We investigate radiative transfer though a homogeneous media, illuminated at its top boundary and a reflective/absorptive surface at its bottom boundary, that contains a uniform mix of particles which may absorb or scatter light particles that are traveling through the media. We will limit this investigation to the plane-parallel situation, in which we only model the depth to which a light particle has penetrated, (increasing as the light particle descends deeper), the relative velocity (the cosine of the angle of travel relative to the vertical), and its azimuth (the angle of travel relative to the medium's “north”). The intensity function is interpreted here as the probability that a light particle is at depth y in the media, and has a relative velocity of v, where v=cos⁡(θ). So (v=1, θ=π/2) is straight down, (v=0, θ=0) is horizontal, and (v=-1, θ=-π/2) is straight up.

The particles may either absorb or scatter a light particle encountering them. The process of a light particle travelling until it is absorbed or scattered out of the beam is known as extinction, and the expected path length of the process is the extinction path length. Upon extinction, the fraction of the particles that are then scattered is the single scatter albedo also known as the scatter fraction. EigenFlux currently does not consider thermal emission or inelastic scattering from the particles. The scatter distribution is determined by a rotationally symmetric phase function k(v) with asymmetry g measuring the expected amount of back scatter vs. forward scatter, g=∫2πvk(v)dv where g=1 is full forward scatter and g=-1 is full backward scatter. The scattering matrix is given by K(v,w;a-b)=kvw+cos⁡(a-b)1-v21-w2

Further information and background can be found in the references .

Given a boundary at the top of the medium, the left-hand side of Eq. () is a first-order linear differential equation in y that is unstable for v<0, i.e. for light traveling up to the top of the medium. To fix this problem, we use the notion of natural reflectance to find canonical solutions that can be combined to find two-boundary solutions to Eq. ().

The natural reflectance of a material is the reflectance of a sample that is optically thick, e.g. adding additional depth does not change the reflectance significantly. In considering atmospheric radiation, this would be the assumption of an infinitely deep atmosphere (without the attendant pressure increase). In considering paints and coatings, this would be the assumption of a essentially opaque film.

Let us write down the standard textbook solution for a first order differential equation with the initial boundary condition s(v). I(v,y,a)=s(v,a)e-y/v+ω(1/v)e-y/v∫η=0y∫w=-11∫b=02πeη/vK(v,w,a-b)I(w,η,b)dbdwdη However, for v<0, e-y/v is unbounded in y, and under the natural reflectance condition, the light exiting the surface, I(v,0) for v≤0 should be solely determined by the light entering the surface, I(v,0) for v≥0.

By using a second boundary at depth M for the light moving up, we can get a two-sided equation in which we specify I(v,0) for v≥0 for light moving from the surface to the bottom, and I(v,M) for v≤0 for light moving from the bottom to the surface. Forv≥0,I(v,y,a)=s(v,a)e-y/v+(ω/v)e-y/v∫η=0y∫w=-11∫b=02πeη/vK(v,w,a-b)I(w,η,b)dbdwdηForv<0,I(v,y,a)=sM(v,a)e(M-y)/v-(ω/v)e-y/v∫η=yM∫w=-11∫b=02πeη/vK(v,w,a-b)I(w,η,b)dbdwdη Now apply the natural reflectance condition by letting M→∞ under the condition that sM(v,a) is bounded, and split out the specific solution for v=0, which results in 2Forv>0,I(v,y,a)=s(v,a)e-y/v+(ω/v)e-y/v∫η=0y∫w=-11∫b=02πeη/vK(v,w,a-b)I(w,η,b)dbdwdηForv=0,I(v,y,a)=ω∫w=-11∫b=02πK(v,w,a-b)I(w,y,b)dbdwForv<0,I(v,y,a)=-(ω/v)e-y/v∫η=y∞∫w=-11∫b=02πeη/vK(v,w,a-b)I(w,η,b)dbdwdη As desired, the light exiting the surface, I(v,0) for v≤0, is determined by the light entering the surface, I(v,0) for v≥0.

