Determination of atmospheric parameters from measurement of polarization of upward radiation by satellite or space probe

ZCARUS 6, 348-359 (1967)

Determination of Atmospheric Parameters from Measurement of Polarization of Upward Radiation by Satellite or Space Probe Z. S E K E R A Department of Meteorology, University of California, Los Angeles, California and The RAND Corporation, Santa Monica, California Received September 29, 1966 Measurements of the polarization characteristics of the radiation emerging from a planetary atmosphere contain significant information about the concentration and vertical distribution of the atmospheric scatterers as well as the reflecting properties of the underlying surface. For an atmosphere with Rayleigh scattering only and a nonreflecting lower boundary, a method is derived that permits four independent estimates of the total optical thickness. If the lower boundary reflects radiation according to Lambert's law, this method permits two estimates of its reflectivity and two independent estimates of the optical thickness. The problem of distinguishing Rayleigh from non-Rayleigh scattering is discussed, and suggestions are made for deriving the aerosol content and the level of maximum concentration of aerosol particles on the basis of radiation measurements. well as intensity. These characteristics can provide up to three times as much informaThe use of the visible and adjacent spec- tion about the scatterers than the measuretral regions for sensing a planetary atmos- m e n t of intensity alone. phere offers three basic advantages. First, Although the idea of using the polarizathese are the regions where solar radiation tion of skylight to determine atmospheric has a broad inaximum, and thus sufficient turbidity is almost a century and a half old, energy is available for sophisticated meas- the quantitative study of radiation emerging uring techniques. Second, scattering pre- upward from the atmosphere is relatively dominates in this region, and absorption recent (Coulson, 1959). The s t u d y of the can be neglected. Finally, as we pass from inversion problem, i.e., the method of detershortwave to near-infrared wavelengths, we mining atmospheric parameters from measfind t h a t the predominance of molecular urements of upward radiation, is even more scattering is gradually replaced b y aerosol recent. Previous a t t e m p t s to solve the inverscattering. This last circumstance is especially sion problem for a scattering atmosphere useful for the evaluation of an atmosphere's have been based on the unrealistic assumpaerosol content. In general, the radiation tion of isotropic scattering, which precludes emerging from a scattering atmosphere is a polarization effects (Bellman et al., 1965). simple consequence of the scattering of In this paper, however, we describe an incident solar radiation b y scatterers that inversion technique in which the polarization are smaller than or comparable to the wave- characteristics of upward radiation are length of the scattered radiation. As the specifically taken into account, and discuss scattering process polarizes the emerging the additional information thereby obtained radiation, it presents the possibility of about the molecules and particulate m a t t e r measuring polarization characteristics as ill the atmosphere. 348 I, INTRODUCTION

DETERMINATION

II.

MEASUREMENT

OF POLARIZATION

OF THE AND THE

OF ATMOSPHERIC

349

PARAMETERS

polarization from the n a d i r on the optical thickness r is illustrated for such a a atmosphere at various zenith distances of the Sun. The neutral points are named in the same way as those for skylight; i.e., the point below the antisun, closer to the nadir, is called the Babinet point; the point above the antisun, close to the horizon, is the Brewster point; and the point on the solar side of the Sun vertical is the Arago point. The thin line represents the position of the Arago and Babinet points when the planetary surface is taken to be a Lambertian

DEGREE POSITIONS

OF NEUTRAL POINTS Studies of atmospheric turbidity from measurements of skylight polarization have been based for almost 150 years on measurements of the degree of polarization at the point of maximum polarization or at the zenith, as well as the positions of neutral points in the Sun vertical (local meridian passing through the Sun) (Sekera, 1951, 1957). Were we to apply an analogous procedure to measurements of the polarizaBABINET BREWSTER POINT 30 °

ARAGO POINT

Z e, 8 4 " ~

~A

• 80

%

170 e

z ZI~ ~ , ~ BAS|NETT~

O

sS

s'%~,"~

,,.*'°8"ABINE T

160"

.-''"

I0*

150"

! O.01

I

i

I

I

I I I I O.i

I

I

I

I

i

!

