15
Advanced Ground-Penetrating Radar for Soil Moisture Retrieval
sequentially with the classical Nelder–Mead simplex algorithm (Lagarias et al. 1998)
for minimizing the function.
2.3.3 PetRoPhySical RelationShiPS
A petrophysical relationship is necessary for translating the optimized dielectric
permittivity ε from GPR data inversion into volumetric soil moisture θ. Reviews
of ε– θ relationships can be found in the works of Huisman et al. (2003), Robinson
et al. (2003), Ponizovsky et al. (1999), Fernandez-Galvez (2008), and Steelman and
Endres (2011). Generally, petrophysical relationships are developed by two main
approaches. The first approach is empiric and uses measurements of dielectric permittivity for a variety of soil types at different water contents to construct the regressive polynomial formulas relating the water content to the dielectric permittivity.
The most frequently used empirical formula is the relationship suggested by Topp
et al. (1980):
θ
ε
ε
ε
= − ×
+
×
−
×
+
×
−
−
−
−
5 3 10
2 92 10
5 5 10
4 3 10
2
2
4 2
6 3
.
.
.
.
.
(2.6)
This equation has been widely applied to predict soil moisture from TDR and GPR
measurements, and its validity was established in many studies (Drungil et al. 1989;
Hallikainen et al. 1985; Roth et al. 1992). However, its applicability appeared to be
poor for organic, clayey, and fine-textured soils (Dirksen and Dasberg 1993; Todoroff
and Langellier 1998; Ponizovsky et al. 1999).
The second approach is more theoretical and derives the water content from
dielectric mixing models of soil. According to this approach, soil is a complex mixture of air, water, and soil particles. The permittivity of soil, therefore, is predicted
from the permittivity of each component weighted by their volume fraction. A general formulation of a commonly adopted dielectric mixing model is the power law
model:
ε θε
φ θ ε
φ ε
α
α
α
α
=
+ −
+ −
(
)
w
a
s
(
)
(
)
,
/
1
1
(2.7)
in which ϕ is the soil porosity; ε a , ε w , and ε s are the permittivities of air, water, and
soil particles, respectively; and α is the empirical power coefficient of the equation,
which holds for the spatial structure of soil mixture and its orientation with respect to
the electromagnetic field. Different power coefficients were proposed based on calibration with empirical data. Birchak et al. (1974) used the coefficient of 0.5, which
is widely known as the complex refractive index model (CRIM). CRIM was also
confirmed by Shutko and Reutov (1982), Roth et al. (1990), and Gorriti (2004) as the
most suitable power law model. By contrast, Dobson et al. (1985) found that α = 0.65
enabled describing the complex permittivity at the frequency range from 1.4 to 18
GHz for different soil types. Compared to empirical formulas, this approach takes
into account the composition of soil materials, and thus, it is expected to better predict the water content. However, in order to estimate the water content, the approach
Advanced Ground-Penetrating Radar for Soil Moisture Retrieval
sequentially with the classical Nelder–Mead simplex algorithm (Lagarias et al. 1998)
for minimizing the function.
2.3.3 PetRoPhySical RelationShiPS
A petrophysical relationship is necessary for translating the optimized dielectric
permittivity ε from GPR data inversion into volumetric soil moisture θ. Reviews
of ε– θ relationships can be found in the works of Huisman et al. (2003), Robinson
et al. (2003), Ponizovsky et al. (1999), Fernandez-Galvez (2008), and Steelman and
Endres (2011). Generally, petrophysical relationships are developed by two main
approaches. The first approach is empiric and uses measurements of dielectric permittivity for a variety of soil types at different water contents to construct the regressive polynomial formulas relating the water content to the dielectric permittivity.
The most frequently used empirical formula is the relationship suggested by Topp
et al. (1980):
θ
ε
ε
ε
= − ×
+
×
−
×
+
×
−
−
−
−
5 3 10
2 92 10
5 5 10
4 3 10
2
2
4 2
6 3
.
.
.
.
.
(2.6)
This equation has been widely applied to predict soil moisture from TDR and GPR
measurements, and its validity was established in many studies (Drungil et al. 1989;
Hallikainen et al. 1985; Roth et al. 1992). However, its applicability appeared to be
poor for organic, clayey, and fine-textured soils (Dirksen and Dasberg 1993; Todoroff
and Langellier 1998; Ponizovsky et al. 1999).
The second approach is more theoretical and derives the water content from
dielectric mixing models of soil. According to this approach, soil is a complex mixture of air, water, and soil particles. The permittivity of soil, therefore, is predicted
from the permittivity of each component weighted by their volume fraction. A general formulation of a commonly adopted dielectric mixing model is the power law
model:
ε θε
φ θ ε
φ ε
α
α
α
α
=
+ −
+ −
(
)
w
a
s
(
)
(
)
,
/
1
1
(2.7)
in which ϕ is the soil porosity; ε a , ε w , and ε s are the permittivities of air, water, and
soil particles, respectively; and α is the empirical power coefficient of the equation,
which holds for the spatial structure of soil mixture and its orientation with respect to
the electromagnetic field. Different power coefficients were proposed based on calibration with empirical data. Birchak et al. (1974) used the coefficient of 0.5, which
is widely known as the complex refractive index model (CRIM). CRIM was also
confirmed by Shutko and Reutov (1982), Roth et al. (1990), and Gorriti (2004) as the
most suitable power law model. By contrast, Dobson et al. (1985) found that α = 0.65
enabled describing the complex permittivity at the frequency range from 1.4 to 18
GHz for different soil types. Compared to empirical formulas, this approach takes
into account the composition of soil materials, and thus, it is expected to better predict the water content. However, in order to estimate the water content, the approach