- Email: [email protected]
Rainfall interstation correlation functions
Journal of Hydrology, 62 (1983) 27--51
27
Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
[4]
RAINFALL INTERSTATION CORRELATION FUNCTIONS VI. Application to the Tri-rectangular Storm Model
Ph.Th. STOL
Institute for Land and Water Management Research (I. C. W.), 6700 AA Wageningen (The Netherlands) (Received May 10, 1982; revised and accepted August 12, 1982)
ABSTRACT Stol, Ph.Th., 1983. Rainfall interstation correlation functions, VI. Application to the trirectangular storm model. J. Hydrol., 6 2 : 2 7 - 5 1 . A method is given to systematically derive the theoretical interstation correlation function for a tri-rectangular, not necessarily symmetric, rainfall storm model. Since this storm model permits distinguishing between three principal storm types and two storm subtypes, there are five different correlation functions to be derived. In the general case, viz. when the total storm width B is less than the length L of the gaged area, each correlation function for this model is a seven-sided polygon. The solution is conveniently obtained by means of a number of tables based upon the analysis published by the same author in a previous paper. Some examples are given to illustrate typical properties of the analytically derived correlation functions for tri-rectangular storm models of different shape.
1. I N T R O D U C T I O N
In Part V (Stol, 1983a) of this series of papers, published in the same issue as the present paper, the general m-rectangular storm model has been treated in order to describe its properties essential to analytically derive its rainfall interstation correlation function. In this paper we will propose a systematic procedure with which these correlation functions can be obtained. For practical application of the theoretical analysis it is suggested to employ a tabular way of solution, which will be given for a general, not necessarily symmetric, tri-rectangular rainfall storm model. Correlation functions for particular rectangular models then can easily be obtained from the general solution. Notational conventions and definitions of specific terms can be found in Part V (Stol, 1983a, this issue).
0022-1694/83/0000--0000/$03.00
© 1983 Elsevier Scientific Publishing Company
28
B1
B2
bo
b~
B3 b2
b3
storm aXIS X
''2
rainfall depth
h
Fig. 1. T h e general n o n - s y m m e t r i c t r i - r e c t a n g u l a r s t o r m m o d e l . T h e c e n t r a l p a r t m a y have zero rainfall ( H 2 = 0). 2. T H E T R I - R E C T A N G U L A R S T O R M M O D E L
T h e t r i - r e c t a n g u l a r s t o r m m o d e l is a p a r t i c u l a r case o f t h e m - r e c t a n g u l a r m o d e l w i t h m = 3 (Fig. 1). Its s t o r m - d e f i n i n g e q u a t i o n s are (see P a r t V, eq. 1 or, in s h o r t , eq. V.1): °f(x)
=
h(x)
x < O,
Ho = 0
H 1,
b 0 ~
b0 = 0
H 2,
b 1
~3f(x)
H3,
b2
b3 = B
I,°f(x)
Ho,
ba
Ho = 0
[ if(x) ~2f(x)
= Ho,
(1)
!
T h e c o n d i t i o n s f o r rainfall d e p t h s Hk t o b e m e a s u r e d at t h e e a r t h ' s s u r f a c e are:
H 1 > 0,
H 2 ~> 0,
Ha > 0
(2)
T h e partial s t o r m w i d t h s o f this m o d e l are: B1 =
[[bo, bllL x = b 1 - - b o ,
>0
B2
=
Ilbl,b2 IIx = b 2 - - b 1,
> 0
B3
=
IIb2,b a Ilx = b a - b
>0
2,
(3)
while t h e t o t a l s t o r m w i d t h is given b y : 3
B=~B~
(4)
k=l
We first c o n s i d e r m o d e l s w i t h p r o p e r t y : B 1 : # B 2 4: B s
(5)
29 The solution of the correlation function is considered to be a general solution if conditions expressed by eqs. 2, 3 and 5 are fulfilled and if the total storm width B in eq. 4 is less than the length L of the gaged area, so if: B
(6)
This storm mode] produces a spatial correlation function whose general structure is an M-sided polygon where we have with m -- 3 in eq. V.4:
M = ( m + 1 ) _}_ 1 = 7
(7a)
2
If B ~> L the distance class conditions of eq. V.13 (see Section 6) come into operation and the correlation function then will be less-sided. This is symbolized by defining the actual number M L of polygon sides by eqs. V.5b and V.5c, namely: ML (B) < M
for
B ~>L
M L (B) = M
for
B < L
(7b)
(general case)
(7c)
where ML
M L
---- distance class K for D, whose upper b o u n d a r y is equal to L = last distance class K for D in the general case that B < L = length of gaged area under investigation (fig. II.1)
In deriving general formulas we have from eq. 7a: M L -- M = VII which will be written with roman numerals. If M L ~ M we only have M L polygon sides. They are obtained from the general solution by only taking the first M L equations of the correlation function in the order I, I I , . . . , M L .
3. S T O R M C H A R A C T E R I S T I C S
Storm characteristics for single r a n d o m points a in the storm (subsections V.2.4 and V.2.6) can be expressed as sums of products. Formulas are given in eqs. V.6 and V.8. In particular, we have with m -- 3: 3
I0 = ~
Bk
(Sa)
~ Bk'H4
(8b)
k=l
3
I1 =
4=1 3
111 = ~
Bk " H k H k
k--1
(For systematic use in matrix equations redefined by eqs. 19.)
(8c)
30 where Bk is proportional to the probability with which values occur of Hk and H 2 assigned to it. So, with eqs. 8 we can derive mathematical expectations and we find for the sum of all probabilities (using eq. 4): B- 1i0 = (unity)
(9a)
For the mathematical expectation we arrive at: 3
P = E ( h a ) = B-111
= B -1
--
(9b)
~, Bk "Hk k=l
The variance is obtained by: 3
02 = E ( h a - - P ) 2 = B - 1 I n
- - B - 2 I ] = B -1
--
~, Bk "HkHk _ p 2
(9c)
k=l
4. THE THEORETICAL CORRELATION FUNCTION 4.1. General
The theoretical correlation function for the storm model considered, consists of seven equations. They will be written as the vector p(D) and we define the entire set of equations by: p(D) = { p l ( D ) . . . . . PK (D),..., PvII (D)} w
(10)
where T denotes the transpose of the row vector. From eq. V.9 we have: pg(D)
= I+(L+B)
KI12(D) - - I l l (L+B)I~--(1--p)I~'
K -- I, I I , . . . , V I I (11a)
where p is the fraction of completely dry days over the entire gaged area (Stol, 1981a, Part II). This expression can also be written as: P K ( D ) -- ( L + B ) K I 1 2 ( D ) - ( 1 - - p ) I ~ , (L + B ) I u -- (1 - - p ) I 2
K = I, I I , . . . VII
(11b)
F r o m these formulas we see that if the fraction p of dry days over the total area is large (p -+ 1), the correlation function becomes independent of the length L of the gaged area since eq. l l b then reduces to P K ( D ) = KIl2 ( D ) / I l l . In these formulas K i12 (D) stands for the sum of products to be determined as part of the covariance (eq. V.7a). The difference KI12 (D) -- I u will be referred to as the reduced sum of products. The d e n o m i n a t o r of eqs. 11 (A, say) can be evaluated by inserting eqs. 8b and 8c, to yield: A = (L+B)
~, B k ' H 2 - ( l - p ) k=l
~'H
(12a)
31 or in alternative form (appendix V.B):
3
3
A = L ~ BkH~ + E k=l
3
~, [BkBl(Hk --H~):] +p
( h~-)[31BkHk\2
(12b)
4=1 l>k
clarifying t h a t A > 0 so that the sign of the fraction in eq. l l b is determined by the sign of its numerator. The expression for A is independent of D and so it is constant with respect to the distance classes K = I, I I , . . . , M and, consequentially, also for all tri-rectangular storm types (Table I).
4.2. Evaluating the covariance The covariance for pairs of points (a, a') in the storm (see fig. II.1) does depend on the interstation distance D and therefore its evaluation needs more concern. Part of the covariance is represented in eqs. l l a and 11b by the sum of products K I12 (D). This sum of products is to be evaluated according to a sequence of formulas described in the next sections. From eq. V.7a we have:
m~
B~:,i*(D)'g*g,i*,
KI12(D ) = ~
K = I, II . . . . . VII
(13)
i*= l
where K
=
/72K BK,i*
D
=
H K,i*
I, II, ...
=
index to indicate an arbitrary distance class of D in which each covariance factor B~, i.(D) can be represented by the same equation in the variable D d u m m y variable to enumerate all (HkH~')-combinations that occur in the current distance class K: the relevant combinations total n u m b e r of (HkHk,)-combinationsoccurring in the current distance class K i*th covariance factor for the current distance class K, being a function of D rainfall gage interstation distance i*th relevant (HkHk')-combination for the current distance class K 1st, 2nd, ... distance class K of D.
The covariance f a c t or B~:,i* is proportional to the probability with which the i*th relevant (HkHk,)-combination, denoted by H ~ , i . , occurs in the required distance class K. The covariance factor B~,i* is numerically equal to the length of the interval over which the end-point a of an interstation distance interval [a,a']x on the x-axis (Fig. 3) and of length D = lla,a' [Ix = a' - a, (a' > a), belonging to distance class K, can be shifted along the x-axis through the storm w i t h o u t changing the i*th relevant (HkHk')-combination to which the end-points a and a' belong (see subsection V.3.8).
32 4.3. Evaluating the (Hk Hk ')-combinations The w a y in which the relevant (HkHk')-combinations are to be selected has been described formally in subsections V.3.9 and V.3.10. Relevant combinations are counted b y the d u m m y variable i* (Fig. 3, left-hand margin, 1st column) for each distance class K mentioned in the right-hand margin. When the last relevant combination in a particular distance class has been encountered we set mK* = i* (Fig. 3, right-hand margin). The total number of all possible (HkH k, )-combinations is, according to eq. V.16: m* = ( m + 2 1) = 6
(14)
4.4. Evaluating the covariance factor The covariance factor in eq. 13 is given b y eq. V.14b which reads: i*
BK, i*(D) = llb~,i*-l,bg,i* llx,
iK
=
1 (1) m~
(15)
= I, II . . . . . VI
where
b~,i*-i
=
b~,i*
=
lower b o u n d a r y of the i*th covariance factor interval for the Kth distance class upper b o u n d a r y of the i*th covariance factor interval for the Kth distance class
The procedure of Subsection 4.3 starts with b~:, 0 = 0 (Fig. 3) and each time a new (HkHk')-combination is encountered by shifting the interval [a,a']x through the storm, the new starting position of a is marked with b*K,i*. When a' reaches the right most point in the storm (a' = B) the procedure is terminated by setting the corresponding location of a to b~, mi~. Each distance class K produces its own set of b*-values (Fig. 3 for K = I, IV, VI, VII). The last class (K = VII) in the general (B ~ L) case, so B ~ D ~ L, has no b*-values with a and a' both in the storm and so it is empty. Values of b~,i* depend on the interstation distance D which causes the covariance factor B~, i* to be a function of D.
4.5. A practical solution It will be shown in Section 7 that the determination of br,i** and mK* in eq. 15 can easily be performed by inspection and a tabular solution will be proposed. All (HkHk')-combinations (Fig. 3, left-hand margin, 3rd column) then are ranked. The ranking numbers (Fig. 3, left-hand margin, 2nd column) are obtained by the index formula of eq. V.19a and derived in appendix V.A. It reads: i(k,k') = ½(k'--k)[2m+l--(k'--k)] +k, {:--l(1)m (16) '
k (I) m
33 which for the combination (k, h') = (1,rn) reduces to eq. 14. Each time that we process a new distance class in the tabular solution, we set, with i given by eq. 16 and i* found b y the procedure o f Subsection 4.3, in general, for all distance classes (Fig. 3, left-hand margin, 2nd column):
H* = HkHk',
i = i(h,h')
(17a)
and, in particular, in the Kth distance class:
B*K,i(D) --
* (D), IBm,i (0,
if
(HkHk') is a relevant combination
if
(HkHk') is n o t a relevant combination
(175)
For all m* = 6 (eq. 14) possible (HkHk,)-combinations we can determine the value of the products HkHk', and so of HI., in advance by eq. 17a. Then we only need to determine from eqs. 15 and 17b the corresponding values of the covariance factor B}, i(D) per distance class K. In connection with this approach we redefine the formula for the sum o f squares (eq. 8c) in the following way: m $
Ii, = E B ; ' H *
(18)
i=I
where now, in general:
H*
=
H k H k,
i = i(k,k)
(19a)
and, in particular: tBi,
if
i ~< m
Bi = t0,
•
if
(m + 1) <~i ~ m *
(19b)
not depending on the interstation distance class K. The reduced sum of products becomes: m $
KI12(D ) - 1 1 1
= ~ [B~:,I(D) - S * ] H*
(20a)
i=l
to be used in the form in*
KI12(D)--Ill
= ~
CK,i(D)H*,
g
= I, II, ..., VII
(205)
i=l
which can be presented b y a matrix × vector product.
