Geoscience Reference
In-Depth Information
Here
() is a nonlinear function of the variables
= Q t −1 ,Q t −2 ,...,u t ı ,u t ı −1 ,...,U t ,U t −1 ,...
t
(B4.3.3)
where the u values are past values of effective rainfall inputs and, for generality, we include
values of other exogenous variables, U , at this and previous time steps. The variable t repre-
sents the stochastic part of the model and is assumed to be a zero mean stochastic variable,
independent of the variables u and U.
The first-order transfer function model above may then be written as
Q t =
x t +
t
(B4.3.4)
where x t is the noise-free output of the model defined as
b o
a 1 z −1 u t ı (B4.3.5)
Let p represent the vector of parameter values b o ,a 1 ,... . In the time variable parameter
case, changes in any individual parameter, p i , are described by the generalised random walk
of the form
x t =
1
G i t (B4.3.6)
where the elements of p i are made up of two components for each parameter value, a changing
level and a changing slope, and t is a vector of zero mean white noise inputs.
p it =
F i p it −1 +
˛ˇ
0
,
i
1
F i =
(B4.3.7)
Some specific examples of this general model are the random walk (RW: ˛
=
0; ˇ
=
=
1;
=
0), the smoothed random walk (SRW: 0 <˛< 1; ˇ
=
=
1; t =
0) and the integrated
random walk (IRW: ˛
0).
Then, given some measurements with which to compare the model outputs, the transfer
function model may be written in what is called a state space form as the two equations:
=
ˇ
=
=
1;
=
Observation equation:
Q t =
Hp t +
t
State equation:
p t =
Fp t −1 +
G t
where, for the simple first-order transfer function above, the parameter vector p contains the
time-varying values and slopes for the changing transfer function parameters a 1 and b o .
The time variable state space equations are readily solved using a form of Kalman filtering
(see Young, 2011a) that involves both forward filtering and backward smoothing passes through
the data. The algorithm is recursive, in that the calculations are carried out time step by time
step so that estimates of the updated parameters are available at every time step. In matrix
notation, the time variable parameter estimation algorithm has the following form:
1. Forward pass filtering
a. Prediction:
p t | t −1 =
F
p t −1
(B4.3.8)
FP t −1 F T
GN r G T
P t | t −1 =
+
(B4.3.9)
b. Correction:
p t = p t | t −1 +
H t P t | t −1 H t −1 Q t
H t p t | t −1
P t | t −1 H t 1
+
(B4.3.10)
 
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