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\begin{center}

{\bf TECHNICAL PROPOSAL}

\vspace{2ex}

ONR BAA Announcement Number 12-001

\vspace{5ex}



ONR Arctic \& Global Prediction Program \\

\vspace{1ex}

and \\

\vspace{1ex}

ONR Applied Computational Analysis Program
     


\vspace{10ex}





{\large {\bf Multiscale Models of Melting Arctic Sea Ice}}


\vspace{8ex}

Principal Investigator: Kenneth M. Golden\footnote{University of Utah, 
Department of Mathematics, 155 S 1400 E, RM 233, Salt Lake City, UT 84112-0090,
Tel: 801-581-6176, Fax: 801-581-4148, Email: golden@math.utah.edu}
%U. of Utah, Math



%, Ph.D.  \\ Professor of Mathematics \\




\vspace{4ex}


Co$-$Principal Investigators: Donald K. Perovich\footnote{ERDC -- CRREL,
72 Lyme Road, Hanover, NH 03755,
Tel:  603-646-4255, Fax: 603-646-4561, Email: donald.k.perovich@usace.army.mil} \\
%CRREL \\
\hspace{26.2ex} Courtenay Strong\footnote{University of Utah,
Department of Atmospheric Sciences,
135 S 1460 E, RM 819,
Salt Lake City, Utah 84112-0110,
Tel:  801-585-0049, Fax:  801-585-3681 , Email: court.strong@utah.edu } \\
%U. of Utah, Atmospheric Sciences\\
\hspace{20.4ex}
Bacim Alali\footnote{University of Utah, Department of Mathematics, 
155 S 1400 E, RM 233, Salt Lake City, UT 84112-0090,
Tel:  585-9113, Fax:  801-581-4148, Email: alali@math.utah.edu}

%, Ph.D. \\ Assistant Professor of Mathematics \\


\vspace{4ex}

Senior Personnel: 
Tolga Tasdizen\footnote{Scientific Computing and Imaging Institute,
University of Utah,
72 S Central Campus Drive, 3750 WEB, Salt Lake City, UT 84112,
Phone: 801-581-3539, Fax: 801-585-6513,
Email: tolga@sci.utah.edu } \\
\hspace{27ex}
Christopher Polashenski\footnote{ERDC -- CRREL,
72 Lyme Road, Hanover, NH 03755,
Tel:  570-956-6990, Fax: 603-646-4561, Email: chris.polashenski@gmail.com} \\
%CRREL \\






\vspace{6ex}

Administrative/Business Contact:  
Shauna Peterson\footnote{1471 East Federal Way, 
Salt Lake City, UT 84112-0090,
Tel: 801-585-0062,
Fax: 801-581-3007,
shauna.peterson@osp.utah.edu}

