ASSESSMENT OF ELECTROMAGNETIC ABSORPTION OF ICE … · 2019. 2. 19. · 3 2 3 0 10 20 30 40 100 80...
Transcript of ASSESSMENT OF ELECTROMAGNETIC ABSORPTION OF ICE … · 2019. 2. 19. · 3 2 3 0 10 20 30 40 100 80...
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Abstract— Ice core drillings have been performed in various zones in Antarctica and Greenland to
obtain climatological information, study ice properties, or analyze air and dust encapsulated in the
ice during the quaternary period. During these procedures, a set of measurements to characterize the
ice and to evaluate its physical and chemical properties are usually performed in situ. In particular,
using known temperature and dielectric profiles (DEP measurements), it is possible to evaluate the
ice electromagnetic power absorption profile, valid at the drilling site. In last decades, bedrock
characterization through Radio Echo Sounding (RES) surveys has been improved by the analysis of
the power of radar echoes. In this way, analysis of the electromagnetic properties of bedrock
interfaces makes it possible to assess the physical characteristics and to distinguish between wet and
dry conditions. Power variation of the received echoes also depends on ice absorption and on bedrock
reflectivity due to specific physical conditions of the ice.
In this paper the propagation of electromagnetic waves through the ice sheet is examined, and in
particular a new method for establishing the electromagnetic absorption profile for ice from core
drilling measurements is proposed and discussed. Variation in the ice absorption is deduced, starting
from analysis of ice core data from EPICA at the Concordia station (Antarctica) and from the GRIP
site (Greenland). This direct method of measurement is proposed with the aim of defining common
characteristics of the ice absorption rate that are valid both in Antarctica and in Greenland.
A. Zirizzotti1, L. Cafarella
1, S. Urbini
1, J.A. Baskaradas
2, A. Settimi
1
1 Istituto Nazionale di Geofisica e Vulcanologia, Roma, Italy
2 School of Electrical and Electronics Engineering SASTRA University, Thanjavur, India
email: [email protected]
ASSESSMENT OF ELECTROMAGNETIC ABSORPTION OF ICE FROM ICE CORE
MEASUREMENTS.
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INTRODUCTION
The Radio Echo Sounding (RES) technique is widely used in polar ice sheet exploration for mapping
bedrock morphologies and ice properties. It remains an indispensable tool for obtaining information about
the physical conditions of the ice mainly from its electromagnetic properties (Plewes and Hubbard, 2001
[36]). During recent years, the scientific interest has been drawn to the possible existence of water
circulation beneath the ice (Remy et al., 2003 [37], Kapitsa et al., 2006 [19]; Wingham et al., 2006 [46];
Bell et al., 2007 [1], Carter et al., 2009 [6], Tabacco et al., 2006 [43]) but, while differences between rock
and lake surfaces can be identified reasonably well in radargrams, this is not always the case for wet and
dry ice-bedrock interfaces. Certain characteristics of the physical condition of the ice-bottom interface have
been deduced in a number of studies using the power of radar echoes (Corr et al., 1993 [7], Bianchi et al.,
2004 [2], Carter et al., 2007 [5], Paden et al., 2005 [33], Oswald and Gogineni, 2008 [32], Fujita et al., 2012
[13]). Briefly, the information about the physical condition of ice-bottom interfaces (assessed from
electromagnetic reflectivity) was obtained starting from the solution of the radar equation using
electromagnetic power variations in echoes received after passing through the ice (Borogosky, 1995 [3]).
Attenuation of the echoes depends mainly on ice conductivity modulated by acidity, due to the quantity of
impurities originating from the presence of sea salt and erupted volcanic elements and in minor contribution
to the crystal fabric. The critical factor for the solution of the radar equation is determining the power
absorption of the ice, which can be obtained from conductivity measurements.
In the past, laboratory-frozen ice measurements were conducted to establish the electrical characteristics of
ice, and in recent decades the connection between the results and electromagnetic wave propagation through
ice has been investigated (Matzler and Wegmuller 1987 [26], Fujita 2000[12]). The aim was to define the
physical properties of ice in order to create models for its electromagnetic absorption, so that these results
could be extended to larger areas covered by RES measurements. Laboratory measurements of ice
conductivity and permittivity at different impurity concentrations, pressures, and temperatures were carried
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out (summarised in Fujita 2000 [12]) and the models were improved, although discrepancies still remain
between laboratory measurements and those conducted on natural ice (Stillman 2013 [41]).