Figure  shows the intensities for a strongly forward scattering media.

Now we turn our attention back to the basic equation Eq. (). This can be decomposed into two parts, the direct component which is the intensity of the source rays before they are scattered or absorbed (also known as extinction), and the diffuse component which is the intensity of the scattered light. Furthermore, we can immediately determine the form of the direct component.

We accordingly define the two components of the intensity as R(v,y,a)=e-y/vs(v,a)= Direct intensity distribution, M(v,y,a)=I(v,y,a)-R(v,y,a)= Diffuse intensity distribution.

with the boundary condition ∀v>0,I(v,0)=s(v). Substituting these definitions into Eq. (), we obtain the basic equation and side condition for the diffuse component. 3M(v,y,a)+v∂yM(v,y,a)-ω∫K(v,w,a-b)M(w,y,b)dbdw=ω∫K(v,w,a-b)R(w,y,b)dbdw∀v>0,M(v,0,a)=0 Figure  shows the internal direct intensity, which depends solely on the distribution of the source radiation and the extinction depth.

Figure  shows the intensities for strongly forward scattering and backscattering media.

We use Galerkin's method to solve the integral equations in Eq. (). This method uses a framework of basis functions and test functions to generate a mesh approximation for the various functions that constitute the full solution.

vi= Approximation points for relative velocity

When seeking an approximate f̃(v)≈g(v), we expand f̃ and integrate both against the test functions to get the linear equation which can be solved for the approximating coefficients. ∀i,∑jfi∫ψi(v)ϕj(v)dv=∫ψi(v)g(v)dv

Any reasonable subdivision of points can be used for vi. In this paper, vi will be the concatenation of two sets of the Chebyshev points, one “arc” for -1≤v≤0 and another for 0≤v≤1. The basis functions ϕi are triangular bump functions centered on the mesh points. We will be using two different choices for test functions in the course of this paper, either

ψi=ϕi where the test functions are the basis functions themselves, or

Applying the Galerkin approximations to the basic equations Eq. (), we get the mesh approximations: K(v,w;a-b)=∑jpcKjpcϕj(v)ϕp(w)ϕ̃c(a-b) is the phase function kernel, p0(v;a)=∑jcp0jcϕj(v)ϕ̃c(a) is the intensity at surface, M(v,y;a-b)=∑jcMjc(y)ϕj(v)ϕ̃c(a-b) is the diffuse intensity distribution, ϕ̃c(a-b)=ϕ̃c(a-b+2π) is the azimuth test function.

From this we derive the finite dimensional linear ODE. 4∑jcAijB̃fc∂yMjc(y)+∑jcBijB̃fcMjc(y)-ω∑jpcBijC̃fdcKjpdBpqMqc(y)=ω∑jpcBijC̃fdcKjpdDpq(y)p0qc where Aij=∫vψi(v)ϕj(v)dv, Bij=∫ψi(v)ϕj(v)dv, Cij=∫ϕi(v)ϕj(v)dv, B̃fc=∫ψ̃f(a)ϕ̃c(a)da, C̃fdc=∫ψ̃f(a)ϕ̃d(a-b)ϕ̃c(b)dadb, Dij(y)=∫e-y/vϕi(v)ϕj(v)dv.

More succinctly, 5AB̃∂yM(y)+(BB̃-ωBC̃KC)M(y)=ωBC̃KD(y)p0

One of the benefits of using Galerkin's Method is that it is an orthogonal projection from the Hilbert space of differentiable functions to the Hilbert space of piece-wise continuous functions, so it preserves conservation properties under the correct weights. This includes energy conservation, reciprocity, and flux closure.

