Abstract

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 κs 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

1. INTRODUCTION

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) [15]. 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.

 figure: Fig. 1.

Fig. 1. Schematic illustration of the model geometry and the individual contributions considered in the calculated intensity.

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To clarify the appearing equations, the representation of the solution to the RTE in terms of a series expansion in the scattering coefficient κs of the covering layer (following Fung [6] and Ulaby et al. [7]), 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 ωτ (or water-cloud) model as used in the remote sensing community [8]. 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 p^ 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 [9], governing the alteration of a beam of specific intensity If(r,Ω) propagating within a scattering and absorbing media described via:

  • • an extinction coefficient κex
  • • a scattering coefficient κs along with
  • • a scattering phase function p^(ΩiΩs) describing the directionality of the scattered radiation,

is given by (neglecting the emission contributions)

cos(θ)If(r,Ω)r=κexIf(r,Ω)+κsΩ=4πIf(r,Ω)p^(ΩΩ)dΩ,
where p^(ΩiΩs) has to obey the normalization condition
02π0πp^(ΩiΩ)sin(θ)dθdφ=1.

Ω=(θ,φ) hereby denotes the pair of polar and azimuthal angles, and dΩ=sin(θ)dθdφ 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) as

θ=0πIf(θ)p^(θθ)sin(θ)dθ=θ=0π/2[If(θ)p^(θθ)sin(θ)+If(πθ)p^(πθθ)sin(θ)]dθ,
one can perform a separation of the RTE into a set of two coupled equations by splitting the specific intensity into an upwelling and downwelling part and introducing the upwelling and downwelling angle as illustrated in Fig. 2
upwellinggradiation:Ifθ[0,π/2]θuθandI+(θu)If(θu),
downwellingradiation:Ifθ[π/2,π]θdπθandI(θd)If(πθd).

 figure: Fig. 2.

Fig. 2. Angles introduced in the separation of the RTE.

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Writing Eq. (1) separately for θ[0,π/2] and θ[π/2,π], 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 μ=cos(θ)):

θ[0,π/2]:μuI+(μu)r+κexI+(μu)=F+(μu),
θ[π/2,π]:μdI(μd)r+κexI(μd)=F(μd),
where the source functions F±(μ) are given by
F±(μ)=κs02π01[I+(μ)p^(μ±μ)+I(μ)p^(μ±μ)]dμdφ.

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 d containing a scattering and absorbing media.

 figure: Fig. 3.

Fig. 3. Illustration of the problem geometry.

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The top of the volume layer is assumed to be uniformly illuminated with an incident intensity Iinc incoming from a single incidence direction Ωi=(θi,φi). Thus, the boundary condition at z=0 can be written as

I(z=0,μd)=Iincδ(μdμ0)δ(φdφi),
where δ(xx0) denotes the Dirac-delta function.

The scattering properties of the surface are described by means of BRDF(ΩiΩs), relating the downwelling intensity incident on the ground surface to the upwelling intensity emerging from the ground surface, i.e.,

I+(z=d,μu)=02π01I(z=d,μ)BRDF(μμu)μdμdφ.

For physical consistency, the BRDF has to be normalized via

02π0π/2BRDF(θiθ)cos(θ)sin(θ)dθdφ=ρ(θi,φi)1,
where ρ(θi,φi) is the directional-hemispherical reflectance [10], 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:

BRDF(θiθs)=BRDF(θsθi),
p^(θiθs)=p^(θsθi).

C. Formulation of the Scattering Series

Since Eqs. (6) and (7) still contain the unknown source terms F±(μ), which contain integrals of the desired upwelling and downwelling intensities I±(μ), solving this set of equations directly is generally not possible. Therefore, we restrict the following discussion to a weakly scattering volume layer (κs1), and assume contributions of O(κs2) to be negligible. Thus, we seek an expansion of the solution in terms of a power series in κs.

In order to generate such a series, we will first generate a formal solution by considering the source terms F+(μ) 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:

I+(z,μu)=I+(d,μu)exp[κexμu(z+d)]+dz1μuexp[κexμu(zz)]F+(z,μu)dz,
I(z,μd)=I(0,μd)exp[κexμdz]+z01μdexp[κexμd(zz)]F(z,μd)dz.

Inserting the boundary conditions Eqs. (9) and (10) for the appearing boundary terms I+(d,μu) and I(0,μd), a first-order expansion of the solution for the upwelling radiation I+(μu) in terms of κs can be found by successively inserting the gained solutions in the source terms Eq. (8) and neglecting all terms of O(κs2) or higher. Performing the calculation, one arrives at the following representation for the upwelling radiation:

I+(z,μu)=Isurf++(Ivol++Iint++Isvs+)termsproportionaltoκs+O(κs2).

