[PP2-2-6-4] Surface roughness slope
The root-mean-square (RMS) slope m of a one dimensional height profile for one random surface is given by (1), with s as the standard deviation of the surface height variation (or RMS height), and p''(0) as the second derivative of the surface correlation function p(ξ), evaluated at ξ=0. Since p(ξ) is an even function, p''(0) is a negative quantity. For modeling of electromagntic scattering at soil surfaces, assumptions of the functional forms of p(ξ) have to be made. The most common known forms are the exponential and Gaussian correlation functions. Additionally, some models allow the assumption of a x-Power correlation function, with x as value between 1 and 2. For the varying surface correlation functions, the RMS slope m is given by (2)-(4). In literature, for L-band, the slope m should be lower than 0.3 or 0.4 in case of single scattering and bare soil surfaces with moderate RMS heights.
Explanation
The root-mean-square (RMS) slope m
of a one dimensional height profile for one random surface is
(1)
with 
as the standard deviation of the surface height variation (or RMS height), and 
as the second derivative of the surface correlation function 
, evaluated at 
. Since is an even function, is a negative quantity.
For modeling of electromagntic scattering at soil surfaces, assumptions of the functional forms of have to be made. The most common known forms are the exponential and Gaussian correlation functions. Additionally, some models allow the assumption of a 
-Power correlation function, with as value between 1 and 2. For the varying surface correlation functions, the RMS slope is given by:
(2)
(3)
(4)
In literature, for L-band, the slope should be lower than 0.3 or 0.4 in case of single scattering and bare soil surfaces with moderate RMS heights.
External resources
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Ulaby, F. T., Long, D.G., Blackwell, W., Elachi, Ch., Fung, A., Ruf, C., Sarabandi, K., Zebker, H. H. and van Zyl, J. J. (2014). Microwave radar and radiometric remote sensing. University of Michigan. Press Ann Arbor. ISBN: 978-0-472-11935-6
Ulaby, F. T., Long, D.G., Blackwell, W., Elachi, Ch., Fung, A., Ruf, C., Sarabandi, K., Zebker, H. H. and van Zyl, J. J. (2014). Microwave radar and radiometric remote sensing. University of Michigan. Press Ann Arbor. ISBN: 978-0-472-11935-6
Learning outcomes
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1587 - Model surface roughness slope
Model surface roughness slope
Self assessment
Completed
Outgoing relations
- [PP2-2-6-4] Surface roughness slope subclassof [PP2-2-6] Surface roughness
Contributors
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Anke Fluhrer
https://www.researchgate.net/profile/Anke_Fluhrer