The original scattering matrix K(v,w,a-b), besides being symmetric in v,w and cyclic in a-b, also satisfies the conservation condition that ∀w,b,∬K(v,w,a-b)dadv=1. In order for the mesh approximate scattering matrix, K(v,w;a,b)=∑jpcKjpcϕj(v)ϕp(w)ϕ̃c(a-b), to satisfy that condition, we must have 6∀p,b,∑jaKjpc∫v=-1v=-1ϕj(v)dv∫a=0a=2πϕ̃c(a-b)da=1 So the tensor K̃jpab=Kjp(a-b) must be doubly stochastic with respect to weights ∫ϕj(v)dv and ∫ϕ̃b(a)da. In general, the simple choice of defining the approximation by simply sampling the kernel at the mesh points, K̃jpc=K(vj,wp,ac) will yield a matrix that is close to, but is not doubly stochastic.

To ensure the problem remains conservative, EigenFlux uses a variation on the Sinkhorn-Knopp algorithm suitable for re-scaling a matrix so that it is doubly stochastic with respect to the weights. The result is that the modified scattering matrix remains conservative in the mesh approximation.

To solve Eq. (), we use an eigenvalue expansion of the linear system generated by the mesh approximation.

The tensor A is near-singular at v=0, but (B-ωBKC) is diagonally dominant and non-singular. So we can look at the eigenvalue decomposition of (B-ωBKB)-1A for eigenvalues νk and eigenvectors Qk. The resulting ODE becomes 7νk∂yμk(y)+μk(y)=ρk(y) where νk,Qk Eigenvalues and eigenvectors, μk(y) Eigenvector decomposition of Mj(y), ρk(y) Transformed boundary data, and AQk=νk(B-ωBKC)Qk, Mj(y)=∑kQkμk(y), ρk(y)=ωQ-1(B-ωBKC)-1BKD(y)s.

Figure  shows the computed eigenvalues for varying parameter values. In general, there will be a continuous spectrum from -1 to 1, plus isolated eigenvalues at v=±1 that correspond to the limiting distribution at increasing depths. Figure  shows the computed eigenvectors for a single set of the parameter values. Note in particular the all-positive eigenvectors for ν=±1.8. These are the limiting distributions of the intensity.

We now have decomposed the problem into a set of one-dimensional first-order linear ODEs which have the general solution form 8μk(y)=μk(0)e-y/νk+(1/νk)∫0ye(η-y)/νk)ρk(η)dη There are additional considerations when the eigenvalue is zero or negative.

One of the eigenvalues will be close to (or equal to) 0. In this case, we go back to Eq. () and use the resulting condition μk(y)=ρk(y).

When the eigenvalue is negative, Eq. () would imply that we have an unstable mode. In this case, we can apply the natural reflectance assumption discussed previously to the new ODEs.

With these considerations, we have the following more specific solution.

9νk>0⇒μk(y)=μk(0)e-y/νk+(1/νk)∫0ye(η-y)/νk)ρk(η)dηνk=0⇒μk(y)=ρk(y)νk<0⇒μk(y)=-(1/νk)∫y∞e(η-y)/νk)ρk(η)dη

Note in particular that this implies that for k:νk≤0, μk(0) is determined by ρk (and hence by p0). For positive νk, determination of μk(0) is slightly more complicated – they are determined by taking the side-condition v>0⇒M(v,0)=0 (see Eq. ), expanding by Mj(0)=∑kQkμk(0) (see Eq. ) and solving for the unknown μk(0).

The eigenvalue decomposition of R̃=(BB̃-ωBC̃KB)-1AB̃ can be computationally costly. However, note that all functions/matrices (other than p0(v,a)) that are dependent on azimuth are cyclically symmetric. As a result, the tensor R̃ has some special properties:

R̃ is block circulant, thus R̃fcij=R̃(f-c)0ij

Because of (1) and (2), the eigenvalues and eigenvectors of R̃ are real and can be determined from the number of blocks and the eigenvalue decomposition of a geometric sum of the individual blocks . Let the blocks of R̃ be given by Ri-j. Let ρ be a primitive root of unity of order n, the number of blocks, so that ρn=1 and such that {ρi} generates a complete list of all roots of unity of order n. Let H(ρ)=∑jρjRn-j. Then the eigenvalues of R̃ are the n⋅m eigenvalues of H(ρk), and the eigenvectors are the outer product of {ρkj} and the eigenvectors of H(ρk).

