ArticlePDF Available

# Frequency, Temperature and Salinity Variation of the Permittivity of Seawater

Authors:

## Abstract and Figures

With the emergence of unmanned marine robots, underwater communication systems have received much attention in recent years. To successfully develop radio wave based communication solutions, it is essential to understand properties of electromagnetic wave transmission in seawater. These properties are determined by the frequency variation of the permittivity of seawater. Existing models for the permittivity of saline water are empirical ones that best fit experimental data. We propose a physically realistic model, similar to the one used in plasma physics, for the variation of the dielectric constant of water with varying frequencies and salinities. Our model is in excellent agreement with existing empirical fits for frequencies between 1 and 256 GHz. We use this model to study the propagation of electromagnetic waves in seawater. We explain that large propagation distances would be possible at MHz frequencies if the conductivity of seawater decreases at small field strengths due to the hydrogen bonding of water molecules. However, we were unable to experimentally verify any reduction in the conductivity of seawater
Content may be subject to copyright.
Frequency, temperature and salinity variation of the
permittivity of Seawater
Ram Somaraju and Jochen Trumpf
Abstract
With the emergence of unmanned marine robots, underwater communication systems have received much
attention in recent years. To successfully develop radio wave based communication solutions, it is essential to
understand properties of electromagnetic wave transmission in seawater. These properties are determined by the
frequency variation of the permittivity of seawater. Existing models for the permittivity of saline water are empirical
ones that best ﬁt experimental data. We propose a physically realistic model, similar to the one used in plasma
physics, for the variation of the dielectric constant of water with varying frequencies and salinities. Our model is
in excellent agreement with existing empirical ﬁts for frequencies between 1 and 256 GHz. We use this model to
study the propagation of electromagnetic waves in seawater. We explain that large propagation distances would be
possible at MHz frequencies if the conductivity of seawater decreases at small ﬁeld strengths due to the hydrogen
bonding of water molecules. However, we were unable to experimentally verify any reduction in the conductivity
of seawater.
Index Terms
Permittivity, Underwater radio propagation, Electromagnetic propagation in plasma media, Attenuation
Ram Somaraju and Jochen Trumpf are with the Department of Information Engineering, Research School of Information Sciences and
Engineering, Building 115, The Australian National University, Canberra, ACT 0200, Australia and National ICT Australia Limited, Locked
Bag 8001, Canberra, ACT 2601, Australia.
National ICT Australia Limited is funded by the Australian Government’s Department of Communications, Information Technology and
the Arts and the Australian Research Council through Backing Australias Ability and the ICT Centre of Excellence Program.
1
Frequency, temperature and salinity variation of the
permittivity of Seawater
I. I
NTRODUCTION
E
XISTING systems for underwater communication
largely depend on acoustic technologies. However,
acoustic communication is riddled with problems in-
cluding time-varying multipath propagation and large
latencies. Therefore, Al-Shammaa et. al. claim that radio
communication is a viable alternative [1]. To understand
the properties of radio wave propagation in seawater it is
essential to know the frequency variation of seawater’s
relative permittivity because the rate of attenuation of
plane electromagnetic waves is a function only of the
relative permittivity of the medium.
However, existing models used for the permittivity of
seawater are empirical ones that best ﬁt experimental
data and are not based on a sound physical model.
We propose a model for the permittivity of seawater
that is similar to the one used for ionic plasmas. In
the following, we brieﬂy review the general theory of
polarization of dielectrics including Debye’s theory of
molecular relaxation. This is followed by a description
of models that are currently being used to determine the
permittivity of fresh and sea water. We then explain our
model, and continue with an evaluation of the model
including implications on electromagnetic wave propa-
gation. As will be shown in section VI the predictions of
this model disagree with the results of Al-Shammaa et.
al. [1] but agree with the measurements we calculated.
II. P
OLARIZATION
Any dielectric substance placed in an electric ﬁeld
undergoes polarization, which involves the appearance
of bound charges on the surface of the dielectric. Polar-
ization is deﬁned as the dipole moment per unit volume
and it may be divided into two categories: induced and
orientation polarization [2], [3].
Debye investigates the contribution of these two forms
of polarizations to the relative permittivity of a dielectric
substance [4]. In section IV we will introduce atomic
polarization leading to a theory more appropriate for
seawater.
A. Debye’s Theory
In the book Polar Molecules [4], Debye explains the
relationship between relative permittivity and the fre-
quency of electromagnetic waves in a dielectric medium.
Debye assumes that the molecules are free and do not
interact with each other [2], [3] and that the polarization
of the dielectric consists of induced and orientation
components.
When a static electric ﬁeld E is applied to a dielectric,
the induced component of polarization is assumed to
have no inertia and almost instantaneously attains a value
of P
i
= ǫ
0
(ǫ
1)E. However, the orientation polar-
ization rises exponentially to reach a maximum value of
P
o
= ǫ
0
(ǫ
s
1)E ǫ
0
(ǫ
1)E at t = . Therefore,
for a static electric ﬁeld E, the total polarization reaches
a maximum value of P = ǫ
0
(ǫ
s
1)E. The time-constant
τ of the exponentially increasing orientation polarization
is called the relaxation time. Also, ǫ
s
and ǫ
are the
static and inﬁnite frequency relative permittivities of
the dielectric and ǫ
0
is the permittivity of free space.
It depends on the temperature of the substance and
is independent of the nature of the electric ﬁeld and
the time of application of the ﬁeld. Note that other
parameters such as viscosity and pressure inﬂuence ǫ
r
.
But, this variation is not studied in the paper. Based on
these assumptions, it can easily be shown (see [2], [3])
that the frequency dependance of the relative permittivity
ǫ
r
may be written as
ǫ
r
(ω) = ǫ
+
ǫ
s
ǫ
1 + jωτ
(1)
Here, ω is the angular frequency of oscillation of the
electric ﬁeld. It should be noted that the terms in the
above equation are functions of the temperature T of the
substance and therefore it is more appropriate to write
equation (1) as
ǫ
r
(ω, T ) = ǫ
(T ) +
ǫ
s
(T ) ǫ
(T )
1 + jωτ(T)
(2)
1) Dielectric properties of real molecules: Debye’s
model is based on the assumption that there are no
intermolecular interactions and this simple model does
not accurately predict the permittivity of real dielectrics.