Electromagnetic ice absorption can be calculated from conductivity measurements on an ice core, using the
temperature profile, obtaining results valid for a particular drilling site. These measurements in the past have
been made at different sites also applying different methods. These include electrical conductivity
measurements (ECM) using direct current (DC) valid exclusively for shallow ice cores (Hammer et al. 1980
[15]). This depends essentially only on the acidity of the ice, even if a large concentration of neutral salt is
present (Moore et al. 1992 [28]). For deep ice cores, the dielectric profiling method (DEP, Moore and Paren,
1987 [27]) is generally used. In this case measurements are made at AC frequencies (below 300 kHz) and the
results depend on acidity, acidic and neutral salt concentrations, and ice ammonium concentrations (Moore
and Paren, 1987 [27]; Moore et al. 1992b [29]; Moore and Fujita, 1993 [30]). Improved DEP instruments
have been developed to simultaneously measure ice core conductivity and permittivity (Wilhelms et. al., 1998
[45]), both profiles being useful in the study of radar signals and propagation velocity through the ice.
Another approach is to evaluate electromagnetic absorption of ice directly, using the amplitudes of RES
measurements (MacGregor et al. 2007 [22]). In this case the radar equation can be solved by assessing ice
absorption using some simplifications. For example, by only considering RES measurements collected over
subglacial lakes, and assuming a constant value for reflectivity. This hypothesis permits assessment of ice
absorption at different lake depths, also in different areas (Zirizzotti et al. 2014 [53]).
Conversely, ice reflectivity can be measured starting from the RES amplitude, assuming a linear trend for ice
absorption (in logarithmic units) (Jacobel et al. 2009 [17], Zirizzotti et al. 2010 [51]). In fact, by assuming a
linear trend for ice absorption (constant ice absorption rate versus ice depth), it is possible to measure
variations in bedrock reflectivity in order to establish a map of wet/dry zones on the bedrock interface (Carter
et al., 2009 [6]; Gades et al., 2000 [14]; Jacobel, 2010 [17]; Langley et al., 2010 [21]; Peters et al. [35], 2005;
Wright et al., 2012 [49]). Furthermore, by analyzing the amplitude of the signal reflected from internal layers,
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it is possible to extend the information about ice absorption to measure this contribution to bedrock
reflectivity (Siegert and Fujita 2001 [39], MacGregor et al. 2007 [22], Zirizzotti 2012 [52]).
In this paper, data from the GRIP Ice Core Project in Greenland, and from the EPICA European Ice Coring
Project at the Concordia station in Antarctica, (Wolf 1995 [47], community members 2004 [8], Wolf 2004
[48]), are presented and compared. A new method is proposed to assess the ice absorption rate from ice core
measurements, applied to the data from the two drilling sites. The results are discussed in detail and a
comparison is made between the two sites.
MODELLING ICE CONDUCTIVITY
Starting from the EPICA ice core measurements performed at the drilling site, it is possible to plot the raw
conductivity values (measured using the DEP method) as a function of depth with the corresponding
temperature profile (red and blue lines respectively in figure 1).
0 1 103
2 103
3 103
0
10
20
30
40
100
80
60
40
20
0
DEP
scaled DEP
Temp
Depth [m]
Con
du
ctiv
ity [
µS
/m]
Tem
p.
[°C
]
Figure 1: Measured ice conductivity (258),
scaled conductivity and temperature at the EPICA
drilling site
0 1 103
2 103
3 103
0
10
20
30
40
100
80
60
40
20
0
Depth [m]
Co
nd
uct
ivit
y [
µS
/m]
Tem
p.
[°C
]
Figure 2: Measured ice conductivity (258), scaled
conductivity and temperature at the GRIP drilling
site
During the drilling process the extracted ice at a defined depth was measured at the relatively constant
temperature range of -20 ± 2° C. The ice conductivity values 258 were measured and scaled to –15° C
(assumed as a reference value) using the Arrhenius model reported in equation (1). The 258conductivity
shows a slow reasonably constant trend with a mean value of 12.8 µS/m shifting gradually from 20 to 10
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µS/m. As a second step the reconstructed in situ conductivity profile (green line shown in figure 1) can be
obtained by scaling these measurements according to the temperature profile T(z) using the equation (1)
(Corr 1993 [7], Paden et al. 2005[33], Stauffer et al. 2004 [40], Kulessa 2007 [20]):
258
258
1 1( ) ( )exp
( )B
Ez z
K T T z
1)
where KB = 8.6173324(78) · 10−5
eV/K is the Boltzmann constant, E=0.22 eV is the applied activation
energy (a parameter that measures the sensitivity of the conductivity to temperature) while T(z) and 258(z)
are the temperature and conductivity measurements at different depths z.