Assuming the interaction coefficients of the volume layer (κs,κex) to be uniform within the volume layer, the first-order contributions to the upwelling radiation at the top of the layer (i.e., z=0) in direction Ωex=(μex,φex) are found to be given by the surface contribution

Isurf+(μ0,μex)=Iincexp[τμ0τμex]μ0BRDF(μ0μex),
the volume contribution
Ivol+(μ0,μex)=Iincωμ0μ0+μex(1exp[τμ0τμex])p^(μ0μex),
and the first-order interaction contribution
Iint+(μ0,μex)=Iincμ0ω{exp[τμex]Fint(μ0,μex)+exp[τμ0]Fint(μex,μ0)},
where the remaining interaction integral Fint is given by
Fint(μ0,μex)=02π01μμ0μ(exp[τμ0]exp[τμ])×p^(μ0μ)BRDF(μμex)dμdφ,
and the following quantities have been introduced:
SingleScatteringAlbedo: ω=κsκex,
Optical Depth: τ=κexd.

The third first-order contribution appearing in Eq. (16) denoted by Isvs 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 κs, 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 [11].)

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 φ-averaged product of the BRDF and the volume scattering phase function p^ exists and that the functions fn of this expansion are known

02πp^(μ0μs)BRDF(μsμex)dφs=n=0fn(μ0,μex)μsn.

Inserting the expansion of Eq. (23) in Eq. (20), the integral simplifies to

Fint(μ0,μex)=n=0fn(μ0,μex)[exp(τ/μ0)A(n+1)B(n+1)],
where the remaining integrals A(N) and B(N) are given by
A(N)=01(μ)Nμ0μdμ,
B(N)=01(μ)Nμ0μexp(τ/μ)dμ.

Since both integrals encounter a singularity at μ=μ0, which is certainly located on the integration path since μ0(0,1), a meaningful solution will be obtained in the following by means of the Cauchy principal value.

For a function f(x) encountering a singularity at x0(a,b), the Cauchy principal value is defined by [12]

Pabf(x)dx=limε0(ax0εf(x)dx+x0+εbf(x)dx).

In order to find a solution to the integrals Eqs. (25) and (26), we notice that we can expand the term (μ)N/(μ0μ) appearing in the integrands as

(μ)Nμ0μ=μ0Nμ0μk=1Nμ0Nk(μ)k1.

Inserting this expansion in Eqs. (25) and (26) leads to

A(N)=01μ0Nμ0μdμk=1N01μ0Nk(μ)k1dμ,
B(N)=01μ0Nμ0μexp(τ/μ)dμk=1N01μ0Nk(μ)k1exp(τ/μ)dμ.

As shown in Appendix A.1, the Cauchy principal values of the remaining integrals are found to be given by

P01μ0Nμ0μdμ=μ0Nln(μ01μ0),
01μ0Nk(μ)k1dμ=μ0Nkk,
P01μ0Nμ0μexp(τ/μ)dμ=μ0N[Ei(τ)exp(τ/μ0)×Ei(τ/μ0τ)],
P01μ0Nk(μ)k1exp(τ/μ)dμ=μ0NkEk+1(τ),
where Ei(x) denotes the exponential integral function, and En(x) denotes the generalized exponential integral.

Inserting Eqs. (31)–(34) in Eqs. (29) and (30), analytic solutions for the integrals A(N) and B(N) are given by

A(N)=μ0N[ln(μ01μ0)k=1Nμ0kk],
B(N)=μ0N[Ei(τ)exp(τ/μ0)Ei(τ/μ0τ)k=1NEk+1(τ)μ0k].

Finally, inserting Eqs. (35) and (36) in Eq. (24), an analytic representation of the interaction integral can be given by

Fint(μ0,μex)=n=0fn(μ0,μex)μ0n+1{exp(τ/μ0)ln(μ01μ0)Ei(τ)+exp(τ/μ0)Ei(τ/μ0τ)+k=1n+1μ0k(Ek+1(τ)exp(τ/μ0)k)}.

4. ON THE EXISTENCE OF THE EXPANSION COEFFICIENTS

The existence of the fn coefficients Eq. (23) as needed to compute Eq. (37) is in general not assured. If, however, the phase function p^ and the BRDF can be expressed in terms of a power series of a generalized scalar product (i^TMj·s^) between an incoming (i^) and an outgoing (s^) vector as stated below, it will be shown in the following that the coefficients can (in principle) always be computed (i^T denotes the transpose of the vector i^).

