977 1018.5 1978 2999.8 1979 2939.8 1980 1609.7 1981 2616.9 1982 2597.4 1983 3771.8 1984 2199.4 1985 4755.3 1986 4822.5 1987 2104.9 1988 716.7 1989 1288.6 1990 1748.5 1991 2484.6 1992 2061.5 1993 4722.4 1994 3680.7 1995 3085.6 1996 2475.8 1997 3893.7 1998 2174.6 1999 3560.2 2000 2650.7 2001 3338.3 RETURN FLOW (CFS) 2 2708.44 5 1735.35 10 1294.12 25 899.69 50 691.88 100 537.05 Analysis of 1940-2001 Data Set COL= 1 YEAR LISTING 1940 1310.5 1941 2592.2 1942 3089.0 1943 3074.9 1944 3359.3 1945 2842.6 1946 2800.3 1947 2646.9 1948 1753.6 1949 2425.1 1950 2302.2 1951 4385.4 1952 3139.7 1953 4229.9 1954 2720.9 1955 2118.0 1956 1790.4 1957 2865.8 1958 1329.7 1959 1708.5

1960 1382.0 1961 913.2 1962 1955.8 1963 1758.5 1964 1762.1 1965 2951.8 1966 2487.1 1967 1561.5 1968 2222.4 1969 1908.0 1970 1362.4 1971 2139. 8 1972 3582.1 1973 2737.5 1974 2002.7 1975 3986.0 1976 787 .1 1977 1018.5 1978 2999.8 1979 2939.8 1980 1609.7 1981 2616.9 1982 2597.4 1983 3771. 8 1984 2199.4 1985 4755.3 1986 4822.5 1987 2104.9 1988 716.7 1989 1288.6 1990 1748.5 1991 2484 .6 1992 2061.5 1993 4722.4 1994 3680.7 1995 3085. 6 1996 2475.8 1997 3893.7 1998 217 4. 6 1999 3560.2 2000 2650.7 2001 3338 . 3 RETURN FLOW (CFS) 2 2454.72 5 1666.19 10 1317.73 25 1000.04 50 825.72 100 689.84

Table 5 Monthly Average Mississippi River Flow Recorded at the MANGP Site (1989 - April 2004)

AM Y Flow (cfs) Obs 1 1 88 2921.6 25 2 1 88 3312.1 26 3 1 88 4096.0 31 4 1 88 6330.7 29 5 1 88 3252.8 31 6 1 88 1373.7 30 7 1 88 867.2 31 8 1 88 2194.2 31 9 1 88 2474.5 29 10 1 88 3201.7 31 11 1 88 2735.7 30 12 1 88 3144.5 30 1 1 89 3717.6 31 2 1 89 4292.1 28 3 1 89 4517.2 31 4 1 89 13495.3 30 5 1 89 9310.5 31 6 1 89 5095.2 30 7 1 89 3722.1 31 8 1 89 1433.4 31 9 1 89 2279.2 30 10 1 89 3574.8 31 11 1 89 3191.9 30 12 1 89 2757.0 31 1 1 90 3242.7 31 2 1 90 3207.2 28 3 1 90 6705.4 31 4 1 90 5745.7 30 5 1 90 7601.9 31 6 1 90 9191.7 30 7 1 90 4362.1 31 8 1 90 2110.8 31 9 1 90 2256.6 30 10 1 90 3564.4 31 11 1 90 3712.4 30 12 1 90 2512.5 31 1 1 91 3631.9 31 2 1 91 3604.2 28 3 1 91 4762.1 31 4 1 91 8350.0 30 5 1 91 11316.4 31 6 1 91 6267.9 30 7 1 91 6336.8 31 8 1 91 3203.2 31 9 1 91 4198.6 30 10 1 91 3327.8 31 11 1 91 4857.5 30 12 1 91 5082.2 31 1 1 92 4598.9 31 2 1 92 3287.1 29

