To begin, we need to understand two concepts: the correlation function and the power spectrum of a surface.

Surfaces in the real world are not perfectly smooth, they're rough. Such surfaces are mathematically represented as realisations of a

*random field*. This means that the height of the surface at each point is effectively sampled from a statistical distribution. Each realisation of a random field is unique, but one can classify surface types by the properties of the random field from which their realisations are drawn. For example, each sheet of titanium manufactured by a certain process will share the same statistical properties, even though the precise surface morphology of each particular sheet is unique.

Let us denote the height of a surface at a point x as h(x). The height function will have a mean <h(x)> and a variance. (Here and below, we use angular brackets to denote the mean value of the variable within the brackets). The variance measures the amount of dispersion either side of the mean. Typically, the variance is calculated as:

Var = <h(x)

^{2}> − <h(x)>

^{2}

Mathematically, the height at any pair of points, x and x+r, could be totally independent. In this event, the following equation would hold:

<h(x)h(x+r)> = <h(x)

^{2}>

The magnitude of the difference between <h(x)h(x+r)> and <h(x)

^{2}> therefore indicates the level of correlation between the height at points x and x+r. This information is encapsulated in the height auto-correlation function:

ξ(r) = <h(x)h(x+r)> − <h(x)

^{2}>

Now the auto-correlation function has an alter-ego called the power spectrum. This is the Fourier transform of the auto-correlation function. It contains the same information as the auto-correlation function, but enables you to view the correlation function as a superposition of waves with different amplitudes and wavelengths. Each of the component waves is called a

*mode*, and if the power spectrum has a peak at a particular mode, it shows that the height of the surface has a degree of correlation at certain regular intervals.

Related to the auto-correlation function is the height-difference correlation function:

C(r) = <(h(x+r)−h(x))

^{2}>

This is essentially the variance in height as a function of distance from an arbitrary point x. This is a useful function to plot graphically because it represents the difference between the auto-correlation function and the overall variance, as a function of distance r from an arbitrary point x:

C(r) = 2(Var−ξ(r))

Which brings us to self-affine fractal surfaces. For such a surface, a typical height-difference correlation function is plotted below, (

*Evaluation of self-affine surfaces and their implications for frictional dynamics as indicated by a Rouse material*, G.Heinrich, M.Kluppel, T.A.Vilgis, Computational and Theoretical Polymer Science 10 (2000), pp53-61).

Points only a small distance away from an arbitrary starting point x can be expected to have a height closely correlated with the height at x, hence C(r) is small to begin with. However, as r increases, so C(r) also increases, until at a critical distance ξ

_{||}, C(r) equals the variance to be found across the entire surface. Above ξ

_{||}, C(r) tends to a constant and ξ(r) tends to zero. ξ

_{||}can be dubbed the

*lateral correlation length*. In road surfaces, it corresponds to the average diameter of the aggregate stones.

To understand what a self-affine fractal surface is, first recall that a self-similar fractal surface is a surface which is invariant under magnification. In other words, the application of a scale factor x → a⋅x leaves the surface unchanged.

In contrast, a self-affine surface is invariant if a separate scale factor is applied to the horizontal and vertical directions. Specifically, the scale factor applied in the vertical direction must be suppressed by a power between 0 and 1. If x represents the horizontal components of a point in 3-dimensional space, and z represents the vertical component, then it is mapped by a self-affine transformation to x → a⋅x and z → a

^{H}⋅z, where H is the

*Hurst exponent*. In the height-difference correlation function plotted above, the initial slope is equal to 2H, twice the value of the Hurst exponent.

Note, however, that road surfaces are considered to be

*statistically*self-affine surfaces, which is not the same thing as being exactly self-affine. If you zoomed in on such a surface with the specified horizontal and vertical scale-factors, the magnified subset would not coincide exactly with the parent surface. It would, however, be drawn from a random field possessing the same properties as the parent surface, hence such a surface is said to be statistically self-affine.

