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[PP2-2-6-5] Single-scale & multi-scale roughness

In reality, one random surface has multiple roughness scales, since the commonly used surface description based on single-scale roughness parameters does not comprise all the properties of natural surfaces relevant for describing wave scattering. Depending on the wavelength λ of the microwave sensor the dimension of the surface roughness parameters s and l correspond to specific roughness scales. In case of multi-scale roughness, the equivalent RMS height is a composite of the individual RMS heights at different roughness scales (1). A three-scale surface, as shown in Fig. 1, for example consists of a small-scale high-spatial frequency variation (c) ‘riding’ on top of the larger scales, the medium-scale perturbation (b) and the large-scale undulation (a). At microwave frequencies, the centimeter scale is the scale of roughness of primary importance, since λ is on the order of centimeters to a few tens of centimeters. For natural surfaces it is very difficult to measure millimeter-scale roughness.

Explanation

In reality, one random surface has multiple roughness scales, since the commonly used surface description based on single-scale roughness parameters does not comprise all the properties of natural surfaces relevant for describing wave scattering. Depending on the wavelength  of the microwave sensor the dimension of the surface roughness parameters  and  correspond to specific roughness scales.

In case of multi-scale roughness, the equivalent RMS height is a composite of the individual RMS heights at different roughness scales:

.                                                               (1)

A three-scale surface, as shown in Fig. 1, for example consists of a small-scale high-spatial frequency variation (c) ‘riding’ on top of the larger scales, the medium-scale perturbation (b) and the large-scale undulation (a).

 

                                                                                                                                  

                                                                      Fig. 1. Multi-scale surface roughness of a three-scale surface. a) large-scale undulation, b) medium-scale perturbation, c) small-scale high-spatial frequency variation [Ulaby et al., 2014].

 

At microwave frequencies, the centimeter scale is the scale of roughness of primary importance, since  is on the order of centimeters to a few tens of centimeters. For natural surfaces it is very difficult to measure millimeter-scale roughness.

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