3 1 92 6529.3 31 4 1 92 7836.9 30 5 1 92 5757.5 31 6 1 92 3482.4 30 7 1 92 4960.7 31 8 1 92 2556.8 31 9 1 92 3641.3 30 10 1 92 2893.4 31 11 1 92 3278.9 30 12 1 92 3141.9 31 1 1 93 4316.8 31 2 1 93 4223.2 28 3 1 93 4121.2 31 4 1 93 9028.4 30 5 1 93 8931.3 31 6 1 93 10478.9 30 7 1 93 14150.6 31 8 1 93 8060.1 31 9 1 93 6457.2 30 10 1 93 5109.7 31 11 1 93 5323.4 30 12 1 93 5276.5 31 1 1 94 4946.4 31 2 1 94 4556.3 28 3 1 94 7647.6 31 4 1 94 12165.6 30 5 1 94 12160.4 31 6 1 94 6375.1 30 7 1 94 8226.3 31 8 1 94 4744.4 31 9 1 94 4319.8 30 10 1 94 7056.3 31 11 1 94 6367.8 30 12 1 94 5132.2 31 1 1 95 4337.0 31 2 1 95 4109.0 28 3 1 95 9791.4 31 4 1 95 10571.1 30 5 1 95 10085.9 31 6 1 95 5313.7 30 7 1 95 6077.4 31 8 1 95 5828.4 31 9 1 95 5240.9 30 10 1 95 12186.2 31 11 1 95 8643.6 30 12 1 95 6181.8 31 1 1 96 5653.7 31 2 1 96 5957.7 29 3 1 96 6621.6 31 4 1 96 18079.4 30 5 1 96 15055.2 31 6 1 96 7075.0 30 7 1 96 6091.6 31 8 1 96 4315.5 31 9 1 96 2696.0 30 10 1 96 4118.8 31 11 1 96 7572.4 30

12 1 96 7139.2 31 1 1 97 8988.3 31 2 1 97 9303.5 28 3 1 97 7134.3 31 4 1 97 30768.0 30 5 1 97 10853.8 31 6 1 97 5847.4 30 7 1 97 10328.6 31 8 1 97 5883.0 31 9 1 97 4594.9 30 10 1 97 5481.5 31 11 1 97 5573.7 30 12 1 97 5332.6 31 1 1 98 4186.3 31 2 1 98 5431.9 28 3 1 98 6398.6 31 4 1 98 8552.7 30 5 1 98 5295.0 31 6 1 98 8061.1 30 7 1 98 7995.1 31 8 1 98 3362.3 31 9 1 98 2432.3 30 10 1 98 5065.4 31 11 1 98 7047.4 30 12 1 98 6209.8 31 1 1 99 5171.3 31 2 1 99 5074.2 28 3 1 99 5672.9 31 4 1 99 11835.0 30 5 1 99 18113.1 31 6 1 99 . 9163.8 30 7 1 99 8176.8 31 8 1 99 8404.6 31 9 1 99 8507.6 30 10 1 99 7415.0 31 11 1 99 6159.3 30 12 1 99 4772.3 31 1 1 0 5189.9 31 2 1 0 8547.1 29 3 1 0 7451.9 31 4 1 0 5686.1 30 5 1 0 6668.2 31 6 1 0 5365.2 30 7 1 0 5575.0 31 8 1 0 3064.3 31 9 1 0 2876.9 30 10 1 0 3491.6 31 11 1 0 8095.8 30 12 1 0 5240.9 31 1 1 1 4769.1 31 2 1 1 4414.4 28 3 1 1 4614.8 31 4 1 1 30844.8 30 5 1 1 20139.4 31 6 1 1 18867.0 30 7 1 1 6567.2 31 8 1 1 4282.1 31

9 1 1 3344.3 30 10 1 1 4174.9 31 11 1 1 4869.1 30 12 1 1 5212.7 31 1 1 2 4788.8 31 2 1 2 4084.0 28 3 1 2 3905.9 31 4 1 2 9257.6 30 5 1 2 8512.8 31 6 1 2 5048.2 30 7 1 2 11229.9 31 8 1 2 6852.5 31 9 1 2 5936.2 30 10 1 2 6494.6 31 11 1 2 5451.6 30 12 1 2 4271.4 31 1 1 3 3501.4 31 2 1 3 2793.3 28 3 1 3 3486.4 31 4 1 3 6230.2 30 5 1 3 8007.3 31 6 1 3 7306.0 30 7 1 3 9766.5 31 8 1 3 3158.4 31 9 1 3 2227.4 30 10 1 3 2133.5 31 11 1 3 2677.1 30 12 1 3 3290.0 31 1 1 4 3074.4 31 2 1 4 3251.4 29 3 1 4 4105.7 31 4 1 4 6893.5 26