5. TRI-RECTANGULAR STORM TYPES
5.1. General To elaborate eqs. 17 the relevant (HkHk')-combinations must be selected b y appropriately chosen values of B~,~(D) in class K of interstation distance
34 B1 bo H0 ~ H1 =H3
B2
B3
b1
b;-
' :
x
b3 ....
:
:,,
, ,:::~
Main Sts1Ormtype
H2
B1 bo
B2
B3
bl
b2
b3
H°
x $2
H1 =H 3
B1
bo
B2
bl
B3 be
b3
/ rainfall depth h
Fig. 2. The three principal storm types for the tri-rectangular storm model obtained by permutation of the sizes of the partial storm widths B1, B2 and B3.
values D. However, the class interval boundaries depend on the magnitude of the partial storm widths and their moving sums. Their ordering defines the type of the storm. They will be discussed first.
5.2. Principal storm types According to subsection V.3.4 there are m!/2 = 3 principal storm types for the tri-rectangular storm model. The principal types (S1, $2 and $3) are given in Fig. 2 where the rectangle with the middle-sized base is heavily shaded for better comparison. The relational expressions between the partial storm widths to define the principal storm types are: principal storm type $1 :
B 1 < B2 < B3
principal storm type $2:
B 1
principal storm type $3:
B 2
(21)
35 TABLE I Interstation distance upper class boundary values B(K) based on partial stormwidths Bk and their moving sums for the three principal storm types of Fig. 2 and two subtypes derived fromthem
Distance class, K
I
II III IV V VI VII
Boundary, B(K)
Storm subtypes SI. 1
S1.2
$2.1
$3.1
$3.2 0
B(°)
0
0
0
0
B(I) B(II) B(III) B(IV) B(V) B(VI) B(VID
B1 B2 B3 (B1 + B2) (B2 + B3) B L
B1 B2 (/31 + B2) B3 (B2 + B3) B L
B1 B3 B2 (B1 + B2) (B2 + B3) B L
B2 B2 B1 B1 B3 (B1 + B2) (B1 + B2) B3 (B2 + B3) (B2 + B3) B B L L
5.3. Storm subtypes Without further knowledge of the actual magnitudes o f B 1, B2 and B 3 in eq. 21, it cannot be decided whether for the principal storm types Sl and $3 the ordering o f the first moving sum of partial storm widths is: B3 <(B 1 +B2)
or
(B 1 + B 2 ) < B
3
(22)
The order of moving sums therefore is not uniquely defined by eqs. 21 only. We have for the principal storm t y p e $1 : subtypeS/./:
0 < B 1 < B 2 < B 3 < ( B 1 +B2) < ( B 2
+Ba)
subtypeSI.2:
0 < B I < B 2 < ( B 1 + B 2 ) < B 3 < ( B 2 +B3)
and an analogous series for principal storm type $3. The second principal type ($2, see Fig. 2), however, has no subtypes since the moving sums of partial storm widths define the ordering for this type uniquely. Note that (BI + B3) is not an element of the set of moving sums of partial storm widths. We have by definition from eq. 21 for $2 the inequality B 1 < B 3 < B2, so logically B 2 < (B2 + B 1) and B2 < (]32 + B3). From BI < B3 we also have (B1 + B2 ) < (B3 + B2 ) which finally gives the principal storm type $2:
principal storm type $2:
0
(23c)
In conclusion we have three principal storm types and two subtypes so that there are five different storm types to be considered. The ordered sets
36
of partial storm widths and their moving sums, viz. B(K ) for K = O, I, II, ..., VII, are given in Table I that implicitly defines all storm types for the trirectangular storm model.
6. I N T E R S T A T I O N DISTANCE CLASSES
Partial storm widths and their moving sums as given in Table I, define the upper class boundaries of the distance classes according to eq. V.13: classK:
BCK_I)~D~min(B(K),L),
K = I, I I , . . . , V I I
(24a)
and, since we assumed B ~ L, class I:
B¢o) ~ D ~ min ( B 1 , B 2 , B 3 )
(24b)
class VII:
B ~D ~ L
(24c)
Since the sum of two terms in the argument of eq. 24b exceeds the magnitude of each single term, the first two upper boundary values are obtained from the smallest and the smallest b u t one partial storm width, to be denoted b y B(I) and BCH~, respectively. This means that the last four terms of the ordered set are: (B--B(II)) ~ (B--B(I)) ~B ~L
(25)
except, however, if either BcI) or B(~I) stands for B 2 since B - - B 2 = B 1 + B 3 does not belong to the set of moving sums. See for instance storm types $3.1 and $3.2 in Table I.
7. SYSTEMATIC P R O C E D U R E
7.1. General
The procedure to determine the reduced sum of products K I12(D) - - I l l in the numerator of eq. l l a and redefined in eqs. 20, will be executed step by step in the following subsections using storm s u b t y p e $1.1 as an example. The final solutions for all storm types distinguished is given in Appendix A. 7.2. Partial storm widths
Rearrange all permutations of partial storm widths and their moving sums in order of increasing magnitude to define all storm types. Example: Subsections 5.2 and 5.3 and Table I.
7.3. (Hk Hk ' )-combinations
Determine all possible (HkHk')-combinations and order them according to the index values obtained with the index formula of eq. 16.
•
,
t
.
(HkHk,)-
combination
H1H 1 H2H 2 Hall 3
HIH 2 H2H3
H1H a
Index
i ( k , k')
1 2 3
4 5
6
B 2 ~D~B
0 (D ((BI +B2) 0 <:D ~ (B2 + B s )
O(D(B l O(D(B2 0 ~ D ~ Bs
Primary conditions for D
D ~ B1 D ~B2
D~B 1 D <~B 2 D>(BI
D ~ O
--
D<(B1
D > O D~>O
---
+B2)
1
1
+B2) D<(B2
D~B 2 D ~B3
--
--
--
TO-code
FROM-code 2
Supplementary conditions for D
+B3)
D>(B2
D ~ B2 D ~ Bs
D~O
D ~ O
D > O
2
+Bs)
bo
b0 bl
bi D b2--D b2-- D
b0 -b 1 ..b 2 --
.-•--
bl
b1 b2
2
1
1
ba - D
b2 - - D b3 D
b 1 --D b2 --D b3 --D
2
FROM-code TO-code
Location of a
Possible values of interstation distances D relative to partial storm widths B k to produce conditions and supplementary conditions for D, and the locations of a that produce the interval of displacement of the left-hand end-pomt of the interval [a,a ] x with length D for each ( H k H k ' ) - c o m b i n a t i o n and for all storm types
TABLE II
38 Storm
storm
~1t
type $11
81
~
~
_
width B -
82
B3
~
partiat storm width bo :
o
b!
b3= B
b2
X 4:,
for o f o r o'
i"
i
H,H k
b'io
H,H.
11
b'11
b~2
a
a
~
b'nz o
b'~
I 4 H~H? 2
6
H~H~
3
5
H2H 3
b',3
b~
b'!s
__
K=I
bier2
b'~. 3
D ~
a" I Q" i
•
1 6
Q
b'z o
H1H 3
b~j 1
a
o
D ~
.1H: :0
~,
1 D ~
K
Fig. 3. Schematical illustration of the procedure of shifting an interval [a, a'] x of length D along the x-axis from the left to the right through a storm (see fig. V.3) (open circles: starting position of a before a shift; black dots: terminal position o f a after the shift).
Example: See first t w o c o l u m n s of Table II, and Fig. 3, left-hand margin, 2nd c o l u m n .
7.4. Primary conditions for D D e t e r m i n e all p o s s i b l e values D can t a k e o n for e a c h
(HkHk')-combination.
Example: For this part of the procedure Fig. 3 is basic. Values can be read from it. For any storm type, taking the c o m b i n a t i o n (HkHk') = (H1H 2 ) as an example, values of the interstation distance D that produce a E B1 and a' @ B 2 range 0 < D < (B1 + B2). (To avoid ambiguity with respect to locations in the n e i g h b o u r h o o d of the class boundaries, the "<"-signs are used instead of the "~<"-signs. This is allowed since the probability of being exactly on a class boundary equals zero.)
39
Values are obtained from Fig. 3 and are collected in Table II, 3rd column and are valid for all storm types. 7. 5. S u p p l e m e n t a r y c o n d i t i o n s for D
Within each range determined in Subsection 7.4 determine the supplementary conditions for D, expressed in partial storm widths, for which the location of a is either constant or a function of D. E x a m p l e : If D ~ B1 (Fig. 3, K = I, i* = 2, i = 4, H k H k' = H 1 H 2 ) the general f o r m u l a for the locations of a (open circles) for arbitrary values of D, before shifting the interval [a, a' ] x to the right, is a = bl -- D and so it is a f u n c t i o n of D. This holds for all values of D that satisfy 0 ~ D ~ B1, thus the s u p p l e m e n t a r y c o n d i t i o n is D ~ B~. If, however, D B 1 (Fig. 3, K = IV, i* = 1, i -= 4, H k H k' = H 1 H 2 ) the location o f a (open circle) before shifting the interval [a,a' Ix to the right is a = b0, constant, for all values of D that satisfy B 1 ~ D ~ (B 1 + B2 ). The s u p p l e m e n t a r y c o n d i t i o n t h e n is B 1 ~ D.
Supplementary conditions of this kind have been collected in Table II under the heading FROM-code 1 or 2, respectively. If D ~ B 2 (Fig. 3, K = I, i* = 2, i = 4, H k H k ' = H 1 H : ) the general f o r m u l a for the l o c a t i o n o f a (black dot) after having shifted the interval [a,a']x to the right, is a = b l , constant, for all values of D given by 0 ~ D ~ B2, the s u p p l e m e n t a r y c o n d i t i o n being D ~ B 2. If, however, D ~ B2 (Fig. 3, K = IV, i* = 1, i ----4, H k H k' = H 1 H 2) the location o f a (black dot) after the shift is a = b2 -- D, and so a f u n c t i o n of D, for all D that satisfy B 2 ~ D ~ (B1 + B2 ). The s u p p l e m e n t a r y c o n d i t i o n t h e n is D ~ B2.
Supplementary conditions of this kind have been collected in Table II under the heading TO-code 1 or 2, respectively. All supplementary conditions for D in general terms, regardless the storm type, have been collected in advance in Table II for further particular use. 7.6. L o c a t i o n o f a
Determine the location of point a b e f o r e as well as in the situation after the shift of the interval [a, a'] x of length D along the x-axis, for all ( H k H k ' ) combinations and for all supplementary conditions for D collected in Table II. E x a m p l e : F r o m Fig. 3 it is seen t h a t the general expression for the o p e n circles, illustrating the c o m b i n a t i o n H k H k' = H l H2, reads w i t h primary c o n d i t i o n s and s u p p l e m e n t a r y c o n d i t i o n s for D c o m b i n e d :
FROM-code/,
(O~D~B1)
FROM-code2,
(B 1 ~ D ~ B
~a 1 -bB2)~a
= b 1 --D
(26a)
= bo
(26b)
= b1
(27a)
= b2 - - D
(27b)
F o r the black dots in Fig. 3 this b e c o m e s :
TO-code 1,
(0~D~B2)
~a
TO-code2,
(B 2 ~ D ~ B
1 +B2)-~a
40
These values o f a produce the sets o f boundary values b~:,i* o f eq. 15 as required to determine the covariance factor B~:,i*. All locations of a, still regardless the storm type and the distance class index, have been collected in advance in Table II for further particular use.
7. 7. Covariance factors in general Evaluate for further particular use the four possible " F R O M - - T O " s i t u a t i o n s ( F R O M - c o d e = 1, 2 ; T O - c o d e ----1, 2) a c c o r d i n g t o c o l u m n n u m b e r s i n T a b l e II f o r e a c h ( H k H k ' ) - c o m b i n a t i o n i n t e r m s o f p a r t i a l s t o r m w i d t h s t o o b t a i n t h e covariance factors t h a t are to be applied. This covariance factor (eq. 1 5 ) is t h e l e n g t h o f d i s p l a c e m e n t o f t h e e n d - p o i n t a u n d e r t h e corresponding "FROM--TO "-situation. Example: With ( F R O M , T O ) - - ( 1 , 2 ) the combination HkH k' ----H1H2 (see Table II) gives as distance conditions D < B 1 and D < B 2 and for locations of a "FROM": a : bl -- D and " T O " : a = b2 -- D with length of displacement equal to:
lib, --D, b 2 - D IIx -- (b= - - D )
-D)
-- (bl
= B2
(28)
TABLE III Collection of all possible covarianee factors for all storm types (FROM,TO) code
(HkHk')-combinations and their index i
H1H1 1
H2H2 2
H3H3 3
H1H2 4
(1,1)
--
--
--
D
(1,2) (2,1) (2,2)
--B , -- D
--B2 -- D
-B2 -BI B 3 - D (BI -b B2 - - D )
H2H3 5
HIH3 6
D
(D --B2)
B3 B3 B2 BI (B2 q- B3 - - D) ( B - - D )
TABLE IV Collection of (FROM,TO)-codes for storm subtype S1.1 Distance class, K
I II III IV V VI VII
(HkHk')-combinations and their index i
m* = 6
H 1H 1 1
H2 H2 2
H 3H 3 3
H 1H2 4
H2 H 3 5
HI 1"/3 6
* mK
(2,2) ---. . .
