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In response to the work of Daly et al. (1994), an attempt was made to filter the DEM to 1000 m cell size. However, due to a misinterpretation of the neighbourhood statistics functions in ArcMap, instead of aggregating 40 x 40 25 m cells (resulting in 1000 m resolution), the elevation derivative produced was a 1000 m resolution but then also smoothed by a 40 x 40 cell neighbourhood function. The new layer had 1000 m grid cells with values smoothed by a 40 km x 40 km filter. Although this had not been the intended result, it had produced an elevation-based variable that displayed a significantly greater degree of correlation with annual and monthly precipitation values than had been displayed against the precise elevation values of the stations. It was evident this degree of filtering better encapsulated the orographic effect than did the original 25 m DEM. The filter applied was a standard low pass filter in which the neighbourhood mean calculated for the moving window is assigned to the central grid cell.
What level of spatial filtering begins to capture a reasonable amount of this so called orographic effect? Is there an optimal filtering level to apply prior to use of the DEM as a predictor variable? To this end, an investigation was conducted to examine how the size of the neighbourhood used for the low pass filter and the adjusted cell resolution affect the correlation between elevation and precipitation.
ArcMap’s Spatial Analyst extension was utilized to generate test layers of varying resolution and neighbourhood influence. Layers were calculated with cell sizes of 250 m, 500 m, and 1000 m then filtered (low pass) according to neighbourhoods of 5, 10 and 15 cells as well as with no filter application. Unfortunately, applying neighbourhood filters of equal cell count rather than actual equal area provided a slight hindrance to intuitive interpretation. The results of this are shown in Table 4 and Figures 6, 7, 8, 9.
The results revealed several important relationships. Firstly, the degree of correlation increases with increasing neighbourhood size for all three resolutions examined. This is best exemplified in Figures 6, 7, and 8. However, the R-squared value appeared to increase at a decreasing rate once the larger neighbourhoods were utilized. Figure 8 displays this trend best as it considers neighbourhoods of 5 km, 10 km and 15 km. The need for a more precise understanding of this decreasing marginal increase seemed to warrant further investigation (discussed below). Secondly, there is almost no difference in R-squared values between different cell resolutions filtered according to the same size neighbourhood. This can be observed in Figure 9 by comparing the 1000_5 and 500_10 models (5000 m neighbourhood), the 500_5 and the 250_10 models (2500 m neighbourhood), or the “Absolute”, 1000_0, 500_0, 250_0, and 25_0 models (no low pass filtering). The correlation coefficient calculated for a comparison between the 1000_5 and 500_10 models for annual data and all months exceeded 0.997.
In response to this first trend and the knowledge of the correlation observed with the 40 km neighbourhood, it appeared that it would be necessary to conduct further investigation to determine if there would be an optimal neighbourhood size to consider. The second trend showed that there was little or no difference in performance among the three main grid cell sizes compared, meaning that the choice would have to be arbitrary. A grid cell size of 500 m was chosen as a trade-off between the resulting precision possible in the final precipitation maps and computing load. The increased precision of 500 m resolution over 1000 m resolution was judged to be important, whereas it did not seem that an increase to the precision of 250 m resolution would yield a meaningful improvement in the final products. Furthermore, 500 m resolution would improve computing speed over the 250 m resolution.
The subsequent investigation compared the effects of the size of neighbourhood filter used with the resulting R-squared value for precipitation and filtered elevation, all at a cell resolution of 500 m. Neighbourhoods of 0, 2.5, 5, 7.5, 10, 15, 20, and 30 km were compared. These results appear in Table 4 and Figure 10.
A common general trend is displayed among all thirteen monthly and annual data sets. The R-squared measure of correlation increases rapidly with increasing neighbourhood size then begins to decrease in rate. The point of inflection appears to roughly occur between 7.5 km and 10 km. The choice was made to use the 10 km neighbourhood for our analysis. This was a trade-off between the increase in performance and the increase in computing demand and potential intuitive complexity for readers. Beyond a 10 km neighbourhood there is not a substantial rate of increase for predictive power.
From the results displayed in Figure 10, it appears that the correlation seems to continue increasing with increasing neighbourhood size, thus increasing predictive power, albeit at a decreasing rate. However, it is likely that the increasing degree of correlation is primarily for stations at which precipitation is predominantly controlled by small scale patterns, at the expense of the ability to resolve large scale differences. For example, large scale variation such as the difference in precipitation between the top and bottom of Burnaby Mountain should be more poorly correlated with elevation filtered over an increasing spatial extent because the modified elevation values will show a much smaller difference between the top and bottom of the mountain. However, the loss of this ability to resolve correlations at a large scale will likely be in exchange for greater correlation with stations whose prime influence operates on a much smaller scale. If there are more precipitation stations affected by small scale variation in elevation than by large scale variation in elevation, it would be expected to see a continual rise in the correlation between elevation and precipitation while increasing the neighbourhood filter size. This seems reasonable since a qualitative examination of station situation and location indicates that the bulk of the stations in this study should be more greatly affected by small scale variation in elevation, that is, the dominant north-south trend. In the Lower Mainland, the most predominant trend in elevation, and consequently precipitation, is the continual small scale increase in elevation from the south to the north. This trend is far greater than the east-west trend, as illustrated in the sections of the Discussion.
Although the investigation of the effects of neighbourhood size did not include neighbourhoods larger than 30 km, it is expected that after the observed gradual increase in correlation values, there will eventually be a decrease. This must occur because at the extreme end of the spectrum, the neighbourhood considered would be so large that every cell would be smoothed to the same value, in which case there would be a single value for elevation that would poorly correlate to the diversity of precipitation values. It is suspected that the peak correlation would occur with a neighbourhood size of 40 – 60 km. Approaching this theoretical maximum, the derived surface would be approaching a planar surface in which the east-west trend would basically be zero but the north-south trend still retained a significant proportion of the original information content but with little or none of the noise. Beyond this theoretical optimum, increasing the neighbourhood filter size would begin to significantly degrade the level of information retained in the north-south trend, to the extreme situation in which the trend has been completely removed and the resulting surface is merely a flat plain (as referred to above). In the current work, this theoretical optimum was not chosen. Instead, the 10 km neighbourhood size was chosen as a compromise between the increasing correlation achieved with greater neighbour size and the ability to retain a reasonable degree of large scale variation.
It is believed that the reason that July and August do not reflect the same trend nearly as well is due to the increasing importance of convective precipitation. Convective precipitation occurs more frequently during the summer months and since precipitation is generally much lower, convective precipitation will represent a greater proportion of total precipitation. Convective precipitation is not strongly dependent on elevation and topography so therefore it is not surprising to see a less pronounced change while modifying the elevation layer.
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