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Time resolution of the hydrologic time-series models
Journal o f Hydrology, 32 (1977) 347--361 347 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
TIME RESOLUTION
OF THE HYDROLOGIC
TIME-SERIES MODELS
M.L. KAVVAS',*, L.J. COTE 2 and J.W. DELLEUR 1
School o f Civil Engineering, Purdue University, West Lafayette, Ind. 47907 (U.S.A.) 2 Statistics Department, Purdue University, West Lafayette, Ind. 47907 (U.S.A.) (Received February 7, 1976; revised and accepted May 7, 1976)
ABSTRACT Kavvas, M.L., Cote, L.J. and Delleur, J.W., 1977. Time resolution of the hydrologic timeseries models. J. Hydrol., 32: 347--361. Hydrologists have constructed time-series models for the daily, monthly, annual and other time intervals for the simulation studies of a water-resource system. The mathematical operations involved in attaining the monthly hydrologic data from the daily hydrologic data and the annual hydrologic data from either the daily or the monthly hydrologic data are standard and well known. Through the utilization of these operations one can find the covariance and the spectral relations among the hydrologic sequences corresponding to different sampling intervals. Once these relations are known, the hydrologist, starting from the smallest interval, can calculate the covariance and the spectral structures of the hydrologic sequences corresponding to larger sampling intervals. One can then construct the time-series models corresponding to these larger intervals from the theoretically calculated covariance and spectral density functions without ever constructing the data for larger intervals or without the computation of the sample covariance and the sample spectral-density functions for larger intervals. This paper establishes the relationships between the covariances and between the spectra, respectively, of the hydrologic time series with time steps which are multiples of each other, such as daily, monthly and annual intervals. The stochastic structure in the time domain of the series with the longer time steps can be derived in terms of the structure of the series with the shorter time steps. The scheme discussed in this paper is an aggregation scheme and describes, in detail, the filters and the many-to-one transformations that are required to go from the smaller to larger time sampling intervals. The relations show that when the hydrologic time-series models are disaggregated from larger to smaller time sampling intervals the transformations involved are one-to-many, indicating that there are m a n y ways for the disaggregation whereas there is only one unique way for aggregation.
INTRODUCTION By the introduction of the spectral model for the periodic hydrologic time series to hydrology (Cote, 1973; Kavvas and Delleur, 1975) it is now possible to make an analytical analysis of the various transformations underlying the formation of the hydrologic sequences corresponding to small time sampling intervals. In this way, a time resolution of the hydrologic time-series models, *Present address: Industrial Engineering Department, MiddleEast Technical University, In~nii Bulvari, Ankara, Turkey.
348
in terms of their covariance and spectral structures, is going to be achieved. First, the relations between the hydrologic sequences corresponding to monthly intervals and the sequences corresponding to annual intervals will be studied. Once the covariance and the spectral relations between the monthly and the annual series is established, they will be checked by the hydrologic data. Then a general aggregation scheme for the covariance and the spectral functions of the hydrologic time series will be developed. AGGREGATION
OF MONTHLY
SERIES TO ANNUAL
SERIES
An annual hydrologic sequence may be formed from a monthly hydrologic sequence by adding the monthly quantities in a year to form the annual value for that year and so on. This operation corresponds to adding consecutive values of twelve monthly hydrologic quantities and then picking up every 12th value to form the annual series. Let {Mj ) be the monthly hydrologic time series, ( Yj ) be the moving sums formed from the addition of twelve consecutive monthly values, and {Aj) be the annual series. The moving sums ( Y j ) will be formed from the monthly series {Mj} by the operation: 11
Yj = ~
Mj÷~
(1)
o~=0
and the annual series {Ak ~ will be obtained by picking up every 12th value of ( Yj ). That is:
Ak = Yl2k
(2)
Covariance relations First the covariance relations for this aggregation scheme will be worked out. The covariance function of the moving sums ( Y t ) may be obtained by working out the expectations: 11
II
E[Yt+r Yt] = ~ ~ E[Mt+r+JMt+k] J=o k--o which yields: 11
Covy(T) = 12 COVM(r ) + ~
s[CovM(7 + 12 - - s )
+ COVM(T --
12 + s)]
s=l
where Covy(T) is the covariance of the moving sums {Yj} at the lag T and COVM(T) is the covariance of the monthly hydrologic series at the lag T. From eq. 2 it follows that:
E[Aj+TAj] = E[Y12j+ 12T Y,2J]
(3)
349
which leads to the covariance relation: COVA(r ) =
Covy(12r)
(4)
where COVA(r) is the covariance of the annual series at lag T. From expressions (4) and (3) it follows that: 11 C O V A ( T ) --
12 COVM(12T) + ~ s[COVM(12T + 12 - - s ) + COVM(12r -- 12 + s)] $=1
(5) Expression (5) gives the relation between the covariance function of the monthly series and the covariance function of the annual series. It shows that twenty-three values of the monthly covariance function are combined to form one value of the annual covariance function. This is a unique transformation which will always give a unique covariance function for the annual series from the covariance function of the monthly series. However, if a disaggregation scheme was used where the covariance function of the monthly series was obtained from the annual series, the transformation, as is seen from eq. 5, is one-to-many and there are many solutions to the disaggregation scheme.
Spectral relations The spectral relations between the monthly and the annual hydrologic sequences can be conveniently worked o u t in terms of the spectral model for a periodic hydrologic time series. A monthly hydrologic time series (Mj}, has periodicities with the fundamental annual frequency. The spectral model may be written as: 1/28
MJ=V~ f - 1/26
6
fM~/-fM-~ei2"~sJdz(w)+X/~~
x//~cos(2~J6w~+cP~)
~=0
(6) where cos = ~/12~, 8 is the sampling interval equal to one month (expressed in the units of time in use), fM(¢O) is the spectral density of the monthly hydrologic series, dZ(w ) are stationary, uncorrelated random increments, l~ is the magnitude of the ~th discontinuity in the spectral distribution function, Q~ are mutually uncorrelated random variables, and ¢P~ are mutually uncorrelated random phases. For more details on this model the reader is referred to Kavvas and Delleur (1975). Consider first the circularly stationary component, Md,j, of the monthly hydrologic time series, Mj, which is the second term on the right side of eq. 6. If the moving sums { Yt } are formed by consecutively adding twelve values of
350
m o n t h l y series, the circularly stationary part Yd,t of Yt is obtained as: 11
11
Yd, t = ~ Md, t . J = ~ J=O
6
~
~/l~Oa c o s [ 2 n ( t + J ) ~ - ~ + ~ 1
(7)
J=O a=O
However, from eq. 7, it is a simple exercise to show that: Yd, t = 1 2 x / l ~ 0 cos ~0
(8)
that is, the moving sums of the periodic part are constants in time. In data this constant is removed by subtracting the average, Y. Ordinarily this is put into the model by assuming that l0 = 0. This is an important result which indicates that the summation operation on the m o n t h l y hydrologic series, to form the annual series, wipes out the cyclicities present in the m o n t h l y hydrologic series. An important point to note is that when a disaggregation scheme is developed, transforming the annual sequences to m o n t h l y sequences, unless the circularly stationary comp o n e n t of the m o n t h l y sequence is known beforehand, there is no way to retrieve the discrete spectral c o m p o n e n t of the m o n t h l y series from the spectrum of the annual series. Consider next the stationary random part Me, j, of the m o n t h l y hydrologic time series M j , which is the first term on the right side of eq. 6. From eqs. 1, 6 and 8 it follows that the moving sums (Yk} can be modeled in the form: 11
1/25
Yk = ~ V~ J= 0
f
~/fM(w) e i2"(k÷J)~°5 dZ(co),
k = ...,--1, 0, 1 . . . .
