Voigt equivalent widths and spectral-bin single-line transmittances: Exact expansions and the MODTRAN®5 implementation

Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

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Voigt equivalent widths and spectral-bin single-line transmittances: Exact expansions and the MODTRANs5 implementation Alexander Berk n Spectral Sciences, Inc., 4 4th Avenue, Burlington, MA 01803, USA

a r t i c l e in f o

abstract

Article history: Received 17 August 2012 Received in revised form 12 November 2012 Accepted 20 November 2012 Available online 7 December 2012

Exact expansions for Voigt line-shape total, line-tail and spectral bin equivalent widths and for Voigt finite spectral bin single-line transmittances have been derived in terms of optical depth dependent exponentially-scaled modified Bessel functions of integer order and optical depth independent Fourier integral coefficients. The series are convergent for the full range of Voigt line-shapes, from pure Doppler to pure Lorentzian. In the Lorentz limit, the expansion reduces to the Ladenburg and Reiche function for the total equivalent width. Analytic expressions are derived for the first 8 Fourier coefficients for pure Lorentzian lines, for pure Doppler lines and for Voigt lines with at most moderate Doppler dependence. A strong-line limit sum rule on the Fourier coefficients is enforced to define an additional Fourier coefficient and to optimize convergence of the truncated expansion. The moderate Doppler dependence scenario is applicable to and has been implemented in the MODTRAN5 atmospheric band model radiative transfer software. Finite-bin transmittances computed with the truncated expansions reduce transmittance residuals compared to the former Rodgers-Williams equivalent width based approach by  2 orders of magnitude. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Voigt line-shape Equivalent-width MODTRAN Band model Transmittance Radiative transfer

1. Introduction The MODTRAN51 atmospheric radiation transfer model [1,2] accurately and efficiently computes narrow band (0.1– 15.0 cm  1) atmospheric transmittances, radiances and fluxes for the 0.2 to 4 500 mm spectral range. Molecular transition data from the HITRAN database [3] is preprocessed to define finite spectral bin temperature-dependent absorption coefficient and line spacing band model parameters along with collision (Lorentz) half-width and line-tail absorption coefficient data for each molecular absorption source. These parameters are combined with Doppler half-widths to statistically characterize the molecular absorption within homogeneous path segments. Path segments are specified by their temperature, pressure, molecular abundances and path length. The finite-bin Voigt single-line equivalent width of the statistically averaged molecular lines is computed and combined with line position and strength distribution assumptions [4–6] to determine in-band line-center molecular transmittance. The band model approach provides a major advantage over the more rigorous line-by-line (LBL) technique in that inband values are computed directly. In LBL calculations, spectral optical depths are determined at a very high spectral resolution (typically, 0.01–0.0001 cm  1) and then transmittances are spectrally integrated to obtain the in-band values. At

n

Tel.: þ 1 781 273 4770; fax: þ1 781 270 1161. E-mail address: [email protected] 1 MODTRANs is a registered trademark owned by the United States Government as represented by the Secretary of the Air Force.

0022-4073/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jqsrt.2012.11.026

A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

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each spectral point, the absorption from all contributing molecular lines must be explicitly computed and summed. In the long wave infrared, hundreds-to-thousands of lines often contribute significantly to a single spectral point. It is for this reason that a statistical approach is beneficial and often crucial for solving real world problems. With the increasing signal-to-noise, spectral resolution and data flow of current and developmental infrared optical sensors, there is a desire and requirement to improve band models so that finer spectral resolution and higher accuracy predictions can be quickly generated. A critical aspect of this development is the calculation of accurate equivalent widths. The standard approach for calculating finite spectral bin equivalent widths involves first computing the total integrated line absorptance (one minus transmittance) known as the total equivalent width, and then subtracting the line-tail contributions. At moderate spectral resolution ( Z1.0 cm  1), the total equivalent width can be computed reasonable well by combining the pure Doppler and Lorentz total equivalent widths via the Rodgers and Williams approach [7], and the line-tail contributions determined from the reciprocal-frequency squared drop off of the Voigt line shape. The development of a higher spectral resolution (0.1 cm  1) band model requires higher accuracy total and line-tail equivalent width calculations. When the frequency displacement from line center to line-tail is comparable to the Lorentz or Doppler half-widths, the magnitude of the two terms being differenced is also comparable. Furthermore, the line shape of the tails exhibits strong Voigt behavior. In this paper, exact modified Bessel functions expansions for the total, line-tail and narrow spectral bin Voigt equivalent widths are derived. From these expressions, exact expansions for the Voigt single-line transmittance within a finite spectral bin are also obtained. Equations for the expansion coefficients are given for the pure Lorentz limit, for the pure Doppler limit and for Voigt line-tails with zero-to-moderate Doppler contributions.

2. The problem A primary objective of this paper is deriving an accurate and efficient algorithm for evaluating the Voigt line shape absorptance WD ¼WD(Su, gc, gd) integrated between spectral frequencies n and N for a molecular transition centered at frequency n0 r n: Z 1   ½1exp Suf n dn: ð1Þ WD  D

The absorptance is defined as one minus the transmittance, and D is the spectral frequency displacement,n  n0, from line center. The Voigt line-shape function fn is the convolution of Lorentz and Doppler line-shape functions:   Z 1   exp t 2 gc ð2Þ f n  f n gc , gd ¼ 3=2  2 dt: p 1 g2 þ ng t d c The area under the absorptance curve, illustrated in Fig. 1, is the Voigt line-tail equivalent width, i.e., the width a black line would have if its total absorptance were equivalent to that of the line-tail. The variables in Eqs. (1) and (2) are

gc Z0, the collision (Lorentz) half-width at ‘‘half’’ maximum [cm  1], gd Z0, the Doppler half-width at ‘‘1/e’’ of the maximum [cm  1], n, the spectral frequency integration variable [cm  1], S, the line strength [atm  1 cm  2], and u, the column density [atm cm]. The Voigt line shape function fn is symmetric in n, so the absorptance W  D equals 2W0 WD. It suffices to restrict the analysis to D Z0. When D 40, WD is the (single-sided) line-tail absorptance.

Absorptance ( 1 - Transmittance )

1.0

Δ

Δ

0.5

0.0 -0.5

Line-Tail 0.0 0.5 -1 Distance from Line Center (cm )

Fig. 1. The right-hand side Voigt line-tail equivalent width, WD, is the area under the absorptance curve a distance D from line center out to infinity (cross-hash). The finite-bin single-line equivalent width, Wsl, is the area under the curve between displacement frequencies (D D) and D (horizontal-hash).

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At the center of a line, the Voigt line shape function simplifies to a complementary error function: !   1 g2c gc : erf c f 0 ¼ pffiffiffiffi exp 2

gd p

gd

gd

This expression is correct in both the pure Lorentz (gd ¼0; gc 40) and the pure Doppler (gc ¼0; gd 40) limits: pffiffiffiffi limgd -0 þ p f 0 ¼ 1=gc ; limgc -0 þ pf 0 ¼ 1=gd :

ð3Þ

ð4Þ

3. The Ladenburg and Reiche function Ladenburg and Reiche [8] derived the exact expression for the total absorptance from a Lorentzian line (gd ¼0, D- N) in terms of exponentially-scaled Bessel functions of the first kind with imaginary arguments, also referred to as exponentially-scaled modified Bessel functions:      Su Su W Lorentz ¼ 2W Lorentz ¼ Su I~0 ð5Þ þ I~1 ; I~n ðzÞ  ez In ðzÞ 1 0 2pgc 2pgc (The exponentially-scaled modified Bessel functions, ~I0(z) and ~I1(z), can be evaluated using the polynomial approximations listed in Section 9.8 of Abramowitz and Stegun [9]). Ladenburg and Reiche also showed that the Lorentzian line-tail Lorentz equivalent width WD in the limit of D bgc, i.e., where fn ffi gc/pn2, can be expressed in terms of an error function: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!    pffiffiffiffiffiffiffiffiffiffi Sugc =p Sugc =p Lorentz D 1exp  WD ¼ Sugc erf ð6Þ ; Dcgc : 2

D

D

Lorentz

In this paper, an expansion is derived for WD

that is exact for any spectral displacement, D.

4. Strategy for determining WD In this section, the Ladenburg and Reiche formula is generalized to provide an exact modified Bessel function expansion for Voigt line-tail absorption. Appendix A provides an analogous derivation for the equivalent width of a narrow finite spectral bin containing a single Voigt line. The line-tail absorption, Eq. (1), can be integrated by parts by setting P¼ 1 exp( Sufn) and dQ¼dn: Z 1    

  df n W D ¼ D 1exp Suf D Su n ð7Þ exp Suf n dn: d n D As n approaches N, n[1  exp( Sufn)] approaches zero because the Voigt line-shape falls off proportional to 1/n2 except in the pure Doppler limit where it falls off even faster. The terms (dfn/dn) and dn in Eq. (7) can be combined to switch the integration variable from the frequency variable n to the Voigt line-shape function itself: Z fD   W D ¼ Su nf exp ðSuf Þdf D½1exp Suf D : ð8Þ 0

Defining angle y by the relationship       y 1cos y 2f ¼ fD ) yðf Þ  cos1 1 f ðyÞ ¼ f D sin2 ; 0 r f rf D , fD 2 2 and substituting into Eq. (8) gives      Z p    

Su y y Su f D cosy dyD 1exp Suf D : cos exp ny sin W D ¼ Suf D exp  f D 2 2 2 2 0

ð9Þ

ð10Þ

The coefficient of the exponential in the integrand, nysin(½y) cos(½y), will now be shown to be bounded for all y between 0 and p. From Eq. (9), f(y ¼ p) equals fD, which implies that np equals D and the sine and cosine product equals zero at y ¼ p. At y ¼0, the situation is more complicated. Since the Voigt function f(y) is zero at y ¼0, the spectral frequency n approaches infinity. However, this is exactly compensated by the sine term of Eq. (9): sffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffi   y y f ðyÞ gc ¼ ¼ lim ny lim ny sin cos : ð11Þ fD 2 2 pf D y-0 y-0 This last equality follows because, as noted above, the Voigt function becomes Lorentzian for large spectral frequencies and varies inversely with the square of the frequency. Even in the case of a pure Doppler line, this expression is correct because the Lorentz half-width gc is zero. Since the coefficient of the exponential in the integrand is continuous, boundedness is established. A generalization of the Ladenburg and Reiche expression is obtained by expanding the coefficient of the

