A New Look at Fundamentals of the Photometric Light Transport and Scattering Theory. Part 1: One-Dimensional Pure Scattering Problems

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S. Persheyev, D.A. Rogatkin
88 ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 5
scattering; but, at the same time, it will be enhanced due to the backscattering of the
opposite flux in Δx. Hence, the system of continuous differential equations for the
multiple scattering case will has the form:
( ) ( ) ( );
( ) ( ) ( ).
dF x SF x SF x
dx
dF x SF x SF x
dx

 

 
  
 
(23)
Evidential solution for the system is
1 2
1 3
( ) ;
( ) .
F x C x C
F x C x C


 
 
(24)
where C 2 and C3 are constants, which should be determined using boundary
conditions, and 1 3 2 ( ). C S C C  At boundary conditions x = 0: F+(0) = F0; x = H0:
F− (H0) = 0, and we can find the fluxes as follows:
0 0
0 0
0 0
1 ( ) ( )
( ) ; ( ) .
1 1
S H x S H x
F x F F x F
SH SH  
  
 
 
(25)
It yields for the transmitted and backscattered fluxes respectively:
0 0
0
1
( ) ,
1
F F H F
SH    

(26)
and
0
0
0
(0) .
1 BS
SH
F F F
SH   

(27)
We do not see the exponential law here! Fluxes at multiple pure scattering do not
obey the exponential law. This is important and well-known result, but it is not all!
Eqs. (25)−(27) tells us nothing about S. This is a key point. We can assume
ln(1 ), S R
    like it was derived at SSA, but it will mean, evidently, that we
assume SSA inside Δx for each fluxes at MSA, and multiple scattering occurs due to
mutual scattering between each Δx. It is the case 1 — the quasi-multiple scattering
process. It, probably, can be acceptable for a major number of turbid media, but here
for the purpose of the investigation of theoretical basics of LT&ST, we have to study
the extreme case of multiple scattering directly inside Δx. This is the case 2 — the
perfect multiple scattering process. How can we do it? We know at least two ways.
The more interesting way is based on the theory of Markov processes.
An assumption of the ''photon'' migration phenomenon in LT&ST is now widely
used in many applications, especially in statistical Monte-Carlo simulation. If this
model is assumed, then the probability of the ''photon'' location at any point xi of the
space can be interpreted like a probability of the state of the Markov N-section chain.
In this case, the magnitudes of the parameter t (time) in a function of conditional