Journal of Hydrology 9 (1969) 405-426 © North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
RAINFALL-RUNOFF RELATIONSHIPS E X P R E S S E D BY D I S T R I B U T I O N P A R A M E T E R S EMIL O. FRIND Department of Civil Engineering, University of Toronto, Canada Abstract: This study is concerned with the statistical parameters of precipitation and of runoff and the interrelations between corresponding parameters. Three processes are considered: the input process (mean annual effective precipitation), the transformation process (basin storage), and the output process (mean annual runoff). The input is assumed to be a pure-random series, with known statistical parameters. The transformation is characterized by an exponential recession curve with one parameter. With these assumptions, equations are developed expressing the statistical parameters of the output for any length of carryover. The validity of these equations is confirmed by means of computersimulated series, which are analyzed for their moments. The type of distribution of the output is also established. 1. Introduction
The section of the hydrologic cycle extending f r o m rainfall to runoff is o f considerable importance to the hydrologist. Precipitation reaches the g r o u n d as rain, snow or in some other form. Part of this precipitation returns to the atmosphere by means o f evaporation and evapotranspiration. A n o t h e r part m a y serve to recharge aquifers extending beyond the boundaries o f any particular drainage basin. The remainder o f the precipitation is collected in the basin and eventually discharges at the basin outlet as runoff. That portion o f the precipitation which contributes entirely to runoff is usually called the effective precipitation. Effective precipitation is a process which can be characterized by a set o f statistical parameters. Similarly, runoff is a process with another set of parameters. These two processes are connected by the transformation resulting f r o m the temporary storage and the transport o f water t h r o u g h the basin. I f the transformation is a linear one, then it is possible to find the relation between the two sets o f parameters. A study concerned with this problem was carried out by Shenl), who used electronic analog methods. He f o u n d that lognormal inputs to a linear reservoir lead to lognormal output with decreased variance and skewness. Matalas 2) transformed series o f normally-distributed input by means 405
406
E.o. FRIND
of a moving-average process and proved that the output also has a normal distribution. Jeng and Yevjevich z) investigated the outflow characteristics of natural lakes when the inflows are lognormally distributed. The present study endeavours to establish the relationships between the corresponding statistical parameters of effective precipitation and runoff in a general way. The results are independent of the type of statistical distribution of the input, but are based on the assumptions that the input is random, and thetransformation is linear. A similar approach may be used for the transformation of non-random input. Only elementary hydrological and mathematical concepts are employed in this paper. The work was carried out by the author 4) as part of his M.A.Sc. program at the Department of Mechanical Engineering of the University of Toronto.
2. The input process Most hydrologic processes, including precipitation, are of a stochastic and ergodic nature. Although precipitation is actually a continuous process (which includes periods of zero intensity), it is usually convenient to discretize it by using the mean values of intensity over some time base. Commonly used time bases are the day, the month, or the year. Daily or monthly precipitation means are subject to the cyclical influence of the seasons, leading to some degree of interdependence among the successive terms in the series. A sequence of annual means, on the other hand, may often be considered pure-random. In the recent past it was thought that sun spots might exert some influence on precipitation, but YevjevichS), in his study on the fluctuations of wet and dry years, has concluded that there is no evidence linking these fluctuations with the eleven-year sun spot cycle. In another paper concerned with monthly and annual precipitation series, Rodriguez and Yevjevich 6) arrived at the same result. Sou6ek 7) is in partial disagreement with these findings. For the purpose of simplicity in deriving the desired relations, it was decided to let the input process be a pure-random series with a mean value of unity. This led to the adoption of the year as a time base. The use of other time bases is of course possible, provided the theory of Section 4 is suitably modified. Many types of frequency distributions have been used in hydrology to represent precipitation. Markovic 8) has found that any one of the normal, lognormal and gamma distributions may be used for this purpose with about equal success. The normal distribution is known to have two parameters, whereas a skewed distribution in its general form has three parameters. In this study, the third parameter will take the form of a location
407
RAINFALL-RUNOFF RELATIONSHIPS
parameter, which is the distance from the origin to the lower limit of the distribution. Markovic has observed that, upon fitting a skewed distribution to a natural series of precipitation or runoff, the location parameter frequently becomes zero, so that the distribution degenerates into one having two parameters only. The density functions of the above-mentioned distributions, along with their most relevant moments, are reproduced in the Appendix.