i i Ill i.O

OPTICAL

THICKNESS

i

~

|

i

I I ! IO.O

"r

FIG. 1. Positions of neutral points of the radiation emerging from the top of a Rayleigh atmosphere. tion of the upward radiation, we would restrict the measurements to the Sun vertical and measure the positions of the neutral points in addition to the degree of polarization at one point (for example, in the direction of the nadir). For a pure molecular atmosphere the measured quantities would depend only on the optical thickness and the characteristics of ground reflection. In Figs. 1 ! and 2, the dependence of the positions of the neutral points and of the degree of z Figures 1 and 2 are based' on computations made by Dr. J. V. Dave, and his permission to use these data is greatly appreciated.

reflector with a reflectivity of 80%. Figure 1 allows the determination of the optical thickness from the known positions of the neutral points. Any uncertainty in this determination b e c a u s e of the unknown reflectivity of the ground is not large. However, for large optical thicknesses the accuracy of the determination Considerably diminishes, and for optical thicknesses greater than 1 the determination is not even unique. In Fig. 2 the dependence of the degree of polarization on the reflectivity of the ground is striking, especially for small optical thicknesses. For a pure molecular

350

Z.

SEKERA

I0 Ze=

09

OB ¢__ 6

z z

N d 55" ~

/

o4

o2

ol

=2-" ....... o.oi

-- I

._

io OPTICAL

1

I

I

I

I

Jo

2

4

8

IG

THICKNESS

T

FIG. 2. Degree of polarization of the radiation emerging from the top of a Rayleigh atmosphere in nadir. atmosphere, Figs. 1 and 2 thus show that the optical thickness and the reflectivity of the ground can be determined reasonably well from the location of the positions of the neutral points and from the measurements of the degree of polarization at the nadir, respectively. However, when atmospheric turbidity increases, the neutral points will be shifted from their positions in the molecular atmosphere (Fig. 1), and the degree of polarization at the nadir will decrease. From the measurements of the position of the neutral points or of the degree of polarization at the nadir only, we will be able with the use of the diagrams in Figs. 1 and 2 to obtain some kind of "effective" molecular optical thickness and "effective" reflectivity, but we will not be able to make any quantitative estimate of the aerosol content or the size distribution of the aerosol particles. It is evident that for this purpose it is necessary to modify the method of measurement. We may use the theory of radiative transfer as a clue to this modification.

III.

MEASUREMENTS STOKES

OF

TIlE

PARAMETERS

If we assume that the particulate m a t t e r in the atmosphere scatters light according to the Mie theory, then the Stokes parameters [as modified by Chandrasekhar (1950)], which are solutions of the radiative transfer equations for the upward radiation, can be expressed as the series Ii(0; #, ~) N

Iz(")(r; #,/~0) cos n ( ¢ It(0; p, 9)

~0) + i~* (1 ")

N

Ir(")(r; #,/~0) cos n(¢ -- ¢0) + It* n~O

U(0; ~, ~) N

U(")(r; #, I~o) sin n(~ - ~0) + U*. n=O

351

DETERMINATION OF ATMOSPHERIC PARAMETERS

where (~, ~), (~0, ~0) are the directional parameters (cosine of the zenith angle, azimuth) of the direction of observation and of the direction of the incident solar radiation at the top of the atmosphere, respectively, and r is the total optical thickness of the atmosphere. The last terms on the right-hand side of Eq. (1) define the contributions of ground reflection to the upward radiation. The Stokes parameter V is omitted from this analysis, since its values are always very small and it is not likely that it can be measured with the accuracy necessary for inversion purposes. If the parameters It, It, and U on the left side of Eq. (1) are measured in several vertical planes (i.e., for several values of ~ - ~0) and in several directions (i.e., for several values of ~), then by harmonic analysis we can derive the values of the coefficients in the series in Eq. (1). The inversion problem is lhen reduced to the determination of r and ground reflectivity from the values of these coefficients and the values of It*, / / , and U*. We shall first discuss the method of inversion for a plane-parallel molecular atmosphere, i.e., for scattering according to Rayleigh's law. IV. INVERSION FOR PLANE-PARALLEL ATMOSPHERE WITH RAYLEIGH SCATTERING WITHOUT GROUND REFLECTION

For Rayleigh scattering (Chandrasekhar, 1950, p: 251; Coulson et al., 1960) we obtain, from the theory of radiative transfer, N = 2, and Ir (1) = U (°) = 0.

