Through-Wall Imaging With UWB Radar System - KEMT FEI TUKE

2.5 Radar Imaging Methods Overview 21
where ∇ 2 φ = � ∇. � ∇φ and ∂φ
∂n = � ∇φ.�n. In the case when φ = φ1φ2, the φ ′ =
φ2φ ′ 1 + φ1φ ′ 2 and (2.5.11) becomes:
or
�b
a
�b
a
[φ2φ ′ 1 + φ1φ ′ 2]dx = φ1φ2| b
a
φ2φ ′ 1dx = φ1φ2| b
a −
�b
a
(2.5.14)
φ1φ ′ 2dx. (2.5.15)
An analogous formula in higher dimensions arises by substituting A = φ2 � ∇φ1 into
(2.5.12). Using the identity � ∇.a� ∇b = � ∇a. � ∇b + a∇2b leads to:
� �
�∇φ2. � ∇φ1 + φ2∇ 2 � �
φ1 dvol =
∂φ1
φ2 dsurf.
∂n
(2.5.16)
V
Similar result can be obtained when A = φ1 � ∇2φ2: � �
�∇φ1. � ∇φ2 + φ1∇ 2 � �
φ2 dvol =
V
V
Finally, subtracting (2.5.17) from (2.5.16) yields to:
�
�
φ2∇ 2 φ1 − φ1∇ 2 �
�
φ2 dvol =
�
∂φ1
φ2
∂n
∂V
∂V
∂V
φ1
− φ1
∂φ2
dsurf. (2.5.17)
∂n
�
∂φ2
dsurf. (2.5.18)
∂n
(2.5.18) is known as Green’s theorem that represents fundamental of the Kirchhoff
migration theory. It is a multidimensional generalization of the integration by
formula from elementary calculus and is valuable for its ability to solve certain
partial differential equation. φ1 and φ2 in (2.5.18) may be chosen as desired to
conveniently expressed solutions to a given problem. Typically, in the solution to
a partial differential equation like the wave equation, one function is chosen to be
the solution to the problem at hand and the other is chosen to be the solution to
the more simple reference problem. The reference problem is usually selected to
have a known analytic solution and this solution is called a Green’s function [96].
In general, the scalar wavefield ψ is a solution to the Helmholtz scalar wave
equation: � ∇ 2 + k 2 � ψ = 0 (2.5.19)
where ∇ 2 is the Laplacian and k is the wavevector. When the wave travels through
the medium from point x0 to the point x this Helmholtz scalar wave equation can
be expressed with the aid of Green’s function G (x0, x) as:
� ∇ 2 + k 2 � G (x0, x) = δ (x0 − x) (2.5.20)