NASA CR-/_/ll / fERIM 190100-24-T

/

" RADIATIVE TRANSFER IN REAL ATMOSPHERES

(NASA-C_-I_I"9) :_ _i._-i_:, Tg:,NSF_ IN N7__31,a73RS_.L _TMOSPEE.;{}:.C,?_c}.:.ical [_eport, IFeu. - 31 Oc[. 1o73 {?[viro_._en_i

Re_arch Inst. of "icLi_:_h) 1_,8 p Uncles: HC $9.50 CSCL C_A G3/13 _7_bZ

by

Robert E. Turner

IMrared and Optics Division

July 1974

ENVIRONMENTAL

R{SEARCHINSTITUTEOFMICHIGAN '

FORMERLY WILLOW RUN LABORATORIES.

THE LJNIVERSITY OF MICHIGAN

NATIONAL AERONAUTICS AND SPACE ADMINISTRAT_ \''''. _ ._

.:.' Johnson Space Center, Houston, Texas 77058 !'.,'- ;i".i_$__b ,,'?'I EarthObservationsDivision "_, "%_',>, ,:%

Contract NAS 9-9784, Task IV _-<. ,,._.Y."(//'_cr.,=c'.,%,'>

{

1974023760

https://ntrs.nasa.gov/search.jsp?R=19740023760 2020-04-01T06:03:24+00:00Z

TECHmCALREPOI_TSTA._;DARDTITLEPAGE

I. R,'p_r!NO.NASA CR-ERIM I 2. GovernmentAce:.sslong,,. 3. Reciplent'sC.,t.do_h,,.190100-24-T L

4. Ttt!c and Subtlth. 5. RepolI Date

December 1973RADIATIVE TRANSFER IN REAL ATMOSPHERW, S 6. performm,,Or_amzat,,mcode

7. Authorls) 8. Performing Or_amzatlon Rrpor! No.

Robert E. Turner ERIM 190100-24-T i9. Performma Org, amzatlon Name and Address 70. Work Umt No.

Environmental Research Institute of Michigan Task IVIntrared and Optics Division H. contractor GrantNo.P.O. Box 618 NAS9-9784Ann Arbor. Michigan 48107 is. ryp_ of aep_t _d Pe_,odCo,'_,_ .

12. Spor.sormg Agency Name and Address Technical Report !_National Aeron.qmtics and Space Administration 1 February 1973 -

Johnson Space Center 31 October 1973 ,_Earth Observations Division t4. SlmnsoringActncyCod_ |Houston, Texas 77058

_5. Supplementary Notes

Dr. Andrew Potter/TF3 is Technical Monitor for NASA./

16. Abstract £

Any complete analysis of multispectral data must include the _fects of the at-mosphere on the radiation f_eld. This report treats the problem of multiply scat-tered r_ diation in an atmosphere characterized by various amounts of aerosol ab-sorption ,rod different particle-size distributions. Emphasis is put on the visiblepart of the sp-,ctrum and includes the effect of ozone absorption. Absorption byother g.ts m._ "'o,-'.,_c_._nts in the infrared region is not treated.

The work also includes an investigation o£ an atmosphere bounded by a non-homogeneous, Lambertian surface. The effect of background on target is studied in -_terms ofvarious atmospheric and geometric conditions.

Results of thisinvestigationshow thatcontaminated atmospheres can change theradiation field by a considerable amount and that the effect of a non-uniform _,,zrface

can significantly alter the intrinsic radiation from a target element_a fact o_ con- iisiderableimportance inthe recognitionprocessing of multispectralremote sensingdata.

17. Key Words IS. Distrihutio01 State,lOOn! '_

Aerosol absorption Initialdistributionislistedat theMu!tiplyscatteredradiation end ofthisdocument.

Ozone absorptionRecognitionprocessingRemote sensing

19. Security Classil (of this report) ] 20. Security Classaf. (of tlus page) 21. No. of I'agt,s { 22. Prh'c

I IUNCLASSIFIED UNCLASS!FIED 106I

i

1974023760-002

|"_ IrO_ti_l['itLY WIL*.OW NUN I.A_RATOIItlir$ TH[[ UNIVI[RslIrY OF M|CHi_AN

1: i.:_

PREFACE -e,_

This report describes part of a comprehensive, continuing research program _

in remote sensing of the environment by aircraft and satellite. The research is ._

being conducted for NASA's Lyndon F Johnson Space Center, Houston, Texas. by

the Environmental Research Institute of Michigan (ERIM), formerly the Willow

Run Laboratories, Institute of Science and Technology, The University of Michigan.

The main objective of t_s program is to develop remote sensing as a practical -'

I tool for obtaining extensive environmental information quickly and economically. _,Remote sensing of the environment involves the transfer of radiation from

objects of interest on the Earth's surface through the atmosphere to a sensor _"

i_ el.ther within or outside the atmosphere. Since this intervening atmospheric me-

P' dium alters the il:t.rinsic radiation from surface elements, the interpretation of

mui_Aspectral scanner data can be significantly affected. This report deals with

_ the definition of the natura! radiation field encountered in realistic remotea sens-

ing environment from the viewpoints of radiative-transfer theory and atmosphericoptics.

_ The research described in this report _s performed under Contract NAS9-

_ _: 9784, Task IV, and covers the period from 1 February 1973 through 31 October

1973. Dr. Andrew Pott_.r _erved as Technical Monitor. The program was

d_rected by R. R. Legault, Vice President of ERIM, J. D. Erickson, Principal In-

vestigacor and Head of ERIM's Information Systems and Analysis Department,

and R. F. Nalepka, Head of ERIM's Multispectral Analysis Section. The ERIM

number of this report is 190100-24-T.

The author wishes to acknowledge the direction provided by R. R. Legault and

J. D. Erickso_. Helpful suggestions were made by R. F. Nalepka. Computer pro-

gramming and other technical analysis were provided by N. A. Contaxes, W. B.

Morgan, and L. R. Ziegler. The author also thanks R. M. Coleman for her secre-

, tarial assistance in the preparation of this report.

,i

' i3

pp.lO

:|i

]974023760-003

I+ ?

+4

; [RIM+S.

IroRMI'$1L_F WILLOW RUN LAIIIOI_ATOIrlIFS THF UNIVIERSlTY O I_ MICHI_A_ -+

• j

•CONTENTS

1. Summary ................................... 13 _

2. Int rcduc:ion ................................. 14 _.

3. The A_mosph +ric State ........................... 16

3.1 Composition of the Atmosphere 16 :_3 L.i Atmospheric Gases 16

• .i.,_ Aerosols 18 i, 3.2 Turbidity 21 ,

• 3.3 Vit.'bility 22

I 4. Optical Par'_mzters ............................. 25

_ 4.1 Scattering Theory 25 :+. 4.1.1 Rayleigh Scattering 25

4.1.2 Mie Scattering 26"_ 4.2 Attenuation Coefficients 28 -_"V

e 4.3 Optical Depth 32 '

4.4 _mgle-Scattering Albedo 33 ;_ 4.5 ;Jc,ttering Phase Functions 45

:: ,, 5. Inhomogeneous Atmospheres ........................ 49 :_ _,t The Double-Delta Approximation 49

• _ 5., Calculations 50 _i

: ": 6. Multidimensional Radiative Traasfer .................... 62 ": _ 6.1 General Theory 62

i G.2 The Uniform Disk Problem 63 .- 6.2.1 Isotropic Scattering 646.2.2AnisotropicScattering 73 +++

6.3 The hffinite Strip Problem 92

7. Conclusionsand Recommendat,:ons ..................... 100 ':

A_pendix: Fourier Transform Derivation ................... 103 _,References ................................... 104Distribution List ................................ 106

• PRECEDING PAGE BLANK NOT FILMED

5

1974023760-004

FIGURE_-#

1. Altitude Profile for Ozone Density ..................... 1"/

2. Haze-Type Distribution Functions Used .................. 20

3. Aerosol and Temperature Profiles for 4-5 November 1964 ...... 23

4. ScatteringEfficiencyFactorforHomogeneousSpheresofComplex RefractiveIndexm ..................... 29

5. _bsorptionEfficiencyFactorforHomogeneousSpheres_fComplex RefractiveIndexm ..................... 30

6. TotalEfficiencyFactorforHomogeneous ,,neresofComplex RefractiveIndexm ..................... 31

7. DependenceoftheAerosolAbsorptivityParameter,f,onWavelengthforHazes L and M wRh Three RefractiveIndices............ 35

8. DependenceofSingle-ScatteringAlbedoon WavelengthforVariousARitudes--Refractive_ndexm = 1.b ........... 37

9. DependenceofSingle-ScatteringAlbedoonWavelengthforVariousARitudespComplex RefractiveIndexm : 1.5- 1.0i .............................. 38

10. ARitudeProfileoftheSingle-ScatteringAlbedowithand WithoutOzone ............................ 39

11. ARRude ProfileoftheSingle-ScatteringAlbedowithCompleteAerosclAbsorption...................... 40

12. ARitudeProfileoftheSingle-ScatteringAlbedowithNo AerosolAbsorption.......................... 41

13. ARitudeProfileoftheSingle-ScatteringAlbedoforHaze L ....... 42

14. ARitudeProfileoftheSingle-ScatteringAlbedointheLower Troposphere........................... 43

15. DependenceoftheSingle-ScatteringAlbedoon VisualRange ...... 44

16. ScatteringPhase Functionsas Calculatedfrom Mie Theory

forHaze LmComplex RefractiveIndexm = 1.5- im2 ......... 46

17. ScatteringPhase Functionsas Calculatedfrom Mie TheoryforHaze L_Complex RefractiveIndexm --1.5- 0.01i......... 47

_' 18. Scattering Phase Functions as Calculated from Mie Theoryfor Haze L and Uaze M........................... 48

19. Dependence of Path Radiance and Total Radiance on Wavelengthfor Haze L--Visual Range -- 23 km .................... 51

20. Dependence of Path Radiance and Total Radiance on Wavelengthfor Haze L_Vtsual Range -_2 km .................... 52

21. Dependence of Path Radiance on Single-Scattering Atbedo forHaze L ................................... 53

22. Dependence of Path Radiance and Total Radiance on Altitudefor Haze L--Visual Range = 23 km .................... 5_i

i

1974023760-005

, n ,FORt4ERLY WILLOW I_UN LABORATORIES, TH[ UNIV[R_*ITY (_F" MICHIGAN

? 23. Dependence of Path Radiance and Total Radiance on Altitudefor Haze L--Visual Range = 2 km .................... 56

"= 24. Dependence of Path RadiaJ_ce and Total Radiance on Altitude_' in the Lower Troposphere for Haze L .................. 57 -:

i 25. Dependence of Path Radiance on Altitude in the LowerTrooosphere for SeveraI Refractive Indices ............... 58

26. Dependehce of Path Radiance. Attenuated Radiance and Total Radianceon Altitude for No Absorption and Heacy Absorption .......... 59

27. Dependence of Path Radiance on Nadir View Angle in the SolarPlane for Several Refractive Indices ................... 60

28. Geometry for Isotropic Scattering with a Uniform Surface ........ 66

29. Disk Geometry from Equation (66) .................... 66

30. Target Reflectances ............................ 67

31. Singly-Scattered Surface Radiances of Colored Disks on GreenVegetation Background .......................... 68

32. Singly-Scattered Surface Radiance for Black Disk with GreenVegetation Background .......................... 69

¢ 33. Singly-Scattered Surface Radiance for White Disk with GreenVegeta.tion Background .......................... 70

34. Singly-Scattered S,lrface Radiance for Red Disk with Green" Vegetation Background .......................... 71

35. Radiance Values for Black Disk with Green VegetationBackground ................................ 72

36. Radiance Values for White Disk with Green VegetationBackground ................................ 74

37. Exact and Approximate Scattering Phase Functions for Haze Lat a Wavelength of 0.4 _tm and Relractive Index of 1.5 76., • , • • ° • , • ,

38. Exact and Approximate Scattering Phase Functions for Haze Lat a Wavelength of 0.9 #m and Refractive Index of 1.5 ......... 77

39. Exact and Approximate Scattering Phase FunctionJ for Haze Lat a Wavelength of 0.4 _m and Complex Refractive Index mof 1.5 - 1.01 ................................ 78

40. Exact and Approximate Scattering Phase Functions for Haze Lat a Wavelength of 0.9 #m and Complex Refractive Index mof 1.5 - 1.0t ................................ 79

41. Wavelength Dependence of Solar Irradlance at Top ofAtmosphere, Green Vegetation Reflectance. and the Productof the Two ................................. 81

¢

42. Dependence of Singly-Scatte2 ed Surface Radiance on Disk RadiusVisual Range = 23 km .......................... 82

43. Dependence of Singly-Seatter_._ Surface Radiance on Disk• Radius _Vtsual Range = 2 km ...................... 83

7

+,

i

1974023760-006

_ER_M FOMMI[RLY WILLOW IR_JN; ABOIRATORIES. THE UNIVERSITY OF MiCHiGAN

44. Dependence ofSingle-ScatteredSurface Radiance on Altitude--

Visu_l Range : 23 km .......................... 84

45. Dependence ofSingly-ScatteredSurface Radiance on AltitudeVisu.tlRange = 2 km .......................... 85

46. Dependcnce ofSingly-ScatteredSurface Radiance on W&velengthfor a Black Disk and Green Vegetation .................. 87

47. Dependence of Singly-Scattered Surface Radiance on Wavelengthfor a White Disk and Green Vegetation Background ........... 88

48. Depe,dence ofSingly-ScatteredSurface Radiance on Wavelength .at 50 km ARitude for Various Disk Radii--Visual Range = 23 km .... 89

49. Dependence ofSingly-ScatteredSurface Radiance on Wavelengthat 50 km ARitude for Various Disk Radii--Visual Range = 2 km .... 90

50. Dependence ofSingly-ScatteredSurface Radiance on WavelengthforVariousDisk Rcflectances--Visual Range = 23 km .......... 91

51. Dependence ofSingly--ScatteredSurface Radiance on Wavelengthfor Various Disk Reflcrt_ces--Visual Range = 2 km .......... 93

52. Dependence of Radiances on Wavelength--Visual Range --23 km ..... 94

53. Dependence of Radiances on Wavelength--Visual Range = 2 km ..... 95

54. Infinit_. Strip Approximation ........................ 96 "J

8

l

/

1974023760-007

i l l

FORMERLY WILLOW RUN LABORATORIES. THr_ UNIVERSITY OF MICHIGAN

SYMBOLS

A _terosol (subscript)

a. e, b, _ distribution parameters

b° brightness factor

C contrast i

CO contrast ratio (1 - Lo/LB)

E° e: _r_terre_trial solar irradiance at the top of the atmosphere !!E" isotropic fie]d _

totaldownward irradiance "_- :._

f aerosol absorptivity factor _

I intensity of radiation

i index for a, s, or t, Eq. (39)

I(o) beam intensity at ori_:in

k* complexpropagatioi_vector

Lat t directly attenuated scattering component

LB background radiance

Lds doubly scattered sky radiance _'

L° intrinsic radiance of the object _

Lp spectral path radiance

L radiance along infinite path through atmospherePA

L(R,_) total_'pectralradiance

LS spectral radiance at surface

L singly scattered surface radiance _sur ,!

Lto t sum of Lsu r + Lss + Lds

L singly scattered sky radiancess

mI real part of refractive index

m2 imaginary part of refractive index

._ refr_tctlveindexof themedium

re(k) refractiveindex

N molecularnumber density

9

|

1974023760-008

n[M FORM,-RLY _VlLCOW RUN LABORkTORIES. THE UNIVERSITY OF MICHIGA,_

N(z. r) particle re, tuber density for ra lu:; r at altitude z

P(cos ×) scattering phase function

Pc(cos X) Legendre polynomialf

p(cos X) scattering phase function

Qs(X. m) scattering efficiency factor

r" coordinate position vector

s particulate size distribution

T spectral transmittance

t time or total

t(A) aerosol turbidity

V visual range

x distance, size parameter

aA(_, m, z) absorption coefficient for polydispersion

_¢A(k. m, z) scattering coefficient .'or polydispe rsion

5 anisotropy parameter "_

K volume extinction coefficient

KA(k, m, z) extinction coefficient for polydispersion

Kd(k) volume extinction coefficient for dust particle

I_R(k) volume extinction coefficient for Rayleigh scattering

_w(_) volume extinction coefficient for water aerosol

_o' _o angular coordinates of the sun

7/(r) anisotropy parameter

sin 7/ _'os %5

v sin _ cos

Di interior Lambertian disk reflectance

_e exterior Lambertian disk reflectance

a R(k) Rayletgh scattering cross section

as(A, r, m) scattering cross section

et(_, r, m) total cross-sechon scattering

(r) optical depth

X angle between the incoming direction and the scattered direction

_ and _f Riccati-Bessel functions :"

10

tt

r •

1974023760-009

_Rl_ "O--'L*W'_OW'O_LA_R.*O_,**TH*ON'V_S,*.O.M,O_,O.'_!. £ solidangle

A A

unitvector inthe directionofthe scatteredphotonat positionR i_

v w angular frequency _.

