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Estimating flood probabilities within specific time intervals
Journal of Hydrology 1 (1963) 265-271; © North.Holland Pubh'shing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ESTIMATING FLOOD PROBABILITIES WITHIN S P E C I F I C T I M E INTERVALS W A R R E N A. HALL and D A V I D T. HOWELL
Water Resources Center, University of California, Los Angeles, U.S.A.
Received 30 September, 1963
Abstract: A technique is presented for computing the probability that a flood of certain magnitude will be equalled or exceeded one or more times in any given time period. Values of this probability for floods of different return period are tabulated. The probabilities are then obtained of one or more occurrences in a given time period of floods in class intervals between magnitudes corresponding to certain return periods. The limitations of the technique are discussed.
1. Introduction There are circumstances in which knowledge is required as to what is the probability of the occurrence of a flood of a certain size in a certain interval of time, and in what way this probability varies with the length of this interval of time. These circumstances may arise in the scheduling of dam construction operations and in the planning of optimal flood reservoir operation as described by Burton x). In practical problems dealing with the handling of floods, the interval of time considered may often be quite short, of the order of a few days, for instance. This interval of time will be short compared to the period at which, on the average, any particular flood magnitude of importance will be equalled or exceeded. Flow magnitudes which are expected frequently receive different engineering treatment from floods which could conveniently be defined as flows of such a large magnitude as to occur, on the average, infrequently.
2. Basis for probability estimates For a particular location on a river, a curve can be prepared which has been called a partial duration curve by Jarvis z), showing flood magnitudes plotted against the period of time at which, on the average, the flood magnitude will be equalled or exceeded. This period of time we will call the return 265
266
W . A. H A L L A N D D. T. H O W E L L
period, as does Gumbela). In effect, for the preparation of such a curve, all flood peaks above a certain magnitude chosen to define a flood for the particular circumstances governing are considered, whether more than one occur in a single year or not. Such a curve may be used as a basis for the estimation of the probability of a flood of any particular magnitude being equalled or exceeded in any particular time interval. From the probabilities for floods of different magnitudes being equalled or exceeded, the probabilities can be determined for floods between certain magnitudes. To the extent that the floods in any particular situation being investigated can be assumed to be independent randomly occurring events, the probability of a flood of any particular magnitude being equalled or exceeded can be estimated from the Poisson distribution expression. It gives the probability P(r) of exactly r occurrences of a randomly occurring event in a time interval as e-~2r P(r) =. , (1) r! where 2 is the mean number of occurrences of the event per time interval. The probability of one occurrence will be P(I) =
\1!]"
(2)
15)
The probability of one or two occurrences will be
P ( 1 , 2 ) = ~ P ( r ) = e -~ ,=,
\1!
+
2.1 "
(3
The probability of one, two or three occurrences will be 3
P(1,2,3) =
~P(r) =
e_;~ (2__
12 +
13) +
(4)
The probability of one or more occurrences, supposing that it were possible to squeeze an infinite number of occurrences into the time interval, would be P(1,2, oo) E P ( r ) e- a 2 1
. . . r.= l .
+-2~ +'"~
i oo
Ir
= e - a ( e +'~ - 1)
(5)
= 1 -
(6)
e-Z
Thus, the probability of there being no occurrence is e-z. For low values of 2, the differences are small between the probability
267
ESTIMATING FLOOD PROBABILITIES
o f one occurrence, o f one or two occurrences, o f one, two or three occurrences, and o f one or more up to an infinite n u m b e r o f occurrences. Table 1 TABLE 1 Variation o f probability coefficients
Coefficient o f e -z Number ofoccurrences 2
e -~
.01 .1 .2 .3 .4 .5 1.0
.99005 .9048 .8187 .7408 .6703 .6065 .3679
1
1 or 2
1,2 or 3
1,2 .... o~
F.)
~ + ~.
