The authors’ aim is to introduce stochastic models allowing arbitrary combinations of fractal dimension

$D$ and Hurst coefficient

$H$ which characterizes long-memory dependence. For self-affine models in

$n$-dimensional space such as fractional Brownian motion one has

$D+H=n+$1. The authors’ key item is the Cauchy class consisting of the stationary Gaussian random fields

${\left(Z\left(x\right)\right)}_{x\in {\mathbb{R}}^{n}}$ with correlation function

$c\left(h\right)={(1+{\left|h\right|}^{\alpha})}^{-\beta /\alpha},$ $h\in {\mathbb{R}}^{n}$, where

$\alpha \in (0,2]$ and

$\beta >0$. This simple model allows any combination of the two parameters

$D$ and

$H\xb7$ Two figures provide displays of profiles and images in which the effects of fractal dimension and Hurst coefficient are decoupled. Special attention is paid to the problem of estimating

$D$ and

$H$ when the equation

$D+H=n+1$ does not hold. Related models able to separate fractal dimension and Hurst effect are also discussed.