Using spherical coordinates, we have

i^=(sin(θi)cos(φi)sin(θi)sin(φi)cos(θi))s^=(sin(θs)cos(φs)sin(θs)sin(φs)cos(θs)),
cos(Θ˜i)=i^TMi·s^withMj=(ai000bi000ci),
and, therefore,
cos(Θ˜i)=i^TMi·s^=aicos(θi)cos(θs)+sin(θi)sin(θs)[bicos(φi)cos(φs)+cisin(φi)sin(φs)].

The diagonal elements (ai,bi,ci) of the weighting matrix Mi are hereby seen as fitting parameters that allow consideration of off-specular and anisotropic effects as proposed in [13].

Assuming that both p^ and the BRDF can be represented as a power series in a generalized scattering angle cos(Θ˜i)n, we have

p^n=0pncos(Θ˜1)nBRDFn=0bncos(Θ˜2)n.

Expanding cos(Θ˜i)n in the above representations, the series can be written as (with b˜i=bicos(φi) and c˜i=cisin(φi))

p^n=0{αn+βn[b˜1cos(φs)+c˜1sin(φs)]n},
BRDFn=0{γn+ηn[b˜2cos(φs)+c˜2sin(φs)]n},
where the coefficients αn,βn,γn, and ηn can furthermore be represented as a series of the form
αn=i[αn]icos(θs)iβn=sin(θs)ni[βn]icos(θs)i,
γn=i[γn]icos(θs)iηn=sin(θs)ni[ηn]icos(θs)i.

For the sake of compactness, the dependencies of the coefficients [αn]i,[βn]i,[γn]i,[ηn]i on the incoming and outgoing directions (θ1,2,φ1,2) are not mentioned explicitly. If the functions are defined as in Eq. (23), the angles would correspond to (θ1,φ1)=(θex,φex) and (θ2,φ2)=(θ0,φ0).

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) [14], i.e., [p=0ap][q=0bq]=n=0cn with cn=k=0n(akbnk),

BRDF·p^=n=0k=0n{αnkγk+βnkγk[b˜1cos(φs)+c˜1sin(φs)]nk+ηkαnk[b˜2cos(φs)+c˜2sin(φs)]k+ηkβnk[b˜1cos(φs)+c˜1sin(φs)]nk[b˜2cos(φs)+c˜2sin(φs)]k}.

Integrating the above representation with respect to φs, we find for the appearing integrals

02π[b˜1cos(φs)+c˜1sin(φs)]ndφs{0ifneven0ifnodd,
02π[b˜1cos(φs)+c˜1sin(φs)]k[b˜2cos(φs)+c˜2sin(φs)]nkdφs{0ifneven0ifnodd.

Applying this result to the representation Eq. (46), one can see that in the φs-integrated product only even coefficients of βn and ηn or products of the form β(even)η(even) or β(odd)η(odd) appear. In terms of the θs dependency of the residual terms, we thus find from Eqs. (44) and (45) that they all consist of either powers of cos(θs) or even powers of sin(θs), which can consequently always be represented in terms of cos(θs) using sin(θs)2n=[1cos(θs)2]n.

This therefore proves that the φs-integrated product of the BRDF and p^ as given in Eq. (41) can always be represented in terms of a series expansion in cos(θs), ensuring the existence of the fn 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 [17] or general approximated phase functions using, for example, the G-δ-L method as proposed in [18]. Therein, a solution for the δ part of the phase function can readily be found since the integral Eq. (20) can directly be solved for p^(μμex)δ(μμex)δ(φφex).

Furthermore, possible analytic BRDF are, for example, given by the ideal diffuse (Lambert) BRDF, arbitrary cosine lobe models [19], or the Lafortune model [13].

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.

5. CONCLUSION

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.

APPENDIX A

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

μ0Nμ0μdμ=μ0Nln(μ0μ)+const.

Inserting this result in the definition of the principal value in Eq. (27), we find

P01μ0Nμ0μdμ=μ0Nlimε0[ln(μ0)+ln(ε)ln(μ01)ln(ε)].

Using the identities ln(a·b±1)=ln(a)±ln(b), we find that the limit can readily be evaluated. Thus, the solution to the integral is given by

P01μ0Nμ0μdμ=μ0Nlimε0[ln(εμ0)ln(ε[μ01])]=μ0Nlimε0[ln(εμ0ε(μ01))]=μ0Nln(μ01μ0).

2. Calculation of Eq. (33)

A solution to Eq. (33) can only be given in terms of the exponential integral function Ei(x), which is defined in [12] as

Ei(x)=Pxexp(t)tdtwithx>0.