_o(r) single-scatteringalbedo .!

11

1974023760-010

|11FORMERLY WILLOW RUN LABORATORIES. THE UNIVERSITY OF MICHIGAN

11' SUMMARY

As part of the continuing development of an atmospheric-radiative-transfer model which

can be used to correct multispectral data, we must col_sider as many atmospheric effects as

i possible that may have Pn influence on the data. In previous studies we have constructed a

radiative-transfer model which included the effect of multiple scattering in homogeneous, non-

absorbing, plane-parallel atmospheres bounded by spatially uniform surface conditions. !

Although variable atmospheric states characterized by visual range were included, we consider-

ed only non-absorbing aerosol particles in the a,,,ospheze. Also, the surLtce dealt with was aspatially uniform Lambertian one.

!_ In the present further work we consider a variety of more realistic atmosphez ic conditionscharacterized by visual range, composition of aerosols, size d(stributiou of aerosol p._rticles,

: _. and o_.one absorption. Hence, we have expalzded our atmospheric-radiative-transfer model to

I includealmosteverykindofatmospheric_tatepossiblewi_1_zegardtothecomposihon_ndw

i structureofaerosols.Excludedare certainstronggaseou:absorptioneffectsintheinfrared,._ withthisomissionjustifiedopthegroundsthatinthedesig:lof _ensors,one usuallyignoresi.

theseregionsanyway.

i Anotherproblemaddressedinthecurrentstudy"..stheo_casionaiznfluenceofbackb'roundt on the target through radiation being scattered frozn eiement_ outskdr the instantaneous field

of view into that field of view. In ff_is report we examine this effect a:td employ the necessary

approximations to solve the problem analytically Also, a soluhon is presented for the case of

an airborne or spacecraft sensor v_.ewing a non-Lambertian surface from any point in or out-

side of the atmos,_here. This solution can be used in an analysis of the changes to be expected

in multispectral data as a sensor moves in a horizontal plane ove_. the terrain being investigated.

J

1974023760-011

'ERIM_BJ FORMERLY WILLOW RUN LABORATORIES. THE UNIVERSITY OF MICHIGAN

+:

2 . ,,

INTRODUC TION

In the remote sensing of terrain features on Earth's surfa, e, discrimination of specific _ +.!

targets from their backgrounds is essenti_,l. One method of _tccompiishing this is to analyze •,P

the radiation received in several wavelength intervals on the assumption that classes of objects

can be defined in terms of their spectral properties. Such analysis depends upon variabilities

in remotely sensed data which can be attributed t3 the complex spectral properties intrinsic tn

materials found on the Earth's surface. Discrimination techniques are by no means limited to

spectral variations alone; other schemes caP. be based upon polarization or goniometric character-

istics of various classes of objects toprovide a "signature"and thus a unique means ot identifica- ti

tion. But remote sensing based upon polarization and/or goniometric properties has not found lwidespread use, probably because (1) very little is known about the intrinsic polarization char-

acteristics of natural materials and even less concerning their goniometric properties, and (2)

the sensor requirements and operational systems needed to acquire meaningful data based ut-on

these properties are rather complex. In this report, however, we deal exclusively w'tb a_laJ tical

approaches applicab.e to the d_velopment and implementation of ¢_iscrimination tech,iques t)ased "

upon sjectral variations.

Aside from the intrinsic properties of natural materials, there exist variations, sach as the

variability of the atmosphere, resulting from elements extrinsic to these materials. Although

one would like to collect data under ideal environmental conditions, the implementation of an

operatiopal remote sensing system necessitates the acquisition of data under conditions which.

unfortunately, are oftentimes far from ideal. For this reason it is important'that the more sys-

tematic variations associated with atmospheric effects be considered in the analysis of remote

sensor data.

The spectral radiance at a sensor is given by the following:

L T = LsT + Lp

where L S is the spectral rad!,auce at the surface

T is the spectral transmittance between the sensor and the object being observed

L p is the spectral path radiance--that is, the radiation along the path between the sensor

and targe_ arising from multiple scattering or intrinsic emission

All three of these quantities depend upon the state of the atmosphere; they can vary considerably

with altitude, wavelength, view angle, sun angle, and surface albedo. A complete atmospheric

radiation model should provide those quantities which determine a unique set of parameters B

appropriate to the environmental conditions which exist at _ particular time.

14

i

1974023760-012

i

i1t

/ ...

FORMERLY WILLOW RUN LABORA _'ORIES. THE UNIVERSITY OF MICHIG q_N '2

i

Previous work on this problem has resulted in an atmospheric-radiative-transfer model

which acco,'nts --assuming multiple scattering in cloudless, plane-parallel atmospheres with

• no absorption---for variations in the radiometric quantities named above. Th._ usual assump- :

tion in atmospheric-radiative-transier problems is that there is little absorption by aerosols !

in the visible part of the electromagnetic spectrum. This is definitely not the case, however,

for the contaminated air commonly found in large urban areas where sulfate, carbonate, and )

other soot particles exist in large quantities. Furthermore, from recent investigations of the ::

atmospheric partieul _te load aro,4nd the world, it is becoming evident that aerosols play an -_" _ _

important role indeterminm_ the overallenergy balance ofEarth. For thisreason we inves-

i tigated the influenc_ of contaminated atmospheres on the natural radiation field.

Another effect treated in this report is the influence of background on t',rget via scatteringt"

inthe atmosphere. Radiationreflectedfrom background objectswilleventuallybe scattered

_ into the field of view of a sensor and, hence, will influence the signal from the target. This

• i, complicated effect requires that the radiative-transfer equation be solved in two or three

dimensions in order to define the radiate.on field anywhere in the atmosphere for non-bomoge-

I neous boundaries. In our study of this phenomenon, we _sume it to be a function of basic pa-f

i rameters describingpertinentopticalcharacteristicsof the atmcsphere.

b1

i

,E,

• 1

1974023760-013

_iM ' ' ' ' FUNMIrRLY WILLOW NUN L_411(:)RATORII[S Tt'II_"UNIVI_RS_TY OF MiC_'tGAN

3THE ATMOSPHERIC STATE

In the remote sen_tng of Earth's surface b:, means of electromagnetic radiation roflected

front terrain objects_ the atmosphere can scatter and absorb t _" r_.diation and, hence, a!ter

the intrinsic radiation characteristics of the object being investigated. The resultant rad._tzon

received by a sensor depead_ upon factors such as viewing geometry, time of year, surface

: conditions, and atmospheric state. It is the state of the atmosphere that we shall discuss in this

section.

3.1 COMPOSITION OF THE ATMOSPHERE

The state of the atmosphere at any location and at a_'ly time can be defined by a ,mique set

of parameters fully describing all aspects of the atmosphere. However, we are not interested

in all aspects of the atmosphere, but only in those of key importance to remote sensing applica-

tions. Thus. micro-fluctuations in density or ion content are of no import, but we are certainly

interested in v,sibility. Our goal, therefore, is a somewhat limited specification of parameters

which can be used to define the state of the atmosphere. For purposes of the present study,

we shal.l be concerned mainly with the scattering and absorbing properties of gases and partic

ulates that exist in our atmosphere.

3.1.1 ATMOSPHEBIC GASES

By definition, the visible portion of the e!ectromagnctic spectrum is that part of it to '_ht,;h

the human eye is most sensitive; thus it is not surprising that this part of the spectrum und_.'r-

goes little absorption by the atwosphere. The major permanent gaseous components of our

atmosphere are molecular oxygen, nitrogen, and argon. Th,'_L _gases absorb very little radia-!lion in the v_sible and near-infrared part of the spectrum: only, variable con,.pop,cots such as

" water vapor, c,_rbon dioxide, ozone, sulfur dioxide, nitrogen compounds, methane, and other

trace gases absorb Juch radiation to any extent. Of these variable components, ozone, water

- i vapor, and carbon _'lioxide are the most important absorbers of radiation in the near-infrared., For the purposes of _his study, we shall consider only ozone in the Chappuis bands in the

! region 0.44 to 0.74 #m and assume that all other radiation in the region 0.40 to -2.0 #m is not

absorbed by gases. Since the gaseous absorption regimes of the various atmospheric compo-

nents are well known, it is relatively easy to select those spectral regions within which absorp-

tion is at a minimum. Mo_'e detailed studies of atmospheric gases have been done by Goody

[ll andzuyevF I.

Throughout the '_isible part of the spectrum, ozone is the primary gaseous absorber. . :

'_ There are variable amounts '.n the troposph,_rc, but the general altitude variation is known [3 I.

;' A profile of the ozone number density is ih _str ted in Fig. 1. Note that the maximum density

occurs at an altitude of about 23 kin.16

t

i

1974023760-014

1974023760-015

FORMERLY WILL(_W RUN LADONATORILS. 1- * UNIVirNSnY OF MICHIGAN

3.1.2 AEROSOLS

O_e mc.y definean aer'_solas a semi-permanent suspension ofsolidor liquidparticlesin

the Earth's atmosphere. Typical aerosol distributions would be hazes, clouds, mists, fogs,

smokes° and dusts. It is the aerosol component of the atmosphere which is quite variable.

Depending upon location and time, aerosols can have many compositions, sizes, shapes, and

densities. In condensation processes, the predominant composition is water (for which the re- '.

L fractive index is well known). Depending upon the history of the distribution, however, various

contaminants can mix with water and water vapor in the formation of droplets to produce a

composition quite different from that of water. The major l_rimary source of atmospheric

t aerosols on a world-wide basis issea spray [4];other primary sources are wind-blown dust _,

and smoke from forest fires. While the composit;o_ of worldwide aerosol distribution can be

estimated, this composition can nevertheless vary over a wide range depending upon its prox-

imity to various local sources. The only ways m which the composition can be determined is

by direct in situ sampling of the air in a particular location or by radiometric techniques.

_ Much in situ sampling has been done by Volz [5, 6], Grams et al. [7], and Flanigan and

Delong [8]. As a result of their work and that of others, the mean world-wide aerosol can be -

considered to have a real refractive index of about 1._ and an imaginary part between 0.01 and

1.0. Hence,

m(_) = m 1 -im 2 (1) ;

• where m 1 and m2, respectively, are the real and imaginary part of the refractive index, and

A is ",he wavelength. For realistic atmospheric conditions,

1.33 _- m 1 _" 1.55 (2)

and

0 < m 2 _- 1.0 (3)

which are roughly independent of wavelength.

I Particles also come in various s_zes. Generally, however, they can be put into threecategories: the Aitken nuclei with radii between 10 -7 and 10-5 cm, the large particles with

radii between 10"5 and 10 .4 cm, and the so-called giant particles with radii greater than 10-4 ii

il cm. In z_ast hazes, the optically active region i_ made up of those particles in the large orgiant categories. Junge [91 showed that most aerosol distributions follow the simple power l

N(z, r) : C(z)r "¢ (4) •

18

r , I -- --:----I I | -- ,-_ I

1974023760-016

_'-_R;M FORMERLY WILLOW RUN LAaORATORIES THE UNIVERSITY OF MICHIGAN

where N(z, r) is the particle number density for radius, r, at altitude, z

v is the exponent of the power law i

v Experimentally, u I,as been found to vary f . _l 2 to 5 for various tropospheric aerosol distribu- "

tions. Also, other investigators have found a better fit to the particulate d_ta by using the mod-

ified gamma distrioution,

N(z, r) = ar a exp (-brY); 0 <-r < _o (5) _

where a, a, b, and y are parameters which describe the distribution.

Deirmendjian [10] has used this function to :haracterize different hazes. The f(,llowing is

a list of the hazes Deirmendjian considered and the relevant parameters.

Haze Type a a Z b ,i

M 5.3333 x 104 1 1/2 8.9443

L 4.9757 x 106 2 1/2 15.1186

- H 4.0000 x 105 2 I 20.0000

Haze M, used to describe a marine or coastal haze, has a peak in the distribution at

r = 0.05_m. The haze L represents a continental distribution and has a peak at r = 0.07#m.

The haze H model can be used to represent a stratospheric aerosol or dust layers: it has a

peak at 0.10_m A graph of the three hazes is illustrated in F;g. 2. We shall make use of these

hazes in the detailed analysis of atmospheric radiation.

For the liquid aerosols it can be. assumed that the particles are spherical or nearly spher-

ical in shape. For solid particles, however, the shape may assume any form. A number of

investigators [11, 12, 13] have studied the influence of particle shape on the scattering of radLa-

tion. The scattering of electromagnetic ,'adiation by odd-shaped particles has a different pat-

tern from that produced by spherical particles; but given a polydisperse collection of odd-

shaped particles, the nature of the difference is not clear. Most of the work on radiation in

atmospheres concerns spherical aerosols; we shall follow suit and neglect the complications

of particle _hape.

Thus far :e have considered the composition, sizes, and shapes of aerosol particles, but i

in order to define the atmospheric state we must also know the concentration of particles.

Wiegand [14] was the first tc measure the vertical profile of an aerosol number concentration _

. of condensation nuclei. As a result of his measurements and many others cited in Ivlev [151 ,

it was found that the concentration of condensation nuclei obeys an exponential law with altitude.

It was also determined that a zone of increased concentration of large particles exists in the _

19 ii

1974023760-017

FOI_MERLY WILLOW RUN LABORATORIES, THE UNIVERf)ITY OF MICHIGAN

102

O_, 101

Z

IOO

IO -1

10 -20,01 0.1 1.0 10

RADIUS (r)

FIGURE 2. HAZE-TYPE DISTRIBUTIONFUNCTIONS USED. Units depend on the

particular model. [10]

20

1974023760-018

-- RIM , i, **..FORMIERLY WILLOW RUN LAI_IOR_,TORIE_, "THEUNIVERSITY OF MICI"II_AN

17 to 23 km altitude range (Junge layer) and in a probable layer under the tropopause at 9 to 10 !

_ km. l"hese layers are relatively stable compared to the lower part of the troposphere. As an

• example of an aerosol model atmosphere, Zuyev [2] has constructed the following alt;'lde-

; dependent, number density-dis*ributi'Jn-functior for aerosols:

N(0_,e-bZ; z <-5 km :-;

i .03:5km-<z-< 15km i

_0 (6) ?N(z) =10.03e0.06z, 15 km -<z -<20 km

IL0.01e'0"09z; z ->20 km

• where N(z) and N(0), respectively, are the number densities at altitude z and the Earth's surface(altitude 0).

I In conclusion, we can say that the most significant aspect of aerosol particles in the atmo- :sphere is their high degree oi variability--in composition, size distribution, and especially inf

number density. All the modeLq in the current literature deal with a highly approximate

_ average-condition from which large deviations can occur in real situations. The man,,.details

of aerosol science will not be considered in this report. For a more complete study of the

physicsand chemistryofaerosols,seeMason [16[,Fuchs [17],Davies[18],or Green and

Lane [19].

3.2 TURBIDITY

Having examined the basic characteristics of _he gases and aerosol particles composing

the atmosphere, we now consider those param_kers which relate to the attenuation of radiation

passing through the atmosphere.

If a collimated bean- of monochromatic radiation is incident upon a scattering and absorb-

ing medium, then the intensity of the beam at distance x is given by

I(x)= I(o)e-Kx (7)

where I(o)isthebeam intensityattheorigin.The quantityK,calledthevolumeextinction

coefficient,isequaltothesum ofthevolumeabsorptionand volumescatteringcoefficients.

