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MODTRAN: A MODERATE RESOLUTION MODEL FOR LOWTRAN

A. Berk, L. S. Bernstein, and D. C. RobertsonSpectral Sciences. Inc.i11 S. Bedford StreetBurlington, MA 01803-5128

O TIC"08 July 1987D

1

Scientific Report No. 1

- Approved for public release; distribution unlimited

AIR FORCE GEOPHYSICS LABORATORYAIR FORCE SYSTEMS COMMANDUNITED STATES AIR FORCEHANSCOM AIR FORCE BASE, MASSACHUSETTS 01731

81i' 10 8 03(

"This technical report has been reviewed and is approved for publication'

/n A-( Signature) ,_

LEONARD W. ABREU GEORGE A. VANASSEContract Manager Branch Chief

FOR THE COMMANDER

(Signature)R. EARL GOODActing Division Director I ,

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SSI-TR-124 AFGL-TR-87-0220NAME OF PERFORMING ORGANIZATION 1 6b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION

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Spectral Sciences, Inc. Air Force Geophysics Laboratory

ADDRESS (Cty, State, and ZIP Code) 7b. ADDRESS (City, State, and ZIP Code)

Il1 South Bedford Street Air Force Systems CommandBurlington, MA 01803-5128 Hanscom AFB, MA 01731-5000

. NAME OF FUNDING/SPONSORING 8b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER

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PROGRAM PROJECT TASK WORK UNITELEMENT NO. NO. NO. ACCESSION NO.

62101F' , 7670 j 09 AZ

I. TITLE (Include Security Claslfication)

MODTRAN: A Moderate Resolution Model for LOWTRAN (U)

2. PERSONAL AUTHOR(S)Alexander Berk, Lawrence S. Bernstein. and David C. Robertson

]a. TYPE OF REPORT 13b. TIME COVERED 14. DATE OF REPORT (Year, Month. Day) 5. PAGE COUNTScientific #l FROM86Mv 12 TO 87Mav 1 87 July 08 40

16. SUPPLEMENTARY NOTATION

17. COSATI CODES . 18. SUBJECT TERMS (Continue on reverse if necessay and identify by block number)FIELD GBOUP ,' SUB-GROUP Transmission, Radiation, Infrared, Band Model

]\.\ /I[Atmospheric, Transmittance, and LOWTRAN

19. ABSTRACT (CucI/on reverse if necesay and Identify by block number)

This interim technical report describes a new band model formulation

for the LOWTRAN 6 atmospheric transmittance/radiation computer code. Band

model parameters for H2 0, C02, 03, N20, CO, CH4-, 02, and N2 were calculated

using the 1986 HITRAN line atlas. They were calculated for I /Xm-F ins

from 0 - 17,900/cm and at five temperatures from 200 to 300K. This

transmittance model and associated subroutines have been integrated into r

LOWTRAN 6. The spectral resolution of this new option is better than 5

cm'-t(FWHM). A preliminary version of the code has been delivered to AFGL

for testing. Validation against FASCOD2 calculations will be the emphasis

for the remainder of this effort.

20. DISTRIBUTION IAVAILABIUTY OF ABSTRACT ' ' 121. ABSTRACT SECURITY CLASSIFICATION-b0"UNCLASSIFIED/UNLIMITED Q' SAME AS RPT. 0]OTIC US'EtS UNCLASSIFIED

22a. NAME OF RESPON5IBLE INDIVIDUAL 22b. TELEPHONE (Include Area Code) 22c. OFFICE SYMBOL rLeonard Abreu (617) 3 7 AFGL/OPI

DD FORM 1473, 4 MAR 83 APR edition may be used until exhausted. SEgURITY CLASSIFICATION OF THIS PAGEAll other editions are obsolete. UNCLASSIFIED

. . . . ... ..,. . .. , .-.. .•. .~,,.ww•d . ." '....-.,%..'•'.'.- .- , , .- ".""w- .-- -- .

TABLE OF CONTENTS

Section Pagc

1.0 INTRODUCTION . . . . . . . . . . . . . . . . . ... .

2.0 THE MODTRAN CODE, AN OVERVIEW ............. ............... 5

3.0 MOLECULAR BAND MODEL PARAMETERS ........... .............. 73.1 Line Center Parameters .......... ................ 73.2 Line Tail Parameters ............... ................. 93.3 Parameter Data File ............ .................. .. 10

4.0 BAND MODEL TRANSMITTANCE FORMULATION ...... ............ .. 124.1 Line Center Transmittance ........ ............... .. 12

4.1.1 Curtis-Godson Approximation ..... .......... 134.1.2 Uniformly Mixed Gases ....... ............. .. 15

4.2 Line Wing Absorption ........... ................. .. 16

5.0 INTEGRATION INTO LOWTRAN 6 .......... ................. .. 175.1 MODTRAN Subroutines ............ .................. .. 17

5.1.1 Subroutine BMDATA .......... ................ .. 175.1.2 Subroutine CONDEN ........ ................ .. 185.1.3 SubrouLine BMOD .............. ................. 195.1.4 Subroutine CALC .......... ................. .. 215.1.5 Subroutine EWIDTH. ......... ................ .. 215.1.6 Subroutine ERFU .......... ................. .. 225.1.7 Subroutine CPFI2 ......... ................ .. 22

5.2 Modifications to LOWTRAN 6 ......... .............. 22

6.0 VALIDATION .................. ......................... .. 246.1 Band Model Parameters .......... ................. .. 246.2 Comparison with FASCOi12 .............. ................ 25

7.0 CONCLUSIONS AND FUTURE WORK ........... ................ 28

8.0 REFERKNCES..... . .......... ......................... ... 29

APPENDIX: Derivation of the Band Model Parameters ..... ... 31

Accesion For

DT!C T,-,9 El

~~ .........

I . . .. .. ... . .......

,' ,

- ~'A-f

LIST OF ILLUSTRATIONS

F i gure Page

1 Lorentz Line Shape at Optical Depths of 0.2, 2.0, and 20.In the Band Model Transmittance Formulation, AbsorptionDue to Lines Whose Centers are Within a Bin (Between theDashed Lines) is Modeled Separately from that Due to LineTail Absorption (Outside the Dashed Lines) ....... .......... 3

2 Transmittance Curves for a 150 from Zenith Slant Pathfrom 5 to 10 lkm Through the US Sta-ndard Atmosphere. TheSolid Curve was Obtained from MOOTRAN by Passing a 5 cm- 1

FWHM Triangular Slit Over the 1 cm- 1 Calculations. TheDotted Curve is the Regular LOWTRAN 6 Result also Obtainedfrom MODTRAN (20 cm- 1 Resolution with DV Set to 5 cm-1.

Comparison of Continuous and Discretized Triangular SlitFunctions. FASCOD2 Uses the Weighting from theContinuous Function, but MODTRAN Uses the DiscretizedApproximation .................. ........................ .. 23

4 Comparison of Band Model LOWTRAN 5 (Dotted Curve) andBand Model LOWTRAN 6 (Solid Curve) Molecular AbsorptionCoefficients for Water Vapor at 300 K ........ ............ .. 25

5 Comparison of i FASCOD2 and a MODTRAN Calculation. N2 0and CO2 are the primary absorbers .......... .............. .. 26

6 Comparison of a FASCOD2 and a MODTRAN Calculation is theMicrowave ..................... .......................... .. 27

A--I Comparison of FASCOD2 and MODTRAN Calculations. TheMODTRAN Curve was Generated with the Old Band ModelParameters, Eqs. (A 4) & (A--) , and no N2 0 ContinuumFunction ..................... .......................... 35

iv

1.0 INTRODUCTION

This interim technical report describes work done in year one (12 May

1986 - 11 May 1987) of a program (Contract No. F19628-86--C-0079) to

increase the spectral resolution of LOWTRAN 6(") by developing a radiative

transfer model and attendant computer subroutines for molecular absorption

effects. Specifically, the goals for this effort are to:

* develop algorithms providing at least 3 cm. 1 resolution (FWHM- full width tit half maximum),

* model molecular absorption of atmospheric molecules as afunction of temperature and. broadening density from 0 to17,900 cm- 1 ,

0 treat the following atmospheric molecules: water vapor,ozone, carbon dioxide, methane, nitrous oxide, carbonmonoxide, molecular nitrogen and oxygen, and

• integrate the new algorithms and subroutines into LOWTRAN 6.

The upper limit of the spectral range, 17,900 cm- 1 , is determined by the

HITRAN line atlas for atmospheric molecules.(2)

1. F. X. Kneizys, E. P. Shettle, W. 0. Gallery, J. H. Chetwynd, Jr., L.W. Abreu, J. E. A. Selby, S. A. Clough and R. W. Fenn, "AtmosphericTransmittance/Radiance: Computer Code LOWTRAN 6," AFGL-TR--83--0187,Air Force Geophysics Laboratory, Hanscom AFB, MA 01731, (August 1983).ADAI137786

2. R. A. McClatchey, W. S. Benedict, S. A. Clough, 1). F. Burch, R. F.Calfee, K. Fox, L. S. Rothman, and J. S. Garing, "AFCRL AtmosphericAbsorption line Parameters Compilations," AFCRL--TR--73-0096, Air ForceGeophysics Laboratory, Hanscom AFB, MA 01731 (January 1973); L. S.Rothman, App]. Opt., 20, 791 (1981). AD762904

A substantial fraction of this work has been completed. Moolecular

transmittance through the atmosphere has been modeled using standard bandI

model techniques.( 3 5 ) In the present approach, the spectral region is

partitioned into 1 cm-1 bins for each molecule. Within each bin,

contributions from lines whose centers are within the bin and from nearby

lines centered outside of that bin are modeled separately with their own

temperature and pressure dependences; see Figure 1. The absorption due to

lines within the bin is calculated by numerically integrating over the

Voigt line shape(5). The Curtis--Godson( 6 ' 7 ) approximation, which is

accurate for the moderate temperature, pressure, and concentration

variations found in the earth's atmosphere, is used to replace multilayered

paths by an equivalent isotropic one. The transmittance algorithms have

been coded and integrated into LOWTRAN 6. Comparisons to "exact"

calculations with FASCOD2( 8 ' 9 ) resulted in a reexamination of and a new

definition for the standard line density band model parameter, l/d; this

new definition lowers the line center absorption by taking the finite bin

widths into account. A complete band model tape has been generated for the

molecules listed above. A preliminary version of MODTRAN, this moderate

resolution LOWTRAN 6, has been delivered to AFGL for evaluation.

