Modeling

How to Discretize, Disaggregate and … Complexity 27
von der Regendauer", (Berichte aus de Institut für Wasserwirtschaft und
Gesundheits-ingenieurwesen Technische Hochschule München Nr. 2, 1969, ca
200pp.). What conclusion can be drawn from Figure 2.2, other than that c varies
widely and unpredictably with duration? Surely, that the given relationship is
poorly structured, or fuzzy, perhaps even a mess, since, in the absence of a
better explanation of the processes involved, evidently any value is as good as
any other.
Reporting this variance of c in those models that use it has to my knowledge
never been done. It seems to be virtually unknown in the English literature on
the subject. In this sense a simplistic sub-model increases a difficulty when
using the model, because its mathematical formulation is likely incorrect
physically, and evidently meaningful values of the empirical coefficient c
cannot be acceptably fudged. Ideally, the inherent scatter in all sub-models
should be incorporated in the computations in the model, and reported (the
uncertainty can probably be uncovered in the original publications).
Complexity arises when there is a large number of processes, when the
relationships are non-linear, and where there are nonholonomic constraints (e.g.
if a pump station in a drainage system switches on or off without regard to the
system as a whole).
We may think of model complexity C as the sum
where:
C
N
N
N
∑∑ ∑
≡ M S pr
m= 1 s= 1 p = 1
r
Np a
p,
s , m
N m = the number of modules active in the model,
N s = no. of sub-spaces modeled in each module,
N pr = no. of processes modeled in each sub-space,
Np a = no. of input parameters required for each process.
Model complexity is related to the total number of uncertain
input parameters.
C is independent of the number of time steps in the input time series (e.g.
rainfall or inflows) that drives the system model. For instance, a rain time series
may be 100 y long at 1 minute time steps, comprising about 3.15x10 9 time
steps, while the system model itself may have just six elements each requiring
(say) ten parameters to describe their geometry. Such a system model may be
said to have a complexity of 60, a relatively small number, even though the
driving input function is extremely large.
There is another hierarchy of complexity that depends upon how wellknown
the processes are, and how they depend upon earlier levels of
understanding. In this approach, physical systems are the simplest. SWMM