-1/25
(9) After some manipulations: 1/25 -
kFSinsin 121r6 ~6 wc° 1 ei1'~5¢° ei2~5 kw dZ(co)
(10)
1/25
From the representation (10) for the moving sums Yk, k = ..., --1, O, 1, ..., the spectral density function gy(~o ), of { Yk } may be obtained in terms of the spectral density function fM(¢O) of the m o n t h l y hydrologic series (Mj} as:
gy(co ) = fM(o~ )
[ sin 12~6w ] sin ~6 co '
--1/26 ~< co < 1/26
(11)
where 6 is the sampling interval equal to one month. The window function W(w) = sin 12~6co / sin ~6co has the period 1 in co5, i.e. --1/25 < w ~< 1/25, and has its peak at co = 0 with the equivalent bandwidth of the peak equal to 1/125 (Jenkins and Watts, 1968). It has zeros at co = k/126, k = +1, +2 . . . . . +6. Therefore, this window is a lo~v-pass filter which filters out all the frequency components higher than ll/1261. Therefore, summation of the m o n t h l y time
351
series not only removes the periodic component, as was seen from expression (8), but also removes all the high-frequency components outside (--1/126, 1/126). The window function W(~) is shown in Fig. 1. The spectral density function fA(cO) of the annual hydrologic time series 120 108
I
Window function
9.4 8.2 6.7 5,5 4.1
-1.2
-24
-0.5 -0.4 -0,3 -0.2 -0.1
0
i
01
i
0.2
i
0.3
i
04
I
0.5
F'recluency
Fig. 1. Window function for the moving monthly sums.
can be represented in terms of its covariance function COVA(r) as: fA(cO)= ~
COVA(T)e- i 2 ~ r ~
(12)
T ~ --oo
where the annual sampling interval A equals twelve monthly sampling intervals 6. From expressions (4) and (12) it follows that: +oo
fA(~) = ~
Covy(12~)e - i 2 ~ ' 2 ~
(13)
T= -~
Introducing the spectral representation for Covy(12T) into eq. 13: +~
fA(¢O) = ~ 7.= - o ¢
6 :
1126
gy(co')e i2n'w'121"5 d¢o 'e -i2nwr126
-1/26
which can be represented in limit form as: fA(~O) = lim m---> °¢
~ f -1/2~
gy(co')
e i2~r12~(~°'-~°) d~o' ;r=-
(14)
352
T h e sum in b r a c k e t s in t h e e x p r e s s i o n (14) is a w i n d o w f u n c t i o n : w(m)[(~
'
-
co)a]
t h a t can be r e p r e s e n t e d as: +?n
ei2,( T= -m
, _ ~ ) a T _ sin ~ ( 2 m + 1 ) ( w ' -- co)A = w(rn)[(co, _ co)A] sin ~(co'--- co)A
(15)
F r o m eq. 15 it follows t h a t w ( m ) [ ( c o ' - - co)A] is periodic with p e r i o d 1 in (co' - - co)A, t h a t is, - - 1 / 2 ~ (co' - - co)A ~ + 1/9. covers o n e period. T h e r e f o r e , - - 1 / 2 A ~ (co' - - w ) ~ 1/2A covers one period. T h u s t h e w i n d o w f u n c t i o n w(m)[coA] is p e r i o d i c in co with p e r i o d 1/A or 1 / 1 2 5 . F r o m this fA(co) is periodic with p e r i o d 1/125 and we m a y limit o u r c o n s i d e r a t i o n t o its values in - - 1 / 2 4 5 < co < 1 / 2 4 5 . T h e graph o f w(m)[(co ' - - co)A] f o r t h e single p e r i o d - - 1 / 2 A < co' - - co < 1/2A will s h o w a c o m m o n central p e a k at co' = co o f h e i g h t (2m + 1) with smaller side peaks and t r o u g h s s y m m e t r i c a l l y arranged a r o u n d it. T h e zeros are at co' = co + k / ( 2 m + 1)A, h = +1, +2, ..., +m so the effective b a n d w i d t h o f the central p e a k is 1 / ( 2 m + 1)A. T h e peaks and t r o u g h s o c c u r n e a r the frequencies: 1 co'=co+(2m+l)A
k+
,
k=-+l,-+2,...,-+m
and are o f absolute magnitudes: (2m + 1)A
1
1
T h e integral o f w(m)[(co ' - - co)A] over a p e r i o d can be w o r k e d o u t f r o m t h e sum f o r m t o be 1/A. F o r large m the peaks and t r o u g h s have widths o f the o r d e r l / m , t h o s e near co' = co have m a g n i t u d e s o f the o r d e r o f m while t h o s e at t h e outside are o f o r d e r 1. I f g y ( c o ' ) is s m o o t h , t h e n f o r large m W( m ) [ (co, _ co ) A ] will act like a delta f u n c t i o n (1/A )5 (co' - - co ) in the interval - - 1 / 2 A < co - - co' < 1/2A. T h e same a r g u m e n t m a y be applied t o o t h e r single periods, e.g. 1/2A < co - - co' < 3 / 2 A , etc. T h e r e f o r e , a train o f delta f u n c t i o n s , e q u i s p a c e d at intervals o f length 1/A will be o b t a i n e d f r o m the w i n d o w f u n c t i o n W ( m ) [ ( w ' - co)A] as m -~ oo. Thus the w i n d o w f u n c t i o n w(m)[(co ' -- co)A] can be expressed in the limit as: 1
limrn.~ w(rn)[(co'
--
+~
co)A] = ~ m =~ - ~
d(w'
- - co - -
m/A)
(16)
w h e r e d(" ) is Dirac's delta f u n c t i o n . C o m b i n i n g expressions (14) and (16), t h e s p e c t r u m o f the annual h y d r o l o g i c series, fA(co ), is o b t a i n e d as:
353
fA( )=a f
(17)
- 1/26
m
= -*
Only those Dirac functions whose peaks are in the region of integration are used. That is, for m such that: --1/26 < ¢o --m/125 < 1/2 or 1 2 5 w - - 6 < m < 125w+ Since we will work only with co between --1/246 and +1/246,126 co is between --1/2 and 1/2. For o) = 0 the sum is from --6 to 6, for co > 0 the sum is from --5 to 6 and for ¢o < 0 the sum is from --6 to 5. If these limits are simply understood: 1
fA(¢O) = -~ ~m gy(¢o + m/125)
(18)
Combining expressions (18) and (11), the spectrum of the annual hydrologic series, fA(¢O), is obtained in terms of the spectrum of the monthly series, f M( ¢O ) , as:
1
_
F sin 12~6(¢o + m/125)32
fA(¢°)="1"2 ~m fM(c° + m/125 )|L s'n 1 ~5(w + m/125) J|
--1/246 ~< ¢o ~< 1/246 On the positive side of the spectrum 1 6 12 ~
fA(w ) is obtained as:
fM(¢O+ m/12) rsin
12n5(¢o
+
/
)n
m'125"/2 [ . s m ~6(¢o + m/125) J
m = -S
fA(co)
0 < w <<.1/246 1
6
(19)
[
(20)
sin ~m
= ~ 3 - 6 fM(m/126) Lsin ~ m / 1 2 ] 2'
¢o=0
Expression (20) has the physical meaning that the spectrum of the annual hydrologic series is aliased with the folding frequency 1/246 where 5 is one month. It shows that each value of the annual spectrum f(¢o) is formed by the combination of twelve to thirteen values of the monthly spectrum fM(¢O). This is a many-to-one transformation and it is unique. That is, there is only one aggregation scheme from the monthly to the annual hydrologic time series. However, following expressions (19) or (20), there are many ways that one can disaggregate the annual series to the monthly series to satisfy the equalities (19) and (20).