3. Transformation from input to output The runoff from a drainage basin is governed by a multitude of influences, which may be broadly classified into climatic and physiographic factors. The former include form, intensity, duration and areal distribution of precipitation, temperature, wind, humidity, etc., while the latter comprises the geometry of the basin, land use, vegetal cover, soil type, the extent of lakes, characteristics of stream channels, and other factors. At the present time no single mathematical expression is known to take account of all these factors, which are virtually unique for each basin. However, it is sometimes reasonable to approximate a basin by a linear reservoir (Chow g), p. 14:27), for which the storage s is directly proportional to the outflow, q, or (1)
s = cq.
It may then be shown that the release of water at time no precipitation, is given by: q = qo e-(t-'°)/c
t,
during a period of (2)
where qo is the discharge at time t o. This equation is in the form of a simple exponential decay curve which is known to govern many natural phenomena. With t o = 0, it may be written in simplified form as: q = qo ebt.
(3)
Equation (3) describes a recession curve, with b (which is negative) as its single parameter. This equation will be employed in the transformation from effective precipitation to runoff. That part of the effective precipitation which is not discharged from the basin during the same year is usually called the carryover. It consists of all values of Eq. (3) for which t # 0. The extent of carryover for a natural river basin depends on the storage capacity available in the soil as well as in lakes and stream channels. Carryover will be greatest for basins with considerable lake storage, such as the St. Lawrence River Basin, while for basins with very little lake storage it may be less than one year. YevjevichS), has shown
408
E.O. FRIND
that carryover is rarely significant beyond about three years. From the practical standpoint it may therefore seem hypothetical to consider a carryover greater than, say, five years. However, longer periods better illustrate the interesting trends in the distribution parameters, hence the plots found in Section 4 extend considerably beyond five years. The relations which will be developed hold for any length of carryover.
P
Effective
F.~..
i=O
Precipitation for year" i
Corresponding Runoff
1 ....
2
3
4
5
6
---~
~o
Time (years) following year i
Fig. 1.
Graphical representation of precipitation-runoff equation.
To find the relation between input and output, let us consider some year i, with effective precipitation Pi- Since in the long run all the water which enters the basin has to leave it, we may write: Effective Precipitation (P) = Runoff (Q).
(4)
Both sides of this equation are shown graphically in Fig. 1. From Eq. (4), we see that (omitting the subscript i):
P = ~ q~
(5)
j=o
or, substituting Eq. (3): P = qo ~ ebJ'.
(6)
j=0
Recalling the properties of a geometric progression, we may write Eq. (6) as:
P
i
qo 1 -
e b"
(7)
Therefore the direct runoff during the year i due to effective precipitation
409
RAINFALL-RUNOFF RELATIONSHIPS
Pi is given by: qoi
=
19/(1 - e")
(8)
and the carryover into succeeding years j is: qjl
=
Pi (1
-
e b) e bj .
(9)
The total runoff Q~ during the year i will consist of the direct runoff plus the carryover from all preceding years (Fig. 2): Qi
(10)
qoi "4- q l ( i - 1 ) + q 2 ( i - 2 ) + " " "
=
Upon substitution of Eq. (9), Eq. (10) becomes: Q,
= (1 - eb)(pi + Pi-, eb + Pi-2 e2b + ' " )
(11)
or
Qi=(1-e
b) ~ Pi-jeJb.
(12)
j=O
Yeor
Yeor
Year
qo;
i
F'I 0
1
2 Time
3
0
1
2 Time
3
4
1
2 Time
3
4
4
(i-1)
~
o~
(i-2)
0
Fig. 2.
~
oo
Accumulation of runoff.