\vspace{6ex}

{\large December 1, 2012 -- November 30, 2015}







\end{center}


\vspace{0ex}


~~~~~~


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\section{Technical Approach}


\subsection{Introduction}









\begin{figure}                \centering
\includegraphics[width=6.5in]{melt_pond_area_critical_behavior.eps}
   \caption{Arctic melt pond coverage vs. calendar date near Barrow, AK in (a) 
          and melt pond coverage vs. days
      since onset of pond formation in (b), 
      for the years 2008--2010 \cite{Polashenski:JGR:2012}.
     Average albedo vs. calender date in 2000 near Barrow is shown in (c).  
     Each data set exhibits critical behavior at the onset of melt pond formation, 
     similar to the behavior of order parameters characterizing phase
     transitions in statistical physics and composite materials. }
%       Images courtesy of Don Perovich.  }
     
    \label{melt_pond_area_critical_behavior}
        
\end{figure}







During the melt season the Arctic sea ice cover 
becomes a complex, evolving composite of ice, 
melt ponds, and open water. Melting is strongly 
influenced by the morphological characteristics 
of the ice cover, such as the size and shape 
of melt ponds and ice floes. Albedo, light 
transmission, and melting are closely connected 
to melt pond characteristics, while floe perimeter 
is a primary controlling parameter for lateral melting. 
We propose to develop multiscale mathematical 
descriptions of the melting sea ice pack, focusing
on the geometry and evolution of surface melt ponds.
Models of phase transitions in 
statistical physics and composite materials will be
used to investigate melt pond structure.
This work will improve 
our ability to model the partitioning of solar 
radiation between reflection, absorption in 
the ice, surface melting, bottom melting, 
lateral melting, and heat storage in the upper ocean.


 
While snow and ice reflect most incident sunlight, 
melt ponds and ocean absorb most of it. The overall reflectance
or albedo of sea ice is determined by the evolution of melt pond geometry 
\cite{Perovich:2002:JGR,Scott:2010:JGR,Polashenski:JGR:2012} and 
ice floe configurations \cite{Toyota:JGR:2006}.
As melting increases, so does solar absorption, 
which leads to more melting, and so on. This 
{\it ice--albedo feedback} has played a significant 
role in the decline of the summer Arctic 
ice pack \cite{Perovich:GRL-2008}, which
most climate 
models have underestimated \cite{Serreze:S-1533,Boe:2009:NG}.
Sea ice albedo is a significant source 
of uncertainty in climate projections
and a fundamental problem in climate modeling
\cite{Flocco:2010:JGR,Scott:2010:JGR,Pedersen:2009:JGR,Polashenski:JGR:2012}.

From the first appearance of
visible pools of water, often in early June,
the area fraction $\phi$ of sea ice covered by melt ponds
can increase rapidly to over 70\% in just a few days
\cite{Polashenski:JGR:2012,Scharien:TGRS:2005},
as demonstrated in Figure \ref{melt_pond_area_critical_behavior}
(a) and (b). 
Moreover, the accumulation of water at the surface
dramatically lowers the albedo where the ponds form.
A corresponding critical drop-off in average albedo 
is displayed in Figure \ref{melt_pond_area_critical_behavior} (c).
The resulting increase in solar absorption in the ice
and upper ocean accelerates melting 
\cite{Perovich:2003:JGR}, possibly triggering {\it ice-albedo feedback}.
Similarly, an
increase in open water fraction $\psi$ lowers albedo,
thus increasing solar absorption and subsequent melting.
The spatial coverage and distribution of melt ponds 
on the surface of ice floes and the open water between the
floes thus exerts primary control of ice pack albedo and 
the partitioning of solar energy in the ice-ocean system
\cite{Eicken:JGR-C08007,Polashenski:JGR:2012}.
%Given the critical role of ice-albedo feedback in the
%recent losses of Arctic sea ice
%\cite{Perovich:GRL-2008,Eisenman:PNAS-2009},
%ice pack albedo and the formation and evolution of melt ponds
%are of significant interest in climate modeling.



           


While melt ponds  
form a key component of the
Arctic marine environment,
comprehensive observations or theories  
of their formation, coverage, and
evolution remain relatively sparse. 
Available observations of melt ponds show that their areal 
coverage is highly variable, particularly for first year ice 
early in the melt season, with rates of change as high as 
35\% per day \cite{Scharien:TGRS:2005,Polashenski:JGR:2012}.
%\cite{SY2005,PPC2011}. 
Such variability, as well as the influence of many competing 
factors controlling melt pond and ice floe evolution, makes the 
incorporation of realistic treatments of albedo into 
climate models quite  
challenging \cite{Polashenski:JGR:2012}. %\cite{PPC2011}.  
Small and medium scale models of melt ponds 
which include some of these 
mechanisms have been developed 
\cite{Flocco:2007:JGR,Skyllingstad:2009:JGR,Scott:2010:JGR},
and melt pond parameterizations are being incorporated
into global climate
models \cite{Flocco:2010:JGR,Hunke:2010,Pedersen:2009:JGR}.





The surface of an ice floe is viewed here as a two phase 
composite \cite{MILTON:2002:TC}
%\cite{M2002,T2002} 
of dark melt ponds and white snow or ice.
The onset of ponding and the rapid increase in 
coverage beyond the initial threshold 
is similar to critical phenomena in statistical
physics \cite{Christensen:2005,Thompson-88}
and composite materials \cite{Stauffer-92}.
Here we ask if the evolution of melt pond geometry exhibits 
universal characteristics which do not necessarily depend on the details 
of the driving mechanisms in numerical melt pond models.
Fundamentally, the melting of 
Arctic sea ice is a phase transition phenomenon, 
where a solid turns to liquid, albeit on large regional scales 
and over a period of time which depends on environmental forcing 
and other factors. We thus look for features of melt pond evolution
which are mathematically analogous to related 
phenomena in the theories of phase transitions and composite materials.
As a first step in this direction, and a key finding which provides
a principal route of investigation in the proposed work, we consider
the evolution of complexity of Arctic melt ponds.







\begin{figure}                \centering
\includegraphics[width=6.5in]{ONR_Figure_1_sea_ice.eps}
   \caption{ (a) Area -- perimeter data for 5,269 Arctic melt ponds,
             plotted on logarithmic scales.
             The slope transitions from about 1 to 2 at a
              critical length scale of around 100 square meters. 
              (b) Melt pond fractal dimension $D$ as a function of area $A$,
               computed from the data in (a), showing the transition
               to complex ponds with increasing length scale.
             Ponds corresponding to the three red triangles in (a),
             from left to right, are
             shown in (c), (d), and (e), respectively. The transitional
               pond in (d) has horizontal scale of about 30 m. }
     
    %\label{meltponds_fractal_dimension}

     \label{area_perimeter_fractal_dimension}
        
\end{figure}











By analyzing area$-$perimeter data from hundreds
of thousands of melt ponds,
we have discovered an unexpected separation of scales,
where the pond fractal dimension 
$D$
exhibits a transition from 1 to 2 around a
critical length scale of 100 square meters in area
\cite{Hohenegger:TC:2012}, as shown in 
Figure \ref{area_perimeter_fractal_dimension}.
Small ponds with simple boundaries coalesce or percolate 
to form larger connected regions, as shown in 
Figure \ref{melt_pond_geometry} (a).
Pond complexity increases rapidly
through the transition region and
reaches a maximum for ponds larger than 1000 m$^2$ 
whose boundaries resemble space filling curves 
\cite{Sagan-94} with $D \approx 2$.
These configurations affect the complex
radiation fields under melting sea ice,
the heat balance of sea ice and 
the upper ocean \cite{Frey:2011:GRL},
under-ice phytoplankton blooms \cite{Arrigo:S:2012}, 
biological productivity, and biogeochemical processes.






\begin{figure}                \centering
\includegraphics[width=5in]{melt_pond_geometry.eps}
   \caption{(a) The complex geometry of well developed Arctic melt ponds, 
             illutrated in an aerial photo taken 
            August 14, 2005 on the Healy-Oden Trans Arctic Expedition (HOTRAX),
            courtesy of Don Perovich. (b) 
          A melt pond drains through a small seal hole (near the arrow tip).
          The length scale over which melt water drains through this hole is 
          strongly influenced by the length scale of melt pond connectivity. 
          Photo courtesy of Chris Polashenski. }
     
    \label{melt_pond_geometry}
        
\end{figure}








Melt pond evolution also appears to exhibit a 
{\it percolation threshold}, where one phase in a composite
becomes connected on macroscopic scales as some parameter 
exceeds a critical value \cite{Stauffer-92,Christensen:2005}.
