From sigmoidal curve family I prefer to use the Logistic Function, which is able to describe quite well the natural probability decreasing.
y = L / 1 + e^k(x-f)
You need to decrease the probability from 1 to 0 between 250 and 1250 meters of distance, so the interval is 1000 m and the midpoint at 500 m.
First you have to compute the Euclidean Distance, which creates a raster file with the distance values from your area of interest.
Then you need to use the Raster Calculator with conditional statement: from 0 to 250 set 1, from 250 to 1250 compute the Logistic Function, beyond 1250 set 0.
The Logistic Function has these parameters:
- L = 1 (the maximum probability value)
- f = 500 (midpoint distance)
- k = 0.01 (curve steepness)
If your distance raster is "EucDist", the Raster Calculator expression is:
Con("EucDist"<=250, 1, Con(("EucDist">250) & ("EucDist"<1250), (1/(1+Exp(0.01*(500-(1250-"EucDist"))))), 0))