OFFICE COPY
BULLETIN NO. 44
STATE OF ILLINOIS
WILLIAM G. STRATTON, Governor
DEPARTMENT OF REGISTRATION AND EDUCATION
VERA M. BINKS, Director
RAINFALL
RELATIONS
ON
SMALL AREAS IN I L L I N O I S
BY
F. A. HUFF AND J. C. NEHL
STATE WATER SURVEY DIVISION
WILLIAM C. ACKERMANN, Chief
URBANA
1957
Printed by authority of the State of Illinois
TABLE OF CONTENTS
Page
INTRODUCTION
9
ACKNOWLEDGMENTS
9
RAIN-GAGING FACILITIES........................................................................................................................11
El P a s o Network
Panther Creek Network
Goose Creek Network
E a s t Central Illinois Network
Boneyard Creek Network
Monticello Network
Airport Network
R A I N F A L L VARIABILITY
Introduction
Rainfall Relative Variability
R e s u l t s of A n a l y s i s
Effect of G a g e D e n s i t y on M a x i m u m R e c o r d e d P o i n t R a i n f a l l
F R E Q U E N C Y OF POINT A N D A R E A L M E A N RAINFALL RATES
Introduction
D a t a Used in
F r e q u e n c y of
F r e q u e n c y of
D i s c u s s i o n of
Analysis
Point Rainfall Rates
A r e a l M e a n Rainfall R a t e s
Results
A R E A - D E P T H RELATIONS
Introduction
A n a l y s i s and R e s u l t s
VARIATION OF P O I N T R A I N F A L L WITH DISTANCE
Introduction
A n a l y s i s of A v e r a g e D i f f e r e n c e s
Confidence L i m i t s
Equation Comparisons
C o n s i d e r a t i o n of A d d i t i o n a l V a r i a b l e s
A R E A L R E P R E S E N T A T I V E N E S S OF P O I N T R A I N F A L L
Introduction
A v a i l a b l e Data
A c c u r a c y of A r e a l R a i n f a l l E s t i m a t e s F r o m a C e n t e r e d Gage
Average E r r o r
Confidence L i m i t s
A r e a l Rainfall E s t i m a t e s F r o m an Off-Center Gage
11
11
12
12
12
13
13
15
15
15
15
20
21
21
21
22
23
23
25
25
25
31
31
31
31
31
32
35
35
35
35
35
38
39
TABLE OF CONTENTS (continued)
Page
RELIABILITY OF STORM MEAN RAINFALL ESTIMATES
43
Introduction
Data Used
Sampling P r o c e d u r e s
Random Start
Combined Sampling Plan
Best-Centered
Analytical P r o c e d u r e
Results of Analysis on 100-Square Mile Network
Random Start
Combined Sampling Plan
Best-Centered
Comparison of Results from Three Sampling Plans
Results of Analysis for Areas of Different Size
Average E r r o r
Confidence Limits
EXCESSIVE RAINFALL RELATIONS ON 100-SQUARE MILE AREA
Gage Density to Observational Frequency Relations
Effect of Gage Density on Observed Maximum
P o i n t - A r e a l Mean Relations
RELATION BETWEEN POINT AND AREAL RAINFALL FREQUENCIES
43
43
43
43
43
45
46
46
46
46
47
47
47
47
50
51
51
51
52
55
A MICROMETEOROLOGICAL STUDY OF RAINFALL VARIABILITY
57
Introduction
Description of Network
Analysis of Data
57
57
57
TABLE OF CONTENTS (continued)
LIST OF ILLUSTRATIONS
Figure
Title
Page
1
Reference map
2
El Paso network, 280 square miles.........................................................................................11
3
Panther Creek network, 100 square miles
11
4
Goose Creek network, 100 square miles
12
5
East central Illinois network, 400 square miles
12
6
Boneyard Creek network, 8.5 square miles
12
7
Monticello network, 400 acres
13
8
Airport network, 19 acres
13
9
Storm rainfall variability
15
10
Examples of rainfall variability
17
11
One-minute mean rates
18
12
Time variations in rainfall pattern
19
13
Effect of gage density on maximum recorded point rainfall
20
14
Distribution of point rainfall rates in individual storms
21
15 Distribution of point rainfall rates during average season
10
22
16
Time distribution of point rainfall
22
17
Point rainfall time-rate relation
23
18
Distribution of areal mean rainfall rates
23
19
Effect of gage density on isohyetal pattern
26
20
Area-depth relations for heaviest storms, 5.5-square mile area
27
21
Area-depth relations for heaviest storms, 100-square mile area
27
22
Area-depth relations for heaviest storms, 280-square mile area
27
23
Average area-depth relations, 5.5-square mile area, mean rainfall = 1.00 inch.
28
24
Average area-depth relations, 100-square mile area, mean rainfall = 1.00 inch .
28
25
Average area-depth relations, 5.5-square mile area, 6-hour storm duration. . . .
28
26
Average area-depth relations, 100-square mile area, 6-hour storm duration. . . .
28
27
Extreme area-depth relations, 5.5-square mile area, mean rainfall = 1.00 inch .
29
28
Extreme area-depth relations, 100-square mile area, mean rainfall = 1.00 inch. .
29
29
Extreme area-depth curve slopes, 5.5-square mile area, selected areal
mean rainfalls
29
TABLE OF CONTENTS (continued)
LIST OF ILLUSTRATIONS (continued)
Figure
30
Title
Page
Extreme area-depth curve slopes, 100-square mile area, selected a r e a l
mean rainfalls
29
31
Variation of point rainfall with distance
32
32
95 per cent confidence range for variation of point rainfall with distance . . . .
32
33
25, 50 and 100-square mile a r e a s , Goose Creek 1954
35
34
Storm relation between point and a r e a l mean rainfall based on average
deviations
36
Weekly relation between point and a r e a l mean rainfall based on average
deviations
36
Monthly relation between point and a r e a l mean rainfall based on individual
deviations
37
37
Storm relation between point and a r e a l mean rainfall based on standard e r r o r s .
38
38
Weekly relation between point and a r e a l mean rainfall based on standard e r r o r s . .
39
39
95 per cent confidence range for s t o r m rainfall
40
40
95 per cent confidence range for weekly rainfall
41
41
Effect of gage location on single gage estimates of mean rainfall on
100-square mile a r e a
42
Groups of gages from which 3 random s t a r t systematic samples of size
16 were selected
43
43
Combined sampling plan for samples of Size 1, 2, 3, 4, 6 and 8
44
44
B e s t - c e n t e r e d sampling plan for Panther Creek network
45
45
Comparison of s t o r m sampling plans on a 100-square mile a r e a
47
46
Relation between sampling e r r o r , s t o r m size, and gage density
on 100-square mile a r e a
48
95 per cent confidence range of s t o r m mean rainfall for various gage
densities
49
48
Effect of gage density on maximum observed excessive r a t e s
51
49
Relation between point and a r e a l mean r a t e s under excessive rate conditions. .
52
50
30-minute excessive r a t e s , July 8, 1951
53
51
Excessive rate area-depth relations, July 8, 1951
53
52
Comparison between point and areal mean rainfall at equal frequencies on
25-square mile a r e a
55
Effect of several meteorological factors on relative variability of
s t o r m rainfall
60
35
36
42
47
53
TABLE OF CONTENTS (continued)
TABLES
Table
Title
Page
1
Storm rainfall variability, 100-square mile a r e a
16
2
S t o r m rainfall variability, 280-square mile area
16
3
Monthly rainfall variability, 100-square mile a r e a
16
4
Frequency of point rainfall rates calculated from different base periods . . . .
22
5
Frequency of a r e a l mean rainfall r a t e s
24
6
Time distribution of a r e a l mean rainfall r a t e s
24
7
Correlation coefficients
30
8
Comparisons between Equations (3) and (4)
31
9
Multiple correlation coefficients
33
Regression constants for storm, weekly and monthly equations on various
areas
35
11
Multiple correlation coefficients
37
12
Comparisons between Equations (4) and (8)
38
13
Comparison between storm, weekly and monthly relations
38
14
Storm groups used in standard e r r o r computations
39
15
Regression constants
47
16
Equations for average and standard deviations
48
17
Relation between excessive rainfall frequency and gage density
51
18
Effect of gage density on observed maximum amounts
51
19
Differences between gage p a i r s
57
20
Relative variability trends for 6-foot gage p a i r s
58
21
Variation of point rainfall with distance
58
22
Monthly and seasonal differences between gage pairs
59
23
Relation between a r e a l mean rainfall and central gage point rainfall
10
. . . . . .
59
9
INTRODUCTION
During the p e r i o d 1 9 4 8 - 5 5 , the S t a t e W a t e r
S u r v e y M e t e o r o l o g y Subdivision o p e r a t e d s e v e r a l
c o n c e n t r a t e d r a i n gage n e t w o r k s o n s m a l l a r e a s
t o c o l l e c t d e t a i l e d p r e c i p i t a t i o n d a t a which can
f u r t h e r our knowledge of the w a t e r r e s o u r c e s
of I l l i n o i s . T h e c o l l e c t e d d a t a have b e e n used in a
n u m b e r of s t u d i e s to obtain i n f o r m a t i o n on p r e c i p i t a t i o n p e r t i n e n t t o h y d r o l o g i c a n a l y s i s , d e s i g n , and
planning. This bulletin p r e s e n t s c u r r e n t r e s u lts of
s e v e r a l of t h e s e s t u d i e s . S o m e of the s t u d i e s a r e
of a continuing n a t u r e and as a d d i t i o n a l d a t a b e c o m e
a v a i l a b l e , r e f i n e m e n t s will b e p u b l i s h e d . H o w e v e r ,
it is felt that a d e q u a t e a n a l y s i s h a s now b e e n c o m pleted to p r e s e n t r e s u l t s that may be beneficial to
h y d r o l o g i s t s and e n g i n e e r s a c t i v e l y e n g a g e d in the
field of w a t e r r e s o u r c e s .
Although t h e s e s t u d i e s a r e b a s e d o n Illinois
d a t a , the r e s u l t s should b e a p p r o x i m a t e l y r e p r e s e n t a t i v e of c o n d i t i o n s in the M i d w e s t and of o t h e r
a r e a s having s i m i l a r c l i m a t e and t o p o g r a p h y . T h e
s t u d i e s to d a t e h a v e b e e n c o n c e n t r a t e d on the a n a l y s i s of r a i n f a l l d u r i n g the s p r i n g to fall t h u n d e r s t o r m season on relatively small a r e a s ranging
f r o m l e s s t h a n one s q u a r e m i l e t o 400 s q u a r e m i l e s .
The extreme variability in precipitation experie n c e d d u r i n g the t h u n d e r s t o r m s e a s o n c r e a t e s difficult p r o b l e m s for the h y d r o l o g i s t c o n c e r n e d with
s m a l l w a t e r s h e d s , which a r e s u b j e c t t o flash floods
and r a p i d l y affected by d r o u g h t c o n d i t i o n s . The
m a j o r p o r t i o n of the s u r f a c e - w a t e r s u p p l i e s for
m u n i c i p a l i t i e s i n the s t a t e a r e o b t a i n e d f r o m l a k e s
having w a t e r s h e d a r e a s u n d e r 400 s q u a r e m i l e s .
Included in this b u l l e t i n a r e r e s u l t s of s t u d i e s
on: the r e l a t i v e v a r i a b i l i t y of s t o r m and m o n t h l y
r a i n f a l l o v e r s m a l l a r e a s : the d i s t r i b u t i o n o f point
and a r e a l m e a n r a i n f a l l r a t e s i n s h o w e r - t y p e p r e cipitation; a r e a - d e p t h relations on s m a l l waters h e d s ; the v a r i a t i o n of point r a i n f a l l with d i s t a n c e ;
t h e a r e a l r e p r e s e n t a t i v e n e s s of point r a i n f a l l in
m e a s u r i n g a r e a l m e a n r a i n f a l l o n a s t o r m , weekly,
and m o n t h l y b a s i s ; the c o m b i n e d effect of s t o r m
s i z e , a r e a , and n u m b e r o f r a i n g a g e s o n the a c c u racy of s t o r m mean rainfall e s t i m a t e s ; relations
d u r i n g p e r i o d s of e x c e s s i v e r a i n f a l l o v e r a 100-
s q u a r e m i l e b a s i n ; the r e l a t i o n b e t w e e n point and
a r e a l m e a n r a i n f a l l f r e q u e n c i e s ; and m i c r o - m e t e orological variations in storm rainfall.
ACKNOW L E D G M E N T S
T h i s bulletin w a s p r e p a r e d u n d e r the d i r e c t i o n
of W i l l i a m C. A c k e r m a n n , Chief of the Illinois
State Water Survey. R e s e a r c h was accomplished
u n d e r the g e n e r a l guidance of G l e n n E. Stout,
Head, Meteorology Subdivision.
The authors a r e indebted to A. M. Buswell,
f o r m e r Chief, and H . E . H u d s o n , J r . , f o r m e r Head
of the E n g i n e e r i n g S u b d i v i s i o n , for s u g g e s t i o n s and
g u i d a n c e in the e a r l y p h a s e s of the r e s e a r c h inc o r p o r a t e d in the b u l l e t i n .
Within the M e t e o r o l o g y S u b d i v i s i o n , s p e c i a l
c r e d i t i s due D o u g l a s M . A . J o n e s , who s u p e r v i s e d
a m a j o r p o r t i o n of the data c o l l e c t i o n and C a r o l
V e r s e m a n , who p e r f o r m e d or s u p e r v i s e d a m a j o r
p o r t i o n of the r o u t i n e a n a l y s i s . D r a f t i n g of the i l l u s t r a t i o n s and c o v e r w a s done by J o h n Wesselhoff.
N u m e r o u s r e s e a r c h a s s i s t a n t s h e l p e d with the ins t a l l a t i o n and s e r v i c i n g of r a i n g a g e s , c o l l e c t i o n
and t a b u l a t i o n of d a t a , and c o m p u t a t i o n .
C r e d i t is due L e s t e r P f i s t e r , P r e s i d e n t of the
P f i s t e r Hybrid C o r n C o m p a n y , the U. S. Soil Cons e r v a t i o n S e r v i c e , and the Civil E n g i n e e r i n g Dep a r t m e n t of the U n i v e r s i t y of I l l i n o i s , who p r o v i d e d
s o m e of the r a i n g a g e s for the s t u d i e s O p e r a t i o n
of the G o o s e C r e e k n e t w o r k and d a t a c o l l e c t i o n
w e r e s p o n s o r e d i n p a r t b y the Signal C o r p s E n g i n e e r i n g L a b o r a t o r i e s , B e l m a r , New J e r s e y u n d e r
Contract DA-36-039 SC-42446. The Agricultural
E n g i n e e r i n g D e p a r t m e n t of the U n i v e r s i t y of Illinois furnished data from its 4 0 0 - a c r e r a i n - g a g e
network.
C e r t a i n p h a s e s o f the a n a l y s e s w e r e g r e a t l y exp e d i t e d t h r o u g h u s e of f a c i l i t i e s at the S t a t i s t i c a l
S e r v i c e Unit and the Digital C o m p u t e r L a b o r a t o r y
at the U n i v e r s i t y of I l l i n o i s .
10
FIGURE 1 REFERENCE MAP
11
RAIN-GAGING F A C I L I T I E S
Rainfall data from seven r a i n - g a g e networks
l o c a t e d i n c e n t r a l Illinois w e r e a v a i l a b l e for s t u d i e s
d e s c r i b e d i n t h i s b u l l e t i n . G e n e r a l l y , the n e t w o r k s
w e r e i n full o p e r a t i o n d u r i n g the w a r m s e a s o n
f r o m A p r i l t h r o u g h O c t o b e r , after which they w e r e
c l o s e d or r e d u c e d to r e l a t i v e l y few g a g e s for the
w i n t e r s e a s o n . The n e t w o r k l o c a t i o n s within the
s t a t e a r e shown on the r e f e r e n c e m a p in F i g u r e 1.
T h e 1948 E l P a s o n e t w o r k which i s not i l l u s t r a t e d
w a s a s o u t h w a r d and w e s t w a r d e x t e n s i o n of the
P a n t h e r C r e e k n e t w o r k . T h e Goose C r e e k n e t w o r k
was l o c a t e d in the s o u t h e a s t e r n p o r t i o n of the p r e s ent e a s t c e n t r a l Illinois n e t w o r k . T h e C r a b O r c h a r d n e t w o r k i n s o u t h e r n Illinois w a s not i n s t a l l e d
until the fall of 19 54. D a t a from it h a v e not b e e n
a n a l y z e d a d e q u a t e l y to be p r e s e n t e d in t h i s b u l l e t i n .
d i f f e r e n c e s e x i s t e d , thus p e r m i t t i n g u t i l i z a t i o n of
c o m b i n e d n e t w o r k d a t a for the d e r i v a t i o n of v a r i o u s
e m p i r i c a l rainfall r e l a t i o n s .
El P a s o Network
The first of these rain-gage networks located
on an a r e a of 280 s q u a r e m i l e s in the v i c i n i t y of
El P a s o , Illinois ( F i g . 2) was e s t a b l i s h e d in the
s p r i n g of 1948. D u r i n g 1948 and 1949 a t o t a l of 17
r e c o r d i n g and 3 1 n o n - r e c o r d i n g g a g e s , o r a n a v e r age of one gage p e r 5.8 s q u a r e m i l e s , w a s o p e r a t e d
on this a r e a .
T h e t e r r a i n within t h e c e n t r a l I l l i n o i s e x p e r i m e n t a l a r e a s i s r e l a t i v e l y flat with n o d i s t i n c t
t o p o g r a p h i c f e a t u r e s t o influence s t o r m b e h a v i o r .
All g a g e s w e r e l o c a t e d with open e x p o s u r e s . The
r e c o r d i n g g a g e s , f r o m which the m a j o r i t y of the
analytical data were obtained, were serviced by
t r a i n e d t e c h n i c i a n s . The n o n - r e c o r d i n g g a g e s w e r e
i n s t a l l e d b y W a t e r S u r v e y t e c h n i c i a n s , and the
m e a s u r e m e n t s w e r e t a k e n b y n e a r b y r e s i d e n t s who
had b e e n c a r e f u l l y i n s t r u c t e d i n the p r o p e r o p e r a tion of the g a g e s . In m o s t c a s e s , t h e s e c o o p e r a t i v e
o b s e r v e r s w e r e f a r m e r s having a l i v e l y i n t e r e s t
in the m e a s u r e m e n t of r a i n f a l l .
Statistical tests were performed to determine
the h o m o g e n e i t y of d a t a obtained f r o m the v a r i o u s
n e t w o r k s . T h e s e t e s t s i n d i c a t e d that n o significant
FIGURE
Panther
FIGURE 2 EL PASO NETWORK, 230 SQUARE MILES
3 PANTHER CREEK NETWORK, 100 SQUARE MILES
Creek Network
T h e n e t w o r k i n the E l P a s o a r e a w a s r e d u c e d
in s i z e at the c l o s e of the 1949 t h u n d e r s t o r m s e a son. B e g i n n i n g in the s p r i n g of 1950, a n e t w o r k of
100 s q u a r e m i l e s within the E l P a s o n e t w o r k was
c o n t i n u e d o n the P a n t h e r C r e e k w a t e r s h e d . T h i s
a r e a had b e e n gaged with 6 r e c o r d i n g and 14 nonr e c o r d i n g g a g e s d u r i n g the 1948 and 1949 s e a s o n s ,
c o r r e s p o n d i n g to an a v e r a g e of one gage p e r 4.75
s q u a r e m i l e s . E m p l o y i n g both r e c o r d i n g and nonr e c o r d i n g g a g e s , the P a n t h e r C r e e k n e t w o r k w a s
e x p a n d e d to 40 g a g e s in 1950. D u r i n g 1 9 5 1 - 5 3 , the
n e t w o r k of 25 r e c o r d i n g g a g e s i l l u s t r a t e d in F i g u r e 3 w a s m a i n t a i n e d . In 1954,
the P a n t h e r C r e e k
n e t w o r k w a s r e d u c e d to 10 r e c o r d i n g g a g e s , s i n c e
the g a g e s w e r e n e e d e d e l s e w h e r e and 10 w a s cons i d e r e d a sufficient n u m b e r for the P a n t h e r C r e e k
study.
12
one r e v o l u t i o n e v e r y s i x h o u r s . T h e G o o s e C r e e k
n e t w o r k w a s e n l a r g e d d u r i n g J u l y 1952 t o i n c l u d e
100 s q u a r e m i l e s ( F i g . 4 ) . A t o t a l of 50 r e c o r d i n g
r a i n g a g e s , equipped w i t h the 1 2 . 6 5 - i n c h d i a m e t e r
c o l l e c t o r and the 6 - h o u r c h a r t d r i v e , w a s o p e r a t e d
o n t h i s a r e a d u r i n g t h e r e m a i n d e r o f t h e 1952
t h u n d e r s t o r m s e a s o n a n d d u r i n g the 1953 t h u n d e r s t o r m season. This network w a s r e o r g a n i z e d
s o m e w h a t i n 1954; h o w e v e r , the a r e a o f t h e n e t w o r k
w a s m a i n t a i n e d at 100 s q u a r e m i l e s . A t o t a l of 48
r e c o r d i n g gages was u s e d during the 19 54 s e a s o n .
E a s t C e n t r a l Illinois N e t w o r k
I n o r d e r t o obtain d a t a o v e r a n a r e a l a r g e r t h a n
100 s q u a r e m i l e s , the G o o s e c r e e k n e t w o r k w a s
r e o r g a n i z e d and e x p a n d e d n o r t h and w e s t d u r i n g
the. s p r i n g of 1955. T h e new n e t w o r k ( F i g . 5) i n c l u d e s an a r e a of 400 s q u a r e m i l e s . A t o t a l of 49
r e c o r d i n g g a g e s was u s e d o n this a r e a d u r i n g the
1955 t h u n d e r s t o r m s e a s o n .
FIGURE 4 GOOSE CREEK NETWORK, 100 SQUARE MILES
Goose
Creek Network
D u r i n g the s p r i n g of 19 5 1 , a n e t w o r k w a s i n s t a l l e d o v e r an a r e a of 50 s q u a r e m i l e s on the
Goose C r e e k w a t e r s h e d in the vicinity of Deland,
I l l i n o i s . T h i s n e t w o r k c o n s i s t e d of 33 r e c o r d i n g
r a i n g a g e s which w e r e e q u i p p e d with 1 2 . 6 5 - i n c h
d i a m e t e r c o l l e c t o r s and with c h a r t d r u m s m a k i n g
FIGURE 6 BONEYARD CREEK NETWORK, 8.5 SQUARE MILES
Boneyard Creek Network
FIGURE 5 EAST CENTRAL ILLINOIS NETWORK, 400 SQUARE MILES
A 1 2 - s t a t i o n n e t w o r k for s a m p l i n g an a r e a of
a p p r o x i m a t e l y 8.5 s q u a r e m i l e s o n t h e B o n e y a r d
Creek watershed at Champaign-Urbana, Illinois is
shown i n F i g u r e 6 . T h i s n e t w o r k h a s b e e n o p e r a t e d
s i n c e the fall of 1948 in c o o p e r a t i o n with the C i v i l
Engineering Department, University of Illinois.
T h e m a j o r i t y o f the r a i n f a l l a n a l y s i s h a s b e e n
confined t o d a t a f r o m a n a r e a o f 5.5 s q u a r e m i l e s
within the w a t e r s h e d ( c r o s s - h a t c h e d i n F i g . 6).
Streamflow m e a s u r e m e n t s are made on this a r e a ,
and the r a i n gage d e n s i t y i s g r e a t e r t h a n o v e r t h e
r e m a i n i n g p o r t i o n of the w a t e r s h e d .
13
s i n c e 1949. D a t a f r o m the n e t w o r k w e r e m a d e
a v a i l a b l e t o the a u t h o r s for a n a l y s i s and i n c l u s i o n
in t h i s b u l l e t i n .
FIGURE 8 AIRPORT NETWORK, 19 ACRES
FIGURE 7 MONTICELLO NETWORK, 400 ACRES
Airport
Monticello Network
T h e M o n t i c e l l o n e t w o r k ( F i g . 7) c o n s i s t i n g of 400
a c r e s (0.6 s q . m i . ) i s o p e r a t e d b y the A g r i c u l t u r a l
Engineering D e p a r t m e n t , University of Illinois.