~ + ~ + ~
(e~ - I)
.01000 .1000 .2000 .3000 .4000 .5000 1.0000
.01005 .1050 .2200 .3450 .4800 .6250 1.5000
.01005 .1052 .2213 .3495 .4907 .6458 1.6667
.01005 .1052 .2214 .3499 .4918 .6487 1.7183
shows values o f e - z and h o w the values o f the coefficient in parentheses o f e - z in the expressions (2), (3), (4) and (5) vary with the number o f occurrences and with 2. It can be seen from the values o f the coefficients that for ). less than 0.2 the probability o f 1,2,... oo occurrences does not differ from the probability o f exactly one occurrence by more than about 1 0 ~ . F r o m the values o f e -z, it can be seen that as 2 becomes small, the value o f the probability o f one occurrence will a p p r o a c h 2. However, as 2 increases, divergence is rapid and the probability o f one occurrence differs from 2 by more than about 10~o for 2 greater than 0.1. Values o f P(1,2... r) are tabulated in Table 2 for the same values o f r and as in Table 1. TABLE 2 Probabilities o f different n u m b e r s o f occurrences. Values o f P(1, 2 . . . r ) are tabulated in t h e b o d y o f the table. ~,
.01 .1 .2 .3 .4 .5 1.0
r 1
2
3
oo
.00990 .09048 .16734 .22224 .26812 .30325 .3679
.00995 .09500 .18011 .25558 .32174 .37906 .55185
.00995 .09518 .18118 .25891 .32892 .39168 .61318
.00995 .09518 .18126 .25921 .32965 .39344 .63216
268
W. A. HALL AND D. T. HOWELL
I n T a b l e 2, it can be seen that the value o f P(1,2... r) for a n y value o f r approaches 2 as 2 becomes very small. The term 2, being the average n u m b e r of events per time period, is equal to the time period At being considered divided by the return period T. T h a t is, with t a n d T being measured in the same units. Then, .'it
P = 1 - e-¥
(7)
P being written instead o f P(1,2... ~ ) . F o r example, if the probability is being sought o f a flood being equalled or exceeded in a period o f one m o n t h , the flood in question being equalled or exceeded once per year o n the average, the r e t u r n period is one year a n d 2 = At/T=
1 T-~"
Values o f P for a range of values of T a n d t are shown in Table 3. It can be seen from Table 3 that the probability of one or more occurrences o f a n event in a time interval equal to its return period is equal, with 2 = ~Jt/T = 1, to 0.632. T h a t is, the probability o f the occurrence in a n y
one year o f one or more floods which can be expected to be equalled or exceeded with a n average period o f one year is 0.632. The p r o b a b i l i t y of exactly one flood is 0.368 which is the same as the probability o f n o floods. The probability o f one or more occurrences o f a flood with a r e t u r n period TABLE 3
Probabilities of one or more occurrences of events with different return periods in different time intervals Values of probability P are printed in the body of the table Time Interval At .1 .2 .4 1 2 4 10 20 40 100 200
Return Period T .1
.2
.4
1
2
4
l0
20
40
100
200
.632 .865 .982 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
.393 .632 .865 .993 1.000 1.000 1.000 1.000 1.000 1.000 1.000
.221 .393 .632 .918 .993 1.000 1.000 1.000 1.000 1.000 1.000
.095 .181 .380 .632 .865 .982 1.000 1.000 1.000 1.000 1.000
.049 .095 .181 .393 .632 .865 .993 1.000 1.000 1.000 1.000
.025 .049 .095 .221 .393 .632 .918 .993 1.000 1.000 1.000
.010 .020 .039 .095 .181 .330 .632 .865 .982 1.000 1.000
.005 .010 .020 .049 .095 .181 .393 .632 .865 .993 1.000
.003 .005 .010 .025 .049 .095 .221 .393 .632 .918 .993
.001 .002 .004 .010 .020 .039 .095 .181 .330 .632 .865
.0005 .001 .002 .005 .010 .020 .049 .095 .181 .393 .632
Note: Both At and T must be measured in the same units of time.