To identify the integrand as an exponential integral, we split the appearing fraction as follows:

1μ0μ=1μ+μ0μ(μ0μ).

Inserting this representation, we find

P01μ0Nμ0μexp(τ/μ)dμ=μ0NP01exp(τ/μ)μdμ+μ0NP01μ0exp(τ/μ)μ(μ0μ).

The first integral can now directly be interpreted as Ei(τ) by substituting t=τ/μ, i.e.,

P01exp(τ/μ)μdμ=|t=τ/μdμ=τ/t2dt=Pτexp(t)tdt=Ei(τ).

In order to find the necessary substitution for the second integral, we notice that

P01exp[f(x)]f(x)f(x)dx=Pf(0)f(1)ettdt=Ei[f(1)]if{f(0)=f(1)R,
a possible candidate for a function f(μ), which can be used to identify the second integral of Eq. (A5), is given by
f(μ)=τμτμ0f(μ)f(μ)=μ0μ(μ0μ).

Using this function, we therefore find

P01μ0μ(μ0μ)exp(τ/μ)=exp(τμ0)P01[μ0μ(μ0μ)]exp(τμ+τμ0)=exp(τμ0)Ei(ττμ0).

Combining the results of Eqs. (A6) and (A8), we thus find

P01μ0Nμ0μexp(τμ0)dμ=μ0N[Ei(τ)exp(τμ0)Ei(ττμ0)].

3. Calculation of Eq. (34)

A solution to Eq. (34) can similarly be given in terms of the generalized exponential integral function En(x), which is defined in [12] as

En(x)=P1exp(xt)tndt.

The identification of the integral as a generalized exponential integral function can readily be performed via

P01μ0Nk(μ)k1exp(τ/μ)dμ=|t=(μ)1dμ=dt/t2=μ0NkP1exp(τt)tk+1dt=μ0NkEk+1(τ)

B. Example

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)

p^(θi,θs)=316π(1+cos(Θ)2),
BRDF(θi,θs)=Max[cos(Θ)5,0]=n=0(2n+1)15π16Γ(7n2)Γ(8+n2)Pn(cos(Θ))
with
cos(Θ)=cos(θi)cos(θs)+sin(θi)sin(θs)cos(φiφs),
cos(Θ)=cos(θi)cos(θs)+sin(θi)sin(θs)cos(φiφs).

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 t=0.7, 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. 58.

p^(θi,θs)=14π1t2[1+t22tcos(Θ)]3/2
=14πn=0(2n+1)tnPn(cos(Θ)).

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.

 figure: Fig. 4.

Fig. 4. Illustration of the scattering distributions in Eqs. (A12), (A13), and (A16) as used in their following examples.

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 figure: Fig. 5.

Fig. 5. Visualization of the resulting contributions in Eqs. (17)–(19) to the scattered intensity in linear scale as a function of the outgoing direction (θex,φex) using the phase functions of Fig. 4’s Example 1 with θ0=45°,τ=0.7, and ω=0.3.

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 figure: Fig. 6.

Fig. 6. Visualization of the resulting contributions in Eqs. (17)–(19) to the scattered intensity in linear scale as a function of the outgoing direction (θex,φex) using the phase functions of Fig. 4’s Example 2 with θ0=45°,τ=0.7, and ω=0.3.

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 figure: Fig. 7.

Fig. 7. Illustration of the backscattering contributions [Eqs. (17)–(19) with θex=θ0 and φex=π] in decibel scale using the phase functions of Fig. 4’s Example 1 with τ=0.7 and ω=0.3.

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 figure: Fig. 8.

Fig. 8. Illustration of the backscattering contributions [Eqs. (17)–(19) with θex=θ0 and φex=π] in decibel scale using the phase functions of Fig. 4’s Example 2 with τ=0.7 and ω=0.3.

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Funding

Seventh Framework Programme (FP7) (606971).

REFERENCES

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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]  

References

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  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]

2011 (1)

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]

2010 (1)

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]

2006 (2)

Q. Liu and F. Weng, “Combined Henyey–Greenstein and Rayleigh phase function,” Appl. Opt. 45, 7475–7479 (2006).
[Crossref]

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]

2005 (1)

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]

2001 (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]

1996 (1)

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]

1983 (1)

M. A. Box, “Power series expansion of the Mie scattering phase function,” Aust. J. Phys. 36, 701–706 (1983).
[Crossref]

1978 (1)

E. P. W. Attema and F. T. Ulaby, “Vegetation modeled as a water cloud,” Radio Sci. 13, 357–364 (1978).
[Crossref]

1975 (1)

B. T. Phong, “Illumination for computer generated pictures,” ACM Commun. 18, 311–317 (1975).
[Crossref]

1941 (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Alvarez-Mozos, J.

