Keywords: Rayleigh scattering, Mie scattering, Aerosols, Lidar, Lidar equation

Transcription

1 CEReS Atmospheric Report, Vo., pp.9- (007 Moecuar and aeroso scattering process in reation to idar observations Hiroaki Kue Center for Environmenta Remote Sensing Chiba University -33 Yayoi-cho, Inage-ku, Chiba-shi, Japan Abstract: Lidar is an important and versatie too for various types of targets in the atmosphere. In this report, we describe the fundamentas of ight scattering due to atmospheric partices, namey the Rayeigh scattering from air moecues and Mie scattering from aeroso partices. The soution of the idar equation is discussed in reation to the derivation of aeroso quantities from the idar data. Keywords: Rayeigh scattering, Mie scattering, Aerosos, Lidar, Lidar equation. Rayeigh scattering When the partice radius is much smaer than the waveength of the scattered radiation, the process can be described by means of the Rayeigh scattering theory. For visibe ight, this scattering mode occurs for air moecues. It is we known that the Rayeigh scattering scheme can ead to the bue coor of the cear sky and reddish sky coor at dusk and dawn. The differentia cross-section of the Rayeigh scattering is given as ~ d σ α + scat k cos θ Ω 4πε d 0 Here, k is the wavenumber (π divided by the waveength λ, α ~ is the poariabiity of air moecues, and ε 0 is the permittivity of vacuum. Athough the moecuar poariabiity is, in genera, a tensor quantity, it can be treated as a scaar quantity for isotropic media. When the radiation with waveength λ μm is transmitted through the dry atmosphere of 5, atm, and containing 0.03% CO, the refractive index n ~ can be written as ( ~ λ λ The reationship between the poariabiityαand ~ the refractive index n ~ is given as α~ n~ - 4πε π N 8 n + + ( 0 ( (3 9

2 Here, N stands for the moecuar number density. This equation, caed the Lorent-Loren formua, describes the reationship between the eectromagnetic property of a singe moecue, and the optica response of moecuar ensembe. Eqs. ( and (3 can be used to evauate the magnitude of the Rayeigh scattering at each waveength λ. The tota cross-section of the Rayeigh scattering, σ scat, can be obtained by integrating Eq. ( over the entire soid ange: dσ σ scat scat sinθdθdϕ dω ~ αk π π (cos + sin 4 θ θdθ (4 0 πε0 ~ 8π αk 3 4 πε0 If we consider the non-sphericity of atmospheric moecues, the differentia cross-section can be given as dσ ext dω α~ k γ 4πε + 0 [{ cos θ+ γ( + sin θ } cos ϕ + ( + γsin ϕ] The depoariation factor, Δ, is defined as the ratio between the p-poariation (parae to the scattering pane and s-poariation (perpendicuar to the scattering pane at the direction of Ѳ π/. We have Δ for air moecues. The parameter γ in Eq. (5 is reated to this depoariation parameter as Δ γ (6 - Δ The tota cross-section, on the other hand, is the same as the spherica case of Eq. (4. When we consider the waveength range in which moecuar absorption can be negigiby sma, the ight extinction through the transmission path can be ascribed to the scattering (extinction scattering + absorption. Thus, the moecuar extinction coefficient α can be described as (5 α ( n( σ scat (7 Here n( stands for the moecuar density profie [moecues/m 3 ] as a function of atitude. 0

3 Substituting σ scat of Eq. (4, we obtain α ( 8π ( ( α~ 4 π n 3 4πε. (8 0 λ For the evauation of n(, the foowing formua is usefu: 5 3 [ ( n0] 0 exp[.306( a + b + c d] 5 n ( 0 exp ξ + (9 a [km -3 ] b [km - ] c [km - ] d These coefficients are obtained by fitting the US standard atmosphere (US76 with the exponentia function given by Eq. (9. The ratio between the moecuar extinction and moecuar back-scattering coefficient is caed the S parameter, which is needed to sove the idar equation as shown beow. The vaue of S is reated to the depoariation factor Δ, and given by 8π Δ S [sr]. (0 3. Mie scattering The Mie scattering formua 3 shoud be empoyed when the waveength λ of the incident radiation is of the order of the partice sie. In contrast to the Rayeigh scattering, generay Mie scattering is characteried with the dominance of the forward scattering as compared with the backward scattering. Since the aeroso partice sie is of the order of μm, ight scattering within and near the visibe waveength is usuay treated in the framework of Mie-scattering theory. In the Mie-theory, the scattering partice is assumed to be a dieectric sphere. The differentia cross-section is given as dσ scat i + i dω k Here, i and i refer to the ight intensity with eectric fied vector perpendicuar and parae, (

