APPENDIX A:  TABLES


TABLE A-1   PROPERTIES OF WATER IN SI UNITS
Temperature
(°C)
Specific
Gravity
Density
(g/cm3)
Heat of
Vaporization
(cal/g)
Viscosity Vapor Pressure
Absolute
(cp)
Kinematic
(cs)
(mm Hg) (mb) (g/cm2)
0 0.99987 0.99984 597.3 1.790 1.790 4.58 6.11 6.23
5 0.99999 0.99996 594.5 1.520 1.520 6.54 8.72 8.89
10 0.99973 0.99970 591.7 1.310 1.310 9.20 12.27 12.51
15 0.99913 0.99910 588.9 1.140 1.140 12.78 17.04 17.38
20 0.99824 0.998211 586.0 1.000 1.000 17.53 23.37 23.83
25 0.99708 0.99705 583.2 0.890 0.893 23.76 31.67 32.20
30 0.99568 0.99565 580.4 0.798 0.801 31.83 42.43 43.27
35 0.99407 0.99404 577.6 0.719 0.723 42.18 56.24 57.34
40 0.99225 0.99222 574.7 0.653 0.658 55.34 73.78 75.23
50 0.98807 0.98804 569.0 0.547 0.554 92.56 123.40 125.83
60 0.98323 0.98320 563.2 0.466 0.474 149.46 199.26 203.19
70 0.97780 0.97777 557.4 0.404 0.413 233.79 311.69 317.84
80 0.97182 0.97179 551.4 0.355 0.365 355.28 473.67 483.01
90 0.96534 0.96531 545.3 0.315 0.326 525.89 701.13 714.95
100 0.95839 0.95836 539.1 0.282 0.294 760.00 1013.25 1033.23
Source:  Linsley, R. K. et al. (1982). Hydrology for Engineers. 3d. ed., New York: McGraw-Hill.


TABLE A-2   PROPERTIES OF WATER IN U.S. CUSTOMARY UNITS
Temperature
(°F)
Specific
Gravity
Density
(lb/ft3)
Heat of
Vaporization
(Btu/lb)
Viscosity 1 Vapor Pressure
Absolute
(lbs/ft2)
Kinematic
(ft2/s)
(in Hg) (mb) (lb/in2)
32 0.99986 62.418 1075.5 3.746 1.931 0.180 6.11 0.089
40 0.99998 62.426 1071.0 3.229 1.664 0.248 8.39 0.122
50 0.99971 62.409 1065.3 2.735 1.410 0.362 12.27 0.178
60 0.99902 62.366 1059.7 2.359 1.217 0.522 17.66 0.256
70 0.99798 62.301 1054.0 2.050 1.058 0.739 25.03 0.363
80 0.99662 62.216 1048.4 1.799 0.930 1.032 34.96 0.507
90 0.99497 62.113 1042.7 1.595 0.826 1.422 48.15 0.698
100 0.99306 61.994 1037.1 1.424 0.739 1.933 65.47 0.950
120 0..98856 61.713 1025.6 1.168 0.609 3.448 116.75 1.693
140 0.98321 61.379 1014.0 0.981 0.514 5.884 199.26 2.890
160 0.97714 61.000 1002.2 0.838 0.442 9.656 326.98 4.742
180 0.97041 60.580 990.2 0.726 0.386 15.295 517.95 7.512
200 0.96306 60.121 977.9 0.637 0.341 23.468 794.72 11.526
212 0.95837 59.828 970.3 0.593 0.319 29.921 1013.25 14.696
1 To obtain values of viscosity, multiply values shown in Table by 10-5.
Source:  Linsley, R. K. et al. (1982). Hydrology for Engineers. 3d. ed., New York: McGraw-Hill.


TABLE A-3   FACTOR p IN BLANEY-CRIDDLE METHOD1
Latitude
North
South
 
Jan
July
 
Feb
Aug
 
Mar
Sep
 
Apr
Oct
 
May
Nov
 
Jun
Dec
 
Jul
Jan
 
Aug
Feb
 
Sep
Mar
 
Oct
Apr
 
Nov
May
 
Dec
Jun
60° 0.15 0.20 0.26 0.32 0.38 0.41 0.40 0.34 0.28 0.22 0.17 0.13
50° 0.19 0.23 0.27 0.31 0.34 0.36 0.35 0.32 0.28 0.24 0.20 0.18
40° 0.22 0.24 0.27 0.30 0.32 0.34 0.33 0.31 0.28 0.25 0.22 0.21
30° 0.24 0.25 0.27 0.29 0.31 0.32 0.31 0.30 0.28 0.26 0.24 0.23
20° 0.25 0.26 0.27 0.28 0.29 0.30 0.30 0.29 0.28 0.26 0.25 0.25
10° 0.26 0.27 0.27 0.28 0.28 0.29 0.29 0.28 0.28 0.27 0.26 0.26
0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27
1 p = [(mean daily daytime hours for a given month) / (total daytime hours in a year)] × 100


