area of concern

Whenever we consider an issue, or wish to manage something, we implicitly choose an area that is appropriate to the situation.   Because we do this so easily, we are not always explicit about the area to be considered.  Most of the time we don’t run into misunderstandings as the others who are involved in the same circumstance are likely to have assumed the same area; as for example a jurisdictional boundary for a government agency.

When we want to simulate something that has a spatial dimension we have to be explicit about the area we are going to represent.   In this case jurisdictional boundaries may not be the appropriate guideline.  What may matter more is the extent or the density of the flows of matter, energy and influence.  That is, although we can always find some flow in our out of any natural system, some areas are more self contained than others.  Populations of plants and animals have ranges, lakes have edges, and geography imposes various other constraints.  So we often simulate populations, or ecosystems, or forest stands or whole regions. 

The choice of area thus depends in part on the extent of the problem, and in part on the degree of variation we are willing to collapse into a generality.  There are no strict rules, but any experienced modeller will ask a series of questions that help reveal an appropriate choice.

Google Earth images

boundary

When an area is chosen as the appropriate representation for a model issue, then a boundary is implicitly also set.  Boundaries in the real world are rarely discrete.  Even the edge of a lake may be difficult to determine, especially if the land is flat and marshy.  Boundaries are conceptual, they are distinctions we make based on some regularity we consider important, and there is something arbitrary about where in the transition between two eco-regions, ecosystems, or stand types we draw a boundary.   Computers however do not operate with ambiguities, so model boundaries are explicit. 

Whatever area is chosen, there are always inputs and outputs.  However not all concerns need to consider all inputs and outputs.  A stand model may not be concerned with the movement of large animals across the stand boundary.   Even for a global scale model will have inputs in the form of solar energy.    Where the inputs are known and their pattern in time can be represented, they are generally treated as driving variables or externalities.  (The word externalities is also used in economics with a slightly different meaning so I avoid it in discussing models)  Things that leave the model are usually simply treated as losses. 

When people talk about a “sustainable region” do they mean that the region could get along without any flows in or out of it?  In order to be considered “sustainable” what can cross the regional boundary and what must be provided internally?
Does it matter whether the surrounding regions are also sustainable?

spatial resolution

In considering spatial scale, one could treat the area as homogenous, or the processes that are being considered could be appropriately treated as averages.  Averages always obscure many underlying processes.  Averages are the statistical description of the results of the more detailed processes that generate the differences that are being averaged.  

Thus for a modeller the question is what level of processes are relevant to the issues being considered?   This is often practically dealt with by considering what processes we might influence, and which ones will continue regardless of anything we do.  For stand growth we do not model photosynthesis, but we may consider the effect of shading on growth, knowing that the underlying mechanism is photosynthesis.

Where the area to be considered is heterogenous in a way that matters for the dynamics we want to simulate, we have to consider how to represent that variation in the model, and hence we come the question of spatial resolution.

For example, if we are simulating water quality due to forest fertilizer treatment in a low hilly area with lots of streams and lakes we may wish to divide the area into sub-basins, and consider the treatment levels separately, and the influences on stream reaches separately.  In this sort of situation we can simply run the same model logic in parallel for a number of model sub units that specify the proper hectares, stream lengths, treatment etc. for each unit.

Where there is no natural or logical large scale boundary to place on the area, we often use a grid, and divide the model into a series of squares.  Each square can then be indexed as to the kind of area it represent, on the average. 

The advantage of grids is that various things that happen at different scales can be considered in this system.  We can specify everything from mouse populations, to grizzly ranges, forest cover, species mix and sub basins memberships with a series of grid attributes.  Each grid element is a mini model, with the same rules for change, but different initial conditions, parameters, and model outputs.  The region-wide results can be summed over the whole area.

How would you go about obtaining initial condition data for hard to measure variables for a very large area with a fine scale grid? 

Under what conditions would you want to do this?

No matter how small a grid element is chosen, there is usually still some variation in it.  For example on a regional model with a one hectare grid, there may be different forest stands.  If the actual spatial arrangement of those stands doesn’t matter (ie. which is next to which other, or how much edge is there between one kind of area and another) then we can simply represent them in the model by creating a table of how much of each area type there is in each grid element. 

Not all grids are generated as uniform squares.  Polygons of various shapes and sizes can be specified using various coordinate systems.  Geographical information systems usually have at least one layer of overlays defined as a polygon system; and these are often used directly in simulation models.

Though there are methods to accommodate spatial issues – for example edge effect could be represented by adding another variable for the length of edge - sometimes it does matter where something is within any given space unit.    As a conceptual example, even though the proportion of different stand types could be seen to change over time, without knowing their actual shape, one could not calculate a change in their edge lengths. 

A more realistic example of a model problem where location matters is a root rot simulation.  To model root rot dynamics, it was necessary to indicate actual locations of infection.  This was done for each grid cell by providing the x/y coordinates for each center of infestation and then making the simplifying assumption that they are basically circles with a given radius. 

Locations of infestation would not have any relevance to simulation if there wasn’t a change.  Simulation is after all about projecting consequences, given particular processes, into the future.  The spread of a root rot infestation leads to a consideration of movement over time.