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\begin{document}


\pagestyle{plain}

\begin{frontmatter}

%%%\begin{center}

%{\Large {\bf Theoretical Bounds and Microstructural Inversion of the 
%Complex Permittivity of Sea Ice from Low Frequency }} \\

%{\Large {\bf Complex Bounds and Microstructural Recovery}} \\

%\vspace{1ex}

%{\Large {\bf from Measurements of Sea Ice Permittivity  }} \\



\vspace{-8ex}




\title{A Network Model for Electrical Transport 
in Sea Ice}

\vspace{2ex}

\author%[utah]
{J. Zhu}\ead{zhu@math.utah.edu},
\author%[utah]
{K. M. Golden}\ead{golden@math.utah.edu},
\author%[utah]
{A. Gully}\ead{gully@math.utah.edu},
and
\author%[utah]
{C. Sampson}\ead{christian.sampson@gmail.com}

%\vspace{1ex}

%{\Large {\bf Complex Permittivity of Sea Ice}}

%\vspace{1ex}

%{\Large {\bf Comparison of Low Frequency Measurements   }} \\

%\vspace{1ex}

%{\Large {\bf of the Complex Permittivity of Sea Ice with }}  \\

%\vspace{1ex}

%{\Large {\bf Theoretical Forward and Inverse Bounds }}


%\vspace{4ex}

\address%[utah]
{University of Utah, Department of Mathematics \\
155 S 1400 E RM 233, Salt Lake City, UT, 84112-0090 USA}


%\noindent
%{\large {\bf A. Gully\footnote
% {Univ. of Utah, Department of Mathematics,
%  155 S 1400 E RM 233, Salt Lake City, UT 84112-0090 USA}, 
% L. G. E. Backstrom\footnote{Geophysical Institute, P.O. Box 757320, 
%903 Koyukuk Drive, 
%Univ. of Alaska, Fairbanks, AK 99775-7320 USA}, 
%H. Eicken$^{2}$, and K. M. Golden$^{1}$  }}


\maketitle


%\end{center}


\vspace{-1ex}


\begin{abstract}


We develop a network model for the 
electrical conductivity of sea ice, and obtain close
agreement with the results of experiments we conducted in 
Antarctica. 
Monitoring the thickness of sea ice is an important tool
in assessing the impact of global warming on Earth's polar regions,
and most methods of measuring ice thickness 
depend on detailed knowledge of its electrical properties.










\end{abstract}
\begin{keyword}
sea ice \sep electrical conductivity \sep network model 
\sep sea ice thickness \\
{\it PACS Numbers:} 92.10.Rw, 
%Sea ice (mechanics and air/sea/ice exchange processes),
77.84.Lf,
%Composite materials 
77.22.Ch,
%Permittivity (dielectric function)   
72.80.Tm, 
%Inverse problems 
%92.70.Mn,
%Impacts of global change; global warming
93.85.Jk,
%Magnetic and electrical methods 
93.30.Li,
%Arctic Ocean 
93.30.Ca
%Antarctica        
\end{keyword}




\end{frontmatter} 



\section{Introduction}


Sea ice is a sensitive indicator of climate change, as well 
as a critical component of Earth's climate system. 
Determining the thickness distribution of the polar sea ice 
packs is a central problem in monitoring the impact of 
global warming. However, there is significant uncertainty
in our knowledge of the ice thickness distribution and how it 
is changing. Not only does this uncertainty affect assessments
of how the changing climate is impacting the polar regions, but it
also affects predictions of global climate models, where accurate
knowledge of sea ice initial conditions is essential for long term
simulations.


Most methods of measuring sea ice thickness, and interpretation
of the data obtained, depend on detailed
knowledge of the electrical properties of the ice. 
Since sea ice is
a composite of pure ice with brine inclusions 
\cite{Weeks:1986:GSP,Eicken:GMP-02},
whose volume fraction
and geometry depend strongly on temperature, understanding 
its electrical properties is a challenging problem in the 
theory of inhomogeneous materials. Here we develop a
network model for the electrical conductivity of sea ice, and 
compare the results with direct measurements of the vertical 
conductivity of first year sea ice we made during the 
2007 Sea Ice Physics and Ecosystem eXperiment
(SIPEX) expedition off the coast of East Antarctica,
from the Australian icebreaker {\it Aurora Australis}. 