The whole procedure was repeated for the GRIP ice core data and the results are reported in figure 2. In the
latter case, conductivity measurements in situ were performed directly on the extracted ice core at the fixed
temperature of -15°C, making the two data sets comparable. As shown in figure 2, 258 conductivity at the
GRIP site exhibits a sudden step at a depth of about 1600 m, changing values from around 20 µS/m to 10
µS/m. A smaller step is also visible at a depth of about 1000 m where conductivity values change from
around 18 µS/m to 22 µS/m.
As shown in equation (1), lower ice temperatures induce lower ice conductivity values. This behavior can
be clearly observed by plotting ice core conductivity (1) against ice temperature as shown in figure 3. The
data were calculated using the known temperature profiles valid at the selected sites. In the same figure,
laboratory measurements obtained from Fujita et al. 2000 [12] are added for comparison as black-dotted
line. These detailed laboratory measurements were made to analyze variations linked to crystal fabric,
density, impurity concentrations, temperature, pressure, air bubbles, and plastic deformation. The black
dots report the measurements performed on pure ice with an additional component of background acidity of
2 µM. Higher acidity values increase ice absorption, as observed in the field in Antarctica, where this value
reaches local peaks up to 10 µM (Fujita et al. 2000 [12]). Laboratory conductivity measurements are
performed at frequencies below 300 MHz using the DEP method. Ice absorption and electromagnetic
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scattering increase at frequencies higher than these, because electromagnetic waves of higher frequencies
penetrate the ice with more difficulty, hampering deep RES measurements. As shown in figure 3, the
reconstructed in situ ice conductivity profiles exhibit similar average values in the common interval
between -30° to 0° C. The difference at temperatures around -32 °C is due to a local surface temperature
influencing the conductivity profile in the first 1600 m. Large variations at the bottom of the ice are instead
probably linked to different ice acidity levels due to catastrophic volcanic events recorded at both polar
sites. It is worth noting that, in spite of these differences, the reconstructed in situ ice conductivity
measured at EPICA and GRIP (and the corresponding scaled conductivities) is very similar, and also close
to the laboratory measurements. This is the evidence of strong dependence of conductivity on temperature,
as highlighted by equation (1).
Figure 3: Ice conductivity at the two sites vs. ice temperature.
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MODELLING ICE ABSORPTION
Evaluation of electromagnetic absorption, in a medium with defined electromagnetic properties, starts
from the solution of the wave equation. In particular, it is possible to evaluate ice absorption from the
solution of the reduced wave equation (Helmholtz differential equation, Ulaby 1981 [44]) in the case of a
sinusoidal electrical field Ex oscillating at the frequency along the direction x orthogonal to the
downward direction of propagation z in the ice, with relative dielectric permittivity εr and relative magnetic
permeability µr,:
)()()( 2
2
2
2
2
zEzncz
zExc
x
,
2)
nc is the complex refraction index, which can be separated into real and imaginary parts (as usual j is the
imaginary unit):
)(
)(
2
)()()(
)()()()()()(
00 z
zzjzz
zjzzzzzn
r
rrrrrcrc
3)
in the case of low-loss media (low conductivity):
)()(
0
zz
r
4)
This condition is valid in the case of ice for frequencies higher than 0.1 MHz. Here c(z) is the complex
relative dielectric permittivity, which is not constant in the ice but depends on depth z (through r(z) and
σ(z) the electrical conductivity), c2= 1/ (ε0µ0) is the velocity of light in the vacuum, and ε0 and µ0 are free
space dielectric permittivity and magnetic permeability respectively. Moreover, in the ice conductivity σ(z)
depends on acidity and temperature, as shown in the previous paragraph. As a consequence of the fact that
the ice temperature varies along the depth, the conductivity profile is not constant. The relative dielectric
permittivity r(z) also depends on the media temperature and on the frequency of the electromagnetic
signal.
