LIDAR REMOTE SENSING PROF. XINZHAO CHU CU-BOULDER, FALL 2012 Lecture 08. Fundamentals of Lidar Remote Sensing (4) Solutions for Lidar Equations

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1 Lecture 08. Fundamentals of Lidar Remote Sensing (4) Solutions for Lidar Equations Lidar Classification with Physical Processes Solution for scattering form lidar equation Solution for Raman lidar equation Solution for differential absorption lidar equation Solution for fluorescence form lidar equation Solution for resonance fluorescence lidar Solution for Rayleigh and Mie lidar in middle atmos Summary 1

2 Review Lidar Equation Lidar equation is to link the expected lidar returns (N S ) to the lidar parameters (both transmitter and receiver), transmission through medium, physical interactions between light and objects, and background/noise conditions, etc. Keep in mind the big picture of a lidar system - " "Radiation source " "Radiation propagation in the medium " "Interaction of radiation with the objects " "Signal propagation in the medium " "Photons are collected, filtered and detected Can you derive a lidar equation by yourself? 2

3 Physical Process Elastic Scattering by erosols and Clouds bsorption by toms and Molecules Inelastic Scattering Elastic Scattering By ir Molecules Resonance Scattering/ Fluorescence By toms Doppler Shift Device Mie Lidar DIL Raman Lidar Rayleigh Lidar Resonance Fluorescence Lidar Wind Lidar Objective erosols, Clouds: Geometry, Thickness Gaseous Pollutants Ozone Humidity (H 2 O) erosols, Clouds: Optical Density Temperature in Lower tmosphere Stratos & Mesos Density & Temp Temperature, Wind Density, Clouds in Mid-Upper tmos Wind, Turbulence Laser Induced Fluorescence Fluorescence Lidar Marine, Vegetation Reflection from Surfaces Target Lidar Laser ltimeter Topography, Target 3

4 Parameters in General Lidar Equation ssumptions: independent and single scattering N S (λ,r) = P L(λ L )Δt β(λ,λ L,θ,R)ΔR hc λ L [ ] N S (R) - expected received photon number from a distance R P L - transmitted laser power, λ L - laser wavelength Δt - integration time, h - Planck s constant, c - light speed β(r) - volume scatter coefficient at distance R for angle θ, ΔR - thickness of the range bin - area of receiver, T(R) - one way transmission of the light from laser source to distance R or from distance R to the receiver, η - system optical efficiency, G(R) - geometrical factor of the system, N B - background and detector noise photon counts. [ ] [ η(λ,λ L )G(R)] + N B R 2 T(λ L,R)T(λ,R) (8.1) 4

5 Review Lidar Equation General lidar equation with angular scattering coefficient N S (λ,r) = N L (λ L ) [ β(λ,λ L,θ,R)ΔR] N S (λ,r) = N L (λ L ) [ β T (λ,λ L,R)ΔR] R 2 T(λ L,R)T(λ,R) [ ] η(λ,λ L )G(R) [ ] + N B General lidar equation with total scattering coefficient 4πR 2 T(λ L,R)T(λ,R) [ ] η(λ,λ L )G(R) [ ] + N B General lidar equation in angular scattering coefficient β and extinction coefficient α form (4.2) (4.12) N S (λ,r) = P L(λ L )Δt [ 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 5

6 Specific Lidar Equations Lidar equation for Rayleigh lidar N S (λ,r) = P L (λ)δt ( β(λ,r)δr) hc λ R 2 T 2 (λ,r)( η(λ)g(r) )+ N B (5.17) Lidar equation for resonance fluorescence lidar N S (λ,r) = P L (λ)δt ( σ eff (λ,r)n c (z)r B (λ)δr) hc λ 4πR 2 T 2 a (λ,r)t 2 c (λ,r) Lidar equation for differential absorption lidar ( ) η(λ)g(r) ( )+ N B (4.14) off N S (λ on,r) = off NL (λ on ) [ off βsca (λ on,r)δr ] z off R 2 exp 2 α (λ on, r ʹ )dr ʹ 0 z off exp 2 σ abs (λ on, r ʹ )nc ( r ʹ )dr ʹ [ 0 η(λ off on )G(R) ] + N B ( ) 6

7 Elastic Scattering Lidar: Rayleigh Lidar and Mie Lidar Strong Mie Scattering above Rayleigh Background Chapter 3 in our textbook 7

