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elementary propagation times along the solid, water and air phases of the soil
expressed by (23) by means of (24):
2L
ε 2Ls
ε s 2La
ε a 2Lw
ε w
t = ts
+ ta
+ tw
= = + +
(25)
c c c c
with: L s , L a , L w are idealized by the presented model parts of the probe inserted
into an elementary soil phase. Rewriting to last two elements of the Eq. (25),
multiplying by the elementary cross-section S and then dividing by the whole
volume of soil in the cylinder defining the sphere of influence of the TDR
measurement method gives:
Vs
Va
Vw
ε = ε s + ε a + ε w = ε s ( 1 −φ
) + ε a ( φ −θ
) + ε wθ
(26)
V V V
or using the parameter α, which summarizes the geometry of the medium with
relation to the applied electric field (Roth et al. [77]):
1
α
α α
( −φ
) ε + ( φ −θ
) ε θ ε ) α
ε = (27)
1 s a + w
which is the 3-phase dielectric mixing model α, expressing the square root of the
apparent dielectric constant of soil in the terms used by soil scientists: φ – soil
porosity and θ – volumetric water content. Whalley [100] presented the similar
discussion.
Following Birchak et al. [9], Alharthi and Lange [3] assumed α = 0.5, which
gives the Eq. (26). After introducing adsorbed water as a forth phase Dobson et
al. [28] determined α by regression data for different frequencies (ranging from
1.4 to 18 GHz) and soil types (ranging from sandy loam to silty clay) and
obtained α = 0.65. The four-phase dielectric mixing model α, accounting for the
adsorbed to soil solid phase bound water has the following form:
α
α α
( θ −θ
) ε θ ε
α
α
s + ( φ −θ
ε a + bw w +
ε = ( 1−φ)
ε )
(28)
where water exists in two phases: as free water, w, like capillary water and bound
water, bw, like molecular or film water. Generally the α-model of soil dielectric
mixture may be presented as follows:
bw
bw
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