Early DC resistivity measurements on sea ice were aimed at determining ice thickness \cite{Fujino:1963:LTS,Thyssen:1974:P,Timco:1979:JG}. Initially all these studies employed surface soundings using 4 electrodes in either the Wenner or Schlumberger configurations, although Timco \cite{Timco:1979:JG} later used vertically arranged electrodes in the side of an ice pit. Later measurements in the Antarctic were also reported \cite{Buckley:1986:ISM}. 
The anisotropic nature of the resistivity of sea ice, due to the preferential vertical alignment and connectivity of brine pores, leads to such measurements significantly underestimating the ice thickness.

More promising determinations of sea ice thickness have been achieved using low frequency electromagnetic (EM) techniques \cite{Haas:1997:G,Haas:1998:CRST,Haas:2004:GRL,Worby:1999:CRST,Reid:2006:AG}. The technique relies on a time varying primary magnetic field (generated by a transmitter coil) inducing eddy currents in the seawater beneath the comparatively resistive ice. The secondary magnetic field produced is sensed by a receiver coil, determining an apparent conductivity which results essentially from an integration over the vertical distance between the instrument and induced currents. The thickness is found using empirical relationships \cite{Haas:2003:DVT}, with good results for smooth ice and underestimates near ridges \cite{Haas:2003:DVT}. The technique is adaptable to continuous measurements being made either from a helicopter or ship \cite{Haas:1998:CRST}. 




Previous measurements of the conductivity of sea ice relied almost 
exclusively on indirect methods which mix the horizontal and 
vertical components. Moreover, these indirect means make it
difficult to accurately recover the dependence of the conductivity
on the properties of the brine microstructure, namely, its
brine volume fraction $\phi$, which depends on the temperature
$T$ and salinity $S$ of the ice 
\cite{Frankenstein:JG-943,Weeks:1986:GSP,Eicken:GMP-02}. 
During the 2007 Australian SIPEX expedition, Golden and Gully
extracted cylindrical cores of sea ice and made vertical conductivity
measurements along these cores using metal probes attached
to a Yew Earth Resistance Tester, as described in 
\cite{Golden:2010:DSR}. We also measured salinity and temperature 
along each core in order to relate the electrical measurements
to microstructural data 
\cite{Perovich:JGR-18327,Golden:S-2238,Golden:2007:GRL,%
Pringle:JGR:2009}. 

Part of our motivation for focusing on the vertical component
of the electrical conductivity is that it is closely related
to the vertical component of the fluid permeability of sea ice.
Fluid transport in sea ice mediates a broad range of processes
such as the growth and decay of seasonal ice, the evolution
of melt ponds which determine ice pack albedo, and bio-mass
build-up \cite{Golden:2007:GRL,Golden:NAMS:2009}. Our work
here will help lay the foundation for electrically
monitoring fluid transport in sea ice. In fact, 
the random resistor network
model we develop here is based on the random pipe network 
we used previously to model the fluid permeability 
of sea ice \cite{Zhu:2006:AG}. Statistical information 
about the brine microstructure 
\cite{Perovich:JGR-18327,Golden:S-2238,Golden:2007:GRL,%
Pringle:JGR:2009}
is used to determine the statistical distributions
of the resistors in the electrical network.





\section{The Network Model for the Effective Conductivity of Sea Ice}

In this model, we consider a piece of sea ice with an averaged brine volume fraction $\phi$, and focus on the effect of brine structure on the electric conduction of the material. The conducting property of a medium can be summarized by its conductivity.
Let $\Phi$ be the electric potential, and $\bsig$ the local 
conductivity tensor, 
which may depend on the local volume fraction, temperature, and salinity. Since the current density $\JJ$ is related to the
electric potential through $\JJ = \bsig \nabla \Phi$, using Kirkoff's law and assuming the material is free of electrical charge, 
the equation for electrical conduction is
\be                \label{conduction}
\nabla \cdot {\bsig} \; \nabla \Phi = 0,
\ee
which is similar to the fluid pressure equation resulting from Darcy's law
\be                \label{permeability}
\nabla \cdot {\kk} \; \nabla p = 0, 
\ee
where $p$ is the the flow pressure and $\kk$ is the permeability tensor. 

Here we intend to define an effective conductivity that describes the behavior of conduction of the sea ice structure 
in the direction that is of importance to us. In particular, we are interested in the effective vertical
conductivity defined through
\be
J^\ast = \sigma_v \; \frac{\Delta \Phi}{\Delta z}
\ee
for the current density in $z$ direction $J^\ast$, and the potential difference $\Delta \Phi$ over a thickness $\Delta z$.