4 respectivey, to the scattering pane. In terms of the ampitude functions S (θ and S (θ (do not confuse these with the idar S parameters, these are given as with S(θ i S (θ i ( + S θ τ } (3 S { ( a π + b ( + + θ { τ }. (4 ( b π + a ( + Functions π and τ that appear in these equations are defined as ( π P sinθ (5 d ( τ P dθ (6 Here ( P are associated Legendre functions. In Eqs.(3 and (4, coefficients a and b are determined from the boundary condition at the surface of the dieectric sphere as ψ '( ~ ~ ( ~ nka ψ '( '( ~ n ( ka nψ nka ψ ka a nka ( ka n~ ( nka ~ (7 ψ ξ ψ ξ '( ka b n n~ ψ '( ~ ( ~ nka ψ ( '( ~ '( ~ ka ψ nka ψ ka n nka ( ka ( nka ~ (8 ψ ξ ψ ξ '( ka Here the foowing functions are empoyed to describe the eectromagnetic fied components: ψ ( ξ ( ξ + d ξdξ ξ ξ sin( (9 χ ( ξ ( ξ + d ξdξ ξ ξ cos( ξ ( ξ ψ ( ξ K + iχ ( ξ ( with ξ ka. The tota cross-section can be derived by integrating Eq. ( over the entire soid ange: (0

5 σ scat π k π k 0 { S( θ + S ( θ } ( + [ a + b ] π dσ scat dω sinθdθdϕ sinθdθ The absorption coefficient, σ abs, is derived from the optica theorem as 4π 4π σ abs Re{ S(0 } ( + Re( a + b (3 k k Here the notation Re indicates the rea part of a compex number. If the partice number density is n, the aeroso extinction coefficient can be cacuated as α ext nσ ext n(σ scat +σ abs. (4 The singe-scattering abedo, ω 0, is defined by the ratio between σ scat and σ ext : ω 0 σ scat / σ ext (5 This parameter equas to unity when no absorption takes pace. The phase function, i.e. the anguar dependence of the scattered radiation, is given by the ratio between the differentia cross-section of Eq.( and the tota scattering cross-section of Eq.(: dσ f (cos θ (6 σ dω scat The asymmetry parameter, g, is given as θ g cosθ f dω (7 This parameter is 0 for the case when the radiation is equay scattered for the forward and backward directions symmetricay, as in the Rayeigh scattering. The idar ratio, or the S parameter used in the idar equation, is defined as σ /( dσ / dω θ π / f θ π S scat (8 ( 3. Refractive index and sie distribution In the Mie-scattering theory, refractive index and sie distribution of partices are fundamenta parameters that govern the ight scattering and absorption properties. The refractive index can be written as n~ n ik (9 3

6 Here n is the u sua refractive index and the imaginary part is reevant to the partice absorption. As for the aeroso sie distribution, beow we describe both the Junge distribution and the Jaenicke mode. 3-. Junge distribution Junge distribution (Junge 955 was introduced on the basis of the observation that the number density of partices with arger than sub-micrometer diameter decreases in accordance with the power aw of the partice radius. The distribution can be written as dn n Cr d n r ν ( r. Here N stands for the partice number density, C is a constant, and the parameter ν is caed the Junge sope. This sope ν is widey used as a parameter characteriing the sie distribution, and we have ν 3 for usua circumstances. The waveength dependence of aeroso extinction (or equivaenty, optica thickness τ a, on the other hand, is often described in terms of the Angstrom exponent α ang ; namey τ a is proportiona to λ α ang. The Junge parameter ν is reated to the Angstrom exponent as ν α ang +. The vaue of ν becomes arger when reativey smaer partices (fine-mode partices dominate in the atmosphere. Care must be taken to the fact that the description of Eq. (30 is vaid ony for partices arger than sub-micrometer diameters. Nevertheess, since partices in this diameter range give more than 90% contribution to the aeroso extinction, Eq. (30 can ead to reasonabe resuts with additiona constraint, for exampe, that the distribution saturates and takes a constant vaue for partices smaer than the sub-micrometer regime. 3-. Log-norma distribution modes after Jaenicke 5 In this mode, it is assumed that the aeroso sie distribution is represented by a superposition of three og-norma distributions as 3 dn( r N i og ( r / r i n ( r exp (3 d og r i π ogσ i og σ i Here r i is the mean radius of the i th mode, N i is the partice number density, and σ i is the width (standard deviation of the mode. These parameters are tabuated for typica aeroso sie distributions of urban, rura, maritime, remote continenta, background, and desert dust storm (30 (3 4