Table 6 Output Listing from Log-Pearson Program for Flow Analysis of Mississippi River Data Recorded at Site Analysis of 1990-2003 Data Set COL= 1 YEAR LISTING 1990 1730.7 Flow (cfs) 1991 2379.0 1992 2166.7 1993 4158.6 1994 3541.6 1995 3048.7 1996 2546.3 1997 3771.7 1998 2314.9 1999 3433.3 2000 2366.3 2001 3062.9 2002 2603.4 2003 1905.0 RETURN FLOW (CFS) 2 2696.39 5 2167.18 10 1933.69 25 1713.02 50 1585.27 100 1480.19

Table 7 Output Listing from Log-Pearson Program for Flow Analysis at MNGP Site Using Mississippi River Data Scaled from USGS Stations, 1990-2001 Analysis of 1990-2001 Data Set COL= 1 YEAR LISTING 1990 1748.5 1991 2484.6 1992 2061.5 1993 4722.4 1994 3680.7 1995 3085.6 1996 2475.8 1997 3893.7 1998 2174.6 1999 3560.2 2000 2650.7 2001 3338.3 RETURN FLOW (CFS) 2 2881.09 3.458 0.128 -0.052 0.009 5 2244.81 3.458 0.128 -0.052 -0.837 10 1965.80 3.458 0.128 -0.052 -1.287 25 1703.84 3.458 0.128 -0.052 -1.771 50 1553.25 3.458 0.128 -0.052 -2.085 100 1430.12 3.458 0.128 -0.052 -2.364

RAY K. LINSLEY, JR.

MAX A. KOHLER JOSEPH L. H. PAULHUS l

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/v I C- -W 112 2-/ ;i2 H ydrology r.

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• :xWJ I Engineers

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I Ih 'I I

- _0000*

PROBABILITY IN HYDROLOGY: A BASIS FOR PLANNING 361 I-

.1-than the period of record. The table shows the uncertainty in the plotting position assigned to the highest event in a series, but for m 2 4 the range of 3-uncertainty is reasonably narrow. If frequency analysis is intended to gain information on floods with return periods less than n/5, one may plot flood 95% 99%o magnitude against plotting position and fit a curve by eye. For longer return I-periods it is better to fit a theoretical distribution (Sec. 13-3) to the data. Note that tests for the best theoretical distribution are greatly influenced by the o 0 in assumed plotting positions. a o o 2 0 S I 13-3 Theoretical Distributions of Floods *1-

'il 51 10 513 100 Statistical distributions are usually demonstrated by use of samples numbering d in the thousands. No such samples are available for streamflow and it is not e possible to state with certainty that a specific distribution applies to flood peaks.

Ti period of the Numerous distributions have been suggested on the basis of their ability to "fit" Eq. (13-2). If.

the plotted [13-151 data from streams (Sec. 13-2). Despite much effort, tests e

'qfloods is less 1161 suggest that there is no best distribution for floods. Intuitively at least, there y is no reason to expect that a single distribution will apply to all streams world-wide. The log-Pearson Type III has been adopted [171 for use by United States d (13-4) federal agencies for flood analysis. The first asymptotic distribution of extreme I-values (EVI), commonly called the Gfumbel Type I distribution, has been tely nances widely used and is recommended in the United Kingdom. These two distribu- d cries is greater tions are described in the following sections. n Chow [18] has shown that most frequency functions can be generalized to ty X =X +Korx (13-5) If n

where X is a flood of specified probability, X is the mean of the flood series, of Al 0.99 ax is the standard deviation of the series, and K, a frequency factor defined by I.

6 200 a specific distribution, is a function of the probability level of X. n 498 996 1990 13-4 Log-Pearson Type III Distribution i-5970 n

The recommended procedure for use of the log-Pearson distribution is to con- I.

5 17.0

) 37.4 vert the data series to logarithms and compute:t I-71.1 Il 138 Mean: log X = Xlg X (f}, (13-6) 408 Fl 7.10 14.1 id 25.6 t Alternatively, 48.6 i-140 / J(log x)2 - (I log x)2/n 4.50 r0" X = V n - I 8.26 14.4 G = n1 (log x? - 3nE log x£ (log x)' + 2(1 log x)2 26.6 n(In - I)(n -)(¢r,,_)3 75.2 where log x = log X - log X.

