use of estimation techniques for flood forecasting
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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
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