(2,2) (2,2) ---
(2,2) (2,2) (2,2) -. . .
(1,1) (2,1) (2,2) (2,2)
(1,1) (1,1) (2,1) (2,2) (2,2)
--(1,1) (i.i) (2,1) (2,2)
5 4 4 3 2 1 0
.
. . .
. .
. .
.
41
w h i c h easily c a n be verified g r a p h i c a l l y f r o m Fig. 2. F r o m t h e figure it is also seen t h a t this p a r t i c u l a r case o n l y c a n a p p l y t o s t o r m t y p e $3.
All results, regardless the storm type, have been collected in advance in Table III for further particular use.
7.8. Storm-depending (FROM,TO)-codes Process Table I and determine for all storm types successively the corresponding (FROM,TO)-codes by means of Table II for each (HkHk')-combination, using the columns headed with "supplementary conditions for D " , until the primary condition for D is violated. E x a m p l e : F o r s t o r m s u b t y p e $1.1 t h e d i s t a n c e class K -----IV reads ( T a b l e I): B 3 ~ D (B 1 4- B 2 ) ~ (B 2 4- B 3 ) ~ B ~ L. C o n s i d e r i n g t h e r e l e v a n t c o m b i n a t i o n H k H k' = H 1 H 3 ( w h e r e i =- 6) we have to use in T a b l e II:
FROM-code = 1
since
D ~ (BI 4- B2)
(29a)
TO-code
since
D ~ (B2 4- B3)
(29b)
= 1
For storm subtype $1.1 all corresponding codes have been collected in Table IV.
7. 9. Covariance factors in particular Decode the (FROM,TO)-codes in Table IV with the aid of Table III into the corresponding length of displacement of point a, thus giving the covariance factor B~,i for KI12 with K = I, II, ..., VII and i = 1 ( 1 ) 6 . According to eq. 17b open places should be zero filled. E x a m p l e : If t h e c o d e is for s t o r m s u b t y p e $1.1, we e n t e r T a b l e IV w i t h H k H k ' = H1 H2 ( w h e r e i ----4) a n d K ----II, t h e c o d e is ( F R O M , T O ) -= (2, 1 ) giving f r o m T a b l e III t h e value BIL 4 -~ B 1 .
For storm s u b t y p e $1.1 the particular values of the covariance factors are collected in Table V, according to instructions laid down in Subsection 4.5.
7.10. Reduced sum o f products Finally subtract I l l (eq. 18, and worked in eqs. 19) from the results obtained in Subsection 7.9 to obtain the reduced sum of products gI12 (D) -Ill (eq. 20a) being the numerator of eq. l l a . E x a m p l e : F r o m eq. 19 a n d T a b l e V it is a p p a r e n t t h a t we o n l y n e e d t o s u b s t r a c t t h e values B1, B2 a n d B 3 f r o m t h e results o b t a i n e d in S u b s e c t i o n 7.9 f o r t h e c o m b i n a t i o n s H k H ~' ----HkHk, k = 1 ( 1 ) m (see T a b l e A--I).
For all storm types the reduced sum of products is given in Appendix A.
42 TABLE V Collection of particular covariance factors BK, i for storm subtype $1.1
(HkH k')-combinations and their index i
Distance class, K
HIH1 1
H2H2 2
H3H3 3
I
(BI--D)
(B2--D)
II III IV V VI VII
0 0 0 0 0 0
(B2 - D ) 0 0 0 0 0
H1H2 4
H2H3 5
HIH3 6
(B3 --D) D (B3 - D ) B I
D D
0 0
(B3 --D) 0 0 0 0
B2 (B 2 + B 3 - - D ) (B 2 + B s - - D ) 0 0
(D--B2) (D--B2) B1 (B -- D) 0
(BI +B2--D) (B1 + B 2 - - D ) 0 0 0
7. I 1. The interstation correlation function W r i t e t h e i n t e r s t a t i o n c o r r e l a t i o n f u n c t i o n o f eq. 1 0 w i t h t h e aid o f t h e t a b l e s in A p p e n d i x A a n d eq. 11a. Solution: We d e f i n e t h e f o l l o w i n g m a t r i c e s a n d v e c t o r s : CK(D)
=
{ C K . , ( D ) , . . . , CK,i(D),...,CK,m* (D)}(1 x 6) K =
C(D) =
(30)
I, I I . . . . , V I I
ICK,i(D)I,
(31) i
1,2 . . . . . 6
(vii x 6)
H = {HIH1, H2H2,• ' ' ,H,H3}T (6 X 1)
(32)
I = {1,1,..., I}T(vII X I)
(33)
From eqs. 10, 12a and 20b we then have for eq. l l a :
p(D) = I + A - I ( L + B ) C(D)'H,
(Vllx 1)
(34)
where C(D) and H can be obtained from the tables in Appendix A.
Example: The interstation correlation function for storm subtype SI,I consists of VII equations. From Table A-I we obtain: I Pi(D)
p(D)=
Pv(D)
---- 1 -- A-I(L + B) H21 + H 2 +H23--H1H 2 .....H2H3]D = 1 -- A-l(L + B )
'
(35a)
]
~BkH~) --HIH3BI --H2H3 (B2 + B3 -- D k=l
(35b) Pvii(D)
= 1--A-I(L +B) k=l
where only the Ist, Vth and VIIth have been written in full as an example. E x a m p l e s o f i n t e r s t a t i o n c o r r e l a t i o n f u n c t i o n s c a l c u l a t e d w i t h eqs. 35 c a n b e f o u n d in P a r t V.
43 coy
factor for
wI1z
storm type $1:1
B:I
I
~b
B~?
i
1 2
H~ H 1 H~ H;
3
H3 H3
.
.
.
®
.
~' 5 5
H H H? H H H3
B~ to be read B~*B !
B~
i
B: I / /
I
\~,
,,, B,
B
B3 B,
B~3
B23
B
' L B
B,
B~
B? B .
B~
823 m t e r s t a t i o n distance [3
interstat,on d i s t a n c e class K
Fig. 4. The f u n c t i o n s BK, i = BK, i(D) that produce the covariance factors for storm subt y p e $1.1: ( A ) for the squares HkH~, k -- 1 ( 1 ) 3 ; and (B) for the p r o d u c t s H k H k ' , k -- 1 ( 1 ) 2 , k' = (k + 1) (1)3.
7.12. Remarks In the described procedure n o use has explicitly been made o f the quantities b ~ , i . since the e x e c u t i o n o f the procedure has b e e n performed by inspection. The covariance-factor f u n c t i o n s BK,i = BK,i (D) t h a t are valid for each (HkHk , ) - c o m b i n a t i o n for storm s u b t y p e $1.1 as given in Table V have been set o u t graphically in Fig. 4 A and B. The figures also explicitly illustrate the range of distance values D (distance classes K) o n w h i c h the covariance factors are defined. Example: F r o m the figure it can be s e e n that (H/~H/~')-combinations w i t h i = 5 and i = 6 are the o n l y relevant c o m b i n a t i o n s in distance class K ---- V (see also Fig. 5).
8. T E S T S O F C O R R E C T N E S S
There are s o m e c o n d i t i o n s that can be used to test the correctness o f the results.
8.1. Sum of intervals constant The s u m o f all intervals [bK,i*-l,br,i* * * ]x adds up t o B -- D as can be seen from Fig. 5, where c u m u l a t e d values of B*K,i(D) have b e e n set o u t . This holds for distance classes K = I, II, . . . , VI. The last class K = VII yields zero. In Tables A-I--A-V o b t a i n e d by subtracting Ill from gI12 , all lines s h o u l d add up t o - - D , t h e last o n e giving - - B . This can readily be checked.
44 cov-foctor l
for KI 12
Z B; ,l:1(1)rn~ storm type $1.1
B
i : H k Hk, 1 : H 1 H1 2 : H2 H2
B23
3 M3H 3 4 : H, H 2 5 H2H 3
B13 ~__
m ~ = 6 H1N 3
5
B~
B0
B~
B2
B3 B12
B~3
B23 interstatton
interstation
F i g . 5. T h e f u n c t m n s
BK, i
=
BK, i(D ) o f
B // L distance D
distance crass K
Fig. 4 cumulated.
8.2. Continuity in vertices T h e i n t e r s t a t i o n c o r r e l a t i o n f u n c t i o n is c o n t i n u o u s at t h e vertices. This can be c h e c k e d by inserting t h e values o f t h e class interval b o u n d a r i e s w h i c h are c o l l e c t e d in Table I, i n t o t h e c o r r e s p o n d i n g variable D in Tables A - I - - A - V .
Example: F o r s t o r m s u b t y p e $ 1 . 1 t h e b o u n d a r y b e t w e e n d i s t a n c e class K --- II a n d dist a n c e class K ---- III is D ----B 2 . T h e v a l u e s o f CK, i in T a b l e A-I f o r t h e r e q u i r e d c l a s s e s are: K = II:
-B 1 --D
--D
B1
D
0
K = III:
--B 1 --B 2
--D
(B1 + B 2 - - D )
B2
(D--B2)
w h i c h are e q u a l f o r D = B 2 .
8.3. The rectangular storm A special case is t h e rectangular s t o r m m o d e l (Stol, 1 9 8 1 a ) w h e r e H I = H 2 = H 3 = H ( c o n s t a n t ) . Partial s t o r m w i d t h s n o w are n o essential q u a n t i t i e s a n y m o r e a n d all s t o r m t y p e s r e d u c e t o this s i m p l e m o d e l . T h u s e a c h line in Tables A - I - - A - V p r o d u c e s :
KI12 - - I l l
= - - D H 2,
gI12--Ill
----
- - B H 2,
K =
I, I I , . . . , VI
K = VII
(36a) (36b)
45 thus giving eq. II.50 where H 2 is cancelled in the fraction. This solution, however, can also be obtained by taking into account the central partial storm width only so with B 1 = B a = 0, B 2 = B and H : = H. Then we obtain eq. II.50 only by means of the third column in Tables A-I--A-V: K II: - - I l l
= - D H : , K = I, . . . , N
(37a)
KI12 - - I l l
---- - - B H 2 , K
(37b)
= N + 1, ..., VII
where N = II for storm subtypes $1.1 and $1.2; N = III for storm subtype $2.1 a n d N = I for storm subtypes $3.1 and $3.2. 9. SOME GENERAL PROPERTIES In order to investigate the properties of the derived correlation function we write for the first derivative of eq. 34 with respect to D: dp (D) _ L + S dD A
d C ( D ) . H, dD
(VII × 1)
(38)
From the five tables in Appendix A we first note t h a t for all storm types the functions PI (D), Pv (D), Pvi (D) and Pvii (D) have the same structure. The first derivatives are (see eqs. 35 and Tables A-I--A-V): dp~(D)/dD = - - A - i ( /
+B)(--H1H 2 --H2H a +HI +H~ THe)
(39a)
dpv (D)/dD = - - A -1 (L + B ) H 2 H a
(39b)
dpvx (D)/dD = -- A-1 (L + B) H1H a
(39e)
dPv u ( D ) / d D = 0
(39d)
which are always negative (eqs. 39a--39c) or zero (eq. 39d) (Appendix B). The interstation correlation function departs from {D, p (D) } = (0, 1) with negative slope. In the general case (B < L), the function has two negativesloped sides followed by a horizontal polygon side at the right most part. The sign of the slope of PK (D) is given by the sign of eqs. 38 since A > 0 (eq. 12b). The matrix of coefficients of first derivatives for storm subtype $1.1, obtained from Table A-I, reads:
dC(D) dD
--1 0 0 0 0 0 0
--1 --1 1 1 0--1 --1 0 1 0 0 --1 --1 0 1 0 0 --1 --1 1 0 0 0 --1 0 0 0 0 0 --1 0 0 0 0 0
(40)
46
p 1.0 .8 5
\
/,
,-.... 0.8
2
O.L.
0 -.2
0
-4
B=1.5 -8
-1.0 0
I .1
I .2
1 3
I .4
5
l 5
I 7
I 8
I 9
10 D
Fig. 6. A n a l y t i c a l l y derived i n t e r s t a t i o n c o r r e l a t i o n f u n c t i o n s for s t o r m m o d e l 5 o f Fig. 7, n o w w i t h t o t a l s t o r m w i d t h B ---- 1.5 (in u n i t s o f L). Curves are labelled w i t h t h r e e values for t h e f r a c t i o n p o f c o m p l e t e l y d r y days over t h e area.
