USE OF ESTIMATION TECHNIQUES FOR FLOOD FORECASTING

N.S. Hsu W. W-G. Yeh

B.J. Williams M.K. Stenstrom

Civil Engineering DepartmentSchool of Engineering and Applied Science

University of CaliforniaLos Angeles, California

CALIFORNIA WATER RESOURCES CENTER

University of California Davis, California 95616

The research leading to this report was supported by the University of California, Water ResourcesCenter, as part of Water Resources Center Project UCAL-WRC-W-605.

Contribution No. 192 February 1985ISSN 0575-4941

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS

ABSTRACT

LIST OF FIGURES

LIST OF TABLES vi

INTRODUCTION 1

PROBLEM FORMULATION 3

MODEL DESCRIPTION AND STATE-SPACE FORMULATION OFCONCEPTUAL MODELS 6

Model Description 6

State-Space Formulation 13

SOLUTION TECHNIQUE

NUMERICAL EXAMPLES 20

Preliminary Procedures 20

Application to Hypothetical Catchment 22

Application to Real Catchment 39

CONCLUSIONS 45

REFERENCES 46

LIST OF SYMBOLS 49

ACKNOWLEDGEMENT

The research leading to this report was supported by the University of

California, Water Resources Center, as part of Water Resoruces Center Project

UCAL-WRC-W-605. The encouragement and continued support provided by Maurice

Roos and Robert Zettlemoyer of the Department of Water Resources are greatly

appreciated. Dr. B.J. Williams participated in the research project while

spending his sabbatical leave at UCLA from the University of Newcastle,

Australia.

ii

ABSTRACT

An Iterated Extended Kalman Filter (IEKF) is implemented for parameter

estimation and flood forecasting. The filter is applied to a conceptual

model. The conceptual model selected for study is a generalized one

dimensional kinematic wave model of a watershed which incorporates storages in

both surface water and groundwater. Linearization of the nonlinear conceptual

model is based upon the Influence Coefficient Method of Becker and Yeh (1972)

which performs linearization externally without changing the structure of the

conceptual model. Four parameters in the conceptual model are recursively

estimated.

A set of numerical experiments are conducted using synthetic, as well as

real, watershed data. Sensitivity and correlation analyses of the estimated

parameters are carried out. The research results indicate that conceptual

models with fewer physically based parameters, such as the kinematic wave

model, are feasible for implementations with IEKF for on—line parameter esti-

mation and flood forecasting.

iii

LIST OF FIGURES

Page

Fig. 1 Definition Sketch 8

Fig. 2 Discretization of x—t Plane 10

Fig. 3 Hypothetical Watershed 23

Fig. 4 Input Hyetograph and True Hydrograph for theHypothetical Catchment 25

Fig. 5a Correlogram of the Residuals at Downstream Stationof the Hypothetical Catchment — Case 1 31

Fig. 5b Correlogram of the Residuals at Downstream Stationof the Hypothetical Catchment — Case 2-1 32

Fig. 5c Correlogram of the Residuals at Downstream Stationof the Hypothetical Catchment — Case 2-2 33

Fig. 5d Correlogram of the Residuals at Downstream Stationof the Hypothetical Catchment — Case 3 34

Fig. 6a Observed, One—Step—Ahead—Forecasted, and EstimatedHydrographs of the Conceptual Model — Case 1 35

Fig. 6b Observed, One—Step—Ahead—Forecasted, and EstimatedHydrographs of the Conceptual Model — Case 2-1 36

Fig. 6c Observed, One—Step—Ahead—Forecasted, and EstimatedHydrographs of the Conceptual Model — Case 2-2 37

Fig. 6d Observed, One—Step—Ahead—Forecasted, and EstimatedHydrographs of the Conceptual Model — Case 3 38

Fig. 7 Catchment Details of Williams River 40

Fig. 8 Observed (3/1976), One—Step—Ahead—Forecasted, and EstimatedHydrographs of Williams River from the Conceptual Model 41

iv

LIST OF TABLES

Page

Table 1 Parameter Values Used for the Hypothetical Catchment 24

Table 2 Covariance Matrix of the True Parameter Values for theHypothetical Catchment 27

Table 3 Correlation Matrix of the True Parameter Values for theHypothetical Catchment 28

Table 4 Parameter Estimations on the Hypothetical Catchment 29

Table 5 Lag 1 Cross Correlation Matrix of the Residuals forCase 2-1 30

Table 6 Covariance Matrix of Estimated Parameters 42

Table 7 Correlation Matrix of Estimated Parameters 43

Table 8 Parameter Estimation on Real Catchment 44

INTRODUCTION

With the advent of mini—computers, the preparation of flood forecasts is

shifting from predominantly manual methods to computerized procedures with

automatically selected data from real—time, telemetry reporting systems.

These data are used as inputs to a rainfall—runoff model, ranging from complex

soil moisture accounting models to time—series types of models, for making

real—time flood forecasts. The parameters imbedded in the model are time—

varying during a flood situation. Estimation techniques, such as the Kalman

filter developed in optimal control, have been used for estimating the states

(floods) as well as the time—varying parameters in a rainfall—runoff process.

Most of the published results deal with linear systems where the dynam-

ics are governed by linear and discrete state equations. A closed—form

representation of the state equation is essential in the application of the

Kalman Filter. Time—series types of models have been used to represent the

rainfall—runoff process, and the Kalman Filter has been applied to time—series

models for parameter estimation and flood forecasting, such as the work of

Todini and Bouillot (1975), Szollosi—Nagy (1976), Szollosi —Nagy et al. (1977).

Wood and Szollosi—Nagy (1978), Todini and Wallis (1978), and others.

Application of the Kalman Filter to nonlinear, conceptual watershed

models for parameter estimation and flood forecasting has been sparse.