An important example of this phenomenon
in the microphysics of sea ice, which is fundamental
to the process of melt pond drainage,
is the percolation transition
exhibited by the brine phase in sea ice,
known as the {\it rule of fives}
\cite{Golden:S-2238,Golden:2007:GRL,Pringle:JGR:2009}.
When the brine volume fraction of columnar sea ice exceeds
about 5\%, the brine phase becomes macroscopically 
connected so that fluid pathways allow flow through
the porous microstructure of the ice.
Similarly, even casual inspection of aerial photos
shows that the melt pond phase of sea ice undergoes a 
percolation transition where disconnected ponds evolve into
much larger scale connected structures with complex boundaries.
Connectivity of melt ponds promotes further melting and 
break-up of floes, as well as horizontal transport
of meltwater and drainage through cracks, leads, 
and seal holes \cite{Scharien:TGRS:2005,Polashenski:JGR:2012},
as illustrated in Figure \ref{melt_pond_geometry} (b).




The rich behavior we observe for melt pond evolution 
is similar to phase transitions in
statistical mechanics \cite{Christensen:2005,Thompson-88,Coniglio:PRL:1989}
and composite materials \cite{Stauffer-92,MILTON:2002:TC}. 
Such systems often exhibit universal critical behavior,
where order parameters like the effective conductivity
of a composite or the magnetization of an Ising ferromagnet
are described near their threshold
by critical exponents depending only on 
dimension and not on the details of the system. 
Developing similar theories for melt ponds 
could provide fundamental new insights into the 
evolution of ice pack albedo and Arctic melting, as
well as a rigorous framework for finding 
large scale, effective properties from local information,
a fundamental issue in climate modeling. 
The application of modern theories 
of homogenization and statistical physics to 
melt pond and albedo evolution will lend insights into
parameterizing sub-grid scale processes
into sea ice and climate models.
For example, the scale of connected
fluid pathways on the surface influences melt pond drainage
through cracks, seal holes, and the
porous microstructure of sea ice \cite{Polashenski:JGR:2012}.
Investigating     
critical behavior as a natural aspect of the polar 
marine environment
will shed light on key questions such as the rapidity
of the sea ice retreat and whether a 
so-called {\it tipping point} or critical transition
has been passed in the decline
\cite{Eisenman:PNAS-2009}. It will also
advance our ability to model the future trajectory
of the Arctic sea ice pack.










\subsection{Objectives and Goals}

We propose to investigate the formation and evolution of Arctic melt ponds.
Models and techniques of statistical physics and 
composite materials will be employed to provide new insights into melt pond structure,
and to explore universal behavior characteristic of phase transition phenomena.
We will also begin to investigate how such findings can impact the computation
of ice pack albedo in sea ice numerical models.


\vspace{-1ex}

\begin{enumerate}
  \item Identify the critical melt pond area fraction $\phi_c$, or percolation
        threshold, where the ponds
        become connected over large length scales. What is the critical exponent
        for the divergence of the correlation length? How does
        $\phi_c$ depend on local characteristics such as 
        snow and ice topography, or whether the ice is first year or multiyear? 

\vspace{-1ex}
  \item Investigate our finding that around a critical length scale, 
        the fractal dimension of melt pond boundaries transitions 
        from 1 (simple, Euclidean shapes) to about 2 (space filling curves)         
        \cite{Hohenegger:TC:2012}. Explore the use of partial differential
        equations of pattern formation and phase change to model
        this striking phenomenon. 


\vspace{-1ex}
  \item Use methods of homogenization and spectral measures to characterize
        melt pond and ice floe configurations, as well as critical transitions in their
        effective properties. This approach provides a rigorous framework for 
        {\it upscaling} local morphological characteristics into larger scale models. 
  

\vspace{-1ex} 
 \item Develop stochastic lattice models of melt pond formation based on 
        simple cellular automata and the Ising model for phase transitions 
        in statistical mechanics.

\vspace{-1ex}    
  \item Explore how our mathematical results
        can be exploited to improve 
         the representation of melt ponds and albedo in 
         global climate models and high-resolution 
         sea ice models.

\vspace{-1ex}
\item
Develop methods of image analysis focused on addressing the 
questions about melt ponds raised above, such as tracking 
how features evolve in time, and mapping melt pond images
to networks of nodes and edges. These techniques will likely
be useful to other ONR investigators studying melt ponds.


\end{enumerate}









\subsection{Mathematical Models of Composites and Phase Transitions}

Here we give a brief overview of some of the mathematical models
and techniques that we will use in studying critical behavior 
of melting in the sea ice pack.





\vspace{2ex}


\noindent
{\bf Percolation models.}
Consider the $d-$dimensional integer lattice ${\mathbb{Z}}^d$, 
and the square or cubic network of bonds joining nearest
neighbor lattice sites. In the percolation model
\cite{Broadbent:PCPS-629,Stauffer-92,Grimmett-89,Bunde:FDS-91},
we assign to each bond a conductivity $\sigma_0>0$ with probability $p$,  
meaning it is open (black),
and with probability $1-p$ we assign a 0, meaning it is closed.
Two examples of lattice configurations 
are shown in Figure \ref{statistical_physics_models}, with 
$p=1/3$ in (a) and $p=2/3$ in (b).
Groups of connected open bonds are 
called {\it open clusters}. In this
model there is a critical probability $p_c$,
$0<p_c<1$, called the {\it percolation threshold}, at which the
average cluster size diverges and an infinite cluster
appears. For the two dimensional bond lattice $p_c=1/2$. 
For $p<p_c$ the density of the infinite cluster $P_{\infty}(p)$
is 0, while for $p > p_c$, $P_{\infty}(p) > 0$ and
near the threshold, $P_{\infty}(p) \sim (p-p_c)^{\beta}$
as $p \rightarrow p_c^+$,
where $\beta$ is a universal critical exponent, that is, 
it depends only on dimension and not on the details of the lattice.
Let $x,y \in {\mathbb{Z}}^d$ and 
$\tau(x,y)$ be the probability that $x$ and $y$
belong to the same open cluster. Then for $p < p_c$,
$\tau(x,y) \sim e^{-|x-y|/\xi(p)}$, and the correlation
length $\xi(p) \sim (p_c-p)^{-\nu}$ diverges 
with a universal critical exponent $\nu$ 
as $p \rightarrow p_c^-$,
as shown in 
Figure \ref{statistical_physics_models} (c).

The effective conductivity $\sigma^*(p)$ of the lattice, now viewed as 
a random resistor (or conductor) network,
defined via 
Kirchoff's laws, vanishes for $p<p_c$ like $P_{\infty}(p)$ since
there are no infinite pathways, 
as shown in Figure \ref{statistical_physics_models} (e).
For  $p>p_c$, $\sigma^*(p)>0$, and near $p_c$,
$\sigma^*(p)\sim\sigma_0(p-p_c)^t, \;p \rightarrow p_c^+,$
where $t$ is the conductivity critical exponent, with 
$1 \leq t \leq 2$ in $d=2,3$ 
\cite{Golden:PRL-2923,Golden:CMP-467,Golden:HPM-27},
and numerical values $t \approx 1.3$ in $d=2$ and 
$t \approx 2.0$
in $d=3$ \cite{Stauffer-92}.
%$\sigma^*(p)$ in $d=2$ is shown in Figure \ref{perc_fig_perm_data} (a).
Consider a random pipe network with effective fluid
permeability $\kappa^*(p)$ exhibiting similar behavior
$\kappa^*(p)\sim \kappa_0 (p-p_c)^e,$ 
where $e$ is the
permeability critical exponent, with $e=t$
\cite{Chayes:CMP-133,Sahimi-95,Golden:HPM-27}. Both $t$ and $e$ are believed to
be universal $-$ they depend only on
dimension and not the lattice. Continuum
models %like the Swiss cheese model, 
can exhibit nonuniversal behavior with exponents different 
from the lattice case and $e \neq t$
\cite{Halperin:PRL-2391,Feng:PRB-197,Stauffer-92,%
Sahimi-94,Kerstein:JPA-3071}.








\begin{figure}                \centering
\includegraphics[width=6.5in]{statistical_physics_models_new.eps}
   \caption{The two dimensional square lattice percolation model below
  its percolation threshold of $p_c=1/2$ in (a) and above it in (b).
  (c) Divergence of the correlation length as $p$ approaches $p_c$.
  The infinite cluster density of the percolation model is shown in (d), and
 the effective conductivity is shown in (e). (f) The magnetization
  of the Ising model. }
     
    \label{statistical_physics_models}
        
\end{figure}






\vspace{2ex}


\noindent
{\bf Homogenization and spectral measures.}
{\it Homogenization} denotes a field of applied mathematics
where the goal is to find a homogeneous medium which behaves
macroscopically the same as a given inhomogeneous medium.
The methods are focused on finding
the effective properties of inhomogeneous media such as composites.