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105

exponential in the integrand of Eq. (10) (multiplied by pfD) in a Fourier series:

y

y

2

2

pf D ny sin cos ¼

1 X VD 0 þ VD n cosðnyÞ, 2 n¼1

ð12Þ

where VD n  Vn



D gc ,

gd gd



¼ fD

Z p 0

ny sinycosðnyÞdy: D

ð13Þ D

It is evident from Eq. (13) that the Vn are dimensionless; as a result, the Vn are not independent functions of the displacement distance D and the Lorentz and Doppler half-widths, gc and gd, but instead 2-dimensional functions of their ratios. Since the exponentially-scaled modified Bessel functions of integer order ~In(z) have the integral representation Z ez p I~ n ðzÞ ¼ cosðnyÞexp ðzcosyÞdy ð14Þ

p

0

[Abramowitz and Stegun, Eq. 9.6.19], Eqs. (10), (12) and (14) can be combined to give the key result: "   X  # 1  

VD Su 0 ~ ~n Su f fD þ VD D 1exp Suf D : I W D ¼ Su I0 D n 2 2 2 n¼1

ð15Þ

This expansion essentially reduces the dimensionality of the problem from one with 3 independent variables WD(Su,gc,gd) to one with 2 independent variables, Vn(D/gd, gc/gd). The dependence of WD/Su on optical depth Su resides solely in the exponentially-scaled modified Bessel functions, ~In(½SufD), and in the line-tail offset term on the right. It is also worthy of note that Eq. (15) is valid not only for Voigt line shape functions, but also any line shape function that is continuous, that decreases monotonically as displacement distance, D, increases from its finite peak value at D ¼ 0, and that falls off as 1/D2 or faster as D approaches infinity. For small z, the modified Bessel functions can be computed from the ascending series [Abramowitz and Stegun, Eq. 9.6.10]: In ðzÞ ¼

 n þ 2k 1 X z=2 : k!ðn þ kÞ! k¼0

ð16Þ

Only the backward recurrence relationship [Abramowitz and Stegun, Eq. 9.6.26], In1 ðzÞ ¼ In þ 1 ðzÞ þ

2n 2n ~ In ðzÞ or I~n1 ðzÞ ¼ I~n þ 1 ðzÞ þ I n ðzÞ z z

ð17Þ

is numerically stable. Within MODTRAN5, the SLATEC Common Mathematical Library DBESI module [10,11] provides 14-place accuracy for the exponentially scaled modified Bessel functions, returning the complete series ~In(z) to ~In þ m(z) (for non-negative integers n and m) in a single call. The Voigt line-shape function at spectral displacement D, fD, is also determined to high accuracy (14 places) within MODTRAN5 via the Poppe and Wijers [12] algorithm. Both of these algorithms are relatively slow. Accuracy and computational efficiency testing is currently underway to determine if the Poppe and Wijers algorithm can be replaced by the recently published and extremely simple Abrarov and Quine [13] expansion, and to determine if the DBESI module can be replaced by the convergent and exact exponential-arc expansion of Borwein et al. [14]. Given all these readily available algorithms for determining the special functions, computation of the Voigt line-tail equivalent width reduces to calculation of the Fourier coefficient integrals Vn(D/gd, gc/gd). Although the focus of this paper is the analytic evaluation of the Voigt equivalent width and transmittance, it should be noted that the Eq. (10) representation of the equivalent width is particularly well suited for evaluation by numerical integration. Both the domain of the integration and the integrand are bounded. Although determining the frequency ny from y is difficult, one can instead select frequencies between D and N, and then easily map each of these to a value for y. To insure convergence, the y intervals can be adaptively sub-divided as needed by selecting intermediate spectral frequencies.

5. Finite-spectral-bin Voigt-single-line transmittance For a finite spectral bin of width D containing a single Voigt line, the equivalent width, Wsl, can be expressed as a function of Voigt line-tail equivalent widths, as illustrated in Fig. 2. Given a line center to nearer bin edge distance of D, the finite-bin Voigt single-line equivalent width expression is given by W sl ¼ 2W 0 W D W DD :

ð18Þ

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A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

Fig. 2. Decomposition of the finite-bin single-line equivalent width in terms of line-tail contributions.

This equation can then be inserted into the definition of the finite-bin Voigt single-line transmittance, tsl: 8 "   X  #   1 DD >   Suf DD D > DD ~ > Su V 0 I~0 Suf DD þ I þ 1 exp Suf DD V > n n > > D 2 2 2 D > n ¼ 1 > > " >  X  # > 1 D    Suf D D W sl < Su V 0 ~ Suf D D~ þ þ exp Suf D Vn In I0 ¼ þ t sl  1 D 2 2 2 D > D > n¼1 > > " >   X  # 1 0 > > Su V Suf Suf 0 > 0~ 0 0~ > þ V nI n I0 > 2 > : D 2 2 2 n¼1

ð19Þ

Traditionally, one of the difficulties with computing the finite bin Voigt transmittance from the equivalent width formalism is that numerical accuracy is compromised in the strong line limit, where the optical depth, Su, is very large. If, for example, the true transmittance is just less than 0.001, then the calculated single-line equivalent width over the bin width equals 0.999; the equivalent width calculation must be accurate to more than three-places. Given the approximations actually used in the past to compute the total and line-tail equivalent widths, absolute transmittance errors as large as 0.03 were not uncommon. Also, computed single-line equivalent widths occasionally exceeded the bin width; in these cases, the transmittance was just approximated as zero. With the exact expansion, Eq. (19), the constant terms cancel. At first it appears as if all remaining terms fall off exponentially. However, this apparent exponential fall off is deceptive because the exponentially-scaled modified Bessel functions do not fall off exponentially as their arguments become large. More detailed analysis of the strong line limit is provided both in Section 8, ‘‘Strong-line limit’’ and in Appendix A, ‘‘Voigt single-line equivalent width and transmittance for narrow spectral bins.’’ In the appendix, an alternative expansion for the finite spectral bin Voigt transmittance is obtained and that expression explicitly illustrates the exponential decay of the finite spectral bin transmittance. 6. Computing the Fourier coefficients D

Determining the Fourier coefficients, Vn , directly from Eq. (13) is complicated by the spectral frequency term. For the pure Doppler and Lorentz limits, inversion of the line shape function to determine spectral frequency is straightforward. However, deriving an expression for nf (or, equivalently, ny) that is valid for Voigt lines is challenging. In general, the calculation of the Fourier coefficients is greatly simplified by reintroducing spectral frequency as the dependent variable. This is accomplished by first substituting for the angle y using Eq. (9): Z fD   2f 2df cosyf ¼ 1 ) sinyf dyf ¼ ) Vn ¼ 2 nf cos nyf df : ð20Þ fD fD 0 D

For notational simplicity, the superscript D has been dropped from the Fourier coefficient term, Vn Vn . Next, integrate by 0 parts with Q¼ nf ¼ n and dPn ¼cos(nyf)df ¼cos(nyfn )df v    Z 1 Z 1 Z fn       2f n¼D Vn ¼ 2 P n dn þ2nP n 9n ¼ 1 ¼ 2 P n f n dn þ2DPn f D ; Pn f n  cos ncos1 1 df : ð21Þ fD D D 0 The boundary term 2nPn approaches zero as n approaches infinity, because the domain of the Pn integral is [0, fn], fv approaches 0, and the integrand is bounded. The Pn(fn) functions can be analytically integrated for each value of nonnegative integer n. The Fourier coefficients for n r7 have the following form: V 0 ¼ /f 0 S þ2Df D ,

ð22aÞ

V 1 ¼ /f 0 S/f 1 S,

ð22bÞ

V 2 ¼ /f 0 S4/f 1 Sþ

8 2 /f S Df D , 3 2 3

ð22cÞ

A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

107

V 3 ¼ /f 0 S9/f 1 S þ 16/f 2 S8/f 3 S,

ð22dÞ

V 4 ¼ /f 0 S16/f 1 Sþ

160 128 2 /f 2 S64/f 3 Sþ /f 4 S Df , 3 5 15 D

ð22eÞ

V 5 ¼ /f 0 S25/f 1 Sþ

400 256 /f 2 S280/f 3 Sþ 256/f 4 S /f 5 S, 3 3

ð22fÞ

V 6 ¼ /f 0 S36/f 1 Sþ 280/f 2 S896/f 3 S þ V 7 ¼ /f 0 S49/f 1 Sþ where /f n S 

2 n fD

Z

1

nþ1

fn

D

6912 2048 2 /f 4 S1024/f 5 Sþ /f 6 S Df 5 7 35 D

and

1568 19712 /f 2 S2352/f 3 Sþ 5376/f 4 S /f 5 Sþ 4096/f 6 S1024/f 7 S, 3 3

dn:

ð22gÞ

ð22hÞ

ð23Þ

These expressions for the Fourier coefficients in terms of Voigt function moment integrals, /fnS, were derived with a substantial amount of algebraic manipulations. However, one of the reviewers of this paper noted that the integrand of the Pn integrals, Eq. (21), is a Chebyshev polynomial:   ð24Þ T n ðxÞ ¼ cos ncos1 x : Knowing this, greatly simplifies calculation of the Pn integrals, and enables the determination of a general equation for the Vn coefficients. For n greater than zero, the Chebyshev polynomial is expressed in terms of derivatives via the equation T n ðxÞ ¼

1=2 d 1=2 d T n þ 1 ðxÞ T n1 ðxÞ, n þ1 dx n1 dx

n ¼ 1,2,. . .:

ð25Þ

Defining x as 1–2 fn/fD, Eqs. (24) and (25) can be inserted into the definition of the Pn integrals, Eq. (21):   f Pn f n ¼ D 2

Z

1 12f n =f D

  f cos ncos1 x dx ¼ D 2

Z

1

T n ðxÞdx ¼ 12f n =f D

     1T n1 12f n =f D f D 1T n þ 1 12f n =f D  , 4 n þ1 n1

n ¼ 1,2,. . .

