An approximate solution to the radiative transfer equation for bistatic scattering from a rough surface covered by a tenuous distribution of particulate (scattering and absorbing) media is derived by means of a series expansion in the scattering coefficient of the covering layer up to the first order. The formulation of the successive orders of a scattering series is reviewed, and an analytic solution to the first-order interaction contribution is given by means of a series expansion of the azimuthally averaged product of the bidirectional reflectance distribution function of the surface and the scattering phase function of the covering layer.
© 2016 Optical Society of America
The formulation of an approximate analytic radiative-transfer model to calculate the radiation scattered by a rough surface covered by a tenuous distribution of particulate media (also referred to as “turbid media”) as illustrated in Fig. 1 is discussed. Our motivation for investigating the possibility for deriving an analytical solution for first-order interaction problems stems from studies in the field of satellite-based microwave remote sensing. The effects induced by a vegetative coverage of a soil surface on the backscatter in the microwave domain are commonly treated via a zero-order approximation of the solution to the radiative transfer equation (RTE) [1–5]. However, first- and higher-order interaction contributions are either added via empirically driven correction terms or assumed to be negligible to omit the high computational effort and furthermore to circumvent the problem of under-determination since the required bistatic scattering properties of the vegetative coverage and the soil surface are generally rarely known. In the following, it is shown that, by using approximate analytic functions to represent the bistatic scattering properties, an approximation of the first-order contributions can be gained with reasonable computational effort, providing a consistent estimate of necessary corrections (applied to microwave backscatter observations) in the retrieval of soil and vegetation characteristics. For the sake of generality, the scattering distributions of the surface and the covering layer are defined as general functions.
To clarify the appearing equations, the representation of the solution to the RTE in terms of a series expansion in the scattering coefficient of the covering layer (following Fung  and Ulaby et al. ), based on the assumption that the covering layer can be considered as a weakly scattering medium, is reviewed. The zero-order approximation to this expansion is the widely known (or water-cloud) model as used in the remote sensing community . In Section 3, the general solution to the first-order interaction contribution is presented in detail. Since the solution is based on an expansion of the azimuthally averaged product of the bidirectional reflectance distribution function (BRDF) and the scattering-phase function of the covering layer, the existence of those expansion coefficients is briefly discussed in Section 4.
2. SUCCESSIVE ORDERS OF SCATTERING APPROXIMATION TO THE RTE
A. Separation of the RTE
The well-known RTE , governing the alteration of a beam of specific intensity propagating within a scattering and absorbing media described via:
- • an extinction coefficient
- • a scattering coefficient along with
- • a scattering phase function describing the directionality of the scattered radiation,
is given by (neglecting the emission contributions)
hereby denotes the pair of polar and azimuthal angles, and denotes the differential solid angle.
In order to increase the readability, the radial and azimuthal dependency will be suppressed in the following as long as their appearance is clearly deducible.
Noting that the integral appearing in Eq. (1) can be written (without loss of generality) as2
Writing Eq. (1) separately for and , inserting the integral representation Eq. (3), and introducing the new angles as defined in Eqs. (4) and (5) as well as the specific notation for upwelling and downwelling radiation, one finds the following separation of the RTE (with the shorthand notation ):
This set of coupled integro-differential equations can now be used as a starting point for calculating the backscattered radiation from a uniformly illuminated rough surface covered by a layer of scattering and absorbing material.
B. Problem Geometry and Boundary Conditions
In the following subsection, the problem geometry and boundary conditions are specified. As illustrated in Fig. 3, we consider a rough surface separating a homogeneous ground layer from a volume layer of depth containing a scattering and absorbing media.
The top of the volume layer is assumed to be uniformly illuminated with an incident intensity incoming from a single incidence direction . Thus, the boundary condition at can be written as
The scattering properties of the surface are described by means of , relating the downwelling intensity incident on the ground surface to the upwelling intensity emerging from the ground surface, i.e.,
For physical consistency, the has to be normalized via10], describing the fraction of radiation being re-scattered into the upper hemisphere.
Furthermore, in the following, both the volume scattering phase function and the BRDF are assumed to obey reciprocity:
C. Formulation of the Scattering Series
Since Eqs. (6) and (7) still contain the unknown source terms , which contain integrals of the desired upwelling and downwelling intensities , solving this set of equations directly is generally not possible. Therefore, we restrict the following discussion to a weakly scattering volume layer , and assume contributions of to be negligible. Thus, we seek an expansion of the solution in terms of a power series in .
