🚀 go-pugleaf

Hi everybody.

I am not familiar with Earth Sciences very much, so this posting may be naive.

I am looking for a statistical model for the distribution and correlation of
terrain elevation.

Looking at  digitized elevation data of some (small number of) sites that were
far from the sea, and not TOO mountainous, I made the following observations:

1. Elevation follows a Gaussian distribution
2. The auto-correlation function is exponential [ c(x)=exp(-|x|/cleangth) ]
3. The correlation length (clength) is direction-dependent. A contour of the
   correlation lengths (or any equ-level contour of the auto-correlation func.)
   is well approximated by an ellipse.

None of the above approximations was universally excellent, but each was
preaty good for most sites (sometimes the approximations were REALLY good).

Knowing nothing of the relevent sciences, but being of a mathematics background,
I made the following "hand-waving" arguments for each of the observations:

1. Gaussian elevation would form if each terrain patch was formed by many
   successive depositions (or pushing from below) acting in an additive fashion.
2. Exponential auto-correlation function would form if the elevation at any
   point is directly correlated ONLY with its neighboring points (Markovian)
3. If the terrain was isotropic (no preferred direction) and homogenous (no
   preferred positions), the correlation lengths contour would be a circle.
   If we remove the isotropic requirement (as there IS a preferred direction)
   but keep the homogenous requirement - in effect stretching the terrain
   along the preferred direction - the correlation contour would become an
   ellipse.


Well, I am sure none of the above is of any news to the experts in the field.
In fact, I would not be surprised if some of what I said is simply wrong.
I would very much thank anyone who would like to point out the state-of-the-art
in terrain elevation statistical models.

- Well, thanks in advance.

Dan Shoham
shoham@ll.mit.edu