ð26Þ Here the identity Tn(1) ¼1 has been used to evaluate the Chebyshev polynomial at the integral upper limit. Next, these expressions for the Pn integrals are inserted into the Fourier coefficient expression, Eq. (21). Noting that Tn(  1)¼(  1)n, the following equation is obtained:

Vn ¼

fD 2

Z

1



D

    1T n þ 1 12f n =f D 1T n1 12f n =f D Df  dn þ D nþ1 n1 2

! 1ð1Þn þ 1 1ð1Þn1  , nþ1 n1

n ¼ 1,2,. . .:

ð27Þ

The Chebyshev polynomials are expanded using the form:

T n ðxÞ ¼ 1 þ n

n n X X 2k ð2k þ1Þnk 1T n ðxÞ 2k ð2k þ 1Þnk ¼ ðx1Þk ) ðx1Þk , ð Þ ð Þ! ð Þ ð Þ! n n þ k nk n þ k nk k¼1 k¼1

ð28Þ

where the Pochhammer symbol [(a)0  1 and (a)j  (a) (a þ1) (a þj  1) for j ¼1, 2, y] is introduced to simplify the notation. Inserting this series into Eq. (27) and simplifying, the following compact expression that is valid for all the Fourier coefficients is obtained: ( n 2Df D X ð4Þk ð2k þ 1Þnk n even, /f k S n2 1 V n ¼ /f 0 Sþ n ð Þ ð Þ ð Þ! k þ 1 n þ k nk 0 n odd k¼1

n ¼ 0,1,2,. . .

ð29Þ

Eqs. (22a)–(22h) easily follow from explicit application of this expression.

7. Weak line limit In the weak line limit, i.e. when Su is small, line absorption is known to be proportional to Su. From Eq. (16), In(½SufD) ¼[(¼SufD)n/n!] {1 þO[(Su)2]}, where, the notation O[(Su)2] denotes terms of order Su raised to the second power

108

A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

and higher. Substituting this relationship into Eq. (15), the line-tail absorption expression reduces to the correct weak line limit:  " D   X  # 1 h i   Su D Su V0 Su Su fD þ f V 0 SuDf D þ O ðSuÞ2 I0 W D ¼ Suexp  f D VD I D 1exp Suf D ¼ D n n 2 2 2 2 2 n¼1 Z 1 h i h i  Su  ð30Þ /f 0 Sþ 2Df D SuDf D þ O ðSuÞ2 ¼ Su f n dn þO ðSuÞ2 : ¼ 2 D In the total line absorption limit, V0(0) is one and W  N ¼2W0 equals Su to leading order. 8. Strong line limit The asymptotic expansion of the exponentially-scaled modified Bessel function for large argument z and for n fixed is given by [Abramowitz and Stegun, Eq. 9.7.1] " #         4  14n2 94n2 14n2 94n2 254n2 1 14n2 : ð31Þ þ I~n ðzÞ  pffiffiffiffiffiffiffiffiffi 1 þ þ þ O z 8z 2pz 2!ð8zÞ2 3!ð8zÞ3 Formally, it is incorrect to simply insert this expression into the expansion of Eq. (15) both because Eq. (31) is an asymptotic expansion and because the sum index n will ultimately approach and exceed any fixed argument z. Empirically, the substitution has been shown to produce some important and useful results. Since the leading term in Eq. (31) is independent of n, the asymptotic expansion produces the following expression for the Voigt line-tail equivalent width: sffiffiffiffiffiffiffiffiffi"  # 1 X  

Su V D 2 0 WD  þ VD þ O D 1exp Suf D : ð32Þ Suf D pf D 2 n ¼ 1 n An expression for the sum of the Fourier coefficients is obtained by taking the limit of Eq. (12) as y approaches zero and inserting the result from Eq. (11): qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X VD 0 þ VD pgc f D n ¼ 2 n¼1

ð33Þ

When the displacement from line center is zero, the sum of Fourier coefficients can be expressed in terms of the complementary error function: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u   1 X upffiffiffiffig  V 00 g2c gc 0 c t erf c þ V0 ¼ p exp ð34Þ 2 2 g g g d d d n¼1 Substituting Eq. (33) into Eq. (32) produces a generalization of the square root curve of growth: sffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffi   4=p W D þ D  Sugc þO ; gc 4 0 3 Suf D

ð35Þ

The sum of the line-tail equivalent width and the displacement from line center grows proportional to the square root of the optical depth in the strong line limit. The finite spectral bin Voigt single-line transmittance, tsl, from its definition, clearly equals one at optical depth Su equal zero, and decreases monotonically towards zero as Su increases. Thus, the finite spectral bin transmittance, tsl  1 Wsl/D, cannot decrease proportional to OSu in the strong line limit even though line-tail equivalent widths increase proportional to OSu. Inserting Eq. (35) into the definition of tsl, Eq. (19), the terms proportional to OSugc all cancel: rffiffiffiffiffiffiffi 9 8 pffiffiffiffiffiffiffiffiffiffi 4=p > > > > þ O g 2 Su > > 3 c = < Suf 0 2W 0 W D W DD 1  rffiffiffiffiffiffiffiffi  rffiffiffiffiffiffiffiffiffiffiffi  1 t sl ¼ 1 p p ffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffi ffi > D> D 4=p 4=p > > > >  Sugc ðDDÞ þ O Sugc D þ O 3 3 ; : Suf D

"



1 2O D

sffiffiffiffiffiffiffiffiffiffi! 4=p 3 Suf 0

þO

sffiffiffiffiffiffiffiffiffiffiffi! 4=p 3

Suf D

þO

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# 4=p 3 Suf DD

 pffiffiffiffiffiffi ¼ O 1= Su :

Suf DD

ð36Þ

It is important to emphasize the analysis leading to Eqs. (35) and (36) should not be construed as proofs that the line-tail equivalent width grows proportional to OSu and that the finite-bin transmittance decreases proportional to 1/OSu or faster in the strong-line limit. These equations are provided here as heuristic arguments of these dependencies. In Appendix A, an alternative Bessel function expansion for tsl explicitly demonstrates that it has an asymptotic expansion that falls off exponentially with Su.

A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

109

For numerical implementations, it has been found that enforcing the strong line limit by adding an mþ1 Fourier coefficient, ! qffiffiffiffiffiffiffiffiffiffiffiffiffi m X VD D 0 ½V D þ   ¼ pg f V ð37Þ c D m þ 1 StrongLine n , 2 n¼1 to a truncated expansion with 0 to m terms will generally significantly improve the convergence. The only exception seems to be in the limit of pure Doppler lines, as discussed in Section 10. Henceforth, inclusion of this extra term in the truncated expansions will be referred to as application of the strong-line limit sum rule. 9. The Lorentz limit In the pure Lorentz limit (gd ¼0; gc 40), the line-shape function has a simple form: Lorentz

fn

gc =p : g2c þ n2

¼

ð38Þ

It is convenient to define parameter a as the ratio of the spectral displacement from line center D to the Lorentz half-width gc:

a  D=gc :

ð39Þ

Inserting Eqs. (38) and (39) into Eqs. (22a)–(22h) and (23), the following analytic expressions are obtained for the first 8 Fourier coefficient integrals:

pV Lorentz 0 2

¼ cot1 a þ 

a

ð40aÞ

1 þ a2



pV Lorentz ¼ a a2 1 cot1 a 1 

ð40bÞ

pV Lorentz ¼ 2a3 acot1 a1 þ 2 

1=3 1þ a2



4 3

 ð40cÞ 



pV Lorentz ¼ a3 5a2 þ a 5a2 þ 3 cot1 a 3 



ð40dÞ



pV Lorentz ¼ 2a3 a 7a4 þ 8a2 þ 2 cot1 a7a4  4 

pV Lorentz ¼ a3 42a6 þ56a4 þ 5 

17 2 11 1=15 a  þ 3 15 1 þ a2

 ð40eÞ

 

301 2 4 a þ a 42a6 þ 70a4 þ35a2 þ 5 cot1 a 15 3





pV Lorentz ¼ 2a3 a 66a8 þ 144a6 þ108a4 þ32a2 þ3 cot1 a66a8 122a6  6 

pV Lorentz ¼ a3 429a10 þ 1012a8 þ 7

ð40fÞ

366 4 538 2 73 1=35 þ a  a  5 35 105 1 þ a2

 ð40gÞ

   4279 6 2168 4 131 2 4 a þ a þ a þ a 429a10 þ 1155a8 þ 1155a6 þ 525a4 þ 105a2 þ 7 cot1 a 5 7 3 3

ð40hÞ Lorentz Vn

3

Lorentz V0

Lorentz V1

For nZ2, each of terms are proportional to a and are small for D 5 gc. In this limit, and approach 1 Lorentz and ½, respectively, and the remaining Vn approach 0, which together lead to the Ladenburg and Reiche’s formula. For Lorentz large a, the expressions for Vn are numerically unstable, because the inverse cotangent terms are exactly canceled by the polynomials in a for all non-negative powers of a. The large a limit is more easily evaluated using the asymptotic series below: 4

8

12

16

20

24

pV Lorentz ¼  3 þ 5 7 þ 9 þ  0 a 3a 5a 7a 9a 11a11 pV Lorentz ¼ 1

4 8 12 16 20 24  þ  þ  þ  3a 15a3 35a5 63a7 99a9 143a11

ð41bÞ

4 8 4 16 20 8 þ  þ  þ þ  15a 21a3 9a5 33a7 39a9 15a11

ð41cÞ

4 8 4 16 20 24  þ  þ  þ  35a 63a3 33a5 143a7 195a9 255a11

ð41dÞ

pV Lorentz ¼ 2 pV Lorentz ¼ 3

ð41aÞ

pV Lorentz ¼ 4

4 232 428 48 76 344 þ  þ  þ þ  63a 3465a3 6435a5 715a7 1105a9 4845a11

ð41eÞ

110

A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

pV Lorentz ¼ 5

4 376 268 304 524 40  þ  þ  þ  99a 9009a3 6435a5 7293a7 12597a9 969a11

pV Lorentz ¼ 6 pV Lorentz ¼ 7

ð41fÞ

4 184 7292 46192 12596 7432 þ  þ  þ þ  143a 6435a3 255255a5 1616615a7 440895a9 260015a11