To simulate the electric field through the conducting microstructure
of sea ice, consider an ice sheet of depth $D$. We will model the medium in the way that is analogous to
our previous work \cite{Zhu:2006:AG}. Take a thin
vertical slice of horizontal thickness $h$ 
and length span $L$. 
We model this ice sheet by a two dimensional
lattice of nodes connected by conducting tubes.
The slice has a rectangular $L \times D$ vertical cross section, 
which is divided into a grid with $m$ equally spaced sections  
in the horizontal direction and $n$ equally spaced sections 
in the vertical direction, so that
$L/m = D/n = h$, for some large integers $m$ and $n$. 
The model parameter $h$ can be viewed
as the dimension of a cell in which a typical brine 
inclusion is contained. 
In this network model, $h$ will be chosen according 
to the sea ice we simulate, its brine volume fraction, 
and our computing capacities.
The tubes are assumed to
have circular shapes with different radii, and the 
current through the medium is induced by
an electric potential drop $\Phi_b  - \Phi_t$, where $\Phi_b$ and $\Phi_t$ are
the potentials at the bottom (liquid) and the top (air) of 
the sea ice, with the assumption that
$\Phi_b > \Phi_t$ so there is an upward current flow in the medium. 
The cross sectional areas of the tubes chosen below  
generate fluid pores comparable to the brine 
inclusions found in young sea ice.
The lattice nodes are the vertices 
$(i,j)$, $0 \leq i \leq m, \; 0 \leq j \leq n,$
of a rectangular grid, as in Figure \ref{pipes}  (a).
Nearest neighbors are 
connected by vertical and horizontal tubes,
with a potential $\Phi_{i,j}$ defined at each node $(i,j)$.
To each node $(i,j)$ with 
$0 \leq i \leq m-1, \; 0 \leq j \leq n-1$,
the horizontal tube to the right of $(i,j)$ 
has radius $R=R_{i,j}^{h}$, and the vertical tube on top of $(i,j)$ 
has radius $R=R_{i,j}^{v}$, as shown in Figure 1.
Along the right edge with $i=m$ the nodes have one vertical
tube, and along the top edge the nodes have one horizontal tube
(except the last).





Since the brine conductivity is substantially higher than the conductivity of the surrounding ice (on an order
of about $10^8$), we can
assume that electrical conduction takes place mostly through the brine tubes. The effect of conduction through pure
ice will be modeled by adding a simple conducting component to the system. Unlike the permeability
model, where the fluid flux depends only on the brine geometry, the electric conduction in the microstructure
would include a temperature dependent local conductivity.
For each tube of radius $R$ connecting two nodes, we assume an uniform conductivity
$\sigma_{tube}$ for the brine, so an electric current 
within can be established based on the voltage drop and the cross sectional area. In our particular case
we have
\be
{\bf I} = \sigma_{tube} \;A \;{\bf E} =  - \sigma_{tube} \;\pi R^2 \;\nabla \Phi,
\ee
where $\Phi$ is the electric potential.
For each tube connecting
two neighboring nodes, the potential gradient can be
well approximated by the difference
between the potentials at these nodes, divided 
by the spacing $h$.
%For each node $(i,j)$ in the network system, 
%there are two vertical (up and down, indexed
%by $(i,j+\half)$ and $(i,j-\half)$)
%and two horizontal (left and right, 
%indexed by $(i-\half,j)$ and $(i+\half,j)$) ducts 
%connected to it, each with a different 
%radius. 
Given the potentials at neighboring nodes,
different fluxes converging to the node $(i,j)$ 
can be easily computed, and they must balance 
due to Kirkoff's law. Let $\sigma_{i,j}^h$ and $\sigma_{i,j}^v$ denote the brine conductivity for the tubes
to the right and on the top of node $(i,j)$, respectively.
This leads to the following equations,
%\be
%R_{i,j+\half}^4 (p_{i,j+1} - p_{i,j}) + 
%R_{i,j-\half}^4 (p_{i,j-1} - p_{i,j}) + 
%R_{i+\half,j}^4 (p_{i+1,j} - p_{i,j}) 
%+ R_{i-\half,j}^4  (p_{i-1,j} - p_{i,j}) = 0,  \label{eq:pres}
%\ee
\begin{align}
\sigma_{i,j}^v (R_{i,j}^{v})^2 (\Phi_{i,j+1} - \Phi_{i,j}) &+
\sigma_{i,j-1}^v (R_{i,j-1}^{v})^2 (\Phi_{i,j-1} - \Phi_{i,j}) \; +   \nonumber   \\ 
\;\;\;\;\; \sigma_{i,j}^h (R_{i,j}^{h})^2 (\Phi_{i+1,j} - \Phi_{i,j}) &+
\sigma_{i-1,j}^h (R_{i-1,j}^{h})^2  (\Phi_{i-1,j} - \Phi_{i,j}) = 0,  \label{eq:cur_bal}
\end{align}
for $ i=1, \ldots, m-1$, and $ j=1,\ldots,n-1$,
with appropriate modifications on the edges of the lattice.
Notice that this equation is similar to the equation derived for the fluid permeability model \cite{Zhu:2006:AG}:
\begin{align}
 (R_{i,j}^{v})^4 (p_{i,j+1} - p_{i,j}) &+
(R_{i,j-1}^{v})^4 (p_{i,j-1} - p_{i,j}) \; +   \nonumber   \\ 
\;\;\;\;\; (R_{i,j}^{h})^4 (p_{i+1,j} - p_{i,j}) &+
(R_{i-1,j}^{h})^4  (p_{i-1,j} - p_{i,j}) = 0,  \label{eq:pres}
\end{align}
here $p_{i,j}$ is the pressure at node $(i,j)$. We comment that the current system has coefficients dependence
on the radius not as strong as that in the permeability case, but they also have an extra dependence on the local brine conductivity,
which varies from one location to another due to temperature and salinity variations.