F o r the S u r v e y s t u d i e s s i x r e c o r d i n g g a g e s w e r e
u s e d f r o m t h i s n e t w o r k , which h a s b e e n o p e r a t e d
Network
D u r i n g 1 9 5 3 - 5 5 , a m i c r o - n e t w o r k of 18 nonr e c o r d i n g g a g e s w a s o p e r a t e d a t the U n i v e r s i t y o f
I l l i n o i s A i r p o r t , six m i l e s south of C h a m p a i g n U r b a n a . T h i s n e t w o r k c o n s i s t e d of p a i r s of g a g e s
s p a c e d six feet a p a r t at e a c h of nine s t a t i o n s .
T h e s e s t a t i o n s w e r e l o c a t e d at i n t e r v a l s of 300
feet to f o r m a s q u a r e g r i d p a t t e r n on 19 a c r e s
(0.03 s q . mi.) a s shown i n F i g u r e 8 .
15
R A I N F A L L VARIABILITY
Introduction
T h e g r e a t v a r i a b i l i t y o f t h u n d e r s t o r m and r a i n s h o w e r p r e c i p i t a t i o n within r e l a t i v e l y s h o r t d i s tances is generally recognized by meteorologists
and h y d r o l o g i s t s . H o w e v e r , i n f o r m a t i o n c o n c e r n ing the d e g r e e o f this v a r i a b i l i t y i s l i m i t e d , e s p e c i a l l y for s m a l l w a t e r s h e d s . O b v i o u s l y , r a i n f a l l
v a r i a b i l i t y i s p e r t i n e n t t o h y d r o l o g i s t s and m e t e o r o l o g i s t s c o n c e r n e d w i t h r a i n gaging of w a t e r s h e d s and the i n t e r p r e t a t i o n of r a i n f a l l d a t a for
application to hydrologic analysis. F o r example,
the a r e a - d e p t h r e l a t i o n and, c o n s e q u e n t l y , the
r a i n f a l l - r u n o f f r e l a t i o n a r e c l o s e l y r e l a t e d t o the
v a r i a b i l i t y within a given w a t e r s h e d .
b a s i n . C o n s i d e r i n g the m u l t i - c e l l u l a r n a t u r e o f
t h u n d e r s t o r m s , i t a p p e a r s l o g i c a l that a n a r e a
c o m e s u n d e r the i n f l u e n c e o f m o r e c e l l s with
heavy rainfall and, consequently, a tendency exists
for g r e a t e r r e l a t i v e u n i f o r m i t y i n the r a i n f a l l
p a t t e r n . It was found on the T h u n d e r s t o r m P r o j e c t ( 1 )
that s i n g l e - c e l l e d t h u n d e r s t o r m s w e r e g e n e r a l l y
w e a k c o m p a r e d t o the m u l t i - c e l l u l a r s t o r m s . The
e x p e c t e d t e n d e n c y for the v a r i a b i l i t y t o i n c r e a s e
with a r e a is a p p a r e n t f r o m a c o m p a r i s o n of a v e r a g e
v a l u e s in T a b l e s 1 and 2. The m a x i m u m a n d m i n i m u m v a l u e s i n t h e s e t a b l e s a r e the e x t r e m e s o b s e r v e d for i n d i v i d u a l s t o r m s i n e a c h m e a n
rainfall c l a s s .
D a t a f r o m two c o n c e n t r a t e d n e t w o r k s , the 280s q u a r e m i l e E l P a s o n e t w o r k (Fig.2) and the 100s q u a r e m i l e P a n t h e r C r e e k n e t w o r k ( F i g . 3 ) have
b e e n u s e d for a l i m i t e d study on t h u n d e r s h o w e r
and r a i n s h o w e r v a r i a b i l i t y . Data for 1948-49 on
the El P a s o n e t w o r k and for 1948-53 on the P a n t h e r C r e e k n e t w o r k w e r e used i n the study.
Rainfall Relative Variability
The r e l a t i v e v a r i a b i l i t y for t h e s e a r e a s w a s
o b t a i n e d by c a l c u l a t i n g t h e coefficient of v a r i a t i o n
in each s t o r m on each network. T h e r e w e r e 48
g a g e s in the El P a s o n e t w o r k and 20 g a g e s on the
P a n t h e r C r e e k n e t w o r k u s e d for c a l c u l a t i o n s . F o r
c o n v e n i e n c e , the v a r i a b i l i t y f a c t o r (coefficient of
v a r i a t i o n ) w a s e x p r e s s e d i n p e r c e n t and w a s o b t a i n e d f r o m the following equation:
w h e r e CV is the coefficient of variation in p e r
c e n t , SD is the s t a n d a r d d e v i a t i o n of all the n e t w o r k
o b s e r v a t i o n s , and P i s the a r e a l m e a n r a i n f a l l .
The v a r i a b i l i t y f a c t o r is a m e a s u r e of the d i s p e r s i o n of point r a i n f a l l v a l u e s about the a r e a l
m e a n r a i n f a l l . In a d d i t i o n to the a c t u a l v a r i a t i o n s in
the s t o r m r a i n f a l l the v a r i a b i l i t y f a c t o r i n c l u d e s
the effects o f i n s t r u m e n t a l o r o b s e r v a t i o n a l e r r o r s
and gage n o n - r e s p r e s e n t a t i v e n e s s . H o w e v e r , n o
d i s t i n c t t o p o g r a p h i c f e a t u r e s e x i s t e d i n the e x p e r i m e n t a l a r e a , and all g a g e s w e r e l o c a t e d with open
e x p o s u r e s . T h e r e f o r e , t h e effect o f g a g e n o n - r e p r e s e n t a t i v e n e s s i s e l i m i n a t e d for all p r a c t i c a l p u r poses.
A c l o s e c h e c k was m a i n t a i n e d o n the
o p e r a t i o n of the r a i n g a g e s , so the v a r i a b i l i t y d a t a
s h o u l d b e r e a s o n a b l y a c c u r a t e and r e p r e s e n t a t i v e .
R e s u l t s of A n a l y s i s
The d a t a w e r e f i r s t a n a l y z e d t o d e t e r m i n e the
g e n e r a l effects o f s t o r m m e a n r a i n f a l l and a r e a
on relative variability. The r e s u l t s of this analys i s a r e s u m m a r i z e d in T a b l e s 1 and 2. Both t a b l e s
c l e a r l y i n d i c a t e that for a given b a s i n the r e l a t i v e
v a r i a b i l i t y t e n d s t o d e c r e a s e with i n c r e a s i n g s t o r m
m e a n r a i n f a l l . T h i s t e n d e n c y i s a c c o u n t e d for b y
the n a t u r e of s h o w e r - t y p e rainfall and the l o c a t i o n
of the h e a v i e r s t o r m c o r e s with r e s p e c t to the
FIGURE 9 STORM RAINFALL VARIABILITY
T a b l e 3 shows the r e l a t i v e v a r i a b i l i t y c a l c u l a t e d on a m o n t h l y b a s i s for the 1 0 0 - s q u a r e m i l e
d r a i n a g e b a s i n . C o m p a r i n g T a b l e s 1 and 3, it is
s e e n t h a t the r e l a t i v e v a r i a b i l i t y t e n d s t o d e c r e a s e
a s t h e r a i n f a l l i n c r e a s e s and the t o t a l i z i n g p e r i o d
is i n c r e a s e d . F i g u r e 9 is a s c a t t e r d i a g r a m for the
1 0 0 - s q u a r e m i l e b a s i n , i l l u s t r a t i n g the r e l a t i o n ship b e t w e e n s t o r m m e a n r a i n f a l l and r e l a t i v e
v a r i a b i l i t y b a s e d on 185 c a s e s . The t r e n d for the
r e l a t i v e v a r i a b i l i t y t o d e c r e a s e with i n c r e a s i n g
m e a n rainfall is apparent. However, a close c o r r e l a t i o n d o e s not e x i s t as shown by the a m o u n t of
s c a t t e r o n the d i a g r a m . O t h e r f a c t o r s , s u c h a s
s t o r m d u r a t i o n , m o v e m e n t o f s t o r m with r e s p e c t
to b a s i n , and the i n t e r n a l s t r u c t u r e of t h u n d e r s t o r m s and r a i n s h o w e r s undoubtedly c o n t r i b u t e d
to the observed s c a t t e r .
16
TABLE 1
STORM R A I N F A L L VARIABILITY
100 SQ. MI. A R E A
Mean
R a i n f a l l (in.)
Av.
0.01
0.21
0.51
1.01
Over
72
48
30
25
14
-
0.20
0.50
1.00
2.00
2.00
Coefficient of V a r i a t i o n (% )
Max.
200
133
100
37
20
Min.
Number
of C a s e s
14
9
7
6
9
75
44
42
18
6
Min.
Number
of C a s e s
23
16
22
14
17
14
15
5
TABLE 2
STORM R A I N F A L L VARIABILITY
280 SQ. MI. A R E A
Mean
R a i n f a l l (in.)
0.01
0.21
0.51
1.01
-
0.20
0.50
1.00
3.00
Av.
81
56
42
31
Coefficient of V a r i a t i o n (% )
Max.
127
78
64
47
TABLE 3
MONTHLY R A I N F A L L VARIABILITY
100 SQ. MI. A R E A
Mean
R a i n f a l l (in.)
0.01
2.01
4.01
6.01
-
2.00
4.00
6.00
8.00
Av.
Coefficient of V a r i a t i o n (% )
Max.
Min.
Number
of C a s e s
31
15
18
12
37
19
26
25
16
9
12
8
6
9
4
7
17
FIGURE 10 EXAMPLES OF RAINFALL VARIABILITY
The tendency for the relative variability to
d e c r e a s e with increasing mean rainfall and with
the length of observation period is illustrated in
Figure 10. This figure shows the isohyetal maps
for storm, weekly, monthly, and seasonal (June-
August) rainfall during the summer of 195i on the
Panther Creek network. In the illustration, is the
mean rainfall and CV is the coefficient of variation.
18
FIGURE 11 ONE-MINUTE MEAN RATES
Figure 11 shows some examples of relative
variability in thunderstorms when mean rainfall is
calculated on a very short time basis. The four
isohyetal maps a r e for one-minute periods m e a s ured on the Goose Creek network during 1953, using
special recording rain gages (see previous section
and Figure 4). Isohyetal values in Figure 11 r e p r e sent one-minute amounts expressed in rainfall rate
in inches per hour. P r e p r e s e n t s the mean rainfall
rate and CV the coefficient of variation. Since the
rainfall values on the maps represent nearly instantaneous values, the isohyetal patterns a r e approximately representative of the instantaneous
pattern of rainfall within thunderstorms. The great
variability existing in the internal s t r u c t u r e of
thunderstorms is apparent and helps explain the
great variability observed in total s t o r m rainfall
in thunderstorms and the scatter in the coefficient
of variation shown in Figure 9.
19
A. 1605-06 CST
C. 1613-14 CST
B. 1609-10 CST
D. 1617-18 CST
FIGURE 12 TIME VARIATIONS IN RAINFALL PATTERN
F i g u r e 12 is a s e r i e s of o n e - m i n u t e i s o h y e t a l
m a p s s p a c e d four m i n u t e s a p a r t i n a n i n t e n s e
t h u n d e r s t o r m and i l l u s t r a t e s the r a p i d c h a n g e s
which m a y take p l a c e in the i n t e r n a l s t r u c t u r e of
thunder-storms.
The e f f e c t s of s t o r m rainfall v a r i a b i l i t y will
b e d i s c u s s e d i n the s e c t i o n o n a r e a - d e p t h c u r v e s ,
w h e r e i t s a p p l i c a t i o n t o h y d r o l o g i c a n a l y s i s and
d e s i g n will be shown.
20
Effect of G a g e D e n s i t y on M a x i m u m R e c o r d e d
Point Rainfall
The p e a k r a i n f a l l o c c u r r i n g w i t h i n b a s i n s t o r m s
is a useful f a c t o r in h y d r o l o g i c a n a l y s i s . O b v i o u s l y ,
the m a x i m u m r e c o r d e d r a i n f a l l i n s h o w e r - t y p e
s t o r m s is c l o s e l y r e l a t e d to the g a g e d e n s i t y on a
given b a s i n .
r a t i o o f m a x i m u m r e c o r d e d t o c e n t r a l gage r a i n fall w a s found to i n c r e a s e f r o m 1.15 at a g a g e d e n s i t y of 50 s q u a r e m i l e s p e r g a g e to 1.38, 1.58 and
1.83 at g a g e d e n s i t i e s of 20, 10 and 5 s q u a r e m i l e s
per gage, respectively.
Using r a i n f a l l d a t a c o l l e c t e d f r o m 70 s t o r m s on
the P a n t h e r C r e e k B a s i n d u r i n g 1 9 4 8 - 5 0 , a s t u d y
was m a d e of the effect of gage d e n s i t y on the o b s e r v e d m a x i m u m point r a i n f a l l within a n a r e a . T o
study t h i s effect, point r a i n f a l l r e c o r d e d a t the m o s t
c e n t r a l l y - l o c a t e d s t a t i o n i n the 1 0 0 - s q u a r e m i l e
b a s i n w a s c o m p a r e d with the m a x i m u m point r a i n fall o b t a i n e d b y i n c r e a s i n g the n e t w o r k p r o g r e s s ively to 3, 5, 10, and 20 g a g e s . T h e s e n e t w o r k s ,
c o r r e s p o n d i n g to gage d e n s i t i e s of 3 3 , 20, 10, and 5
s q u a r e m i l e s p e r g a g e , w e r e c h o s e n t o give the
m o s t uniform distribution of gages possible.
F o r e a c h of the four gage d e n s i t i e s in e a c h
s t o r m , the r a t i o n o f n e t w o r k m a x i m u m t o s i n g l e
gage a m o u n t was d e t e r m i n e d f r o m the r a i n - g a g e
c h a r t s . T h e s e d a t a w e r e t h e n g r o u p e d , b a s e d upon
c l a s s i n t e r v a l s for s i n g l e g a g e r a i n f a l l of 0.01 0.20, 0.21 - 0.50, 0.51 - 1.00 and 1.01 - 2.00 i n c h e s . The g r o u p e d d a t a w e r e t h e n u s e d t o o b t a i n a n
empirical relation between network m a x i m u m r a i n f a l l , single g a g e r a i n f a l l , and g a g e d e n s i t y . T h e e m p i r i c a l equation obtained is
w h e r e R m i s the m a x i m u m r e c o r d e d point r a i n f a l l
( i n . ) , R s is the c e n t r a l g a g e a m o u n t (in.) and G is
the g a g e d e n s i t y ( s q . m i . / g a g e ) . T h e d e v e l o p e d r e l a t i o n i s i l l u s t r a t e d i n F i g u r e 13.
R e f e r e n c e to F i g u r e 13 s h o w s the e x p e c t e d
t r e n d for t h e m a x i m u m r e c o r d e d point r a i n f a l l t o
i n c r e a s e s i g n i f i c a n t l y a s the g a g e d e n s i t y d e c r e a s e s ( n u m b e r o f g a g e s i n c r e a s e s ) . The a v e r a g e
FIGURE 13 EFFECT OF GAGE DENSITY ON
MAXIMUM RECORDED POINT RAINFALL
21
F R E Q U E N C Y OF POINT AND A R E A L
MEAN R A I N F A L L RATES
Introduction
D u r i n g 19 5 2 - 5 3 , one m i n u t e r a i n f a l l t o t a l s for
a n u m b e r of s t o r m s w e r e t a b u l a t e d in conjunction
with a p r o j e c t c o n c e r n e d with t h e i n v e s t i g a t i o n of
the utility o f r a d a r for q u a n t i t a t i v e r a i n f a l l m e a s u r e m e n t s . ( 2 ) Use o f s p e c i a l r e c o r d i n g r a i n g a g e s
with 12.6 - i n c h d i a m e t e r o r i f i c e s and s i x - h o u r
c h a r t s enabled r e c o r d i n g of o n e - m i n u t e amounts
o v e r an a r e a of 100 s q u a r e m i l e s . I n f o r m a t i o n on
the d i s t r i b u t i o n o f rainfall r a t e s w i t h i n s t o r m s
should be of i n t e r e s t to h y d r o l o g i s t s , e s p e c i a l l y to
those concerned with sedimentation. A r e a l d i s t r i b u t i o n s o f o n e - m i n u t e r a t e s , which c l o s e l y a p p r o x i m a t e the a c t u a l p a t t e r n o f r a i n f a l l r a t e s within
s t o r m s , a r e p e r t i n e n t i n s t u d y i n g the p h y s i c a l
s t r u c t u r e of s t o r m s . Information on s h o r t - p e r i o d
r a t e s i s a l s o v a l u a b l e i n e v a l u a t i n g r a d a r for q u a n titative precipitation m e a s u r e m e n t s a n d other
meteorological u s e s , since precipitation attenuation is p r o p o r t i o n a l to the r a i n f a l l r a t e s w i t h i n a
s t o r m , and a t t e n u a t i o n p r e s e n t s a n o u t s t a n d i n g
p r o b l e m in a p p l y i n g r a d a r to the m e a s u r e m e n t of
rainfall
intensity.(3)
Consequently, an analysis
w a s m a d e of the f r e q u e n c y of o c c u r r e n c e of point
and a r e a l m e a n r a i n f a l l r a t e s . T h e a v a i l a b l e d a t a
w e r e restricted to shower-type rainfall occurring
d u r i n g the t h u n d e r s t o r m s e a s o n f r o m l a t e s p r i n g t o
e a r l y fall.
D a t a Used in A n a l y s i s
One-minute rainfall amounts f r o m 19 s t o r m s
d u r i n g 1 9 5 2 - 5 3 , t a b u l a t e d in c o n j u n c t i o n with the
r a d a r - r a i n f a l l project mentioned previously, w e r e
u s e d i n the s t u d y . T h e o n e - m i n u t e a m o u n t s w e r e
o b t a i n e d f r o m 50 r e c o r d i n g g a g e s l o c a t e d on t h e
100-square mile Goose C r e e k Network(Fig.4).*
T h e d i s t r i b u t i o n of g a g e s on the n e t w o r k was a p p r o x i m a t e l y u n i f o r m , the u n i f o r m i t y b e i n g l i m i t e d
only b y e x i s t i n g r o a d s y s t e m s and s u i t a b l e e x p o s u r e s . The n e t w o r k t e r r a i n i s r e l a t i v e l y flat with
no distinct topographic features to influence s t o r m
behavior.
Careful a t t e n t i o n w a s given to s y n c h r o n i z i n g the
g a g e s s o . t h a t o n e - m i n u t e r a i n f a l l p a t t e r n s and a r e a l
m e a n s could b e r e l i a b l y r e p r e s e n t e d . All c l o c k s
w e r e c h e c k e d for a c c u r a t e t i m i n g b e f o r e i n s t a l l a tion e a c h s p r i n g . B e f o r e p r o c e e d i n g t o the n e t work to change c h a r t s , o b s e r v e r s synchronized
t h e i r w a t c h e s with a m a s t e r c l o c k , which w a s
c h e c k e d d a i l y with t i m e s i g n a l s f r o m r a d i o s t a t i o n
WWV, the N a t i o n a l B u r e a u of S t a n d a r d s S t a t i o n
a t W a s h i n g t o n , D . C . When i n s t a l l i n g and r e m o v i n g
c h a r t s f r o m the d r u m , the o b s e r v e r m a d e a v e r t i c l e t i m a c h e c k m a r k with the pen o n the c h a r t and
e n t e r e d the e x a c t " o n " o r "off" t i m e f r o m h i s w a t c h .
C h a r t s w e r e c h a n g e d a f t e r e a c h s t o r m and t w i c e a
w e e k d u r i n g p e r i o d s of no r a i n to m i n i m i z e any
c l o c k t i m i n g e r r o r s . When a r a i n - g a g e r e c o r d i n d i c a t e d the g a g e c l o c k had gained or l o s t a s m a l l
a m o u n t o f t i m e , the e r r o r was p r o r a t e d o v e r the
p e r i o d of r e c o r d . As a f u r t h e r c h e c k , a f t e r i s o h y e t a l m a p s w e r e p l o t t e d for e a c h m i n u t e of a s t o r m ,
g a g e c h a r t s w e r e r e e x a m i n e d i f doubtful v a l u e s
a p p e a r e d at any s t a t i o n s on a o n e - m i n u t e m a p .
FIGURE 14 DISTRIBUTION OF POINT RAINFALL RATES IN INDIVIDUAL STORMS
* O p e r a t i o n o f t h e n e t w o r k and d a t a r e d u c t i o n w e r e s p o n s o r e d i n p a r t b y the Signal C o r p s E n g i n e e r i n g
L a b o r a t o r i e s , B e l m a r , New J e r s e y u n d e r c o n t r a c t D A - 3 6 - 0 3 9 S C - 4 2 4 4 6
22
F r e q u e n c y of P o i n t R a i n f a l l R a t e s
The f r e q u e n c y with which v a r i o u s point r a i n f a l l
rates o c c u r r e d in each s t o r m was calculated f i r s t ,
u s i n g the d a t a f r o m all 50 g a g e s . F i g u r e 14 s h o w s
c u m u l a t i v e f r e q u e n c y c u r v e s for the ten 1953
s t o r m s . The graph shows cumulative p e r c e n t a g e s
plotted a g a i n s t r a i n f a l l r a t e i n i n c h e s p e r h o u r ,
b a s e d upon o n e - m i n u t e a m o u n t s f r o m all 5 0 g a g e s .
F o r e x a m p l e , i n the s t o r m for J u l y 2 , 5 0 p e r c e n t
of the r a i n f a l l o c c u r r e d at r a t e s e x c e e d i n g 0.90
inch p e r h o u r , while for the s t o r m of J u n e 25, 50
p e r cent o f the total r a i n f a l l o c c u r r e d a t r a t e s e x c e e d i n g 3.10 i n c h e s p e r h o u r .
E x a m i n a t i o n of the 19 s t o r m s i n d i c a t e d that t h e
g r o u p s h o u l d be a p p r o x i m a t e l y r e p r e s e n t a t i v e of
c o n d i t i o n s e x p e r i e n c e d d u r i n g the w a r m s e a s o n i n
the M i d w e s t . C o n s e q u e n t l y , the d a t a w e r e c o m b i n e d
to d e t e r m i n e the f r e q u e n c y of point r a i n f a l l r a t e s
under a v e r a g e conditions during a w a r m s e a s o n .
R e s u l t s of t h i s a n a l y s i s a r e shown in c o l u m n s 1 a n d
2 of T a b l e 4. It will be noted that on the a v e r a g e ,
b a s e d o n o n e - m i n u t e r a t e s i n the 1 9 - s t o r m s a m p l e ,
50 p e r c e n t of the w a r m s e a s o n r a i n f a l l o c c u r s at
r a t e s e x c e e d i n g 1.45 i n c h e s p e r h o u r . H o w e v e r ,
the m e d i a n r a t e i n i n d i v i d u a l s t o r m s i n the 1 9 - s t o r m
s a m p l e w a s highly v a r i a b l e , r a n g i n g f r o m 0.10 t o
3.10 i n c h e s p e r h o u r .
F o r c o m p a r i s o n p u r p o s e s , the p e r c e n t o f t o t a l
s t o r m rainfall occurring at various rainfall r a t e s
when t h e s e r a t e s a r e b a s e d o n 5 - , 1 0 - , 1 5 - , a n d
30-minute summation periods was determined.
Obviously, the averaging p r o c e s s s u p p r e s s e s d e t a i l s o f the high r a t e s o c c u r r i n g d u r i n g the a v e r aging p e r i o d . Most s t u d i e s i n the p a s t , such a s
t h o s e of Y a r n e l l ( 4 ) and the U . S . W e a t h e r B u r e a u ( 5 )
o n which e n g i n e e r i n g d e s i g n h a s b e e n b a s e d , w e r e
c a l c u l a t e d on the b a s i s of p e r i o d s f r o m five m i n u t e s
u p w a r d . H y d r o l o g i s t s h a v e found t h a t r a i n f a l l r a t e s
for p e r i o d s l e s s than 10 to 15 m i n u t e s s e l d o m h a v e
p r a c t i c a l s i g n i f i c a n c e with r e s p e c t to runoff f r o m a
drainage a r e a . However, as mentioned previously,
s h o r t e r period r a t e s have application in s e d i m e n tation s t u d i e s , in gaining an u n d e r s t a n d i n g of the
p h y s i c a l s t r u c t u r e of s t o r m s , and in the e v a l u a t i o n
o f r a d a r for m e t e o r o l o g i c a l p u r p o s e s . I n the p r e s e n t study, a n a t t e m p t w a s m a d e t o a s c e r t a i n t h e
effect of the a v e r a g i n g p r o c e s s on the o b s e r v e d
f r e q u e n c y of r a i n f a l l r a t e s .
