A Brownian surface is a

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

generated via a

fractal In mathematics, a fractal is a 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 scales, as illu ...

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-oriente ...

. As with

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

, Brownian surfaces are named after 19th-century biologist Robert Brown.

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, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...

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 Gauss ...

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, i.e. e ...

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 effici ...

method. A more efficient method is Stein's method, which generates an auxiliary stationary Gaussian process using the

circulant embedding In linear algebra, a circulant matrix is a square matrix in which all row vectors are composed of the same elements and each row vector is rotated one element to the right relative to the preceding row vector. It is a particular kind of Toeplitz ...

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 expon ...

. 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