Several dielectrics may be better modelled using the
Cole-Cole model [5] which states that the relative per-
mittivity is given by
ǫ
r
(ω, T ) = ǫ
(T ) +
ǫ
s
(T ) ǫ
(T )
1 + (jωτ(T))
1h
(3)
2
where 0 h 1. This is an empirical model and does
not really have a physical basis. There is however an
interpretation of the Cole-Cole model as the result of
a distribution of relaxation times rather than a single
relaxation time. Several ﬁts have been proposed for
the permittivity of both sea and fresh water based on
both Debye and Cole-Cole models as explained in the
following section.
III. E
XISTING MODELS FOR SEA AND FRESH WATER
PERMITTIVITY
An extensive set of experimental measurements [6]–
[9] are available for the permittivity of fresh water. The
dielectric properties of fresh water may be modelled us-
ing equ. 1 for frequencies up to 100 GHz [10]. However,
for higher frequencies a double-Debye model is found to
be more appropriate. The double-Debye model is based
on the assumption that there exists a second polarization
process with a different relaxation time and is given by
the equation
ǫ
r
(ω, T ) = ǫ
(T ) +
ǫ
s
(T ) ǫ
1
(T )
1 + jωτ
1
(T )
+
ǫ
1
(T ) ǫ
(T )
1 + jωτ
2
(T )
(4)
The widely used equations of Liebe et. al. [10] are based
on such a double-Debye model. Liebe et. al. [10] claim
that their model may be used for frequencies up to 1THz
and may be extended up to 30THz by the inclusion of
two Lorentzian terms. Double-Debye ﬁts for fresh water
are found in several other papers including Stogryn et.
al. [11] and Meissner et. al. [12].
Until recently, comprehensive models based on exten-
sive experimental measurements were not freely avail-
able for seawater. Descriptions used for seawater until
the early 1990s consisted of works of Stogryn [13]
and Klein [14]. However, in the last decade and a
half several single and double Debye type models were
developed by Ellison et. al. [15], Stogryn et. al. [11] and
Meissner et. al. [12]. The double-Debye model used by
Meissner et. al. [12] and Stogryn et. al. [11] is similar
to the fresh water model with the addition of the effect
of conductivity on the dielectric constant and may be
written as
ǫ
r
(ω, T, S) = ǫ
(T, S) +
ǫ
s
(T, S) ǫ
1
(T, S)
1 + jωτ
1
(T, S)
+
ǫ
1
(T, S) ǫ
(T, S)
1 + jωτ
2
(T, S)
+ j
σ(T, S)
ǫ
o
ω
(5)
Here, S is the salinity of seawater in parts per thousand
(ppt). Ellison et. al. [15] use a single Debye model to
ﬁt to experimental data. All these models evaluate the
functional dependance of the terms in equation (5) on the
salinity and temperature by ﬁtting polynomial, rational or
exponential functions to experimental data. For example,
Meissner et. al. [12] use the ﬁt
ǫ
s
= ǫ
s
(T, 0) · exp(b
0
S + b
1
S
2
+ b
2
T S) (6)
for the static relative permittivity of seawater. Here,
ǫ
s
(T, 0) is the static relative permittivity of fresh water
and the constants b
i
are evaluated by ﬁtting the best curve
to experimental data. By the author’s own admission,
there is no physical basis for the model (equation (6))
used. In addition to using the dielectric model of fresh
water Ellison et. al. [15], Stogryn et. al. [11] and
Meissner et. al. [12] respectively use 30, 13 and 12
parameters that are determined from experimental data
to predict the variation of all the terms in equation (5)
with temperature and salinity. In contrast, our model
is not only physically realistic but also uses only two
additional parameters to describe the dielectric behavior
of seawater.
IV. M
ODEL OF SEAWATER PERMITTIVITY
Seawater has several dissolved salts and is therefore
a good conductor. However, increased conduction is
not the only phenomenon that occurs when salts are
dissolved in water. The ions are hydrated to varying
extents (see [16]–[18]).
The hydration number is deﬁned as the number of
water molecules in the immediate vicinity of the ion. It is
based on the dynamical behavior of the water molecules
in solution that move with the ion as a unit [17]. This
should be distinguished from the coordination number
of the ion which is the number of molecules in the
immediate neighborhood of the ion. The coordination
number depends on the distance of the water molecules
from the ion [17]. It is bigger than the hydration number
and includes all the molecules that are hydrogen bonded
to the molecules in the immediate vicinity of the ion.
The model we propose here assumes that the Debye
model of Stogryn et. al. [11] is adequate for fresh
water. However, we develop a physically realistic model
for the variation in the permittivity of seawater with
varying salinities and temperatures. There are three basic
differences between sea and fresh water that need to be
considered in order to develop this model.
The conductivity of water increases with the ad-
dition of ions and the increase in conductivity is
approximately proportional to the number of ions.
The extent of polarization due to the displacement
of bound charges (i.e. induced and orientation po-
larization) in seawater depends on its salinity due
3
to the presence of ions. Therefore, ǫ
s
, ǫ
and τ are
functions of seawater’s salinity.
The static relative permittivity ǫ
s
, of seawater re-
duces because all the water molecules that are
in the vicinity of an ion orient themselves with
respect to the ion. We assume that these molecules
do not contribute to the orientation polarization of
seawater. We further assume that the number of
water molecules that orient themselves about the
dissolved ions is directly proportional to the number
of ions. Hence, we would expect ǫ
s
to decrease
linearly with increasing salinity. This assumption is
in accordance with the model of Ellison et. al. [15]
and furthermore seems reasonable based on the
physical intuition given above.
The effect of the ions on the induced polarizability
is difﬁcult to analyze. Firstly, each ion will have a
different absorption spectrum in the infrared region
and will contribute different amounts to induced
polarization. Further, the ions will affect the mag-
nitude of induced polarization of water molecules.
However, if the concentration of ions is small, these
effects may be ignored.