3. D. Anding, "Band Model Methods for Computing Atmospheric Slant-PathMolecular Absorption," Rpt. No. 7142-12-T, Willow Run Laboratories,Environmental Research Institute of Michigan, Ann Arbor, MI (1967).

4. A. J. LaRocca and R. E. Turner, "Atmospheric Transmittance andRadiance: Methods of Calculation, Rpt. No. 107600--10--T, EnvironmentalResearch Institute of Michigan, Ann Arbor, MI (June 1975).

5. W. L. Wolfe and G. J. Zissis, "The Infrared Handbook: RevisedEdition," The Infrared Information and Analysis (IRIA) Center,Environmental Research Institute of Michigan, Ann Arbor, MI (19851,

6. A. R. Curtis, Q. J. R. Meteor. Soc, 78, 165 (1952).7. W. L. Godson, J. Meteor., 12. 123 (1955).8. S. A. Clough, "Preliminary User Instructions for FASCOD2," Letter,

AFGL/OPI, Hanscom AFB, MA 01731 (1 October 1985).9. H. J. P. Smith, D. J. Dube, M. E. Gardner, S. A. Clough, F. X. Kneizys

and L. S. Rothman, "FASCODE - Fast Atmospheric Signature Code\Spectral Transmittance and Radiance)," AFGL-TR-78--G1B1, Air ForceGeophysics Laboratory, lHanscom AFB, MA 01731 (1978). ADA!57506

-2 -

- Ii I

"+-a- Sw I I

0

C-0

Cf)

-3 -U -t 2 3

WQvenumber (cm- 1 )

Figure 1. Spectral Absorptivity for Lorentz Line Shapes.Curves Generated for Products of Line Strength andAbsorber Amounts Equal to 0.1, 1.0 and 10.0 cm-1The Halfwidth has been fixed at 0.1 cm'-1. In theBand Model Transmittance Formulation, Absorption Dlueto Lines Whose Centers Are Within a Bin (Between theDashed Iiines) is Modeled Separately from That Due toLine Tail Absorption (Outside the Dashed Lines).

An overview of the MODTRAN code is presented in Section 2. In the

.•ubsequent two sections, calculnt ion of the band model parameters wnd the

transmittance formulation are descrihed. Modifications and addit.ionls to

LOWTRAN 6 are presented in Section 5. In Section 6, MODTRAN output is

analyzed for the purpose of validnt on. Conclusions and future plans are

-3-l

The following indexing convention is used throughout the report:

Symbol Indexed Quantity

i spectral binm mcleculen temperaturea speciesft atmospheric layer

The species symbol a. will refer to either an individual species such as

water or ozone, or a mixture of molecules such as the uniformly mixed gases

(1ING). The atmospheric layers I are characterized by a fixed temperature,

pressure, and molecular constituency. Constants used include:(5)

c = speed of light (2.997924580x10 1 0 cm sec 1)

k = Boltzmann constant (1. 3 8 066 2 xlO-16 erg K-1)h = Planck constant (6.626176xi0- 2 7 erg-sec)N = Avogadro constant (6.022045xI0 2 3 mole- 1 )

Ts = Standard temperature used in the HITRAN line atlas (296 K)To = Standard temperature used in LOWTRAN (273.15 K)

4-

2.0 THE MODTRAN CODE - AN OVERVIEW

MODTRAN is the name given to the LOWTRAN 6 code affixed with a new

moderate resolution radiative transport algorithm. The new subroutines

have been written in portable ANSII standard FORTRAN and constructed so

that their interfacing with LOWTHAN 6 minimizes coding changes. These

additional elements do not Interfere with the regular operation of

LOWTRAN 6; rather they represent an additional capability for higher

spectral resolution. The only modification to the input data sequence is

5 - 10 KM AT 15 -_ US Std Rtm

1--

4-- -ciC:

C-

w .-

5 cm' 1 MOOTRRN

20 .OWII)WN .

Wovenumber (cm -1)

Figure 2. Transmittance Curves for a 150 from Zenith Slant Pathfrom 5 to 10 km Through the US Standard Atmosphere. TheSolid Curve was Obtained from MODTRAN by Passing a 5cm- 1 FWHM Triangular Slit Over the 1 cm- 1 Calculation,The Dotted Curve is the Regular IOWTRAN 6 Result alsoObtained from MODTRAN (20 cm- 1 Resolution with DV Set to5 cm- 1 ).

5 -

I.III 10(lti it j 1041, r-11 ~ c JBMOIl. Wir it ii ;W added til tilt' t'ud of' CARDI The

L ~ ~ ~ ok1 who i-, not ju rs diii tis u'1' tht- MOiDllN oIpt i (mi S iflip l Iji~5 ti i

pl~liimel ule anld pre'pires hlis i li1o111 i I e ill thlt Usual wil tO lit!' p1rogra t henl

t m I: I ionsii i u it im ri'na I1 ninth. aiiid doet's no t reqol i Icit( ss t o mn c x 1. vim r1u

dat ;I r i Iv (If bailid mode I pa rame t Oers Whenl is i Ill' thet( baoin notle- I opt ji ul

(.limkUII I) atccss t o t lIIi " dlt a t, ie Is v e i 1ed.

A 'onip~iii-isou io' MOIYI'RAN mid 1,OWITAN U denoittdralt-s t he, imrporit an'- of

thc 11 ihe I- 'o I olm Ii i o li ;I o ri it lim F. lJ'iorv 2 shiow., t Katisnmi t taonce. cil('ulattt '!

ait !) and 21 v'i les I o il fo r itt l ow it'. it itude s1 ilit ptit Ii t Itigh9 the( Us

St ;noiid At mospherv. vit. ,i crar (FWIM ) curive i'e-sti ts from oh1 iltnt erna

dv),rdtio1 it i11,I iit tra~tismli t t aimcv Wit i aI ti-iaaii. 1 ;iih slit ti lict 1(111

mid tlh 20 um Icorve- is thet rcgluo r IOWWkAN G rvsol t it,, ('alui l itvd Ity

tnvlA ithe 5 C in maI I cuIi at i ow, we re del' riode d t v I lie- 20 rin LOWMFIAN Ii

ualci l at.oimi, t lit ttvrallII~tttei would be quito )d. Howe-ver , at lii ihier'

rt'solIki killt, liti'-t (1v di ot~ ionis fr'om thet- 1 owvv resilti o -.i 1''I Mlt imiis- (t('tuli

I ii par1 1 t ti la r. tw h.' MTh1IZAN cal cil at ion resulves 1 'stroctujre (it iv;itt r

I j Ies Ie l ow "10 no aud( thett haliti (tiit er of, t i. :'lm'i at 2220

i'6

3.0 MOLIRCULAR BlANJO MOD)EL PA) WTFRS

Tlli, laiti.ic assxumllit ioni of tilt Iaiud muide Icontillep is thi1w sp~vl , 1.1

I1'kI)FVSVIli ed hy at sijuhli e athrmnait ical mode I !I iIC tal jiilIiLi

so thatit liehil of ['ilft I iiitve mo11 I.I iis m ditt d- Ollj 'I thheiihe i

We i-i.' chonsei a hliln widith .)I out, wavclnumli'r, 6v I e-II Lim-j diiata

tr oni, t he IiI 1Ul(AN t jI ' I I lie AFI( . Iilit, at ItI ts, I s '.~I1 % cd Cit Icli il ct t Ie -lid -I

mudi, I pair.Inet1 els -4 i lice i t Ilia (tit it for I i Iles i Ii t Ill! Icviptielcy !'&iii1 : o)t~

1 7h00 c ir 3 7900)l) bins mxii -t 1W Cel iiit( t cFi i7.1d, Fo luz t ei'I cott I ihtit iill'

Riol 1cu I e, t vilflh, l'dt ilt.r: detuip iideuu t Mli-iorI)t j oti ctit- fl c utg t iic 1 n i Ill (it! IIF i t i es

ill Lill' ?. i. il it I i liv wid It~ l Jimr aiiit-1 er illt. miis' It o 1 ) I-i alm k k - IA z t i -~ ia he

1[it i; Whotie (cell ri cls I I W i t huti a1 ~it Venl 1i ii; a S Ill; Iv tvilpeiat olic di-livildv~il

ahsou-lit ionl (-nf .1i c i cut piirairne eI v(let erm I lie"u- t lit! eult ,i ibut i noi to at hinl 1-om

It( 1i , I t iI s (11 I I liis c:tuli t i:ed i it nke.1-by bhin s.