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In the case of constant refractive index values, nc=const. (with all the electromagnetic parameters
constant and independent on the z axis), equation (2) can be resolved in incoming and backwards waves
with the damping and oscillating parameters (Ulaby 1981 [44]):
20
0
r
r
rrc
5)
6)
The general case is a linear combination of the two solutions with coefficients that depend on the initial
boundary conditions.
In the case of ice, a media contaminated with impurities, the constant refractive index is no longer valid,
and neither are the parameters (5) and (6). In this context, the equation (2) was solved using the WKB
method, proposed to solve the Schrödinger equation (applied to the Planck parameter ħ adopting ħ 0) and
also applied in optics (for the wavelength parameter =2πc/ω, adopting 0), as proposed in Settimi et al.,
2003 [38]. Using a WKB-like approximation, the solutions can be written as:
z
c dzznc
j
x eEzE 0
')'(
0)(
7)
As shown in the appendix A of Settimi et al., 2003 [38] the condition of validity of the two solutions (7)
is:
4
)(
1)(2
zndz
zdn
c
c.
8)
where is the wave-length in the vacuum. For ice this condition is generally valid, and the left side of
equation (8) in the case of the EPICA and GRIP data are always less than 10-5
m-1
calculated for the whole
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ice core depth. Moreover, in ice the condition (8) is always valid throughout the frequency range of validity
for condition (4).
Considering only the minus solution (incoming wave travelling along the positive z axis) and using
equation (3) gives:
)()(
0
')'(
00)( zjz
dzznc
j
x eEeEzE
z
c
.
9)
Here, a damping term is multiplied by an oscillating term, so that:
zz
dzz
r
zrz
dzzzZz00
0
0)'(
)'(
2
)'(')'()'(
2
1)(
10)
z
rr
z
dzzzc
dzznc
z00
')'()'(')'()(
11)
where Z(z)2=µ(z)/(z) is the intrinsic impedance, and n(z) is the real refraction index of the ice. Equation 10
and 11 are important because they allow us to calculate amplitude and phase of a propagating
electromagnetic wave in a media with non constant electromagnetic parameters (r, σ, µr ). This is a general
case when we consider natural non uniform material like ice in glaciers, water in oceans and lakes or
stratified terrain. In the specific case of constant conductivity and constant dielectric permittivity and
permeability, this can be expressed as:
zzzr
r
2)(
0
0
12)
10
zzc
z rr
)(
where and are the factors calculated with Eqs. (5) and (6), generally used in the constant solution of
equation (2) (Ulaby et al., 1981 [44]). These equations are also used in amplitude analysis of radar signals
and also in cases of non constant electromagnetic parameters of a media.
The electromagnetic absorption of ice L[dB]
(z), i.e. the attenuation of a radar impulse passing through an
ice column in dB, can be evaluated from the attenuation of the radar signal using the following equation:
)(686.820)( )(][ zeLogzL zdB , 13)
In order to verify this equation and the solution (9), the ice absorption L[dB]
was compared, considering
three different cases. In the first case a numerical solution to evaluate the amplitude attenuation of a wave
propagating in a medium with physical parameters εr(z) and σ(z) was solved numerically using a specific
algorithm of the commercial software “Mathematica”. The conductivity profile was used to calculate the
propagation of an electromagnetic wave at both EPICA and GRIP. It is possible to extract the wave
amplitude from the numerical solution of equation (2), which changes as function of the distance z, in order
to obtain the absorption rate of a selected medium. In the second case the exact solution (13) of the ice
absorption using equation (10) was calculated with the same conductivity and permittivity parameters used
in the numerical solution. Finally, in the third case, the solution was calculated using the equation (12)
which is only valid in the case of constant electromagnetic parameters. The three solutions are plotted and
compared in figure 4 for the EPICA and GRIP data. The red lines are the numerical solutions of the
amplitude variation for a short sinusoidal pulse (50 ns pulse length at f = 60 MHz). Edge effects are visible
and they are due to the initial boundary conditions of the partial derivative equation (initial amplitude and
speed of the transmitted pulse). Below the red line, a black dashed line represents the exact solution of the
ice absorption (equation 13). As shown in the figures, there is no difference between these two cases. The
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blue lines report the constant solution of absorption rate (equations 12). As is clear, there is an error up to
10 dB using the constant solution compared to the exact one, which is more evident in the deepest zone.
