Uniform error bounds for fast calculation of approximate Voigt profiles
UUniform error bounds for fast calculation ofapproximate Voigt profiles
Sven Nordebo ∗ , ∗ Department of Physics and Electrical Engineering, Linnæus University, 351 95 V¨axj¨o, Sweden. E-mail: [email protected]
Abstract —The broadband line-by-line analysis of radiativetransfer in the atmosphere is extremely demanding with regardto computational resources. As a remedy, we present here thecalculation of uniform error bounds for approximating theclassical Voigt profile. A new “full” Voigt profile, which canbe expressed as a combination of two Faddeeva evaluations, isalso presented and included in the analysis. The uniform boundscan be used to rigorously determine the domains on which theVoigt profiles can be approximated by the corresponding Lorentzprofiles to any desired accuracy. The bounds can furthermorebe employed to make a fast and efficient estimate of the mostsignificant lines to be included in a subband adaptive lineselection strategy. By using a realistic numerical example ofradiative transfer in the atmosphere, we demonstrate that theseapproximation approaches are able to reduce the computationaltime of the associated line-by-line analysis by several orders ofmagnitude with little loss of accuracy.
Index Terms —Radiative transfer in the atmosphere, spectralline shapes, Voigt function, uniform approximation.
I. I
NTRODUCTION
Even though there are advanced theories of molecular lineshapes taking line mixing and velocity changes into account[1], [2], [3], [4], [5], [6], [7], most databases and radiativetransfer codes are still using the classical Voigt profile, see e.g. ,[8], [9], [10], [11]. A simple explanation for this is probablythe fact that the broadband line-by-line analysis of radiativetransfer in the atmosphere is extremely demanding with regardto computational resources. Quoting Liou [12, p. 126]: “Thecomputer time required for line-by-line calculations, even withthe availability of a supercomputer, is formidable. This isespecially true for flux calculations in which an integrationover all absorption bands is necessary.” Typically, hundreds ofthousands of spectral lines must be resolved to a resolutioncorresponding to millions of frequency points at each atmo-spheric level. And at the same time, most of these calculationscould have been performed based on the simple Lorentzianprofile rather than by using the more complex Voigt profile,and most of the spectral line contributions could have beendisregarded with very little impact on the final accuracy ofthe computation. Much of these problems have been overcomewith recent line-by-line algorithms based on simplifications,approximations and pre-computed line data, see e.g. , [8]. As anaid in the future development, improvement and extension ofsimilar algorithms, this paper is providing rigorous bounds forapproximating the Voigt profile by the much simpler Lorentzprofile. The Voigt profile is the convolution between the Lorentzianprofile and the Gaussian. And even though there exists com-puter codes for efficient calculation of the associated Faddeevafunction, see e.g. , [13], [14], [15], [11], it is much fasterjust to compute the corresponding Lorentz profile. Moreover,the superexponential Gaussian profile is a highly localizedfunction, a property that can be exploited to significantlyaccelerate the calculation of an approximate Voigt profile. Inthis paper we formulate and prove a theorem which quantifiesthe fact that the Voigt profile converges uniformly towards theLorentzian profile in two different regards: (1) with respectto the Lorentzian half-width normalized by the Gaussian half-width, and where the approximation error is measured overthe whole frequency axis, and (2) with respect to the linewidth normalized by the Gaussian half-width, and beyondwhich the far wing approximation error is measured. Thus,(1) if the Lorentzian is sufficiently broad in comparison tothe Gaussian, it approximates the Voigt profile everywhere.And (2) in the far wings, the Voigt profile can always beapproximated by the Lorentzian, provided that the beginningof the far wing is suitably defined. In this paper we will showhow the Voigt (or Faddeeva) function can be used to pre-calculate the associated threshold values in order to achieveany desired approximation accuracy. The related computationsbased on thresholding and sorting can readily and efficientlybe implemented in a computer code. In a typical applicationof radiative transfer in the atmosphere, most of the frequencypoints to be evaluated will fall into the category where theseapproximations are applicable, and hence the potential to savecomputation time is almost the same as replacing the Voigtcomputation for the Lorentzian.However, the line-by-line computation of broadband radia-tive transfer in the atmosphere is still huge. To further reducethe computational complexity, a subband (or block) adaptiveline selection procedure is presented here. For each spectralline exterior to a fixed and relatively narrow subband, the newerror bounds can be used to determine precisely where theLorentz approximation is applicable, and then very fast andeffectively estimate its in-band contribution to any desiredaccuracy. Thus, the “exterior” spectral lines that are estimatedto have a negligible in-band contribution can be excludedfrom the computation. In-band lines are always included, andthe computation can then proceed with the next subband.Even though this is a suboptimal approach, it is a simple,pragmatic and very effective way to include only the lines a r X i v : . [ phy s i c s . a o - ph ] J a n hat are the most significant, and the procedure can be tunedwith very few threshold parameters to control the accuracy.By using realistic numerical examples of radiative transfer inthe atmosphere, we will demonstrate that the combination ofthe two approximation approaches described above can reducethe computational time of the line-by-line analysis by severalorders of magnitude with very little loss of accuracy.In this work we do not attempt to compare the performanceof our algorithm with other highly sophisticated commercialcodes for radiative transfer such as the MODTRAN 6 line-by-line algorithm reported in [8]. However, a few general remarkscan be made as follows. First, the purpose of our numericalexample has been to evaluate and compare the proposedapproximation methods by using an algorithm that is as simpleand straightforward and possible. A layer-recursive algorithmfor radiative transfer in a one-dimensional plane-parallel andnon-scattering atmosphere has therefore been developed forthis purpose. The frequency resolution has been chosen basedon the (frequency dependent) Doppler broadening in the upperatmosphere at
