Brownian surface

For instance, in the three-dimensional case, where two variables X and Y are given as coordinates, the elevation function between any two points (x1, y1) and (x2, y2) can be set to have a mean or expected value that increases as the vector distance between (x1, y1) and (x2, y2).

For instance, the fractional Brownian motion variable may be used, or various rotation functions may be used to achieve more natural looking surfaces.

[2] Efficient generation of fractional Brownian surfaces poses significant challenges.

[4] Since the Brownian surface represents a Gaussian process with a nonstationary covariance function, one can use the Cholesky decomposition method.

The figure below shows three typical realizations of fractional Brownian surfaces for different values of the roughness or Hurst parameter.