Lecture 08. Solutions ofLidar 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

HWK Report #1

T ( L ,R)

NL ( L )

T ( ,R)

( , L , ,R)

A

R2

( , L )G(R)

General Lidar Equation

NS - expected photon counts detected at and distance R; 1st term - number of transmitted laser photons; 2nd 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.

Assumptions: independent and single scattering

NS ( ,R) = NL ( L ) ( , L , ,R) R[ ]A

R2T ( L ,R)T ( ,R)[ ] ( , L )G(R)[ ] +NB

More in General Lidar Equation

NS (R) - expected received photon number from a distance RPL - transmitted laser power, L - laser wavelengtht - integration time,

h - Planck’s constant, c - light speed(R) - volume scatter coefficient at distance R for angle ,R - thickness of the range bin

A - area of receiver,T(R) - one way transmission of the light from laser source todistance R or from distance R to the receiver, - system optical efficiency,

G(R) - geometrical factor of the system,NB - background and detector noise photon counts.

NS ( ,R) =PL ( L ) t

hc L

( , L , ,R) R[ ]

A

R2T ( L ,R)T ( ,R)[ ] ( , L )G(R)[ ] +NB

Solution forScattering Form Lidar Equation

Solution for scattering form lidar equation

( , L ,z) =NS ( ,z) NB

PL ( L ) t

hc L

z

A

z2

T ( L ,z)T ( ,z)[ ] ( , L )G(z)[ ]

Scattering form lidar equation

NS ( ,z) =PL ( L ) t

hc L

( , L ,z) z[ ]

A

z2

T ( L ,z)T ( ,z)[ ] ( , L )G(z)[ ] +NB

Solution forFluorescence Form Lidar Equation

nc(z) =NS ( ,z) NB

PL ( ) t

hc

eff ( )RB ( ) z( )

A

4 z2

( )Ta

2( ,z)Tc2( ,z)G(z)( )

Solution for fluorescence form lidar equation

NS ( ,z) =PL ( ) t

hc

eff ( ,z)nc(z)RB ( ) z( )

A

4 z2

Ta2( ,z)Tc

2( ,z)( ) ( )G(z)( )+NB

Fluorescence form lidar equation

Differential Absorption/Scattering Form

NS ( on ,z) = NL ( on ) sca ( on ,z) z[ ]A

z 2

exp 2 ( on , z )d z

0

z

exp 2 abs( on , z )nc ( z )d z 0

z

( on )G(z)[ ] + NB

For the laser with wavelength on on the molecularabsorption line

NS ( off ,z) = NL ( off ) sca ( off ,z) z[ ]A

z 2

exp 2 ( off , z )d z

0

z

exp 2 abs( off , z )nc ( z )d z 0

z

( off )G(z)[ ] + NB

For the laser with wavelength off off the molecularabsorption line

Differential Absorption/Scattering Form

NS ( on ,z) NB

NS ( off ,z) NB

=NL ( on ) sca ( on ,z)

NL ( off ) sca ( off ,z)

( on )

( off )

exp 2 ( on , z ) ( off , z )[ ]d z 0

z

exp 2 abs( on , z ) abs( off , z )[ ]nc ( z )d z 0

z

The ratio of photon counts from these two channels is afunction of the differential absorption and scattering:

= abs( on ) abs( off )

Solution forDifferental Absorption Lidar Equation

nc (z) =1

2

d

dz

lnNL ( on ) sca ( on ,z)

NL ( off ) sca ( off ,z)

( on )

( off )

lnNS ( on ,z) NB

NS ( off ,z) NB

( on , z ) ( off , z )[ ]

= abs( on ) abs( off )

Solution for differential absorption lidar equation

Solution forResonance Fluorescence Lidar Equation

NS ( ,z) =PL ( ) t

hc

eff ( ,z)nc(z)RB ( ) z( )

A

4 z2

Ta

2( )Tc2( ,z)( ) ( )G(z)( )+NB

NR ( ,zR ) =PL ( ) t

hc

R ( , )nR (zR ) z( )

A

zR2

Ta2( ,zR ) ( )G(zR )( )+NB

nc(z)

nR (zR )=NS ( ,z) NBNR ( ,zR ) NB

z2

zR2

4 R ( , )

eff ( ,z)RB ( )

Ta2( ,zR )G(zR )

Ta2( ,z)Tc

2( ,z)G(z)

nc(z) = nR (zR )NS ( ,z) NBNR ( ,zR ) NB

z2

zR2

4 R ( , )

eff ( ,z)RB ( )

1

Tc2( ,z)

Resonance fluorescence and Rayleigh lidar equations

Rayleigh normalization

Solution for resonance fluorescence

Solution forRayleigh and Mie Lidars

NS ( ,z) =PL ( ) t

hc

R (z) + aerosol (z)( ) z

A

z2

Ta2( ,z) ( )G(z)( )+NB

NR ( ,zR ) =PL ( ) t

hc

R (zR ) z( )

A

zR2

Ta2( ,zR ) ( )G(zR )( )+NB

R (z) + aerosol (z)

R (zR )=

NS ( ,z) NBNR ( ,zR ) NB

z2

zR2Ta2( ,zR )G(zR )

Ta2( ,z)G(z)

Rayleigh and Mie (middle atmos) lidar equations

Rayleigh normalization

For Rayleigh scattering at z and zR

R (z)

R (zR )= R (z)natm (z)

R (zR )natm (zR )=natm (z)

natm (zR )

Solution (Continued) Solution for Mie scattering in middle atmosphere

Solution for relative number density in Rayleigh lidar

aerosol (z) = R (zR )NS ( ,z) NBNR ( ,zR ) NB

z2

zR2

natm (z)

natm (zR )

RND(z) =natm (z)

natm (zR )= R (z)

R (zR )=NS ( ,z) NBNR ( ,zR ) NB

z2

zR2

R ( ,zR , ) = 2.938 10 32 P(zR )

T (zR )

14.0117

m 1sr 1( )

R (z)

R (zR )=

NS ( ,z) NBNR ( ,zR ) NB

z2

zR2Ta2( ,zR )G(zR )

Ta2( ,z)G(z)

Rayleigh normalization when aerosols not present

Summary Solutions of lidar equation can be obtained by solving

the lidar equation directly if all the lidar parameters andatmosphere conditions are well known.

Solutions for three forms of lidar equations are shown:scattering form, fluorescence form, and differentialabsorption form.

However, system parameters and atmosphere conditionsmay vary frequently and are NOT well known toexperimenters.

A good solution is to perform Rayleigh normalization tocancel out most of the system and atmosphere parametersso that the essential and known parts can be solved.