ML19248D259
| ML19248D259 | |
| Person / Time | |
|---|---|
| Site: | Crane  |
| Issue date: | 05/18/1979 |
| From: | Wallis J AMERICAN GEOPHYSICAL UNION |
| To: | Hendrie J NRC COMMISSION (OCM) |
| Shared Package | |
| ML19248D254 | List: |
| References | |
| NUDOCS 7908150463 | |
| Download: ML19248D259 (24) | |
Text
5
&=== JV =E International Business Machines Corporation Thornas J. Watson Research Center P.O. Box 218 Yorktown Heights, New York 10598 914/945-3000 May 18, 1979 Joseph M. Hendrie Director Nuclear Regulatory Commission Malomic Building 1717 H Street NW Washington, D.C.
20555
Dear Mr. Hendrie:
While watching the Three Mile Island, 3MI, debacle on television I was struck with the appearance of the island.
It appears that 3MI is a comparatively flat, low-lying island situated in the middle of a large river, and probably comprised of alluvial sands and gravels.
The geomorphology literature attests to the fact that this is the sort of topography that gets redistributed during the really extreme, say 1/1,000 or 1/10,000 year flood events.
This raises two obvious questions, first what happens to a nuclear power plant when it is submerged in flood waters, and second what is the flood risk at the 3MI site?
To evaluate the second of the above questions I have investigated the Susquehanna river data base, and asked a few questions here and there.
There are 87 years of streamflow measurements for the Susquehanna river at Harrisburg, a fortunate circumstance as this is only a little upstream of the 3MI site.
The highest instantaneous flow fer each year can be obtained from the published records, ranked, and plotted on cumulative probability paper (see Figure 1).
A log. Pearson III curve following the official U.S. Water Resources Council guidelines (Bulletin 17a) has also been shown on the figure.
One might feel happier with this flood risk estimate (the fitted curve) if the procedures were not known to be totally fallacious.
The original design for the 3MI site estimated that "the maximum possible flood", MPF, for the site was 1.1 million cfs (cubic feet per cecond).
In 1972 hurricane Agnes arrived and subsequently a new MPF of 1.6 million cfs.
Will the next big event raise the MPF another 45%?
Do you know the mechanics of how such estimates are made?
From the name it sounds like the procedure developed in the thirties that has little to do with science, nothing to do with 908150V675 96 076
1 PAGE 2 probability theory, and with uncertainties at every step of the calculation.
The cumulative error is unknowable.
The use of MPF does not appear to be the sort of decision theoretic approach to estimating geophysical hazards that one might have expected of an NRC.
It is to be hoped that by now the NRC has a sufficiently large competent in-house staff that thev do not have to rely upon the log. Pearson III or similarly technically unsound and hap-hazard flood risk appraisals that may be generated by the more tradition-bound Federal agencies.
The risk of flooding does seem to be rather high for the 3MI site.
One can't help but wonder why 3MI was selected, when the use of Hill Island would have removed this consideration.
It is understood that a levee has been constructed that is meant to maintain the integrity of the 3MI site from an event the size of the latest estimated MPF.
All well and good, but levees have been known to fail, and MPF's are not decreed by God, but estimated by fallible man.
Are the geophysical hazards at other nuclear plants of similar magnitude, and are risk evaluations, as at 3MI, always based upon a single number?
Sincerely,
(, e G.
L J.
R. Wallis President-elect Section of Hydrology American Geophysical Union enc.
ain lll
t SUSQUEH ANNA RIVER AT HARRISBURG, PA.
1891 - 1977
= 2341 cr = 1296 y = 2.70 A = ' 3.87 b
g 1600 - POST 1972 MPF = 1,6G0,000 cfs o
en S
1400 5
8 f
1200 l
z PRE 1972 M PF = 1,100,000 e f s
/
Y 1000 1972 EVENT 2
/
l o
800 8
+
/
_J
/
u.
/
600
+/
g
+!
log PEARSON III o
z FOLLOWING 400 f
W.R.
C. PROCEDURE
S s
m 200 Z_
do O
g 50 70 90 98 99.5 99.9 60 80 95 99 99.8 99.99 CUMULATIVE PROBABILITY
$96 078
U
'4
)
4 January,1979 i
6
^
\\
l d
i Dr Leo Eisel, Director United States Water Resources Counci!
2120 L street, N.W i
Washington, D.C. 20037 1
Dear Leo,
i a
it was great to see you and Vicki once again, and to be brought up to'date on your respective careers and other doings. Sha'l we make it an annual event, or just continue to hase it just once every hydrologic decade '
i Since meeting with you a couple of papers have been published (reprints enclosed) that i
have implications for flood frequency estimates made using the WRC's flood frequency 1
guidelines tBulletin 17-A). In the WRR paper it is pointed out (figure 7, and appendix C) i that if flood sequences were distributed as Weibuif and positne!y skewed in real space, yg then infinite samples wculd always yield estimates of skew in log space. yt, of -1.14. Of a
1 course, small finite samples, n, would in expectation hase other values of yt, but these estimates would not be primary functions of yg, but rather of n, and of the means, u,, and J
standard deviations, o., of the individual records in real space. In view of this it would seem i
be neces.sary that we be absolutely sure that none of the records used to estimate regional yt Weibull distobuted m re.il space. 1:urther, if one proceeds.siihout such an assur.mee then i
j there appears to be no possible statistical justdication f or atic:ncting to find a new reg:onal yt.
I I
or for that matter to use any pre-eust:ng yt I
l
'd e
i J
O 9
=
^
A si 4
Y t.
In real space it is easy to see that flood hazard is, other thines being equal, a function of yg Howeser, we have yet to identify a distribution for which yt # f(n.g.o ).
Hence, it new becomes dif ficult if not impossible to justify any flood estimating procedure that depends upon yt, and ail past estimates made using such techmques must automatically be suspect. In particular, because log Pearson III with widely different yt 's can lead to identical yg (see figure 9, and table Al o
'/Rn paper; a -oum awa.
earson III flood fregency e
sN anal - > would be the most suspect of all.
y J
i.
I These are hardly new ideas. They were dicussed at the December,1977 AGU symposium a
dedicated to the foibles of using the WRC flood freqency procedures. At that meeting there I
was an interesting spitt displayed between representatives of the Federal bureaucracy and those e
i in the private sector. The latter group were mostly afraid that if they used WRC procedures for engineering consulting work, that they would be making themselves liable for civil suits charging professional incompetence. They hase a point, the gulf between the scientific literature and the concepts of Bulletin 17-A does seem to be ;.;etting excewively large.
e u.. Mad several invitations to lecture on the statistic frequency analysis, and how current knowledgc relates to WRC procedures. Presumably your hydrology committee is concerned with such matters. Please keep me posted on all late breaking developments because it would be nice if my lectures could end on an up-beat note.
llope to see yi n ag.un one of these d.iys.
Sincerely, 6t Jia
~
e
! !. 0 0O TT
-)S,r.
s
N OL.14. NO: $
W ATER RESOLRCES RESEARCH OCTOBER WM Some Comparisons of Eood Statistics in Real and Log Space J. NI ACICNAS LANDWEHR ASD N. C. h14Tu ss LS Geolopcal Surcer. National Center. Reston. Hrpma ::09:
J. R. W ALUS IBM Corporation. Watson Research Laboratory. Yorktown llemhts. New York 10598 Some statistics of histoncal and simulated dood sequences were examined in real and log space. It was found that seseral star stical properties of doods m real space could not be inferred from those in log space without extensne knowledge of the distnbuuon of doods in real space as well as information about their sa.nphng charactensucs. It is shown that the construction and use of regional skew maps in log space are most likely counterproductne.
What is reasonable is real; that which is real is reasonable.-George titioned into 14 regions, and in Table I the distnbution of the Hegel 101.
histoncal dood records over the regions is gnen. as well as the number 4(n) of nonoverlapping sequences of length n.
INTxootcTros Let Y denote any of the parameters C,[
], t[
), and A[
].
The US Water Resources Counct/ [1976) recently issued and let y denote a moment estimate of Y derived from a guidehnes for dood frequency analysis stating that 'the Pear-sequence of length n. Given the k(O values of y, the mean f son Tyre ill distnbution with log transformation of the dood and standard deviation &ty) were calculated and are gnen in data (log-Pearson Type 111) is recommended as the basic dis.
Tables 2 (RS) and 3 (LS).
tnbution for dehnmg the annual dood senes.' The guidelines.
Watalas et al. [1975] noted that the increase in afyl in RS howeser. are not strictly binding inasmuch as 'in those cases with n suggests that doods are charactenzed by high values of w here the procedures of this guide are not followed, desiations t (>2). The values of C, and A. as well as t. presented in must be supported by appropriate study and accompanied by Tables 2 and 3. will serve as a guide for further analysis.
a companson of results usmg the recommended procedures.-
Bnefy, it is noted that the vanations of f and &f y)with n in LS In this work we show that such cases are more likely to be the are somewhat inconsistent with those m RS. For F = C. f t
rule than the exception and &(y) are not very sensitne to n in either ' in real spaa This work has been extended O for n s 30 in all regions. Howeser. in LS the tendency is into log space and m addition to the distnbutions formerly for i < 0. except m regions I. 2. and 14. For a gnen region.
treated. two further distnbutions base been added. One is the @
). the proportion of sequences pelding est: mates of y m log Pearson type 111 datnbunon. recommended by the Water RS and LS of hke or opposite signs. :s gnen m Table t It is Resources Council and the other is the Wakeby distnbution. noted that G(-- > tends to decrease with n for all repons Jehi.ed 5 H 1 Thomas a personal commumeation.1976) and escent I and 4. For regions 3-11. Gt --) < 0.50 for all n. G(-
exammed by Houdrion [1977}.
+ i = 0 for all repons and n. That is. no sequence y telded a The extent to which skewness in log space is atfected by the negatne esumate of y in RS and a posune estimate m LS.
lower bound of the distnbution of doods is esammed. The Escept for reg:en I. G(--i tends to increase with n. and m lmplication of these results tn the construction of reponal general for n 2 20. G(--) > 0.50. For all regions. [Gt --)
skew maps in log space as advocated by the Water Resources G('-)] tends to increne with n. For all regions. [G(--) -
Council s detai!cd.
G(--i] tends to merease with n. w here for n - 30. [G(- p -
More mathematical aspects of this work are mcluded in
@ - - )] 2 OMO Thus as n becomes !arge, dood sequentes mpend m pe!d dommantly poutne estimates of y m RS and negatne esumates :n LS Rarely do dood secuences peld negatne Fl ooo ST ulsrics esumates of - m both RS and LS and nen more rareh.
Hoods are denoted as Y m real snaceiRS: and 2 = In Y m seemms neser. ao they yield negatne esumates of m RS foy,rateil Si Ts andom s a abies the dntnbunons of A and and routne esumates.n LS
/ may He ;naracten/ed. chough not necewanh antauen. by Cmomm of SFP sR ATtoN the weFc:ents of.ananon. C,[
uew ness.1 [. and kur-tosh. y, To mes, eme sampling procernes of est mates of Preuously. ifatalaa et al. D 975linsesneated some samp.me
(~,
' and y ' m RS and LS. seauences of!enztn n = properun of t. w here F =, m RS and noted that the relanon M 20. and 31 derned trom i351 hntonca! dood records vre between i and du ) derned from the hntoncal dood sequences
- onudered in Fnzure I the area of the United States
- s par.
was not in accord with that derned tr m Monte Carlo everi-menM o* so ml we nown dntnhution funcuans For the ib, ger s mt wect to L S wpy net. Punhsned m A n me \\merm Gennu un th, Monte Carlo results. let. denote an estimate of y Based on W)2 q
W (f $ 'O f f 1 (/ r.
