Data-Based Mechanistic Modelling and the Emulation of Large Environmental System Models - Environmental Modelling: Finding Simplicity in Complexity - page 118

t , while there are nominally an infinite number of
discrete-time models, as their parameters are a function
of t and have to change when it is changed. Second, the
uniquely defined parameters of the differential equation
model often have direct physical meaning since most
natural laws (e.g. mass and energy conservation) are
normally posed in terms of such differential equations.
For example, the partition-percentage and residence-
time parameters of the first-order parallel pathway
components of the Canning River model, as shown in
Figure 7.3, are derived directly from the model (7.2a) and
are easily interpreted in mass-conservation terms. There
are also statistical estimation advantages associated with
the continuous-time model: it is much superior when
the time-series data are rapidly sampled because the
eigenvalues of the resulting model then lie close to the
unit circle in the complex z domain and discrete-time
model estimation is seriously impaired in this situation.
Moreover, the continuous-time methods can be adapted
to handle the case of irregularly sampled data or
non-integral time delays that are often encountered in
the modelling of real systems. 4

DBM/CAPTAIN tools that are able to identify the most
important parameters in the complex simulation model
and examine how they affect its dynamic behaviour. These
tasks are in the domain of uncertainty and sensitivity
analysis; analysis that has been revolutionized in recent
years by our ability to apply MCS-based methods to
complex simulation models (see e.g. Saltelli et al ., 2000;
Beven et al ., 2000; and Thiemann et al ., 2001). A typical
example of such Monte Carlo analysis is described in
Parkinson and Young (1998), where MCS and the related
technique of generalized (or regional) sensitivity analysis
(GSA: see Spear and Hornberger, 1980) are used to
assess the effect of input and parametric uncertainties
(as defined by climate scientists) on the behaviour of the
global carbon-cycle simulation model. This analysis helps
to reveal the most important physical parameters in the
model and enhances further studies aimed at exposing the
modes of dynamic behaviour that dominate its response
characteristics.
One DBM technique that is able to extract these domi-
nant modes of dynamic behaviour is known as dominant
mode analysis (DMA: see Young, 1999a), which employs
the same optimal RIV methods of model identification
and estimation used for the DBM modelling of systems
from real data. This analysis is able to identify simple,
reduced order representations of the large simulation
model that are characterized by the dominant modes and
can often reproduce the dynamic behaviour of the large
computer model to a remarkably high degree: e.g. over
99% of the large model output variance explained by
the reduced order model. The large model can be deter-
ministic or stochastic but a typical deterministic example
is the reduced order modelling of the 40 th order ocean
heat upwelling diffusion model used by Eickhout et al .,
2004) and shown diagrammatically in Figure 7.5. Here,
the vertical mixing processes in the ocean are represented
by vertical diffusivity and upwelling, with 40 ocean layers,
each of 100 m depth, under a mixed layer of depth 90 m.
The diffusivity and upwelling processes occur between
each ocean layer and, in order to implement the ther-
mohaline circulation in this one-dimensional model, a
downwelling process is added from the mixed layer to the
bottom layer.
The estimated DBM emulation model is typified by the
transfer function for the 8 th

7.7 The reduced-order modelling of
large computer-simulation models

This chapter has concentrated so far on inductive, data-
based modelling and analysis applied to real-time series
data. However, when real data are scarce or insufficient,
many environmental scientists and engineers, including
the present authors, use the alternative hypothetico-
deductive approach to modelling and construct more
speculative and complex computer-simulation models.
Although extremely difficult, if not impossible, to vali-
date in a strict statistical sense, such models can provide
a very good method of extending our 'mental models'
of environmental systems, often as a valuable prelude
to the design of experimental and monitoring exercises
or, more questionably, as an aid in operational control,
management and planning exercises.
When speculative simulation models are exploited in
these ways, however, it is important to ensure that their
construction and use is preceded by considerable critical
evaluation. In these situations, it is possible to exploit

layer:

− 1 . 094 × 10 − 5 s 2 + 0 . 000153 s + 8 . 5911 × 10 − 7

T ( t ) =

s 3 + 0 . 06575 s 2 + 0 . 00103 s + 1 . 0981 × 10 − 6

4 Although the CAPTAIN routines do not yet have these options.

× Q ( t − 7) (7.6)