Equation (12) gives the total runoff during any year in terms of the effective precipitation in the same year and the preceding years, and the parameter of the recession curve. The first term in the summation represents the direct runoff. The infinite series of Eq. (12) may be truncated at a suitable point according to the degree of accuracy desired. If we define carryover as having ceased when its magnitude drops below one percent of the direct runoff, and we suppose that this occurs after k years, then qk = qo ebk ---- 0.01 qo
(13)
410
E.O. FRIND
from which it follows that b-
In 0.01 k
(14)
Equation (14) defines the parameter of the recession curve in terms of the length of carryover. Whenever Eq. (12) is used for the term-by-term transformation of a computer-generated input series into an output series, which is then analyzed for moments, the summation should be carried over at least 2k terms for the sake of accuracy. On the other hand, when the output parameters are calculated directly by means of the equations developed in Section 4, the limit of summation should be extended somewhat farther. For future reference, the limit of summation, m, will be defined by m -- 4k.
(15)
This multiple was found to yield the best results. Hence Eq. (12) becomes: Qi -- (1 - e b) ~ P~_j e gb .
(16)
j=O
With the help of this equation, a series of mean annual effective precipitation may be easily transformed into a series of mean annual runoff.
4. The output process The series of mean annual runoff (output) from the basin, obtained by transformation of the input, will possess some statistical distribution, which is as yet unknown. The mean of this distribution must be the same as the mean of the input, since in the long run input must equal output. Variability and skewness, however, will not remain unchanged, but will decrease, since the transformation is linear. It is unknown whether any of the commonly used density functions might fit the output. 4.1. VARIABILITY The variability of a hydrologic process is usually expressed by its coefficient of variation, Cv, defined as: Cv = trip.
(17)
Since the mean was fixed at unity, the coefficient of variation numerically equals the standard deviation. In the following development the variance a 2 will be used.
411
RAINFALL-RUNOFF RELATIONSHIPS
The variance of a statistical sample of size n is defined as:
tr2 _
_1
~ ( x , - p)2.
n
(18)
li=l
Assuming that n is large relative to unity, and expanding, we obtain the alternative form: . . . . . n
xi
i=1
n
.
(19)
i=1
If the mean is equal to unity, this expression simplifies to: a2=l
~ xlz _ 1.
(20)
ni=l
Therefore, if P is the input and Q is the output, 1
"
O'p = n i = l
2
aq
Z p2 _ 1
(21)
=1 ~ Q~_I. n i=1
(22)
Substitution of Eq. (16) into Eq. (22) yields: :
(1--eb)2 ~ ( ~
pi_jeJb) 2 -
O'q - -
n
i= 1
1.
(23)
j=0
The squared inner sum may be expanded and the resulting terms summed individually. Those terms which are sums of squares are expressed in terms of the input variance by means of Eq. (21). The crossproducts in the expansion may be evaluated after recalling the definition of the kth serial correlation coefficient of a series of n variates x (where n is again large): 1 Pk = ?l(;r~xxi= 1
(x, - p) (X,-k - p)
(24)
which may be expanded to:
P k = n a 2 i=lX~X~-k--n-
i=lX i
.
(25)
Since the input P was assumed to be pure-random, all serial correlation coefficients of this series are zero. Hence it follows that: i=1
P,P,-k --
P, n
i=1
= 0
(26)
412
E.O. FRIND
for all k. The mean is at unity, therefore:
1 ~ P~P,-k =1.
(27)
ni=l
The expansion of Eq. (23) now readily simplifies to:
a2=(l_eb)2F L1 ~2 -
e 2 b -I- s o 2 -~- s / 3 3
]
-
1
(28)
where
Sv 2= ~ (j + l)e jb
(29)
j=O 2m
Sv 3=
~
(2m-j+l)
eJb
(30)
j=m+l
and m is defined by Eq. (15). Equation (28) gives the variability of the output in terms of the variability of the input and the parameter of the recession curve or, by virtue of Eq. (14), the carryover. This variability function was plotted for several values of input variance. It was found that it is possible to arrive at a single function by dividing all values by Cvo, the coefficient of variation of the input, a fact which is not readily apparent from Eq. (28). A plot of the variability function is shown in Fig. 3. The curve shows that the variability of the output decreases with increasing 1.2
t
+
r
,
1
I
~"
'
I
I
1.0
0.8
o •~. 0.6
0.4
0.2
0.0 0
I
5
I 10
I
I 20 Carryover
I (years)
Fig. 3. Variability function.