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]

Arfken, G.

G. Arfken, H. Weber, and F. Harris, Mathematical Methods for Physicists, 7th ed. (Elsevier, 2013).

Attema, E. P. W.

E. P. W. Attema and F. T. Ulaby, “Vegetation modeled as a water cloud,” Radio Sci. 13, 357–364 (1978).
[Crossref]

Barros, A. P.

R. Bindlish and A. P. Barros, “Parameterization of vegetation backscatter in radar-based, soil moisture estimation,” Remote Sens. Environ. 76, 130–137 (2001).
[Crossref]

Bindlish, R.

R. Bindlish and A. P. Barros, “Parameterization of vegetation backscatter in radar-based, soil moisture estimation,” Remote Sens. Environ. 76, 130–137 (2001).
[Crossref]

Boisvert, R. F.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1st ed. (Cambridge University, 2010).

Box, M. A.

M. A. Box, “Power series expansion of the Mie scattering phase function,” Aust. J. Phys. 36, 701–706 (1983).
[Crossref]

Casali, J.

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]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).

Clark, C. W.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1st ed. (Cambridge University, 2010).

Crow, W. T.

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]

Emblanch, C.

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]

Foo, S.

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.

Fung, A.

A. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, 1994).

F. Ulaby, R. Moore, and A. Fung, Microwave Remote Sensing (Artech House, 1986), Vol. 3.

Ginsberg, I. W.

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).

Gonzalez-Audicana, M.

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]

Greenberg, D.

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.

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Harris, F.

G. Arfken, H. Weber, and F. Harris, Mathematical Methods for Physicists, 7th ed. (Elsevier, 2013).

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Hsia, J. J.

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).

Lafortune, E.

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.

Liang, P.

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]

Lievens, H.

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]

Limperis, T.

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).

Liu, Q.

Lozier, D. W.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1st ed. (Cambridge University, 2010).

Luo, S.

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.

Moghaddam, M.

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]

Moore, R.

F. Ulaby, R. Moore, and A. Fung, Microwave Remote Sensing (Artech House, 1986), Vol. 3.

Naeimi, V.

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]

Nicodemus, F. E.

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).

Normand, M.

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]

Olver, F. W.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1st ed. (Cambridge University, 2010).

Phong, B. T.

B. T. Phong, “Illumination for computer generated pictures,” ACM Commun. 18, 311–317 (1975).
[Crossref]

Pierce, L. E.

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]

Richmond, J. C.

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).

Taconet, O.

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]

Torrance, K.

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.

Ulaby, F.

F. Ulaby, R. Moore, and A. Fung, Microwave Remote Sensing (Artech House, 1986), Vol. 3.

Ulaby, F. T.

E. P. W. Attema and F. T. Ulaby, “Vegetation modeled as a water cloud,” Radio Sci. 13, 357–364 (1978).
[Crossref]

Verhoest, N.

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]

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]

Vidal-Madjar, D.

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]

Wagner, W.

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]

Weber, H.

G. Arfken, H. Weber, and F. Harris, Mathematical Methods for Physicists, 7th ed. (Elsevier, 2013).

Weng, F.

Yin, Q.

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.

ACM Commun. (1)

B. T. Phong, “Illumination for computer generated pictures,” ACM Commun. 18, 311–317 (1975).
[Crossref]

Appl. Opt. (1)

Astrophys. J. (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Aust. J. Phys. (1)

M. A. Box, “Power series expansion of the Mie scattering phase function,” Aust. J. Phys. 36, 701–706 (1983).
[Crossref]

IEEE Geosci. Remote Sens. Lett. (2)

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]

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]

IEEE Trans. Geosci. Remote Sens. (2)

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]

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]

Radio Sci. (1)

E. P. W. Attema and F. T. Ulaby, “Vegetation modeled as a water cloud,” Radio Sci. 13, 357–364 (1978).
[Crossref]

Remote Sens. Environ. (2)

R. Bindlish and A. P. Barros, “Parameterization of vegetation backscatter in radar-based, soil moisture estimation,” Remote Sens. Environ. 76, 130–137 (2001).
[Crossref]

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]

Other (8)

S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).

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).

A. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, 1994).

F. Ulaby, R. Moore, and A. Fung, Microwave Remote Sensing (Artech House, 1986), Vol. 3.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1st ed. (Cambridge University, 2010).