?!O 2 f
')
3 P tu jg 7
c am y
LANDwEHR ET AL Flooo Stansncs 903 l
"7 ' % (
5 lf Q
2 i
(14 (
ry s,,j
' 1 e,,/
'y c
T/
_e)
~
y w
t q
7
('
f l
@s$,b ekf'n-.,s-e W-M,,%
i y
- ' p'i f f ppy
\\
3 1 g7
'/
11 \\
r e
s s
,, g 7M5a f
y W \\,Q7p+, D i pa d
~ ~~
~
%.+
[' ^~ ~
y g
is1y C
i
)
8 &~
IL. >
I
\\f' h
b sc(
i F i g. 1.
Regional hydrologic dnision of the United St.ites.
100,000 generated 'tlood' sequences cf tength n, the mean J(w) separation can be accounted for by spatial mixing of values of and standard desiation &(w) were determined for feasible val-y within a region and by nonstationanty in y [Wallis et al.,
ues of y in the range [0,100}. The discordance between the 1977]. There is. of course, the possibility that a distribution historical floods and the generated floods was defmed by what which is not an element of 4 may explain the separation.
was called the condition of separation; that is, for f = sis),
Some insight into the nsure of such a distribution may be
&(y) > &(s) for each element of the distnbution set 4, w here gained by reference to Figure 2 depicting the condition of the elements of a were the uniform (U), normal (N), log separation for n = 10 for the distribution set 4. From Figure 2 normal (LN), Gumbel extreme value type I (G), Pearson type it is seen that the points (f. &(f)) for all regions lie above the til (PIII), Weibull (W), and Pareto (P) distnbutions. The cun es formed by the points (A(w), &(w)) for each element of 4.
separation is a function of a and tends to become more pro-The upper bound on the curves is LN except for f greater than nounced as n increases.
about 1.3 (corresponding to y 215). where the upper bound is The separation cannot be explamed by the small number seemingly formed by the cune for P. It has previously been 1(n) of historical flood sequences relative to the 100.000 gener-observed [.t/ata/as et al. 1975] that the condition of separation ated tlood sequences, by autocorrelation [.t/aralar et al.,1975],
becomes more pronounced as n ir creases.
or by cross correlation [Wallis et al,1977]. dowever, the in reference to Table 5. giving the relation between skewness y and kurtosis A for the distnbution set t,it is noted that the relative positions of the curves for gnen values of y can be TABLE I.
Distnbution of Histoncal Records and Sequences explained as functions of A. Gnen two elements of b, say oi of Lergth n and o, the curse for oi will be aoose that for c, oser that r
range of values for which A of o,is larger than A of on. Thus it No. of Sequences A(5) seems that a necessary but perhaps not sutlicient condition for Region Records a = 10 g = 20 m = 30 minimizing the separation is that floods follow a distnbution with A larger than that for LN for y s 15 and larger than that i
193 691 2S6 184 for P for y > 15. Two distributions, log Pearson [11 and f
th Wakeby, not previously in the set b. are considered below as p ssibie emlanations of the condition of separation.
4 50 N:
7:
46 5
130 414 160 104 Lg Pearwn Du.tributwn s
- 9 y
9, u
~
100 314 12I
~8 The random vanable 1 ttlood in RS) is said to be lee
)
Pearson type ill (LP) distnbuted if Z = In (X - u)(thod m 10 35 133 55 30 LS) is distnbuted as Pearson type III (Pill), where the con-11 31 3:1 135 4
stant a G s. The density furction of X is 12 N
311 1:3
'O 13 41 144 61 34
[( t ) = ( 3 / f X - v ilff )
(l) 14 356 145 91 w here b:). the density function of Z, is b
9G4 LANDw EHR ET AL.; Ftooo Si, rics TABLE 2.
Flood Statistics in RS n = 10 a = 20 a = 30 C,
y A
C, y
A C,
y A
Region I i
0.53 0.94 3.40 0.60 1.43 5.29 0.62 1 67 6 60
&O) 0.24 0.73 1.61 0.27 0 88 3.36 0.27 0.96 4 63 Region 2 0
0.58 0.86 3 24 0.65 1.35 5.09 0.69 1.56 6 32
& (y )
0.23 0.74 1.60 0.22 0.90 3 45 0.22 1 00 4 69 Region 3 0
0 49 0 '6 3.09 0 53 1.10 4 51 0.54 1.28 5.41
&O>
0 21 0.7 I I.46 0.22 0.87 3.09 0.21 0.01 4.06 Region 4 i
0.41 0.59 2.79 0.44 0.36 3.75 0 45 1.08
- 4. '0 4 (y )
0 16 0 66 1.24 0.16 0 '2 2.26 0.15 0 85 3.70 Region 5 i
0.62 0.78 3.14 0.70 1.26 4.96 0.70 1.41 5.92
& ty) 0 27 0.75 1.55 0.31 0 93 3.44 0.29 I.N 4 95 Region 6 i
0 63 0 84 3.17 0.71 1.24 4 68 0.75 1.50 6.24
&fr) 0 34 0.*6 1.54 0.40 0 89 3.26 0.43 1.14 5.51 Region 7 i
0 67 0 84 3 22 0.77 1.29 4.89 0.79 1.47 5.97
& (y )
0.32 0 77 I.59 0.39 0.94 3.42 0.36 1.07
- 5. I$
Region 8 i
0 8l 1.00 3 45 0.94 1.54 5.*6 0 98 1.97 8 07
&(r) 0 34 0.'8 1.79 0.50 1.04 4.1 I 0.40 1.12 5.39 Reron 9 i
0 A6 0.53 2.82 0.50 0.73 3 66 0 $1 0.95 4 38 4(v) 0 24 0 76 1.28 0.25 0 86 2.34 0.26 0 86 2 99 Reg:on 10 v
0 56 0 68 3 01 0 61 0.91 424 0 60 1.02 4.73 d 'y l 0 34 0 '$
l.38 0.42 0 99 3.21 0.43 1.01 4.19 Region 11 0
0.85 1.12 3 62 0 95 1.61 5.75 0 99 1.95
',70
&nt 0.35 0.76 1.78 0 39 0 94 3.64 0.43 1.06 5.40 Region 12 i
0 39 0 59 2.79 0 42 018 3 69 0.42 0 93 3 97
& ty p 0 15 0 64 1.14 0.14 0.68 3.69 0.12 OM 2.10 Region 13 0
0 41 0 57 2.76 0 43 0 82 3.52 0.47 0 94 4.53 4 01 0 21 0 71 1.30 0.17 0.70 2.13 0.19 0 93 3.51 Region 14 i
0>,
O SI 3 13 0 43 1.04 4 09 0.47 1.11 4 52
&(v) 0.20 0 69 1.42 0.16 0.72 2.41 0 15 0.69 2.56
[ ~, _ y) i
- /,_g)-
where m LS the distnbution is Pill. Supenmposed in Figure 3
&(w) were originally ir.terpreted as being in RS [Matalas er al..
d
-~(~
(2) 1975). Howeser, they also can be interpreted as being in LS.
'^t:) =
esp "I
d The density functions f(x ) and f(:) are de6ned for a
- 9 and b are the points (!f!,4(y))in LS for each of the regions for n =
> 0. If a > n. then c 5 : s r, and m s x 5 r. and if a < 0.
- 10. All but one of the regional points (region 12) lie abose the then -r 5 : s c. and 0 s x s m w here m =, - e.Thus if a N-Pill curve. indicatmg that in LS. hi>torical tloods do not
< 0 there is a ' maximum certain llood' of salue m. The accord well with PIII or N.
moments and some properties of the distnbutions of X and Z Values of 4(w) and &(w) in RS derned from 50.000 LP are given in Appendit A. It is noted that if a > 0. then y in sequences for each of several values of y are given in Table 7 both RS and LS is positive; if a < 0, then y in LS is negatne. The algonthm for generating LP sequences. insohmg ex-but r in RS may be positive or negaine.
ponentiation of the algonthm for generating Pill sequences In Appendit A. tables of C,. t. and A m RS and LS are [/ohnk.1964: Berman.1971l. is described in \\ppendit A. The gnen as functions of a and h conditioned on c = 0 and m = 0.
points asia. hta) are shown m Figure 4 in relation to the LN for w hich y = - t. Values of y and A in RS and LS depend only curse. From Figure 4 it is seen that most of the LP po'nts lie upon a and b. In RS. as a necomes large, there are some s alues sery near the LN curse, in the case w here the LP A is less than of r v th associated values of A larger than those for LN. For the LN A for gnen salues of y the points lie below the LN example, with a = 015 and b = 2.5. y = 2.99(Table A5). and curve. Of those points for which the LP A is greater than the A = 25 03 (Table -\\7). In the case of LN. A s 22.4 for y = 2.99 LN A for gnen salues of y. 6 lie on or slightiv abose the LN
< Table 5 L This is further illustrited m Table 6. Thus LP may curse. and 2 somewhat below it. Thus LP otters at best only potentially mmimize the condition of separation.
marginal improvement oser LN with respect to lessenmg the In Figure 3 a portion of the curve denned by the pomts condition of separation. Large kurtosis m.3y be necessary. but G(w L N)): for PIII and N is shown. The v" lues of 56:) and it is ne'sutScient for explammg the condition of separation.
n37
- f. 0 ;
'J
- v. l9 L:
L4NDw tHR ET AL.. Flood STansucs 905 TABLE 3.
Flood Statistics in LS n = 10 n = 20 n = 30 C.
t A
C, t
A C.