Kitanidis and Bras (1980a) and Kitanidis and Bras (1980b) used a linearization

technique to effectively linearize the internal structure of the Sacramento

model (Burnash, et al., 1973), and the Extended Kalman Filter was used for

making real—time flood forecasts. Georgakakos and Bras (1982) also utilized a

statistical linearization technique to internally linearize a nonlinear

1

kinematic wave model (Mein et al., 1974). A similar algorithm was then used

to make real—time flood forecasts. Moore and Weiss (1980) proposed a simple

nonlinear rainfall—runoff model and formulated the model in a different

state—space form, that is, the parameters constitute the state vector and the

dynamics of the system behavior are represented by the measurement equation.

The Extended Kalman Filter was then applied to the linearized measurement

equation to obtain the parameter estimates recursively. The flood forecasts

are computed in turn based on those parameters.

Some comparisons have been made of the performance of flood forecasting

models. Bolzern, et al., (1980) compared the flood forecast performances of

an ARMAX model with Kalman Filter of differing complexities. O'Connel and

Clarke (1982) made comparisons among four flood forecasting models, ranging

from conceptual hydrological models that use unsophisticated parameter estima-

tion procedures to simple time—series types of models that use relatively

sophisticated state filtering and parameter estimation procedures.

The objectives of this research are: (1) to develop an algorithm to

perform on—line parameter estimation and on—line flood forecasting by applying

the Iterated Extended Kalman Filter to conceptual models (e.g., Field and Wil-

liams, 1983), (2) to use the Influence Coefficient Method of Becker and Yeh

(1972) to linearize externally the nonlinear structure of conceptual models

without changing the structure of the models, and (3) to demonstrate the use-

fulness of the proposed algorithm.

2

PROBLEM FORMULATION

It has been shown by Szollosi—Nagy (1976) that a general state—space

representation of the following form can be used to describe the behavior of a

water resource system:

Xk+1

= 4(1Ck, U

k, W

k)

(1)

Zk = m(Xk , Vk)(2)

where "lc = the nx—vector of the states of the

system at discrete time k, k=0, 1, 2, ...,

U1 = the nu—vector of the deterministic inputs,

Wk

= the nw—vector of the system noise,

Zk = the n —vector of the measurements on theZ

system at discrete time k, k=1, 2, 3, ...,

Vk

= the nv—vector of the measurement noise,

4, n = certain functionals characterizing the system.

In modeling the rainfall—runoff processes for a watershed in state—space

formulation, such as the work of Toyoda, et al. (1960), Todini and Bouillot

(1975), Szollosi—Nagy, et al. (1977), Wood and Szollosi—Nagy (1978), and

Kitanidis and Bras (1980a), Eqs. (1) and (2) can be rewritten as:

Xk+1 = F(%ki U1 ) + r(%k )Wk

(3)

7.1 = H(Xk) + Vk(4)

3

where

Wk • zero mean, independent white Gaussian process,

i.e., WkN(O,Qk) and E[Wk ,W i] = 0 for k#i,

Qk • E[Wk,Wkl, covariance matrix of W k'

Vk • additive measurement error which is assumed to

be zero mean, independent white Gaussian processes,

i.e.,VkN(0,Rk) and E[Ark,Vi] = 0 for k#i,

RkE[Vk,Vkl, covariance matrix of Vk .

k'k ir(Xk) and h(Xk) are vector functions which are assumed to be

continuous and differentiable. In this paper, F(Xk'k) corresponds to a

watershed simulation model. Furthermore, (Vk) and (Wk) are assumed to be

independent processes, i.e., E[Vk W i ] = 0 for all k and i.

The linear form of Eqs. (3) and (4), known as a Gauss—Markov model, can

be written as

Xk+1 = fak + G Uk + rwk (5)

Zk = HXk + Vi(6)

where I is a (nX x n

X) matrix,

G is a (nX x n ) matrix,

is a (nx x nW) matrix, and

• is a (nz x nX) matrix.

This model has been extensively proposed for flood forecasting (e.g., Todini

and Buillot, 1975, Szollosi—Nagy, et al., 1977, Wood and Szollosi—Nagy, 1978,

4

et al.)

Given the above—mentioned problem, the objective is to obtain the best

estimate of state Xk as we observe the output Zk , Zk_i , Z1 . Among all

possible criteria (Poljak and Tsypkin, 1980), let us choose X k , the estimate

of the state Xk

, as a function of the observations up to and including Zk,

such that

Xk = Min E[(Xk — X:) T (Xk — X:)/Ykl

(7)

where Yk

E

iT[Zi , Z2 , ....

any estimate of state X k ,

• the statistical expectation,

• conditioning notation, meaning that Yk is given.

It can be shown (Nahi, 1969) that such an estimate is

Xk = E[1(k/Yk]

(8)

The additional requirements for the best estimate are:

1. It must be observed sequentially. At each observation we want to update

the estimate without having to reprocess all the data from the beginning

of the observation, and

2. It must be computationally feasible; preferably, it should be implement—

able by a mini—computer or microprocessor.

5

MODEL DESCRIPTION AND STATE—SPACE FORMULATION OF CONCEPTUAL MODELS

Model Description

The conceptual model selected for study in this paper is a generalized

one—dimensional kinematic wave model of a catchment (Field and Williams, 1983)

which incorporates storages in both surface and groundwater. This model has

been presented by Field (1982) and Field and Williams (1983). The present

form of the model assumes that rainfall intensity varies with time, but not

spatially in the catchment. The excess rainfall is routed through a nonlinear

storage to provide a 'surface supply' and the infiltrated rainfall is routed

through a linear storage (with a considerably higher storage coefficient) to

provide a 'groundwater supply.' The two supplies together are routed down the

main channel using a kinematic wave equation (Field, 1982). The governing

equations which describe the above—mentioned processes are:

1. The continuity equation for flow through surface storage:

dh(p-0) - s =s dt

where p • rainfall intensity,

0 • infiltration index,

sssurface supply (discharge per unit area),

• volume per unit area,

• time,

2. The continuity equation for flow through groundwater storage:

6

(9)

