To correct for the attenuation in the VIL signal, the optical depth
between the VIL and each point in the profile has to be known
(Equation 1). To determine the optical depth between the
VIL and each data point, the extinction cross section at each point has
to be determined. Since the VIL data was converted into aerosol
backscatter cross sections, these values were used to calculate the
extinction cross sections. The aerosol backscatter cross sections were
first converted into aerosol scattering cross sections using
Equation 2 assuming no absorption and additional knowledge of
a bulk 
for the mesoscale volume. The
average bulk 
calculated with the
HSRL was used as the bulk 
for the mesoscale
volume. Since no absorption at visible wavelengths by the
cirrus ice crystals was assumed, Equation 14
relates the aerosol scattering cross sections to the extinction cross sections.
A forward integration was performed on the resulting extinction cross
sections to determine the attenuation
between the VIL and each data point. The attenuation to each point was used
to correct the existing VIL 
for signal loss.
In this analysis, a multiple scattering correction was included.
The attenuation along each profile was corrected by a multiple
scattering factor of 0.5 which decreased the attenuation by a factor of
2 (Eloranta and Shipley (1982)). This correction factor was
a result of half of the light attenuated by the cirrus cloud
ice particles being
diffracted in the forward direction. This diffraction peak stayed close to the
initial beam and further scattering of this light by other particles
resulted in greater backscatter at the receiver.
The bulk 
for cirrus cloud particles was
calculated by the HSRL for each cirrus profile. These bulk
were averaged over the 3 hour time period
resulting in a 
of 0.0499 sr
.
This value fell within the expected range of 
as described by Takano and Liou (1989). Their results give values of
for thin plates (0.025 sr
), ice columns
(0.038 sr
), and thick plates (0.087 sr
).
(Plates were detected at the tropopause as noted by the specular reflection
described previously.)
This average 
was used to correct all of the VIL
data for attenuation.
The method of a forward integration of the backscattered signal was first used to correct radar backscatter for attenuation. Hitschfeld and Bordan (1954) were one of the first to test the forward integration method. Klett (1981) showed the instability of this forward integration for large optical depths. In this study, if the one way optical depth of the cirrus cloud along each VIL profile became greater than 0.7, then the attenuation correction at further ranges (or larger optical depths) was considered to be unstable. This choice in the one way visible optical depth was chosen on the assumption of a 10% error in the VIL aerosol backscatter cross sections. The attenuation correction (using an optical depth of 0.7) of the data with a 10% error would result in a 40% error in the attenuation corrected signal.