TABLE A-4   FACTOR K IN THORNTHWAITE METHOD1
Latitude Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
60°N 0.54 0.67 0.97 1.19 1.33 1.56 1.55 1.33 1.07 0.84 0.58 0.48
50°N 0.71 0.84 0.98 1.14 1.28 1.36 1.33 1.21 1.06 0.90 0.76 0.68
40°N 0.80 0.89 0.99 1.10 1.20 1.25 1.23 1.15 1.04 0.93 0.83 0.78
30°N 0.87 0.93 1.00 1.07 1.14 1.17 1.16 1.11 1.03 0.96 0.89 0.85
20°N 0.92 0.96 1.00 1.05 1.09 1.11 1.10 1.07 1.02 0.98 0.93 0.91
10°N 0.97 0.98 1.00 1.03 1.05 1.06 1.05 1.04 1.02 0.99 0.97 0.96
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
10°S 1.05 1.04 1.02 0.99 0.97 0.96 0.97 0.98 1.00 1.03 1.05 1.06
20°S 1.10 1.07 1.02 0.98 0.93 0.91 0.92 0.96 1.00 1.05 1.09 1.11
30°S 1.16 1.11 1.03 0.96 0.89 0.85 0.87 0.93 1.00 1.07 1.14 1.17
40°S 1.23 1.15 1.04 0.93 0.83 0.78 0.80 0.89 0.99 1.10 1.20 1.25
50°S 1.33 1.19 1.05 0.89 0.75 0.68 0.70 0.82 0.97 1.13 1.27 1.36
1 K = Constant to correct PET for latitudes other than 0° (Eq. 2-46).


TABLE A-5   VALUES OF F (z) (AREAS UNDER HALF OF THE NORMAL PROBABILITY DENSITY FUNCTION, Eq. 6-11) VERSUS FREQUENCY FACTOR z (DOUBLE-ENTRY TABLE)
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.0000 0.0040 0.0080 0.0120 0.0159 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224
0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2518 0.2549
0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852
0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133
0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621
1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830
1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015
1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177
1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319
1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4430 0.4441
1.6 0.4452 0.4463 0.4474 0.4485 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545
1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633
1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706
1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4762 0.4767
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857
2.2 0.4861 0.4865 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890
2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916
2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936
2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952
2.6 0.4953 0.4955 0.4956 0.4757 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964
2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974
2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4980 0.4980 0.4981
2.9 0.4981 0.4982 0.4983 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990
Source:  Weatherburn, C. E. (1957). Mathematical Statistics. Cambridge University Press.


TABLE A-6   FREQUENCY FACTORS K FOR PEARSON TYPE III DISTRIBUTIONS VERSUS SKEW COEFFICIENT CS AND RETURN PERIOD T (OR PROBABILITY OF EXCEEDENCE P)
Skew Coeff.
Cs
Return Period T (y)
1.05 1.11 1.25 2 5 10 25 50 100 200
  Probability of Exceedence (percent)
95 90 80 50 20 10 4 2 1 0.5
3.0 -0.665 -0.660 -0.636 -0.396 0.420 1.180 2.278 3.152 4.051 4.970
2.8 -0.711 -0.702 -0.666 -0.384 0.460 1.210 2.275 3.114 3.973 4.847
2.6 -0.762 -0.747 -0.696 -0.368 0.499 1.238 2.267 3.071 3.889 4.718
2.4 -0.819 -0.795 -0.725 -0.351 0.537 1.262 2.256 3.023 3.800 4.584
2.2 -0.882 -0.844 -0.752 -0.330 0.574 1.284 2.240 2.970 3.075 4.444
2.0 -0.949 -0.895 -0.777 -0.307 0.609 1.302 2.219 2.192 3.605 4.398
1.8 -1.020 -0.945 -0.799 -0.282 0.643 1.318 2.193 2.848 3.499 4.147
1.6 -1.093 -0.994 -0.817 -0.254 0.675 1.329 2.163 2.780 3.388 3.990
1.4 -1.168 -1.041 -0.832 -0.225 0.705 1.337 2.128 2.706 3.271 3.828
1.2 -1.243 -1.086 -0.844 -0.195 0.732 1.340 2.087 2.626 3.149 3.661
1.0 -1.317 -1.128 -0.852 -0.164 0.758 1.340 2.043 2.542 3.022 3.489
0.8 -1.388 -1.166 -0.856 -0.132 0.780 1.336 1.993 2.453 2.891 3.312
0.6 -1.458 -1.200 -0.857 -0.099 0.800 1.328 1.939 2.359 2.755 3.132
0.4 -1.524 -1.231 -0.855 -0.066 0.816 1.317 1.880 2.261 2.615 2.949
0.2 -1.586 -1.258 -0.850 -0.033 0.830 1.301 1.818 2.159 2.472 2.763
0.0 -1.645 -1.282 -0.842 0.000 0.842 1.282 1.751 2.054 2.326 2.576
-0.2 -1.700 -1.301 -0.830 0.033 0.850 1.258 1.680 1.945 2.178 2.388
-0.4 -1.750 -1.317 -0.816 0.066 0.855 1.231 1.606 1.834 2.029 2.201
-0.6 -1.797 -1.328 -0.800 0.099 0.857 1.200 1.528 1.720 1.880 2.016
-0.8 -1.839 -1.336 -0.780 0.132 0.856 1.166 1.448 1.606 1.733 1.837
-1.0 -1.877 -1.340 -0.758 0.164 0.852 1.128 1.366 1.492 1.588 1.664
-1.2 -1.910 -1.340 -0.732 0.195 0.844 1.086 1.282 1.379 1.449 1.501
-1.4 -1.938 -1.337 -0.705 0.225 0.832 1.041 1.198 1.270 1.318 1.351
-1.6 -1.962 -1.329 -0.675 0.254 0.817 0.994 1.116 1.166 1.197 1.216
-1.8 -1.981 -1.318 -0.643 0.282 0.799 0.945 1.035 1.069 1.087 1.097
-2.0 -1.996 -1.302 -0.609 0.307 0.777 0.895 0.959 0.980 0.990 0.995
-2.2 -2.006 -1.284 -0.574 0.330 0.752 0.844 0.888 0.900 0.905 0.907
-2.4 -2.011 -1.262 -0.537 0.351 0.725 0.795 0.823 0.830 0.832 0.833
-2.6 -2.013 -1.238 -0.499 0.368 0.696 0.747 0.764 0.768 0.769 0.769
-2.8 -2.010 -1.210 -0.460 0.384 0.666 0.702 0.712 0.714 0.714 0.714
-3.0 -2.003 -1.180 -0.420 0.393 0.636 0.660 0.666 0.666 0.667 0.667
Source:  U.S. Interagency Committe on Water Data, Hydrology Subcommitee (1983). "Guidelines for Determining Flood Flow Frequency," Bulletin No. 17B, issued 1981, revised 1983.