• =g• dt

(1 0)

where 0 • infiltration index which is set to be equal to

p when 0 > p and s s = 0,

• groundwater supply (discharge per unit area),

3. The relationship between storage volume and discharge:

h=b sY*

(11)

where b and y are constants, and s * is either surface supply or ground-

water supply,

4. The relationship between b and storage coefficient Kc

b = Kc p1

ey

(12)

where

Pe is a characteristic rainfall intensity, which can be

taken as the average intensity for the storm duration, and

K0is either surface water storage coefficient, Ks

or groundwater storage coefficient, K 8 ,

5. The nonlinear form of the kinematic wave equation:

at 'B' gxBs= — (m+1) a m+1 tal 111+1 [PS. —

(13)

subject to the boundary condition q(0,t) = 0 and prescribed initial

values of q(x,0),

where q • discharge,

B • width of the elemental strip (Fig. 1),

7

(a)

0

L

Fig. 1. Definition Sketch

8

x and t

a and m

• the volumetric rate of total supply from surface

storage and groundwater per unit area

to channel flow,

• spatial and temporal variables,

• the dimensional coefficient and the exponent

respectively in the channel velocity—depth

relationship,

v = a ym

(14)

For convenience, the preceding equations are non—dimensionalized before

they are solved. The resulting equations are of identical form. The dimen-

sionless quantities are defined by Field and Williams (1983). Referring to

Fig. 2, letting xj = j As, where Ax is a difference step, tk = k At, writing

q(jAx, kAt) = q. a. = a(jAx), sk

= s (kAt), etc., and using the Lax—WendroffJ' J

solution technique (Kibler and Wolhiser, 1979), which employs the second—order

Taylor's expansion of the dependent variable q and the subsequent substitu-

tion of spatial derivatives for temporal derivatives using Eq. (13), the fol-

lowing recurrence formula results (Field and Williams, 1983)

= + 8 + e

where 8 = —1 x2 x3

1 _ mA

(1) a.m+1

B. m+1

xl = Atx

9

(15)

(16)

(17)

t •

•At

k+1k+1

qj

k •k

qj-1k

qi •

kqj+1

k-1..Ax...... -.11F... Ax--•-

j + 1j - 1-..-0 L

Fig. 2. Discretization of z—t Plane

k m+1m

2• = (q.)

• -k

2

q. - q. ($ + sk+1 )1+1 1-1 As B.2

(a. - a. )1 ,2 ,2 m f 1 8 = 112' ( (m+1)ai 2

(m+1)q.i

(qk q. ) (B - B. )j+1 1-1 j+1 1-1

m+1)( 2 (m+1)B. 2qJ

(B - B ) 1+1, ,-+ 1_1" ( k _ 2 k + k 4-5

x3 L qj+1 qj qj-1' 2 2

(18)

(19)

(20)

1k+2

(-33k + 4s

k+1 - s

2 Az B. 2)

At the catchment outlet (i.e., j = N), the term in the recurrence equa-

tion (20) is

(3aN - 4aN_1 + aN_2 )= 1-(a )

2 (r4 ) 2 ( • 1 ( 12 1 2 k (m+1)aN2

(m+1)qN4

(3qk - 4qk + qk )N-1 14-2 m (3BN - 4BN-1

+ BN-2

)

2 (m+1)BN2(m+1)qkN

11

(21)

(3BN - 4B + B ) k 1+1% . k N-1 N-2 (s +s 1/ -(2q - 5q + 4qN,2 - qN,3)

x4N N-1 2 2

A. 1 1._k , k+1

(x1 'a) 2 Ax BN

4 (x)2

4

11:21±21:.sk+2).)

where

2 k a k akk k+1%'N-2 (s + s

Nx4 - 2B

2

To be consistent with earlier developments, a second-order finite

difference scheme is required for the solution of the supply from surface

storage of Eq. (9). Replacing the derivative of the right-hand side of Eq.

(9), substituting for h from Eq. (11) and substituting for b from Eq.

(12) yields

3Ks (sk+1 ) y sk+1s (4(

s k)7 - (sk-109) (pk+1 pk) - 20 sk

At s At s(23)

The recurrence formula, Eq. (23), requires values of sk and sk-1 to generate

the solution for sk+1 .

The supply from groundwater storage is computed in the same way using

Eqs. (10), (11) and (12) with y = 1. The recurrence equation is then

3k(sk+1 )sk+1

4(sk) - sk-11 4. 20 sk

At g At g

Thus at any instant in time the model can predict the surface and groundwater

supplies and discharges at a predetermined number of increments (N) in the

main channel.

12

(22)

(24)

k+1cl*

k+1s s

sk

s

sk+1a

sI

- g

The parameters which determine the size of these supplies and discharges

are: (1) the time of concentration, t, which is used to non–dimensionalizec

the dimension 'time' of all quantities, (2) the surface water storage coeffi-

cient, Ks, (3) the groundwater storage coefficient, g , and (4) the infiltra-

g

tion index, p. The only other parameters in the model are the exponents in

the storage–discharge equations and the velocity–depth relationship, and these

parameters have been assumed fixed from physical considerations (Field and

Williams, 1983).

State–Space Formulation

In order to progress from one time step to the next, the model requires

information about the current and next rainfall intensity, the current level

of the two supplies and also their level at the preceding time step, in addi-

tion to N discharges in the main channel. Using the state–space formula -

tion, Eqs. (15) – (24) can be rearranged to form the following set of non-

linear equations with additive errors

f (qk• s

k sk-1 , sk , sk-1 )

1 sss gg

k k-1 k+1 kf2 (s s ' s s

' pk+l, p )

s s+W

k

(25)

f (s • sk-1 sk, s

k-1 )3 g g s s

sk

- g

13

k k kwhere q* [ (12 ,T

fl

f2'

f3

k k-1s , ss s

I k-1s , sg g

k+1 kP P

N—vector function,

scalar functions,

• discharge at time k at the jth spatial point,

surface water supply at time k and k-1

respectively,

• groundwater supply at time k and k-1 respectively,

and

rainfall intensity at time k+1 and

k respectively.

Corresponding to the system equation (3), the state vector and the input

vector can be defined specifically as:

k k k k k-1 k k-1 T

KIL = [qv q2 , qN, s s , s s , sg , sg

U = [pk+1

• p]T

k

The noise vector Wk is a (N+4)—vector which is subject to the same assump-

tions as that in Eq. (3) andr(xk) is assumed to be an identity matrix.