65 km height, and is therefore extremely densewith a resolution starting at · − cm − in the lower
100 cm − band. All profiles are calculated on-the-fly, basedon line parameters that have been retrieved from the HITRANdatabase [9], [10]. Similar as with the MODTRAN algorithm,our algorithm is computing Voigt profiles with their linecenters within certain narrow subbands. The main difference isthe way in which the exterior tail contributions are selected andcomputed. Our algorithm is employing the adaptive line selec-tion procedure as described above, whereas the MODTRANalgorithm is employing a fixed 25 cm − maximal distancefor their line selection [8, p. 544–545]. The MODTRANalgorithm is furthermore employing pre-computed line shapedata which is stored on a fine temperature and coarse pressuregrid, and which can then readily be scaled for the adequatepressure on-the-fly. The latter procedure, which is based onPad´e approximations of Voigt sums, is no doubt an effectiveway of making the algorithm very fast.We will also introduce and incorporate in the analysisthe “full” Voigt profile. The full Voigt profile is based onthe convolution between the Gaussian profile and the “full”Lorentzian profile which is just the classical Lorentz modelwithout performing the traditional resonance approximation.It will be shown that the full Voigt profile can be computedbased on two Faddeeva calculations, instead of just one aswith the traditional Voigt profile. However, since most of thecomputations in a typical application can be performed underthe full Lorentz approximation in very much the same wayas have been described above, the approximation of the fullVoigt profile is only slightly more complex in comparison tothe approximation of the traditional Voigt profile.The rest of the paper is organized as follows. In SectionII is given a brief account on the “full” Lorentz profile,and its uniform convergence properties with respect to thetraditional resonance approximation. In Section III is giventhe corresponding Voigt profiles, and in particular the “full”Voigt profile which can be expressed as a combination of two Faddeeva calculations. The details of the derivation are givenin the Appendix A. In Section IV is given the theorem whichquantifies the uniform approximation properties of the Voigtprofiles with regard to the corresponding Lorentz profiles. Thedetails of the proof are given in the Appendix B. The subbandadaptive line selection strategy is described in Section V andthe numerical examples are given in Section VI. The paper isconcluded with a short summary and a discussion.II. T HE CLASSICAL L ORENTZ PROFILE
In a nonscattering atmosphere in the long wave regime, theabsorption cross section k ν of a molecule is the same as itsextinction (or total) cross section σ t . Based on the classicaloptical theorem [16, pp. 227–230, 272], [17, pp. 18-20] theabsorption (extinction) cross section k ν can then be modeledas k ν = Im { πνα } , (1)where α is the polarizability, ν = λ − is the wavenumber and λ the wavelength in vacuum. This means that we can obtainthe absorption coefficient as k ν = Im { h ( ν ) } where h ( ν ) is aHerglotz function in the complex variable ν , and rememberthat it is really the extinction coefficient that we are modeling.Basic properties of Herglotz functions can be found in e.g. ,[18], [19], [20], [21], [22]. Perhaps the most simple, and themost widely used model of the molecular polarizability is theclassical Lorentz model α = S π ν − ν − i2 γν , (2)where S is the line strength, ν the transition frequency(including the pressure shift) and γ the Half-Width-Half-Maximum (HWHM) parameter modeling the collision inducedline broadening. Notably, the expression (2) can be derivedby using a simple classical phenomenological model, as wellas by using the more comprehensive perturbation techniquesof quantum mechanics, see e.g. , [23, pp. 232-233], [24], [25,pp. 117], [26, pp. 104-108] and [27, pp. 1808-1809].The Herglotz function modeling the line shapes in (1) isnow given by h ( ν ) = 2 π νν − ν − i2 γν , (3)and its imaginary part is f FL ( ν ) = Im { h ( ν ) } = 4 π γν ( ν − ν ) + 4 γ ν . (4)Here, we will refer to (4) as the “full Lorentz” profile todistinguish it from the usual resonance approximation givenbelow. The asymptotic behavior of (3) is readily found as h ( ν ) = (cid:40) a ν + o ( ν ) as ν → ,b − ν − + o ( ν − ) as ν → ∞ , (5) A Herglotz function h ( z ) is a holomorphic function defined on the openupper half-plane C + = { z ∈ C | Im { z } > } and where its imaginary partis non-negative, i.e. , Im { h ( z ) } ≥ for z ∈ C + . o ( · ) denotes the small ordo [28, p. 4] and where a = 2 /πν and b − = − /π . The Herglotz function in (3)is symmetric, and the following two sum rules [21] apply π (cid:90) ∞ Im { h ( ν ) } d ν = − b − = 2 π , (6)and π (cid:90) ∞ Im { h ( ν ) } ν d ν = a = 2 πν . (7)The first sum rule in (6) establishes that the full Lorentz profileis normalized over R + = [0 , ∞ ) , and the second gives asum rule relating the extinction cross section σ t to the staticpolarizability of the line, (cid:82) ∞ d νσ t /ν = π α (0) = S/ν , cf. ,[24], [29], [30], [21].The classical Lorentz profile used in most radiative transferanalysis is given by f L ( ν − ν ) = 1 π γ ( ν − ν ) + γ , (8)see e.g. , [31, p. 73], [12, p. 21-23], [32, pp. 263-266] and [10,Eq. (A14)]. The classical Lorentz profile above can readilybe obtained as an approximation of (4) valid for ν close toresonance at ν . It is emphasized, however, that (8) can also bederived from first principles, as in [5, pp. 77-78]. It is noticedthat the convergence of the aforementioned approximation issomewhat subtle, in particular for small ν and small γ since f FL (0) = 0 and f L ( − ν ) = γ/π ( ν + γ ) . To analyse thissituation we consider the relative error E FL γ,ν ( ν ) = f FL ( ν ) − f L ( ν − ν ) f L ( ν − ν )= ( ν − ν ) (3 ν + ν )( ν + ν ) ( ν − ν ) + 4 γ ν , (9)yielding the upper bound (cid:12)(cid:12) E FL γ,ν ( ν ) (cid:12)(cid:12) ≤ | ν − ν | (3 ν + ν )( ν + ν ) . (10)By assuming that | ν − ν | ≤ B , it is readily seen that (cid:12)(cid:12) E FL γ,ν ( ν ) (cid:12)(cid:12) ≤ Bνν ≤ B ( ν + B ) ν for ν > ν > , (11)and where