TABLE A-7   OUTLIER FREQUENCY FACTORS Kn FOR LOGPEARSON DISTRIBUTIONS
Record
Length
(y)
Kn Record
Length
(y)
Kn Record
Length
(y)
Kn Record
Length
(y)
Kn
10 2.036 45 2.727 80 2.940 115 3.064
15 2.247 50 2.768 85 2.961 120 3.078
20 2.385 55 2.804 90 2.981 125 3.092
25 2.486 60 2.837 95 3.000 130 3.104
30 2.563 65 2.866 100 3.017 135 3.116
35 2.628 70 2.893 105 3.033 140 3.129
40 2.682 75 2.917 110 3.049 145 3.140
Source:  U.S. Interagency Committe on Water Data, Hydrology Subcommitee (1983). "Guidelines for Determining Flood Flow Frequency," Bulletin No. 17B, issued 1981, revised 1983.


TABLE A-8   MEAN yn AND STANDARD DEVIATION σn OF GUMBEL VARIATE (y) VERSUS RECORD LENGTH n
n yn σn n yn σn n yn σn
8 0.4843 0.9043 35 0.5403 1.1285 64 0.5533 1.1793
9 0.4902 0.9288 36 0.5410 1.1313 66 0.5538 1.1814
10 0.4952 0.9497 37 0.5418 1.1339 68 0.5543 1.1834
11 0.4996 0.9676 38 0.5424 1.1363 70 0.5548 1.1854
12 0.5035 0.9833 39 0.5430 1.1388 72 0.5552 1.1873
13 0.5070 0.9972 40 0.5436 1.1413 74 0.5557 1.1890
14 0.5100 1.0095 41 0.5442 1.1436 76 0.5561 1.1906
15 0.5128 1.0206 42 0.5448 1.1458 78 0.5565 1.1923
16 0.5157 1.0316 43 0.5453 1.1480 80 0.5569 1.1938
17 0.5181 1.0411 44 0.5458 1.1499 82 0.5572 1.1953
18 0.5202 1.0493 45 0.5463 1.1519 84 0.5576 1.1967
19 0.5220 1.0566 46 0.5468 1.1538 86 0.5580 1.1980
20 0.5236 1.0628 47 0.5473 1.1557 88 0.5583 1.1994
21 0.5252 1.0696 48 0.5477 1.1574 90 0.5586 1.2007
22 0.5268 1.0754 49 0.5481 1.1590 92 0.5589 1.2020
23 0.5283 1.0811 50 0.5485 1.1607 94 0.5592 1.2032
24 0.5296 1.0864 51 0.5489 1.1623 96 0.5595 1.2044
25 0.5309 1.0915 52 0.5493 1.1638 98 0.5598 1.2055
26 0.5320 1.0961 53 0.5497 1.1653 100 0.5600 1.2065
27 0.5332 1.1004 54 0.5501 1.1667 150 0.5646 1.2253
28 0.5343 1.047 55 0.5504 1.1681 200 0.5672 1.2360
29 0.5353 1.1086 56 0.5508 1.1696 250 0.5688 1.2429
30 0.5362 1.1124 57 0.5511 1.1708 300 0.5699 1.2479
31 0.5371 1.1159 58 0.5515 1.1721 400 0.5714 1.2545
32 0.5380 1.1193 59 0.5518 1.1734 500 0.5724 1.2588
33 0.5388 0.1226 60 0.5521 1.1747 750 0.5738 1.2651
34 0.5396 1.1255 62 0.5527 1.1770 1000 0.5745 1.2685
Source:  Gumbel, E. J. (1958). Statistics of Extremes. Irvington, New York: Columbia University Press.