7 modes. Tabe shows the parameters and Fig. shows the resutant sie distributions. Here we consider the dependence of the S parameter (i.e. the ratio between the extinction and back-scattering coefficients of aeroso partices on the waveength and the refractive index. The resuts for Urban and Maritime aeroso modes are shown in Fig.. Generay the vaue of S parameter increases with the waveength λ. The decrease of S for λ shorter than about 500 nm, on the other hand, is ascribed to the increase in the absorption. Figures 3 (for Maritime mode and 4 (for Urban mode indicates that the S parameter widey varies between around 0 and 00 sr, showing compicated behavior against the variation of the rea and imaginary parts of the refractive index. dn (r/d ogr (cm URBAN MARITIME R.C RURAL B.G D.D Radius (μm Fig. Log-norma sie distributions for various aeroso modes reported by Jaenicke (993. R.C. stands for Remote Continenta, BG, Background, and D.D., Desert Dust Storm. Tabe Parameters of og-norma sie distribution for various aeroso modes: after Jaenicke (993. Aeroso mode Mode i N i (cm 3 r i (μm og σ i Urban Rura

8 Maritime Remote continenta Background Desert dust storm S Urban mode Maritime mode Waveength (nm Fig. Waveength dependence of the S parameter for Urban and Maritime sie distribution modes. The rea and imaginary parts of the compex refractive index are n.55 and k

9 Fig. 3 Compex refractive index dependence of S parameter cacuated for the Maritime sie distribution mode. 7

10 Fig. 4 Compex refractive index dependence of S parameter cacuated for the Urban sie distribution mode. 8

11 4. Lidar equation 4. Extinction coefficient The extinction coefficient describes the rate at which the ight intensity decreases with the propagation in the medium. The microscopic mechanism of extinction is either scattering or absorption. When the initia ight intensity is I 0, the decrease di associated with the traveing distance of dr is given as di α I0 dr. (33 The proportionaity constant α [m - ] in this equation is caed the extinction coefficient. In the form of differentia equation, Eq. (33 becomes di dr α. (34 I 0 When the tota cross-section of aeroso scattering is σ scat and the number density is N, we have αnσ scat (35 Integrating Eq.(30, we obtain the ight intensity at a distance R as R I( R I exp 0 ( r dr I0 exp[ τ ( R] α 0 (36 Here the non-dimensiona quantity τ is caed the optica thickness, and Texp(-τ gives the transmission coefficient. 4. Back-scattering coefficient Back-scattering coefficient, β, indicates the fraction with which the incident ight is scattered to the backward direction. Let us consider the radiance I (r, θ, φ [Wm - sr - ] of the scattered radiation observed at the direction specified by the poar coordinate of (r, θ, φ: I0 dσscat I (37 r dω Here I 0 [Wm - ] is the incident irradiance and dσ scat /dω [m /sr] is caed the differentia cross-section, which can be cacuated by the Mie-theory. The back-scattering coefficient β [m - sr - ] is reated to the differentia cross-section (at the direction of θπ rad as dσscat β N dω θ π Here N [m -3 ] is the partice number density. (38 9

