Geoscience Reference
In-Depth Information
The formula of the derivative is specially converted to form (5.12), (5.13).
Written in this way, it looks as integral (2.20) directly calculated with the
Monte-Carlo method according to (2.21).
Ψ
a
(
u
)
B
a
(
u
)
W
a
(
u
)
du
=
Ψ
a
(
ξ
ξ
M
(
)
W
a
(
))
(5.14)
ξ
That is to say, the calculation of the derivatives according to (5.14) is reduced
to the multiplying of the value written to the counter by a certain “weight”
function
W
a
(
ξ
) (Marchuk et al. 1980).
To construct the concrete algorithm of calculating
W
a
(
ξ
)thederivative
explicit form of the right part of series (5.10) is obtained. For that we are using
the known expression of the derivative of the product through the sum of
logarithm derivatives (
xyz
...)
=
(
xyz
...)(
x
|
x
+
y
|
y
+
z
|
z
+...).Thefollowing
is obtained:
Ψ
a
(
u
)
q
a
(
u
)
Ψ
a
(
u
)
,
Ψ
a
(
u
)
+
q
a
(
u
)
Ψ
a
q
a
)
=
(
q
a
(
u
)
...
dudu
1
...
du
n
Ψ
a
K
a
q
a
)
=
Ψ
a
(
u
)
q
a
(
u
1
)
K
a
(
u
1
,
u
2
)...
K
a
(
u
n
,
u
)
(
Ψ
a
(
u
)
.
Ψ
a
(
u
)
+
q
a
(
u
)
q
a
(
u
)
+
K
a
(
u
1
,
u
2
)
K
a
(
u
1
,
u
2
)
+...+
K
a
(
u
n
,
u
)
×
K
a
(
u
n
,
u
)
(5.15)
After writing (5.15) to form (5.14) as it is more convenient for the Monte-Carlo
method, finally derive:
...
dudu
1
...
du
n
Ψ
a
K
a
q
a
)
=
Ψ
a
(
u
)
q
a
(
u
1
)
K
a
(
u
1
,
u
2
)
(
...
K
a
(
u
n
,
u
)
W
a
(
u
,
u
1
,
u
2
,...,
u
n
) ,
(5.16)
=
Ψ
a
(
u
)
Ψ
a
(
u
)
+
q
a
(
u
)
q
a
(
u
)
+
K
a
(
u
1
,
u
2
)
K
a
(
u
1
,
u
2
)
+...+
K
a
(
u
n
,
u
)
W
a
(
u
,
u
1
,
u
2
,...,
u
n
)
.
K
a
(
u
n
,
u
)
As it follows from (5.16), in the Monte-Carlo method the derivatives could be
calculated using the same algorithms as desired values with multiplying value
Ψ
ξ
ξ
(
)byspecialweight
W
a
(
)duringeachwritingtothecounter.Inaddition,
Ψ
ξ
ξ
if value
(
) depends on the current magnitude of random value
only,i.e.of
ξ
the current coordinates of the photon, then
W
a
(
) is the sum and depends on
the whole history of its trajectory.
Thus, to compute the derivatives of the irradiances, it is enough to dif-
ferentiate the explicit expressions of functions
Ψ
a
(
u
),
q
a
(
u
)and
K
a
(
u
,
u
)with
respect to the retrieved parameters. Then the following elementary changes are
introduced to the algorithm of irradiance calculations described in Sect. 2.1:
the counting of values
W
a
(for entire set of parameters) at every modeling of
theelementofthephotontrajectorywiththewritingtothespecialcounters
of the derivatives simultaneously with writing to the counters of the irradi-
ances. Although the irradiances are calculated as integrals with respect to
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