This discussion provides a framework for an
effective albedo of the ice pack, as well as the effective
horizontal flow of melt water on the sea ice surface,
where the inhomogeneities are melt ponds or topography.
We will see that the {\it spectral measure} provides a powerful
tool for upscaling geometrical information about a composite 
into calculations of effective properties.



We now briefly describe the {\it analytic continuation method}
for studying the effective properties of composite materials
\cite{Bergman:PRL-1285,Milton:APL-300,Golden:CMP-473,Golden:IMA-97}.
This method has been used to obtain rigorous bounds on effective
transport coefficients of composite materials from partial knowledge
of the microstrucure, such as the relative volume fractions
of the phases. 
For simplicity we choose the electrical conductivity 
of a two phase composite, although the method applies to any classical 
transport coefficient. This framework 
has been extended to finding the effective diffusivity of a passive tracer
advected by a fluid velocity field
\cite{Avellaneda:PRL-753,Avellaneda:CMP-339}.

We consider a two-phase random medium with
$\sigma(x,\omega)$ the local conductivity,
a spatially stationary random field
in $x\in {{\mathbb{R}}}^d$ and $\omega \in \O$, where $\O$ is the
set of realizations of the medium. %with probability measure $P$.
Assume
$\sigma(x) = \sigma_1 \chi_1 + \sigma_2 \chi_2$,
where $\chi_j(x,\omega)$ is the characteristic function of medium $j=1,2$,
equaling 1 for $\omega \in \O$
with medium $j$ at $x$, and 0 otherwise. 
Let $E(x)$ and $J(x)$ be the stationary
random electric and current fields, related by 
$J= \sigma E$, satisfying
%\begin{equation}                                 \label{gold1_1}  
$\nabla \cdot J = 0$ and $\nabla \times E = 0$,                               
%\end{equation}
where the average $\langle E (x) \rangle = e_k$,                                   
and $e_k$ is a unit vector in the $k^{th}$ direction.
%, for some 
%$k = 1, \ldots, d$, and $\langle \cdot \rangle$ means ensemble average over 
%$\O$ or spatial average over  all of ${\mathbb{R}}^d$.
The effective conductivity tensor 
$\bsig^*$ is defined as
%\begin{equation}                                    \label{gold1_2}
$\langle J \rangle=  \bsig^* \langle E \rangle$.
%\end{equation}
We focus on one diagonal coefficient
$\sigma^* = \sigma^*_{kk}$, with
$\sigma^* =  \langle \sigma E_k \rangle$, and since
$\sigma^*$ depends on 
$h = \sigma_1/\sigma_2$, we define
$m(h) =\sigma^*/\sigma_2$, which is 
a Stieltjes function. 
It is analytic
off $(- \infty, 0]$ in the $h$--plane, and 
maps the upper half plane to the 
upper half plane \cite{Bergman:PRC-377,Golden:CMP-473}.



The key step \cite{Golden:CMP-473,Bergman:PRC-377,Milton:APL-300,MILTON:2002:TC} 
is to obtain an integral representation for $\sigma^*$.  
Consider $F(s) = 1 - m(h), s = 1/(1-h),$
which is analytic off $[0,1]$ in the $s$--plane. Then \cite{Golden:CMP-473}
\vspace{-1ex}
\begin{equation}                                     	\label{gold1.eq.3}
    F(s)=1 - \frac{\sigma^*}{\sigma_2} =\int_0^1 {\frac{d\mu(\lambda)}{s-\lambda}}\;,
   \qquad s = \frac{1}{1-\sigma_1/\sigma_2}\;,
\vspace{-1ex}
\end{equation}
where $\mu$ is a positive measure on $[0,1]$. 
This formula arises from the resolvent
representation  of the electric field 
%obtained via manipulation of (\ref{gold1_1}) and (\ref{gold1_2}),
%\begin{equation}                              	\label{gold1.eq.3aa}
$E = (s + \Gamma \chi_1)^{-1}e_k$,
%\end{equation}
where $\Gamma  = \nabla (- \Delta )^{-1} \nabla \cdot$ and $\Delta=\nabla^2$ is the Laplacian, 
yielding
%\vspace{-1ex}
%\begin{equation}                              	\label{gold1.eq.3a}
$F(s) = \langle \chi_1 [(s + \Gamma \chi_1)^{-1} e_k] \cdot e_k \rangle.$
%\vspace{-1ex}
%\end{equation}
%The operator $\Gamma$ acts as a
%projection onto curl-free fields, and is based on
%convolution with the 
%free-space Green's function associated with $-\Delta$. 
In the Hilbert space $L^2 (\Omega, P)$ with weight $\chi_1$ in
the inner product, $\Gamma \chi_1$ is a bounded self adjoint
operator. % with $\parallel \Gamma \chi_1 \parallel \leq 1$.
Formula (\ref{gold1.eq.3})
is the spectral representation of the resolvent, 
and $\mu$ is a spectral measure of $\Gamma \chi_1$, in the 
$e_k$ state.
Formula (\ref{gold1.eq.3})  
separates the component parameters 
in $s$ from the geometrical information in $\mu$.
(Extensions to multicomponent media involve several complex variables
\cite{Golden:JSP-655,Golden:JMPS-333,Milton:CPAM-647,Milton:CMP-281,DellAntonio:JMP-2688}.)
Information about the geometry enters
through the moments 
$
\mu_n = 
\int^1_0 z^n d \mu (z)  =
(-1)^n \langle \chi_1 [( \Gamma \chi_1 )^n e_k ] \cdot e_k \rangle.
$
%\end{equation}
Then $\mu_0 = \phi$, where $\phi$ is the volume or area fraction
of phase 1, such as the melt pond coverage, and
$\mu_1 = \phi(1-\phi)/d$ if the material is statistically isotropic.  
In general, $\mu_n$ depends on the $(n+1)$--point correlation function of the
medium.



Computing the spectral measure $\mu$ for a given composite microstructure first
involves discretizing a two phase image of the composite into a square lattice filled with 
1's and 0's corresponding to the two phases.
Then the key operator $\Gamma \chi_1$, which depends on the geometry of the 
network via $\chi_1$, becomes a self adjoint matrix. 
%On this square lattice
%the action of the differential operators $\nabla$ and $\nabla\cdot$ are 
%defined in terms of the
%forward and backward difference operators
%\cite{Golden:CMP-467}.
The spectral measure may be calculated directly from the
eigenvalues  $\{\lambda_i\}$ and eigenvectors $\{v_i\}$ of this matrix
via %the formula
\begin{equation}                    \label{Spectral_Measure}
  d\mu(\lambda)=\sum_{i=1}^{n} m_i \delta(\lambda-\lambda_i) d\lambda,\quad
  m_i=\langle  e_0^T v_i v_i^T e_0  \rangle, 
\end{equation}
where $\delta(\lambda)$ is the Dirac delta function and $e_0$ is a vector of
ones. In \cite{Golden:JB:2010} we computed 
the spectral measure for samples of healthy and osteoporotic bone
to distinguish them via the connectivity of the trabecular architecture.
%We also developed regularized inversion schemes to recover $\mu$
%from data on the effective complex permittivity of bone samples.


As the system size $N$ increases, the eigenvalues become
increasingly dense in the spectral interval [0,1]. 
The presence or absence of gaps in the spectrum 
near the endpoints of [0,1],
and the details of how large a gap is or how large the spectral
values $m_i$ are, give important information pertaining to
the connectivity and effective transport properties of the system. 


\vspace{2ex}






\begin{figure}                \centering
\includegraphics[width=5.5in]{melt_pond_percolation_July_2012.eps}
   \caption{Evolution of melt pond connectivity. (a) Disconnected
ponds, (b) transitional ponds, (c) fully connected melt ponds. The bottom row shows
the color coded connected components for the corresponding image above: (d) no single color
spans the image, (e) the red phase just spans the image, (f) the connected red phase dominates
the image. The scale bars represent 200 m for (a) and (b), and 35 m for (c). }
     
    \label{connected_components}
        
\end{figure}









\noindent
{\bf Phase transitions and the Ising model of a ferromagnet.}
The Ising model of a ferromagnet in a magnetic field $H$ and 
at temperature $T$ is perhaps the most studied
example of a phase transition in statistical mechanics
\cite{Thompson-88,Christensen:2005,Golden:JMP-5627}.
We consider
a finite box $\Lambda \subset \mathbb{Z}^d$ containing $N$ sites.
At each site there is a spin
variable $s_i$ which can take the values $+1$ or $-1$. 
We consider a Hamiltonian with ferromagnetic pair interaction 
$J \geq 0$ between
nearest neighbor pairs $<i,j>$,
\begin{equation} \label{ising_hamiltonian}                              
{\cal{H}}_\omega=-H\sum_i{s_i}\;-\;J\sum_{< i,j >}{s_is_j}\; ,                 
\end{equation} 
for any configuration $\omega \in \Omega=\{-1,1\}^N$ of the spins. 
%The canonical partition function
%$Z_N$ is given by
%\begin{equation}                                                        
%$Z_N(T,H)=\sum_{\omega \in \Omega}\exp(-\beta {\cal{H}}_\omega) = 
%         \exp(-\beta N f_N)$,      
%\end{equation}
%where  $\beta=1/kT$, $k$ is Boltzmann's constant, 
%and $f_N(T,H)=\frac{-1}{\beta N}\log{Z_N(T,H)}$ 
%is the free energy per site.
%We are interested in the infinite volume limit 
%$f(T,H) = \lim_{N\->\infty} f_N(T,H)$.  
The average magnetization, which serves as the principal 
{\it order parameter} in the system, 
$
M(T,H) =\lim_{N\->\infty}     
\frac{1}{N}\langle \sum_{i=1}^{N}{s_i} \rangle,
$
where $\langle \cdot \rangle$ in this context
denotes averaging over $\omega \in \Omega$ with
Gibbs weights, can be expressed in terms of the
free energy (per unit site) $f$ as 
%\begin{equation}     \label{mag_free}
$M(T,H)=-\frac{\partial f}{\partial H}$.