12 4.3 Lidar equation A Mie-scattering idar receives photons that are back-scattered from targets, namey atmospheric moecues and aerosos. The ight intensity received from a target at an atitude can be written as cτ G( P + p ( P0 AK β ( exp d Pb α( ' ' (39 0 Here P 0 is the emitted aser power [W] c is the speed of ight, τ p is the puse width [s], A is the effective area of the teescope system [m ], K is the efficiency of the optica system ( 0 K, G( is the overap function between the teescope fied-of-view and the aser beam ( 0 G(, β( [m- sr - ] is the back-scattering coefficient α( [m - ] is the extinction coefficient, and P b is the bias. Normay this bias eve in the idar signa must be determined from the signa eve at a far boundary, or aternativey, from the eve of the pre-trigger signa. 4.4 Soution of the idar equation In the rea atmosphere, one has to consider both moecuar and aeroso contributions in soving Eq. (39. When the aeroso contribution is predominant, however, one can use the Kett method. In this method, it is assumed that the range-corrected signa, X(, can be given as X ( Cα( exp α( ' d' 0 Here C is a constant. By differentiating this equation, it is easy to obtain (40 α ( d d α( n X ( (4 By introducing a new variabe u(x(/α(, one obtains du/d X(. Thus, u( u( C X ( X ( α( α( C C C X ( ' d' (4 Here C stands for the distance (atitude at an arbitrary-chosen far end boundary. By soving this equation in terms of α(, we obtain α( X ( X ( C C + X ( ' d' α( C (43 0

13 In this Kett approach, it must be noted that the soution is independent of the idar ratio S, as far as the parameter C can be assumed to be constant aong the idar observation path. Next, we consider a more genera method appicabe to the two-component (i.e. moecue pus aeroso atmosphere. For the moecuar part, we have aready shown the required parameters in Eqs. (-(9. The chemica composition of air moecues is more or ess constant up to about 80 km atitude, and the vaue of the moecuar idar ratio, S 8.5 sr can be empoyed for a the cases. In contrast, optica properties of aeroso partices are quite variabe, and the vaue of aeroso idar ratio, S α /β, is known to vary between and 00 sr. Nevertheess, if we can assume that this S parameter is constant regardess of the atitude, we can make use of the so-caed Fernad s method to anayticay sove the idar equation. When the aeroso and moecuar contributions are separated expicity, Eq. (39 can be rewritten as K P ( [ β + ] [ + ] ( β( exp α ( ' α ( ' d' (44 0 Here K indicates the system constant. Given the vaues of S and S, it is straightforward to sove this equation in terms of the aeroso extinction coefficient, α. The resut is summaried as foows: S( S( X ( exp I( α ( α ( + (45 S X ( c + J ( α ( c α ( c + S ( S Here, X(, I(, and J( are defined as X ( J ( P( c S ( ' I ( α S c S ( ' X ( ' c exp ( ' d ' I ( ' d ' In Eqs. (45 and (46, c referes to a far distance at which the boundary vaues of the extinction coefficient shoud be given a priori. Since generay aeroso partices exist up to a imited atitude (e.g. 3-5 km, depending on the situation, it is possibe to assume that at the atitude of c, the idar signa arises ony from the moecuar back-scattering. This means that in Eq. (45, one can put α ( c 0, and the moecuar extinction coefficient, α ( c, can easiy be evauated using Eqs. (46

14 (9 and (0. In Fig. 5, we show an exampe of aeroso distribution derived from the CEReS muti-waveength idar system Atitude (m nm 53nm 756nm 064nm Extinction coefficient (km - Fig. 5 Vertica profie of aeroso extinction coefficient derived from muti-waveength measurement at CEReS, Chiba University. References R.M. Goody and Y.L. Yung, Atmospheric Radiation Theoretica Basis, Oxford University Press, 989. Cooquium Spectroscopicum Internationae, H.C. van de Hust, Light Scattering by Sma Partices, Dover, New York, C. Junge, The sie distribution and aging of natura aerosos as determined from eectrica and optica data on the atmosphere, J. Meteoroogy,, -35, R. Jaenicke, Tropospheric aerosos, in Aeroso-Coud-Cimate Interactions, ed. by PV. Hobbs, Academic Press, 993. CEReS Atmospheric Report, Vo., pp.9- (007 Copyright: CEReS, 007