1 30
40
413
RAINFALL-RUNOFF RELATIONSHIPS
carryover, tending asymptotically to zero. It is obvious that the change in variability depends in no way on the type of the input distribution. 4.2. SKEWNESS A measure of the skewness of a statistical distribution is its third moment about the mean. Alternatively, skewness may be defined by the coefficient of skewness: Cs = #3/tr 3 . (31) may be positive or negative. An expression for the skewness of the output distribution may be derived in a similar manner as in the preceding subsection. The third moment of a distribution about its mean is defined as: Cs
1 #3 - n
(x, - #)3.
(32)
li=i
The sample size n is again large. Expansion yields: 1 #3----n
x3 i=1
3
--ni=l
x~
xi + ~
xi
i=1
.
(33)
i=1
With the mean fixed at unity, this simplifies to: #3 = n
x 3 - 3a~ - 1.
(34)
i=i
If P is the input and Q is the output, we then have:
#3p
=
1 ~ pi3_ 3a 2 _ 1 n i= 1 1
(35)
"
#aq = n i~1 Q/3 _ 3tr2 _ 1.
(36)
Substituting Eq. (16) into Eq. (36) yields: #aq-
~ n
i=1
~ P , _ j e jb
- 3a~-l.
(37)
j=o
The cubed inner sum will again be expanded. The sums of cubes lend themselves to easy evaluation by means of Eq. (35): 1 ~ V 3 = #3p + 3tr2 + 1.
ni=l
(38)
To evaluate the various crossproducts, we recall that the expected value of a product of independent random variates is equal to the product of the
414
E.O. FRIND
expected values of the individual functions, or: E {fl (x~) f2 (x2)}
=
E {f~ (x~)} E {f2 (x2)} •
(39)
For pure-random input, this leads to: 1
ni=l 1
ni=l
Pi2Pi_k = ~p + 1
PiPi_tPi_k
for all k
1 for all l
(40)
and all k
(41)
Now the expansion of Eq. (37) simplifies to:
= (1 - eb)3 L[ l -~-eab 1,3p + 3 Ss2 +
+
d - 3tr2 - 1
(42)
where m+l
Ss2 = ~
m+l
E e(ZJ+i-3)b
j=l
i=1
m+l
m+l
SS3 = E j=l
E i e(J+i-2)b
(43) (44)
i=1
m+ 1 2m+ 1
Ss4= E j=l
(2rn - i + 2) e (j+i-2)b
E
(45)
i=m+2
and the limit of summation rn is defined by Eq. (15). The coefficient of skewness of the output is then:
CSq - -
t/3q3 " O'q
(46)
We now have an expression for the skewness of the output in terms of skewness and variance of the input, and the parameter of the recession curve. This expression is valid for any skewed distribution. If the two-parameter version of either the lognormal or the gamma distribution is used to describe the input process, then the following holds: Cs = 3Cv + Cv 3
for lognormal
(47)
Cs = 2Cv
for gamma.
(48)
These equations may be used to evaluate the first term in Eq. (42). Several sets of curves were generated, corresponding to various values of input skewness. It was again found that by dividing all output values by Cso, the coefficient of skewness of the input, a single curve for each type of distribution may be adopted to approximate the skewness function. The resulting curves for the lognormal and gamma distributions are shown in Figs. 4 and 5. These two curves are very similar in shape.