Sinceattenuationofradiationintheatmosphereiscausedprimarilyby molecularscattering,

scatteringandabsorptionbywaterdroplets,and alsoby dustparticles,Linke[20]proposed

I thattheattenuatLonbe measuredby takingtheratioofthetotalintegratedattenuationcoeffi-

cientsto the pure RRyleigh integrated coefficients. Thus,i

21

1974023760-019

FOAMERLY WILLOW RUN LA_RATORIES. THE UNI_*F'RSITY OF MICHIGAN

OO ¢_ OO

W

=- (8) ,

fK R(_)dz

;- where KR(;_), _w(;_), and Kd(A) are the volume extinction coefficients for Rayleigh, water aerosol,and dust particle scattering respectively. Thus, the turbidity, t(A), is a measure of the depar-

ture of a real atmosphere from the ideal pure Rayleigh atmosphere. Equation (8) can also be

written as

t=I+W+R (9)

where W is the humid turbidity factor and R is the resid,Jal turbidity factor. For a pure _t,_c-

sphere frt:e of water and dust, t = 1. Values of the turbidity vary from 3.59 for a continental

tropical air mass in the summer months to 2.16 for a sea arctic air mass in winter. Tables

: of typical turbidity values are given by Kondratyev [21] for many locations and weather condi-

tions. The turbidity as a function of altitude is shown in Fig. 3 [22 ]; this figure was constructed

from data collected by Elterman [23] in optical searchlight measurements. The maximum near

the tropopause is the result of convective activity causing particles to concentrate at that stable

position.

3.3 VISIBILITY

Visibility in meteorology refers to the transparency of the atmosphere to visible radiation.

Usually characterized by a quantity, V, called visual range, it depends upon the optical prop-

erties of the atmosphere, the properties of both the object sighted and its background, and

illumination conditions. Middleton [24] derives the so-called air-light equation given by

L = Loe'Kx + Lp(1-e "Kx) (10)

• where I_ is the intrinsic radiance of the .ct

Lp is the radiance along an infinite path through the atmosphere7

; K is the volume extinction coefficient

Tverskoi [25] derives a contrast equation:

Co tc = (11)1 - 1)o 1

where CO is the ratio of the difference between background and intrinsic radiances to background

radiances, and botS a brlghtnes', factor. This equation can be simplified:

22

1974023760-020

I ! I ' ] , i !

I 1 i I i !

FORMERLY WILLOW RUN L.ABOHATOWIES, THE UNIVERSITY OF MICHIGAN

_4

!!

!

' ! ' ' " ' " ! " -' " ' " 5'0 " 0 " 7*0 'TEMPERATURE(OC)+10 0 -10 -20 -30 -40 - -6 - '_

25 :i

2o

" _ Tropical

15 Tropopause i

_ InversionLayer

0 , J , _A_._....______ i ,TURBIDITY FACTOR !_0 1 2 3

FIGURE 3. AEROSOL AND TEMPERATUREPROFILES FOR 4-5NOVEMBER1964 [22]

23

1974023760-021

i I ' !t i f i 1= ' ' J _ I I

FORMERLY WILLOW kUN LA_ORATOltlIES. TIlE tJNIVlEflsrr¥ OF MICHIGAN

C -- C e "'_x (12)O

The visual range in km is that distance at which the relative contrast is 2%, i.e.,Q

f n(C/C o) = { n 0.02 = e"KV (13)

or

g = { n 0.02 = 3.912 (14)K K

: Equation (14) is usually taken as the defining equation for visual range at a wavelength ofi

0.55#m, which is near the peak of the human visual-response curve.

24

i

1974023760-022

!

1 I I I I ....,....

i' ' ' FORMERLY WILLOW RU_;'"I,ABORAT_RII[S. TI-_E UNIVERSITY OF M,CHIGAN _

.5

i OPTICAL PARAMETERS _''iSection 3 considered the physical state of the atmosphere. In this section we deal with

the optical properties of ihe atmosphere; specifically, we define and calculate the appropriate _

atmosphericopticalparametersusedinradiative-transfermodels. _

4.I SCATTERING THEORY _

Radiation which passes through a homogeneous medium fr _e of any discontinuities will ::• causewaves tointerfereinsuchaway thatno scatteringtakesplace.Realisticmedir,attem-

peraturesabove absolute zero, however, t,,ave discontinuities :such as crystal impurities, den-

sity fluctuations,and varioussizedparticles;Eartn'satmosphereissucha medium. We

assume, moreover, that all scatterings are independent, that Ls, that there is no phase relation i

betweenscatteredwaves. For thistobe true,thedistancebetweenscatteringcentersmust

beatlea_tseveral times the size of the scattering centers tl',emselves. This condition is

certainlyfulfilledforatmosphericImzesand evenfordenseEogs.Giventhis.r"vs;.:alsetting,

we willnow determinethebasicopticalquantitiesneededforradiative-transferanalysis.

4.1.1RAYLEIGH SCATTERING _'

The theory of light scattering in the atmosphere originated when Lord Rayleigh [26] found "_

an explanationforthebl,;ecoloroftheclearsky. Itisne'_known thatthishue istheresultof

thescatteringofradiationfrom densityfluctuations,ratherthanfrom moleculesas bad been

assumed earl._er.Tfwe h_.vea planeelectromagneticwave withelectricfieldstrength _:

_(_',t)= _oei_'r''_t) (15) _iL"

where k"isthecomplexpropagationvector

r isa coordinat_positionvector

is the angular frequency _

thenthiswave impingingupona scatteringcentelwhosesize,ismuch smallerthanthewave-

lengthc_theradiati'mwillproducescatteredwaves i._an efl'ectcalledRayleighscattering.

The scatteringcross-sectioncan be obtainedby integra.t,ng_he Poyntingvectorovera period I

of oscillation. The result is

8_3tm 2- 1)2 6+35 '_

R(k)= 3N2 ._----. _ (161

11

1974023760-023

_R_M ' FORMERLY 'WILLOW BUN LABORATORIES. THE UNIVERSITY OF MICHIGAN

where N is the molecular number density

k is the wavelength

5 is an anisotropy parameter t_ account for polarization effects

m is the refractive index of the medium

Of importance here is the 1/k 4 dependence of the cross-section, a characteristic of radiation

from an oscillating dipole field.

It can also be shown that the scattering pattern is described by the following formula:

_ p(ccs X) = 3(1 + cos 2 X) (17)

I in which X is the angle between th_ incoming direction and the scattered direction. The function

i p(cos ×), which is called the scattering phase function, is normalized to unity over all 4zr ste r-

adians, that is,

jp(cos×)d_=1 (._8)

I "where _2is the solid angle.

f 4.1.2MIE SCATTERING

-_ The scatteringof electromagneticradiationby homogeneousdielectricspheresiscall_d

Mie scattering.Inprinciple,determiningthescatteringand absorptiftncross-sectionsa,ac'

thescatteringphasefunctionisstraightforward.First,thewave equationissolvedinside

i and outsidethesphere;thenthesolutionsare matchedattheboundarytodeterminethecon-

stants.The intensityoftheradiationscatteredintoan angleX isgivenby

x2 IS12+ IS212

I-. 4rt2R2-- 2 Io (19)

where R isthedistancefrom thesphere,and I° istheincidentintensity.The scattc-_ng

amplitudesS112and 133 are givenby

_2_+lrt _s1 =/., _-(TT-iTt,_ _+_ 1 !

(20)

P=I

InEq. (20)a_ and b_,whichare theMie coefficients,are givenby

i

1974023760-024

t

iIM +*+

IrORM_'RLY WILLOW RUN [ A+('I+ATO_IFq; THE + UNIVERSITY OF MICHIGAN ,_

++'i+

+_ (mx)_V+(x) - m_V+(mx)+_ (x) +

a = +_(mx)_p(x) - m,_(mx)+_(x) (21)

• - ,+(mx),i(x)b = mg,_ (mx)_t(x) - _(mx)_ _ (x) (22)

.+

where m and x, respectively, are the complex refractive index and size parameters--that is. '_

m = m1 - im2 and x = 2_r/X when r is the particle radius. The functions _V_and _ are the

Riccati-Bessel functions, and a prime indicates differentiat:on with respect to the argument. _,_

The +_ and r_ functions are given by I

dP_(cos X) (2_)_(cos X) = d cos X

and

d+r_(cosX)TI(COS_)= COS X+;_(COSX)- sin× d cos X (24)

where Pi(cosX)isa Legendrepolynomial,The scattcrir.,gcross-sectionis

, as(_, r, m)= _r2Qs(X, m) = _-___1(2_ + 1) a_12 + Ib_ (25)

where Os(X, m) is called the scattering efficiency factor. The total (scattering plus absorption)cross-sectionis

h2 co

at(k.r,m) = :rr2Qt(x,m) = _--__"_(2_+ l)Re(af+ bp) (26)P--I

where Ot(x m) isthetotaleificlencyfactor,Likewise,theabsorptioncross-sectionisthen

aa(_, r, m) =at(X, r, m) - as(k, r, m) (27)

with a co,'resvonding efficiency factor Qa(X, m). As in the Rayleigh case, a scattering phase ifunction can be defined:

12P(cos x) = 2_X_Qs(X' m)

Calculating all the functions given above for various refractive indices and size param-eters is quite involved. Nevertheless, computer programs have been written which allow the

performa:,ce a this analysis. Having obtained a program written by J. V. Dave. we used it to

27

1974023760-025

..... ii_R_M FORMIr R_Y WILLCI_N RUN LAB(_RATORIES. THE UNIVERSITY OF I_IICHtGAN

calculate the cross-sections and phase functions. For example, we have calculated the scatter-

ing efficiency iactor for homogeneous spheres of refractive i,,Jices m = 1.29, !.29 - 0.0465i,

and 1.28 - 1.37i. The results are shown in Fig. 4. It should be noted that the efficiency factorJ

is greatest for the real index (that is. when there is no absorption) and decreases wi_h increasing

imaginary index. The absorption efficiency factor has also been calculated for m -_ 1.28 - 0.0465i

and 1.28 - 1.37i. Here the efficiency factor is large for a high imaginary index and small parti- ,q

cles. as illustrated in Fig. 5. Finally, the total efficiency factor is shown in Fig. 6 for the same

set of refractive indices. Thus, it ca_,_be seen that efficiency factors vary strongly for different i

refractive indices and size parameters, i

4.2 ATTENUATION COEFFICIENTS

Knowing the cross-sections, one can then calculate the scattering, absorption, and extinc-

tion coefficients by multiplying the cross-sections by the particle number density. For real

atmospheric conditions characterized by b.Rze, fogs, and dusts, there is a distribution of p_,rticle

sizes. (One distribution, characterized in Section 3.1.2, is the modified g_tmma distribution.)

Thus. for a polydispersion, one must integrate over particle size to obtain the absorption,

scattering, and extinction coefficients:

_A(X. m. z) = _Oa,A(k, m, r)n(z, r)dr (29)

00

_A(_. m, z) = J_s,A(X, m, r)n(z, r)dr (30)0

KA(X, m, z) = I et,A (k' m, r)n(z, r)dr (31): 0

where _, t_, and K denote absorption, scattering, and extinction coefficients

A designates aerosol

n(z, r) is the aerosol-particle number density at altitude z for parH_tes in size range

Ar at size r

The number density is normalized as follows:

N(z) = _,(z, r)dr (32)M

0

where n(z) is just the total p,%_loer density. Likewise, the corresponding coefficients for mo-

lecular scatteringe_-,,oe found. The scatteringcross-sectionisgiven by Eq. (16),and the ab-

sorptioner_s-section is usuallytaken tobe thatfor ozone inthe visiblespectralregion. Thus,

we have for the complete atmosphere: 28

1974023760-026

|

H FORMI=RLY WILLOW RUN LAOORATORIE_ Tile UNIVERSITY OF MICHIGAN "_!

4 •

m= 1.29 !

g

a

1 :'

0 0 5 I0 15 20

SIZE PARAMETER (x = 2_,r/_)

FIGURE 4. SCATTERING EFFICIENCY FAC fOR FOR HOMOGENEOUS '_SPHERES OF COMPLEX REFPACTI_E INDEX m '_

: ii_r 29 ,

.,|i

197402:3760-027

!

.t0_ 3 --

z

5 I0 15 20SIZE PARAMETFI_. (x = 2_r/_.)

FIGURE 5. ABSORPTION EFFICIENCY FACTOR FOR HO._._OGENEOUSSPHERES

I OF COMPLEX REFRACT._VE INDEX m

3O

L

1974023760-028

I 1

FI_MI_PI_Y WILLOW RUN LAmO_ATOilhI[$.. riG' UNIVl[l_SITy OF MiCHIgAN

m : 1.29

m = 1.Y.9- 0.0465i3

Z 1m = 1.29- 1.3"/i

o _

?

0o s 1o is 20

SlZE PARAMETER (x= 2_r/;t)

FIGURE 6. TOTAL EFFICIENCY FACTOR FOR HOMOGENEOUS SPHERES OFCOMPLEX REFRACTIVE INDEX m

,i

M

,|

1974023760-029

i

, ; I _ ] , 1

"I gramL"_J _- _ _ FORMI[RLY 'WILLOW RUN LAaORPTORtES. _HE UNIVERS_V OF MICHIGAN

a(x. m. s. z) : cR(x. z) +OA(X.m, s. z) (33)

j;(_., m, s, z_ : _]R(_.. z) :_A(_, m, s, z) (34)

K(X. ,n.s.z) : KR(_.,z)* _A()t,m, s, z) (35)

where we have shown the explicitdependence on a complex refractiveindex m and P particular

stze distr;butions.

' Itis sometimes use._ultodeal with _n average cross-sectionfor aerosols. Assuming

ttmtthe size distributionisaltitude-independent,we ca:iwrite Eqs. (29),(30),and (31)as

GA_t, m. z): N(z)_a,A(k,m) (36)

_A(_. m, z) : i_(Z)_s.A(_,m) (37)

KA(X, m, z) : N(z)_t,A(_, m) (38)

where

_i,A(X,m) = lai,A(_, m, r)_(r)dr (39)0

inwhich the function_(r) isnormalized to I a,, the index ican indicatea, s,or t. We have

computed many values ofthe average cross-sectionsfor homogeneous spheres of various re-

fractiveindicesand size distributions.The resultsofthisanalysiswillbe presented later.

4.3 OPTICAL DEPTH

For radiative-transfer calculations it is usually more meaningful to deal with the dimen-

sionless quantity, optical depth in a medium, rather than with actual distances. We define

optical depth as

7(_., h) : _ K(X, z)dz (40)h

and opticalthicknessas

o0

Vo(_)= tK(X, z)dz (41)

where h issome definitealtitude.Thus, atthe top ofthe atmo,_phereT(_,h-_) = 0.while at

the bottom _(k,h=0) = To(_).Opticaldepth can be thoughtofas h_e distanceintoa medium

exrreFsed inunitsof mean free photonpaths. A small T° indicatet_thatlittleattenuationtakes

32

1974023760-030

! 1 !

%

J

IrORM£RLY WILLOW PtUN LABORATORIES. THr UI_iVERSrrY OF MICHIGAN '+

place, whereas a large TO means that the atmosphere is either strongly absorbing or scatters

much of the radiation. Optical depths can easily be obtained for R_y!e_.g_ atmospheres. +_

Elterman{27]has tabulatedtheresultsforallaltitudesfrom 0 to50km and forselectedwave-

• lengthsfrom 0.27_ to4.00_rn.The opticalthicknessvariesfrom 1.928at0.Z7_ to0.001

at 1.67 _m. "_

Likewise, if the extinction coefficient for aerosols is available, the corresponding aerosoloptical depths can be determined. By analyzing many experimentally determined aerosol pro-

files, Elterman [28] has calculated the extinction coefficients and optical depths for realistic -f

atmospheric conditions. For a 2-km visual range, the aerosol optical thickness is 2.521 at ++,

0.36 _rn and 1.053 at 0.90 _m. These resrlts indicate that multiple scaLtering occurs in Earth's

atmosphere since the mean free photon paths are short compared to the actual distances

traveled.

4.4 SINGLE-SCATTERING ALBEDO

A very important parameter in radiative-transfer analysis is the single-scattering albedo,definedas

_]R(_,z) m, s,z)%(x,m,s,z)=- Kix,m,s,z) i42)

?

where K(X,m, s,z)isthetotaliRayleighplt _rosol)extinctioncoefficient.The albedo

_o(_,m, s,z)isthefractionofscatteringwl_chcanoccur.Thus,ifthereisneitheraerosol

absorptionnor ozoneabsorption,then_o(X,m, s,z)= 1 andwe havea purescatteringatmo-

sphere.For stronglyabsorbingaerosols,however,_]A(k,m, s,z)issmalland Wo(k,m, s,z)

I can be as smallas 0.09.

Inordertoconstructmodelatmospheressire11atingvariousdegreesofcontamination

resultingfrom pollutedairmasses or varyingweatherconditions,we shalldo thefollowing:

(I)use Elterman'svaluesofthetotalextinctioncoefficientKiX,z)correspondingtovarious

visibilityconditions;(2)calculatefrom theMie scatteringtheorytheaveragecross-sections

forseveralrealisticrefractiveindicesand sizedistributions,and (3)use theseresultsto

calculatethesingleoscatieringalbedo.