t A
Region I i
0 06 0.19 2 60 0.07 0 31
).01 0.07 0 30 3 22
& (y) 0 02 0.63 0 84 0.02 0 57 1.10 0 02 0 53 1.06 Region 2 i
0.07
- 0.01 2.54 0.07 0 07 2.85 0 08 0 01 2,92
&(n 0 02 0 64 0 86 0.02 0.56 0.89 0 02 0.49 0.78 Region 3 i
0 06
- 0.01 2.59 0 06
- 0.09
- 3. I 3 0 06
-0. I 1 3 25
&M ON 0 66 0.90 0.04 0.66 1.13 0.04 0 58 1.08 Region 4 i
0 05
- 0.08 2.55 0.06
- 0.14 2.98 0 06
-0 03 3 16
& ty) 0.02 0.64 0.84 0.02 0 60 1.19 0.02 0.62 1.33 Region 5 i
0 09
- 0.19 2.57 0.09
-0.23 3.16 0.09
- 0.29 3.37
& (y )
0.05 0.65 0.93 0.05 0.68 1.44 0.05 0.69 I 84 Region 6 i
0.10
- 0.06 2.57 0.10
- 0.08 2.94 0.11
-C 09 3.06
& (yl 0 07 0 69 0.87 0 06 0.65 0.94 0.08 0.66 0.M Region 7 i
0 09
-O le 2 60 0.10
-0.17 2.97 0.10
- 0.21 3.23 hu) 0.10 Ot9 0.94 0.07 0.63 1.08 0.10 0 66 1.38 Region 8 i
0.12
-017 2 49 0.13
-0.18 2.84 0 13
- 0.18 3 02
& 01 0 06 o c2 0.79 0 05 0.56 0.97 0.05 0.51 0.95 Region 9 i
0 07
- 0.16 2.50 0 07
- 0.29 2.98 0.07
- 0.24 3.14
& (3 )
0.03 0.63 015 0.03 0 62 1.16 0.03 0.59 1.23 Region 10 i
0.10
- 0 14 2.47 0.10
- 0 34 3.09 0.10
- 0 40 3 38
& (y )
0 06 0 62 0.89 0.06 0.67 1.21 0 05 0.57 1.35 Region 11 i
0.13 0 01 2.53 0.13
-0 M 2.86 0.13
- 0 04 3 07
&b) 0.08 0 68 0.97 0.07 0 64 0.98 0.07 0 57 1.25 Region 12 i
0.05
-0 02 2.44 0.05 0.04 2.65 0 05
-004 2.81
& (y )
0.02 0.57 0 '5 0.02 0.47 0 65 0 02 0.57 0 69 Region 13 i
- 0 06
- 0 01 2 40 0F
-002 2.63 0.07
- 0.14 2.'7 4 (y )
ON 0 63 0 83 0 03 0.51 0.77 0.03 0.46 0.70 Region 14 i
0 06 0 12 2.52 0 06 0 06 2.78 0 06 0 01 2.33 Jhn 0 03 0.62 0 86 0 02 C 56 0 90 0 02 0.49 0 67 TABLE 4 Propornon Qt ) of Estirnates of t in RS and '_S of Like and Oppsite Signs Value Region of m
i 2
3 4
5 6
8 9
10 11 12
!3 14 q...;
!0 0 80 0 51 0 48 0.45 0.37 0.45 0 41 0.38 0.36 0 44 0 48 0.50 0.52 0.; 9 20 0 'l 0 47 0 41 0 43 0 31 0 43 0 34 0.34 0.36 0 26 0.46 0 52 0.4A 0 52
'O 3 74 0 $0 0 38 0 49 0 28 0 43 0.28 0.37 Oj) 0 27 0 40 0 43 0 3R 0 53 gi--,
!O 0.10 0 14 0.12 0 ! ; 13 0.12 0 14 0 09 0.24 0 19 0 06 0 !s 0.19 0 12 20 0 02 0 02 0 04 0 11
- 0. 0' 0 05 0.06 0 02 0.21 0.14 0 02 0.07 0.0*
0.05 30 0 01 0 01 0 01 0 01 0 02 0 04 0 03 0 01 0.10 0 07 0.00 0 06 0 06 0 03 Q'-:
10 0.30 0 35 0 40 OJ6 0 50 0.43 Or 0 53 0 40 0.37 Ou a"
^ 29 0 29 20 02' 0 52 0 55 0 46 0 62 0 52 OM 0 64 0 43 O M) 0 52 0 41 0 45 04) 10 0.25 0 49 0 61 0 50 00 155 0.69 0 62 0 57 0 66
- 0. % 0.51 0.56 0 44
,~
.. [t For G l- - i. no eases w ere omened.
O O
906 LoDwEHR Er At : Ftooo Simsncs 1.2 (-
1.0 C
- P w,,7'o 3.' g '.ii ja 4
~
o
~
I
- s 0.6 jr s
i A
u g 0.4 O
0.2 t
t t
I I
t t
i 00 0.2 04 06 0.8 1.0 1.2 1.4 1.6 i.8 2.0 Mean Sk.w. F. en R5 Fig. 2.
Mean skew f sersus standard deuation of skew, d(v) for n = 10 histora:al and umulated data in RS [see.tfatalas er al. 1975).
Although they are not shown, similar results in LS and RS separation may be accounted for by WA with b > 1 but not were oMained for n = 20 and 30.
with b < l. In Appendix B it is shown that if a, c, and d > 0 and 0 < b < 1, then the density function of WA monotonically W, a. by Dismbun.on decreases with x.
Following H. A. Thomas (personal communication,1976),
in Table 8 the values of h) and 3M for estimates of y in the random variabie X is said to be distnbuted as Wakeby RS based on 10,000 WA sequences oflength n = 10 are given (WA ) if for selected values of b and d and for (c/a) = 5 with a = 1. The x = m + a[! - (1 - F)*l - c[I - (1 - F)-*]
(3)
Ig rithm for gerierating WA sequences is given in Appendix B. Also in Table 8 the values of y and A corresponding to those where F = F(x) = P[X s x] and r 2 m. The moments and of (c/a), b, and d are given. For each value of t, A is larger some properties of X are gisen in Appendix B. Also in Appen-than that for LN. From F,gure 5, showing the points (W),
dix B. tables are gnen for values of the coedicients of skewness &(w)) relative to the LN curse, it is rioted that those points nd kurtosis for various values of (c/a), b, and d, as well as conditioned on b = 8 lie above the LN surse, whereas those values of the coefficient of 5ariation conditioned on m = 0.
conditioned on b = OA lie below it. Thus for a given value of y, (Note that the ratio (c 'a)is used rather than indisidual values A larger than that.or LN is not sutEcient to account for the of c and a, to allow for concise tabular presentation.)
condition of separation.
- cughton [1977] has developed a technique for fitting WA A WA with a = 1, b = 16. c = 5 and d = 0.19 can be to observed seyuences and has show n that WA can account for transformed so that u[X1 - it ' ano a[X] = 660, i.e., C,[X] =
the condition of separation. Thus WA orTers an alternatise to 0.66. Estimates of C,[Xl densed from 10.000 sequences of accounting for the condition of separation by the mixing of length a = 10 hase nean 0.55 and standard desiation 0.20, values of y within a region as suggested by ' Vallis er al. [1977].
which agree clo<.iy with the mean and standard deviation of Hewever, all WA's do not account for the condition of separa-estimates vi u[Y) for region 1 (see Table 2). Furthermore, in tion ese..f for a gisen value of y, A is larger than that for LN.
RS for n = 10 this WA yields estimates of y with L) = 0.95 Exploratory work has indicated that for given values of y, and JM = 0.75, which are in close agreement with i = t,H where A for WA is larger than that for LN, the condition of and &(y) = 0.73 for region 1. The corresponcing WA values of TABLE 5.
Values of Skewness y 4:':1 Kartesis A for Elements of 4 Value of A Value ofy U
N G
LN Pill W
P 0
M 3
2.72
}
- 3. I I 3 00
- 2. 7'
}
3 45 3.33 3.03 j' 8 3 90 375 3.40 1
4 43 4 50 4.16 1.14 54 5.39 4.95 4.62 2' 8 6.75 6 00 5.73 2
1036 0 00 9 00 3
22.40 I6 50 17F 20.72 4
41 00 27 00 30 60 44 67 5
6C6 40 50 44 25 48.44 10 3877i 153 00 218 43 15 1139 73 340 50
.Mn?1
- Here A does not eust.
,e v $
TD s
g c1D J
LoowmR ET AL.: Flood SrMistics 907 T A B L E ',,. Selected Values for y and A for LP in RS and LS and LN in RS LP Pararneter Values LS RS LN in RS a
b y
A Y
A y
A
- 0.10 15
- 0.52 3 40 0.55 3.21 0 50 3 45
-0 15 15
- 0.52 3 40 1 04 4.41 1 00 4 33 0.01 4
1 'JO 4.50 1.08 4 82 0.09 10 0 63 3 60 1.93 I1.06 2 00 10 86
-0 08 4
-:.00 4 50 2.fM 7 96 0 Oi 1
2.00 9.00 2.06 9.50 0.14 5
0 89 4 20 2.84 23 60 3 00 22.40 0.12 20 0 45 3,30 3.72 43.42 4 00 41.00 0.18 3
1.15 5.00 3 81 50.19 0.16 10 0.63 3 to 4.55 82.41 5.00 68 26 0.20 1
2.00 9 00 4 65 73.80 0.24 1
2.00 9 00 6.39 467.33 10 00 387 78
- 0. I 6 20 C 45 3.30 3 23 509.47 Note that if Y is distributcd as LN. then log Y is dmributed as N with y = 0 and A = 3.
o[X]Fil - leb) y and A are 3.95 and 51.94, respectnel). From Tabie 5 it is
- ",}~
noted tl at A = 51.94 is larger than A = 41 for LN with y = 4
[F( 1 - 2/ b ) - F2( 1 - l< b))' '
and hence may satisfy the condition of separation. For region
= u[XJ - a[X]Wb)
(5)
I the y miung algorithm [Walhs er al.,1977} prosided a regional salue of y a 4.5. which is similar to that of the aforementioned WA. Thus both the WA and the mixing al-where a[X] and a[X] denote the mean and standard deviation of X, If C[XJ = ~[X[ u[X] = d 4b), then m = 0, and if C [X}
gorithm suggest that for floods, y is large, at least for region 1.
< 6-'(b), then ra > 0. It is mathematically possible. but not y m RS and LS phy sically reasonable, to have m < 0, in w hic. case C,[X] > 9" It was noted abose that historical riood sequences vielded (b ).
estimates of y w hich tended to be positne in RS and n'egatne To examine the effect of sample size n on the relation be-tween the skews in RS and LS as a function of the lower bound in LS. It w as further noted that if X is distributed as LP, then y may be positne in RS and negatne in LS. But if y in LS is m in RS. Monte Carlo experiments were conducted. To gener-negatise, then X distributed as LP is bounded abose. Other ate W sequences with a > 0 (y > 0 in RS). th: sariate x was distributions, which are unbounded above may be character-expressed as ired by positne y in RS and negatne y in LS This point i<
x = m - a[
u.
1 - u l},
(6) illustrated below.
Consider the case where X is distributed as Weibull. The where u = f(x) = f.,'f(x) dx is distributed uniforml> on the density function of X is interval [0 []. For convenience. gtx;was 7t equal to 1000. As
,~
p previously defined,5(s } represents the mean of estimates of f(x ) = I * ~
exp t4) skew derived from simulated s;quences. Values of Jis) for a
a a
estimates of 3[Z} where Z = In (X), are gnen in Tables 9 atd w h ere b > 0. I f a - ) (< 0), then in RS, y > 0 (< 0),.ir.J m 10 for y[1] = 1(b a 2.~7) and 3[1] = 2 (b = 1), respectacl>.
denotes the lower (igpers bound. From the moments of Y, The values of Afs) were obtained from 10.000 sequences of gnen in Appendis C, with a > 0. m is de6ned as length n = 10, 20, and 30 and from to sequences ailength n =
a 3.7 e I
+7 I
3
.s
.2 4
i
- is
.i eia ao se
- 0. S L
=f\\
- u 3
N i
0 0.0 s 01 0i's 02 veon Skew lE m Ls hg. 3.
\\ lean skew i senus standard devimon of skew,10 L for n = 10: histancal and sama!ated Pill data in LS.
/
n>
O,b
\\.. U b
908 Lasowrun ET 4t.. Ft ooo STansncs T A BL E 7.
Mean J(w) and St;ndard Desistion Ju) of Estimates.TA B L E 8.
Values of y, A. J(w), and &(w t w = y, in RS for WA of y in RS for 5000 Sequences of Length a = 10 Generated as LP Sequences of Length n = 10 and 4ccas - 5 WRh Parameters a and b Value of d a
b d(w )
At 0.16 0 24 0.32 LP Charaarn:ed by Smatter A Than LM
- 0.80 4
1.15 0 61 b - 0.40
- 0.15 15 0.56 0 53 y
2.99 5 52 35.43
- 0.10 15 0.30 0.50 A
23.78 346.25 0 01 4
0 54 0 60 Ju) 1.05 1 17 1.24 0 01 1
0 93 0 67
&'w )
0 64 0.67 0 69 LP Charassen:ed be Larger A Tian LN*
b=3 t
0 09 10 0.73 0.66 y
2.9 I 5.78 37.70 0 14 5
0.38 0 63 A
24.49 404 46 0.12 20 0 97 0 69 J!w) 0.75 0.98 1.13
- 0. I 8 3
0.9%
0 68
&(w )
0.76 0.75 0.74 0 20 I
l.01 0.69 0.16 10 1.18 0.67
- Here A is not detined, see Appendit B.
