Copyright 2018 Society of Photo-Optical Instrumentation Engineers (SPIE). One print or electronic copy may be made for personal use only.

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1 Copyright 018 Soiety of Photo-Optial Instrumentation Engineers (SPIE) One print or eletroni opy may be made for personal use only Systemati reprodution and distribution, dupliation of any material in this paper for a fee or for ommerial purposes, or modiation of the ontent of the paper are prohibited Alexander Berk and Robert Sundberg MODTRAN and the GrossDoppler line-shape funtion," SPIE 10768, Imaging Spetrometry XXII: Appliations, Sensors, and Proessing, I (18 September 018) DOI: See next page for full paper

2 Distribution A: Approved for Publi Release MODTRAN and the GrossDoppler Line-shape Funtion Alexander Berk and Robert Sundberg* Spetral Sienes, In, 4 Fourth Avenue, Burlington, MA, USA ABSTRACT MODTRAN models the moleular absorption for the entire 0 to 50,000 m -1 spetral range Typially, radiative transfer models define distint line-shape funtions depending on the spetral region This an produe spetral anomalies at the transitions A 3-parameter GrossDoppler line-shape funtion is defined that provides a spetrally-universal model for omputing moleular absorption Key Words: Voigt line-shape, Weiskopf line-shape, Gross line-shape, MODTRAN, moleular absorption 1 INTRODUCTION The MODTRAN6 radiative transfer model enjoys widespread use within the remote sensing, data analysis, sene simulation and limate foreasting ommunities Muh of MODTRAN s appeal derives from its statistial band model algorithm, whih enables the radiative transfer equations to be rapidly solved over narrow spetral hannels without requiring finely gridded monohromati alulations Reently, a line-by-line (LBL) algorithm was developed for MODTRAN6 1 At first, it may seem as adding this apability is ounter to the fundamental MODTRAN philosophy of providing rapid in-band radiative transfer However, statistial algorithms, by their very nature, require methods or metris for evaluating their auray The MODTRAN LBL method was introdued to provide users of the model the ability to evaluate the statistial unertainties of MODTRAN results for their spei appliations Moleular absorption in the MODTRAN LBL algorithm is modeled using the Van-Vlek Huber (VVH) line-shape funtion 1 This three parameter line-shape funtion defines the ontribution of moleular absorption for frequenies extending 5 m -1 from line enter; the parameters are the transition frequeny,, the ollision-broadened half-width at half-maximum (HWHM), γ, and the Doppler half-width at 1/e of maximum line-shape value, γ d Temperature and pressure dependent ontinua oeffiients are used to model the more distant moleular line ontributions The VVH spetral form provides a smooth transition from the Voigt line-shape funtion in the infrared to the Van-Vlek Weiskopf, or simply Weiskopf, line-shape funtion in the mirowave However, MODTRAN models the absorption arising from the full suite of HITRAN moleular transitions thus, its spetral range extends to even longer radio waves, where the VVH line-shape is not appliable In this paper, an effort is initiated to define a spetrally-universal, 3-parameter lineshape funtion *rob@spetralom; phone fax ; spetralom