ν < ν + B , and (cid:12)(cid:12) E FL γ,ν ( ν ) (cid:12)(cid:12) ≤ Bν ν ≤ B ( ν + B ) ν for 0 < ν < ν , (12)and where ν < ν + B . For a fixed bandwidth B γ (related tothe half-width γ ) and | ν − ν | ≤ B γ , it is now readily seenthat E FL γ,ν ( ν ) converges uniformly to zero as ν → ∞ . It isemphasized that this result is merely a mathematical propertyof the line shapes which we will need later, and we do notintend to let the limit of large transition frequencies ν violatethe assumption about a non-scattering atmosphere.For most practical purposes the analytically more tractableLorentz profile (8) is an extremely good approximation of (4),except for some far wing calculations in the infrared windowswhere the density of spectral lines are very sparse. It mayalso be of interest to observe that the classical Lorentz profile (8) is not symmetric, and hence does not correspond to asymmetric Herglotz function as would be physically expectedin relation to the optical theorem (1), cf. , [30], [21], [29]. Inparticular, since f L ( − ν ) (cid:54) = 0 there is no sum rule in the formof (7) relating the ν − moment of f L ( ν − ν ) to the staticpolarizability of the line.III. V OIGT PROFILES
The classical Voigt profle is defined by the convolutionintegral f V ( ν ) = (cid:90) ∞−∞ f G ( t ) f L ( ν − t )d t, ν ∈ R , (13)where f G ( ν ) is the Gaussian profile f G ( ν ) = (cid:114) ln 2 π α e − ν ln 2 /α , ν ∈ R , (14)modeling the Doppler broadening and where α is the corre-sponding HWHM parameter. Notice that the absorption lineassociated with the line center frequency ν is given by thetranslation f V ( ν − ν ) .It is well known that the Voigt profile (13) can be computedby means of the Faddeeva function w ( z ) = i π (cid:90) ∞−∞ e − t z − t d t, (15)for which there exists many efficient numerical codes, see e.g. ,[13], [14], [15], [11]. In particular, by extending the integrandabove with the complex conjugate of its denominator, it canbe readily shown that f V ( ν ) = (cid:114) ln 2 π α Im { i w ( x + i y ) } , (16)where x = ν √ ln 2 /α and y = γ √ ln 2 /α . It may be noticedthat i w ( z ) is a symmetric Herglotz function generated by theGaussian density, whereas f V ( ν − ν ) is the density of a non-symmetric (shifted) Herglotz function.Now, we define also the “full Voigt” profile by f FV ( ν ) = (cid:90) ∞−∞ f G ( t ) f FL ( ν − t )d t, ν ∈ R , (17)where f FL ( ν ) is the full Lorentz profile given by (4). InAppendix A is shown that the full Voigt profile can becomputed by means of two Faddeeva calculations as f FV ( ν ) =Im { h FV ( ν ) } where h FV ( ν ) = (cid:114) ln 2 π α (cid:32)(cid:16) − γa + i (cid:17) w (cid:32) ( ν + a + i γ ) √ ln 2 α (cid:33) + (cid:16) γa + i (cid:17) w (cid:32) ( ν − a + i γ ) √ ln 2 α (cid:33)(cid:33) , (18)and where a = (cid:112) ν − γ and ν > γ .3V. U NIFORM APPROXIMATION OF V OIGT PROFILES
A. The classical Voigt profile
We define the relative error between the classical Voigt andLorentz profiles as E V α,γ ( ν ) = f V ( ν ) − f L ( ν ) f L ( ν ) , ν ∈ R , (19)and which can be manipulated to read E V α,γ ( ν ) = (cid:114) ln 2 π α (cid:90) ∞−∞ e − t ln 2 /α t (2 ν − t ) γ + ( ν − t ) d t, (20)which is an even function of ν . It is readily seen that E V α,γ ( ν ) = O{ ν − } for fixed values of α and γ and where O{·} denotes the big ordo [28, p. 4]. Hence, for given valuesof α and γ the absolute error | E V α,γ ( ν ) | will be arbitrarilysmall for sufficiently large ν . The superexponential Gaussianprofile is furthermore highly localized and converges to theDirac delta function as α → . Hence, we certainly have lim α → f V ( ν ) = f L ( ν ) when γ and ν are fixed. However, asshown in the Appendix B, the convergence is in fact uniformover ν ∈ R as α → and γ is fixed. From the propertiesmentioned above it is now very close at hand to formulate thefollowing theorem providing two simple statements, or criteria,for achieving a prescribed uniform error tolerance. Theorem 4.1:
For any (cid:15) > and n > there exists positivereal numbers n and n such that the following two statementsregarding the line parameters ( α, γ ) hold (cid:40) γ/α > n ⇒ max ν ∈ R | E V α,γ ( ν ) | < εγ/α > n ⇒ max | ν | >n α | E V α,γ ( ν ) | < ε. (21)The lower limit relating to γ/α > n above is probably notneeded as n can be chosen to be arbitrarily small, but isincluded here with the purpose of simplifying the proof of thetheorem. In addition, in a practical application the parameter γ/α is always bounded from below by a nonzero number n .The Theorem 4.1 will be proved below, but let us first give afew comments on its application.For a given set of line parameters ( α, γ ) the first statementin (21) is the stronger one, and it can be used as a criterion todetermine whether the Voigt profile can be approximated bythe Lorentz profile within the given error tolerance ε on thewhole of the frequency axis ν ∈ R . If the condition in the firststatement is not satisfied, then the second (weaker) statementasserts that it is sufficient to calculate the Voigt profile withinthe (usually very small) limited frequency range | ν | ≤ n α ,and hence that the Lorentz profile can be used for | ν | > n α .The practical usefulness of the theorem stems from the factthat it is numerically much more efficient to calculate theLorentz profile in comparison to the Voigt profile, and thatsimple comparative conditions such as | ν | ≤ n α above canreadily be implemented in software by using sorting routinesat very low computational cost. The application of Theorem4.1 can now be summarized as follows. Assume that ε > and γ/α > n . Then there are positive real numbers n and n such that the following criteria can be applied to the lineparameters ( α, γ ) γ/α > n ⇒ Use the Lorentz profile for all νγ/α < n ⇒ Use the Lorentz profile for | ν | > n α and the Voigt profile for | ν | < n α, (22)which guarantees that the approximation error | E V α,γ ( ν ) | < ε for all ν ∈ R .To prove Theorem 4.1 it is convenient to normalize (20)using the substitution t √ ln 2 /α ↔ t to yield E V α,γ ( ν ) = 1 √ π (cid:90) ∞−∞ e − t t (2˜ ν − t )˜ γ + (˜ ν − t ) d t = E V˜ α, ˜ γ (˜ ν ) , (23)and where we have introduced the dimensionless parameters ˜ α = √ ln 2 , ˜ γ = γα √ ln 2 , ˜ ν = να √ ln 2 . (24)In Appendix B below is shown that the expression (23)converges to zero uniformly over ˜ ν ∈ R as ˜ γ → ∞ , andhence that lim γ/α →∞ max ν ∈ R | E V α,γ ( ν ) | = lim ˜ γ →∞ max ˜ ν ∈ R | E V˜ α, ˜ γ (˜ ν ) | = 0 . (25)The result (25) asserts the