269
ESTIMATING FLOOD PROBABILITIES
of l0 years in an interval of one year is, with 2 = 0.1, equal to 9.5%. The probability of one occurrence is 9.0 %. 3. Probabilities in short time intervals In Table 4 are presented values of the probability of occurrence of one or more floods with a range of return periods from 0.1 to 100 years in short periods of time up to 35 days. These ranges of variables have been found useful in some flood reservoir operation studies. TABLE4 Probabilities of one or more occurrences of events with different return periods in different time intervals Values of P are printed in the body of the table Time Interval At (days)
Return period T (years) 0.1
0.5
1
2
5
10
100
5 10 15 20 25 30 35
.1280 .2396 .3370 .4219 .4960 .5604 .6167
.0270 .0533 .0789 .1038 .1280 .1916 .1745
.0136 .0270 .0403 .0533 .0662 .0789 .0914
.0068 .0136 .0203 .0270 .0337 .0403 .0468
.0027 .0055 .0082 .0109 .0186 .0163 .0190
.0014 .0027 .0041 .0055 .0068 .0082 .0095
.0001 .0003 .0004 .0005 .0007 .0008 .0010
Because the return periods of floods in the partial duration curve represent periods at which, on the average, certain flood magnitudes are equalled or exceeded, the probabilities for any flood magnitudes calculated from such a curve represent cumulative probabilities. The probability of a flood between certain magnitudes, that is, in a certain class interval of magnitude, occurring once or more in a given time interval may be obtained by subtracting the probability of the magnitude which is the higher boundary from the probability of the magnitude which is the lower boundary. The boundary values of flood magnitude may be made to correspond to selected values of return period. Values of the probability of one or more occurrences in various time intervals of floods or other similar events in different class intervals of return period are presented in Table 5. These probability values, AP, have been obtained as the differences between values in successive columns of Table 4. For various partial duration curves different values of return period may be found suitable for marking off the class intervals.
270
W. A. HALL AND D. T. HOWELL TABLE 5
Probabilities of one or more occurrences of events with return periods between different values Values of AP are plotted in body of the table Time Interval At (days)
Return period T (years) 0.1 to 0.5
0.5 to 1
1 to 2
2 to 5
5 to 10
10 to 100
5 10 15 20 25 30 35
.1010 .1863 .2581 .3181 .2680 .4088 .4422
.0034 .0263 .0386 .0505 .0618 .0727 .0831
.0068 .0134 .0200 .0263 .0325 .0386 .0446
.0041 .0081 .0121 .0161 .0201 .0240 .0378
.0013 .0038 .0041 .0054 .0068 .0081 .0095
.0013 .0024 .0037 .0050 .0061 .0074 .0085
The procedure described uses the tabulated values of probability in conjunction with a single partial duration curve. It is effective in situations where floods are equally likely to occur at any time of the year. I f the likelihood of flooding is different in different seasons of the year, then partial duration curves for the separate seasons will have to be used. It is recognized that renewed flooding in short time intervals will not meet the requirement of randomness needed by this analysis. Partial saturation of the soils of the watershed, amongst other things, will result in a flood flow of a given magnitude with substantially less rainfall than required for the initial flooding. The effect will be to increase the probabilities of floods of given sizes following shortly after other floods. To this extent, the methods described here which are based on independence of successive flood flows are limited in their applicability. This limitation is felt most severely in one field where the results would be most useful, in the estimation of the probability of renewed flooding.
Acknowledgement This study forms part of work being carried on in the Department of Engineering, University of California, Los Angeles, with the support of the Water Resources Center, University of California. The probability tables presented here are extracted from tables prepared by Richard L. Viersen using the services of the Western Data Processing Center and the U C L A Computing Facility.
ESTIMATING FLOOD PROBABILITIES
271
References 1) J. R. Burton, W. A. Hall and D. T. Howell, Proc. 10th Congress I.A.H.R., London (September, 1963) 2) C. S. Jarvis et al., Water Supply Paper 771, U.S. Department of the Interior, Geological Survey, Washington, D.C. (1936) 3) E. J. Gumbel Annals of Mathematical Statistics, 12 (1941) 163-190
ESTIMATING FLOOD PROBABILITIES WITHIN S P E C I F I C T I M E INTERVALS W A R R E N A. HALL and D A V I D T. HOWELL
Water Resources Center, University of California, Los Angeles, U.S.A.
