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WATER RESOURCES RESEARCH, VOL. 28, NO. 12, PAGES 3175-3191, DECEMBER 1992

A Nonlinear Disaggregation Method With a Reduced Parameter Set

for Simulation of Hydrologic Series

DEMETRIS KOUTSOYIANNIS

Department of Civil Engineering, National Technical University of Athens, Zografou, Greece

A multivariate dynamic disaggregation model is developed as a stepwise approach to stochastic

disaggregation problems, oriented toward hydrologic applications. The general idea of the approach is

the conversion of a sequential stochastic simulation model, such as a seasonal AR(1), into a

disaggregation model. Its structure includes two separate parts, a linear step-by-step moments

determination procedure, based on the associated sequential model, and an independent nonlinear

bivariate generation procedure (partition procedure). The model assures the preservation of the

additive property of the actual (not transformed) variables. Its modular structure allows for various

model configurations. Two different configurations (PAR(l) and PARX(1)), both associated with the

sequential Markov model, are studied. Like the sequential Markov model, both configurations utilize

the minimum set of second-order statistics and the marginal means and third moments of the

lower-level variables. All these statistics are approximated by the model with the use of explicit

relations. Both configurations perform well with regard to the correlation of consecutive lower-level

variables each located in consecutive higher-level time steps. The PARX(1) configuration exhibits

better behavior with regard to the correlation properties of lower-level variables with lagged

higher-level variables.

1. INTRODUCTION

Disaggregation models are widely used tools for the sto-

chastic simulation of hydrologic series. They differ from

sequential stochastic simulation models in that instead of

generating events sequentially, they divide known higher-

level values (e.g., annual) into lower-level ones (e.g., sea-

sonal) which add up to the given higher-level values. Thus,

they have the advantage of providing the ability to transform

a time series from a higher time scale to a lower one,

resulting in a multiple time scale preservation of the sto-

chastic structure of the time series. However, disaggregation

models are more complicated than sequential models and

they have some other drawbacks, as discussed below.

The linear disaggregation model in its initial structure was

developed by Valencia and Schaake [1972, 1973]. Their

model assures the resemblance of variance and covariance

properties between historical and generated lower-level vari-

ables in the interior of a higher-level time step. However, it

makes no effort to preserve covariances of the lower-level

variables belonging to consecutive higher-level time steps.

In fact, this model does not assume any connection between

lower-level variables of different higher-level time steps.

Different model structures and parameter estimation pro-

cedures intended to preserve the lagged covariance proper-

ties among lower-level variables belonging to consecutive

higher-level times steps have been suggested by Mejia and

Rousselle [1976], Hoshi and Burges [1979] and Stedinger

and Vogel [1984]. However, as shown by Stedinger and

Vogel [1984], the reproduction of the exact historical serial

correlations of the lower-level variables can be an impossible

task within a disaggregation framework because of structural

constraints imposed by that framework, thus giving rise to

an inconsistency.

In cases where all variables are normally distributed, the

Copyright 1992 by the American Geophysical Union.

Paper number 92WR01299.

0043-1397/92/92 WR-01299505.00

basic linear generating scheme of the Valencia-Schaake

model assures the preservation of higher-order moments and

distribution functions as well; otherwise, the model in its

primary formulation fails to maintain such properties. Sug-

gested modifications of the model, permitting the represen-

tation of nonnormal distributions, can be classified in two

categories. First are methods oriented toward the preserva-

tion of merely the skewness of the lower-level variables [Tao

and Dellcur, 1976; Todini, 1980]. They generate independent

random components having skewed distributions with the

skewness coefficients being determined in terms of the third

moments of the historical data. Second are methods that

utilize nonlinear transformations of the variables, so that the

transformed variables have normal distributions [Valencia

and Schaake, 1972; Hoshi and Burges, 1979; Stedinger and

Vogel, 1984]. However, as Todini [1980] notes, this means

that the additive property, which is one of the main at-

tributes of the original disaggregation scheme, is lost. To

overcome this problem a correction procedure is usually

suggested [Lane and Frevert, 1990; Stedinger and Vogel,

1984; Grygier and Stedinger, 1988, 1990].

Another drawback of disaggregation models is their exces-

sive number of parameters, because of the large number of

cross correlations that they attempt to reproduce. Two

different procedures to reduce the required number of pa-

rmeters have been developed: the staged disaggregation

models (SDMs) and the condensed disaggregation models

(CDMs). The former [Lane, 1979, 1982; Sa!as eta!., 1980;

Stedinger and Vogel, 1984; Grygier and Stedinger, 1988;

Lane and Frevert, 1990] disaggregate higher-level variables

at one or more sites to lower-level variables at those and

other sites in two or more steps. The latter [Lane, 1979,

1982; Pereira et al., 1984; Oliveira et al., 1988; Stedinger and

Vogel, 1984; Stedinger et al., 1985; Grygier and Stedinger,

1988] reduce the number of required parameters by explicitly

modeling fewer of the correlations among the lower-level

variables.

Recently, another approach aiming mainly toward the

3175

3176 KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES

reduction of the parameter set required for a disaggregation

procedure, as well as toward providing the ability to repre-

sent non-Gaussian distributions, was introduced. The

model, called the dynamic disaggregation model (DDM)

[Koutsoyiannis, 1988; Koutsoyiannis and Xanthopoulos,

1990] has been formulated as a generalized stepwise ap-

proach to stochastic disaggregation problems, allowing for a

variety of configurations. DDM is closely connected with an

associated particular sequential generating model (e.g., a

PAR model) and uses the same parameter set as in this

sequential model. With the DDM approach, lower-level

variables are generated one at a time, given the total amount

to be allocated across the remaining periods. Thus the

disaggregation of a higher-level variable into its components

(lower-level variables) is split into equivalent sequential

steps, each corresponding to a specific lower-level time

period. The method involves two separate procedures in

each step. First the parameters of the generation equations

for this step are determined so as to preserve the specified

marginal and conditional moments given the total amount

remaining to be allocated and the already generated values at

previous steps. Second, these parameters are used to gener-

ate the step's amount (lower-level variable) and to update

the remaining amount to be allocated in next steps. These

two separate procedures are called the moments determina-

tion procedure and the partition procedure.

Each model configuration depends on the particular par-

tition procedure and moments determination procedure uti-

lized. The former is affected mainly by the marginal distri-

butions of consecutive lower-level variables, while the latter

is influenced by the type of their stochastic dependence, as

described by the associated sequential model. The first

studied forms of the model [Koutsoyiannis and Xanthopou-

los, 1990] concerned only single-site problems described by

Markov sequences with Gaussian or gamma marginal distri-

butions.

In this paper the model has been generalized for the

multivariate Markov case, i.e., it assumes a Markov depen-

dence of the lower-level variables. The Markov case is

selected because of its simplicity and the minimum set of

parameters that it involves. In addition, the explicit treat-

ment of third-order moments is supported by the present

model form. The model also approximates the lag one

correlation of the lower-level variables belonging to consec-

utive higher4evel time steps. Finally, a model configuration

capable of preserving of correlations of lagged higher- and

lower-level variables, in light of the inconsistency reported

by Stedinger and Vogel [1984] is discussed.

The model has been used in several hydrologic applica-

tions, such as the disaggregation of annual flows, rainfall

depths and lake evaporation into monthly quantities, as well

as the disaggregation of monthly into hourly rainfall depths.

2. ESSENTIAL ELEMENTS OF THE DYNAMIC

DISAGGREGATION MODEL

Consider a specific higher-level time step, or period (e.g.,

1 year), and a subdivision of the period in k lower-level time

steps or subperiods (e.g., 12 months), each denoted with a

time index t = 1, ..- , k. We denote by Z = (Z•, Z2, "'

Z,) r the higher-level variables of that period at n sites and

by X t = (X•, X•, --. , X,t) r the lower-level variables of

subperiod t at the same n sites. Higher- and lower-level

variables satisfy

X •+X 2+ '" +X t+ '-' +X k=Z (1)

We define a partial sum of lower-level variables, referred to

as amount still to go, by

S t = X t + X t+l + ..- + X k (2)

also expressed by

S t = Z - X •- X 2 ..... X t-• (3)

With the help of the above notation we discuss now some

key elements of the multivariate dynamic disaggregation

model. These elements should not be considered as model

steps but rather as an introductory summary of the key

features of the proposed model to be discussed in detail in

the next sections.

1. The disaggregation of the higher-level variable, Z, into

its k components X •, X2, ... , X k is split into n(k - 1)

sequential steps, each referring to the generation of one

lower-level variable at a single site. (For one site we need k

- 1 steps, since the last lower-level variable can be calcu-

lated from (1)).

2. At the beginning of step t at site j, we have at our

disposal knowledge of the previously generated information,

consisting of (1) the whole sequence of higher-level values

and (2) the lower-level values of the previous steps. Conse-

quently, from (3), the amount still to go, sJ, is also known.

3. Based on (2) and a specific assumption about the

stochastic dependence of the X t sequence in each step, the

distribution function of (xJ, s]), conditional on previously

generated information, is determined, or, in fact, approxi-

mated via conditional moments. Thus, in each step, we have

a moments determination procedure. A mathematically con-

venient assumption about the stochastic dependence of the

X t, allowing for the calculation of conditional moments, can

be that X t is an autoregressive sequence. This specific

autoregressive sequence of X t will be referred to as the

associated sequential model of the disaggregation model. It

must be emphasized that, given an assumption for the

dependence of X t, the dependence between X t and Z t (orX t

and S t) is fully determined by (1) (or (2)).

4. Given the joint and marginal moments of (X], SJ),

conditional on previously generated information, the gener-

ation of XJ can be viewed as an entirely isolated problem

with three variables, XJ, SJ +• and S J, satisfying

Jj + sj +'-. sj (4)

This isolated procedure, called the partition procedure,

ignores all other variables of the problem. It divides the

known amount SJ into two parts xj and S] +'. Its connec-

tion to the other part of the model is that it is fed by the

conditional moments of (X], sJ), which contain the stochas-

tic structure of the whole sequence of variables. The parti-

tion procedure can take several forms, depending on the

particular marginal distribution of the lower-level variables

and the statistics that are to be preserved. This procedure is

based on a generating expression,

XJ = h(SJ, wj) (5)

where h( , ) is generally a nonlinear function to be

discussed later, and W] is a random variable independent of

sJ and not necessarily normal.

KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES 3177

5. After the execution of the partition procedure the

previously generated information is updated with the known

XJ and the remaining quantity SJ + • is transferred to the next

step of the same site.

The main novelty of this approach is the different treat-

ment of the dependence across the sequence of low-level

variables and the dependence implied by the additive prop-

erty (1). The former is studied through the moments deter-

mination procedure and the latter with the partition proce-

dure. The resulting modular structure of the model is flexible

and can give rise to several model configurations.

The present model apparently has many differences from

other disaggregation models. Unlike the totally linear type of

disaggregation models [Valencia and Schaake, 1972, 1973;

Mejia and Rousselle, 1976; Tao and Delleur, 1976; Hoshi

and Burges, 1979; Todini, 1980; Stedinger and Vogel, 1984],

DDM is a combination of a linear part (moments determina-

tion procedure) and a nonlinear one (partition procedure,

based on (5)). Furthermore, the linear part is not at all like

the linear disaggregation equations used by other models.

Like Todini's [1980] model, DDM is capable of explicitly

treating the skewness of the lower-level variables, without

loss of the additive property, but there are no other similar-

ities between the two models. DDM maintains only a subset

of the joint second-order moments among the lower-level

variables, as do condensed disaggregation models (CDMs);

however, there are essential differences between DDM and

CDMs, to be discussed later.

3. THE PARTITION PROCEDURE

General Considerations

Since the partition procedure is isolated from the other

parts of the disaggregation model, it is studied separately in

this section. For reasons of convenience the notation in this

section will be as simple as possible, avoiding time and site

indexes which are not necessary here. The partition proce-

dure may be considered as a simple disaggregation problem

with two lower-level variables at a single site. It must be

emphasized that the extension of the partition procedure to

the multivariable and multisite case is not necessary. What is

needed in order to have a multivariate disaggregation model

is the development of a separate moments determination

procedure and the connection of the two procedures.

Assume, therefore, the case of two lower-level variables,

X and Y adding up to the higher level variable S, i.e.,

X+ Y=S (6)

In order to generate X we need to determine the following

generating expression, similar to (5):

X = h(S, W) (7)

where W is a random variable independent of S and not

necessarily Gaussian. Given a specific expression h( , )

and a specific distribution function F(w), X can be directly

generated from (7) and Y can then be obtained from (6). The

determination of h( , ) and F(w) aims toward the pres-

ervation of the joint distribution of (X, S). However, with

the exception of some special cases, the preservation of the

complete distribution is an impossible task. For practical

purposes though, the preservation of the first three moments

may suffice. The introduction of the third moment is neces-

sary if we want to cope with nonnormal distributions.

Since S is already known at the beginning of the partition

procedure, the preservation of its marginal moments, i.e., ,X'•

= E[S] and A i = E[S - X'•)i], i = 2, 3, '-' , is presumed.

The complete preservation of first, second, and third mo-

ments of X, Y and S requires that the following six quantities

are maintained:

Marginal moments of X

• • = E[X] r• 2 = Var [X] = E[(X- '1 '•)2] (Sa)

V3 '• /i'3[ X] -' E[(X- •1•1) 3]

Joint moments of (X, S)

Sr•l = Coy IX, S] = E[(X- r/'•)(S - X '•)] (SO)

sr•2 = tx•2[X, S] = El(X- ,q'•)(S- X'•) 2] (8c)

= s] = El(X- (8a)

where /z with one or two sub scripts denotes marginal or

cross central moments. It is easily shown that any marginal

or joint moment of order <3 between (X, Y, $) can be

determined in terms of the above six quantities and the

marginal moments of S, i.e., X•, X 2, ,X 3 (see, as examples,

(21)-(24) below). Thus, given that ,X are preserved outside of

the partition model and Y is derived from (6), the preserva-

tion of the ,/and •'implies the preservation of any moment of

order -<3, marginal or joint. In cases where only the marginal

moments of X and Y are of interest, the preservation of

merely the difference •'•2 - •'2•, instead of their particular

values, is required.

The Linear Scheme

Let us first examine the linear form of h(

disaggregation scheme), defined by ) (the linear

X = aS + b W (9)

where a and b are parameters to be estimated. Assuming

with no loss of generality that Var [W] = 1, the other two

parameters of the model are E[W] and/x3[W]. This scheme

is capable of preserving first and second moments, but not

nonzero third moments of X and ¾. Indeed, a, b and E[W]

can be determined so as to preserve the first and second

moments (a = •'•/X2, b = (*/2 - •'•/X2)•/2, E[W] = (*l'•

- a,•)/b). Then Ix3[W] can be determined such that */3 =

/z 3 [X] is preserved. Since there is no other parameter, there

is no way to preserve •'12 and •'21. Consequently, tz3[ Y] could

not be preserved by this scheme. The same is pointed out

elsewhere [Koutsoyiannis and Xanthopoulos, 1990] and is

implied by Todini [1980]. In fact, Todini [1980, p. 203] uses a

linear disaggregation scheme with more than two lower-level

variables, where the situation is different, if only marginal

third moments are to be preserved.

The preservation of first- and second-order moments is

probably the only task which can be faced in an exact way.

The linear model is sufficient for this task. For the special

case where all variables are normal the linear model can also

preserve the complete distributions of all variables. Below

some nonlinear schemes are examined, which can cope with

higher moments for the general case, but are not exact

3178 KOUTSOY!ANNIS: SIMULATION OF HYDROLOGIC SERIES

schemes in that they do not preserve any statistic exactly but

only approximately, as will be discussed later.

The Use of Nonlinear Transformations

It is a common practice in stochastic models, even in

disaggregation models, to use nonlinear transformations of

variables with nonsymmetrical distributions in order to

achieve normal distributions, which are easier to handle. Let

us examine the logarithmic transformation, i.e.,

X' = In (X- Cx) (10)

Y' = In (Y- cr) (11)

S' = In (S - Cs) (12)

where C x, c r and Cs are parameters to be estimated.

Apparently, because of (6) and since lognormal distribution is

not regenerative, X', Y' and S' cannot all be normal. Conse-

quently, there is no case where a linear generating rule like

X' = axS' + bxW• (13)

along with (6) could be exact with respect to the complete

distribution of the three variables. Furthermore, if we reduce

the requirements to the moments preservation, the situation

will be similar to that of the linear scheme. Indeed, the

scheme in (13) again has four parameters to be determined:

ax, bx, Cx and E[Wk]. Note that Cs is not a parameter of

the generating scheme itself, since it refers to the presumed

distribution of $. Furthermore, t•3[W•c] is not a parameter;

it should be zero because W• should be assumed normal;

otherwise there do not exist analytical relations between the

statistics of X and X'. Thus the above scheme, because of a

lack of a sufficient number of parameters, is not capable of

preserving the third-order joint moments •'•2 and •r2•.

Another method, which is mostly used in disaggregation

models, initially ignores (6) and generates independently X'

from (13) and Y' from a similar relation, i.e.,

Y' = arS' + brW¾ (14)

By taking antilogarithms in (13) and (14) we find the follow-

ing first approximations of X and Y:

X* -- (S - cs)axwj• x d- c X (15)

Y* = (S- cs)arw br + c (16)

Y Y

where Wx = exp (W•) and Wr = exp (W'r). Apparently,

X* and Y* do not add up to $. In order to regain the additive

property (6), X* and Y* must be corrected. Lane and

Frevert [1990, p. V-22], $tedinger and Vogel [1984] and

Grygier and Stedinger [1988, 1990] have developed several

adjusting procedures. The so-called proportional adjust-

ments define the simplest procedure, which is

S S

"' X* Y = . Y*

X = X* +"" Y* ' X* -•' r* (17)

Combining (15), (!6) and (17), we get the following final

solution for X, or the final generating rule:

x = [(s - c +

ß [(s - ß + cx + (s- + c

(18)

which can be combined with (6) for the generation of Y.

The last scheme has a sufficient number of independent

parameters (eight) in order to assure the preservation of the

statistics in (8). However, its complexity prohibits the ana-

lytical determination of any statistic, even of the mean. In

that respect the scheme could not be exact, since no exact

parameter estimation procedure can be established.

Somewhat simpler is the situation if an adjustment scheme

based on the standard deviations of the lower-level variables

is used [Lane and Frevert, 1990, p. V-22]:

X= X* + (S - X* - Y*)l• x, (19)

r= ¾* + (s- x*- r*)(1 - •x)

where 8x = •rx/(rrx + rrr) and rr x and rr r are the standard

deviations of X and Y. Combining (15), (16) and (19) we get

the following explicit generating rule:

X = (1 - 8x)[(S- c$)axw• x + Cx]

- xE(s - c + c r] + (20)

which can be combined with (6) for the generation of Y. By

using (20) one can get explicit but too complicated relations

for the marginal moments of X (r/terms) and for the joint

moments of X and S (• terms). These relations may be

solved for the unknown parameters only numerically.

Moreover, (18) and (20) have another source of inaccuracy

arising from the fact that the lognormal distribution is not

regenerative. Even if there would exist an exact parameter

estimation procedure, this would be based on an assumption

that S, Wx and W r are lognormal. But then, X and Y could

not be lognormal. Thus in the next execution of the partition

procedure, in which the updated S is equal to the previous

Y, the above assumption is no longer valid.

Consequently, the requirement for exactness involves

insurmountable difficulties. In order to establish an approx-

imate procedure, the usual assumption is that the statistics of

X equal those of X* and the statistics of Y equal those of Y*.

This approximation has performed well in many models.