\begin{figure}                \centering
\begin{frontmatter}
\includegraphics{resistor_network.eps}
   \caption{(a)  (b)}
      
    \label{resistor_network}
   
\end{frontmatter}     
\end{figure}







At the top of the region ($j=n$), we assume the boundary condition
\be
\Phi_{i,n} = \Phi_t,
\ee
and at the bottom ($j=0$) of the region
\be
\Phi_{i,0}=\Phi_b.
\ee
Let $I_{i,j}$ be the current through the vertical duct on top
of the $(i,j)$ node.
The total current through the brine network system is therefore
\be
I_{brine} \; = \; \sum_{i=0}^m I_{i,n-1} \; = \; \pi\;
\sum_{i=0}^m \; \sigma_{i,n-1}^v (R_{i,n-1}^v)^2 \;\; \frac{\Phi_{i,n-1} - \Phi_t}{h} \; ,
\ee
where $R_{i,n-1}^{v},\ i=0,\ldots,m$ are the radii 
of the tubes connected to the top of
the sea ice, and $\Phi_{i,n-1}$ are
the potentials at the nodes just below the top surface. 

The effect of conduction through pure ice can be modeled as an additional current flow through
another medium, in parallel to the brine network. The current through such a medium is
\be
I_{pure\ ice} = \sigma_{ice} \; L h (1-\beta(\phi))  \frac{\Phi_t - \Phi_b}{D}.
\ee
Here we introduce a function $\beta(\phi)$ that describes the loss of ice surface for conduction due
to the brine inclusions. 

When the piece of sea ice is viewed as a composite, 
the effective vertical conductivity $\sigma_v$ 
can be defined through
\be
J^\ast =  -\sigma_v \; \frac{\Phi_t - \Phi_b}{D}
\ee
Where $J^\ast$ is the average current density through the medium. 
We make the connection between
$J^\ast$ and the total current through
\be
J^\ast= \frac{I_{brine} + I_{pure\ ice}}{L h} 
\ee
where $L h$ is the horizontal cross sectional area of the slice.  
%From now on we choose
%the thickness $d=h$, the grid size. 
Based on these assumptions, we
have the effective conductivity
\be
\sigma_v \; = \; \frac{\pi D}{ L h^2 } \;  
\sum_{i=0}^m \; \sigma_{i,n-1}^v (R_{i,n-1}^v)^2 \; \frac{\Phi_t - \Phi_{i,n-1}}{\Phi_t - \Phi_b} + (1-\beta) \sigma_{ice}.
\label{eq:eff_num}
\ee
The cross section reduction form for the ice should depend on the microstructure of the sea ice under study.
One possible model is to assume that half of the tubes are vertical and contributing to the vertical conduction.
This leads to the following reduction factor:
\be
\beta = \hf \phi.
\ee

The multigrid algorithm to solve the system Eq.(\ref{eq:pres}) can be easily modified to solve the system
Eq.(\ref{eq:cur_bal}), and the convergence is faster due to the coefficient dependence change from $R^4$ to
$R^2$.