Received 30 September, 1963
Abstract: A technique is presented for computing the probability that a flood of certain magnitude will be equalled or exceeded one or more times in any given time period. Values of this probability for floods of different return period are tabulated. The probabilities are then obtained of one or more occurrences in a given time period of floods in class intervals between magnitudes corresponding to certain return periods. The limitations of the technique are discussed.
1. Introduction There are circumstances in which knowledge is required as to what is the probability of the occurrence of a flood of a certain size in a certain interval of time, and in what way this probability varies with the length of this interval of time. These circumstances may arise in the scheduling of dam construction operations and in the planning of optimal flood reservoir operation as described by Burton x). In practical problems dealing with the handling of floods, the interval of time considered may often be quite short, of the order of a few days, for instance. This interval of time will be short compared to the period at which, on the average, any particular flood magnitude of importance will be equalled or exceeded. Flow magnitudes which are expected frequently receive different engineering treatment from floods which could conveniently be defined as flows of such a large magnitude as to occur, on the average, infrequently.
2. Basis for probability estimates For a particular location on a river, a curve can be prepared which has been called a partial duration curve by Jarvis z), showing flood magnitudes plotted against the period of time at which, on the average, the flood magnitude will be equalled or exceeded. This period of time we will call the return 265
266
W . A. H A L L A N D D. T. H O W E L L
period, as does Gumbela). In effect, for the preparation of such a curve, all flood peaks above a certain magnitude chosen to define a flood for the particular circumstances governing are considered, whether more than one occur in a single year or not. Such a curve may be used as a basis for the estimation of the probability of a flood of any particular magnitude being equalled or exceeded in any particular time interval. From the probabilities for floods of different magnitudes being equalled or exceeded, the probabilities can be determined for floods between certain magnitudes. To the extent that the floods in any particular situation being investigated can be assumed to be independent randomly occurring events, the probability of a flood of any particular magnitude being equalled or exceeded can be estimated from the Poisson distribution expression. It gives the probability P(r) of exactly r occurrences of a randomly occurring event in a time interval as e-~2r P(r) =. , (1) r! where 2 is the mean number of occurrences of the event per time interval. The probability of one occurrence will be P(I) =
\1!]"
(2)
15)
The probability of one or two occurrences will be
P ( 1 , 2 ) = ~ P ( r ) = e -~ ,=,
\1!
+
2.1 "
(3
The probability of one, two or three occurrences will be 3
P(1,2,3) =
~P(r) =
e_;~ (2__
12 +
13) +
(4)
The probability of one or more occurrences, supposing that it were possible to squeeze an infinite number of occurrences into the time interval, would be P(1,2, oo) E P ( r ) e- a 2 1
. . . r.= l .
+-2~ +'"~
i oo
Ir
= e - a ( e +'~ - 1)
(5)
= 1 -
(6)
e-Z
Thus, the probability of there being no occurrence is e-z. For low values of 2, the differences are small between the probability
267
ESTIMATING FLOOD PROBABILITIES
o f one occurrence, o f one or two occurrences, o f one, two or three occurrences, and o f one or more up to an infinite n u m b e r o f occurrences. Table 1 TABLE 1 Variation o f probability coefficients
Coefficient o f e -z Number ofoccurrences 2
e -~
.01 .1 .2 .3 .4 .5 1.0
.99005 .9048 .8187 .7408 .6703 .6065 .3679
1
1 or 2
1,2 or 3
1,2 .... o~
F.)
~ + ~.