10. E X A M P L E S
Examples o f theoretical correlation functions for tri-rectangular nons y mmetr ic storm models have been given by Stol (1983a). Some further typical examples for symmetric equidistant (B 1 = B 2 = B 3 ) storm models are given below. Th e first example concerns the influence of t he fraction p of com pl et el y d r y days over t he gaged area and is based on a tri-rectangular symmetric equidistant model with B = 1.5L, L = 1, H 1 = H 3, H 2 = ½H 1 (Fig. 6). Since B > L th e correlation f u n c t i o n only consists o f two pol ygon sides defined on the intervals 0 ~ D ~ ~ B and 12B < D ~ L. It can be com pared with No. 5 in Fig. 7. When the fraction p of d r y days over the area increases, so does the correlation coefficient (fig. II.7). This is the first model of those considered thus far t h a t produces, at a certain distance f r om the origin, positive correlation coefficients t h a t are c ons t ant until D = L is reached. Th e second example (Fig. 7) pertains a tri-rectangular symmetric equidistant storm model. St or m parameters are: storm width B = 0.75L, H1 = H 3 = 10 (arbitrary units) while t he length o f the gaged area, w i t h o u t loss of generality, has been taken L = 1. Details are given in the figure caption.
47 storm model
storm model
...-.
-/"0 / 0 storm 0
tO0 h
J I I .1 ,2 .3 modet tB=O.?5) 1
I ~,
I .5 2
I .7
5 3
I 8
I .9
4
1.0
0 5
.1
.2
.3 6
.4
.5
.g
.7
.8
.9
1.0 D
?
H2
Fig. 7. E x a m p l e s of theoretically derived interstation correlation f u n c t i o n s for seven s y m m e t r i c a l equidistant tri-rectangular s t o r m models with total storm width B ---- 0.75 (in units o f L). Models could be characterized as follows: a. (1) E x t r e m e l y peaked, p s e u d o - e x p o n e n t i a l ; ( 2 ) m o d e r a t e l y peaked; (3) pseudo-triangular; (4) rectangular. b. (4) Rectangular; (5) m o d e r a t e l y double-peaked; (6) double-peaked; (7) bi-cellular. Partial storm widths are B1 = B2 = B3 = 0.25. In all models H 1 = H 3 = 10. Rainfall depths in arbitrary dimension, storm widths and interstation distance D in units of L (length of gaged area). Range o f interstation distances 0 ~ D ~ 1.
Rainfall depth in the center of the storm has been taken variable thus obtaining a series of models from extremely peaked (No. 1 ) to bi-cellular ones (No. 7). The theoretical interstation correlation functions for these models show a variety of shapes. Correlation functions in Fig. 7a can be compared to those derived for exponential and triangular storm models (figs. II.5, II.6 and V.6). As found earlier (Stol, 1981b) the trend of the storm model seems to be more decisive for the shape of the correlation function than its actual form. Correlation functions in Fig. 7a are monotonically decreasing. Storm models Nos. 1 and 2 produce correlation functions that are of a structure according to those found for an exponentially shaped storm with parameter values 5 ~ b ~ 10 (Stol, 1982, fig.3, lower part and diagrams for B = 0.5 and B = 1). Model No. 3 is of a shape that can be compared with an exponential storm model with 1 < b < 3 (Stol, 1982, ibid.) or with the triangular model (fig. II.6). Model No. 4 represents the rectangular storm model (fig. II.4) while models Nos. 5 - 7 introduce the property of non-monotonousness in the
48 c o r r e l a t i o n f u n c t i o n . This p r o p e r t y will be discussed in detail in Part VII (Stol, 1 9 8 3 b ) . When rainfall d e p t h at the c e n t e r o f t h e s t o r m (H2) b e c o m e s less t h a n half the rainfall d e p t h at t h e edges ( H 2 < ½H 1, H 1 = H3 ), the c o r r e l a t i o n f u n c t i o n t e n d s t o f u n c t i o n s t h a t s h o w a s e c o n d m a x i m u m at D = ~ B. This s e c o n d a r y m a x i m u m is m o s t e x t r e m e if H 2 = 0.
APPENDIX A -- MATRICES TO PRODUCE THE REDUCED SUMS OF PRODUCTS FOR FIVE STORM MODELS In Tables A-I--A-V t h e e l e m e n t s o f the m a t r i x o f r e d u c e d sums o f products, C(D) = ]CK,i(D)I, eqs. 20b, 30 and 31, have been c o l l e c t e d f o r the s t o r m t y p e s d e f i n e d in Table I. T h e t r a n s p o s e T o f t h e c o l u m n v e c t o r o f ( H k H k ' ) - c o m b i n a t i o n s is d e f i n e d b y H I = ( H 1H I , H2H2 . . . . . H 1H 3 ) according t o eq. 32. T h e r e d u c e d sum o f p r o d u c t s f o r each distance class K is o b t a i n e d b y tile p r o d u c t C ( D ) H , a (VII x 1)-vector, t h a t p r o d u c e s expressions f o r KI12(D ) - - I l l , K = I, II . . . . , VII, in eq. 11a.
Example: For storm subtype $1.1 (Table A-I) the result is given in eqs. 35. APPENDIX B -- NEGATIVE SLOPE OF FIRST POLYGON SIDE T h e slope o f t h e first p o l y g o n side o f the i n t e r s t a t i o n c o r r e l a t i o n f u n c t i o n for the five s t o r m s u b t y p e s p r o d u c e d b y the tri-rectangular s t o r m m o d e l is given by:
d p i ( D ) / d D = A - I ( L + B) ( H I H 2 + H 2 H 3 --H21 -H22 - H 2 3 )
(B-I)
w h i c h is o b t a i n e d f r o m eq. 39a. T h e sign o f this e x p r e s s i o n d e p e n d s on the sign of:
--(H21 + H 2 + H~ - H 1 H 2 - H 2 H 3 )
= t,
say,
(B-2)
since A > 0 (eq. 12b). Eq. B-2 can be w r i t t e n in d i f f e r e n t ways to clarify its s t r u c t u r e . We choose:
t I = -- [H3(H 3 - H 2 )
+ H 2 ( H 2 - - H l ) + H~ ]
(B-3)
w h i c h o b v i o u s l y is negative for H 3 > H 2 > H 1 . T h e n :
t2
=
-
[H3 (H3
-- H2)
+ H1 (H1 -- H2 ) + H ~ ]
(B-4)
w h i c h is negative f o r s t o r m models with ( H 3 > H 2) A ( H 1 > H 2 ) so f o r o r d e r e d rainfall d e p t h s H 3 > H 1 > H 2 or H 1 > H 3 > H 2 . S u b s e q u e n t l y , we have: t3 = _ [ ( H I
__H2 + H 3 ) 2 + H I ( H 2 _ H 3 ) + H 3 ( H
2 _H1)]
(B-5)
49
T A B L E A-I V a l u e s o f t h e r e d u c e d s u m o f p r o d u c t s CK, i(D) in eq. 31 for s t o r m s u b t y p e $1.1 Distance class, K
HT
HI H 1
H2 H2
Ha H3
H1 H2
H2 H3
H1 H3
i=1
i=2
i=3
i=4
i=5
i=6
I II nI IV
-- D --B1
-- D --D
-- D -- D
D BI
D D
0 0
--B~
-- B 2
-- D
(B1 + B 2 - - D )
B2
(D -- B2)
-- B 1
-- B 2
--B 3
(B 1 + B 2 - - D)
V
-- B1
-- B2
--B 3
0
VI VII
--B1 -B1
--B2 -- B 2
--B 3 -- B 3
0 0
(B2 + B3 - - D) (B2 + B3 - - D) 0 0
(D -- B 2 ) B1 (B - - D) 0
T A B L E A-II V a l u e s o f t h e r e d u c e d s u m o f p r o d u c t s CK, i(D) in eq. 31 for s t o r m s u b t y p e $ 1 . 2 Distance class, K
I II III IV V VI VII
Hw
H1 H1 i=l
H2H2
H3 H3
H1 H2
H2 H3
H1 H3
i=2
i=3
i=4
i=5
i=6
--D --B 1 --B1 - - B1 --B 1 --B 1 - - B1
--D --D --B2
--D --D --D
D B1 (231 + B 2 - - D )
D D B2
0 0 (D - - B 2 )
-- B 2
-- D
0
B2
B1
--B 2 -- B 2 - - B2
--B 3 -- B 3 -- B 3
0 0 0
(B 2 + B 3 - D ) 0 0
B1 (B - - D) 0
T A B L E A-III V a l u e s o f t h e r e d u c e d s u m o f p r o d u c t s CK, i(D) in eq. 31 f o r s t o r m s u b t y p e $2.1 Distance class, K
I II III IV V VI vn
HT
HIHI
H2H2
H3H 3
H1H2
H2H3
H1H3
i=1
i=2
i=3
i=4
i=5
i=6
--D --B1 --B1 --B1 --B 1 --B 1 --B1
--D --D --D --B2 --B 2 --B 2 --B2
--D --D --B3 --B3 --B 3 --B a --B 3
D B1 B1 (Bt +B2--D) 0 0 0
D D B3 (B2 + B 3 - - D ) (B 2 + B 3 - - D ) 0 0
0 0 0 (D--B2) B1 (B - - D ) 0
50
T A B L E A-IV Values o f t h e r e d u c e d s u m o f p r o d u c t s Distance class, K
CK, i(D) in
eq. 31 for s t o r m s u b t y p e
S3.1
HT
HIH1
H2H2
H 3 H 3 HIH2
H2H3
H1H3
i=1
i=2
i=3
i=4
i=5
i=6
I
-- D
-- D
-- D
D
D
0
II III IV V VI VII
-- D
-- B2
--D
--B 1 --B 1 -- B 1 -- B 1 - - B1
--B 2 --B 2 -- B 2 -- B2 - - B2
--D --B 3 --B a -- B 3 --B3
B2 (BI + B2 --D) (B 1 + B 2 - - D) 0 0 0
B2 B2 (B 2 + B 3 - - D) (B 2 + B 3 - - D) 0 0
(D --
B2)
(D --B2) (D - - B2) B1 (B - - D) 0
T A B L E A-V Values o f t h e r e d u c e d s u m o f p r o d u c t s Distance class, K
CK, i(D) in
eq. 31 for s t o r m s u b t y p e
$3.2
HT
HIH1
H2H2
H 3 H 3 H1H2
H2H3
HtH3
i=1
i=2
i=3
i=4
i=5
i=6
I
-- D
-- D
-- D
D
D
0
II III IV V VI VII
-- D
--B2
--D
--B2 - - B2 --B 2 --B2 - - B2
--D -- D -- B 3 --B3 -- B 3
B2 B2 B2 (B 2 + B 3 - - D) 0 0
(D --B2)
--BI - - B1 -- B 1 --B1 - - B1
B2 (B 1 + B 2 - - D) 0 0 0 0
w h i c h is n e g a t i v e f o r s t o r m m o d e l s w i t h ( H 2 ) H 3 >H t4
--- -
1 orH 2 >H
1 >H
(D - - B2) B1 B1 (B - - D ) 0
Ha ) A (H2 ~ HI ) so for H 2
3.Finally:
[H 1 (H 1 - H 2 ) + H 2 (H 2 -- H a ) + H 2 ]
(B-6)
w h i c h is n e g a t i v e f o r s t o r m m o d e l s w i t h t h e p r o p e r t y : H 1 > H 2 > H 3 . T h i s c o m p l e t e s t h e p r o o f f o r all 3! = 6 p e r m u t a t i o n s of relational expressions between
rainfall depths H1, H2 and H 3 .
REFERENCES Stol, Ph.Th., 1981a. Rainfall i n t e r s t a t i o n c o r r e l a t i o n f u n c t i o n s , II. A p p l i c a t i o n t o t h r e e s t o r m m o d e l s w i t h t h e p e r c e n t a g e o f dry days as a n e w p a r a m e t e r . J. H y d r o l . , 50: 7 3 - 104 (in this p a p e r r e f e r r e d t o as Part II).
51
Stol, Ph.Th., 1981b. Rainfall interstation correlation functions, III. Approximations to the analytically derived equation of the correlation function for an exponential storm model compared to those for a rectangular and a triangular storm. J. Hydrol., 52: 2 6 9 - 289 (in this paper referred to as Part III). Stol, Ph.Th., 1982. The exponential storm model and its interstation correlation function. Hydrol. Sci. J., 27(1/3): 35--52. Stol, Ph.Th., 1983a. Rainfall interstation correlation functions, V. Analysis of mrectangular storm models. J. Hydrol., 6 2 : 1 - 2 6 (this issue; in this paper referred to as Part V). Stol, Ph.Th., 1983b. Rainfall interstation correlation functions, VII. On non-monotonousness. J. Hydrol., 64 (in press; in this paper referred to as Part VII).