I~ (2) =

-~I,

(~) = ½ ~ U (~).

i¢o) = [ I t ° ) ]

3~o

.

[L(°)J = 1 6 ( ~ ~ go) [K(T, ~).]~(r, u0) - L(r, ~)'l.(r, u0)]'F, (3) where the two-by-two matrices K and L satisfy the integral equations K ( r , ~) = M ( , )

-{- g

[K(,).t(,')

-- L(~).L(~')].M(~')

# ud~' -[- ~'

(4a)

L(r, #) = M(~)e-~/~ + 3g --/o1 EL(u).t(~') -- K ( ~ ) . I , ( / ~ ' ) ] . M ( u ' )

/~d~' _ ,,

(4b)

(10 ]

(5)

where M(~)

= /1,

and zF is the matrix of the net flux of parallel solar radiation illuminating the atmosphere from above. As this radiation is neutral (unpolarized), then if 7rF0 represents the net flux,

(2a)

The seven remaining coefficients in Eq. (1) are not, however, all independent, since the following relationships exist between them: I { 1) = ~ U (1)

with such relations, then the atmosphere might contain non-Patyleigh scatterers. The foregoing statements provide useful tools for the analysis of the data and for the inversion problem. Because of the relations in Eq. (2), there remain only four coefficients that can be used for the inversion: It(°)(r, #, #o), I,(°)(r, ~, #o), U(1)(r,/~,/~o), and I(2)(r, #, go). The first two are the elements of a column matrix I (°), which according to theory (Sekera, 1963) is given by

(2b) (2c)

If these relations are satisfied by the coefficients of the series in Eq. (1), computed from the measurements, then the atmosphere consists of Rayleigh scatterers only. If the computed coefficients are incompatible

F = ~F0

.

If we write in Eq. (4) go for ~, g for ~', and transpose, we obtain ~(#0) = lVI(g0) -{- 3g jo [ 1 ~(~). [K(~).~t(.0) -- L ( g ) . I , ( g o ) ]

godg + ~,,.

(6)

Multiplying Eq. (3) by l~(~)d~, integrating from 0 to 1, and substituting from Eq. (6),

352

z.

SEKERA

~C~o= 1 . 0 0

. . .."'"

020

A = 025 5

[

"'""

--

I ,10

0

001

[ 1.0 OPTICAL

I

~

2

4

THICKNESS

I 8

P'o : 0 ~ 0

I

16

T

FIO. 3. Dependence of the function MI(T,/ao) on the optical thickness T for different values of the parameter #o (for ~0 = 1.00 also for different values of the ground reflectivity A). we obtain

~0

1 M(~).I(°)(r,

#, ~ o ) d ~ 1

= ~ []~(~0) -- l~(u0)]-F = ~ If we write K(r, ;L) =

LM,(T, ~o)l"

(7)

[K1

K3 v ~ K 4 ]

a n d if we denote the nth m o m e n t of the function F(u) b y

T h e functions in the right-hand side of Eqs. (8) and (9) are known functions 2 of the optical thickness r and ~o [their values are t a b u l a t e d ill C h a n d r a s k e h a r and Elbert (1954), Sekera and Blanch (1952), and Sekera and K a h l e (1966)]. T h e values of the functions M~(T, ~o) and Mr(r, ;~0) are plotted in Figs. 3 and 4 as functions of the optical thickness r for several values of the p a r a m eter ~0. It is evident t h a t M l and Mr are monotonically increasing functions of r, and thus for a given u0 t h e y uniquely determine the optical thickness. Similarly, we h a v e (Chandrasekhar, 1950, p. 253; Coulson et al., 1960, p. viii) U ° ) ( "r, #, #o) = --~(1 - #2)1/2(1 - #02) '/2

m,[F(~)] = fo I F(~)~"d;~

X uo2FoW(')(T, ~, ~o)(~ + ~o) - t

(10)

t h e n a f t e r substitution a n d multiplication we obtain f r o m Eq. (7) the relations

A(2)(r, ~, ~0) = ~ u 0 F o ( 1 -- ~o 2) × W~)(~, ~, ~o)(~ + ~o)-',

(1~)

m~[/z(°)(r, #)] + mo[A(°)(T, u)]

where

=

(Fo/4)[K~(~',

= F o M z ( r , ~o)

no) + Ua(T, ~o) - - 1 - - ~o 2] (8)

m0[Iz(0)(T, ~)] -- m~[Iz(0)(T, ~)] = (Fo/4)[K~(T, ~o) -t- K4(~', g0) = FoM,(T, ~o).