415
RAINFALL-RUNOFF RELATIONSHIPS 1.2
1.0
0.8
o e (J
0.4
0.2
0.0
i
~
I
r
0
i
I
5
10
I
I
I
I
20 Carryover
30
40
(years)
Fig. 4. Skewness function for lognormal input. 1.2
, ,
, ,
,
1
t
I
I
5
10
I
i
I
I
I
1.0
0.8
0 "-0.6
04I
0.2
0.0
, L , ,
0
20 Carryover
I
30
40
(years)
Fig. 5. Skewness function for gamma input. Since the skewness of a distribution may also be negative, several sets of curves (not included with this paper) for negative input skewness were generated as well. When rotated about the horizontal axis, these curves were found to resemble closely the corresponding positive curves. In the vicinity of zero input skewness (that is, normally or almost distributed input) the use o f Eq. (42) is difficult because of accuracy problems. It is, however,
416
E.o. FRIND
known in a qualitative way (as pointed out by Kisiel 1°) that a skewed distribution will respond to a linear transformation with a reduction in skewness. A normal distribution, when similarly transformed, will remain normal. Thus we have sufficient information to predict the skewness function for any input. A study of this function over the full range of input skewness leads to the conclusion that the skewness of a distribution decreases in absolute value as carryover increases, tending asymptotically to zero. 4.3 STATISTICALDISTRIBUTIONOF THE OUTPUT It is of interest to know what type of theoretical distribution function may be used to represent the output process. For this purpose the relation between Cs and Cv (discussed in detail by KIeme~ and Jones 11) is useful. The functions defined by Eq. (47) and (48), for the region Cs>O, are shown in Fig. 6. These two curves represent the loci of all points correspond-
" ~ c'q
t C5
Fig. 6. Domains of two skewed distributions. ing to the two-parameter gamma and lognormal distributions respectively. On the same diagram we may also plot Cv versus Cs for any of the output distributions under consideration. If the resulting point falls on one of the two-parameter lines, then obviously the corresponding two-parameter distribution may be used to represent the output. In general, however, this is not likely to occur; we may obtain a plot anywhere in the region Cv>O. The useful domain is further reduced after realizing that the statistical distribution of a hydrologic process must be wholly positive (since negative precipitation or runoff is physically impossible). For a skewed distribution, the smallest variate is equal to the location parameter, which is the distance from the origin to the beginning of the distribution. Considering the gamma distribution, we see from Eq. (A9) given in the
RAINFALL-RUNOFF RELATIONSHIPS
417
a = # - aft.
(49)
Appendix that: The coefficients C s and C v may also be written in terms of ct and p: 2 cs
(50)
-
cv -
(51)
Y Substituting for e and fl in Eq. (49) we obtain for the location parameter:
a = #
1
Cs/Cv
"
In a similar way a more general expression for the location parameter of any skewed distribution may be derived: a =y(1
_R;)
(53)
where R = Cs/Cv
(54)
R o --- C s o / C v o .
(55)
and Equation (53) reveals that we must have R o / R <<.1, or R/> Ro, in order to obtain a wholly positive distribution. For the gamma distribution, it follows that C s / C v ~>2, or Cs>~ 2Cv. Consequently, a distribution must plot to the right of (or below) the two-parameter line if it is to have physical significance. Similar reasoning holds for the lognormal distribution. Thus the useful domains are as indicated in Fig. 6. With the aid of the variability and skewness functions developed previously we may now synthesize any number of output distributions. If we take as input a two-parameter distribution with C v = 1.0 then, from Eqs. (47) and (48), the input skewness will be Cs=4.0 or Cs=2.0 for lognormal or gamma respectively. Hence we can multiply the skewness function (Figs. 4 and 5) by the appropriate constant and then plot C v versus C s for the output derived from lognormal or gamma input. The resulting curves are shown in Fig. 7. It is evident that the output distributions plot entirely within the respective three-parameter domains. This allows the conclusion that the transformation has changed the two-parameter input into a three-parameter output. In other words, the output possesses a statistical distribution which is in a more general form than that of the input.
418
E.O. FRINO
1.5
1.0
0.~=
,
0.0
~
,
I
i
i
1.0
0.0
,
,
I
,
i
i
2.0
i
I
I
I
I
i
3,0
I
4.0
I
I
I
O.O
CS
Fig. 7. Behaviour of skewed output.
