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ε α = ∑V
α
iε
i
(29)
i
where V i is the volumetric concentration of i-th phase and ε i is the dielectric
constant of this phase.
The four-phase model proposed by de Loor [25,26] and used by Dirksen and
Dasberg [27] and Dobson and Ulaby [28] is:
( ε − ε ) + 2θ
( ε − ε ) + 2θ
( ε − ε )
3ε
s + 2θ
w w s bw bw s a a s
ε =
(30)
⎛ ε s
⎞ ⎛ ε s
⎞ ⎛ ε s
⎞
3 + θ w
⎜ −1
⎟ + θbw
⎜ −1
⎟ + θ a
⎜ −1
⎟
⎝ ε w ⎠ ⎝ ε bw ⎠ ⎝ ε a ⎠
where the indexes s, w, bw and a refer to solid phase, water, bound water and air,
respectively. This model describes heterogeneous mixture where foreign granules
with dielectric constant ε i are imbedded in a homogeneous and isotropic dielectric
with dielectric constant ε. The imbedded foreign granules in Eq. (30) represent
free water, bound water and air. The host medium is the solid phase.
The empirical models ε (θ ) and ε (θ,ρ), where ρ is the soil bulk density, form
the formula of the appropriate regression curves [58,59,78,90]. The formula (31)
presented below and discussed in Skierucha [78] will be used in this paper as the
reference:
( 0,573
+ 0,582ρ
+ (7,755 0,792ρ)
θ ) 2
ε = +
(31)
where ρ [g/cm 3 ] is the soil bulk density.
The verification of the discussed 3-phase and 4-phase models is presented in
the following sections.
5.1. Material and methods
Sixteen mineral soils of different texture, volumetric water content and bulk
and particle density were investigated (Table 6).
The following variables were measured: dielectric constant, ε, bulk density, ρ,
particle density, ρ s , and gravimetrically determined volumetric water content, θ.
Dielectric constant, ε, was determined using TDR (LOM/m from Easy Test, Ltd.
[29]). The remaining variables were measured with the application of standard
methods.
The solid phase dielectric constant, ε s , was calculated from the formula
(derived from the formula (27)):
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