8 Solution for Scattering Form Lidar Equation Scattering form lidar equation N S (λ,r) = P L(λ L )Δt β(λ,λ L,R)ΔR hc λ L [ ] Solution for scattering form lidar equation R 2 T(λ L,R)T(λ,R) [ ] [ η(λ,λ L )G(R)] + N B (8.2) β(λ,λ L,R) = P L (λ L )Δt hc λ L N S (λ,r) N B ΔR R 2 T(λ L,R)T(λ,R) [ ][ η(λ,λ L )G(R)] (8.3) 8

9 Inelastic Scattering Lidar: Raman Lidar Raman spectra for O 2, N 2 and H 2 O under 355nm laser wavelength H 2 O in atmosphere 9

10 Solution for Raman Lidar Equation Raman lidar equations for gas of interest (8.4) and reference gas (8.5) P cra (R,λ cra ) = E oη λcra R 2 O(R,λ cra )β cra (R,λ 0,λ cra )exp R [ α(r,λ 0 ) +α(r,λ cra ) 0 ]dr Solution for Raman lidar equations { } { } P RefRa (R,λ RefRa ) = E oη λrefra R 2 O(R,λ RefRa )β RefRa (R,λ 0,λ RefRa )exp R [ α(r,λ 0 ) +α(r,λ RefRa ) 0 ]dr P cra (R,λ cra ) P RefRa (R,λ RefRa ) = η λ cra η λrefra N cra (R) dσ cra dω (π,λ 0,λ cra ) N RefRa (R) dσ RefRa dω (π,λ 0,λ RefRa ) Mixing Ratio = N cra(r) N RefRa (R) (8.4) (8.5) exp{ R α(r,λ cra )dr} 0 exp{ R α(r,λ RefRa )dr} 0 (8.6) 10

11 Differential bsorption Lidar (DIL) This is a nice schematic, but in reality, light cannot turn as drawn! Collis and Russell in Laser monitoring of the atmosphere 11

12 Differential bsorption/scattering Form For the laser with wavelength λ on on the molecular absorption line [ ] N S (λ on,r) = N L (λ on ) β sca (λ on,r)δr R 2 exp 2 σ abs (λ on, r ʹ )n c ( r ʹ )dr ʹ exp[ 2 R α (λ on, r ʹ 0 )dr ʹ ] [ R ] 0 η(λ on )G(R) [ ] + N B (8.7) For the laser with wavelength λ off off the molecular absorption line [ ] N S (λ off,r) = N L (λ off ) β sca (λ off,r)δr R 2 exp 2 σ abs (λ off, r ʹ )n c ( r ʹ )dr ʹ exp[ 2 R α (λ off, r ʹ 0 )dr ʹ ] [ R ] 0 η(λ off )G(R) [ ] + N B (8.8) 12

13 Differential bsorption/scattering Form The ratio of photon counts from these two channels is a function of the differential absorption and scattering: N S (λ on,r) N B = N L(λ on )β sca (λ on,r) N S (λ off,r) N B N L (λ off )β sca (λ off,r) R 0 η(λ on ) η(λ off ) { } exp 2 [ α (λ on, r ʹ ) α (λ off, r ʹ ) ]d r ʹ (8.9) { } R 0 exp 2 [ σ abs (λ on, r ʹ ) σ abs (λ off, r ʹ ) ]n c ( r ʹ )dr ʹ Δσ = σ abs (λ on ) σ abs (λ off ) (8.10) 13

14 Solution for Differential bsorption Lidar Equation Solution for differential absorption lidar equation n c (R) = 1 2Δσ d dr ln N L(λ on )β sca (λ on,r) N L (λ off )β sca (λ off,r) η(λ on ) η(λ off ) ln N S(λ on,r) N B N S (λ off,r) N B R 2 [ α (λ on, r ʹ ) α (λ off, r ʹ )]dr ʹ 0 (8.11) Δσ = σ abs (λ on ) σ abs (λ off ) (8.10) 14

15 Resonance Fluorescence Lidar Resonance Fluorescence Profile on an Fe meteor trail measured by lidar [Chu et al., 2000] 15

16 Solution for Fluorescence Form Lidar Equation Fluorescence form lidar equation N S (λ,r) = P L (λ)δt ( σ eff (λ,r)n c (R)R B (λ)δr) hc λ 4πR 2 T 2 a (λ,r)t 2 c (λ,r) Solution for fluorescence form lidar equation ( ) η(λ)g(r) ( ) + N B (8.12) n c (R) = P L (λ)δt hc λ N S (λ,r) N B ( σ eff (λ)r B (λ)δr) 4πR 2 η(λ)t 2 a (λ,r)t 2 c (λ,r)g(r) ( ) (8.13) 16