TABLE A-9   NRCS 24-HR RAINFALL TABLES (AT HALF-HOUR INCREMENTS)
Time
(h)
Fraction of 24-h rainfall depth
Type I Type IA Type II Type III
0.0 0.00000 0.00000 0.00000 0.00000
0.5 0.00871 0.01000 0.00513 0.00500
1.0 0.01745 0.02000 0.01050 0.01000
1.5 0.02621 0.03500 0.01613 0.01500
2.0 0.03500 0.05000 0.02200 0.02000
2.5 0.04416 0.06600 0.02813 0.02519
3.0 0.05405 0.08200 0.03450 0.03075
3.5 0.06466 0.09800 0.04113 0.03669
4.0 0.07600 0.11600 0.04800 0.04300
4.5 0.08784 0.13500 0.05525 0.04969
5.0 0.09995 0.15600 0.06300 0.05675
5.5 0.11234 0.18000 0.07125 0.06419
6.0 0.12500 0.20600 0.0800 0.07200
6.5 0.13915 0.23700 0.08925 0.08063
7.0 0.15600 0.26800 0.09900 0.09050
7.5 0.17460 0.31000 0.10925 0.10163
8.0 0.19400 0.42500 0.12000 0.11400
8.5 0.21900 0.48000 0.13225 0.12844
9.0 0.25400 0.52000 0.14700 0.14575
9.5 0.30300 0.55000 0.16300 0.16594
10.0 0.51500 0.57700 0.18100 0.18900
10.5 0.58300 0.60100 0.20400 0.21650
11.0 0.62300 0.62400 0.23500 0.25000
11.5 0.65550 0.64500 0.28300 0.29800
12.0 0.68400 0.66400 0.66300 0.50000
12.5 0.70925 0.68300 0.73500 0.70200
13.0 0.73200 0.70100 0.77200 0.75000
13.5 0.75225 0.71900 0.79900 0.78350
14.0 0.77000 0.73600 0.82000 0.81100
14.5 0.78625 0.75281 0.83763 0.83406
15.0 0.80200 0.76924 0.85350 0.85425
15.5 0.81725 0.78529 0.86763 0.87156
16.0 0.83200 0.80096 0.88000 0.88600
16.5 0.84625 0.81625 0.89119 0.89838
17.0 0.86000 0.83116 0.90175 0.90950
17.5 0.87325 0.84569 0.91169 0.91938
18.0 0.88600 0.85984 0.92100 0.92800
18.5 0.89825 0.87361 0.92969 0.93581
19.0 0.91000 0.88700 0.93775 0.94325
19.5 0.92125 0.90001 0.94519 0.95031
20.0 0.93200 0.91264 0.95200 0.95700
20.5 0.94225 0.92489 0.95844 0.96336
21.0 0.95200 0.93676 0.96475 0.96944
21.5 0.96125 0.94825 0.97094 0.97523
22.0 0.97000 0.95936 0.97700 0.98075
22.5 0.97825 0.97009 0.98294 0.98598
23.0 0.98600 0.98044 0.98875 0.99094
23.5 0.99325 0.99041 0.99444 0.99561
24.0 1.00000 1.00000 1.00000 1.00000
Source: Roger G. Cronshey, USDA Natural Resources Conservation Service, Engineering Division, Washington, D.C., June 1988.


TABLE A-10   STANDARD NRCS DIMENSIONLESS TEMPORAL RAINFALL DISTRIBUTION FOR EMERGENCY SPILLWAY AND FREEBOARD-POOL DESIGN
Fraction of storm duration Fraction of storm depth Fraction of storm duration Fraction of storm depth
0.00 0.0000    
0.02 0.0080 0.52 0.7240
0.04 0.0162 0.54 0.7420
0.06 0.0246 0.56 0.7590
0.08 0.0333 0.58 0.7750
0.10 0.0425 0.60 0.7900
0.12 0.0524 0.62 0.8043
0.14 0.0630 0.64 0.8180
0.16 0.0743 0.66 0.8312
0.18 0.0863 0.68 0.8439
0.20 0.0990 0.70 0.8561
0.22 0.1124 0.72 0.8678
0.24 0.1265 0.74 0.8790
0.26 0.1420 0.76 0.8898
0.28 0.1595 0.78 0.9002
0.30 0.1800 0.80 0.9103
0.32 0.2050 0.82 0.9201
0.34 0.2550 0.84 0.9297
0.36 0.3450 0.86 0.9391
0.38 0.4370 0.88 0.9483
0.40 0.5300 0.90 0.9573
0.42 0.6030 0.92 0.9661
0.44 0.6330 0.94 0.9747
0.46 0.6600 0.96 0.9832
0.48 0.6840 0.98 0.9916
0.50 0.7050 1.00 1.0000
Source: USDA Natural Resources Conservation Service. (1983). "Computer Program for Project Formulation: Hydrology," Tecnical Release No. 20 (TR-20), Engineering Division, Washington, D.C., May.



APPENDIX B:  DERIVATION OF THE NUMERICAL DIFFUSION COEFFICIENT
OF THE MUSKINGUM-CUNGE METHOD


Space-time discretization of kinematic wave equation

Figure B-1  Space-time discretization of kinematic wave equation.