Finding a good approximation mesh for the depth y can be problematic because y has an unbounded range, so instead we use the exponential depth z where z=e-y, 0≤z≤1 so that for a function of depth f(y) we have f(y)=f(-ln⁡(z)).

Similar to Step One, we apply the following mesh approximation to Eq. (), (but now with ψ¯i(z)=δ[z=zi], option (2) in the discussion on Mesh Approximation).

The boundary conditions for a two-boundary radiative transfer problem specify the downward sources of illumination at the top boundary, and the upward sources of illumination at the bottom boundary.

I(wj,0)=r0(wj)= downwards illumination at top boundary (wj≥0)

The general solution for radiative transfer for a medium with finite depth and two boundaries, top and bottom, can be expressed as the linear combination of the one-boundary solution for the top boundary with the one-boundary solution for the bottom boundary.

Let Φ be the fundamental solution for Eq. (), so that M=Φp0. To simplify the notation, we replace the indices by the underlying mesh points e.g. Mi(y)=M(vi,y). We then note that the solution for a system with a bottom boundary is given by a depth-reversal of the solution for a system with a top boundary. 10M̃top(vi,y)=∑wj≥0Φ(vi,wj,y)r0(wj)M̃bot(vi,y)=∑wj≤0RΦ(vi,wj,b-y)rb(wj)RΦ(vi,wj,b-y)=Φ(-vi,-wj,b-y) where b is the depth of bottom boundary, M̃top is the solution to top-boundary problem under natural reflectance assumption, M̃bot is the solution to bottom boundary problem under natural reflectance assumption, RΦ is the depth-reversed fundamental solution, r0(wj) is the downwards illumination of top boundary (wj≥0), rb(wj) is the upwards illumination of bottom boundary (wj≤0).

Both the downwards and upwards system satisfy the base integro-differential equation Eq. (). A linear combination of the two sets of solutions has the same degrees of freedom as the two-boundary problem, so a linear combination of the two will span the solution set.

To find that solution, first let's decompose the systems into downwards flows (v≥0) and upwards flows (v≤0).

Iy↑Iy↓=RΦb-y↓Φy↑RΦb-y↑Φy↓rb↑r0↓ where Iy↑ is the Intensity of flow upwards at depth y, Iy↓ is the Intensity of flow downwards at depth y.

From this we can determine the boundary input and output flows. Sb↑S0↓=RΦ0↓Φb↑RΦb↑Φ0↓.rb↑r0↓

O0↑Ob↓=RΦb↓Φ0↑RΦ0↑Φb↓.rb↑r0↓

And so we have found the two-boundary system. O0↑Ob↓=RΦb↓Φ0↑RΦ0↑Φb↓.RΦ0↓Φb↑RΦb↑Φ0↓-1.Sb↑S0↓ where Sb↑ is the Intensity of source input at bottom boundary at depth b, S0↓ is the Intensity of source input at top boundary at depth 0, O0↑ is the Intensity of output from top boundary, Ob↓ is the Intensity of output from bottom boundary boundary.

For the numerical experiments, the mesh approximation for the relative velocity v have 61 sample points distributed as two arcs of shifted Chebeyshev points, to give additional resolution around the three points v=-1,0,1. The mesh approximation for the scaled depths 0≤z≤1 has 67 sample points distributed as shifted Chebeyshev points to provide additional resolution at y=0,∞. The actual scaling used is z=e-y/y^05 where y^05 is the two-stream estimate of the depth at which the transmission was reduced to 5 % of the original source. This provides better resolution by the numerical algorithm when the test case has strong forward scattering or a significant backward scattering component.