-

-

1 H- g0 2] (9)

W " ) b ' , u, ~,o) = X ( % , ) X ( % , o ) - Y(~)(#)Y(O(~o)(i = 1, 2)

In Chandrasekhar's notation, Kx ~ ~I,, K2 ~ ~,, K3----~x, K 4 - - - - - - t , L ~ , L 2 - - - - - - - 7 ,

L= ~ a, L, ----- 0.

DETERMINATION

0.25

OF ATMOSPHERIC

353

PARAMETERS

I'"~'~'~'~.~. , A • 0 . 8 0 "', ,, ,, /"/'o = I . O 0

0.20

[Lo* 0 . 4 0

0.15 A • 0.50

::t.

O.IO

P'o" O. I 0

A=025

0.05

!~ - " -

0 O,

A=

I I .lO

l.lo

OPTIGAL

I

I

I 8

I 16

T

THICKNESS

Fxo. 4. Dependence of the function M, (r, #o) on the optical thickness Tfor different values of the parameter #0 (for #0 = 1.00 also for different values of the ground reflectivity A).

and the X and Y functions satisfy the wellknown equations (Chandrasekhar, 1950, p. 183) X")(.) = 1

+

~0 1

udu'

W(*(T, ~, ~,')¢.")(~') I" + ,

Y")(#) = e-,/, +

(12)

fO1[Y(1)(,)X(i)(# ')

-- X(~)(u) y(')(~')]~b(o(~') ~ #d/ff _ ~, with ¢(i)(~) = -~(1 - ~ ) ( 1 + 2 v 9 , ¢(2)(~) = A ( 1 + ~2)~.

Multiplying Eq. (10) by (1 - ~)1~(1 + 2 v g d v

and integrating from 0 to 1 we obtain, with respect to Eq. (12) ~01 (1

$z2)1/2(1 "~- 2/~2)U(1)(T, ~ , / ~ o ) d ~

= ~o(1 #o2)l/2Fo[X(i)(T, #o) -- 1] = FoM(l)(r, uo). -

-

(13)

Similarly, multiplying Eq. (11) by (1 + ~2)2d~

and integrating from 0 to 1 we obtain, with respect to Eq. (12), o1 (1 + ~)%(2)(~, ~, ~o)d~ = 1_. (1 -

2

~o~)Fo[X(~)(~, ~o) -- 11

= FoM(2)(r, ~o).

(14)

354

Z. SEKERA. C 28

024

020

J~o= 0 6 0 ( Z e = 53")

%

~L. 0 1 6 i.o"

~-

012 /z ° 0.40 ( Ze= 66 ° 0 08

0 04

I 00.001

0.01

O. I OPTICAL

I tILe=0 I0 ( z l l = 84"

1.0 THICKNESS

I0

I00

T

Fro. 5. Dependence of the function :1I(1~ (r, uo) o11 the optical thickness r for different values of the parameter # 0 . The functions M(i)(r, tt0)(i = 1, 2) are where plotted in Figs. 5 a n d 6 as functions of the r~,,(r ' u) 1 optical thickness r for several values of g0. r ( r , t~) = / These are again monotonically increasing L%(r, U) functions of r, and thus uniquely determine -- l~.e-'f. -[- ~ [L(~) .l~(u') the optical thickness for a given value of #o. Substituting in the left side of Eqs. (8), ____ (9), (13), and (14) the measured values _ E(u).fj(u,)].lt. u'du', (16) (expressed in units of Fo) of the coefficients with Iz (°), Ir (°), U (x), I/2), we obtain the values °fMk(r,#°),M(°(r,~°) (k=l,r;i= 1'2)" E= [:]. From the diagrams in Figs. 3, 4, 5, and 6 we can then obtain four independent esti- The matrices K and L satisfy the integral mates of the optical thickness r. equations in Eq. (4) and

