232.

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		1.5.5— Example Where the Variance Does Not Exist: The Roughness of Rocks

		The edges of rocks are self-similar. Small sharp peaks over short distances are reproduced as ever larger peaks over ever larger distances.

		The root mean square (RMS) of the heights along the rock profile is the square root of the average of the square of the heights along the rock. As the root mean square of the heights is measured over longer distances on the rock, ever larger peaks are included. Thus the root mean square of the heights increases with the distance measured along the rock. It does not approach a finite, limiting value.

		The root mean square (RMS) of the heights along the rock is related to the variance of the fluctuations in height along the edge of the rock.

		For a non-fractal object, as the variance is measured over longer distances, it approaches a finite value. Thus the variance exists and we identify that limiting value as the "real" value of the variance.

		However, for the fractal edges of rocks the variance increases with the distance measured. The variance does not approach a finite value. Thus the variance does not exist.

 

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