The glued photocount profile for each of the channels (one for parallel, one for perpendic- ular, and one for Rayleigh Elastic), with their associated uncertainties, are the end goal for this chapter. These profiles are used in all subsequent chapters of this thesis when a “counts profile” is indicated.
For this thesis, the glued profiles are used in the calculation of two main derived data products: Depolarization Ratio and Depolarization Parameter.
Traditional Depolarization Method
The purpose of the CANDAC Rayleigh-Mie-Raman lidar (CRL)’s new depolarization chan- nel is to provide measurements of linear depolarization parameter in tropospheric Arctic clouds, to help differentiate ice particles from water particles in the backscattered lidar data.
With the new hardware installed at CRL (see chapter 4), the first task was to carry out depolarization calibration and calculations based on the data, using the traditional methods already in use in the community.
6.1
Traditional Depolarization Method Theory
Chapter 2 detailed several expressions for the depolarization of the lidar’s laser beam as a result of the microphysical properties of the particles in the cloud. Three equations were given:
Equation 2.23 for the depolarization ratio in terms of measured signals was: δ=kS⊥
Sk
, (6.1)
Equation 2.25 for the depolarization parameter in terms of the depolarization ratio was:
dG1 =
2δ
1+δ, (6.2)
and Equation 2.27 for the depolarization parameter in terms of measured signals was: dG1 = 2kS⊥ Sk 1+kS⊥ Sk , (6.3) in which:
S⊥is the signal measured by the perpendicular channel
Skis the signal measured by the parallel channel
k= Gk
G⊥ is the depolarization calibration constant, in which:
Gkis the gain (attenuation) of the parallel channel
G⊥is the gain (attenuation) of the perpendicular channel
A different pair of expressions fordis discussed in the following section, so thedfrom Equations 6.2 and 6.3 has been labelleddG1here to differentiate these expressions from the
others. Calculations ofdare the primary quantity of interest in this thesis.
6.1.1
Lower uncertainty expression for depolarization parameterd
Lower uncertainty (σd) results for the depolarization parameter are possible ifdG1is recast
by dividing numerator and denominator byδ=kS⊥
Sk. The new expressions fordare labelled
dG2. Equation 6.2 becomes: dG2= 2 1 δ+1 , (6.4)
and Equation 6.3 becomes:
dG2= 2 1 k Sk S⊥ +1 . (6.5)
The newly expressed d in Equations 6.5 and 6.4 are labelled dG2 in this chapter to
differentiate them fromdG1. dG2 is not mathematically different fromdG1. In the case that
dG1 anddG2 will also be identical. However, in the real-world case in which these signal
and calibration uncertainties are greater than zero, the uncertainties associated withdG1and dG2will be different.
In both thedG1 anddG2 cases,d and its uncertaintyσd can be calculated starting from
depolarization ratio δ and its uncertainty σδ. To determine the best expression to use,
the one with the final lower uncertainty is selected. Unsurprisingly, the expression which includes an uncertainty term only once (dG2, Equation 6.4) was usually more desireable
than that which included the uncertainty term in both numerator and denominator (dG1,
Equation 6.2).
To determine quantitatively which would be the superior expression, each type of depo- larization parameter with its associated uncertainty was calculated for many combinations of depolarization ratio and depolarization ratio uncertainty, each ranging from 0.01 to 1. This simple model allows us to see the circumstances for which each expression is better. To demonstrate the advantage of the second expression, Figure 6.1 shows the difference between the uncertainties for each expression ofd. The uncertainties fromdG2 have been
subtracted from those from the expression given directly in Gimmestad (2008), dG1; Dif-
ference= σdG1 −σdG2. Hence, red (positive) values indicatedG2 being the better choice,
and blue (negative) values indicatedG1being the better choice.
At very low depolarization ratio values (below 0.1) with high uncertainty (greater than 0.5, in the units of depolarization ratio), dG1 fares better, with lower overall uncertainty
thandG2. Everywhere else,dG2has the lower uncertainty. As most of the interesting clouds
take place with depolarization ratio values higher than 0.1, it is advantageous to use thedG2
expression.
Recall that the depolarization parameter values calculated for both expressions are iden- tical; only the final uncertainty differs. If the expressions using signals and k calibration factor are used, the result is the same;dG2 still outperformsdG1. That test is just harder to
σd G1−σd G2 D iffe r e nc e be twe e n
D e polar iz at ion Par ame t e r Unc e r t aint ie s
Depolar ization Ratio
D ep o la ri za ti o n R a ti o U n ce rt a in ty 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
Figure 6.1: Plot in units of depolarization pa- rameter which compares the absolute uncer- tainty propagated through both expressions for depolarization parameter d. The x-axis gives the depolarization ratio test input val- ues, δ. The y-axis gives the depolarization ratio uncertainty test input values, σδ. For
each combination, uncertainties were propa- gated through the equations 6.2 and 6.4 for
d. The colourbar indicates the difference be- tween these two uncertainties: σdG1 − σdG2.
The many positive (red) values indicate that uncertainties for dG2 are most often smaller
than those for dG1. Therefore, dG2 is the
preferable expression for the traditional cal- culation method ofd.
From this point on, any references to d or to “the traditional method depolarization parameter” refer to calculations made using thedG2expression, Equation 6.5.