354 The practical aspect of the relations (5) and (19) is that once the covariance and the spectral structure of the monthly hydrologic series are obtained t h e covariance and the spectral structure of the annual hydrologic series are already known through the relations (5) and (19). They can be calculated through these theoretical relations and the time-series models can be fitted to these calculated covariance or spectral functions.
Prediction of the covariance function and the spectrum of the annual hydrologic series from monthly hydrologic series The aggregation eqs. 5 and 20 for the monthly to annual aggregation of . the hydrologic time series were tested by comparison of the predicted and the sample covariance and spectral functions for the runoff series in the USGS Station No. 3-2535 on the Licking River at Catawba, Kentucky. The autocovariance function and the spectrum of the m o n t h l y runoffs on the Station No. 3-2535 are shown on Fig. 2. Using the covariance and spectral function values for the monthly runoffs the covariance and the spectral functions of the moving sums were theoretically calculated by eqs. 3 and 11. The sample covariance and the sample spectral functions were then calculated and com3000
£ 20oo E ~
IO.O0
8
g b
0.00
o -1QOC
).00
5.29
10.57
15.86
21.14
2643
31.71
i 37.00
L 0,1200
i 01400
Lag (months)
4GGD
z "F ~' 3 0 0 . 0
8 E
~ 2oo0
O0 00000
00200
0.0400 0 0 6 0 0
~ ~ 0 0 8 0 0 01000
Frequency (cycles/month)
Fig. 2. Autocovariance function and spectrum of the monthly runoffs at station No. 3-2535, Licking River at Catawaba, Kentucky.
355
pared to their theoretically calculated counterparts. This comparison is shown in Fig. 3. The results show very close agreement between the predicted values and the sample estimates. Then the covariance and spectral functions for the annual runoffs were theoretically computed by eqs. 5 and 20 from 300.Ox~
-x
\
Sample covariance for moving sums Theoreticcll coveriQnce for moving sums
x
:~ ~ooo co
x~
oo
-~°°%.oo 8~
~ x- - ~ . ~
~6'.oo ~Ioo j.oo ~o'oo ~81oo ~'oo Log ( m o n t h s )
19000 h
E
i
-x
Sample spectrum for moving sums Theoreticcll spectrum for moving sums
9500 x
L
0.( DO00 0 . 0 1 ~ 00371
0.0557
0.0"743
0,0929
0.1114
0.1~
Frequency (cycles/month)
Fig. 3. Theoretical and sample eovariance and spectrum for moving sums, station No. 3-2535.
the sample covariance and the sample spectral functions of the monthly runoffs. The sample covariance and the sample spectral functions for the annual runoffs were computed and then compared to their theoretical counterparts. The correspondence of the theoretical values and the sample estimates for the covariance function are very good. For the spectrum the agreement is very good for the low frequencies but, due to the sampling errors and finiteness of the data, is only fair for the high frequencies. This comparison is seen in Fig. 4. THE GENERAL AGGREGATION SCHEME
This section will study the general aggregation scheme where the covariance and the spectral functions of the hydrologic time series, corresponding to large sampling intervals, will be obtained from the covariance and the spectral functions of the hydrologic time series, corresponding to small sampling intervals.
356 400.0
- - S a m p l e covariance fo~ annual runoff x Theoretical covariance for annual runoff
I
~
i
X
0.0
x
U - 200.(
-400(
O0
1,86
3. 1
5 7
Z43
g.29
11.14
13.00
Log ( y e a r s )
2000
- - S a m p l e spectrum f o r annual r u n o f f x Theoretical spectrum f o r annual r u n o f f
x 1500
100.0 50.0 0.0 00000
L
01000
i
I
0.2000
Frequency
03000
i
_
0.4000
~
0.5000
(cycles/year')
Fig. 4. Theoretical and sample eovarianee and spectrum for annual runoff. Any time series can be aggregated in two steps: (1) Addition of r prior values of the series {Xj} (hourly, daily, weekly, monthly, etc.) to form the series of moving sums { Yt}. That is: r - 1
Yt = ~
Xt+J,
t = . . . , - - 1 , 0, + 1 , . . .
(21)
J=O
(2) Picking up every rth value of { Yt} to obtain the desired hydrologic series {Ak }. The covariance and the spectral relations in the aggregation scheme will be studied by the above two steps.
Covariance relations in the general aggregation scheme The covariance function of the moving sums at lag T, Covy(r) is obtained from the covariance function, Covz(r), of the series (Xj} by taking:
357
r- r - 1
r-1
-J=o
k=o
-i
r-1
r-1
E [ X t ÷ r +jXt+k] J=o k=o
= rCovX(T ) + (r -- I)[Covx(T + I) + Covx(~ -- 1) + ... +
+ COVX(T + r - - I) + Covx(7 - - r + I)] which yields:
r-I COVy(T) = rCovx(T) + ~ s[Covx(T + r -- s) + Covx(T -- r + s)]
(22)
s=l
If every rth value of (Yt) is picked up to obtain the desired hydrologic series ( A k ) the corresponding mathematical expression is simply: Aj = Yl+(J-,)r,
J = . . . , - - 1 , 0, +1, ...
(23)
Taking the expectations: E[Aj
+r A j ]
=
E [ Y,
+( j +r - t ) r Y I + ( J -
1)r]
(24)
If the (Yt} series is weakly stationary, then: E [ A j +r Aj] = E[ Yrr Y0]
(25)
from which it follows that:
COVA(T)
(26)
= COvy(rT)
Combining eqs. 26 and 22, the covariance function CovA(r) of the aggregated series (A k ) is obtained from the covariance function, Covx(T), of the original series ( X j ) as:
r-i COVA(T ) =
rCovx(rT ) + ~ .
s[Covx(rT + r -- s) + Covx(rr -- r + s)]
(27)
s=l
In the derivation of relation (27) the weak stationarity of the moving sums was assumed. This assumption corresponds to assuming that the circularly stationary c o m p o n e n t of the original hydrologic time series is removed in the formation of moving sums. Since the circularly stationary c o m p o n e n t is represented through the discrete spectral c o m p o n e n t in the spectral representation of the periodic hydrologic time series the conditions of the weak stationarity of (Yk) series will be developed in the spectral relations calculations. The covariance relation (27) of the general aggregation shows the many-toone transformation in an aggregation scheme. This is a unique operation in the sense that the covariance function of the aggregated series will be unique. On the other hand, if a disaggregation scheme was used, there would be many ways to disaggregate the original hydrologic time series so as to satisfy the relation (27).