J

V. INVERSION FOR AN ATMOSPHERE WITH RAYLEIGH SCATTERING WITH GROUND REFLECTION ACCORDING TO

LAMBERT'S LAW

If the ground reflection is governed by Lambert~s law with the reflectivity A, then only the functions M/°) and M J °) are modified, since the terms I~* and I / o n the right-hand side of Eq. (1) are azimuthindependent and U* = 0. The expressions for these terms have been derived by Chandrasekhar (1950, p. 279) in the form (of. Sekera, 1966, p. 51) [ L * ( r , . , ,o)] I*

ApoFo

= [L*(r, u, ,o)] - 411 -- ~ ( r ) ] X [Tl(r, ,o) -F %(r, #o)]r(r, u),

8(g) = 1 -- ;1 lYz(r, ~) + %(r, U)]udu. The functions 7~(r, u)(i = l, r) can be easily computed front the elements of the matrices K and L (see Appendix). They are usually tabulated together with the elements K6 L~ (Chandrasekhar and Elbert, 1954; Sekera and Blanch, 1952). For Lambertian ground reflection the right-hand side of Eq. (7) must be augmented by the term f/ ju l~(u)-I*(r, u, uo)d~

411

AuoFo -A~(r)]

X b',(r, uo) -I- %(r, uo)]O(r), (15)

(17)

where from Eq. (16) we obtain, after replacing, by ~' and vice versa,

DETERMINATION OF ATMOSPHERIC PARAMETERS

355

0.14

01~

/..c • 0 . 4 0 ( Z e , 66"1 = 0 . 6 0 (Ze= 53 ° )

~

olo

~

o.08

0.06

--

004

p.o= 0 . 1 0 ( Z ~ ,

64°]

0.0~ I

0

0.001

0.01

0.1

1.0

OPTICAL

THICKNESS

I0

I00

"Z-

Fzo. 6. Dependence of the function M (~) (r, ~0) on the optical thickness r for different values of the parameter ~o.

O(~) =

then

fo 1 1~(~') • F(T, g')d~'

[M**(r,

=)(01

Mr*(r,

l~(v') • {e -~/# + 3 ~01 [ L ( t )

• l~(g) -- K(g').~,(g)]

~,&, ) ~-~_ ~E

dg'.

(18)

L(~) = l~(~)e--/. + ~3~01 l~I(d)

A)

411

--

~r(r, ~0)]

A~(r)]

= [mo[Ll(,O) + La(,o)]], Lmo[L2(#o) + L4(#o)]J

(21)

m2[I~(°)(r, #)] + mo[I/°)(r, /~)] (19)

If in the last term on the right-hand side of Eq. (18) the order of integration with respect to ~ and ~' is reversed, then from Eq. (19) we have

O(r) = { ~o1~'l(~')e-r/~'d~' + ~ol [I,(~) -- 1VI(~)e-'/"]d~}.E = /o I ~,(g).Edg.

#o;

where Li(#)(i = 1, 2, 3, 4) are the elements of the matrix L. With this notation, Eqs. (8) and (9) assume the form

From~Eq. (4) we have

• [K(g').T.(g) -- L(g').I~(g)] # gdg' _ ,.

#o;A)]=A~o['~,(r,#o)+

(20)

If we write

fot M(~).l*(r, ~, ~o)d~

.

]- M, (r, Vo, A)

= Fo [ %/-2Mr*(r, ~o," A)

]

= Fo'[M~(r, ~o) +/z*(r, ~0; A)] mo[I~(°)(r, ~)] - m2[Iz(O)(r,~)] -- Fo'[Mr(r, /.to) + i,*(r, g0; A)].

(22a) (22b)

In Figs. 3 and 4 the curves labeled A -- 0.25, 0.50, and 0.80 represent the variations of the functions Ms(r, go) + Mz*(r, ~0; A) and Mr(r, g0) + Mr*(r, ~0; A) with r for g0 = 1. These curves indicate very clearly that the monotonic increase of these functions disappears rapidly with increasing values of A, and they can no longer be used for a unique determination of the optical thickness. For larger values of A we have to use only the functions M(1)(r, go) and M(2)(r, #o) for the determination of the optical thickness. We can then use the diagrams in Figs. 7 and 8 for deriving the values of the ground reflectivity A. If the ground reflection follows Lambert's law and the atmosphere contains

356

z. SEKERA

05

~6.i

/,/.o=0.40

0.2

2.0

ol ~

i/o.,,

p..=O.lO o

0.05

°.0%

o,.~

d4

-

15 --4.O

o . o , ~ ~ ~

0'..