1.4
0
,
,
,
,
I
,
I 10
,
I
'
i
~
I
0
¢n >
1.2
1.1
1.0 0
5
20 Carryover
I
30
40
(years)
Fig. 8. Skewness/variability ratio for lognormal input. Figures 8 and 9 show the change in the ratio of skewness/variability with increasing carryover, for the lognormal and gamma distributions, plotted in the form of (Cs/Cso)/(Cv/Cvo). It is evident that while both Cs and Cv decrease with increasing carryover, the ratio of these coefficients increases, being always greater than unity and tending, in the limit, to some finite
RAINFALL-RUNOFF
419
RELATIONSHIPS
value. This implies that a skewed distribution will always retain some degree of skewness, regardless of the length of carryover. The skewness can only vanish if the variance also vanishes.
1.41
I
I
,
~
I
13
(JU 1.2 ¢n >
1,1
I
1.0 0
5
I
t
10
I
20 Carryover
t
30
40
(years)
Fig. 9. Skewness/variability ratio for gamma input.
5. Testing the theory by simulation To verify the analytical relations between the respective statistical parameters of effective precipitation and runoff, an input series consisting of 10000 random numbers was simulated on the IBM 7094-1I computer of the University of Toronto. The random numbers were modified to yield three different inputs, namely normal, lognormal and gamma input, each with two parameters. These series were transformed, term by term, by means of Eq. (16), for carryover from one to twenty-five time units. The resulting output series were analyzed for their first three moments. The changes in variability with respect to carryover were found to be in excellent agreement with those given by Eq. (28), this confirming the validity of this relation. The same agreement was not achieved with respect to skewness; however, the discrepancy was traced to accuracy problems. After allowing for this shortcoming, the results of the simulation were judged to be sufficient to confirm the validity of Eq. (42) as well. Along with the first three moments, the simulation runs also produced the first twenty-five serial correlation coefficients for each of the output
420
E.O. FRIND
series. These coefficients were used to plot output correlograms (Fig. 10) for the various lengths of carryover. It was found that each of these correlograms could be represented reasonably well by a theoretical correlogram defined by: Pk = (p,)k (56) where Pl, is the first serial correlation coefficient for some particular output series. This equation is valid for the first-order Markov process (Chow g), pp. 8: 87 and 8 : 93), which is frequently used in hydrology to generate series of runoff. Hence if the first serial correlation coefficient for a particular 1.0
o.g
0.8
~0.7 Q~
Simulation
ID
"G 0.6
-----
Equation
Runs (5 6 )
Q) 0
o
tO
0.5
13
k0 0.4
o
t~
o.3 t/1
0.2
~5
Years
of
Carryover
0.1
0.0 10
Fig. 10.
15 Subscript
20 k
Output correlograms.
25
30
421
RAINFALL-RUNOFFRELATIONSHIPS
output series is known, the complete correlogram may be synthesized with good accuracy. To confirm the findings of Section 4.3, each set of output moments was used to define three theoretical distributions of the kind described in Section 2. These distributions were fitted onto the actual distributions of runoff obtained from the simulation runs. The fits were tested by means of a series of z2-tests, performed on the IBM 1620/1710 computer of the Department of Electrical Engineering at the University of Toronto. The results are qualitatively summarized in the following table. Quality of fit
Output fitted by:
Input (2 parameters) Normal Lognormal Gamma
I
_
_
Normal
Lognormal (3 parameters)
Gamma (3 parameters)
good poor poor
fair good good
~ir good good
In addition to the above fits, some two-parameter lognormal and gamma fits were performed, with very poor results, as may be expected. It was also found that a normal fit, applied to output derived from skewed input, becomes increasingly acceptable as carryover increases, suggesting a trend to normality. Thus, it is feasible that a skewed input, after transformation by a long recession curve, may be represented equally well (at some level of significance) by either a symmetrical or an asymmetrical distribution. The fitting process supported the results of Section 4.3, with the additional information that any output obtained from skewed input may be represented by either a lognormal or a gamma distribution. Hence we are not confined solely to the distribution corresponding to the input (which is useful particularly if the input distribution is unknown) and we have some liberty in selecting a suitable distribution function. However, it is important that for output a general three-parameter form be used, unless a two-parameter fit is justified as a result of the test described in Section 4.3. A description of the computer programs used in this study may be found in the M.A.Sc. Thesis prepared by the author4). 6. Conclusions