Expanding the grid function Q( jΔx,nΔt ) (Fig. B-1) in Taylor series about point ( jΔx,nΔt ) leads to:

                                  ∂Q                   1      ∂2Q
Q j n+1  =  Q j n  +  [ _____ ] j  Δt  +  ___ [ ______ ] j  Δt 2  +  ot 3)
                                  ∂t                     2       ∂t 2
(B.1)

                                       ∂Q                       1      ∂2Q
Q j+1n+1  =  Q j+1 n  +  [ _____ ] j+1  Δt  +  ___ [ ______ ] j+1  Δt 2  +  ot 3)
                                        ∂t                        2       ∂t 2
(B.2)

                                 ∂Q                     1      ∂2Q
Q j+1n  =  Q j n  +  [ _____ ] n  Δx  +  ___ [ ______ ] n  Δx 2  +  ox 3)
                                 ∂x                      2      ∂x 2
(B.3)

                                       ∂Q                         1      ∂2Q
Q j+1n+1  =  Q j n+1  +  [ _____ ] n+1  Δx  +  ___ [ ______ ] n+1  Δx 2  +  ox 3)
                                        ∂x                         2       ∂x 2
(B.4)

Substituting Eqs. B.1 to B.4 into Eq. 9-61 and neglecting third-order terms yields:

           ∂Q                    1         ∂2Q
X  { [ ____ ] j  Δt  +   ____   [ ______ ] j  Δt 2 }
            ∂t                     2          ∂t 2
 

                         ∂Q                        1         ∂2Q
+  (1 - X )   { [ _____ ] j+1  Δt  +   ____   [ _____ ] j+1  Δt 2 }
                          ∂t                         2          ∂t 2
 

       C          ∂Q                       1          ∂2Q
+  ____  { [ _____ ] n   Δx  +   ____   [ ______ ] n  Δx 2 }
       2           ∂x                        2          ∂x 2
 

       C          ∂Q                           1          ∂2Q
+  ____  { [ _____ ] n+1   Δx  +   ____   [ ______ ] n+1  Δx 2 }  = 0
       2           ∂x                            2          ∂x 2
(B.5)

in which C = c (Δtx) is the Courant number.

Expressing the derivatives at grid point [( j + 1)Δx, (n + 1)Δt ] in terms of the derivatives at grid point ( jΔx, nΔt ) by means of Taylor series:

    ∂Q                     ∂Q                  ∂2Q
[ _____ ] j+1  =   [ _____ ] j    +  [ ______ ] j,n   Δx   +  ox 2)
     ∂t                       ∂t                  ∂xt
(B.6)

    ∂Q                      ∂Q                     ∂2Q
[ _____ ] n+1  =    [ _____ ] n    +  [ _______ ] j,n   Δt   +  ot 2)
     ∂x                       ∂x                    ∂xt
(B.7)

    ∂2Q                      ∂2Q                   ∂3Q
[ ______ ] j+1  =    [ ______ ] j    +  [ _______ ] j   Δx   +  ox 2)
     ∂t 2                      ∂t 2                  ∂t 2x
(B.8)

    ∂2Q                       ∂2Q                    ∂3Q
[ ______ ] n+1  =    [ ______ ] n    +  [ _______ ] n   Δt   +  ot 2)
     ∂x 2                      ∂x 2                   ∂x 2t
(B.9)

Substituting Eqs. B.6 to B.9 into B.5 and neglecting third-order terms:

            ∂Q                      1         ∂2Q
X  { [ _____ ] j   Δt  +   ____   [ ______ ] j  Δt 2 }
             ∂t                       2          ∂t 2
 

                         ∂Q                        ∂2Q                            1         ∂2Q
+  (1 - X )   { [ _____ ] j   Δt  +   [ ______ ] j,n  Δx Δt  +  ____   [ ______ ] j  Δt 2 }
                          ∂t                        ∂xt                            2          ∂t 2
 

       C          ∂Q                       1          ∂2Q
+  ____  { [ _____ ] n   Δx  +   ____   [ ______ ] n  Δx 2 }
       2           ∂x                        2          ∂x 2
 

       C          ∂Q                         ∂2Q                            1         ∂2Q
+  ____  { [ _____ ] n   Δx  +  [ ______ ] j,n  Δx Δt  +  ____   [ ______ ] n  Δx 2 }  = 0
       2           ∂x                         ∂xt                           2          ∂x 2
(B.10)

In Eq. B.10, dividing by Δt and simplifying:

    ∂Q                     ∂Q                Δt       ∂2Q             c Δx      ∂2Q
[ _____ ] j  +   c [ _____ ] n    +  ____ [ ______ ] j  +  ______ [ ______ ] n
     ∂t                      ∂x                  2        ∂t 2                2         ∂x 2
 

                               C           ∂2Q
+ Δx  { ( 1 - X ) +  ____ }  [ ______ ] j,n  = 0
                               2           ∂xt
(B.11)

The first two terms of Eq. B.11 constitute the kinematic wave equation, Eq. 9-18. The remaining terms are the error R of the first-order-accurate numerical scheme:

         Δt       ∂2Q             c Δx      ∂2Q                                           C            ∂2Q
R =  ____ [ ______ ] j  +  ______ [ ______ ] n  + Δx  { ( 1 - X )  +  _____ }  [ ______ ] j,n  = 0
          2        ∂t 2                2         ∂x 2                                           2           ∂xt
(B.12)