However, besides the nonexactness of the procedure, an-

other problem arises: The number of parameters is no longer

sufficient to preserve all the statistics listed in (8). Indeed,

four parameters, namely ax, bx, cx and E[W•r], are used to

preserve the three marginal moments of X* plus the covari-

ance of X* and S. The remaining parameters are a r, b r, c r

and E[ W'r]. Since the generation of Y* is independent of X*

(X* + Y* % S), the first and second moments of Y* are not

automatically preserved, as they are in the case where Y is

generated from (6). Thus one has to use three parameters to

preserve these moments, namely E[ Y*], Var [ Y*] and Coy

[ Y*, S], and there is one remaining parameters, which has to

be used to preserve the third marginal moment of Y*. The

result of this parameter allocation is that the difference •12 -

G• is preserved and not the values of •12 and •2]. (Note that

t•3[ Y] is given by (23) below, X 3 is preserved outside of the

partition model, r/3 is approximately preserved by the gen-

eration of X*, and thus the preservation of t•3[ Y] is equiv-

alent to the preservation of •12 - •2].) Of course, this

weakness is not major: The joint third moments of X and ¾

are not preserved, but the marginal ones are preserved

approximately. It is emphasized that the above difficulties in

establishing an exact model are not specific to the present

KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES 3179

model, but are also shared by other disaggregation models

that use the logarithmic transformation of variables.

The realization of an approximate parameter estimation

procedure suiting the requirements of this specific partition

procedure is simple. First, calculate the moments of Y in

terms of the moments in (8), by using (6), i.e.,

ElY] = X '• - r/'• (21)

Var [ Y] = X 2 + r/2- 2st l l (22)

/'t'3[ Y] --- '•'3- •/3- 3Srl2 + 3•21 (23)

Coy [ Y, S] = X 2 - f it (24)

Second, assume equality of the moments of X* and Y* with

those of X and Y. Third, determine the moments of X' and

Y' (plus Cx and c r) in terms of the moments of X* and Y*

by the method of moments [e.g., Charbeneau, 1978; Kotte-

goda, 1980, p. 137; Stedinger, 1980]. (Due to the moments

orientation of the procedure other estimation methods dis-

cussed by Stedinger [1980] are not applicable here.) Fourth,

calculate the parameters of the linear schemes (13) and (14)

by standard methods.

The generation part of the procedure consists of the

following: First, generate WSc and W'y from the normal

distribution. Second, calculate X' and Y' by (13) and (14)

and take antilogarithms to determine X* and Y* (also, add

cx and c r)- Third, use (17) or (19) or any other correction

procedure to determine X and Y.

It is noted that the above partition procedure based on the

logarithmic transformation of variables has not been used in

a real application in the framework of the proposed model.

In a similar way other procedures, based on different

nonlinear transformations, such as Wilson-Hilferty, can be

developed. The main problems of such procedures are the

same as those described for the case of the logarithmic

transformation.

The Quadratic Scheme

Finally we shall examine a quadratic generation rule,

without transformations of variables, defined by

x = a(s) + f(s) w (25)

g(S) = ao + a lS + a2 S2 (26)

f(S) = bo + biS + b2 S2 (27)

where a i and b i, i = 0, 1, 2, are parameters to be estimated

and W is a random variable not necessarily normal, indepen-

dent of S, with zero mean and unit variance. This scheme,

though it seems not physically reasonable, is introduced as

the most convenient one with respect to the moments

calculation, having a sufficient number of parameters to

cover the relevant restrictions. However, as explained be-

low, it has certain structural problems. Due to these prob-

lems it cannot be an exact scheme, but an approximate one.

In the following analysis, for mathematical convenience

and without loss of generality it will be assumed that all

random variables have zero mean. This scheme contains

seven parameters, the six coefficients a i and b i, plus the

skewness of W, 03 = E[ W3]. So, the number of parameters

exceeds the number of constraints in (8), thus giving one

degree of freedom.

By making the proper transformations in (25) and then

taking expected values, the following equations can be easily

obtained:

•[X] = •r[a(S)] (28)

E[XS] = E[Sg(S)] (29)

E[X 2] = E[g2(S)] + E[f2(S)]E[W 2] (30)

E[XS 2] = E[S2•7(S)] (3 I)

E[X2S] = E[S#2(S)] + E[Sf2(S)]E[W 2] (32)

E[X 3] = E[93(S)] + 3E[g(S)f2(S)]E[W 2]

+ E[f3(S)]E[W 3] (33)

It is obvious that these equations are explicit relations

between the moments of X and S and the unknown model

parameters. In particular, (28), (29) and (31) determine

completely g(S), yielding simple expressions for the calcu-

lation of the a i. Somewhat more complicated is the deriva-

tion off(S), obtained from (30) and (32), by setting E[W 2 ] --

1. Finally, 03 is obtained from (33). All derivations and the

final equations are given in Appendix A.

There are two limiting cases where this scheme is exact

with regard to the preservation of the complete distribution

functions (not only moments). First, if the variables X and Y

(or, equivalently, X and S) are jointly normal, then the

quadratic scheme downgrades to the linear one and (25)--(27)

reduce to (9), as is theoretically anticipated (see Appendix

A). Apparently, in the normal case W is also normal and the

statistical resemblance is extended to the complete distribu-

tion of (X, Y). Second is the case where X and Y are

deviations from means of two independent, two-parameter

gamma-distributed variables • and ? with con-tmon scale

parameter. In this case if •' = .,• + ? then the variable

W = .•/•' (34)

is independent of • and beta distributed [Johnson and Kotz,

1972, p. 234]. Consequently, the deviations from means are

related by

X = (E[œ] + •[•]•r[g]) + E[½]S + (•[•] + S)W (35)

where the random variable W = 1• - E[I•] is independent

of S. Equation (35) is a special case of (25) where both •/(S)

andf(S) have downgraded to linear functions (a 2 -- b 2 = 0)

and the complete distributions of variables are preserved.

Let us now consider the problems resulting from the

quadratic scherne in the general case. A first problem results

from the quadratic forms of f(S) and g(S), which have a

single maximum or minimum. Due to this, not all combina-

tions of A, r/, and s r values yield solutions of (28)-(33). The

relevant constraints concerning the existence of functions

f(S) and #(S) are given in Appendix A (equations (A19)-

(A21)). The limitations arising from that problem are illus-

trated further in section 6.

The second problem may be seen in (30), (32) and (33)

where high moments of S indirectly appear, i.e., A4, X5 and

'•6- This is confirmed by the expressions of Appendix A used

for the computation of the a and b terms. These high

3180 KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES

moments must be known in order to evaluate these expres-

sions and preserve lower moments of X and Y. If the

complete marginal distribution of S is known, then these

moments can be determined exactly in some way. However,

this is not the case here, given that the partition procedure is

iteratively executed across the disaggregation steps. More-

over, there is no way to compute exactly in each step the

high moments of the updated amount still to go (of the next

step, which here is represented by Y), since any calculation

would introduce even higher moments of S.

For practical purposes, one could assume a typical distri-

bution function for S and calculate high moments analyti-

cally for this distribution. Apparently, however, the exact

solution is then lost. The use of cumulants may be helpful for

numerical purposes, i.e.,

2

A4 = K4 + 3X2 (36)

X 5 -- K 5 + 10A3A 2 (37)

3

A 6 = K 6 q- 15K4A 2 4- 10A} + 15A 2 (38)

where K4, Ks and K6 are the cumulants of S, of orders 4-6

[Kendall and Stuart, 1963, p. 70]. Two typical cases will be

emphasized (for both, see Kendall and Stuart [1963, p. 70]).

These are the normal distribution with

K4 = *:5 = K6 = 0 (39)

and the gamma distribution with

K r = (r- 1)Kr_lX3/2A2 r > 3 (40)

where K 3 = A3. Either (39) or (40), along with (36)-(38) may

also be used for estimation when the distribution of S is not

accurately determined, but can be approximated by the

Gauss or the gamma probability distribution function.

To inspect the influence of an assumption about cumulants

on the moments of the generated series a numerical investi-

gation was made. A gamma-distributed S was assumed and

then disaggregated into two lower-level variables. Two dif-

ferent assumptions for the cumulants were made: first, that

t< r is given by (40) (true assumption) and second, that Kr =

0 as in the normal distribution (equation (39), false assump-

tion). The second assumption does not indicate normality; A3

in (36)-(38) was still nonzero. From this investigation it was

found that the results are quite similar in both cases (an

example is given in section 6 and Table 2). This is an

indication that the model although not exact is not very

sensitive to the assumption about the cumulants.

In the applications of this study the cumulants of the

gamma distribution (equation (40)) were adopted for vari-

ables with distribution approximately gamma. Furthermore,

the beta distribution has been used to approximate the

distribution of W (after a linear transformation).

4. THE MAIN DISAGGREGATION MODEL

General Considerations

A disaggregation method may be viewed as a technique for

generating lower-level variables satisfying the additive prop-

erty (1). The lower-level variables of the current period

X • , Xk are a subset of a stochastic sequence expanded

1 0 ! k

in both time directions, i.e., (..-, X- , X , X ,'", X ,

Xk+•, .. .). Also the higher-level variable of the current

period Z is a term of another stochastic sequence (- ß ß, Z -!

Z ø Z 1 m Z, Z 2 ... Z * ...). Note the different time

indexes used in the notation of X t and Z •, in order to avoid

dual indexes; if the number of subperiods in each period

(i.e., k) is the same for all periods then Z

+ ß ß ß + X •. Also note the notational simplification Z -- Z

Due to the independence of the generation procedures of

higher- and lower-level variables, it can be assumed that the

higher-level variables are all known (. ß., Z ø = z ø Z

Z 2 = z 2, - ß -) at the beginning of the disaggregation. Also, it

is assumed that the disaggregation procedure has already

been completed at the previous periods; thus all the previous

X t have known values (. , X-2 x-2 X-2 -2 X 0

Consider the disaggregation step t, at the current period

and at the site j. The generation of the lower-level variable

XJ is characterized by

xj + sj+' = sj

where the amount still to go, S J, has a known value given by

(3), since the previous steps have been completed.