We can definean aerosolabsorptivityparameterfas follows:

aA(X,m, s,z) Niz)a"a A(X,m, s) aa,A0_,m, s)f- -- -- _ ' = (43)KA(X,m, s,z) N(z)_t,A()t,m, s) _t,A(k,m, s)

Also,since?

_A(a,m,s,z)

f= I - _A(_,m, s,_) (44)

] 974023760-03]

I i, i I

Y[RIMFORMERLY WILLOW RUN LABORATORIES THE UNIVERSrrY QF MiCHiGAN

we have

_R(X. z) + (I - f)gA(_, m, s, z)%(x.m.s.z)-- _(_,v,z) (4_)

whichreducesto

fi_R(x,z)-(1-fk_(x,z)_o(X.m, s, V. z) = x(x. V, z) + 1 - f (46)

Thus. we have the single-scattering albedo for a realistic atmosphere in terms of wavelength

k. refractive index m, size distribution parameter s, horizontal visual range V, and altitude z.

It is interesting to consider special cases. If there is no aerosol absorption, aA --0 and f = 0 or

aR(x,z)%(x,V,z): I _(x,V,Z) (47) ,

Ifthereisnog,_seousabsorotion,aR =0 and

z)%(_,m.s,v,z):;-f K-_.V,z-)) (4s)

Ifthereisnoaerosolscattering,_A --0,f= 1 and

_R(x,z)%(_,v, z): K_, v,_ (49)

Finally, if there is no gas :ous scattering, _R =0 and

_o0_,m, s,V, z)= (1- f) -x(_,V,z)) (SO)

The Mie scatteringcomputerprogram was runtoobtainvaluesofaa,A(X,m, s)and

(_t,A()_,m, s),fromwhichtheaerosolabsorptivityfwas calculated.Thiswa: donefor

Deirmendjian'shazeL (contine,tal)and hazeM (maritime),as wellas forcomplex refractive

I indicesof thefollowing:¢'1.5

1.5- 0.01i

m = 1.5-0.10i

1.5- 1,00i

1.5- _oi

These correspondtoalldegreesofabsorptionby aerosols,from no absorptionatm = 1,5to

completeabsorptionatm =1.5- col.For theformer,f= 0,and forthelatter,f: I. Figure7

illustratesthespectraldependenceofabsorptivityforvariousrefractiveindicesandhazes.

To get"exact"valuesofcrosssectionsusingtheMie scattcringprogram,onewouldhaveto

34

1974023760-032

, J J' i I i I

.+.,_II+IM !FORMLrRI+Iy WILLOW RUN LABORATORIES 1' H I_ UNa +ERSlTY OF MICHIGAN !_

-+_

++i

0.6

_._e_t_..,._. __m= 1.5 - 1.Ot _.

0.4 -_

m = 1.5 - O.lOt

_ o.3 -

_ 0.2 -

0.1 _ m - 1.5 - 0.01i

I I [ I I I I I00.4 0.5 0.6 0.7 0.8 0.9 1.0 I.I 1.2 1.3

WAVELENGTH (11m) !

FIGURE 7. DEPENDENCE OF THE AEROSOL ABSORPTIVITY -+

PARAMETER, [,ON WAVELENGTH FOR HAZES L AND M !9WITH THREE REFRACTIVE INDICES _

3s ,_

'l ii

1974023760-033

I I ! 1

' i i 1

FORMERLY' WtLLOW _UN t-At_ORATORI_'$. THE UNtVEASFrY OF iblICHtGAN

run the program inhnitely long. However, sufficiently accurate values can be obtained after a

reasonable amount of calculation since the values tend to an asymptote. Our program was run

until the (ixed values had only a few percent change as a function of the size parameter x. De-

tails of the computational procedure can be found in Deirmendjian [10].

We can now calculate the single-scattered albedo as a function of wavelength for various

size distributions, refractive indices, altitudes, and visual ranges. Figure 8 illustrates the

spectral dependence of wo for five altitudes. This is for a refractive index of 1.5, that is, no

aerosol absorption. At the surface, _o is unity because the ozone contribution is very small.However, as we go up in the atmosphere, the ozone absorption band near _).6 _'n becomes

quite evident. Figure 9 illustrates the same eftect for a heavily contaminated atmosphere with I

high absorption. Here wo is only about 0.5 near the surface, increasinf; slightly as we go higher

into the atmosphere. The low value of _o near the surface is the result of strong aerosol ab-sorpt'on in the dense lower troposphere.

To s_e the effects of aerosol and ozone absorption throughout the entire atmosphere, we

will look at altitude profiles of _o" Figure I0 shows the profile for a hypothetical atmosphere,both with no aerosol absorption and with the maximum absorption. Aerosol absorption is

especially pronounced in the lower troposphere, and the relatively strong ozone absorption

occurring at higher altitudes is quite evident.

The profiles for various wavelengths are portrayed in Fig. 11 for a dense haze. This case

represents the maximum amount of absorption (aerosol and ozone) 'rhich can occur. It should

be noted that in the lower troposphere most absorption occurs _.t the longer wavelengths, since

aerosol absorption increases with wavelength beyond a certain point (as was seen in Fig. 7).

At higher altitudes, however, the ozone band near 0.6_n dominates. Figure 12 shows the same

effect except that the case is _ne of no aerosol absorption.

We now turn to the more realistic conditions of partial aerosol absorption. Figure 13 il-

lustrates the altitude profile for three refractive indices: m = 1.5 - 0.01i, or little absorption:

m --1.5 - 0.10i, moderate absorption; and m = 1.5 - 1.0i, heavy absorption. Note that there is

quite a variation in wo in the lower troposphere.

The altitude profile for wo in the first 5 km is shown in Fig. 14 for the two hazes L and M

and for various amounts of absorption. Varying the size distribution seems to have only a

minor effect on the profile, as opposed to changes in the imaginary part of the refractive in,_,

Finally, we can see the effects of a change in visual range on the single scattering albedo

_o' Figure 15 shows that _o is essentially col,=tant a_ a function of visual range for a givenhaze. increasing rapidly as ti_e amount of aeroso_ decreases. As any haze (contaminated or

36

i

1974023760-034

1N'" - iFC_MERLY WILLOW NUN LABORATORIES. TH I_ UNIVI_RSITY OF" _ICNIGAN

o I I I I_km

3°

o I I I 135kin

<

o I I I 120kin

Z

o I I I 11okm i

o I I I I_.Ikm0.4 O. 5 0.6 0.7 0.8

• WAVELENGTH (/_m)

FIGURE 8. DEPENDENCE OF SINGLE-SCATTERING ALBEDO ONWAVELENGTH FOR VARIOUS ALTITUDES--REFRACTIVE IN-

DEX m = 1.5. Visual range = 2 kin,haze L.

37

_._, ......."........................ _ -I'IIIIIII ,

1974023760-035

i I , ! j I I

FORMERLY WILLOW WUN LABORATORIF_.. THE UNIVERSITY OF MICHIGAN

1

0 I I I I I0 km

I ] I I Okra !00.4 0.5 0.6 0.7 0.8

WAVELENGTH (_m) " 1!

FIGURE 9. DEPENDENCE OF SINGLE-SCATTERING ALBEDO ONWAVELENGTH FOR VARIOUS ALTITUDES--COMPLEX REFRAC-

TIVE INDEX m = 1.5 - 1.01.Visual range = 2 kin,haze L.

38

1974023760-036

i i ' !

= , j i I ....._._ .

;iFORMERLY WILLOW RUN LABORA'rORIES. THE UNIVERSITY OF MICH GAN "_

R

°145 i/

40 -- ( Ozone No Ozone

I

' !._._-- Complete Aerosol35 -- % .;

30- I

%

25 \

< No Aerosol Absorption ..-_1

20 _

15 \\ _

%

%\

II

5 %%

Io I., I I L I I I0 0.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

SINGLE-SCATTERING ALBEDO

FIGURE I0. ALTITUDE PROFILE OF THE SINGLE-SCATTERING ALBEDO WITH "4_

AND.WITHOUT OZONE. Wavelength = 0.6/Jm.

1

39

!

1974023760-037

! I

+ t I 1

"........... _,- I _ j !

_-_RIM +o__.+,,,Lo,,,.,+,.o,,,,o.,+,+,....,+,+,o+++,0,

I 0 I I 1 1 l 1 I, 0 O.I 0.2 0.3 0.4 0,5 0.6 0.7 0.8 0.9 l.O

, SINGLE-SCATTERING ALBEDO (_o)

FIGURe: 11. ALTITUDE PROFILE OF THE SINGLE-SCATTERING ALBEDO WITHCOMPLETE AEROSOL ABSORPTION. Visoal range = 2 kin.

4O

ii

1974023760-038

I H

FORMERLY WILLOW RUN LABORATORIES. THE UNIVERSITY OF NIICHIGA_

50 _ /// /"

/ //45 / /

/ // /"

k = 0.60_m / (_401 |X = 0.70/_m . = 0.50/_m

\ I !

3s \ _,.\

30 \ \

\ \25 \ \

_ ,_ _.

_.

\\15 \_

10

1o I I I I [ I I I0.I 0,2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

SINGLE-SCATTERING ALBEDO (coo)

FIGURE 12. ALTITUDE PROFILE OF THE SINGLE-SCATTERING ALBEDOWITH NO AEROSOL ABSORPTION. VisualL.-.nge= 2km.

41

1974023760-039

i

_,, flLABORATORIES, "THE UNtVERSI'TV OF MICHIGAN ,

50 '!

!

;_ = 0.60gin k = 0.40gin

'/3O

N m2= 0

20 _

: '/IX _m_ _ _

m 2 = 1.0 _.

10 ....

,;g

0.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ._

SINGLE-SCATTERING ALBEDO (tOo) i

13. ALTITUDE PROFILE OF THE SINGLE-SCATTER.ING ALBEDOFIGURE

FOR HAZE L, Visual range = 2 kin, complex refractive index rn - 1 5 - im 2. i_

'2 1ji

J

I

1974023760-040

1 1 1! I I i !

,?

_IMFORMFRLY WILLOW RUN LABORATORIES. THE UNIVERSITYO'Jmm'm=w_MICHIGA'mmmmmwm_ '¢,_

, !!

IHaze L I

------ Haze M I

, i;| I

3--. m2: I _ I

I m 2 = 0.10 J _[,-' j ,_• ,-,1

I

m 2 ::0.bl

i

I--. I 4

l ii

I 1 1 " I..... ,, J. I , ._.1 0.2 0.3 0,,L 0. 0.1 0.7 0.8 0.9 1.0

"iiSINGL;,,-3CA_ITERING ALBEDO (o_o)

FIGURE 14. ALTITUDE PROFILE OF THE SINGLE-SCATTERING ALBEDO

IN THE LOWER TROPOSPHERE. Wavelength=0.gpm,complexrefractive ,_

indexm = 1.5- im2,visualrange= 2kin.

1

1974023760-041

_RIM .o.._._. wiLLow nun ualaOnaTO_(S t.l[ UniV£RS,.Y 0 ..... _a.

" RfM_ m m FOMMIr_LYIW|LLOW RUN L&OOR&TORf_[5 TPII[ U/_liV&-_bl_ • _Lr mOIC_._jAL/

not) decreases, the _lbedo _o approaches one cimr_ ;terizir_, a pure, aerosol-free l_yleigh

atmosphere. It is clear that the aominant effect results from a change in refractive index.

4.5 SCATTERING PHASE FUNCTIONS

Besides optical depth and the single-scattering albedo, the radiation field within the atmo-

sphere depends strongly on the single-scattering phase function. Assuming spherical particles,

we have used th_ Mie theory to calculate phase functions for various refractive indices, wave-

lengths, and size distributions. Figure 16 illustrates the dependence of the phase function on

the imaginary part of the refractive index. It is interesting to note that for little or no ab-

sorption there is the asual peak in the backward direction. For strong abscrption, however.

the backward peak disappears and the distribution becomes more anisotropic.+

The wavelength-dependence of the phase function is exhibited in Fig. 17 for a weakly ab-

sorbing aerosol. The dependence is as one might expect from elementary scatte_.ing theory.

For shorter wavelengths the function is more peaked than for longer wavelengths. This is

also true for the backward part of the function.

TI,_ effect of varying particle size distribution can be seen in Fig. 18 for aerosols with no

absorption. As in the ca,.e v.f single-scatte,-ing albedo, the predominant effect comes not from

size distribution but from cowposition.

Tables of the scattering phase functions for 13 wavelengths, 10 visual ranges. 4 refractive

indices, 2 hazes, and a complete range of altitudes from 0 to 50 km have been formulated for

use in computer programs to calculate the radiation (ield. The implementation of these results

in our radiative-transfer studies will be presented in Section 5.

To sum up. we can calculate the important basic optical parameters for an atmosrhere

characterized by scattc,_ing and absorbing, spherical aerosol-particles, and by ozone absorption.

The single-scattering albedo can be c,'tlc_dted for any refractive index and particle size dis-

tr,bution, and we can model any kind of vertically inhomogeneous atmosphere. Likewise,

single-scattering phase functions can be calculated for any composition and size distribution.

However. for inhomogeneous, non-spherical particles, actual experimental data on the atten-

uation coefficients and phase functions must be used.

45

L I

1974023760-043

102_ I

fx\

lO-I I I I I I0 30 60 90 _,0 _so _80SCATTERINGANGLE(deg)

FIGURE 16. SCATTERING PHASE FUNCTIONS AS CALCULATEDFROM MIE THEORY FOR HAZE L--COMPLEX REFRACTIVE

INDEX m = 1.5 -Im 2. Wavelength = 0,55_m. .;,t

:6 ._

1 t

]97402:3760-044

i liRa , FOR_RILV WILLOW mUN LA_IRATORIF.$. THF UNIV£Rsn'Y 0_" Idh_MIGAq

102 I

X = 6.40p.m

="--"-_X = 0.55p.m

_"_,X :: O.'TOp.rn

- 1.06_ mm t

lo1 -:Cm.

r_L_

<

Z _

100 _r.,j

t

_ j_ OOOOoooo 1

¢

x

0 30 60 90 120 150 180 ,_SCATTERING ANGLE (deg)

FIGURE 17. SCATTERING PHASE FUNCTIONS AS CALCULATED ._FROM MIE THEORY FOR RAZE L--COMPLEX REFRACTIVE

INDEX m = 1.5 - 0.01i.

!

1974023760-045

1

• IroRme'Rt.+ WILLOW RUN LA_RATt[_IqII[S. THE U_IVS'RS,_Y OF MICH_&m i

|

-1102

- i

I Haze L

Haze M

]

X_0

o 101O,

Z _

0Z

[., i00<

/

lo-1 T I0 30 60 90 12(; 1,50 180

SCATTERING ANGLE (deg)

FIGURE 18. SCATTERING PHASE FUNCTIONS AS CALCULATEDFROM MIE THEORY FOR HAZE L AND HAZE M. Wavelength --

0.55_m, complex refractive index m = 1.5.

48

• __ J

i I i I i

1974023760-046

2

FORMIrRL¥ WILLOW RUN LABORATORIf.'S THI[ UNIVERSITY OF MICHIGAN

5INHOMOGENEOUS ATMOSPHERES ._

iWe now consider the case ofa layeredatmosphere inwhich each layerishomogeneous

with regard to atmospheric optical parameters. More exact formulations [2g I involving per-

turbationtechniqueshave dealtwith the generalinhomogeneous l_'lyleighatmosphere, but no .:

one has solvedthe generalproblem for anisotropicscattering.