\section{Sea Ice Microstructure and Numerical Results}

 
In this model, the microstructure of the sea ice slice is described
by an averaged brine conductivity $\sigma_b$, and a collection of tubes with cross sectional area
 sampled from a log-normal distribution, with parameters based 
on measurements of brine inclusions in first year sea ice
\cite{Perovich:JGR-18327,Light:2002:ETM,Bock:2005}. 
Specifically, we sample the radius $R$ so that
$
\log A = \log ( \pi R^2 ) 
$
is normally distributed with mean $\mu$ and variance $\sigma^2$. We also assume that all the random radii 
are independent from each other. 
Given a particular sample of the tube radii, the volume 
fraction $\phi$ of the slice can be readily
computed by
\be
\phi = \frac{\pi}{L D} 
\left( \sum_{i=0,j=0}^{m-1,n} (R_{i,j}^h)^2 + \sum_{i=0,j=0}^{m,n-1} 
(R_{i,j}^v)^2 \right). \label{eq:phi_struct}
\ee
%where the $i=m$ and $j=n$ cases in the sum are treated appropriately.
The goal of this study is to investigate the dependence 
of the effective vertical conductivity $\sigma_v$, and the form factor $\sigma_v/\sigma_b$,
on the porosity $\phi$, which is connected to the microstructure through Eq.(\ref{eq:phi_struct}).
For consistency, it is necessary to choose the parameters $\mu$ and $\sigma$ such
that the desired volume fraction is arrived, and that the statistical properties of the actual
sea ice are reasonably matched. To this end, we first notice that the expected value of the cross sectional area
\be
E[ A ] = e^{\mu + \hf \sigma^2},
\ee
given our assumption about the distribution of $\log A$. This expected value for the model should
be matched to an interpolation of measured averages for the cross sectional
area $A$ as a function of brine volume fraction $\phi$ \cite{Golden:2005}:
\be              \label{avg_area}
\langle A \rangle \; = \; \theta(\phi) =  \pi (7 \times 10^{-5} + 
1.6 \times 10^{-4} \phi)^2 \;\;\; \mbox{m}^2.
\ee
This function approximates the dependence of the mean cross sectional area on the brine
volume fraction $\phi$ observed by Perovich and Gow (1996) \cite{Perovich:JGR-18327} in horizontal thin sections of
young, primarily columnar sea ice. Since we have two parameters for the model to be determined, 
the matching condition leaves us with one free parameter $\sigma$, which
we will choose several different values to compare with the measured sea ice data. It should
be pointed out that the tomography information on the microstructure is rather insufficient and the distribution of the
brine inclusions as reflected through $\mu$ and $\sigma$ still requires substantial modeling. Therefore we will 
leave the parameter $\sigma$ open for further modeling. As observed in \cite{Perovich:JGR-18327}, values of $\sigma$
between $1$ and $2$ seem
to be reasonable for the available topography information.
%As a crude way to model observed anisotropy in the medium
%\citep{Freitag:1999:HPA}, we scale the electric currents in the 
%horizontal ducts by a factor of 1/10, although for 
%the current model the results are not very sensitive
%to this factor. 

Also as observed in \cite{unknown}, brine channels in the sea ice become connected to a substantial level only when
the brine volume fraction reaches above $5\%$. To reflect this behavior, we allow some randomly selected
tubes to be disconnected from the system in an effort to simulate the disconnection of brine 
inclusions along certain directions. The capability of the multigrid algorithm makes it
possible to study the general percolation problem from a new perspective.
For the numerical algorithm, we choose to develope a multigrid algorithm especially targeted at 
this unusual linear system. Multigrid methods in general are very powerful in dealing with large 
linear systems resulting from elliptic equations, and they are particularly appropriate here  to 
address different length scales involved in the microstructure. Our numerical examples demonstrate
that this algorithm is robust, accurate, and efficient.

Similar to the work of fluid permeability, we will focus on the form of dependence of the effective vertical 
conductivity of the sea ice on the brine volume
fraction. In particular, we hope to establish a scaling law that relates the form factor $\sigma_v/\sigma_b$
 to $\phi^m$ for certain $m>0$.
As an initial attempt, we assume a simple salinity dependence on the temperature, and express the
local brine volume fraction as a function of the local temperature alone. By assuming a temperature profile
across the depth of the sea ice slab, we obtain a variable brine volume fraction as a function of the depth.
Given a brine volume fraction for a slab of
sea ice, we will impose the structure of the local brine inclusions: whether they are mostly
vertically or horizontally connected, by assigning a probability of disconnection to each tube. 
This extra variable is clearly temperature dependent and we will model this by varying this probability
according to the overall brine volume fraction. 