FIGURE 15
TABLE 4
FREQUENCY OF POINT RAINFALL RATES
CALCULATED FROM D I F F E R E N T BASE PERIODS
Rainfall Rate ( i n / h r ) Equaled o r Exceeded F o r
S e v e r a l Base P e r i o d s (Min.)
Cumulative
P e r Cent
1
10
5
15
Max.
13.80
7.50
5.72
4.20
2.45
5
5.35
4.10
3.25
2.64
1.62
10
4.20
3.35
2.80
2.34
1.41
20
3.25
2.58
2.13
1.93
1.15
30
2.45
2.02
1.75
1.42
0.88
40
1.90
1.52
1.34
1.07
0.69
50
1.45
1.16
0.97
0.82
0.56
60
1.05
0.83
0.70
0.62
0.44
70
0.68
0.54
0.46
0.41
0.33
80
0.37
0.28
0.25
0.23
0.21
90
0.15
0.12
0.11
0.11
0.10
95
0.08
0.07
0.07
0.07
0.06
P e r Cent of 1-Min. Rate for S e v e r a l P e r i o d s (Min.)
Cumulative
P e r Cent
5
10
15
30
Max.
54
41
30
18
5
77
61
49
30
10
80
67
56
34
20
79
66
59
35
30
82
71
58
36
40
80
71
56
36
50
80
67
57
39
60
79
67
59
42
70
79
68
60
48
80
76
68
62
57
90
80
73
73
67
95
87
87
87
75
R e s u l t s a r e i l l u s t r a t e d in F i g u r e 1 5 and T a b l e 4.
In Table 4 (upper part) it is s e e n that when o n e minute totals a r e used, 50 p e r cent of the rain was
f o u n d t o o c c u r a t r a t e s e x c e e d i n g 1.45 i n c h e s p e r
hour. When a five-minute b a s e period was used, the
5 0 p e r c e n t v a l u e d r o p p e d t o 1.16 i n c h e s p e r h o u r
and then to 0.97 inch p e r h o u r for a 1 0 - m i n u t e b a s e
p e r i o d , 0.82 inch p e r hour for a 15-minute b a s e
p e r i o d , and 0.56 inch p e r h o u r for a 3 0 - m i n u t e b a s e
period. A l s o , in Table 4 note the rapid d e c r e a s e in
the m a x i m u m o b s e r v e d rainfall r a t e when this r a t e
is based on various averaging p e r i o d s .The averaging effect i s f u r t h e r i l l u s t r a t e d i n the l o w e r p a r t
of Table 4, w h e r e the longer period values have
b e e n e x p r e s s e d a s p e r c e n t a g e s o f the o n e - m i n u t e
values.
DISTRIBUTION OF P O I N T R A I N F A L L
RATES DURING AVERAGE SEASON
30
FIGURE 16
TIME DISTRIBUTION OF POINT R A I N F A L L
23
T h e t i m e d i s t r i b u t i o n of point r a i n f a l l was
s t u d i e d next, c o m b i n i n g d a t a f r o m all 19 s t o r m s .
R e s u l t s of this a n a l y s i s a r e shown in F i g u r e 16
w h e r e p e r cent of t o t a l t i m e of r a i n f a l l in the 19
s t o r m s has b e e n plotted a g a i n s t p e r c e n t o f t o t a l
r a i n f a l l i n t h e s e s t o r m s . F r o m this c u r v e i t i s
s e e n t h a t 10 p e r c e n t of the s e a s o n r a i n f a l l , b a s e d
on the 1 9 - s t o r m sample, o c c u r s in approximately
one p e r cent of the t i m e it is r a i n i n g , 50 p e r c e n t
of the r a i n o c c u r s d u r i n g about 9 p e r c e n t of the
t i m e t h a t r a i n is falling, and 90 p e r c e n t of the
total r a i n o c c u r s in 50 p e r cent of the t i m e that it
is raining. Thus, it is apparent that w a r m season
r a i n f a l l t e n d s to o c c u r in s t r o n g b u r s t s within a
s t o r m p e r i o d ; that i s , m o s t o f the r a i n t e n d s t o
o c c u r in a s m a l l p o r t i o n of the s t o r m .
m i l e n e t w o r k was s u b d i v i d e d to give a r e a s of 10,
2 5 , and 5 0 s q u a r e m i l e s for c o m p a r i s o n s with a r e a l
m e a n s t o r m rainfall r e l a t i o n s o n the 1 0 0 - s q u a r e
mile network. One-minute areal mean rates were
c a l c u l a t e d for e a c h m i n u t e i n each s t o r m for e a c h
of the a r e a s m e n t i o n e d a b o v e .
FIGURE 18 DISTRIBUTION OF AREAL MEAN RAINFALL RATES
FIGURE 17 POINT RAINFALL TIME-RATE RELATION
F i g u r e 17 shows a t i m e - r a t e r e l a t i o n for point
r a i n f a l l d e r i v e d f r o m the c o m b i n e d d a t a f o r the 19
s t o r m s . This g r a p h f u r t h e r e m p h a s i z e s t h e b u r s t
n a t u r e of shower-type r a i n f a l l .
Reference to
F i g u r e 17 shows that r a t e s equalling or e x c e e d i n g
1.45 i n c h e s p e r h o u r , which a c c o u n t e d f o r 50 p e r
cent of the t o t a l r a i n f a l l a c c o r d i n g to F i g u r e 15,
o c c u r r e d only 9 p e r cent of the t i m e it w a s r a i n i n g .
S i m i l a r l y , r a t e s e x c e e d i n g 4.25 i n c h e s p e r h o u r
o c c u r r e d only one p e r c e n t of the t i m e r a i n w a s
f a l l i n g , while r a t e s e x c e e d i n g 0.14 inch p e r h o u r
w e r e r e c o r d e d d u r i n g 50 p e r cent of the t i m e it w a s
raining.
F r e q u e n c y of A r e a l Mean Rainfall R a t e s
T h e foregoing d i s c u s s i o n h a s been c o n c e r n e d
with t h e f r e q u e n c y of point r a i n f a l l r a t e s . Often,
h y d r o l o g i s t s a r e m o r e c o n c e r n e d with a r e a l than
with point r e l a t i o n s h i p s . C o n s e q u e n t l y , d a t a f r o m
the 5 0 - g a g e n e t w o r k o n the 1 0 0 - s q u a r e m i l e a r e a
w e r e f u r t h e r a n a l y z e d t o obtain a r e a l f r e q u e n c y
r e l a t i o n s h i p s . F o r t h i s p u r p o s e , the 1 0 0 - s q u a r e
All 19 s t o r m s w e r e c o m b i n e d to o b t a i n a v e r a g e
r e l a t i o n s h i p s for the v a r i o u s a r e a s . T h e r e s u l t s a r e
s u m m a r i z e d in T a b l e s 5 and 6 and F i g u r e 18.
R e f e r e n c e t o T a b l e 5 s h o w s the e x p e c t e d t r e n d . F o r
e x a m p l e , 50 p e r c e n t of the point r a i n f a l l w a s
found to o c c u r at a r a t e e x c e e d i n g 1.45 i n c h e s p e r
h o u r , while 50 p e r c e n t of the a r e a l m e a n r a i n f a l l
o c c u r r e d at r a t e s of 1.02, 0.96, 0.78 and 0.66 i n c h e s
p e r h o u r , r e s p e c t i v e l y , for the 1 0 - , 2 5 - , 5 0 - , and
1 0 0 - s q u a r e m i l e a r e a s . Table 6 i l l u s t r a t e s the
v a r i a t i o n in the t i m e d i s t r i b u t i o n of r a i n f a l l r a t e s
a s the a r e a i s i n c r e a s e d . F o r e x a m p l e , 5 0 p e r c e n t
of the point r a i n f a l l w a s found to o c c u r d u r i n g 9 p e r
c e n t of the t i m e it w a s r a i n i n g . At 25 s q u a r e m i l e s ,
50 p e r cent of the a r e a l m e a n r a i n f a l l o c c u r r e d in
11 p e r c e n t of the t i m e , while for 100 s q u a r e m i l e s
50 p e r cent o c c u r r e d in 12 p e r cent of the r a i n t i m e .
T h i s t a b l e shows a t e n d e n c y for t h e t i m e d i s t r i b u tion to change only v e r y slowly with i n c r e a s i n g
area.
D i s c u s s i o n of R e s u l t s
Although the r e s u l t s i n this study a r e b a s e d upon
a r e l a t i v e l y s m a l l s a m p l e , the v a r i e t y of s t o r m s
i n c o r p o r a t e d i n the 1 9 - s t o r m s a m p l e s h o u l d m a k e
the r e s u l t s a p p r o x i m a t e l y r e p r e s e n t a t i v e o f a v e r age w a r m s e a s o n c o n d i t i o n s . I t would h a v e b e e n
d e s i r a b l e t o have i n c l u d e d m o r e s t o r m s ; h o w e v e r ,
the g r e a t a m o u n t of h a n d tabulation r e q u i r e d did not
p e r m i t it. I t i s hoped t h a t the r e s u l t s w i l l p r o v e
useful in e v a l u a t i n g the a c t u a l d i s t r i b u t i o n of r a i n fall r a t e s i n w a r m s e a s o n r a i n f a l l i n I l l i n o i s and
the M i d w e s t . A t the p r e s e n t t i m e , m a c h i n e p r o c e s s ing of the r a i n - g a g e c h a r t s is b e i n g i n v e s t i g a t e d ,
and if it is f e a s i b l e , t h e p r e s e n t i n v e s t i g a t i o n c a n
be g r e a t l y expanded s i n c e a l a r g e a m o u n t of unp r o c e s s e d data a r e a v a i l a b l e which w e r e c o l l e c t e d
from 1951-54.
24
TABLE 5
F R E Q U E N C Y O F A R E A L M E A N RAINFALL RATES
Cumulative
P e r Cent
of R a i n f a l l
Max.
5
10
20
30
40
50
60
70
80
90
95
Point
Rainfall Rate ( i n / h r ) Equaled or
E x c e e d e d for Given A r e a (Mi. 2 )
10
25
50
13.80
5.50
4.20
3.25
2.45
1.90
1.45
1.05
0.68
0.37
0.15
0.08
4.08
3.48
3.18
2.28
1.80
1.26
1.02
0.72
0.48
0.25
0.11
0.07
4.02
2.82
2.55
2.22
1.77
1.26
0.96
0.72
0.48
0.26
0.12
0.07
3.18
2.52
2.22
1.74
1.32
1.02
0.78
0.54
0.36
0.18
0.10
0.05
P e r Cent
100
10
2.88
2.40
1.86
1.35
1.11
0.90
0.66
0.48
0.28
0.16
0.08
0.04
30
63
76
70
73
66
70
69
71
68
73
87
TABLE 6
T I M E DISTRIBUTION O F A R E A L MEAN
RAINFALL RATES
A r e a (Mi. )
Point
10
25
50
100
P e r Cent of Total R a i n Time Accounting
for Given P e r Cent of T o t a l R a i n f a l l
10
25
50
75
90
1
1
1
1
1
3
3
4
4
4
9
10
11
12
12
25
26
27
28
30
48
51
53
55
58
of Point
Rate
25
50
100
29
51
61
68
72
66
66
69
71
70
80
87
23
46
53
54
54
54
54
51
53
49
67
62
21
44
44
42
45
47
46
46
41
43
53
50
25
AREA-DEPTH RELATIONS
Introduction
F o r e n g i n e e r i n g d e s i g n p u r p o s e s , a knowledge
o f d e t a i l e d a r e a - d e p t h r e l a t i o n s for s m a l l a r e a s
i s often e s s e n t i a l . F e w d e t a i l e d d a t a for a r e a l units
u n d e r 300 s q u a r e m i l e s h a v e b e e n p u b l i s h e d . R a i n gage n e t w o r k s of sufficient d e n s i t y to define a r e a depth r e l a t i o n s a c c u r a t e l y f r o m i s o h y e t a l m a p s for
such s m a l l a r e a s a r e u n c o m m o n . While p u b l i c a t i o n s
of the M i a m i C o n s e r v a n c y D i s t r i c t ( 6 ) and t h e U.S.
C o r p s of E n g i n e e r s ( 7 ) i n c l u d e d a t a on s m a l l wat e r s h e d s , the d a t a a r e b a s e d o n r e l a t i v e l y s p a r s e
spacing of rain g a g e s . The concentrated networks
e m p l o y e d in t h i s study w e r e d e s i g n e d for c o l l e c t i n g
m o r e precise data.
To obtain d e t a i l e d i n f o r m a t i o n on the c h a r a c t e r i s t i c s o f the a r e a - d e p t h c u r v e for s m a l l w a t e r s h e d s , t h u n d e r s t o r m and r a i n s h o w e r d a t a c o l l e c t e d
on four d e n s e l y - g a g e d n e t w o r k s in c e n t r a l Illinois
have b e e n a n a l y z e d . T h e s e t h r e e n e t w o r k s c o n s i s t
o f the 2 8 0 - s q u a r e m i l e E l P a s o n e t w o r k ( F i g . 2 ) ,
the 1 0 0 - s q u a r e m i l e P a n t h e r C r e e k ( F i g . 3 ) and
G o o s e C r e e k ( F i g . 4) n e t w o r k s , and the B o n e y a r d
C r e e k n e t w o r k of 5.5 s q u a r e m i l e s ( F i g . 6). The
n e c e s s i t y for c o n c e n t r a t e d n e t w o r k s , which can b e
u s e d t o obtain d e t a i l e d i s o h y e t a l p a t t e r n s and t h e r e b y m i n i m i z e i n t e r p o l a t i o n e r r o r s i n the c o n s t r u c t i o n
of a r e a - d e p t h c u r v e s , is i l l u s t r a t e d by the s e r i e s of
i s o h y e t a l m a p s i n F i g u r e 19. T h e s e m a p s w e r e
d r a w n f o r the s a m e s t o r m o n the 1 0 0 - s q u a r e m i l e
P a n t h e r C r e e k w a t e r s h e d u s i n g the v a r i o u s gage
densities indicated.
A n a l y s i s and R e s u l t s
Most previous investigations of a r e a - d e p t h rel a t i o n s have b e e n c o n c e r n e d with r e l a t i v e l y l a r g e
w a t e r s h e d s , e m p l o y i n g l e s s d e n s e n e t w o r k s than
t h o s e u t i l i z e d in t h i s s t u d y . It h a s been o b s e r v e d
that a r e a - d e p t h c u r v e s d e r i v e d f r o m i s o h y e t a l
m a p s s o m e t i m e s a p p r o a c h s t r a i g h t l i n e s o r flat
c u r v e s when the l o g a r i t h m o f a r e a i s plotted
against average
rainfall.(8)
T h i s often r e s u l t s
f r o m a l i b e r a l e n v e l o p m e n t of p o i n t s . C u r v e d lines
w e r e obtained with the I l l i n o i s a r e a - d e p t h d a t a i n
m o s t c a s e s b y plotting the d a t a o n s e m i - l o g a r i t h m i c
p a p e r with depth on the l i n e a r s c a l e and a r e a on the
logarithmic scale.
Since a straight-line r e p r e s e n t a t i o n is d e s i r a b l e ,
s e v e r a l h y p o t h e s e s w e r e i n v e s t i g a t e d i n a n effort
to obtain an easily computed straight-line relation.
Within the l i m i t s t e s t e d for the four s m a l l a r e a s , i t
was found that the data c o n f o r m e d c l o s e l y to the
g e n e r a l equation(9)
Y = a + b X
1 / 2
( 1 )
w h e r e Y is a v e r a g e r a i n f a l l depth in i n c h e s , X is
the a r e a enveloped in s q u a r e m i l e s , and a and b a r e
c o n s t a n t s r e p r e s e n t i n g m a x i m u m s t o r m point r a i n fall and m e a n r a i n f a l l g r a d i e n t , r e s p e c t i v e l y .
The a r e a - d e p t h curve is a two-dimensional
r e p r e s e n t a t i o n of rainfall distribution o v e r a r e a ,
in w h i c h v a l u e s of a v e r a g e r a i n f a l l within an i s o h y e t
a r e plotted a g a i n s t the a r e a e n v e l o p e d . T h e a r e a -
d e p t h c u r v e i s c o n s t r u c t e d a s though the h i g h e s t
value of r a i n f a l l o c c u r r e d at one point, and l o w e r
v a l u e s a p p e a r e d (in a n i s o h y e t a l r e p r e s e n t a t i o n )
r o u g h l y c o n c e n t r i c a l l y a r o u n d the h i g h e r v a l u e s .
Chow ( 1 0 ) h a s p o i n t e d out that E q u a t i o n (1) ind i c a t e s that the t h r e e - d i m e n s i o n a l r e p r e s e n t a t i o n
of the r a i n f a l l i s o h y e t s , which he c a l l s the " s t o r m
pile," of thunderstorms or rainshowers is very
c l o s e to the f o r m of a c o n e . He i n d i c a t e d t h a t a d d i t i o n a l t e r m s of h i g h e r o r d e r s of X' m u s t be cons i d e r e d when the s t r a i g h t - l i n e r e l a t i o n s h i p of the
d e p t h plotted a g a i n s t the s q u a r e root of t h e a r e a
d o e s not apply t o s t o r m s c o v e r i n g l a r g e r a r e a s ,
that i s m o r e than 300 s q u a r e m i l e s .
Within the " s t o r m p i l e , " point r a i n f a l l i s a l s o
l i n e a r l y r e l a t e d to the d i s t a n c e f r o m the c e n t e r of
t h e s t o r m a s r e p r e s e n t e d b y the following e q u a t i o n
y = A + Br
(2)
w h e r e y is the point r a i n f a l l at a d i s t a n c e , r, from
the c e n t e r of the s t o r m , and for a given s t o r m , A
and B a r e c o n s t a n t s , r e p r e s e n t i n g m a x i m u m point
r a i n f a l l and point r a i n f a l l g r a d i e n t .
Analysis(11)
h a s shown that A = a in E q u a t i o n (1) and b = 0 . 3 7 6 B ,
so that
Y = A + 0.376B X 1 / 2
(3)
E q u a t i o n s (1) to (3) c a n be u s e d to c o n s t r u c t
h y p o t h e t i c a l a r e a - d e p t h c u r v e s for d e s i g n p u r p o s e s
o n s m a l l b a s i n s , when t h e m a x i m u m point r a i n f a l l
o f the s t o r m c e n t e r c a n b e obtained f r o m f r e q u e n c y
data, such as those obtained by Yarnell,(4) provided
that the r a i n f a l l g r a d i e n t o r m e a n r a i n f a l l o n the
b a s i n can b e e s t i m a t e d f r o m a l o c a l r a i n - g a g e netw o r k . D e v e l o p m e n t of a r e l a t i o n s h i p for r a i n f a l l
g r a d i e n t will b e d i s c u s s e d l a t e r .
E q u a t i o n (1) w a s t e s t e d o n s t o r m s r a n g i n g f r o m
one to 24 h o u r s d u r a t i o n on the four a r e a s . Only
s t o r m s i n which the a r e a l m e a n r a i n f a l l e q u a l e d o r
e x c e e d e d 0.50 inch w e r e u s e d . T h e s e i n c l u d e d 126
s t o r m s on the two 1 0 0 - s q u a r e m i l e n e t w o r k s , 72
on the 5 . 5 - s q u a r e m i l e a r e a , and 19 on the 280s q u a r e m i l e n e t w o r k . R e s u l t s for the h e a v i e s t
s t o r m s i n each a r e a a r e illustrated i n F i g u r e s
20-22.
T o m a k e the a r e a - d e p t h r e l a t i o n d e v e l o p e d i n
E q u a t i o n (1) m o r e a p p l i c a b l e for e n g i n e e r i n g d e s i g n
p u r p o s e s , data f r o m the 5.5- and 1 0 0 - s q u a r e m i l e
a r e a s w e r e a n a l y z e d f u r t h e r t o d e t e r m i n e the r e l a t i o n s h i p b e t w e e n t h e c u r v e slope ( r a i n f a l l g r a d ient) and o t h e r s t o r m p a r a m e t e r s . A n a l y s i s w a s
not a c c o m p l i s h e d for the 2 8 0 - s q u a r e m i l e a r e a
d u e to the l i m i t e d d a t a a v a i l a b l e . E x a m i n a t i o n of
the d a t a for the two s m a l l e r a r e a s i n d i c a t e d that
the b e s t fit would be obtained with a r e l a t i o n s h i p
of the f o r m
b = K P1 T m
(4)
w h e r e b is the m e a n r a i n f a l l g r a d i e n t in E q u a t i o n
(1), P is a r e a l m e a n r a i n f a l l , T is s t o r m d u r a t i o n ,
and K, 1, and m a r e c o n s t a n t s .
26
2.4 sq.mi. per gage
4.75 sq. mi. per gage
9.5 sq.mi. per gage
19.5 sq. mi. per gage
U.S. Weather Bureau Climatological Network, 225 sq. mi. per
gage.
FIGURE 19 EFFECT OF GAGE DENSITY ON ISOHYETAL PATTERN
27
FIGURE 20 AREA-DEPTH RELATIONS FOR HEAVIEST STORMS,
5.S SQUARE MILE AREA
FIGURE 21 AREA-DEPTH RELATIONS FOR HEAVIEST STORMS,
100 SQUARE MILE AREA
R e s u l t s of this study y i e l d e d E q u a t i o n s (5) and
(6) for the 100- and 5 . 5 - s q u a r e m i l e b a s i n s , r e spectively,
FIGURE 22 AREA-DEPTH RELATIONS FOR HEAVIEST STORMS,
280 SQUARE MILE AREA
b = 0.094
P.83
T-.28
(5)
b = 0.169
P.89
T-.15
(6)
A s e x p e c t e d , the d a t a i n d i c a t e d c o n s i d e r a b l e v a r i a b i l i t y in the r a i n f a l l g r a d i e n t a m o n g s t o r m s of
s i m i l a r m e a n r a i n f a l l and d u r a t i o n . O b v i o u s l y ,
t h e r e a r e o t h e r f a c t o r s i n addition t o s t o r m s i z e
and s t o r m d u r a t i o n , that a r e not e a s i l y r e d u c e d t o
m a t h e m a t i c a l e x p r e s s i o n , which affect the c a l c u l a t e d r a i n f a l l g r a d i e n t within s t o r m s . F o r e x a m p l e ,
the a r e a - d e p t h c u r v e for a given b a s i n will be i n fluenced by the l o c a t i o n of the s t o r m c o r e with r e s p e c t to the b a s i n , m o v e m e n t of the s t o r m w i t h
r e s p e c t to the a x i s of the b a s i n , and the i n t e r n a l
s t r u c t u r e of the s t o r m . A m u l t i p l e c o r r e l a t i o n c o efficient of 0.65 w a s o b t a i n e d f r o m E q u a t i o n (5)
and a coefficient of 0.55 for Equation (6). T h e s e
r e l a t i v e l y low coefficients r e f l e c t the high v a r i a n c e
of the d a t a .
28
FIGURE 23 AVERAGE AREA-DEPTH RELATIONS, 5.5 SQUARE MILE
AREA, MEAN RAINFALL = 1.00 INCH
Applying the r e l a t i o n s for the c u r v e s l o p e o r
r a i n f a l l g r a d i e n t in E q u a t i o n s (5) and (6) to Equa-.
tion (1), a v e r a g e a r e a - d e p t h r e l a t i o n s w e r e d e v e l oped for the two a r e a s and a r e i l l u s t r a t e d i n F i g u r e s 23 to 26. F i g u r e s 23 and 24 show a v e r a g e
a r e a - d e p t h r e l a t i o n s for v a r i o u s s t o r m d u r a t i o n s
b a s e d on an a r e a l m e a n r a i n f a l l of one i n c h . F i g u r e s 2 5 and 2 6 show a v e r a g e a r e a - d e p t h r e l a t i o n s
for s t o r m s o f v a r i o u s m a g n i t u d e ( m e a n rainfall)
for a s t o r m d u r a t i o n of six h o u r s . O b v i o u s l y , the
d e v e l o p e d e q u a t i o n s can be u s e d in c a l c u l a t i o n s
for o t h e r s t o r m s i z e s and d u r a t i o n s .