¢5. I THE DOUBLE-DELTA APPROXIMATION

The one-dimensional, radiative-transfer equation can be approximately ._olved for highly

anisotropically scattering atmospheres by using the so-called double-delta approximation

[30, 31, 32] . The resulting equations, however, are different for Wo_ I, becoming

dE_(r)dr = g(T)E_(T)- h(z)E'(z)- /_oh(r)Es(Z) (51)

#

dE'(r) :_-

dr : g(r)E'.(_-)- h(r)Z_(T)+ [.og(r)- liEs(7) (52) -'i

for the anisotropic field. The functions g(r) and h(r) are _.

i

g(r) = [1- _0o(r)))(r)1/._ (53) _?,a

h(r) = ¢_o(r)[1 - rl (r)]/% (54) :

where )7(r) is an anisotropy parameter. Similarly, the isotropic field E" can be represented

by _

dE_(T)

dr = 2/_o[g(r)Z+(r) - h(T)E_(T)] (55)

dE_(T) -.

dr : 2fto[h(r)E+(_') - g(T)E"(T)I (56) ._

Solving these two sets of coupled differential equations with the appropriate botuldary conditions _.

allows us tc define the fiz st "iterative" solution. Knowing the irradiances, we can form the 'g_

functton I

49

I ,t I J J 1 _ I

1974023760-047

_jRJM FORMERLY WILLOW RUN LABORATORIES THE UNIVERSITY OF MICHIGAN

1

h(_, /_,_) : _ [E'+(7)5(_ - _o)5 (_ - _o - ¢')�E'_(T)5 (_ + _o)a (_ - _o)I

+ 2_ (57)

and insert it into the transfer equation. Representing both the single-scattering albedo _{7)

and the scattering phase function, respectively, by the following step functions:

n

_(7) =_-aWiX(Ti,Ti_l)(T) (58_I=I

n

p(7,iz.P. /_',P') = pi(/i,_b,/I _')X(_ _ _('r) (59)'i"i-l'i=l

we lhen can easily solve the radiative-transfer equation for each layer.

5.2 CALCULATIONS

We now present some results of our experience in using the inhomogeneous radiattve-tr,t_,_-

fer equation to find aerosol absorption. Figure 19 shows the spectral dependence of path radi-

ance and total radi_mce for two extreme values of refract;re indices. Path radiance for the

contaminated atmosphere is about one-half that for the uncontaminated atmosphere, although

spectral dependence is nearly the same. Likewise, the total radiance exhibits the same behavior.

In Fig. 20 we see the spectral dependence of path radiance and total radiance for the same

conditions as those in Fig. 19, exceot that the visual range is 2 km instead of 23 kin. In this

c_se, the effect of aerosol absorption is quite pronounced in that the "clean" atmosphere has

radiance values many times greater than those for the contaminated atmosphere.

Figure 21 illustrates the d_pendence of path radiance on the single-scattering albedo cco.

Strictly speaking, the curves should coincide at w = 1, which implies pure scattering. Becauseo

ofa l'_mitingprocedure in the program, however, we useti:'_o= 0.999. The differencecan be

accounted for inthe differentphase functionsat m = 1.5and m = 1.5- 1.0i. Nevertheless,

the main effect,the strongdependence ofpath radiance on O:o,isquiteevident.

The dependence ofboth path radiance and totalradiance on altitudeisexhibitedin Fig.22

for the two refractiveindices For allaltitudesboth the path radiance and the totalradiance

are greaterforthe "clean"atmosphere thanfor the dirtyone.

50

1974023760-048

Total Radiance ?

<

¢\

\

= . _ /m 2 = 1.0

1 -- m2 00 \.\_, -

o ,I I I I I "I0.4 0.5 0.6 0.7 0.8 0.9

WAVELENGTH (# m)

FIGURE 19. DEPENDENCE OF PATH RADIANCE AND TOTAL RADIANCEON WAVELENGTH FOR HAZE L--VISUAL RANGE = 23 kin. Altitude :- 1

kin, solar zenith angle = 30°, nadir angle = 0°, target reflectance : 0.1, back-

ground albedo -- 0.1, complex reiractive index m = 1.5 - im 2.

51 _

• '.... i I J

1974023760-049

FORMEtQL Y WILLO_P_'RVN L.ABORATORti[_.. THE UNIV_'RSfTY OF- MICHtGAN

10 - _ Total Radiance

9 -- _ -- _ Path Radiance

7 -, / _ "_2 :o.o_ 6- •

t

\

3 -- _ _ 'i

m2 = 1.0

Im 2 = 1.0 " "-' --, ..._ --- ..-.

t .... I I I I _,.,0

0.4 0.5 0.6 0.7 0.8 0.9WAVELENGTH(_ m)

FIGURE 20. DEPENDENCE OF PATH RADIANCE AND TOTAL

RADIANCE ON WAVELENGTH FOR HAZE L_VISUAL RANGE = 2kin. Altitude = 1 kin, solar zenith angle = 30°, nadir angle =0o, targetreflectance =0.1, backgroundalbedo = 0.1, complex refractive indc.

m = 1.5 - im 2.

52

1974023760-050

,3 t

1974023760-051

FOR'MERLY WILLOW RUN LABORATORIES. THE'UNIVERSiTY OF MICHIGAN

5

4 _ Totai Radiance

o I I I I I0 10 20 30 40 50 60

ALTITUDE (kin)

FIGURE 22. DEPENDENCE OF PATH RADIANCE AND TOTAL RADIANCE

ON ALTITUDE FOR HAZE L--VISUAL RANGE = 23 kin. Wavelength =0.55pro, solar zenith angle = 30°, nadir angle = 0 ( , target reflectance = 0.1,

background albedo = 0.1, complex re[ractive index m = 1.5 -im 2.

54

L _ ' _ I "Ill I ' l '' ........

i l _ ,' '1 ,- I ,_ .j ! t

J i 1 J ii

1974023760-052

___jI FORMEHLY WlLLL_ RUN LABORATGi_JES THE ;.,f%IIVERSITY OF MtCHIGAN

The same s;_.uation is considered in Fig. 23,except that the visual range is 2 km instead of _i

23 kin. In this case, wo note a large difference between the path radiances and total radiances

for the two refractives indices. Also, one should notice the rapid change in radiance with

altitode, an easily understandable effect evident in passage through our heavily contaminated

lower trot, osphere. Figure 24 shows the same conditions as in Fig. 23, except that we consider

the variation within the first 5 kin.

Figure 25 illustrates the dependence of path radiance on altitude in the lov:er troposphere ._

for several refractive radices. It is clear that even a small amount of absorption can change

the path radiance by a significant amount. The results are qu.'_e sensitive to the i ._..aginary

part of the refractive index, especially when 0.01 "_ m 2 _ 0.10.

Figure 26 pox'trays the variations with altitude in path radiance, attenuated radiance, and

total radiance for the cases of no absorption and heavy absorption. Recall that

L T = LA + Lp (60)

• where LA, the attenuated radiance, is the product of surface radiance and transmittance. In

this case of a heavy haze (that is, for a visual range of 2 km), path radiance increases rapidly

with altitude for a "clean" atmosphere, and attenuated radiance decreases rapidly. The com-

binati¢,n produces a total radiance which at first decreases, then reaches a minimum, and

• finally increases to its asymptotic limit. In the case of heavy absorption, all radiances are

lower.

Finally, Fig 27 illustrates the behavior of path radiance as a function of the nadir view-

angle for different refractive indices. The peak occurs at an antisolar angle of 30 °. Again,

contaminated air produces small values of radiance.

These results clearly show the large differences between radiances for "clean" and "dirty"

atmospheric conditions. The index of 1.5 represents a silicate haze, not uncommon in various

pa_s of the world. The index of 1.5 - 1.0i is quite similar to that for soot particles found in

the contaminated air of urban areas. Typical refractive indices for various aerosols are pre-

sented in the work of Kondratyev et al., [ 33] for the 0.40- to 15.0-/_m wavelength region;

they confirm our original assumption of a very weak spectral variation m the refractive ,ndex,

at least for the visible and near-infrared regions.

In conclusion, we now have the ability to model any kind of contaminated atmosphere char-

acterized by aerosol and ozone absorption. The radiation field can vary considerably, depend-

ing upon the degree of contamination as represented by refractive index and particle size.

55 ,_

I

1974023760-053

t FORMERLY WILLOW RUN LABJRA"roRI*- S THE eJNIVERSI1 _"OF MICHIGe_N

56

' !

1974023760-054

i

, .i ,i ! ,' j

FORMERLY W{LLOW RUN LABORATORIES THE UNIVERSITY OF MICHIGAN

-:}{

?

II

I _ _ ,:z;o"_

I o_7I! se a

• o _ _ ._II

.'_ n '._

, 0_\ .- ° _ _, _,,

----2-/-d-..-..:'I_ _ _" _ _ "_ _ _ _ o _ ,._

,'.,5't i

c

g°

l .... 1 I 1 t

1974023760-055

TM |L , iiIroRM_mLY WILLOW r<UN LAIBORATORIIr_ THE IJNIVERSIT_ (_F J4I(_I-I_&N

I

FORMERLY WILLOW RUN LA_RATORI[S THE IJN;VI[k_)IT¥ OF MJ('_L*';,_4

Z

I1 :_

L T _10 :

-i

9 ""

m=l.5 \\

_ .

? 7 -

Z 5<

i •/ ,3 ;

1 m_.._ 1.5- 1.0i ---- i

o° I I I . I 1 i I I.! 1 2 3 4 s 6 7 8 9 iALTITUDE (kin) !

FIGURE 26. DEPENDENCE OF PATH RADIANCE, ATTENUATF.D RADIANCE

_. AND TOTAL RADIANCE ON ALTITUDE FOR NO ABSORPTION AND HEAVYABSORPTION. Wavelength = 0.55_m, solar zenith angle = 30°, nadir view ._angle = 0°,targetreflectance= 0.I,background reflectance= 0.I,visual ,,

range = 2 kin. "._i

59 i

, j_ _,, , ,, ,, , m m m m

1974023760-057

lO-1--L_L__2' I I [ I I 1 I I90so 6_ 40 20 0 20 40 60 8090NADIRANGLE(deg)

FIGURE 27. DEPENDEN _E OF PATH RADIANCE ON NADIR VIEW ANGLEIN THE SOLAR PLANE Ft_R SEVERAL REFRACTIVE INDICES. Wave-

length= 0.55#m, altitude= 9+kin,solar zenithangle = 30°, targetreflec-tance --0.1,background reflectance= 0.1,visualrange = 2 kin.

6O

iT

I j J £ I i i --

1974023760-058

4FORMERLY WILLOW RUN LABOPtATONIr$ TM[ UNIVE:R$ITY OF MICHIGAN

For remote sensing applications, in situ measurements of the air would aid in the definition

of aerosol content and, hence, would _11ow us to calculate the basic radiometric quantities, _

transmittance, irradiance, and radiance. If such measurements are not feasible, then we could 1

look at multispectral scanner data corresponding to the anti-solar direction in order to obtain i "

some knowledge of the phase function. This would help in *_e estimation of aerosol composi-

tion. Also, laser scanning or lidar techniques could be employed to determine the s_me qu`mtity.

=t

)

- ._.

?

Z"

4

]#

5

I

i 1 _ I 1 j I

1974023760-059

_nJl Ir4_Rlli[llL¥ WILLOW RUN L&BOR;,.I'ORt([$. TME uNrvEItsn'v OF MI_H_AN

6MULTIDIMEN_"_.ONAL RADIATIVE TRANSFER

In considering the transfer of r'_dlation, we can assume a plane-parallel atmosphere if we

exclude satellite remote sensing at very high altitudes. Almost all analysis in radiative-

transfer theory has dealt with a one-dimensional plane-parallel a.mosphere, that is, one in

which the optical parameters, _o and _, and phase function, p(cos X), vary only in the direction

perpendicular to the surface. It has been further assumed that the lower surface is a perfectly

diffuse one which is homogeneous in a horizontal plane. These simplifications are sometimes

necessary in order to solve the radiative-transfer equation. Although assuming that wo, _, and

phase functions vary only in the vertical direction, we will consider the surface to be a non-

homogeneous Lambertian type. This set of assumptions will enable us to calculate the influence

of background reflectances on the target radiance.

6.1 GENERAL THEORY

The general, three-dimensional, time-independent, radiative-tralJsfer equation for an

atmosphere which is plane-parallel, scattering and absorbing, but non-emitting is as follows:

_(_) f ^ ^__ A ^= _l. fl)L(R, _l') dG' (61)jr,,

^

where fl' is the unit vector in the direction of the scattered photon at positiQn R, and ]:(R, f_) is

the total spectral radiance, t.hat is, the sum of the direct solar component and the diffuse com-

ponent.

The first term on the right-hand side represents a loss of energy along a vath, whereasA

the second term represents the sum of all energy scattered into the direction ft. For _. one-

dimensional atmosphere, this can be written as

, ,2n 1¢O_T) f c

dL j]/J _-.': L(T, _o *) - -_-- plr, _, _, _', _')L(T, _', e')d_'d_'0 -I

%(r) -r/. °4_' Eoe P(_' q_' "_o' eo) (62)

where _, _ define the angular coordinates of the scattered radiation

p(T, _, _, _', _') is the single-scattering phase function

' E° is the extra-terrestrial solar irradiance of the top of the atmosphere '_i

_o' _o define the angular coordinates of the sun

One can immediately find a formal solution to the integro-diffcrential equation above by

using an integrating factor; the solution is

]974023760-060

I I

FOI_MER .1"WILLOW RUPI LABORATORJE,_ THE UNIVIERSfTY Or MICHIGAN

f

-(ro-r)/_}_ " LD(r. IJ,_1 = LD(7o, IJ. _)e

"72_ 1 o

i- +4,_._I_°o Ip(p,_./j,,_,)ie (,- -'-",)_LD_,T,,/j,_,)d.r,d_,d_,/ ,i 0 -I T

+ 4_(/J+ _o) - e e 163)

fortheupwellmgradiance.The downwellingradianceisgivenby

LDlr.-/J._#): LD(O,-_._#)e-T/_

2_ 1 v

+_--_- p(-,,,,/j'.,') _',,')dr'd_'d_'0-I 0

" _°P°E°P('ll'_'-"°'_°)(e-7'/u -v/"°) (64)+ 4_(p - _o ) -e ;_*Po

There are two general boundary conditions:

L(O. -_,,9)= o

2:;1

Liro./j,_b)= ! I_'O'(_,_,-p',_')[L(_o,-_'._')+Ls(rc,,-_',_')]d/;d_b (65)

wherep'(l_. _, /;, _') is the bidirectional reflectance of the surface. Thus, Eq. (62). to-

gether with the bou',dary conditions, can be used to calculate the complete radiation field in

a plane-parallel atmosphere having horizontal homogeneity.

6.2 THE UNIFORM DISK PROBLEM

We can study the effects of background on target by considering a uniform disk with a

! perfectly diffuse reflectance which is surrounded by a Lambertian background surface with a

i differentreflectahce.First,we shallconsideran isotropicscatteringlaw and calculatethesingly-scatteredsurfaceradiance,the singly-scatteredsolarradiance,thedoubly-scattered

! solarradiance,and theattenuatedradiance.Secondly,we shallcalculatethesame quantities

fsran anisotropicscatteringlaw.

-|

_ f 63

l '

]97402:3760-06]

YERIM_-I.i_ -- FORMERLY WILLOW RUN LA_ORATORIE._,. THI[ UNIVERSITY OF MICNIGAN

6.2/. ISOTROPIC SCATTERING

Using the one-dimensional se'ution of Eqs. (63) and (64), we can impose a coordinate sys-

I tern and obtain a solution for the case of a uniform disk surrounded by a spatially infinite, uni-

form background surf.ace. The appropriate geometry is illustrated in Fig. 28. Let us assume

that an observer is located at altitude h corresponding to optical depth _"and is viewing the

origin of the coordinate system, i.e., the center of the disk. Radiation emanates from point S

on the surface, is attenuated as it propagates to point Q. is scattered into direction QP, and

finally is attenuated to the point of observation at P. Mathematically, we iterate the integral

equation once for single scattering. Assuming isotropic scattering that corresponds to a very

light haze condition, we have for the result

L(r' l)-'- L(r°' l)e-('r°-r) t°°l_°E° i-v/IJ° -'°/_° 1+ 4_ (I + _o ) _ - e e -(T°-T

I

% - -(7o-_'),'.c

+ _--_[,eE_(Oe)- DiE <_li ,ce - (,o - _")EI [(,o - T')/Sic d

% - -(7e-T)

+_-_OeE_(Oe)_ +(TO-T- i)e -(TO-T)EI(" O- T)1

7 7 ]_20/i°E° -r°/_° + e -('r°-r) 1 - (r ° r + lie)El(re r) /_oEi r+_ oin _----_ - - - ]l

+ _in +_oE1 ro- 7) + _oEiL° V-ol-T 0

+ e - e + (r ° - _o)El(ro) -- (r - _o)El(r (66)

where /Jc = z2+z2

E 1 and l_i are exponential integrals

The first term represents the radiation from the center of the disk attenuated to point P. The

second represents the singly-scattered solar radiation. The next two terms represent the

surface radiation being singly scattered and attenuated to point P. And the last term is the

doubly-scattered solar radiation.

