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Chapter 4:
Using Rayleigh-Rice to Calculate Smooth-Surface Statistics from the BRDF
Author(s): John C. Stover
Published: 2012
DOI: 10.1117/3.975276.ch4

Statistics are no substitute for judgment. - Henry Clay

Chapters 2 and 3 have revealed the surface PSD function as the logical path to move back and forth between surface topography and scatter generated by roughness. This chapter concentrates on application of the Rayleigh-Rice relationship to the inverse-scatter problem: the calculation of reflector surface statistics from measured scatter data. This is important because a number of industrial surfaces meet the smooth, clean, front-surface-reflective requirements introduced in Chapter 1, and in these cases, scatter measurement can be used as a fast, noncontact method of microroughness characterization. The special cases of one-dimensional grating-like surfaces and isotropic two-dimensional surfaces receive most of the attention. The conversion of the Rayleigh-Rice diffraction result to the Davies-Bennett TIS relationship is also reviewed. Other than the scatter measurement geometry, the details of how the scatter data is obtained is left for Chapter 7. Chapters 1 - 3 are used as source material.

4.1 Practical Application of the Rayleigh-Rice Perturbation Theory

The use of scatter data as a means of specifying reflector surface quality is a powerful noncontact inspection technique. This chapter discusses the inverse-scatter problem, where BRDF data is used to calculate the PSD and the various surface parameters. Eq. (3.43), introduced in Section 3.3, gives the general relationship between the PSD of an arbitrary, smooth, clean, front-surface reflector and the corresponding scatter pattern, or BRDF. In Eq. (4.1), the terms have been rearranged so that the BRDF is given directly in terms of measurement and sample parameters:


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Bidirectional reflectance transmission function


Scatter measurement



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