Inverse Scattering Fundamentals - Introduction to Imaging from Scattered Fields - page 46

Equation 4.11 can now be substituted back into Equation 4.9, and the entire
equation can then be integrated over a small volume ∇ V about the origin as
shown:

Ce
r

ikr

∫ ∫

(4.12)

d Vk ( ∇+

2 2

) =− = δ () r d Vxyz (()( )( )) − δ δδ

∇ v ∇ v

where the right-hand side can be simplified due to the identity of the delta
function as follows:

Ce
r

ikr

∫

(4.13)

d Vk ( ∇+ ) =− 1

2 2

v

Now the left-hand side of Equation 4.13 can be simplified by applying the dis-
tributive property as follows:

Ce
r

ikr

Ce
r

ikr

∫ ∫

(4.14)

∇

2

d Vk +

2

d V =− 1

∇ v ∇ v

As V becomes smaller and smaller, the second term in Equation 4.14 gradu-
ally goes to zero due to the fact that d V in this term is defined as

d Vr = 2 sinθθφ d dd r (4.15)

and as V gets smaller so does the r 2 term in Equation 4.15 which causes the
second term to go to zero as stated earlier. This further simplifies Equation
4.14 as follows:

Ce
r

ikr

∫

∇

2

d V = − 1 (4.16)

v

Equation 4.16 can then be rearranged as follows:

Ce
r

ikr

� Ce

r

ikr

-
‚ =−

∫ ∫

∇

2

d V = d V ∇⋅ ∇ 1 (4.17)

∇ v ∇ v

This now allows the divergence theorem to be applied to Equation 4.17 as
follows:

� Ce

r

ikr

-
‚ =∇ =−

Ce
r

ikr

∫ ∫∫

d V ∇⋅ ∇ d S 1 (4.18)

v

S

which can be expanded as

� ∂

∂

Ce
r

ikr

-
‚

∫∫

dr S

2

sinθθφ dd =− 1 (4.19)

r

S