Due to the similarity of (41) to (6), the generation of XJ can

be executed by using a specific scheme of the partition

procedure described in the previous section. From the point

of view of the main disaggregation model, the partition

procedure may be considered as a black box procedure. The

input from the main model to this procedure is the value of

$J and a list of statistical marginal and joint moments of (X,

S). The output received by the main model consists of the

values of X• and SJ +2 . Each time the partition procedure is

executed, no information about previous lower-level vari-

ables and their stochastic dependence is utilized. However,

there is a possibility of indirectly introducing the dependence

of previous lower-level variables into the black box partition

procedure without changing its structure. This could be done

by sending as input the conditional moments of (X, S) given

the previously generated information. Namely, these mo-

ments and their corresponding moments of the partition

procedure are as follows:

Conditional marginal moments of SJ

t t

,•'• = E[SJ llJ], x 2 = Var [sj llj], x3 =/•

(42a)

Conditional marginal moments of XJ

,/'1 = E[Xj]Ilj], ,/2 = Var [Xj l•J], (42b)

= 3[xjlnj]

Conditional joint moments of (XJ, S J)

c,, Cov IX, Sjlaj] c12 • 12[ X, t ,

(42c)

where •J is an abbreviation for the already generated

information. Depending on the succession of time steps •d

sites in each period, two d•erent courses of the disaggrega-

tion procedure may be considered: the horizontal co•se,

where each step follows the previous step at the same site,

KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERiES 3181

and the vertical one, with each step following the same step

of the previous site. In the former case, which is adopted in

the present study, !• generally consists of the following

items' the full sequence of higher-level variables (.. ß, Z 2 =

z 2, Z • -- z 1 Z ø = z ø ..-)' the lower-level variables of

previous periods at all sites (X ø = x ø, X -1 -- x -1, ...); the

lower-level variables of the current period and current

site/previous subperiods (XJ -• = x•-•, .-. , X! = x/),

previous site/all subperiods (Xf-1 xf_•,... X/_• ---

x.• tc

•-1), and so forth through the first site/all subperiods (X1 =

The determination of conditional moments could be an

impossible task if all the information contained in 11• is

considered. Thus, simplifications are necessary. These can

be based on an assumption for merely the sequence of

lower-level variables. A convenient simplifying assumption

about this sequence could be a linear autoregressive struc-

ture. This structure (or model) will be used here not for the

generation of lowerrevel variables but for the determination

of conditional moments.

The Associated Sequential Model

The selection of a sequential linear model for the lower-

level variables depends on which of their statistics are to be

maintained. Here, in order to build a parameter parsimoni-

ous model, the minimum set of statistics is considered. This

set is the same as in the sequential Markov model [Matalas

and Wallis, 1976, pp. 60, 63], i.e., it contains the following

groups: (1) mean values of X](•:J); (2) variances of XJ; (3)

skewhess coefficients of XJ; (4) lag one autocorrelation

coefficients between XJ and XJ -• (same site); and (5) lag

zero cross-correlation coefficients between X• and Xr t, for j

• r (same period).

The third-order joint moments, which can be also sup-

ported by the partition procedure, are not independent

parameters in a linear model of lower-level variables. Their

values, which are required as input items to the partition

model, can be obtained from the assumed linear structure as

expressions of the above groups of statistics (see section 5

and Appendix B).

The preservation of the five groups of statistics enumer-

ated above is possible with a seasonal AR(1) (PAR(l)) model

for the process X t, which is the selected associated sequen-

tial model (not the disaggregation model):

X t '- atX t-1 + btV t (43)

or equivalently

J

XJ _ tx..t-1

aj•j q- E t t

= bjrVr,

r=l

j= 1,--., n (44)

where a t can be assumed (n x n) diagonal, i.e., a t = diag

(a f, a•, -.- , a•t), whereas b t = [b/•] is a (n x n) lower

triangular matrix of coefficients, and V t = [V}] is a vector of

n random variables, completely independent both in time

and location. We assume that •t = E[X t] and [•t = E[V t]

are not necessarily zero; we set (for mathematical conve-

nience) Var IV t] = 1, and also denote •/t = /z3[Vt]. The

process stationarity is not necessary and hence not implied

in the notation. However, the assumption of a seasonal

stationarity may be helpful in some of the following analy-

ses.

Note that the model does not make any distinction for the

first lower-level variable XJ, where (44) is also applied with

Xj ø, the last lower-level variable of the previous period.

The groups of parameters a t , b t and h, t are computed from

the groups of statistics 2-5 listed above by the following

relations which are extensions of those of the stationary

Markov model given by Matalas and Wallis [1976, p. 63]

coy xj -1]

t (45)

aj = Var [XJ -1]

bt(bt) T '- 0 't- ato. t-la t (46)

j-1

/'t'3[X•]- (a•)3/'t'3[X; -1] -- E (b•

k=l

t . . (47)

=

where

o 't-- Coy [X t, X t] = E[(X t- •t)(xt - •t)T] (48)

and the lower triangular b t is obtained from its Gramian

b t (b t) T by decomposition.

Connection of the Sequential Model

to the Disaggregation Model

It should be emphasized that the model (43) with param-

eters determined from (45)-(47) concerns only lower-level

variables and does not form a disaggregation model. How-

ever, (43) can be combined with (1) or (2) in order to describe

joint properties between lower-level and higher-level vari-

ables or, more generally, between lower-level variables and

amounts still to go. The required combination should be

oriented toward the determination of the moments listed in

(42) and not toward the generation of any variable, since the

generation is executed independently by the partition proce-

dure. Thus, what remains to complete the disaggregation

model is the development of the moments determination

procedure, which can be supported by the associated se-

quential model. This development is done in two different

versions in sections 5 and 7.

Finally, note that the set of independent parameters for

the disaggregation model is the same parameter set as for the

associated sequential model. No additional independent

parameter is required for the determination of the moments

listed in (42) (see sections 5 and 7) and the partition proce-

dure is not fed by any external parameter.

5. THE MOMENTS DETERMINATION PROCEDURE:

PAR(l) CONFIGURATION

Consider the items of the already generated information in

the general case described in the previous section. First, we

note that higher-level variables of previous periods, (i.e.,

Z -1 = z -l , Z -2 = z -2 ß ß -) can be omitted because they are

contained (due to (1)) in the sequence of the corresponding

lower-level variables. With the approach followed, the item

(Z = z) of the current period can be substituted by the known

amount still to go, given the known values of the previous

lower-level variables. Furthermore, the handling of the

3182 KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES

amount still to go is left to the partition procedure and

ignored at the moments determination procedure. A major

simplification of the disaggregation model is obtained if the

information containing the higher-level variables of the next

periods (i.e., Z 2 = z 2, Z 3 = z 3, ß ß .) is deleted from f•J. This

is the case in all disaggregation models. The consequences of

this deletion are discussed in section 6.

Consequently, in this section all items concerning the

information of higher-level variables are omitted to simplify

the determination of conditional moments. Later, in section

7 the question of how the item (Z 2 = z 2) can be taken into

consideration will be discussed. In the Markovian case

examined, f•J is further simplified, since the information

of only one previous step is affected. Thus f•J reduces to

: - -' (x/_ , , ß

= ) .. . , x/_ = x: x -

-1] ['J '' U (Xl k - x1, ''' , X• - x•, X =

The following analysis is very helpful to the moments

determination procedure. Let e t be the (n x n) diagonal

matrix, whose (j, j)th element is eJ = bJ•.; it is then easily

shown that the (n x n) matrix d t defined by

d t = I - et(bt) -1 (49)

with I the unit matrix, is lower triangular with zeros on the

diagonal. By means of d t and e t (43) becomes

X t = atX t-1 4- dt(X t - atX t-l) + etv t (50)

and consequently

j-1 dir(X r - a )+ ei VJ

r--1

(51)

The latter is a modification of (44) in which only one

innovation term (V J) appears. (It is similar to equations

(10)-(12) of Stedinger et al. [1985]). Equation (51) may be

also written as

where

j-1

Uj = •'• djr(Xr t - a}Xr t-•)

r=l

(54)

T] and UJ are expressions of lower-level variables contained

in 11• while WJ is a random variable independent of TJ and

UJ. Specifically, TJ is related to the information of the

current location, while U] is related to previous locations.

Thus, conditional moments of XJ of order greater than 1,

given 11•, are expressed in terms of the moments of W] only.

Furthermore, in each disaggregation step the conditional

mean of XJ can be easily determined by using these equa-

tions.

By iteratively using (52) and (53) each Xf , r = t, ... , k

may be expressed in terms of XJ -• , Uf (items contained in

l•J) and Wf. Then by adding the expressions of each Xf we

get

where

k k

sJ = aj txt--1 r=t r"'t

(56)

k

•'f= 1 + • af+•a: +2--'a? (57)

u"'r+l

In a manner similar to the case of conditional moments of

X], described above, the conditional moments of S J, given

f•J, can be easily derived with the use of (56). Specifically,

the moments of order > 1 are expressions of the moments of

Wj only. Furthermore, (52) and (56) may be combined by

multiplication to allow the calculation of conditional joint

moments of (XJ, S J). A systematic algorithm for the calcu-

lation of all needed moments is easily constructed from these

equations and given in Appendix B.

6. MODEL OVERVIEW

General Considerations

As described above, the proposed model consists of two

isolated procedures, a partition procedure and a moments

determination procedure. Three forms of the former and one

of the latter were studied above. Another improved form of

the moments determination procedure is studied in the next

section.

Regardless of the configuration used, the model perfor-

mance is similar to that of the sequential PAR(l) (Markov)

model, except that the lower-level variables add up to the

already known values of the higher-level ones. This property

makes the disaggregation model superior to the sequential

one because there is no loss of statistical resemblance for the

higher-level variables. Instead of deriving the higher-level

variables as sums of the generated lower-level variables

(sequential way), the use of the disaggregation model per-

mits the separate generation of the former, based on the

preservation of their own statistics.

It is important to note that, with the exception of the

special cases discussed in section 3, the statistical resem-

blance achieved by the model is not strict. Theoretically, the

synthetic sequences generated by the model have moments

which may be good approximations of their theoretical

values but do not equal them. This nonexactness of the

model is due to the structural inconsistencies of the partition

procedure, described in section 3 and is encountered in other

disaggregation models as well.