Ill thet liiext Iwo stihtsectt iobus Ol tic lcl iiit lo of th lie in-c 1cciill. blana

moodeic parianet ers is d-sr i bed, unid thlt lft,iliit t iitug of tilie (tiltit f ile is

d i scis~ i ii the thuti d sit15t'et mu o. Fotiit mis uised t Ii terli'. tile. piluliliet I's.

iPcef el aloill w W;li : I dsusd15 itl ofI I lier(l'ltih't kill ti-m~ltivritu 11t'1ai

I~ rt~-n _ c~tr P r-imt erns

tLael w~i~iiiiiiiltt' II bill~ U~l~iO to it 1-i I'Jfl ~ lan cnail

p~iiaintlm,111- lo m(.leclil es WithI lieis ill that iiittc viva I . ~ forin cu1is 1

Whic i li hiil lfodlOh p lramv1.vI'ci jilt ill icluded ait'.

Iic' meWl Nc (I, COirp ('If 0'1.' ft Nji ~l (m ai'a .1iii

Ir

(r51 j20() K, 225 K, 250 K. 275 K, 300 K) .(A

I it t he' lniid motiIn' siihrout ines it I i near. i uit er'po 1,11 ifll is tuscd to calunilate

ilbsoript ion coc Cf iuicints at. temperatures betweein 200 andI 300 K. F C)r'

t vnlpvi';a 1ills lei W 200 Z1nd abiove 300 K, the extrecme vilii OtiS (S/d)Ian

S/d)' , tuspect i vely . are Iused. 'The mo lecul ar' absorpt ion band modelI

1) i I alcti t's aiI-(- cill c~il I ut ed I r'oi I lic i Iid ivi (I la 1 i lie st r Iitls.

(S d 2- Sjn(l (2)

j

liev uSj ('1'11 is t hie i nt egitit (!(1 I i i(, st reiipl-. It ut temlierat ure T, (if' thei j It

1ille oft Moiecuilet M ill il lii . 'Fihe I Iit( sI leigth it t ani arbi 1 i'iiy I empei'at he

(-.ln he scaled3 from the III TRAN lin e s trenogth Iila. i ts st andir d temuperat ur'e

Q(,Q v(' Ii)'

Q'k r ('I'n Q v*Ir,

e-xi( i(-v E j Ta Ts4

enp ey*T)exp(-! I' ) ','j (T,,) . 3

whiere Q,, and( Q, re thev rot at i onal and v lu'itit 1114 partit1ion fuin-tions, and

I"( t .1'eherl'y of t he lower t raolsit ion sta~te.

A ee(il Iision lirolcid-iid 0w L~ireiitz i. iLi width piutramet en (f,),) is' del iielf-

at Si P (1T - To - 2 71. 15 K . I PO =~ 10 13. 25 nuhuir's) A singfle vali- *:an hI

Si nee(d lo'eU11Si' thti' m)L S;til'v .iid ti'ipiriat il'ý thi'jwndiwiti'iiii of w L'loriruit ,Ii me.

widtlhI is vilslS ii Ylundi' Iid .

C P

Wlive1- 1t. lo. ixpoiiiiit X,,, hls hlnen "(.I to 1/2 (to ;1I nn ieo' except ('2 byol

wh i ( h x o T I w Cy Ia l I II I T1 le It1 )i Ihr itrM t E.l c Cit (711i l ut ud (I a I i nv

st ri'vligt 11 wt. iglited aVul ;i1g( liv(.- It(li t aill flilt ed l'Vureilt z I ill(- widt his (Y .) j cr s

Cyo (T /To 0 )xM I~Y~(I, Cy( 5) s)~jIQ7- j~(')s(ý;] tlfl

i, iw ke ite aibs(3tpt olou couff icieuit s I hev Ii i] (it-Its itI y baind modelI

p~ironete I /(li )D ( rn) ar'e ,itIi cul ated at. the 5 r eie eiprtile i

jo II (lpol' ill ed Wle-il liserl by I ht Ioitld iiiodt.l sour 0111i i Iews. Th(,. I tille (It.[Is i tY

(11I!1 Ii ied by

N N

(I~ ~ /d s

fhisi uef illit ioit fori the Iii te bSfiiic)t.", WhichI is d(letiVt(l, ill thf lie teildtx,

diffet- fromt the usuwil def iii it ionl i [ivo)lv lg i skli over the squiiir&' root of

the Ii ille st ref'li" O .41011

I. 3.2 LineTail Parameters

'Ihle 1 joe li al Iloirilmtnicts colnsist of 1 i Itie coilt r jhot iolis f romtI I [ie's

locýatucd cut ::d of a 1-f 'Jeri hilt but wit~hiti t 25 cm The I tite tall

asr iolV couff ic ent bauid mode-lI piwircimt er3cis(R mlit ir

detIe rmi tied by i nt egraut iig the Lo tent. I i tii shape. over tt iisi l it e l'Vit

C. 2. (1 t ) Sd 1 1,~ 'k Ifi( i )~ AV (7)k i-251/

where t li del t.i Bitict mul SerIves, to exc:] dt-1 tlii, k- i terit from Othe stin

the Iiitie ce titer cootilr i t ioi) .For mnolecules olhor I.halt 1120 anld C02 1 tail

cut V Ihutj(ioii5 ht-yoti(l '15 ciw -' frtont a liltie cenIter- hive bweln assitiiie'l to iloul(f. it

tieg'ijjf iulec0t1010011tinn to the. ttoi-r'n.jltctare aitd withinl theo 25 cm'' limit ;I

1), C. Roburt oi x I .S. Ilrit ii wid R . "ajm s Add it jolt if it 5

CMI S pie ra I Re so lut iit-iiilbu Model Opt ioni to I OWTIIAN !)," ARI IMP 2,12AResearch , I n'a.c , It ill er i ci MA ( October- 1 'J80).1cial

11 1) C.~~ ~ ~ ~ Rlwr sol iti S. 11 c.1t iii, P. Ilto lnes J. W11uitiulet 1 ilit .UMg~ 5 ii iitud Modec I Opt i oitt to IOWITAN !i, ApiJ) I. 01t 3z 2.318

(l~t~l%

C arn faic t r , f I (k i ) AV equii I to I has hec(It ass ilmed . The( usua I LOWTRAN 6

fiW;l 1 r CMI t i ouurrr coni'si3ts o f tIt(- tailI conritib Ut ionis beyond 25 cm 1 lu

eI raoae vil(- of tlt! cont r'iution withli 25 cm " (for -;o1Its'

1For C0 2 . the continuum from thet FASCOD2f"O has been added to Cii to

wxwunhIt lot' thle lil uicnt i ihutin Ofl5Cromi linez bteyond 25 cm

hcv TSCI 4*C vitatnh(j~-1 y-- C(V 1 (6

Hiere ,C ( u) i s t iet( frequielicy inv jti I ii id vaiii fiCr'om lASCOIWs i blok ditt a

/F0 Iu oh120adC 2' the( valtic of ('1 hiii, also i 5tIeiii reduccf2( by an

I illifhl l1it e1;i 1 lo the 25 cm 1 f rom 1 ine cente(r cont r jibut i oi s i ne thIi s

cont. I- hintt ion i s i it(-cI udeld a I ready in thIto cant i nutum (ati it 2) 1'h C1 art:c

take~n to hov prlipor~t ionll to p~ressure

3.3 Parameter Data File

F li-callrs 11! thre liaite irmoonti i (it (lit t i t he I ic i It forI the' bitrid marie~v

Iil'imt't is an~ cXUI~l'~Id fill( wr'itte eli i binairy formal which alilows for

qipiIci'r ;wcvs55 duiiring the cal culat ion. Each ent ry corrusepnnids to itI cm1

iii eivoi 1 lul cord iiills, t nirolecuri ii pilraruni .r'set .ZerIOed( dat; aT- Wi'ont

bt ore'1 for' mii Iecu I v:i cont. -i ju t.o inpli no I ; it(,- to it IT ivteir I it erva I . Tit!rerefort-,

ilit ol'm~it 11111 witui cli I ilbel I he tt pararneit 01- set milst als I e "'m it( ored.

I. ii. Ii '.~ ,ti (It ;1 ra -inieiii eiti Se I,, the 1 ji Ii n tl wrfifti i . rom the iijii

liirniei . lii lt- (lit ii) i lit )f' tirel it- eri./i-i- is ca18i ly cii ciii atvoIl

Irlid itIi I I i iocs whont. (:0111 ei; lii) I Iii iti I thl I tI peneill i lit a'v~I- I I v V A /2 V'

t V runt r i huit v t oi )i it Ii

I .S. A. C-:II tiiihi I'I-SOii il I toinirrirri i vl.i Oh .

