Transcript Document
Particle Scattering Single Dipole scattering (‘tiny’ particles) – Rayleigh Scattering Multiple dipole scattering – larger particles (Mie scattering) Extinction – Rayleigh particles and the example of microwave measurement of cloud liquid water Microwave precipitation Scattering phase function – radar/lidar equation backscattering properties e.g. Rayleigh backscatter & calibration of lidar, radar reflectivity Analogy between slab and particle scattering Insert 13.10/ 14.1 slab particle Slab properties are governed by oscillations (of dipoles) that coherently interfere with one another creating scattered radiation in only two distinct directions - particles scatter radiation in the same way but the interference are less coherent producing scattered stream of uneven magnitude in all directions Radiation from a single dipole* Scattered wave is spherical in wave form (but amplitude not even in all directions) Scattered wave is proportional to the local dipole moment (p=E) Basic concept of polarization Key points to note: • parallel & perpendicular polarizations • scattering angle * Referred to as Rayleigh scattering Any polarization state can be represented by two linearly polarized fields superimposed in an orthogonal manner on one another Scattering Regimes From Petty (2004) Scattering Geometry Rayleigh Scattering Basics 8 m -1 Qs » x 4 2 3 m +2 2 2 é m 2 -1 ù Qa » 4xÁm ê 2 ú »Qext ëm + 2û wº Qs µ x3 Qs + Qa Single-particle behavior only governed by size parameter and index of refraction m! Rayleigh “Phase Function” Vertical Incoming Polarization Horizontal Incoming Polarization Incident Light Unpolarized Polarization by Scattering Fractional polarization for Rayleigh Scattering The degree of polarization is affected by multiple scattering. Position of neutral points contain information about the nature of the multiple scattering and in principle the aerosol content of the atmosphere (since the Rayleigh component can be predicted with models). Rayleigh scattering as observed POLDER: Radiance Strong spatial variability Scattering angle 0.04 Pol. Rad Smooth pattern 650 nm Signal governed by scattering angle 0 Proportional to Q (Deuz₫ et al., 1993, Herman et al., 1997) Radiation from a multiple dipole particle r ignore dipole-dipole interactions rcos At P, the scattered field is composed on an EM field from both particles 2r (1 cos) E E1e i E2 e i size parameter I E E 2E1 E2 cos 2 1 2 2 P For those conditions for which =0, fields reinforce each other such that I4E2 Scattering in the forward corresponds to =0 – always constructively add Larger the particle (more dipoles and the larger is 2r/ ), the larger is the forward scattering The more larger is 2r/, the more convoluted (greater # of max-min) is the scattering pattern Phase Function of water spheres (Mie theory) Properties of the phase function g High Asymmetry Parameter 1 1 P(cos) cosd cos 2 1 asymmetry parameter g=1 pure forward scatter g=0 isotropic or symmetric (e.g Rayleigh) g=-1 pure backscatter • forward scattering & increase with x • rainbow and glory • Smoothing of scattering function by polydispersion Low Asymmetry Parameter Particle Extinction Particle scattering is defined in terms of cross-sectional areas & efficiency factors Geometric cross-section r2 σext = effective area projected by the particle that determines extinction Similarly σsca, σabs The efficiency factor then follows s ext,sca,abs Qext,sca,abs = p r2 Particle Extinction (single particle) =1 Note how the spectrum exhibits both coarse and fine oscillations Implications of these for color of scattered light How Qext2 as 2r/ extinction paradox ‘Rayleigh’ limit x 0 (x<<1) Extinction Paradox shadow area r2 Qext combines the effects of absorption and any reflections (scattering) off the sphere. shadow area 1 ?? 