In on—line flood forecasting, the parameters of the system must also be

estimated during each time step. For this purpose, they can be added to the

state vector. The augmented state vector becomes

r k k k k k-1 k k-1 T= ql , q2 , qN, s s , s s , s g , sg tc, K s , Kg , 0]L

where tc

the time of concentration,

Ks

• the surface water storage coefficient,

14

(26)

(27)

(28)

Kg= the groundwater storage coefficient,

0

the infiltration index.

In this paper, the observations are discharges, which in fact are some

of the states containing measurement errors. Hence, the measurement equation

is of the form of Eq. (6), i.e.,

where H • observation matrix of dimension n x nZ X

with entries of values 0 and 1,

number of observation stations,

nx • number of states of augmented systems.

(6)

15

f l ood forects.+ 4 -- mw initial states, the

(30)

(31)

(32)

SOLUTION TECHNIQUE

The Iterated Extended Kalman Filter (e.g. Wishner, et al., 1969) is

chosen and implemented for state and parameter estimation because of its gen-

erality. Linearizing F(Xk' Uk) about a reference state vector K

k yields

*Xk+1 = Falk, Uk) + F (Kk,Uk)(Xk — + Wk

where F is the Jacobian matrix which can be computed by the technique of

influence coefficients of Becker and Yeh (1972). The reference state vector,

Kk, from which point the linear trajectory is projected, is updated itera-

tively at each time increment.

Let X* be

, the updated reference state vector. The Iterated—k/k+1

Extended Kalman Filter can be represented by a set of recursive equations as:

State Prediction

= F(X11/k+1' Uk) F

,

(X114

y

+1' Uk )(Xk/k ^k/k+1'

2. Covariance prediction

, , 4Pk+1/k F (K iU )P [F (X' U )] T +

k+1/k k/k+11 k k/k k/k+1' Uk)]

Kalman Aain

i T 113-1k+1 k+1/k

K HT (Hpi= p

16

(29)

4. Covariance update

P i=i

[I — Kk+1H] P

k+1/kk+1/k+1

5. State update

;i+1 Qi—k+1/k+1 = —k+1/k 4+1 [71+1 — H4+1/k3

%k/k+1 4. pi [F 1 uHT. %k/k k/k (%k/k+1' k

[Pi 1-1 f;i+1L'k+1/1" ' —k+1/k+1 X11+1/k)

where the superscript i represents the ith iteration, "^" represents esti-

mation and k-1/k represents the value at time k-1 based on the information

from the initial stage up to time k. Also, we will assume, for each time I,

i0—k+1/k = ik/k

;()—k+1/k+1 = Xk+1/k

Given the input, e.g., precipitation and the initial conditions

E [X]0/0

E[(%0— 110/0)(X0 — 10/0)] = P0/0

in which P0/0 is the covariance matrix of the error of the initial states, the

flood forecasting and parameter estimation can be carried out by repeated

17

(33)

(34)

(35)

(36)

(37)

(38)

(39)

applications of the eqs. (30)–(35).

Note that if i takes the value 0 the filter reduces to the Extended

Kalman Filter.

In the application of the IMF, the true value of the system noise

covariance Qk and the measurement noise covariance Rk are unknown. However,

these values can be updated sequentially, from arbitrary initial estimates,

with adaptive estimation algorithms. The measurement noise covariance can

be adaptively estimated based upon the estimated measurement error according

to the following approximation which is similar to that of Szollosi–Nagy

(1976)

i i T T iRk =

[(k -1 " k(\)k ) Pk/k–linik

where

= Z –-k k k/k-1

Similarly, the system noise covariance Q1 can be estimated from the following

approximation

Qk = [(k-1)Qk-1 + wk (w) T + Pi Pil/k

k/k k/k-1

where

wi = K iv ik I

In addition, Qk and Rk are assumed to be nonnegative definite. Hence, if they

are not, an arbitrary small number X can be artificially added to their diago-

18

(40)

(41)

(42)

(43)

nal entries, i.e.,

= Qk + XI

(44)

and

1211 = RI + I

(45)

such that Qk and Rk are nonnegative definite.

The adaptive procedures for Qk and Rk are continued until convergence is

reached.

19

NUMERICAL EXAMPLES

To test the proposed state—space formulation and estimation technique,

applications are made to hypothetical cases for the chosen conceptual model.

A real catchment is then used to demonstrate the performance of the algorithm.

Preliminary Procedures

In order to have a better performance of the Iterated Extended Kalman

Filter, several preliminary procedures have to be carried out.

1. Sensitivity analysis

The purpose of a sensitivity analysis is to investigate the identifia-

bility of the parameters. As pointed out by Dawdy and O'Donell (1965), the

more sensitive the parameters are, the more rapidly the parameters will con-

verge. The index of sensitivity used in this study is the covariance matrix

of the estimated parameters, which can be shown (Bard, 1974 and Yeh and Yoon,

1976) as

COV(e) = M—L(46)

where J( 6 ) = summation of the squares of the differences

between computed values (e.g. discharges) at

0 and observations,

A(0) • [Ja(0)] (;)]T,

ja(6) • the Jacobian matrix evaluated at 0,0 the parameter estimates,

• the number of observations,

20

the number of parameters.

The Jacobian matrix can be approximated by the influence coefficients (Becker

and Yeh, 1972). The covariance matrix provides the information regarding the

sensitivity of the estimated parameters. The larger the variance, the less

sensitive the corresponding parameter will be. A correlation matrix (C) which

would indicate the degree of interdependence among the parameters with respect

to the objective function at the estimated parameters can also be obtained

from the covariance matrix as

C(6) = [c..]

a..cij

—

da ieji(4 8)

where c.. the entry of C(0) at (i,j),

a.. the entry of COV (e) at (i,j).

2. Generalized Least Squares Optimization (GLSO)

In order to obtain better initial estimates of the parameters, the GLSO

algorithm (Williams and Yeh, 1983) can be used to perform parameter estima-

tions using historical observations.