27
Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
[4]
RAINFALL INTERSTATION CORRELATION FUNCTIONS VI. Application to the Tri-rectangular Storm Model
Ph.Th. STOL
Institute for Land and Water Management Research (I. C. W.), 6700 AA Wageningen (The Netherlands) (Received May 10, 1982; revised and accepted August 12, 1982)
ABSTRACT Stol, Ph.Th., 1983. Rainfall interstation correlation functions, VI. Application to the trirectangular storm model. J. Hydrol., 6 2 : 2 7 - 5 1 . A method is given to systematically derive the theoretical interstation correlation function for a tri-rectangular, not necessarily symmetric, rainfall storm model. Since this storm model permits distinguishing between three principal storm types and two storm subtypes, there are five different correlation functions to be derived. In the general case, viz. when the total storm width B is less than the length L of the gaged area, each correlation function for this model is a seven-sided polygon. The solution is conveniently obtained by means of a number of tables based upon the analysis published by the same author in a previous paper. Some examples are given to illustrate typical properties of the analytically derived correlation functions for tri-rectangular storm models of different shape.
1. I N T R O D U C T I O N
In Part V (Stol, 1983a) of this series of papers, published in the same issue as the present paper, the general m-rectangular storm model has been treated in order to describe its properties essential to analytically derive its rainfall interstation correlation function. In this paper we will propose a systematic procedure with which these correlation functions can be obtained. For practical application of the theoretical analysis it is suggested to employ a tabular way of solution, which will be given for a general, not necessarily symmetric, tri-rectangular rainfall storm model. Correlation functions for particular rectangular models then can easily be obtained from the general solution. Notational conventions and definitions of specific terms can be found in Part V (Stol, 1983a, this issue).
0022-1694/83/0000--0000/$03.00
© 1983 Elsevier Scientific Publishing Company
28
B1
B2
bo
b~
B3 b2
b3
storm aXIS X
''2
rainfall depth
h
Fig. 1. T h e general n o n - s y m m e t r i c t r i - r e c t a n g u l a r s t o r m m o d e l . T h e c e n t r a l p a r t m a y have zero rainfall ( H 2 = 0). 2. T H E T R I - R E C T A N G U L A R S T O R M M O D E L
T h e t r i - r e c t a n g u l a r s t o r m m o d e l is a p a r t i c u l a r case o f t h e m - r e c t a n g u l a r m o d e l w i t h m = 3 (Fig. 1). Its s t o r m - d e f i n i n g e q u a t i o n s are (see P a r t V, eq. 1 or, in s h o r t , eq. V.1): °f(x)
=
h(x)
x < O,
Ho = 0
H 1,
b 0 ~
b0 = 0
H 2,
b 1
~3f(x)
H3,
b2
b3 = B
I,°f(x)
Ho,
ba
Ho = 0
[ if(x) ~2f(x)
= Ho,
(1)
!
T h e c o n d i t i o n s f o r rainfall d e p t h s Hk t o b e m e a s u r e d at t h e e a r t h ' s s u r f a c e are:
H 1 > 0,
H 2 ~> 0,
Ha > 0
(2)
T h e partial s t o r m w i d t h s o f this m o d e l are: B1 =
[[bo, bllL x = b 1 - - b o ,
>0
B2
=
Ilbl,b2 IIx = b 2 - - b 1,
> 0
B3
=
IIb2,b a Ilx = b a - b
>0
2,
(3)
while t h e t o t a l s t o r m w i d t h is given b y : 3
B=~B~
(4)
k=l
We first c o n s i d e r m o d e l s w i t h p r o p e r t y : B 1 : # B 2 4: B s
(5)
29 The solution of the correlation function is considered to be a general solution if conditions expressed by eqs. 2, 3 and 5 are fulfilled and if the total storm width B in eq. 4 is less than the length L of the gaged area, so if: B
(6)
This storm mode] produces a spatial correlation function whose general structure is an M-sided polygon where we have with m -- 3 in eq. V.4:
M = ( m + 1 ) _}_ 1 = 7
(7a)
2
If B ~> L the distance class conditions of eq. V.13 (see Section 6) come into operation and the correlation function then will be less-sided. This is symbolized by defining the actual number M L of polygon sides by eqs. V.5b and V.5c, namely: ML (B) < M
for
B ~>L
M L (B) = M
for
B < L
(7b)
(general case)
(7c)
where ML
M L
---- distance class K for D, whose upper b o u n d a r y is equal to L = last distance class K for D in the general case that B < L = length of gaged area under investigation (fig. II.1)
In deriving general formulas we have from eq. 7a: M L -- M = VII which will be written with roman numerals. If M L ~ M we only have M L polygon sides. They are obtained from the general solution by only taking the first M L equations of the correlation function in the order I, I I , . . . , M L .
3. S T O R M C H A R A C T E R I S T I C S
Storm characteristics for single r a n d o m points a in the storm (subsections V.2.4 and V.2.6) can be expressed as sums of products. Formulas are given in eqs. V.6 and V.8. In particular, we have with m -- 3: 3
I0 = ~
Bk
(Sa)
~ Bk'H4
(8b)
k=l
3
I1 =
4=1 3
111 = ~
Bk " H k H k
k--1
(For systematic use in matrix equations redefined by eqs. 19.)
(8c)
30 where Bk is proportional to the probability with which values occur of Hk and H 2 assigned to it. So, with eqs. 8 we can derive mathematical expectations and we find for the sum of all probabilities (using eq. 4): B- 1i0 = (unity)
(9a)
For the mathematical expectation we arrive at: 3
P = E ( h a ) = B-111
= B -1
--
(9b)
~, Bk "Hk k=l
The variance is obtained by: 3
02 = E ( h a - - P ) 2 = B - 1 I n
- - B - 2 I ] = B -1
--
~, Bk "HkHk _ p 2
(9c)
k=l
4. THE THEORETICAL CORRELATION FUNCTION 4.1. General
The theoretical correlation function for the storm model considered, consists of seven equations. They will be written as the vector p(D) and we define the entire set of equations by: p(D) = { p l ( D ) . . . . . PK (D),..., PvII (D)} w
(10)
where T denotes the transpose of the row vector. From eq. V.9 we have: pg(D)
= I+(L+B)
KI12(D) - - I l l (L+B)I~--(1--p)I~'
K -- I, I I , . . . , V I I (11a)
where p is the fraction of completely dry days over the entire gaged area (Stol, 1981a, Part II). This expression can also be written as: P K ( D ) -- ( L + B ) K I 1 2 ( D ) - ( 1 - - p ) I ~ , (L + B ) I u -- (1 - - p ) I 2
K = I, I I , . . . VII
(11b)
F r o m these formulas we see that if the fraction p of dry days over the total area is large (p -+ 1), the correlation function becomes independent of the length L of the gaged area since eq. l l b then reduces to P K ( D ) = KIl2 ( D ) / I l l . In these formulas K i12 (D) stands for the sum of products to be determined as part of the covariance (eq. V.7a). The difference KI12 (D) -- I u will be referred to as the reduced sum of products. The d e n o m i n a t o r of eqs. 11 (A, say) can be evaluated by inserting eqs. 8b and 8c, to yield: A = (L+B)
~, B k ' H 2 - ( l - p ) k=l
~'H
(12a)
31 or in alternative form (appendix V.B):
3
3
A = L ~ BkH~ + E k=l
3
~, [BkBl(Hk --H~):] +p
( h~-)[31BkHk\2
(12b)
4=1 l>k
clarifying t h a t A > 0 so that the sign of the fraction in eq. l l b is determined by the sign of its numerator. The expression for A is independent of D and so it is constant with respect to the distance classes K = I, I I , . . . , M and, consequentially, also for all tri-rectangular storm types (Table I).
4.2. Evaluating the covariance The covariance for pairs of points (a, a') in the storm (see fig. II.1) does depend on the interstation distance D and therefore its evaluation needs more concern. Part of the covariance is represented in eqs. l l a and 11b by the sum of products K I12 (D). This sum of products is to be evaluated according to a sequence of formulas described in the next sections. From eq. V.7a we have:
m~
B~:,i*(D)'g*g,i*,
KI12(D ) = ~
K = I, II . . . . . VII
(13)
i*= l
where K
=
/72K BK,i*
D
=
H K,i*
I, II, ...
=
index to indicate an arbitrary distance class of D in which each covariance factor B~, i.(D) can be represented by the same equation in the variable D d u m m y variable to enumerate all (HkH~')-combinations that occur in the current distance class K: the relevant combinations total n u m b e r of (HkHk,)-combinationsoccurring in the current distance class K i*th covariance factor for the current distance class K, being a function of D rainfall gage interstation distance i*th relevant (HkHk')-combination for the current distance class K 1st, 2nd, ... distance class K of D.
The covariance f a c t or B~:,i* is proportional to the probability with which the i*th relevant (HkHk,)-combination, denoted by H ~ , i . , occurs in the required distance class K. The covariance factor B~,i* is numerically equal to the length of the interval over which the end-point a of an interstation distance interval [a,a']x on the x-axis (Fig. 3) and of length D = lla,a' [Ix = a' - a, (a' > a), belonging to distance class K, can be shifted along the x-axis through the storm w i t h o u t changing the i*th relevant (HkHk')-combination to which the end-points a and a' belong (see subsection V.3.8).
32 4.3. Evaluating the (Hk Hk ')-combinations The w a y in which the relevant (HkHk')-combinations are to be selected has been described formally in subsections V.3.9 and V.3.10. Relevant combinations are counted b y the d u m m y variable i* (Fig. 3, left-hand margin, 1st column) for each distance class K mentioned in the right-hand margin. When the last relevant combination in a particular distance class has been encountered we set mK* = i* (Fig. 3, right-hand margin). The total number of all possible (HkH k, )-combinations is, according to eq. V.16: m* = ( m + 2 1) = 6
(14)
4.4. Evaluating the covariance factor The covariance factor in eq. 13 is given b y eq. V.14b which reads: i*
BK, i*(D) = llb~,i*-l,bg,i* llx,
iK
=
1 (1) m~
(15)
= I, II . . . . . VI
where
b~,i*-i
=
b~,i*
=
lower b o u n d a r y of the i*th covariance factor interval for the Kth distance class upper b o u n d a r y of the i*th covariance factor interval for the Kth distance class
The procedure of Subsection 4.3 starts with b~:, 0 = 0 (Fig. 3) and each time a new (HkHk')-combination is encountered by shifting the interval [a,a']x through the storm, the new starting position of a is marked with b*K,i*. When a' reaches the right most point in the storm (a' = B) the procedure is terminated by setting the corresponding location of a to b~, mi~. Each distance class K produces its own set of b*-values (Fig. 3 for K = I, IV, VI, VII). The last class (K = VII) in the general (B ~ L) case, so B ~ D ~ L, has no b*-values with a and a' both in the storm and so it is empty. Values of b~,i* depend on the interstation distance D which causes the covariance factor B~, i* to be a function of D.
4.5. A practical solution It will be shown in Section 7 that the determination of br,i** and mK* in eq. 15 can easily be performed by inspection and a tabular solution will be proposed. All (HkHk')-combinations (Fig. 3, left-hand margin, 3rd column) then are ranked. The ranking numbers (Fig. 3, left-hand margin, 2nd column) are obtained by the index formula of eq. V.19a and derived in appendix V.A. It reads: i(k,k') = ½(k'--k)[2m+l--(k'--k)] +k, {:--l(1)m (16) '
k (I) m
33 which for the combination (k, h') = (1,rn) reduces to eq. 14. Each time that we process a new distance class in the tabular solution, we set, with i given by eq. 16 and i* found b y the procedure o f Subsection 4.3, in general, for all distance classes (Fig. 3, left-hand margin, 2nd column):
H* = HkHk',
i = i(h,h')
(17a)
and, in particular, in the Kth distance class:
B*K,i(D) --
* (D), IBm,i (0,
if
(HkHk') is a relevant combination
if
(HkHk') is n o t a relevant combination
(175)
For all m* = 6 (eq. 14) possible (HkHk,)-combinations we can determine the value of the products HkHk', and so of HI., in advance by eq. 17a. Then we only need to determine from eqs. 15 and 17b the corresponding values of the covariance factor B}, i(D) per distance class K. In connection with this approach we redefine the formula for the sum o f squares (eq. 8c) in the following way: m $
Ii, = E B ; ' H *
(18)
i=I
where now, in general:
H*
=
H k H k,
i = i(k,k)
(19a)
and, in particular: tBi,
if
i ~< m
Bi = t0,
•
if
(m + 1) <~i ~ m *
(19b)
not depending on the interstation distance class K. The reduced sum of products becomes: m $
KI12(D ) - 1 1 1
= ~ [B~:,I(D) - S * ] H*
(20a)
i=l
to be used in the form in*
KI12(D)--Ill
= ~
CK,i(D)H*,
g
= I, II, ..., VII
(205)
i=l
which can be presented b y a matrix × vector product.
