4  LMie Outputs

r21, r22

The actual minimum and maximum radii in the size distribution. Same as the inputs r1 and r2 for all but the mono (disperse) and power law size distributions.

norm

Normalization constant C of the size distribution.

reff

Effective radius of the size distribution:

reff = 1 Gavg ⌠ ⌡ r2 r1  n(r) πr2 dr,

(8)

where G_avg is defined under input gavg.

veff

Effective variance of the size distribution:

veff = 1 Gavg r2eff ⌠ ⌡ r2 r1  n(r) (r − reff)2 πr2 dr,

(9)

where G_avg is defined under input gavg.

gavg

The average projected area per particle of the size distribution:

Gavg = ⌠ ⌡ r2 r1  n(r) πr2 dr.

(10)

vavg

The average volume per particle of the size distribution:

Vavg = ⌠ ⌡ r2 r1  n(r) 4 3 πr3 dr.

(11)

ravg

The average radius of the size distribution:

ravg = ⌠ ⌡ r2 r1  n(r) r dr.

(12)

rvw

The volume-weighted average radius of the size distribution:

rvwa = ⌠ ⌡ r2 r1  n(r) r 4 3 πr3 dr.

(13)

cext

Ensemble averaged extinction cross section per particle.

csca

Ensemble averaged scattering cross section per particle.

cbak

Ensemble averaged backscattering cross section per particle.

g

Ensemble averaged asymmetry factor.

theta

Angles at which the elements of the ensemble averaged scattering phase matrix are computed as an array of size n_angles. Only valid if the input flag calc_pf is set.

pf

Elements of the ensemble averaged scattering phase matrix at the angles indicated by output theta as a (6 × n_angles) array. Only valid if the input flag calc_pf is set.

The scattering phase matrix has the form appropriate for a macroscopically isotropic and mirror symmetric scattering medium given by

F(Θ) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ a1(Θ) b1(Θ) 0 0 b1(Θ) a2(Θ) 0 0 0 0 a3(Θ) b2(Θ) 0 0 −b2(Θ) a4(Θ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,

(14)

where a₁(Θ) is the scattering phase function. For the case of homogeneous isotropic spheres a₁(Θ) = a₂(Θ) and a₃(Θ) = a₄(Θ) but for compatibility reasons the output is for all six elements. The elements map to the output array pf (in C syntax) as

a1(Θi)	=	pf[0][i]
a2(Θi)	=	pf[1][i]
a3(Θi)	=	pf[2][i]
a4(Θi)	=	pf[3][i]
b1(Θi)	=	pf[4][i]
b2(Θi)	=	pf[5][i]

where

Θ_i = theta[i]

and 0 ≤ i < n_angles.

n_coef

The number of expansion coefficients (and associated derivatives) output in the gc and lc arrays, and when derivatives are to be computed, the gc_l and lc_l arrays. This is the number of coefficients required to accurately represent the scattering phase matrix F(Θ) to the accuracy given as the input accuracy.

gc

Coefficients for the expansion of the ensemble averaged scattering phase matrix (equation 4.7) in terms of generalized spherical functions as a (6 × n_coef) array. Only valid if the input flag calc_gc is set.

The expansion has the form given by

a1(Θ) = N ∑ l = 0  βl Pl0,0(cosΘ), a2(Θ) + a3(Θ) = N ∑ l = 2  (αl + ζl) Pl2,2(cosΘ), a3(Θ) − a3(Θ) = N ∑ l = 2  (αl − ζl) Pl2,−2(cosΘ), a4(Θ) = N ∑ l = 0  δl Pl0,0(cosΘ), b1(Θ) = N ∑ l = 0  γl Pl0,2(cosΘ), b2(Θ) = N ∑ l = 0  εl Pl0,2(cosΘ),

where P^l_m,n(cosΘ) are generalized spherical functions. It is worth noting that a₁(Θ) is equal to the so called scattering phase function P(cosΘ) and the generalized spherical function P^l_0,0(cosΘ) is equal to the Legendre polynomial P_l(cosΘ). The expansion coefficients are the so "Greek constants" [Siewert, 1982] of the matrix

Bl = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ βl γl 0 0 γl αl 0 0 0 0 ζl −εl 0 0 εl δl ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,

(15)

which is a common form of input for many vector radiative transfer models. The expansion coefficients map to the output array gc (in C syntax) as

βl	=	gc[0][l]
αl	=	gc[1][l]
ζl	=	gc[2][l]
δl	=	gc[3][l]
γl	=	−gc[4][l]
εl	=	−gc[5][l]

where 0 ≤ l < n_coef.

lc

Coefficients for the expansion of the ensemble averaged scattering phase matrix (equation 4.7) in terms of Legendre polynomials as a (6 × n_coef) array. Only valid if the input flag calc_lc is set.

The expansion has the form given by

a1(Θ) = N ∑ l = 0  β1,l Pl(cosΘ), a2(Θ) = N ∑ l = 0  β2,l Pl(cosΘ), a3(Θ) = N ∑ l = 0  β3,l Pl(cosΘ), a4(Θ) = N ∑ l = 0  β4,l Pl(cosΘ), b1(Θ) = N ∑ l = 0  β5,l Pl(cosΘ), b2(Θ) = N ∑ l = 0  β6,l Pl(cosΘ),

where P_l(cosΘ) are Legendre polynomials of degree l. It is worth noting that a₁(Θ) is equal to the so called scattering phase function P(cosΘ). The expansion coefficients map to the output array lc (in C syntax) as

β1,l	=	lc[0][l]
β2,l	=	lc[1][l]
β3,l	=	lc[2][l]
β4,l	=	lc[3][l]
β5,l	=	lc[4][l]
β6,l	=	lc[5][l]

where 0 ≤ l < n_coef.

r21_l, r22_l, reff_l, veff_l, gavg_l, vavg_l, ravg_l, rvw_l, cext_l, csca_l, cbak_l, g_l

Linearized outputs indicated by a "_l" appended to the name for the associated output base value. These outputs are one dimensional arrays of size n_derivs where each element is associated with one of n_derivs derivatives.

pf_l, gc_l, lc_l

Linearized outputs indicated by a "_l" appended to the name for the associated output base value. These outputs are three dimensional arrays of size (n_derivs ×6 × n_angles) for pf_l and (n_derivs ×6 × n_coef) for gc_l and lc_l, where each subarray of the first dimension is associated with one of n_derivs derivatives.