0 16 20 1.16 0?0 0.24 I
I 22 0 67
- For gnen salues of y.
determined for each X/ based on 10.000 WA sequences of lengths n = 10,20, and 30 and 10 WA sequences oflength n =
10,000. The values of 5(w) are gnen in Tables 12-16 for X ',
i 10,00i1 From Tables 9 and 10 it is noted that for m = 0,4(w)is
, X ', respectis ely.
=
negatne for all n, ano as,n increases, in which case C,[x]
From Tables 12 and 16 it is noted that for X,' with m e 0, decreases,4(w) increases and becomes positne. Morcoser,it is 4(w) is negative for all i, and as m imeases and C,[X'] de-noted that for m = 0,4(eln = 10,000) = -1.14 for both values creau,4(w) increases and becomes positive. To this extent, of y[X]. In Appendit C it is show n that if X is distnbuted as W X ' distnbuted as WA behaves as though it wcre distnbuted as i
with y[Vj > 0 and m = 0, then Z = In X is distnbuted as G W. The errect of increasing A[X/] is that for given values of with y (Z) = - 1.139 -
for all y[X] > 0. and thus 4(W n)-
d(X/], C,[X/], and y(X/], AM decreases and becomes nega-
- 1.139 -
as n m.
tae. The tables illustrate that although y[Z/], approximated To assess the effect of A[X] on estimates of y[Zl, where Z =
by 4(w) wi,h n = 10,000, may be positne,4(w; with n small in X, with X unbounded above and distnbuted with y[X] > 0, may be negative but will b come positne and approach y[2/]
X was assumed to be distnbuted as WA. Consider fhe random as,a m.
variabies X,
, J., each distributed as WA with y[X] s 2 If X is unbounded abos e and dstnbuted with y(X) > 0, then i
such that A(X ] <
< A[X.]. The values of the WA parame-Z = In X distnboted with y[Z) < 0 is realizable. It does not i
ters a, b, c, and d for X,, ( = 1.
5, and the corresponding follow that this is so for all such X. If X is distnbuted as LN values of y[X,] and A[X,] are gnen in Table 11 (see Appendix with y[X] > 0, then 2 = In x is distributed with r[Z] 2 0 form B;. Note that X has y[X ] = 1.91 and A[X ] = 10.73 and 2 0. Thus if f distnbuted as WA is distributed approximately i
i i
therefore X, is distnbuted approumately as LN (see Table 5 ).
', as in the case of X/, the sampling properties of 2 = In a
Each X, was transformed to X/ = a[X/]X, + u[X/] where X do not accord with those where X is indeed distnbuted as u[X/] = 1000 for all i. Note that y(X,] = y[X/] and A[X,] =
LN. In general, the value of y[Zj depends upon C,[X], y[X),
A[X/]. For y[Z/], where Z/ = In X/, salues of AM were and A[X]. Estimates of y[2] arc in expectation depenoent upon 1.2 l
1OY 3.I I
e o r E
06 J
I b g4L O-- LP chorocter :ad by sinal er A than LN for geven volves of 7 02 x-- LP choroctorized by iorger A
'han LN for given,oives of 7 0'
o
- 0. 2
- 0. 4 0.6 0.8 i.0
- 1. 2 i4 1.6 i8 2.0 Moon show 2( w). in #s Fy. 4
\\tean new Jui sersus standard deuauon of skew. A), for n = !O: simulated LN and LP sequence in RS
} [>
h 9, i a.
LODw f HR ET AL.: FlouD Sr4Tistics 909 TABLE 10. Values of JM for Estimates of 312]
i 2h
- g, j io*
C,[4j m
a = 10 n = 20 n = 30 a = 10.000
)..
1 0
- 0.52
- 0.74
- 0 43
- 1.14 09 in0
-0.10
- 0 11
- 0 11
- 0 11 t
07 0
0 18 3 !
06 AM 0 37 0 46 0 49 0 55
""'
- 1 a4 0$
$00 0 48 0 59 0.63 0 73 3 8 "-
O4 600 0 58 0.73 0.?8 0 40 oa-03
'00 0 66 0 35 0 92 1.07 02 800 0.77 1.00 1.04 1 29 1
0 Y is distnbuted as W with u(1) = 1000 and y[1} = 2 (b = lt o
oa o.
on as to ia i.
is is ao a h. as Frg. 5.
Mean skew A i sersus standard deviation of skew, A p, for n = 10 umul.ited LN and W A data in RS 3[2]. howeser smooth oi rugged the contours of the estimates of y[2] may be.
n as well as CJX], y[1], and y[Y}. Thus X bounded abose is Consider the case w here X is distributed as W with y[X] > 0 not a necessary condition for y[2} < 0 to be realized.
and m = 0. For a hypothetical region, depicted m Figure 6, there is a north to south tiowing stream fed by hrst-order REGION AL SKEW M APs tributaries flowing eJst to west. One of three hydrologies is The biases and large sampling errors inherent in estimates of assumed to exist, where the hydrologies are as follows. Fo.
y[X} from histoncal flood sequences hase led the US Water hydrology I (H ), y[X] = 2.5 at all sites. For hydrology 2 (Ib),
i Resources Council (1976l to suggest that if n < 100, the use of y[X] decreases from 4 to 1 in a north to south direction but is a regional skew maps is preferable to the estimate of y[X] ob-constant along any tributary. For hydrology 3 (Hz),3[X] does tainabic from using only the specific record. In particular, the not sary from one tributary to another, bat along any tri-Cour.ul suggested the construction of regional skew maps in butary, y(X) decreases from 4 to I m the downstream direc-LS. Howeser, since histoneal records in the United States are tion. Thus for each bydrology the mean regional skew in RS is almost always less than 100 years. the Water Resources Coun.
- 2. 5.
c:1 a effectnely adsocaung the use of regional skew maps at all The regional skew maps in RS and LS for the three hydrol-sites.
ogies are depicted in Figure 7. It is noted that although the The abose discussions indicate that the usefulness of re-skew contours in RS are distinctly dirTerent for the three gional skew maps in LS is quest;onable. It w as noted that y[2]
hydrologies, th contours in LS are identical. Thus gisen the is not simpiy related to 3[X] but depends upon C,[X}, A[Xl, LS reg:enal skew map, the RS contours cannot be uniquely and the distnbution of X as well. Thus it is ditEcult to mfer speci6ed. Esen if it is known that X is distnbuted as W and a properties of X from those of Z without considerable know!-
perfect regional skew map in LS is asailable. the contours m edge about the distnbution of X and its sampimg character-RS still cannot he uniquely mferred.
istics.
For the hypothetical region, consider a fourth hydrology A regional skew map consists of a set of contours of equi-which is derined as follows. For hydrology 4 (H.), X ts distnb-skew. 3moothmg out the sanability ir :he estimates of skew at uted as W A with y[X} = 2 and a[X] = 61 at ail sites. w here X a large number of sites in the region. Quite similar regtonal n Yi in the abose WA discussions. Along any tnbutary, m maps in LS may be realized whether y[X] is a constant for all increases m the downstream direction from 0 to 10(L Con-sites u a region or sanes among the site. But esen if distmetly sequently, a[X] increases in the downstreani direction from ditTerent LS maps were realized. :t would be ditlicult to deter-100 to 680.
mme if a particular LS map was generated by constant or in Figure 8 the regionai skew map m RS is shown in telation sarying salues of [X]. Gisen only the estimates of y[Z} at the to LS maps based on expected values of estimates of y[Z] for 3
sanous sites, isttle can be said with confidence about y[X] or three cases: (1) n = 10 at all sites, (2) n increases along each tnbutary in the downstream direction from 10 to 30, and (3) n
= at all sites. Expected values of estimates of 3[2] for n =
TABLE 4
% alues oi ai for Estimaics of 3[2]
are approximated by those for e = 10J01 From Figure i it is noted that the skew contours are dis-u C, p l m
er=10 n = 20 n = 30 a = 1020 TABLE 11. Charactenstws of E Distnbuted as W \\
0 34 0
-051
- 0 73
-053
- 1. ! 4 a 24 110
- 0 41
-0 56
- 0 61
-0 5
a 8
3 ly, }
ug.)
0:4 60
- 0 31
- 0 42
-0 45
- 0 53 0.24 340
-o :3
-0 20
- 0 33
- o 37 1
1 i
5 q 12 1 91
- 0 73 0 19 510
- 0.15
-0 14
- 0 ',0
-0:3 i
- 1. 5 25 O l?
i 96 15.14 0 14 640
- 9 07
- 0 09
-0 10
- 0 08 3
1 4
15 0 17 2 00 15 45 0 09 "O
O 01 0 01 0 02 0 04 4
1 5.5
!.25 n :1 242 33 ~
5 1
4 0 33 0:3 105 87 ?<
t :s ontmeo as W with ap j = ou) and,l t ] = ib - 2.77)
/~ n./,
F09
. > ~
910 LasowEHa ET 4U Ftooo Srrristics TABLE 12. Satues cf JM for Estimates of 3{Zi]
TABLE 14. Values of h) for Esumates of y[Zij EM M
C, [lil m
n = 10 a = 20 n = 30 n = 10.000 C,[1/}
m
- i. = 10 m = 20 n = 30 n - 10.000 0 87 0
-061
-087
- 0.9 %
-1 32 0.61 0
- 0.76
- 1.16
-l.36
- 2.00 07' I!5
- 0 23
-0 26
- 0.26
-0 27 0.51 164
-O 44
- 0.57
- 0 61
-0 66 0 67 230
- 0.03
-001 0 00 0.03 0.41 323
-024
- 0.26
-0.26
-0 21 0 57 345 0.13 0 18 0.20 0.25 0 31 493
- 0 06
- 0.01 0.02 0 16 0 47 460 0.26 0 34 0.37 0.45 0.;l 657 0.10 0.22 0 28 0.50 0.37 575 0.37 0 48 0 54 0.66
- 0. I I 821 0 25 0 44 0.55 1 03 0 27 690 0 49 0M 0.70 0 83 0 09 985 0.40 0.6A 0 86 1.99 n 17 805 0.54 0.78 0 88 1.11 0 07 920
- 0. 'O 0 96 1 07 1.48 Xi is distnbuted as WA with u(Yi] = 1000 and y[Yi] = 2.06.
Ai is htnbuted as WA with a[L'] = 1000 and y[C] = l.91.
By means of Nionte Carlo experiments, Matalas et al. [1975]
showed that the condition of separation can be explained tinctly ditTerent in RS and L3. Morcoser, m LS the contours neither by the small number of histoncal 8vod sequences vary considerably with n, though the:r general structures are relative to the scry large number of generated sequences condi-the'same. Thus n finite and vanab!c from site to site adds tiened on the specific distnbution functions nor by autocorre-
~
confusion to the already confusmg task of mierrma the RS lation. Wallis et al. [1977] noted that cross correlation cannot skews from the LS skews or for that matter inferrm'g the LS espl m the condition of separation but that separation can be contour that would apply wi h n = m accounted for both by spatial mixmg of salues of y[X) withm a t
For the same hypothetical region,,:onsider a tifth and a region and by nonstationanty in y[X).
snth hydrology defined as follows. For hydrology 5 (H.), X Analysis of histoneal Good data in addition to further with lower bound equal :o zero is distnbuted as LP with a =
Monte Carlo expenments m real space (RS) and log space
-1 and b = 15 such that at a!! sites, y[X] = 0.55 and y[Z] =
(LS) led to the follow mg conclusions.
1.
In RS, estimates of y(X) derised from dood sequences
-0.52. For hy drology 6 (H.), X with low er bound equal to 1 is distnnuted as LP with a = 0.01 and b = 25 such that at all are dominantly positie and become more so as n increases, sites, y[X] = 0 56 and y[Z] = 0.40.
whereas m LS, estimates of y[Z) are dommantly negatne and For H. and H. the regional skew maps in RS are sen nearls become more so as n increases for most of the U.S. regions.