From Eq. 9-18:

 ∂Q                 ∂Q
____    =  - c   ____
 ∂t                   ∂x
(B.13)

Therefore:

  ∂2Q                   ∂2Q
______    =  - c   ______
 ∂x ∂t                   ∂x 2
(B.14)

 ∂2Q                   ∂2Q
______    = c 2   ______
  ∂t 2                    ∂x 2
(B.15)

Substituting Eqs. B.14 and B.15 into B.12 and simplifying:

                           1        ∂2Q
R = c Δx ( X  -  ___ )  _____
                           2         ∂x 2
(B.16)

Comparing Eq. B.16 with the right-hand side of the diffusion wave equation, repeated here:

  ∂Q             ∂Q              ∂2Q
 ____  + c   _____  = νh  _______
   ∂t              ∂x                ∂x 2
(B.17)

it follows that the numerical diffusion coefficient of the Muskingum-Cunge method is:

                    1
νh = c Δx ( ___  -  X )
                    2
(B.18)




APPENDIX C:  U.S. GEOLOGICAL SURVEY FLOOD HYDROLOGY REPORTS


TABLE C-1   USGS REPORTS FOR ESTIMATING RURAL FLOOD PEAKS USING WATERSHED AND CLIMATIC CHARACTERISTICS
State Reference
Alabama Olin, D. A. (1984). "Magnitude and frequency of floods in Alabama," U.S. Geological Survey Water-Resources Investigations 84-4191.
Alaska (r) Curran, J. H., D. F. Meyer, and G. D. Tasker. (2003). "Estimating the magnitude and frequency of peak streamflows for ungaged sites on streams in Alaska and conterminous basins in Canada," U.S. Geological Survey Water-Resources Investigations 84-4191.
Arizona Eychaner, J. H.(1984). "Estimation of magnitude and frequency of floods in Pima County, Arizona, with comparisons of alternative methods," U.S. Geological Survey Water-Resources Investigations Report 84-4142.

U.S. Geological Survey Fact Sheet 111-98. (1999). "The National Flood-Frequency Program - Methods for Estimating Flood Magnitude and Frequency in Rural Areas in Arizona."
California Gotvald, A. J., N. A. Barth, A. G. Veilleux, and C. Parrett. 2012. "Methods for determining magnitude and frequency of floods in California, based on data through water year 2006," U.S. Geological Survey Scientific Investigations Report 2012 - 5113.

Waananen, A. O., and J. R. Crippen. (1977). "Magnitude and frequency of floods in California," U.S. Geological Survey Water-Resources Investigations Report 77 - 21.
Colorado Kircher, J. E., A. F. Choquette, and B. D. Richter. (1985)."Estimation of natural streamflow characteristics in Wesern Colorado," U.S. Geological Survey Scientific Investigations 85-4086.

Livingston, R. K. and K. Russel. (1981)."Rainfall-runoff modeling and preliminary regional flood characteeristics of small rural watersheds in the Arkansas River Basin in Colorado," U.S. Geological Survey Water-Resources Investigations 80-112.

Livingston, R. K., and D. R. Minges. (1987). "Tecniques for estimating regional flood characteristics of small rural watersheds in the plains regions of eastern Colorado," U.S. Geological Survey Water-Resources Investigations 87-4094.
Connecticut U.S. Geological Survey Fact Sheet 014-01. (2001). "The National Flood-Frequency Program - Methods for Estimating Flood Magnitute and Frequence in Connecticut."
Delaware U.S. Geological Survey Fact Sheet 013-01. (2001). "The National Flood-Frequency Program - Methods for Estimating Flood Magnitute and Frequence for Non-Tidal streams in Delaware."
Florida Verdi, R. J. and J. F. Dixon. (2006)."Magnitude and Frequency of Floods for Rural Streams in Florida,"U. S. Geological Survey Water-Resources Investigations Report 2011-5030.
Georgia Feaster, T. D., A. J. Gotvald, and J. C. Weaver. (2011). "Methods for Estimating the Magnitude and Frequency of Floods for Urban and Small, Rural Streams in Georgia, South Carolina, and North Carolina," U.S. Geological Survey Scientific Investigations Report 2014-5030
Hawaii Oki, D. S., S. N. Rosa and C. W. Yeung .(2010)."Flood-Frequency Estimates for Streams on Kauai, Oahu, Molokai, Maui, and Hawaii, State of Hawaii,"U. S. Geological Survey Water-Resources Investigations Report 2010-5035.
Idaho Kjelstrom, L. C., and R. L. Moffatt. (1981)."Method of estimating flood-frequency parameters for streams in Idaho,"U. S. Geological Survey Open-File Report 81-909.
Illinois Curtis, G. W. (1987). "Technique for estimating flood-peak discharges and frequencies on rural streams in Illinois,"U. S. Geological Survey Open-File Report 87-4207.
Indiana Glatfelter, D. R. (1984). "Techniques for estimating magnitude and frequency of floods in Indiana,"U. S. Geological Survey Open-File Report 84-4134.
Iowa Lara, O. (1987). "Methods for estimating the magnitude and frequency of floods at ungaged sites on unregulated rural streams in Iowa,"U. S. Geological Survey Open-File Report 87-4132.
Kansas Rasmussen, P. R., and C. A. Perry. (2000)."Estimation of Peak Streamflows for Unregulated Rural Streams in Kansas"U. S. Geological Survey Open-File Report 00-4079.
Kentucky Choquette, A. F. (1987). "Regionalization of peak discharges for streams in Kentucky"U. S. Geological Survey Open-File Report 87-4209.
Louisiana The National Flood-Frequency Program - Methods for estimating flood magnitude and frequency in rural areas in Louisiana. (1999).U. S. Geological Survey Fact Sheet 099-01.