validity of the first statement in The-orem 4.1. From the error bound (62) derived in Appendix Bit is furthermore seen that | E ˜ α, ˜ γ (˜ ν ) | is uniformly bounded byan arbitrary ε > for ˜ γ > n √ ln 2 where n > is fixedand the normalized frequency | ˜ ν | is sufficiently large. This isalso consistent with the observation that E ˜ α, ˜ γ (˜ ν ) = O{ ˜ ν − } for large ˜ ν , and which can be seen directly from (23). Hence,there exists a positive number n such that | E V˜ α, ˜ γ (˜ ν ) | < ε for | ˜ ν | > n √ ln 2 , (26)or equivalently | E V α,γ ( ν ) | < ε for | ν | > n α, (27)and where γ/α > n . The result (27) finally asserts thevalidity of the second statement in Theorem 4.1.The uniform error bounds (61) and (62) derived in Ap-pendix B are useful to prove the validity of the two statementsin (21), but the bounds are not particularly sharp. However,since the normalized form of the relative error defined in (23)can be represented by the single parameter ˜ γ , we can readilyfind approximate values of n and n yielding sharp errorbounds by direct numerical calculation of (19) for variousvalues of γ/α .In Fig. 1 is shown a computation of the relative error | E α,γ ( ν ) | defined via (19) and (24), plotted here as a functionof the normalized frequency ν/α for γ/α = 10 − , , , .The Voigt profile is computed based on (16) and where theFaddeeva function (15) has been implemented in Matlab usingthe software described in [13], [14]. As can be seen in thisplot, there is a local maximum around ± for small valuesof γ/α < , and around ± γ/α (in fact around ± . γ/α )for large values of γ/α > . This means that we nowhave full numerical control over the localization of extremal4 −
15 0 15 5010 − − − − Normalized frequency ν/α | E V α , γ ( ν ) | Relative error | E V α,γ ( ν ) | γ/α = 10 − γ/α = 1 γ/α = 10 γ/α = 30 Fig. 1. Computation of the relative error | E V α,γ ( ν ) | between the Voigtprofile and the Lorentz profile. The error is plotted here as a function ofthe normalized frequency ν/α for γ/α = 10 − , , , . values when executing the numerical study. Each plot inFig. 1 is made with 2000 frequency points in the interval ν/α ∈ [ − , , which is a sufficient resolution for ourpurposes here.In a typical application of broadband radiative transfer in theatmosphere (see the numerical examples below), the normal-ized pressure broadening is γ/α > − . With reference to theTheorem 4.1, we can now infer from Fig. 1 that the parameterchoices n = 10 − together with ( n , n ) = (10 , and ( n , n ) = (30 , will guarantee a relative error less than (cid:15) = 10 − and (cid:15) = 10 − , respectively. B. The full Voigt profile
Similar as above we can now define the relative errorbetween the full Voigt and the full Lorentz profiles as E FV α,γ,ν ( ν ) = f FV ( ν ) − f FL ( ν ) f FL ( ν ) , ν ∈ R , (28)and which can be manipulated to read E FV α,γ,ν ( ν ) = (cid:114) ln 2 π α (cid:90) ∞−∞ e − t ln 2 /α t (2 ν − t )( ν − ν − ν t (2 ν − t ))(( ν − ( ν − t ) ) + 4 γ ( ν − t ) ) ν d t, (29)which is an even function of ν . The expression (29) isobviously more involved than (20), but it could of course beanalyzed in a similar manner as with the classical Voigt profiledescribed in the Appendix B. However, this is not necessaryat this point, as we can now employ more simple argumentsto establish its convergence as follows. First, it is noticed thatthe expression (29) is singular at ν = 0 , which of course isnatural since f F L (0) = 0 but f F V (0) (cid:54) = 0 . Hence, we cannot have uniform convergence on the whole of R . Secondly,we have now one more parameter to consider, the line center frequency ν . We therefore proceed as above, and rewrite (29)by using the substitution t √ ln 2 /α ↔ t , yielding E FV α,γ,ν ( ν ) = 1 √ π (cid:90) ∞−∞ e − t t (2˜ ν − t )(˜ ν − ˜ ν − ˜ ν t (2˜ ν − t ))((˜ ν − (˜ ν − t ) ) + 4˜ γ (˜ ν − t ) ) ˜ ν d t = E FV˜ α, ˜ γ, ˜ ν (˜ ν ) , (30)where we have introduced the dimensionless parameters ˜ α = √ ln 2 , ˜ γ = γα √ ln 2 , ˜ ν = να √ ln 2 , ˜ ν = ν α √ ln 2 . (31)The scaling introduced above means that it is sufficient toconsider the uniform approximation properties of (30) overthe normalized frequencies | ˜ ν − ˜ ν | < ˜ B , and study how theapproximation error behaves for a range of values of ˜ γ as ˜ ν → ∞ . As a mathematical argument for this procedure, wecan now employ the convergence of the full Lorentz profile f F L ( ν ) → f L ( ν − ν ) which is uniform for | ν − ν | < B as ν → ∞ , cf. , the derivation of (11) and (12) in Section II. Dueto the convolution with the Gaussian profile, we immediatelyhave also that f F V ( ν ) → f V ( ν − ν ) as well as E FV α,γ,ν ( ν ) → E V α,γ ( ν − ν ) , and which are uniform for | ν − ν | < B as ν → ∞ .The uniform approximation procedure is illustrated inFigs. 2 and 3. As can be seen in these plots, the full Voigterror E FV α,γ,ν ( ν ) is locally very similar to the Voigt error E V α,γ ( ν − ν ) of Fig. 1 already at ν /α = 100 , and almostidentical at ν /α = 300 (excluding the region close to theorigin). In a typical application of broadband radiative transferin the atmosphere (see the numerical examples below), thenormalized center frequency is in the order of ν /α > .Hence, assuming (or checking) that ν /α is sufficiently largewith reference to the analysis illustrated in Figs. 2 and 3above, it is safe to employ the same Theorem 4.1, the sameapproximation procedure (22) and the same criteria parameters n and ( n , n ) in connection with the full Voigt profile,as with the classical Voigt profile. The only difference isthat the approximation is now valid over some finite regionwith bandwidth B , including the transition frequency ν andexcluding a suitable neighborhood of the origin.V. A DAPTIVE LINE SELECTION
Even when using the fast approximation of Voigt profilesas expressed in (22) above, the broadband line-by-line calcu-lations of interest in radiative transfer analysis may still becomputationally huge and therefore impractical. Hence, it isof interest to make the computations faster and more effectiveby reducing the number of spectral lines that are included inthe computations and only employ the lines that are necessaryand relevant at each frequency. There is no simple rule to makethis line selection as efficient as possible in order to achievesome required accuracy. However, based on the approximationtheory that has been developed above we will outline herea simple, pragmatic and readily programmable criterion tomake an adaptive line selection using only a few adjustableparameters to control the accuracy. The method is obviously5