3 The normalized Voigt line-shape funtion, f V ( γ γd ) BACKGROUND ( t ) γ exp dt, ; =, 3 γ + γ π ( t) d is ommonly used to model the spetral dependene of moleular absorption in the infrared, visible and ultraviolet spetral regions The Voigt line-shape is a funtion of the spetral displaement from line enter, the dferene between the measurement frequeny,, and the transition frequeny The Voigt form arises from the spetral onvolution of the normalized Lorentz and Doppler line-shape funtions: ( ) ( ) γd exp L γ π D f ( γ; ) = and f ( γd ; ) = + γ γ π d It is well known that the Voigt lineshape does not aurately apture the spetral dependene of moleular absorption in the mirowave, where the line-shape begins to exhibit asymmetri absorption about the line enter In this region, ontributions from Doppler shting are small and the Weiskopf line-shape funtion is most ommonly used: ; ( ) ( ) + γ + + γ (3) W γ π γ π f ( γ ) ( ) + This form is somewhat deeptive in that the maximum of the line-shape funtion does not our exatly at the transition frequeny,, and the HWHM is not exatly equal to γ However, the disrepanies between these paired values is proportional to the square of the ratio of the half-width to the transition frequeny, ie (γ / ) and γ (γ / ) For a 30 GHz (1 m) line, this ratio at 1 atm pressure is typially small: (γ / ) (006) 0004 Thus, the Weiskopf line-shape an be evaluated at the measured transition frequeny and HWHM in the mirowave spetral region Optimally, one would reate a smooth transition from the mirowave to the infrared by spetrally onvolving the Weiskopf and Doppler line-shape funtions This is not possible beause the Weiskopf form is not integrable (normalizable) Instead, line-by-line models often transition smoothly between the mirowave and infrared spetral regions by simply replaing the Lorentzian ontributions with their Voigt ounterpart: W V V f ( γ, γd ; ) f ( γ, γd ; ) + f ( γ, γd ; ) Note that this form for the Weiskopf line-shape inludes dependene on the Doppler half-width As the transition frequeny inreases from the mirowave into the infrared, the oeffiient of the braketed term approahes one near line enter ( 5 m -1 ) and the negative resonane term (dependent on ) beomes small Thus, the Eq (4) lineshape approahes the Voigt form of Eq (1) As one transitions in the opposite diretion, to even longer radio waves, the auray of the Weiskopf form deteriorates At these wavelengths, an alternate line-shape funtion, attributed to Gross, is ommonly used: G 4 γ π f ( γ; ) + 4 γ ( ) This form has a number of appealing features It approahes zero at zero frequeny; it is integrable sine it falls off as 1/ for large ; it is normalized; and it does not beome exeedingly large when is small (<< 1 m -1 ) and > (1) () (4) (5)

4 Furthermore, the maximum of f Gross ours at =, and although the line-shape is asymmetrial, the half-width at halfmaximum (HWHM) is equal to γ Finally, the value of the Gross line-shape at line enter, 1/(π γ ) equals the Lorentz line-shape transition frequeny value It is natural to ask what the relationship is between the Gross line-shape funtion and the mirowave line-shape funtions To make this omparison, the Gross line-shape funtion for γ, a ondition appliable in the mirowave, an be rewritten as follows: G γ γ + γ f ( γ; ) = +, γ π ( ) + γ ( + ) γ γ + γ + γ π 3γ π + for >> γ ( ) + γ ( + ) γ (6) When >> γ, the term on the left approahes the term on the left in the Weiskopf line-shape expression, Eq (3) The primary dferene between the Gross and Weiskopf line-shape funtions arises from the ontributions to the signiantly smaller negative resonane terms; the numerator of the right-hand side term in Eq (6) approahes 3 [not 1] Also, the γ term in the denominator is subtrated, not added, but this term is small ompared to ( ) 3 A SPECTRALLY UNIVERSAL LINE-SHAPE FUNCTION Sine the Gross line-shape funtion is normalized and it approahes the Lorentz line-shape in the infrared, it an be onvolved with the Doppler line-shape to produe a single, normalized GrossDoppler (GD) line-shape funtion appliable for all radio wave through UV wavelengths For < γ, Doppler ontributions will always be extremely small and the Gross form, Eq (5), an be used diretly When the transition frequeny exeeds the Lorentz half-width, the spetral onvolution an be shown to be given by the following expression: GD D G ( γ, γd ; ) = t ( γd ; ) t ( γ; ) f f f dt where yγ γ yγ K x+, y, + L x+ y 1 + γ γ + γ = γ d π + γ γ + γ + K x +, y L x +, y γd γ γ d ( ) ( ) ( X ) exp( ) ( ) γ, γ exp Y τ dτ 1 τ τ x, y, K ( X, Y ) = and L( X, Y ) = dτ γ γ π Y + X τ π Y + X τ d d (7) (8)