.dentical, but m LS the5 are distinctly ditTerent (see Fiaure 95.
2.
The log Pearson type !!! distnbution (LP), reco m-Thus with X distnbu:ed as LP, inference problems are'no less mended by the U.S. Water Resources Counal[1976) for use by confusmg than tacy are with X distnbuted ditTerentiv.
all federa! agencies in conductmg dood frequency studies, can
~
accommodate y[X] > 0 and y[Z) < 0. Howeser,if y[Z] < 0, X Stante oo Coscu sloss is bounded abose regardless of the sign of y[Xj.
3 The LP distnbution is unkkely to explain the condition Flood statistics in both real space, X, and log space. Z = In X, were assessed with respect to specific distnbution functions. of separation, smce it fails to do so for representatne parame-ter salues of the distnbution.
Based on a partitioning of the Umted States into 14 regior's, 4.
In LS, all but one of the 14 regional points (f. &qy)) lie the mean f and standard deviation 4(y) of estimates of y[X]
derned from available ! bod sequences of length n acn n a abose the 44) versus dW) eune for the Pearson type 111 region wcre compared with the mean su and standard devia.
distnbution (P!li), indics'ing that the condition of separation tion su of estimates of y[X] derned from Monte Carlo cannot be accounted for by LP.
5.
For a distabution to 3: eld a less pronounced condition expenaents conditioned on specitic distnbution ft.nctions.
For sescral wc!!-known distnbution functions defined bs three f separation relatne to LN it is apparently necessary but not or few er parameters, Matalas et al. [1975] showed that in real su!!icient that for a gnen value of y[X] the distnbution should space there exists, for each region, what was called a condition have a value of.\\[X] larger than that for LN.
6.
Although for certam LP, A[X] a larger than that for L N of separation; that is, for f = AM, hty) > JM. Among the distnbution functions the condition of separation was less f r a gnen value of y[X], the condition of separation is only pronounced for the log normal distnbution LN and for the margmally less pr n unced than that for LN.
7 Houghton [1977] has shown that the Wakeby distnbu-Pareto distnbution P at the higher salues of y[X].
tion (WA), dedned as TABLE 13. Vaiues of L for Estimates of y{2i]
x = m + a[1 - ( 1 - FM - @ - (1 - FM lw T ABLE 15 Values of Ju for Estimates of 3(li]
C [1,i m
n = 10 e = 20 e = 30 n = 10.000 uwt 0N 0
- 0 69
-099
- 1.12
- 1.52 0M 128
- 0 33
- 9 39
- 0 40
- o 41 C,[ril m
n = 10 e = 20 n - 30 n = 10.000 0.5 s 2r
- 0 13
- 0.12
- 0. l l
- 0 09 0 44 345 0 04 OM 0 :0 0 17 0 49 0
-0 33
-1.35
- 1.61
-2.61 134 513 017 1 26 09 03%
0.39 202
-0 55
- 0.76
-0 C
- 0 93 0 28
$42 0 29 0 41 04' n 65 0 29 405
- 0 35
-041
- 0 42
- 0.36 0.13 "O
04%
0.58 0 67 0 95 0 19 607
-016
- 0.12
- 0 08 0 14 0 03 s99 0 52
- 0. ? '
0 46 1 33 0 09 810 d0l 0.15 0 2J 0 16 t :s Jtstrit uted as WA with a[Y/] = 1000 and 3 { ri! = 1.96.
Yi is distrmuted as WA with a[Yi] = 1000 and il = 0.02.. el O i
p J V, i.' D !
911 TABLE 16.
Values of h) for Estimates of y[z.')
- 13. The construction and use of regional skew inaps are most likely to be counterproductne.
u6)
APPESDl4 A: CostMENTs ON THE Loc PE ARSON C. [ A.']
m n = 10 a = 20 a = 30 a = 10.000 TYPE 111 DisTaint TtON 0.40 0
-0 'i2
-1.44
-1.78
-3 2f Deftmtion 0 30 244
- 0 57
- 0 85
- 0.97
.-1.16 Let a be a constant and X > r a andom sanable distnbuted 0 20 495
- 0.36
- 0 48
-0 $1
- 0 41 as log Pearson type _ ll (LP), such that 2, detined as I
0 10
'43
-0 17
- 0.15
- 0.10 0 33 ON NO 0 00 0.17 0.20 1.34 Z = In (X - y)
(Al)
A.' is datnbuted as W A with u(1.'] = 1000 and 3[ t."* = 2.05.
is a random s ariable distributed as Pearson type ill(Pill). The probability density function of X can be wntten as
- ~#
where F = f(x) = P[X 5 r}. ean esplain the condition of u paration. Among those WA's where A[1] is larger than that w here for LN for a gnen value of y[X]it seems that the condition of I
separation can be explained with b > I but not with b G l f(. 3 esp _-
w hen a. b, c, and d > 0.
!al fib) a a.
(A33 3.
WA otters an alternatise to spatial miung of salues of ial > 0 y[T ] within a region as an explanation for the condition of separation. A particular WA yielding sa!ues of 46) and &W) The ranges of variation of X and Z are in close agreement u ith i and &ty) for region I has y[X] = 3 95.
>0
/+r5x5 m This is eiose to the value 4.5 obtained by Wallis et al. [1977] by spatial miung of 5aices of y[X] within region I condiuoned on a<0 y5xs/+r LN. Whether WA and spatial miung of y[X] would sield
~
a>0 c 5 :5 I results in close agreement for the other regions remains to be determined.
a<0
-m5:5c 9 \\1onte Carlo e,nments with WA indicate that for a gnen salue of y[X] > 0. y[Zj < 0 is realizahle. For gnen When a > 0, there custs a lower bound m on the values of x values of a[X], C,[1], and y[X] the espected salue AL) of such that estimates of y[Z) decreases as.\\[X} increases.
,,g.,
(35)
- 10. If X is unbounded abose and distributed with y[X] >
0, then y[2] may be negatne, particularly if m, the lower and x = m when : = c. Consersely, when a < 0, m is an upper bound on X, is small. If y[2] < 0 with m small then y[2] will bound on x; again, x = m a hen : = c. If c = 0, then m = 1 + r.
- ncrease and become positne as m increases. Thus the property and if both c = 0 and a = 0, then m = 1.
of y[Y) > 0 but y[Z] < 0 is realizable with X unbounded abose. Howeser. X would be bounded abose ifit wcre distnb-uted as LP.
Consider only the case where a > 0. By definition. the 4th-
- 11. From values of y[2] it is dirlicult to infer y[X]. Dis-order moment E[X'] is tinctly ditierent y[X] contours may gise nse to identical r[2]
E[1*] = f",Pfix ) dx contours and sice sersa. Thus, in general, the skew map in RS (Ae) cannot ne uniquely mferred from that in LS.
l'.
W ith small sampies it is ditlicult to infer y[1] from so that estimates of y[X], and these didiculties are only compounded by attempting inference frorn estimates of y[2].
g[y.)
{
- y. E (1-iaf i A7)
I where ( 1 - ia) > 0 for i = 0,1.
,4. Thus the 4th moment of t
=
l X custs only if a < 1/k. For 2.
t i
E[Z * } =
- f(:) d:
(A8) so that E[Z'] =
ca' ~ f t k - b - 0 (A91 exists for all a
- 0. The mean a[ J the standard des tation c[
], and the coedicients of skewncss 1[ ] kurtosis. A[
}.
and vanation. C,[
}, for X and Z are defined m Table -\\ 1.
It is noted that if c 2 L (, then the moments of order 4 e 4, t
are not detined for X. although the moments of order k cu.st for Z 7 k. Thus if a 2 f. A[X]is not defined;if a 2 f, y[X] n not defincd: if a 2 t a[Y} is not defined; and if a 21. a[X] is not F3 stream neturk inr Motnewal repn.
oc:ined.
o 00
912 lod %)HR ET AL.: Ftono Srinsncs H,
R$
H : R$
H : RS 3
3 y
(1) (27 13J (4)
N 8 '
A u
___m n
ll l
l
_ ____2)<
I l
I I
___33
'd 14) i I
i l
ir y
7(x) = 2 5 et oil sites I
I stream ne, work
- - - shew coamor (etvolves of show H,: LS H,: LS H : LS 3
N N
N si s,
sL ir 1r if 7 (Z)=.i.14 av oil saes T (Z) =. l.14 at a41 sites T(2)= 1.14 oe oil sees Fig. 7.
Skew maps in RS and LS for hypothetical region: A datributed as W with m = 0.
H,: RS N
i O
JL u
^
0 9_-m=0 a --m m 50 m
O -
m= 100 m
Stream network 7(X):2 of oil sites
- ~ - - Skew contour (e) Volves of show H.:LS H,: LS H.: LS l (O 84) 00 01) FO 76) N I
( t.99)
(0.16) ( 2 00) N to aci t o 06, ( -o 76) N l
t e
e r
t n
\\
t e
g n
a l
I I
I I
I I
I I
I I
l i
i i
i l
l I
l l
l l
I l
i I
I l
l l
1 I
I I
f.
1 f
I t,
I, i
t I
I n::10 n=10 n= t o n.
30 n=20 n=to n:ce nm n=co 1r y
,r Fig. 4 %ew maps :n RS and LS for hypothethal repon. Y distributed as % \\.
\\, l )t
() %
J)
, '] * )
Lonw ma er 4t.. Ftoon Srsristin 913 H g RS H,; RS mereases (b' decreasest (4)If a < 0, then y[X] may be either positne or negaine. H ow ever, gn en b = b*. i.e., for y[2] = 3
- there custs an a = a' < 0 such that y[X] = 0. If a = a*
and b < b' i.e., for y[2] < y'. then y[Xj < 0; consersely, if y[2] > y', then y[X] > 0. Furthermore, as t' decrea3es in salue, the associated a' < 0 also decreases.
Thus suppose that floods X are dntributed as LP. If X has no upper bound, then a must be positne, so that the skews of hoth X and 2 are posune. Alternatnely, if there is a "masi-mum certain floodJ so that X is bounded abose, then a and r
y y[2] are negative, although y[X} may be poune or negatne.
Consersely, if y[2] > 0, then y[X] > 0. and floods are un-Uu)=o ss e mi w n n:- a u..n bounded abose in real space. Howeser, if y[:] < 0, then y[X]
may be positne, zero. or negatise, but there is a maumum 5""#=*"'
Hi. LS H,. LS CCrtJin dood m both log and real space.
The followmg obsersations are made from Tables A6 and 4
A7 with respect to the 3[ J. (1) As b mcreases. A(Zj ap-proaches 3. ( 2) For a gnen value of a, there eusts some b = b*
that 13, some A[Z] = A*. such that A(X]is a mimmum, Further-i more, as.at mcreases. A* increases and b' deercases. (3) For J a < 0 and any gnen b, that is, A(Zl, A[Y) decreases as a decreases (4) For a > 0 and any gnen b that is. A[Z), A(X)
I mereases to mtinity as a approaches 0.25 and decreases.n a approaches 0.
l From Tables A5 and A' one can choose pairs of a and b salues such that for X distributed as LP detined by these wt= o sa..a w m= o.a..o.
parameters. A[X] > A[F), esen though y[X] s y[T] for F bg 9
%ew mars in RS 2nd LS f or h> puthet: cal reg:an.t dninb-distributed as log normal.
uted as LP Log Pearson Random VartaNes Consider the case where a < 0. The equations for the mo.
To generate raridom numbers which are distributed as LP.
ments of X and 2 are the same as those for the case w here a >
the following algorithm (Johnk 1964 Berman,1971l can be
- 0. Howeser, since 4 2 0. (l - Aa) 21 for a < 0. Therefore il a used. Let
( 0, then all moments, for both X and Z are defined.