~ + ~ + ~
(e~ - I)
.01000 .1000 .2000 .3000 .4000 .5000 1.0000
.01005 .1050 .2200 .3450 .4800 .6250 1.5000
.01005 .1052 .2213 .3495 .4907 .6458 1.6667
.01005 .1052 .2214 .3499 .4918 .6487 1.7183
shows values o f e - z and h o w the values o f the coefficient in parentheses o f e - z in the expressions (2), (3), (4) and (5) vary with the number o f occurrences and with 2. It can be seen from the values o f the coefficients that for ). less than 0.2 the probability o f 1,2,... oo occurrences does not differ from the probability o f exactly one occurrence by more than about 1 0 ~ . F r o m the values o f e -z, it can be seen that as 2 becomes small, the value o f the probability o f one occurrence will a p p r o a c h 2. However, as 2 increases, divergence is rapid and the probability o f one occurrence differs from 2 by more than about 10~o for 2 greater than 0.1. Values o f P(1,2... r) are tabulated in Table 2 for the same values o f r and as in Table 1. TABLE 2 Probabilities o f different n u m b e r s o f occurrences. Values o f P(1, 2 . . . r ) are tabulated in t h e b o d y o f the table. ~,
.01 .1 .2 .3 .4 .5 1.0
r 1
2
3
oo
.00990 .09048 .16734 .22224 .26812 .30325 .3679
.00995 .09500 .18011 .25558 .32174 .37906 .55185
.00995 .09518 .18118 .25891 .32892 .39168 .61318
.00995 .09518 .18126 .25921 .32965 .39344 .63216
268
W. A. HALL AND D. T. HOWELL
I n T a b l e 2, it can be seen that the value o f P(1,2... r) for a n y value o f r approaches 2 as 2 becomes very small. The term 2, being the average n u m b e r of events per time period, is equal to the time period At being considered divided by the return period T. T h a t is, with t a n d T being measured in the same units. Then, .'it
P = 1 - e-¥
(7)
P being written instead o f P(1,2... ~ ) . F o r example, if the probability is being sought o f a flood being equalled or exceeded in a period o f one m o n t h , the flood in question being equalled or exceeded once per year o n the average, the r e t u r n period is one year a n d 2 = At/T=
1 T-~"
Values o f P for a range of values of T a n d t are shown in Table 3. It can be seen from Table 3 that the probability of one or more occurrences o f a n event in a time interval equal to its return period is equal, with 2 = ~Jt/T = 1, to 0.632. T h a t is, the probability o f the occurrence in a n y
one year o f one or more floods which can be expected to be equalled or exceeded with a n average period o f one year is 0.632. The p r o b a b i l i t y of exactly one flood is 0.368 which is the same as the probability o f n o floods. The probability o f one or more occurrences o f a flood with a r e t u r n period TABLE 3
Probabilities of one or more occurrences of events with different return periods in different time intervals Values of probability P are printed in the body of the table Time Interval At .1 .2 .4 1 2 4 10 20 40 100 200
Return Period T .1
.2
.4
1
2
4
l0
20
40
100
200
.632 .865 .982 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
.393 .632 .865 .993 1.000 1.000 1.000 1.000 1.000 1.000 1.000
.221 .393 .632 .918 .993 1.000 1.000 1.000 1.000 1.000 1.000
.095 .181 .380 .632 .865 .982 1.000 1.000 1.000 1.000 1.000
.049 .095 .181 .393 .632 .865 .993 1.000 1.000 1.000 1.000
.025 .049 .095 .221 .393 .632 .918 .993 1.000 1.000 1.000
.010 .020 .039 .095 .181 .330 .632 .865 .982 1.000 1.000
.005 .010 .020 .049 .095 .181 .393 .632 .865 .993 1.000
.003 .005 .010 .025 .049 .095 .221 .393 .632 .918 .993
.001 .002 .004 .010 .020 .039 .095 .181 .330 .632 .865
.0005 .001 .002 .005 .010 .020 .049 .095 .181 .393 .632
Note: Both At and T must be measured in the same units of time.