362 HYDROLOGY FOR ENGINEERS Table:

Standard deviation: aggx = ( A -oX (13-7)

Skew coefficient: G (n -I)(nl - 2 )(-ogX) 3 (13-8)

The value of X for any probability level is computed from Eq. (13-5) modified Skew coefficier log X = log X + KqIlgx (13-9) where K is taken from Table 13-4. The cumulative frequency distribution will i 3.0 plot as a straight line on log-normal paper when the skew coefficient G = 0. 2.8 2.6 The Type III distribution is one of a family of distributions suggested by 2.4 Pearson [19]. There are no theoretical arguments for the application of this 2.2 distribution io hydrologic data. It is a skew distribution bounded on the left and *. 2.0 therefore of the general shape of most hydrologic distributions. When skew is 1.8 zero, the Pearson Type III is identical to the log-normal distribution which was 1.6 once widely used in hydrology. With a third parameter, skew, the distribution t- 1.4 1.2 can be "fitted" to most data sets. Reliable estimates of skew require very large 1.0 samples, however. In lieu of using Eq. (13-8) to compute skew, regional average A 0.8 values are often used [17]; although this may not be more reliable it does lead to a- 0.6

.. '4' more consistency between values for various streams in the region. 0.4 0.2 The probability density function for Type III is 0

-0.2 owl..

P(.A) = Po (I - )Ce-cxI2 (13-10) -0.4

-0.6 4 _ I -0.8 I-,"!

w-here (13-11) -1.0

- 1.2

- 1.4 2C 13 (13-12) -1.6

2 112 1.8 n cC+I -2.0 (13-13) -2.2 P°=a e r'(c + 1) -2.4

-2.6 (13-14) ff-2 .

= 2

-3.0 where H2 is the variance, Jimis the third moment about the mean = cr6G, ris the Source:

gamma function, and e is the base of napierian logarithms.

13-5 Extreme-Value Type I Distribution where p is Fisher and Tippett [20] found that the distribution of the maximum (or mini- base of na mum) values selected from n samples approached a limiting form as the size of t bility (Tab the samples increased. When the initial distributions within the samples are exponential, the Type I distribution results. This distribution is given by where X i:

p = I - e _- (13-15) equation i.

P

.A 414 PROBABILITY IN HYDROLOGY: A BASIS FOR PLANN I4NG 363

.. Z/t----/- -.

-/ 7, Table 13-4 K Values for the log-Pearson Type III distribution (13-7)

Recurrence interval, years I

1.0101 1.2500 2 5 10 25 50 200

, ~(1 3-8)

Skew vS Percent chance q2 _ ' jAd%

m Eq. (13-5) modified coefficient II (13-9)

G 99 80 ,. , 50 -. 20, .:;  !°0 ., ', 4 ~.': 2 -)' I C

or iency distribution will 3.0 -0.667 -0.636 -0.3% 0.420 2.180 2.278 3.152 4.051 2.8 -0.714 -0.666 -0.384 0.460 1.21 2.275 3.114 3.973 'C w coefficient G = 0. 2.6 -0.769 -0.696 -0.368 0.499 1.238",I 2.267 3.071 3.889 ibutions suggested by 2.4 -0.832 -0.725 -0.351 0.537 1.262 2.256 3.023 3.800 D'

he application of this 2.2 -0.905 -0.752 -0.330 0.574 1.284 2.240 2.970 3.705

)unded on the left and 2.0 -0.990 -0.777 -0.307 0.609 1.302 2.219 2.912 3.605 1.8 -1.087 -0.799 -0.282 0.643 1.318 2.193 2.848 3.499 utions. When skew is 1.6 -1.197 -0.817 -0.254 *.675 1.329 2.163 2.780 3.388 istribution which was 1.4 -1.318 -0.832 -0.225 0.705 1.337 2.128 2.706 3.271

kew, -the distribution 1.2 - 1.449 -0.844 -0.295 0.732 1.340 2.087 2.626 3.149 ew require very large 1.0 - 1.588 -0.852 -0.164 0.758 1.340 2.043 2.542 3.022 c

cew, regional average 0.8 -1.733 -0.856 -0.132 0.780 1.336 2.993 2.453 2.891

)C eliable it does lead to 0.6 -1.880 -0.857 -0.099 0.800 1.328 1.939 2.359 2.755 0.4 -2.029 -0.855 -0.066 0.816 2.317 1.880 2.261 2.615 the region. 0.2 -2.178 -0.850 -0.033 0.830 1.30n 1.818 2.159 2.472 N.