The ice absorption at EPICA (left plot in the figure) slowly and regularly descends through the ice
thickness while, the GRIP measurements show a faster and regular linear trend up to a first constant step,
clearly visible at about 1600 m (right plot in figure 4). Below 1600 m the ice absorption plots in the two
cases (EPICA and GRIP) have about the same trend with different values.
EPICA site
0 0.8 1.6 2.4 3.240
35
30
25
20
15
10
5
0
Numerical solution
Constant solution
Exact solution
Depth [km]
EM
ice
ab
sorp
tio
n [
dB
]
GRIP site
0 0.8 1.6 2.4 3.240
35
30
25
20
15
10
5
0
Numerical solution
Constant solution
Exact solution
Depth [km]
EM
ice
ab
sorp
tio
n [
dB
]
Figure 4: Electromagnetic absorption of ice compared at two drilling sites
The total absorption rate passing through the ice column is:
)(][][ hLL dBdB
TOT 14)
where h is the ice thickness. The total electromagnetic absorption of the ice calculated by means of the
ice core measurements of conductivity and temperature profile at EPICA is -17.4 dB while the same value
at GRIP is -26.2 dB. The electromagnetic power loss is greater in Greenland because of the higher surface
and inner glacier temperatures compared to Antarctica. The new solution proposed for the electromagnetic
absorption gives reduced ice loss as the result, justifying the excessive received signal obtained from the
radar equation, as noted by several authors (see for example Fujita 2012 [13]). Moreover, this new solution
gives higher values for interface reflectivity than those obtained using other solutions.
Another important parameter for the media is the electromagnetic absorption rate A[dB]
(z) in dB/m which is
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defined as:
)(
)(
2
)(686.8
)(686.8
)()(
0
0
][][
zr
zrz
dz
zd
dz
zdLzA
dBdB
,
15)
where the minus sign takes into account that A is an absorption loss rate, negative in logarithmic unit. In
these equations there is no difference between exact and constant solution demonstrating the strictly
connection between absorption rate in dB and the media parameters.
0 0.5 1 1.5 2 2.5 320
15
10
5
0
GRIP
EPICA
Depth [km]
EM
ab
sorp
tio
rate
[d
B/k
m]
Figure 5: Ice Absorption rate at the two sites.
In figure 5 the ice absorption rate at the two sites has been plotted using the scaled conductivity profile. It
is clear that the electromagnetic absorption of ice is quite different at the two sites in the first part. This is
probably due to the different surface temperatures and the corresponding difference in temperature profile.
At GRIP a constant absorption rate with an average value of -9.4 ± 0.7 dB/km is observed from 0 to about
1.6 km, while at EPICA a slowly diminishing trend with an average value of -3.2 ± 0.8 dB/km is observed
for the first 1.5 km. From 1.6 km to the bottom, a net diminishing trend is visible, at both the sites. This
particular shape is due to the values of corrected conductivity and temperature that at those ranges have
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similar values.
In the proposed solutions the dielectric permittivity was assessed using a constant value of r = 3.18 for
the ice, but it would also have been possible to use the more general expression r = 3.1884 + 9.10-4
·T (here
the temperature T is in Celsius) (Mätzler and U. Wegmüller, 1987 [26]). The influence of temperature on
dielectric permittivity, and thus on the electromagnetic absorption of ice, was evaluated for the whole
temperature range at both sites and its contribution was found to be negligible. At EPICA, for example, it is
less than 0.2 dB/km, less than the difference between the constant and exact solutions.
The calculated absorption rate can be tested more accurately by selecting and analyzing RES
measurements collected over subglacial lakes located in the area of the ice core, in the hypothesis of a
constant ice absorption rate. This has been done in the area nearby the Concordia Station (EPICA drilling)
considering subglacial lakes at depths between 2960 m and 4500 m. Taking into account the widespread
locations of the lakes and their wide depth range, the averaged ice absorption rate is Am= -7.2 ± 1.4 dB/km
(Zirizzotti et al. 2014 [53]). Furthermore, these results are also comparable with the value A[dB]
=-8.1 ± 2.4
dB/km obtained using equation 13 from 2900 m (see figure 5). It is also similar to the average value of -7.2
± 0.7 dB/km related to ice absorption from bedrock reflections obtained in the same area by Zirizzotti et al.,
2010 [51]. These values are in quite close agreement, within the errors range, even with the ice absorption
rate value of -4.9 ± 2.6 dB/km (by Zirizzotti et al., 2012[52]) calculated from RES measurements in the
Dome C area also taking into account the attenuation of echoes due to internal layers located at depths
between 500 and 2600 m (from figure 4: A[dB]
(500)= -3.0 dB/km and A[dB]
(2600)= -9.9 dB/km).
