A Brownian surface is a fractal surface

Fractal landscape

A fractal landscape is a surface generated using a stochastic algorithm designed to produce fractal behaviour that mimics the appearance of natural terrain...

 generated via a fractal

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

 elevation function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

.

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

Expected value

In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

 that increases as the vector distance between (x₁, y₁) and (x₂, y₂). There are, however, many ways of defining the elevation function. For instance, the fractional Brownian motion variable may be used, or various rotation functions may be used to achieve more natural looking surfaces.