To illustrate this, we consider a node $(i,j)$ with vertical depth $z_j$ (the distance from the sea ice surface).
The local brine conductivity $\sigma_{i,j}$ is calculated according to the following procedure:
first the temperature is determined from a measurement $T(z)$ taken from one of our data sets, 
which in turn determines a local brine conductivity via a relation derived by Stogryn and Desargant \cite{unknown}:
\be
\sigma_{i,j} = - T(z_j) \exp \left(0.5193+0.08755\;T(z_j) \right), \ \  \ T(z_j) \geq -22.9 \ ^\circ C.
\ee
The overall brine conductivity $\sigma_b$ is taken to be the average of all the node values. We note that this
particular temperature profile corresponds to one particular value of $\sigma_b$, at one particular volume fraction
$\phi$. In order to compute the effective conductivity at various values of $\phi$, temperature and salinity profiles specific
to the value of $\phi$ are required. Unfortunately, temperature and salinity information measured so far do not allow for
such specific studies. For the current work, we fix this temperature profile, which in turn fixes the local brine conductivity
profile. At the same time, Eq.(\ref{eq:cur_bal}) is invariant with a constant factor in $\sigma_{i,j}$, so the use of the same
$\sigma_{i,j}$ for different calculations with different values of $\sigma_b$ does not lead to contradiction, as long as the
assumption of the local brine conductivity variation resulting from that particular temperature profile is reasonable.

We also use an average pure ice conductivity value of $\sigma_{ice} = 1.1 \times 10^{-8}$ at a temperature $T=-10 \ ^\circ C$.
Here all the conductivity quantities have the unit $\mbox{Ohm}^{-1} \cdot \mbox{m}^{-1}$. In Table 1, the probabilities of
vertically  disconnected tubes are chosen for different brine volume fractions, based on the tomography information.
The horizontal tubes all stay connected in this paper, for the reasons that conduction in horizontal direction is less important
and that the numerical algorithm will encounter difficulties in some of the random configurations.

\begin{table}
\caption{Tube Disconnection Probability}

\vspace{1ex}

\begin{center}
\begin{tabular}{|c|r|r|r|r|r|} \hline
$\phi$ & 0.025 & 0.05 & 0.075 & 0.1 & 0.0125 \\ \hline 
$Prob$(disconnection) & 0.7 & 0.5 & 0.3 & 0.15 & 0 \\ \hline
\end{tabular}
\end{center}
\end{table}






In Figure 2, we show the calculated values of the effective vertical conductivity according to Eq.(\ref{eq:eff_num}),  with two
different choices for the brine inclusion distribution parameter $\sigma$, as compared to
the measured vertical conductivity of the sea ice. Obviously the numerical results from the choice $\sigma=1$ agree
much better with the measurements than that of $\sigma=0.5$. This seems to agree with the findings about the observed 
statistics of brine inclusions of Perovich and Gow \cite{Perovich:JGR-18327}.

A more fundamental quantity for our study is the form factor, the relative conductivity defined as $\sigma_b / \sigma_b$. In Figure 3, we plot the values of the form factor for our 
computed vertical conductivity and our measured sea ice conductivity in a log-log graph. 
Of major interest from the graph is an estimate of the exponent $m$ and to
see how our model agrees with the measured quantities. From this graph,
with the choice of disconnect probability, we have achieved an excellent agreement for $m$ around $2$. A fine tuning of
the various parameters to reflect the microstructure if more detailed tomography information is available should lead to
more accurate predictions of the exponent $m$.





\begin{figure}                \centering
\begin{frontmatter}
\includegraphics{data_comparisons.eps}
   \caption{(a)  (b)}
      
    \label{data_comparisons}
   
\end{frontmatter}     
\end{figure}



































\section{Conclusions}






\vspace{4ex}

\noindent 
{\bf Acknowledgements:} 
We are grateful for the support
provided by the Division of Atmospheric
Sciences and the Division of Mathematical Sciences 
at the National Science Foundation,
through the Collaboration in Mathematical Geosciences
(CMG) Program. Adam Gully 
gratefully acknowledges
the support of the NSF Research Experiences 
for Undergraduates (REU) Program and an
NSF VIGRE graduate fellowship. 



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