5 The funtions K(X, Y) and L(X, Y) are the real and imaginary parts of the omplex probability funtion (CPF), respetively Only the first term in Eq (7) is signiant in the infrared (IR), visible (VIS) and ultraviolet (UV), and this term essentially equals to the Voigt line-shape funtion Voigt 1 1 yγ > m and f ( γ, γd ; ) = K( xy, ) K x+, y 1 γd π γd π + γ < 5 m (9) Here, the frequeny offset, x 1 off ½ γ (γ / ), is a small fration of the Lorentz half-width, and the Voigt line-shape does not vary appreiably on this sale 31 Examples of the GrossDoppler Line-Shape Funtion The Gross Doppler line-shape funtion is examined in this setion for three moleular transitions: one in the thermal infrared, one in the mirowave and a radio wave transition long of the mirowave HITRAN 016 parameters define the hosen transitions, whih were all seleted to be relative strong within their spetral sub-region Table I lists the HITRAN data The line-shapes were all omputed at 96 K Table 1 HITRAN016 Data for Examined Moleular Transitions CH 4: E-0 3E F A1 4A1 1 3A NH 3: E-3 558E A" A1' 3 3 a A1"A' 3 3 s A"A" OH: E E X1/ 0 X1/ 0 40 QQ 45ef Thermal Infrared CH4 Moleular Transition The first moleular transition examined was a CH 4 line at m -1, 7535 µm Sine this line is in the IR, the GrossDoppler line-shape funtion is expeted to equal the Voigt line-shape The pressure was set to 01 atm so that the Lorentz (γ = m -1 ) and Doppler (γ d = m -1 ) ontributions to the line-shape would be omparable With pressure shting, the transition frequeny is m -1 The GrossDoppler, the Voigt and the Gross line-shape are all plotted on the left in Figure 1 Figure 1 (left) The Voigt, GrossDoppler and Gross line-shape funtions for a thermal infrared CH4 moleular transition (right) The relative ontribution of the CPF terms in the GrossDoppler line-shape

6 The dferene between the Gross and GrossDoppler line-shapes in Figure 1 attest to the importane of the Doppler ontribution The Voigt and GrossDoppler line-shape funtions are essentially idential The relative ontributions of the four CPF omponents to the GrossDoppler line-shape are plotted on the right in Figure 1 The first term in Eq (7), whih is denoted as K_1, ompletely dominates The seond term, L_1, is the primary soure of the residual between the Voigt and GrossDoppler line-shape funtions The K_ and L_ terms have a negative sum and the magnitude of their ontributions is exeedingly small at infrared and shorter wavelengths Voigt, GrossDoppler, Weiskopf and Gross line shape funtions are plotted on the left in Figure for an OH moleular transition entered near 89 MHz ( m -1 ) at a pressure of 007 atm and at 96 K For this line at this pressure, the half-width (0008 m -1 ) is not muh smaller than the transition frequeny This is evident in Figure 1, sine the symmetri Voigt line shape funtion does not drop to muh more than half the peak value at 0 m -1 Sine the line-shape should drop to 0 m at 0 m -1, the Voigt form learly does not provide a good representation of the spetral dependene The absurdity of using the Weiskopf form, Eq (3), parameterized with the measured transition frequeny and ollision half-width, in this spetral region is also demonstrated Not only are the peak frequeny and value too large by more than a fator of, the half-width is infinite That is, the short wavelength side of the line-shape never dereases to one-half of the peak value Figure (left) The Voigt, GrossDoppler, Weiskopf and Gross line-shape funtions for an 89 MHz OH moleular transition (right) The relative ontribution of the CPF terms in the GrossDoppler line-shape Sine Doppler ontributions are essentially zero in this region, and Gross and GrossDoppler line-shapes overlap Certainly the Gross line-shape funtion, Eq (5), is muh simpler to ompute than the GrossDoppler, Eq (7) But it is interesting to observe on the right in Figure that all four CPF values ontribute signiantly to the total line-shape value 33 Mirowave NH3 Moleular Transition An NH 3 moleular transition entered at 0796 m -1, 3863 GHz, was used to ompare line-shapes in the mirowave Figure 3 ontains plots of the Voigt, GrossDoppler, Weiskopf and Gross line-shape funtions for this moleular transition with both a log and linear based vertial sale The pressure was set to 05 atm, produing a Lorentz half-width of γ = m -1 The Doppler ontribution is negligible in this region, so the GrossDoppler and Gross lineshape funtions are essentially idential The Voigt line-shape does not derease to zero at 0 m -1, as is evident on the left in Figure 3 On the other hand, Weiskopf form does not fall off proportional to one over frequeny squared for >> and produes too muh absorption at large displaement frequenies (not shown) The Gross and GrossDoppler lineshape funtions have neither of these problems However, previous omparisons to measurements illustrate that the Weiskopf form is most aurate near line enter The Voigt line-shape is symmetri about the line enter The linear sale plot on the right in Figure illustrates that the Weiskopf line-shape asymmetry about line enter is more pronouned than that observed for the Gross line-shape The inset illustrates the fat that the Weiskopf form displaes the transition frequeny and intensity, but these effets are small in this region

