I t now behooves us t o establish, with r e s p e c t t o t h e problem of u n c e r - t a i n t y , a viewpoint, a perspective, a m e t h o d of approach, t h a t h a s h i t h e r t o received i t s principal development a n d application outside t h e boundaries of environmental modeling. Such prior development a n d application, however extraneous t o o u r chief line of i n t e r e s t h e r e , m a y very well be i n a position t o profit by t h e precedents established i n m e t h o d s , in conclusions, a n d , m o s t particularly, in h a b i t of thought*.
Uncertainty m a y have two basically different sources: m e r e "ignorance"
i n one of i t s n u m e r o u s manifestations, or t r u e i n d e t e r m i n i s m of t h e s y s t e m taminated"
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or, a s I would say, perhaps even incommensurable.The t h e o r y , or model, will in t u r n have t o be responsive t o t h e problems.
Science, a n d m o s t obviously applied science, s t a r t s from t h e problems, n o t f r o m the observations o r m e a s u r e m e n t s . Yet observations m a y give rise t o
*The well read will recognize these sentences, which I could not resist adapting from p.41 of t h e 1956 Dover edition of A.J. Lntka's Elements of Mathemtieai (Physical) &logy.
problems, especially if they were u n e ~ p e c t e d , contradictory t o o u r expecta- space technology: satellite reconnaissance, aerial photography, radar-based weather observation, etc.; s e e , e.g., Lillesand a n d Kiefer 1979, Salomonson a n d based on macroscopic conceptualizations. The basic problem, a s discussed above, is i n t h e i r reliance on d a t a usually collected on a microscopic scale a s
technological computation rules and, in fact, algorithms and computer pro- grams. We do not use t h e m in the search for objective t r u t h , but rather to make sufficiently useful predictions. "A theory is a tool we t e s t by applying i t , and which we judge as to its fitness by t h e results of its applications." This
"Darwinian" instrumentalism, cited from Popper (1959, p.108), is criticized by Popper himself only a few pages later. It is, however, in keeping with the best tradition of t h e Vienna Circle of Mach, Wittgenstein, and Schlick.
This, however, will lead t o some more problems for the credibility and applicability of models, and a slightly different interpretation of the testing process as compared with "pure" scientific theories. Theories are tested by attempts t o refute t h e m . For models a s i n s t r u m e n t s , we can usually always find a "test to destruction" (see the application example in Section 3.1). In t e r m s of the above approach, we will almost always be able, for any even only moderately complex nonlinear simulation model, to find an "allowable" input combination t h a t results in unacceptable model response ( a t least from a boundaries within which your model behaves properly (Meadows 1979), which is easy enough by means of numerical experimentation on the computer, e.g.
Monte-Carlo-based trial and error. I t should be made clear t h a t Popper's falsificationism (from naive to sophisticated, as labeled a n d criticized by Lakatos (1978, p.93ff.)) can only strictly be applied t o the lowest operational level of theoretical constructs, i.e. the individually testable hypothesis. In environmental applications, which, as a rule, a r e a t t h e intersections of ecol- and numerical simulation models in particular (and even such supposedly elementary precise and well established models as Schrodinger's equation describing t h e hydrogen atom), are (pragmatic) simplifications, and t h u s include uncertainty, we have to be aware of t h e implications and conse- quences in r e l a t i o n t o the p r o b l e m to be s o l v e d .
4.2 Uncertainty Analysis: Alternative Approaches
Uncertainty inherent in environmental modeling is inevitable
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sto- chastic variability, heterogeneity, rjch behavioral repertoires, and time- varying structural and functional attributes a r e all basic features of environ- mental systems. Thus, i t seems unlikely t h a t any moderately complex environmental system can be well defined in t h e traditional physicochemicalsense (Hornberger and Spear 1981). In fact, environmental systems have been described a s being "poorly" or "badly defined" (Young 1978, 1983).
For a considerable t i m e , t h i s u n c e r t a i n t y and its inevitable conse- from t h e regular, orderly, and highly predictable "clocks" - they a r e "clouds,"
highly i r r e g u l a r , disorderly, and more o r less unpredictable (Popper 1979). tainty, is one approach to rational modeling u n d e r uncertainty.