. -.-. o

k.0

" '. .". .s .. . . . .. .-

A iumber" m ~etwueeo 1 and 8 desigmnat:-s thiet mno irecle Except for N2Z We"

follow tlie conventionit set by Ihlie II'TRAN tape (2)

i 1 2 3 4 Ili 6 7 8

mo I ecu I e K20 002 03 NO (20 COI 02 N2L ' L

The. next entries in tilth paramete.r set are thlet molectular atlsorptitonl

Cttjfkieents (. I a(tm 1imagits ) calctilet.d at t he five referente lice

t1 PII(t'ltlt li',"es T'hlu entrcies are followed by the STP 1,0iclitz hltf widtl,

(-Ic)o, mutltillied by 10 and stored as an ilntvc-p2. I.i tie sjnici og pa t'met ers

11d toe1- the f iVt' 1'Aet t'rCllCV tempeur-"t res toinlpiete the I ille cenitert' larfimt. er

se vs1

E.1ch pýaT'amelt • set foir line tails contains informatiton on onei. or two "

molecules. These line tail parameter sets usk. the same format as tlie. lint!

ceitt 'r lltriute + c` setpst. Agatinl, thti. first entrty is the binl number i atnd tilth

second entry is the molecule designation m. To recogn zc, that thesev

pa-ametet sets denot c. I ine t!t: i I toltrihihut ion. , the ir molecule ]1s aeI-t' ]u i

offset by 8

m 9 10 11 12 13 14 15 16

mo I ecu Ie 1120 C02 0 N2 0 CO cif 02 N_,

Tile con0t iimui,, parameters, ChI, are stored In place of the (S/i lln The

moletu ulra i,:sigelttttit anutd euolilillpn parameters for a second molecul. caru hle .l

unkl, red ill pl tre (if the hIalf width mid (l/d)n pInral~meters.

.1

' -i"

4.0 BAND MODEL TRANSMITTANCE FOR~MULATION

4.1 Line Center Transmittance

The haind mode 1 t.riinsmni ttaricu formu 1 t ion("-101) deve lopedl tor t li 5

CeR opt ion to 1.OW'1RAN 5"'~ hitzz beent used to cruirttv a itiodet ate resoluit i n

opjt ion I or LOWIllAN Tl. lhlxws in u;ed tfo CiltIC11:11C 11104(lecl]itl

trarim ittdlne (110 luIinte ceniters, ib basied on at stati st ; !o mo~t:it iir a

fiII itv 11untTTi. (fit ' li-S il it spectrlill finter'il . anld is; 1ivenl by

where -i is the trainsmilt aiice, cWt li- thle piath ilvvle il jut'v

equivilenit Width lot (it'll line tregt di st rHurt ion ill ia spedi'al interval,

anid <[I,> is the path averilged effet- lye number of' I ines in th I( hii

<11> -AV <lid? .

<lid-' iS the' jlatilvrae line- qpacinig.

For' large In(S /d) andl All f ixed], lilthe t rasmi ttillice -implif ifes to at

more- ree-(og,,ii ahlc exponential form given by

I e.xp( <WS1> <i/d> ) foun <11> larige (1)

Fur t hec mel I cu I it spec loes and rel I t t yeVk I y I ow t einjiveriit tLlE' ('ico.ulit er #'(t i 11

t ll! ear I- t s a t [Do',[Iiere the ave ragt! 111irumer o f I i leiii i nt at h iti is usuirI I ly

Ninal 'if 111t;1t lilte powerl lilvw t rihism it Utnet! ap'rox imat ion i s p re 1''r red.

'I i-ire ai-o many iipprTox imiiationb oivaitilab l for- calvieaat i rij, the ie(q1ivolt Irt

W idthI of at 'Jo it I int. shcl1w: ifeei onles are( voll il fun dIi f lo-el lIt

13 . F. X. Klre-iZYs, 1:1. P. siet t It', W. 01. Galle Iry .-I. It. cltetwyiid. r, It.L.W . Abi--i, .J . F . A . Se Ilby, H . W . l".i-nn, andl R. A. Mcd hi Ot crt'y."At mosplivr' ii u Triii~sTT itIt ilrTU v/Paid i once(:( Comput or Code~ LOWl IýAN 5 ,"1 .

lF ItR 30) 00h7, Air, la nce Geophys ic-, Larhornaliw y , lb uscorn Alit. MA(01731 ( Ut-liruary 19B)i0). AI)AMtt,215

12

rej~ines . )op pIe r oi coIlI is ioti bro i~derinrg. weak Ii ine ort sI l'o gI~ 1i iut' t c

oweve r no s inrglIe approx i milt io aizi adequate for 1he rangei (it p resitir'cs and

opt i cii path liIengthis encoun l.ere~i in ittmosplIttr i cIrrrlri olC cii at io mrs

Rdhet- than incorp()rati ril, di ffereift approxinrat.ion's' we directly eva 1 irate.

thre exitct express ion for the cqriivul Ir!t. W-*,(tl oi(f a single I ira' with at Va i 't

iline Slape; <W1 SO is givell by

1 V j ed> ( ( Sit/di -V1rr2/iT F(XYV(<YU/d>)dX (/a

0

CD

F (X'Y) Y exp T'qd 2) (IT (141))ir 2 , (0(T)2

V = 4112 0 c/(1>/('Yd/d> (41

Whe~re F ( X Y) is I ht P vo it~ Ii tie shape func t oll ,ind <ld/id> Bi( ' c' > ar

pa t h ave raged Dopp 1) 1e r anid co' i Ii s i oil broadened I ine shil[iC pa rame t v rss

resp~ctI i ye. I Th 'ie facto r of 11/2 ini Eq . (14 c) cor1responrds to tire huit1f w idth i

4.1.1 Cturt is Godsuit Approximation

11;thi civu rages al ['e ca I cul I iatve w Iii t 11 ite Curvt is uodsonl

aprpjoX imat i nit. ((.7 14) 1,11'i is lpprl_1x iR t it,,, rupi acts alln ririrrrrtigerw1ous pathi

wit h a homogeneous one by us intg ave'rage valu es for the varci urs ihanrd modle

14. C. Ii. ILudwi1g. W . WMitImrs. J. F.. Rteurdor arnd J. A. 1, Thomrnson,lianildiook 0 f I r1f Frared Rild i alt io01 f ruOm Combus t i onr Gasez;, "NASA Re~portL

parameters. The Curtis Godson approximation is very I,.)d for pllhs where

lilt ' t (illlpeIit ur'e or species gra'dieni is not laliticularly st eep1). This is

certainly the case for u i-mosspher i c paths whtiere the lempcrature vari at ions

f our ait irt ra -y pt Its f ill wit ii n tic range uof 200 to 300 K. The totnl

Ioptlicl depleth is ;I sum over con. ribut ions from tho individual I ay•,s and is

iSo/d/ (1 t

I

Ltwhere (bu ), i s t ihe ieni.enwtal absorber amounint f'romn 1v I -;lknl kS/d)J is

the 1i1i1in lioi ( t f -ilcue.fici l c ililt mbodel piollf';111am ter Ht the " eni rt' Ut e die ittV(d

byt ltie layer index, A. Note, frequency nid species indices are implicitly

assumend for t.his and t he stibseqmuet equatl ionss

Tih total opt ica I depth is used a.s thet( weightt i ng funnt ion in

(il Iciu ;lli n I 14 tu llell p h Iiverages

<lid> i Soildld) SdI(A) 1;i. ~i

0Y 1.d> !(','1 (!/d) (S/d) WOut (17)

<"Tdld> tSl/dtCy () 1 tid)tI (S/d)m (ul) (18)I

'liu'SC boiild iiui(l p;lil;ali(cter, We're defill.d in ti lpe'reviotus s 'cl iou., and1( ('Td) l

ij, I h, al •tip, I Dol I eV w lilt (rc. i.

Cy (1) (•,.1) i.c ('r)

-..- . -j2 (1n i,2)N kT/ tM (it .))C

With M equll Ito 1liet Tit • lecul I - nmlass ( g/miolte) .ii~d a lIhv slpe ies in(-udex.

4

4. 1 .2 t1nii formlIy Mixed Gases,

'Emis far * we. havye imp!lici tly as stimed thain t he spe!c-i(s ain 1,0W'IUAN are.

Simply mole~cules_ n. lit actuality, LOWTRAN b assumes a untiformly mixedga

(lMG) Witlli th-1k molecules C02 N2 ) , co, c!I. anld (12 comb itind accordilug to

&±tnitsplueIric mixinig ratios,X ()

11101 cuiciv C02 N2 0 CO C'11 4 0

xm 3. 3X10 4 2.$x10'7 7.5x10-8 1.ixIOt-C 2.095x10

IBalid modelW pa r'imi' t' rs for th le 11olurm Eu ril y mixed gases, * I"2' anid 03, were(

calculated ind~ividua~lly accor-dinlg to thle erjuat jotas from the subsect ionl 3.1.

