Inversion of Multi-Angle Radiation Measurement Page: 3 of 8
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Fifteenth ARM Science Team Meeting Proceedings, Daytona Beach, Florida, March 14-18, 2005
The question then is whether we can provide a practical method for retrieving the particle sizes in a
vertically inhomogeneous cloud that is consistent with all the multi-angle, multi-spectral polarization
and reflectance measurements that are available. That is, can we fit the data using a simple model that is
consistent with the typically observed vertical variation in cloud droplet size distributions.
The equation of transfer for polarized light can be written in operator form as LI= S where the transport
L = si.VI + 6ext(ri)N(ri) f8(st .st)- -r 4 t dfdi
I is the Stokes vector and S is the source term. The equation for the Greens function of this operator can
formally be expressed as LG = 5(1,2) where 6 (1,2) is a Dirac delta function in both space and angle
variables. If we can determine the Green's function then the Stokes vector can be evaluated from the
expression I= GS. If we now perturb the transport operator by altering the single scattering properties
of the particles present, or the number density of scatterers and/or absorbers then the effects of this
perturbation on the Stokes vector can be expressed as L+ ALj'= S The perturbed Stokes vector can
be expressed as the sum of its unperturbed part I, and a perturbation Al. If we now neglect terms that are
second order in the perturbation (i.e. the term ALAI) we obtain the final result for the effects of
perturbing the radiative transfer equation on the observed Stokes vector
I= I- G(0, )AL(T)G(T,O)Sdr+ O(A2).
where the optical depth variables indicate that this expression is specific to reflection and the two
Green's functions are different. One (G(r,0)) allows the diffuse (and direct) radiation within the layer
to be evaluated while the other (G(0, z)) is the (Mueller matrix generalization of the) "escape" function
introduced by Twomey (1979).
Calculating Green's Functions
So, what are the Green's functions in terms of the more usual quantities that are calculated by radiative
U(r;p,p;( - (0');p> O,p'<0
G(,r,0;pAp;p-p')= Drd(-('+; exp(-zd / (();g)0a 0
D(rp~';p ' +8p-p )(p- ');p ,W<0
Here’s what’s next.
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Cairns, B.; Alexandrov, M. Lacis, A. & Carlson, B. Inversion of Multi-Angle Radiation Measurement, article, March 18, 2005; New York, New York. (digital.library.unt.edu/ark:/67531/metadc778858/m1/3/: accessed September 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.