The test cases used the Henyey-Greenstein phase function parametrized by g the asymmetry parameter, which in this case is equal to the average scatter angle relative to an incident angle.

The 957 test cases used all pairs of 33 values of the asymmetry parameter (g= average scattering angle) and 29 values of the scattering fraction (ω= probability of scatter given extinction =1- probability of absorption given extinction).

Table  shows the parameters for the test cases. The Fortran implementation took 0.0742 s per test case, the Python implementation took 2.45 s per test case, and the Mathematica implementation took 23.5 s per test case. More detailed timing studies are presented below.

Observations of radiance in deep waters show that the shape of the radiance distribution approaches a limiting distribution, and the rate of decay of that distribution is exponential . This is the asymptotic radiance distribution, and the exponent of decay is the diffusion exponent . The max eigenvalue of Eq. () is the diffusion exponent.

As shown in Figs.  and  and discussed below, each material has a unique largest eigenvalue ν greater than 1 and a unique positive eigenvector. The rate of attenuation for each eigenvector is the reciprocal of the corresponding eigenvalue, so this positive eigenvector has the slowest rate of attenuation among the diffuse components of the intensity. Since the largest eigenvalue is also greater than 1, it's rate of attenuation is also slower than the rate of attenuation of the direct component.

Hence the positive eigenvector represents the asymptotic limit of the distribution of the intensity as the optical depth increases, and the rate of convergence is given by the reciprocal of the eigenvalue. As shown in Fig. , the maximum eigenvalue (and hence the slowest decay in diffusion) increases as absorption decreases (ω→1) and as the forward scattering ratio increases (g→1). So the diffuse intensity decays slower than the extinction decay of the direct intensity.

However the maximum eigenvalue is close to one on the majority of the region. Such materials have a decay rate of diffusion very close to the extinction decay rate of the direct component, so are in a strong sense effectively transparent in that there is always a component of direct intensity relative to the diffuse intensity. Given the direct component source, Fig.  shows the convergence of an opaque material to the diffusion limit, while Fig.  shows a semi-transparent material where the direct component remains larger than the limiting diffuse component.

The two-stream/Kubelka-Munk formulation provides simple analytic formulas for computing g and ω from R and η, and for the inverse. We use it as a reference baseline in our investigation of the properties of the multi-stream formulation.

The Inherent Optical Properties (IOP) are the properties of a media that characterize its optical behavior . For Eq. () they consist of the optical depth, the scatter fraction, and the scattering matrix. Since the numerical investigations here assume normalization by the optical depth and use the Henyey-Green one-parameter phase function, it is sufficient to use the scattering asymmetry g and the scattering fraction ω. This is also the common practice when using a two-stream approximation such as the Kubelka-Munk formulation. The Apparent Optical Properties (AOP) are the observed optical properties of a media . Following common practice, we look at the measured reflectance R and attenuation coefficient η.

Figure  shows the reflectance and the attenuation coefficient at an optical depth of 0.75 for (a) g=0.85 as ω varies from -1 to 1, (b) ω=-1 as g varies from 0 to 1, and (c) ω=1 as g varies from 0 to 1, for the numerical simulation and for the two-stream analytic approximation.

This illustrates the following:

The two-stream approximation agrees with the full system on the four boundaries, ω=±1 and g=0,1,

If we look at the scatter diagrams at differing transmission loss depths in Fig. , we see that the difficulty of invertibility increases as the loss increases, the data lets us determine g and ω at loss depths of less than 50 %, or for g>-0.5, while inversion becomes problematic for loss depths greater than 75 % and g<-0.7.

A choice was made as to which Radiative Transfer Models (RTMs) to compare against EigenFlux (See Table ). The major concerns were: (1) Does the RTM explicitly model scattering? (2) Does the RTM include the visible spectrum? (3) Is it available as open access, and is it still supported? (4) Does it use its own solver or does it use a different RTM? (5) Is it mature (version >1.0)?