Fo r I li M ING ANe ind i vi dual aiid model ptz came I r s we-re. comb I ted accord i tg

to their ratios3

(S/d) 1 UM6 7 X11! Sd) 1 1 (20)

I d) 1UMG (S/d),11I U 0 2 /m Im 2 (S/tlLI~m2 W ldn 2)

Ini

I 1G Xjp MI (S/d)3 (22(S/ti) n'1G 2-

1 15 . R. A. McC I it L divy , R. W. ~~~~I,'-iiii ' . S!Ib . F OIz, ai

AFT 2 0 97 i o ce CtIi 11 'ý(- RPS tUC I Ii 0 - k-)t H11 12Nk/ I(23)OAFB, MA 10731 ~~~(24Ags.92.A737

iti'the( sui-scr iplnm ilndexoes mu lectil,-s (sk ippi ng It (), 0 rd i h

sum); mibý;crip zero lias added to ( S/d) for the stanidard teMper~atkUre

ii ' it er ci 1 hse r~d ilh~p oim (:n f f i c. i vi t.arnd tIt( he iperse r i lit ING6 1 ilbe, s t ice

Ulii furn'ily MiXed gaISeS spucies. Agaiin, we have suppressed %isage of a bill

42 Line Witiq AbsorpjA on ,ae -Icou t~ll:e

wit i ch 0 or i 9i ct e w i th; n -, g i voit ictraI iitteIv n d I .tes es

onily thicll fraictilon of' i( it r f l.w ih f l t ih l the Iit l-a ie

hinc iidcod Iil the concpctat, ion of thit. equnivaleýnt widcth. This approx imatio iu

lrklilsonaic I e in t he stronjly alisorhincg reirion (if t bcand; however, because the(

ialhsorpt Vit ly is expressed iii terms of the- local line zitrength dist-ribution,

it tjoco(iflt-5 it poor itHptox ititit i oilIII e ieiols iS wher the ta il (co~iuc ' ut io011

from iviidrhy i ni-s doem inat e the cont-rl ut ilons fromI Iwt-k ori nonexistent lines

Wit hiii a giveni cit icvi- 1 (h jut] This I yp i ccal y Occurs i ri 11th ceniti-l acid fuit-

w ings of it bondi ( i e.. past the hand heca'i), and also inl spectral int ervals

c() i t o i 1i lic uccc 1 i Ile ,; wit jci ( It a r ill thell vi c~i tiiy of 'sl r Coil ll IinI es Focr t ee

si ~uiltimns, lithe locial ablsorpt ion is dominated by the accu~mkia~ted tailis of

lilest- -ronlgec 11i als or il'ilatiutIg outs5idie tlie( I nterIval 1,11v ef fed of, linet

w ing aibsorpjt icoil i s i Ili, 1 tide~d in the t ransun i t ta;ince by adjo iin i nl all

IXcuu~ iW tl 25e.

where [Cu] is, tlt(! lct al cont inuum opdtical depthl

Thel'1ayeru slill~crcuipt R oil C labvhe ll othi thle prues-s~urk anid t.umpI'Ilt iiIur

16A

5.0 INTEGRATION INTO LOWTRAN 6

I Itt 'i'at io (il tilt- MOD'lTHAN subrout i ies into L (W'ITAN 6I has lice ii

aiccompli shed w ithi m iniuim~ chainges to thu or'ifyi nait codr.'1111e intiierfaice

he tv1 teit I.li ret',iil. LiItWIL{AN f; andr tilie MUDI)RAN ( 1 tIoIL S ttio' is mao houpl ci calls-

to two~ subroutin tes ill I.lie LOWMRAN 6 subriout ine T'RANS . A s i ige it , i to

mburotitim B eIMDtATA reads tilt f ir'st nicCU 5fatty wnvit-nimbtl c hiock of hand modelI

~d ~iliI.'lS tili calcl COI i.Ps watvelitiiitlr i Iidept-Iitvielt. qioieitti es. F'or each

waivetimtiirn caills to siihiout lie BMOfl are mode ruict toi- hin iajl i zo I m and(

t-hunr ardit i onri;11y Ini lie loop over axt fluS)hc i c l ayers timht calrkl t es- the,

MTolIecularii t eatlIistiit tIucce

5.1 MOLOTRAN Subroutineg

Thei MODTHAN stibroitt itines are descr ibed inI Ithis sec t. olt. lit tot a .

these zieven siuheoutttiiie: conitalin less than 750) lnt-is- of code.

5. 1 A Sibr out ille lIftAItf

Stihr'uu it I ilIML)A'I'A is call1ed tnce cit,-.t en Iculat i oit by -itibout i ne ITANS .

This subroutilne reads thlt binary band modti4 tape lvad'ru. m;-kes the iniitial1

c~i iI to -uitborit item UCNII)N, anidci calv ates W~iveititrider inde-peitdeiit qutn tiitIe s

for Subscquimtvl. use' by Suhroo t.iii: htMOD)

Aft er read Iiig thle t'ile. header , BDM)'IA cite kN the wavei mmlw raneiit~

b 1 y tht ý' to --e( if it fallis withlin the rangev tii t he file. I f

1tit( request ed jutex'va I is totatl y (utilhide of the blaid mode i parameter

range, t he p cogaina ýslops wit h ano erroe mcs sage. Hiowever ci f the requiestt ed

w~iveioumbfc I inttervalI is only pactki a Iy (tilts i (l of Ilt 1w t owe-d ei' thle

progra1,;m readjalsts the~ upper.1 and/onr lower Waivcnui trrmle to thtilt-fe t' ; Vil Iltes

autd ploceeds w jiti the cot cutlrtionl. Th' lie. vt i-tad bt itt eniit'it sk ips ovvt

waveinimliec hin;cks, in the(, fill- unti ii t Ireachest" thle w;)vei111mhcc[ bloul%

couitajiitg"F VI. t ilt- ut it i a IWiivellittliret

I 17

If the option 1RPT = 4 is used, the next calculation has the same

atmospheric and path parameters but a new wavenumber range. in this case,

the call to BMDATA only rewinds the file and performs the initial reads.

The common block /CARD5/ wa.s added to pass the value of IRPT to BMDATA.

When using the MODTRAN option, the array WPATtl contains the

Incremental absorber amounts for each layer. However, WPATH is not

calculated by subroutine PATH when the path lies completely within one

atmospheric layer, so it must be specified In subroutine BMUATA for use by

subroutine BMOD. Finally, several wavenumber independent quantities for

each layer are calculated in BMDATA and stored in the appropriate arrays

for latur use by BMOV. These are FT. the pressure normalized to one

atrmosphere (P/P1), and the Cemperature indices for interpolation of the

band model parameters.

5 1.2 Subroutine CCGNDEN

Subroutine CONDEN makes binary file reads of the band model pLrameter

tape. The tape contains information on each molecule separately. To form

u ..... pr.........r. +.. reqas ft.-,d h, th TA.OWTRAN 6 structure, subroutine

CONDEN also condenses the molecular information

Data is only stored for molecules contributing to a spectral interval;

therefore, It is convenient to use recurrence formulas to sum over

molecules fox, defining the. UMG band model parameters:

(S/d)n(m'l) ( Sldln(m) ÷Xm+1 (S/d)n,m+1 (27) i

(i/d)11n1' ) I (S/d) 1 1 m( l) 12 / (S/d)n(m) 12 / (1/d)n(m) b

Y, 5m2 (S/d)nim'1 2 / (l/d)nm~i } (28)

- 13 -

c (y f I; ('0S/ o 1

m On IH/d) (m 'I) MIR'~ iSd (S/(l) 1) i, ( I

whiere lh SIljI'vsII jilit s demliit vt h I h v' I 'Ve at' rtecurvi'ic.te i v. 011 ii) aid Oni

arviil the ol d and new Vailue . re(spi.et ive.'ly (if th11 U.MG banid mode'l aiter

NaiI.te it -tse t -(I u I aiols it I-( diriivvd fropi EqN. (20)) t hrutighi(4

I.. Sutirokt i lie IMOD)

S iiokitt ilie 11MOl co Icilat t.. thet( iim . tralB wtlice tis j o thle sitiltist i oll

biand intdi-e dti'scr'iht'd in Sct jAolt 'I. Ani initial izat ion callI is mado olltck

evei'v w;ltvIt'tiuriihr' troin s;ulroulit il orHANS , ott .1 t~tbsiqtlieii col 1s v'rc madel tar

eaIch aityer- tO calclt I lieZio 11 tran-TSM it t oil~' O , Ith tirstI cat.l'i cil I I

CONIIEN il thle iwx~t tJOC of btelL (IIliiO mol !oijiii 1m !-Ied h rad (.1W Is

It he cokni Ill, var i ahit Iv 1n ;11( hent, zerois out qoiolit it ir's tar vlie I rnifsfli it t ance~t

c~i I (:t l lit 1(1115. 01h. t hie stihs.eqiieitI cai11 I I MODI c~i leo: I at I estlt 1, oi t at.t I olt i. ci i I

depthi and (ransmittimoce for each Ipe lls. Th colittIitilum cuiit, ilint ions due.

to1 1ai Is of I i ries or-i gina ti ng outs idt! thle I (-m Iiii. erva I ioreq joll ided ill

li ct'eilcaultit ion. itMOl) retui-lII. ilte t ijiismlut tai oce I ot the it. 11lfrttl;I slirc tif's

to IR~S. T'I. rutis'm it t. U1it!S ure s. IL ored inl Iit-' i't'ea, I i I- LIITRAN 6

Iratismi ailt:t arraty(')

T[he call(s) made 1)y TRANS to ItMODl for calcuololt mijlp th tt' II.i'nnI itt alice is

I.'Mlddeti inl a loopl tvt'r- at inspdit'r I layers. Whenl only tr.tismil it 11 in' ' is

to leoI iat!d, 11 hI'M1 loopi ilt TRANS eons ists ot ante piass . Wilkeii itiilliosphici''iI

i-Ad iiit i ull i S cil -It Ietll th.e,14 TKI.%NS l oot is over' oil layers, tit'aoist' tI i''

r'ad i~it iotlt dtelilfds Ol t lie! i itre'iniiti.It hI clikmtit)' iii I liiii mfi t ! mic fun LI- c-it i

I ayi'r . T1h1 Cuort i s hudsonl appetix Ima t i on for th lit -3t t i st I calI I'nit miod'. I aI I so

t'rpqui I tes t le ccitt r uit bil o' 1 f'Inn Ii'eht I tyler feor t ri'isnt i t t~niwc cal I oi I' i til.