FIGURE 25 AVERAGE AREA-DEPTH RELATIONS, 5.5 SQUARE MILE
AREA, 6-HOUR STORM DURATION
All d a t a for both a r e a s a r e for s t o r m d u r a t i o n s
u p t o 2 4 h o u r s ; c o n s e q u e n t l y , t h e i r c u r v e s should
not b e u s e d for l o n g e r p e r i o d s t o r m s . I t should
a l s o b e pointed out t h a t the m e a n r a i n f a l l s t e s t e d
f o r the 5 . 5 - s q u a r e m i l e a r e a did not e x c e e d 3.5
i n c h e s . E x c e p t for two s t o r m s , a l l s t o r m s o b s e r v e d o n the 1 0 0 - s q u a r e m i l e n e t w o r k s had l e s s
than 5-inch m e a n s . Consequently, upper l i m i t s of
3 - i n c h and 4 - i n c h m e a n s have b e e n u s e d i n F i g u r e s
25 and 26.
FIGURE 24 AVERAGE AREA-DEPTH RELATIONS, 100 SQUARE MILE
FIGURE 26 AVERAGE AREA-DEPTH RELATIONS, 100 SQUARE MILE
AREA, MEAN RAINFALL = 1.00 INCH
AREA, 6-HOUR STORM DURATION
29
FIGURE 29 EXTREME AREA-DEPTH CURVE SLOPES, 5.5 SQUARE
MILE AREA, SELECTED AREAL MEAN RAINFALLS
FIGURE 27 EXTREME AREA-DEPTH RELATIONS, 5.5 SQUARE MILE
AREA, MEAN RAINFALL = 1.00 INCH
FIGURE 28 EXTREME AREA-DEPTH RELATIONS, 100 SQUARE MILE
AREA, MEAN RAINFALL = 1.00 INCH
The c u r v e s in F i g u r e s 23 to 26, of c o u r s e , a r e
for a v e r a g e v a l u e s of rainfall g r a d i e n t ( c u r v e
s l o p e ) . F o r both E q u a t i o n s (5) and (6), 9 5 p e r cent
confidence bands were determined. The upper 95
p e r cent confidence band i n both c a s e s h a s b e e n
u s e d in F i g u r e s 27 and 28 to show e x t r e m e a r e a d e p t h r e l a t i o n s for the two b a s i n s with a m e a n
r a i n f a l l of one i n c h . F i g u r e s 29 and 30 show e x t r e m e c u r v e s l o p e s for d u r a t i o n s of 1 to 24 h o u r s
and for v a r i o u s m e a n r a i n f a l l s . T h e s e f i g u r e s m a y
be u s e d in conjunction with E q u a t i o n (1) to o b t a i n
FIGURE 30 EXTREME AREA-DEPTH CURVE SLOPES, 100 SQUARE
MILE AREA, SELECTED AREAL MEAN RAINFALLS
30
e x t r e m e a r e a - d e p t h c u r v e s under v a r i o u s condit i o n s . Whether a v e r a g e o r e x t r e m e v a l u e s should
be used depends upon the purpose for which a r e a depth r e l a t i o n s a r e to be utilized. Again, the l a r g e
d i f f e r e n c e s between the a v e r a g e c u r v e s in F i g u r e s
23-24 and the e x t r e m e c u r v e s in F i g u r e s 27-28 r e sult from the l a r g e v a r i a b i l i t y in rainfall g r a d i e n t s
among s t o r m s of s i m i l a r size and d u r a t i o n .
In Equations (5) and (6) for the slope of a r e a depth c u r v e s , it is i n t e r e s t i n g to note the r e l a t i v e
weight of the two p a r a m e t e r s , m e a n r a i n f a l l and
s t o r m d u r a t i o n . The r e l a t i v e i m p o r t a n c e of m e a n
rainfall is g r e a t e r in the equations for both b a s i n s
as i l l u s t r a t e d by the exponents on P and T. R e s u l t s
indicate that, a s the a r e a d e c r e a s e s , s t o r m d u r a tion b e c o m e s l e s s i m p o r t a n t in d e t e r m i n i n g r a i n fall g r a d i e n t s within the b a s i n . C h a r a c t e r i s t i c s of
the s t o r m p a s s i n g o v e r the basin a r e p e r h a p s of
greatest importance h e r e .
The r e l a t i v e weights of P and T in d e t e r m i n i n g
the c h a r a c t e r i s t i c of the a r e a - d e p t h c u r v e a r e furt h e r i l l u s t r a t e d in Table 7 where p a r t i a l c o r r e l a tion coefficients between b and P and between b and
T a r e given for the two a r e a s .
TABLE 7
CORRELATION COEFFICIENTS
Type
Items
Partial
Partial
Multiple
b vs. P
b vs. T
b vs. P, T
Basin Areas
5.5 Sq. Mi.
100 Sq. Mi.
0.45
0.21
0.55
0.60
0.38
0.65
While the r e s u l t s of the a r e a - d e p t h study a r e
b a s e d on Illinois d a t a , the relations should be
g e n e r a l l y applicable to s h o w e r - t y p e rainfall in the
Midwest or other a r e a s having s i m i l a r c l i m a t e and
topography.
L i m i t e d t e s t s m a d e on published a r e a - d e p t h
data for the 8 0 0 0 - s q u a r e m i l e Muskingum Basin ( 8 )
indicated that Equation (1) is not applicable to
l a r g e b a s i n s . Chow ( 1 0 ) has suggested that an equation of the form
1/2
Y = a + b X
3/2
+ cX + dX
+. . .
(7)
maybe more applicable to larger basins, the number of higher terms depending on the basin size.
31
VARIATION O F POINT R A I N F A L L
WITH DISTANCE
Introduction
Confidence Limits
The v a r i a t i o n of r a i n f a l l with d i s t a n c e is an
i m p o r t a n t f a c t o r i n h y d r o l o g i c and a g r i c u l t u r a l
s t u d i e s w h e r e a n a l y s i s of p r e c i p i t a t i o n effects is
r e q u i r e d . Due to the o r d i n a r y s p a c i n g of r a i n g a g e s ,
e s t i m a t e s of point r a i n f a l l a r e f r e q u e n t l y n e e d e d
f o r l o c a t i o n s which m a y b e s e v e r a l m i l e s f r o m the
n e a r e s t r a i n g a g e . T h e U.S. W e a t h e r B u r e a u ( 8 ) h a s
p r e s e n t e d s o m e l i m i t e d r e s u l t s for d i s t a n c e s u p t o
f o u r m i l e s , b a s e d upon d a t a from five s t a t i o n s on
L i t t l e Mill C r e e k i n O h i o . O t h e r w i s e , l i t t l e i n f o r m a t i o n on the subject is a v a i l a b l e .
F o r practical application, an estimate of s t o r m
r a i n f a l l at s o m e point at a d i s t a n c e D f r o m the
c o m p a r i s o n gage is m o r e useful if an i n t e r v a l or
r a n g e about the e s t i m a t e can b e d e t e r m i n e d with
s o m e m e a s u r e o f confidence that the e s t i m a t e
i s within t h i s r a n g e . The 9 5 p e r cent confidence
b a n d s a r e f r e q u e n t l y used for t h i s p u r p o s e . T o
e s t a b l i s h confidence l i m i t s , an e s t i m a t e of the
s a m p l i n g s t a n d a r d deviation i s r e q u i r e d . C o n s e q u e n t l y , e x p r e s s i o n s for an e s t i m a t e of the s t a n d a r d d e v i a t i o n s of the d i f f e r e n c e s b e t w e e n the
c o m p a r i s o n gage r a i n f a l l and that o c c u r r i n g at
v a r i o u s d i s t a n c e s f r o m the c o m p a r i s o n gage w e r e
o b t a i n e d f r o m the s a m e data u s e d i n d e r i v i n g
E q u a t i o n s (3) and (4). The e x p r e s s i o n s o b t a i n e d a r e
As a p a r t i a l s o l u t i o n to the p r o b l e m of d e t e r m i n i n g rainfall v a r i a b i l i t y with d i s t a n c e , a study
w a s u n d e r t a k e n , u s i n g s t o r m data c o l l e c t e d o n two
concentrated rain-gage networks. Data from 25
g a g e s o n the 1 0 0 - s q u a r e m i l e P a n t h e r C r e e k n e t w o r k ( F i g . 3) for 1950-54 and f r o m 50 g a g e s on
the 100 s q u a r e m i l e G o o s e C r e e k n e t w o r k ( F i g . 4)
for 1952-54 w e r e u s e d in the study. T h e s e n e t w o r k s p r o v i d e d d a t a for d i s t a n c e s up to 1 1 m i l e s
f r o m a total of 186 s t o r m s having point r a i n f a l l
a m o u n t s r a n g i n g f r o m 0.01 inch to 9.15 i n c h e s and
a r e a l m e a n r a i n f a l l s v a r y i n g from a t r a c e to 7.21
i n c h e s . A n a l y s i s was confined t o w a r m s e a s o n o r
s h o w e r - t y p e r a i n f a l l o c c u r r i n g from s p r i n g t o fall.
A n a l y s i s of A v e r a g e D i f f e r e n c e s
The r a i n gage n e a r e s t the c e n t e r of the a r e a
i n e a c h n e t w o r k was d e s i g n a t e d a s the c o m p a r i s o n
gage. For each s t o r m on each network, differences
w e r e c a l c u l a t e d b e t w e e n the total s t o r m r a i n f a l l
at the c o m p a r i s o n gage and that r e c o r d e d at e a c h
of the o t h e r n e t w o r k g a g e s . The d a t a w e r e f i r s t
g r o u p e d a c c o r d i n g t o d i s t a n c e from the c o m p a r i s o n g a g e . A t e a c h d i s t a n c e , the d i f f e r e n c e s b e t w e e n
the v a r i o u s n e t w o r k g a g e s and the c o m p a r i s o n
gage were further grouped in class intervals a c c o r d i n g to point r a i n f a l l a m o u n t s ( s t o r m size) at
the c o m p a r i s o n g a g e .
G r a p h i c a l p l o t s of the grouped d a t a i n d i c a t e d
o n e of the two following g e n e r a l e q u a t i o n s would
p r o v i d e the b e s t data fit:
Log E = k + 1 Log P + m Log D
(1)
Log E = k + 1 P n + m Log D
(2)
In E q u a t i o n s (1) and (2), E r e p r e s e n t s the a v e r a g e
d i f f e r e n c e b e t w e e n the t o t a l s t o r m r a i n f a l l at the
c o m p a r i s o n gage and at points l o c a t e d at d i s t a n c e
D f r o m the c o m p a r i s o n g a g e , when t h e point r a i n fall r e c o r d e d a t the c o m p a r i s o n g a g e i s P . The
r e g r e s s i o n c o n s t a n t s a r e r e p r e s e n t e d by k, 1, m
and n.
The data f r o m both n e t w o r k s w e r e then c o m b i n e d t o obtain the e m p i r i c a l e x p r e s s i o n s :
Log E =
- 0 . 7 9 6 + 0.49 L o g P + 0 . 3 2 Log D
Log E =
-1.359 + 0.51 P
where
E
0 5
(3)
+ 0.31 Log D(4)
and P a r e in i n c h e s and D is in m i l e s .
Log s = - 0 . 3 9 8 + 0.49 Log P + 0.32 Log D
Log s = -0.961 + 0.51
P 0.5
+ 0.31 Log D
(5)
(6)
T h e 9 5 p e r cent confidence b a n d s for P w e r e e s t i m a t e d b y taking two t i m e s the s t a n d a r d d e v i a t i o n
and adding and s u b t r a c t i n g f r o m the c o r r e s p o n d i n g
v a l u e of P.
Equation Comparisons
Multiple c o r r e l a t i o n c o e f f i c i e n t s of 0.74 and
0.78 w e r e obtained f r o m E q u a t i o n s (3) and (4),
r e s p e c t i v e l y . T h e s e c o r r e l a t i o n coefficients indic a t e t h e r e is l i t t l e d i f f e r e n c e in the g o o d n e s s of
fit p r o v i d e d by t h e s e two e q u a t i o n s . G r a p h i c a l
p l o t s of the g r o u p e d d a t a i n d i c a t e d t h a t E q u a t i o n
(4) g i v e s a s o m e w h a t b e t t e r fit at high and low
v a l u e s of P, while E q u a t i o n (3) fits b e s t at the int e r m e d i a t e v a l u e s of P.
A c o m p a r i s o n is given in T a b l e 8 of the E v a l u e s
o b t a i n e d for v a r i o u s P v a l u e s at a d i s t a n c e (D) of
five m i l e s . Only s m a l l d i f f e r e n c e s in E a r e obt a i n e d f r o m the two e m p i r i c a l e q u a t i o n s below P
v a l u e s o f t h r e e i n c h e s . Above t h r e e i n c h e s , E q u a tion (4) p r o d u c e s a p p r e c i a b l y h i g h e r v a l u e s of E
t h a n E q u a t i o n (3). H o w e v e r , only a s m a l l p o r t i o n
of the point r a i n f a l l o b s e r v a t i o n s f r o m which the
e m p i r i c a l equations were derived exceeded three
inches.
TABLE 8
COMPARISONS BETWEEN EQUATIONS (3) AND (4)
P
0.10
0.25
0.50
1.00
2.00
3.00
5.00
(in.)
E (in.) at 5 Mi.
Eq. (3)
Eq. (4)
0.08
0.14
0.19
0.27
0.38
0.45
0.59
0.10
0.13
0 17
0.23
0 38
0.55
1.00
F r o m the h y d r o l o g i c s t a n d p o i n t , a t l e a s t , the
v a r i a t i o n of rainfall with d i s t a n c e in h e a v y s t o r m s
is usually of g r e a t e r interest than rainfall v a r i a t i o n s in light s t o r m s . Since E q u a t i o n s (3) and (4)
p r e d i c t s i g n i f i c a n t l y different v a l u e s of E above P
v a l u e s of t h r e e i n c h e s , an effort was m a d e to d e -
32
t e r m i n e which e q u a t i o n a p p e a r s m o s t a p p l i c a b l e
at high v a l u e s of P. Although the data u s e d in the
study i n d i c a t e d E q u a t i o n (4) p r o v i d e d a b e t t e r fit
at the high P v a l u e s , it was d e s i r e d to f u r t h e r t e s t
the r e l a t i o n with i n d e p e n d e n t d a t a .
F o r t h i s p u r p o s e , data w e r e used f r o m two
u n u s u a l l y heavy I l l i n o i s s t o r m s : the J u l y 9 , 1951
s t o r m i n North C e n t r a l
Illinois,(12)
a n d the
J u l y 1 8 - 1 9 , 1952 s t o r m in R o c k f o r d and v i c i n i t y . ( 1 3 )
The W a t e r S u r v e y conducted e x t e n s i v e field s u r v e y s following t h e s e s t o r m s which p r o v i d e d
a d e q u a t e r a i n f a l l data for c o n s t r u c t i n g d e t a i l e d
i s o h y e t a l m a p s . A r e c t a n g u l a r g r i d o v e r l a y was u s e d
with the i s o h y e t a l m a p s f r o m t h e s e two s t o r m s t o
obtain d a t a o n r a i n f a l l v a r i a t i o n s with d i s t a n c e .
E a c h 3 - m i l e i n t e r v a l o n the g r i d o v e r l a y was c o n s i d e r e d a n o b s e r v a t i o n point and rainfall d i f f e r e n c e s
w e r e o b t a i n e d f r o m the u n d e r l y i n g i s o h y e t a l m a p
at d i s t a n c e s of t h r e e , s i x , and nine m i l e s f r o m
e a c h o b s e r v a t i o n point in e a c h of four d i r e c t i o n s n o r t h , e a s t , south and w e s t . T h i s method p r o d u c e d
a total of 760 o b s e r v a t i o n s in the two s t o r m s .
P o i n t rainfall v a l u e s a t the o b s e r v a t i o n p o i n t s
r a n g e d f r o m 2.0 to
10.6 i n c h e s . E q u a t i o n s ( 3 )
and (4) w e r e then t e s t e d by d e t e r m i n i n g how m a n y
of the 760 o b s e r v a t i o n s fell within the 95 p e r cent
confidence b a n d s for e a c h e q u a t i o n . It was found
that 94 p e r cent of t h e s e o b s e r v a t i o n s fell within
the 95 p e r cent confidence b a n d s of E q u a t i o n (4)
c o m p a r e d to 80 p e r cent for E q u a t i o n (3). T h e s e
r e s u l t s i n d i c a t e that E q u a t i o n (4) i s m o r e r e l i a b l e
for p r e d i c t i n g d i f f e r e n c e s in point r a i n f a l l with
d i s t a n c e i n heavy s t o r m s , a s s u g g e s t e d e a r l i e r b y
g r a p h i c a l p l o t s of the o r i g i n a l d a t a upon which the
two e m p i r i c a l e q u a t i o n s w e r e b a s e d .
FIGURE 32 95 PER CENT CONFIDENCE RANGE FOR VARIATION
OF POINT RAINFALL WITH DISTANCE
A g r a p h d e r i v e d f r o m E q u a t i o n (4) for p r e d i c t i n g
the a v e r a g e d i f f e r e n c e i n s t o r m r a i n f a l l b e t w e e n
points a t v a r i o u s d i s t a n c e s i s given i n F i g u r e 3 1 .
The 9 5 p e r cent c o n f i d e n c e b a n d s , b a s e d upon
E q u a t i o n (6), for s e l e c t e d v a l u e s of D a r e p r e s e n t e d i n F i g u r e 32. The g r e a t v a r i a b i l i t y i n s h o w e r type r a i n f a l l i s well i l l u s t r a t e d i n F i g u r e 32. F o r
e x a m p l e , within a d i s t a n c e of one m i l e f r o m a gage
r e c o r d i n g a rainfall of one i n c h , d i f f e r e n c e s up to
0.35 inch or 35 p e r c e n t a r e i n d i c a t e d by the 95 p e r
cent l i m i t s ; at five m i l e s , d i f f e r e n c e s up to 0.55
inch o c c u r ; and at 10 m i l e s , d i f f e r e n c e s r e a c h
0.70 i n c h .
C o n s i d e r a t i o n of Additional V a r i a b l e s
In an e a r l i e r , l i m i t e d study u s i n g data for 33
s t o r m s o v e r the G o o s e C r e e k n e t w o r k d u r i n g
1 9 5 3 ( 1 4 ) a n a l y s i s was p e r f o r m e d to d e t e r m i n e if a
m o r e efficient equation m i g h t b e o b t a i n e d b y s u b s t i t u t i n g o t h e r i n d e p e n d e n t v a r i a b l e s into E q u a t i o n
(1) in a d d i t i o n to P and D. The d a t a had not b e e n
fitted to E q u a t i o n (2) at the t i m e of this a n a l y s i s .
FIGURE 31 VARIATION OF POINT RAINFALL WITH DISTANCE
The only p r a c t i c a l v a r i a b l e s which can b e u s e d
in an e x p r e s s i o n s u c h as E q u a t i o n (1) a r e t h o s e
which can b e r e a d f r o m the r a i n - g a g e c h a r t a t the
o b s e r v a t i o n g a g e . I t w a s r e a s o n e d that c o n s i d e r a tion of s t o r m d u r a t i o n ( T ) , m e a n r a t e of r a i n f a l l
( R a ) , and m a x i m u m r a t e of r a i n f a l l (Rm) m i g h t
33
improve the prediction efficiency of expressions
such as (3) and (5) which utilize only P and D. The
maximum 5-minute rainfall rate for each storm
was used to represent the maximum storm rate.
Multiple correlation coefficients between standard deviation (s) and various combinations of P,
D, T, R a , and Rm were computed to determine
whether the correlation would improve from consideration of the three additional variables. The
various coefficients are shown in Table 9. The
squares of the coefficients, expressed in per cent,
are also shown. These values indicate the relative
efficiency of the various combinations of independent variables.
C o n s i d e r i n g the r e l a t i v e l y s m a l l i m p r o v e m e n t ,
in efficiency shown by the m o r e complex e x p r e s sions and the additional w o r k involved in getting
d a t a t o u s e w i t h t h e m , E x p r e s s i o n s (3) t o (6) a p p e a r
m o r e p r a c t i c a l for p r e d i c t i o n p u r p o s e s . Undoubtedly t h e r e a r e o t h e r f a c t o r s which would be useful in
i n c r e a s i n g t h e e f f i c i e n c y o f E q u a t i o n s (3) t o ( 6 ) .
An i m p o r t a n t factor, for i n s t a n c e , is the s t o r m
path with r e f e r e n c e to the location of the m e a s u r e ment gage. However, it is impossible to estimate
the s t o r m path f r o m a single gage, and only f a c t o r s
which could be obtained f r o m a single r e c o r d i n g gage
w e r e investigated in this study.
TABLE 9
M U L T I P L E CORRELATION COEFFICIENTS
The results presented in Table 9 indicate that
the three additional variables are about equal in
efficiency. Since the product of R a a n d T i s equivalent to P, their use as variables in an expression
already containing P appeared doubtful. However,
to more thoroughly explore the problem, tests
involving these two variables in conjunction with P
were made. Table 9 shows that including Ra and T
with P increased the efficiency by only eight per
cent (42 to 50). The same efficiency was obtained
by using Ra and T without P (last line of Table 9).
Using T, Ra and R m i n c o n j u n c t i o n w i t h P a n d D
i n c r e a s e d the p r e d i c t i o n efficiency by only 10 per
cent.
Independent
Variables
Multiple
C o r r e l a t i o n (C)
C 2 (% )
PD
0.65
42
PDRm
PDT
0.71
0.70
50
49
PDR a
PDR m T
0.70
0.71
49
50
PDR m R a
PDTRa
0.71
0.71
50
50
PDR m T R a
0.72
DTR
0.71
a
52
50
35
AREAL REPRESENTATIVENESS OF
POINT R A I N F A L L
Introduction
The a c c u r a c y with which a point r a i n f a l l m e a s u r e m e n t r e p r e s e n t s t h e m e a n r a i n f a l l for a r e a s o f
v a r y i n g s i z e s in the vicinity of the point o b s e r v a tion is p e r t i n e n t to the d e s i g n of r a i n - g a g e n e t w o r k s
for v a r i o u s p u r p o s e s and in the i n t e r p r e t a t i o n of
data from existing networks. Information regarding
the r e l a t i o n b e t w e e n point rainfall and a r e a l m e a n
s t o r m rainfall i s e s p e c i a l l y a p p l i c a b l e t o h y d r o l o gic p r o b l e m s .
The lack of concentrated r a i n - g a g e networks
o v e r a r e a s o f v a r i o u s s i z e , which a r e n e c e s s a r y
for d e r i v i n g p o i n t - a r e a l m e a n r e l a t i o n s , has i n the
past limited development of e m p i r i c a l relations.
S o m e l i m i t e d d a t a for a r e a s of 375, 1500 and 8000
s q u a r e m i l e s h a v e b e e n p r o v i d e d b y the U.S.
Weather Bureau.(8)
As a p a r t i a l s o l u t i o n to the a b o v e p r o b l e m ,
rainfall data collected on several concentrated
n e t w o r k s i n c e n t r a l I l l i n o i s have b e e n u t i l i z e d t o
d e r i v e e m p i r i c a l r e l a t i o n s b e t w e e n point and a r e a l
m e a n r a i n f a l l for a r e a s ranging f r o m 0.03 t o 400
s q u a r e m i l e s . P o i n t r a i n f a l l was m e a s u r e d a t the
c e n t e r o f e a c h a r e a and e m p i r i c a l e q u a t i o n s d e v e l oped for s t o r m , w e e k l y , and m o n t h l y p e r i o d s . In
a d d i t i o n , s t o r m d a t a for a r e a s of 10 to 100 s q u a r e
m i l e s w e r e u s e d t o obtain a n e m p i r i c a l r e l a t i o n
b e t w e e n point and a r e a l m e a n r a i n f a l l for c o n d i t i o n s when the point r a i n f a l l is m e a s u r e d at a gage
d i s p l a c e d f r o m the a r e a l c e n t e r b y v a r i o u s d i s t a n c e s .
