Lecture 08. Solutions of Lidar Equations

Transcription

1 Lecture 08. Solutions of Lidar Equations HWK Report #1 Solution for scattering form lidar equation Solution for fluorescence form lidar equation Solution for differential absorption lidar equation Solution for resonance fluorescence lidar Solution for Rayleigh and Mie lidar Summary

2 HWK Report #1 (, L,,R) T( L,R) T(,R) N L ( L ) (, L )G(R) R

3 General Lidar Equation ssumptions: independent and single scattering N S (,R) = N L ( L ) [ (, L,,R)R] R T( L,R)T(,R) [ ] (, L )G(R) [ ]+ N B N S - expected photon counts detected at and distance R; 1st term - number of transmitted laser photons; nd term - probability that a transmitted photon is scattered by the scatters into a unit solid angle at angle ; 3rd term - probability that a scatter photon is collected by the receiving telescope; 4th term - light transmission during light propagation from laser source to distance R and from distance R to receiver; 5th term - overall system efficiency; 6th term - background and detector noise.

4 More in General Lidar Equation N S (,R) = P L ( L )t [ (, L,,R)R] hc L R [ T( L,R)T(,R) ][ (, L )G(R)]+ N B N S (R) - expected received photon number from a distance R P L - transmitted laser power, L - laser wavelength t - integration time, h - Planck s constant, c - light speed (R) - volume scatter coefficient at distance R for angle, R - 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, - system optical efficiency, G(R) - geometrical factor of the system, N B - background and detector noise photon counts.

5 Solution for Scattering Form Lidar Equation Scattering form lidar equation N S (,z) = P L ( L )t [ (, L,z)z] hc L [ T( L,z)T(,z)][ (, L )G(z)]+ N B Solution for scattering form lidar equation (, L,z) = P L ( L )t hc L z N S (,z) N B T( L,z)T(,z) [ ][ (, L )G(z)]

6 Solution for Fluorescence Form Lidar Equation Fluorescence form lidar equation N S (,z) = P L()t ( eff (,z)n c (z)r B ()z) hc 4 ( T a (,z)tc (,z) )( ()G(z) )+ N B Solution for fluorescence form lidar equation n c (z) = P L ()t hc N S (,z) N B ( eff ()R B ()z) 4 ()T a (,z)tc (,z)g(z) ( )

7 Differential bsorption/scattering Form For the laser with wavelength on on the molecular absorption line [ ] N S ( on,z) = N L ( on ) sca ( on,z)z exp N S ( off,z) = N L ( off ) sca ( off,z)z exp 0 exp 0 ( on, z )dz abs ( on, z )n c ( z )dz [ ( on)g(z) ]+ N B z For the laser with wavelength off off the molecular absorption line [ ] 0 exp z 0 ( off, z )dz abs ( off, z )n c ( z )dz [ ( off )G(z) ]+ N B z z

8 Differential bsorption/scattering Form The ratio of photon counts from these two channels is a function of the differential absorption and scattering: N S ( on,z) N B = N L ( on ) sca ( on,z) ( on ) N S ( off,z) N B N L ( off ) sca ( off,z) ( off ) exp exp z 0 z 0 ( on, z ) (, [ z )]dz off abs ( on, z ) (, [ z )]n abs off c ( z )dz = abs ( on ) abs ( off )

9 Solution for Differental bsorption Lidar Equation Solution for differential absorption lidar equation n c (z) = 1 d dz ln N L ( on ) sca ( on,z) ( on ) N L ( off ) sca ( off,z) ( off ) ln N S( on,z) N B N S ( off,z) N B [ ( on, z ) ( off, z )] = abs ( on ) abs ( off )

10 Solution for Resonance Fluorescence Lidar Equation Resonance fluorescence and Rayleigh lidar equations N S (,z) = P L()t ( eff (,z)n c (z)r B ()z) hc N R (,z R ) = P L ()t ( R (,)n R (z R )z) hc Rayleigh normalization n c (z) n R (z R ) = N S (,z) N B N R (,z R ) N B z R n c (z) = n R (z R ) N S(,z) N B N R (,z R ) N B 4 z R 4 R (,) eff (,z)r B () Solution for resonance fluorescence ( T a ()Tc (,z) )( ()G(z) )+ N B T a (,z R )( ()G(z R ))+ N B T a (,zr )G(z R ) T a (,z)tc (,z)g(z) z 4 R (,) R eff (,z)r B () 1 T c (,z)

11 Solution for Rayleigh and Mie Lidars Rayleigh and Mie (middle atmos) lidar equations N S (,z) = P L()t ( R (z) + aerosol (z))z hc T a (,z) ( ()G(z) )+ N B N R (,z R ) = P L()t ( R (z R )z) hc T a (,z R )( ()G(z R ))+ N B z R Rayleigh normalization R (z) + aerosol (z) R (z R ) = N S (,z) N B N R (,z R ) N B For Rayleigh scattering at z and z R R (z) R (z R ) = R (z)n atm (z) R (z R )n atm (z R ) = n atm(z) n atm (z R ) T a (,zr )G(z R ) z R T a (,z)g(z)

12 Solution (Continued) Solution for Mie scattering in middle atmosphere N aerosol (z) = R (z R ) S (,z) N B N R (,z R ) N n atm (z) B z R n atm (z R ) R (,z R,) = P(z R ) T(z R ) ( m1 sr 1 ) Rayleigh normalization when aerosols not present R (z) R (z R ) = N S (,z) N B N R (,z R ) N B T a (,zr )G(z R ) z R T a (,z)g(z) Solution for relative number density in Rayleigh lidar RND(z) = n atm(z) n atm (z R ) = R (z) R (z R ) = N S(,z) N B N R (,z R ) N B z R

13 Summary Solutions of lidar equation can be obtained by solving the lidar equation 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. 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.