Ilit keepl i hig w it 11 h i' philI usuoilly of min1 in un i z i jg cli.ijigi's I o I hir l;is ic uLOWiRAN Gi

plir tn Iiis I iye!- loop11 is 'hate ill lIMOD tof. r~imisni ittac. onti tily

N:

,iii u ~iat i 'ls ~iid inl TRANS ItorIit C(I iait ion CitI icuI at ions. Thuss t III(. e loop

ill IMOD conrisi si s of just onev pass whvin 'I ANS is loop ilill Over a i lOY"IrS ilnd

vi (t' vursiu. Sulrout ince ItMOl dt-etiiiiues whe-tliur the loui' ill TRANS is (Jvi'

uiiP.- I ~iyer or all l Iayers ucnd 1.heni adjusts its ownl loopl ar iaible accord inu;I y.

Thw abslus'iii deiis it j es I(PATIII ) caiicu lat 'd in iA)WITAN arv C-oIVo i-te l( o

111 kie oi t:i of 1rn (imag~Jts in BAMOO to be~ consist ciii With t het uniti s of II[ he liull

mmolt p~iitram't cr's. lThv alsmoihci. dt-isi ty (oliVe.-is l IOU1actors' Irom the 1h 1 I

ILOWTHAN G kiiit.s to t hoske use~d ini MOOlTRAN are,

S1E I F 11S 1 OWTRAN 6 MiiI)TRAN

11201 E/tli1 .25 x 10 (m I ouIiuuuihit

IINI(~~ I r.0 x 1 0 anu lgatl~

0 *(.I t1 amaigt I c H! am iin l'a

wiut'. It 1 ilti rlt 1 'ilt il .t SITP'

Thli t\(-v ofI Pýrowt 1i hi~eo ill t Ilve N 1,it i s I i ra I modt.Il IlS bse 141 o he

cku11 ivletIr 1t w idth 1 olf a1 S lg i ', k!a c iao i 11ie iii1 all i [tt!rva I The OqIiivijlent

wiIth ;iunld I ilue Cciit cr t rillsiulit taunres are c vleuill;ltu lI I y su11111u ''i ill- 'w i )Wi' i i1

us I riq, :1 'v' ie !t I jli shiape' Io uomb in th- le 0bhpl I v arid Iort'ii II I im bie shapes

[TIII- Cuirt i 5 C(hil~tii .iluprox jnoutl i (1 i s ised I. (IctI cal ti;It c thei 1.or'iit Y. iit Ift w i (it1

(Si j hio1pp ci' alf w i dlhi (S2) and I liet denusit y ( S"') for t hue veqivii vi ri

Iuoino'iieuuit-s pii lli. The vte ctij e riimiuluer- of 1 i ien is also cai i li ated.

Ill add itionn to ithe molecl'uir uonitr uhl jons from liitis Wit hiin cot-ch I

cml 5I teUCvirii inter'vail coTtrltIut joltS troli titii oS(f lile jit' 'XtvrlijI~ tou haltili' '-Ivol .1re1 alsot ;ohi llated Sines'th-s I hisli ils hiive- i sinuitut; spit-utruil

is, 11sed to ccdi uiuiitut t hie Iitrousm it I one. 'I'ltesvt' Lii colia r iluit ionls aoid tilhl.

'Pt Ct(1l11ll01l0ls'i jita'o (('tlli) l- ( 111 ( 'utlil l 1 I Od I opel Ii~ IV P) and St _WVII ill t 'W

Siltt'ix TX? for muist-quenliuse- by -,ittrout iine TRANS.

20'

5 . I1 4 Solirouut i it CAIX

CAt .t i s en III bi~y biubrtiut inc 13M01 to de t erlinile thi- hanid modelI

par'ame ters fok arbi t ra ry t emperat ures . The tabul1at ed band modelI pa rame ters(

are l inearlIy inuterpol1atIed overF t empehit ure for ehchlIaye r. iIf tie

atmospher i tvi emperat-ure is out side of the temperaut re range of the

t almIl t ed paramieters, the enitry fur eithter- the l owest or hiighest

EW I Will Is cal led by subrout inle IIMO11 to enlcleoIatIe thte vq i va I emt w idthI

of a sinogle aver-age line fur t he I um int erval1 in add it ion to the line

ce!nt er transmit tance. As discussed in Suibsect.ion 3.1 , the Mo gt 1line shape

is numer icallty i iittgrated over- the full int erval. Whien the opticlnI dept i

at the liint center- is less than XMIN./¶, Beteers law is usetd. Otheivrw ise

VW I flhl compaire-, tit( relatlive valu res of t he D~opplecr and j rent z a if w i dths.

and1 spe its tie proiper formula fur the iiomerivai lteio

Muveral Inuinvrici n approx imat i ons itre tised ill ra I cuI itt i rig' t he

euvalent' width, The Vo igt linp shape funct ion is the real1 part of 1,lthe

conqilv err'ior funtii~on" un iid is criiirulnt eu ini subiroutin ICPFI'2. The

region out to three half widths is SCalCUltetd biy using a linear

alppro~ximat ion in the fintegraion iinijt erval . Frm-r ai study o)f various

combinat ions for the numtwr of inttegrantion iinter'vu s for vni:1i halif width

MOT 1 nnp id the niinlir- of hanlfI wi tis. from thle I ine cenqt er INAL.F)

integraLting out to thrmee half widths in six steps was found to represent :1

reasonabltl1e t radeofft btieIwen nmimr ic non Iarucnty and ve~cco ion t i me. 'The

region beyond threei hialf widths of5ci the I Wet Cuvtvir is ca lcul ated usinog no

16. M . Abrnumow it z and I . A. St Cgu,'i "Hanldb~ook nf Mathemleii Wil Furt ioins,

withI Furmut ah, Graphs, and Mathlimait i l Tables," NtIS Auiil i dMathtemmit ius Serlies 55 (lWeMOiibe I 96i) .

21

5. 1 .6 Sll)rolti tiou IFliU

L"i-, mSite(i by Lw ir1m1 t 0(; citima h t ie it Ft V I unict ion. A polyno om ial

al 1 iprnx imit- lon wit ic(h is accurate to better than 2. 5 x 10 5is used. ( 16)

('lF'l2 is cal led by EWIFITI? to calculate the complex error I'Lnctionii t V)

midi~ it-, f ivzst de-i Ivittlye. The real part is the( Vo ig' l inte shiape fuictintoll,

anid the. first. der' vat ivu is used to approximate its frequtency dependence

Wlticn per torni i t the. j 11t cgra t i Otti ;1 LWIDH 'rim1 Th uomplex e!rroc furict ionl

subirou t ine, .Wh~i ch wit-, developed by J . Horn]i cek,"6 i-s based onl a 2 'thi

order flernitii e poly~nomtial aipproximatiion. An 8' ob(rder approx imart.ion was

jinves t iated, hilt thle results were not suifficientlIy itccuralIe.(10)

5 2 Moc.lj~ications to LOWTRAN 6

An att1.empt hats heeri made to miniimijze niod ificationis to LOWTRAN 6. As

inclititomid ini the hiej i :1in iitg ol this ;ectioln, the! Nw iteli .BMOI) has bleln aldded

to / CARD I / OnltIy ifC .J DM01 erqullIs 1 are any change-s i n iti ated.

A 1 ew vou1l~tins have rqijit i ed niii i no I. li(ti fji(:111 iotis. Ini thli MAI1N , i IMOl)

is readl mid thle ham:d niode I pacitmetver fille is openued. Also(, MA IN has been

mod ifet je o I h;; a div idt. by Zero error- does not resul It when ALAMI is

ditet tin'ned at aI frequiency of 0 cm- . Thk- molectiltir coltlimit dens iiy iii

S Mi ' len ildul tiltii. essilie Itli t11 at it luits i it, iltaye liit GEur0a 01 tiImbv sl tpvitt

".I' tIIlItIT I htlouk for' iiis' bty the: MOD~'IT'AN mtbrout ines.

'11i1. st riaI i-'ti forward ipp I iriit i on of the hanoi modeltIL traism i Itiance

t~lt o1111 ýlit ioil t oit hu it si I)!', I sca-t t tkr i rill modelI inl I.OWIRAN 6 woiuIld rttpi iire

doutiI rigi oh ilt I layer. d imri:;i ions" and) storiage of too manty puitit (tatia poi litES

17. R. M. Good, 'Atinspher ic. Rad jilt Iriti LIXo.( oil Uiver1sitIy Press t196').1 11 . J . Ilium I ie (-- , 'Ant IFrf jct-Wn~ Met hod f or "vtil I tui.t ion (if tI lie CoimtI t ox

l'rT01,I i I i t N'IMI lun lou: 'The Voirtlott ion ;Irid it-, lIeniv~it iv.S.Ijiinit .Spe IRatvi. T~ronsfer, 21I . 31)') (197k0

22

Th is proble m was Lvo idtied 1y def i iii i a s ing I e layl r for thie pr imary sol il

path. Till single 1jyeir approx imat ion , which is onl1y uste(I by the MODYWAN

slibrutotill1 es' requ it res t he cill cu l ilt i oil o)f Cutrt is Goidsonl ;aver~liged peoe

and St enp a ie nSG( i gol i f i call t mod if icat ions were requIl i red f or

THANS. 2Q(! 1 ties ()f code have bee~n atdded to.. t his s ubrouut ite, hitlt it s hia~sit

Sog ir rema iris timi ti-t'rd, When .J IN10i) i; set. t-iual to ow-,' THANS

"* sets freuImiitty stief) size to I c m

"* byis ses ClI)TA. C21Y1A , ;rId C31V1'A and callIs tw lie P)FHAN

*i Oturppo1 tes tia'ism it tances ca leo L1i tted ill 5 CMW1 Steps) lot-pe. Sotlici 01him 1120, 11MG' and 0-), an

C MplIOY, it diSGcret ir('ed t -imingkililt slit fuiitrc tiol Withi FWIIM setto INT(l)V) to ~iwtounaticil ly deerade the 1 cm' b~in results.