3. Determination of g and R

Although adaptive procedures do not require accurate initial estimates

of Q and R, a better estimate of R (assuming that Q = 0) can be easily

obtained from the result by the GLSO algorithm.

21

(47)

and

4. Determination of initial state and its covariance matrix

The initial state is assumed to be the same as the true observations at

the starting point. The covariance matrix associated with the initial state

can then be found by a trial—and—error procedure based upon some physical con-

siderations about the state. The goodness of the assumed values can be veri-

fied by a post test of the satisfaction of the assumption, i.e., the whiteness

of the residuals of the observations.

When the preliminary procedures have been carried out and the• required

information for the IMF has been obtained, the IEKF can then be implemented.

Application to Hypothetical Catchment

A hypothetical catchment as shown in Fig. 3 is considered. The parame-

ter values associated with the catchment are listed in Table 1. The catchment

is subject to an arbitrarily chosen storm to generate discharges at specified

stations. A storm with its duration of 20 hours and variable intensity, as

indicated in Fig. 4, is considered. The resulting "true" hydrograph is also

shown in Fig. 4.

Three ways of generating observations were used. The first case assumes

that both system and measurement are subject to White Gaussian Noise (WGN).

In this case, two sequences of random numbers were generated with zero mean,

given variance (time invariant) and normal distribution. They were then added

to the system response and observations which were simulated using true

(known) parameter values. In the second case, the measurement error is

assumed to be time variant. In other words, noise was added to each generated

22

Qi012 13 100Z4 0

Q5 <6 11.1>

-1

7 w 0 I-•-10 km-1 0 50

DISTANCE FROM TOP OF WATERSHED (km)

i - - -

50 km

.law. •••••• •••■•••

Table 1. Parameter Values for the Hypothetical Catchment

Parameters True Value

Initial

Estimates

Time of concentration, t (hrs) 15.0 12.0

Surface Water Storage Coefficient, Ks (hrs) 12.0 10.0

Groundwater Storage Coefficient, K (hrs) 40.0 50.0

Infiltration index, 0 (mm/hr) 3.0 1.0

24

OO

O

CD0

O_CO

OO

O_CO

OO

O

OO

O __(N)

O

Cb.00 1'0. nn 20.00 30.001TIME (h)

50.90 0n. nn 70.09 80.09

Fig. 4. Input Hystonraph and 'True" Hydrograph for the Hypothetical

Catchment

observation by an amount equal to the random number multiplied by 5% or 1096 of

the generated values. In the third case, the measurement error was generated

by perturbing the parameter randomly at each time step (which can be con-

sidered as a noise with time—variant covariance). The system errors of cases

2 and 3 are assumed to be corrupted by a sequence of WGN which is generated by

the same procedures as case 1. The generated measurements have a time base of

100 hours with a time interval of 0.2 hours.

In this hypothetical case study, the preliminary procedures were applied

to the first 200 observations to obtain the required information for IEKF.

The IEKF was then applied to the rest of the data points. In performing sen-

sitivity analysis, true values of parameters were used. Since the variance of

the measurement errors, J(4)/(N—L), does not affect the relative relationship

among the parameters, it is set to be unity for comparison purposes. The

covariance and correlation matrices are shown in Tables 2 and 3. A similar

result was obtained by Williams and Yeh (1983) and some physical interpreta-

tions were also illustrated. The true values of parameters, their estimates

from GLSO and their estimates from the IEKF are listed in Table 4. The post

analysis of verification of the whiteness of the residuals of the observations

has been done. The lag—one cross correlation matrix of the residuals is shown

in Table 5. The correlogram of the observations at the downstream station is

also plotted in Figures 5a, 5b, 5c and 5d. Figures 6a, 6b, 6c and 6d show the

observations, the one—step—ahead forecasts and the estimations of the

discharges at the outlet of the watershed for three cases. The results show

that the proposed algorithm provides a favorable performance of flood fore-

casting and parameter estimation. At each time step the IEKF takes at most 2

iterations after several time steps have been taken.

26

Table 2. Covariance Matrix of the True Parameter Values

for the Hypothetical Catchment

tc

Ks 0

tc

0.54 —1.15 —0.11 0.15

Ks

—1.15 5.03 1.18 —1.02

I8

—0.11 1.18 1.71 —0.33

0.15 —1.02 —0.33 I 0.24

27

Table 3. Correlation Matrix of the True Parameter Values

for the Hypothetical Catchment

tc

IIIKBS I

I0g

I I I

tc 1.00

1—0.70

1—0.12 0.43

I I I

Ks

—0.70 1.00 0.40 —0.94

1

11

—0 .12 0.40 1 1.00 —0.52

111

6 0.43 —0.94 1 —0.52 1.00

1

28

Table 4. Parameter Estimations on the Hypothetical Catchment

True Initial

Case Parameter Guess GLSO IEEF-4- -I

* **SD= 0.4

1t c 15.0 12.0 14.998 15.023

1 SD2 = 0.4 ks 12.0 10.0 12.016 11.979

kg

40.0 50.0 39.992 40.005

____4___4____0 3.0 1.0 2.996

--1----1------12.995

SD = 0.4 15.0 12.0 15.109 15.087*** c

2-1 SD2 = 5% ks 12.0 10.0 11.625 11.625

kg

40.0 50.0 39.794 39.794

0 3.0 1.0 3.083 3.078

= 0.4 I t c 15.0 12.0 15.235 15.201

2-2 SD2 = 1096 i ks 12.0 10.0 11.260 11.260

Ik 40.0 50.0 39.825 39.823

0 3.0 1.0 3.148 3.143

SD = 0.4 tc1 15.0 12.0 14.507 14.550

3 SD2 = 15% I ks 12.0 10.0 12.614 12.612

I kg40.0 50.0 39.401 39.410

0 3.0 I 1.0 I 2.852 I 2.852

* SD1 = System noise and SD 2 = measurement noise.

