APPROACH TO A SOLUTION
Now, that the problem of defining a discrete treeline is solved, the treeline data can be used to answer to questions that are sought to be answered by this project: "What is the elevation at treeline?" and "What are the factors determining this limit?"
"What is the elevation at treeline?"
It will be straightforward to answer the first question. The treeline model will extract only those pixels from the DEM that are part of the treeline. As a result, an average elevation for these pixels can be calculated. Then, the aspect of slopes will be calculated from the DEM, and a similar extraction procedure will result in an image with aspect values only at treeline pixels. These pixels will be divided in 4 categories, namely north, east, south, and west. The treeline is essentially cut in four, so the initial process of calculating an average treeline elevation can be repeated for each of these 4 treelines. As a result, we will know the elevation of the treeline depending on aspect. The quantitative results of this analysis are very meaningful and accurate to an acceptable degree. You can consider this to be the crucial part of my analysis.
Theoretically, answering the second question is an incredibly difficult and diverse procedure. The complexity of factors that influence the altitudinal limit of tree growth is astonishing. Using GIS to produce quantitative results, which can be used to predict a change of treeline altitude with a change in environmental factors, is something suitable for a PhD thesis, not this project. I cannot emphasize enough that no accurate quantitative results can be expected from my analysis, as the model I created is highly simplistic. The model is only used to show broad general trends in terms of altitudinal temperature and precipitation variation. Read more about this in the section Methodological Problems.
"What are the factors determining this limit?"
After extensive study of bioclimatic literature, and temperature data of the study area, I found that there is a seasonal difference in the amount of temperature decrease with elevation. I decided on a temperature lapse rate of 5 degrees Celsius per km in the winter, and 7 degrees Celsius per km in the summer. This means, for example, that temperature decreases by 7 degrees in the summer with an increase of 1000 m elevation. The average global lapse rate is 5.7 degrees Celsius per km. The average lapse rate for the BC Rockies is between 5.5 and 6.5 degrees. I derived sample lapse rates from the collected temperature climatic data, incorporated the average lapse rates found in research literature, and decided on appropriate lapse rates that were identical for both study areas in order to make the results comparable. The continuous surface of temperature values was created with the following formula:
(temperature in deg C) = (temperature at climate station in deg C) - ( (lapse rate per meter) * ( [DEM] - (elevation at climate station in meters) ) )
I used the usual extraction procedure (using OVERLAY A covers B except where 0) to show only temperature values for treeline pixels. I could then calculate the average temperature at treeline level both for January and July, and compare between the two study sites. As the result of this is very significant (you will see), I developed a way to represent the temperature surface in a 3-dimensional image (using DEM elevation for the z-value) with the treeline superimposed on it. I had to reclassify the treeline, so it was fit for OVERLAY. I then used the STRETCH operation in order to transform temperatures into byte values from 0 to 255, now suitable as a drape image. The ORTHO operation creates the 3D image.
Altitudinal variation of precipitation is also very complex, and not as well researched as temperature variation. I could identify 12 different stations that would supply useful data for the region (9 in the Yoho area, and 3 in the Whistler area). After preparation of that data, I ran a regression on it, resulting in linear equations, describing change of precipitation with an increase in elevation. The correlation coefficient r was 0.89 for Yoho, and 0.98 for Whistler. Running a regression on the Whistler data is questionable, because the data was only collected at two locations, twice in Whistler, and once on Whistler Mountain. However, the close proximity of the two locations, and the fact that one is actually in the valley, and the other one is up on a mountain neighbouring the valley, makes is possibly even more meaningful than the results for Yoho. There, I could run a regression on 9 stations, which has meaningful results. However, all of those stations are in the valley (at different elevations in the valley, but still: always at low points of the terrain). As a result of the lack of climate data, the precipitation surface is only suitable for determining approximate information, such as high vs. low precipitation. The usual extraction technique was used to create an image with precipitation values only at treeline level. Average winter precipitation, which is entirely snow, taking into account the freezing temperatures at that treeline level, could then be calculated.
Finally, I employed the usual extraction technique to create an image that shows only the elevation at pixels that are covered by glaciers. I could then estimate an approximate minimum elevation of those glaciers, and compare that information to treeline elevation, in order to see how much those two phenomena correlate.