%\end{equation}
The magnetic susceptibility $\chi(T,H)$, which is the analog
of the effective conductivity in the models above, is given by
$\chi(T,H)=\frac{\D M}{\D H}=-\frac{{\D}^2f}{\D H^2}\geq0.$  
When $H=0$, $M(T) \sim(T_c-T)^{\beta}$ as $T\->{T_c}^-$, 
as shown in Figure \ref{statistical_physics_models} (f),
and $\chi \sim (T-T_c)^{-\gamma}$ as $T\->{T_c}^+$. 
The universal exponent $\beta$ here plays a similar role as $\beta$
for the percolation model, but has a different numerical value.
Below we will discuss how this framework can be used to 
model melt ponds.








\subsection{Principal Scientific Investigations}







\subsubsection{Melt pond percolation}


Melt pond connectivity, as shown in 
Figure \ref{connected_components}, plays a fundamental role 
in the evolution of the melting Arctic sea ice pack.
Such processes as horizontal melt water transport,
illustrated in Figure \ref{melt_pond_geometry} (b),
facilitate drainage through
macro-pores like seal holes, cracks, and leads. 
The break-up of ice floes as they melt is strongly influenced 
by the connectivity of the ponds.
For example, as ponds deepen they weaken an ice floe, making it more
susceptible to breakage and fracture. Connected ponds spanning an ice floe 
are more efficient at promoting fracture than a series of disconnected
ponds. We therefore propose to examine the key issue 
of melt pond connectivity, as illustrated in 
Figure \ref{connected_components}.


%This enhances melting of the pack, since
%many smaller floes are easier to melt than a smaller number of
%larger floes of the same area.











For melt ponds, we ask:
what is the percolation threshold for melt pond evolution.
More precisely, if $\phi$ is the area fraction of melt pond coverage
on a large floe, what is the critical threshold $\phi_c$ for 
large scale connectivity? This can be addressed by using image analysis
to study how the correlation length $\xi(\phi)$ grows as the ponds evolve. 
As outlined above, the percolation threshold can be characterized by
where $\xi(\phi)$ ``diverges,'' or in the case of large floes, where
the scale of the correlation length spans the floe size. 
Does this threshold value depend on floe topography or the age of the floe?
What is the variability of $\phi_c$ from floe to floe?
In mathematical percolation theory, the threshold value depends on the
details of the lattice model, such as square vs. triangular lattice.
However, the critical exponent $\nu$ characterizing the divergence of
$\xi$ is universal, depending only on dimension. Does the value
of $\nu$ for melt ponds match the universal lattice value of 4/3
in two dimensions?

We also expect that our investigations of melt pond connectivity 
will be aided by
continuum percolation models based on the intersection of
a level set with a random surface \cite{Isichenko:RMP-1992}.
Here, the level set represents the melt water level and 
the random surface the ice or snow topography. A 
configuration for a Gaussian model is shown in 
Figure \ref{continuum_percolation_stochastic_model} (a).
Questions of 
percolation thresholds and critical exponents have been 
studied in this context, and we expect that such studies
will lend insights into the melt pond problem.




\begin{figure}                \centering
\includegraphics[height=1.5in]{ONR_Figure_3.eps}
   \caption{(a) Connected clusters in a continuum percolation model 
      where a level set (water level) 
        intersects a surface defined by a Gaussian random field 
       with covariance determined by  
       snow topography data \cite{Scott:2010:JGR}. The resulting 
      configurations resemble melt ponds on first year sea ice.     
     (b) A simple, stochastic lattice growth model
         yields fairly realistic ``ponds,'' and also 
         exhibits a transition in pond fractal 
         dimension with length scale. } 
                    
     
    \label{continuum_percolation_stochastic_model}
        
\end{figure}







Since one aerial photo generally does not span a large floe, 
to address the above questions we must develop 
techniques for stitching together
many images to create a mosaic. 
We are also interested in the time evolution of melt
ponds and their geometry during the melt season, 
so being able to track the growth of individual ponds and 
their connectivity will drive some of the imaging work. 
%which is discussed below. 
Moreover, mapping melt ponds onto
networks (graphs) of vertices and edges facilitates analysis
of pond connectivity and the correlation length.  










%\begin{figure}                \centering
%\includegraphics[width=5in]{areaperimeter.eps}
%   \caption{Area-Perimeter data for August 22 (HOTRAX) shows a transition
%in melt pond fractal dimension. The three black triangles, from left to right,
%correspond to the ponds shown in (a), (b), and (c).
%Small ponds have $D=1$, while much larger ponds 
%have $D=2$. The transitional 
%pond in (b) has a horizontal length scale of about 29 m. }
     
    %\label{fig:areaperimeter}
     
  
%\end{figure}



\vspace{-1ex}


\subsubsection{Transition in the fractal geometry of Arctic melt ponds}







For simple objects like circles and polygons, 
the perimeter $P$ scales like the square root of the area $A$, that is,
$P \sim \sqrt{A}$.
However, for complex planar regions with fractal curves as their boundaries,
\begin{equation}
P \sim \sqrt{A}^{\; D}
\end{equation}
where the exponent $D$ is the fractal dimension of the boundary curve. 
By analyzing area-perimeter data, Lovejoy
\cite{Lovejoy:S:1982} found that clouds have a fractal dimension
of $D \approx 1.35$. 

Viewing images of complex melt ponds, such as in Figures \ref{melt_pond_geometry}
and \ref{connected_components},
suggests similar fractal behavior. However, after developing automated techniques for
obtaining data on the area $A$ and perimeter $P$ for melt ponds from helicopter photos, 
and analyzing the data from thousands of melt ponds, we found something
unexpected. In particular, we discovered a bend in the $P \; {vs}. \; A$ data displayed
on logarithmic scales around a critical length scale of about 100 square meters
in terms of area, as shown in Figure \ref{area_perimeter_fractal_dimension} (a).
We then devised an algorithm to statistically compute the derivative of this data set,
yielding the graph for $D(A)$ in Figure \ref{area_perimeter_fractal_dimension} (b), 
the fractal dimension $D$ as a function of 
melt pond length scale, as measured by the area $A$.
These data showed that melt ponds exhibit a transition in 
fractal dimension from about 1 for $A$ less than 10 m$^2$
to about 2 for $A$ greater than 1000 m$^2$.
We remark that $D=2$ is the upper bound for the 
fractal exponent, since it corresponds to curves which 
fill two dimensional space, such as the famous Peano curve or Brownian motion. 
 


A principal goal of the proposed research 
is to develop mathematical models of melt pond evolution. 
We will explore deterministic models involving
partial differential equations (PDE) as well as stochastic models.
In particular, we will look for models which help explain
the transition in fractal dimension we observe in actual melt ponds,
and the value of the critical length scale. 
In \cite{Hohenegger:TC:2012} we proposed an argument which relates the
critical length scale to the onset of
self-similarity, as illustrated in 
Figure \ref{area_perimeter_fractal_dimension}.
However, we seek a quantitative understanding of how
this scale is set through appropriate mathematical models.



One natural approach is to explore the use of phase field equations,
such as the Ginzburg--Landau model
\cite{Caginalp:PRA:1989}. It was originally developed 
in the context of superconductivity, yet has been used in the
study of many phase transition and pattern formation problems.
The two main functions of interest are the order parameter 
$\psi = \psi(x,y,t)$ (not to be confused with open water fraction)
and the temperature $u=u(x,y,t)$, for $(x,y) \in \mathcal{R}$
for some region $\mathcal{R} \subset \mathbb{R}^2$, and $t \geq 0$.
In the case of melt ponds, we have two 
extreme states, ice and water.  
The frozen state is characterized by $\psi=-1$
and the melted state by $\psi = +1$.
We use the following coupled equations for 
$\psi$ and $u$,
\vspace{-2ex}
\begin{equation}
u_t= K \Delta u - \frac{b}{2} \psi_t,
\label{dom1g}
\end{equation}
\begin{equation}
\alpha \xi^2 \psi_t= \xi^2 \Delta \psi + a^{-1}g(\psi) + 2(u-\theta),
\label{dom2g}
\end{equation}
where $g=c(\psi - \psi^3)$, $\theta$ is the melting temperature, 
$K$ is the coefficient of thermal conductivity, the parameters
$a, b, c, \alpha$, and $\xi$ are determined from the specific
physics of the problem, and appropriate thermal boundary conditions
are applied at the phase interface. We propose to explore how this
system applies to the melt pond problem, by studying the evolution
of the interface separating the melted and frozen states, and
incorporating relevant physics, such as
