Second-order small perturbation method for transmission from dielectric rough surfaces

Abstract

Although the small perturbation method (SPM) for rough surface scattering has been studied extensively in problems in optics, remote sensing and propagation, there are fewer studies on rough surface transmission by the SPM. In this paper, from Huygens' principle and the extinction theorem, the SPM is used to derive the transmitted field to the second order, and expressions for the bidirectional transmission coefficient and the total surface transmittance to the second order are developed for the random rough surface. The refined expressions can be applied to the situations where the transmission characteristics of a random rough surface need to be more accurately calculated. For example, to calculate the brightness temperature of stratified rough media by the incoherent method, we have to know the bidirectional transmission coefficient or transmittance of random rough surface as accurately as possible. The accuracy of the presented expressions is verified through the conservation of energy. It is shown that the transmission characteristics calculated by SPM to the first order violate conservation of energy, whereas solutions to the second order conform to energy conservation much better. This is particularly important for the calculation of transmittance or emissivity.

Brightness Temperature Calculation of Lunar Crater: Interpretation of Topographic Effect on Microwave Data From Chang'E

“…In addition, the unconsidered factors, including surface roughness and subsurface rock fragment, may be the reasons for the departure between simulation and observation. For example, surface roughness on small scales may increase the TB value [53]. Volume scattering and high thermal inertia, due to subsurface rock fragment, can darken the emission [7] and cause unusual physical temperature of lunar surface [54].…”

Effects of slightly rough surfaces on the brightness temperature of the lunar regolith

“…Since the distribution of local elevations for lunar soil is typically well described by Gaussian statistics [ Helfenstein and Shepard , ], then the lunar surface of small‐scale roughness is assumed Gaussian here, i.e., the Gaussian height distribution and Gaussian autocorrelation function W (⋅), as used in Appendix. We utilize the second‐order SPM to calculate the bidirectional transmission coefficient [ Chen et al ., ] . It can be divided into zero‐order contribution , first‐order contribution and , and second‐order contribution , where θ 2 c , ϕ 2 c correspond to the direction of coherent transmission, determined by k 1 sin θ 1 = k 2 sin θ 2 c , ϕ 2 c = ϕ 1 .…”

“…The expressions of , , , and are given in the Appendix, and the derivation is given by Chen et al . [].…”

“…t 12 pq θ 1 ; f 1 ; θ 2 ; f 2 ð Þ is the bidirectional transmission coefficient for q polarization incident of layer 2 in the direction (θ 2 ,f 2 ) and the p polarization transmitted wave of layer 1 in the direction (θ 1 ,f 1 ), where p can be h or v. Since the distribution of local elevations for lunar soil is typically well described by Gaussian statistics [Helfenstein and Shepard, 1999], then the lunar surface of small-scale roughness is assumed Gaussian here, i.e., the Gaussian height distribution and Gaussian autocorrelation function W(Á), as used in Appendix. We utilize the secondorder SPM to calculate the bidirectional transmission coefficient [Chen et al, 2011]…”

“…[9] In the paper, the small perturbation method (SPM) is utilized to study the effect of the small-scale roughness of the lunar surface on TBs, since the SPM is valid when the standard deviation of the rough lunar surface is much smaller than the wavelength. To increase the accuracy of the emission, the second-order SPM transmittances are used because the second-order solution is necessary to ensure energy conservation [Chen et al, 2011]. Meanwhile, the noncoherent method is utilized to calculate the emission of the nonuniform regolith which can be discretized to a multilayer planar layered media.…”