7 Figure 3 The Voigt, Gross Doppler, Van Vlek Weiskopf and Gross line-shape funtions for a mirowave NH3 moleular transition Log sale is shown on left, and a linear sale on the right Line-shape residuals relative to GrossDoppler are plotted on the left in Figure 4 Even with this expanded sale, there is no appreiable dferene between the Gross and Gross Doppler line-shape funtions for this mirowave line The Voigt and Weiskopf line-shape residuals have a dominant anti-symmetri omponent about the transition frequeny However, the magnitudes of the Weiskopf line-shape residual peaks dfer substantially with values of 096 m at 0758 m -1 and m at 0839 m -1 The question arises as to whether the deviation from a true anti-symmetri shape is primarily an artat of the Weiskopf slow drop off with inreasing frequeny? Can an anti-symmetri orretion to the Gross line-shape be defined that eliminates the predominant Weiskopf residual? Figure 4 (left) The Voigt (blak), Van Vlek Weiskopf (blue) and Gross (green) line-shape funtion residuals for a mirowave NH3 moleular transition (right) The total relative ontribution of the CPF terms from Eq (8) Notie the break in the vertial sale at 04 m, and the zero residual line is plotted as a dashed orange line The relative ontributions CPF terms to the Gross Doppler line-shape are plotted on the right in Figure 4 A break in the vertial sale at 0 m is introdued to expand the plot for the smaller terms This expanded vertial sale enables one to observe the deviation of both the K_ and L_ funtions from zero The K_1 funtion ontinues to be dominant, but the ontribution of L_1 is signiant approximately 4% of the total near line enter Furthermore, the shape of the L_1 funtion resembles the shape of the Weiskopf to GrossDoppler line-shape residual (both urves are blue), although the magnitude of the residual is ~70% larger than that of the L_1 funtion This suggests that a orretion to the GrossDoppler form, appropriate for the mirowave, ould be made by introduing a half-width, γ, and transition frequeny,, dependent oeffiient to the L_1 term For near γ, the oeffiient should approah 1 (no orretion) In the mirowave, where γ is muh less than, the oeffiient would be defined to math the near line enter Weiskopf line-shape At shorter wavelengths, the L_1 term drops off preipitately, so that a onverged mirowave value for the oeffiient would not ontribute signiantly to the IR through UV lineshape

8 4 CONCLUSION AND FUTURE WORK The presene of spetral branh points in radiative transfer modeling is problemati In partiular, system studies an be adversely affeted by spetral disontinuities in the modeling of moleular absorption In this paper, a spetrally universal 3-parameter GrossDoppler line-shape funtion is introdued In its most general form, the alulation of this lineshape requires approximately twie the effort required to ompute the Voigt lineshape However, the spetral regions where this generality is needed are quite limited In the IR through UV, the GrossDoppler and Voigt forms onverge At long wavelengths, Doppler ontributions beome negligible and the Gross line-shape is appropriate In the transition between these regions, GrossDoppler provides a smooth transition This issue remains that the GrossDoppler and Gross line-shape funtions are not as aurate as the Weiskopf form in the mirowave Work is underway to define a modied GrossDoppler line-shape funtion that mathes Weiskopf near the enter of mirowave lines This work is expeted to be ompleted in the next few months 5 ACKNOWLEDGEMENT Alexander Berk wishes to aknowledge support from Spetral Sienes, In Internal Researh and Development (IRAD) The ontinuing maintenane support for MODTRAN6 from the Air Fore Researh Laboratory (Contrat No FA C-009) is also aknowledged 6 REFERENCES [1] Berk A, and F Hawes, "Validation of MODTRAN 6 and its line-by-line algorithm," J Quant Spetros Radiat Transfer 03, (017) [] Gordon, IE, LS Rothman, C Hill et al, "The HITRAN016 Moleular Spetrosopi Database," J Quant Spetros Radiat Transfer 03, 3-69 (017)