Lee, F. N. (1985), "Floods in Louisiana, Magnitude and frequency,"Fourth Edition, Department of Transportation and Development, Water Resources Technical Report No. 36.
Maine Hodgkins, G. (1999). "Estimating the magnitude of peak flows for streams in Maine for selected recurrence intervals"U. S. Geological Survey Open-File Report 99-4008.
Maryland Dillow, J. J. A. (1996). "Tecnique for Estimating Magnitude and Frequence of Peak Flows in Maryland," U.S. Geological Survey Water-Resources Investigations Report 95-4154.
Massachusetts Wandle, S. W. (1983)."Estimating peak discharges of small, rural streams in Massachusetts" U.S. Geological Survey Water-Supply Paper 2214
Michigan Holtschlag, D. J., and H. M. Croskey. (1984). "Statistical models for estimating flow characteristics of Michigan streams," U.S. Geological Survey Water Resources Investigations Report 84-4207
Minnesota Jacques, J. E., and D. L. Lorenz. (1988). "Techniques for estimating the magnitude an frequency of floods in Minnesota," U.S. Geological Survey Water Resources Investigations Report 87-4170
Mississippi Landers, M. N., and K. Wilson, Jr. (1991). "Flood characteristics of Mississippi streams" U.S. Geological Survey Water Resources Investigations Report 91-4037
Missouri Hauth, L. D. (1974). "Technique for estimating the magnitude and frequency of Missouri floods," U.S. Geological Survey Open-File Report 91-89.

Alexander, T. W., and G. L. Wilson. (1995). "Technique for estimating the 2 -to 500 - year flood discharges on unregulated streams in rural Missouri" U.S. Geological Survey Water Resources Investigations Report 95-4231

Southard, R. E. (2010). "Estimation of the magnitude and frequency of floods in urban basins in Missouri" U.S. Geological Survey Water Resources Investigations Report 2010-5073
Montana Omang, R. J., Parrett, C., and J. A. Hull. (1986). "Methods for estimating magnitude and frequency of floods in Montana based on data through 1983" U.S. Geological Survey Water Resources Investigations Report 86-4027
Nebraska Beckman, E. W. (1976). "Magnitude and frequency of floods in Nebraska" U.S. Geological Survey Water Resources Investigations 76-109
Nevada "The National Flood-Frequency Program - Methods for estimating flood magnitude and frequency in rural areas in Nevada." (1999). U.S. Geological Survey Fact Sheet 123-98
New Hampshire LeBlanc, D. R. (1978). "Progress report on hydrologic investigations of small drainage areas in New Hampshire - Preliminary relations for estimating peak discharges on rural, unregulated streams," U.S. Geological Survey Water-Resources Investigations 78-47.
New Jersey Watson, K. M., and R. D. Schopp. (2009). "Methodology for estimation of flood magnitude and frequency for New Jersey streams," U.S. Geological Survey Water-Resources Investigations 2009-5167
New Mexico Waltmeyer, S. D. (1986). "Techniques for estimating flood-flow frequency for unregulated streams in New Mexico" U.S. Geological Survey Water-Resources Investigations Report 86-4104

"The National Flood - Frequency Program - Methods for estimating flood magnitude and frequency in rural areas in New Mexico." (2000). U.S. Geological Survey Fact Sheet 055-00
New York Zembrzuski, T. J., and B. Dunn. (1979). "Techniques for estimating magnitude and frequency of floods on rural unregulated streams in New York, excluding Long Island," U.S. Geological Survey Water-Resources Investigations 79-83
North Carolina Pope, B. F., G. D. Tasker, and J. C. Robbins. (2001)."Estimating the magnitude and frequency of floods in rural basins of North Carolina - Revised," U.S. Geological Survey Water-Resources Investigations Report 01-4207

Feaster, T. D., A. J. Gotvald, and J. C. Weaver. (2011). "Methods for Estimating the Magnitude and Frequency of Floods for Urban and Small, Rural Streams in Georgia, South Carolina, and North Carolina," U.S. Geological Survey Scientific Investigations Report 2014-5030
North Dakota Sether, T. W. (1992)."Techniques for estimating peak-flow frequency relations for North Dakota streams," U.S. Geological Survey Water-Resources Investigations Report 92-4020
Ohio Koltun, G. F. (2003). "Techniques for estimating peak-flow discharges of rural, unregulated streams in Ohio" U.S. Geological Survey Water-Resources Investigations Report 03-4164
Oklahoma Tortorelli, R. L., and D. L. Bergman. (1984). "Techniques for estimating flood peak discharges for unregulated streams and streams regulated by small floodwater retarding structures in Oklahoma" U.S. Geological Survey Water-Resources Investigations Report 86-4358

