Wedge neighborhood on Focal Statistics calculation

Question

I am using Focal Statistics in Spatial Analyst of ArcMap10.2, and use the wedge type of neighborhood for a radius of 10 km around each processing pixel. I want to calculate the maximum elevation for four sectors of aspect which are: Northeast, Southeast, Southwest, and Northwest, which in degrees would mean 0-90, 90-180, 180-270, 270-360 respectively, If 0 degrees were on the north and the wedge would extend clockwise. Of course, as I have read from the ArcGIS Help, "The wedge extends counterclockwise from the starting angle to the ending angle. Angles are specified in arithmetic degrees from 0 to 360, where 0 is on the positive x-axis (3:00 on a clock)". Nevertheless, what I have observed is that 0 degrees are placed on the west (9:00 on a clock) and the wedge extends counterclockwise (90 degrees placed on 6:00 o clock, 180 placed on 3:00 o clock, 270 placed on 12:00 o clock). To be more precise, the images below depict a part of my region. The first image has a wedge with start angle at 0 degrees and end angle at 90, the second has an angle of 90-180, the third 180-270, and the fourth image has an angle of 270-360 degrees.

Answer 1

Here is a dataset in which one grid cell has a nonzero value, shown as a blue dot:

Here is a wedge-shaped neighborhood oriented at 0 degrees north of east, shown in orange:

Its left tip is meant to be centered at each point. It represents a collection of displacements relative to the left tip: it is not yet located anywhere in particular.

Let us consider some points within the map. At each one we will construct a neighborhood of this size and shape, such as at this point x:

Because this neighborhood includes the blue point, the blue point's value will contribute to the focal statistic at x.

The neighborhood of the next point y does not overlap the blue point.

Therefore the value at the blue point does not contribute to the focal statistic at y.

Let's outline all the points like x whose neighborhoods cover the blue point. I show just a few of their neighborhoods (in white). The set of all points whose neighborhoods cover the blue point is also shown in blue:

The effect of the blue point has been spread throughout a neighborhood that is the geometric inverse of the original wedge. (In two dimensions it has been rotated by 180 degrees. In three dimensions--when you are doing true 3D analyses--it is rotated and reflected.)

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We have merely rediscovered what subtraction means geometrically. If you are still in doubt, here is a rigorous mathematical demonstration of the foregoing. "A" neighborhood is defined to be a set N of displacements (dx,dy). "The" neighborhood of any point (x,y) consists of all points of the form (x,y) + (dx,dy) where (dx,dy) is in N. This particular neighborhood is written N + (x,y).

The inverse of a neighborhood, written -N, is the set of opposite displacements (-dx,-dy) where (dx,dy) is in N.

In any focal operation involving N, a point at location (x0,y0) contributes a value to another point at location (x,y) if and only if (x0,y0) lies within the N neighborhood of (x,y); that is, (x0,y0) lies in N + (x,y). This means there is a displacement (dx,dy) in N for which (x0,y0) = (dx,dy) + (x,y). That is algebraically equivalent to (x,y) = (-dx,-dy) + (x0,y0). But that merely means that (x,y) is in the -N neighborhood of (x0,y0).

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These two demonstrations--visual/intuitive, mathematical/rigorous--imply the same thing: When a grid contains prominent values that contribute to focal statistics, those values will appear to be propagated within the -N neighborhood of each prominent value. The visual impression is one of a mosaic of -N neighborhoods: all directions appear to have been reversed.