0.6 0,020

01.;' I

i

o.,

0'.6

o.B

o!2

o!4

o'.6

o!8

REFLECTIVITY A

Fro. 7. Mz -t- Mr* as a function of the reflectivity A for different values of the optical thickness r and the parameter ~o.

scatterers of molecular size only, the values of the reflectivity derived from the diagrams in Figs. 7 and 8 should be identical.

tions in Eqs. (2a,b,c) should be satisfied and the independent determinations of the optical thickness from Eqs. (13) and (14) should lead to the same value. Moreover, for ground reflection following closely the Lambert law of isotropic and unpolarized reflected radiation, Eqs. (22a) and (22b) should give the same value of the ground reflectivity A when the optical thickness

V I . DETERMINATION OF AEROSOL SCATTERING CONTRIBUTION AND ATMOSPHERIC TURBIDITY

For a pure molecular atmosphere that exhibits only Rayleigh scattering, the condiF 0.2

Tffi

oo

~ ~

'

p.o=1.00 :

T= oo -160

~o=0.60

.15

i

.

__

o.t

05

I

I ///

:o,o

= - - - -

/ 003 0 15

00, 0

003

"

0.2

0.4

0.6

0

0.2

0.4

"

!0.6

o'.2

0

/

I

i

I

J

0.2

0.4

0.6

0.8

o',

d6

o'8

0.8

RE:FLECTIVITY A

FIG. 8. Mr -{- Mr* as a function of the reflectivity A for different values of the optical thickness T and the parameter ~o.

DETERMINATION OF ATMOSPHERIC PARAMETERS

determined from Eqs. (13) and (14) is used. If any one of these requirements is not satisfied, there are two possibilities: (a) The actual law of ground reflection is different from Lambert's law, or (b) non-Rayleigh scattering is also present in the atmosphere. The separation of these two effects is the most immediate (and most difficult) problem in the theory of radiative transfer. As in the case of Lambert reflection, it is possible to evaluate theoretically the coefficients of the harmonic series I~*(n), I, *(n), and U *(n) for specular reflection (Fresnel's law). However, the natural terrain reflects radiation in a manner different from both Lambert's law and Fresnel's law, as demonstrated by studies of reflection from various ground samples, conducted by Coulson et al. (1965). In the general case, the law of reflection is given by a four-by-four matrix, and thusl in order to determine the general law of reflection by a particular area of the Earth's surface, it is necessary to determine 16 elements of this matrix as functions of the direction of the incident and reflected radiation. It is obvious that such a determination would require a vast amount of observational data that would be very difficult to gather in the short time during which a particular area of the Earth's surface comes under the view of satellite instruments. It seems that the only way to avoid this difficulty lies in undertaking an extensive study of ground reflection with the specifc goals of determining the law of reflection for various types of terrain and of evaluating the effects of these laws on the emerging radiation fields. We encounter a similar basic difficulty in determining atmospheric turbidity, i.e., the effect of non-Rayleigh scattering by particulate matter in the atmosphere. There are two reasons for this difficulty: (a) the lack of a theoretical evaluation of the radiation emerging from a turbid atmosphere with mixed (i.e., molecular and aerosol) scattering, and (b) the great variability of the parameters determining aerosol scattering. Although several approximations have

357

been suggested for the theory of the turbid atmosphere (de Bary, 1964; Irvine, 1965), very little theoretical background is available for immediate use in the inversion theory. However, the development of the method of forward peak separation, which allows us to overcome the main difficulty in the analytical treatment of aerosol scattering, greatly supports the expectation that the transfer problem in a turbid atmosphere will be solved in the very near future. Local variations in the size distribution and optical characteristics of aerosol particles can be treated by adopting an approach similar to that outlined in the previous paragraph. To determine all the parameters that would completely define the locally variably size distribution of aerosol particles would require an excessive quantity of observational data. We are therefore forced to resort to typical models that reduce the number of parameters required by the inversion method. We can consequently expect that the inversion method will provide only "effective" parameters (e.g., an "effective" turbidity factor, "effective" size distribution, etc.); that is, if the aerosol content has the characteristics of the model, then the parameters defining the model would have the "effective" values obtained from the measurements. Since such a determination of these parameters is far from unique, it is necessary to take measurements at different wavelengths. In the visible and adjacent spectral regions, the ratio of the optical thickness of the aerosol component (rL) to the optical thickness of the molecular component (re), or the so-called turbidity factor [T~ = 1 + (~L/~)],

varies considerably with the wavelength. Consequently, the theoretical values that are critically dependent on this ratio will be sufficiently different for different wavelengths. This feature ensures independent determinations of the "effective" parameters. The problem is then to select one particular model that will give the same "effective" parameters for all wavelengths. The method of obtaining these "effective" parameters from the measurements is generally suggested by the existing studies of