Also, the time constant τ should not be affected
by the addition of ions. This is because τ is based
on the inertial properties of orientation polarization
and we are assuming that the water molecules that
are oriented about the ions do not contribute to the
orientation polarization. Also, the inertial forces on
the water molecules that are not near the ions should
not be effected signiﬁcantly by the presence of ions.
In addition to induced and orientation polarization,
there exists a third kind of polarization in seawater.
Non-uniform distribution of free ions in the water
will result in atomic polarization, P
f
. The contri-
bution of P
f
to polarization has to be taken into
account in calculating the relative permittivity.
A. Polarization of seawater
The model we propose here is based on the one used
for gaseous plasmas which is composed of positive and
negative ions, electrons and also neutral atoms [19].
The total polarization of seawater, P may be written
as P = P
b
+ P
f
. Here, P
b
is the polarization due to
the displacement of bound charges in water molecules
(i.e. induced and orientation polarization) and P
f
is due
to the displacement of ions inside water (i.e. atomic
polarization). We can write P
b
= ǫ
0
χE, where χ = ǫ
b
1
and
ǫ
b
(ω, T, S) = ǫ
(T ) +
ǫ
1
(T ) ǫ
(T )
1 + jωτ
2
(T )
+
ǫ
s
(T )(1 α(T )S) ǫ
1
(T )
1 + jωτ
1
(T )
(7)
This is similar to equation (4) that is used for fresh
water with the small but signiﬁcant additional term
α(T )S in accordance with the assumption that the
static relative permittivity of seawater decreases linearly
with increasing salinity. The remaining terms in this
equation are assumed to be the same as the one used
by Stogryn et. al. [11] to model fresh water.
1) Evaluation of P
f
: We make the following assump-
tions in deriving a model for the variation of atomic
polarization with frequency.
Seawater is composed of water and several dis-
solved ion types, indexed by i, with mass m
i
and
charge q
i
. m
i
is the total mass of the i
th
type of ion
and all the water molecules in the hydration shell
of this type of ion.
The drift velocity of the water molecules is zero and
the drift velocities of all other ions are measured
with respect to water.
The density of ions is small and so collisions
between ions may be ignored and only collisions
between neutral water molecules and ions are sig-
niﬁcant.
If collisions are ignored, the rate of change of the drift
velocity of the i
th
type of ion v
i
may be written as
N
i
m
i
µ
v
i
t
+ v
i
· v
i
= N
i
q
i
E + N
i
q
i
v × B
+ N
i
m
i
g p
i
(8)
where N
i
is the number of ions of type i per unit
volume, p
i
= N
i
kT
i
is the pressure and N
i
m
i
g gives the
gravitational force. Though not shown explicitly for ease
of notation, both N
i
and m
i
are functions of temperature
and salinity. For wavelengths that are large compared to
atomic dimensions, the pressure gradient and the non-
linear v
i
· v
i
terms may be ignored [19]. Furthermore,
gravitational force is small compared to the force due to
the electric ﬁeld and can therefore be ignored. Collisions
are incorporated into equation (8) by adding a damping
term that is proportional to v
i
and an effective collision
rate ω
eff
i
and we get,
N
i
m
i
µ
v
i
t
= N
i
q
i
E + N
i
q
i
v
i
× B
N
i
m
i
ω
eff
i
v
i
(9)
If we deﬁne the drift displacement r
i
of the i
th
ion by
v
i
=
r
i
t
(10)
4
then we can write
P
f
=
X
i
N
i
q
i
r
i
(11)
Also, equation (8) may be re-written in terms of drift
displacement as
N
i
m
i
2
r
i
t
2
= N
i
q
i
E + N
i
q
i
r
i
t
× B
N
i
m
i
ω
eff
i
r
i
t
(12)
For waves with exponential dependance of the form
exp{j(kr ωt)}, this equation may be written as
(jω)
2
µ
i
r
i
= q
i
(E + jωr
i
× B) (13)
where µ
i
= m
i
{1 + j(ω
eff
i
)}. Substituting equa-
tion (13) into (11), and ignoring the contribution of the
magnetic ﬁeld, which tends to be small compared to that
of the electric ﬁeld in non-magnetic materials, we get
P
f
=
X
i
N
i
q
2
i
µ
i
ω
2
E (14)
B. The relative permittivity
Now, the total displacement ﬁeld D = ǫ
0
E+P
f
+P
b
.
Using equations (7) and (14) we get
ǫ
r
(ω, T, S) = ǫ
(T ) +
ǫ
s
(T )(1 α(T )S) ǫ
1
(T )
1 + jωτ
1
(T )
+
ǫ
1
(T ) ǫ
(T )
1 + jωτ
2
(T )
X
i
N
i
q
2
i
ǫ
0
µ
i
ω
2
(15)
Substituting the value of µ
i
from equation (13) we get
ǫ
r
(ω, T, S) = ǫ
(T ) +
ǫ
s
(T )(1 α(T )S) ǫ
1
(T )
1 + jωτ
1
(T )
+
ǫ
1
(T ) ǫ
(T )
1 + jωτ
2
(T )
X
i
c
i
ǫ
0
ω
2
(1 + jω
eff
i
)
(16)
where c
i
=
N
i
q
2
i
m
i
.
It is however difﬁcult to calculate and measure ω
eff
i
for the individual ions. Therefore, we assume that the
effective collision rate is the same for all the ions and is
equal to ω
eff
. We can then rewrite equation (16) as
ǫ
r
(ω, T, S) = ǫ
(T ) +
ǫ
s
(T )(1 α(T )S) ǫ
1
(T )
1 + jωτ
1
(T )
+
ǫ
1
(T ) ǫ
(T )
1 + jωτ
2
(T )
c(T, S)
ǫ
0
ω
2
(1 + jω
eff
(T, S))
(17)
where c(T, S) =
P
i
c
i
. Both c(T, S) and ω(T, S) are
functions of both Salinity and temperature because the
number N
i
of ions of type i in solution and the mass
m
i
of i
th
type of ion along with its hydration shell are
functions of temperature and salinity.