269
ESTIMATING FLOOD PROBABILITIES
of l0 years in an interval of one year is, with 2 = 0.1, equal to 9.5%. The probability of one occurrence is 9.0 %. 3. Probabilities in short time intervals In Table 4 are presented values of the probability of occurrence of one or more floods with a range of return periods from 0.1 to 100 years in short periods of time up to 35 days. These ranges of variables have been found useful in some flood reservoir operation studies. TABLE4 Probabilities of one or more occurrences of events with different return periods in different time intervals Values of P are printed in the body of the table Time Interval At (days)
Return period T (years) 0.1
0.5
1
2
5
10
100
5 10 15 20 25 30 35
.1280 .2396 .3370 .4219 .4960 .5604 .6167
.0270 .0533 .0789 .1038 .1280 .1916 .1745
.0136 .0270 .0403 .0533 .0662 .0789 .0914
.0068 .0136 .0203 .0270 .0337 .0403 .0468
.0027 .0055 .0082 .0109 .0186 .0163 .0190
.0014 .0027 .0041 .0055 .0068 .0082 .0095
.0001 .0003 .0004 .0005 .0007 .0008 .0010
Because the return periods of floods in the partial duration curve represent periods at which, on the average, certain flood magnitudes are equalled or exceeded, the probabilities for any flood magnitudes calculated from such a curve represent cumulative probabilities. The probability of a flood between certain magnitudes, that is, in a certain class interval of magnitude, occurring once or more in a given time interval may be obtained by subtracting the probability of the magnitude which is the higher boundary from the probability of the magnitude which is the lower boundary. The boundary values of flood magnitude may be made to correspond to selected values of return period. Values of the probability of one or more occurrences in various time intervals of floods or other similar events in different class intervals of return period are presented in Table 5. These probability values, AP, have been obtained as the differences between values in successive columns of Table 4. For various partial duration curves different values of return period may be found suitable for marking off the class intervals.
270
W. A. HALL AND D. T. HOWELL TABLE 5
Probabilities of one or more occurrences of events with return periods between different values Values of AP are plotted in body of the table Time Interval At (days)
Return period T (years) 0.1 to 0.5
0.5 to 1
1 to 2
2 to 5
5 to 10
10 to 100
5 10 15 20 25 30 35
.1010 .1863 .2581 .3181 .2680 .4088 .4422
.0034 .0263 .0386 .0505 .0618 .0727 .0831
.0068 .0134 .0200 .0263 .0325 .0386 .0446
.0041 .0081 .0121 .0161 .0201 .0240 .0378
.0013 .0038 .0041 .0054 .0068 .0081 .0095
.0013 .0024 .0037 .0050 .0061 .0074 .0085
The procedure described uses the tabulated values of probability in conjunction with a single partial duration curve. It is effective in situations where floods are equally likely to occur at any time of the year. I f the likelihood of flooding is different in different seasons of the year, then partial duration curves for the separate seasons will have to be used. It is recognized that renewed flooding in short time intervals will not meet the requirement of randomness needed by this analysis. Partial saturation of the soils of the watershed, amongst other things, will result in a flood flow of a given magnitude with substantially less rainfall than required for the initial flooding. The effect will be to increase the probabilities of floods of given sizes following shortly after other floods. To this extent, the methods described here which are based on independence of successive flood flows are limited in their applicability. This limitation is felt most severely in one field where the results would be most useful, in the estimation of the probability of renewed flooding.
Acknowledgement This study forms part of work being carried on in the Department of Engineering, University of California, Los Angeles, with the support of the Water Resources Center, University of California. The probability tables presented here are extracted from tables prepared by Richard L. Viersen using the services of the Western Data Processing Center and the U C L A Computing Facility.
ESTIMATING FLOOD PROBABILITIES
271
References 1) J. R. Burton, W. A. Hall and D. T. Howell, Proc. 10th Congress I.A.H.R., London (September, 1963) 2) C. S. Jarvis et al., Water Supply Paper 771, U.S. Department of the Interior, Geological Survey, Washington, D.C. (1936) 3) E. J. Gumbel Annals of Mathematical Statistics, 12 (1941) 163-190