Available Data
Data u s e d in t h i s study w e r e c o l l e c t e d on the
networks illustrated in Figures 2-8, comprising
a r e a s r a n g i n g f r o m 0.03 t o 400 s q u a r e m i l e s .
I n a d d i t i o n , the G o o s e C r e e k and P a n t h e r C r e e k
networks
a r e a s of
networks
which a
available
sampling
the 1954
w e r e subdivided t o p r o v i d e d a t a for
10, 2 5 , and 50 s q u a r e m i l e s . T h e v a r i o u s
p r o v i d e d d a t a for nine a r e a l s i z e s f r o m
t o t a l of 1900 s t o r m o b s e r v a t i o n s w e r e
for d e r i v a t i o n of e m p i r i c a l r e l a t i o n s . T h e
s c h e m e i s i l l u s t r a t e d i n F i g u r e 33, u s i n g
Goose C r e e k n e t w o r k .
All d a t a used in the study w e r e c o l l e c t e d d u r i n g
s p r i n g , s u m m e r , a n d e a r l y fall; c o n s e q u e n t l y , the
developed relations a r e most applicable to showertype r a i n f a l l . While the r e s u l t s a r e b a s e d o n
I l l i n o i s d a t a , they should b e a p p r o x i m a t e l y r e p r e s e n t a t i v e for the M i d w e s t , in g e n e r a l , and for o t h e r
a r e a s having s i m i l a r c l i m a t e and t o p o g r a p h y .
A c c u r a c y o f A r e a l Rainfall E s t i m a t e s F r o m a
C e n t e r e d Gage
Average E r r o r . Various graphical plots were
m a d e o f data for e a c h a r e a l s i z e , r e l a t i n g point
r a i n f a l l at the c e n t e r of each a r e a to the d i f f e r e n c e
b e t w e e n the point o b s e r v a t i o n and the a r e a l m e a n
r a i n f a l l a s m e a s u r e d b y all g a g e s within t h a t
p a r t i c u l a r a r e a . Inspection of these graphical plots
i n d i c a t e d that one of the following g e n e r a l e q u a t i o n s would fit the data b e s t :
Log E = k + 1 Log P
Log E = k + 1 P
(1)
m
(2)
In t h e s e e x p r e s s i o n s , P is the point r a i n f a l l for
e i t h e r the s t o r m , w e e k l y , o r m o n t h l y o b s e r v a t i o n
at the c e n t e r of a given a r e a ; E r e p r e s e n t s the
d i f f e r e n c e b e t w e e n point and a r e a l m e a n r a i n f a l l
in t h e given a r e a ; and k, 1 and m a r e r e g r e s s i o n
c o n s t a n t s . V a l u e s for k and 1 for s t o r m , w e e k l y ,
and m o n t h l y r a i n f a l l a r e shown in T a b l e 10 for
e a c h n e t w o r k b a s e d o n E q u a t i o n (1).
TABLE 10
REGRESSION CONSTANTS FOR STORM, WEEKLY AND
MONTHLY EQUATIONS ON VARIOUS AREAS
Area
(sq. mi.)
0.03
0.60
4
10
25
50
100
280
400
Storm
k
1
-2.155
-1.538
-1.377
-1.377
-1.222
-1.167
-1.131
-0.914
-0.932
0.46
0.46
0.47
0.36
0.52
0.51
0.50
0.45
0.53
Weekly
k
J
-2.097
-1.469
-1.260
-1.229
-1.051
-1.060
-1.065
-0.783
0.61
0.61
0.73
0.62
0.68
0.78
0.66
0.69
Monthly
k
1
-2.301
-1.347
-1.252
-1.268
-1.252
-1.174
-1.131
0.72
0.38
0.86
0.84
0.78
0.80
0.74
The v a l u e s of 1 v a r y at r a n d o m , w h e r e a s the
k v a l u e s show an i n c r e a s i n g t r e n d with a r e a . Cons e q u e n t l y , it w a s felt that the d a t a f r o m a l l n e t w o r k s
could be c o m b i n e d into a s i n g l e e x p r e s s i o n of the
form:
Log E = k1 + l1 Log P + n Log A
FIGURE 33 25, 50 AND 100 SQUARE MILE AREAS, GOOSE CREEK 1954
(3)
r e l a t i n g the m e a s u r e m e n t e r r o r o f a r e a l m e a n
r a i n f a l l t o the p o i n t r a i n f a l l ( P ) r e c o r d e d a t the
36
FIGURE 34 STORM RELATION BETWEEN POINT AND AREAL MEAN RAINFALL BASED ON AVERAGE DEVIATIONS
FIGURE 35 WEEKLY RELATION BETWEEN POINT AND AREAL MEAN RAINFALL BASED ON AVERAGE DEVIATIONS
37
FIGURE 36 MONTHLY RELATION BETWEEN POINT AND AREAL MEAN RAINFALL BASED ON INDIVIDUAL DEVIATIONS
c e n t e r of the a r e a and to t h e s i z e of t h e a r e a (A).
T h e c o n s t a n t s k 1 , 1 1 , and n w e r e d e t e r m i n e d by t h e
m e t h o d o f l e a s t s q u a r e s , a n d the following e m p i r i c a l e q u a t i o n s for s t o r m , w e e k l y , and m o n t h l y r a i n fall w e r e o b t a i n e d .
L o g E = - 1 . 3 4 1 + 0.50 L o g P + 0 . 2 9 L o g A
(Storm)
L o g E = - 1 . 2 8 5 + 0.58 L o g P + 0.20 L o g A
(Weekly)
L o g E = - 1 . 5 5 9 + 0.71 L o g P + 0 . 2 6 L o g A
(Monthly)
(4)
(5)
(6)
F o r c o m p a r i s o n p u r p o s e s , the d a t a w e r e next
fitted to an e x p r e s s i o n of t h e f o r m :
Log E = k 2 + 1 2 P
m
+ n1 L o g A
(7)
i n w h i c h E , P , and A r e p r e s e n t the s a m e q u a n t i t i e s
a s i n E q u a t i o n (3), and k 2 , l 2 , m , and n 1 a r e a g a i n
regression constants.
The
resulting empirical
equations are:
L o g E = - 2 . 0 1 1 + 0.54 0.5
P
L o g E = - 1 . 8 5 3 + 0.51 P0.5
Log E = -1.889
+ 0.37 P0.5
(10) a n d a r e p r e s e n t e d i n T a b l e 1 1 . T h e s e c o r r e lation coefficients indicate there is no significant
d i f f e r e n c e i n t h e g o o d n e s s o f fit p r o v i d e d b y E q u a t i o n s (3) a n d ( 7 ) . T h e s m a l l e r c o r r e l a t i o n c o e f f i c i e n t
o b t a i n e d with the m o n t h l y data is due p r i m a r i l y to
the fact that a n a l y s i s w a s p e r f o r m e d on u n g r o u p e d
data, w h e r e a s the s t o r m and weekly a n a l y s e s w e r e
p e r f o r m e d on grouped data. The n u m b e r of individual m o n t h l y o b s e r v a t i o n s did not p e r m i t g r o u p i n g .
T h e r e w e r e only 255 m o n t h l y v a l u e s c o m p a r e d t o
663 w e e k l y t o t a l s and 1900 s t o r m o b s e r v a t i o n s .
Graphical plots of the grouped data used in
E q u a t i o n s ( 4 ) , ( 5 ) , ( 8 ) , a n d (9) i n d i c a t e d t h a t E q u a t i o n s ( 8 ) a n d (9) p r o v i d e a s o m e w h a t b e t t e r fit a t
high and low v a l u e s of P. A s i m i l a r o c c u r r e n c e
was
noted b e t w e e n t h e s e two g e n e r a l t y p e s of
equations in the study of the v a r i a t i o n of point
rainfall with d i s t a n c e d i s c u s s e d in the p r e v i o u s
s e c t i o n of this bulletin.
TABLE 11
MULTIPLE CORRELATION COEFFICIENTS
+ 0.29 Log A
(Storm)
(8)
+ 0.18 Log A
(Weekly)
(9)
+ 0.25 Log A
(Monthly) (10)
Multiple
correlation
coefficients
were
then
d e t e r m i n e d for E q u a t i o n s (4), (5), ( 6 ) , (8), ( 9 ) , and
Equation
Number
Correlation
Coefficient
Rainfall
Type
(4)
(8)
(5)
(9)
(6)
(10)
0.89
0.90
0.85
0.85
0.62
0.60
Storm
Storm
Weekly
Weekly
Monthly
Monthly
38
A c o m p a r i s o n of t h e E v a l u e s obtained for v a r i ous P v a l u e s f r o m E q u a t i o n s (4) and (8) for s t o r m
r a i n f a l l on an a r e a of 100 s q u a r e m i l e s is p r e s e n t e d
in T a b l e 12. Only s m a l l d i f f e r e n c e s in E a r e o b t a i n e d f r o m the two e m p i r i c a l e q u a t i o n s b e l o w P
v a l u e s o f t h r e e i n c h e s . Above t h r e e i n c h e s , E q u a tion (8) p r o d u c e s a p p r e c i a b l y h i g h e r v a l u e s of E
than E q u a t i o n (4). A s m e n t i o n e d p r e v i o u s l y ,
i n d i c a t i o n s a r e that e m p i r i c a l equations d e r i v e d
f r o m E x p r e s s i o n (7) fit h e a v y rainfall d a t a b e t t e r .
Since h e a v y r a i n f a l l is of p r i m e i n t e r e s t to the
h y d r o l o g i s t , f u r t h e r a n a l y t i c a l efforts h a v e b e e n
c o n c e n t r a t e d on t h i s f o r m of equation. E q u a t i o n s
(8), (9) and (10) a r e r e p r e s e n t e d g r a p h i c a l l y in
F i g u r e s 34, 35, and 36.
s e n t e d b y s t o r m , w e e k l y and m o n t h l y v a l u e s . T h i s
t r e n d b e c o m e s m o r e p r o n o u n c e d a s the point r a i n fall a t the a r e a l c e n t e r i n c r e a s e s . I t would a p p e a r ,
t h e r e f o r e , that a c e n t e r e d gage m e a s u r e s a r e a l
m e a n r a i n f a l l with a n i n c r e a s i n g d e g r e e o f a c c u r a cy as the s a m p l i n g p e r i o d i n c r e a s e s . Such a t r e n d
i s t o b e e x p e c t e d . The w e e k l y r a i n f a l l i s u s u a l l y
the r e s u l t o f s e v e r a l s t o r m s and the m o n t h l y t o t a l
o r d i n a r i l y c o n t a i n s m o r e s t o r m s than t h e w e e k l y
t o t a l . In the a b s e n c e of significant t o p o g r a p h i c or
c l i m a t i c f e a t u r e s , the a r e a l d i s t r i b u t i o n o f r a i n fall will t e n d t o b e c o m e m o r e u n i f o r m a s the n u m b e r of s t o r m s i n c r e a s e s , because individual s t o r m s
tend to t r a v e l in v a r i o u s p a t h s a c r o s s a g i v e n
area.
TABLE 13
TABLE 12
COMPARISON BETWEEN STORM, WEEKLY AND
MONTHLY RELATIONS
COMPARISONS BETWEEN EQUATIONS (4) AND (8)
P (i".)
E (in.) for 100 Square
Mile Area
Eq. (4)
Eg. (8)
0.10
0.04
0.06
0.25
0.07
0.07
0.50
0.10
0.09
1.00
0.14
0.13
2.00
0.20
0.22
3.00
0.25
0.32
5.00
0.32
0.60
A c o m p a r i s o n of E v a l u e s obtained f r o m the
s t o r m , w e e k l y and m o n t h l y e q u a t i o n s is shown in
T a b l e 13. A s e x p e c t e d , all t h r e e e q u a t i o n s show
that t h e a v e r a g e d i f f e r e n c e b e t w e e n point and a r e a l
m e a n r a i n f a l l (E) i n c r e a s e s a s the s t o r m s i z e , r e p r e s e n t e d b y the point m e a s u r e m e n t a t t h e a r e a l
c e n t e r , i n c r e a s e s . F o r a point rainfall m e a s u r e m e n t of a given m a g n i t u d e , a t r e n d e x i s t s for E to
d e c r e a s e with i n c r e a s i n g s a m p l i n g t i m e a s r e p r e -
Av. Difference (in.) Between Point and
Point
Mean Rainfall for 100-Square Mile Area
Rainfall (in.)
Storm
Weekly
Monthly
0.25
0.07
0.06
0.06
1.00
0.13
0.10
0.10
2.00
0.22
0.17
0.14
4.00
0.45
0.34
0.22
Confidence L i m i t s . An e s t i m a t e , P, of the
a r e a l m e a n rainfall b e c o m e s m o r e meaningful
when s o m e m e a s u r e i s m a d e o f the p o s s i b l e e r r o r
in the e s t i m a t e . It is d e s i r a b l e to d e t e r m i n e an
i n t e r v a l a b o u t P with s o m e m e a s u r e of c o n f i d e n c e
that the a r e a l m e a n i s i n that i n t e r v a l . T o e s t a b l i s h c o n f i d e n c e l i m i t s , an e s t i m a t e of the s a m p l i n g
standard e r r o r was r e q u i r e d . This was m a d e from
14 g r o u p s of s t o r m s having about the s a m e P
values. These classes, expressed in inches, a r e
shown in T a b l e 14 A s t a n d a r d d e v i a t i o n w a s c o m puted for e a c h of the c l a s s i n t e r v a l s in T a b l e 14.
The c o n s t a n t s k, 1, m and n w e r e a g a i n d e t e r -
FIGURE 37 STORM RELATION BETWEEN POINT AND AREAL MEAN RAINFALL BASED ON STANDARD ERRORS
39
FIGURE 38 WEEKLY RELATION BETWEEN POINT AND AREAL MEAN RAINFALL BASED ON STANDARD ERRORS
m i n e d for e q u a t i o n s of t h e f o r m r e p r e s e n t e d by
E x p r e s s i o n (7). The r e s u l t i n g e q u a t i o n s for s t o r m
and w e e k l y rainfall a r e :
Log s = - 1 . 8 4 3 + 0.48
P 0.5
Log s = - 1 . 7 3 1 + 0.46
P 0.5
+ 0.27 Log A
(Storm)
+ 0.19 Log A
(Weekly)
(11)
(12)
In t h e s e e q u a t i o n s , s ( i n c h e s ) r e p r e s e n t s the
b e s t e s t i m a t e of the s t a n d a r d e r r o r , P ( i n c h e s )
r e p r e s e n t s the' m i d - p o i n t o f the c l a s s i n t e r v a l s ,
and A r e p r e s e n t s the a r e a i n s q u a r e m i l e s . M u l t i ple c o r r e l a t i o n coefficients for E q u a t i o n s (11) and
(12) w e r e 0.85 and 0.82, r e s p e c t i v e l y . F i g u r e s 37
and 38 show a g r a p h i c a l r e l a t i o n b e t w e e n A and s
for s e v e r a l P v a l u e s for s t o r m and w e e k l y r a i n f a l l .
An insufficient n u m b e r of m o n t h l y t o t a l s p r e v e n t e d
the d e t e r m i n a t i o n of a m o n t h l y r e l a t i o n s h i p .
T h e 9 5 p e r cent c o n f i d e n c e l i m i t s for s t o r m and
weekly a r e a l m e a n rainfall w e r e e s t i m a t e d b y c o m puting p ± 2 s f r o m E q u a t i o n s (11) and (12). T h e
r e s u l t i n g l i m i t s a r e p r e s e n t e d for s e v e r a l v a l u e s
of A in F i g u r e s 39 and 40. O t h e r confidence l i m i t s
m a y b e d e t e r m i n e d i f d e s i r e d . F o r e x a m p l e 70,
80, 90 and 99 p e r cent c o n f i d e n c e l i m i t s for s t o r m
TABLE 14
STORM GROUPS (IN.) USED IN STANDARD
ERROR COMPUTATIONS
0.01-0.10
0.11-0.20
0.21-0.30
0.31-0.40
0.41-0.50
0.51-0.60
0.61-0.70
0.71-0.80
0.81-0.90
0.91-1.00
1.01-1.50
1.51-2.00
2.01-3.00
3.01-4.00
and w e e k l y a r e a l m e a n r a i n f a l l m a y b e e s t i m a t e d
by m u l t i p l y i n g s f r o m E q u a t i o n s (11) and (12) by
1.05, 1.30, 1.67 and 2 . 6 6 , r e s p e c t i v e l y , and adding
and s u b t r a c t i n g f r o m P .
A r e a l R a i n f a l l E s t i m a t e s f r o m a n OffC e n t e r Gage
F r e q u e n t l y , r a i n g a g e s a r e not l o c a t e d a t o r
n e a r the c e n t e r of the a r e a of i n t e r e s t . The n e a r e s t gage m a y even be o u t s i d e the b o u n d a r y of the
a r e a . It is logical to expect t h a t a r a i n gage which
is l o c a t e d at a d i s t a n c e f r o m the a r e a l c e n t e r will
on the a v e r a g e give a l e s s a c c u r a t e e s t i m a t e o f t h e
a r e a l m e a n than a gage l o c a t e d at the c e n t e r . The
following p a r a g r a p h s s u m m a r i z e a n a n a l y s i s t o
d e t e r m i n e the a c c u r a c y o f s t o r m a r e a l m e a n r a i n fall e s t i m a t e s f r o m g a g e s l o c a t e d a t v a r i o u s d i s t a n c e s f r o m the c e n t e r of s e v e r a l a r e a s of d i f f e r ent s i z e s .
Data f r o m a r e a s of 10, 2 5 , 50, and 100 s q u a r e
miles w e r e analyzed. Analysis techniques were
s i m i l a r to those used in c o m p a r i n g a r e a l m e a n
r a i n f a l l with the r a i n f a l l a m o u n t obtained f r o m a
c e n t r a l l y - l o c a t e d g a g e . An e x a m i n a t i o n of g r a p h i cal plots of the d a t a i n d i c a t e d that an e x p r e s s i o n
of the f o r m
Log E = a + b P c + d Log A + e Log D
(13)
followed t h e t r e n d of t h e d a t a s a t i s f a c t o r i l y . In t h i s
e x p r e s s i o n , E r e p r e s e n t s the a v e r a g e of the d i f f e r e n c e s , A is the n e t w o r k a r e a , D is d i s t a n c e f r o m
the a r e a l c e n t e r , ' P r e p r e s e n t s r a i n f a l l a m o u n t s a t
40
FIGURE 39 95 PER CENT CONFIDENCE RANGE FOR STORM RAINFALL
41
FIGURE 40 95 PER CENT CONFIDENCE RANGE FOR WEEKLY RAINFALL
the observation gage; a, b, c, d, and e a r e r e g r e s sion constants.
The equation obtained by fitting Expression
(13) to the data is:
Log E = -1.411 + 0.51 P 0 . 5
-0.01 Log A + 0.37 Log D
(14)
In this equation E and P a r e in inches, A is in
square miles and D is in miles. The relation is
illustrated for an a r e a of 100 square miles in
Figure 41.
Within the limits of the data used, the magnitude of the coefficient of Log A suggests that a r e a
was not an important variable in the analysis. This
42
r e s u l t i s r a t h e r difficult t o r a t i o n a l i z e . H o w e v e r ,
it m a y be due to the fact t h a t a gage l o c a t e d at any
given d i s t a n c e f r o m t h e c e n t e r of two a r e a s of diff e r e n t s i z e i s r e l a t i v e l y c l o s e r t o the c e n t e r o f the
l a r g e r a r e a . C o n s e q u e n t l y , the gage m a y b e i n a n
e q u a l l y good p o s i t i o n for s a m p l i n g the m e a n r a i n fall i n b o t h a r e a s .
A n e m p i r i c a l e q u a t i o n f o r the s t a n d a r d d e v i a tion w a s d e t e r m i n e d for u s e i n e s t i m a t i n g c o n f i d e n c e
limits. This equation is:
Log s =
-1.179
+ 0.41
P 0.5
- 0.01 Log A + 0.30 Log D
(15)
w h e r e s is in i n c h e s and P, A, and D h a v e the s a m e
d e f i n i t i o n and u n i t s of m e a s u r e m e n t as they h a d in
E q u a t i o n (14). T h e 9 5 p e r c e n t confidence l i m i t s
for v a r i o u s v a l u e s of P, A, and D m a y be c o m puted f r o m P ± Z s.
FIGURE 41 EFFECT OF GAGE LOCATION ON SINGLE GAGE
ESTIMATES OF MEAN RAINFALL ON 100 SQUARE MILE AREA
Since A c o n t r i b u t e s v e r y l i t t l e to the value of E
and s in E q u a t i o n s (14) and (15), the a r e a v a r i a b l e
could b e o m i t t e d without s i g n i f i c a n t e r r o r w i t h i n
the l i m i t s o f a r e a t e s t e d i n t h i s study. T h e s e e q u a t i o n s would t h e n be of the s a m e f o r m as E q u a t i o n s
(4) and (6) in t h e v a r i a t i o n of r a i n f a l l w i t h d i s t a n c e
study d i s c u s s e d in the p r e v i o u s s e c t i o n of t h i s
b u l l e t i n . The coefficient of P 0 . 5 is the s a m e and
the coefficient of Log D n e a r l y equal in both s t u d i e s .
T h e s e r e s u l t s s u g g e s t that the e s t i m a t o r s P and
D give p r a c t i c a l l y t h e s a m e m e a s u r e m e n t e r r o r
in both a n a l y s e s .
43
R E L I A B I L I T Y O F S T O R M MEAN
RAINFALL ESTIMATES
Introduction
In the p r e v i o u s s e c t i o n , the a c c u r a c y of a r e a l
r a i n f a l l e s t i m a t e s o b t a i n e d f r o m a single gage at
o r n e a r t h e c e n t e r o f a n a r e a was i n v e s t i g a t e d .
H o w e v e r , a n e s t i m a t e o f a v e r a g e rainfall m a y b e
r e q u i r e d f o r a n a r e a o n which m o r e than one g a g e
o c c u r s . C o n s e q u e n t l y , a n a n a l y s i s was m a d e t o
d e t e r m i n e a n e s t i m a t e o f both the a v e r a g e e r r o r
and the s t a n d a r d e r r o r involved i n m e a s u r i n g a r e a l
m e a n r a i n f a l l b y m e a n s o f v a r i o u s gage d e n s i t i e s
within s e v e r a l a r e a s o f d i f f e r e n t s i z e .
D a t a Used
Data f r o m the P a n t h e r C r e e k , Goose C r e e k , E l
P a s o and e a s t c e n t r a l I l l i n o i s n e t w o r k s w e r e i n cluded in the a n a l y s i s . Only s h o w e r and t h u n d e r s t o r m r a i n f a l l w e r e c o n s i d e r e d . The 1 0 0 - s q u a r e
m i l e G o o s e C r e e k n e t w o r k w a s subdivided i n t o
n e t w o r k s c o m p r i s i n g 25 and 50 s q u a r e m i l e s f o r
the 1952, 1953, and 1954 s e a s o n s . Two 2 0 0 - s q u a r e
m i l e n e t w o r k s w e r e c h o s e n ; one from the 2 8 0 square m i l e El P a s o network (Fig. 2) operated in
1948 and 1949, and the o t h e r f r o m the 4 0 0 - s q u a r e
mile east central Illinois network (Fig. 5) o p e r a t e d in 1955 and 1956. T h e s e d a t a p e r m i t t e d an
a n a l y s i s f o r a r e a s o f 2 5 , 50, 100, 200, a n d 400
square m i l e s .
Sampling P r o c e d u r e s
More t h a n one s a m p l i n g plan was c o n s i d e r e d
t o a s c e r t a i n the m o s t a p p l i c a b l e plan for s a m p l i n g
a r e a l m e a n rainfall. T h r e e sampling plans w e r e
t r i e d on a 1 0 0 - s q u a r e m i l e n e t w o r k . T h e s e i n cluded the r a n d o m s t a r t , a p l a n which c o m b i n e d
c e n t r a l l y l o c a t e d s a m p l e s with r a n d o m s t a r t
s a m p l e s , and a s i n g l e b e s t - c e n t e r e d s a m p l i n g p l a n .