1) Evaluation of α(T ), c(T, S) and ω
eff
(T, S):
α(T ) · S is equal to the fraction of water molecules
that are oriented towards ions in solution. Let the con-
centration of the i
th
type ion in water be ρ
i
parts per
thousand. Now, some fraction, say β
i
, of these ions will
be dissociated and these are the only ions that contribute
towards the reduction of the static permittivity of water.
The number of such ions N
i
in 1 Kg of solution is given
by
N
i
=
β
i
ρ
i
ν
i
moles (18)
Here, ν
i
is the atomic mass of the i
th
type ion. If the
coordination number of this ion is k
i
, then the total
number of water molecules that are oriented about the
i
th
type of ion is N
i
k
i
. Therefore the fraction of water
molecules, α(T ) · S that are oriented about all the ions
in solution is given by
α(T ) · S =
X
i
N
i
k
i
1/0.018
=
X
i
0.018β
i
ρ
i
k
i
ν
i
(19)
Note that we used the fact that the molecular mass of
water is 18 g/mole in deriving the last equation.
To evaluate c(T, S), we need to know the hydration
number and not the coordination number of an ion. Let
the hydration number of the i
th
type of ion be h
i
and
A
v
= 6.023e23 be Avogadro’s number. Then the mass
of the i
th
type of ion and all the water molecules in its
hydration shell is m
i
= (ν
i
+ 0.018h
i
)/A
v
. Using this
and equation (18) we get
c(S, T ) =
X
i
c
i
=
X
i
N
i
q
2
i
m
i
=
X
i
A
2
v
β
i
ρ
i
q
2
i
ν
i
(ν
i
+ 0.018h
i
)
(20)
Finally, because the ionic conductivity of water
σ(T, S) c(T, S)
eff
(T, S) we can calculate the
ratio of ω
eff
(T, S) to c(T, S) using the well established
values of ionic conductivity of water.
However, the hydration number, coordination number
and percentage of dissociation are difﬁcult to ascertain
accurately from experimentation. For instance, the ex-
perimental values for the hydration number of Na+ ions
varies from 4-8 [16]. Therefore it was decided that α(T )
and c(T, S) should be determined using experimentally
5
Ion Type Density (ppt)
Cl
19.135
Na
+
10.76
SO
2
4
2.712
Mg
2+
1.294
TABLE I
DENSITY AND MOLALITIES OF COMPONENTS OF SEAWATER WITH
SALINITY S = 35 PPT
measured values of the permittivity and then the resulting
value for the hydration and coordination numbers be
compared to existing predictions.
V. RESULTS
It is difﬁcult to experimentally measure the permittiv-
ity of seawater for a varying range of frequencies because
it is extremely lossy. Therefore, it was decided that
the validity of the model be ascertained by generating
pseudo-data from the empirical ﬁts to experimental data
mentioned in section III. Figure 1 compares the real and
imaginary parts of permittivity of seawater of our model
with the ﬁts of Stogryn et. al. [11], Meissner et. al. [12],
Ellison et. al. [15] and Wentz et. al. [20] for various
values of salinity and temperature.
It is clear from these ﬁgures that our model is in ex-
cellent agreement with these ﬁts for frequencies between
1 and 256 GHz. The maximum deviation of our model
from the ﬁts of Stogryn et. al. [11] and Meissner et.
al. [12], is 7.6% and 6.9% respectively in the frequency
range 1-256GHz. The ﬁts of Stogryn et. al. [11] and
Meissner et. al. [12] differ by as much as 11.5% in the
same frequency range. Further veriﬁcation of the model
comes from the fact that the values of α(T ) and c(T, S)
obtained by ﬁtting best curves to pseudo-data are the
right order of magnitude. We get,
α(T ) = 0.00314 ppt
1
(21)
c(T, S) 1 × 10
12
· S C
2
/Kg (22)
It should be noted that no temperature dependance was
detected for both c(T, S) and α(T). To check that
these values are as expected, we need to know the
composition of seawater. Table I shows the densities
of four components of seawater that have the highest
concentrations [21].
We note that the only signiﬁcant elements are Sodium
and Chlorine. Because we are only interested in the order
of magnitude of the hydration numbers we assume that
seawater only consists of these two components and from
the calculated value of α we get
β(k
Cl
+ k
Na
) = 11.3 (23)
Here, β is the degree of dissociation of NaCl and
k
Cl
and k
Na
are the coordination numbers of Cl and
Na respectively. These numbers are as expected [17].
Similarly it can be shown that c(T, S) is of the right
order of magnitude.
However, more experimental results over a wider
range of frequencies are required to completely validate
the model and get an accurate value for c(T, S). Such
experiments are particularly important for frequencies
greater than a few hundred gigahertz. High frequency
measurements are needed to accurately compute c(T, S)
because ω
eff
is of the order of 1THz and we require
measurements in this frequency range to accurately de-
termine c(T, S).
Also, at smaller frequencies seawater behaves as a
conductor and this can easily be seen by taking the low
frequency limit of equation (17). Then the permittivity
reduces to
ǫ
r
(T, S) =
c(T, S)
jǫ
0
ωω
eff
(T )
=
σ(T, S)
jǫ
0
ω
(24)
This is what one would expect for a good conductor.
In fact the low frequency limit of the empirical equa-
tion (5) is identical to the one used in this model. It is
therefore safe to assume that the model is valid for low
frequencies.
VI. I
MPLICATIONS FOR THE PROPAGATION OF
ELECTROMAGNETIC WAVES
Our model for the permittivity of seawater assumes
that it is independent of the applied electric ﬁeld strength
and is only a function of the temperature and salinity of
seawater and the frequency of the electromagnetic wave.
Therefore the rate of attenuation of an electromagnetic
wave in seawater, which depends only on its permittivity,
is not a function of the distance from a transmitting an-
tenna. This is in accordance with the classical literature:
the articles [22]–[26] indicate that no such change in
attenuation occurs as the distance from the transmitting
antenna increases even if the antennas are insulated [26],
[27]. Therefore, we expect the range in seawater to be
comparable to the skin depth, which is of the order of 0.3
m at a frequency 1 MHz if we assume the conductivity
of seawater to be 4 s/m [28].