~
Tables: Coettictents of i,anatwn.
.- In ] u. - B In u t = exp c+a
+a
( A 10 )
St ew. and Kr.rrosn where u - uniform (C) on [0,1], [b] denotes the greatest Tables A: -A 7 present salues for tne eocthe:ents of saria-mteger less than or equal to b, and B n defined as follows. (1) tion, skew, and kurtosis for a and b with c = 0 and m = 0. The Set r = b - [b} and 3 = 1 - r = ! - b - [bl. (2) Generate ui, ai follow mg obsers ations are made from Tables A2 and A3 with
~ L (0, l }. ( 3) Set f = u ' ' and i = u/ '. (4) If f - i > 1. return respect to C,[
- l. (I) As b increases. C,[Z] approaches 0. (2i to step 2. otherwi3e, proceed as follow s. (5 ) Set B = f ti - i).
For a gn en s alue of b. C,[X] increases as. a, increases ( 3 i For if b n an in'eger B = 0, and therefore a gnen salue of a. C.[X] mcreases as y[2]l decreases.
{ In { u. (+y
( All)
)
The following obsersations are made from Tables A4 and r = esp sc-a A5 with respect to yi
). t 1) As b increases. r {Z] approaches O.(:, For a gnen value of 6. tFat is, for a gnen y[Z], y[X}
that is, nereases as.a! mereases. ( 3) If a > 0. then y[X] > 0. G nen a
= a' > 0, y[X] is a minimum for some b = b* i c., for some h ln u.
(Al2)
.t = esp c+a
-r y[2] = y' As a increases m value, the associated y
- also
.T
- l T\\BLE Al. Statisn at Pmperties of i Distnbuted as LP and 2 Datnbuted as Pill Propert)
Detin: tion afAl y -nl-ai m:
siil e k l - 2a e ' - i l - a n Hil
<'91 tai ' - h l - n 11 - 2a i ' - :(I - a i % y t l a l l - a i ' a i 11 - 3a i * - m l - a i 91 - :a i o ep qg) 4 11 a
e C. { ll
< M I -- a i ' - il - n 'T yy -ni-ai ~}>
J71
- ah n/'
aM*
Hr
'a[ a y n -
42]
3-Ab' C {/}
a h' N - Jb} '
/. O o o6 6
- / '
/_
914 Loow rHa ET 4t.: Flooo Strimics TABLE Al Coerficient of Vanation in RS (X) and LS (2) for Log Pearson Distnbution With m = 0 and c = 0 C,[1]
a.
a-a-
a-a.
a-a.
a.
a.
a-h
- 2.00
-1 50
-1.00
- 0 80
- 0.50
- 0.30
-015
-010
- 0.05
- 0 01 C,[2]
0.05 0 17 0.15 0.12 0.11 0.08 0.05 0 03 0.02 0 01 0 00
- 4 X, 0 10 0 25 0 21 0.17 0.15 0.11 0 07 0 04 0 03 0 02 0 00
-316 0 25 0 40 0.34 0 27 0.24 0.17 0.12 0 07 0 05 0 02 0 00
- 2 00 0.75 0.74 0 63 0 49 0 42 0.30 0.20 0 11 0.08 ON 0 01
- 1.15 1 00 0 89 0.75 0.58 0.50 0 35 0.~ 1 0 13 0 09 0.05 0 01
-1 00 1 00 1 50 1.20 0 88 0.74 0.52 04 0.19 0.13 0 07 0 01
-071 2.50 1 83 1.43 1.03 0 86 0 59
- 0. 8 0.21 0 14 0 03 0.02
- 0 63 3.00 2.20 1.68 1.17 0.97 0.65 0.42 0.23 0.16 0 08 0 02
- 0 58 3M0 2.61 1.94 1 32 1.08 0.71 0.46 0.25 0.17 0 09 0.02
- 0 53 4 00 3.08 2.23 1.47 1.19 0.78 0.49 0.27 0.18 0.10 0.02
- 0.50 5 00 4 23 2xl 1.79 1.42 0W 0.56 0.30 0 21
- 0. l l 0 02
- 0 45 10 00 18 87 9 26 4 09 2.33 1.50 0.85 0.43 0.29 0.15 0 03
-032 15 00 Cl3 28 40 8.59 5.11 2.20 1.13 0.54 0 36
(' ' 9 ON
- 0.26 20 00 357 05 46.73 17 73 8.94 3 09 1.41 0.64 0.42 d2 0 04
-022 25 00 264 70 36 44 15 62 4.24 1.71 0 73 0.48 0 24 0 05
-020
' Greater than 999 99 If b <
I, then [b] = 0, so that f = dF/dx = [abt ! - F)" + cdt 1 - F)-8-'j-'
(B2) x = esp !c - a[-B la u]l + r
( A 13 )
If F = 0, then x = m and f = idab + cd). Note that sincef 2 Equation ( All) presents an algonthm which is faster compu- 0 v x. (ab + cd) 2 0. For F = 1 the salues of x and f depend upon the values of the parameters of the dntribution, the tationally than ( A 12). Howeser, for b large it becomes impos-sible to use i A 11) c14 computer, s:nce the product of the u.
upper bound on x being either + r or (m + a - c). F.urther-more. the definition of f precludes certain parameter combina-rapidls becomes too small to express on the machine. This tions, as is shown in Table Bl.
problem is avoided by using the sam of the logarithms rather than the logarithm of the product of the u..
Sfoments Aeenoix B: CowENTs ON W4KEnv DisTRIBt; TION The moments are detinea es E(1*} = f.'x' dF. where x is gisen by (BI). E(X'} cannot be computed if d 21/4 or if b d -l/4. since the integral would not be defined properly.
The Wakeby distribution (H. A. Thomas, persona
,mm u-Thus if d 21/k or if b s -l/k then the moments of order k nica00n.19761is defined in the following manner. Lt ' be a and higher do not exist. That is, the mean, g[s]. eusts only if random sariable such that d < 1 and b > -1; the standard desiation. a[xl. eusts if d < f and b > -i the coetficient of skew.1[r}. exists if d < i and
= m - a[I - (I - F)'] - c[1 - (1 - F)-d]
(BI) t b a -t; and the coetficient of kurtosis. A(xJ, exists if d < i and w here F e f(x) = P(Y L x) and x 2 m. The density function b>-1
/ = f( t > is defined as To allow concise tabular presentation of these properties T ABLE A3. Coerficient of Vanation in RS (Y) and LS (2) for Lug Pearson Distnbution With m = 0 and c = 0 Cd1]
a-
.1 -
a=
a.
a-a-
a-a-
a.
a-6 0 01 0.05 0 09 0.10 0.15 0.18 0 22 0 24 0.28 0 32 C,[2]
0 05 O f Y) 0 01 0 02 0.02 0 04 0 05 0 06 0.07 0 09
- 0. l l 4 47 O !O ON 0 02 0 03 0 04 0 06 0.07 0 09 0 10 0.13 0.16 3 16 0 25 0 01 0 03 0 05 0 06 0 09 0 11 0.14 0.16 0 20 0 25 2(0 0 75 0 01 0 05 0 09 0 10 0 15
- 0. !9 0 25 0 29 0 36 0 45 1.15 l'O 9 01 0 05 O !O 0 11 0.18 0 23 0 29 0 33 0 42 0.53 1.00 2.f 0 0 01 0 07 0 14 0 16 0 26 0 32 0 42 0 44 0 62 0 81
- 0. ' l
. 50 0 02 0 08 0 16 0.18 0.29 0.36 0 44 0.55 0.71 0 13 0 63 3.00 0 02 0 09 0.17 0.19 0 32 0 40 0 53 0 61 0 50 f.On 0.54 3 50 0 02 G id 0.19 02!
0 34 0 43 0.58 0 67 034
- 1. : 8 0 53 4 00 0.02 0 11 0 20 0.23 0 37 0 47 0.o)
- 0. '2 0.96 1 31 0 50 5 u0 0 02 0.12 0 22 0 25 0 41 0 $3 0.72 0 83 1 13 1.58 0 45 10rn 0 03 0 17 0 32 0 36 0 61 0 80 1.14 1.36 2 04 3 35 0 32 15 oo 0.04 0.21 0 40 0 45 Of8 1.05 1.57 1.96 3 27 6 46 0 26 20 W 0 05 0 24 0.47 0 53 0 94
' 30 1 06
- 2. M 5 06 l11M 0 22 25 00 0 05 02' O $3 0 n0 1.10 I fn 2.64 3.58
'. 'O 21 0 0.20
./ ",
A' 7
e s
p.
E af
LODw FHR E T AL.. Ftooo SirrisTics 915 TABLE A4. Coctficient of Skew in RS (1) and LS (2) for Log Pearson Distribution M1)
J' d*
G=
Q*
G*
0*
Q*
G*
Q*
G*
b
- 2.00
- 1.50
-1.00
-080
-050
- 0.30
- 0.15
-0.10
- 0 05
-001 u21 0 05
- 3 81
-4 16
- 4.70
- 5 02
-5.73
- 6.50
- 7 40
-781
- 8 32
-8di
- 8 94 0.10
- 2.49
- 2.75
-3 17
-341
- 3 94
- 4.51
- 5 18
- 5 49
- 5 86
-622
- 6 32 0.25
-1.17
-1.38
-1 69
-1.88
-2.27
- 2 69
-318
- 3 40
-3.67
- 1 93
- 4 00 0 '5 0 22
- 0 02
- 0.34
- 0 52
-0 88
- 1.24
-I.64
- 1.82
- 2.04
- 2.25
- 2.31 1.00 0 64 0 36
- 0 00
- 0 19
-0.57
- 0 93
-I.34
- 1.52
-1.73
-I 94
- 2.00 2.00 2.13 1.59 0.97 0 69 0.19
- 0.25
- 0. 'O
- 0.90
- 1.12
-1.35
- 1.41 2.50 2.97 2.19 1.39 iN 0 45
-0 04
- 0.52
- 0.73
- 0 97
-1 20
-1.26 3 00 3.94 2M l.s0 1.37 0 68 0 14
-0 38
- 0.59
-0 85
-1 09
- 1 15 3.50 5.12 3 60 2.23 1.70 0.90 0.30
-0 25
- 0.48
- 0 75
-l 00
-1.07 4 00 6.55 4 46 2.68 2N 1.11 0.44
- 0.15
- 0 39
- 0.66
- 0 93
-1.00 5.00 10 55 6.64 3.71 2.'6 1.50
- 0. 'O 0.03
- 0.23
- 0 53
- 0 82
-0 89 10 00 108.33 42 81 14 18 8 59 3 67 1.75 0 62 0.24
-0 16
- 0.53
-0 63 276 41 51.30 23 91 6 98 2.s0 IN 0.55 0 05
- 0 40
-0 52 15 00 20 00 187 98 66 91 12.70 4.03 1 41 0.79 0 :0
- 0.31
- 0.45 693 40 189.13 23 01 5 58 1.76 1.00 0 32
- 0.25
- 0 40 25.00
- Greater than 999.99.
isee Tables BI-B29 of the microtiche supplement 4, the piram-n;; a and c ea i be ohtained as functions of b and d, gisen a[X},
eter c is espressed as a funcion of a: i.e., c = ra, w here r = c a.
a[1], nd m. How es er, gisen a, s. a[X}, a[X), and m. explicit The detinition of WA as gisen in (B1)is rewritten as expressions for b and d cannot be readily obtained The value of r is a function, expheith, of m,2 b. and J. as
~
r = m + a r[1 - ( 1 - Ff] - r[I - ( 1 - FP* p' (B3) well as just c and a:
and the aforement:oned s'atistical properties are gnen in m/ a - [ I - F(a )},
c (B4)
Table B,.
r=-=a 1 - [I -- F(a ll-*
From Table B2 it a seen that the statistical properties are functions of all four parameters a, b.rtc), and J. Property u[X] so that if m = 0, is also a function of m. Howeser, if m = 0, then C,[X} = a[X}-
[I - F(a n]"'
u[1]is only a funetton of b, d, and r. Regard! css of the 5alue of 7
aB5) m, y[1] and A[1] are only functions of b, d, and r.