358

z. SEKERA

skylight polarization, and especially by the comparison of the measured values with the computed values (Sekera, 1957). These studies have demonstrated that the effect of aerosol scattering can be regarded as a perturbation of the polarization field of a pure molecular atmosphere. Since daily and local variations of the molecular content of the atmosphere (i.e., of the molecular optical thickness rR) are negligible (Deirmendjian, 1955), the optical thickness rR for a given wavelength can be regarded as known. The coefficients I (~), I, (~), and U (~) in Eq. (1) can then be computed for this value of rR and compared with the values derived from the measurements. The differences between the measured values and those derived from the theory for Rayleigh scattering provide a set of values from which the "effective" parameters have to be determined. The success of this determination nmst be assessed after the transfer theory for a turbid atmosphere is completely developed. So far, it has been possible to obtain only qualitative measures of atmospheric turbidity. As turbidity increases, we can expect that the conditions satisfied by Rayleigh scattering in Eq. (2a, b, c) will be decreasingly satisfied, and the optical thickness obtained from the four functions Mk, M (i) (k = l, r; i = 1, 2) will disagree more and more. Therefore, the magnitude of this disagreement can be regarded as the first indication of the existence and magnitude of turbidity. VII. CONCLUDING REMARKS Although for a plane-parallel Rayleigh atmosphere the inversion method outlined above allows four (or two in the case of Lambert~s reflection) independent deterniinations of the optical thickness, the determination of the atmospheric turbidity and its parameters from satellite measurements will require considerable effort and extensive preparatory studies. However, it would appear that the inversion problem for diffusely reflected radiation is not likely to be radically different from the corresponding problem for diffusely transmitted radiation (skylight). It is therefore possible to study the accuracy and other pertinent

characteristics of the inversion method quite extensively by ground-based measurements before adopting the method for satellite observatio~)s. The theory of the inversion method for skylight, together with results of actual measurements, will be presented in a forthcoufing paper. ACKNOWLEDGMENTS The research leading to this paper was partly supported by a grant from the National Aeronautics and Space Administration, No. NGR-05-007-041, to the Space Science Center, Institute of Geophysics and Planetary Physics, University of California, Los Angeles, and partly by Project R A N D (RPN 1162). The author gratefully acknowledges the useful suggestions of Dr. D. Deirmendjian and Professor S. Venkateswaran.

APPENDIX DERIVATION OF THE EXPRESSIONS FOR FUNCTIONS TI(#) AND 5'r(tZ)

The functions ~',(t,) and ~'r(t~), which are the elements of the matrix F(t0, are given t)y the equation r ( , ) = [~'r(~)

~

•]~(~') -- K(g).I,(g')]

[L(,)

g'dg',.E.

/z--#

(AI)

If we introduce the following notation

1 0 - - [1/%/~ ],

mk[K] = fol "fC(r, tt)#kdt,;

then it follows from Eq. (4) that

M(#)-C = E.

(A2)

Since ill the last term of Eq. (A1) we can write tt' # /z -- #' - # -- Izt

1

(A2a)

we obtain, after substituting from Eqs. (A2) and (A2a) in Eq. (A1), r ( . ) = Ee -~" + g

[L(.).Ii(.')

-- K ( . ) . L ( . ' ) ] . M ( . ' )

.d.'

/~--/z

!

.C

3 8 L(t~)m0[I~].E ~- 3~ K(u)m0[ I.].E.

(A3)

DETERMINATION OF ATMOSPHERIC PARAMET~ll~S

COULSON, K. L. (1959). Characteristics of the radi-

F r o m Eq. (4) we then have r(~) = L(~){C -

bn0[~t].v.}

-4- K(tt).~m0[?-].E.