17 Solution for Resonance Fluorescence Lidar Equation Resonance fluorescence and Rayleigh lidar equations N S (λ,z) = P L(λ)Δt ( σ eff (λ,z)n c (z)r B (λ)δz) hc λ N R (λ,z R ) = P L (λ)δt σ R (π,λ)n R (z R )Δz hc λ ( ) Rayleigh normalization n c (z) n R (z R ) = N S(λ,z) N B N R (λ,z R ) N B z 2 z R 2 4πz 2 z R 2 4πσ R (π,λ) σ eff (λ,z)r B (λ) Solution for resonance fluorescence 2 2 T a (λ)tc (λ,z) ( ) η(λ)g(z) ( )+ N B T a 2 (λ,z R )( η(λ)g(z R ))+ N B 2 T a (λ,zr )G(z R ) 2 2 T a (λ,z)tc (λ,z)g(z) (8.12) (8.14) (8.15) n c (z) = n R (z R ) N S (λ,z) N B N R (λ,z R ) N B 2 z 4πσ R (π,λ) R σ eff (λ,z)r B (λ) 1 2 T c (λ,z) z 2 (8.16) 17

Rayleigh and Mie Lidar for Middle tmosphere Detection ltitude (km) 100 90 80 70 60 50 [Chu et al.

18 Rayleigh and Mie Lidar for Middle tmosphere Detection ltitude (km) [Chu et al., GRL, 2001] PMC PMC Courtesy of Pekka Parviainen Polar mesospheric clouds (PMC), also noctilucent clouds (NLC), are thin scattering layers of nanometer-sized water ice particles, occurring around km in the high-latitude summer mesopause region Volume backscatter coefficient β (10 9 m -1 sr -1 ) Polar mesospheric clouds Noctilucent clouds Chapter 5 in our textbook 18

19 Solution for Rayleigh and Mie Lidars Rayleigh and Mie (middle atmos) lidar equations N S (λ,z) = P L(λ)Δt 2 ( β R (z) + β aerosol (z))δz hc λ z 2 T a (λ,z) ( η(λ)g(z) )+ N B N R (λ,z R ) = P L (λ)δt ( β R (z R )Δz) hc λ 2 z T a 2 (λ,z R )( η(λ)g(z R ))+ N B R Rayleigh normalization β R (z) + β aerosol (z) β R (z R ) = N S (λ,z) N B N R (λ,z R ) N B z 2 2 z T 2 a (λ,zr )G(z R ) 2 R T a (λ,z)g(z) (8.17) (8.18) (8.19) For Rayleigh scattering at z and z R β R (z) β R (z R ) = σ R (z)n atm (z) σ R (z R )n atm (z R ) = n atm (z) (8.20) n atm (z R ) 19

20 Solution (Continued) Solution for Mie scattering in middle atmosphere N β aerosol (z) = β R (z R ) S (λ,z) N B z 2 N R (λ,z R ) N 2 B z n atm (z) R n atm (z R ) (8.21) β R (λ,z R,π) = P(z R ) T(z R ) 1 λ ( m 1 sr 1 ) (5.14) Rayleigh normalization when aerosols are not present β R (z) β R (z R ) = N S (λ,z) N B N R (λ,z R ) N B z 2 2 z T 2 a (λ,zr )G(z R ) 2 R T a (λ,z)g(z) (8.22) Solution for relative number density in Rayleigh lidar RND(z) = n atm (z) n atm (z R ) = β R (z) β R (z R ) = N S (λ,z) N B N R (λ,z R ) N B z 2 z R 2 (8.23) 20

21 Rayleigh Backscatter Cross Section It is common in lidar field to calculate the Rayleigh backscatter cross section using the following equation dσ m (λ) dω = λ ( m 2 sr 1 ) (5.16) where λ is the wavelength in nm. The Rayleigh backscatter cross section can also be estimated from the Rayleigh backscatter coefficient β Rayleigh (λ,z,θ = π) = P(z) T(z) 1 dσ m (λ) dω = β Rayleigh (λ,z,π) n atmos (z) λ ( m 2 sr 1 ) ( m 1 sr 1 ) where λ is the wavelength in meter, P in mbar, T in Kelvin. (8.24) (5.14) 21

22 Summary Physical processes in the interaction between radiation and objects classify lidars into different kinds, each possessing unique features and lidar equations along with certain applications. Solutions of lidar equations can be obtained by solving the lidar equations directly if all the lidar parameters and atmosphere conditions are well known. Solutions for three forms of lidar equations are shown: scattering form, fluorescence form, and differential absorption form, along with a few specific lidar types. However, system parameters and atmosphere conditions may vary frequently and are NOT well known to experimenters. good solution is to perform Rayleigh normalization to cancel out most of the system and atmosphere parameters so that the essential and known parts can be solved. 22

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