20 0 20 40 60 80 100 12010 − − − Normalized frequency ν/α | E F V α , γ , ν ( ν ) | Relative error | E FV α,γ,ν ( ν ) | γ/α = 10 − γ/α = 1 γ/α = 10 γ/α = 30 Fig. 2. Computation of the relative error | E FV α,γ,ν ( ν ) | between the full Voigtprofile and the full Lorentz profile. The error is plotted here as a function of thenormalized frequency ν/α for γ/α = 10 − , , , . The center frequencyis ν /α = 100 . − − − Normalized frequency ν/α | E F V α , γ , ν ( ν ) | Relative error | E FV α,γ,ν ( ν ) | γ/α = 10 − γ/α = 1 γ/α = 10 γ/α = 30 Fig. 3. Same plot as in Fig. 2, except here the center frequency has beenincreased to ν /α = 300 . suboptimal, but it is able to significantly reduce the number ofspectral line calculations that are included based on simple andcomprehensible criteria while at the same time maintaining ahigh computational accuracy.We consider the calculation of the absorption coefficient k ν expressed as k ν = (cid:88) j S j f j ( ν ) , (32)where S j are the line strengths and f j ( ν ) the approximateVoigt profiles as described in (22). An individual spectralprofile in (32) is denoted k ν j ( ν ) = S j f j ( ν ) , and which isassociated with a specific transition frequency ν j (includingpressure shift), and line parameters γ j and α j , all of which de-pend on height (temperature and pressure) in the atmosphere.It is assumed that γ j /α j > n for all j and where n isthe parameter defined in Theorem 4.1. We will adopt herea block processing approach where the whole computationaldomain is divided into subdomains of relatively small andmanageable sizes. Hence, we will consider the calculation ofthe absorption coefficient k ν over a fixed and relatively narrowsubinterval Ω = [ ν a , ν b ] , and estimate the contribution fromall the individual spectral lines where k ν j ( ν ) = S j f j ( ν ) for j = 1 , . . . , J . We start by defining the minimum distance between thetransition frequency ν j and the set Ω as D j = (cid:40) min {| ν a − ν j | , | ν b − ν j |} if ν j / ∈ Ω , if ν j ∈ Ω , (33)for all j = 1 , . . . , J . Now, it is readily seen that for D j > n α j the following implication holds ν ∈ Ω ⇒ n α j < D j ≤ | ν − ν j |⇒ S j π γ j γ j + D j > S j π γ j γ j + | ν − ν j | ≈ k ν j ( ν ) , (34)where ν j is outside of Ω and where the last approximation isdue to the Theorem 4.1 and the criterion in (22).For subintervals Ω which contain at least one spectral line j , we define the largest self-contribution over Ω as k intmax = max ν j ∈ Ω k ν j ( ν j ) = max ν j ∈ Ω S j f j ( ν j ) , (35)which is a reasonable computational task based on the ordinaryVoigt profile implemented by using e.g. , [13]. Based on (34)we can also estimate the largest contribution to k ν fromspectral lines j outside of Ω as k extmax = max D j >n α j S j π γ j γ j + D j , (36)and where the condition D j > n α j guarantees that theLorentz approximation is valid for ν ∈ Ω . An estimate ofthe largest contribution from any individual spectral line to k ν is now given by k max = max { k intmax , k extmax } . If Ω does notcontain any spectral lines j , we simply choose k max = k extmax .It is now readily seen that k ν j ( ν ) ≤ k max for ν ∈ Ω and forall j such that ν j ∈ Ω or D j > n α j .We now decide to exclude all the spectral lines with D j > n α j and S j π γ j γ j + D j < k max A, (37)where A is a small positive number. From (34), we see that forthose lines it will hold that k ν j ( ν ) < k max A for all ν ∈ Ω . Itis finally observed that the criterion in (37) is also equivalentto include all the spectral lines satisfying D j < n α j or S j π γ j γ j + D j > k max A. (38)It is noted that lines satisfying D j < n α j are alwaysincluded, i.e. , all lines inside or in a close neighborhood of Ω . It is also very useful to put an upper bound K (cid:28) J onthe number of lines satisfying both D j > n α j as well as thesecond criterion in (38) organized in descending order. Thecalculations in (33), (35) and (36) as well as the criterion (38)can now be readily implemented in a computer code.The extension of the procedure outlined above to the casewith the full Voigt profile is straightforward. This means e.g. ,that the sequence ˆ f j = 1 π γ j γ j + D j (39)6hould be replaced for the sequence ˆ f j = π γ j ν ( ν − ν j ) + 4 γ j ν for ν j < ν a π γ j ν ( ν j − ν ) + 4 γ j ν for ν j > ν b , (40)which will guarantee that S j ˆ f j > k ν j ( ν ) for ν ∈ Ω and D j >n α j , similar as in (34). In practice, this modification of ˆ f j isscarcely needed in view of the very small approximation errorbetween the Voigt and the full Voigt profiles. The importantmodification here is to replace the calculation of the classicalLorentz profile with the calculation of the full Lorentz profilein the implementation of (22).VI. N UMERICAL EXAMPLES
As a simple benchmark problem for comparing the accuracyand computational effort associated with the proposed profileapproximations, we consider here a broadband line-by-lineanalysis of radiative transfer in the atmosphere. The computercode is implemented in Matlab and executed on a standardlaptop. It is emphasized that the code has not been optimizedfor speed. It is merely a straightforward implementation ofa simple recursive algorithm with the aim of comparing thedifferent approximation methods on equal terms.We consider the computation of the outgoing monochro-matic irradiance F ν transmitted by the Earth at
65 km heightin a representative, plane-parallel and piece-wise homogeneousatmosphere, as depicted in Fig. 4. The pressure profile isimplemented as the exponential law of an isothermal at-mosphere [33, Eq. (4.24)] based on the mean value of thetemperature profile shown in the figure. A recursive solutionof the corresponding transfer equation for thermal IR radiationin a non-scattering atmosphere [12, Eq. (4.2.2)] is given by I ν ( z i +1 , µ ) = I ν ( z i , µ )e − d i (cid:80) s N ( s ) ( z i ) k ( s ) ν ( z i ) /µ + B ν ( T ( z i )) (cid:16) − e − d i (cid:80) s N ( s ) ( z i ) k ( s ) ν ( z i ) /µ (cid:17) , (41)for i = 1 , . . . , . Here, I ν ( z i , µ ) are the radiances at height z i and direction µ = cos θ , z i = ( i − · , d i = z i +1 − z i , N ( s ) ( z i ) the number density of each species ( s ), k ( s ) ν ( z i ) the corresponding absorption coefficients and B ν ( T ( z i )) thePlack function of blackbody radiation at temperature T ( z i ) .The iteration is started at z = 0 with