snow topography. Interestingly, using asymptotic methods,
the classical Stefan and Hele-Shaw systems can be obtained
from the Ginzburg--Landau model in the limit of a sharp
interface \cite{Caginalp:PRA:1989}, 
which we believe should be quite relevant for the melt pond problem.


We will also develop simple stochastic models on lattices to
simulate the evolution of melt ponds. In 
Figure \ref{continuum_percolation_stochastic_model} (b)
we show the results of one such very simple growth model.
Here we specifically wanted to begin to understand why
the fractal dimension undergoes a transition. We made the
following simple observation, that adding a melted square
connected only by a corner to an existing pond increases
the perimeter more than a square connected along an edge.
The probability of this occurring is significantly increased
for pond sizes above some critical size, and in fact, we
see a transition in the fractal dimension around that length
scale, with data quite similar to 
Figure \ref{area_perimeter_fractal_dimension} (a).
We plan to build on this initial success to construct
more realistic models which develop a transition in 
fractal geometry on their own, now that we begin to 
understand one possible mechanism.













\begin{figure}                \centering
\includegraphics[height=1.5in]{Ising_configurations_melt_ponds.eps}
   \caption{ Islands of like spins in the Ising model
    of a ferromagnet. The configurations resemble melt ponds. 
    (a) Early melt ponds. In this simulation 
      the heat content $H>0$ is small.
     (b) Well-developed melt ponds. In this simulation 
        the heat content $H>0$ is large.}
     
    \label{Ising_configurations_melt_ponds}
        
\end{figure}










\vspace{-1ex}


\subsubsection{Ising model for melt pond formation}
We consider an Ising model for melt ponds 
where the ``spins'' $s_i$ are assigned either
$-1$ for ice or $+1$ for melt water.
We will explore various Hamiltonians that model 
the energy associated with melt pond configurations. 
The Ising model of statistical mechanics is a feasible
model for melt pond evolution because it possesses 
the following two properties:
1. Disconnected small scale same-phase regions tend to 
cluster and favor the formation of larger regions
when the system is not dominated by random fluctuations.
2. The system favors
same-phase regions to grow or 
shrink their boundaries as opposed to, for example, shifting or 
being deformed.






The role of the applied magnetic field in Equation
(\ref{ising_hamiltonian}) can be played by the
external forcing of the sea ice layer principally
by net solar radiation and air temperature \cite{Andreas:JAS:1982}.
Then $H$ can be thought of as external heating of the system.
For $H>0$, the system is driven toward more
widespread melting. 
For $H<0$, 
the system is driven toward less melting.
The average magnetization $M(H)$ is then closely related
to the sea ice albedo. Two realizations of the model
configurations are shown in Figure \ref{Ising_configurations_melt_ponds},
which bear striking resemblances to actual melt ponds.
Moreover, the model yields an average albedo displaying
the critical behavior shown in 
Figure \ref{statistical_physics_models} (f), which is similar to
the albedo in Figure \ref{melt_pond_area_critical_behavior}.



The model can be enhanced further 
%and become more realistic 
by assuming an initial  surface topography for the ice cover. 
For a given topography, represented by a 
height function $h$, we can define the interaction 
field $J=J_h(i,j)$ in such a way that the system 
favors water moving in the direction of 
steepest topographical descent. The surface 
topography used to initialize the model 
can ideally be obtained from measurements by
our colleagues (C. Polashenski and D. Perovich), 
or calculated using topography models; 
see for example \cite{Scott:2010:JGR}.







\begin{figure}                \centering
\includegraphics[height=7.5in]{sea_ice_spectral_measures.eps}
   \caption{{\it Spectral analysis of sea ice structures.} 
   Increasingly connected configurations
    of the percolation model (a)-(c),
    melt ponds (d)-(f), and leads in the ice pack
    (g)-(i), with corresponding spectral measures
    below. The area under each spectral function
    in (d)-(f) is the melt pond area fraction $\phi$, and 
    the open water fraction $\psi$ in (g)-(i).
    For the network, the width of the
    gap in the spectrum near
    $\lambda = 1$ for $p=0.1$ shrinks to 0 with
    increasing connectedness  
     as the percolation threshold $p=0.5$ is
    approached, with similar behavior for melt ponds
    and the ice pack.  }
     
    \label{spectral_measures_RRN_meltponds_icepack}
        
\end{figure}




\vspace{-1ex}


\subsubsection{Homogenization theory and spectral measures}
As discussed above, 
homogenization theory \cite{MILTON:2002:TC,Torquato:RHM-02} 
provides a mathematical
framework to analyze the effective transport and rheological
properties of complex, heterogeneous
systems. The analytic continuation method \cite{Golden:CMP-473}, 
in particular, is based on powerful integral representations
for effective properties, and involves the {\it spectral measures}
which depend on the composite
geometry, such as the volume or area fractions of the phases,
as well as the connectivity and percolation properties.
The moments of the spectral measure are related to the
correlation functions of the microstructure. 
We propose to use this framework to {\it upscale} the local
geometry of the ice pack, such as melt pond and ice floe 
configurations, into calculations of their effective
properties. 
In Figure \ref{spectral_measures_RRN_meltponds_icepack}
we show a series of spectral measures computed
from discretizations of melt pond and 
ice pack images, which raises the following questions.
Does the evolution of the melt pond spectral measures
obey universal laws of critical behavior, such
as the collapse of a spectral gap with the onset of 
large scale connectivity? Can critical behavior in the
local geometry be related to critical behavior in 
effective rheological and transport properties?
In view of Figure \ref{melt_pond_geometry} (b), which
illustrates the importance of horizontal flow of meltwater
on the sea ice surface, we observe that the calculation
of melt pond spectral measures immediately gives a
rigorous formula for the effective behavior
of such flows, such as the large scale, horizontal surface
fluid permeability considered in the next section.
Sinks representing seal holes and cracks, 
which are relevant to regional sea ice models and albedo evolution, 
can be incorporated into homogenized PDE models.





\vspace{-1ex}


\subsubsection{Calculating sea ice albedo in climate models}




Modern global climate models (GCM's) account for melt pond albedo effects by explicitly
calculating meltwater volume from sea ice or snow and then
parameterizing the associated pond fractional coverage $\phi$ and
depth $h_p$ \cite{Pedersen:2009:JGR,Holland:JC:2011}. The various parameterizations
in use for $\phi$ and $h_p$ are based on observed relationships
\cite{Perovich:2003:JGR}, curves fit to output from high-resolution mass balance simulations
\cite{Luthje:JGR:2006}, or the assumption that meltwater 
travels horizontally to cover the ice of lowest surface height
\cite{Flocco:2007:JGR,Flocco:2010:JGR}. In higher-resolution frameworks with
explicit sea ice topography, spatial
patterns of $\phi$ and $h_p$ are simulated by 
transporting meltwater horizontally according to Darcy's Law for
flow through porous media \cite{Luthje:JGR:2006}. 

We will investigate the potential for applying the mathematical models
and techniques developed in this proposal to
the simulation of melt ponds in modeling
frameworks including GCM's. 
Recall that GCM's with physically based melt pond schemes calculate
meltwater volume $V_p$, and then the albedo calculation
involves specifying the associated pond
fractional coverage $\phi$. Pond depth $h_p$ is often assumed constant on a
given sea ice thickness class, so the governing equation 
\begin{equation}
V_p=\phi h_p 
\label{eq:Vp}
\end{equation}
is typically closed by assuming that $\phi$ is a known 
function of $h_p$ \cite{Pedersen:2009:JGR,Holland:JC:2011}. 

We expect that the critical transition in pond complexity
discovered using pond perimeter and area
\cite{Hohenegger:TC:2012} will project strongly into the GCM variable
space $(V_p,\phi,h_p)$, and that this projection can be
exploited to formulate a closure for (\ref{eq:Vp}) that obeys the
observed universal scaling. For example, when meltwater volumes reach
levels conducive to complex networking of ponds, it may be more natural to
specify $\phi$ from $V_p$ rather than from $h_p$. To enable the required multivariate analyses,
extraction of $\phi$ will be a straightforward addition to the
perimeter-area analysis, and 
more sophisticated image processing will be used to make color-based
estimates of $h_p$ and hence $V_p$ \cite{Hohenegger:TC:2012}. 
