#016 - The logarithmic waterfront

Highly skewed distributions are those that have values clustered at one end of their range. Direct visualization using a colormap doesn’t work, because only a fraction of the total color spectrum will be used by the data. Two things can fix this, either the colormap is transformed to match the distribution, or the distribution is transformed so that clustering is reduced.

Transforming colormaps is not a good idea, because they are purposefully built with human vision in mind. Perceptually uniform colormaps such as viridis, inferno, magma, and plasma are loaded from a pre-set vector, tailor-made to be color-blindness and luminosity proof. Transforming these colormaps will distort them.

In contrast, transforming data is a routine pre-processing technique, usually through logarithm or square root. Logarithmic transform is useful for continuous data, especially clustered around zero. The logarithmic scale becomes vertically asymptotic towards negative infinity at zero, but in the vicinity of zero the resolution of numbers increases dramatically. However, if the data has actual zeroes this transform will not work, to fix this a constant must be added to “slide” the data across the horizontal axis.

In this case we are interested in the hidden patterns around the tidal margin of downtown Vancouver, the most urbanized region of Burrard Inlet. Here data is truncated, anything below 0 meters (including 0) and anything above 15 meters (including 15) is discarded. What remains is transformed but avoiding the intense scaling of near-zero logarithms, so a constant of 100000 is added. This results in a largely uniform colormap application that enhances subtle shoreline details.



The figure above uses the inferno perceptual colormap and is exported in CMYK colorspace at 600 dpi. Three shorelines are overlaid in white, the 2020 low tide mark at zero meters, the 2020 high water mark, and the buried shorelines from old maps dating between 1880 and 1890. The code below renders truncates and transforms the elevation data and then renders the figure above.

load ENC2020_Polygon
load HighTide2020_lakes
load XYZ_DepthZeroJitter 
load geoXY_Oldest
load geoXY_1890North
load ElevationModelX
load ElevationModelY
load ElevationModelZ
figure('Position',[10,10,1420,780],'Resize','off','Color','k');hold on;
colormap(flipud(inferno));axis off;drawnow;



There are three regions that just before 1880 were shallow enough to be drowned at high tide, marked A, B, and C in the figure above. The logarithmic colormap reveals the shallowest points between water bodies, with bright yellow closest to sea level. To explore further, the figures below are high-contrast renders with a synthetic high-tide that stops as soon as a connection between water bodies occurs at either A, B, or C.

Site A is where Lost Lagoon connects to Second Beach, which occurs when water rises to 4.3 meters above sea level (lowest tide, or 0 meters). Site B is today Hastings, which in 1880 was a much narrower isthmus. This region today would connect False Creek Flats with Inner Harbour if the water rose 8.11 meters. Interestingly, at 8.01 the connection is broken, yet the area is quite large, meaning that the flooded region is today remarkably flat with an elevation difference of 10 centimeters. Finally, site C is a lost creek that once extended northward off False Creek Flats, where Raymur Avenue is today. This narrow meander connects at 9.1 meters, but remains non-flat and creek-shaped even today, as shown in the logarithmic colormap above.




Many thanks for the council of Michelle George and Mike George for sharing Tsleil-Waututh knowledge about the pre-contact state of Burrard Inlet.