GIS Change Detection Math

I do a lot of topographic\geomorphic change detection, mostly with raster digital elevation models but I've also started getting into point cloud differencing too. It's simple subtraction, but the way you do your subtraction can make a big difference in how you interpret the data. I've had several students ask about this, so I thought a short write up was in order. The examples here are topographic (erosion and deposition), but the concepts apply to any raster-based change detection (i.e. land use/land cover change).


Conventions:

• T₁ = Time 1 (usually the older dataset)
• T₂ = Time 2 (you guessed it...the newer data)
• We'll use blue to represent negative numbers and red to represent positive numbers

• Some people have strong feelings about what colors represent which end of the number line, but we can discuss that at another time...

Change = T₂ - T₁

This is the most common way to do topographic change detection (hint - use this one if you're unsure). It works out nicely that decreases in elevation (erosion) are calculated as a negative numbers and increases in elevation (deposition) are calculated as positive numbers.
In the ArcGIS Python Console:

from arcpy.sa import *
diff = Raster("Time2.tif") - Raster("Time1.tif")

Change = T₁ - T₂

By switching the times, we also switch the signs. Positive numbers switch to represent decreases in elevation (erosion) and negative numbers represent increases in elevation (deposition). There are probably good reasons to use this type of change calculation, but I can't think of any off the top of my head...I'll do this by accident sometimes and get really confused.