358
Spectral relations in the general aggregation scheme In the formation of moving sums { Yt } the stationary random and the circularly stationary components of the spectral model will be handled separately. The stationary random c o m p o n e n t of the original hydrologic time se~es {Xj} has the spectral representation: 1126
X j =Vr~ f -'{/26
V ~ ( ~ o ) e i2~J~6 dZ(w)
(28)
where f(¢~) is the spectrum of {Xj}. From eq. 21 it follows that the contribution of the stationary random c o m p o n e n t of {Xj} to the moving sums { Yt} will be: r - 1
1126
f
Zt = x/6 ~
V~X-(W)ei2"(t÷d)~6 dZ(¢o)
(29)
-1126
J=o
which reduces to the form:
Yt =V~ f -
1/26 ~ s i n
x/fx(¢o)
1126
:
7rcoSr einO~6(r_1
s i n 7rco8
) dZ(¢o)
(30)
From eq. 30 the spectrum of the moving sums,,gy(¢o ), is obtained in terms of the spectrum of the original hydrologic series, fx(w), as:
gy(~ ) = fx(co)[sin ~¢o6r/sin ~¢o6]2
(31)
where 6 is the time sampling interval of the original hydrologic series. However, it should be remembered that eq. 31 shows only the contribution of the stationary random c o m p o n e n t of {Xj} to { Yt}. The contribution of the circularly stationary part may be studied through the discrete spectral representation of the original hydrologic time series {Xj}. If the discrete c o m p o n e n t of X j is denoted by Xd,J, and if it is assumed that there are [r/2] = k, k = 0, 1, 2, ..., significant harmonics in (Xd,J}, t h e n Xd,J has the spectral representation: k (~=0
so that the circularly stationary c o m p o n e n t contribution Yd, t to Yt may be expressed as: r-1
=
r-1
k
J=0
~=0
= J=0
--+
(32)
r
The right side of eq. 32 becomes zero (or constant if l0 ¢ 0) if 1/r~ is the fun-
359 damental significant frequency. This is the case when the aggregation scheme goes from dally to annual, weekly to annual, and m o n t h l y to annual hydrologic series. Therefore, in these three schemes the circularity drops out and the time resolution computations become tractable. In the following discussion only these three schemes will be considered. Assuming t h a t only daily to annual, weekly to annual, and m o n t h l y to annual aggregation schemes are considered, the spectrum of the moving sums ( Yt} is represented by expression (31). The spectrum fA(co ) of the aggregated hydrologic series {Ak ), formed by picking up every rth value of the moving sums ( Yt}, has the covariance representation:
fA(w) = ~
COVA(T)e-i2"~T~,
A=
r6
(33)
Using eq. 26: -I-VO
fA(co)= ~
Covy(rT)e -i2~wTr6
(34)
Introducing the spectral representation of Covy(r~) into eq. 34: +m
1/25
fA(co)=lim m~
5 f oo
fy(co')
-1/25
~
e i2~rr~(w'-~)
dco'
(35)
T= -/'r/
wh ere: +m
ei2~rA
(w ' -w)
T= -m
is a window function denoted by w(m)[(co ' -- co)A ]. Using the arguments to those after eq. 15: lim m-~
~
1 *~ d(co' -- co -W(m)[(co ' -- co)A] = A--m~ = -~
m/A)
(36)
]
(37)
Introducing eq. 36 into eq. 35:
fA(co) =
?° -1/26
fY(co
"
~ d(co' -- co -m = -oo
m/rS) dco'
where d(co' -- co -- m/rS) ¢ 0 only when co' = co + rn/r6. From this property and eq. 37 the spectrum fA(co ) of the aggregated hydrologic series (Ak) can be obtained in terms of the spectrum fr(co) of the moving sums (Yt) as:
360
1 fA(w) =--~fy(w r
+ m/rS),
--1/2r5 <~ co < 1/2r8
(38)
m
The range of m in the sum is all the integers between --r/2 + ¢or6 and r/2 + wrS. If we limit our attention to w between 0 and 1/2r6, wr8 will be between 0 and 1/2. For values between these two limits, --r[2 < m <~ r/2. If w = 0 and r even, the range of m has the additional integer --r/2. If w = 1/2r6 and r is odd, then the range has the additional integers (r + 1)/2. We will omit these complications in what follows. Combining eqs. 38 and 31, fA(w ), the spectrum of the larger time sampling intervals, is obtained in terms of the spectrum f x ( w ) of the original series of smaller time intervals, as:
fA( ) =l'-'fx(¢°~ rm
!r sinn(w + m/rS)r8 ]2 + mlrS) L sin~r(w + m/rS)6 J
(39)
where--1/2r6 <~ ¢o <~ 1/2r6. The spectral relation (39) shows a many-to-one transformation in the aggregation of smaller time intervals in the formation of larger sampling intervals. This is a unique operation in the sense that the spectrum of the aggregated series will be unique. In a disaggregation scheme there would be many ways to satisfy the relation (39) since the transformation will be one to many. The spectrum of the aggregated series is aliased with the folding frequency 1/2r6 where r is the number of smaller sampling intervals aggregated to form one larger sampling interval. Through the use of the relations (27) and (39) and recognizing the limitations, the hydrologist can obtain the covariance and spectral structure of the hydrologic time series corresponding to a larger sampling interval from the covariance function and spectrum of the hydrologic time series corresponding to smaller sampling intervals. There is no need to compile the data and estimate the sample covariance and spectral functions for the hydrologic time series corresponding to larger sampling intervals once the covariance and spectral functions for the smaller intervals are known. CONCLUSIONS
The covariance and, spectral function relations among the hydrologic time series with time steps which are multiples of each other (e.g., daily, weakly, monthly and annual intervals) are derived. The relations between the hydrologic sequences corresponding to monthly intervals and the sequences corresponding to annual intervals are verified by the data. Through the utilization of the covariance and spectrum relations for the general aggregation scheme, the hydrologist, starting from the smallest interval, can calculate the covariance and spectral structures of the hydrologic sequences corresponding to larger sampling intervals. One can then construct the time series models correspond-
361 ing to these larger intervals from the theoretically calculated covariance and spectral functions without ever constructing the data for larger intervals. ACKNOWLEDGEMENTS The work reported in this paper is based on a research supported in part by the office of Water Resource and Technology under matching fund grant OWRT-B-036-IND, in part by the Victor M. O'Shaughnessy Scholarship Fund and in part by Purdue University.
REFERENCES Cote, L., 1973. Lecture notes on Time Series Analysis. Purdue University, West Lafayette, Ind. (unpublished). Jenkins, G.M. and Watts, D.G., 1968. Spectral Analysis and Its Applications. Holden-Day, San Francisco, Calif., 525 pp. Kavvas, M.L. and Delleur, J.W., 1975. Removal of periodicities by differencing and monthly mean subtraction. J. Hydrol., 26: 335--353.
