Environmental Engineering Reference
In-Depth Information
And the simplest lumped parameter model will
relate specified inputs to specified outputs.
In general, the system disturbances
z
k
actually
affect the states and are important in defining the
underlying, 'real' behaviour of the dynamic sys-
tem (in the present context, they are normally
disturbances such as unmeasured input flows or
losses to groundwater); while
j
k
and
g
k
are mea-
surement or other noise sources that obscure our
observation of this real dynamic behaviour. In this
setting, the function of a model and forecasting
system is to 'filter' off the measurement noise
effects and reconstruct the true, underlying
behaviour, including the stochastic input distur-
bance effects.
Of course, the model (Equation 9.1) has to be
obtained in some manner, normally by reference
to the measurement vectors:
In this digital age, all the models will be solved
in a digital computer and so we will consider them
within the following generic, discrete-time, sto-
chastic structure:
x
k
¼
F
ð
x
k
1
;
u
k
; z
k
; u
Þ
y
k
¼
G
ð
x
k
;
u
k
; j
k
Þ
ð
9
:
1
Þ
v
k
¼
ð
u
k
; g
k
Þ
H
where
x
k
is a state vector composed of state vari-
ables that are affected dynamically by an input
vector
u
k
and a 'system' noise vector
z
k
that nor-
mally represents unmeasured, stochastic distur-
bances to the system;
y
k
is an observation or
'output measurement' vector that is some combi-
nationof the state and input variables, aswell as an
output measurement noise vector
j
k
; and
n
k
is a
measurement of the input vector that is a function
of the true input vector
u
k
contaminated in some
manner by an input measurement noise vector
g
k
.
Finally,
u
is a p
T
T
y
k
¼
y
1;
k
;
y
2;
k
...
y
p
;
k
;
v
k
¼
v
1;
k
;
v
2;
k
...
v
m
;
k
;
k
¼ 1; 2; ...;
N
ð
9
:
2
Þ
1 vector of (normally unknown)
parameters that define the inherent static and
dynamic behaviour of the system. There may be
other parameters that define the characteristics of
the measurement equations but these have been
omitted here both for simplicity and because it is
often assumed that they will be known a priori.
Normally, the state variables x
1,k
, x
2,k
...
where the superscript T denotes the vector trans-
pose and N is the total sample size. In the present
context, the elements of these vectors will nor-
mally take the formof telemeteredmeasurements
from remote, suitably located rainfall and flow/
level gauges in the catchment. This will normally
involve the specification or 'identification' of the
detailed model structure and the 'estimation' or
'calibration' of the parameter vector
u
that char-
acterizes this structure. Such an identification and
estimation procedure presents a considerable
challenge, and many different approaches have
been suggested in the hydrological literature.
These will not be reviewed in the present chapter
since our object here is to consider how these
model parameters and states can be updated on-
line and in real-time after the model structure has
been satisfactorily identified and a 'nominal' esti-
mate
u
0
of
u
has been obtained from prior, off-line
studies.
One very important aspect of model parameter
estimation is that the model being used as the
basis for the forecasting system design must be
'identifiable' from the available data, in the sense
x
n,k
that comprise the state vector
x
k
of this n
th
order,
stochastic dynamic system will relate to the dy-
namic behaviour at specified spatial locations (in
the present context, flow or level variations at
stations along the river system) but, again for
simplicity, the spatial index has been omitted
here. If the model is of a distributed parameter
form, then there will be many such spatial loca-
tions defined by whatever method of spatio-tem-
poral discretization has been used to convert the
partial difference equations to a discrete space-
time form. In the case of a 'quasi-distributed',
lumped parameter model, the locations will nor-
mally be far fewer and will be defined by those
spatial locations that are most important (in the
present context, the flow or level variations at
various specified gauging stations along the river).
