My county ( Pima, Arizona ) uses a number called “average cross slope” which they define as

ACS = (I * L)/A


I = contour interval  
L = length of contour lines in the area   
A = area

Given a unit square with corners (0,0) (1,1), a contour interval of 1 and two contour lines (0,0)(0,1) and (1,0)(1,1), I get L=2,I=1,A=1 and ACS = (I*L)/A = 1*2/1 = 2

Using the same unit square, a contour interval of 0.5 and three interval lines (0,0)(0,1), (0.5,0)(0.5,1) and (1,0)(1,1) I get

ACS = (I*L)/A = (0.5*3)/1 = 1.5 

Since these two examples represent the same topography, I would expect the result to be the same.

Also, the county refers to average cross slope as a percentage. Intuitively, since the topography is a 45 degree angle running left to right, I would think the average slope would be 1. (slope = rise/run), or 100%

So I have three different answers for the same topography, 2, 1.5 and 1.

I’m not a GIS person, I’m a software guy and owner/builder.

Is this a common equation in GIS?

  • The answer is 1. You need half of the lengths of 2 extreme contours to compute their total length.
    – FelixIP
    Nov 4, 2019 at 3:03
  • In a real life, i.e. multiple contours the correction above will have little effect on computed slope.
    – FelixIP
    Nov 4, 2019 at 3:07

1 Answer 1


Contours do not cross each other, so I cannot explain it using your example. Instead, let me illustrate what this ACS does by a picture below.

enter image description here

Looking at the pictures [1] and [2], you will find steep slope is represented by dense contours [1], hence the total length of contours gets longer than in gentle slope area [2].

What this ACS is good at is seen in the [3] (bumpy area) example. Overall slope of [3] is even less than that of [2], but internally it has complex topography. ACS can pick up such complicated topographical change as far as it is captured by contours.

In one word, it is an indicator to show topographical complexity in given area. I do not know if it is commonly used. Probably developed before DEM (digital elevation models) age.

  • 2
    That's a very interesting metric.. can it be calculated in a focal statistic to indicate roughness (rockyness/undulation perhaps) to show areas where vehicular access is likely to be difficult (or more fun, depending on what vehicle the observer has)?. Nov 4, 2019 at 3:36
  • 1
    @MichaelStimson I wonder how we can replicate this approach using raster data. Maybe we will filter out high-frequency changes in the elevation first, then calculate statistics (focal and/or block)...not sure. Outputs would help us to understand the big picture of terrain surface, and indeed useful for vehicles as you say.
    – Kazuhito
    Nov 4, 2019 at 8:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.