The stepwise approach of the model permits the use of

parallel procedures adjusting properly the generated values

in each step, without loss of the additive property. The case

of positive lower-level variables is an example when a proper

parallel procedure may handle (reject or modify) the possibly

generated negative values. Note, though, that such parallel

procedures introduce additional bias to the simulated series

and, thus, may affect the validity of the procedure.

Due to the stepwise course of the dynamic model there are

no difficulties in handling the dependence between the

lower-level variables associated with higher-level ones of

consecutive periods. In fact, .as shown in section 4, the

mathematical formulation is the same as within a particular

higher-level variable.

KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES 3183

At first view, the model exhibits a similar performance

with the condensed disaggregation models, since both

schemes do not use the all-at-once generation approach of

the Valencia-Schaake class of models and utilize only a

subset of the parameters of the latter. However, there are

essential differences between the two schemes:

1. The stepwise course of DDM is applied not only to

different time steps (CDM case) but also to different sites.

2. In each step DDM generates not only the correspond-

ing lower-level variable, but also the amount still to go of the

next step. CDMs do not use the amount still to go.

3. In CDMs each lower-level variable is expressed as a

linear function of the higher-level variable. DDM uses the

amount still to go instead.

4. CDMs use nonlinear transformations of the actual

hydrologic quantities as lower-level variables while DDM

uses the actual quantities instead. If nonlinear transforma-

tions are utilized by DDM, these are internal to the partition

procedure and invisible from the point of view of the main

disaggregation model.

5. The third moments are treated by DDM with explicit

analytical relations while in CDMs they are not.

6. CDMs, like all Valencia-Schaake type models, are

totally linear. DDM eventually utilizes a nonlinear part

(partition procedure).

7. One element of a CDM's parameter set is the covari-

ance matrix between lower-level and higher-level variables.

DDM does not introduce this element as an independent

parameter, thus reducing its parameter set to that of a typical

sequential model (PAR). DDM, in order to evaluate covari-

ances between lower- and higher-level variables or, more

generally, covariances between lower-level variables and

amounts still to go, uses the properties of the associated

sequential model and computes such covariances using

analytical expressions.

Model Limitations

The essential limitations of the disaggregation model are

related to the evaluation of the parameters b t and are met

also in all sequential and disaggregation model [Matalas and

Wallis, 1976, p. 69; Bras and Rodriguez-Iturbe, !985, p. 150;

Grygier and Stedinger, !990, p. 31].

The matrix c t = b t(b t) r may be written as

C t= COV [(X t-- atXt-1), (X t- atXt-1)] (58)

and consequently its elements must satisfy the inequality

t

Cjk

< < ¾j (59)

-- t t 1/2 -- '

(cftc)

In addition, the existence of b t requires that c t is positive

definite. It is possible that these two structural constraints

are not satisfied for certain hydrologic data, utilized to

determine c t by means of (45) and (46). Furthermore, (47)

may yield unreasonable skewness coefficients, e.g., of the

magnitude reported by Todini [1980] (y > 30). In fact, there

is no limit for yJ, since the denominator (b•.)3 in (47) may

take too small values. These problems are encountered

mostly in cases where the historical records do not refer to a

common time period.

It is emphasized that these problems are related to the

sequential PAR model [Matalas and Wallis, 1976, p. 69] and

not to the disaggregation model itself. A practical solution to

overcome the problems may be the reduction of the cross-

correlation or autocorrelation coefficients, or even the skew-

ness coefficients of the historical data. Such a reduction of

the characteristics of the historical data is a major problem.

However, in the case of a disaggregation model, such a

modification does not influence the characteristics of the

higher-level variables.

In a similar situation concerning records with different

lengths at different sites Grygier and Stedinger [1990, p.

31-33] discuss practical solutions to relevant problems,

suggesting modifications to the items of a matrix to be

decomposed, if the decomposition fails. Here another sim-

plified technique was used. The decomposition of c t =

b t (b t) r can fail at the point where a diagonal element b j} is

to be calculated by the equation

(60)

If the right-hand side of this becomes negative there is no

real solution for b3. To avoid this, one can impose a lower

limit on each diagonal element b3. such as

h tminh 2 t

..;• , = pcj• (61)

where p is a constant (0 < p -< 1). If b• is set equal to this

limit then the nondiagonal elements of the row (bJk, k =

1, ..., j - 1) can be corrected by multiplication with a

single factor • determined so as to regain the validity of (60).

In this way the preservation of Var [XJ] is assured while a

minimum positive Var [WJ] is imposed. An indirect benefi-

cial consequence is that the skewness of WJ (or VJ) cannot

take unreasonably high values. However, the cross correla-

tions of XJ with the lower-level variables of other sites are

apparently reduced with this algorithm.

Another solution to the same problem could be the reduc-

tion of the matrix b t in a manner such that only the highest

concurrent cross-correlation coefficient is preserved, thus

having one off-diagonal nonzero element per row of b t . This

will have as a consequence the further reduction of the

model parameter set (see next subsection). Other methods of

parameter estimation for the PAR(l) model could be more

effective, but they have not been examined in the framework

of this study.

Some secondary limitations result from the constraints of

the quadratic partition procedure, i.e., inequalities (A19)-

(A21), if this particular procedure is selected for the model

configuration. A relevant illustration of the ranges of appli-

cability of the quadratic partition procedure is given in

Figure !. This is done by numerical applications of the model

for several combinations of parameters and selection of

those combinations which yield a limiting fulfillment of the

constraints. One can observe that if the lower-level variables

are quite skewed and strongly correlated then a problem

might arise for the model. However, it was observed from

the applications that these constraints are generally fulfilled

if the skewness coefficients yJ have reasonable values. A

solution to situations of very large yJ could be the invocation

of the lognormal scheme described in section 3, which

theoretically has no limits of applicability.

3184 KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES

z

1,0

0.8

O. LI

0.2

-0.0

-0.2

-0.6

-0.8

-1.0

1.0

Cs. x -'6 ... "+6

(8}

0.0 O.t 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

VarEX] / ¾arEY]

C•i.X = 0

f_

f_

o

:z

0.2

-J

m: -0.2

• -O.tl

-0.6

-0.8

-1.0

ñ1

:t:6

0.0 O.t 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

VarEX3 / VarET3

Fig. 1. Illustration scheme of the ranges of applicability of the

quadratic partition procedure. The curves represent upper and

lower limits of correlation coefficients of lower-level variables (Corr

[X, Y]) that lead to real solutions of the equations. These limits are

given versus the ratio of the variances (Var [X]/Var [ Y]) and for two

combinations of the skewness coefficients of the lower-level vari-

ables: (a) Cs,x = Cs, r and (b) Cs,x = 2Cs,r.

Model Parameters

As stated in section 4 the present configuration of DDM

uses the minimum set of parameters, equal to that of the

sequential Markov model. A further reduction of the param-

eter set is possible (but not necessary) by defining the b t

matrices with only one nonzero element in each row, in such

a manner that only the highest concurrent cross-correlation

coefficient is preserved.

Table 1 summarizes the number of parameters needed for

the model, in comparison with other well-known models. It

is obvious that DDM has the fewest parameters. Note that

the results given in Table 1 concern only second-order

statistics. To these results the number of means and skew-

:ness coefficients of the lower-level variables and of the

parameters of the model used for the generation of the

higher-level variables must be added.

Apparently, the parsimony of parameters of DDM in-

volves a cost (omission of terms of the full covariance

matrices), but this cost is not associated with the preserva-

tion of the additive property or of the correlation between

consecutive lower-level variables of consecutive periods.

Preservation of Marginal Distributions

Figures 2a, 2b and 3 give examples for the model perfor-

mance concerning the preservation of characteristics of the

marginal distributions of lower-level variables. They origi-

nate from two applications of the model for the study of the

water supply system of Athens, Greece [Koutsoyiannis et

al., 1990]. Application 1 involves the simulation of concur-

rent rainfall and runoff in three basins supplying the water

system of Athens. Application 2 refers to the simulation of

lake evaporation of the three relevant reservoirs. In both

applications, synthetic annual data of 5000 years were dis-

aggregated by the model into monthly data. The gamma

distribution function and the quadratic partition procedure

were adopted for monthly rainfall and runoff data. The

Gauss distribution function and the linear partition proce-

dure were adopted for monthly evaporation data. The figures

indicate the good performance of the model in both cases. As

shown in Figure 2, though the model is a stepwise one,

errors in skewness do not build up as we progress through

the 12 months.

An application of a previous specific version of the model

in hourly rainfall generation, where the marginal distribution

of lower-level variables was J-shaped gamma or Weibull, can

be found elsewhere [Koutsoyiannis, 1988; Koutsoyiannis

and Xanthopoulos, 1990].

Table 2 illustrates a comparison of DDM to the Valencia-

Schaake model. A gamma higher-level variable, Z, is disag-

gregated into two correlated exponential lower-level vari-

ables, X 1 and X 2, and it is assumed that there is no

correlation between subsequent higher-level time steps.

DDM was directly applied to the actual (not transformed)

variables using the quadratic partition procedure with a

parallel procedure rejecting negative values (series 1 and 2).

In specific, series 1 is generated by using the exact cumu-

lants of the known gamma distribution of Z. For the gener-

ation of series 2 the false assumption that the cumulants K4

= •<5 = •<6 = 0 was used in order to investigate the effect of

such a false assumption on the statistics of the generated

series. The results indicate that this effect is not significant.