2 r shadow area area filledby diffraction r 2 r 2 r 2 2 2 r Qext insert 14.10 Poisson spot – occupies a unique place in science – by mathematically demonstrating the non-sensical existence of such a spot, Poisson hoped to disprove the wave theory of light. Mie Theory Equations • Exact Qs, Qa for spheres of some x, m. • a, b coefficients are called “Mie Scattering coefficients”, functions of x & m. Easy to program up. • “bhmie” is a standard code to calculate Q-values in Mie theory. • Need to keep approximately x + 4x1/3 + 2 terms for convergence Mie Theory Results for ABSORBING SPHERES 4 3 r 3 r 10 m 10 -3 cm V No Volumes containing clouds of many particles N0 100 droplets per c.c V 100 Extinctions, absorptions and scatterings by all particles simply add- volume coefficents L-1 L-4 ext,abs,sca n(r)r 2 Qext,abs,sca (r, )dr L 3 4 10 -3 3 V 10 -7 half of 14.9 0 L2 n( r)= the particle size distribution # particles per unit volume per unit size n(r) = const e-r / a n(r) = const r 1-3b r b ab e Exponential distribution (rain) Modified Gamma distribution (clouds) æ (ln r - ln r ) 2 ö 0 ÷÷ n(r) = const expçç2 2s è ø Lognormal distribution (aerosols, sometimes clouds) r Effective Radius & Variance ¥ r = ò r n(r) dr 0 ¥ Mean particle radius – doesn’t have much physical relevance for radiative effects ò n(r) dr 0 ¥ reff = ò rp r 2 n(r) dr 0 ¥ 2 p r ò n(r) dr For large range of particle sizes, light scattering goes like πr2. Defines an “effective radius” 0 ¥ n eff = ò (r - r eff 0 ) 2 p r 2 n(r) dr “Effective variance” ¥ r 2 eff ò pr 0 2 n(r) dr n(r) = const r 1-3b r b ab e Modified Gamma distribution a = effective radius b = effective variance Polydisperse Cloud: Optical Depth, Effective Radius, and Water Path (visible/nir ’s) t = ò s ext (z)dz Cloud Optical Depth s ext = ò n(r)p r 2Qext dr Volume Extinction Coefficient [km-1] Dz Qext ® 2 t = 2 ò dz ò n(r)p r 2 dr Cloud Optical Depth Dz 4p r water ò n(r)r 3dr 3 ò n(r)r 3dr rcloud = re = Local Cloud Density [kg/m3] Cloud Effective Radius [μm] ò n(r)r dr t » 2Dz 2 3 p ò n(r)r 3dr 4pr water re » 3 2 r water L re ρcloudz t» 3L 2 rwater re 1st indirect aerosol effect! (Twomey Effect) Variations of SSA with wavelength Somewhat Absorbing NonAbsorbing! Satellite retrieve of cloud optical depth & effective radius Non-absorbing Wavelength (~1): Absorbing Wavelength (<1): Reflectivity is mainly a function of optical depth. Reflectivity is mainly a function of cloud droplet size (for thicker clouds). • The reflection function of a nonabsorbing band (e.g., 0.66 µm) is primarily a function of cloud optical thickness • The reflection function of a near-infrared absorbing band (e.g., 2.13 µm) is primarily a function of effective radius – clouds with small drops (or ice crystals) reflect more than those with large particles • For optically thick clouds, there is a near orthogonality in the retrieval of tc and re using a visible and near-infrared band • re usually assumed constant in the vertical. Therefore: LWP 2 3 ret Cloud Optical Thickness and Effective Radius (M. D. King, S. Platnick – NASA GSFC) Cloud Optical Thickness 1 10 Ice Clouds King et al. (2003) >75 1 Cloud Effective Radius (µm) 10 Water Clouds >75 6 17 28 Ice Clouds 39 50 2 9 16 23 Water Clouds 30 Monthly Mean Cloud Effective Radius Terra, July 2006 Liquid water clouds –Larger droplets in SH than NH –Larger droplets over ocean than land (less condensation nuclei) Ice clouds –Larger in tropics than high latitudes –Small ice crystals at top of deep convection Aerosol retrieval from space- the MODIS aerosol algorithm Uses bi-modal, log-normal