It is now in order to summarize briefly the results obtained in this study. The work is based on the assumptions that:
422
E.O. FRIND
1. The input series is pure-random, 2. The basin may be approximated by a linear reservoir. The results, in a qualitative way, are: 1. The variability of any distribution of annual precipitation means will decrease with increasing carryover, tending asymptotically to zero. 2. The skewness of the same distribution will decrease in absolute value, also tending asymptotically to zero. 3. An input process having some asymmetrical two-parameter distribution will emerge as output with the same type of distribution, but changed to a more general form. For practical purposes, one skewed distribution may be replaced by another, which must have similar flexibility. 4. Theoretically, the output derived from a skewed input distribution will always remain skewed, regardless of the length of carryover. However, for large carryover, skewness may become sufficiently small for the distribution to be regarded as Gaussian. The relationships developed in this paper may be used to determine completely the statistical distribution of mean annual runoff from some basin for which runoff records are insufficient or non-existent, provided the statistical distribution of precipitation is known. Effective precipitation may be determined by means of established methods for the estimation of evaporation and evapotranspiration. The recession curve parameter can be obtained by trial from Eq. (16) if a small number of annual runoff means are available. This information is sufficient to obtain the desired distribution.
Appendix The following is a compilation of the density functions of the normal, the lognormal, and the gamma distributions, along with their first three moments, as they are used in this paper. The first moment is understood to be about the origin, while the second and third moments are about the mean. This list in its compact form is thought to be useful for quick reference, but more comprehensive treatments may be found in most texts on theoretical statistics. Very useful in particular are Aitchison and Brown12), Fraser la), and Kendall 14).
A. Normal Distribution The density function in its classical form is:
f(x)=
a xf2~ e
-oo~< x ~ o o .
(A1)
423
RAINFALL-RUNOFF RELATIONSHIPS
The variate x is normally distributed with arithmetic mean/~ and standard deviation a.
B. Lognormal Distribution The density function may be written as:
f (x) =
1 e _½(,,~x-,)-u,]2 " ~" / ( x - a ) a,x/2n
a ~ x ~
(A2)
This distribution may be considered as a transformed normal distribution in which the variate has been replaced by its logarithmic value. The variate x is related to the normally-distributed variate y by: y = In (x - a).
(13)
The three distribution parameters are: /~y, the mean of the normal transformation, %, the standard deviation of the normal transformation, ~, the location parameter. The first three moments of the lognormal distribution are: /a = e u" +½~2 + a
(A4)
/t 2 = e2U'+~'2(e'2 -- 1) = a z
(A5)
/~a = e a " ' + ~ " ( e ' ' - 1 ) ( e ' " + 2).
(A6)
The lognormal distribution is positively skewed for all choices of the parameters.
C. Gamma Distribution The density function is given by:
y(x) =
1
(x--a~e-le-(X-#a) \-F- /
a
x .
(A7)
The three parameters are: e, the shape parameter, /~, the scale parameter, a, the location parameter. The g a m m a function F (a) is defined by the integral: oO
F (~) = f x'-X e - x dx. o
(A8)
424
E.O. FRIND
This integral is seldom used to compute specific values of the gamma function. The Mathematical Handbook by Abramovitz and Stegun 15) contains excellent approximation formulas which will yield results to any desired degree of accuracy. The first three moments of the gamma distribution are: /~ = ufl + a
(A9)
# 2 = (Zfl 2 = 0"2
(A10)
/23 = 2c¢fl3 .
(All)
The gamma distribution is positively skewed for all values of the three parameters. The two-parameter gamma distribution may be written in the form of the well-known chi-square distribution by making the following substitutions: ct = ½v
(A12)
fl = 2a 2
(A13)
where each chi-square variate x is the sum of the squares of a series of v normally-distributed random numbers y, with a mean of zero and a standard deviation of as, such that: x
~
(Yi') 2
=
.