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.

G. Arfken, H. Weber, and F. Harris, Mathematical Methods for Physicists, 7th ed. (Elsevier, 2013).

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.

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Figures (8)

Fig. 1.
Fig. 1. Schematic illustration of the model geometry and the individual contributions considered in the calculated intensity.
Fig. 2.
Fig. 2. Angles introduced in the separation of the RTE.
Fig. 3.
Fig. 3. Illustration of the problem geometry.
Fig. 4.
Fig. 4. Illustration of the scattering distributions in Eqs. (A12), (A13), and (A16) as used in their following examples.
Fig. 5.
Fig. 5. Visualization of the resulting contributions in Eqs. (17)–(19) to the scattered intensity in linear scale as a function of the outgoing direction ( θ ex , φ ex ) using the phase functions of Fig. 4’s Example 1 with θ 0 = 45 ° , τ = 0.7 , and ω = 0.3 .
Fig. 6.
Fig. 6. Visualization of the resulting contributions in Eqs. (17)–(19) to the scattered intensity in linear scale as a function of the outgoing direction ( θ ex , φ ex ) using the phase functions of Fig. 4’s Example 2 with θ 0 = 45 ° , τ = 0.7 , and ω = 0.3 .
Fig. 7.
Fig. 7. Illustration of the backscattering contributions [Eqs. (17)–(19) with θ ex = θ 0 and φ ex = π ] in decibel scale using the phase functions of Fig. 4’s Example 1 with τ = 0.7 and ω = 0.3 .
Fig. 8.
Fig. 8. Illustration of the backscattering contributions [Eqs. (17)–(19) with θ ex = θ 0 and φ ex = π ] in decibel scale using the phase functions of Fig. 4’s Example 2 with τ = 0.7 and ω = 0.3 .

Equations (67)

