Geoscience Reference
In-Depth Information
different parts of the study period, as well as the changes in the parameters themselves. The
noise variance ratio (NVR) matrix,
N
r
,
is normally assumed to be a diagonal matrix with
elements for each of the parameters to be estimated. The NVR controls the effective memory
of the algorithm. Large values allow rapid changes in the parameter values at each time step;
small values mean that those changes are damped over a larger number of time steps. The
NVR may be optimised to achieve the best overall forecasting performance.
In the study of Young and Beven (1994) that is described in the case study of Section 4.4,
this type of FIS algorithm was used to define the time variable gain values that were used
to define the nonlinear power law filter on the rainfall inputs that is shown in Figure 4.6a.
In later work, this was extended by Peter Young to a more general technique that he called
“state dependent parameter” (SDP) estimation. He realised that, when it was expected that the
nonlinearity might be related to some index of the state of the system (wetness, in the case of
a catchment system), the time variable gain values could be filtered in the dimension of that
index rather than in time. This is quite easy to achieve by ordering the gain estimates by the
value of the index and using FIS on the ranked values (Young, 2000). This SDP methodology
actually provides a form of non-parametric function between the gain and the index variable
(as shown, for example, in Figure B4.3.1). In prediction, this can either be used directly (in the
form of a look-up table) or it can be represented by a simple parametric form (the power law
Figure B4.3.2
Predicted discharge from a DBM model for Coweeta using the input nonlinearity of Figure
B4.3.1 (after Young, 2000, with kind permission of Wiley-Blackwell). Peter Young also shows in this paper
how this model can be improved even further by a stochastic model of a seasonal function of temperature,
representing a small effect of evapotranspiration on the dynamics of runoff production.