temperature T = 288 K .Five different species ( s ) are included in the computationcomprising a total of 430070 transitions in the range 0-3000 cm − , cf. , Table I. The line parameters are taken fromthe HITRAN data base [9], [10], [34] and the following sixdifferent profile computations are considered: The Voigt (V)profile (13) and the full Voigt (FV) profile (17) are basedon the Faddeeva function as in (16) and (18), respectively,and implemented by using [13]. The corresponding approx-imations, the fast Voigt (fV) profile and the fast full Voigt(fFV) profile are based on (22), and their combinations withthe adaptive line selection procedures (fV+LS and fFV+LS)are described in Section V. The parameter setting for both Height z (km) T e m p e r a t u r e ( K ) Temperature profile − − − − Height z (km) M i x i n g r a t i o l og ( M R ) Mixing ratios H OCO O CH N O Fig. 4. Representative vertical profiles of temperature and mixing ratios forthe five most important greenhouse gases in midlatitude regions according to[12, Figs. 3.1 and 3.2].
Constituent Isotopologue lines cm − Water vapor H O C O O CH N O Total lines 430070
TABLE IT
HE ISOTOPOLOGUES OF THE SPECIES THAT HAVE BEEN INCLUDED INTHE MODELING AND THE CORRESPONDING NUMBER OF TRANSITIONSTHAT ARE AVAILABLE IN THE
HITRAN
DATABASE OVER THE BANDWIDTH cm − . of the fast Voigt procedures implemented as in (22) are here n = 10 − and ( n , n ) = (10 , for a maximum of 1 % relative approximation error. The adaptive line selection isimplemented with parameters A = 10 − and K = 1000 , seeSection V.The full bandwidth of the computation is 100-2000 cm − and which is divided into blocks of 2000 frequency pointseach. The frequency resolution of each block is set by theDoppler broadening of the heaviest species (ozone) at thelowest temperature (220 K ) yielding a total of 1955 blocksand . · frequency points over the whole bandwidth. Foreach frequency block Ω , the iteration in (41) is implementedas an array of × values of radiances evaluated at 2000Gauss-Legendre nodes ν ∈ Ω and 10 Gauss-Legendre nodes µ ∈ (0 , . The integration of the total irradiance of eachblock is then conveniently executed by using the correspondingGauss-Legendre weights. The resulting monochromatic irra-diance at z = 65 km is then finally evaluated as the totalirradiance per frequency block, as depicted in Fig. 5. Thefrequency resolution of the monochromatic irradiance shownin the figure is hence varying from 0.15 cm − to 3.1 cm − .Note however that the resolution of the computation in (41) is2000 times more dense, i.e. , the resolution varies here between0.000075 cm − and 0.0015 cm − . The result shown in Fig. 5is based on the fast Voigt profile with adaptive line selection(fV+LS) as expressed in (16), (22) and (38). The computationtime on a standard laptop is about 3-4 hours .In Figs. 6 and 7 are shown the corresponding irradiancecalculations for some of the profiles as mentioned above,7nd which are evaluated here in the two narrow bands 667-668 cm − and 900-901.4 cm − , comprising 2000 frequencypoints each. The Voigt (V) and full Voigt (FV) profiles arenot shown here since they are indistinguishable from the fastapproximations (fV) and (fFV) and which are hence veryaccurate in these examples. In Fig. 6 we can see that thereare many closely spaced absorption lines in the 667 cm − band, and there is therefore virtually no difference between theVoigt and the full Voigt profiles, and the latter are thereforenot shown in this plot. As we can see, there is only a smalldeviation between the fast Voigt (fV) and the fast Voigt withadaptive line selection (fV+LS) in this band.In Fig. 7 we can see that there are very few absorption linesin the 900 cm − band, and the far wings are therefore of moreimportance. There is therefore a small difference between theVoigt and the full Voigt profiles, and there is also a smalldeviation using the line selection procedure. The computertime and relative errors (relative V and FV, respectively) ofthese calculations are summarized in Table II. The accuracyin these calculations can readily be improved by tuning theconvergence parameters ( n , n ) , A and K described above,and which hence can be traded against the correspondingincrease in computer time.The total number of line calculations that are available inthe numerical algorithm described above is ·
65 (heights) · ≈ . · , each absorption lineevaluation comprising 2000 frequency points. Furthermore,each of the 1955 block calculations at each of the 65 heightsalso consists of an evaluation of the radiance at the 2000frequency points in 10 different directions, and then followedby the associated Gauss-Legendre integration. Based on thetimings presented in Table II, it can be estimated that acomputation of the spectra as shown in Fig. 5 based on theVoigt profile without approximations will take about 220 days,and about twice this time using the full Voigt profile since itrequires two Faddeeva evaluations instead of just one. By usingthe fast Voigt (fV) approximation (22) the computer time forcalculating all lines reduces to about 7 days. Finally, by usingthe adaptive line selection criteria (fV+LS) (38) the numberof line calculations reduces from a total of . · to about . · corresponding to about 0.88 % of all the available linecalculations. The final computation time on a standard laptopis now 3-4 hours.VII. S UMMARY AND CONCLUSIONS
This paper presents uniform error bounds for fast calculationof approximate Voigt profiles. The purpose is to acceleratethe computationally huge broadband line-by-line analysis ofradiative transfer in the atmosphere. The main idea is toreplace the relatively demanding Voigt calculations for themuch simpler and faster Lorentz calculations whenever thiscan be done within a given error tolerance. In addition, itis also demonstrated how the uniform bounds enable a fastand efficient subband adaptive line selection strategy thatincludes only the spectral lines that give the most significantcontribution to the absorption coefficient. Numerical examples , , , , , Wavenumber ν (cm − ) F ν ( W / m / c m − ) Monochromatic irradiance F ν at TOA (65 km ) πB ν (288) (15 ◦ C ) Fig. 5. A computation of the outgoing monochromatic irradiance F ν transmitted by the Earth at
65 km height. The computation is based on thefast Voigt approximation and the adaptive line selection procedure (fV+LS).The dashed line shows the corresponding blackbody radiation πB ν ( T ) attemperature T = 288 K .