A r a n d o m s a m p l i n g p r o c e d u r e (not u s e d in t h i s
study) a l l o w s the s e l e c t i o n of g a g e s in each s a m p l e
to be d e t e r m i n e d e n t i r e l y by c h a n c e . A s t r a t i f i e d
r a n d o m s a m p l i n g plan p r o v i d e s a m o r e c o n s i s t e n t ly u n i f o r m d i s t r i b u t i o n of g a g e s in e a c h s a m p l e
than that obtained by a p u r e l y r a n d o m plan, but
a l l o w s the s e l e c t i o n of g a g e s to be d e t e r m i n e d
m o r e b y c h a n c e than d o e s a r a n d o m s t a r t s a m p ling p l a n , for e x a m p l e . S a m p l i n g p l a n s which i n volve the s e l e c t i o n of c e n t r a l l y l o c a t e d g a g e s in
c o n t i g u o u s a r e a s r e q u i r e the o m i s s i o n of a c o n s i d e r a b l e n u m b e r o f o b s e r v a t i o n s from the a n a l y s i s . A random start systematic sampling procedure
p r o v i d e s a plan for s p r e a d i n g the s a m p l e o b s e r v a t i o n s o v e r the n e t w o r k , and a t the s a m e t i m e ,
p e r m i t s t h e u s e o f d a t a f r o m all g a g e s .
R a n d o m S t a r t . T h e s a m p l i n g p r o c e d u r e for the
r a n d o m s t a r t s y s t e m a t i c s a m p l i n g plan can b e
i l l u s t r a t e d by r e f e r e n c e to F i g u r e 42. When t h e
total n u m b e r of g a g e s in a n e t w o r k is 48 and it is
d e s i r e d to t a k e s a m p l e s of 3 g a g e s , the 48 g a g e s
a r e divided into 16 c o n t i g u o u s g r o u p s of 3 g a g e s
e a c h . A c c o r d i n g t o the r a n d o m s t a r t plan, a s t a r t ing p o s i t i o n is s e l e c t e d in one g r o u p of g a g e s . One
gage i n a p p r o x i m a t e l y the s a m e l o c a t i o n i s a u t o m a t i c a l l y d e s i g n a t e d in e a c h of the o t h e r g r o u p s to
c o m p l e t e the o b s e r v a t i o n s in e a c h s a m p l e .
In a
FIGURE 42 GROUPS OF GAGES FROM WHICH 3 RANDOM START
SYSTEMATIC SAMPLES OF SIZE 16 WERE SELECTED
network of 48 gages, there a r e three possible
s a m p l e s of 16 which a r e t r e a t e d as h a v i n g e q u a l
p r o b a b i l i t y of b e i n g an a c t u a l s a m p l e of 16 g a g e s .
Other s a m p l e sizes may be designated in a s i m i l a r
manner.
C o m b i n e d S a m p l i n g P l a n . The r a n d o m s t a r t
s a m p l i n g p l a n d o e s not p r o v i d e for a s i n g l e g a g e
s a m p l e . When the s a m p l e s i z e i s d e c r e a s e d t o one
g a g e , the r a n d o m s t a r t plan of dividing the n e t w o r k into g r o u p s o f c o n t i g u o u s g a g e s b e c o m e s t h e
s a m e a s s e l e c t i n g one o b s e r v a t i o n a t r a n d o m . T h e
v a r i a n c e of t h e s e e s t i m a t e s would be e q u i v a l e n t
t o the p u r e l y r a n d o m s a m p l i n g v a r i a n c e for s i n g l e
g a g e s a m p l e s . T h e r e i s a l s o a t e n d e n c y for the
r a n d o m s t a r t s y s t e m a t i c s a m p l i n g plan t o a p p r o a c h
r a n d o m s a m p l i n g for o t h e r s m a l l s a m p l e s , b e c a u s e
the s p r e a d of o b s e r v a t i o n s in each s a m p l e b e c o m e s
l e s s u n i f o r m o v e r the a r e a s a s the s a m p l e s i z e d e c r e a s e s . T h i s f e a t u r e o f the r a n d o m s t a r t plan a l lows s a m p l i n g e r r o r s for the s m a l l s a m p l e s i z e s
which a r e s o m e w h a t l a r g e r t h a n would b e o b s e r v e d
n o r m a l l y i n a c t u a l p r a c t i c e , for the r e a s o n t h a t
gages in an operating network usually approach
a uniform distribution.
When one is d e s i g n i n g a n e t w o r k , e a c h r a i n
gage would l o g i c a l l y be p l a c e d r e l a t i v e l y c l o s e to
the c e n t e r of the a r e a to be gaged. A c e n t r a l l y l o c a t e d gaging plan should p e r m i t s a m p l i n g e r r o r s
of a m a g n i t u d e l e s s than a p l a n in which g a g e l o c a t i o n s w e r e left p a r t i a l l y o r wholly t o c h a n c e . T h i s
a r g u m e n t m a y b e a d v a n c e d for r a n d o m s t a r t s y s t e m a t i c s a m p l e s i z e s of 2, 3, 4, and p o s s i b l y 6 and 8
g a g e s in 100 s q u a r e m i l e s . C o n s e q u e n t l y , a s a m p ling plan w a s t r i e d which involved s a m p l e s i z e s of
1, 2, 3, 4, 6, and 8 which w e r e o b t a i n e d f r o m a
c e n t r a l l y l o c a t e d plan a s i l l u s t r a t e d i n F i g u r e s
43-A through 4 3 - F .
44
FIGURE 43 COMBINED SAMPLING PLAN FOR SAMPLES OF SIZE 1, 2, 3, 4, 6 AND 8
45
FIGURE 44 BEST-CENTERED SAMPLING PLAN FOR PANTHER CREEK NETWORK
F o r s a m p l e s o f s i z e 1 , five g a g e s w e r e s e l e c t e d
within 1 - 1 / 4 m i l e s of t h e g e o g r a p h i c c e n t e r of the
n e t w o r k t o r e p r e s e n t t h e p o s s i b l e s a m p l e s for t h i s
s i z e . F o r s a m p l e s of s i z e 2, 3, 4, 6, and 8 the n e t w o r k w a s divided into 2, 3, 4, 6, and 8 s e c t i o n s ,
r e s p e c t i v e l y . G a g e s l o c a t e d n e a r the c e n t e r s o f
these sections were designated as observation
s t a t i o n s . A s the s a m p l e s i z e i n c r e a s e d beyond
eight, i t b e c a m e i m p o s s i b l e t o s e l e c t s a m p l e s
which w o u l d b e m o r e c e n t r a l l y l o c a t e d than the
r a n d o m s t a r t s a m p l e s . C o n s e q u e n t l y , the r a n d o m
s t a r t s y s t e m a t i c p l a n w a s u s e d for d e s i g n a t i n g
s a m p l e s h a v i n g m o r e t h a n eight g a g e s .
B e s t - C e n t e r e d . T h e t h i r d s a m p l i n g plan w h i c h
w a s s e l e c t e d was t h a t defined a s the one b e s t s u b -
s a m p l e f o r e a c h gage d e n s i t y , t h a t i s , a p p r o x i m a t e l y a u n i f o r m grid o r c e n t e r e d s y s t e m a t i c
s a m p l e . T h i s s a m p l i n g p l a n h a s one a d v a n t a g e
o v e r e i t h e r of the o t h e r two d e s c r i b e d , b e c a u s e it
m o r e c l o s e l y a p p r o a c h e s the p r a c t i c a l n e t w o r k
d e s i g n . T h e b e s t - c e n t e r e d plan h a s the a d v a n t a g e
of p r o v i d i n g d a t a t h a t a l l o w for e a s y c a l c u l a t i o n of
e i t h e r the a v e r a g e o r the s t a n d a r d e r r o r o f m e a n
r a i n f a l l . H o w e v e r it h a s t h e d i s a d v a n t a g e of b e i n g
d e p e n d e n t on a s i n g l e s u b j e c t i v e s e l e c t i o n of a
b e s t s a m p l e . T h i s plan d o e s not u t i l i z e i n f o r m a t i o n
t h a t w a s g a t h e r e d at all the o t h e r g a g e s of the n e t work except in the estimation of the network a v e r a g e . Gage l o c a t i o n s u s e d for t h i s s a m p l i n g p l a n a r e
shown in F i g u r e s 44A t h r o u g h 44D.
45
Analytical Procedure
T h e m a i n p r o b l e m i n the a n l y s i s w a s the e s t i m a t i o n of the t r u e v a r i a n c e of the e s t i m a t e d a v e r a g e p r e c i p i t a t i o n , P , about the t r u e a v e r a g e p r e c i p i t a t i o n for the n e t w o r k a r e a u n d e r c o n s i d e r a t i o n .
An e s t i m a t e of t h i s v a r i a n c e l e a d s to an e x p e c t e d
m e a s u r e o f the e r r o r involved i n s a m p l i n g with
e a c h d i f f e r e n t gage d e n s i t y . The c o m p u t a t i o n cons i s t e d m a i n l y o f two s t e p s . T h e f i r s t s t e p w a s the
d e t e r m i n a t i o n of the d e v i a t i o n s f r o m the n e t w o r k
m e a n by the following f o r m u l a
v a r i a t i o n in the d i s t r i b u t i o n of r a i n f a l l o v e r the
network among s t o r m s of s i m i l a r m e a n rainfall.
A p r a c t i c a l q u a n t i t a t i v e m e a s u r e of a r e a l rainfall
d i s t r i b u t i o n which c a n b e i n c l u d e d i n E x p r e s s i o n
(2) is difficult to o b t a i n . It was t h o u g h t that the
d u r a t i o n o f r a i n f a l l m i g h t p a r t i a l l y r e f l e c t the a r e a l
d i s t r i b u t i o n , s i n c e t h e d i s t r i b u t i o n of point r a i n f a l l
a m o u n t s m a y d e p e n d upon the s t o r m d u r a t i o n f a c t o r . C o n s e q u e n t l y , the d u r a t i o n f a c t o r , T , w a s
a d d e d to E x p r e s s i o n (2) and the g o o d n e s s of fit of
an e x p r e s s i o n of t h e f o r m
L o g s = a1 + b1 L o g P + c1 Log N + e1 Log T
w h e r e d r e p r e s e n t s the a b s o l u t e v a l u e of the d i f f e r e n c e b e t w e e n P , the s a m p l e m e a n r a i n f a l l f r o m a
s a m p l e of N g a g e s , and
the a r e a l m e a n r a i n f a l l
b a s e d on the total n u m b e r of g a g e s in the n e t w o r k .
T h e s e c o n d s t e p in d e t e r m i n i n g an e s t i m a t e of
the e r r o r involved t h e fitting of a r e g r e s s i o n s y s t e m t o the d e v i a t i o n s . The r e g r e s s i o n l i n e s p r o v i d e
v a l u e s which a r e d e s i g n a t e d a s the e s t i m a t e o f the
e r r o r of P about
A d i s c u s s i o n of the r e g r e s s i o n
s y s t e m follows.
F r o m g r a p h i c a l plots o f the d a t a i t w a s o b s e r v e d , in g e n e r a l , t h a t the d e v i a t i o n of the s a m p l e
estimate from
increased as P i n c r e a s e d , although t h e r e was c o n s i d e r a b l e fluctuation i n t h i s
u p w a r d t r e n d . A l s o , the d e v i a t i o n s g e n e r a l l y inc r e a s e d a s the n u m b e r o f o b s e r v a t i o n s ( r a i n g a g e s )
i n the s a m p l e d e c r e a s e d . U n d o u b t e d l y , t h e r e a r e
m a n y o t h e r f a c t o r s which c o n t r i b u t e t o the v a r i a bility of these deviations from s t o r m to s t o r m ,
s u c h a s the (1) m e t e o r o l o g i c a l f a c t o r s c a u s i n g the
s t o r m , (2) location o f the s t o r m c o r e with r e s p e c t
to the c e n t e r of the n e t w o r k , (3) d u r a t i o n of the
s t o r m , and (4) r a t e of r a i n f a l l . H o w e v e r , it is
difficult to e x p r e s s (1) and (2) q u a n t i t a t i v e l y and
P is a function of (3) and (4).
G r a p h i c a l plots i n d i c a t e d that the a v e r a g e
m a g n i t u d e of the d e v i a t i o n s was a l s o a function of
N. M a t h e m a t i c a l e x p r e s s i o n s of the f o r m
Log s = a + b Log P + c Log N
(2)
and
Log s = a + b P
m
+ c Log N
(3)
a p p e a r e d to p r o v i d e t h e b e s t fit to the d a t a . In t h e s e
e x p r e s s i o n s , s r e p r e s e n t s the s t a n d a r d d e v i a t i o n
of the a b s o l u t e v a l u e s of d f r o m E x p r e s s i o n (1).
T h e g o o d n e s s of fit of E x p r e s s i o n (2) was d e t e r mined in preliminary analyses.
R e s u l t s of A n a l y s i s on 1 0 0 - S q u a r e Mile N e t w o r k
R a n d o m S t a r t . When E x p r e s s i o n (2) w a s fitted
t o the d e v i a t i o n s o b t a i n e d b y the r a n d o m s t a r t s y s t e m a t i c s a m p l i n g p l a n , the m u l t i p l e c o r r e l a t i o n
coefficient was 0 . 5 2 . Although this c o r r e l a t i o n i s
s i g n i f i c a n t , a c o r r e l a t i o n of g r e a t e r d e g r e e is
d e s i r a b l e for p r e d i c t i n g the e x p e c t e d d e v i a t i o n
b e t w e e n the s a m p l e and the a r e a l m e a n r a i n f a l l .
It w a s felt that a p a r t of the v a r i a t i o n not a c counted for by the r e g r e s s i o n s y s t e m w a s d u e to
(4)
w a s d e t e r m i n e d . F i t t i n g E x p r e s s i o n (4) t o the d a t a
r e s u l t e d in a m u l t i p l e c o r r e l a t i o n of 0 . 5 3 . It is
e v i d e n t that the d u r a t i o n v a r i a b l e did not signific a n t l y i n c r e a s e t h e efficiency i n p r e d i c t i n g the
e x p e c t e d d e v i a t i o n . C o n s e q u e n t l y , it w a s felt that
a d d i n g the d u r a t i o n f a c t o r t o the r e g r e s s i o n s y s t e m
did not w a r r a n t the e x t r a w o r k i n v o l v e d .
C o m b i n e d S a m p l i n g P l a n . When E x p r e s s i o n (2)
w a s fitted to the d a t a for the d e v i a t i o n s obtained
f r o m the c o m b i n e d s a m p l i n g p l a n , t h e r e s u l t i n g
m u l t i p l e c o r r e l a t i o n coefficient w a s 0 . 7 0 . This i n c r e a s e i n t h e c o r r e l a t i o n coefficient f r o m 0.52 for
the random start plan apparently reflects a general
i n c r e a s e i n s t a b i l i t y o f the s a m p l e e s t i m a t e s f r o m
t h e m o r e u n i f o r m d i s t r i b u t i o n of g a g e s for the
small sample sizes.
Another independent variable was then introd u c e d into the p r e v i o u s r e g r e s s i o n s y s t e m i n o r d e r
to e x a m i n e m o r e t h o r o u g h l y the e s t i m a t e of the
s a m p l i n g v a r i a t i o n . An e x p r e s s i o n of t h e following
form
w a s fitted to the d a t a , w h e r e s
is t h e s t a n d a r d
d e v i a t i o n of the e n t i r e n e t w o r k of g a g e r e a d i n g s ,
e is a c o n s t a n t , and the o t h e r s y m b o l s r e p r e s e n t
t h e s a m e q u a n t i t i e s a s t h e y did i n p r e v i o u s d i s c u s s i o n s . T h e s variable r e p r e s e n t s the best availa b l e e s t i m a t e o f v a r i a t i o n i n the a r e a l p r e c i p i t a t i o n p a t t e r n . T h e s a m p l i n g e r r o r for v a r i o u s s a m p l e s i z e s should d e p e n d c o n s i d e r a b l y upon the
v a r i a b i l i t y of the p r e c i p i t a t i o n p a t t e r n . When E x p r e s s i o n (5) was fitted to the d a t a the m u l t i p l e
c o r r e l a t i o n w a s i n c r e a s e d f r o m 0.70 t o 0.84. T h e
s i m p l e c o r r e l a t i o n c o e f f i c i e n t s b e t w e e n log s and
log P , log N , and l o g w e r e 0.40, 0 . 5 8 and 0 . 6 5 ,
respectively.
F r o m the p r e c e d i n g c o r r e l a t i o n c o e f f i c i e n t s i t
i s evident that the e x p e c t e d s a m p l i n g e r r o r i s
h i g h l y d e p e n d e n t upon the v a r i a t i o n in r a i n f a l l , as
m e a s u r e d by the s t a n d a r d d e v i a t i o n of a r e l a t i v e l y
d e n s e network of r a i n g a g e s . It is a l s o evident, as
would be e x p e c t e d , t h a t the s a m p l i n g e r r o r has a
h i g h e r d e g r e e of c o r r e l a t i o n with v a r i a t i o n of r a i n fall than with e i t h e r m e a n r a i n f a l l o r g a g e d e n s i t y .
It should be noted that the i n s e r t i o n of the s t a n d a r d d e v i a t i o n o f point r a i n f a l l t o t a l s f r o m all g a g e s
i n t o the e q u a t i o n p r o d u c e s a n e s t i m a t e that i s not
p r a c t i c a l f r o m the h y d r o l o g i c s t a n d p o i n t . D e n s e
n e t w o r k s o f r a i n g a g e s a r e s e l d o m u s e d except i n
r e s e a r c h . The hydrologist frequently n e e d s a means
o f e s t i m a t i n g the s a m p l i n g e r r o r for m e a n rainfall
47
a m o u n t s which have b e e n obtained f r o m a s p a r s e
n e t w o r k of g a g e s . C o n s e q u e n t l y , the s t a n d a r d d e v i a t i o n , sp of point r a i n f a l l a m o u n t s i n c l u d e d in e a c h
s a m p l e was i n s e r t e d into E x p r e s s i o n ( 5) in p l a c e of
F o r e x a m p l e , if five g a g e s w e r e i n c l u d e d in a
s a m p l e , sp would be the s t a n d a r d d e v i a t i o n of the
five point r a i n f a l l a m o u n t s . The m a g n i t u d e of s P is
a m e a s u r e of the p r e c i s i o n of a s a m p l e m e a n r a i n fall v a l u e . T h i s v a r i a b l e should have a n influence
upon e s t i m a t e s o f the s a m p l i n g e r r o r for a r e a l
m e a n r a i n f a l l v a l u e s . I n s e r t i n g this v a r i a b l e into
the e x p r e s s i o n r e s u l t e d in a m u l t i p l e c o r r e l a t i o n
of 0 . 7 1 . T h i s is a v e r y s m a l l i n c r e a s e o v e r the
0.70 coefficient which w a s o b t a i n e d f r o m E x p r e s s i o n
(5). C o n s e q u e n t l y , i t m u s t b e concluded that the s p
v a r i a b l e is not useful as a t h i r d i n d e p e n d e n t v a r i a b l e . The inefficacy of s P p r o b a b l y r e f l e c t s the fact
t h a t the v a r i a t i o n of s a m p l e s t a n d a r d d e v i a t i o n s
w i t h i n s t o r m s a p p r o x i m a t e d the v a r i a t i o n i n the
s t a n d a r d d e v i a t i o n s of a p a r t i c u l a r s e t of gage
readings from s t o r m to s t o r m . Effectiveness of
the s a m p l e s t a n d a r d d e v i a t i o n v a r i a b l e i n c r e a s e d
a s s a m p l e s i z e i n c r e a s e d . H o w e v e r , the l a r g e r
v a r i a t i o n a s s o c i a t e d with s m a l l s a m p l e s i z e s offs e t m o s t of the effect of t h e s t a n d a r d d e v i a t i o n s
f o r l a r g e r s a m p l e s i z e s . T h e efficiency o f s t a n d a r d deviation as a third independent variable
r e a c h e d a m a x i m u m when all g a g e s w e r e i n c l u d e d ,
a s w a s the c a s e when the
v a r i a b l e was u s e d .
B e s t - C e n t e r e d . A s t a n d a r d d e v i a t i o n of d v a l u e s
f r o m E x p r e s s i o n (1) w a s o b t a i n e d b y a n a d j u s t m e n t
s i m i l a r t o t h a t u s e d b y L i n s l e y and K o h l e r i n t h e i r
a n a l y s i s of d a t a f r o m a c o n c e n t r a t e d n e t w o r k n e a r
Wilmington,
Ohio.(15)
S t o r m d a t a w e r e grouped
i n t o c l a s s e s o n the b a s i s o f s a m p l e m e a n r a i n f a l l
as m e a s u r e d by a s a m p l e of N g a g e s . A s t a n d a r d
d e v i a t i o n w a s c o m p u t e d for e a c h c l a s s . E x p r e s s i o n
(2) w a s t h e n fitted to the d a t a , using t h e s t a n d a r d
d e v i a t i o n for e a c h c l a s s a s the d e p e n d e n t v a r i a b l e .
T h e s a m p l e s i z e and the m i d - p o i n t of the rainfall
for e a c h c l a s s w e r e the independent v a r i a b l e s . A
m u l t i p l e c o r r e l a t i o n coefficient of 0.86 was o b t a i n e d with t h i s r e g r e s s i o n s y s t e m .
C o m p a r i s o n of R e s u l t s from T h r e e Sampling P l a n s
A c o m p a r i s o n of sampling e r r o r equations for
the t h r e e sampling plans can be made by r e f e r e n c e
to F i g u r e 45. It is evident that the random s t a r t
plan will predict considerably higher sampling
e r r o r s than either of the other p l a n s . The other
two sampling plans resulted in e r r o r s which a r e
v e r y s i m i l a r in magnitude within the range of data
used. The a and c values for the b e s t - c e n t e r e d and
the combination plan a r e a l m o s t identical. The
differences between the b values could be attributed
to s a m p l i n g v a r i a t i o n .
R e s u l t s of Analysis for Areas of Different Size
Average E r r o r .
The b e s t - c e n t e r e d sampling
plan a p p r o a c h e s the p r a c t i c a l situation which hyd r o l o g i s t s must contend with in using the r e s u l t s
of sampling e r r o r a n a l y s e s . In addition, this plan
ranked well in the c o m p a r i s o n with other sampling
plans which were considered. Consequently, the
b e s t - c e n t e r e d s a m p l i n g plan was chosen for a
m o r e complete a n a l y s i s of data involving a r e a s of
25, 50, 100, 200, and 400 s q u a r e m i l e s .
As noted p r e v i o u s l y in the s e c t i o n s on variation
of point rainfall with distance and a r e a l r e p r e s e n t a t i v e n e s s of point rainfall, graphical plots of the data
indicated that an equation for a v e r a g e e r r o r , E, of
the form
Log E = a 3 + b 3 P0.5 + c 3 Log N
(6)
would provide a b e t t e r fit to the data at high and
low values of P than an e x p r e s s i o n of the form
given in (2). Consequently, an e x p r e s s i o n of the
f o r m shown in (6) was applied to data for all five
a r e a s . The values of a 3 , b 3 , and c 3 a r e tabulated in
Table 15.
TABLE 15
REGRESSION CONSTANTS
Area
(Sq. M i . )
25
50
100
200
400
Log E
a3
-1.720
-1.597
-1.439
-1.305
-1.356
b3
0.65
0.56
0.59
0.49
0.54
Log S
c3
-0.73
-0.55
-0.65
-0.56
-0.48
a4
-1.556
-1.421
-1.275
-1.220
-1.198
b4
0.60
0.53
0.49
0.50
0.50
c4
-0.63
-0.60
-0.63
-0.58
-0.54
More data should be added to the a n a l y s i s to
s u b s t a n t i a t e the t r e n d s in b 3 and c 3 The Water
Survey has rain-gage networks in operation which
will furnish these data.
Since gage density is a more commonly used
variable than the sample size variable, the constants a3 and c3 presented in Table 15 were adjusted to express the sample size factor in terms
of gage density by the substitution
Log N = Log A - Log G
(7)
where G is in square miles per gage. Equations
for Log E with the adjusted constants are shown in
Table 16.