]it Subisectiontti 5.1 .1 and 5.1 .3, the callIs to MOhDlRAN subrout ints werk:

d iscussed . Note that the- di screLi zed triangular sli t funict ion when DV i s

set to I cm is at I clIi reel anguilar sl it ftitict ion, as ziliuwi inl F igurt 3.

FWHM CONTINUOUS DISCRETIZEDTRIANGULAR TRIANGULAR

SLIT SLIT

F ilgore 3 . Cow par i son of Cu lit inuous and P i sure I.i ed riiuuIaS I it 1`111l1 't i uris . VASCOI))2 Uses the Weip1 fllI ig from the-C o it Ii nIrolkoos Funt n u hI u i11 1t MO DIRAN Usevs the 1)i s reý iz v

Al pI rox Ima i t oi ii.

L 3

6.0 VALIDATION

Val1i dait ion of 1.t li m iiid miodlel Ilrtliflie t ori'31 l i (1h 3 dat tape3 11.1 beell

F-comp I t- ed. lloweve-r only a limitmi ai onuont of test iniý of' M0)I)THAN's

t ranisiuit I iiiC Formulhnat ion hats lo~f.ii per forimed thols 1I1nr All1 v 1itIidt ion ol'

hIIe (;ode will be. toy (Tililjr i Silo to l'AS(,0[)2- Cie3iAItz -: inuv~ it -is the liniai i-it

f- C(Sg) Iti~ii 1(131- 1 l ly )j o m. cill nct ioos of FAS.CI1)? whichii MOl)TRAN is at templintg

0- 3 Band Modu'1Parameters

Tlie fle.w 1,131(1 modf-I par.Inwet 01 r 5 Pva1i~li~tU( algai tinst thle iiIri(

ri'so ut i on 1.01W'RtAN ( i)v~ii1 oCi. 71we I cm i;(ttam sl it band modu]

paraet us cani lot- dIegradied to ', cm Isquare slit btIind nmoeit pranet r (JI

(111() losinig the 'qua in:

(S d n, ( S/d ) 1

j I~

j iil (I j fCi) l O

hini ii : ~iIII(I Ill. ~iii l3 i1 .t g (1 ) oiml (.) r-1lt(I. to the' I cm andl . iA k 11

hil1~1t m ith' 1 ; Ii;Ir~nwi c riii- rci p.c t i'.jr I (Niut v t h.1t Iho. lim- ta I 1 ihsfoipt, i o(il

1. 21

awn" - -. N

H20 Vopor oi- 300K

L I. -

07

zr 1c' V. I !

1. 0 220.4 26 18 8

.K.

6. C Cmarsnwt ACD

IlFigure 4 5. ComTarisondo BAuCd2 Modellsitic LOWTRAN 5rl (Dotted'- fore ra lo alttud, "andl piaud Mhodel-1 tilt!A US So llidaCrve ) Mo Itipiueu il tilt-it-cira regiont 2160 2260t cm 'I ent to11 n! w tere Vuapor wit h a0 I ".,

coe f f igui shn t1ol overl ais app ing fote ar MOT Na nd over lappio n g for i. ts spC tr)I

K ogadt o i nt ei iva Ie ) InF g ic ,t.hespcta srutue2f5h

1-0 KMATI15'- US Std Atm

(I)

oip ]/ (iNý)((I~ r u

WI. LvcrIum cr (m-1

Fiue 5. Cma io (i a M D2adaMDRNC ILltil. N?0aIdC2ae h rmryAsres

221() rn, I it; ;Ill ('Xu rpio il / c m ilrilNlIJý twoi~ Ir-"1 il il thid p

rusciullbcit(cK 1

S"i~r oh i iIlio iSOD n -a t MODTAN Cl Io 0 cmi t N-v111 andiCO moo pmhletr Wremry Ahorenu ,M.TA

callh f 1hsorp tion perftorim Il-cgi onIs r ibimp ol rune y (be t om il th N 0 au citturo led

aPosuts~~ Im ier 0r1 iumo Itv- pmh uscath ~io tin: jursi Stnlard I i ne.livre i ar

2211)ared to I:A except)' i on Fi ore 11 C.) Tf is good but i iij I c h I ai p.

SjiniI i ai Iysi I'I e ai 1iei o uiW ~diSct~ai ie dll iml i26ta

evident A series of 1 ines at 7, 11, 13, anwl 15 cm 1 are Mi-iss.ing in tilw

MOI)TRAN cal cuIit ion. Further invest itt.tt ion is :cc.;ssary to det erliii I, the

origin of these discrepancies.