"The National Flood - Frequency Program - Methods for estimating flood magnitude and frequency in rural and urban areas in Oklahoma." (2001). U.S. Geological Survey Fact Sheet 008-01
Oregon Harris, D. D., and L. E. Hubbard. (1982). "Magnitude and frequency of floods in eastern Oregon" U.S. Geological Survey Water-Resources Investigations 82-4078

Harris, D. D., and L. E. Hubbard. (1979). "Magnitude and frequency of floods in western Oregon," U.S. Geological Survey Water-Resources Investigations 79-553
Pennsylvania Filippo, H. N., Jr. (1977). "Floods in Pennsylvania: a manual for estimation of their magnitude and frequency," U.S. Geological Survey Water-Resources Investigations Report 76-391
Puerto Rico Gines, O. R. (1999). "Estimation of magnitude and frequency of floods for streams in Puerto Rico: New empirical models," U.S. Geological Survey Water-Resources Investigations Report 99-4142
Rhode Island Johnson, C. G., and G. A. Laraway. (1976). "Flood magnitude and frequency of small Rhode Island streams," U.S. Geological Survey Open-File Report 76-883
South Carolina Whetstone, B. H. (1982). "Techniques for estimating magnitude and frequency of floods in South Carolina" U.S. Geological Survey Water-Resources Investigations Report 82-1

Feaster, T. D., A. J. Gotvald, and J. C. Weaver. (2009). "Magnitude and frequency of rural floods in Southeastern United States, 2006: Volume 3, South Carolina," U.S. Geological Survey Water-Resources Investigations Report 2009-5156

Feaster, T. D., A. J. Gotvald, and J. C. Weaver. (2011). "Methods for Estimating the Magnitude and Frequency of Floods for Urban and Small, Rural Streams in Georgia, South Carolina, and North Carolina," U.S. Geological Survey Scientific Investigations Report 2014-5030
South Dakota Becker, L. D. (1974). "A method for estimating magnitude and frequency of floods in South Dakota," U.S. Geological Survey Open-File Report 74-35

Becker, L. D. (1980). "Techniques for estimating flood peaks, volumes and hydrographs on small streams in South Dakota," U.S. Geological Survey Scientific Investigations Report 80-80

Becker, L. D. (1982). "Magnitude and frequency of floods from selected drainage basins in South Dakota," U.S. Geological Survey Scientific Investigations Report 82-31
Tennessee Law, G. S., and G. D. Tasker. (2000). "Flood-frequency prediction methods for unregulated streams of Tennessee," U.S. Geological Survey Water-Resources Investigations Report 03-4176
Texas Schroeder, E. E. and B. C. Massey. (1977). "Technique for Estimating the Magnitude and Frequency of Floods in Texas," U.S. Geological Survey Water-Resources Investigations Report 77-110

Land, L. F., E. E., Schroeder, B. B. Hampton. "Technique for Estimating the Magnitude and Frequency of Floods in the Dallas-Fort Worth Metropolitan Area, Texas," U.S. Geological Survey Scientific Investigations Report 82-18
Utah Thomas, B. E., and K. L. Lindskov. (1983). "Methods for estimating peak discharge and flood boundaries of streams in Utah," U.S. Geological Survey Water-Resources Investigations Report 83-4129

"The National Flood-Frequency Program - Methods for estimating flood magnitude and frequency in rural areas in Utah." (1999). U.S. Geological Survey Fact Sheet 124-98
Vermont Olson, S. A. (2002). "Flow-frequency characteristics of Vermont streams," U.S. Geological Survey Water-Resources Investigations Report 02-4238
Virginia Bisese, J. A. (1995). "Methods for estimating the magnitude and frequency of peak discharges of rural, unregulated streams in Virginia," U.S. Geological Survey Water-Resources Investigations Report 94-4148
Washington Sumioka, S. S., D. L. Kresch, and K. D. Kasnick. (1998). "Magnitude and frequency of floods in Washington," U.S. Geological Survey Water-Resources Investigations Report 97-4277

"The National Flood-Frequency Program - Methods for estimating flood magnitude in Washington." (2001). U.S. Geological Survey Fact Sheet 016-01
West Virginia Runner, G. S. (1980). "Technique for estimating magnitude and frequency of floods in West Virginia," U.S. Geological Survey Water-Resources Investigations Report 80-1218

Wiley, J. B., and J. T. Atkins, Jr. (2010). "Estimation of flood-frequency discharges for rural, unregulated streams in West Virginia," (2001).U.S. Geological Survey Water-Resources Investigations Report 2010-5033
Wisconsin Walker, J. F., and W. R. Krug. (2003). "Flood-frequency characteristics of Wisconsin streams," U.S. Geological Survey Water-Resources Investigations Report 03-4250
Wyoming Lowham, H. W. (1976). "Techniques for estimating flow characteristics of Wyoming streams," U.S. Geological Survey Water-Resources Investigations Report 76-112

Miller, K. A. (2003). "Peak-flow characteristics of Wyoming streams,"U.S. Geological Survey Water-Resources Investigations Report 03-4107

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