** 0.4 means (0.4)X(N(0,1))

*** 5% means (5%)MmagnitudeMN(0,1))

29

Table 5. Lag 1 Cross Correlation Matrix of the Residuals for Case 2-1

1 2 3 4 5 6 7

1

2

3

4

5

6

7

0.0517

(sym)

0.0426

0.0481

0.0784

0.0141

—0.0124

0.1374

0.0989

0.1293

0.0604

0.0585

0.0325

0.0938

0.0953

—0.0068

0.1767

0.118

0.0012

0.0602

—0.0365

—0.0491

0.0373

—0.0135

0.0636

—0.1263

—0.0487

0.0412

—0.0887

30

o

I 0■1

iJ- ryU_ • -., 0 0C 0 m 0U1 C) 0 0 0- c.CD 0 0

00 oo°0 0oon

0° 0 0 ° 0 oo 00 0 000Li ° 0 TO o 1 co'zcO.00 10.00 e0.00 30.00 SO.::

040.00

D LAG0■■••1

Fig. Sa. Correiogram of the Residuals at Downstream Station of theHypothetical Catchment - Case 1

31

OO-ED

Oto

•-•

LIJ11-4

U

Li_ r‘jLI_ • - 0W° 0 0CDLi 0 °Op Oc,-, 00 0

0 0 r) ?CDC) 00 m ° 1 •r, 0 ('20mOe cZ c0 oq,0 'oeld.o, cgo. 0 30. l .:) 0 (tt-w . 02 - 0 A.00fm r‘i 0 G—. c,_

Fig. 5b. Correiograu of the Residuals at Dovuatreau Station of theHypothetical Catchment - Case 2-1

32

C 0CD 0

,

C.) o 000 000 00 cC 0 rr, poT °I o o , 0 r-,, co

Z c0 0 00Q C . 0 Cc3 0 . W) 0 30 . L0 C d& 0c rC) C 5 . C ,...

CA-

ED r‘' ü LRG.D.]

0

Fig. Sc. Correlogram of the Residuals at Downstream Station of the

Hypothetical Catchment — Case 2-2

CJ

LLn

L1.1

33

CIO

00000000000 00

0000000000°0000000000000000000000000-

5b. c:

W

410.00 30.00

LL•

20.00

L AG

LJJED

110.00

z

0 0

cpc‘j.

cc c°_

Fig. 5d.Correlogram of the Residuals at Downstream Station of the

Hypothetical Catchment — Case 3

34

O-r70.00 An. fin3b. on 60.0001) 2h. on —T —r

40.00 5(1. 1111

TIME (h)

Co.no

TRUE VALUES

OBSERVATIONS

ONE-STEP-AHEAD FORECASTS

0 y y K ESTIMATIONS

000_

00OCO

000

Fig. 6a. Observed. One—Step—Ahead—Forecasted, and Estimated Hydrographsof the Conceptual Model — Case 1

OO_

CO

OO_

CD

OOO_

OO

O

cb.00 1 0 . 00 20.00 3h. no 'In. nnTIME (h)

0n_ _

70. 11 00.00

O

_ TRUE VALUES

OBSERVATIONS

I + + 4 ONE-STEP-AHEAD FORECASTS ESTIMATIONS

w

LL.

Fig. 6b. Observed, One—Step—Ahead —Forecasted. and Estimated Bydrograph•

of the Conceptual Model — Case 2-1

TRUE VALUES

--- OBSERVATIONS

--1---4---1---4- ONE-STEP-AHEAD FORECASTS

xxxxESTIMATIONS

CD

0

CD

O_

Fig. 6c. Observed, One—Step—Ahead—Forecasted, and Estimated Bydrographs

of the Conceptual Model — Case 2-2

TRUE VALUES

OBSERVATIONS

I' + +. ONE-STEP-AHEAD FORECASTS

X x x x ESTIMATIONS

OOO_O

OC3

tD_

OO

-r-

hoo ph.no 30.on 40.00 50.nn nnTIME (h)

Fig. 6d. Observed, One—Step—Ahead—Forecasted, and Estimated Hydrographsof the Conceptual Model — Case 3

Rn.nn70. on

Application to Real Catchment

The catchment of the Williams River in the Hunter Valley in N.S.W., Aus-

tralia (Fig. 7) was selected for study. The storm of March 1976 was chosen to

test the proposed algorithm. The hyetograph with a duration of 25 hours and

its corresponding hydrograph at the outlet of the catchment with a time base

of 138 hours are shown in Fig. 8. Fifty—two (52) data points of the hyeto-

graph and 46 data points of the hydrograph were recorded.

For the conceptual model, the first 20 discharges and the parameter

values estimated by Field and Williams (1982) were used to estimate the ini-

tial values of parameters for the IEKF algorithm. A sensitivity analysis was

made prior to the GLSO procedures. The results were similar to those of Wil-

liams and Yeh (1983) in that the groundwater storage coefficient is not sensi-

tive. Therefore, it has been set to be a fixed value in this study. Tables 6

and 7 show the covariance matrix and correlation matrix of the estimates of

the parameters. The parameter estimates by Field and Williams, by GLSO, and

by IEKF are listed in Table 8. The proposed IEKF algorithm has been carried

out to obtain the parameter estimates and the flood forecasts. Figure 11

shows the observations, the one—step ahead forecasts and the estimations of

the discharges at the outlet of the watershed.