I.1 IKM P9TH at 0 tKM ALT US Std Attm A.

~~~~~~ .......... .... i D I 1Nt "

",........ I& tlUMlMfN

I cmI [0tS)20 no corli inuurme_

cJ

C)

Wavenumber (cm-1)

Figure 6. Comparison of a FASCOD) and a MODTRAN Calculation Inthe Micr:Iowave.

40

,7

- 27,

"...r

1.0 CONCLUSIONS AND FUTURE WORK

A radiaitive transfer model appl icable to upper altitudes lids- been

* ~duvelI ped tor- LOWTAN 6. Th is mode ratt ti pect~ral reso I tli art model ihas beeni

coiled and jut egralccl intoa LOWTIiAN 6 . Ttes ti g hats begun ort thle niew codle

i-ferr ed to a s NOIY'IAN, and al IrCligiifii -I nay yei-si ha l~s Iceit tie live red to a theI ~ ~A ir Fe rcv Git-ophysies Laboraiitory.

Ili thle toillowling ~-t most of our, effort will conceit trate (ill

vaii idat ion and opti m iza tfit oif thle code. For 1thle f reclue icy range from 0 to

1 7,900 cm I , MOI)WAN mnd FASCOII2ý to ou lot Ions will bef comparted f or- 1low arid

high altitulde pat its. Sign if icait d iscrepanc ies will be ident if ied and

P.corrected . At tert val1i dot ion i s campletedr the MOiPIltAN code wil 1 boi made

more. vfi icicot . Mi-thlods whit i-d may be, fruitful far opt imizring the code

t ic I uldt

* dcif i [Iii lip, cut off va 1 11C.5 fat absar-pi bit

coef-f icu i eiit s,

a -it s itti oire iieiiii i, i i i, ri ofns aht ri'''p

tlmiit det er-in i niziigl I ras;I t ittce o f eaclt sni vc- iesNS(!1ii r zI t e I y

* deve 1 (11)i rig a mot-e ef fi icient roo t i tie faor i nt egrail i ngIli- Voiii lIiiit shape! and/az'

* allow inig far- thle sped rtial resolutit tl. of' tilt! iratidmodle pa ront- t etr to hIe deg ratted i nt e rnally.

To I tic ext '-itt t In tie jIr-n it c ticti t tIhese ariitroiiciits wi 1 bI eIt i Itvs* I v t -rI I

j14

8.0 REFERENCES

1. F. X. Knvizys, E. 1. Shettle, W. 0. Gullery. J. IH. Chetwy'id, Jr. I,.W. Abreo, .1. 1. A. Selby. S. A. Clough and R. W. Fenn, "AtmosphericTrutism iltit nc/Rad i a CompunItcr Code LOWTRAN 6 ," AFG(. TR 83 0187.Air For'ce ;eophysics Lahoratory, Hlanscom AFI3, MA 01731. (August198).. A11A137786

2. R. A. McClatchey, W. S. Beltudict, S. A. Clough, D.. F. lhutch, 1. F.('alfee. K. Fox. L. S. Rothman, and .1. S. Gu1aring, "AFCRI, Almtnsplher''i c

Absorrpt ion li ne Par'ameters Compilations," AFCRI, TR 73 0096., Air Fo'vc(

Geophysics Laboratory, Hanscom AFB. MA 01731 (January 1973); L. S.

RI t tii n, Appl]. Opt , 20, 791 (1981). AI)762904

3. 1). Andiug-;, "Banid Model Methods for Compnting Atmosphvric Slanit .Path -

Molecular Absorption," Rpt. No. 7142-12-T, Willow Run Laboratories,

Unvironmenilja Research Institute of Michigan, Ann Arbor, M• (1967).

4. A. J. LaRocca and R. F. Turner, "Atmospheric Transmit tance andRildiincv: Methods of Ctlculation, Rpt. No. 107600-10-T, Envi'ronimentalResearch Institute of Michigan, Ann Arbor, MI (June 1975).

5. W. L,. Wo ite and t]. J. ZisIb,, "The Infrared Handbook: Rev ised

Edition. " The Infrared Itnformation and Analysis (IRIA) Center.I.inv i I-noi n it Re1.s-e;trch I iI st itA ute f M! ct i iga n, Anni A1-1 i- , Mk s (.1% 1 -

6. A. R. Curtis, Q. J. R. Meteor. Soc, 78, 165 (1952).

7. W, L., Godson, .J, Meteor., 12, 123 (1955).

8. S. A. Clough, "Preliminary User Instructions for FASCOD2," Ltter,AFGI,/OPI , limiscom AFI,. MA 01731 (1 October 1985).

9. H. J. P. Smith, U. J. Oube. M. F.. Gardner, S. A. Clough, F. X.Kne izys and 1.- S . Rhohlmi, '"AASCOUDZ ;-ast Atmosphe, ric Si gnatuore Cod.

(Spectral Truansmittance kind Radiaace})," AVGL TR 78 OO-R! Air ForceGeophysics Inxboratory, ltanscom AFBI, MA 01731 (1978). Ai)A057506

10, I). C. Robertsoin. L. S. Bernstein, end R. Haimvs, "Add t ion of a 5m I Spectral Retisolution Band %odel Option tro 1,OvWIAN 5," ARI RR 2.)32

Ac rodyni- Ve e.;arch. Inct., 1itllerlca, MA (October 1900).

11 . D). C. R~obertson, L,. S. Iternstuin, R, Ilaimes, 3. Wkinderlich, and 1.,Vega, '5 cm-I Band Model Opt im to I.OWTRAN 5," 0ppl __t. . 20, 3218

(1981).

12. S. A. Cl ough. p)QrsVIUjl CoImnuI icatinnj.

29-

13. F. X. Kiteizys. E. P. Sheltle, W. 0. Gallery, J. H. Chetwynid, Jr., L.

W. Abreu, J. E. A. Selby, R. W. Fenn, and R. A. McClatchey.

"Atmospheric Transmittz,nce/Radiance: Computer Code LOWTRAN 5,"

AFGL-TR-80 0067, Air Force Geophysics L•boratory, Hanscom APB, MA

01731 (Felbruary 1980). ADA088215

14. C. B. Ludwig, W. Malkmus. J. F. Reardon and J. A. L. Thcmson,

"iHandbook of Infrared Radiation from Combustion Gases," NASA Report

SI, 3080 (1980).

15. It. A. McClatchey, H. W. Fenn. J. V. A. Selby, F. F. Volz, and .. S.

Garing, "Opt i-al Properties of the Atmospherp, " 3rd ed. ,

A"CRI, 72 -0497. Air Force Cambridge Research Laborat ories, Hunscom

AFIB, MA 01731 (24 August 1972). AD753075

16. M. Abranowitz rind 1. A. St-gun , "Handbook of Mat hemat , Ic ,' Functions

with Formulas, Graphs, and Mathematical Tables," NBS Applied

M;tthlminiiic. Series - 55 (December 1965).

17. R. M. Goody, "Atmospheric Radiation," Oxford University Press (1964).

18. .1. Blumlicek, "An EIfficient Method for Evalunt ion of the Complex

Probability Function: The Voigt Function and its Derivatives."

.1. Quantt. Spoecol. Rud. Transfe'r, 21.. 309 (1978).

- 30 -

- - - - - - - - -- - - - - - - ----

APPENDIX: DERIVATION OF THE BAND MODEL PARAMETERS

In MQIA*RAN, tile molecular traisnmittuice, T, fron, elins origi 11ti lug

within a spectral bin, tu, is determined from an exprvss ion of the for-i

av/21 2= ( .i e-Sulv) dv )n (A 1) ,

0

why're 6.' i' ) W is ai line shape furctiton, * 1 is the alis?)t' itmotint, illd S at!d 11

are function.s of the absorptiout coefficient (S/d) anrd line dit-sity (1/:t .l

h1til moth'I 1 )arimet -vs

(S/d)

S (l/d) (A 2) A

I n (1/d) Av. (A-)

-AI E'q. (A 1) results from Eqs. (11) and (14a).A staudarid method for determining buid model partame ters wits l11iId out

by Gootdy (A 1) The weýak and at runlg lin.e limits of a two parameter

transmittance expressior ire determined. These results are equated to the,

warik and strong line limits of thie Ladethbur' -Re icht. ClauntioIL, ard thie

resultirig equations are solved for the unknown parameters. When Goody

i11)plI ed llis a ppi ro'ttcth to Iwo tt'itl1 t ritsllm it tiulic. fto u-mull s, both !'AiveC

expressions for 1 }itlenu lode .. ."band.. ,d.f the fotrm"

-(S/d) S- 5i (A 4)

I -'-

!

A I . M - . Goody, "Atmospheric RadIitat ion, ()Oxturd 1 l ii IV 'S ity l'ress ".(196C4).

-31

-- P P -' "Lt p"

N NV- 4 -.ISi )2 / 2 S (A 5)

'I v 2 -P4= i-I

tlere, N is the number of line, and Si is tihe integratted line strength of

]Jline i (the line spacing formula assumezi all N lines have the sallue half

width).

It is imnportnlt to noti e that Goody's approach i% inappropri'ate for the

tramismittaucee expression in MODTRAN. Eq. (A-i) models the transmittance

throul 'h ii set ofi lines whose centers fall withiii a givein finite •pectralr

bi n. The Ladenhiirg-Reiche function, on the other hand, mode•is the total

absorption due to those lines - not just their contribut ion withitt a finite

spectrual bin.

lnstend of equating the weak kind strong ]ine! limit (if Eq. (A 1) to the

Ludenhurg Re iche fxinction, we equate the weak and strong line limit of Eq.

(A I) to P;tss' expression(A 2) for the transmission due to lines whose

centerms ir,' randomly distributed in a given spectral interval,

Av/2 Au/22• r 2¢

TV - , v) dii Sn l n { -j (V expi-Siult(u)] dv ( (A 6i)

0 0

here, all N line•s are assumed to hlve the same half w i.ith, but the line

Shitll). fulmil ion iS irbitzary-fl'-fi;'..d !: b th . i

-- b(al-)] /ji di', (A- 7)

0

I'hen, in the weak I ine limit, :q. (A 6) heromesn

A 2. G. N. P•ass, "Mode].s for Siec trail lhuind Absolrpt ion, 3. O1 , . Soc.Am., 48. 690 (!95B).

-32

221 ,, o (B1 n S) u2 (I • ( ) (B2 --- '•--) " J2 ~ ''c.a)=

1 (A22 22 N

1ZNB I12 Z N 2 2 N ~ 3 '( • ,'

t Ok 2 t)s i) 11u2 + y ( 1s2 2 sE i2 (i)(A8

i -I i - i -I

Tit f irst orde'' C'jutil i on ill u I'CeeS tab Ii~lk ~1 :-(Iq. (A 4)p.

N

n S = Cs i (A 9)2..

To detrnminke the strong line limit of Eq. (A 6) we as.ume 1(v) to bit

a non il(ctra.sinsg bounded positivye fulictiotl oil O,D) . Consider a geIItt x'n 1 l,

iit-jgrr.l of th, form'.

AVI2..

edv (A 10)

0

It follows that.

0 < u< .Sul)(t/V/2) < 1 (A 11"

T'his inequdlity re(quiras thait I fdils off eXpontentiUlly witt) larg(! u and 11(

t hut th.t ex pot ui is I intear in u . The c x ponnt til dcay• fltct oi', o c, is vivc 1 .ye

by the limit as u -4 of the negative logarithm of I divided by it

C rnlir u lI(1)/uI (A 12)

Z6

AI'p'lyinlt; Idhl n , one obtains U._

'S

C S <t)(v)> (A 13)

3e

33 - ',

Ie

LL

I. wnere

av/2 Av/2

<1(V)> - li., U J b Nv) e-Sub(- d r •-Sub() dv. (A 14)

0 0

Note that this limit is independent of the value of S as long as S > 0.

Taking the logaritthm of both sides cC' Eq. (A6), dividing by u, and

r .quilt ing the asymptotic limits, one r)tins

N

n S <b(u)> N Si <b(v)>. (A 15)SiZ-I il

(.ine <b(v)> is Iine independ-nt, the surprising result is that Eq. (A 9)

the weak line limit, is also thlt strong line limit!

To determine n and S. we return to the weak line limit, Eq. (A 8), and

consider the equation obtained from the coefficient of the quadratic terms

in u. This e(qIut ion givesN

n c 22 Si 2

(A 16)

TIh, result in g expressions for the banmd Inodel parameters are

N

(S/d) Si (A 4)i--

N N(lid) -- ( Si )2 / si2 (A 17)

Ii •1i :1

("/d() has the simine form as before, but a smallier valuet for (I/d) is

obtained, because thhe weaker I ini'-s are weighted 1less in Eq. (A 17) thaln in

F,'l. (A 5)

34 -

I. Figures 5 and A-1 compare a 1 cm-'1 non-overlappitng rectangular slit

MODTRAN calculation with the new and old band model parameters to a 1 cm 1

overlappinrg rectangular slit FASCOI)2 calculation. The results with the new

formulation are much improved. Further investigation and validation are

planned.

1 5'- I KM RT 150 -- US Std Rtm

C'

I i "IE| _

U)

1o Fft3COD?

old U/du€nd noN20 COW€I IULnJu

1 .. ,_a]s (I _ i] d_.1 S..oc,

F160 P17- 21 2Imar 190 o2f 2210 a2n IPR3 ?Aqo ,)cuti ?o.

S~Wavenumber (cm-i)

Figure A-1. Comparison of FASCOD2 and MODTRAN Calculations.

The MOD)TRAN Curve was Generated( w#ith the Old Band

Model "or.=.-,tertý, Eqs. (A-4) (A-5) and no N20

Continuum Fun•ctilon.

RIEF.ERENCES

A 1. R(. M. Goody, "Atmeosph,;r-ic Raidiation," Oxford Unive:rsity IPrrs%, 1964.

A-2. 0. N. Plass, "Models for Spectr'al Band Absorpt ion, " Jl Opt.- Soc'

Am., 48, 690 (i9,58).

-35 -