For the application of the Valencia-Schaake model in this

example the Wilson-Hilferty transformation of the actual

variables was used, followed by the inverse transformation

of the generated data. Since there are only two lower-level

variables the Valencia-Schaake model is equivalent to a

partition procedure using the Wilsoh-Hilferty transforma-

tion, in a way similar to the logarithmic transformation

described in section 3. In the case of the Va!encia-Schaake

model (series 3) the additive property apparently is not

preserved. By the adjustment procedure described by (!9)

and applied to the inverse-transformed data the additive

property was then regained and, in addition, negative values

were corrected (series 4). For reasons of comparison the

KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES 3185

TABLE 1. Comparison of the Number of Second-Order Statistics Used in Various Disaggregation Models

Number of Second-Order Parameters

Fork = 12 Fork-- 12

Model Type In General and n = 1 and n = 3 For k = 12

andn = 6 For k = 12

andn = 10

Valencia-Schaake

Mejia-Rousselle

LAST model (from Grygier and

Stedinger [1988])

Stedinger-Pe6Cohn model

[Stedinger etal., 1985]

Stedinger and Vogel [1984]

Full size (all lag zero covariances)

Reduced size (one lag zero covariance)

Full All-at-Once Disaggregation Models

kn(kn + 2n + 1)/2 90

kn(3kn + 2n + 1)/2 234

Condensed Disaggregation Models

kn(5n + 1)/2 - n 2 35

kn(7n + 1)/2- 3n 2 45

774 3,060 8,460

2,070 8,244 22,860

279 1,080 2,960

369 1,440 3,960

Staged Disaggregation Models

kn(n + 5)/2- n + 90 125 231 480 980

Dynamic Disaggregation Model (Based on a Sequential Markov Model)

kn(n + 3)/2 24 108 324

k(3n - 1) 24 96 204 780

348

direct Wilson-Hilferty transformation was again applied to

the data of series !, 2 and 4 and the calculated relevant

statistics are also shown in Table 2.

One can observe the generally good performance of DDM

in the preservation of the statistics of the actual variables.

The Valencia-Schaake model exhibits larger deviations from

the anticipated values (mainly in the variance and third

moment of X2) but this is not necessarily a major flaw of the

model, since it was not supposed to reproduce these param-

eters. Concerning the transformed variables, DDM does not

z

-• ½ o historical

•= q_ - - + s•mulate

z

• 0.6

• o.•

= 0.2

I I

0.0 0 N D J F M g H J g S

•O•TH

Fig. 2. Comparison of monthly skewness •d co.elation coef-

ficients of generated data: (a) skewness coefficient, applicgtio• 1,

location 3 (Morass mnoffi, (b) skewhess coefficient, application 2,

location 2 (Morass reservoir evaporation), (c) lag one autoco.ela-

tion coefficient of mon•ly v•ues for applicatio• 1, location 3

(•omos mnoffi.

exactly fit the theoretical values and it was not supposed to.

The transformed data of series 3 of the Valencia-Schaake

model are in agreement with the anticipated values, as was

expected. However, this agreement disappears when the

correction to regain the additive property is made as seen in

series 4. Obviously, the transformed corrected data are no

longer Gauss distributed (they have nonzero third moments)

and they do not have unit variance as they should. The bias

has been introduced by the variance of the adjusting factors

for regaining the additive property.

Preservation of Correlations

Figure 2c, originating from the case study discussed in the

previous subsection, indicates the preservation of the lag

one correlation coefficients achieved by the model. A similar

level of performance was found for the preservation for the

concurrent correlation coefficients for different sites. Note in

Figure 2c that the good performance of the model is ex-

tended also to the correlation of the first monthly runoff of

the current period and the last monthly runoff of the past

period. A more specific view of this topic is obtained from

- + historical ,,

..... simula(ed ,,'/

- gamma diskr•bukion

_ - -- gauss dis(rlbu(ion ,.///......

_

_ +

_

_

0.05 0.2 1 2 5 tO •0 30 40 58 BO 70 80 90 9S 98 99.5 99.9 99.99

DISTRIBUTION FUNCTION F (X)

Fig. 3. Comparison of empirical and theoretical distribution

functions of monthly values for application 1, location 3, subperiod

2 (Mornos runoff in November). The two-parameter gamma distri-

bution was adopted for the simulation.

3186 KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES

TABLE 2. An Example for the Comparison of the Dynamic Disaggregation Model (DDM) to the

Valencia-Schaake Model (VSM) for Non-Gaussian Variables

Simulated Values From

DDM/Quadratic Partition

Procedure Simulated Values From

VSM/Wilson-Hilferty

Gamma Gauss Corrected

Theoretical Cumulants Cumulants Raw Data Data

Statistic Values (Series 1) (Series 2) (Series 3) (Series 4)

Actual Variables

E[Z] 3.000 3.067 3.069 3.031 3.034

E[X 1 ] 1.000 1.024 1.024 1.006 1.006

E[X 2 ] 2.000 2.043 2.045 2.019 2.028

Var [Z] 8.000 8.242 7.998 8.508 8.489

Var [X 1 ] 1.000 1.053 ! .023 0.997 0.958

Var [X 2] 4.000 4.037 3.939 4.351 4.427

/23 [Z] 40.503 44.910 41.890 65.665 65.667

/x3[X• ] 2.000 2.311 2.326 2.061 1.807

/x3[X • ] 16.000 16.417 14.744 31.148 29.723

Con' [X 1 , X 2 ] 0.750 0.765 0.756 0.753 0.754

Corr [X•, Z] 0.884 0.893 0.888 0.853 0.880

Con' [X 2, Z] 0.972 0.973 0.972 0.978 0.975

Transformed Variables (Wilson-Hilferty)

E[Z] 0.000 0.053 0.057 0.010 0.023

E[X • ] 0.000 0.060 0.073 0.008 -0.030

E[X2] 0.000 0.030 0.019 0.010 -0.051

Var [Z] 1.000 0.882 0.873 1.004 0.948

Var IX • ] 1.000 0.866 0.827 0.996 1.180

Var [X 2] 1.000 0.984 1.036 0.998 1.269

/-•3 [Z] 0.000 0.271 0.237 O. 045 O. 191

/2, 3 [X' 1 ] O. 000 0.299 0.287 O. 019 --0.505

/x 3[X2] 0.000 0.005 --0.114 0.071 --0.486

Con' [X • , X 2] 0.782* 0.797 0.777 0.783 0.781

Corr [X 1 , Z] 0.872* 0.881 0.866 0.873 0.892

Con' [X 2, Z] 0.981' 0.982 0.980 0.981 0.956

The Valencia Schaake model uses the Wilson-Hilferty transformation and is combined with a

correction procedure for regaining the additive property. The size of all synthetic series is 16,000.

*Estimations obtained by simulation.

Table 3, which refers to all runoff and evaporation series of

applications 1 and 2 (the rainfall autocorrelation coefficients

were negligible). Recall from section 4 that to achieve these

results no special treatment is required, since the model is

applied without any modification for the first lower-level

variable X• using as previous information (Xj ø = x?) the last

lower-level variable of the previous period.

From a practical point of view the model performance

concerning the preservation of the correlations between the

consecutive lower-level variables is sufficient. However, theo-

retically we cannot speak about exact preservation, but just an

TABLE 3. Correlation Coefficients of the Lower-Level

Variables Between the First Subperiod of the Present

Period and the Last Subperiod of the Previous Period,

for Applications 1 and 2

Distribution Historical Generated

Variable/Site Function Data Data

Application 1

Runoff/Evinos gamma 0.32 0.29

Runoff/Mornos gamma 0.16 0.09

Runoff/Yliki gamma 0.59 0.54

Application 2

Evaporation/Evinos Gauss 0.75 0.76

Evaporation/Mornos Gauss 0.02 0.03

Evaporation/Yliki Gauss 0.07 0.07

approximation. A systematic investigation of the model's be-

havior with regard to this is depicted in Figure 4, summarizing

the results of a series of simulations which concern the single-

site disaggregation of a higher-level variable into two compo-

nents. The correlation coefficient between consecutive lower-

level variables of consecutive periods ranges between -1 and

+ 1. The results of this figure are explained below.

Stedinger and Vogel [1984] showed that disaggregation

models do not preserve the correlation between the lower-

level variables of the previous period and the higher-level

variables of the current period, or equivalently and following

the notation of the present study, the quantities Corr [X t,

Z2], t = 1,.--, k. Since such quantities are used as

parameters in the Mejia-Rousselle model this model is totally

affected by this shortcoming. The original Valencia-Schaake

model does not use these quantities, but it does not repro-

duce at all the correlation between consecutive lower-level

variables of consecutive periods. The dynamic disaggrega-

tion model, like the Valencia-Schaake model, does not use

these quantities as parameters, but it performs well in

reproducing the latter correlation. However, its behavior is

similar to that of the Mejia-Rousselle model. In the examples

of Figure 4 the two models gave exactly the same curves

which are not distinguishable from each other (dashed lines

represent curves of both models). One can observe in Figure

4 significant deviations of the simulated Corr IX t, Z 2 ] from

their anticipated values. The deviations of Corr [X • , X 2 ] are

KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES 3187

1.0

0.8

0.6

0.2

0.0

-0.2

-0.q

-- OorrEX•,Z 2] - t:heoret:ical

-- OorrEX2,Z2'l - •heore•[cal

+- + OorrEXt,Z23 - s[mulat:ed

• - e CorrEX 2, Z23 - s i mu 1 a ted

.0 4.8 -0.6 -O,q -0,2 -0.0 0.2 O.tt 0.6 0.8 1.0

CORRELClTION: al=œorrEX ø,

'• o.8

• o.• /'

• -0.0

c• •orr X Xm3- •heore•ca]

-0.2 [C -- EorrEX • Z] -

-0.• W• • • CorrEXø,X•3 -

/• • -x CorrEX •, X•3 - simulated /

-0.6 •/ •--e Corr[X•,Z• - s•mula•ed /

-l.0 -1,0 -0.8 4.8 -0,4 -0.2 -0.0 0.2 0.• 0.6 0.8 .0

CORRELRT ION:

Fi•. 4. Illustration scheme for the pc•o•ancc of •hc PA•(1)

model configuration wkh re•ard to •he prostration of the co•ela-

fion o• (•) •he lowcr-lewl variables of •hc cu•cm period with the

•h•r-levcl variables of •hc ncx• period, (8) •he lower-!•wl and

hi,her-level variables of •h• cu•en• period and (c) consecutive

lower-level variables of the same or consecutive periods. The cu•cs

o• •he •jia-Rouss•ll• mod•l ar• •xactly th• sam• as •hos• of DD•

(dashed lin•s). The assumptions for the const•cfion of •hc scheme

• (1) disa•rc&afion of a hi•her-level variable into two compo-

n•ms, • and • , (Z) Var [• ] = Var [• ] = 1 and (3) Co• [•0,

•] = Co• [•, •z]. The siz• of •hc synthetic series is 16,000.

negligible but there is a small difference in Corr [X ø, X1],

which is maximum near _+0.5.