39

A = 969 km2.1 = 104.5 kmPLUVIOMETER

SITE

10 km1

CLARENCETOWN •

1500

13 10001O

to 5001

ELEV 201

t-o101

— 01 201 401 601 801 Igo-1DISTANCE (km)

Fig. 7. Catchment Details of Williams River

40

O

V)o

rik)

O

O

_—

20.00 4'0.00 60.00 Bo.00 1 on. 00 I ,Dn. (11)TIME (h)

Orz)

cb.on 1Flqn.nn 15 11 .11n

OBSERVATIONS

ts A A ONE-STEP-AHEAD FORECASTS

ESTIMATIONS

O

OCO

O

0CO

(3/1976), One-Stop-Ahead Forecasted, and EstimatedFig. 8. ObservedHydrographs of Williams River from the Conceptual Model

I

4

Table 6. Covariance Matrix of Estimated Parameters

IItcKsK 0

g IIII2.961'10 5 —9.847'10 5 —1.241'101 1 1.498x10 5

tc

4------4--

Ks4.761'104

___I__

K (sym)g

6

I

1 I

6.864110 I —8.013x10-5

iL i________II —11.186'103 I —1.323x10I

_I__IiI 1.508x10-5i1 _L._ _____L

42

Table 7. Correlation Matrix of Estimated Parameters

tc

Ks

K 08

tc 1.0 —0.829

-I---4----

-V ------4

I—0.662 1 0.709

1

Ks

0.9131 .0 —0.946

( sym) 1.0 —9.893

1 .00

Kg

43

Table 8. Parameter Estimations on Real Catchment

Method tKS K;

Initial Guess (Field 4 Williams) 8.0 23.0 500.0 0.5

GLSO 8.45 23.97 500.0 0.447

IEKF 8.45 23.97 500.0 0.447

* K is fixed to be 500.0

44

CONCLUSIONS

In terms of the hypothetical data which have been examined, the Iterated

Extended Kalman Filter seems to be an effective technique for parameter esti-

mation and flood forecasting in nonlinear conceptual models.

In the application of the IEKF on the chosen catchment, the performance

of the conceptual model is fairly good in that the peaks for both observations

and one-step ahead prediction are very close. Unfortunately, the post

analysis of the prediction error could not be done due to the lack of data.

The proposed approach has the advantage that practical nonlinear concep-

tual models can be used, in conjunction with the IEKF for real-time parameter

estimation and flood forecasting.

45

REFERENCES

1. Bard, Y., Nonlinear Parameter Estimation, Academic Press, New York,1974.

2. Becker, L., and W. W—G. Yeh, 'Identification of Parameters in UnsteadyOpen Channel Flows,' Water Resources Research, Vol. 8, No. 4, pp. 956-

965, August 1972.

3. Bolzern, P., M. Ferrario, and G. Fronza, 'Adaptive Real—Time Forecast ofRiver Flow—rates from Rainfall Data,' Journal of Hydrology, Vol. 47, No.3/4, pp. 251-267, July 1980.

4. Burnash, R.J.C., R.L. Ferrel, and R.A. McGuire, 'A Generalized Stream-flow Simulation System: Conceptual Modeling for Digital Computers,'U.S. Dept. of Commerce, Nat'l. Weather Serv./State of Calif., Dept. ofWater Resources, Sacramento, Calif., 1973.

5. Dawdy, D.R., and T. O'Donnell, 'Mathematical Models of CatchmentBehavior,' Journal of Hydraulics Division, ASCE, Vol. 91, No. HY4, pp.123-137, July 1965.

6. Field, W.G., 'Kinematic wave Theory of Catchment Response With Storage,'Journal of Hydrology, Vol. 55, No. 1/4, pp. 279-301, February 1982.

7. Field, W.G., and B.J. Williams, 'A Generalized One—Dimensional KinematicCatchment Model,' Journal of Hydrology, Vol. 60, No. 1/4, pp. 25-42.January 1983.

8. Georgakakos, K.P., and R.L. Bras, 'Real—time, Statistically Linearized,Adaptive Flood Routing', Water Resources Research, Vol. 18, No. 3, pp.523-524, December 1980.

9. Hydrologic Engineering Center, Real—Time Flood Forecasting and ReservoirControl in the Kanawha River Basin, Special Projects, Memo No. 83-10,Davis, CA, December 1983.

10. Klemes, V., Empirical and Causal Models in Hydrology, in ScientificBasic of Water—Resource Management, pp. 95-104, National Academy Press,Washington, D.C. 1982.

11. Kibler, D.F., and D.A. Woolhiser. 'The Kinematic Cascade as a Hydrolo-gic Model' Colorado State University, Fort Collins, Colo., Pap. No. 39,pp. 27, 1970.

12. Kitanidis, P., and R. Bras, 'Real—Time Forecasting With a ConceptualHydrologic Model: 1. Analysis of Uncertainty,' Water ResourcesResearch, Vol. 16, No. 6, pp. 1025-1033, December 1980a.

13. Kitanidis, P., and R. Bras, 'Real—Time Forecasting With a ConceptualHydrologic Model: 2. Application and Results', Water Resources

46

Research, Vol. 16, No. 6, pp. 1034-1044, December 1980b.

14. Mein, R.G., E.M. Laurenson, and T.A.McMahon, 'Simple Nonlinear Model forFlood Estimation,' Journal of Hydraulic Division, ASCE, Vol. 100, No.HY11, pp. 1507-1518, November 1974.

15. Moore, R.J., and G. Weiss, 'A simple nonlinear Conceptual Model,' inReal—time Hydrological Forecasting and Control, edited by P.E. O'Connel(from Proceedings of 1st International Workshop, July, 1977), Instituteof Hydrology, Wallingford, 1980.

16. Nahi, N.E., Estimation Theory and Application, John Wiley, New York,

1969.

17. O'Connel, P.E., and R.T. Clarke, 'Adaptive Hydrological Forecasting — AReview,' Hydrological Sciences Bulletin, Vol. 26, No. 2, pp. 179-205,1981.

18. Poljak, B.T., and J.Z. Tsypkin, 'Robust Identification', Automatica,Vol. 16, pp. 53-63, Pergamon Press, 1980.

19. Szollosi—Nagy, 'An Adaptive Identification and Prediction Algorithm forthe Real—Time Forecasting of Hydrological Time Series', Hydrological Sciences Bulletin, Vol. 21, No. 1, pp. 163-176, March 1976.

20. Szollosi—Nagy, A., E. Todini, and E.F. Wood, 'A state space Model forReal—Time Forecasting of Hydrological Time Series', Journal of Hydrolog-ical Sciences, Vol. 4, No. 1, pp. 61-76, 1977.

21. Todini, E., and D. Bouillot, 'An Rainfall—Runoff Kalman Filter Model' inSystem Simulation in Water Resources, ed. G.C. Vansteenkiste, pp. 69-80,1976, North—Holland Publishing Company, Amsterdam.

