Geoscience Reference
In-Depth Information
τ
µ
ϕ
(
) for the conformity with definitions (1.55). According to its meaning,
the probability density is the product of three probability densities: the density
ofthephotonfreepathofdistance
,
,
τ
according to (2.8), density of
the non-absorption of the photon in the atmosphere
∆τ
=
τ
−
ω
0
(
τ
), and the density
of the scattering of the photon with change of the direction from (
µ
,
ϕ
)to
µ
ϕ
τ
χ
)
|
π
(
) as per (2.19). However, this product is
exactly equal to
K
according to (1.55)! Taking into account that as per (2.6) and
(2.11) the photon probability within the directional interval [−
,
), which is equal to
x
(
,
(4
µ
,0]isequalto
µ
)
=
µ
)
|π
τ
=
τ
0
,
P
(
2 arccos(−
the following condition is added to (1.55) for
µ
<
0 to consider the surface albedo in the Monte-Carlo method:
µ
2
exp
−
τ
.
τ
A
−
τ
,
µ
,
ϕ
)
=
τ
µ
ϕ
=
µ
1−
K
K
(
,
,
,
−
(2.23)
µ
π
2
Ψ
Now to find operator
remember that the variables noted in the definition of
the
K
operator (1.55) as (
τ
µ
ϕ
), later become the integration variables them-
selves when the desired values are calculated using (2.20). For example, during
the calculation of the radiance according to source function (1.52)
,
,
τ
is a vari-
τ
able noted in equations of the source function (1.53)-(1.55) as
. Therefore,
ϕ
) at (2.10). After
the radiance calculation with (1.52), the irradiance is computed according to
relation (1.6) and factor 1
τ
µ
ϕ
τ
,
µ
,
coordinates of the point (
,
,
) are to be noted as (
|µ
is canceled out. The integrating is accomplished
τ
,
µ
,
ϕ
)andforoperator
Ψ
over all three variables (
it yields the expression
exactly equal to simple local estimation (2.18). When the radiance is computed
with (1.52) the integration variable is
τ
only, so there is no dependence of the
ϕ
). Actually, the probability density of
transition
K
is written accounting for the change of the notions for coordinates
(
µ
,
source function upon coordinates (
τ
,
µ
,
ϕ
) and for the radiance computation, using (1.52) coordinates (
τ
,
µ
ϕ
)
are applied. Hence, the scattering angle, which the photon trajectory is sim-
ulated with, in the
K
operator according to the Monte-Carlo method, differs
from the operator defined by (1.55) in the transfer equation. Therefore, the
probability density of the scattering to direction (
,
ϕ
) has not yet accounted
for. To account for it we are accomplishing the multiplication by the phase
function in the equation for local estimation (2.19). Thus, there is a complete
correspondence between (2.19) and (1.52)-(1.55) also during the consideration
of the radiance.
Thecaseofsimulatingthephotontrajectorieswithweights
w
corresponds
to the coordinated transformation of operators
K
and
µ
,
Ψ
taking into account
that they are used in solution (2.22) only as a convolution of
K
with
Ψ
.In
ω
0
(
τ
this case, the multiplication by probability of the quantum surviving
)
Ψ
is passing from operator
K
to
. It corresponds to the changing of photon
weight
w
when the powers of the
K
operator are calculated in (2.22), and then
to the multiplication of the local estimation to photon weight
w
in (2.18) and
(2.19) (Kargin 1984). Analogously it is concluded that the direct modeling of
the irradiances otherwise corresponds to the passing from the exponential
factor (the local estimation (2.18)) to the
K
operator. Similar transformations,
many of which are difficult to present from the physical point of view, are
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