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cos ( θ ) I f ( r , Ω ) r = κ ex I f ( r , Ω ) + κ s Ω = 4 π I f ( r , Ω ) p ^ ( Ω Ω ) d Ω ,
0 2 π 0 π p ^ ( Ω i Ω ) sin ( θ ) d θ d φ = 1 .
θ = 0 π I f ( θ ) p ^ ( θ θ ) sin ( θ ) d θ = θ = 0 π / 2 [ I f ( θ ) p ^ ( θ θ ) sin ( θ ) + I f ( π θ ) p ^ ( π θ θ ) sin ( θ ) ] d θ ,
upwelling gradiation : If θ [ 0 , π / 2 ] θ u θ and I + ( θ u ) I f ( θ u ) ,
downwelling radiation : If θ [ π / 2 , π ] θ d π θ and I ( θ d ) I f ( π θ d ) .
θ [ 0 , π / 2 ] : μ u I + ( μ u ) r + κ ex I + ( μ u ) = F + ( μ u ) ,
θ [ π / 2 , π ] : μ d I ( μ d ) r + κ ex I ( μ d ) = F ( μ d ) ,
F ± ( μ ) = κ s 0 2 π 0 1 [ I + ( μ ) p ^ ( μ ± μ ) + I ( μ ) p ^ ( μ ± μ ) ] d μ d φ .
I ( z = 0 , μ d ) = I inc δ ( μ d μ 0 ) δ ( φ d φ i ) ,
I + ( z = d , μ u ) = 0 2 π 0 1 I ( z = d , μ ) BRDF ( μ μ u ) μ d μ d φ .
0 2 π 0 π / 2 BRDF ( θ i θ ) cos ( θ ) sin ( θ ) d θ d φ = ρ ( θ i , φ i ) 1 ,
BRDF ( θ i θ s ) = BRDF ( θ s θ i ) ,
p ^ ( θ i θ s ) = p ^ ( θ s θ i ) .
I + ( z , μ u ) = I + ( d , μ u ) exp [ κ ex μ u ( z + d ) ] + d z 1 μ u exp [ κ ex μ u ( z z ) ] F + ( z , μ u ) d z ,
I ( z , μ d ) = I ( 0 , μ d ) exp [ κ ex μ d z ] + z 0 1 μ d exp [ κ ex μ d ( z z ) ] F ( z , μ d ) d z .
I + ( z , μ u ) = I surf + + ( I vol + + I int + + I svs + ) terms proportional to κ s + O ( κ s 2 ) .
I surf + ( μ 0 , μ ex ) = I inc exp [ τ μ 0 τ μ ex ] μ 0 BRDF ( μ 0 μ ex ) ,
I vol + ( μ 0 , μ ex ) = I inc ω μ 0 μ 0 + μ ex ( 1 exp [ τ μ 0 τ μ ex ] ) p ^ ( μ 0 μ ex ) ,
I int + ( μ 0 , μ ex ) = I inc μ 0 ω { exp [ τ μ ex ] F int ( μ 0 , μ ex ) + exp [ τ μ 0 ] F int ( μ ex , μ 0 ) } ,
F int ( μ 0 , μ ex ) = 0 2 π 0 1 μ μ 0 μ ( exp [ τ μ 0 ] exp [ τ μ ] ) × p ^ ( μ 0 μ ) BRDF ( μ μ ex ) d μ d φ ,
Single Scattering Albedo :  ω = κ s κ ex ,
Optical Depth :  τ = κ ex d .
0 2 π p ^ ( μ 0 μ s ) BRDF ( μ s μ ex ) d φ s = n = 0 f n ( μ 0 , μ ex ) μ s n .
F int ( μ 0 , μ ex ) = n = 0 f n ( μ 0 , μ ex ) [ exp ( τ / μ 0 ) A ( n + 1 ) B ( n + 1 ) ] ,
A ( N ) = 0 1 ( μ ) N μ 0 μ d μ ,
B ( N ) = 0 1 ( μ ) N μ 0 μ exp ( τ / μ ) d μ .
P a b f ( x ) d x = lim ε 0 ( a x 0 ε f ( x ) d x + x 0 + ε b f ( x ) d x ) .
( μ ) N μ 0 μ = μ 0 N μ 0 μ k = 1 N μ 0 N k ( μ ) k 1 .
A ( N ) = 0 1 μ 0 N μ 0 μ d μ k = 1 N 0 1 μ 0 N k ( μ ) k 1 d μ ,
B ( N ) = 0 1 μ 0 N μ 0 μ exp ( τ / μ ) d μ k = 1 N 0 1 μ 0 N k ( μ ) k 1 exp ( τ / μ ) d μ .
P 0 1 μ 0 N μ 0 μ d μ = μ 0 N ln ( μ 0 1 μ 0 ) ,
0 1 μ 0 N k ( μ ) k 1 d μ = μ 0 N k k ,
P 0 1 μ 0 N μ 0 μ exp ( τ / μ ) d μ = μ 0 N [ E i ( τ ) exp ( τ / μ 0 ) × E i ( τ / μ 0 τ ) ] ,
P 0 1 μ 0 N k ( μ ) k 1 exp ( τ / μ ) d μ = μ 0 N k E k + 1 ( τ ) ,
A ( N ) = μ 0 N [ ln ( μ 0 1 μ 0 ) k = 1 N μ 0 k k ] ,
B ( N ) = μ 0 N [ E i ( τ ) exp ( τ / μ 0 ) E i ( τ / μ 0 τ ) k = 1 N E k + 1 ( τ ) μ 0 k ] .
F int ( μ 0 , μ ex ) = n = 0 f n ( μ 0 , μ ex ) μ 0 n + 1 { exp ( τ / μ 0 ) ln ( μ 0 1 μ 0 ) E i ( τ ) + exp ( τ / μ 0 ) E i ( τ / μ 0 τ ) + k = 1 n + 1 μ 0 k ( E k + 1 ( τ ) exp ( τ / μ 0 ) k ) } .