667 667 . . . . . . . Wavenumber ν (cm − ) F ν ( W / m / c m − ) Monochromatic irradiance fV+LSfV
Fig. 6. The outgoing monochromatic irradiance F ν as in Fig. 5, evaluatedhere in the band 667-668 cm − . The computations are based on the fast Voigt(fV) approximation as well as the combination with the adaptive line selectionprocedure (fV+LS), respectively.
900 900 . . . . . . . . Wavenumber ν (cm − ) F ν ( W / m / c m − ) Monochromatic irradiance fV+LSfVfFV+LSfFV
Fig. 7. The outgoing monochromatic irradiance F ν as in Fig. 5, evaluated herein the band 900-901.4 cm − . The computations are based on the fast Voigt(fV) as well as the fast full Voigt (fFV) approximations, and as combinationswith the adaptive line selection procedure (fV+LS, or fFV+LS), respectively. and (cm − ) Profile Time (s) rel. error . · − . · − . · − . · − . · − . · − . · − . · − TABLE IIR
EQUIRED COMPUTER TIME ON A STANDARD LAPTOP AND RELATIVEAPPROXIMATION ERRORS IN THE COMPUTATION OF THE TOTALIRRADIANCE BASED ON THE
FREQUENCY POINTS SHOWN IN F IGS . 6
AND AND THE
LINES LISTED IN T ABLE
I. T
HE REFERENCECOMPUTATION CORRESPONDING TO THE V OIGT (V)
AND FULL V OIGT (FV)
PROFILES ARE BASED ON [13]. are included to illustrate that the two approaches can acceleratethe line-by-line computations by several orders of magnitudewith very little loss in accuracy.A new “full” Voigt profile is also presented based onthe “full” Lorentz profile, and which is obtained withoutmaking the traditional resonance approximation. The fullLorentz profile can be very well motivated from the classicalprinciples of molecular polarizabilities, optical theorems andassociated sum rules connecting to the static polarizability. Onthe other hand, the classical Lorentz resonance approximationcan also be derived directly from first principles based onquantum mechanics, as in [5, pp. 77-78]. Hence, more researchis needed to explore the potential application of this new(old!) profile. In particular, the full Voigt profile is potentiallysuitable for far wing (off resonance) calculations in spectralregions where line mixing effects can be ignored. Based onthe numerical examples of broadband radiative transfer in theatmosphere that are presented here, we can see that there is aslight discrepancy in the computed irradiances at high altitudewhen using the Voigt and the full Voigt profiles, respectively.These discrepancies are furthermore most significant in theso called “infrared windows”, where the density of spectrallines is sparse and the far wing contributions are of moreimportance.As a future potential of employing the approximation tech-niques that have been presented in this paper, it may be ofinterest to explore their use to increase the accuracy andefficiency of existing line-by-line algorithms, see e.g. , [8].It may also be of interest to explore their extension to beused with existing line mixing methods [1], [2], [3], [4],[7], as well as with the new line shapes based on partialcorrelation, speed dependency and velocity changes that havebeen developed recently, see e.g. , [6], [11]. It can be expectedthat, while advanced methods are employed for narrow inbandcalculations, it may be possible to approximate the far wingcontributions by using adequate asymptotics based on simple functions, error estimates as well as pre-calculated and storeddata. The aim of these studies would be to render the new,more accurate line shapes and line mixing formulas moreapplicable for computationally huge broad-band line-by-lineanalysis of radiative transfer in the atmosphere.A
PPENDIX
A. The full Voigt profile
To show the result (18) we will employ the analytic Fourier(Laplace) transform F ( ω ) = F{ f ( t ) } = (cid:90) ∞−∞ f ( t )e − i ωt d t (42)with inverse f ( t ) = F − { F ( ω ) } = 12 π (cid:90) ∞ +i y −∞ +i y F ( ω )e i ωt d ω, (43)where ω = x +i y and the contour integral is carried out insidethe region of analyticity of F ( ω ) . We will make use of thefollowing standard transforms F{ sin( at ) u ( t ) } = aa − ω (44) F{− i cos( at ) u ( t ) } = ωa − ω (45) F{ (cid:114) bπ e i ω t e − bt } = e − ( ω − ω ) / b (46)where a and b are positive real constants, ω a complexvalued constant and u ( t ) the unit step (Heaviside) function.The region of analyticity of (44) and (45) is Im { ω } < andfor (46) it is the whole of C . We will furthermore employ thefollowing analytic form of the Parseval’s relation (cid:90) ∞−∞ f ( t ) g ∗ ( t ) d t = 12 π (cid:90) ∞ +i y −∞ +i y F ( ω ) G ∗ ( ω ∗ ) d ω, (47)which can be derived directly by using (43) and where ( · ) ∗ denotes the