Lecture 06. Fundamentals of Lidar Remote Sensing (4) Physical Processes in Lidar Physical processes in lidar (continued) Doppler effect (Doppler shift and broadening) Boltzmann distribution Reflection from target or surface Multiple scattering Polarization in scattering Fundamentals behind physical processes Atomic structure and energy levels Molecular structure and energy levels General equations for light transmission Summary 1
Doppler Shift and Broadening Doppler Effects - Doppler linewidth broadening and Doppler frequency shift are temperature-dependent and wind-dependent, respectively (applying to both Na, K, Fe resonance fluorescence and molecular scattering) σ rms = ω 0 c k B T M = 1 λ 0 k B T M Δω = ω ω 0 = k v v cosθ (6.1) = ω 0 (6.2) c 2
Doppler Shift and Broadening in Atmospheric Scattering Signals Doppler Broadening 3
Boltzmann Distribution Maxwell-Boltzmann distribution is the law of particle population distribution according to energy levels (under thermodynamic equilibrium) N k N 2, g 2 ΔE N 1, g 1 N = g k exp( E k /k B T) g i exp( E i /k B T) i E 2 E 1 (6.3) N 2 = g 2 exp E 2 E 1 N 1 g 1 T = { ( ) k B T} ΔE /k B ln g 2 N 1 g 1 N 2 (6.5) (6.4) N 1 and N 2 - particle populations on energy levels E 1 and E 2 g 1 and g 2 - degeneracy for energy levels E 1 and E 2, ΔE = E 2 - E 1 k B - Boltzmann constant, T - Temperature, N - total population Population Ratio Temperature 4
z 5 F 0 3d 6 4s4p a 5 D 3d 6 4s 2 LIDAR REMOTE SENSING PROF. XINZHAO CHU CU-BOULDER, FALL 2012 372nm 100% Fe Boltzmann Technique 374nm 91% ΔE 416cm 1 368nm 9% Atomic Fe Energy Level J'=1 J'=2 J'=3 J'=4 J'=5 J=0 J=1 J=2 J=3 J=4 [Gelbwachs, 1994; Chu et al., 2002] Example: Fe Boltzmann N(J = 4) N(J = 3) = g 1 g 2 exp ΔE k B T { } g 1 = 2 * 4 +1= 9 g 2 = 2 * 3 +1= 7 ΔE = 416(cm 1 ) = hc 416 100(J) ΔE /k B = 598.43K For T = 200, N(J = 4) N(J = 3) = 9 7 e598.43/200 = 25.6 5
LIDAR REMOTE SENSING PROF. XINZHAO CHU CU-BOULDER, FALL 2012 Example Lidar Signals from Antarctica Taken from [Chu et al., Geophysical Research Letters, 2011] 6
Reflection from Target or Surface Specular reflection from water or ice or target surface; Laser non-specular reflection from ground; Laser non-specular reflection from target surface; Solar reflectance from ground or target surface. For laser altimeter, laser scattering from ice sheet 7
Multiple Scattering Lidar equations developed in Lecture 4 were developed under the single-scattering assumption. That is, all scattered photons except those scattered in the direct backward direction (θ = π) are permanently removed from the transmitted and received lidar beams. N S (λ,r) = P L (λ L )Δt A [ β(λ,λ L,θ,R)ΔR] hc λ L R 2 R exp α(λ L, r ʹ )dr ʹ 0 exp R 0 α(λ, r ʹ )d r ʹ η(λ,λ L)G(R) (4.17) [ ] + N B Multiple scattering is a natural phenomenon, occurring in the lower atmosphere, especially in fog, cloud and rain. Multiple scattering has three major effects on the lidar return signals: #1) increasing the received power, #2) causing larger depolarization, and #3) delaying the time of flight. 8
Multiple Scattering Multiple scattering increases the lidar received power by two major factors: Forward scattered photons never leave the lidar beam, and some photons that are scattered out of the lidar beam may later be scattered back in. Thus, the actually received lidar signal includes photons that have been scattered more than once. As a result, the received power somewhat exceeds the value given by Eq. (4.17) or the other lidar equations given in Lecture 04. Equivalently, the effective value of the scattering extinction coefficient is somewhat smaller than the single-scattering values discussed in Lecture 05. Figure taken from Lidar and multiple scattering by Bissonnette in the book of Lidar 9