Alternative approaches involve t h e d i r e c t apriori use of t h e probability ecosystem. Nevertheless, practical implementation of this type of approach requires a fair arnount of m a t h e m a t i c a l sophistication and a willingness t o be quite r u t h l e s s a b o u t model simplification (Silvert 1983b). Although "proba- bilistic model s t r u c t u r e s " m a k e i t possible t o c a r r y o u t stochastic modeling without extensive Monte Carlo simulations, t h e covariance calculations add a s u b s t a n t i a l computational burden. For a c e r t a i n class of problems, where
techniques and first-order variance propagation to e s t i m a t e overall model variance (or uncertainty) originating f r o m uncertain initial conditions, p a r a m e t e r s , or driving variables. They also require a fair degree of mathematical and statistical sophistication, and may involve considerable computational burden. And most i m p o r t a n t of all, t h e y require several assumptions to be made about t h e model a s well as about t h e s e t of d a t a used for comparison. First-order e r r o r propagation employs a first-order lineariza- tion of t h e model, t h a t is, t h e original nonlinear model is linearized and replaced by its first-order Taylor series approximation. This may eventually t u r n o u t to be inadequate. Since t h e second-order propagation equation complete nonlinear simulation model - may produce even bimodal distribu- tions for certain state variables, indicating bifurcations (Section 3.3). Clearly, applied in the field of modeling complex aquatic ecosystems, for example by Lewis and Nir (1978), Jdrgensen e t al. (1978), Di Toro and van S t r a t e n (1979), criterion chosen is free from subjective elements. For example, in problems with s t a t e variables with different physical dimensions, some (subjective)
extremely difficult t o find s u c h a unique vector if t h e n u m b e r of p a r a m e t e r s t o be estimated is l a r g e r t h a n , say, six t o t e n . If, however, s u c h a best param- e t e r vector exists - by definition - a n d c a n be identified by whatever method, its meaning and interpretation would still be problematic.
One way of comparing such approaches looking for a "best" solution with the methods described above is t h e following: if a "best" (by whatever s e t of Sections 3.1 and 3.2 is one such possible extension of the basic procedure.
As s t a t e d in t h e introduction, t h e Monte Carlo method is nothing more problem-specific exploitation of all t h e available information.
The method requires the formal definition of an acceptable model
"not acceptable." The classification is discrete, a:nd once t h e constraint condi- tions a r e formulated t h e r e is n o m o r e ambiguity, n o gradual or partial agree- m e n t o r disagreement between t h e model response and t h e observations, cal- ling for arbitrary judgments. How small would t h e s u m of squared e r r o r s have
to be for a given state variable to make a model acceptable? Although a observations is described as "acceptable or of reasonably good fit," ignoring t h e fact t h a t severe discrepancies exist between parts of the model response
Uncertainty in ecological modeling is certainly a n inevitable element in t h e method a s well a s in t h e object of study, which is most obvious when one tries to predict t h e future on t h e basis of a fuzzy present. The analysis of model uncertainty together with appropriate methods for model calibration under uncertainty, and of its consequences, i.e. i t s "inverse," prediction accuracy, is certainly a t an early stage of development. However, being aware of model and especially prediction uncertainty and the t h u s obvious limits of predictability, i.e. t h e range within which a given model may reason- ably be applied, might well help to avoid too naive a t r u s t in numerical models. Analysis of the various sources of model uncertainty and their rela- tions and interdependences will be necessary to improve model applicability.
And t h e least impact from model e r r o r analysis on model application should be a critical reevaluation of the questions t h a t can reasonably be addressed and answered by means of numerical models.
The implications of uncertainty are many: t h e r e are implications for t h e testability of hypotheses, which, in t e r m s of simulation modeling, is primarily on model development. This may cast new light on t h e principle of parsimony ever, since t h e uncertainty is a basic characteristic of t h e systems dealt with, we have t o live with it, and exploit i t wherever possible (Holling 1978). One possibility, as demonstrated above, is to estimate over which time span and over which range of conditions useful predictions