Inverse Scattering Fundamentals - Introduction to Imaging from Scattered Fields - page 52

change induced by the target or scattering object must be very small in terms
of the unit wavelength. Equation 4.46 implies that this approximation begins
to break down as V ( r ) approaches zero. A global condition for the validity of
the Rytov approximation is

∫ ∫ ∇

kV () r ′ Ψ () ( r ′ G rr r , ′ ) d ′ [ Φ () r ] Ψ () ( r ′ G r rr , ′ ) d

2 2

(4.47)

inc 0 s inc 0

D D

So, when this criterion is satisfied and the Rytov approximation is valid,
the complex scattered phase can be expressed as

ik

2

∫

Φ () r = V () r ′ Ψ () ( r ′ G rr r , ′ ) d ′

(4.48)

s

Ψ () r

inc 0

inc

D

This can now be substituted back into Equation 4.40 to compute the total
field as follows:

�

∫

e i k

(

2

/ Ψ ( r )) V () r ′ Ψ ( r ′ )( G r rr , ′ ) d ′

(4.49)

·

inc inc 0

Ψ ( rr , ) = Ψ ( ) r

D

inc inc

If the argument of the exponent is now isolated by dividing by the incident
field and the logarithm applied, the resulting form of this equation is

·

Ψ
Ψ

( r r

rr

, )

·

� =− k D

Ψ ( rr , ) ln

inc 2

() r ′ Ψ () ( r ′ G 0 r rr , ′ ) d ′

(4.50)

inc inc

( , · )

in c

inc inc

This equation is comparable to Equation 4.35 from the Born approxima-
tion analysis in that it basically defines an inverse Fourier relationship or
procedure to recover V ( r ). Equation 4.50 can be very difficult to evaluate due
to the nature and challenges of dealing with the multivalued issues of the
natural logarithm (Fiddy et al., 2004). We will encounter the same difficulty
in Chapter 8.
Also, comparable with the Born approximation, when the conditions for
the Rytov approximation are not valid, V RA ( r ) is recovered in lieu of V ( r ) where

1
2

·

V ( rr , ) = V ( ) r − [ ∇ Φ () r ]

2

(4.51)

RA inc s

k

ReFeRenCeS
Avish, C. K. and Slaney, M. 1988. Principles of Computerized Tomographic
Imaging. New York: IEEE Press.
Chew, W. C. 1995. Waves and Fields in Inhomogeneous Media. Piscataway:
IEEE Press.
Darling, A. M. 1984. Digital Object Reconstruction from Limited Data
Incorporating Prior Information. Thesis, University of London.