A yet further adaptation is necessary to make the self-affine model applicable to road surfaces. Roads are known to be characterised by two distinct length-scales: the macroscopic one determined by the size of aggregate stones, and the microscopic one determined by the surface properties of those stones, (see diagram below).

One attempt to adapt the self-affine model to road surfaces introduces two distinct Hurst exponents, one for the micro-roughness and one (purportedly) for the macro-roughness, as shown below, (

*Investigation and modelling of rubber stationary friction on rough surfaces*, A.Le Gal and M.Kluppel, Journal of Physics: Condensed Matter 20 (2008)):

This, however, doesn't seem quite right. The macro-roughness of a road surface is defined by the morphology of the largest asperities in the road, the stone aggregate. Yet, as Le Gal and Kluppel state, a road surface only displays self-affine behaviour "within a defined wave length interval. The upper cut-off length is identified with the largest surface corrugations: for road surfaces, this corresponds to the limit of macrotexture, e.g. the aggregate size."

It's not totally clear, then, whether the macro-roughness of a road surface falls within the limits of self-affine behaviour, or whether it actually defines the upper limit of this behaviour.

So whilst the notion that a road surface is statistically self-affine appears, at first sight, to have been empirically verified by the correlation functions and power spectra taken of road surfaces, perhaps there's still some elbow-room to suggest a generalisation of this concept.

For example, consider

*mounded surfaces*. These are surfaces in which there are asperities at fairly regular intervals. In the case of road surfaces, this corresponds to the presence of aggregate stones at regular intervals. Such as surface resembles a self-affine surface in the sense that it has a lateral correlation length ξ

_{||}. However, there is an additional length-scale λ defining the typical spacing between the asperities, as represented in the diagram below, (

*Evolution of thin film morphology: Modelling and Simulations*, M.Pelliccione and T-M.Lu, 2008, p50).

In terms of a road surface, whilst ξ

_{||}characterizes the average size of the aggregate stones, λ characterizes the average distance between the stones.

In terms of the height-difference correlation function C(r), a mounded surface resembles a self-affine surface below the lateral correlation length, r < ξ

_{||}. However, above ξ

_{||}, where the self-affine surface has a constant profile for C(r), the profile for a mounded surface is oscillatory (see example plot below, ibid. p51). Correspondingly, the power spectrum for a mounded surface has a peak at wavelength λ, where no peak exists for a self-affine surface.

The difference between a mounded surface and a genuinely self-affine surface is something which will only manifest itself empirically by taking multiple samples from the surface. Individual samples from a self-affine surface will show oscillations in the height-difference correlation function above the lateral correlation length, but the oscillations will randomly vary from one sample to another. In contrast, the samples from a mounded surface will have oscillations of a similar wavelength, (see plots below, from

*Characterization of crystalline and amorphous rough surface*, Y.Zhao, G.C.Wang, T.M.Lu, Academic Press, 2000, p101).

Conceptually, what's particularly interesting about mounded surfaces is that they're generalisations of the self-affine surfaces normally assumed in tyre friction studies. Below the lateral correlation length-scale ξ

_{||}, a mounded surface

*is*self-affine (M.Pelliccione and T-M.Lu, p52). One can say that a mounded surface is

*locally*self-affine, but not

*globally*self-affine. Note that whilst every globally-affine surface is locally self-affine, not every locally self-affine surface is globally self-affine.

A self-affine road surface will have aggregate stones of various sizes and separations, whilst a mounded road surface will have aggregate stones of similar size and regular separation.

In fact, one might hypothesise that many actual road surfaces in the world are indeed locally self-affine, but not globally self-affine. For this to be true, it is merely necessary for there to be some regularity in the separation of aggregate within the asphalt. If the distance between aggregate stones is random, then a road surface can indeed be represented as globally self-affine. However, if there is any regularity to the separation of aggregate, then the surface will merely be locally self-affine. If true, then existing academic studies of tyre friction have fixated on a special case which is a good first approximation, but which does not in general obtain.

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