TIME RESOLUTION
OF THE HYDROLOGIC
TIME-SERIES MODELS
M.L. KAVVAS',*, L.J. COTE 2 and J.W. DELLEUR 1
School o f Civil Engineering, Purdue University, West Lafayette, Ind. 47907 (U.S.A.) 2 Statistics Department, Purdue University, West Lafayette, Ind. 47907 (U.S.A.) (Received February 7, 1976; revised and accepted May 7, 1976)
ABSTRACT Kavvas, M.L., Cote, L.J. and Delleur, J.W., 1977. Time resolution of the hydrologic timeseries models. J. Hydrol., 32: 347--361. Hydrologists have constructed time-series models for the daily, monthly, annual and other time intervals for the simulation studies of a water-resource system. The mathematical operations involved in attaining the monthly hydrologic data from the daily hydrologic data and the annual hydrologic data from either the daily or the monthly hydrologic data are standard and well known. Through the utilization of these operations one can find the covariance and the spectral relations among the hydrologic sequences corresponding to different sampling intervals. Once these relations are known, the hydrologist, starting from the smallest interval, can calculate the covariance and the spectral structures of the hydrologic sequences corresponding to larger sampling intervals. One can then construct the time-series models corresponding to these larger intervals from the theoretically calculated covariance and spectral density functions without ever constructing the data for larger intervals or without the computation of the sample covariance and the sample spectral-density functions for larger intervals. This paper establishes the relationships between the covariances and between the spectra, respectively, of the hydrologic time series with time steps which are multiples of each other, such as daily, monthly and annual intervals. The stochastic structure in the time domain of the series with the longer time steps can be derived in terms of the structure of the series with the shorter time steps. The scheme discussed in this paper is an aggregation scheme and describes, in detail, the filters and the many-to-one transformations that are required to go from the smaller to larger time sampling intervals. The relations show that when the hydrologic time-series models are disaggregated from larger to smaller time sampling intervals the transformations involved are one-to-many, indicating that there are m a n y ways for the disaggregation whereas there is only one unique way for aggregation.
INTRODUCTION By the introduction of the spectral model for the periodic hydrologic time series to hydrology (Cote, 1973; Kavvas and Delleur, 1975) it is now possible to make an analytical analysis of the various transformations underlying the formation of the hydrologic sequences corresponding to small time sampling intervals. In this way, a time resolution of the hydrologic time-series models, *Present address: Industrial Engineering Department, MiddleEast Technical University, In~nii Bulvari, Ankara, Turkey.
348
in terms of their covariance and spectral structures, is going to be achieved. First, the relations between the hydrologic sequences corresponding to monthly intervals and the sequences corresponding to annual intervals will be studied. Once the covariance and the spectral relations between the monthly and the annual series is established, they will be checked by the hydrologic data. Then a general aggregation scheme for the covariance and the spectral functions of the hydrologic time series will be developed. AGGREGATION
OF MONTHLY
SERIES TO ANNUAL
SERIES
An annual hydrologic sequence may be formed from a monthly hydrologic sequence by adding the monthly quantities in a year to form the annual value for that year and so on. This operation corresponds to adding consecutive values of twelve monthly hydrologic quantities and then picking up every 12th value to form the annual series. Let {Mj ) be the monthly hydrologic time series, ( Yj ) be the moving sums formed from the addition of twelve consecutive monthly values, and {Aj) be the annual series. The moving sums ( Y j ) will be formed from the monthly series {Mj} by the operation: 11
Yj = ~
Mj÷~
(1)
o~=0
and the annual series {Ak ~ will be obtained by picking up every 12th value of ( Yj ). That is:
Ak = Yl2k
(2)
Covariance relations First the covariance relations for this aggregation scheme will be worked out. The covariance function of the moving sums ( Y t ) may be obtained by working out the expectations: 11
II
E[Yt+r Yt] = ~ ~ E[Mt+r+JMt+k] J=o k--o which yields: 11
Covy(T) = 12 COVM(r ) + ~
s[CovM(7 + 12 - - s )
+ COVM(T --
12 + s)]
s=l
where Covy(T) is the covariance of the moving sums {Yj} at the lag T and COVM(T) is the covariance of the monthly hydrologic series at the lag T. From eq. 2 it follows that:
E[Aj+TAj] = E[Y12j+ 12T Y,2J]
(3)
349
which leads to the covariance relation: COVA(r ) =
Covy(12r)
(4)
where COVA(r) is the covariance of the annual series at lag T. From expressions (4) and (3) it follows that: 11 C O V A ( T ) --
12 COVM(12T) + ~ s[COVM(12T + 12 - - s ) + COVM(12r -- 12 + s)] $=1
(5) Expression (5) gives the relation between the covariance function of the monthly series and the covariance function of the annual series. It shows that twenty-three values of the monthly covariance function are combined to form one value of the annual covariance function. This is a unique transformation which will always give a unique covariance function for the annual series from the covariance function of the monthly series. However, if a disaggregation scheme was used where the covariance function of the monthly series was obtained from the annual series, the transformation, as is seen from eq. 5, is one-to-many and there are many solutions to the disaggregation scheme.
Spectral relations The spectral relations between the monthly and the annual hydrologic sequences can be conveniently worked o u t in terms of the spectral model for a periodic hydrologic time series. A monthly hydrologic time series (Mj}, has periodicities with the fundamental annual frequency. The spectral model may be written as: 1/28
MJ=V~ f - 1/26
6
fM~/-fM-~ei2"~sJdz(w)+X/~~
x//~cos(2~J6w~+cP~)
~=0
(6) where cos = ~/12~, 8 is the sampling interval equal to one month (expressed in the units of time in use), fM(¢O) is the spectral density of the monthly hydrologic series, dZ(w ) are stationary, uncorrelated random increments, l~ is the magnitude of the ~th discontinuity in the spectral distribution function, Q~ are mutually uncorrelated random variables, and ¢P~ are mutually uncorrelated random phases. For more details on this model the reader is referred to Kavvas and Delleur (1975). Consider first the circularly stationary component, Md,j, of the monthly hydrologic time series, Mj, which is the second term on the right side of eq. 6. If the moving sums { Yt } are formed by consecutively adding twelve values of
350
m o n t h l y series, the circularly stationary part Yd,t of Yt is obtained as: 11
11
Yd, t = ~ Md, t . J = ~ J=O
6
~
~/l~Oa c o s [ 2 n ( t + J ) ~ - ~ + ~ 1
(7)
J=O a=O
However, from eq. 7, it is a simple exercise to show that: Yd, t = 1 2 x / l ~ 0 cos ~0
(8)
that is, the moving sums of the periodic part are constants in time. In data this constant is removed by subtracting the average, Y. Ordinarily this is put into the model by assuming that l0 = 0. This is an important result which indicates that the summation operation on the m o n t h l y hydrologic series, to form the annual series, wipes out the cyclicities present in the m o n t h l y hydrologic series. An important point to note is that when a disaggregation scheme is developed, transforming the annual sequences to m o n t h l y sequences, unless the circularly stationary comp o n e n t of the m o n t h l y sequence is known beforehand, there is no way to retrieve the discrete spectral c o m p o n e n t of the m o n t h l y series from the spectrum of the annual series. Consider next the stationary random part Me, j, of the m o n t h l y hydrologic time series M j , which is the first term on the right side of eq. 6. From eqs. 1, 6 and 8 it follows that the moving sums (Yk} can be modeled in the form: 11
1/25
Yk = ~ V~ J= 0
f
~/fM(w) e i2"(k÷J)~°5 dZ(co),
k = ...,--1, 0, 1 . . . .