4 152 8660 6736 2012 11608  þ  þ  þ  195a 7293a3 415701a5 323323a7 96577a9 557175a11

ð41gÞ

ð41hÞ

In the Lorentz limit, the expression for the sum of the Fourier coefficient integrals, Eq. (33), reduces to 1 X V Lorentz 1 0 þ V Lorentz ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi : n 2 1 þ a2 n¼1

ð42Þ

At a ¼0, this condition is exactly met by the first two terms in the expansion with all remaining terms being zero. For large a, the series truncated at n¼7 approximately equals 1.0028/a þO(1/a3). As noted above, compliance with the strong line Lorentz limit sum rule generally greatly improves convergence, so that V8 should be set as follows ! 7 Lorentz X 1 V0 þ StrongLine ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi  V Lorentz ½V Lorentz : ð43Þ 8 n 2 1þ a2 n¼1 This is quite an important condition. The general approach for calculating the absorption integrals involves expanding the coefficient of the exponential in Eq. (10), ny sin(½y) cos(½y), in a Fourier series. However, in the strong line limit, the exponential in the integrand is dominated by values near y ¼0. The Fourier series fits the entire domain of y values, giving no special preference to the part of the domain that most significantly contributes for strong lines. The strong line limit sum rule forces a better fit near y ¼0. In Fig. 3, the in-band transmittance is plotted as a function of spectral band half-width for a band-centered single Lorentzian line of half-width gc ¼0.06 cm  1. The plot includes a break in the in-band transmittance scale at 0.016 to facilitate viewing of the curves for small transmittances. The thick solid curves show the exact results, integrated numerically, for values of the line-center optical depth ranging from Su/pgc ¼0.04 to Su/pgc ¼625. In-band transmittance Lorentz increases as bandwidth increases. The results from the Bessel function series truncated at n¼ 8 with V8 computed from the strong-line limit sum rule overlap the numerical integration curves with residuals less than 0.000026; the symbols in Fig. 3 are sample points. If the full bandwidth is a factor of 10 times larger than the Lorentz half-width (2D ¼0.6 cm  1) and the line center optical depth is less than 25, then the line-tails can be reasonably well represented by the reciprocal frequency squared relationship (thin curves), which leads to the line-tail equivalent width expression of Eq. (6). For narrower band models, this relationship breaks down and the exact expansion is required. The convergence of the Lorentz in-band transmittance expansion for band-centered Lorentzian lines of width gc ¼0.06 cm  1is illustrated by Fig. 4. The thick solid curves show the exact transmittances, obtained from numerical integration, plotted as a function of line center optical depth; curves are defined for band widths between 0.05 and 12.8 cm  1 (D between 0.025 and 6.4 cm  1). Also included in Fig. 4, as thin lines with symbols, are the results from the Lorentz Bessel function series truncated at n¼8 with V8 computed from the strong-line limit sum rule. The largest error for any transmittance greater than 0.001 is less than 4  10  6. For transmittances above 10  4, the truncated expansion and numeric integration curves of Fig. 4 overlap. However, below 10  4, the limitations of the n ¼8 truncation are apparent. An alternative form for the Bessel function expansion of the finite spectral bin Voigt transmittance is derived in Appendix A,

In-Band Transmittance 1 - (W0 - WΔ ) / Δ

1.0

0.04

0.8

0.2

Su/

0.6

c

= 1.0 c

5.0

0.4

= 0.06

25.0

0.2 125.0 0.016 0.008 0.000

0.0

0.2

0.4

0.6

0.8

625.0 1.0 -1

One-Half Band-Width, Δ (cm ) Fig. 3. In-band transmittance plotted as a function of half bandwidth for a band-centered pure Lorentzian line with half-width 0.06 cm  1. Results are illustrated for the labeled line center optical depths, Su/pgc. The thick curves are the exact results from numerical integrations. The results from the modified Bessel function expansion truncated at n¼ 8 with the last term defined by the strong-line limit sum rule are plotted as symbols and are in full agreement with the exact curves. The thin lines without symbols are results computed assuming line-tails are inversely proportional to frequency squared.

A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

111

In-Band Transmittance 1 - (W0 - WΔ ) / Δ

1 0.1 Δ

0.01 c

= 0.06

1E-3 1E-4 1E-5

-1

(cm ) 6.400 1.600 0.400 0.100 0.025

0 10 20 30 40 50 250 500 750 1000 Line Center Optical Depth, Su / c

Fig. 4. In-band transmittance plotted as a function of line center absorption for a band-centered pure Lorentzian line with half-width 0.06 cm  1. Differences between the exact results (thick lines) and the Bessel function expansion truncated at n¼ 8 with the last term defined by the strong-line limit sum rule (thin lines with symbols) are only evident when the transmittance drops below 10  4.

and this expansion is expected to be accurate in the small transmittance limit. In the current MODTRAN implementation of Eq. (15), the in-band transmittances are restricted to be between the minimum transmittance of the spectral bin, exp(–Suf0), and the maximum, exp(–SufD). With this requirement, the largest transmittance error is less than 6  10  5 (for the D ¼0.4 cm  1 curve with a line-center optical depth of 442.6, the exact transmittance is  2.7  10  6, but the truncated expansion predicts 5.89  10  5). For the terrestrial atmosphere applications of MODTRAN5, computing molecular transmittances with an absolute accuracy of better than 6  10  5 is more than adequate. For applications that require high accuracy when transmittances drop below 10  4, the approach laid out in Appendix A should be implemented. 10. The pure Doppler limit In this section, the expression for the Voigt line-tail absorptance, Eq. (15), is applied to a pure Doppler line (gc ¼0). In this limit, the line shape function has the following form: Doppler

fn

¼

exp ðbn Þ pffiffiffiffi ;

gd p

bn 

n2 : g2d

ð44Þ

Inserting this definition into Eq. (23), the Voigt function line-tail moment integrals have a simple form: D E exp ðnb Þ erf cpðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n þ 1ÞbD Doppler D pffiffiffiffiffiffiffiffiffiffiffi : fn ¼ nþ1

ð45Þ

The Doppler Fourier coefficients are computed by directly substituting these moment integrals into Eqs. (22a)–(22h) or Eq. (29). In the limit of total Doppler line absorptance (D ¼0 ) bD ¼0), the moment integrals are simply the reciprocal of the square root of n þ1: Doppler

/f n

1 SD ¼ 0 ¼ pffiffiffiffiffiffiffiffiffiffiffi nþ1

ð46Þ

The following approximation for the total Doppler equivalent width is obtained:    2  3 I~ 0 2gSupffiffipffi þ 0:58578644I~1 2gSupffiffipffi 0:57765281I~ 2 2gSupffiffipffi 0:25271345I~3 2gSupffiffipffi d d d d 6 7     ¼ 2W Doppler ¼ Su4 W Doppler 5 1 0 0:14605219I~ 4 2gSupffiffipffi 0:09628129I~ 5 2gSupffiffipffi 0:06867590I~6 2gSupffiffipffi 0:05165314I~7 2gSupffiffipffi     d

d

d

d

ð47Þ The coefficients in this expansion can be represented analytically via the following equations: V Doppler ðD ¼ 0Þ ¼ 1, 0

ð48aÞ

1 ðD ¼ 0Þ ¼ 1 pffiffiffi , V Doppler 1 2

ð48bÞ

pffiffiffi 8 ðD ¼ 0Þ ¼ 12 2 þ pffiffiffi , V Doppler 2 3 3

ð48cÞ

9 16 ðD ¼ 0Þ ¼ 3 pffiffiffi þ pffiffiffi , V Doppler 3 2 3

ð48dÞ

112

A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

1.0

0.6

Exact Simple Up to I1

0.4

Up to I3

0.2

Up to I5

2 W0

Doppler

/ Su

0.8

0.06 0.04 0.02 0.00 0.01

Up to I7 Residuals 0.1

Su f0

1 Doppler

10

= Su /

100

1/2

d

Fig. 5. Doppler total equivalent width predictions as a function of line center optical depth. The modified Bessel function expansions truncated with n¼ 1, 3, 5 and 7, and a simple weak- to strong-line interpolation formula, Eq. (50) are each compared to exact results obtained from numerical integration. The curves include a break in the ordinate axis at 0.075 in order to accentuate the residuals.

pffiffiffi 160 128 V Doppler ðD ¼ 0Þ ¼ 318 2 þ pffiffiffi þ pffiffiffi , 4 3 3 5 5

ð48eÞ

25 400 256 256 ðD ¼ 0Þ ¼ 139 pffiffiffi þ pffiffiffi þ pffiffiffi  pffiffiffi , V Doppler 5 5 3 6 2 3 3

ð48fÞ

pffiffiffi 280 6912 1024 2048 ðD ¼ 0Þ ¼ 44718 2 þ pffiffiffi þ pffiffiffi  pffiffiffi þ pffiffiffi V Doppler 6 5 5 7 7 3 6

and

ð48gÞ

561 1568 5376 19712 4096 ðD ¼ 0Þ ¼ 1175 pffiffiffi þ pffiffiffi þ pffiffiffi  pffiffiffi þ pffiffiffi : V Doppler 7 5 7 2 3 3 3 6

