## Grand Challenge Scaling Problem

An important problem in ecological and biogeochemical modeling is related to scale. To illustrate this 'scaling problem', consider the illustrative case of nutrient-limited grass growth on a field. The soil nutrient concentration was measured at four locations, roughly equally spaced and representative of an equally large portion of the field, as shown in Figure 9 (top left). Two different models are used to estimate the growth rate. Model A is linear and has the form p = pMAXS/Km, and model B is the nonlinear Monod equation introduced above. Both models are applied at two different scales or resolutions. The high-resolution application (middle column) has four segments each corresponding to one measurement, and the low-resolution application (right column) has one segment with a nutrient concentration equal to the average of the four measurements. When model A is applied, the growth rates are different, but average out to the same value, regardless of the resolution of the model (middle row). This is an important point:

Data

High resolution

Low resolution

— | |||

+ |
30 |
270 | |

26 + |
1 |
15 |
26 |

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(ave = |
85) | |

< |
0.90 |
8.10 | |

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0.45 |
0.78 | ||

te | |||

rat th wt |
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m el |
0.60 |
1.47 |

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Figure 9 Illustration of scaling problem. Nutrient concentration, S (nmol l-1); specific growth rate, ß (d-1); model A: ß = ßMAXS/Km; model B: ß = ßMAXS/(Km + S); maximum specific growth rate, ßMAX = 1.8 d-1; half-saturation constant, linear models are scale insensitive and the resolution can be chosen freely to suit other needs, like the availability of input data, desired resolution of output data, computing resources, etc. The output from model B for the low- and high-resolution applications does not average out to the same value (bottom row). That is because nonlinear models are scale sensitive. Applying them to the same data at different resolutions will produce different results, and the model resolution can not be chosen freely. The underlying mathematical theory is known as 'Jensen's inequality'. In future, the scaling problem will become increasingly important as (1) the spatial and temporal resolution of data increases as a result of advances in in situ and remote-sensing technology, and (2) process models are becoming increasingly functionally complex and nonlinear. The heterogeneity of terrestrial and soil environments is well known, but increasingly recognized as important in the aquatic environment. Two-dimensional imaging fluorom-eters capture spatial phytoplankton variability of almost an order of magnitude at subcentimeter scale. Moored sensors record fluctuations in phosphate concentration of over an order of magnitude within the course of a day. Phytoplankton models simulate intracellular speciation and transformation among multiple nitrogen species. Models consider luxury uptake of nutrients and trace elements, their intracellular transformation and excretion. These two trends in spatial and temporal resolution and functional complexity are on a direct 'collision course' with the scaling problem. This is one of the grand challenges of integrated ecological and biogeochemical modeling. |

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