A Brownian surface is a

fractal surface A fractal landscape or fractal surface is generated using a stochastic algorithm designed to produce fractal behavior that mimics the appearance of natural terrain. In other words, the surface resulting from the procedure is not a deterministic, ...

generated via a

fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...

elevation

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

. The Brownian surface is named after

Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...

Example

For instance, in the three-dimensional case, where two variables ''X'' and ''Y'' are given as coordinates, the elevation function between any two points (''x''₁, ''y''₁) and (''x''₂, ''y''₂) can be set to have a mean or

expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...

that increases as the vector distance between (''x''₁, ''y''₁) and (''x''₂, ''y''₂). There are, however, many ways of defining the elevation function. For instance, the

fractional Brownian motion In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gaus ...

variable may be used, or various rotation functions may be used to achieve more natural looking surfaces.

Generation of fractional Brownian surfaces

Efficient generation of fractional Brownian surfaces poses significant challenges. Since the Brownian surface represents a

Gaussian process In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...

with a nonstationary covariance function, one can use the

Cholesky decomposition In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for eff ...

method. A more efficient method is Stein's method, which generates an auxiliary stationary Gaussian process using the circulant embedding approach and then adjusts this auxiliary process to obtain the desired nonstationary Gaussian process. The figure below shows three typical realizations of fractional Brownian surfaces for different values of the roughness or

Hurst parameter The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. Studies involving the Hurst expone ...

. The Hurst parameter is always between zero and one, with values closer to one corresponding to smoother surfaces. These surfaces were generated using
Matlab implementation
of Stein's method. Fractional Brownian surfaces for different Hurst parameter

Fractional Brownian surfaces for different Hurst parameter

References

{{Reflist Fractals

Example

Generation of fractional Brownian surfaces

See also

References