Relevant Angles for CIPS Observing Geometry

Solar Zenith AngleSun-SV-Zenith

Scattering Angle(angle between original and scattered path)

Viewing AngleNadir-CIPS-SV

SV=Scattering Volume

Decrease in pressure with height:

Some Useful Descriptions of the Atmosphere I

Ideal gas law: kB=Boltzmann’s constant, T=temperature

Pressure is the force per unit area exerted by the atmosphere

Therefore pressure could be expressed as the weight of a column of air molecules:

Tnkp B=

)(')'()( zmgNdzznmgzpz

== ∫∞

N(z) is the “column density”, the number of molecules in a column of unit area extending from altitude z to the top of the atmosphere

')'()( ∫∞

=z

dzznzN

Hp

Tkpmgzmgndzznmg

dzd

dzdp

Bz

−=−=−=⎥⎦⎤

⎢⎣⎡= ∫

∞)(')'(

H is called the “scale height”Hp

dzdp

−=mgTkH B=

p=pressure, n=number density, m=mean mass of individual molecules (~.8*mN2 + .2mO2 in trop.)

Relate pressure to height:

Some Useful Descriptions of the Atmosphere II

H is the distance over which the atmospheric pressure decreases by a factor e.

Hp

dzdp

−=mgTkH B=

'100∫∫ −=z

z

p

pdz

Hpdp

)(1lnlnln 00

0 zzHp

ppp −−==−

Hzz

epp )(

0

0−−

= Hzz

epzp)(

0

0

)(−

−=

Hzz

BB TeknTkzn)(

0

0

)(−

−=

Hzz

enzn)(

0

0

)(−

−=

H is also then the distance over which the atmospheric number density of molecules decreases by a factor e.

Solve assuming constant temperature

How is column density related to scale height? (assuming constant temperature):

Some Useful Descriptions of the Atmosphere III

H is also the height the atmosphere would be if it were collapsed to a layer of uniform density

Hzz

enzn)(

0

0

)(−

−=

HzneeHzndzezndzznzN Hz z

Hzz

)()(')(')'()( 0)'(

=⎥⎦⎤

⎢⎣⎡

−−===∞

−∞ ∞−

−

∫ ∫

HznzN )()( =

These relationships are useful for getting a feel for how the atmosphere behaves. Because they assume that temperature is constant, their quantitative utility is limited. They should only be applied in small altitude intervals. In practice its usually better to calculate column density by its defining equation. This is for a vertical column.

')'()( ∫∞

=z

dzznzN

Sometimes, what is needed is the slant column densityfor a path at angle relative to vertical:(this version works for < ~70 degrees)

')cos()'()( ∫

∞=

zdzznzN

')'()(),( ∫∞

=z

dzznEzE στ

Optical Depth

τ is called optical depth and describes how far a photon is likely to travel through a column of gas. As a flux travels a distance such that τ equals unity, the flux is reduced by a factor of e.

Z),(

'))'()(

)(),(

)(),(

')()'(),(),(

)()(),(),(

)()(),(),(

zE

dzznE

z

z

eEFzEF

eEFzEF

dzEznzEFzEdF

dzEznzEFzEdF

EznzEFzEFdzd

z

τ

σ

σ

σ

σ

−∞

∫−∞

∞

∞

=

=

=

=

=

∞

∫∫

F(,z)

F is irradianceσ is cross section

cm2

Cross Sections - have units of area – represent the “size of the target” for photons colliding with atoms, molecules, or ions

Dependent upon photon energy

There is one total cross section describing the area presented by the target atom or molecule in a collision, this cross section is the sum of many individual cross sections that represent probabilities or efficiencies of all the individual possible processes (scattering, absorption, etc.)

Cross sections therefore represent the efficiency of a given process

Cross Sections

Ozone Absorption

Rayleigh Scattering

CIPS Observing Geometry for a Single Observation

A beam of solar photons travels along a path to the scattering volume and then to CIPS. Along the way photons are removed from the beam due to absorption by ozone.

Note that there are contributions to the Rayleigh scattered signal from all points along the path (these are not shown).

The observed albedo from Rayleigh scattering may be written according to the single scattering formula:

( )[ ]∫ +−Θ==1

0

)(exp)( ppXSdpPFIA R λλλλ

λλ βαβ

I = atmospheric radianceF = solar irradiance = Rayleigh scattering coefficientPR = Rayleigh phase functionΘ= scattering angle p = pressure in mb = absorption coefficient of ozone

S ≈ 1/cos() + 1/cos() = viewing angle = solar zenith angleX = ozone density as a function of pressure level = wavelength (265 nm for CIPS)

CIPS Algorithm Overview

We have generalized the result of McPeters et al. [1980] and shown that by assuming ozone density varies exponentially with altitude and that the ratio of the ozone scale height to that of the background atmosphere is constant, then:

where,Nair = the air vertical column density above 1 mbCO3 = ozone column density above 1 mb = cos()0 = cos()

And σ = the ratio of the ozone scale height to that of the background atmosphereHALOE observations have shown that s does not deviate significantly from 0.7.

( )σσ

σ

)(11

)()1()(

30

zC

zNPA

O

airRR

⎟⎟⎠⎞

⎜⎜⎝⎛+

+ΓΘ=

Relative contribution to nadir viewing Rayleigh scattered radiance

Ozone concentration

),,( Θ= RPAA MPMCoσerved

where,APMC = is the nadir viewing albedo of the cloud if observed at Θ = 90°PM = Mie Phase Function

Note that although currently not implemented, this equation may need to be scaled by = cos(), this should be a topic for consideration

Cloud Albedo

Phase function is the fraction of radiance emitted per unit solid angle

Phase function for Gaussian particle distribution with width 14 nm

Mie Phase function of mean particle size 0,10,20,30,40,50, 60 nm

Rayleigh Phase Function

),,(),,,(),( 3 Θ+Θ=Θ σ RPACAA MPMCORoσerved

Interpreting CIPS Scattering Profile

The unknowns are CO3, σ, APMC, and R

Option 1: Non linear least squares fit and use 7 data points to retrieve 4 unknowns

- experience says this is prone to significant error bars

Option 2: Assume an σ, use NLSfit to retrieve CO3, APMC, and R

Option 3: Have an indicator of cloud presenceif cloud not thought to be present, solve only for CO3,

σif a cloud is thought present, assume σ, solve for CO3,

APMC, and R(assumed σ could be taken from observations in cloud

free regions)

How can we determine the presence of a cloud?

For the case of no cloud, y should be a simple linear function of x (with a slope of σ)

If a cloud is present, the slope is changed, the effect is different for small scattering angles versus large ones.

For example, calculate slope at small scattering angles and compare to same calculation for large scattering angles, if ratio is significantly different from 1, then a cloud is likely to be present.

( )σσ

σ

)(11

)()1()(

30

zC

zNPA

O

airr

⎟⎟⎠⎞

⎜⎜⎝⎛+

+ΓΘ=

⎟⎟⎠⎞

⎜⎜⎝⎛

+=0

11lnμμ

x ⎟⎟⎠⎞

⎜⎜⎝⎛

Θ=

)(ln

rPAy μ

PMC (forward scattering)

PMC (backward scattering)

Rayleigh Background