64

it

.j[

I i I' 1 j ',

l i ,t

1974023760-062

_--_jRiM F,3;_MERLY '_/ICCOW RUN LABORATORIES, THE UNIVERSITY OF MICIJIGAN

A more advancedcasehas alsobeendevelopedforbotha finitesolidangleofviewand

i| smallscanangles.The correctionterm tobe addedtoEq.(66)istooinvolvedtopresenthere,

- butresultsoftheanalysiswillbe. The geometryisillustratedinFig.29. The viewangleof

thesensorisP.steradians,altitudeish (inkm),visualrangeisV, interiorLambertiandisk

reflectanceis0i,and exteriorLambertianbackgroundreflectanceisPe" Inouranalysisthe

solidangleflofthesensorisa._sameutobe 2.5mrad: thereflectancesusedare illustrated

inFig.30. [nFig.31 we seethesingly-scatteredsurfaceradianceforblack,blue,red,and

whitediskson a greenvegetationbackground.Radiancewiththeblackdiskisessentially

thatforgreenvegetation(asexpected);radiancewiththebluediskisslightlyhigherfor

0.4-<;t<-0.65_n becauseofthehigherreflectanceofthedisk.For a redpanelwitha peak

inthereflectancecurvenear0.65_n,theradianceisslightlyhigherthan_ora blackdiskat

longerwavelengths.For shortwavelengths(i.e.,0.4< ;t<0.55_m) itisessentiallyzero.

Finally,fora whitedisktheradianceismuch higherthanfora blackor coloreddisk,butit

generallyfollowsthespectralshapeforgreenvegetation.

We can now examinetheeffectofever-increasingdisksizeon singly-scatteredsurface

radiance.Figure32 illustratesthiseffectfora blackdiskon greenvegetationfordiskradii

ol10m, 500m, 1000m,and 5000m. For thesmallestdisktheradianceissimilartothatfor

greenvegetation,whereasfora diskof100timesthatradius,theradiancedecreasesbya fac-

torof8. Thisillustratesthestrongdependenceofsurfaceradianceon disksize.Visual

range is 2 km. _.

Figure 33 is the same situation, except that we now have a white disk on a green vegetation

background. It should be noted that the effect is just the opposzte from that of the black disk.

As disksizeincreases,theradianceincreasesbecauseitzsdominatedby thehighreflectance

ofthewhitediskrelativetothelowerreflectanceofthebackground.Also,forsmalldisksthe

curvesare similartothoseforgreenveget_ttion,whileforlargedisksthecurvesapproach

thespectrallyflatwhitereflectance,biased(ofcourse)bytheintrinsicspectralpropertiesof

theatmosphere.

These effectsare illustratedina more meaningfulway inFig.34. Here thesingly-

sc,ttteredsurfaceradianceispresentedas a functionofdiskradiusforseveralwavelengths

bythelowerreflectanceofthedisk.At thepeakofred reflectance(k=0.7/_n),however,

radianceincreasesbecausethediskreflectanceishigherthanthatforgreenvegetationatthat

wavelength.Aftera radiusofabout5 kin,theinterferenceeffecthardlychanges.

We can now consider the relative magnitudes of various components of the total spectral ,_, ','_radiance received by a sensor. Figure 35 illustrates the relative magnitude for a black disk

on a greenvegetationbackground.Lsur isthesingly-scatteredsurfaceradiance,I,ssisthe

singly-scatteredsky radiance,Lds isthedoubly-scatteredsky-radiance,and Ltotisthesu_

6s i

i!

i J

1974023760-063

FORMERLY WILLOW RUN LABOe_A rORllrS, THE UNIVERSITY O1- MICHIGAN

Z

_Q

s

x

FIGURE 28. GEOMETRY FOR ISOTROPIC SCATTERING WITH AUNIFORM SURFACE

Pi _e/ ib_---R----d

FIGURE 29. DISK GEOMETRY FROM EQUATION (66)

i

1974023760-064

:¢

'4

FORMERLY WILLOW RUN LABORATORIES IHE UNIVERSITY OF MICHIGAN _

.2

i?

i

I00 "_"""_White Disk !i_,

96 - .I

80 -

70 -

tJ=-, 60-

Green Vegetation Y_

z 50-

Red Disk .._ .:''._9£d

4Old

30 - ./_ Blue Disk ..::. /".....

00.4 6.5 0.6 03 0.8 0.9

WAVELENGTH (_m) ,

FIGURE 30. TARGET REFLECTANCES

|

6'/ •

I ': ,k.,.,,m i - ,m,._.wn_" "m" I : ....... _,=.,w,.! .......:_":r_ IrJ.LI I ml _ i i i IIIIL II" i L _ i i

1974023760-065

_1_ FORMERLY WILLOW RUN LABORATORI(5, THE UNIVERSITY OF MICHIGAN

2.5

Disk

o I I I0.4 0.5 0.6 0.7 0.8 0.9 1.0

WAVELENGTH (_ m)

FIGURE 31. SINGLY SCATTERED SURFACE RADIANCES OF COLORED DISKS

ON GREEN VEGETATION BAC}.'GROUND. Disk radius = 10m, visualrange = 2krn,altitude= 1 kin.

68

I

1974023760-066

FORMERLY WILLOW RUN t ABORATORIE5 THE UNIVERSITY OF MICHIGAN ,_

i7_

"A

2.0 I

_ g

Z _

-. _: !o_ _.c__) 1.0

?

_o._- __ _:_OOOm0 _0,4 0.5 0.6 0.7 0.8 0.9 :"

WAVELENGTH (_ m) i"

FIGURE 32. SINGLY SCATTERED SURFACE RADIANCE FOR

BLACK DISK WITH GREEN VEGETATION BACKGROUND. Visual

range = 2 kin, altitude = 1 kin.

69 :1

1............................. I i isis it-: r=:-::z :'- :: r_e,, --_ .m_'?'_-- ..... .'T'" p, -;;Zr ........... - 7 m -'- Illl [ [ _ "_'_"

" J i' j' i i

]974023760-067

RIM .... FORMERLY WILLOW RUN LA6ORATORIIE$. THE UNIVER_r' ¢ O' MICHIGAN

R-- 5000m

5.0 -- f""-_. "

_ 4.0 R = 500m

3.0

2.0-

0.4 0.5 0.6 0.7 0.8 0.9

WAVELENGTH (#m)

FIGURE 33. SINGLY SCATTERED SURFACE RADIANCE FORWHITE DISK WITH GP.EEN VEGETATION BACKGROUND. Visual

range.= 2 km, altltud= 1kin.¢

70

2

iI

I i I ' ' J J J I

1974023760-068

'Y[HIM.-,,--,--I FORMERLY WJLLO'W RUN LABORATORII_S. THE UNIVERSITY OF MI_ 41GAN

I

I

FORMERLY WILLOW RUN LA_K)RATORII_$. THE UNIVIrRSITy OF MICHIGAN

I

O.Ol I l I 1 !0.4 0.5 0.6 0.7 0.8 0.9

WAVEI,ENGTH (_ m)

E 'FIGURE 35. RADIANCE VALUES FOR BLACK DISK WITH GREEN

VEGETATION BACKGRDUND. Disk radius = 10m, visual range = " i2 kin, altitude= 1 km.

1 ,

1974023760-070

-" --(RIM FC}NIll4ERLY WILL(%W RUN LABORATORIIr,_ THE UNIVI[[ASfTY OF MICH_--_AN

of these components. Lat t. the directly attenuated component, is zero in this case because the

resolutior, element lies entirely within the black di,_k. The peak at 9.5_n resuits primarily .-

from the peak in extraterrestrial irradiance at the top of the atmosphere. Dominating every-thing is the singly-scattered sky radian"e.

Now we can examine the case of a white disk on a green vegetation background 'see Fig. 36).

:: " Here Lat t is not negligible: in fact. it is the dominaut term. The singly-scattered surface radi- .

ance is also not negligible, compared to sky radiance (at least for long wavelengths).

In all o! the analysis se far. only isotropic scattering has been considered wwhich prob-

ably tends to overemphasize the surface contribution. In a more advanced treatment we con-

sider anistropic scattering, as well as various geometries.

6.2.2 ANISOTROPIC SCATTERING

We can now consider the mor_ realistic, but also more complicated, case of aaisoL,-oplc

scattering. Iterating the formal solution for upwelling radiance produces the following:

-{ro-r)/_ _0o/JoEo _ _)Le" '//JO e-TO/POe-(7o -T)/pL(r... _,)= 1.(ro. _,._)e +4_f,_,+Uo')p(_'"_' _o' "

J2- 1 o

+ _ p(/_, _, :£, _') /_', _')e dr' d/£ d_"0-I _"

T2._1 o

�_P(tJ._. #',_') 4,7-_o) P(_'_" - /_o'_o) ×O-

-'rO/_ O -(rO-r' /g'- e e dr" j,_' d*' (671

.J

.o first term on the right-hand side represents the directly attenuated surface radiance" the

second is the singl$ scattered solar radiance: the _hird is the singly-scattered surface radi-

ance: and the last represents the doubly-scattered solar radiance. We can immediately evalu-

ate the first and second terms by using actual phase furctions: for the last term we can make

use of the isotropic scattering law. For the third term. however, we must consider an approxi-

mation for the phase function.

I,et us represent the phase function by the following:

n-I -ai//_

p_) C,e + cn (_s) _i--l

]974023760--07]

IrOl_141rNLy WILLOW RUN LADOIqrATOI_[S. Tit[ IJNIVlrR_ICfY OF kilCHI@AN

100.0

I

i

-- Lto t _

10.0i

_ -_ -

? - _

_ _

_ -

__° - J '..2 .

1.0 ___ _

..o.I [ ! l [ I0.4 0.5 0.6 0.7 o.s 0.9WAVELE_;GTH (_m)

FIGURE 36. RADIANCE VALUES FOR W;!ITE DISK WITH GREENi VEGETATIOI_ BACKGROUND. Disk radius = 10m, visual range =

2 kin, altitude = 1 kin,f

74

i

J J J _ I l ,

1974023760-072

_ml FORMERLY W;LLOW NUN Ldt_K)R&TORIfS THE UNIVFR$1Yy OF IIIICHIGA_

where C i Cn. and a i are constants to be determined. We will attempt to fit the phase function

in the region 0° to 90 ° with the first part of this function and the last half from 90 ° le 180°

with the constant term Cn. This is reasonable since (1] only the first half of the phase function

-' is used in the disk problem, and (2) the last half of the phase funct_nn is almost constant ar,yway.

At the origin, or when 4 = 0o -_

n-1 -a.

p(1) po =E Cie l+ Cn (69) _i=l

Since the phase function is azimuthally symmetric, we have the normalization ._

I, 2 _ p(p)d_ = I (70)

P

..°

This becomesJ

n-1 :_

I/2 )"aCiE2(a i) + Cn = 1 (71)i=I

when E2(a i) is the exponential integral, that is,

_CC -zt

1

The a i can be determined by using an iteration scheme. The results of this fitting procedure

are illustrated in Figs. 3'/-40 for _ -- 0.40 and 0.901_nand for m = 1.5: and m = 1.5 - 1.0i.

Taking 0 = 0 °, or g = 1, we have for the singly-scattered surface radiance:

Lsur(7. 1 .) _O n !i[ '-'" e'(ai+T°-")//_c• --"_ E-(r°)_-'1_Ci(°Ei=l- °I) ""e'( ' "_c "

}- �r- r')E 1 a i+r - r')/iJ dro o7

H° f+ _ _._lro))-_Cio I la i �r° - r - 1)El(a t + r ° - r)

i 1)El(ai)e "(r°-r) �e'(ai+ro-r)fn_ai + r o - r1" a. ('/3) _,

: !

1974023760-073

LuSh m , IrORIdI[RLV _00LI.b'. "O,N LAIIOIIAv'':.,.=. I H;" II%jv|l_Sll"f OP' MICHIGAN

100.0

I p(O)= 1.547 × 1032 e "70"42/p

+ 57.01 e -2"58//'t" + 0.216

when d _<90 °

--i

10.0

_ °o '_ ,,

i r_ 1.0

• _ _ ctZ

[..,

0 Approximate --_-

0.1

o.o_- I I I I I I I I I I I0 1:30 4b 60 15 90 105 120 135 150 165 180

SCATTERING ANGLE (deg)

FIGURE 37. EXACT AND APPROXIMATE SCATTERING PHASE

FUNCTIONS FOR HAZE L AT A WAVELENGTH OF 0.4proANDREFRACTIVE INDEX OF 1.5

76

' 1974023760-074

.... & .... 4

FORMIrRLy WILLOW RUN LA6OIqlATO_IES THE UNIVEIQSIT'f OF MICHIGAN ._

,%,

I00.0

-- p(O) = 3.19 × 1011 e -23"94/1" i_"

:| -- • 37.72 e"2"22'/_ + 0.223; -- when 0 < 90°_

10.0 a

tQO¢J

_ --

z

' 1.0_

o \z --

f _ _ Approximate<L)

0.1_

t -

- 5

is

_ 1 I i t. I I I 1 I I I _, k

0"010 15 30 45 60 75 90 105 120 135 150 165 180 !_SCATTERING AI_GLE (deg) q

FIGURE 38. EXACT AND APPROXIMATE SCATTERING PHASE jFUNCTIONS FOR HAZE L AT A WAVELENGTH OF 0.9_m AND

REFRACTIVE INDEX OF 1.5

• tf,

• 77

l'" t"' : _-i _- • - |' i ,r -...... - ---" - ._i.i___*a __J

1974023760-075

IM ,'OmMERLY WILLOW _UN L,A_RATHRIE_. THE UNIVERsrrY OF MICHIGAN

o.o_ I i i . I i I .I I I I ,,10 15 30 45 60 75 90 105 120 1,?,5 150 165 180 i

SCATTERING ANGLE (de_) i

FIGURE 39. EXACT AND APPROXIMATE SCATTERING PHASE iFUNCTIONS FOR HAZE L AT A WAVELENGTH OF 0.4/Jm AND

COMPLEX REFRACTIVE INDEX m OF 1.5 - 1.01

78 'i

t

i '

1974023760-076

FIGURE 40. EXACT AND APPROXIMATE SCATTERING PHASE

FUNCTIONS FOR HAZE L AT A WAVELENGTH OF 0.9_m ANDCOMPLEX REFRACTIVE INDEX m OF 1.5- 1.01

|' i

]974023760-077

,r

FORMERLY WILLOW RUN LABORATORIES, THE UNIVERSITY OF MICHIGAN£

z (74)where

Equation (73)has been u._edtocaIculatethe singly-scatteredsurfaceradiance for combinations

ofspectrallyblack,gray. and white diskswith black,gray, white,and green vegetationalback-

.: grounds. A black surface isassumed tohave a reflectanceof0 atallwavelengths,the gray

surfacea reflectanceof 0.5.and the white surfacea reflectanceof 1.0. The green vegetation

background has a reflectanceas givenin Fig, 41[30]. For comparison, Fig. 41 also shows the

spectraldependence ofthe extraterrestrialsolar irradianceatthe top ofthe atmosphere and

the product of thatirradiancewith the reflectanceofgreen vegetation.The resultingcurve

represents the spectral dependence of the intrin:,ic radiance for a green vegetation surface t

w_.th no atmosphere. As expected, it ',s dominated by the spectral cb_tracter nf the vegetation, ] •

at least for the visible part of the spectrum.

, We will now consider the case of black and white surfaces for varying disk ,-adii. It willi

be assum,-d "hat the instantaneous field o: view is a clrcle subtending a FalI angle of 2.5 mrad

at the detector. Thus. at an altitude of 1 km the __ield oi view on the su:'face is a circle with a

radius of 2.5m. The variation in singly-scattered surface radiance with disk radius is illus-

trated in Fig. 42 for an atmosphere with a light haze. For the case o," a white disk and a black

background, the sensor "sees" both a small white disk at R =Im and also a larger circular

ring of black surface; in addition, the radiance is low. As disk radius increa--_,;, radiance

increases until it reaches a value appropria te to a co.nplete white surface. This occurs at a

radius of I kin. The opposite case is also presented, that is when there is a biack disk and a

white background. At smaU radii the strong influence of the white background is quite evident,

even when the disk has a radius of 10re,or four times the field of view.

The same situation is shown in Fig. 43 for the case of a very hav y atmot:phr, re. Note that

the radiances here are almost an order of magnitude greater.