(A4)

If we write for the 0-th m o m e n t s of the elements of the K a n d L matrix

ki = ~m0[K~(T, tL) -4- Ki+2(T, tt)] li = ~m0(Li(r, •) -{- Li+~(7, #)]

(i = 1, 2)

then, after multiplication of the matrices in Eq. (A4), we have

"~(t~) = l~Kl(r, ~) -4- 21sK2(T, ~t) -4- [1 -- k~]L~(T, tt) -4- [1 -- 2k~)L2(T, tt) ~(tt) = l~K3(r, it) A- 2/2K4(r, tt) -4- [1 -- k~]L3(r, g) -4- [1 -- 2k2]L4(T, tt).

(A5) F r o m the t h e o r y of the K a n d L matrices (Sekera, 1966) it follows t h a t ll -4- kl = 1,

12 -4- k2 - ½.

Substituting these relations in Eq. (A5), Vt(D) = /I[K,(~', p) -~- LI(T, ~)] + 2/2[Ks(T, ~) + 52(7, U)]

"y,(~) = l,[K3(-r, ~) -4- L3(T, t~)] -4- 2/~[K4(r, tt) -4- L4(r, ~)1.

(A6)

Adding the equations in Eq. (A6), multiplying b y tLdtt, and integrating f r o m 0 to 1, we get

s = 1 -- llml[K1 "4- K~ -4- L1 "-k L3] 2l,m~[K2 -4- K4 iv L2 "4- L,]. -

-

359

(A7)

REFERENCES

BELLMAN,A..E., KA~IWADA,N. N., KAt,AaA,R. E., AND USNO, S. (1965). Inverse problems in radiative transfer: layered media. Icarus 4, 119-126. CHANDRASEKIIAR, S. (1950). "Radiative Transfer." Clarendon Press, Oxford, England. CHANDRASEKHAR,S., AND ELBERT, D. D. (1954). The illumination and polarization of the sunlit sky on Rayleigh scattering. Trans. Am. Phil. Soc. 44 (Part 6), 693-728.

ation emerging from the top of a Rayleigh atmosphere (I). Planetary Space Sci. 1, 265-276. COULSON,K. L., DAVE,J. V., ANn SEKERA,Z. (1960). "Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering." Univ. of California Press, Berkeley, California. COULSON, K. L., Boumc~us, G. M. B., ArCDGaAY, E. L. (1965). Optical reflection properties of natural surfaces. J. Geophys. Res. 70, 4601~1611. DE BARV, E. (1964). Influence of multiple scattering on the intensity and polarization of diffuse sky radiation. Appl. Opt. 3, 1293. DEIRMEN1)JLAN,D. (1955). The optical thickness of the molecular atmosphere. Arch. Meteorol., Geophys., Bioklimatol., Ser. B 6, 452-461. IaVINE, W. M. (1965). Multiple scattering by large particles. Paper presented at Symposium on Radiation Processes, Leningrad, 1964; Intern. Union Geodesy Geophys. Monograph No. 28, p. 11, and Astrophy.v.J. 142, 1563. SEKERA, Z. (1951). Polarization of skylight. "Compendium of Meteorology," pp. 79-89. Am. Meteorol. Soc., Boston, Massachusetts. SEY*ERA, Z. (1957). Polarization of skylight. Encyclopedia o] Physics 48, 288--328 (SpringerVerlag, Vienna). SEKERA, Z. (1963). Radiative transfer in a planetary atmosphere with imperfect scattering. R A N D Rept. B413-PR. The RAND Corporation, Santa Moniea, California. SEt,ERA, Z. (1966). Reduction of the equations of radiative transfer for a plane-parallel, planetary atmosphere, Part II. R A N D Rept. RM-5056-PR. The RAND Corporation, Santa Monica, California. SEKERA, Z., AND BLANCH, G. (1952). "Tables Relating to Rayleigh Scattering of Light in the Atmosphere." Scientific Report No. 3, U. S. Air Force Contract No. Af 19(122)-239, Department of Meteorology, University of California, Los Angeles, California. SEKERA, Z., .AND KAHLE, A. B. (1966). Scattering fimctions fl)r Rayleigh atmospheres of arbitrary thickness. R A N D Rept. R-452-PR. The RAND Corporation, Santa Monica, California.