However, Al-Shammaa et. al. claim that radio com-
munication over a distance of 100m is possible at MHz
frequencies in seawater [1]. Al-Shammaa et. al. [1]
further claim that as the distance from the transmitting
antenna increases, the rate of attenuation of electromag-
netic waves reduces greatly. In fact, ﬁgures 13 and 14
in [1] indicate that there is minimal attenuation once the
distance from the transmitting antenna increases beyond
6
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
80
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
log
2
f (GHz)
ℜ{ǫ}
ℜ{ǫ} vs log
2
(f) (T = 15°C, S = 35 ppt)
(a)
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
log
2
f (GHz)
ℑ{ǫ}
ℑ{ǫ} vs log
2
(f) (T = 15°C, S = 35 ppt)
(b)
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
80
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
log
2
f (GHz)
ℜ{ǫ}
ℜ{ǫ} vs log
2
(f) (T = 25°C, S = 35 ppt)
(c)
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
80
90
100
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
log
2
f (GHz)
ℑ{ǫ}
ℑ{ǫ} vs log
2
(f) (T = 25°C, S = 35 ppt)
(d)
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
80
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
log
2
f (GHz)
ℜ{ǫ}
ℜ{ǫ} vs log
2
(f) (T = 15°C, S = 25 ppt)
(e)
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
ℑ{ǫ}
log
2
f (GHz)
ℑ{ǫ} vs log
2
(f) (T = 15°C, S = 25 ppt)
(f)
7
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
80
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
log
2
f (GHz)
ℜ{ǫ}
ℜ{ǫ} vs log
2
(f) (T = 25°C, S = 25 ppt)
(g)
0 1 2 3 4 5 6 7 8
0
10
20
30
40
50
60
70
80
Meissner et. al.
Stogryn et. al.
Ellison et. al.
Wentz et. al.
Model
ℑ{ǫ}
log
2
f (GHz)
ℑ{ǫ} vs log
2
(f) (T = 25°C, S = 25 ppt)
(h)
Fig. 1. Real and Imaginary parts of permittivity as a function of frequency
3-4 meters. Al-Shammaa et. al. [1] explain this reduction
in attenuation with increased distance by claiming that
the conduction current losses may be ignored once the
distance from the transmitting antenna becomes large.
We believe that such a change in attenuation of
electromagnetic waves in seawater could only occur
if seawater behaves differently at small electric ﬁeld
strengths and hence at large propagation distances from
a transmitting antenna. If the conductivity of seawater
decreases at small electric ﬁeld strengths, then, as the
distance from the transmitting antenna increases, the
amplitude of the transmitted electromagnetic wave would
reduce and therefore we would see a reduced rate of
attenuation. One possible explanation as to why seawater
might be a poor conductor at small ﬁeld strengths is as
follows.
A positive and a negative ion may be bonded to each
other through water molecules that are hydrogen bonded
to each other (See ﬁgure 2). These bonds, if they do
exist, will be extremely weak and easy to break apart.
Therefore at high electric ﬁeld strengths, with forces
acting in opposite directions on positive and negative
ions, these bonds might be broken apart and we would
get free positive and negative ions. However, for small
electric ﬁeld strengths, there would be no free ions to
conduct and therefore the conductivity might decrease
drastically. This would particularly be the case at higher
frequencies because with alternating ﬁelds, the time
available to break these weak bonds would be shorter.
It is well known that the conductivity of seawater is
not constant above a certain frequency. Gabillard et.
al. [25] show that if the conductivity was constant for
+
_
Two water molecules hydrogen
bonded to each other
Hydrogen
Bond
Ions and their
hydration shells
Fig. 2. Ions bonded to each other through hydrogen bonded water
molecules
high frequencies then yellow light would only be visible
up to 29 cm underwater. But we can see a yellow lamp
much further than 29 cm in seawater.
It is hence conceivable that the rate of attenuation
decreases with decreasing ﬁeld strength. It could well be
that such a reduction has not been detected previously
because one could not measure extremely small electric
ﬁeld strengths until recently. However, with better mea-
suring equipment available now one might be able to
detect small ﬁeld strengths.
We decided to experimentally verify if the conductiv-
ity of seawater changes by using the setup shown in
ﬁgure 3. We measured the amplitude of V 1 and V 2
using a lock-in ampliﬁer to calculate the impedance
of salt water from the ratio V 1 and V 2. We used a
8
R1
B
A
V2
W
V1
Lock−in Amplifier
Fig. 3. Experimental setup to measure the conductivity of saltwater
Princeton Applied Research EG & G 5210 ampliﬁer at a
frequency of 50 KHz and a Stanford Research Systems
SR844 ampliﬁer at a frequency of 1 MHz. It was decided
that it was unnecessary to use a Wheatstone bridge
circuit because it is not essential that the impedance
be measured accurately. We were only interested in
measuring large changes in impedance as only this would
explain the large differences in the rates of attenuation.
As a control experiment, at 1 MHz the water cell was
also replaced by a 820 carbon resistor.
The electric ﬁeld strength applied to the water cell
was reduced by increasing the resistance of the variable
resistor R1. All the components were shielded inside
grounded metal boxes to reduce the effects of external
noise. We detected no change in the conductivity of salt
water at 50 KHz. The smallest voltage applied to the
water cell was 600nV at 50 KHz. The cell is 5 cm
long and if we assume a uniform electric ﬁeld then the
smallest ﬁeld applied was 12µV /m.
However, at 1 MHz we initially detected a change in
the ratio of the voltages V 1 and V 2. Exactly the same
change in ratio was also present in the control experi-
ment, where we replaced the water cell with a resistor.
We concluded that this effect was due to capacitative and
inductive coupling between poorly shielded wires. After
shielding the wires properly we did not detect a change
in the ratios of the two voltages and hence no change in
the conductivity of seawater down to a voltage of 30 µV
at 1 MHz. If we assume a uniform electric ﬁeld then the
smallest ﬁeld strength applied was 1.5 mV /m.
We can conclude from this experiment that the con-
ductivity of seawater does not change for electric ﬁeld
strengths as small as 12µV /m at a frequency of 50 KHz
or 1.5mV/m at a frequency of 1 MHz and hence the rate
of attenuation does not change for these ﬁeld strengths.