i ^ = ( sin ( θ i ) cos ( φ i ) sin ( θ i ) sin ( φ i ) cos ( θ i ) ) s ^ = ( sin ( θ s ) cos ( φ s ) sin ( θ s ) sin ( φ s ) cos ( θ s ) ) ,
cos ( Θ ˜ i ) = i ^ T M i · s ^ with M j = ( a i 0 0 0 b i 0 0 0 c i ) ,
cos ( Θ ˜ i ) = i ^ T M i · s ^ = a i cos ( θ i ) cos ( θ s ) + sin ( θ i ) sin ( θ s ) [ b i cos ( φ i ) cos ( φ s ) + c i sin ( φ i ) sin ( φ s ) ] .
p ^ n = 0 p n cos ( Θ ˜ 1 ) n BRDF n = 0 b n cos ( Θ ˜ 2 ) n .
p ^ n = 0 { α n + β n [ b ˜ 1 cos ( φ s ) + c ˜ 1 sin ( φ s ) ] n } ,
BRDF n = 0 { γ n + η n [ b ˜ 2 cos ( φ s ) + c ˜ 2 sin ( φ s ) ] n } ,
α n = i [ α n ] i cos ( θ s ) i β n = sin ( θ s ) n i [ β n ] i cos ( θ s ) i ,
γ n = i [ γ n ] i cos ( θ s ) i η n = sin ( θ s ) n i [ η n ] i cos ( θ s ) i .
BRDF · p ^ = n = 0 k = 0 n { α n k γ k + β n k γ k [ b ˜ 1 cos ( φ s ) + c ˜ 1 sin ( φ s ) ] n k + η k α n k [ b ˜ 2 cos ( φ s ) + c ˜ 2 sin ( φ s ) ] k + η k β n k [ b ˜ 1 cos ( φ s ) + c ˜ 1 sin ( φ s ) ] n k [ b ˜ 2 cos ( φ s ) + c ˜ 2 sin ( φ s ) ] k } .
0 2 π [ b ˜ 1 cos ( φ s ) + c ˜ 1 sin ( φ s ) ] n d φ s { 0 if n even 0 if n odd ,
0 2 π [ b ˜ 1 cos ( φ s ) + c ˜ 1 sin ( φ s ) ] k [ b ˜ 2 cos ( φ s ) + c ˜ 2 sin ( φ s ) ] n k d φ s { 0 if n even 0 if n odd .
μ 0 N μ 0 μ d μ = μ 0 N ln ( μ 0 μ ) + const .
P 0 1 μ 0 N μ 0 μ d μ = μ 0 N lim ε 0 [ ln ( μ 0 ) + ln ( ε ) ln ( μ 0 1 ) ln ( ε ) ] .
P 0 1 μ 0 N μ 0 μ d μ = μ 0 N lim ε 0 [ ln ( ε μ 0 ) ln ( ε [ μ 0 1 ] ) ] = μ 0 N lim ε 0 [ ln ( ε μ 0 ε ( μ 0 1 ) ) ] = μ 0 N ln ( μ 0 1 μ 0 ) .
E i ( x ) = P x exp ( t ) t d t with x > 0 .
1 μ 0 μ = 1 μ + μ 0 μ ( μ 0 μ ) .
P 0 1 μ 0 N μ 0 μ exp ( τ / μ ) d μ = μ 0 N P 0 1 exp ( τ / μ ) μ d μ + μ 0 N P 0 1 μ 0 exp ( τ / μ ) μ ( μ 0 μ ) .
P 0 1 exp ( τ / μ ) μ d μ = | t = τ / μ d μ = τ / t 2 d t = P τ exp ( t ) t d t = E i ( τ ) .
P 0 1 exp [ f ( x ) ] f ( x ) f ( x ) d x = P f ( 0 ) f ( 1 ) e t t d t = E i [ f ( 1 ) ] if { f ( 0 ) = f ( 1 ) R ,
f ( μ ) = τ μ τ μ 0 f ( μ ) f ( μ ) = μ 0 μ ( μ 0 μ ) .
P 0 1 μ 0 μ ( μ 0 μ ) exp ( τ / μ ) = exp ( τ μ 0 ) P 0 1 [ μ 0 μ ( μ 0 μ ) ] exp ( τ μ + τ μ 0 ) = exp ( τ μ 0 ) E i ( τ τ μ 0 ) .
P 0 1 μ 0 N μ 0 μ exp ( τ μ 0 ) d μ = μ 0 N [ E i ( τ ) exp ( τ μ 0 ) E i ( τ τ μ 0 ) ] .
E n ( x ) = P 1 exp ( x t ) t n d t .
P 0 1 μ 0 N k ( μ ) k 1 exp ( τ / μ ) d μ = | t = ( μ ) 1 d μ = d t / t 2 = μ 0 N k P 1 exp ( τ t ) t k + 1 d t = μ 0 N k E k + 1 ( τ )
p ^ ( θ i , θ s ) = 3 16 π ( 1 + cos ( Θ ) 2 ) ,
BRDF ( θ i , θ s ) = Max [ cos ( Θ ) 5 , 0 ] = n = 0 ( 2 n + 1 ) 15 π 16 Γ ( 7 n 2 ) Γ ( 8 + n 2 ) P n ( cos ( Θ ) )
cos ( Θ ) = cos ( θ i ) cos ( θ s ) + sin ( θ i ) sin ( θ s ) cos ( φ i φ s ) ,
cos ( Θ ) = cos ( θ i ) cos ( θ s ) + sin ( θ i ) sin ( θ s ) cos ( φ i φ s ) .
p ^ ( θ i , θ s ) = 1 4 π 1 t 2 [ 1 + t 2 2 t cos ( Θ ) ] 3 / 2
= 1 4 π n = 0 ( 2 n + 1 ) t n P n ( cos ( Θ ) ) .

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