The required algorithms can build on the {\it color--gradient} technique
developed in \cite{Hohenegger:TC:2012}.
A validation and sensitivity analysis of the new specification of $\phi$ 
will be performed using the Los
Alamos CICE model \cite{Hunke:2010}. 
%focusing on the SHEBA period. 




Finally, we will also explore developing 
a predictive modeling framework that
captures the observed critical transition in melt pond
complexity. We will begin with the high-resolution two-dimensional model introduced by
Luthje and collaborators \cite{Luthje:JGR:2006}. This framework's horizontal
transport scheme is based on
Darcy's Law for flow through porous media, and thus does not simulate
more rapid horizontal movements of water across the surface of the ice in
``outflow pathways'' \cite{Polashenski:JGR:2012}. 
We will replace this scheme with a coupled framework
that uses Darcy's Law for the porous subsurface flow and uses Navier
Stokes equations for the surface flow
\cite{Cesmelioglu:JNM:2008}. Incorporating an
explicit mechanism for surface outflow pathways will enable the model 
to develop more realistic networks of
interconnected ponds and reproduce observed critical transitions in
pond complexity \cite{Hohenegger:TC:2012}. 
The spectral calculations discussed above for the effective flow behavior
based on melt pond geometry and connectivity will be employed here to 
make the computations more rigorous and efficient.






















%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{table}


\begin{center}

\footnotesize{


\begin{tabular}{l|llll}


Region & Type & Year & Reference & Comment\\
\hline\hline
Beaufort Sea &    Aerial & 1998 & Perovich & Time series \\
&&& et al., 2002 & May--October \\
\hline
Trans-Arctic &    Aerial & 2005 & Perovich &
Spatial survey in \\
&&& et al., 2005 & August--September \\
\hline
Several years, & Satellite &
Several & Fetterer and & National Technical \\
several sites &&& Untersteiner, 2002 & means \\
\hline
Chukchi, & Satellite &    2011 & Frey,
personal & Images from \\
Beaufort Seas &&& communication & Quickbird \\
\hline
Chukchi, & Satellite & 2013, &
Results from &    Images from \\
Beaufort Seas && 2014 & Polashenski et al. & Quickbird \\
&&& proposal & \\
\hline
Arctic Ocean, & Aerial & 2012, &
Results from &   Spatial survey in\\
north of Svalbard && 2013 & Granskog and Pedersen & July-August \\

\hline
 \end{tabular} \\

}


\end{center}


\caption{Sources of images for melt pond analysis.}

\end{table}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%











\subsection{Image Analysis}



We will first compile and then analyze a library of aerial and satellite images of sea ice during the melt season. Table 1 summarizes some of the sources for the image library. The spatial resolution of the images is approximately 1 m. In many cases a preliminary processing of these images has occurred that we will extend.


For the analysis in \cite{Hohenegger:TC:2012} we had access 
to image sets captured during two measurement campaigns: 
Healy Oden TRans Arctic EXpedition (HOTRAX, 2005) and Surface Heat Budget 
of the Arctic Ocean (SHEBA,1998). Helicopter photographic survey flights
gave extensive high resolution imagery of the polar marine environment
consisting of ice, melt ponds and open water.  
To obtain melt pond area--perimeter and
connectivity data, it was necessary to develop {\it segmentation
algorithms} capable of distinguishing between ponds and open water.
%which is challenging for deep ponds. 
We developed
a method based on computing the directional derivative in color
space, where ponds have at least slight color gradients while
open water generally does not. The segmenation algorithm
is illustrated in Figure \ref{fig:segmentation}.










\begin{figure}                \centering
\includegraphics[width=5in]{segmentation1.eps}
   \caption{Image segmentation for August 14th (HOTRAX): (a) original image; (b) initial four color threshold with ice, shallow ponds, deep ponds and ocean/deep ponds colored as white, green, red and black, respectively; (c) segmented image with ice, ponds and ocean colored as white, blue and black, respectively. }
     
    \label{fig:segmentation}
        
\end{figure}






\begin{figure}                \centering
\includegraphics[width=6.5in]{ice_mosaic.eps}
   \caption{A series of image tiles corresponding to a horizontal pass of the helicopter over the ice aligned into a strip using {\em ir-tweak}. The result is shown after $10 \times $ downsampling. }
     
    \label{fig:align}
        
\end{figure}




In our investigations of large scale connectedness and area--perimeter
relations, individual ponds can be much larger than one image, thus it
is critical to be able to {\it stitch} images together.
Tasdizen \emph{et al.} \cite{Tasdizen10a} have developed a suite 
of automatic and interactive software tools for mosaicking of 
microscopy image tiles. More specifically, {\em ir-tweak} is an 
interactive graphical user interface for image alignment. It 
allows users to drop and drag fiducial control points on either 
the moving or the stationary image. As the user moves the control 
points, the resulting warping of the moving image is shown in 
real time to the user allowing for a smooth interaction. The 
user utilizes information in overlapping areas of the moving 
and stationary images to bring them into alignment.
Currently, {\em ir-tweak} does not support rigid rotation 
because it was designed to handle microscopy images which 
typically require deformable maps. In this project, 
we will add a rigid rotation capability into {\em ir-tweak} 
which is needed for better alignment of the ice images into strips.




Hogrebe \emph{et al.} \cite{Hogrebe11a} have  developed an automatic 
landmark based algorithm to align light microscopy sections of the retina 
with deformable transformations.
In this project, we will modify the Hogrebe algorithm to track melt ponds 
in time. After segmentation of individual melt ponds, we will use their 
properties such as color, size, location and other shape properties to 
match them across time using bipartite matching. This matching will 
drive a deformable warping of the image at time $t+1$ to the 
image at time $t$. This calculated warp, iterated until convergence, 
allows refinement of the landmark matching. %The process will be iterated until convergence.


As discussed above, a key tool which will facilitate the
structural analysis of melt ponds is mapping them onto graphs, 
or networks of vertices and edges.
In general, a melt pond consists of simpler round ponds, 
which correspond to vertices of a graph, connected through 
thin elongated channels, which correspond to its edges. 
For each melt pond, the thin elongated channels can be separated
from the round convex parts by applying appropriate mathematical 
morphological operations. A graph which captures the essential
geometrical and connectedness characteristics of a melt pond
can be obtained using such methods. 
Viewing melt ponds as graphs allows for their classification 
as complex, simple, and transitional, as in 
Figure \ref{area_perimeter_fractal_dimension},
and is essential to any studies of percolation
in melt pond evolution.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{-2ex}

\subsection{Linkages to Other Programs}




\noindent
{\bf 1. ONR Melt Pond Proposals.} 
Following the suggestion of Dr. Jeffries, the Principal Investigators from the proposals “Sunlight, sea ice, and the ice albedo feedback in a changing Arctic sea ice cover” (Perovich and Light), “Multiscale models of melting Arctic sea ice” (Golden et al.), and “Developing remote sensing capabilities for meter-scale sea ice properties” (Polashenski et al.) have had several discussions about our projects and how best to integrate them and reduce costs. Describing melt ponds is a major focus in all three proposals. We believe that the proposals nicely complement each other and are synergistic. For example, the analysis of the high-resolution images proposed by Polashenski et al. could be implemented into the Golden et al. mathematical pond description. It could also be applied as input data for the Perovich and Light solar partitioning calculations. The results of these calculations would in turn provide context for the Polashenski work. Floe size distributions from Polashenski et al. could be used in the lateral melting computations of Perovich and Light, and in
fractal analyses of Golden et al. Through these discussions, we identified several areas of collaboration where our efforts dovetail together. We also found a few places where there was some overlap. By eliminating the overlap, we were able to reduce our budgets.

Our project is more theoretically oriented, however,
after developing mathematical descriptions of pond evolution, 
we will relate them to the physical processes that govern ponds,
and to the other ONR melt pond projects.
Melt pond formation and evolution is related to surface topography, 