In Greenland, based on RES measurements, absorption rate values of 14 dB/km and 24 dB/km have been
found at ice thickness of 3000 m and 1000 m respectively (Oswald and Gogineni 2008 [32]). In particular,
at GRIP camp, using ice absorption electromagnetic measurements, a value of 14 dB/km for the first 100 m
of ice was obtained (Paden 2005 [33]). This last value is quite similar to the value of 9 dB/km obtained by
DEP conductivity measurements at GRIP station (figure 5).
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SUMMARY AND CONCLUSIONS
The electromagnetic absorption of ice is an important physical parameter that plays a significant role in
the definition of the physical conditions at ice bottom, through RES data analysis. In this paper some
consideration about ice conductivity are briefly reported. The propagation of electromagnetic waves
through the ice sheet is examined and a new method to assess the electromagnetic absorption rate profile of
ice from ice core drilling measurements is proposed and discussed. The rate of variation of the ice
absorption is deduced starting from the analysis of ice core data from two sites where the electromagnetic
properties are known (Concordia station in Antarctica, and the GRIP site in Greenland). First, ice
conductivity profiles coming from the EPICA and GRIP drilling sites were used to calculate the exact
solution of electromagnetic wave propagation using the WKB approximation (generally valid for polar ice
conditions). The exact solution was compared to the numerical solution of the wave equation, and to the
generally used solution, valid only in the case of constant ice electromagnetic parameters. The comparison
revealed no differences between the numerical and exact solutions, while appreciable differences were
observed from the comparison with the generally used solution. The ice absorption obtained using this new
method gives lower attenuation values. Moreover, due to the dependency of ice absorption on temperature,
the attenuation of the radar signal results higher values in Greenland. This can be justified taking into
account that here the ice surface temperature is higher than in Antarctica, maintaining a similar linear trend
between 0 and 1.6 km at both sites, but with a difference in rate absorption of about 10 dB. The total
electromagnetic absorption of the complete ice column is -17.4 dB at the Concordia site, while at the GRIP
site the absorption is -26.2 dB.
Since the research of “oldest ice” (Fisher et al., 2013 [10]) in Dome C area could represent a new important
scientific challenge, this method could be helpful to define the characteristics of the ice absorption, to
establish ice-bottom reflectivity, and to obtain information about dry and wet bedrock interfaces needed for
the ice core site selection.
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Acknowledgements: This research was conducted and funded as part of the framework of the "Progetto
Premiale ARCA".
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21
FIGURES:
0 1 103
2 103
3 103
0
10
20
30
40
100
80
60
40
20
0
DEP
scaled DEP
Temp
Depth [m]
Con
du
ctiv
ity [
µS
/m]
Tem
p.
[°C
]
Figure 1: Measured ice conductivity (258), scaled conductivity and temperature at the EPICA drilling site
0 1 103
2 103
3 103
0
10
20
30
40
100
80
60
40
20
0
Depth [m]
Con
du
ctiv
ity [
µS
/m]
Tem
p.
[°C
]
Figure 2: Measured ice conductivity (258), scaled conductivity and temperature at the GRIP drilling site
22
55 50 45 40 35 30 25 20 15 10 5 01
10
100
EPICA Concordia
GRIP
Lab. meas.
temp. [°C]
Con
du
ctiv
ity [
µS
/m]
Figure 3: Ice conductivity at the two sites vs. ice temperature.
23
0 0.8 1.6 2.4 3.240
35
30
25
20
15
10
5
0
Numerical solution
Constant solution
Exact solution
Depth [km]
EM
ice
abso
rpti
on
[d
B]
0 0.8 1.6 2.4 3.240
35
30
25
20
15
10
5
0
Numerical solution
Constant solution
Exact solution
Depth [km]
EM
ice
ab
sorp
tio
n [
dB
]
Figure 4: Electromagnetic absorption of ice compared at two drilling sites
24
0 0.5 1 1.5 2 2.5 320
15
10
5
0
GRIP
EPICA
Depth [km]
EM
ab
sorp
tio
rate
[dB
/km
]
Figure 5: Ice Absorption rate at the two sites.