Characterizing surface smoothness via estimation of effective fractal dimension.

*(English)*Zbl 0804.62079Summary: The fractal dimension \(D\) of stationary Gaussian surfaces may be expressed very simply in terms of behaviour of the covariance function near the origin. Indeed, only the covariance of line transect samples is required, and that fact makes practical estimation of \(D\) relatively straightforward. The case of non-Gaussian surfaces is more poorly understood, but we might define ‘effective fractal dimension’ in terms of the covariance function, as though the surface were Gaussian.

We suggest simple practical methods for estimating effective fractal dimension, based on the variogram. Like techniques proposed recently by C. C. Taylor and S. J. Taylor [ibid. 53, 353-364 (1991)] ours are founded on log-linear regression. However, the context of our problem is more clearly defined, and so we can develop significantly more detailed theory than Taylor and Taylor could. The problem of the choice of smoothing parameter is addressed, and our methods are applied to real data on the smoothness of the polished metal surface.

We suggest simple practical methods for estimating effective fractal dimension, based on the variogram. Like techniques proposed recently by C. C. Taylor and S. J. Taylor [ibid. 53, 353-364 (1991)] ours are founded on log-linear regression. However, the context of our problem is more clearly defined, and so we can develop significantly more detailed theory than Taylor and Taylor could. The problem of the choice of smoothing parameter is addressed, and our methods are applied to real data on the smoothness of the polished metal surface.

##### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62M09 | Non-Markovian processes: estimation |

62M99 | Inference from stochastic processes |