Environmental Engineering Reference
In-Depth Information
simplest form is, of course, the completely linear
equivalent of this model with Gaussian normal
stochastic disturbances. In the present context,
however, it makes sense to allow the parameters
in the model to vary in some manner over time, in
order to allow for changes in the catchment dy-
namics or inadequacies in themodel. The standard
non-stationary, linear, state space model for a n
th
-
order single input-single output system, with no
input noise, of the kind described later in the
illustrative tutorial example, can be written in the
following discrete-time form:
that themodel parameters must be well estimated
from these data and there should be no ambiguity
about their values. As discussed later, parametric
identifiability requires that the model is not 'over-
parameterized' or, if it is, that many of the para-
meters' values are assumed 'well known' prior to
updating and only a small, and hopefully identi-
fiable, subset have to be estimated. This latter
approach is possible, either deterministically, by
simply constraining the parameters to the as-
sumed values; or stochastically, by imposing tight
prior distributions on their values. Although such
an approach is often used in practice, it is rather
difficult to justify because it can imply an unrea-
sonable level of trust in the prior assumptions.
The situation can be improved, however, by
performing sensitivity analysis (see, e.g., Ratto
et al. 2007b; Saltelli et al. 2000) or by using some
form of model 'emulation', as discussed later un-
der 'Dynamic emulation modelling', where the
large, over-parameterized model is emulated by
a much smaller and identifiable 'dominant mode'
model (Ratto et al. 2007a; Young and Ratto 2008).
x
k
¼
F
k
x
k
1
þ
G
k
u
k
d1
þ z
k
z
k
¼
N 0
ð
;
Q
k
Þ
ð
9
:
3
Þ
h
T
y
k
¼
x
k
þ
g
i
u
k
d
þ j
k
j
k
¼
N
0; s
k
where F
k
is a n
n state transition matrix; G
k
is a
n
1 observation vector;
and g
i
is a scalar gain that is presentwhen the input
has an instantaneous effect on the output variable
y
k
. In order that this model can be used for state
updating and forecasting, it is necessary that the
state variables in the vector
x
k
are stochastically
observable (Bryson and Ho 1969) from the input
and output measurements u
k
and y
k
. In simple
terms, this means that the model is such that an
optimal least squares estimate x
k
of
x
k
(see below)
can be computed from these measurements and
that the variance of the state estimation error
x
k
x
k
is reduced by these operations. Note that
observability does not require that the state vari-
ables are measured directly; it requires only that
the structure of the model is such that the unam-
biguous estimates of the state variables can be
reconstructed fromthemeasured input and output
variables u
k
and y
k
.
The subscript k on F
k
and G
k
is introduced to
allow the elements of these matrices (the model
parameters) to change with time. However, be-
cause the linear relationship between the scalar
flow measurement and the state variables is nor-
mally time-invariant, the observation vector h is
not made a function of the sampling index k. Also,
it is likely that the nominal model will be esti-
mated off-line before this is utilized on-line, in real
1 input matrix; h is a n
Recursive State and Parameter Updating
Before proceeding to consider the general case of
nonlinear, stochastic dynamic models, it is in-
structive to consider the special, but practically
very useful, case of a linear dynamic model with
stochastic input disturbances and measurement
noise that have Gaussian normal amplitude dis-
tributions (or at least can be described sufficiently
well by their first two statistical moments). As we
shall see later, although this is a quite simple
model, it has considerable practical relevance. In
particular, it is quite common for rainfall-flow
data to have simple nonlinear dynamic character-
istics that allow for the utilization of linear fore-
casting methods, such as those discussed below.
Linear or near-linear stochastic systems
Normally, the generic, nonlinear, stochastic, dy-
namic model (Equation 9.1) can be simplified
substantially for use in flood forecasting. The