(M4)
i= 1 kay./
Alternatively, by shifting the origin to the mode and changing the scale, the gamma distribution may be converted to a Pearson Type III distribution.
Acknowledgements The author is indebted to Dr. V. Kleme~, Associate Professor, and to Dr. L. E. Jones, Professor, both of the Department of Mechanical Engineering at the University of Toronto, who supplied much valuable advice and active encouragement during the course of the work. The financial support provided by the Ford Foundation is most gratefully acknowledged.
List of Symbols a - location parameter of a distribution function b - exponent of recession curve c - a constant Cv - coefficient of variation Cs - coefficient of skewness E - expected value, or expected frequency
RAINFALL-RUNOFF RELATIONSHIPS f (x) i, j k m n P q R Q S s t x, y
-
ct fl F (ct) tt p2 -
a function of x arbitrary values of time t a specific value of time t limit for s u m m a t i o n s size of a statistical sample or n u m b e r of class intervals effective precipitation runoff due to a single year's precipitation the ratio Cs/Cv runoff stand-in symbol for s u m m a t i o n s storage time variates
vp tr tr ~ X~ -
shape parameter of g a m m a distribution function scale parameter g a m m a function of alpha first m o m e n t a b o u t zero, or arithmetic m e a n second m o m e n t a b o u t the mean, or variance third m o m e n t a b o u t the m e a n n u m b e r of degrees of freedom population serial correlation coefficient population standard deviation p o p u l a t i o n variance chi-square
i, j k p q s v x y
counting subscripts the kth m e m b e r of a family of variables precipitation runoff skewness variability variate x variate y
pa -
425
Subscripts -
References
1) J. Shen, Use o f A n a l o g Models in the Analysis o f F l o o d runoff. U.S. Geological Survey, Professional Paper 506-A (1965) 2) N. C. Matalas, Statistics o f a runoff-precipitation relation. U.S. Geological Survey, Professional Paper 434-D (1963) 3) R. I. Jeng and V. M. Yevjevich, Effects of lakes on outflow characteristics. Civil Engineering D e p a r t m e n t , Colorado State University, F o r t Collins, Colorado (1966) 4) E. O. Frind, The relationship between effective precipitation and runoff as expressed in the distribution parameters. M.A.Sc. Thesis, University of T o r o n t o , D e p a r t m e n t o f Mechanical Engineering (1967) 5) V. M. Yevjevich, Fluctuations o f wet and dry years. Hydrology Papers, Colorado State University, F o r t Collins, Colorado (1964) *6) I. Rodriguez and V. M. Yevjevich, Sunspots and hydrologic Time series. C o l o r a d o
426
E.O. FRIND
State University, Fort Collins, Colorado (1967) *7) V. Sou~ek, Cyclic fluctuations of variability in hydrologic phenomena. Water Conservancy Board, Prague, Czechoslovakia (1967) 8) R. D. Markovic, Probability functions of best fit to distributions of annual precipitation and runoff. Hydrology Papers, Colorado State University, Fort Collins, Colorado (1965) 9) V. T. Chow, Handbook of applied hydrology. McGraw-Hill Book Co., Inc., New York (1964) *10) C. C. Kisiel, Transformation of deterministic and stochastic processes in hydrology. University of Arizona, Tucson, Arizona (1967) I1) V. Kleme~ and L. E. Jones, The effect upon regulating-reservoir performance characteristics due to the form of distribution model fitted to the mean annual flows University of Toronto, Department of Mechanical Engineering, Technical Publication Series, TP 6703, (September, 1967) 12) J. Aitchison and J. A. C. Brown, The lognormal distribution. Cambridge University Press (1957) 13) D. A. S. Fraser, Statistics, an introduction. John Wiley and Sons, Inc., New York (1958) 14) M. G. Kendall and A. Stuart, The advanced theory of statistics, Vol. 1. Charles Griffin and Co., London (1963) 15) A. Abramovitz and I. A. Stegun, Handbook of mathematical functions. United States Department of Commerce, National Bureau of Standards, Washington (1964) *) Papers presented at the International Hydrologic Symposium at Fort Collins, Colorado, in September, 1967.