Next, we can see the effect of changing the surface field of view by varying the altitude

and keeping the disk radius constant at 12,5m. Consider the case of a white disk with a black

background (see Fig. 44): as the altitude increases the radiance increases, since more surface

is being observed. However, at an alt',tude vl J km the sensor "sees" the beginning of a black

surface and the radiance decreas,.s. ,kt an altitude of 10 km the black area is four times the

white area and the radiance falis to a very low value. For the case of a black disk and a white

background, the radiance gradually increases as a function of altitude up to an altitude of 5 kin,

indicating the influence of the white background outside the field _f view. Beyond 5 km altitude,

the sensor "sees" a white surface and the radiance increase,, r:,_ioly.

In Fig. 45, the same situati,Jn is represented, except that the atmosphere is quite hazy

An interesting effect becomes evident in this case. As the altitude increases, the radiance8O

i ...... _'T'mmi t i

J

1974023760-078

_BIIdll FORMERLY WILLOW RUN LABORATORIES, THE UNIVER51¥ f OF MICHIGAN _

i?

240 60

i_"220 ,_

Solar Irradiance atTop of Atmosphere ........

200 /__ (LeftScale) 50

I _\ " '18o I \ /

'_ / \ l#160 l 40

t

J ,---

_= _ [ Product of Lhe Two

_ : (RightScale) _,° ,{140 ,

Z 120 30 Z

< . \ <_

100 _

aJ 200 80 I

u_ mI

60 / % , Green Vegetation; _ e Reflectance

_" _ (RightScale)I • #

40 I %"°• 10

oooooo "°.'.'_

20 :

o _ I I I I o 1!0.4 0.5 0.6 0.7 0.8 0.9 1.0

WAVELENGTH (g.m)

. _tFIGURE 41. WAVELENGTH DEPENDENCE OF SOLAR IRRADIANCE AT TOP OF {ATMOSPHERE, GREEN VEGETATION REFLECTANCE, AND THE PRODUCT OF

THE TWO •

i

,, !i

1974023760-079

£

IM FORMERLY WILL(.)W RU_ LABORATORIES. THE UNIVERSITY OF MICh.%_N

101

t

......... •.¢ _- :...- :. - _. .........................

' T 1o0,_ : ,.,_ -

y,

r_z

"L: , _ I0-I

:- r_

• - -- White Disk, Black Background----- Black Disk, White Background

: : ...... White Disk, White Background

i

10 "2 l ! I I , ,,,i ' ' ' ' ' '_i[ l I l I l _'' ' !

io0 IoI 1o2 1o3 i_' i

DISK RADIUS (m) : ,,!

FIGURE 42. DEPENDENCE OF SINGIX SCATTERED SURFACE RADIANCE ON 1DISK RADIUS. Visual rarlge= 23 km, altitude= 1 kin,wavelength = 0.55/Jm,solar

_ zenith angle = 45 °, nadir view angle = 0°" " i

]I'I

8_.

I j ' i _ , {{d

1974023760-080

(FORMERLY WILLOW RUN LABORATORIES. THE UNIVEHSITY OF MICHIGAN

102 "

-----.i- White Disk, Black Background

Black Disk,White Background...... White Disk, White Background :4

m N

,_ ............. .,..-.:. :.:"_..............................

,_ _ lo1_ -.... _ :" I

10° / _ i

r,.) % ,;

1'1

iO-I •i I l I I till i i ! i lltit I I I I IP ml _

tO0 l01 102 10 3 ':

. DISK RADIUS (m)

: FIGURE 43. DEPENDENCE OF SINGLY SCATTERED SURFACE RADIANCE ON

DISK RADIUS. Visual range = 2 km, altitude= 1 kin,wavelength = 0.55pro, solar :.zenithangle = 45°,nadir view angle = 0°. _!

Q ,,

83

m

1974023760-081

_-[RIM FORMERLY WILLOW RUN LABORATORIES. THE UNIVERSITY OF MICHIGAN

-RIM u._.,,.o..c.,o..

from a white disk qcreases up to ,ui altitude of :_bout 0.9 kin. after which it gradually decreases.

This :_s due to t_.e fact that as altitude increases, more radiatir, n ,s included in the acceptance

cone of the set sor. However. attenuation is also taking place. As a result, after a certain

altitude mo_'e radiation is lost by scatterin_ out of the field of view than is being scattered into

the field of vLew. At =,n altitude of 5 kin, for example, the sensor "sees" the bl'Jck surface and

the radL, nc.; decreases more rapidly. A similar effect is noted for the case of a black disk with

a white bac _ground.

We no_¢ turn to ti',e cases of black, gray, and white disks with a background of green vege-

tation. F'gurt 46 illustrates the variation iN singly-scattered radiance with wavelength for the

c':.oe of a black disk. Referring to Fi T. 41. we see that the radiance essentially has the spectral

dependence of the product curv, [e, all disk radii. For radii of 10m and 100m. the _ctual

scatterin- phase [uaction i_ _. ,_er tha the isotropic phase function for all angles appropriate

to those disks: henc _ the rac,anccs are greater. For a disk radius of 1000m, however, the

maximum angq ,ff sf. terint at the edge of the disk is 45°. At this angle the actual phase func-

tion is almo._t the same as the isotropic phase function: therefore, the radiances for the _so-

tropic and the anisotroric cases are almost equal.

Figure 47 illustrates the same situation, except for a white disk, Here the curves are

reversed as far as radius is concerned. For large radii the curves have a general behavior

characteristic of the solar spectrum because the effect of the vegetation background is small:

• for a radius " 10m. however, the effect of the vegetatio:, becomes noticeable. The anisotropic

case ' _s mo pec',ral variation than tlw isotropic because the _ctual scattering phase lunc-

h,, -" the iJrmer are dependent upon wavelength, wht _as the isotropic phase function of the

1: .s no spectral dependence.

We can now simulate the case of a sensor in sp,Lce or at an altitude of 50 kin, which for

all practic,d purposes i_ at the top of the atmospht.re. Figure 48 illustrates the spectral L:e-

pendel.ce of :.ingly-scattered surface radian=e as a function of disk radius for a gray disk.

Here the effect c ¢background on target is clear. The field of view i_ a circle w_th a radius of

125m. For radii smaller than this, green vegetation as well as the gray disk can be seen; in

addition, the curve has the spectral dependence of green vegetation. At a radius of 1000m the

dupendence is more closely related to the s_'ar spectrum.

Figu_'e 4_ shows the same effect as [or the pz -wous figure, except that the atmosphere zs

very hazy. In this ca:2 the rad,.'ance is lower at shorter wv,velengths because of the greater

probability of sc" "erinF uut of the field of view.

In Fig. 50 we s,:e the variation o! singly-scattered surface radiance . h wavelent_th ".or

black, gray. and wh_.te disks, each with a radius of 1000m. For a black disk the only spectral

wtri_:tion than can arise is trom the green vegetation ba_.kground. The rather large 50-percent

_6

&l

1974023760-084

_mimil fORM(Rt.V WILLOW RUN t.AOORATORIE$. TNE UNIVERSITY OF MICHJGAtl

?

_ 101

Isotr 3pic ScatteringAnisetropic Scattering

S--, _,oO !1 ,""---

: //,,,,"_E

: I0 #I / #,"'_._ .,/ - .__

i #) R = 100m /

lO - l/

I0"2 ] "_0.4 0.5 0.6 0.7 0.8 0.9

. WAVELENGTH (IXm)

FIGURE 45. DEPENDENCE OF SINGLY SC _ 1":RED SURFACE RADIANCE "_ON WAVELENGTH FOR A BLACK DISKAND ;.I'_l'lVEGETATION. No "_absorption,solar zenithangle-- 45°,n:tdlrvie,,,_r.i --0°, visualrange= 2

kin,altitude= 1k',,. J

87)

I

1974023760-085

_R _mM , • ,L ,FORMIEMLY WILLOW RUN LABOR&'IOlilIWS. TH| UNIVI[RSI'r_ OF MtCHIG It',

102

" Isotropic Scattering" Anisotropic Scattering

R :- lO00m

lo1/_.e, _

Z.<

<

o I e = loo_ .- ...... /. _" I

,oOk---- ./" iR = lOre I

I

1o"I I I , . I . I .0.4 0.5 0.6 0.7 0.8 0.9

WAVELENGTH (_m)

FIGURE 47. DEPENDENCE OF SINGLY SCATTERED SURFACE RADL_.NCEON WAVELENGTH FOR A WHITE DISK AND GREEN VEGETATION BACK-GROUND. No absorption, solar zevith angle = 45°, nadir view angle = 0 °,

visual ranRe = 2 kin, altitude = I kin.

88

t

I

1974023760-086

_RIM ..... FORMERLY WILLOW RUb, LA_)ORATORIE$, THE U_IVER_n'V OF MICHIGAN

101 .

_11 _ _'_

f _ _. -,. _ R = lO,O00m

° / .:,oom.,.. ,/-_ / ,'/N. ,_/ Io looL- -i _, ?"

l-, X', ..'/< ;_ z

10"1 I 1 I I0.4 o.s o.e 0.7 o.s 0.9 ,_:

WAVELENGTH (pm) :;

FIGURE 48. DEPEr,DENCE OF SINGLY SCATTERED SURFACE RADIANCE _.ON WAVELENGTH AT 50 km ALTITUDE FOR VARIOUS DISK RADI'. Visual "_.range = 23 km, no absorption, solar zenith angle -- 45 °, nadir view angle = 0°, _

disk reflectance = 0.5, background reflectance = green _egetatton.

|

1974023760-087

|

_RJM FORMIrRI.Y WILLOW RUN _aORATORII_S ThE UNIVI[ ;tSITV OF MI_H_AN

= Im

1_-I I I0.4 0.5 0.6 0.7 0.8 0.9

WAVELENGTH (/zm)

FIGURE 49. DEPENDENCE OF SINGLY SCATTERED SURFACE RADIANCEON WAVELENGTH AT 50 km ALTITUDE FOR VARIOUS DISK RADII. Visua!

ra'_ge= 2 kin,no absorption,soL'trzenithangle ---45°, nadirview P,ngle = 0°,disk reflectance= 0.5,background reflectance= green vegetation.

90

J

1974023760-088

_-ERINdm_aul F_eJ_'_'_[_l-.v Wtllr_W I_Ukl { &Rt"_R'ATQRIES THE UNIVERSIIV OF MICHIGAN

101

•_......_._VChite Dis,c, Vegetat,nrl Backgroundr

Gr_ _,(50%) Disk, Vegetation Background _

-7- lo 0 _

°' !;- .

Black Disk, Vegetation Background•

<

10-1 _

io-_ I I I l0.4 0.5 0.6 0.7 "3.8 0.9

WAVFLENGTH (pro)

FIGURE 50. DEPENDENCE OF SINGLY SCATTERED SURFACE RADIANCE

ON WAVELENGTH FOR VARIOUS DISK REFLECTANCES. Visualrange=23kin,no absorption,altitude= 53kin,solarzenithangle: 45o,nadirviewangle :

0°, disk radius = lO00m. _.

91

1974023760-089

gray disk removes this spectral variation and therefore the radiance is essentially the same

as that for a white disk except for magnitude.

Figure 51 illustrates the same situation, except for a very hazy atmosphere. Again, the

radiance at shorter wavelengths is lower due to high attenuation.

Finally. we can compare th¢. various radiance components (that is, the singly-scattered sky

radiance, the doubly-scattered sky radiance, the singly-scattered surface radiance, and the

sum of these three comr,_nents) for a black disk and a green vegetation background (Fig. 52).

! The singly-scattered sky radiance is calculated using the actual scattering phase function; the

i i doubly-scattered sky radiance is determined using the isotropie phase function: and the singly-

I scattered surface radiance fom)d via the exponential phase-function approximation. The only

spectral variation in the sky radiances arises from the solar spectrum. The surface term is

small compared to the other radiances, but this is true only because the black disk is so large.

For smaller disks the surface radiance increases (as was seen in Fig. 46). Thus. the influence

of outside rePectance is significant for various disk reflectances and sizes.

Figure 53 illustrates the same condition, except for a vary hazy atmosphere. In this case,

the surface term is less important--an effect which is surprising. However, this is for a sen-

sor at a 50 km-altitude, where the sky radiances are larger than at lower altitudes.

In conclusion, we can say that background surface definitely affects target radiance, the

amount depending upon the relative size of the target and the field of view. For most cases,

the anisotropy of scattered radiation is not too important as far as magnitudes of radiance are

concerned; it is important, howcver, for the consideration of spectral dependence.

6.3 THE INFINITE STRIP PROBLEM

Only inthe lastfew years have investigatorstriedtofindsolution_tothe multidimensior.tl

transportequation. Approximate methods using Fourier transforms have been developed by

Williams [34[,Garrettson and Leonard [35],K._per[36],Erd" "-nn and Sotoodehnia[37I,and

Rybicki[38I. We shalluse a similartechniqueto solvethe t -dimensionalr,'diative-trans[er

equationfor an infinitestrip.

Consider an infinitestripwith a r¢,flectancep and a background with reflectancep,as il-

lustratedin F_g. 54. The positionvectorR is

:(x,y, z)= R(smTlcos_, sin _/ sin_, cos_?) (75)

A #

and the unitvector flis

i'_= (_, _, flz) = (sin _ cos _, sin I_sin _, cos i_)= (i,u, _) ¢76)

The radiative-transferequationbecomes

92

1974023760-090

93

1974023760-091

_!M ........ F_MC'RLY WILLOW RUN LdtaJJORATOI_I_S. THF UNIVIrR_;ITY OF M(CHI_AN

101

i TotalRadiance

22:2::,"._.__._-- Singly Scattered

7 "_..._. Sky Radiance

100 _ / DoublyScattered ,_

: -- Sky Radiance -'_ _--"-------_l,t, t" ----.

L SinglyScatteredSurfaceRadiance

i0-1 -_

I I I I0.4 0.5 0.6 0.7 0.8 0.9

WAVELENGTH (_tm)4 "

FIGURE 52. DEPENDENCE OF RADLA,NCES ON WAVELENGTH. Visaal

range = 23 kin, no absorption, solar zenith angle = 45°, nadir view angle = 0°,disk refl,;ctance = O, background reflectance = green vegetation, disk radius =

: 1000m,altitude= 50kin.

! 94

i

I

i

#

1974023760-092

" FORMERLY WILLOW RUN ¢.ABO_ATORIES THE UNIVERSITY OF MICHIGAN

,jlo1

- I"I"--" • T_, Radiance _

DoublyScatteredSky _

• *°'°*'" ii!°°° °,o

"°° _

100 "" ,:__ ".......,_.------SinglyScatteredSky Radiance ?

_ "'.....**Oo

._. - / _,-. _:

/z_ _ / _!,

10-1 / _._J- /

, _ Singly Scattered Surface Radiance

't

:y

L. _;'

i0"'_ I 1 1 l !;0.4 0.5 0.6 0.7 0.8 0.9 _

WAVELENGTH (_m) !i:P_

FIGURE 53. DEPEND.'.NCE OF RADIANCZS ON WAVELENGTH. Visual {i

r_nge= 2 kin,no absorption,solarzenithangle= 45°,nadirviewangle=0o,diskreflectance:0,backgroundreflectance=greenvegetatlon,disk

radius_ 1000m,altitude= 50 kin.

,_9S .;

1974023760-093

Z

:; x

FIGURE 54. INFINITE STRIP APPROXIMATION

1974023760-094

FORMERLY WILLOW RUN LABORATORIES. THE UNIVERSITY OF MICHIGAN

/

_(z)_ -T/_o,g0L aL -K(z)L(x, z, /J, 4A _ -_-_-_oe P_' 4" -_o' 4'o )-_+.gz= .. ._

2_I '_

_(z)f .,. _, ,)+--_--_-.! p(_, ¢,, _')L(x, z, _ d/;d_' (_,) -_

0-1

We can now define a Fourier transfer for the x coordinate,

P

"_ l,(k. z, U, _) = |eikXL( x, z, /J, _)dx (78)

-_ !

with an inverse :,

t'

__# }

Taking the Fourier transform of Eq. (77) leads to the following equations: ,_:.

-(r o- r-_ z)/_I-(k.z, _, ,) = i(k, O, I_,_)e

z

t. 2n',ii6(k)I e[T-T' _(Z -z')]/_L(oo,z', _, _)dz'o

Z_(z,)e-r'/_Oe[T-T'+_(z-z')]/_ dz,6(k) Eon(N, _. _o)I+-_- -1%,

o

2n 1 z

+ "_,- p(_. _, :a', ¢#'_, _(z')e[T-r'+e(z-z')]/l_L(k, z', 11', _')d_'dC_'dz' (80)o o

oo

2_e 6(k)I e['r"r+e(z'-z)]/_L¢°°" z', -_, _)dz'i(_._.-_,):"-'Vz

_[_(z,)e-r'/_Oe[r'-r )+ 6(k)E°P('_'2,u_' - I_°'_°) I ]/_dz' :!