ð48hÞ Doppler

The strong line limit sum rule, Eq. (33), if applied to pure Doppler limit would dictate that the Vn to zero: 1 X V Doppler 0 þ V Doppler ¼ 0: n 2 n¼1

coefficients sum

ð49Þ

Doppler

The calculation of [V8 ]Strong  Line produces a value near  0.39, much larger in magnitude than any of the previous few Doppler Vn coefficients. Experience suggests that the strong line limit sum rule works well when the previous few Vn coefficients generally decrease in magnitude with increasing n and have varying signs (often alternating). Under these conditions, the sum rule tends to produce a m þ1 coefficient with the same sign as the actual m þ1 coefficient, but with a smaller, intermediate magnitude. In the pure Doppler limit, the n ¼2 through n¼ 7 coefficients are all negative and the Doppler magnitude of the sum rule coefficient is much larger than one expects for the true V8 . This produces much too small of Doppler a value for the total Doppler equivalent width for large values of the line center optical depth, Suf0 . Fig. 5 illustrates the accuracy of the truncated total Doppler equivalent width expansion without the strong line limit sum rule term. For comparison, results are also presented for an often-used simple interpolation formula between the weak and strong line Doppler limits: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 1 W Doppler Su= 2ln 1þ g g ð50Þ d d 1 2 If the Doppler line center optical depth Su/(gdOp) is less than 10, the expansion of Eq. (47) truncated after 6 terms (up to V5) produces a normalized equivalent width residual of less than 0.0005. Errors increase rather dramatically for even higher line center optical depths. At a line center optical depth of 100, the 8 term expansion (up to V7) has a 0.0037 residual, which corresponds to a 14.4% error. In this limit, asymptotic expansions [15] or even Eq. (50) is preferable. There were reasons to suspect that the truncated series for calculation of the total Doppler equivalent, Eq. (47), would not produce well converged results for the full range of line center optical depths, Suf0. The expansion after 8 terms has yet to settle into producing Fourier coefficients of varying sign. Furthermore, the total Lorentz equivalent width is fully converged at n ¼1; the pure Doppler limit represents the complete opposite end of the parameter space. 11. Voigt line-tails with moderate Doppler dependence An ultimate objective of this manuscript is to provide an accurate algorithm for computing tsl, the finite-bin Voigt single line-transmittance, Eq. (19), for the MODTRAN5 atmospheric radiation transport model. This requires calculation of the D D 0 DD DD 0 Fourier coefficients Vn, Vn and Vn . Both Vn and Vn are functions of two variables, while Vn ¼Vn(0, gc/gd) is a function of

A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

113

0

a single variable. A table of Vn coefficients has been generated using an adaptive bisection integration method to explicitly compute the entries with better than 9 place accuracy. The numerical integration technique was validated by 0 demonstrating that the V0(0, gc/gd) coefficients equal one for each value of gc/gd. For each n from 1 to 20, Vn coefficients were computed for a grid of 1201 gc/gd values from 0.00002 to 20. i.e., 200 grid points per decade. The pure Doppler limit coefficients Vn(0, 0) were also computed for interpolating between gc/gd ¼0 and gc/gd ¼0.00002. For gc/gd greater than 20, an analytic based extrapolation procedure is used. The completed interpolation/extrapolation algorithm provides an 0 accurate method for computing the Vn coefficients for arbitrary values of the argument. D DD This section focuses on obtaining analytic expressions for the Fourier coefficients Vn (and Vn ) applied to Voigt lines with at most moderate Doppler dependence. To be more precise, the analysis is restricted to computing line-tail equivalent widths WD (and WD  D) for which 9ZD9 (and 9ZD  D9) is somewhat large, certainly greater than  2. The parameter Zn is defined as follows: Xn 

n g g2 þ n2 2 , Y  c , and Z n  X n þiY ) 9Z n 9 ¼ c 2 : gd gd gd

ð51Þ

In the analysis below, all terms of order less than O(9Zn9  8) are retained. This work was initiated for use within the MODTRAN5 atmospheric model, which uses a statistical band model approach to solve the radiative transfer equations. The finest MODTRAN5 band model spectral resolution is D ¼ 0.1 cm  1. To compute the average absorption of lines within the 0.1 cm  1 intervals, an effective line is defined and centered ¼ of a spectral bin from bin edge, D ¼0.025 cm  1. Doppler 1/e half-widths, gd, are computed from the following formula: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gd ¼ 4:3  107 n0 T ðK Þ=mðamuÞ, ð52Þ where n0 is the molecular transition line center frequency in cm  1, T(K) is the atmospheric temperature in degrees Kelvin, and m (amu) is the molecular mass in atomic mass units. If, for convenience, one assumes a relatively high atmospheric temperature of 300 K and a relatively low molecular mass of 12, so that the square root in Eq. (52) equals 5, the Doppler half-width equations becomes gd ¼ 2.15  10  6n0. In the middle of the long-wave infrared, at say 10 mm, n0 equals 1000 cm  1. This implies a Doppler half-width of 0.00215 cm  1 and a value of 9ZD9 greater than 10 before the Lorentz half-width contribution has even been added in. Lorentz half-widths, gc, are generally about 0.06 cm  1 at standard atmospheric pressure, but they decrease proportional to pressure. At 2 mm, n0 equals 5000 cm  1and the Doppler half-width has increased to  0.011 cm  1. This is still more than a factor of 2 less than the 0.1 cm  1 band model displacement distance and more than a factor of 5 less than the standard atmospheric pressure Lorentz half-width. At wavelengths short of 2 mm, MODTRAN5 is generally run with one of its coarser spectral resolution (1, 5 or 15 cm  1) band models. The coarser resolution is chosen for 2 reasons: (1) modeling of most visible through shortwave infrared (vis/ SWIR) optical instruments (other than lasers which should be modeled using a LBL radiative transfer algorithm) does not require finer than 1 cm  1 spectral resolution and (2) solar multiple scattering, which is the dominant source of radiation in the vis/SWIR, is time consuming to calculate, especially if the spectral grid size is too fine. With the 1 cm  1 band model, MODTRAN5 centers the effective molecular absorption line D ¼0.2 cm  1 from the spectral bin edge. The shortest wavelength transition included in the HITRAN molecular line database [3] and used by MODTRAN5 is a H2O line at 25,232 cm  1 with a Doppler width of  0.054 cm  1. This width is almost 4 times less than the 1 cm  1 band model displacement distance, D. With all terms of order less than O(9Zn9  8) retained, the expressions derived in this section are appropriate for MODTRAN5 use in essentially all its applications. If one needed to run the MODTRAN5 0.1 cm  1 band model in the near infrared or visible spectral regions, substantial errors could arise in the finite spectral bin transmittance calculations. This potential problem will be fixed in a future version of MODTRAN by implementing the procedures outlined in Appendix A. D The general approach for computing the Vn Fourier components for Voigt lines with at most moderate Doppler dependence is similar to the approach used to compute the pure Lorentz limit Fourier components. Initially, an analytic expression for the line-shape function, fn, is determined. This form is then used to analytically evaluate the Fourier coefficient integrals for n r7. The analytic form for these integrals turns out to only be stable numerically if the parameter a (  D/gc), Eq. (39), is not too large. Therefore, asymptotic expressions are derived for each of the Fourier coefficients for larger values of a. The derivation of these expressions required an immense amount of algebraic manipulations. Although the author derived the first few Fourier coefficient integrals and asymptotic expansions by hand, Mathematica software [16] was utilized to determine most of the expressions in Appendix B. In Eqs. (22a)–(22h), or, (25), equivalently, Eq. (29), the calculation of the Fourier coefficients reduces to determining the Voigt function line-tail moment integrals of Eq. (23), /fnS. A Voigt line-shape large 9ZD9 expansion is required to derive an analytic expression for the integrand of these integrals. It follows from Eq. (2) that the Voigt line shape is proportional to the real part of the complex probability function w(Zn) [Abramowitz and Stegun, Eq. 7.4.13]: fn ¼

Re½wðZ n Þ Re½wðX n þ iY Þ pffiffiffiffi  pffiffiffiffi :

gd p

gd p

ð53Þ

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A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

From the continued fraction representation for w(Zn) [Abramowitz and Stegun, Eqs. 7.1.4 and 7.1.15], the Voigt integral can be written     1 1 1=2 1 3=2    ; ðYa0Þ, ð54Þ Im f n gc , gd ¼  Zn  Zn  Zn  Zn  pgd which leads to the following Pade´ approximant and asymptotic series:     2 3 15 1 9 Y2 1 21 Y2 1 1 10 þ3 þ þ 2 2 4 2 6 2  2 4 8 jZ n j jZ n j 6 9Z n 9 9Z n 9 9Z n 9 Y 8 7 6 7       fn ¼ þO 9Z n 9 4 5 pgc 9Z n 92 1 12 2 1  Y 2 2 þ 12 4 7  Y 2 2 þ Y 4 4  9 6 1  Y 2 2 þ 9=168 2 8 2 9Z n 9

and

9Z n 9

9Z n 9

9Z n 9

9Z n 9

9Z n 9

9Z n 9

ð55aÞ

9Z n 9

    9 8 > > 1 3 2Y 2 3 5 5Y 2 4Y 4 > > 1 þ   þ þ > > 2 2 4 2 4 2 4 = < 9Z n 9 9Z n 9 9Z n 9 9Z n 9 9Z n 9 Y2   fn   8 > 15 7 7Y 2 14Y 4 8Y 6 > > pgc 9Z n 92 > > > þO 9Z n 9 6 2 þ 4  6 ; :þ 8 9Z n 9

9Z n 9

9Z n 9

ð55bÞ

9Z n 9

The asymptotic expansion for fn can now be raised to the nth power, expanded in powers of 1/9Zn92, and integrated. In Appendix B, expressions for the Voigt line-shape function line-tail moment integrals for n r7 are explicitly listed along with the asymptotic expansions for large a. In the implementation of these equations, the analytic integrals are used when parameter a is less than or equal to 3, and the asymptotic expressions are used for a greater than 3. All of the moment integral expressions from Appendix B have been integrated in MODTRAN5 for calculation of the Fourier coefficient integrals and, in turn, for calculation of finite-bin Voigt single-line transmittances, Eq. (19). Once the absolute value of consecutive terms of the form

     