5. TRI-RECTANGULAR STORM TYPES
5.1. General To elaborate eqs. 17 the relevant (HkHk')-combinations must be selected b y appropriately chosen values of B~,~(D) in class K of interstation distance
34 B1 bo H0 ~ H1 =H3
B2
B3
b1
b;-
' :
x
b3 ....
:
:,,
, ,:::~
Main Sts1Ormtype
H2
B1 bo
B2
B3
bl
b2
b3
H°
x $2
H1 =H 3
B1
bo
B2
bl
B3 be
b3
/ rainfall depth h
Fig. 2. The three principal storm types for the tri-rectangular storm model obtained by permutation of the sizes of the partial storm widths B1, B2 and B3.
values D. However, the class interval boundaries depend on the magnitude of the partial storm widths and their moving sums. Their ordering defines the type of the storm. They will be discussed first.
5.2. Principal storm types According to subsection V.3.4 there are m!/2 = 3 principal storm types for the tri-rectangular storm model. The principal types (S1, $2 and $3) are given in Fig. 2 where the rectangle with the middle-sized base is heavily shaded for better comparison. The relational expressions between the partial storm widths to define the principal storm types are: principal storm type $1 :
B 1 < B2 < B3
principal storm type $2:
B 1
principal storm type $3:
B 2
(21)
35 TABLE I Interstation distance upper class boundary values B(K) based on partial stormwidths Bk and their moving sums for the three principal storm types of Fig. 2 and two subtypes derived fromthem
Distance class, K
I
II III IV V VI VII
Boundary, B(K)
Storm subtypes SI. 1
S1.2
$2.1
$3.1
$3.2 0
B(°)
0
0
0
0
B(I) B(II) B(III) B(IV) B(V) B(VI) B(VID
B1 B2 B3 (B1 + B2) (B2 + B3) B L
B1 B2 (/31 + B2) B3 (B2 + B3) B L
B1 B3 B2 (B1 + B2) (B2 + B3) B L
B2 B2 B1 B1 B3 (B1 + B2) (B1 + B2) B3 (B2 + B3) (B2 + B3) B B L L
5.3. Storm subtypes Without further knowledge of the actual magnitudes o f B 1, B2 and B 3 in eq. 21, it cannot be decided whether for the principal storm types Sl and $3 the ordering o f the first moving sum of partial storm widths is: B3 <(B 1 +B2)
or
(B 1 + B 2 ) < B
3
(22)
The order of moving sums therefore is not uniquely defined by eqs. 21 only. We have for the principal storm t y p e $1 : subtypeS/./:
0 < B 1 < B 2 < B 3 < ( B 1 +B2) < ( B 2
+Ba)
subtypeSI.2:
0 < B I < B 2 < ( B 1 + B 2 ) < B 3 < ( B 2 +B3)
and an analogous series for principal storm type $3. The second principal type ($2, see Fig. 2), however, has no subtypes since the moving sums of partial storm widths define the ordering for this type uniquely. Note that (BI + B3) is not an element of the set of moving sums of partial storm widths. We have by definition from eq. 21 for $2 the inequality B 1 < B 3 < B2, so logically B 2 < (B2 + B 1) and B2 < (]32 + B3). From BI < B3 we also have (B1 + B2 ) < (B3 + B2 ) which finally gives the principal storm type $2:
principal storm type $2:
0
(23c)
In conclusion we have three principal storm types and two subtypes so that there are five different storm types to be considered. The ordered sets
36
of partial storm widths and their moving sums, viz. B(K ) for K = O, I, II, ..., VII, are given in Table I that implicitly defines all storm types for the trirectangular storm model.
6. I N T E R S T A T I O N DISTANCE CLASSES
Partial storm widths and their moving sums as given in Table I, define the upper class boundaries of the distance classes according to eq. V.13: classK:
BCK_I)~D~min(B(K),L),
K = I, I I , . . . , V I I
(24a)
and, since we assumed B ~ L, class I:
B¢o) ~ D ~ min ( B 1 , B 2 , B 3 )
(24b)
class VII:
B ~D ~ L
(24c)
Since the sum of two terms in the argument of eq. 24b exceeds the magnitude of each single term, the first two upper boundary values are obtained from the smallest and the smallest b u t one partial storm width, to be denoted b y B(I) and BCH~, respectively. This means that the last four terms of the ordered set are: (B--B(II)) ~ (B--B(I)) ~B ~L
(25)
except, however, if either BcI) or B(~I) stands for B 2 since B - - B 2 = B 1 + B 3 does not belong to the set of moving sums. See for instance storm types $3.1 and $3.2 in Table I.
7. SYSTEMATIC P R O C E D U R E
7.1. General
The procedure to determine the reduced sum of products K I12(D) - - I l l in the numerator of eq. l l a and redefined in eqs. 20, will be executed step by step in the following subsections using storm s u b t y p e $1.1 as an example. The final solutions for all storm types distinguished is given in Appendix A. 7.2. Partial storm widths
Rearrange all permutations of partial storm widths and their moving sums in order of increasing magnitude to define all storm types. Example: Subsections 5.2 and 5.3 and Table I.
7.3. (Hk Hk ' )-combinations
Determine all possible (HkHk')-combinations and order them according to the index values obtained with the index formula of eq. 16.
•
,
t
.
(HkHk,)-
combination
H1H 1 H2H 2 Hall 3
HIH 2 H2H3
H1H a
Index
i ( k , k')
1 2 3
4 5
6
B 2 ~D~B
0 (D ((BI +B2) 0 <:D ~ (B2 + B s )
O(D(B l O(D(B2 0 ~ D ~ Bs
Primary conditions for D
D ~ B1 D ~B2
D~B 1 D <~B 2 D>(BI
D ~ O
--
D<(B1
D > O D~>O
---
+B2)
1
1
+B2) D<(B2
D~B 2 D ~B3
--
--
--
TO-code
FROM-code 2
Supplementary conditions for D
+B3)
D>(B2
D ~ B2 D ~ Bs
D~O
D ~ O
D > O
2
+Bs)
bo
b0 bl
bi D b2--D b2-- D
b0 -b 1 ..b 2 --
.-•--
bl
b1 b2
2
1
1
ba - D
b2 - - D b3 D
b 1 --D b2 --D b3 --D
2
FROM-code TO-code
Location of a
Possible values of interstation distances D relative to partial storm widths B k to produce conditions and supplementary conditions for D, and the locations of a that produce the interval of displacement of the left-hand end-pomt of the interval [a,a ] x with length D for each ( H k H k ' ) - c o m b i n a t i o n and for all storm types
TABLE II
38 Storm
storm
~1t
type $11
81
~
~
_
width B -
82
B3
~
partiat storm width bo :
o
b!
b3= B
b2
X 4:,
for o f o r o'
i"
i
H,H k
b'io
H,H.
11
b'11
b~2
a
a
~
b'nz o
b'~
I 4 H~H? 2
6
H~H~
3
5
H2H 3
b',3
b~
b'!s
__
K=I
bier2
b'~. 3
D ~
a" I Q" i
•
1 6
Q
b'z o
H1H 3
b~j 1
a
o
D ~
.1H: :0
~,
1 D ~
K
Fig. 3. Schematical illustration of the procedure of shifting an interval [a, a'] x of length D along the x-axis from the left to the right through a storm (see fig. V.3) (open circles: starting position of a before a shift; black dots: terminal position o f a after the shift).
Example: See first t w o c o l u m n s of Table II, and Fig. 3, left-hand margin, 2nd c o l u m n .
7.4. Primary conditions for D D e t e r m i n e all p o s s i b l e values D can t a k e o n for e a c h
(HkHk')-combination.
Example: For this part of the procedure Fig. 3 is basic. Values can be read from it. For any storm type, taking the c o m b i n a t i o n (HkHk') = (H1H 2 ) as an example, values of the interstation distance D that produce a E B1 and a' @ B 2 range 0 < D < (B1 + B2). (To avoid ambiguity with respect to locations in the n e i g h b o u r h o o d of the class boundaries, the "<"-signs are used instead of the "~<"-signs. This is allowed since the probability of being exactly on a class boundary equals zero.)
39
Values are obtained from Fig. 3 and are collected in Table II, 3rd column and are valid for all storm types. 7. 5. S u p p l e m e n t a r y c o n d i t i o n s for D
Within each range determined in Subsection 7.4 determine the supplementary conditions for D, expressed in partial storm widths, for which the location of a is either constant or a function of D. E x a m p l e : If D ~ B1 (Fig. 3, K = I, i* = 2, i = 4, H k H k' = H 1 H 2 ) the general f o r m u l a for the locations of a (open circles) for arbitrary values of D, before shifting the interval [a, a' ] x to the right, is a = bl -- D and so it is a f u n c t i o n of D. This holds for all values of D that satisfy 0 ~ D ~ B1, thus the s u p p l e m e n t a r y c o n d i t i o n is D ~ B~. If, however, D B 1 (Fig. 3, K = IV, i* = 1, i -= 4, H k H k' = H 1 H 2 ) the location o f a (open circle) before shifting the interval [a,a' Ix to the right is a = b0, constant, for all values of D that satisfy B 1 ~ D ~ (B 1 + B2 ). The s u p p l e m e n t a r y c o n d i t i o n t h e n is B 1 ~ D.
Supplementary conditions of this kind have been collected in Table II under the heading FROM-code 1 or 2, respectively. If D ~ B 2 (Fig. 3, K = I, i* = 2, i = 4, H k H k ' = H 1 H : ) the general f o r m u l a for the l o c a t i o n o f a (black dot) after having shifted the interval [a,a']x to the right, is a = b l , constant, for all values of D given by 0 ~ D ~ B2, the s u p p l e m e n t a r y c o n d i t i o n being D ~ B 2. If, however, D ~ B2 (Fig. 3, K = IV, i* = 1, i ----4, H k H k' = H 1 H 2) the location o f a (black dot) after the shift is a = b2 -- D, and so a f u n c t i o n of D, for all D that satisfy B 2 ~ D ~ (B1 + B2 ). The s u p p l e m e n t a r y c o n d i t i o n t h e n is D ~ B2.
Supplementary conditions of this kind have been collected in Table II under the heading TO-code 1 or 2, respectively. All supplementary conditions for D in general terms, regardless the storm type, have been collected in advance in Table II for further particular use. 7.6. L o c a t i o n o f a
Determine the location of point a b e f o r e as well as in the situation after the shift of the interval [a, a'] x of length D along the x-axis, for all ( H k H k ' ) combinations and for all supplementary conditions for D collected in Table II. E x a m p l e : F r o m Fig. 3 it is seen t h a t the general expression for the o p e n circles, illustrating the c o m b i n a t i o n H k H k' = H l H2, reads w i t h primary c o n d i t i o n s and s u p p l e m e n t a r y c o n d i t i o n s for D c o m b i n e d :
FROM-code/,
(O~D~B1)
FROM-code2,
(B 1 ~ D ~ B
~a 1 -bB2)~a
= b 1 --D
(26a)
= bo
(26b)
= b1
(27a)
= b2 - - D
(27b)
F o r the black dots in Fig. 3 this b e c o m e s :
TO-code 1,
(0~D~B2)
~a
TO-code2,
(B 2 ~ D ~ B
1 +B2)-~a
40
These values o f a produce the sets o f boundary values b~:,i* o f eq. 15 as required to determine the covariance factor B~:,i*. All locations of a, still regardless the storm type and the distance class index, have been collected in advance in Table II for further particular use.
7. 7. Covariance factors in general Evaluate for further particular use the four possible " F R O M - - T O " s i t u a t i o n s ( F R O M - c o d e = 1, 2 ; T O - c o d e ----1, 2) a c c o r d i n g t o c o l u m n n u m b e r s i n T a b l e II f o r e a c h ( H k H k ' ) - c o m b i n a t i o n i n t e r m s o f p a r t i a l s t o r m w i d t h s t o o b t a i n t h e covariance factors t h a t are to be applied. This covariance factor (eq. 1 5 ) is t h e l e n g t h o f d i s p l a c e m e n t o f t h e e n d - p o i n t a u n d e r t h e corresponding "FROM--TO "-situation. Example: With ( F R O M , T O ) - - ( 1 , 2 ) the combination HkH k' ----H1H2 (see Table II) gives as distance conditions D < B 1 and D < B 2 and for locations of a "FROM": a : bl -- D and " T O " : a = b2 -- D with length of displacement equal to:
lib, --D, b 2 - D IIx -- (b= - - D )
-D)
-- (bl
= B2
(28)
TABLE III Collection of all possible covarianee factors for all storm types (FROM,TO) code
(HkHk')-combinations and their index i
H1H1 1
H2H2 2
H3H3 3
H1H2 4
(1,1)
--
--
--
D
(1,2) (2,1) (2,2)
--B , -- D
--B2 -- D
-B2 -BI B 3 - D (BI -b B2 - - D )
H2H3 5
HIH3 6
D
(D --B2)
B3 B3 B2 BI (B2 q- B3 - - D) ( B - - D )
TABLE IV Collection of (FROM,TO)-codes for storm subtype S1.1 Distance class, K
I II III IV V VI VII
(HkHk')-combinations and their index i
m* = 6
H 1H 1 1
H2 H2 2
H 3H 3 3
H 1H2 4
H2 H 3 5
HI 1"/3 6
* mK
(2,2) ---. . .