We currently have no plausible explanation for the large
propagation range observed by Al-Shammaa et. al. [1].
VII. C
ONCLUSION
In this paper we derived a physically realistic model
for the frequency variation of the relative permittivity
of seawater for varying salinities and temperatures. The
model derived is in excellent agreement with existing
empirical ﬁts to experimental data. Also, the model uses
only two parameters that need to be determined from
experimental data as opposed to more than 10 parameters
used by most empirical ﬁts. Furthermore, the remaining
parameters in our model have a physical interpretation
and could hence theoretically be determined by indepen-
dent experiments. Moreover, because our model has a
physical foundation, we are conﬁdent that it is valid over
a wider parameter (frequency, temperature and salinity)
range and can be used for extrapolation in regions where
no experimental data is available.
This model however does not predict large propagation
distances for electromagnetic waves in seawater in the
frequency range of a few Megahertz as measured by
Al-Shammaa et. al. [1]. We believe that the only pos-
sible explanation for these large propagation distances is
that the conductivity of seawater changes at small ﬁeld
strengths due to hydrogen bonding in water. However, we
measured no change in conductivity for electric ﬁelds as
small as 12µV /m and 1.5mV/m at frequencies of 50
KHz and 1 MHz respectively.
R
EFERENCES
[1] A. I. Al-Shammaa, A. Shaw, and S. Saman, “Propagation of
electromagnetic waves at mhz frequencies through seawater,
IEEE Trans. Antennas Propagat., vol. 52, no. 11, pp. 2843–
2849, Nov. 2004.
[2] A. Chelkowski, Dielectric Physics. Elsiver Scientiﬁc Publish-
ing Company, 1980.
[3] E. H. Nora, W. E. Vaughan, A. Price, and M. Davies, Dielectric
properties and molecular behaviour, T. Sugden, Ed. Van
Nostrand Reinhold Company Ltd., 1969.
[4] P. Debye, Polar molecules. Dover, 1929.
[5] K. S. Cole and R. H. Cole, “Dispersion and absorption in
dielectrics, Journal of chemical physics, vol. 9, pp. 341–351,
April 1941.
[6] J. Hasted, S. Hussain, A. Frescura, and J. Birch, “The tem-
perature variation of the near millimetre wavelength optical
constants of water, Infrared Physics, vol. 27, no. 1, pp. 11–
15, 1987.
[7] D. Lide, Ed., Handbook of Chemistry and Physics, 74th ed.
CRC Press, 1993, ch. 6, p. 10.
[8] D. Archer and P. Wang, “The dielectric constant of water
and debye-huckel limiting law slopes, Journal of Physical
Chemical Referrence Data, vol. 19, p. 371, 1990.
[9] R. L. Kay, G. Vidulich, and K. S. Pribadi, A reinvestigation of
the dielectric constant of water and its temperature coefﬁcient,
Journal of Chemical Physics, vol. 73, no. 2, pp. 445–447,
February 1969.
[10] H. J. Liebe, G. A. Hufford, and T. Manabe, A model for
the complex permittivity of water at frequencies below 1thz,
International Journal of Infrared and Millimeter Waves, vol. 12,
no. 7, pp. 659–675, 1991.
[11] A. Stogryn, H. Bull, K. Rubayi, and S. Iravanvhy, “The mi-
crowave dielectric properties of sea and fresh water, GenCorp
Aerojet, Azusa, Ca. 91702, Tech. Rep., 1995.
9
[12] T. Meissner and F. Wentz, “The complex dielectric constant
of pure and sea water from microwave satellite observations,
IEEE transactions on geosciense and remote sensing, vol. 43,
no. 29, Sept 2004.
[13] A. Stogryn, “Equations for calculating the dielectric constant
of saline water, IEEE transactions on microwave theory and
Techniques, vol. 19, pp. 733–736, August 1971.
[14] L. A. Klein and C. T. Swift, An improved model for the
dielectric constant of sea water at microwave frequencies,IEEE
transactions on antennas and propagation, vol. 25, no. 1, pp.
104–111, January 1977.
[15] W. Ellison, A. Balana, G. Delbos, K. Lamkaouchi, L. Eymard,
C. Guillou, and C. Prigent, “New permittivity measurements of
seawater,Radio Science, vol. 33, no. 3, pp. 639–648, May-June
1998.
[16] D. T. Richens, The chemistry of aqua ions. John Wiley &
Sons, 1997.
[17] E. Clementi, Determination of liquid water structure, coordi-
nation number for ions and solvation for biological molecules.
Springer-Verlag, 1976.
[18] J. Burgess, Ions in solution. Horwood Publishing, 1999.
[19] H. G. Booker, Cold plasma waves. Martinus Nijhoff Publish-
ers, 1984.
[20] F. Wentz and T. Meissner, Amsr ocean algorithm (version 2),
Remote Sensing Systems (http://www.remss.com), Santa Rosa,
CA, Tech. Rep., 1999.
[21] K. Turekian, Oceans. Englewood Cliffs, NJ: Prentice-Hall Inc,
1976.
[22] R. Dunbar, “The performance of a magnetic loop transmitter re-
ceiver system submerged in the sea, The Radio and Electronic
Engineer, vol. 42, no. 10, pp. 457–463, October 1972.
[23] I. Bogie, “Conduction and magnetic signalling in the sea -
a background review, The Radio and Electronic Engineer,
vol. 42, no. 10, pp. 447–452, October 1972.
[24] R. King and G. Smith, Antennas in Matter. The M.I.T. Press,
1981, ch. 9.
[25] R. Gabillard, P. Degauque, and J. Wait, “Subsurface electro-
magnetic telecommunication- a review, IEEE Trans. Commun.
Technol., vol. COM-19, no. 6, pp. 1217–1228, 1971.
[26] J. Wait, “Insulated loop antenna immersed in a high conduc-
tivity medium, J. Res. Nat. Bur. Stand., vol. 59, no. 2, pp.
133–137, August 1957.
[27] R. King, “Theory of the terminated insulated antenna in a
conducting medium, IEEE Trans. Antennas Propagat., vol. 12,
no. 3, pp. 305 318, May 1964.