complex conjugate and y > is fixed. The contourintegral in (47) extends along the real line ( d ω = d x ) in theupper half of the complex plane and requires (at least) that F ( ω ) is analytic in a neighborhood of the line ω = +i y in theupper half plane and that G ( ω ) is analytic in a neighborhoodof the line ω = − i y in the lower half.In order to derive (18), we start from (17) and write f FV ( ν ) = f G ( ν ) ∗ f FL ( ν )= (cid:114) ln 2 π α e − ν ln 2 /α ∗ Im (cid:26) π νν − ν − i2 γν (cid:27) (48)where ∗ denotes convolution and where (3), (4) and (14)have been used. This means that we can write f FV ( ν ) =Im { h FV ( ν ) } where h FV ( ν ) = (cid:114) ln 2 π πα (cid:90) ∞−∞ e − ( ν − x ) ln 2 /α x d xν − x − i2 γx . (49)9y introducing ω = x + i γ , ω = ν + i γ , a = ν − γ and b = α / , the integral above can be written I = (cid:90) ∞−∞ e − ( ν − x ) ln 2 /α x d xν − x − i2 γx = (cid:90) ∞ +i γ −∞ +i γ e − ( ω − ω ) ln 2 /α ω − i γa − ω d ω = (cid:90) ∞ +i γ −∞ +i γ e − ( ω − ω ) / b ωa − ω d ω − i γa (cid:90) ∞ +i γ −∞ +i γ e − ( ω − ω ) / b aa − ω d ω, (50)and where it is assumed that ν > γ > and hence a > . Byusing the Parseval’s relation (47) and the standard transforms(44) through (46), it now follows that I = 2 π (cid:90) ∞ (cid:114) bπ e i ω t e − bt ( − i) ∗ cos( at ) d t − i γa π (cid:90) ∞ (cid:114) bπ e i ω t e − bt sin( at ) d t. (51)By rewriting the trigonometric functions using exponentialsthe integral above can also be written in the more genericform I = √ πα √ ln 2 (cid:18) i (cid:90) ∞ e − bt +i( ω + a ) t d t + i (cid:90) ∞ e − bt +i( ω − a ) t d t − γa (cid:90) ∞ e − bt +i( ω + a ) t d t + γa (cid:90) ∞ e − bt +i( ω − a ) t d t (cid:19) . (52)All the integrals above are now in a form where we cansubstitute and complete the squares inside the exponent to get (cid:90) ∞ e − α t / − ct d t = 2 √ ln 2 α e c ln 2 /α (cid:90) ∞ e − ( t + c √ ln 2 /α ) d t = 2 √ ln 2 α e c ln 2 /α (cid:90) ∞ c √ ln 2 /α e − t d t = 2 √ ln 2 α e c ln 2 /α √ π c √ ln 2 /α )= √ π ln 2 α w (i c √ ln 2 /α ) , (53)where c = − i( ω ± a ) , erfc( z ) is the complementary errorfunction and w ( z ) = e − z erfc( − i z ) the Faddeeva function,see [35, Eq. 7.2.1–7.2.3]. Collecting all the results above, weget finally h FV ( ν ) = (cid:114) ln 2 π α (cid:32)(cid:16) − γa + i (cid:17) w (cid:32) ( ν + a + i γ ) √ ln 2 α (cid:33) + (cid:16) γa + i (cid:17) w (cid:32) ( ν − a + i γ ) √ ln 2 α (cid:33)(cid:33) . (54) B. Uniform convergence of the Voigt profile
We consider the uniform converge of the error term E V˜ α, ˜ γ (˜ ν ) = 1 √ π (cid:90) ∞−∞ e − t t (2˜ ν − t )˜ γ + (˜ ν − t ) d t, (55)where ˜ α = √ ln 2 , ˜ γ = ( γ/α ) √ ln 2 and ˜ ν = ( ν/α ) √ ln 2 asdefined in (23). Assume that ˜ ν > , let < a < be asuitable chosen parameter and make the following estimatesof | ˜ ν − t | in the respective subintervals t ≤ ⇒ | ˜ ν − t | ≥ ˜ ν ≤ t ≤ a ˜ ν ⇒ | ˜ ν − t | ≥ (1 − a )˜ νa ˜ ν ≤ t ≤ ν ⇒ | ˜ ν − t | ≥ t ≥ ν ⇒ | ˜ ν − t | ≥ ˜ ν. (56)An error estimate based on (55) is now readily obtained as | E V˜ α, ˜ γ (˜ ν ) | ≤ √ π (cid:90) −∞ e − t ( − t )(2˜ ν − t )˜ γ + ˜ ν d t + 1 √ π (cid:90) a ˜ ν e − t t (2˜ ν − t )˜ γ + (1 − a ) ˜ ν d t + 1 √ π (cid:90) νa ˜ ν e − t t (2˜ ν − t )˜ γ d t + 1 √ π (cid:90) ∞ ν e − t t ( t − ν )˜ γ + ˜ ν d t. (57)By making the substitution t ↔ − t in the first integral, andthen repeating the whole procedure for negative ˜ ν , it is foundthat | E V˜ α, ˜ γ (˜ ν ) | ≤ √ π (cid:90) ∞ e − t t (2 | ˜ ν | + t )˜ γ + ˜ ν d t + 1 √ π (cid:90) a | ˜ ν | e − t t (2 | ˜ ν | − t )˜ γ + (1 − a ) ˜ ν d t + 1 √ π (cid:90) | ˜ ν | a | ˜ ν | e − t t (2 | ˜ ν | − t )˜ γ d t + 1 √ π (cid:90) ∞ | ˜ ν | e − t t ( t − | ˜ ν | )˜ γ + ˜ ν d t. (58)The right hand side of (58) can now be evaluated by using thefollowing integrals √ π (cid:90) R R e − t t d t = 12 √ π (cid:16) e − R − e − R (cid:17) , (59)and √ π (cid:90) R R e − t t d t = 12 √ π (cid:16) R e − R − R e − R (cid:17) + 14 (erf( R ) − erf( R )) , (60)10nd where erf( · ) is the error function [35, Eq. 7.2.1]. Bycarrying out the integrations above the following upper boundis obtained | E V˜ α, ˜ γ (˜ ν ) | ≤ γ + ˜ ν (cid:18)
12 + | ˜ ν |√ π −
14 erf(2 | ˜ ν | ) (cid:19) + 1˜ γ + (1 − a ) ˜ ν (cid:18) | ˜ ν | √ π (1 − e − a ˜ ν )+ a | ˜ ν | √ π e − a ˜ ν −
14 erf( a | ˜ ν | ) (cid:19) + 1˜ γ (cid:18) (2 − a ) | ˜ ν | √ π e − a ˜ ν −
14 (erf(2 | ˜ ν | ) − erf( a | ˜ ν | )) (cid:19) . (61)Finally, the bound in (61) can be further relaxed and simplifiedas | E V˜ α, ˜ γ (˜ ν ) | ≤ γ + ˜ ν (cid:18)
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