In order to generate such a series, we will first generate a formal solution by considering the source terms to be known functions. Doing so, one can directly solve Eqs. (6) and (7) by using the method of variation of constants, which leads to the following formal solutions for upwelling and downwelling radiation:
Inserting the boundary conditions Eqs. (9) and (10) for the appearing boundary terms and , a first-order expansion of the solution for the upwelling radiation in terms of can be found by successively inserting the gained solutions in the source terms Eq. (8) and neglecting all terms of or higher. Performing the calculation, one arrives at the following representation for the upwelling radiation:
Assuming the interaction coefficients of the volume layer () to be uniform within the volume layer, the first-order contributions to the upwelling radiation at the top of the layer (i.e., ) in direction are found to be given by the surface contribution
The third first-order contribution appearing in Eq. (16) denoted by would describe radiation that has been scattered once by the volume layer and twice by the surface. Even though this contribution is also directly proportional to , its contribution will not be considered in the following since it is a second-order surface scattering contribution. Validation of the negligibility of this contribution has to be done with respect to the considered problem specifications using, e.g., numerical simulations. (For microwave scattering from forest canopies, this can be seen, for example, from the simulations in .)
3. GENERAL ANALYTIC SOLUTION TO THE FIRST-ORDER INTERACTION CONTRIBUTION
In the following, it will be shown that an analytic solution to the remaining integral of the interaction contribution can be found by assuming that the following series expansion of the -averaged product of the and the volume scattering phase function exists and that the functions of this expansion are known
Since both integrals encounter a singularity at , which is certainly located on the integration path since , a meaningful solution will be obtained in the following by means of the Cauchy principal value.
For a function encountering a singularity at , the Cauchy principal value is defined by 
As shown in Appendix A.1, the Cauchy principal values of the remaining integrals are found to be given by
4. ON THE EXISTENCE OF THE EXPANSION COEFFICIENTS
The existence of the coefficients Eq. (23) as needed to compute Eq. (37) is in general not assured. If, however, the phase function and the can be expressed in terms of a power series of a generalized scalar product between an incoming () and an outgoing () vector as stated below, it will be shown in the following that the coefficients can (in principle) always be computed ( denotes the transpose of the vector ).
Using spherical coordinates, we have
The diagonal elements of the weighting matrix are hereby seen as fitting parameters that allow consideration of off-specular and anisotropic effects as proposed in .
Assuming that both and the can be represented as a power series in a generalized scattering angle , we have
Expanding in the above representations, the series can be written as (with and )
For the sake of compactness, the dependencies of the coefficients on the incoming and outgoing directions are not mentioned explicitly. If the functions are defined as in Eq. (23), the angles would correspond to and .
Using the above representations Eqs. (42) and (43), the product between the BRDF and the volume scattering phase function can be expanded by rearranging the double series (also referred to as Cauchy-product formula) , i.e., with ,
Integrating the above representation with respect to , we find for the appearing integrals
Applying this result to the representation Eq. (46), one can see that in the -integrated product only even coefficients of and or products of the form or appear. In terms of the dependency of the residual terms, we thus find from Eqs. (44) and (45) that they all consist of either powers of or even powers of , which can consequently always be represented in terms of using .
This therefore proves that the -integrated product of the and as given in Eq. (41) can always be represented in terms of a series expansion in , ensuring the existence of the coefficients needed to compute Eq. (37).
A few well-known analytic phase functions obeying this criterion are the isotropic phase function, the Rayleigh phase function, the Henyey–Greenstein and combined Henyey–Greenstein–Rayleigh phase function [15,16], as well as the Mie scattering phase function in terms of a power series expansion as proposed in  or general approximated phase functions using, for example, the G--L method as proposed in . Therein, a solution for the part of the phase function can readily be found since the integral Eq. (20) can directly be solved for .
Examples using a Rayleigh and a Henyey–Greenstein phase function for the volume scattering phase function as well as a cosine lobe for the BRDF are given in Appendix A.2.
It has been shown that the first-order correction to the scattered signal originating from a rough surface covered by a tenuous distribution of particulate media can be evaluated analytically by using approximate functions to represent the surface BRDF and the scattering-phase function of the covering layer.
The gained solution remains applicable for arbitrary choices of surface and covering layer properties as long as the azimuthally averaged product of the used BRDF and the scattering-phase function can be represented in terms of a power series in the scattering angle. Moreover, in Section 4, it was proven that such a representation is always possible as long as the BRDF and the scattering-phase function can be expressed as a power series in a generalized scalar product between an incoming and an outgoing direction.
Therefore, the method is capable of providing a consistent, analytical solution to the first-order interaction contribution of the successive orders of scattering approximation to the RTE for a wide range of possible choices for the scattering characteristics of both the surface and covering layers.