Consequently, the failure in the reproduction of the quan-

tities Corr [X t, Z 2] affects to a degree the entire correlation

structure of the model, regardless of their presence in the

model formulation. But what is the reason of this influence,

if it is not the presence of quantities like Corr [X t, Z 2] in the

model formulation? The answer may be expressed as fol-

lows: Disaggregation models are techniques for "thicken-

ing" sample functions of stochastic processes, already

known in lower resolution (determined by the higher-level

time step). The original Valencia-Schaake model does not

see outside of each higher-level step, and thus it is known to

be weak in the reproduction of the autocorrelation. The

Mejia-Rousselle model and the above configuration of DDM

both have a way to see to the past, but not to the future;

hence they exhibit a similar behavior. However, the future is

known at the higher-level resolution; thus a better configu-

ration capable of seeing in both directions is possible. This is

attempted in the configuration of the next section.

7. PARX(1) CONFIGURATION

Retaining the Markovian structure of the sequence of

lower-level variables (equations (43)-(47)), we try to modify

the parameter determination procedure in order to take

account of the higher-level variables of the next period, Z 2.

This may be based on the following equation where the Z 2 is

an exogenous input:

X t = ftxt-1 + gtZ2 + htQ t (62)

or, equivalently

J

xJ =f•xJ-' + gJzf + • hJ•Qr t j = 1,'", n (63)

r-1

where ft and gt are assumed (n x n) diagonal, i.e., ft __ diag

(f[,f•, ... ,fnt), gt = diag (g•, g•, --- , g•t), whereas h t =

[h/•.] is a (n x n) lower triangular matrix of coefficients. Qt

= [Qj] is a vector of n random variables independent in time

and location and also independent of Zf. We denote gt =

E[Qt], •1 t - p,3[Qt], and set (for mathematical conve-

nience) Var [Qt] = 1.

The parameters ft, gt and h t can be computed from the

parameter set of the associated sequential PAR(i) model,

without introducing any other independent parameter, by

the following equations derived in Appendix C:

O'jj lf jt q_ Or j-1 t-10•; _ t-1 t- cr:j ß aJcrjj (64)

a j-1 t- t t crjj lfj + vjigj = ajo-jj (65)

ht(ht) T= o 't- fto't-lat-- grotto 't (66)

where

a t=diag(a•, a•,---, a• t) = a •---a 1 a •---a t+•

/=1 (67)

(68)

•, =Cov [z, z]

with {r t and a t defined by (48) and (45), respectively.

In particular, (64) and (65) when solved simultaneously

give j)t and #J, while h t can be derived by using (66) as the

lower triangular matrix.

The remaining quantities for the complete evaluation of

the model are [t and •l t. The former can be easily deter-

mined by taking expected values in (62). The computation of

the latter is complicated, since third-order joint moments of

Z 2 and X t-1 are needed; this can be carried out in a way

analogous to that used for the derivation of similar second-

order moments in Appendix C. Then the following equation

resulting from (63) may be utilized:

--(,qJ)3•3[Zj2 ] - 3(L.t) 2,qj/_6 l[/J -1 Z] 2]

2 ,

j-1

-- 3fjt(gJ )2• 12[/• -1' ZJ 2] - E (hJr)3r/•}

r:l

(69)

3188 KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES

In a way similar to that used for the PAR(l) configuration

we can proceed to the equations that form the basis for the

moments determination in each step. Here the diagonal

matrix e t is defined by eJ = h•. and the lower triangular

matrix d t (with zeros on the diagonal, similar to (49)) by

d t= I - et(ht) -1 (70)

By means of dt and et, (62) becomes

X t = ftxt-1 + gtZ2 + dt(X t- ftxt-1 _ gtZ2 ) + etQ t

(71)

This form is similar to (50) and again results in (equation (52))

xJ= rJ + + w;

where the TJ, UJ and WJ are now defined in a different way,

i.e.,

rj = t t-1 4.Xj + g;Zj 2 (72)

j-1

UJ = • dJr ( Xrt - f •Xr t - ' - g •Zr 2 )

r=l

(73)

W j_ t t

eiQ s (74)

By using (52) and (72)-(74), SJ can be expressed by an

equation analogous to (56)'

k k k

SJ = .et t ,rr t-1 r

Jj•j.A.j + E _.r r•2

r=t r=t r=t (75)

where

k

ß r;--l+ Y. fjr+lfjr+2'''4q

u=r+l

(76)

Equations (52) and (75) may be applied for the calculation of

conditional marginal and joint moments of (XJ, S J) and a

systematic algorithm similar to the one given in Appendix B

for PAR(l) configuration can be constructed from those

equations.

A minor problem related to the above configuration is the

existence of the term aj ø in (64) when it is applied for t = 1.

This term represents the covariance of the lower-level vari-

able Xj ø, located in the previous period, with the higher-level

variable Zj 2 of the next period. Since the model preserves

correlations only in consecutive periods, af will not be

preserved. However, the effect of this disparity is not

important. Practical solutions to overcome this problem

include the following: (1) Consider a ø = 0 for the typical

range of the correlation of higher level variables. (2) For

significantly high values of this correlation calculate a 0 from

(67). (3) An approximation making use of an average et 'ø

instead of a0 'gave good results in the total range of the

correlation of higher-level variables; a 'ø is defined by

0

o•,o = 1

-- Ot r (77)

k r=l -k

1.0

0.8

0.6

0.2

0.0

-0.2

-0.q

t.0

'E

•, o.•

• o.2

-- CorrEXt,Z•'l - theorelical

-- œorrEX•,Z•3 - •heore•ica[

• + [orrEXt,Z• - simulated

o - m CorrEX •, Z •] - s i mu 1 a led

, I , I , I , I ,

-1.0 -0.8 -0.6 -O.q -0.2 -0.0 0.2 O.q 0.6 0.8 1.0

CORRELATION: a•=Corr[X ø, X t]

/ CorrEX•,Z] - theoretical

F / +- • Cørr[X I' Xt, ]-s imula led

I/ ,•__,, Corr[X•,Z] - simulated

-0,2

-0,4

-0,6

-0.8

-t.0 -1,0 -0.8 -0.6 -0,4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0

CORRELATION: a 1= Corr[Xø,X t]

Fig. 5. Illustration scheme for the performance of the PARX(1)

model configuration with regard to the preservation of the correla-

tion of (a) the lower-level variables of the current period with the

higher-level variables of the next period, (b) the lower-level and

higher-level variables of the current period and (c) consecutive

lower-level variables of the same or consecutive periods. The

assumptions are similar to those of Figure 4.

where ot r is calculated from (67), with the observation that

am-k __ am.

The PARX(1) configuration is more complicated than the

PAR(l), particularly in the handling of the skewness, though

they have a common background. However, it performs

pretty well with all kinds of lagged covariances of higher-

and lower-level variables, as indicated in Figure 5 (similar to

Figure 4), which refers to the numerical investigation of the

previous section and was obtained by the use of the

PARX(1) configuration. Note that the developed method

does not require any assumption about the model used for

the generation of the sequence of higher-level variables. In

that respect it is structurally different from a recent method

by Lin [1990] which depends on the specific generating

model of the higher-level variables and aims toward the

modification of the Mejia-Rousselle parameter estimators for

preserving the covariance properties of lower-level vari-

ables.

KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES 3189

8. CONCLUSIONS

Based on a nonlinear partition procedure, linked with an

appropriate implementation of the sequential Markov model,

a multivariate dynamic disaggregation model was developed.

Important features of the model are (1) the assurance of the

preservation of the additive property, (2) the stepwise ap-

proach, (3) the modular structure (composed of two parts

studied separately), and (4) the explicit analytical re!ations

composing its structure.

The mode] is parameter parsimonious, since the minimum

essential set of statistics of the lower-level variables is

maintained (first-, second-, and third-order marginal mo-

ments, lag zero cross-correlation coefficients and lag one

autocorrelation coefficients). The model parameters are es-

timated directly from the historical data with the usual

statistical methods of the sequential Markov mode!.

Various configurations of the model, resulting from the

use of either a different partition procedure, a different

moments determination procedure or a different stochastic

structure of the lower-level variables, are possible. Two

configurations of the mode] are studied, both based on a

Markovian structure of the lower-level variables. The first

(PAR(I)) configuration implements the associated Markov

model in its initial form, while the second (PARX(1)) uses

the known higher-level variable of the next period as an

exogenous input at the current period. The PAR(l) configu-

ration is simpler than the PARX(!), particular!y in handling

the third-order moments, and gives good approximations of

the correlations of lower-level variables located at consecu-

tive higher-level time steps. However, the PARX(1) config-

uration exhibits better behavior with regard to the correla-

tion properties of lagged lower- and higher-level variables.

Obviously, the model has a strong restriction concerning

the Markovian structure assumed for the lower-level series;

this is the price paid for obtaining the parsimony of param-

eters. However, this restriction may be not so important,

given that the errors are not accumulated, since the higher-

level series are generated independently. With the exception

of special cases, the model, similar to other disaggregation

models, is not exact in a strict statistical sense but gives good

approximations of the important statistics of interest.