(2,2) (2,2) ---
(2,2) (2,2) (2,2) -. . .
(1,1) (2,1) (2,2) (2,2)
(1,1) (1,1) (2,1) (2,2) (2,2)
--(1,1) (i.i) (2,1) (2,2)
5 4 4 3 2 1 0
.
. . .
. .
. .
.
41
w h i c h easily c a n be verified g r a p h i c a l l y f r o m Fig. 2. F r o m t h e figure it is also seen t h a t this p a r t i c u l a r case o n l y c a n a p p l y t o s t o r m t y p e $3.
All results, regardless the storm type, have been collected in advance in Table III for further particular use.
7.8. Storm-depending (FROM,TO)-codes Process Table I and determine for all storm types successively the corresponding (FROM,TO)-codes by means of Table II for each (HkHk')-combination, using the columns headed with "supplementary conditions for D " , until the primary condition for D is violated. E x a m p l e : F o r s t o r m s u b t y p e $1.1 t h e d i s t a n c e class K -----IV reads ( T a b l e I): B 3 ~ D (B 1 4- B 2 ) ~ (B 2 4- B 3 ) ~ B ~ L. C o n s i d e r i n g t h e r e l e v a n t c o m b i n a t i o n H k H k' = H 1 H 3 ( w h e r e i =- 6) we have to use in T a b l e II:
FROM-code = 1
since
D ~ (BI 4- B2)
(29a)
TO-code
since
D ~ (B2 4- B3)
(29b)
= 1
For storm subtype $1.1 all corresponding codes have been collected in Table IV.
7. 9. Covariance factors in particular Decode the (FROM,TO)-codes in Table IV with the aid of Table III into the corresponding length of displacement of point a, thus giving the covariance factor B~,i for KI12 with K = I, II, ..., VII and i = 1 ( 1 ) 6 . According to eq. 17b open places should be zero filled. E x a m p l e : If t h e c o d e is for s t o r m s u b t y p e $1.1, we e n t e r T a b l e IV w i t h H k H k ' = H1 H2 ( w h e r e i ----4) a n d K ----II, t h e c o d e is ( F R O M , T O ) -= (2, 1 ) giving f r o m T a b l e III t h e value BIL 4 -~ B 1 .
For storm s u b t y p e $1.1 the particular values of the covariance factors are collected in Table V, according to instructions laid down in Subsection 4.5.
7.10. Reduced sum o f products Finally subtract I l l (eq. 18, and worked in eqs. 19) from the results obtained in Subsection 7.9 to obtain the reduced sum of products gI12 (D) -Ill (eq. 20a) being the numerator of eq. l l a . E x a m p l e : F r o m eq. 19 a n d T a b l e V it is a p p a r e n t t h a t we o n l y n e e d t o s u b s t r a c t t h e values B1, B2 a n d B 3 f r o m t h e results o b t a i n e d in S u b s e c t i o n 7.9 f o r t h e c o m b i n a t i o n s H k H ~' ----HkHk, k = 1 ( 1 ) m (see T a b l e A--I).
For all storm types the reduced sum of products is given in Appendix A.
42 TABLE V Collection of particular covariance factors BK, i for storm subtype $1.1
(HkH k')-combinations and their index i
Distance class, K
HIH1 1
H2H2 2
H3H3 3
I
(BI--D)
(B2--D)
II III IV V VI VII
0 0 0 0 0 0
(B2 - D ) 0 0 0 0 0
H1H2 4
H2H3 5
HIH3 6
(B3 --D) D (B3 - D ) B I
D D
0 0
(B3 --D) 0 0 0 0
B2 (B 2 + B 3 - - D ) (B 2 + B s - - D ) 0 0
(D--B2) (D--B2) B1 (B -- D) 0
(BI +B2--D) (B1 + B 2 - - D ) 0 0 0
7. I 1. The interstation correlation function W r i t e t h e i n t e r s t a t i o n c o r r e l a t i o n f u n c t i o n o f eq. 1 0 w i t h t h e aid o f t h e t a b l e s in A p p e n d i x A a n d eq. 11a. Solution: We d e f i n e t h e f o l l o w i n g m a t r i c e s a n d v e c t o r s : CK(D)
=
{ C K . , ( D ) , . . . , CK,i(D),...,CK,m* (D)}(1 x 6) K =
C(D) =
(30)
I, I I . . . . , V I I
ICK,i(D)I,
(31) i
1,2 . . . . . 6
(vii x 6)
H = {HIH1, H2H2,• ' ' ,H,H3}T (6 X 1)
(32)
I = {1,1,..., I}T(vII X I)
(33)
From eqs. 10, 12a and 20b we then have for eq. l l a :
p(D) = I + A - I ( L + B ) C(D)'H,
(Vllx 1)
(34)
where C(D) and H can be obtained from the tables in Appendix A.
Example: The interstation correlation function for storm subtype SI,I consists of VII equations. From Table A-I we obtain: I Pi(D)
p(D)=
Pv(D)
---- 1 -- A-I(L + B) H21 + H 2 +H23--H1H 2 .....H2H3]D = 1 -- A-l(L + B )
'
(35a)
]
~BkH~) --HIH3BI --H2H3 (B2 + B3 -- D k=l
(35b) Pvii(D)
= 1--A-I(L +B) k=l
where only the Ist, Vth and VIIth have been written in full as an example. E x a m p l e s o f i n t e r s t a t i o n c o r r e l a t i o n f u n c t i o n s c a l c u l a t e d w i t h eqs. 35 c a n b e f o u n d in P a r t V.
43 coy
factor for
wI1z
storm type $1:1
B:I
I
~b
B~?
i
1 2
H~ H 1 H~ H;
3
H3 H3
.
.
.
®
.
~' 5 5
H H H? H H H3
B~ to be read B~*B !
B~
i
B: I / /
I
\~,
,,, B,
B
B3 B,
B~3
B23
B
' L B
B,
B~
B? B .
B~
823 m t e r s t a t i o n distance [3
interstat,on d i s t a n c e class K
Fig. 4. The f u n c t i o n s BK, i = BK, i(D) that produce the covariance factors for storm subt y p e $1.1: ( A ) for the squares HkH~, k -- 1 ( 1 ) 3 ; and (B) for the p r o d u c t s H k H k ' , k -- 1 ( 1 ) 2 , k' = (k + 1) (1)3.
7.12. Remarks In the described procedure n o use has explicitly been made o f the quantities b ~ , i . since the e x e c u t i o n o f the procedure has b e e n performed by inspection. The covariance-factor f u n c t i o n s BK,i = BK,i (D) t h a t are valid for each (HkHk , ) - c o m b i n a t i o n for storm s u b t y p e $1.1 as given in Table V have been set o u t graphically in Fig. 4 A and B. The figures also explicitly illustrate the range of distance values D (distance classes K) o n w h i c h the covariance factors are defined. Example: F r o m the figure it can be s e e n that (H/~H/~')-combinations w i t h i = 5 and i = 6 are the o n l y relevant c o m b i n a t i o n s in distance class K ---- V (see also Fig. 5).
8. T E S T S O F C O R R E C T N E S S
There are s o m e c o n d i t i o n s that can be used to test the correctness o f the results.
8.1. Sum of intervals constant The s u m o f all intervals [bK,i*-l,br,i* * * ]x adds up t o B -- D as can be seen from Fig. 5, where c u m u l a t e d values of B*K,i(D) have b e e n set o u t . This holds for distance classes K = I, II, . . . , VI. The last class K = VII yields zero. In Tables A-I--A-V o b t a i n e d by subtracting Ill from gI12 , all lines s h o u l d add up t o - - D , t h e last o n e giving - - B . This can readily be checked.
44 cov-foctor l
for KI 12
Z B; ,l:1(1)rn~ storm type $1.1
B
i : H k Hk, 1 : H 1 H1 2 : H2 H2
B23
3 M3H 3 4 : H, H 2 5 H2H 3
B13 ~__
m ~ = 6 H1N 3
5
B~
B0
B~
B2
B3 B12
B~3
B23 interstatton
interstation
F i g . 5. T h e f u n c t m n s
BK, i
=
BK, i(D ) o f
B // L distance D
distance crass K
Fig. 4 cumulated.
8.2. Continuity in vertices T h e i n t e r s t a t i o n c o r r e l a t i o n f u n c t i o n is c o n t i n u o u s at t h e vertices. This can be c h e c k e d by inserting t h e values o f t h e class interval b o u n d a r i e s w h i c h are c o l l e c t e d in Table I, i n t o t h e c o r r e s p o n d i n g variable D in Tables A - I - - A - V .
Example: F o r s t o r m s u b t y p e $ 1 . 1 t h e b o u n d a r y b e t w e e n d i s t a n c e class K --- II a n d dist a n c e class K ---- III is D ----B 2 . T h e v a l u e s o f CK, i in T a b l e A-I f o r t h e r e q u i r e d c l a s s e s are: K = II:
-B 1 --D
--D
B1
D
0
K = III:
--B 1 --B 2
--D
(B1 + B 2 - - D )
B2
(D--B2)
w h i c h are e q u a l f o r D = B 2 .
8.3. The rectangular storm A special case is t h e rectangular s t o r m m o d e l (Stol, 1 9 8 1 a ) w h e r e H I = H 2 = H 3 = H ( c o n s t a n t ) . Partial s t o r m w i d t h s n o w are n o essential q u a n t i t i e s a n y m o r e a n d all s t o r m t y p e s r e d u c e t o this s i m p l e m o d e l . T h u s e a c h line in Tables A - I - - A - V p r o d u c e s :
KI12 - - I l l
= - - D H 2,
gI12--Ill
----
- - B H 2,
K =
I, I I , . . . , VI
K = VII
(36a) (36b)
45 thus giving eq. II.50 where H 2 is cancelled in the fraction. This solution, however, can also be obtained by taking into account the central partial storm width only so with B 1 = B a = 0, B 2 = B and H : = H. Then we obtain eq. II.50 only by means of the third column in Tables A-I--A-V: K II: - - I l l
= - D H : , K = I, . . . , N
(37a)
KI12 - - I l l
---- - - B H 2 , K
(37b)
= N + 1, ..., VII
where N = II for storm subtypes $1.1 and $1.2; N = III for storm subtype $2.1 a n d N = I for storm subtypes $3.1 and $3.2. 9. SOME GENERAL PROPERTIES In order to investigate the properties of the derived correlation function we write for the first derivative of eq. 34 with respect to D: dp (D) _ L + S dD A
d C ( D ) . H, dD
(VII × 1)
(38)
From the five tables in Appendix A we first note t h a t for all storm types the functions PI (D), Pv (D), Pvi (D) and Pvii (D) have the same structure. The first derivatives are (see eqs. 35 and Tables A-I--A-V): dp~(D)/dD = - - A - i ( /
+B)(--H1H 2 --H2H a +HI +H~ THe)
(39a)
dpv (D)/dD = - - A -1 (L + B ) H 2 H a
(39b)
dpvx (D)/dD = -- A-1 (L + B) H1H a
(39e)
dPv u ( D ) / d D = 0
(39d)
which are always negative (eqs. 39a--39c) or zero (eq. 39d) (Appendix B). The interstation correlation function departs from {D, p (D) } = (0, 1) with negative slope. In the general case (B < L), the function has two negativesloped sides followed by a horizontal polygon side at the right most part. The sign of the slope of PK (D) is given by the sign of eqs. 38 since A > 0 (eq. 12b). The matrix of coefficients of first derivatives for storm subtype $1.1, obtained from Table A-I, reads:
dC(D) dD
--1 0 0 0 0 0 0
--1 --1 1 1 0--1 --1 0 1 0 0 --1 --1 0 1 0 0 --1 --1 1 0 0 0 --1 0 0 0 0 0 --1 0 0 0 0 0
(40)
46
p 1.0 .8 5
\
/,
,-.... 0.8
2
O.L.
0 -.2
0
-4
B=1.5 -8
-1.0 0
I .1
I .2
1 3
I .4
5
l 5
I 7
I 8
I 9
10 D
Fig. 6. A n a l y t i c a l l y derived i n t e r s t a t i o n c o r r e l a t i o n f u n c t i o n s for s t o r m m o d e l 5 o f Fig. 7, n o w w i t h t o t a l s t o r m w i d t h B ---- 1.5 (in u n i t s o f L). Curves are labelled w i t h t h r e e values for t h e f r a c t i o n p o f c o m p l e t e l y d r y days over t h e area.