FIGURE 45 COMPARISON OF STORM SAMPLING PLANS
ON A 100 SQUARE MILE AREA
Although the computed E tends generally to increase with area, the individual equations permit
48
T A B L E 16
EQUATIONS F O R A V E R A G E AND STANDARD DEVIATIONS
Area
(Sq. Mi.)
Log E
=
a5
25
50
100
200
400
=
=
=
=
=
-2.740
-2.531
-2.739
-2.594
-2.605
Combined
=
Log s
=
-2.642
a
+
=
=
=
=
=
-2.395
-2.440
-2.535
-2.555
-2.603
Combined
=
-2.506
p0.5
4
+
0.65
0.56
0.59
0.49
0.54
+
A-.07
5
P
P0.5
+
0 . 5
+
0.60
0.53
0.49
0.50
0.50
+
c5
Log G
0.73
0.55
0.65
0.56
0.48
0.794
+ b
6
25
50
100
200
400
b
0.692
0.966
c
6
A-.12
Log
Log G
G
0.63
0.60
0.63
0.58
0.54
A-.06
P0.5
+
0.746
A-.05
FIGURE 46 RELATION BETWEEN SAMPLING ERROR, STORM SIZE, AND GAGE DENSITY ON 100 SQUARE MILE AREA
Log G
49
FIGURE 47 95 PER CENT CONFIDENCE RANGE OF STORM MEAN RAINFALL FOR VARIOUS GAGE DENSITIES
50
r a n d o m f l u c t u a t i o n s in t h i s t r e n d . A c o m b i n e d e q u a tion which would r e p r e s e n t all five a r e a s w a s
d e s i r e d for the p u r p o s e of s m o o t h i n g and f o r int e r p o l a t i n g b e t w e e n a r e a s . Since the c o n s t a n t s
r e p r e s e n t e d by a 5 v a r y at r a n d o m , an a v e r a g e of
t h e s e v a l u e s m a y be used in a c o m b i n e d e q u a t i o n .
As p r e v i o u s l y n o t e d , t h e r e is a t e n d e n c y for the
v a l u e s of b 4 a n d C 5 to d e c r e a s e in m a g n i t u d e as
the a r e a i n c r e a s e s . C o n s e q u e n t l y , the v a l u e s o f
the b 4 a n d c r w e r e e x p r e s s e d a s a function o f a r e a
by e x p r e s s i o n of the following f o r m :
b 4 = k A1
(8)
and
c5 =
k1A1
(9)
T h e c o m b i n e d e q u a t i o n f o r L o g E which r e p r e s e n t s
all five a r e a s in the a n a l y s i s is shown in T a b l e 16.
R e l a t i o n s for a 1 0 0 - s q u a r e m i l e a r e a o b t a i n e d f r o m
the c o m b i n e d e q u a t i o n a r e p r e s e n t e d i n F i g u r e 46.
Confidence L i m i t s . T o e s t a b l i s h c o n f i d e n c e
l i m i t s , a n e s t i m a t e o f the s a m p l i n g s t a n d a r d d e v i a tion was r e q u i r e d . T h i s e s t i m a t e w a s obtained b y
r e p l a c i n g Log E in E x p r e s s i o n (6) of Log s and
fitting the r e s u l t i n g e x p r e s s i o n t o the d a t a , a s w a s
done for L o g E.
S t a n d a r d d e v i a t i o n s w e r e c o m p u t e d for the s a m e
c l a s s i n t e r v a l s u s e d i n d e t e r m i n i n g Log E . T h e
c o n s t a n t v a l u e s a r e shown i n T a b l e 15. The v a l u e s
of a 4
and c 4 w e r e a d j u s t e d for the gage d e n s i t y
f a c t o r a n d the r e s u l t i n g e q u a t i o n s for i n d i v i d u a l
a r e a s and for all a r e a s c o m b i n e d a r e shown i n
T a b l e 16.
The c o m b i n e d a r e a l e q u a t i o n for Log s c a n b e
u s e d t o e s t a b l i s h c o n f i d e n c e l i m i t s for a r e a l m e a n
r a i n f a l l . T h e 95 p e r cent confidence l i m i t s f o r P
w e r e e s t i m a t e d s i m p l y b y t a k i n g 2 s and a d d i n g
and s u b t r a c t i n g f r o m the c o r r e s p o n d i n g value o f P .
T h e r e s u l t i n g b a n d s a r e r e p r e s e n t e d for s e l e c t e d
v a l u e s o f G for the 1 0 0 - s q u a r e m i l e a r e a i n
Figure 47.
51
EXCESSIVE RAINFALL RELATIONS ON
100-SQUARE MILE AREA
Data collected from 20 recording rain gages
d u r i n g 19 5 0 - 5 3 on the 1 0 0 - s q u a r e m i l e P a n t h e r
C r e e k w a t e r s h e d (Fig. 3) w e r e used in a limited
a n a l y s i s of e x c e s s i v e rainfall relations. The following e x c e s s i v e rainfall definition of the U . S . W e a t h er B u r e a u w a s used in the s t u d y
R = T + 20
(1)
w h e r e R is the rainfall depth in inches and T is the
period of observation in minutes. Excessive rates
f o r p e r i o d s o f 3 0 , 6 0 , a n d 120 m i n u t e s w e r e i n v e s t i gated. The above definition gives lower limits of
0 . 5 0 , 0 . 8 0 , a n d 1.40 i n c h e s f o r t h e 3 0 - , 6 0 - , a n d
120-minute periods, respectively.
In addition to
the P a n t h e r C r e e k data, data f r o m 48 g a g e s on the
100-square
mile Goose C r e e k network (Fig. 4)
w e r e available for 19 53-54, and these have been
utilized to a l i m i t e d extent in the a n a l y s i s .
Gage Density to Observational Frequency
Relations
T h e effect of gage d e n s i t y upon the f r e q u e n c y of
o b s e r v a t i o n of e x c e s s i v e rainfall values in a 100s q u a r e m i l e a r e a was e x a m i n e d first. F o r this
p u r p o s e , the P a n t h e r C r e e k network was subdivided
into n e t w o r k s of 10, 5, 2, a n d 1 g a g e s , k e e p i n g t h e
distribution in each set as uniform as p o s s i b l e .
S i m i l a r l y , on the Goose C r e e k a r e a , n e t w o r k s of
24, 12, 6, 3, and 1 gages w e r e s e l e c t e d . T h e n u m b e r of e x c e s s i v e amounts r e c o r d e d by one o r m o r e
g a g e s for e a c h of the v a r i o u s n e t w o r k s within b o t h
a r e a s was then tabulated. Use of special recording
rain gages having 12.6-inch d i a m e t e r orifices and
6-hour charts enabled recording of 15-minute excessive
a m o u n t s o n the G o o s e C r e e k a r e a , i n
addition to t h o s e for the 3 0 - , 6 0 - , and 1 2 0 - m i n u t e
periods.
i n d i c a t e s that the o p t i m u m gage d e n s i t y for a 100s q u a r e m i l e a r e a , considering all f a c t o r s , is p r o b ably b e t w e e n 20 and 30 g a g e s . It a p p e a r s that
s t o r m s of adequate intensity to provide e x c e s s i v e
v a l u e s for the t i m e p e r i o d s i n v e s t i g a t e d a r e o f
sufficient a r e a l extent and of sufficient d u r a t i o n to
be a l m o s t always o b s e r v e d by a n e t w o r k of 20 to
30 g a g e s .
Effect
of
Gage
Density
on
Observed
Maximum
The effect o f gage d e n s i t y o n the m a x i m u m
o b s e r v e d excessive a m o u n t s within a 1 0 0 - s q u a r e
m i l e a r e a was investigated next, using the 1950-53
data for P a n t h e r C r e e k . F o r this p u r p o s e , n e t w o r k s
o f 2 0 , 1 0 , 5 , a n d 2 g a g e s w i t h i n t h e 100
square
m i l e s w e r e used. F o r each s t o r m , a ratio of the
m a x i m u m amount o b s e r v e d on the n e t w o r k to the
amount observed at the central gage in the n e t w o r k
w a s c o m p u t e d . T h i s w a s done for e a c h n e t w o r k and
for 3 0 - , 6 0 - , and 1 2 0 - m i n u t e p e r i o d s w h e n e v e r
e x c e s s i v e values w e r e r e c o r d e d by one or m o r e
of the 20 gages m a k i n g up the n e t w o r k of m a x i m u m density. Average ratios were then calculated
for e a c h of the s e v e r a l n e t w o r k s within the 100s q u a r e m i l e a r e a . The r e s u l t s of this study a r e
s u m m a r i z e d in Table 18 w h e r e the a v e r a g e r a t i o s
for each network within the 1 0 0 - s q u a r e m i l e a r e a
a r e p r e s e n t e d for 3 0 - , 6 0 - , and 1 2 0 - m i n u t e p e r i o d s .
The relation is further illustrated in F i g u r e 48.
R e s u l t s of this first p h a s e of the study a r e
s h o w n i n T a b l e 17. T h e e x p e c t e d t r e n d for the o b servational
frequency of
excessive amounts to
i n c r e a s e with increasing gage density is apparent.
Of p a r t i c u l a r i n t e r e s t a r e the data for 24 and 48
gages in the Goose C r e e k network. The v e r y slight
i n c r e a s e in frequency in going from 24 to 48 g a g e s
TABLE 17
RELATION BETWEEN EXCESSIVE RAINFALL
FREQUENCY AND GAGE DENSITY
Number
of
Gages
15
Number of C a s e s for
Given T i m e P e r i o d (Min.)
30
60
120
Total
FIGURE 48 EFFECT OF GAGE DENSITY ON MAXIMUM
OBSERVED EXCESSIVE RATES
Goose C r e e k Network. 100 Sq. Mi.. 1953-54
1
3
14
22
14
19
5
14
2
3
35
58
6
24
22
14
3
63
12
28
25
18
5
76
24
48
33
34
28
28
21
22
6
7
88
91
P a n t h e r Creek. 100 Sq. Mi.. 1950-53
1
2
5
10
20
------
21
34
41
49
52
14
17
25
31
33
5
10
14
15
18
40
61
80
95
103
TABLE 18
E F F E C T OF GAGE DENSITY ON OBSERVED
MAXIMUM AMOUNTS
Number
of
Gages
20
10
5
2
Av. Ratio, Maximum O b s e r v e d / C e n t r a l Gage
Amount for Given Time P e r i o d s (Min.)
30
60
120
1.89
1.70
1.50
1.27
1.75
1.62
1.43
1.21
1.52
1.40
1.30
1.13
52
A s e x p e c t e d the r a t i o t e n d s t o i n c r e a s e with inc r e a s i n g gage d e n s i t y . T h e r a t i o a p p e a r s t o b e s t i l l
i n c r e a s i n g a t a n a p p r e c i a b l e r a t e when the m a x i m u m network of 20 gages is reached.
P o i n t - A r e a l Mean R e l a t i o n s
N e x t , the r e l a t i o n b e t w e e n point e x c e s s i v e and
a r e a l m e a n e x c e s s i v e a m o u n t s was i n v e s t i g a t e d .
The a r e a l mean w a s obtained in each s t o r m by
a v e r a g i n g the h e a v i e s t a m o u n t o b s e r v e d a t e a c h o f
the 20 r a i n - g a g e s t a t i o n s f o r the p e r i o d s of 30, 60,
and 120 m i n u t e s . T h a t i s , the i n d i v i d u a l a m o u n t s
m a k i n g u p the m e a n m a y h a v e o c c u r r e d a t different
t i m e s a t the v a r i o u s n e t w o r k s t a t i o n s a l t h o u g h a l l
o c c u r r e d i n the s a m e s t o r m p e r i o d . T h i s definition
of the mean was c o n s i d e r e d m o r e d e s i r a b l e than
that o b t a i n e d by c a l c u l a t i n g m e a n s at a given t i m e
d u r i n g the s t o r m f o r all g a g e s . T h e c o r e o f h e a v y
rainfall, especially in t h u n d e r s t o r m s , frequently
c o v e r s a r e l a t i v e l y s m a l l a r e a ; c o n s e q u e n t l y , the
heaviest s t o r m r a t e s o c c u r at different t i m e s in
m o v i n g a c r o s s a n a r e a o f 100 s q u a r e m i l e s . Since
s u c h f a c t o r s a s s o i l e r o s i o n and s e d i m e n t a t i o n a p p e a r t o b e c l o s e l y r e l a t e d t o the s h o r t - p e r i o d ,
h e a v y r a i n f a l l r a t e s e x p e r i e n c e d i n s t o r m s , the
m e t h o d u s e d for c a l c u l a t i n g the m e a n s w a s c o n s i d e r e d t o h a v e the m o s t p r a c t i c a l a p p l i c a t i o n .
i s c o n s i d e r e d . F o r e x a m p l e , a s s u m e two t h u n d e r s t o r m s o f the s a m e s i z e and s a m e i n t e n s i t y , one
p a s s i n g o v e r the n e t w o r k with i t s c o r e p a s s i n g
t h r o u g h the c e n t e r o f the n e t w o r k , w h i l e the c o r e
o f the o t h e r s t o r m p a s s e s a l o n g o r n e a r t h e b o r d e r
o f the n e t w o r k . O b v i o u s l y , the h e a v i e s t r a t e s will
b e o b s e r v e d a t the c e n t r a l g a g e when a given s t o r m
p a s s e s t h r o u g h o r n e a r the c e n t e r o f t h e n e t w o r k
a n d t h e s e s t o r m s w i l l a l s o p r o v i d e the m o s t f r e quent a r e a l m e a n e x c e s s i v e r a t e s . A c o r r e l a t i o n
c o e f f i c i e n t of 0.96 w a s obtained b e t w e e n t h e 20g a g e m e a n and the c e n t r a l - g a g e point r a i n f a l l .
O b s e r v a t i o n s u s e d in the p r e c e d i n g a n a l y s i s of
p o i n t - a r e a l m e a n r e l a t i o n s included a c o n s i d e r a b l e
n u m b e r i n which n e i t h e r the c e n t r a l - g a g e a m o u n t
n o r the a r e a l m e a n rainfall r e p r e s e n t e d an exc e s s i v e a m o u n t , s i n c e only one e x c e s s i v e value
among the 20 gages w a s required for inclusion in
the a n a l y s i s . To d e t e r m i n e whether the inclusion of
v a l u e s b e l o w t h e e x c e s s i v e l e v e l m a t e r i a l l y affected
the p o i n t - a r e a l relation, another analysis was made
i n which only t h o s e c a s e s with a n
excessive
a m o u n t a t the c e n t r a l gage w e r e i n c l u d e d . T h i s
a n a l y s i s r e s u l t e d in d e v e l o p m e n t of a r e g r e s s i o n
e q u a t i o n given b y
Y = 0.11 + 0.78 X
(2)
w h e r e Y is the a r e a l m e a n r a i n f a l l in i n c h e s and X
i s the c e n t r a l g a g e a m o u n t i n i n c h e s . T h i s e q u a t i o n
i s i n s i g n i f i c a n t l y d i f f e r e n t f r o m the o n e i l l u s t r a t e d
in F i g u r e 49. A c o r r e l a t i o n coefficient of 0.97 w a s
o b t a i n e d c o m p a r e d t o 0.96 for the c u r v e i n F i g u r e
4 9 . R e s u l t s i n d i c a t e t h a t F i g u r e 4 9 c a n b e u s e d for
e s t i m a t i n g the a r e a l m e a n m a x i m u m o r a r e a l m e a n
e x c e s s i v e r a i n f a l l o n a 1 0 0 - s q u a r e m i l e b a s i n for
s h o r t p e r i o d s within s t o r m s , when a point r a i n f a l l
o b s e r v a t i o n i s a v a i l a b l e a t o r n e a r the c e n t e r o f
the a r e a .
FIGURE 49 RELATION BETWEEN POINT AND AREAL MEAN RATES
UNDER EXCESSIVE RATE CONDITIONS
F i g u r e 4 9 i l l u s t r a t e s t h e r e l a t i o n s b e t w e e n the
point r a i n f a l l o b s e r v e d a t the c e n t r a l gage and
a r e a l m e a n r a i n f a l l for 3 0 - , 6 0 - , and 1 2 0 - m i n u t e
p e r i o d s . T h e t h r e e p e r i o d s w e r e c o m b i n e d into one
relation after s e p a r a t e graphical plots indicated
i n s i g n i f i c a n t d i f f e r e n c e s a m o n g the r e l a t i o n s for
individual p e r i o d s . Points in F i g u r e 49 include all
c a s e s i n which a n e x c e s s i v e a m o u n t w a s r e c o r d e d
b y one o r m o r e g a g e s i n t h e 2 0 - g a g e n e t w o r k . T h e
r e g r e s s i o n e q u a t i o n i n d i c a t e s that the p o i n t rainfall
o b s e r v e d a t the c e n t e r o f a 1 0 0 - s q u a r e m i l e a r e a
i s s l i g h t l y a b o v e t h e a r e a l m e a n r a i n f a l l when e x c e s s i v e a m o u n t s a r e e x p e r i e n c e d (above 0.50 i n c h ) .
T h i s t r e n d i s t o b e e x p e c t e d when s t o r m m o v e m e n t
A s t u d y of t h e d a t a i n d i c a t e d the c e n t r a l gage
g i v e s a v e r y good m e a s u r e of the f r e q u e n c y of e x cessive areal mean rainfall. F o r example, on Pant h e r C r e e k d u r i n g 1 9 5 0 - 5 3 the n u m b e r o f a r e a l
m e a n e x c e s s i v e values was practically the s a m e as
t h e n u m b e r o b s e r v e d a t the c e n t r a l g a g e . F o r a
30-minute period, 21 excessive amounts w e r e obs e r v e d o n both a p o i n t and a r e a l b a s i s . F o r a 6 0 m i n u t e p e r i o d t h e r e w e r e 1 4 point e x c e s s i v e
a m o u n t s and 1 3 a r e a l e x c e s s i v e a m o u n t s , w h i l e for
a 120-minute period t h e r e w e r e 5 e x c e s s i v e values
f o r both c a s e s . C o m b i n i n g a l l t h r e e p e r i o d s , t h e r e
w e r e 40 excessive amounts observed at the central
gage c o m p a r e d to 39 c a s e s of excessive a r e a l m e a n
r a i n f a l l . H o w e v e r , t h i s d o e s not m e a n t h a t the point
a n d a r e a l m e a n r a t e s all o c c u r r e d i n the s a m e
s t o r m s . B y c o m p a r i s o n , i t w a s found t h a t e x c e s s i v e
a r e a l m e a n s o c c u r r e d 17 out of the 21 t i m e s that
e x c e s s i v e amounts w e r e observed at the central
g a g e . S i m i l a r l y , 11 out of 14 point e x c e s s i v e a m o u n t s
o c c u r r e d i n c o n j u n c t i o n with e x c e s s i v e a r e a l m e a n
v a l u e s f o r the 6 0 - m i n u t e p e r i o d , w h i l e a l l five of
t h e point and a r e a l e x c e s s i v e a m o u n t s c o i n c i d e d
f o r the 1 2 0 - m i n u t e p e r i o d .
F i g u r e 50 is an i l l u s t r a t i o n of t h e o c c u r r e n c e
of excessive rainfall amounts over the network
d u r i n g a n e x c e p t i o n a l l y h e a v y s t o r m d u r i n g the
e v e n i n g of J u n e 8, 1 9 5 1 . H e a v y e x c e s s i v e a m o u n t s
w e r e o b s e r v e d a t a l l s t a t i o n s within t h e n e t w o r k
53
during the storm period. An a r e a l average of 7.21
inches fell during the entire s t o r m which lasted
about four hours. Figure 51 illustrates the d i s t r i bution of excessive rainfall in the area by means
of area-depth curves. It is interesting to note that
the square-root relation for total storm rainfall
discussed in the section on area-depth relations
provides a good fit in these cases of unusually
heavy short-period amounts within a storm.
FIGLRE SO 30-MINUTE EXCESSIVE RATES, JULY 8, 1951
FIGURE 51 EXCESSIVE RATE AREA-DEPTH RELATIONS, JULY 8, 1951
55
RELATION B E T W E E N POINT AND AREAL
RAINFALL FREQUENCIES
Although m u c h i n f o r m a t i o n has b e e n p u b l i s h e d
o n point r a i n f a l l f r e q u e n c i e s , little d a t a a r e p r e s e n t l y a v a i l a b l e i n the l i t e r a t u r e o n a r e a l r a i n f a l l
f r e q u e n c i e s . The following p r e s e n t s l i m i t e d i n f o r m a t i o n on the s u b j e c t , o b t a i n e d f r o m a 2 0 - g a g e
n e t w o r k o n the 1 0 0 - s q u a r e m i l e P a n t h e r C r e e k
w a t e r s h e d d u r i n g the p e r i o d 1 9 4 8 - 5 3 .
of 25 and 50 s q u a r e m i l e s within the 1 0 0 - s q u a r e
m i l e n e t w o r k . B e c a u s e of m i n o r c h a n g e s in s o m e of
the g a g e l o c a t i o n s f r o m y e a r t o y e a r , i t w a s not
p o s s i b l e to obtain a 6 - y e a r c o m p a r i s o n for the 100s q u a r e m i l e a r e a . H o w e v e r , d a t a w e r e a v a i l a b l e for
a 3 - y e a r c o m p a r i s o n and t h e s e r e s u l t s a r e a l s o
given.
U s i n g d a t a f r o m this n e t w o r k , c o m p a r i s o n s h a v e
b e e n m a d e b e t w e e n point a n d a r e a l m e a n r a i n f a l l
f r e q u e n c i e s for the above 6 - y e a r p e r i o d for a r e a s
Within e a c h a r e a , the point r a i n f a l l f r e q u e n c i e s
w e r e o b t a i n e d f r o m t h e r a i n gage in the c e n t e r of
the a r e a . A r e a l m e a n r a i n f a l l was c a l c u l a t e d f r o m
FIGURE 52 COMPARISON BETWEEN POINT AND AREAL MEAN RAINFALL AT EQUAL FREQUENCIES ON 25 SQUARE MILE AREA
56
the a r i t h m e t i c a l a v e r a g e o f all gage o b s e r v a t i o n s .
Within the 2 5 - and 5 0 - s q u a r e m i l e a r e a s s t u d i e d ,
t h e r e w e r e 6 and 12 g a g e s , r e s p e c t i v e l y . To obtain
c o m p a r i s o n s b e t w e e n point and a r e a l f r e q u e n c i e s ,
all s t o r m s for e a c h a r e a w e r e r a n k e d b y m e a n
r a i n f a l l . S i m i l a r l y , s t o r m point r a i n f a l l t o t a l s a t
the center of each a r e a w e r e ranked. The rankings
t h e n p r o v i d e d a f r e q u e n c y d i s t r i b u t i o n of point and
a r e a l m e a n r a i n f a l l within a 6 - y e a r p e r i o d for the
2 5 - and 5 0 - s q u a r e m i l e a r e a s and w i t h i n a 3 - y e a r
p e r i o d for the 1 0 0 - s q u a r e m i l e a r e a . R e g r e s s i o n
e q u a t i o n s and c o r r e l a t i o n c o e f f i c i e n t s w e r e o b t a i n e d b e t w e e n point and a r e a l m e a n r a i n f a l l f r e q u e n c i e s for e a c h a r e a , t r e a t i n g equal p o s i t i o n s o f
r a n k a s p a i r s o f o b s e r v a t i o n s . T o i l l u s t r a t e the
p a i r i n g p r o c e d u r e , the h i g h e s t point r a i n f a l l o b s e r v e d a t the c e n t e r o f the 2 5 - s q u a r e m i l e a r e a ,
8.22 i n c h e s , w a s p a i r e d with the h i g h e s t m e a n r a i n fall for t h i s a r e a d u r i n g the 6 - y e a r c o m p a r i s o n
p e r i o d , which w a s 8.16 i n c h e s . O b v i o u s l y , the
paired observations in many cases occurred in
d i f f e r e n t s t o r m s , s i n c e the d a t a w e r e p a i r e d o n a
frequency distribution b a s i s .