A. EXPLICIT SOLUTIONS FOR THE APPEARING INTERACTION INTEGRALS
1. Calculation of Eq. (31)
In order to find the principal value of Eq. (31), we first notice that the antiderivative of the integrand is given by
Inserting this result in the definition of the principal value in Eq. (27), we find
Using the identities , we find that the limit can readily be evaluated. Thus, the solution to the integral is given by
2. Calculation of Eq. (33)
To identify the integrand as an exponential integral, we split the appearing fraction as follows:
Inserting this representation, we find
The first integral can now directly be interpreted as by substituting , i.e.,
In order to find the necessary substitution for the second integral, we notice thatA5), is given by
Using this function, we therefore find
3. Calculation of Eq. (34)
The identification of the integral as a generalized exponential integral function can readily be performed via
In the following, two examples are shown. First, the volume scattering phase function and the BRDF are given by the Rayleigh phase function in Eq. (A12) and a cosine lobe implemented using a 10 coefficient Legendre series approximation given in Eq. (A13)
For the second example, the scattering distribution of the volume has been changed from the equally forward- and backward-scattering Rayleigh distribution to a primarily forward-scattering Henyey–Greenstein phase function in Eq. (A16) with an asymmetry factor of , which has been implemented using a 20 coefficient Legendre series given in Eq. (A17). The used functions are illustrated in Fig. 4, and the resulting distributions are shown in Figs. 5–8.
In the first example, one can see that the interaction contribution is much more significant for the backscattered radiation since the volume contribution is merely negligible for a primarily forward-scattering coverage, and the surface contribution of a cosine lobe decreases rapidly with increasing incidence angle.
Seventh Framework Programme (FP7) (606971).
1. R. Bindlish and A. P. Barros, “Parameterization of vegetation backscatter in radar-based, soil moisture estimation,” Remote Sens. Environ. 76, 130–137 (2001). [CrossRef]
2. O. Taconet, D. Vidal-Madjar, C. Emblanch, and M. Normand, “Taking into account vegetation effects to estimate soil moisture from C-band radar measurements,” Remote Sens. Environ. 56, 52–56 (1996). [CrossRef]
3. J. Alvarez-Mozos, J. Casali, M. Gonzalez-Audicana, and N. Verhoest, “Assessment of the operational applicability of RADARSAT-1 data for surface soil moisture estimation,” IEEE Trans. Geosci. Remote Sens. 44, 913–924 (2006). [CrossRef]
4. H. Lievens and N. Verhoest, “On the retrieval of soil moisture in wheat fields from L-band SAR based on water cloud modeling, the IEM, and effective roughness parameters,” IEEE Geosci. Remote Sens. Lett. 8, 740–744 (2011). [CrossRef]
5. W. T. Crow, W. Wagner, and V. Naeimi, “The impact of radar incidence angle on soil-moisture-retrieval skill,” IEEE Geosci. Remote Sens. Lett. 7, 501–505 (2010). [CrossRef]
6. A. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, 1994).
7. F. Ulaby, R. Moore, and A. Fung, Microwave Remote Sensing (Artech House, 1986), Vol. 3.
8. E. P. W. Attema and F. T. Ulaby, “Vegetation modeled as a water cloud,” Radio Sci. 13, 357–364 (1978). [CrossRef]
9. S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).
10. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometric Considerations and Nomenclature for Reflectance, (National Bureau of Standards, 1977).
11. P. Liang, L. E. Pierce, and M. Moghaddam, “Radiative transfer model for microwave bistatic scattering from forest canopies,” IEEE Trans. Geosci. Remote Sens. 43, 2470–2483 (2005). [CrossRef]
12. F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1st ed. (Cambridge University, 2010).
13. E. Lafortune, S. Foo, K. Torrance, and D. Greenberg, “Non-linear approximation of reflectance functions,” in Proceedings of Conference on Computer Graphics and Interactive Techniques (SIGGRAPH) (ACM/Addison-Wesley, 1997), pp. 117–126.
14. G. Arfken, H. Weber, and F. Harris, Mathematical Methods for Physicists, 7th ed. (Elsevier, 2013).
15. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941). [CrossRef]
16. Q. Liu and F. Weng, “Combined Henyey–Greenstein and Rayleigh phase function,” Appl. Opt. 45, 7475–7479 (2006). [CrossRef]
17. M. A. Box, “Power series expansion of the Mie scattering phase function,” Aust. J. Phys. 36, 701–706 (1983). [CrossRef]
18. Q. Yin and S. Luo, “An approximation frame of the particle scattering phase function with a delta function and a Legendre polynomial series,” in Light, Energy and the Environment (Optical Society of America, 2015), paper EW3A.3.
19. B. T. Phong, “Illumination for computer generated pictures,” ACM Commun. 18, 311–317 (1975). [CrossRef]