[28] M. Tucker, “Conduction signalling in the sea, The Radio and
Electronic Engineer, vol. 42, no. 10, pp. 453–456, October
1972.
Jochen Trumpf (M’04) received the Dipl.-
Math. and Dr. rer. nat. degrees in mathematics
from the University of W
¨
urzburg, Germany, in
1997 and 2002, respectively. He is working as
a Research Fellow for the Department of Infor-
mation Engineering in the Research School of
Information Sciences and Engineering at The
Australian National University, Canberra, Aus-
tralia, and is currently seconded to National
ICT Australia Ltd. His research interests include observer theory and
design, linear systems theory and optimisation problems in digital
communication.
Ram Somaraju received B.Sc. and B.E. de-
grees in Physics and Computer Systems en-
gineering from the University of Auckland,
New Zealand, in 2001 and 2003, respectively.
He is currently pursuing a Ph.D. degree in
the Department of Information Engineering in
the Research School of Information Sciences
and Engineering at The Australian National
University, Canberra, Australia. His research
interests include underwater radio communications, MIMO commu-
nication and uncertainty principles for communication channels.
... The beginning of passive microwave remote sensing of the oceans in the 1970's [1,2] increased the importance of a good model. A major milestone was the development of a model for saltwater by Klein and Swift [3] based on laboratory measurements at L-and S-band and employing a functional dependence on frequency based on the response of polar molecules [4,5]. Most of the work on the dielectric constant of sea water afterward was done to extend This paragraph of the first footnote will contain the date on which you submitted your paper for review, which is populated by IEEE. ...
... The dielectric constant of sea water consists of two parts, a contribution due to orientation and/or distortion of the water molecule (called "polarization" [21,5]) and a contribution due to motion of charge (current). Theory for the polarization of an ideal polar molecule in a viscous (i.e., damping) medium was developed by Debye [4,5] and experimental evidence supports this solution for water [22,23]. ...
... The dielectric constant of sea water consists of two parts, a contribution due to orientation and/or distortion of the water molecule (called "polarization" [21,5]) and a contribution due to motion of charge (current). Theory for the polarization of an ideal polar molecule in a viscous (i.e., damping) medium was developed by Debye [4,5] and experimental evidence supports this solution for water [22,23]. All the models discussed here employ this form for the frequency dependence. ...
Article
Full-text available
The model expressing the dielectric constant of sea water at microwave frequencies as a function of salinity and temperature is an important element in remote sensing of sea surface salinity. It is also important independently as a description of the physical properties of salt water. A major milestone was the development in the late 1970’s by Klein and Swift of a model based on laboratory measurements at L- and S-band and a functional form supported by theory for polar molecules and previous work on freshwater. Much of the subsequent work has focused on measurements at higher frequency and determining model parameters tuned to apply for applications such as remote sensing of sea surface temperature. Interest in the dielectric constant at 1.4 GHz (L-band) increased again with the development of SMOS and Aquarius to measure salinity from space. But there have been few new measurements at L-band and often confusion regarding the applicability of new models at 1.4 GHz. The objective of this manuscript is to compare available models in the context of how well they represent the dielectric constant of sea water at 1.4 GHz. Among the criteria applied will be the recent measurements at the George Washington University of the dielectric constant at 1.4 GHz.
... In the frequency range of interest here (from 0 to 1 GHz), the relative permittivity and electrical conductivity of pure liquids and homogeneous dielectric materials are more or less frequency-independent [44,45]. For example, this is true for pure water [46][47][48] and seawater [49,50]. Therefore, it is often assumed that the dielectric properties of blood plasma, PBS, erythrocyte membrane, and erythrocyte cytoplasm are also frequency-independent over a wide-bandwidth frequency range. ...
Article
Electrochemical impedance spectroscopy of whole blood and blood cells has great potential for assessing the health of patients. This potential has not been fully exploited for a detailed examination of the erythrocyte interior. The erythrocyte interior (cytoplasm) is often considered as continuous media. However, cytoplasm is a colloidal suspension of hydrated hemoglobin molecules. This study mainly focuses on the effect of hemoglobin hydration on the dielectric properties of erythrocyte cytoplasm and whole blood. The impedance spectra of separated cytoplasm and whole blood were measured at frequencies from 1 kHz to 110 MHz. The effective medium theory was used to analyze the experimental data. The cytoplasm was found to be best described as a colloid of hemoglobin cores surrounded by double hydration shells. The dielectric properties of the hemoglobin core, hydration shells and free intracellular fluid, as well as the thickness of the shells, were numerically determined from the impedance spectrum. An approach was developed to determine the intracellular fluid's viscosity and the density of bound water in the hydration shells. It shows that the hemoglobin hydration has a significant effect on the physical properties of blood. The detailed electrical properties of the cytoplasm can be detected from the impedance spectrum of whole blood.
... Estimates of water depth are fundamental for most studies of water bodies and based on recalculating the time delays of the registered GPR signals using the permittivity value ε= 81. Nevertheless, some loss of accuracy may occur, not only due to the dependence of permittivity on water mineralization and temperature [Owen et al., 1961;Archer and Wang, 1990;Somaraju and Trumpf , 2006;Catenaccio et al., 2003] but also due to inhomogeneities in the near-bottom layer, which usually accumulate plant material, mud and silt in the natural environment. ...
Article
Ground-penetrating radar profiling on the surface of water bodies is applied in various geological and engineering studies. Here, we present the results of numerical simulation of the propagation of a video pulse electromagnetic signal in a freshwater body with gradients of the permittivity and electrical conductivity in the near-bottom layer. The method of numerical solutions of Maxwell's equations in the time domain is applied, in the general setting for rapidly changing processes, without restrictions on the magnitude of the change in the parameters of the medium. The results make it possible to explain the apparent decrease in water depth according to GPR data in comparison with the true depth and the appearance of additional reflecting boundaries on radargrams in the bottom layer.
... The complex wavenumber for the submerged dipole source is formulated from the double Debye Model as given in Eq. 7, where parameters like relative permittivity, frequency, temperature (T), and salinity (S) of seawater are presented [47]. Since in this paper, only the inline electric field is focused, electrical permittivity will be incorporated for obtaining the complex wavenumber: ...