3 - II - f(Gl}#
If u(1], a[X], and m are known, then a and b can be derised Speaal Cases explicitly as functions of c and J. S.imdarly, c and d can be derised explicitly as functions of a and b plus u[X}, a[X), and Three special cases of the WA distribution are highlighted, i.e., t I) a = 0,(2) b = r, and (3)c = 0. with not more than one of these conditions holding. CdX], y[Xi, and AiX] for these
'These tables are available with the entire article on micronche.
cases are given in Table B3.
Order from Amencan Geopnysical L nion.1909 K Street. N W N te that if b = 0, the distribution is equnalent to that of.
Washmgton, D. C. 20006. Document C8 007. 51.00. Payment must accompany order.
case I, and if J = 0, it is the saru as case 3. If d =
x, the TABLE \\5 Coedic;ent of Skew in RS (1) and LS (Z) for Log Pearson Distnbution ul) a=
a=
a=
a=
a=
.=
a=
a=
a=
a=
b 0 01 0.05 0 09 0.10 0.15 0.18 0.22 0 :4 0.28 0 32 uz) 0 05 9 09 9.75 10.59 10 83 12.38 13 69 16.25 18.15 24 86 50 10 8 94 0 10 6 43 6.92
' 54 7 72 8 87 9 85 11.'6 13 14 18 25 37.90 6 ;2 0 25 4 08 4 43 4 84 5 01 5.84 6.55 7.96 90:
1:29 29 29 4 00
- 0. *5 2.37 2.66 3 02 3 13 3 il 4 41 5.64 66l 10 46 32.'6 2.31 1.00
- 2. %
2 34
- 2. 'O 2.81 3 50 4 10 5 37 639 10 61 39 M) 2 00 2 no i 44 1.7 2.17 2.28 3 n4 3.75 5.32 6 69 I3.56 i12 34 1 41 2.50 1 33 1 65 2.05
- 2. I' 299 3.
5.53
'.15 1606 209 13 1.26 3 00 1 23 1 55 I 94 2.11 2.97 3.81 5 SI
' '3 19 32 a06 63 1 15 3 50 1 !4 1 44 1.93 2.07 2.09 3#
6.16 4 41 23 54
>t l 7. 5 7 1.07 4 00 1 08 1.43 1 90 2.04 3 02 4 01 6 55 9 21 28 95 IW 5 n0 0 98 1.36 1 36 2.02 3.12 4 28 7.49
! ! 14 44 90 O 39 10 00 0 '4 1.24 1.93 2.16 3 05 6.28
!! 81 32.~9 562.20 4 63
!$ 00 0 64 1.24
- .42 5 09 9 44 36.18 111.49 0.52 20 00 0 '4 12x 2.33 2 71 6 54 14 36
" 33 al' 56 O 45 0 41 25 00 0.56 1 33 2.57 3.03 4 41 22.1%
26.70
- Greater than 994 99 p'
rq 4 m j.1 ti
916 L ODw E HR E T 4L.: Ftooo STartsitcs TABLE A6 Coct'icient of Kurtous in RS (1) and LS (2) fur Lcit Fearson Distnbution MX) a=
a=
a=
a=
a=
a=
a=
a=
a=
a=
b
- 2 00
-1 50
-1 00
-080
-050
- 0.30
- 0.15
-010
- 0.05
-001 A(2) 0 05 17.73 21.18 27.40 31.58 42 21 56.20 76 52 87 41 102.12 118 16 123 00 0 10 3 42 10.17 13 35 15 49
.0 94 28.19 38 74 44 40
$ 2.08 60.47 63 IX) 0 25 3N 3.74 5.05 5 95 8 29 11.43 16 09 18 of 22.06 25 85 27.00 0 75 1 66 1 66 1 90 2.15
- 2. 4 4.14 6 09 7 19 8 73 10 46 l 100 1(U 2.14 1 47 I 80 1.90 2.40 3.31 4 87 5.78 7.07 8.54 9.00 2 00 7.56 5 06 3 15 1 60 2.15 2.33 3.14 3.72 4 59 5.66 6 fC 2.50 13 32 R.19 4.52 3 46 2.42 2.23 2 85 1 33 4 11 5 08 5.40 3 00 22 76 12 87 6 38 4 62 2.82 2.33 2.68 3 09 3.79
- 4. *0 5 00 3 50 34 22 19 83 8 48 6.10 3.34 2.46 2.59 2.93 3.56 4 42 4 71 4 00 63.54 30 15 12.17 7.96 M6 2.64 2.55 2.83 3.40 4 22 4 50
$ 00 174 (*
64 36 22 25 13 23 5 53 3.10 2.54 2.71 3.18 3 93 4 20 10 00 386 71 134 66 24.42 7.30 3.23
- 2. '9 2 83 3.36 3 60 15 00 94 40 15.72 4 41 3 21 2.82 3 18 3 40 20 00 367.78 3110 5 98 3.77 1 90 3 09 3 30 25 00 64 :4 7.98 4 44 3 01 3N 3 24
- Grc irer than 999 99
=x ~ F > 0 and x = m,f F = 0.
for all d and f(x)is a monotonically decreasing function with Jegenerate Jistribution, t
- x. the WA detinitien is almost the f(0) = l 'cd.
resuits. Howeser, if h
=
same as that in case 1. ewept that a is meluded in the derin: tion Consider case 3 with m = 0 and b > 0. The range of x is of the lower cound. As a - O. Cd1] for case 2 approaches that from 0 (F = 0) to e (F = 1), and the probability density of sase 1: f[1] and A[1] are alway the same for sese two function f a dF, dx. is 5 nee the C,[T }. r[1]. and A(X) for these three cases denote f(x ) =
(Bs>
bounds for W A statistical pioperties, they aie importzot for 80 d
tabular presentation ofinformation about W A (see Tables B4-w here Uta Consider case I with m = 0 and d > 0 hence c > 0. The
- ~*
Th)=1-(89) range of X is from 0 (F = 0) to x (F = 11. and tne probabiht) a density function ipdf). ' = JF d.i. is if b = 1. then X is uniformly distributed. If 0 < b < l. then I.
- i..
.f(x) is a monotonically decreasing function with its maumal 1: )=
(B6i s alue I. ab at the lower bound. x = 0. Howes er, if b > I. then
- ~ #~
f(x) is j shaped, approaching infimty at the upper bound of w here
- x. a.
[
Note that the definitiori of WA can be considered a combi-(.
Eb)= 1-[,
(87) nation of eases I and 3. Thus the salue of the parameter b T \\BLE A7 Coe:L ient of Kt.rtows in RS tX).ind LS 12) for Lug Peanon Distnhution M 1) a" a=
a=
a=
a=
a=
a=
a=
6 0 01 0 05 0 09 0 10 0.15 0.1 %
0 22 0.24 MZ) 0 05 121 :^
l54 67 194 44 207 e7 311. 9 4 R'O 905 09 1:1 00 0 10 85 '5
'9 62 100 54 107.57 162.35 230'M 4x4 29 6 3.ilu a 25 28 25 34 60 44 30 47 56
'3 'l 106 63 235.30 541 09 27 00 o '5 11.59 14 62 19 41 21 A6 34 95 53 53 139 94 415 98 l 1.00
!W 9 50 1113 16 b 17 43 30.50 4A37 136 57 467.33 9 00
- W 6 34 4 44 11 97 13 25 25 28 44 35 173 49 6 00 150
'o 1I 14 12 47 25 03 46 90 224 15 5 40 3N 4 34
' :5 10 ':
12 03 25 36 50 la 290 61
$ 00 3 50 5 04 6 92 l0 43 II N 26t%
$4 51 345 32 4 'l 4 ' W1 4 92 hM IG 6 l i.no 2' 44 59 '9 5:110 4 50 5 00 4 fl n '
10 13 ll 64
'9 64
'3 44 40 10 W 3 s9 4 94
! l On 13 45 54 23 252 17 3 t<)
15'l0 3 69 6 03 12 97 16 62 108.34 I 40 20 m 3 60 6 25
! ? 46
- 0.36 227 47 3.30 25 00 3 55 6 54 18 52 26 33 494.37
- 3. 4
- Grea:cr 9an M 9e qC l (/j O jjj r
Leow mA ET 41.. flood STansncs 917 TABLE Bl. Valid and Insahd Parameter Combinations for
- 4. Property A(X] exists for 0 < 4. < 0.25 and b > 0.
Wakeby Distnbution
- 5. If r and 0 < d are held constant and b is allowed to increase,in general. (1) there custs a minimum salue for S[YJ.
Surn of Parameter Valid Distnbution?
(2) there cust local mmima for A[X),(3) the b salue at which a
b c
d Yes Niay be No the minimum skew occurs need not be the b at w hich the local mmimum kurtoses occur,(4) as d increases for a gisen r the b X
salue at whish the minimum skew occurs decreases tthe same
[
[
^
^
is true if r decreases for a given d) (5) similarly, as J increases x
+
x for a gisen r, the b values at w hich the local minimum kurtoses occur decrease (the same is true if r decreases for a gtsen d).
+
+
+
~
+
X and 4 6) C [Y] decreases.
[
[
[
- 6. If r and b are fixed and d increases in salue,in general,(l)
^
as d approaches i, y[X) approaches m,(2) A[X] mcreases, and g
+
X (3) C,[X] increases.
+
+
X
- 7. If b and d are fixed and r increases (1) C [X] increases,
+
~
~
[
~
(2) y[X] increases, and (3) A[X] decre:ises to a mmimum and x
then increases.
X X
Ehtreme Pamts To find the extreme points of f. that is, the salues of r at
'Vahd if ab - cd > 0, i.e., vand pdf.
which the derivative of f sanishes, Jf dx is set equal to 0, so
+Vand if astensked footnote holds and a > c
- Vahd if astensked footnote holds W cither b' < df or c > a that den b; = dl.
jVahd if astemked footnote holds and either bi > d' or a > c
?,0 bib ~ \\ N I ~ fY ' cd( ~d - t X l - FY', 9
'dl when b
=
dx
[abt ! - F)'-' - cd( 1 - F)-i-']'
strongly mtluences the shape of the distribution. If 0 < b < l-(BIO) then the probability density function ofI distributed as W A is a monotonically decreasing function;if I < b, then f(x) has a Th.is will occur if the numerator is zero, and the numerator will mode at some salue other than the lower bound.
be zero if any one of the following three conditions holds:(I)
Tabedar Presentation of C,[X].
F(x) = 1. (:) Either ab = 0 or b = 1. and either cd = 0 or J =
y[X].and A[X]
- 1. ( 3) ab(b - 1)(1 - F)* -' = cd(d - I W I - F)- * -'; that is.
Tables B1-D29 of the microtiche supplem-1t present salues of C,[X] (with m = 0), y[X], and A(X] for 0 s d,0 s b 5 m-II - F) *,
- cd(d - I n (B1i) and 0 s r s x (c > 0). As was noted in the presious section, g _ ;)
special cases 1-3 form the bounds on the salues in these tables.
The following specific observations are also made.
Since the roots of this equation should be real and smce F(x ) s
- l. Properts g X] custs for o G J < 1 and b > 0.
- 1. if an extreme point is to occur, then
- 2. Properth a([X] custs for 0 5 d < 0.5 and b > 0.