Multiple Scattering The paths of received multiply-scattered photons can lie on many different planes, and thus can transfer the plane of the electric vector. As a result, multiple scattering causes the received lidar signals partially depolarized, even when all the scatters are spherical. The radiation backscattered exactly 180 by spherical particles conserves the linear polarization of the original laser beam. However, within 2 off the exact backscatter direction, the depolarization could jump from 0 to as much as 60%. Multiple scattering causes depolarization mainly via the near 180 (but not exactly) single scattering. Figure taken from Lidar and multiple scattering by Bissonnette in the book of Lidar 10
Multiple Scattering The effects of multiple scattering on the received lidar signals depend on the atmospheric scattering and absorption coefficients, the size of scattering particles, the time delay, the field of view (FOV) of the lidar receiver, and the divergence of the lidar transmitted beam. Larger FOV allows more chances of receiving multiply scattered photons. Thus, reducing FOV can help reduce the probability of receiving multiple scattering photons. However, the FOV is never small enough to avoid multiple scattering signals. Larger scattering particles focus a much larger fraction of scattered photons in near-forward directions. The theoretical treatment of multiple scattering in pulsed lidar applications is complex. Monte Carlo approach of numerical simulation can be used to predict the received power for various aerosol/cloud models. The unwanted time delay caused by multiple scattering is of course an error term to laser altimeter or any laser range finders. Have you seen multiple scattering effects even in the STAR lidar? 11
Polarization in Scattering Depolarization is the phenomenon that the scattered photons possess polarizations different than that of the original incident laser photons. Usually the incident laser beam has a linear polarization. Depolarization means that the return photons may have polarization other than this linear polarization, e.g., linear polarizations in other directions, or elliptical polarization. Note that every photon possesses certain polarization, i.e., every photon is polarized. Non-polarized light or depolarized light is talked about on the statistical point of view of a large number of photons. The range-resolved linear depolarization ratio is defined from lidar or optical observations as below in many lidar applications. Detailed sophisticated treatment of lidar depolarization needs to consider the optical matrix for polarization. δ(r) = P P = I I (6.6) where P and I are the light power and intensity detected. 12
Polarization in Scattering Rayleigh scattering from air molecules can cause depolarization but only a few percent or less. Mie scattering from spherical particles at exactly 180 will not introduce depolarization. Depolarization can be resulted from (1) Spherical particle scattering at near 180 but not exact (2) Non-spherical particle shape (true for both aerosol/cloud and atmosphere molecules) (3) Inhomogeneous refraction index (4) Multiple scattering inside fog, cloud or rain. 13
Atomic Structure and Energy Levels Interactions inside and outside an atom determine the atomic energy levels that are usually concrete levels. A class on Fundamentals of Spectroscopy for Optical Remote Sensing can be found at http://cires.colorado.edu/science/groups/chu/classes/spectroscopy2009/ Electrostatic interaction between electron and nucleus (Coulomb force) Electron-electron Coulomb force Electron spin and angular momentum coupling L+S = J Nuclear spin, mass and volume influences External magnetic and electric field influences 14
Example: Na D 2 Lines http://superlidar.colorado.edu/classes/spectroscopy2009/lecturechapter05.pdf 15
Absorption Cross-Section for Atoms Atomic absorption cross section σ abs #for Na 589-nm D2 line is about 10-15 m 2 #for Fe 372-nm line is about 10-16 m 2 How do you derive the backscatter cross section? dσ m dω (θ = π) = σ abs 4π = 10 15 m 2 4π = 7.95 10 16 m 2 sr 1 = 7.95 10 12 cm 2 sr 1 (6.7) Why so? Because resonance fluorescence is isotropic in the 4π solid angle. 16