APPENDIX A: DERIVATION OF PARAMETERS IN THE

QUADRATIC PARTITION PROCEDURE

By expressing the right-hand sides of (28), (29) and (31) in

terms of the a i and then solving the system, we get

(A 4-- A22)g!l-- A3g12

_ 2 (A1)

al= (x-x)x2 x3

A2•12-- •-3•11

_ 2 (A2)

a2 = ('•.4 -- '•.22)X2 A3

a0 = -a2A2 (A3)

The equations for the b i, determining f($), are obtained

from (30) and (32), after a number of operations. In the given

set of equations, the one degree of freedom has been given to

b2, as explained further below.

b0 = P

fl•A n + 2•1•2A3 + (• + 2•2)A 2 + 1 (A4)

(A5)

b 1 = f{ lbo

b2 = fl2b 0 (A6)

where

/92=0 r0>-0

/•2--'-- (--rl + A21/2)/7'2, 'r0 < 0, r, >-- 0 (A7)

• 1/2x•

f•2 = (--7'1 -- /'X2 J/r2, 7'0 < 0, r 1 < 0

--(A 4 -- qA3)• 2 -- A 2 ----- A•/2

/•1 = A3 -- qA2 • 0

A3 - qA2 (AS)

2

1 q- (A•- qX4)/•2

=- , A3 - qA2 =0

/31 2 A2 + (An- qA3)/•2

2

r 0 = (A 3 - qA 2)q + A 2 (A9)

rl = (An- qA3)A2 - (A3 - qA2) 2 (A10)

r2=(An-qA3) 2-(A3-qA2)(AS-qAn) (All)

A2 r 2

= 1 - for2 (A12)

A1 = •0 + 2•fl2 + r2fl• (A13)

q = [;2• - a•A5 - 2a•a2(A4 - A•) - (a• - 2a•A2)A3]/p

(A4)

p = n2- - - (Al)

The meaning of (A7) is that f12 (and hence b2) is set to zero

ff possible (that is, ff it does not result in nonreal solutions for

b l) thus downgrading the quadratic equation to a linear one.

Othe•ise, it takes the absolutely minimum value which

assures A 1 m 0 and hence a reM solution of b l-

It is noted that (A8) normMly yields two d•erent vMues of

ill, and consequently two couples (b o, b 1), both v•id; it

was assumed that the one resulting in the smMlest absolute

value of skewness 03 is finally selected. The vMue of 03 is

obtained from (33) which may be w•tten in the fo•

03 = [•3- •(a, a, a) - 3•(a, b, b)]/•(b, b, b) (A16)

where the symbol • , , ) is an abbreviation for the

foRowing expression, where k, l, m denote t•plets of

parameters such as k = (ko, k l, k2) , etc.'

•(k, l, m) = kolomo + (kolom 2 + kolim • + kol2m 0

+ kilom • + kilim 0 + k210mo)A 2 + (kolim 2 + kol2m•

+ kilore2 + kllimi + kll2mo + k2loml + k21imo)A 3

+ (kol2m2 + kilim2 + kil2mi + k210m2 + k2liml

+ k212mo)An + (kll2m2 + k2lim2 + k212mi)A5

+ k212m 2A • (A17)

It can be proved that

p = •[X 2] - •[a2(S)] (A•8)

and because of (30) it should be the case that

3190 KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES

p > 0 (A19)

It is possible that certain combinations of the initial param-

eters listed in (8) yield values of the a i not obeying (A 19). In

that case, though the at can be determined, g(S) as defined

with (25) and (26) does not exist; thus (A19) is a necessary

condition for the existence of g(S). A similar necessary

condition for the existence off(S) is

/•22%4 + 2/• 1/•2% 3 + (]•12 + 2]•2)% 2 + 1 >0 (A20)

which follows from (A4). Further, the existence of f(S)

demands that

r 0>-0 or A 2>0 (A21)

which is related to (A7).

If the variables X and Y (or, equivalently, X and S) are

jointly normal, then the quadratic scheme downgrades to the

linear one and (25)--(27) reduce to (9). Indeed, the above set

of equations reduce to

a 0 = a 2 = b 1 = b 2 = 0 (A22)

a• = •rll/A 2 (A23)

bo = (r/2 - •r•21/A 2)1/2 (A24)

as was theoretically anticipated. This is also true in any case

where A3 = •q2 = •r21 = 0.

APPENDIX B: BASIC EQUATIONS

FOR THE PAR(l) ALGORITHM

The stepwise algorithm involves the computation of the

conditional moments listed in (42) that are required for the

partition procedure. The computation is based on the follow-

ing relations, obtained from (52) and (56). The proofs are

given below at the end of this appendix.

aj•rjxj + rj (B1)

Var [SJ nJ] : ½J (B2)

EtXylnj] = ' '-'

ajx) + 3 d (B4)

Var ' ' =(e•)Z

x;Inj = (e J) ' (B6)

Sj nj]: •j(eJ) 2 (BY)

y• (B8)

where

j-!

SJ:E[U]InJ]+E[W]]: ' ' '

djr(X r - arX r ) q-

r=l (BlO)

and the following symbols are defined by the backward

recurslye relations'

t j+l t+l k+l

,r i = I + a •j qrj = 0 (Bll)

t t t •.;+1 k+l

q'j -- •Jj7Yj q- ß q'j : 0 (B12)

' •J) 2(½j) 2 t+l k+l (pj -- ( q- (pj (pj = 0 (B13)

*;: (7rd)3(d) 3 T; q- *J+l ½f+1 _ 0 (B14)

Note that (B11) is equivalent to (57), but simpler for pur-

poses of application.

Proof of(BlI). From (57) we have •rJ = I + aJ +• +

+la.t+2 + .--+ at+la t+2 .-. a .• = 1 + aJ+•(1 aJ +2

i

aJ . .a+ a j+2 ' af)= 1+ aJ +l•jt+l +

+ .... . Also, from (57),

= 1. This is compatible with the backward recursive relation

if we set •rf +• = 0.

Proof of(B4). From (52)we have E[XJIgl j] = E[UI j]

+ E[UJ glJ] + E[WJ]. Recall that U and Uj are com-

pletely known because they are expressions of variables

contained in glj. From (53)we get E[ U glJ] = aJxJ -• , from

• j-1 t t t t-1

(54) E[ujInJ] - Y-r=l djr(Xr -- arXr ) and from (55)

E[WJ] = eJl3J. Hence, if we introduce 3J defined by (B10)

we get (B4).

Proof of(B1) and (B•2). r From (56) we have E[SJlnJ] =

aJqrJxJ -1 q- WrZrk=t q'Q(Vj q- Wi) •'•J] '- __t t_ ,-1 - x•k

aj vrj xj -r

•rf 3f. Note that, because of the horizontal course followed,

all Uf for r > t are known since they are expressions of

lower-level variables of previous sites. Also, note that E[(UJ

+ wJ)nj] = E[uJlnJ] + E[wJ] = aJ. Thus, if we denote

•'•k _rer

r=t 7rj oj '- TJ we get (B1). Then it is obvious that this

definition of rJ leads to the backward recursive relation

(B12).

Proof of(B5) and (B6). By abstracting conditional means

from (52) (given ll]) and then squaring and taking condi-

tional expected values we get Var [xJlJ] = Var [wj] =

(eJ) 2 Var [V]] (because of (55)) and since Var [VJ] = 1,

(BS) has been proved. Note that TJ and uJ do not contribute

to the conditional variance because they are known. The

proof of (BG) is quite similar.

Proof of (B2), (B13), (B3) and (B14). By abstracting

conditional means from (56) (given glJ) and then squaring

and taking conditional expected values we get Var [sJ

k Var [Wf] : •.Lt (rrf)2(ef) 2 If we denote

-' •'r=t (qTf) 2 ß

the last sum by •pJ then it is easily shown that •pJ is also given

by the recursire relation (B!3). Note that XJ -• and Uf in

(56) do not contribute to the conditional variance because

they are known. Also recall that Wf, r = t, .'- , k, are

independent. The proof of (B3) and (B14) is quite similar.

Proof of (B7), (B8) and (B9). By abstracting conditional

means from each of (52) and (56) (given ftj) and then

multiplying them and taking conditional expected values we

get Cov [xy, SJlnJ] = =y Vat [wJ] = =y(,y)2. The proof

of (B8) and (B9) is similar.

APPENDIX C, DERIVATION OF EQUATIONS

OF THE PARX(1) CONFIGURATION

By abstracting means from (63) and then multiplying

successively by (XJ-' - se] -1) and (Zj 2 - E[ZJ]) we obtain

respectively

Var [Xj-1]fJ +Cov [Zf, xJ-]gJ =Cov [xj, xj

(C1)

KOUTSOYIANNIS: SIMULATION OF HYDROLOGIC SERIES 3191

Coy [Z•, xJ-•]fJ + var [z•]gJ-- Coy [zj 2, (c2)

By abstracting means from (62), then postmultiplying by (X t

_ •t)r, taking expected values, and also considering that

Coy [Qt, X t] = (ht)T, we obtain

ht(ht) T= Cov IX t, X t] - ft Coy IX t-1 X t]

_ gt COV [Z 2, X t] (C3)

The term Coy [X t-• , X t] in (C3) (also present in (C1))

equals {r t-• a t as is easily derived from (43), which is still

valid. The presence of terms like Cov [Z 2, X t] in (C1)-(C3)

does not append any parameter to the set discussed in

section 4. Indeed, the'covariance matrix of Z 2 = X k+•

+ ... + X 2k and X t can be determined by iterative use of

(43) in terms ofa • l= 1 .-. k and{r t(notethata t+•-

at), since it can be shown that

COV [Z 2, X t] -' (a 1''' a 1) a k''' at+llY t (C4)

t=l

or, with the use of •x t defined in (67)

COV [Z 2 X t] -- otttl t (C5)

By making notational simplifications to (C!), (C2), (C3),

with the use of (48), (68), and (C5), we get (64)-(66).

Acknowledgments. The research leading to this paper was per-

formed within the frame of the project Appraisal of Existing

Potential for Improving the Water Supply of Greater Athens, project

8576710, sponsored by the Greek Ministry of Environment, Plan-

ning and Public Works. The scientific director of the project, Th.

Xanthopoulos, and the members of the research team who assisted

in the preparation of the data are gratefully acknowledged. The

author wishes to thank the anonymous reviewers and the Associate

Editor and the Editor for their comments and their detailed reviews,

as well as E. Foufoula-Georgiou for the comments and the help she

provided.

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D. Koutsoyiannis, Division of Water Resources, Department of

Civil Engineering, National Technical University of Athens, 5 Iroon

Polytechniou, GR-15700 Zografou, Greece.

(Received April 20, 1992;

revised May 20, 1992;

accepted June 2, 1992.)