10. E X A M P L E S
Examples o f theoretical correlation functions for tri-rectangular nons y mmetr ic storm models have been given by Stol (1983a). Some further typical examples for symmetric equidistant (B 1 = B 2 = B 3 ) storm models are given below. Th e first example concerns the influence of t he fraction p of com pl et el y d r y days over t he gaged area and is based on a tri-rectangular symmetric equidistant model with B = 1.5L, L = 1, H 1 = H 3, H 2 = ½H 1 (Fig. 6). Since B > L th e correlation f u n c t i o n only consists o f two pol ygon sides defined on the intervals 0 ~ D ~ ~ B and 12B < D ~ L. It can be com pared with No. 5 in Fig. 7. When the fraction p of d r y days over the area increases, so does the correlation coefficient (fig. II.7). This is the first model of those considered thus far t h a t produces, at a certain distance f r om the origin, positive correlation coefficients t h a t are c ons t ant until D = L is reached. Th e second example (Fig. 7) pertains a tri-rectangular symmetric equidistant storm model. St or m parameters are: storm width B = 0.75L, H1 = H 3 = 10 (arbitrary units) while t he length o f the gaged area, w i t h o u t loss of generality, has been taken L = 1. Details are given in the figure caption.
47 storm model
storm model
...-.
-/"0 / 0 storm 0
tO0 h
J I I .1 ,2 .3 modet tB=O.?5) 1
I ~,
I .5 2
I .7
5 3
I 8
I .9
4
1.0
0 5
.1
.2
.3 6
.4
.5
.g
.7
.8
.9
1.0 D
?
H2
Fig. 7. E x a m p l e s of theoretically derived interstation correlation f u n c t i o n s for seven s y m m e t r i c a l equidistant tri-rectangular s t o r m models with total storm width B ---- 0.75 (in units o f L). Models could be characterized as follows: a. (1) E x t r e m e l y peaked, p s e u d o - e x p o n e n t i a l ; ( 2 ) m o d e r a t e l y peaked; (3) pseudo-triangular; (4) rectangular. b. (4) Rectangular; (5) m o d e r a t e l y double-peaked; (6) double-peaked; (7) bi-cellular. Partial storm widths are B1 = B2 = B3 = 0.25. In all models H 1 = H 3 = 10. Rainfall depths in arbitrary dimension, storm widths and interstation distance D in units of L (length of gaged area). Range o f interstation distances 0 ~ D ~ 1.
Rainfall depth in the center of the storm has been taken variable thus obtaining a series of models from extremely peaked (No. 1 ) to bi-cellular ones (No. 7). The theoretical interstation correlation functions for these models show a variety of shapes. Correlation functions in Fig. 7a can be compared to those derived for exponential and triangular storm models (figs. II.5, II.6 and V.6). As found earlier (Stol, 1981b) the trend of the storm model seems to be more decisive for the shape of the correlation function than its actual form. Correlation functions in Fig. 7a are monotonically decreasing. Storm models Nos. 1 and 2 produce correlation functions that are of a structure according to those found for an exponentially shaped storm with parameter values 5 ~ b ~ 10 (Stol, 1982, fig.3, lower part and diagrams for B = 0.5 and B = 1). Model No. 3 is of a shape that can be compared with an exponential storm model with 1 < b < 3 (Stol, 1982, ibid.) or with the triangular model (fig. II.6). Model No. 4 represents the rectangular storm model (fig. II.4) while models Nos. 5 - 7 introduce the property of non-monotonousness in the
48 c o r r e l a t i o n f u n c t i o n . This p r o p e r t y will be discussed in detail in Part VII (Stol, 1 9 8 3 b ) . When rainfall d e p t h at the c e n t e r o f t h e s t o r m (H2) b e c o m e s less t h a n half the rainfall d e p t h at t h e edges ( H 2 < ½H 1, H 1 = H3 ), the c o r r e l a t i o n f u n c t i o n t e n d s t o f u n c t i o n s t h a t s h o w a s e c o n d m a x i m u m at D = ~ B. This s e c o n d a r y m a x i m u m is m o s t e x t r e m e if H 2 = 0.
APPENDIX A -- MATRICES TO PRODUCE THE REDUCED SUMS OF PRODUCTS FOR FIVE STORM MODELS In Tables A-I--A-V t h e e l e m e n t s o f the m a t r i x o f r e d u c e d sums o f products, C(D) = ]CK,i(D)I, eqs. 20b, 30 and 31, have been c o l l e c t e d f o r the s t o r m t y p e s d e f i n e d in Table I. T h e t r a n s p o s e T o f t h e c o l u m n v e c t o r o f ( H k H k ' ) - c o m b i n a t i o n s is d e f i n e d b y H I = ( H 1H I , H2H2 . . . . . H 1H 3 ) according t o eq. 32. T h e r e d u c e d sum o f p r o d u c t s f o r each distance class K is o b t a i n e d b y tile p r o d u c t C ( D ) H , a (VII x 1)-vector, t h a t p r o d u c e s expressions f o r KI12(D ) - - I l l , K = I, II . . . . , VII, in eq. 11a.
Example: For storm subtype $1.1 (Table A-I) the result is given in eqs. 35. APPENDIX B -- NEGATIVE SLOPE OF FIRST POLYGON SIDE T h e slope o f t h e first p o l y g o n side o f the i n t e r s t a t i o n c o r r e l a t i o n f u n c t i o n for the five s t o r m s u b t y p e s p r o d u c e d b y the tri-rectangular s t o r m m o d e l is given by:
d p i ( D ) / d D = A - I ( L + B) ( H I H 2 + H 2 H 3 --H21 -H22 - H 2 3 )
(B-I)
w h i c h is o b t a i n e d f r o m eq. 39a. T h e sign o f this e x p r e s s i o n d e p e n d s on the sign of:
--(H21 + H 2 + H~ - H 1 H 2 - H 2 H 3 )
= t,
say,
(B-2)
since A > 0 (eq. 12b). Eq. B-2 can be w r i t t e n in d i f f e r e n t ways to clarify its s t r u c t u r e . We choose:
t I = -- [H3(H 3 - H 2 )
+ H 2 ( H 2 - - H l ) + H~ ]
(B-3)
w h i c h o b v i o u s l y is negative for H 3 > H 2 > H 1 . T h e n :
t2
=
-
[H3 (H3
-- H2)
+ H1 (H1 -- H2 ) + H ~ ]
(B-4)
w h i c h is negative f o r s t o r m models with ( H 3 > H 2) A ( H 1 > H 2 ) so f o r o r d e r e d rainfall d e p t h s H 3 > H 1 > H 2 or H 1 > H 3 > H 2 . S u b s e q u e n t l y , we have: t3 = _ [ ( H I
__H2 + H 3 ) 2 + H I ( H 2 _ H 3 ) + H 3 ( H
2 _H1)]
(B-5)
49
T A B L E A-I V a l u e s o f t h e r e d u c e d s u m o f p r o d u c t s CK, i(D) in eq. 31 for s t o r m s u b t y p e $1.1 Distance class, K
HT
HI H 1
H2 H2
Ha H3
H1 H2
H2 H3
H1 H3
i=1
i=2
i=3
i=4
i=5
i=6
I II nI IV
-- D --B1
-- D --D
-- D -- D
D BI
D D
0 0
--B~
-- B 2
-- D
(B1 + B 2 - - D )
B2
(D -- B2)
-- B 1
-- B 2
--B 3
(B 1 + B 2 - - D)
V
-- B1
-- B2
--B 3
0
VI VII
--B1 -B1
--B2 -- B 2
--B 3 -- B 3
0 0
(B2 + B3 - - D) (B2 + B3 - - D) 0 0
(D -- B 2 ) B1 (B - - D) 0
T A B L E A-II V a l u e s o f t h e r e d u c e d s u m o f p r o d u c t s CK, i(D) in eq. 31 for s t o r m s u b t y p e $ 1 . 2 Distance class, K
I II III IV V VI VII
Hw
H1 H1 i=l
H2H2
H3 H3
H1 H2
H2 H3
H1 H3
i=2
i=3
i=4
i=5
i=6
--D --B 1 --B1 - - B1 --B 1 --B 1 - - B1
--D --D --B2
--D --D --D
D B1 (231 + B 2 - - D )
D D B2
0 0 (D - - B 2 )
-- B 2
-- D
0
B2
B1
--B 2 -- B 2 - - B2
--B 3 -- B 3 -- B 3
0 0 0
(B 2 + B 3 - D ) 0 0
B1 (B - - D) 0
T A B L E A-III V a l u e s o f t h e r e d u c e d s u m o f p r o d u c t s CK, i(D) in eq. 31 f o r s t o r m s u b t y p e $2.1 Distance class, K
I II III IV V VI vn
HT
HIHI
H2H2
H3H 3
H1H2
H2H3
H1H3
i=1
i=2
i=3
i=4
i=5
i=6
--D --B1 --B1 --B1 --B 1 --B 1 --B1
--D --D --D --B2 --B 2 --B 2 --B2
--D --D --B3 --B3 --B 3 --B a --B 3
D B1 B1 (Bt +B2--D) 0 0 0
D D B3 (B2 + B 3 - - D ) (B 2 + B 3 - - D ) 0 0
0 0 0 (D--B2) B1 (B - - D ) 0
50
T A B L E A-IV Values o f t h e r e d u c e d s u m o f p r o d u c t s Distance class, K
CK, i(D) in
eq. 31 for s t o r m s u b t y p e
S3.1
HT
HIH1
H2H2
H 3 H 3 HIH2
H2H3
H1H3
i=1
i=2
i=3
i=4
i=5
i=6
I
-- D
-- D
-- D
D
D
0
II III IV V VI VII
-- D
-- B2
--D
--B 1 --B 1 -- B 1 -- B 1 - - B1
--B 2 --B 2 -- B 2 -- B2 - - B2
--D --B 3 --B a -- B 3 --B3
B2 (BI + B2 --D) (B 1 + B 2 - - D) 0 0 0
B2 B2 (B 2 + B 3 - - D) (B 2 + B 3 - - D) 0 0
(D --
B2)
(D --B2) (D - - B2) B1 (B - - D) 0
T A B L E A-V Values o f t h e r e d u c e d s u m o f p r o d u c t s Distance class, K
CK, i(D) in
eq. 31 for s t o r m s u b t y p e
$3.2
HT
HIH1
H2H2
H 3 H 3 H1H2
H2H3
HtH3
i=1
i=2
i=3
i=4
i=5
i=6
I
-- D
-- D
-- D
D
D
0
II III IV V VI VII
-- D
--B2
--D
--B2 - - B2 --B 2 --B2 - - B2
--D -- D -- B 3 --B3 -- B 3
B2 B2 B2 (B 2 + B 3 - - D) 0 0
(D --B2)
--BI - - B1 -- B 1 --B1 - - B1
B2 (B 1 + B 2 - - D) 0 0 0 0
w h i c h is n e g a t i v e f o r s t o r m m o d e l s w i t h ( H 2 ) H 3 >H t4
--- -
1 orH 2 >H
1 >H
(D - - B2) B1 B1 (B - - D ) 0
Ha ) A (H2 ~ HI ) so for H 2
3.Finally:
[H 1 (H 1 - H 2 ) + H 2 (H 2 -- H a ) + H 2 ]
(B-6)
w h i c h is n e g a t i v e f o r s t o r m m o d e l s w i t h t h e p r o p e r t y : H 1 > H 2 > H 3 . T h i s c o m p l e t e s t h e p r o o f f o r all 3! = 6 p e r m u t a t i o n s of relational expressions between
rainfall depths H1, H2 and H 3 .
REFERENCES Stol, Ph.Th., 1981a. Rainfall i n t e r s t a t i o n c o r r e l a t i o n f u n c t i o n s , II. A p p l i c a t i o n t o t h r e e s t o r m m o d e l s w i t h t h e p e r c e n t a g e o f dry days as a n e w p a r a m e t e r . J. H y d r o l . , 50: 7 3 - 104 (in this p a p e r r e f e r r e d t o as Part II).
51
Stol, Ph.Th., 1981b. Rainfall interstation correlation functions, III. Approximations to the analytically derived equation of the correlation function for an exponential storm model compared to those for a rectangular and a triangular storm. J. Hydrol., 52: 2 6 9 - 289 (in this paper referred to as Part III). Stol, Ph.Th., 1982. The exponential storm model and its interstation correlation function. Hydrol. Sci. J., 27(1/3): 35--52. Stol, Ph.Th., 1983a. Rainfall interstation correlation functions, V. Analysis of mrectangular storm models. J. Hydrol., 6 2 : 1 - 2 6 (this issue; in this paper referred to as Part V). Stol, Ph.Th., 1983b. Rainfall interstation correlation functions, VII. On non-monotonousness. J. Hydrol., 64 (in press; in this paper referred to as Part VII).