22. Todini, E., P.E. O'Connel and D.A. Jones, Basic Methodology: KalmanFilter Estimation Problems, in Real—Time Hydrological Forecasting andControl, pp. 95-104, National Academy Press, Washington, D.C. 1982.

23. Todini, E., and J.R. Wallis, 'A Real Time Rainfall—Runoff Model For anOn Line Flood Warning System,' in Applications of Kalman Filter toHydrology, Hydraulics, and Water Resources, edited by C.L. Chin, pp.355-368, University of Pittsburgh, Pittsburgh, Pennsylvania, May 1978.

24. Toyoda, J., N. Toriumi, and Y. Inoue,'An Adaptive Predictor of RiverFlow for On—line Control of Water Resource Systems,' Automatics, Vol. 5,pp. 175-181, Pergamon Press, 1969.

25. Williams, B.J., and W. W—G. Yeh, 'Parameter Estimation in Rainfall—runoff Models,' Journal of Hydrology, Vol. 63, No. 3/4, pp. 373-393,1983.

26. Wishner, R.P., Tabaczynski, and M. Athens, 'A Comparison of ThreeNon—linear Filters,' Automatica, Vol. 5, pp. 487-496, Pergamon Press,

47

1969.

27. Wood, E.F., and A. Szollosi—Nagy, 'An Adaptive Algorithm for AnalyzingShort Term Structural and Parameter Changes in Hydrologic PredictionModel,' Water Resources Research, Vol. 14, No. 4, pp. 577-581, August1978.

28. Yeh, W. W—G., and Y.S. Yoon, 'A Systematic Optimization Procedure forthe Identification of Inhomogeneous Aquifer Aquifers,' in Advances inGroundwater Hydrology, Edited by Z.A. Saleem, pp. 72-82, AWRA, Minneapo-lis, Minnesota, 1976.

48

LIST OF SYMBOLS

A (6) • [Ja(6)] [Ja(6)1);ARMAX • AutoRegressive Moving—Average with

exogenous input;

a • the dimensional coefficient in therelationship between p velocity, v and depth, y;

• width of the elemental strip (Fig. 1);

• the constant in the relationship between

storage volume, h and discharge, s;

C (6) correlation matrix about parameter 0 ;cove) • covariance matrix about parameter 0 ;C. • the entry of C( 6 ) at (i,j);

• differential with respect to t;dt• the statistical expectation;

F(%k ,Uk) • a vector function which corresponds to thewatershed simulation model in this paper;

• the Jacobian of F(%k,Uk);

f1 • N—vector function;

f2, f3 • scalar functions;• a(n x n ) matrix;X IF

GLSO Generalized Least Squares Optimization;

• observation matrix of dimension nz x nXwith entries of values 0 and 1;

H(Xk

) • a vector function;• volume per unit area;

• the identity matrix;

• superscript which represents the ith iteration;

J(6) • summation of the squares of the differencesbetween computed values (e.g. discharges) at

4 and observations;

jaa° the Jacobian matrix evaluated at 4;• indices of increments of spatial variable, x;

• the gain matrix in Kalman filter;

49

Kcstorage coefficient which could be

surface water storage coefficient, Ks or

groundwater storage coefficient, K g ;

Kg = groundwater storage coefficient;

Ks • surface water storage coefficient;

k—i/k • subscript representing the value at timek-1 based on the information from the initial

stage up to time k;

• the number of parameters;

• the number of observations;

• the exponent in the relationship between

channel velocity, v and depth, y;

• number of increments in the main channel;

nu • number of inputs to the system;nV dimension of the measurement noise;

nw • dimension of the system noise;nX • number of states of the system;

number of observation stations;

covariance matrix used in Kalman filter;

• precipitation or rainfall intensity;k+1 kP P • rainfall intensity at time k+1 and

k respectively;

pe • a characteristic rainfall intensity, whichcan be taken as the average intensity for

the storm duration;k-1 ,

p2p1 2• the rainfall inputs at time k-1 to subsystems

1 and 2 respectively;

Qk • E[Wk Wk ] covariance matrix of Wk'' '• discharge;

qk • the runoff (discharge) at time k,k T

'•, ..... n ] •q

• discharge a time k at the jth spatialqj

point in the main channel;

Rk • E[Vk,Vi] , covariance matrix of V •k'the volumetric rate of total supply from

50

Xk =

v* =

x =

Yk

Y =

Zk =

,

I

0

r

surface storage and groundwater per unit

area to channel flow;

• groundwater suply (discharge per unit area);

sf sk1 groundwater supply at

g gtime k and k-1 respectively;

surface supply (discharge per unit area);

1 k-1

s ,s • surface water supply

at time k and k-1 respectively;

s *supply which could be surface supply or ground-

water supply;

• symbol denoting transpose of a matrix;

temporal variable;

Ukthe nu—vector of the deterministic inputs;

Vkthe nv—vector of the measurement noise which

is zero mean, independent white Gaussian

processes, i.e., Vk — N(0,Rk) and

E[Vk '

Vk1=0 for k#i;

velocity;

• the nw—vector of the system noise which is zero

mean, independent white Gaussian process, i.e.,

Wk — N(O,Qk) and E[Wk,Wk] = 0 for k#i;

the nx—vector of the states of the system

at discrete time k, k=0, 1, 2, ...;

any estimate of state Y.k;

spatial variable;

T[Z2 , Z2 Zk] T;

water depth in the main channel;

the n—vector of the measurements on the

system at discrete time 1, k=1, 2, 3, ...;

certain functionals characterizing the

system.

a (n x nX) matrix;

infiltration index which is set to be p when

0 > p and s s= 0;

a (nx x nw) matrix;

51

3 (Xk

) • a vector function;• the constant in the relationship between storage

volume, h and discharge, s;

• symbol denoting estimated value;

• an arbitrarily small positive number;

O • parameters;O • estimates of 0;Ax difference step in x — direction;

At • difference step in t — direction;o.

j • the entry of OVV(0) at (Li);

innovation, i.e., vki

to • K v , and/ = conditioning notation.

= Z-111(kik_1;

52