Molecular Structure and Energy Levels A molecule contains more interactions than an atom because of the interaction among atoms and motions within a molecule. http://superlidar.colorado.edu/classes/spectroscopy2009/lecturechapter08.pdf 17
Absorption and Fluorescence Lidar λ ON λ OFF Absorption Line ON OFF λ 18
Scattering by Molecules in Atmosphere Pure Rotational Raman Rayleigh Cabannes Line 19
Rayleigh Backscatter Coefficient β Rayleigh (λ,z,θ = π) = 2.938 10 32 P(z) T(z) 1 λ 4.0117 ( m 1 sr 1 ) P in mbar and T in Kelvin at altitude z, λ in meter. β(θ) = β T 4π P(θ) = β T 4π 0.7629 (1+ 0.9324 cos2 θ) (5.14) (5.15) Rayleigh Backscatter Cross Section dσ m (λ) dω = 5.45 550 λ 4 10 32 ( m 2 sr 1 ) (5.16) where λ is the wavelength in nm. For Rayleigh lidar, λ = 532 nm, 6.22 x 10-32 m 2 sr -1 20
Backscatter Cross-Section Comparison Physical Process Backscatter Cross-Section Mechanism Mie (Aerosol) Scattering 10-8 - 10-10 cm 2 sr -1 Two-photon process Atomic Absorption and Resonance Fluorescence Elastic scattering, instantaneous 10-13 cm 2 sr -1 Two single-photon process (absorption and spontaneous emission) Delayed (radiative lifetime) Molecular Absorption 10-19 cm 2 sr -1 Single-photon process Fluorescence from molecule, liquid, solid Rayleigh Scattering (Wavelength Dependent) Raman Scattering (Wavelength Dependent) 10-19 cm 2 sr -1 Two single-photon process Inelastic scattering, delayed (lifetime) 10-27 cm 2 sr -1 Two-photon process Elastic scattering, instantaneous 10-30 cm 2 sr -1 Two-photon process Inelastic scattering, instantaneous 21
General Case of Light Transmission I 0 (ν) z 1 = 0 z 2 = z I(ν,z) ν 2 ν 1 I 0 (ν)dν Radiation propagation through an medium ν 2 ν 1 I(ν,z)dν Transmission for single-frequency radiation propagation through an inhomogeneous medium T(ν,z) = I(ν,z) I 0 (ν) = exp z1 z 2 α(ν,z)dz = exp z 2 σ(ν,z)δn(z)dz z1 (6.8) Transmission for the general case of radiation propagation through an inhomogeneous medium in the spectral interval [ν 1, ν 2 ]: T(z) = ν 2 ν 1 ν 2 ν 1 I(ν,z)dν I 0 (ν)dν = ν 2 ν 1 z I 0 (ν)exp 2 σ(ν,z)δn(z)dz z1 dν ν 2 ν 1 I 0 (ν)dν (6.9) 22
General Case of Light Transmission Extinction/Attenuation for the general case of radiation propagation through an inhomogeneous medium in the spectral interval [ν 1, ν 2 ]: A(z) =1 T(z) = ν 2 ν 1 z I 0 (ν) 1 exp 2 σ(ν,z)δn(z)dz z1 dν ν 2 ν 1 I 0 (ν)dν (6.10) In the case of homogeneous spectral dependence of the source intensity over the spectral interval [ν 1, ν 2 ]: ν z T(z) = exp 2 2 σ(ν,z)δn(z)dz z1 dν (ν 2 ν 1 ) (6.11) ν 1 In the case of homogeneous absorbing medium: T(z) = ν 2 ν 1 I 0 (ν)exp[ ΔNσ(ν)L ]dν ν 2 ν 1 I 0 (ν)dν (6.12) In the case of homogeneous absorbing medium and homogeneous source spectral dependence: ν T(z) = 2 exp[ ΔNσ(ν)L]dν (ν 2 ν 1 ) (6.13) ν 1 23
Summary Doppler effects, Boltzmann distribution, polarization properties, etc, all can be utilized in lidar applications to infer atmospheric parameters like wind, temperature, aerosol properties, etc. Multiple scattering should be considered in lower atmosphere studies, especially for cloud and aerosol research. Fundamentals to understand atomic structures and energy levels are related to the interactions inside and outside an atom, including electrostatic interaction, electron spin, angular-spin momentum coupling, nuclear spin, external field, etc. Molecular structures and energy levels further include interactions among atoms within a molecule. Our Textbook Chapter 3 for elastic scattering and polarization Chapter 4 for differential absorption Chapters 5 & 7 for resonance fluorescence, Boltzmann, Doppler Chapter 6 for laser-induced fluorescence Laser monitoring of the atmosphere (Hinkley) Ch. 3 and 4 Laser Remote Sensing (Measures) Chapter 4 Lidar (Ed. Weitkamp) Chapters 2 and 3 24