R e l a t i o n s w e r e d e v e l o p e d for s t o r m r a i n f a l l ,
i n which a s i n g l e s t o r m w a s defined a s a l l r a i n f a l l
u n s e p a r a t e d by a b r e a k in p r e c i p i t a t i o n of six h o u r s
or more. Consequently, several showers were
f r e q u e n t l y i n c l u d e d in a s t o r m as d e f i n e d in t h i s
s t u d y . The r e s u l t s a r e p e r h a p s a l s o i n d i c a t i v e o f
d a i l y r a i n f a l l . T h e study w a s confined t o s h o w e r t y p e r a i n f a l l , the t y p e of s t o r m f r o m which flash
floods o n s m a l l a r e a s o c c u r . T h e r e s u l t s a r e c o n -
s i d e r e d i n d i c a t i v e a n d not definitive of p o i n t - a r e a l
frequency relations.
The r e g r e s s i o n e q u a t i o n s o b t a i n e d for the 2 5 and 5 0 - s q u a r e m i l e a r e a s w e r e Y = 0.96X and
Y = 0.94X, r e s p e c t i v e l y . The 6 - y e a r p o i n t - a r e a l
c o m p a r i s o n for the 2 5 - s q u a r e m i l e a r e a i s i l l u s t r a t e d i n F i g u r e 52.
R e s u l t s of the s t u d y i n d i c a t e t h a t an e x c e l l e n t
r e l a t i o n s h i p e x i s t s b e t w e e n point and a r e a l r a i n f a l l
f r e q u e n c i e s . A c o r r e l a t i o n coefficient of 0.99 was
o b t a i n e d b e t w e e n p o i n t and a r e a l r a i n f a l l for both
t h e 2 5 - and 5 0 - s q u a r e m i l e a r e a s . F o r t h e 100s q u a r e m i l e a r e a , a c o r r e l a t i o n coefficient of 0.99
w a s o b t a i n e d and t h e r e g r e s s i o n e q u a t i o n was
Y = 0.97X, v e r y c l o s e to the e q u a t i o n s o b t a i n e d for
the 2 5 - and 5 0 - s q u a r e m i l e a r e a s . I n d i c a t i o n s a r e
that a r e a l m e a n r a i n f a l l f r e q u e n c i e s a r e c l o s e to,
but s l i g h t l y l e s s t h a n , equivalent point r a i n f a l l f r e q u e n c i e s for s m a l l a r e a s .
A 3 - y e a r c o m p a r i s o n for a n o t h e r s e t of 2 5 - and
5 0 - s q u a r e m i l e a r e a s within the 1 0 0 - s q u a r e m i l e
w a t e r s h e d p r o d u c e d r e g r e s s i o n s of Y = 0.97X for
b o t h a r e a s , a g r e e i n g c l o s e l y with the 6 - y e a r c o m p a r i s o n s . A 3-year frequency c o m p a r i s o n was also
m a d e u s i n g d a t a o n l y for e x c e s s i v e r a i n f a l l r a t e s ,
b a s e d on the d e f i n i t i o n of e x c e s s i v e r a i n f a l l given
in the p r e v i o u s s e c t i o n of this b u l l e t i n . C o m b i n i n g
d a t a for e x c e s s i v e 3 0 - and 6 0 - m i n u t e a m o u n t s , a
r e g r e s s i o n e q u a t i o n , Y = 0.92X, w a s o b t a i n e d .
57
A M I C R O M E T E O R O L O G I C A L STUDY OF
R A I N F A L L VARIABILITY
Introduction
Analyses of rainfall data from s e v e r a l concent r a t e d r a i n - g a g e networks in central Illinois during
r e c e n t y e a r s have r e v e a l e d the f r e q u e n t p r e s e n c e
of r a i n f a l l g r a d i e n t s e x c e e d i n g 0.25 i n c h p e r m i l e in
shower-type rainfall. The relation between rainfall
v a r i a b i l i t y and e i t h e r d i s t a n c e o r a r e a i s p e r tinent in the d e t e r m i n a t i o n of gage d e n s i t y r e q u i r e m e n t s for s p e c i f i c p u r p o s e s , such a s : the c o l l e c t i o n
of s m a l l scale climatological data, hydrologic p r o j e c t s involving runoff and s e d i m e n t a t i o n i n v e s t i g a t i o n s , a g r i c u l t u r a l r e s e a r c h , and s t u d i e s c o n c e r n e d
with e s t a b l i s h i n g the u t i l i t y of r a d a r for q u a n t i t a t i v e
p r e c i p i t a t i o n m e a s u r e m e n t s . A l s o , the m a g n i t u d e
of r a i n f a l l v a r i a b i l i t y with d i s t a n c e should be cons i d e r e d i n d e t e r m i n i n g the s e n s i t i v i t y and c a l i b r a tion a c c u r a c y r e q u i r e d for r e c o r d i n g r a i n g a g e s ,
especially those employed in routine observational
p r o g r a m s w h e r e c o n c e n t r a t e d n e t w o r k s a r e not
feasible.
As p a r t of a p r o g r a m to i n v e s t i g a t e the q u a n t i tative relations existingbetween rainfall variability
and d i s t a n c e , a m i c r o - n e t w o r k of r a i n g a g e s w a s
e s t a b l i s h e d i n 1953. S i n c e the g r e a t e s t v a r i a t i o n i n
r a i n f a l l with d i s t a n c e n o r m a l l y o c c u r s with s h o w e r type r a i n f a l l , data w e r e c o l l e c t e d d u r i n g the s p r i n g ,
s u m m e r and fall s e a s o n s of 19 5 3 - 5 4 . S i m i l a r o b servations on a m e s o m e t e o r o l o g i c a l scale w e r e
a v a i l a b l e for c o m p a r i s o n p u r p o s e s f r o m o t h e r n e t w o r k s d e s c r i b e d i n the s e c t i o n o n r a i n - g a g i n g
facilities.
D e s c r i p t i o n of N e t w o r k
A m i c r o - n e t w o r k of 18 g a g e s w a s e s t a b l i s h e d
on l e v e l m e a d o w l a n d at the U n i v e r s i t y of Illinois
A i r p o r t . Its d e s i g n w a s b a s e d upon a v a i l a b l e i n f o r -
mation concerning rainfall gradients obtained from
rainfall studies on several concentrated watershed
n e t w o r k s d u r i n g 1 9 4 8 - 5 2 . The n e t w o r k c o n s i s t e d o f
p a i r s o f 8 - i n c h n o n - r e c o r d i n g g a g e s s p a c e d six
feet a p a r t a t e a c h o f n i n e s t a t i o n s w h i c h w e r e l o c a t e d at i n t e r v a l s of 300 feet to f o r m a s q u a r e g r i d
p a t t e r n ( F i g . 8). I n f o r m a t i o n o n s t o r m d u r a t i o n a n d .
r a i n f a l l i n t e n s i t y w a s obtained from a r e c o r d i n g
r a i n gage i n s t a l l e d a d j a c e n t t o the m i c r o - n e t w o r k .
Wind d a t a w e r e o b t a i n e d f r o m a n A e r o v a n e s y s t e m
l o c a t e d about 0.5 m i l e f r o m the n e t w o r k .
T h e c a l i b r a t i o n o f e a c h gage w a s c h e c k e d b e f o r e i n s t a l l a t i o n . T h e level of e a c h gage w a s
c h e c k e d p e r i o d i c a l l y and the g a g e s k e p t as f r e e of
d i r t , i n s e c t s and o t h e r f o r e i g n m a t e r i a l a s p o s s i b l e .
M e a s u r e m e n t s o f r a i n f a l l w e r e m a d e for e a c h o c c u r r e n c e of precipitation. These m e a s u r e m e n t s
w e r e m a d e a s soon a s p o s s i b l e a f t e r the end o f
each s t o r m to minimize evaporation effects. C a r e ful a t t e n t i o n w a s given t o gage e x p o s u r e s , m a i n t e n a n c e , and o b s e r v a t i o n a l t e c h n i q u e s . T o m i n i m i z e the r e a d i n g e r r o r , e x p e r i e n c e d o b s e r v e r s
r e a d the r a i n - g a g e a m o u n t s t o the n e a r e s t .001 i n c h .
A n a l y s i s of D a t a
Data w e r e collected from 93 s t o r m s from March
t o N o v e m b e r , d u r i n g 1953 and 1954. F o r e a c h s t o r m ,
average precipitation differences were calculated
f o r the v a r i o u s s e t s of gages l o c a t e d 6, 300, and
600 feet a p a r t on the g r i d p a t t e r n . R e s u l t s of t h i s
a n a l y s i s , c l a s s e d b y s t o r m s i z e , a r e shown i n
T a b l e 19. T h e m e a n m a x i m u m d i f f e r e n c e s shown
in T a b l e 19 a r e t h e a v e r a g e s for the h i g h e s t d i f f e r e n c e o b s e r v e d i n e a c h s t o r m . The a b s o l u t e m a x i m u m i s the h i g h e s t r e a d i n g o b t a i n e d d u r i n g the
1953-54 t e s t p e r i o d for e a c h c l a s s o f s t o r m s i z e ,
a s m e a s u r e d b y the n e t w o r k m e a n r a i n f a l l .
T A B L E 19
D I F F E R E N C E S B E T W E E N GAGE P A I R S
D i f f e r e n c e s (in.) for G i v e n S t o r m S i z e s (in.)
0.01-0.19
0.20-0.49
6-Foot
Average
Mean Maximum
Absolute Maximum
N u m b e r of S t o r m s
0.003
0.009
0.015
46
0.005
0.013
0.046
19
300-Foot
Average
Mean Maximum
Absolute Maximum
N u m b e r of S t o r m s
0.005
0.013
0.026
46
0.009
0.025
0.049
19
600-Foot
Average
Mean Maximum
Absolute Maximum
N u m b e r of S t o r m s
0.006
0.015
0.034
46
0.012
0.027
0.047
19
0.50-0.99
1.00-1.99
Pairs
0.009
0.022
0.041
15
0.020
0.047
0.110
13
Pairs
0.013
0.034
0.048
15
0.027
0.079
0.165
13
Pairs
0.016
0.034
0.070
15
0.038
0.087
0.231
13
58
R e f e r e n c e to T a b l e 19 s h o w s that a v e r a g e 6foot d i f f e r e n c e s for s t o r m s h a v i n g m e a n r a i n f a l l u p
t o 0.50 inch a r e r a t h e r i n s i g n i f i c a n t . H o w e v e r ,
appreciable differences
were usually observed
with the h e a v i e r s t o r m s and o c c a s i o n a l l y with
light s t o r m s . Since the d i f f e r e n c e s h a v e a d e f i n i t e
t e n d e n c y t o i n c r e a s e with s t o r m s i z e , they c a n n o t
be attributed solely to reading e r r o r . Another fact o r which m a y h a v e c o n t r i b u t e d t o the d i f f e r e n c e s
o b s e r v e d b e t w e e n the 6-foot p a i r s i s the t u r b u l e n c e
c r e a t e d i n the wind flow about t h e g a g e s . T h e r e w a s
no way in which to s e p a r a t e t h e effects of r e a d i n g
and t u r b u l e n c e e r r o r s f r o m t h e r e a l d i f f e r e n c e s
o c c u r r i n g i n the s u r f a c e r a i n f a l l p a t t e r n . C o n s i d e r i n g the c a r e f u l a t t e n t i o n which w a s g i v e n t o
gage e x p o s u r e s , gage m a i n t e n a n c e , and o b s e r v a tional t e c h n i q u e s , i t i s b e l i e v e d that the d i f f e r e n c e s
which w e r e obtained b e t w e e n the 6-foot p a i r s r e p r e sent the m i n i m u m t o b e e x p e c t e d i n s h o w e r - t y p e
r a i n f a l l . F o r the 6-foot p a i r s a c o r r e l a t i o n c o e f f i c i e n t of 0.79 w a s o b t a i n e d b e t w e e n a v e r a g e s t o r m
d i f f e r e n c e s and s t o r m s i z e .
The t r e n d for the a v e r a g e d i f f e r e n c e t o inc r e a s e
with i n c r e a s i n g s t o r m s i z e a n d with d i s t a n c e i s
f u r t h e r i l l u s t r a t e d b y the s u m m a r i z e d d a t a f o r the
300-foot and 600-foot p a i r s in T a b l e 19. A c o r r e l a t i o n coefficient of 0.80 w a s o b t a i n e d b e t w e e n
a v e r a g e d i f f e r e n c e and m e a n r a i n f a l l for the 300foot p a i r s . S i m i l a r l y , a c o r r e l a t i o n c o e f f i c i e n t of
0.72 was obtained with the 600-foot p a i r s .
The d a t a w e r e f u r t h e r e x a m i n e d for d i f f e r e n c e s
b e t w e e n 6-foot g a g e p a i r s a r i s i n g f r o m m e t e o r ological factors other than m e a n rainfall. T h e s e
factors include s t o r m duration, m e a n rainfall r a t e ,
and wind. R e s u l t s of t h i s p h a s e of the s t u d y a r e
p r e s e n t e d i n T a b l e 20, w h e r e t h e r e l a t i v e v a r i a b i l i t y o f the a v e r a g e d i f f e r e n c e s h a s b e e n r e l a t e d
t o e a c h m e t e o r o l o g i c a l f a c t o r . The v a r i a b i l i t y
f a c t o r w a s obtained for e a c h s t o r m b y d i v i d i n g the
a v e r a g e d i f f e r e n c e for p a i r s a t a given d i s t a n c e
b y the a r e a l m e a n r a i n f a l l and m u l t i p l y i n g b y 100
t o c o n v e r t the f a c t o r t o p e r c e n t .
TABLE 20
RELATIVE VARIABILITY TRENDS FOR
6-FOOT GAGE PAIRS
Av. Variability (%) for Given Size of Event
Mean Rainfall, in.
0.01-0.09
0.10-0.19
0.20-0.49
0.50-0.99
1.00-1.99
6.1
2.5
1.9
1.4
1.3
Storm Durations, h r s .
0.1-2.0
2.1-4.0
4.1-6.0
6.1-12.0
12.1-24.0
5.8
1.9
2.1
1.9
2.0
Mean Rainfall Rate, in/hr
E x a m i n a t i o n of T a b l e 20 shows t h a t the r e l a tive v a r i a b i l i t y t e n d s t o d e c r e a s e with i n c r e a s i n g
m e a n r a i n f a l l , this t r e n d being m o r e p r o n o u n c e d
for the light s t o r m s . A s i m i l a r t r e n d w a s o b s e r v e d
for s t o r m d u r a t i o n u p t o four h o u r s , a f t e r w h i c h ,
t h e r e a p p e a r e d to be no s i g n i f i c a n t c h a n g e in the
a v e r a g e r e l a t i v e v a r i a b i l i t y with i n c r e a s i n g s t o r m
d u r a t i o n . M e a n r a i n f a l l r a t e p r o d u c e d its g r e a t e s t
effect upon the r e l a t i v e v a r i a b i l i t y at low v a l u e s .
A s l i g h t t r e n d was o b s e r v e d for the r e l a t i v e v a r i a b i l i t y to d e c r e a s e with i n c r e a s i n g m a g n i t u d e of
wind s p e e d , the a v e r a g e r e l a t i v e v a r i a b i l i t y t e n d i n g
to l e v e l off with high v a l u e s of wind s p e e d .
B e c a u s e of the m a g n i t u d e of the c o m p u t a t i o n s
i n v o l v e d , n o a t t e m p t was m a d e t o c o m b i n e a l l the
p o s s i b l e v a r i a b l e s into a m u l t i p l e r e g r e s s i o n e q u a t i o n , e s p e c i a l l y s i n c e a r e l a t i v e l y high c o r r e l a t i o n
w a s o b t a i n e d b e t w e e n the v a r i o u s d i f f e r e n c e s and
storm size of mean rainfall. Scatter d i a g r a m s r e l a t i n g the 6-foot r e l a t i v e v a r i a b i l i t y t o m e a n r a i n f a l l , wind s p e e d , s t o r m d u r a t i o n , and m e a n
r a i n f a l l r a t e a r e i l l u s t r a t e d i n F i g u r e 53.
F o r c o m p a r i s o n with the s m a l l s c a l e d i f f e r e n c e s
i n T a b l e 19, s o m e r e s u l t s f r o m the s t u d y d i s c u s s e d in the s e c t i o n on t h e v a r i a t i o n of point r a i n fall with d i s t a n c e a r e s u m m a r i z e d i n T a b l e 2 1 .
T h e r e l a t i o n s shown i n t h i s t a b l e w e r e o b t a i n e d
from observations in shower-type rainfall collected
o n the P a n t h e r C r e e k and G o o s e C r e e k n e t w o r k s
shown in F i g u r e s 3 and 4.
T h e m a g n i t u d e of the v a r i a b i l i t y b e t w e e n 6-foot
300-foot and 600-foot p a i r s on a m o n t h l y and s e a s o n a l b a s i s i s i l l u s t r a t e d i n T a b l e 22. A t r e n d for
a v e r a g e d i f f e r e n c e s b e t w e e n gage p a i r s t o i n c r e a s e
with d i s t a n c e and with a r e a l m e a n r a i n f a l l w a s
found for both m o n t h l y and s e a s o n a l p r e c i p i t a t i o n .
T h i s t r e n d w a s l e s s p r o n o u n c e d t h a n that found
b e t w e e n a v e r a g e d i f f e r e n c e s and s t o r m m e a n r a i n f a l l . C o r r e l a t i o n c o e f f i c i e n t s of 0 . 6 0 , 0 . 5 8 , and
0.48 w e r e o b t a i n e d b e t w e e n m o n t h l y r a i n f a l l and
a v e r a g e d i f f e r e n c e s of 6-foot, 300-foot and 600-foot
p a i r s , r e s p e c t i v e l y . Although a m a x i m u m monthly
d i f f e r e n c e of 0.27 inch w a s o b s e r v e d b e t w e e n 600foot p a i r s o n one o c c a s i o n , a v e r a g e m o n t h l y d i f f e r e n c e s w e r e g e n e r a l l y l e s s t h a n 0.05 i n c h .
C o m p u t a t i o n s w e r e m a d e o f the a c c u r a c y with
which the c e n t r a l gage w i t h i n the g r i d p a t t e r n
m e a s u r e d the s t o r m m e a n r a i n f a l l for the e n t i r e
area. Results are summarized in Table 23 where
t h e a v e r a g e , s t a n d a r d and m a x i m u m e r r o r s h a v e
b e e n r e l a t e d t o s t o r m s i z e . Mean r a i n f a l l w a s o b t a i n e d f r o m t h e a r i t h m e t i c a l a v e r a g e o f all g a g e s .
T h e e r r o r s w e r e found t o i n c r e a s e with i n c r e a s i n g
s t o r m s i z e , although remaining relatively s m a l l .
TABLE 21
VARIATION OF POINT RAINFALL WITH DISTANCE
Storm Size (in.)
0.01-0.05
0.06-0.10
0.11-0.20
0.21-0.30
Over 0.30
5.8
2.7
2.3
2.2
1.8
Wind Speed, mph
0-5
6-10
11-15
16-20
21-30
3.6
3.7
2.8
2.8
2.4
Av. Difference (in.) at Given Distance
At Given Point
1 Mile
3 Miles
0.10
0.25
0.50
1.00
1.50
2.00
0.06
0.08
0.10
0.14
0.18
0.23
0.09
0.11
0.14
0.20
0.26
0.33
59
TABLE 22
MONTHLY AND SEASONAL DIFFERENCES BETWEEN GAGE PAIRS
1953
Mean
Rainfall, in.
Difference (in.) for Given Distance
6-Ft
300-Ft
600-Ft
Av.
Max.
Av.
Max.
Av.
Max.
April
May
June
July
Aug.
Sept.
Oct.
1954
1.30
1.53
3.39
3.15
0.59
0.41
1.74
0.022
0.022
0.016
0.036
0.012
0.006
0.028
0.067
0.047
0.034
0.057
0.032
0.016
0.061
0.023
0.028
0.032
0.032
0.016
0.008
0.039
0.054
0.078
0.100
0.080
0.036
0.031
0.120
0.032
0.028
0.039
0.036
0.021
0.014
0.055
0.094
0.059
0.079
0.066
0.040
0.039
0.123
March
April
May
June
July
Aug.
Sept.
Oct.
1953
1.18
4.61
4.34
2.59
2.79
4.51
0.30
4.51
0.013
0.043
0.026
0.050
0.020
0.028
0.007
0.032
0.034
0.083
0.053
0.112
0.036
0.055
0.014
0.057
0.023
0.083
0.025
0.044
0.037
0.027
0.013
0.028
0.060
0.234
0.081
0.122
0.090
0.088
0.032
0.076
0.023
0.145
0.029
0.055
0.054
0.025
0.025
0.044
0.044
0.270
0.074
0.102
0.125
0.083
0.036
0.110
12.11
0.053
0.092
0.102
0.234
0.167
0.327
24.83
0.113
0.219
0.175
0.429
0.200
0.482
Apr.-Oct.
1954
Mar.-Oct.
TABLE 23
RELATION BETWEEN AREAL MEAN RAINFALL AND
CENTRAL GAGE POINT RAINFALL
Storm Mean Rainfall, in.
Average E r r o r
Standard E r r o r
Maximum E r r o r
Number of Storms
0.01-0.09
0.10-0.19
0.003
0.004
0.013
28
0.003
0.004
0.011
18
0.20-0.49
0.005
0.007
0.017
19
0.50-0.99
0.007
0.009
0.025
15
1.00-1.99
0.013
0.017
0.044
13
60
FIGURE 53 EFFECT OF SEVERAL METEOROLOGICAL FACTORS ON RELATIVE VARIABILITY OF STORM RAINFALL
61
REFERENCES
1.
The Thunderstorm, Report of the Thunderstorm Project (Joint Project of Air F o r c e , Navy, NACA,
Weather Bureau), U. S. Dept. of Commerce, Weather Bureau, Washington, D. C, June 1949.
2.
Neill, J. C, Analysis of 19 52 Radar and Rain Gage Data, Report of Investigation No. 21, State
Water Survey Division, Urbana, Illinois, July 1953.
3.
Huff, F. A., Neill, J. C, and Spock, M., Evaluation of Low-Powered 3-cm Radar for Quantitative
Rainfall Measurements. Research Report No. 4 under U. S. Army Signal Corps Contract DA-36-039
SC-64723 (State Water Survey Division Report of Investigation No. 29), Urbana, Illinois, 1956.
4.
Yarnell, D. L., Rainfall Intensity-Frequency Data, U. S. Dept of Agriculture, Misc. Pub. No. 204,
Washington, D. C, August 1935.
5.
Rainfall Intensity-Duration-Frequency Curves, Tech. Paper No. 25, U. S. Dept. of Commerce, Weather Bureau, Washington, D. C, December 1955.
6.
Storm Rainfall of Eastern United States, Miami Conservancy District, Dayton, Ohio, 1936.
7.
Storm Rainfall in the United States, Depth-Area-Duration Data, U. S. Corps of Engineers, Office of
Chief of Engineers, Washington, D. C, 1945.
8.
Thunderstorm Rainfall. Hydromet.
Washington, D. C, 1947.
9.
Huff, F. A., and Stout, G. E., "Area-Depth Studies for Thunderstorm Rainfall in Illinois," T r a n s .
AGU, 33, 4, (State Water Survey Division Circular No. 39) August 1952.
10.
Chow, V. T., Discussion of "Area-Depth Studies for Thunderstorm Rainfall in Illinois," T r a n s . AGU,
34, 4, August 1953.
11.
Hudson, H. E. J r . , Stout, G. E., and Huff, F. A., "Rainfall Studies Using Rain Gage Networks and
Radar," Proc. ASCE, Sep. 178, March 1953.
12.
The Storm of July 8, 19 51 in North Central Illinois. Report of Investigation No. 14, State Water Survey Division, Urbana, Illinois, 19 52.
13.
Larson, B. O., Hiser, H. W., and Daniels, W. S., The Storm of July 18-19, 1952 in Rockford, Illinois
and Vicinity, Report of Investigation No. 24, State Water Survey Division, Urbana, Illinois, 19 55.
14.
Huff, F. A., and Neill, J. C, "Variation of Point Rainfall With Distance", P r o c . ASCE, 82, SA1,
Feb. 1956.
15.
Linsley, R. K. and Kohler, M. A., "Variations in Storm Rainfall Over Small A r e a s , " T r a n s . AGU,
32, 2, April 1951.
Report No. 5, U. S. Dept. of Commerce, Weather Bureau,