-1/25
(9) After some manipulations: 1/25 -
kFSinsin 121r6 ~6 wc° 1 ei1'~5¢° ei2~5 kw dZ(co)
(10)
1/25
From the representation (10) for the moving sums Yk, k = ..., --1, O, 1, ..., the spectral density function gy(~o ), of { Yk } may be obtained in terms of the spectral density function fM(¢O) of the m o n t h l y hydrologic series (Mj} as:
gy(co ) = fM(o~ )
[ sin 12~6w ] sin ~6 co '
--1/26 ~< co < 1/26
(11)
where 6 is the sampling interval equal to one month. The window function W(w) = sin 12~6co / sin ~6co has the period 1 in co5, i.e. --1/25 < w ~< 1/25, and has its peak at co = 0 with the equivalent bandwidth of the peak equal to 1/125 (Jenkins and Watts, 1968). It has zeros at co = k/126, k = +1, +2 . . . . . +6. Therefore, this window is a lo~v-pass filter which filters out all the frequency components higher than ll/1261. Therefore, summation of the m o n t h l y time
351
series not only removes the periodic component, as was seen from expression (8), but also removes all the high-frequency components outside (--1/126, 1/126). The window function W(~) is shown in Fig. 1. The spectral density function fA(cO) of the annual hydrologic time series 120 108
I
Window function
9.4 8.2 6.7 5,5 4.1
-1.2
-24
-0.5 -0.4 -0,3 -0.2 -0.1
0
i
01
i
0.2
i
0.3
i
04
I
0.5
F'recluency
Fig. 1. Window function for the moving monthly sums.
can be represented in terms of its covariance function COVA(r) as: fA(cO)= ~
COVA(T)e- i 2 ~ r ~
(12)
T ~ --oo
where the annual sampling interval A equals twelve monthly sampling intervals 6. From expressions (4) and (12) it follows that: +oo
fA(~) = ~
Covy(12~)e - i 2 ~ ' 2 ~
(13)
T= -~
Introducing the spectral representation for Covy(12T) into eq. 13: +~
fA(¢O) = ~ 7.= - o ¢
6 :
1126
gy(co')e i2n'w'121"5 d¢o 'e -i2nwr126
-1/26
which can be represented in limit form as: fA(~O) = lim m---> °¢
~ f -1/2~
gy(co')
e i2~r12~(~°'-~°) d~o' ;r=-
(14)
352
T h e sum in b r a c k e t s in t h e e x p r e s s i o n (14) is a w i n d o w f u n c t i o n : w(m)[(~
'
-
co)a]
t h a t can be r e p r e s e n t e d as: +?n
ei2,( T= -m
, _ ~ ) a T _ sin ~ ( 2 m + 1 ) ( w ' -- co)A = w(rn)[(co, _ co)A] sin ~(co'--- co)A
(15)
F r o m eq. 15 it follows t h a t w ( m ) [ ( c o ' - - co)A] is periodic with p e r i o d 1 in (co' - - co)A, t h a t is, - - 1 / 2 ~ (co' - - co)A ~ + 1/9. covers o n e period. T h e r e f o r e , - - 1 / 2 A ~ (co' - - w ) ~ 1/2A covers one period. T h u s t h e w i n d o w f u n c t i o n w(m)[coA] is p e r i o d i c in co with p e r i o d 1/A or 1 / 1 2 5 . F r o m this fA(co) is periodic with p e r i o d 1/125 and we m a y limit o u r c o n s i d e r a t i o n t o its values in - - 1 / 2 4 5 < co < 1 / 2 4 5 . T h e graph o f w(m)[(co ' - - co)A] f o r t h e single p e r i o d - - 1 / 2 A < co' - - co < 1/2A will s h o w a c o m m o n central p e a k at co' = co o f h e i g h t (2m + 1) with smaller side peaks and t r o u g h s s y m m e t r i c a l l y arranged a r o u n d it. T h e zeros are at co' = co + k / ( 2 m + 1)A, h = +1, +2, ..., +m so the effective b a n d w i d t h o f the central p e a k is 1 / ( 2 m + 1)A. T h e peaks and t r o u g h s o c c u r n e a r the frequencies: 1 co'=co+(2m+l)A
k+
,
k=-+l,-+2,...,-+m
and are o f absolute magnitudes: (2m + 1)A
1
1
T h e integral o f w(m)[(co ' - - co)A] over a p e r i o d can be w o r k e d o u t f r o m t h e sum f o r m t o be 1/A. F o r large m the peaks and t r o u g h s have widths o f the o r d e r l / m , t h o s e near co' = co have m a g n i t u d e s o f the o r d e r o f m while t h o s e at t h e outside are o f o r d e r 1. I f g y ( c o ' ) is s m o o t h , t h e n f o r large m W( m ) [ (co, _ co ) A ] will act like a delta f u n c t i o n (1/A )5 (co' - - co ) in the interval - - 1 / 2 A < co - - co' < 1/2A. T h e same a r g u m e n t m a y be applied t o o t h e r single periods, e.g. 1/2A < co - - co' < 3 / 2 A , etc. T h e r e f o r e , a train o f delta f u n c t i o n s , e q u i s p a c e d at intervals o f length 1/A will be o b t a i n e d f r o m the w i n d o w f u n c t i o n W ( m ) [ ( w ' - co)A] as m -~ oo. Thus the w i n d o w f u n c t i o n w(m)[(co ' -- co)A] can be expressed in the limit as: 1
limrn.~ w(rn)[(co'
--
+~
co)A] = ~ m =~ - ~
d(w'
- - co - -
m/A)
(16)
w h e r e d(" ) is Dirac's delta f u n c t i o n . C o m b i n i n g expressions (14) and (16), t h e s p e c t r u m o f the annual h y d r o l o g i c series, fA(co ), is o b t a i n e d as:
353
fA( )=a f
(17)
- 1/26
m
= -*
Only those Dirac functions whose peaks are in the region of integration are used. That is, for m such that: --1/26 < ¢o --m/125 < 1/2 or 1 2 5 w - - 6 < m < 125w+ Since we will work only with co between --1/246 and +1/246,126 co is between --1/2 and 1/2. For o) = 0 the sum is from --6 to 6, for co > 0 the sum is from --5 to 6 and for ¢o < 0 the sum is from --6 to 5. If these limits are simply understood: 1
fA(¢O) = -~ ~m gy(¢o + m/125)
(18)
Combining expressions (18) and (11), the spectrum of the annual hydrologic series, fA(¢O), is obtained in terms of the spectrum of the monthly series, f M( ¢O ) , as:
1
_
F sin 12~6(¢o + m/125)32
fA(¢°)="1"2 ~m fM(c° + m/125 )|L s'n 1 ~5(w + m/125) J|
--1/246 ~< ¢o ~< 1/246 On the positive side of the spectrum 1 6 12 ~
fA(w ) is obtained as:
fM(¢O+ m/12) rsin
12n5(¢o
+
/
)n
m'125"/2 [ . s m ~6(¢o + m/125) J
m = -S
fA(co)
0 < w <<.1/246 1
6
(19)
[
(20)
sin ~m
= ~ 3 - 6 fM(m/126) Lsin ~ m / 1 2 ] 2'
¢o=0
Expression (20) has the physical meaning that the spectrum of the annual hydrologic series is aliased with the folding frequency 1/246 where 5 is one month. It shows that each value of the annual spectrum f(¢o) is formed by the combination of twelve to thirteen values of the monthly spectrum fM(¢O). This is a many-to-one transformation and it is unique. That is, there is only one aggregation scheme from the monthly to the annual hydrologic time series. However, following expressions (19) or (20), there are many ways that one can disaggregate the annual series to the monthly series to satisfy the equalities (19) and (20).