Article
Full-text available
Marine Controlled-Source Electromagnetic as an established alternative method for offshore hydrocarbon prediction is a successful technique to detect the electrically resistive anomalies in deep waters. However, in shallow water, its certainty of hydrocarbon delineation is arguable due to airwave presence when using a conventional Horizontal Electric Dipole (HED). In this paper, we propose a structural alteration in the HED source that uses conduction current method. The modification is in the form of a synthetic incurvation of the source to focus the down-going wave and disperse the airwave. We examined modified sources of different incurvature factors, a, using synthetic data generated by 1D numerical solution based on integral equation and simulation models based on Finite Element Method. The numerical calculation and simulation’s correlation coefficient that varies from 0.80 to 0.96 (depending on the incurvation factor) validate the agreement between the two methods. For a hydrocarbon target located at a depth of 1 km, we observed a contrast of 34% between with and without hydrocarbon models using modified source with curvature factor a = 0.4, whereas using the HED, the contrast was 22%. For a 4 km deep target, this modified source showed 19% contrast, whereas the delineation using the HED was almost negligible. In addition, we processed a shallow water (150 m) analysis with a deep target (4 km) in which the proposed design produced a maximum contrast of 11% compared to the HED that featured only 4% delineation. These remarkable responses using the synthetic incurvation of HED can be a breakthrough for the offshore hydrocarbon exploration industry.
... Les évolutions des parties réelles et imaginaires des modèles les plus couramment utilisés sont représentées sur la figure 3-3 en fonction de la fréquence et de la température [163]. Nous avons choisi l'expression de la permittivité intrinsèque du milieu la plus fréquemment utilisée pour les fréquences inférieures à 1THz [162] [163]. Cette relation dépend de la pulsation angulaire (< = 2>. ...
Thesis
La surveillance de l’environnement sous-marin nécessite le déploiement de capteurs et d’infrastructures dédiées dont le coût et l’impact sur la faune et la flore doivent être réduits. L’application cible vise des zones géographiques inférieures à 1km2 dans lesquelles les transmissions de flux vidéo et de mesures, prélevés par des capteurs immergés, doivent être réalisées sans-fil sur des distances supérieures à 10m avec un débit minimum de 80kbps pour des puissances d’émission d’une dizaine de Watts. Une étude comparative des méthodes de communication acoustiques, optiques et électromagnétiques en eau de mer est présentée. Cette analyse est introduite en définissant un ensemble de critères de performances destinés à évaluer et sélectionner la technique la mieux adaptée aux besoins applicatifs. Les méthodes électromagnétiques, dont les coûts de déploiements et l’impact environnemental sont minimaux, présentent toutefois des limitations de portée pour le débit de données souhaité. La suite de cette thèse présente les travaux de recherche qui ont été menés pour lever ces verrous technologiques. Un premier modèle simplifié de propagation des champs électromagnétiques en milieu subaquatique a été développé pour différencier les modes de propagation favorisant les pertes par conduction de celles engendrées par les propriétés diélectriques de l’eau de mer. Des prototypes d’antennes ont été développés pour tenter d’exciter le milieu en favorisant l’un ou l’autre mode. Finalement, l’étude détaillée d’un modèle de couplage magnéto-inductif a permis de réaliser et d’évaluer les performances d’une telle liaison en utilisant des techniques originales d’élargissement de bande passante qui ont été implémentées avec succès dans un prototype de MODEM sous-marin.
Article
This article reports an improvement in the model for the dielectric constant of seawater used to fit laboratory measurements of the dielectric constant at the L-band. The new model (dielectric constant as a function of salinity, temperature, and frequency) is based on the response of a polar molecule proposed by Debye and fits the same measurement as reported in earlier work but uses a functional form for conductivity, $\sigma (S,T)$ , that is given by the definition of salinity. The new version of this model fits the data well and has the advantages that the relaxation time constant is allowed to be a function of temperature and salinity and is well behaved when extrapolated to high salinities.
Article
Electrochemical impedance spectroscopy (EIS) is a powerful probe of the processes taking place at an electrode. Depending on frequency, it is sensitive to the solid-liquid interface as well as to processes taking place in the solution further from the electrode. In principle, shrinking electrode dimensions allows probing these processes on the nanometer scale. In practice, however, this represents a formidable challenge. Signals resulting from the stray capacitance of the interconnects can dramatically exceed those from the electrode itself. Furthermore, miniaturized electrodes exhibit faster dynamics, and thus necessitate working at higher frequencies in order to achieve comparable performance. Here we discuss recent advances in nanoscale impedance measurements. We begin with a theoretical discussion of the main concepts and inherent tradeoffs, followed by a review of recent experimental efforts. As this field remains in its infancy, we place particular emphasis on the conceptual and technical aspects of the approaches being developed.
Article
The book aims to present current knowledge concerning the propagation of electro­ magnetic waves in a homogeneous magnetoplasma for which temperature effects are unimportant. It places roughly equal emphasis on the radio and the . hydromagnetic parts of the electromagnetic spectrum. The dispersion properties of a magnetoplasma are treated as a function both of wave frequency (assumed real) and of ionization density. However, there is little discussion of propagation in a stratified medium, for of collisions is included only which reference may be made to Budden [1] . The effect in so far as this can be done with simplicity. The book describes how pulses are radiated from both small and large antennas embedded in a homogeneous magneto­ plasma. The power density radiated from a type of dipole antenna is studied as a function of direction of radiation in all bands of wave frequency. Input reactance is not treated, but the dependence of radiation resistance on wave frequency is described for the entire electromagnetic spectrum. Also described is the relation between beaming and guidance for Alfven waves.
Article
The advent of precision microwave radiometry has placed a stringent requirement on the accuracy with which the dielectric constant of sea water must be known. To this end, measurements of the dielectric constant have been conducted at S-band and L-band with a quoted uncertainty of tenths of a percent. These and earlier results are critically examined, and expressions are developed which will yield computations of brightness temperature having an error of no more than 0.3 K for an undisturbed sea at frequencies lower than X-band. At the higher microwave and millimeter wave frequencies, the accuracy is in question because of uncertainties in the relaxation time and the dielectric constant at infinite frequency.