These considerations led to two RTMS: DISORT and MYSTIC, of which DISORT was chosen due to its long maturity and the multiple RTMs that use it as their solver.

Of the tools considered, the snow and ice models are intended for highly forward scattering problems. Since they use the DO method, they are subject to the instability problems that come with using Legendre expansions of a highly asymmetric phase function. They ameliorate those problems with the use of variations on a specific technique, called the δ-N method in , which treat forward scattering light particles as non-scattering transmitted particles. But as we see below in Fig. , the effect of this is limited.

report that while snow grains have asymmetry g=0.9, brine pockets have asymmetry g=0.997. Their tool, CASIO-DISORT, derived from DISORT, was limited practically to gmax≤0.9. The higher g required for brine-snow mixtures was achieved through a reduced forward scattering transformation which is essentially an externally applied δ-N method.

We can see some of the advantages of the mesh approximation over the Legendre decomposition in managing phase functions. Figure  shows the approximations of the cloud phase function defined in , which contains both strongly forward scattering and a significant backward scattering component. Approximations used are those used in the case of n=56 streams.

To establish the accuracy of the EigenFlux algorithm, numerical investigations were conducted on a MacBook Pro with 2.6 GHz 6-Core Intel Core i7, 16Gb of RAM, running Tahoe V26.3.1. The final algorithm was coded in GNU Fortran 2008. The accuracy of the implementation was first verified by hand comparisons to published tables (see Table 35 in ).

The distribution for DISORT4 was downloaded from the Light and Life Lab website , along with its standard test suite, and compiled with the same compiler and options as Eigenflux was.

To provide a more general coverage of accuracy over a range of conditions, the error between the computed solution and the reference equation was computed. The integral formulation Eq. () was used as the reference equation because it is a numerically stable exact form. A Schlick phase distribution was used as the reference phase kernel for two reasons (1) it is very similar to the popular Henyey-Greenstein phase distribution, (2) its kernel function K(v,w) has a simple analytic form, allowing the exact kernel to be used in the reference equation.

Runs were performed for varying number of streams ranging from 14 to 280, for a Chebyshev mesh and for a linear mesh. Each run included 49 test cases varying scattering fractions from 0.012 to 0.987 and for asymmetry varying from -0.975 to +0.975.

Figure  shows the accuracy as a function of the number of streams for the Chebyshev mesh and for the linear mesh. It shows that the Chebyshev mesh is more accurate by a range of 1 to 2 orders of magnitude. There is significant variance in the accuracy across the 49 test cases of differing scattering fraction and asymmetry. Increasing the number of streams by a factor of 3.77 results in an average increase in accuracy of one magnitude for the Chebyshev mesh.

The change in accuracy for EigenFlux across the test cases is largely a function of the scatter fraction- the lower the absorption, the lower the accuracy as in Fig. . Figure  shows that there is a weak tendency for the method to be more accurate at the extreme points of asymmetry. The change in accuracy for DISORT across the test cases is similar in behavior to EigenFlux as a function of the scatter fraction- the lower the absorption, the lower the accuracy as in Fig. . However unlike EigenFlux, Fig.  shows DISORT is less accurate at the extreme points of asymmetry. The same input parameters were used to directly compare the EigenFlux and DISORT results and timing. The Henyey-Greenstein phase function was used for these comparisons.

One issue in the comparison is that EigenFlux always computes across a spanning set of source functions, whereas DISORT computes a single source function at a time. As a result, the DISORT times in Fig.  were scaled to correspond to the EigenFlux usage. The result shows that DISORT is faster for lower numbers of streams, whereas EignFlux is faster for higher numbers of streams, with a crossover at n=56.

The difference between the intensities computed by EigenFlux versus DISORT show considerable variance, as seen in Fig. . Figures  and  show that this is largely a factor of the asymmetry- DISORT's solution deteriorates for strongly scattering problems.