aerosol size distributions. • 5 small - accumulation mode (.04-.5 m) • 6 large - coarse mode (> .5 m) Look up table (LUT) approach • 15 view angles (1.5-88 degrees by 6) • 15 azimuth angles (0-180 degrees by 12) • 7 solar zenith angles • 5 aerosol optical depths (0, 0.2, 0.5, 1, 2) • 7 modis spectral bands (in SW) Ocean retrievals • compute IS and IL from LUT • find ratio of small to large modes () and the aerosol model by minimizing e = 1 n n å j =1 Im - Ic I m + 0.01 where I c = h I S + (1 - h ) I L and Im is the measured radiance. • then compute optical depth from aerosol model and mode ratio. Land retrievals • Select dark pixels in near IR, assume it applies to red and blue bands. • Using the continental aerosol model, derive optical depth & aerosol models (fine & course modes) that best fit obs (LUT approach including multiple scattering). • The key to both ocean and land retrievals is that the surface reflection is small. “Deep Blue” MODIS Algorithm works over Bright Surfaces • Uses fact that bright surfaces are often darker in blue wavelengths • Uses 412 nm, 470nm, and 675nm to retrieve AOD over bright surfaces. • Still a product in its infancy “Deep Blue” MODIS Algorithm works over Bright Surfaces • Uses fact that bright surfaces are often darker in blue wavelengths • Uses 412 nm, 470nm, and 675nm to retrieve AOD over bright surfaces. • Complements “Dark Target” retrieval well. • Still being improved! MAIAC Scattering phase function æ E sca, ö æ S2 ç ÷=ç E è sca,r ø è S4 æ S2 where ç è S4 S3 ö e-ikr +iw t æ E 0, ö ç ÷ ÷ S1 ø kr è E 0,r ø S3 ö ÷ is the amplitude scattering matrix S1 ø Polarized light is expressed by 4 Stokes parameters, and the phase function is acordingly a 4 ´ 4 matrix. The structure of this phase matrix depends on shape & orientaion of particles. Each element of the matrix is a quadratic function of S1,S2, etc For particles with certain basic symmetry, the phase function becomes : æ S11 S12 0 0ö ç ÷ 0 0÷ 1 ç S12 S22 P(Q) = 2 0 S33 S34 ÷ k Csca ç 0 ç ÷ 0 -S34 S44 ø è0 where S11 = S22 , S 33 = S44 For spheres. If the rayleigh limit holds, then S12 = 0. spheres spherical æ Isca ö æ S11 0 0 ç ÷ ç Q 0 1 sca ç ÷= ç 0 S11 çU sca ÷ k 2 R 2 ç 0 0 S33 ç ÷ ç 0 -S34 è Vsca ø è0 0 ö æ I0 ö ÷ç ÷ 0 ÷ ç Q0 ÷ S34 ÷ çU 0 ÷ ÷ç ÷ S33 ø è V0 ø Non spherical with plane of symmetry Isca Q 1 sca 2 U k R2 sca V sca S11 0 0 S22 0 0 0 0 Isca Q 1 sca U k 2R 2 sca V sca S11 S12 0 0 0 0 S33 S34 S12 0 S22 0 0 0 S33 S34 0 I0 0 Q0 S34 U0 S44 V0 0 I0 0 Q0 S34 U0 S44 V0 non spheres Particle Backscatter Cd()I0 is the power scattered into per unit solid angle Differential cross-section Cd () Csca P() 4 Bi-static cross-section Cbi () 4Cd () Backscattering cross-section Cb 4Cd ( 180) CbI0 is the total power assuming a particle scatters isotropically by the amount is scatters at =180 Polarimetric Backscatter: LIDAR depolarization • Transmit linear • Receive parallel/perpendicular I measured MI sca 1 1 1 1 1 1 M , M r 2 1 1 2 1 1 ( S12 ) S I sca 11 I0 ( S12 ) S 22 I 1 I0 0 Q0 1 I measured , S11 S 22 Water/Ice/Mix I measured ,r S11 S 22 linear depolarization ratio I S11 S 22 measured ,r I measured , S11 S 22 (2 S12 ) Ice =0 for spheres Polarimetric Backscatter: RADAR ZDR • Transmit both horizontal & vertical • Receive horizontal & vertical for spheres, ZDR~0 Lidar Calibration using Rayleigh scattering Laser backscattering Crossection as measured During the LITE experiment For Rayleigh scattering Cb b (m1ster 1 ) 8 1 Csca sca (m ) 3 Lidar Calibration using Rayleigh scattering Rayleigh scattering is wellunderstood and easily calculable anywhere in the atmosphere! 3 2 4 (ns2 1)2 63 R 2 4 2 2 Ns (ns 2) 67 ns = 1 + a * (1 + b λ-2) Stephens et al. (2001)