14
Multiscale Hydrologic Remote Sensing: Perspectives and Applications
The solution of Maxwell’s equations for electromagnetic waves propagating in
multilayered media is well known. Following the approach of Lambot et al. (2004a),
the analytic expression for the zero-offset Green’s function in the spectral domain
(2-D spatial Fourier domain) is found to be
G
R
R
h
xx
n n
n
n n
n
n n
↑
=
−
−
Γ
Γ
Γ
TM
TE
η
ξ
exp(
),
2
(2.3)
where the subscript n equals 1 and denotes the first interface and first layer (in practice, the air layer); R n
TM and R n
TE are, respectively, the transverse magnetic (TM) and
transverse electric (TE) global reflection coefficients (Slob and Fokkema 2002)
accounting for all reflections and multiples from surface and subsurface interfaces;
Γ n is the vertical wave number defined as Γ n
n n
k
=
+
ρ
ξ η
2
; k ρ is a spectral-domain
transform parameter; ξ n = jωμ n ; η n = σ n + jωε n ; and j = −1.
The transformation of Equation 2.3 from the spectral domain to the spatial
domain is carried out by employing the 2-D Fourier inverse transformation:
G
G dk
xx
xx
↑
↑
+∞
=
∫
1
4
0
π
ρ
,
(2.4)
which reduces to a single integral in view of the invariance of the electromagnetic
properties along the x- and y-coordinates. We developed a specific procedure to
properly evaluate that singular integral using an optimal integration path (Lambot
et al. 2007). In addition to avoiding the singularities (branch points and poles), the
path allows for minimizing the oscillations of the complex exponential part of the
integrand, which makes the integration faster.
2.3.2 inveRSion of gPR data
Inversion of Green’s function is formulated by the complex least squares problem as
follows:
min
,
1
φ(b) G
G
C G
G
xx
xx
xx
xx
=
↑
↑
−
↑
↑
−
−
∗ ∗
∗ ∗
T
(2.5)
where G xx
*
↑
↑
= G xx ( )
ω and G
b
xx
↑
↑
= G xx ( , )
ω
are vectors containing, respectively, the
observed and simulated radar measurements, from which major antenna effects
have been filtered using Equation 2.1; C is the error covariance matrix; and b is
the parameter vector containing the soil electromagnetic parameters, which are the
soil relative dielectric permittivity ε and electrical conductivity, and layer thicknesses to be estimated. As function ϕ(b) has usually complex topography, we use the
global multilevel coordinate search algorithm (Huyer and Neumaier 1999) combined
Multiscale Hydrologic Remote Sensing: Perspectives and Applications
The solution of Maxwell’s equations for electromagnetic waves propagating in
multilayered media is well known. Following the approach of Lambot et al. (2004a),
the analytic expression for the zero-offset Green’s function in the spectral domain
(2-D spatial Fourier domain) is found to be
G
R
R
h
xx
n n
n
n n
n
n n
↑
=
−
−
Γ
Γ
Γ
TM
TE
η
ξ
exp(
),
2
(2.3)
where the subscript n equals 1 and denotes the first interface and first layer (in practice, the air layer); R n
TM and R n
TE are, respectively, the transverse magnetic (TM) and
transverse electric (TE) global reflection coefficients (Slob and Fokkema 2002)
accounting for all reflections and multiples from surface and subsurface interfaces;
Γ n is the vertical wave number defined as Γ n
n n
k
=
+
ρ
ξ η
2
; k ρ is a spectral-domain
transform parameter; ξ n = jωμ n ; η n = σ n + jωε n ; and j = −1.
The transformation of Equation 2.3 from the spectral domain to the spatial
domain is carried out by employing the 2-D Fourier inverse transformation:
G
G dk
xx
xx
↑
↑
+∞
=
∫
1
4
0
π
ρ
,
(2.4)
which reduces to a single integral in view of the invariance of the electromagnetic
properties along the x- and y-coordinates. We developed a specific procedure to
properly evaluate that singular integral using an optimal integration path (Lambot
et al. 2007). In addition to avoiding the singularities (branch points and poles), the
path allows for minimizing the oscillations of the complex exponential part of the
integrand, which makes the integration faster.
2.3.2 inveRSion of gPR data
Inversion of Green’s function is formulated by the complex least squares problem as
follows:
min
,
1
φ(b) G
G
C G
G
xx
xx
xx
xx
=
↑
↑
−
↑
↑
−
−
∗ ∗
∗ ∗
T
(2.5)
where G xx
*
↑
↑
= G xx ( )
ω and G
b
xx
↑
↑
= G xx ( , )
ω
are vectors containing, respectively, the
observed and simulated radar measurements, from which major antenna effects
have been filtered using Equation 2.1; C is the error covariance matrix; and b is
the parameter vector containing the soil electromagnetic parameters, which are the
soil relative dielectric permittivity ε and electrical conductivity, and layer thicknesses to be estimated. As function ϕ(b) has usually complex topography, we use the
global multilevel coordinate search algorithm (Huyer and Neumaier 1999) combined