- 3. Property y[?] exists for 0 s d < i. (1) For 0 s b s 10, cdid - I )
y[X] > 0. (2) For 1.0 < b 5 m,3[X] 3 0 and as r decreases.
- y abf b - 1 ) > 0 tat
- )
the negative skew region mcreases.
TABLE B2. Statist 2 cal Procert:es of 1 D"tnbuted as WA Denmtion Prorerty u(Al m-
^
I b l
w 1
~ u - a m - :s > - o-ni-ao-b-an
~
'Pl di u t - wo - :3 >
Enm - 3Emnim :cm NU vp]
Efi 9 - 4EIL1EfI'1 - APmE{l9 - 3E'!Il en]
r 1 n - m
. /.
t Q p+
..a_%,,_a Em O
G O / f(t t"7/
-O
918 LANDw f MR ET 4L.. f t.ooD ST4ristics TABLE B3.
C,[.t]. y[t], and Ap] for 1 Dutributed as WA Special Cases Case 1.
Case 2.
Case 3.
a=0 b==
c=0 i
m - dl - (1 - F)-*]
m + a - cli - (1 - F)-']
m + a[I - (1 - F)*]
C,p ]
fi ff \\
JMt - 1)
I
\\
1 (m = 0)
\\ l - 2d /
\\ l - 2d / (a/c) + d. ( t - d) k I + 2b /
(1+2b)f-il+b)8 (1
- den ( I - d)* - (1 - 24) " J b8
( (1 +3b)
(1 + 2bl - 'J
( 1 - d)"
( 1 + b )'
d' (1 - 34) q
_ 4((1 - 34) +6't-2J) \\
I I ^ 2h '*! ' ' ' h }'
(1-2'
( 1 - d)*
I - 41'
' - #I*
1 + b )*
^#
(1 - 4d>
t J
6' l(I+4b) - 4((1 - 3b) + 6('1 + 2b)
-3
-3 If condition (Bl:) is sati lied, then s ab[I - F(xi)]*" + cd[I - F( ri)]- "d 5 ab(1 - F(xi)]'" + cd[I - F(xi)]- '"
(Bl7)
.cd(d + 1) i***
(Bl3)
F* s.ab(b - 1)-
but The first condition must be treated separately, since it results
!IX I
- I"#I I - F)' ' ' ^ C#I I - I)" ' 'I ' '
in an indelerminate form for df dx. Note that F = 1 when x attains its upper bound. If the second condition occurs, either so that f(m) 2 f(x,) 2 f(x ) w hen 2 5 x i < r,.
/(x) is improperly detired. or Y is ur,iformly distnbuted and
'has no extreme point. Therefore if condition (812)is satistied, I"#"No" N"
thece is generally only one finite s alue of x at w hich an extreme If f(x ) has indection points, the second deris ative of f( r) = (
point can occur, and that point is is zero at these points. The second dernative is
- _ c d(I + d) d'ff (dx )* = [ab(1 - F)* " + cd(1 - F)-d -']-'
a bib - 1)
- [(ab )'(b - 1 )( b - 1 )( 1 - F)88 "
c A l + J)
-d *"
(B14) - abcd(b' + d' + 6bd + 3b - 3d - 2)( I - F) -d "
-c I-a cto - 1 )
~
^
~
Note that if a, c, and d > 0. then it is necessary (although not sufficient) that b > I if(B12)is to be satis 6ed. If 0 < b s I and For F to be defined properly, only / values between 0 and I are a, c. and d > 0. it can be shown that f(x ) will be a monotonic-considered. Furthermore, unless the sign of the second deriva-ally decreasing function such that the maumum salue of f(x) tise changes at /, the value will not denote an indection point.
occurs at x = m, where f(m) = 1,(ab + cd). Let m s xi < xs The slope of f w hen F = 0, that is, x = m, is then 0 = F(m) s F(xil s f(xi) s 1. Since d > 0, then 1 - d >
1 and I
- ' ~
~
P' (m ) =
(B19)
=
dx,.
(ab + cd)
[' - F(m )]- " d' s [I - F(xi)]- " s [1 - F(x )l- '"
If a, b, c and d > 0. so that ab + cd > 0. the following (BI5) conjecture is proposed. (I) If ab(b - 1) < cd(d + 1), then f(m )
Since 0 < b < 1, then -I < b - I < 0. and
< 0.and no indection points exist. (2)lf ab(b - 1) = cd(d + 1),
then f(m) = 0, and the extreme point occurs at x* = m. (3)If
[1 - F(m )]* -' s [1 - F(xill* s [I - F(xi)]'-'
(B16) ab(b - 1) 2 cd(d + 1), then f(m) 2 0, and there exists at least so that since a. b, c, d > 0, one indection point. (4) If b < I, then f(m) < 0, f is *J shaped',
and there is no extreme point x*. although the peak occurs at ab(1 - F(m )]'" + cd[I - F(m)] ~ "d x = m.
TABLE B4 Bounds on C.[i j \\ aives for Fned J. c O. and Y Dntnbuted as W A q
T \\BLL Bf Bounds on 3[t l Values for E ned J. c 0.and i 3
3 Datnoured as W A
\\
i i
i
~
\\ -
'3 3
sr j
,N 6.
~
t
~.
f s x-l_.
,m t
men
~ 1.
w
. 4 a
- l
- 0th *01 t* t l $ t
\\r
>/r pg7
,, o I
r
\\ ;
L4 ow ensi Er at.: Floop Snnsno 919 TABLE B6 Bounds on A[1} Values for Fixed J. c > 0. and Y where u is distnbuted as uniform (U) on [0,1]. For the special Distnbuted as W A case w here m = 0 and a > 0, o<
In x = In a + (1/b)In [-In (1 - u)]
(Cb) i Specia/ CWF sE l s f ow t,)
o If a random vanable Z is distnbuted as Gumbel(G) with,1 0'
+
(
I probability density function la ccaw s) i(x) t uaw i.n l
l
- - m' I
f(:) = a-esp a,
I' * *-
..tc. c 1,:)
9
.t
[
u u - net emt (C7) exp)-exp a,
then random numbers distnbuted accordingly can be gener-Wakeby Random.Vumbers ated using the algorithm To generate random variables distributed as WA, let
- = m' + al-in [-In u'])
(CS) 1 - F(r) = u, where u is distnbuted uniformly on (0,1). Then x = m + a( t - u" ) - c( 1 - u " )
(B20) where u a U(0,1). It is noted that if(l)u' = I - u,(2)m' = in a, and (3) a' = -l, b, then 2, distributed as G, equala in X, for APPE NDix C; COMMENTS ON THE WE!Bt LL filSTRIBLTION X distnbuted as W with m = 0 and a > 0.
The distribution G has a skew equal to :1.14, depending Derimnon "I "
E" E##"*#"#
- P"'#*# ## #
Let X be a random sanable distnbuted as Weibull(W ), so EI E
~ ~
that tne probabilit) density function of X is Furthermore, since the sign of y(X) for X distnbuted as W is 9X I
- 7 [\\ r - m) * '
- [ x - m) -
equal to that of a > 0, then 3(X) > 0. Therefore if a random e
b ICl) sanable X is distributed as W with a lower beund of 0 and is
/
esp -(
)_
a a
positiselv skewed, then Z = In X is distributed as G with a w here ai > 0 and b > 0.
,ge, gr '- 1.14 7 % X) > 0.
it is further noted that if the Wetbull parameter b = f X is
.t/omerns also distobuted as Pearson type 111 (Pill) wi h the Pill para-t By detinitien, the moment of order A, Ei3 *], is meter b = 1 and HX) = 2. (Refer to Appendix A for the form of the Pill distnbution.) Thus if a random variable X is E[X'} = /'= s *f(x ) dx (C2) distr buted as Pill with y(X) = 2 and lower bound 0, then 2 =
In X is distnbuted as G w:th a skew of -1.14.
so that for X distnbuted as W.
No rrTioN E[X'] = f
' m a"F(k + b - 1)
(C3)
RS real space.
LS log space.
X ranuom vanable in RS.
w here E[l') < n - k 2 0.
x vanate salue of X.
The mean, a[X], standard desiation, a(X], and coetticients of In loganthm to base e.
skew ness. S [X], kurtosis. A(X), and s anation, C,[X], are gnen in TaNe Cl.
Z random sanable in LS.
- vanate salue of 2.
Define M b) = C. "[X] w hen m = 0. It is noted that m. the lower t'ound on X, can be detined as C,
coctficient of sar:ation.
y coctficient of skewness.
m = sl,X] - aF(1 - 1. b) = u[X) - a[X]$b) (C4:
A coetlicient of kurtosis.
length of histoneal or simulated tiood sequence.
It is also noted that the sign of y[X] is the same as that of a.
n A(n) number of historical :lood sequences of length n.
Weibull Random Numbers Y equnalent to C., y, or L To generate Weibull-distnbuted random numbers. the fol-y moment estimate of Y densed from a sequence lowing algonthm can be used:
length n.
f mean salue of y.
x = m - a(-ln t 1 - u)P *
(C5) dh) standard desiation of y.
TABLE Cl. Stausucal Prope-ties of i Dntributed as W Propert',
Defininon ai A ]
m - aDI - I b) d t]
ain i - 1 hi - rll
' 6@ 8 yu ayni - a - 3rn - : bico - 1 hi - :r'n - 1,blie q vi Mt) a*!D I - a hi - ar(i - I b irfl - 3 h p - er'(1 + 1. hin t - b ) - 31 *t I + l.b @ 1 t l C (U]
a [D i - M - I'f l - I b @ 9m - a M I - I b t h
(t
)
G
_o
920 L4 sow emt Er AL.: FLOOD Statistirs g( ) proportica of historical flood sequences yielding X'
tiansform cf the randorp variable X.
estimates of y of hke or opposite signs in RS and Z'
transform of the random sariable 2.
LS.
Ell *]
Ath-order moment of Y.
momen estimate of y derned from simulated V random sariable ( Appenda B).
w tlood sequence of length n.
4tw) mean of s.
Ju) standard desiation of w.
R t F E st t sc E S
+ set of datribution functions.
Herman. \\1. B, Generating gamma Jntr buted sariates for computer a element of t.
umulation models. Ry R-MI./ R. 4.) pp. Rand Corp, Santa U uniform distribution
""" C dl'f.. E eb. 1971.
N normal distribution.
Houghton. J C., Robust estimation of the frequency of estreme esents in'a ilood frequency contest. Ph D. dissertation. Harvard Unn,
LN
'ag normal distribution.
Cambridge. \\t ass. 1977.
G tiumbel extreme salue type i distribution.
Johnk. M. D., Eszeugung son tetaserreilten und gamma-serte:lten Pill Pearson tspe 111 dmribution.
Zufallstahlem, Merrr4 L.4 l ), 5-15,194.
W W es bull distribution atalas. A C., J R. Slack, and J. R. W albs, Regional skew in search of a parent. li's.ter Rewur Res. IIIh p. s l 5-A:6.1975 P Pareto distribution.
U S. W ater Resources Council. Guidehnes for determining flood flow LP leg Pearson type 111 distribution.
frequency. Bull /7 H)drol. Comm. W ashington D. C., March W& Wak(bs distribution.
19'6.
a.
sariat value of C.
W'alks, J. R., N C M atatas, and 1 R. Sla.k. \\ pparent regional new.
Il atrr Resour Res.. l.h I ), 159 182, 1977 u,a,/,, d, m parameters of distribution functions.
= f( x > probability density function.
r F = f(r) cumulatne distribution function.
(Retened J ane 15.197' l'{ } gamma funClion, resised Februar) 27.197' 4s h ) reciproc.d et C,. for W with lower bound m = 0.
accepted March 22,19's >
s.. -
,,, -/
- s. / :)
s,
,