Su

DD Suf D In Suf2 D 2V 0n exp  Suf2 0 In Suf2 0

V n exp  Suf2DD In Suf2DD þV D ð56Þ n exp  2

D 0

D

DD

sum to less than 0.0001, convergence is declared and one additional term is added with the Vn þ 1, Vn þ 1, and Vn þ 1 coefficients all determined from the strong line limit sum rule. When the convergence criterion is not met after the n¼7 term is computed, the finite-bin Voigt single-line transmittance is still set to the truncated expansion (with V8 from the strong line limit sum rule included). However, the magnitude of the Eq. (56) n ¼7 term is compared to earlier n¼7 terms from tsl calculations that did not meet the convergence criterion. At the end of the program, a warning message is produced that states the magnitude of the largest of these terms. Generally, the last few terms of the truncated series includes both positive and negative terms, and the strong line limit sum rule accelerates convergence so that the magnitude of the true residual is actually many times less than the magnitude of the Eq. (56) term. Often, no warning appears at end of a MODTRAN5 run, i.e., convergence occurs for every call to the Voigt transmittance routine. For the  80 MODTRAN5 test cases, which collectively call the Voigt transmittance routine many millions of times, the largest magnitude of Eq. (56) with n ¼7 is less than 0.025. Introduction of the Voigt finite spectral bin single-line transmittance algorithm into MODTRAN5 increased run times for transmittance-only runs by less than a factor of 4. Obviously, the equations of Appendix B are long, but they essentially involve calculation of polynomials that can be evaluated rapidly. Most of the computation time (495%) is spent computing of the Voigt line shape function at D and D  D, and calculating the modified Bessel functions. Faster algorithms that retain the necessary radiometric accuracy are currently being tested [13,14]. However, for MODTRAN5 DISORT [17,18] scattered radiance calculations, the increase in computation time resulting from the new transmittance algorithm is completely insignificant because the time required to solve the radiative transfer equation dwarfs any time spent computing transmittances. The faster special function routines are being sought to lessen the impact of the finite spectral bin Voigt transmittance calculations on the overall computation time for MODTRAN transmittance-only runs and MODTRAN radiance calculations that do not use the DISORT scattering model. These are the only calculations for which the time penalty of the new method is significant. In addition to the analytic implementation of truncated expansions for transmittance, MODTRAN5 includes a second extremely slow option to numerically integrate the Fourier coefficients for n up to 20. This option was used to determine the transmittance residuals from the expansion truncated at or before n ¼8. A zenith calculation from ground to space through the mid-latitude winter model atmosphere was run, using the MODTRAN5 1.0 cm  1 band model between 0.2 and 3.0 mm (3333.0–50,000 cm  1) and the MODTRAN5 0.1 cm  1 band model between 3.0 and 32.0 mm (312.5–3333.0 cm  1). The calculation calls the tsl algorithm over 120,000 times. In Fig. 6, the truncated expansion residuals are compared to tsl residuals obtained using the Rodgers-Williams total Voigt equivalent width approximation along with subtraction of Lorentzian line-tails—the algorithm used in earlier versions of MODTRAN. In general, the truncated Bessel function expansion reduces transmittance residuals by 2 orders of magnitude. Absolute residuals for the Bessel function expansion never exceed 0.0004 for the test problem, and over 97% of the transmittances are accurate to better than 5 decimal places.

A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

115

Fig. 6. Comparison of finite-bin Voigt single-line transmittance residuals. Results are shown for both a Rodgers-Williams equivalent width algorithm (solid black) and the Bessel function expansion truncated at n¼ 8 with the last term defined by the strong-line limit sum rule (diagonal hash).

12. Summary and conclusions Almost 100 years ago, Ladenburg and Reiche [8] derived an exact expression for the total equivalent width of a Lorentzian line and along with an approximation for the total absorptance of Lorentzian line-tails. On the centennial of their original work, the equivalent width formulas have been generalized, Eq. (15), for application to the Voigt line shape function. This generalization also leads to an exact expansion of the Voigt single-line transmittance in a finite spectral bin, Eq. (19). The Ladenburg and Reiche total Lorentzian equivalent width is the sum of two exponentially-scaled modified Bessel functions with constant coefficients. The generalization is applicable to not only the total Voigt equivalent width but also narrow spectral bins (Appendix A) and Voigt line-tails from any chosen displacement from line center. The form of the expansion is an infinite sum over exponentially-scaled modified Bessel functions of integer order. The coefficients, Vn(D/gd, gc/gd), are defined as Fourier integrals and are completely independent of optical depth. The optical depth dependence resides solely in the argument of the scaled Bessel functions and in a line-tail offset term. Analytic expressions have been derived for the Fourier coefficients up to n ¼7 in the pure Lorentz limit, in the pure Doppler limit, and for Voigt lines with at most moderate Doppler dependence. In addition, an algorithm for numerically integrating the Fourier coefficient integrals up to n¼ 20 has been developed for use in validation studies. The exact expansion truncated at or before the 8th term (with a final Fourier coefficient determined from the strong line limit sum rule) is now integrated into the MODTRAN5 radiative transfer model to compute band model spectral bin transmittances. The results are compared to the traditional approach using the Rodgers-Williams Voigt equivalent width calculations. The truncated expansion reduces transmittance residuals by two orders of magnitude.

Acknowledgments This work represents the culmination of over 25 years of study by the author on the Voigt equivalent width. Within the last few years, the software algorithms were implemented, tested and embedded into MODTRAN5. The author wishes to acknowledge support for this more recent work from the Air Force Research Laboratories (Contract no. FA8718-07-C0048), the Department of Energy (Contract no. DE-FG02-08-ER85132) and the National Aeronautics and Space Administration (NASA Contract no. NNX12AH71G). The author also wishes to thank the reviewer of the original submitted manuscript for noting that the Fourier coefficients could be more easily evaluated by exploiting the well-known properties of the Chebyshev polynomials.

Appendix A. Voigt single-line equivalent width and transmittance for narrow spectral bins The focus of this manuscript has been determining Voigt finite spectral-bin single-line transmittances, tsl, for the conditions required by the MODTRAN radiative transfer model. The basic approach has involved computing the finite spectral bin equivalent width, Wsl, as the difference between the total equivalent width, W  N ¼2W0, and the line-tail contributions, WD þWD  D. This mimics the general approach that has been used in MODTRAN since its inception [19]; in the original 1.0 cm  1 spectral resolution band model, the line-tail subtraction was viewed as a correction to the total equivalent width. In Section 10, an asymptotic expression for the Voigt line-shape function, Eq. (55b), is used to evaluate 2 2 the line-tail equivalent width for gc þ n2 b gd. However, if the spectral frequency displacement D is sufficiently small, it is preferable to define the finite spectral-bin equivalent width, W0  WD, directly in terms of an expansion of the Voigt lineshape function about n equal 0. The goal of this section is to derive an expression for W0 WD that is accurate whenever fD is well represented by an expansion of fn about n equal 0.

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A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

As it turns out, the derivation used to define the Voigt line-tail equivalent width, Eq. (15), can be generalized to directly determine the (0, D) spectral-bin equivalent width. In this generalization, the Fourier coefficients, Eq. (29), which were defined in terms of line-tail moment integrals, are instead defined in terms of moment integrals over the (0, D) spectralbin. The new derivation is completely analogous to the prior derivation. In fact, equations are provided here without any D additional text since it would be largely redundant. Note, however, that quantities analogous to y, Vn , Pn and /fnS have a D different definition in the following, so these terms are replaced by y~ , V~ n , P~ n and /f n S, respectively. To help follow the logic, the equation numbers from the main text are reused, preceded by an A. Z f0   nf exp ðSuf Þdf ðA8Þ W 0 W D ¼ D½1exp Suf D  þSu fD

   f y~ ¼ f D þ f 0 f D sin2

y~

!

2

¼

   2 f þf D f 0 þf D f 0 f D  cos y~ ) y~ ðf Þ  cos1 1 ; f 0 f D 2 2

f D rf r f 0

! !    Z p  

    Su  y~ y~ Su  f 0 þf D f 0 f D cosy~ dy~ cos exp ny~ sin W 0 W D ¼ D 1exp Suf D þ Su f 0 f D exp  2 2 2 2 0 



y~

y~

2

2

p f 0 f D ny~ sin cos

¼

D 1  X V~ 0 D þ V~ n cos ny~ 2 n¼1

ðA10Þ

ðA12Þ

  Z    p D D gc , ny~ siny~ cos ny~ dy~ ¼ f 0 f D V~ n  V~ n

gd gd

ðA9Þ

ðA13Þ

0

( D   X  ) 1  

  V~ 0   Su  D ~ Su  ~ ~ f f f f W 0 W D ¼ D 1exp Suf D þSuexp Suf D þ V n In I0 2 0 D 2 0 D 2 n¼1

ðA15Þ

8 ( D   X  )! 1 >   D Su V~ 0     > > ~ D I~n Su f f ~ 0 Su f f >  þ exp Suf V I D > n < 2 0 D 2 0 D D D 2 n¼1 t sl ¼ (     X  )! DD > 1 >     D Su V~ 0 ~ Su  DD Su  > > ~ ~ > f f  f f 1 þ þ exp Suf V I I n 0 DD : n D 2 0 DD 2 0 DD D 2 n¼1

ðA19Þ

cosy~ f ¼

V~ n ¼ 2

f 0 þf D 2f 2df  ) siny~ f dy~ f ¼ ) V~ n ¼ 2 f 0 f D f 0 f D f 0 f D

Z

D 0

  P~ n f n ¼

Z

/f n S  

 cos ncos1

fD

2 f 0 f D

n

Z

D



0

f0

fD





nf cos ny~ f df

f n f D

0

n þ 1

dn

h i 8 9 RD  2ðf n f D Þ > dn > < n þ1 1 0 1T n þ 1 1 f 0 f D = f f  h i , V~ n ¼ 0 D R 2ðf f Þ 2 > :  1 D 1T ; dn > 1 n D V~ n ¼ /f 0 Sþ n

n1

ðA21Þ

ðA23Þ

  R1 f 0 f D R 1 1   f 0 f D 12ðf n f D Þ=ðf 0 f D Þ cos ncos x dx ¼ 2 12ðf n f D Þ=ðf 0 f D Þ T n ðxÞdx ~  Pn f n ¼ 1T n þ 1 ½12ðf n f D Þ=ðf 0 f D Þ 1T n1 ½12ðf n f D Þ=ðf 0 f D Þ 2 D , n ¼ 1,2,. . .: ¼ f 0 f  nþ1 n1 4

n1 0

ðA20Þ

D

    P~ n f n dn2DP~ n f D    2 f f D 1 df f 0 f D

n¼D P~ n dn2nP~ n 9n ¼ 0 ¼ 2 fn

Z

Z

n ¼ 1,2,. . .:

ðA26Þ

ðA27Þ

f 0 f D

n X ð4Þk ð2k þ 1Þnk /f S, ðk þ 1Þðn þ kÞðnkÞ! k k¼1

n ¼ 0,1,2,. . .:

D 1 X V~ 0 D V~ n ¼ 0 þ 2 n¼1

    pffiffiffiffiffiffi   pffiffiffiffiffiffi    D   D þ O 1= Su þ exp Suf DD þ O 1= Su t sl  exp Suf D 1 D D

ðA29Þ

ðA33Þ

ðA36Þ

A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

D

½V~ m þ 1 StrongLine ¼ 

D m X V~ 0 D V~ n þ 2 n¼1

117

! ðA37Þ

The key expressions from this list are those that define

the Voigt equivalent width for the (0, D) spectral interval, Eq. (A15),

the Voigt transmittance for a single line centered within a spectral bin of width D at a distance D rD/2 from the bin edge, Eq. (A19),

the Fourier coefficients, Eqs. (A23) and (A29), and

the Voigt transmittance asymptotic expansion, Eq. (A36). The asymptotic expansion, in particular, establishes that the in-band transmittance falls off exponentially with bin edge optical depth. Further investigation is required to







derive expressions for the moment integrals, Eq. (A23), implement these expressions in software, determine whether use of the strong-line limit Fourier coefficient, Eq. (A37) improves convergence, evaluate the computational speed and accuracy of the method, and establish the regime for which the transmittance expression of Eq. (A19) is preferred over that of Eq. (19).

This is all work the author plans for the near future. Appendix B. Voigt line-shape function moment integrals for n r7 The large 9Zn9 asymptotic expression for the Voigt line-shape function, Eq. (55b), can be raised to the nth power, expanded in powers of 1/9Zn92, and integrated to give the following rather lengthy expression: Z  1 1 1  a n þ 1 pDf n n þ 1 dn ¼ 2  D D Y A 9 8 h  2 i  n n þ 1   nðn þ 7Þ 1 6 > > þ A  Y 2 ðn1þ 4Þ 16 n þ 32 þ n þA 1  n þ2 1 n 2þðn8nþþ3Þ18  n þ > > > > ð n þ 3 Þ n þ 2 4 A A > > >  3> > > 2 4 3 2 > > nðn þ 26n þ 251n þ 874n þ 1008Þ  ðn þ 1Þðn þ 5=2Þ  > > ð n þ 1 Þ 2 > > n þ 3=2 þ þ = < 2 2 32ðn þ 2Þ4 A A 6 7 0 1 6 7 4 3 2 7n þ 144n þ 1127n þ 3942n þ 5400 1 > 6 7> > 8ðn þ 4Þ2 > > þ 3Y 4 ðn þ 6Þ  6 n þ 1 @ 7> > A > > 4þ 3 5> 3 2 2 > > 5n þ 99n þ 706n þ 1800 n þ 17n þ 90 > > A > >  þ2 2 > > 2 ð n þ 5 ÞA A ; : n h  io R 1 n n þ 1=2Þ ðn þ 3=2Þðn þ 2Þðn þ 7Þ ðn þ 5=2Þðn4 þ 26n3 þ 251n2 þ 874n þ 1008Þ þ 1 ð 2 1  1 an 0cot a sin2n ydy 2Y ðn þ 2Þ 6Y 2 ðn þ 2Þðn þ 5Þ 4Y 2 ðn þ 3Þ 2

Z

cot1 a

sin2n

0

8 R 1 a 2n1 cot a > sin2n2 ydy > >  2nAn þ 2n 0 > " >  k # > n1 > X > k! 4 > < ðn þn 1Þn cot1 a a A 4 n! ðk þ 1Þk þ 1 A ydy ¼ k¼0 > >  k > 1 > 2n þ 1 X > ðkþ 1Þn 1=a2 > > 1=a > > : n! 2ðn þ kÞ þ 1 k¼0

ðB1Þ

3

all a all a ðB2Þ large a

In these equations, parameter a, Eq. (39), is reintroduced, parameter A is defined as one plus a2, A  1þ a2 ,

ðB3Þ

and the Pochhammer symbol is used to simplify the notation. Expression for the Voigt line-shape function moment integrals for n r7 are explicitly listed below along with the asymptotic expansions for large a. These are the actual equations that have been incorporated into the Voigt single-line transmittance algorithm within MODTRAN. In the implementation of these equations, the analytic integrals are used when parameter a is less than or equal to 3, and the asymptotic expressions are used for a greater than 3. ( "    #) Z 1 1 1 a 1 3 1 1 5 3 4 4 1  þ  þ 2 pDf n dn ¼ cot a þ 1þ ðB4Þ 2 4 D D A 29Z92 9Z9 2 A 9Z9 4 A A

118

A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

1

D

Z

1



pDf n

D

a

2

9 8   1 1 1 2 1 >     Z cot1 a þ 3A  27 2 þ 143  Y 4 16 þ 20A = < Y12 12 a 2 > 20A 5A 1 3 5 2  dn ¼ 1 2 1 2 1 2 a sin ydy þ 1 5 1 3 59 145 144 > > A :þ 6 4Y 4Y 4Y 0 ; 5 2 þ 3  4 þ 64 þ 16A þ Y

Z

cot1 0

8  > < a2 cot1 a 1 þaa2 a  sin2 ydy ¼ > : a12 13  52a2 þ 73a4  94a6 þ 115a8  136a10 þ    2 

143A

39Z9

D

1



D





143A

Z

cot1

0

large a

2431A

46189A

2

56A

A

7A

 8  R 1 > a 3a cot a sin2 ydy a2 > < a 2 0 4 ð1 þ a2 Þ sin4 ydy ¼  > 1 1 3 2 10 15 > : a3 5  7a2 þ 3a4  11a6 þ 13a8  5a710 þ    1 1 2 5  35A  105A2



16 1155A3



32 3003A4



all a large a

5 

128 15015A

3

6  7 6 7 6 þ 1 2 9  31 þ 2 2 þ 20 3 þ 16 4 þ 32 5    7 6 9Z9 14 42A 231A 7 7293A 3003A 3003A 6 Z  7 7 3 1 1 a6 1 65 pDf n dn ¼ 2 6 þ 571 2  4 3  56 4  896 5    7 6 þ 4 2 7 230945A 715A 12155A D D A 6 9Z9 0 11A 143A 7 1 6 7 153 5670 92349 388877 6 7  13A þ 2  3 2 130A 1105A 6 7 1 @ A 4þ 5 6 119Z9 þ 1328 4 þ 7968 5    20995A

D

Z

1



D





   Z cot1 a 75 1529 1 1 a3 sin6 ydy 20Y 2 56Y 2 900Y 2 0 9 8   1  45 3 4 5 > þ 5A  172 þ 363  Y 4 224 þ 28A = < Y12 20 a 4 > 7A 7A  þ 1 1529 1529 139 221 412 400 > A > ; : þ Y 6 4480 þ 5040A þ 2 þ 3  4 þ 9A5

pDf n 4 dn ¼ 1

Z

cot1

0

ðB6Þ

146965A

21

504A

a3

ðB5Þ

5

2

1

7A

   Z cot1 a 21 83 1 1 a2 sin4 ydy 2 2 8Y 20Y 56Y 0 8  9  1  21 3 9 >  192 þ 43  Y 4 160 þ 80A < Y12 18 þ 4A = a 3 > 10A A  þ 1 249 747 83 397 228 32 > A > : þ Y 6 1280 þ 4480A þ ; 2 þ 3  4 þ 5

pDf n 3 dn ¼ 1

560A

a2

7A

all a

3 1 1 5A  4 2  8 3  64 3003A 35A 105A 1155A 6  7 6 7 6 þ 1 2 3  23 þ 2 2 þ 16 3 þ 32 4 þ 128 5    7 Z 1 5 35A 6 7 45045A 315A 3465A 9009A  2 1 1 6 9Z9  7 pDf n dn ¼ 6 7 1 39 349 646 8 32 64 7 A6 þ 4 D D 6 9Z9 28  84A þ 231A2  3003A3  15015A4  36465A5    7 6  7 4 5 1 25 1645 17 713 8824 16 256 þ 5  6 2  3 þ 4 þ 2  22A þ Z

14A

128 5  4 

1 3

1

56A

21A

9A

 8  R cot1 a 3 > > sin4 ydy a 2 3 < a6 5a2 0 a ð1 þ a Þ sin6 ydy ¼  > 1 1 4 10 20 > : a4 7  9a2 þ 11a4  13a6 þ 37a8  1756 a10   

all a large a

2 

3 1 10 1 9A þ 1 2 þ 8 3 þ 16 4 þ 64 5    7293A 33A 429A 1287A 67  7 6 7 6 þ 2 1  8 þ 57  2  8  112 5    7 6 9Z92 3 11A 143A2 2145A3 12155A4 230945A 7 6 Z 1  7 7  4 1 1 6 1 57 1455 3637 12498 60 320 7 pDf n dn ¼ 2 6 6 þ 119Z94 2  13A þ 26A2  221A3 þ 4199A4 þ 29393A5    7 D D A 6 7 0 1 6 7 267 4326 323811 2827373 6 7 2  5A þ 170A2  1615A3 6 7 1 @ A 4þ 5 6 19594592 4448 139Z9 þ 5  4  33915A

