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A Multiscale Approach to Science

In understanding and modeling ecological and economic systems exhibiting considerable biocomplexity, the issues of scale and hierarchy are central (Ehleringer and Field 1993; O'Neill et al. 1989). The term “scale” in this context refers to both the resolution (spatial grain size, time step, or degree of complexity of the model) and extent (in time, space, and number of components modeled) of the analysis. The process of “scaling” refers to the application of information or models developed at one scale to problems at other scales. The scale dependence of predictions is increasingly recognized in a broad range of ecological studies, including: landscape ecology (Meentemeyer and Box 1987), physiological ecology (Jarvis and McNaughton 1986), population interactions (Addicott et al. 1987), paleoecology (Delcourt et al. 1983), freshwater ecology (Carpenter and Kitchell 1993), estuarine ecology (Livingston 1987), meteorology and climatology (Steyn et al. 1981) and global change (Rosswall et al. 1988). However, “scaling rules” applicable to biocomplex systems have not yet been adequately developed, and limits to extrapolation have been difficult to identify (Turner et al. 1989). In many of these disciplines primary information and measurements are generally collected at relatively small scales (i.e. small plots in ecology, individuals or single firms in economics) and that information is then often used to build models and make inferences at radically different scales (i.e. regional, national, or global). The process of scaling is directly tied to the problem of aggregation, which in complex, non-linear, discontinuous systems (like ecological and economic systems) is far from a trivial problem.

Aggregation

Aggregation error is inevitable as attempts are made to represent n-dimensional systems with less than n state variables, much like the statistical difficulties associated with sampling a variable population (Bartel et al. 1988, Gardner et al. 1982; Ijiri 1971). Cale et al. (1983) argued that in the absence of linearity and constant proportionality between variables – both of which are rare in ecological systems – aggregation error is inevitable. Rastetter et al. (1992) give a detailed example of scaling a relationship for individual leaf photosynthesis as a function of radiation and leaf efficiency to estimate the productivity of the entire forest canopy. Because of non-linear variability in the way individual leaves process light energy, one cannot simply use the fine scale relationship between photosynthesis and radiation and efficiency along with the mean values for the entire forest to represent total forest productivity without introducing significant aggregation error. Therefore, strategies to minimize aggregation error are necessary.

Jarvis and McNaughton (1986) explain the source of aggregation error shown by Rastetter by highlighting the discrepancy in transpiration control theory between meteorologists and plant physiologists. The meteorologists believe that weather patterns determine transpiration and have developed a series of equations that successfully calculate regional transpiration rates. The plant physiologists believe in stomatal control of transpiration and have demonstrated this with leaf chamber experiments in the field and laboratory. Therefore, it seems that different processes control transpiration at different scales, and aggregation from a single leaf to regional vegetation is impossible without accounting for this scale-dependent variability in transpiration control. One must somehow understand and embed this variability into the coarse scale.

Turner et al. (1989) list four steps for predicting across scales:

1. identify the spatial and temporal scale of the process to be studied;

2. understand the way in which controlling factors (constraints) vary with scale;

3. develop the appropriate methods to translate predictions from one scale to another; and

4. empirically test methods and predictions across multiple scales.

Rastetter et al. (1992) describe and compare four basic methods for scaling that are applicable to complex systems:

1. partial transformations of the fine scale relationships to coarse scale using a statistical expectations operator;

2. moment expansions as an approximation to 1;

3. partitioning or subdividing the system into smaller, more homogeneous parts (see the resolution discussion further on); and

4. calibration of the fine scale relationships to coarse scale data.

They go on to suggest a combination of these four methods as the most effective overall method of scaling in complex systems. (Rastetter et al. 1992).

 
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