354 The practical aspect of the relations (5) and (19) is that once the covariance and the spectral structure of the monthly hydrologic series are obtained t h e covariance and the spectral structure of the annual hydrologic series are already known through the relations (5) and (19). They can be calculated through these theoretical relations and the time-series models can be fitted to these calculated covariance or spectral functions.
Prediction of the covariance function and the spectrum of the annual hydrologic series from monthly hydrologic series The aggregation eqs. 5 and 20 for the monthly to annual aggregation of . the hydrologic time series were tested by comparison of the predicted and the sample covariance and spectral functions for the runoff series in the USGS Station No. 3-2535 on the Licking River at Catawba, Kentucky. The autocovariance function and the spectrum of the m o n t h l y runoffs on the Station No. 3-2535 are shown on Fig. 2. Using the covariance and spectral function values for the monthly runoffs the covariance and the spectral functions of the moving sums were theoretically calculated by eqs. 3 and 11. The sample covariance and the sample spectral functions were then calculated and com3000
£ 20oo E ~
IO.O0
8
g b
0.00
o -1QOC
).00
5.29
10.57
15.86
21.14
2643
31.71
i 37.00
L 0,1200
i 01400
Lag (months)
4GGD
z "F ~' 3 0 0 . 0
8 E
~ 2oo0
O0 00000
00200
0.0400 0 0 6 0 0
~ ~ 0 0 8 0 0 01000
Frequency (cycles/month)
Fig. 2. Autocovariance function and spectrum of the monthly runoffs at station No. 3-2535, Licking River at Catawaba, Kentucky.
355
pared to their theoretically calculated counterparts. This comparison is shown in Fig. 3. The results show very close agreement between the predicted values and the sample estimates. Then the covariance and spectral functions for the annual runoffs were theoretically computed by eqs. 5 and 20 from 300.Ox~
-x
\
Sample covariance for moving sums Theoreticcll coveriQnce for moving sums
x
:~ ~ooo co
x~
oo
-~°°%.oo 8~
~ x- - ~ . ~
~6'.oo ~Ioo j.oo ~o'oo ~81oo ~'oo Log ( m o n t h s )
19000 h
E
i
-x
Sample spectrum for moving sums Theoreticcll spectrum for moving sums
9500 x
L
0.( DO00 0 . 0 1 ~ 00371
0.0557
0.0"743
0,0929
0.1114
0.1~
Frequency (cycles/month)
Fig. 3. Theoretical and sample eovariance and spectrum for moving sums, station No. 3-2535.
the sample covariance and the sample spectral functions of the monthly runoffs. The sample covariance and the sample spectral functions for the annual runoffs were computed and then compared to their theoretical counterparts. The correspondence of the theoretical values and the sample estimates for the covariance function are very good. For the spectrum the agreement is very good for the low frequencies but, due to the sampling errors and finiteness of the data, is only fair for the high frequencies. This comparison is seen in Fig. 4. THE GENERAL AGGREGATION SCHEME
This section will study the general aggregation scheme where the covariance and the spectral functions of the hydrologic time series, corresponding to large sampling intervals, will be obtained from the covariance and the spectral functions of the hydrologic time series, corresponding to small sampling intervals.
356 400.0
- - S a m p l e covariance fo~ annual runoff x Theoretical covariance for annual runoff
I
~
i
X
0.0
x
U - 200.(
-400(
O0
1,86
3. 1
5 7
Z43
g.29
11.14
13.00
Log ( y e a r s )
2000
- - S a m p l e spectrum f o r annual r u n o f f x Theoretical spectrum f o r annual r u n o f f
x 1500
100.0 50.0 0.0 00000
L
01000
i
I
0.2000
Frequency
03000
i
_
0.4000
~
0.5000
(cycles/year')
Fig. 4. Theoretical and sample eovarianee and spectrum for annual runoff. Any time series can be aggregated in two steps: (1) Addition of r prior values of the series {Xj} (hourly, daily, weekly, monthly, etc.) to form the series of moving sums { Yt}. That is: r - 1
Yt = ~
Xt+J,
t = . . . , - - 1 , 0, + 1 , . . .
(21)
J=O
(2) Picking up every rth value of { Yt} to obtain the desired hydrologic series {Ak }. The covariance and the spectral relations in the aggregation scheme will be studied by the above two steps.
Covariance relations in the general aggregation scheme The covariance function of the moving sums at lag T, Covy(r) is obtained from the covariance function, Covz(r), of the series (Xj} by taking:
357
r- r - 1
r-1
-J=o
k=o
-i
r-1
r-1
E [ X t ÷ r +jXt+k] J=o k=o
= rCovX(T ) + (r -- I)[Covx(T + I) + Covx(~ -- 1) + ... +
+ COVX(T + r - - I) + Covx(7 - - r + I)] which yields:
r-I COVy(T) = rCovx(T) + ~ s[Covx(T + r -- s) + Covx(T -- r + s)]
(22)
s=l
If every rth value of (Yt) is picked up to obtain the desired hydrologic series ( A k ) the corresponding mathematical expression is simply: Aj = Yl+(J-,)r,
J = . . . , - - 1 , 0, +1, ...
(23)
Taking the expectations: E[Aj
+r A j ]
=
E [ Y,
+( j +r - t ) r Y I + ( J -
1)r]
(24)
If the (Yt} series is weakly stationary, then: E [ A j +r Aj] = E[ Yrr Y0]
(25)
from which it follows that:
COVA(T)
(26)
= COvy(rT)
Combining eqs. 26 and 22, the covariance function CovA(r) of the aggregated series (A k ) is obtained from the covariance function, Covx(T), of the original series ( X j ) as:
r-i COVA(T ) =
rCovx(rT ) + ~ .
s[Covx(rT + r -- s) + Covx(rr -- r + s)]
(27)
s=l
In the derivation of relation (27) the weak stationarity of the moving sums was assumed. This assumption corresponds to assuming that the circularly stationary c o m p o n e n t of the original hydrologic time series is removed in the formation of moving sums. Since the circularly stationary c o m p o n e n t is represented through the discrete spectral c o m p o n e n t in the spectral representation of the periodic hydrologic time series the conditions of the weak stationarity of (Yk) series will be developed in the spectral relations calculations. The covariance relation (27) of the general aggregation shows the many-toone transformation in an aggregation scheme. This is a unique operation in the sense that the covariance function of the aggregated series will be unique. On the other hand, if a disaggregation scheme was used, there would be many ways to disaggregate the original hydrologic time series so as to satisfy the relation (27).