1

D

Z

1

D







260015A

   Z cot1 a 363 377 1 1 a4 sin8 ydy 2 2 2Y 224Y 198Y 0 9 8   1 121 5 55 >  1652 þ 253  Y 4 448 þ 224A < Y12 16 þ 6A = a 5 > 56A 4A  þ 1885 145 523 544 58 > A > : þ Y16 4147 ; 2 þ 3  4 þ 5 8064 þ 4032A þ

pDf n 5 dn ¼ 1

3

2

336A

36A

9A

A

ðB7Þ

A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

a4

Z

cot1 a 0

 8  R cot1 a 4 > > sin6 ydy a 2 4 < a8 7a3 0 ð1 þ a Þ sin8 ydy ¼  > 126 > : a15 19  115a2 þ 1315a4  37a6 þ 1770a8  19 a10   

119

all a large a

2 

3 1 1 112  4 þ 1 2 þ 2 3 þ 8 4 þ 46189A 5  143A 429A 2431A 6 3 3 11A  7 6 7 15 215 713 4 56 128 6þ 1 7  þ       6 119Z92 2 13A 78A2 221A3 4199A4 12597A5 7 6 7 Z 1  6 7   1 a6 5 1 165 161 13657 52471 44 792 7 pDf n dn ¼ 3 6 þ  þ  þ þ    4 2 3 4 5 7 4 A 52003A 68A 646A 2261A 139Z9 D D A 6 7 0 1 6 7 113 6057 499779 1196842 6 7  þ  2 3 8 68A 2584A 6783A 6 7 1 @ A þ 4 5 6 3016058 464 9Z9 þ 4  5  52003A

1

Z

D

1



D



260015A

   Z cot1 a 91 647 1 1 a5 sin10 ydy 2 2 28Y 48Y 308Y 0 9 8   1  65 5 6 5 > þ 7A  83 2 þ 223  Y 4 192 þ 16A = < Y12 28 a 6 > 24A 3A  þ 42055 3235 800 > A > ; : þ Y16 59136 þ 4928A þ 647 2 þ 12613  8434 þ 11A 5



pDf n 6 dn ¼ 1

55

2

1056A

a5

Z

cot1 a

ðB8Þ

sin10

0

66A

11A

 8  R 1 > a 9a4 cot a sin8 ydy a5 > < 10 5 0 ð1 þ a2 Þ ydy ¼  > 1 1 6 7 56 126 84 > : a6 11  13a2 þ 5a4  17a6 þ 19 a8  7a10   

all a large a

 3 1 27 8 1 13A þ 71 2  1 3  12 4  4199A 5  11 65A 221A 4199A 6 7  6 7 29 529 3597 10 160 6þ 1 7 6 139Z92 9 A þ 17A2  323A3 þ 2261A4 þ 52003A5    7 6 7 Z 1  7  6 1 1 6 1 15 1245 42231 32893 54476 12 7 pDf n dn ¼ 3 6  þ  þ     þ 4 6 7 37145A5 1292A2 1292A3 7429A4 D D A 6 9Z9 40 68A 7 1 6 7 42885 812475 24295595 6 7 315 19A þ 2  3 133A 3059A 6 7 1 @ A 4þ 5 6 76657326 34038092 179Z9 þ 4  5  2

15295A

1

Z

D

1



D



27531A

   Z cot1 a 13 15725 1 1 a6 sin12 ydy 16Y 2 6Y 2 6864Y 2 0 9 8   1 13 3 7 91 > þ 8A  1192 þ 423  Y 4 32 þ 240A = < Y12 16 a 7 > 30A 5A  þ 1 15725 22015 1295 2135 3115 266 > A > ; : þ Y 6 16896 þ 25344A þ 2 þ 3  4 þ 3A5



pDf n 7 dn ¼ 1

39

1584A

a6

Z

cot1 a 0

sin12

88A

33A

 8  R 1 > a 11a5 cot a sin10 ydy a6 > < 12 6 0 ð1 þ a2 Þ ydy ¼  > 1 1 7 28 84 10 462 > : a7 13  15a2 þ 17a4  19a6 þ a8  23 a10   

all a large a

2  3 1 31 55 1 2 32 13 1 15A þ 51A2  323A3  969A4  22287A5    6  7 6 7 7 77 1575 565 2 36 6 þ 1 2 10 7  þ  þ þ    5 2 3 4 34A 6 9Z9 7 185725A 646A 646A 7429A 6 7 0 1 6 7 Z 1 147 13587 24211 433367  þ  6 7   2 3 1 a6 2 38A 7 38A 874A 7 A pDf n dn ¼ 4 6 þ 1 4 @ 1560012 7 104 D D þ  19665A5    A 6 179Z9 7 10925A4 6 7 0 1 6 7 1785 12585 775791 1002377 6 7  þ  2 3 4 4A 6 7 92A 92A a @ A 4þ 5 6 30338977 199Z9 þ 8504593 5  4  1242A

1

D

Z

1

D





ðB9Þ



18009A

   Z cot1 a 1071 1159 1 1 a7 sin14 ydy 2 2 12Y 440Y 468Y 0 8  9  1  833 7 8 49 > þ 9A  2462 þ 1043  Y 4 1760 þ 110A < Y12 36 = a 8 > 55A 11A  þ 56791 2989 12868 4436 1376 > A > : þ Y16 965447 ; 5 2 þ 3  4 þ 823680 þ 51480A þ

pDf n 8 dn ¼ 1

35

2

2860A

429A

39A

13A

ðB10Þ

120

A. Berk / Journal of Quantitative Spectroscopy & Radiative Transfer 118 (2013) 102–120

 8  R 1 > a 13a6 cot a sin12 ydy a7 > all a < 14 7 0 2 ð1 þ a Þ a7 sin14 ydy ¼  > 1 330 792 0 > : a18 15  178a2 þ 1936a4  740 large a a6 þ 23a8  25a10 þ     2 3 1 1 52 202 484 1 16 3 5  85A þ 323A2  2261A3 þ 7429A4 þ 185725A5    6 7  6 7 242 1154 18728 24761 14 6 þ 4 2 3 19A 7 þ 5  2  3 þ 4  6 179Z9 7 57A 1311A 6555A 58995A 6 7 0 1 6 7 Z 28826 820068 93 544 þ  7 8 2 3 1 1 1 6 A 23A 575A 6 7 1 @ A pDf n dn ¼ 4 6 þ 4 7 1386089 1797244 D D 199Z9 þ     A 6 7 5 4 1725A 10005A 6 7 0 1 6 7 5306 424186 19014752 100854577 6 7 29 23A þ 2  3 þ 4 6 7 575A 15525A 90045A 1 @ A 4þ 6 5 3250038992 9Z9  1499337446 þ    5 6 Z

cot1 a

2791395A

ðB11Þ

30705345A

References [1] Berk A, Anderson GP, Acharya PK, Bernstein LS, Muratov L, Lee J, et al. MODTRAN5: 2006 update. Proceedings of SPIE, 6233; 2006. p. 62331F. [2] Berk A, Acharya PK, Bernstein LS, Anderson GP, Lewis P, Chetwynd JH, et al. Band model method for modeling atmospheric propagation at arbitrarily fine spectral resolution, US patent #7433806, issued October 7, 2008. [3] Rothman LS, Gordon IE, Barbe A, ChrisBenner D, Bernath PF, Birk H, et al. The HITRAN 2008 molecular spectroscopic database. J Quant Spectrosc Radiat Transfer 2009;110:533–72. [4] Plass GN. Models for spectral band absorption. J Opt Soc Am 1958;48:690–703. [5] Robertson DC, Bernstein LS, Haimes R, Wunderlich J, Vega L. 5 cm  1 band model option to LOWTRAN 5. Appl Opt 1981;20:3218–26. [6] Goody RM, Yung YL. Atmospheric radiation: theoretical basis. 2nd ed.New York: Oxford University Press; 1989. [7] Rodgers CD, Williams AP. Integrated absorption of a spectral line with the Voigt profile. J Quant Spectrosc Radiat Transfer 1974;14:319–23. ¨ [8] Ladenburg R, Reiche F. Uber selektive absorption. Ann Phys 1913;42:181–209. [9] Abramowitz M, Stegun IA. Handbook of mathematical functions with formulas, graphs, and mathematical tables.New York: Dover Publications; 1965. [10] Amos DE, Daniel SL, Weston MK. CDC 6600 subroutines IBESS and JBESS for Bessel functions I(NU,X) and J(NU,X), X.GE. 0, NU.GE. 0. ACM Trans Math Software 1977;3:76–92. [11] Olver FWJ. Tables of Bessel functions of moderate or large orders, NPL mathematical tables, vol. 6. London: Her Majesty’s Stationery Office; 1962. [12] Poppe GPM, Wijers CMJ. Algorithm 680: evaluation of the complex error function. ACM Trans Math Software 1990;16:47. [13] Abrarov SM, Quine BM. Efficient algorithmic implementation of the Voigt/complex error function based on exponential series approximation. Appl Math Comput 2011;218:1894–902. [14] Borwein D, Borwein JM, Crandall RE. Effective Laguerre asymptotics. SIAM J Numer Anal Arch 2008;46:3285–312. [15] Ludwig CB, Malkmus W, Reardon JE, Thomson JAL. In: Goulard R, Thomson JAL, editors. Handbook of infrared radiation from combustion gases. NASA SP-3080; 1973. [16] Wolfram S. The mathematic book, 5th ed.; Cambridge University Press, Cambridge, UK, 2003. [17] Stamnes K, Tsay S-C, Wiscombe W, Jayaweera K. Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media. Appl Opt 1988;27:2502–9. [18] Stamnes K, Tsay S.-C., Laszlo I. DISORT, a general-purpose FORTRAN program for discrete-ordinate-method radiative transfer in scattering and emitting layered media: documentation of methodology. Available from: /www.met.reading.ac.uk/  qb717363/adient/rfm-disort/DISORT_docu mentation.pdfS; 2000. [19] Berk A, Bernstein LS, Robertson DC. MODTRAN: a moderate resolution model for LOWTRAN 7, GL-TR-89-0122. Geophysics Directorate, Phillips Laboratory, Hanscom AFB, MA 01731, April 1989, ADA214337.