0 -2.326 -0.842 0. 0.842 1.282 1.751 2.054 2.326 a r

-0.2 -2.472 . -0.830 0.033 0.850 1.258 1.680 1.945 2.178 AC (13-10) -0.4 -2.615 -0,816 0.066 0.855 1.231 1.606 1.834 2.029

-0.6 -2.755 -0.800 0.099 0.857 1.200 1.528 1.720 1.880

-0.8 -2.891,.- -0.780 0.132 0.856 1.166 1.448 1.606 1.733 (13-1 1) - 1.0 -3.022 -0.758 0.164 0.852 1.128 1.366 1.492 1.588

-1.2 -3.149 -0.732 0.195 0.844 1.086 1.282 1.379 1.449

-1.4 -3.271 -0.705 0.225 0.832 1.041 1.198 1.270 1.318 (13-12) -1.6 -3.388 -0.675 0.254 0.817 0.994 1.116 1.166 1.197

-1.8 -3.499 -0.643 0.282 0.799 0.945 1.035 1.069 1.087

-2.0 -3.605 -0.609 0.307 0.777 0.895 0.959 0.980 0.990 (13-13) -2.2 -3.705 -0.574 0.330 0.752 0.844 0.888 0.900 0.905 er

-2.4 -3.800 -0.537 0.351 0.725 0.795 0.823 0.830 0.832 S

-2.6 -3.889 -0.499 0.368 0.696 0.747 0.764 0.768 0.769 20 (13-14) -2:8 - 3.973 -0.460 0.384 0.666 0.702 0.712 0.714 0.714

-;0' -4.051 -0.420 0.396 0.636 . 0.660 0.666 0.666 0.667 g.

= c 6G, ris

,an the Source: Adapted~from 1101. at

/Yget ' ^ , C ' _- I'd A 2nt

'7 In!

where p is the probability of a given flow being equaled or exceeded, e is the yd 1.

base of napierian logarithms, andy, the reduced variate, is a function of proba-iaximum (or mini-bility (Table 13-5). Then form as the size of n the samples are X = X + ((0.7797' -0.45)ax (13-16) Ity

I is given by 31 where X is the mean of the data series and ax is its standard deviation. This I,,

(13-15) equation is equivalent to Eq. (13-5) with K equal to the term in parentheses.

364 1YIDROLOGY FOR ENGINEERS Table 13-5 Values of K [Eq. (13-5)1 for the 1.01 extreme-value (Type I) distribution 120 Reduced 100-Return period, variate 90 80 years Probability y K 70 1.58 0.63 0.000 -0.450 60 2.00 0.50 0.367 -0.164 0 50 2.33 0.43 0.579 0.001 5 0.20 1.500 0.719 40 10 0.10 2.250 1.30 0 LL 20 0.05 2.970 1.87 ._?

30P7, 50 0.02 3.902 2.59 100 0.01 4.600 3.14 200 0.005 5.296 3.68 400 0.0025 6.000 4.23 20-I2 1 Table 13-5 gives values of K for various return periods. When using the Grin- 99 gorten plotting position [Eq. (13-3)], no correction for record length is con-sidered necessary. Two or more computed values of X define a straight line on Figure 13-1 F extreme-value probability paper. (See Figs. 13-1 and 13-2.)t on log-normal From Eq. (13-15) Eq. (13-2).

y = -In[-ln(l - p)] (13-17)

Equation (13-3) is a very good approximation to the unbiased mean value of 1.001 1.01 120r probability associated with a given point assuming the extreme-value distribu-tion [12]. 110 The solution by Eq. (13-16) utilizes the method of moments. The method of 100 maximum likelihood is generally superior but more difficult to solve. Maximiz-ing log likelihood requires the determination of a and ,3 in 90 80 YX = ar(X - /3) (13-18) 0 0

C) 70 -

where /3 = I In ( ) (i3-19) CE i

60

.s L.

50 and FRa) = O = E XmeXm 1e - (X - Ia) 1 e -xn (13-20) 40 30.

20 W

, 99 1 Extrapolation of a straight line is easier than extrapolation of a curve. Hence, frequency analysis is simplified by using special plotting papers scaled so that a specific distribution will plot as a straight line. On such probability paper, the ordinate is normally flowrate, and the abscissa is Figure 13-2 Fk probability or return peridd. For some distributions the logarithm of flow is the ordinate. The value paper ant

. I. . abscissa scale is warped to achieve the straight-line plot.

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