2nl _o !

'If I+ _ p_- _, _. _', _') l_(z')e [T''_'+e(z''z)l/l_ _,(k, z', _', _')d_' d_' dz' (81) 'i

0 -1 z ,i_

97 i :

<

1974023760-095

FOR_ 'LY Wg_l-OW RUN LABORATORIES. THi[ UNIVERSITY OF MICHIGI,_

l where e =ik_ = ik_- u2cos_,and L(_. z, _,_)and I,( , z,-_,_)aretheasymptot.ic so- . j

"_ lutions which coincide with solutionz for the one-dimensional problem. By iterating tho above

equations,applyingthe boundary conditions,and then by takingthe inverse Fourier transform,

one obtainsthe solutionfor the two-dimension_l equation.This has been done,but the resultingi

mathematical expressionsare too involvedtobe presented here, except for the most interesting

par'.dealingwith the surface term. Let us conslder thatterm.

For a Lam_ertian surface, L(k. e, _, _ = L(k, 0), and the iterated (first) term beco,::es

2r, 1 z

'i! +_,(k,z, !,_1 =_-_ p(_,_, _') _(z')exp[r-r' .__(z- z')l/_L(k,0) ._O0 0

× ex_.-l"rr_ - T' - e'z'_/_'dlJ'd_'dz' (82)

The integratio,_ over _ ' is

exp(c'z'/_') d_ ' = 2_J o0

where J is the zero-th order Bessel function. Nuw, let us represent the scattering phaseofunction by

n e'_'j/u'

P(U, _, _', _') =i___C.--j #, , p - 1 (84),

Hence,

L(k, z, 1, _)= ['(_ ')),k-n-_C._,_(-')e r-z' - '

j--1 0,

where _ -- k+j+ ro - _" _:

The integration over _' can easily be accomplished by :st.ng a Laplac_ transform:

L(k, z, 1, _)= _ dz' (86) "i�k+'+'

It can be shown* that "+

,g

++*See Ap?endix A. ++

.+

i:i,s

1974023760-096

FORMERLY WILLCIW RUN LAI:IORATORI£S. THE UNI_ I[RSITY OF MICH - I_N

L(k. O_ -_.,7__E(_)5(k)+ foE(o)- pE(0)]---_ (87) ;

Taking the inverse Fourier t_ansform of Eq. (86) then leads to the following:

_.(x,,. _,_,- _ c e -dz':-.-)-C-._, _ j ),.+r -?'j=l 0 ) o

(Z I

C [M(z', x o) - M(z', x - -:o)) dz' (88)j---1o !

where

l

in which K is the modified Bessel function. The ).imats for M are )o

M(z', 0) = 0; M(z', oo', = _e "'_z'-_ (9o)

" Therufore, the numerical integration ofEq. (88) will g_ve the spectral radiance for a downward-

looking observer at any poiat in the atmosphere. In particu]-tr, we would be able to simulate

the response of a sensor as ;c passes over a natural boundary between two dissimilar surfaces

!l

1974023760-097

!

+'S -"RIM_--_ FOI_IEIILY WILLOW RUN LAIIOAATOItlt_ TklE UNIVE_ITY OF MICHIGAN

7CONCLUSIONS AND RECOMMENDATIONS

As explained in Section 3, Earth's atmosphere has a high degree of variability. In the

:_,er part of the atmosphere (that is, for altitudes less than 5 km, aerosol density can be very

high and quite variable in both composition and distribution of particl_ sizes. Visual range .

allows estimation of the aerosol content of the atmosphere but not of composition nor distribu-

tion of particle sizes. It is recommended that the atmosphere's spectral opUcal depth at the alti-

t, tde of the s nsorbe measured and that the spectraloptical thickness alsobe measured. This can

bedone by m +nitoring solar irradiance at the sensor altitude and the surface. Optical depth is a

+,ore mean +_,zulparameter than visual range in radiative-transfer studies. In investigations of

I' "_e-scale remote sensing by satellites, another quantity which can be used is atmospheric turbidity

•alues of v_tch are becoming available from many measurtne sites in the United States.

In Section 4 we investig+.ted the effect of composition and size distribution of aerosols on

the single-scattering albedo and the single-scattering phase function. Changes in aerosol com-

position characterized by the imaginary part of the refractive index can considerably alter the

single-scattering albedo. Of less importance is the change in size distribution. The single-

scattering albedo is relatively insensitive to visual range up to +bout 20 km. This is true

regardless of the degree of contamination (as expressed in terms of the refractive index).

Similar effects were found for the singie-scattertng phase functions. The change in size

distribution did not have much effect on the shape of the phase function, but changes in the

imaginary part of the refractive index did produce significant differences in the phase function

at large angles. This suggests that lid_tr techniques might be employed to measure the hack-

sc,_tter coefficient (or the phase function at 180°). In turn, this would be useful in estimating

the composition of aerosols.

In Section 5 we extended the current radiative-transfer model to include absorption. The

single scattering albedos and phase functions calculated in Section 4 were then used in the

radiative-transfer equations to determine spectral path radiance and total spectral radiance

for various degrees of contamination. Radiances can change by at least a factor of six between

a "clean" atmosphere and a strongly absorbing one. The actual atmosphere probably liesbetween these limiting cases. The variation in path radiance with nadir vie_ angle is not great- ,_

ly affected by de_ree of contamination, but the magnitude c_n differ by factors of four or five. +

In Section 6 we considered the effect of background reflectance on radtance from a target + _

First, a number of colored disks were chosen as the target with green vegetation aselement.

the background. Since this was done assuming Isotroplc scattering, the spectral variationactually was not realistic.

100 i

...... _' " _l___i_+m+'_ - __ . I_1111........ _

1974023760-098

i ji ii i iFolittlrRI.Y WILLOW RUN LAIK)IiATORI[$. TH[ UNIV[ltSffY OF* MICHIGAN

* We also considered the more realistic case of anisotropic scattering by fitting actual cal-

culated phase functions to an a_proximate exponential formula. The fit was surprisingly good,

particularly for the angles of importance. These phase functions were then used in the single-

scattering formulas to calculate singly-scattered surface radiance for a variety of situations.

We found that bacXground definitely influences the target and that the amount of influence de-

pends uponthe relative size of the target and thefield of view, the relative reflectances of target

and background, and the visual range. Although this influence of surface radiation is usually of

lesser importance than that caused by radiation from the sky, in the case of large surface

reflectances the effect is of equal importance. The result is that signatures based upon an

analysis which excluded surface interaction could lead to inaccuracies in the recognition pro-

cessing of multispectral data. Specific algorithms can be developed, however, to account for

this effect and correct the data.

Finally, we investigated the more general problem of determining the radiance at any point

in the _tmosphere at which the surface. _ a non-un':form Lambertian reflector. For simplicity

we assumed all :'_finite strip with one reflectance and a background with another. Thus, we

" simulatecl _ne case ot an aircraft or spacecraft moving across the boundary between two dis-

similar surfaces. A m_thematical expression was obtained for the singly-scattered surface

• radiance as a function o:"distance from the strip, altitude of the sensor, r_flectances of the

surfaces, and atmospheric parameters. A numerical integration of the equation is necessary

for realistic atmospheric conditions.

We highly recommend that specific flights be made to collect multispectral data for sit°

uatlons approximating the geometric and physical conditions used in the analysis of this report.

Simple experiments will allow us to estimate the degree of contamination, and hence determine

the variance in radiance. Likewise, the effect of background on targe_ __diance should be inves-

tigated for the simple geometric conditions we have described.

Strong gaseous absorption effects in the infrared have been excluded in this analysis, but

they can be considered by unifying the results obtained here with the ERIM band model for gas-

eous absorption. The unification of these two models would be particularly advantageous for

remote sensing applications. Since there is a high degree of variability in the reflectiw and

emissive properties of natural materials in the infrared region, a unified model which accounts /

for scattering and absorption by aerosols and gases could be used to correct multispectral data

over the complete visible and infrared spectral regions.

In view of the development and use of meterological satellites, we now can obtain meaning-

ful information on the actual state of the atmosphere at various locations and times over the

entire globe. If such infoz mation were readily available to investigator_ they would have a more

101

1974023760-099

_--'--'J -- IrOAM_ItlLY WIt.LOW NUN LABONATONII[S. THE UNIVERSITy OI r ilICHli_AN

efficient and economic means of applying atmospheric conditions in the processing of multi-

spectral data. Therefore, a closer relationship between meteorological investigators and the

earth-resources remote sensing community is highly desirable.

As a result of the knowledge and insights gained in this investigation we can now interpret

real multispectral data in a more meaningful way. By means of techniques developed in this _.

report, calculations based upon our present radiative-transter model can be implemented to t

generate algorithms for the correction of real data for systematic atmospheric variations.

Also, variances in muliispectral data resulting from atmospheric effects can be simulated in

a more realistic way.t

102

1974023760-100

_IN ,o..,.,,w,_Low.u._-..,o.E_,._o,,v_,-,o..o....

1AppendixFOURIER TRANSFORM DERIVATION

The F_urier transform of intrinsic radiance t the surface is derived as follows.

j L(k, 0) = [e ikx L(x, 0) dx

where 0 implies zero altitude. Now, by definition the radiance is

23 1.m

=/ / _'p'(x' _' _' -_'' ¢')L(x, 0,-/_', _b')d/_'d_L(X, 0)

U 0

which, for a Lambertian surface, becomes

Hence,

-- eO

L(k, 0)=_EG) e ikx dx + e ikx dx + _ J-oo "X X

o o

-E@_(')f2 sin kM 2sinkxo] E_) 2 sin kx o != k k-J+ _ k . M-_

i

103

1974023760-101

REFERENCt S

1. R. M. Goody, Atmospheric Radiation, Oxford University Press, New York,1964.

2. V. Ye. Zuyev, Propagation of Visible and IrLrared Waves in the AtmGsphere,NASA TTF-707, NTIS, U.S. Department of Commerce, Springfield, Va., 1970.

3. U.S. Standard Atmosphere Supplements, U.S. Government Printing Office,Washington, 1966.

, 4. G. M. Hidy and J. R. Brock, An Assessment of the Global Sources of Tropo-spheric Aerosols, Academic Press, New York, 1971, p. 1088. -:

5. F. E. Volz, Infrared Absorption by Atmospheric Aerosol Substances, J. C-eo-phys. Res., 77, No. 6, 1972, p. 1017.

6. F. _.. Volz, Infrared Optical Constants of Ammnuium Sulfate, Sahara Dust,Volcanic Pumice, and Flyash, Appl. Optic_ ., No. 3, 1973, p. 564.

7. G. W. Grams, I. H. Blifford, Jr., B. G. Schuster, and J. J. DeLuisi, Com-plex Index of Refraction of Airborne Fly Ash Determined by Laser Radar andCollection of Particles at 13 kin, J. of Atmos. Sciences, 29, 1972, p. 900.

8. D. F. Flanigan and H. P. DeLong, Spectral Absorption Characteristics of the tMa_or Components of Dust Clouds, Appl. Optics, 10, No. 1, 1971, p. 51. t'

9. C. E. Junge, Journal Meteorol., 12, 19_5, p. 13. !_

10. D. Deirmendjian, Electromagnetic Scattering on Spher;cal Polydispersions,Elsevier, New York, 1969, p. 78.

" 11. A. C. Holland and G. Gagne, The Scattering of Polarized Light by Polydis-perse Systems of Irregular Particles, Appl. Optics, 9, No. 5, 1970, p. 1113.

12. R. S. Powell, R. R. Circle, D. C. Vogel, P. D. Woodson IH, and B. Donn,Optical Scattering from Non-Spherical Randomly Aligned, PolydisperseParticles, Plainer. Space 8ci., 15, 1967, p. 1641.

13. D. M. h_tb,:en and K. F. Loughlin, Effects of Particle Shape and Size Dis-tribution on tile Transient Solution of the Diffusion Equation, Nature Phys.Sci., 230, 1971, p. 69.

14. C. Junge, Chemical Composition and Radioactivity of the Atmosphere, WorldPublishing Co., Cleveland, 1963.

15. L.S. Ivlev, Aerosol Model of the Atmosphere, Prob. Fiz. Atmos. Leningrad,No. 7, 1967, pp. 125-160.

le. B.J. Mason, The Physics of Clouds, Clarendon Press, Oxford, 1971.

17. N.A. Fuchs, The Mechanics of Aerosols, Pergamon, Elms_ord, N. Y.,1964.

18. C.N. Davies, Aerosol Science,Academic Press, New York, 1966.

19. H.L. Green and W. R. Lane, Particulate Clouds, Dusts, Smokes and Mists, 2ndEdition, D. Van Nostrand, Inc., Princeton, 1964.

20. F. Ltnke, Handbuch der Geophystk, Vol. VIII, Sprlnger-Verlag,Berlin, 1943.

21. K, Ya. Kondratyev, Radiation in the Atmosphere, Academic Press, New York,1969.

104 _

1974023760-102

!

I I | _ t

22. D. T. Chang and R. Wexler, Relation of Aerosols to Atmospheric Features,Final Report AFCRL-68-0360, AD 674 629, Air Force Cambridge ResearchLaboratortes, Bedford, Mass., 1968.

23. L. Elteyman, An Atlas of Aerosol Attenuation and Extinction Profiles for ti_eTroposphere and Stratosphere, Environmental Research Papers No. 241,Air Force Cambridge Research Laboratories, Bedford, Mass., 1966.

24. W. E. K. Middleton, Vision Through the Atmosphere, University of TorontoPress, Toronto, 1952.

25. P. N. Tverskoi, Physics of the Atmosphere, A Course in Meteorology, NASATTF-288, TT65-50114, U. S. DepartnLent of Commerce, Washington, 1965.

26. Lord Rayleigh, Phil. Mag., 47, 375, 1899.

27. L. Elterman, U. V., Visible and IR Attenuation for Altitudes to 50 km, ReportNo. AFCRL-68-0153, Air Force Cambridge Research Laboratories, Officeof Aerospace Research, Bedford Mass., 1970.

28. L. Elterman, Vertical-Attenuation Model with Eight Surface MeteorologicalRanges 2 to 13 Kilometers, Report No. AFCRL-70-0200, Air Force CambridgeResearch Laboratories, Office of Aerospace Research, Bedford, Mass., 1970.

29. A. L. Fymat and K. D. Abhyankar, Theory of Radiation Transfer in Inhomoge-neous Atmospheres. I. Perturbation Method, Ap. J.Vol. 158, 1969, p. 315.

30. W. A. Malila, R. B. Crane, and R. E. Turner, Information Extraction Tech-niques for Multispectral Scanner Data, Report No. 31650-74-T, Willow RunLaboratories, Ann Arbor, 1972.

31. R.E. Turner and M. M. Spencer, Atmospheric Model for Corzection of Space-craft Data, Proceedings of the 8th International Symposium on Remote Sens-ing of Environment, Center for Remote Sensing Information and Analysis,Willow Run Laboratories, Ann Arbor, 1972.

32. R. E. Turner, Remote Sensing in Hazy Atmospheres, Proceedings of ACSM/ASP Meeting, Washington, March 1972.

33. K. Ya. Kondratyev, O. B. Vasilyev, L. S. Ivlev, G. A. Nikolsky, andO. I. Smokty, Influence of Aerosol on Radiative Transfer: Possible ClimaticConsequences, University of Leningrad Press, Leningrad 1973.

34. M. M, R. Williams, Approximate Solutions of the Neutron Transport Equationin Two and Three Dimensional Systems, Nukleonik, Band 9, Heft 7, 305, 1967.

35. G. A. Garrettson and A. Leonard, Green's Functions for Multidin'.ensionalNeutron Transport in a Slab, Trans. Am. Nucl. Soc., Vol. 12, 159, 1969.

36. H. G. Kaper, Elementary Solutions of the Reduced Three-Dimensional Trans-port Equation, J. of Math. Phys., Vol. 10, No. 2, 1969, p. 286.

37. R. C. Erdmann, and A. Sotoodehnia, On the Solution of the Boltzmann T.'ans-port Equation in Two Dimensions, J. Of Math. Analysis and App., Vol. 31, 1970,p. 603,

_- 38. G. B. Rybicki, The Searchlight Problem with Isotropic Scattering, J. Quant.Spectrosc. Radiat. Transfer, Vol. 11, No. 6, 1971, p. 827.

105

1974023760-103