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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=
),,( 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