358
Spectral relations in the general aggregation scheme In the formation of moving sums { Yt } the stationary random and the circularly stationary components of the spectral model will be handled separately. The stationary random c o m p o n e n t of the original hydrologic time se~es {Xj} has the spectral representation: 1126
X j =Vr~ f -'{/26
V ~ ( ~ o ) e i2~J~6 dZ(w)
(28)
where f(¢~) is the spectrum of {Xj}. From eq. 21 it follows that the contribution of the stationary random c o m p o n e n t of {Xj} to the moving sums { Yt} will be: r - 1
1126
f
Zt = x/6 ~
V~X-(W)ei2"(t÷d)~6 dZ(¢o)
(29)
-1126
J=o
which reduces to the form:
Yt =V~ f -
1/26 ~ s i n
x/fx(¢o)
1126
:
7rcoSr einO~6(r_1
s i n 7rco8
) dZ(¢o)
(30)
From eq. 30 the spectrum of the moving sums,,gy(¢o ), is obtained in terms of the spectrum of the original hydrologic series, fx(w), as:
gy(~ ) = fx(co)[sin ~¢o6r/sin ~¢o6]2
(31)
where 6 is the time sampling interval of the original hydrologic series. However, it should be remembered that eq. 31 shows only the contribution of the stationary random c o m p o n e n t of {Xj} to { Yt}. The contribution of the circularly stationary part may be studied through the discrete spectral representation of the original hydrologic time series {Xj}. If the discrete c o m p o n e n t of X j is denoted by Xd,J, and if it is assumed that there are [r/2] = k, k = 0, 1, 2, ..., significant harmonics in (Xd,J}, t h e n Xd,J has the spectral representation: k (~=0
so that the circularly stationary c o m p o n e n t contribution Yd, t to Yt may be expressed as: r-1
=
r-1
k
J=0
~=0
= J=0
--+
(32)
r
The right side of eq. 32 becomes zero (or constant if l0 ¢ 0) if 1/r~ is the fun-
359 damental significant frequency. This is the case when the aggregation scheme goes from dally to annual, weekly to annual, and m o n t h l y to annual hydrologic series. Therefore, in these three schemes the circularity drops out and the time resolution computations become tractable. In the following discussion only these three schemes will be considered. Assuming t h a t only daily to annual, weekly to annual, and m o n t h l y to annual aggregation schemes are considered, the spectrum of the moving sums ( Yt} is represented by expression (31). The spectrum fA(co ) of the aggregated hydrologic series {Ak ), formed by picking up every rth value of the moving sums ( Yt}, has the covariance representation:
fA(w) = ~
COVA(T)e-i2"~T~,
A=
r6
(33)
Using eq. 26: -I-VO
fA(co)= ~
Covy(rT)e -i2~wTr6
(34)
Introducing the spectral representation of Covy(r~) into eq. 34: +m
1/25
fA(co)=lim m~
5 f oo
fy(co')
-1/25
~
e i2~rr~(w'-~)
dco'
(35)
T= -/'r/
wh ere: +m
ei2~rA
(w ' -w)
T= -m
is a window function denoted by w(m)[(co ' -- co)A ]. Using the arguments to those after eq. 15: lim m-~
~
1 *~ d(co' -- co -W(m)[(co ' -- co)A] = A--m~ = -~
m/A)
(36)
]
(37)
Introducing eq. 36 into eq. 35:
fA(co) =
?° -1/26
fY(co
"
~ d(co' -- co -m = -oo
m/rS) dco'
where d(co' -- co -- m/rS) ¢ 0 only when co' = co + rn/r6. From this property and eq. 37 the spectrum fA(co ) of the aggregated hydrologic series (Ak) can be obtained in terms of the spectrum fr(co) of the moving sums (Yt) as:
360
1 fA(w) =--~fy(w r
+ m/rS),
--1/2r5 <~ co < 1/2r8
(38)
m
The range of m in the sum is all the integers between --r/2 + ¢or6 and r/2 + wrS. If we limit our attention to w between 0 and 1/2r6, wr8 will be between 0 and 1/2. For values between these two limits, --r[2 < m <~ r/2. If w = 0 and r even, the range of m has the additional integer --r/2. If w = 1/2r6 and r is odd, then the range has the additional integers (r + 1)/2. We will omit these complications in what follows. Combining eqs. 38 and 31, fA(w ), the spectrum of the larger time sampling intervals, is obtained in terms of the spectrum f x ( w ) of the original series of smaller time intervals, as:
fA( ) =l'-'fx(¢°~ rm
!r sinn(w + m/rS)r8 ]2 + mlrS) L sin~r(w + m/rS)6 J
(39)
where--1/2r6 <~ ¢o <~ 1/2r6. The spectral relation (39) shows a many-to-one transformation in the aggregation of smaller time intervals in the formation of larger sampling intervals. This is a unique operation in the sense that the spectrum of the aggregated series will be unique. In a disaggregation scheme there would be many ways to satisfy the relation (39) since the transformation will be one to many. The spectrum of the aggregated series is aliased with the folding frequency 1/2r6 where r is the number of smaller sampling intervals aggregated to form one larger sampling interval. Through the use of the relations (27) and (39) and recognizing the limitations, the hydrologist can obtain the covariance and spectral structure of the hydrologic time series corresponding to a larger sampling interval from the covariance function and spectrum of the hydrologic time series corresponding to smaller sampling intervals. There is no need to compile the data and estimate the sample covariance and spectral functions for the hydrologic time series corresponding to larger sampling intervals once the covariance and spectral functions for the smaller intervals are known. CONCLUSIONS
The covariance and, spectral function relations among the hydrologic time series with time steps which are multiples of each other (e.g., daily, weakly, monthly and annual intervals) are derived. The relations between the hydrologic sequences corresponding to monthly intervals and the sequences corresponding to annual intervals are verified by the data. Through the utilization of the covariance and spectrum relations for the general aggregation scheme, the hydrologist, starting from the smallest interval, can calculate the covariance and spectral structures of the hydrologic sequences corresponding to larger sampling intervals. One can then construct the time series models correspond-
361 ing to these larger intervals from the theoretically calculated covariance and spectral functions without ever constructing the data for larger intervals. ACKNOWLEDGEMENTS The work reported in this paper is based on a research supported in part by the office of Water Resource and Technology under matching fund grant OWRT-B-036-IND, in part by the Victor M. O'Shaughnessy Scholarship Fund and in part by Purdue University.
REFERENCES Cote, L., 1973. Lecture notes on Time Series Analysis. Purdue University, West Lafayette, Ind. (unpublished). Jenkins, G.M. and Watts, D.G., 1968. Spectral Analysis and Its Applications. Holden-Day, San Francisco, Calif., 525 pp. Kavvas, M.L. and Delleur, J.W., 1975. Removal of periodicities by differencing and monthly mean subtraction. J. Hydrol., 26: 335--353.