melt rates, and ice permeability \cite{Polashenski:JGR:2012}. We will 
determine how changes in sea ice conditions, 
such as melt onset and ice permeability, relate to changes in 
pond fractal dimension. We will examine whether the variation 
in topography between first year and multiyear ice is 
manifested in the mathematical properties of ponds in these two ice types.

Ultimately, we will apply the mathematical description of melt 
ponds to examine the heat and mass balance of the ponds. 
The pond area affects the partitioning of incident solar 
radiation on the ice cover. The pond perimeter to area ratio 
influences the total amount of melting along the pond edges. 


\vspace{1ex}

\noindent
{\bf 2. NSF Math Climate Research Network (MCRN).} 
Golden is a Co-PI on a 5 year NSF DMS funded project to bring young mathematicians
into climate research, through a network of 12 hub institutions, including the 
Math Department at the University of Utah.
The MCRN grant is structured to 
facilitate interaction between core research programs, 
and the project proposed here would dovetail
perfectly with the objectives of the MCRN.
This linkage will
broaden the expertise available to us in this project.
For example the network would bring in additional expertise in
dynamical systems, ice--albedo feedback, 
bifurcation theory, data assimilation, 
time series analysis, and climate modeling.
It will also broaden the impact of our results, and
speed the dissemination of the results of our work.














\eject



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%










%\pagestyle{empty}



\section{References}


%\addcontentsline{toc}{section}{References}

\vspace{-5ex}


\bibliographystyle{plain}
\bibliography{golden,cmg,opp08,vert}


\eject


\section{Project Schedule and Milestones}


\noindent
{\bf Year 1:} 
Collect melt pond images.
Begin analysis of existing images
using percolation theory and image processing techniques.
Specifically address the issue of what determines the
critical length scale where melt pond geometry changes
from Euclidean to space filling behavior.
Explore appropriate theories in statistical physics 
and composite materials to understand critical behavior
of melt pond evolution. Begin development of image
analysis techniques.


\vspace{3ex}

\noindent
{\bf Year 2:}
Begin development of PDE models for melt pond evolution.
Continue percolation and statistical mechanics investigations.
Investigate how the mathematical models can aid in 
representing melt ponds in climate models. Begin publishing
results on mathematical models and image analysis.


\vspace{3ex}



\noindent
{\bf Year 3:}
Continue model development and tune the models with results
from new images. Continue work on including our
theoretical results in climate models.
Continue data analysis, model development and synthesis
with other investigators. Focus on producing group publications
and disseminating results to the broader community.


\vspace{5ex}

\noindent
{\bf Image software development:}
In years one and two, we will focus on developing the image processing
software and testing it on the available data sets. By the end of the
project, we plan to have a user-friendly software package for the analysis of
melt pond and floe images. The software will be made available to
other groups and we will pay particular attention to the packaging of
the software including an easy to use graphical interface and help
documentation.






\eject




\section{Management Approach}


\noindent
{\bf University of Utah:}
The PI Golden will be responsible for the overall success of the
project, for providing mathematical direction and undertaking
investigations, for supervising the post-doc and students.
The co-PI's Strong and Alali will be responsible for scientific
computation and physical modeling of the systems of interest,
and co-advising students.
Tasdizen and Alali will be responsible for developing 
methods of image analysis.


\vspace{3ex}



\noindent
{\bf ERDC--CRREL:}
Perovich and Polashenski will be responsible for obtaining 
melt pond imagery that satisfies the requirements of the
principal modeling objectives. They will also be 
closely involved in relating the mathematical results to
the geophysics of melt ponds, sea ice, and the upper ocean. 
Perovich will also contribute to formulating relationships 
between pond properties and melting on sides and bottoms of ponds.






\vspace{4ex}


\noindent
{\bf Facilities and Equipment at the University of Utah}

\vspace{1ex}


\noindent
Department of Mathematics:

\vspace{1ex}

\noindent
Computer \\
The Department provides outstanding computing facilities for use by faculty,
students and staff. The fully internetworked workstation and microcomputer
configuration includes almost 300 systems in a range of models from these
architectures: Apple Macintosh, DEC Alpha, Intel IA-64, Itanium, Sun AMD64
Opteron, Intel IA-32 Pentium and AMD Athlon, Power PC GNU/Linux, Silicon
Graphics, Sun Ray stations and SUN SPARC.
These include at least one file server from each UNIX architecture.


\vspace{1ex}

\noindent
Office \\
The Mathematics Department is housed in two buildings containing administrative,
graduate student and private faculty offices. Also, there is a student tutoring center,
computer labs and a mathematics library.


\vspace{1ex}

\noindent
Equipment obtained by the PI for sea ice field research:


\begin{enumerate}



\item
Agilent N9923A Field Fox Network Analyzer.
Provides full two port transmission and reflection parameters (S21, S11, S12, S22) for frequencies from 2MHz to 4GHz.  The network analyzer is water-resistant, rugged, and has built-in QuickCal technology that enables field calibration without a kit.  The N9923A is the only handheld network analyzer compliant with Class 2 MIL PRF 28800F.



\item
Agilent 85070E High-Temperature Dielectric Probe Kit.
    The dielectric probe provides complex permittivity measurements from 200MHz to 50GHz.  The probe is hermetically sealed making it resistant to corrosive and abrasive chemicals.  Additionally the probe can withstand a temperature range of -40°C to +200°C.  The probe has a 3.5mm aperture, providing a larger sensing volume than other available probes.



\item
Two AEMC 6470-B Earth Resistivity Meters.
Capable of 3 and 4 point fall-of-potential measurement, 3-point earth coupoling measurement,   automatic resistivity measurement using 4-point wenner and schlumberger method, and 2 and 4-wire DC resistance measurement with automatic polarity reversal.  Additionally the device performs automatic frequency scan from 40-513Hz for electrically noisy environments and can store up to 512 complete test results in internal memory.



\item
Two BK Precision Model 886 Synthesized LCR / ESR Meter
Measures compleximpedance providing magnitude and phase, 
inductance, capacitance, dc resistance, equivalent series resistance 
with a .2\% basic accuracy.  Additionally the meter provides information 
allowing calculation of meter error for each test measurement.  
The meter tests at 100Hz, 120Hz, 1kHz, 10kHz, 100kHz, 
and is capable of measuring at 1Vrms, 0.25Vrms, and 0.05Vrms.


\item 
Six Omega Model 866C thermistors, three WTW Cond 3110 salinometers, 
one Kovacs Mark II coring system,
and a selection of drills and saws, custom made PVC packers for blocking
horizontal flow in core holes during fluid permeability measurements, 
two custom made sea ice core platforms for conducting field experiments, 
and other items.


\end{enumerate}


\vspace{3ex}

\noindent
Department of Atmospheric Sciences:

\vspace{1ex}

\noindent
Co-PI Strong's research group owns a 72-core Linux cluster built on Intel
Nehalem processor technology configured with Infiniband Quad Data Rate
linking and 3GB of RAM per core. This system is maintained and supported
by the University of Utah Center for High Performance Computing (CHPC). PI
Strong additionally has access to the 1000+ core metaclusters maintained
by CHPC, and owns an 8-core workstation suitable for developing and
testing parallel code and performing expensive data analysis operations
such as large matrix inversion.

\vspace{1ex}

\noindent
The Department maintains a comprehensive meteorological archive on
hard disks and tapes, including resources such as GOES satellite imagery.
The Computation and Visualization Laboratory (CVL) is a state of the art
10-workstation facility for education and research in numerical weather
prediction, atmospheric visualization, and dynamic meteorology.



















%\section*{Resources and environment}

\subsection*{SCI Institute Resources}

The SCI institute has over 25,000 square feet of functional space
allocated to its research and computing activities within the new John
and Marva Warnock Engineering Building.  Laboratory facilities for the
researchers listed in this project include personal office spaces and
access to a variety of common working areas.

\subsection*{Computing}

The SCI computing facility, which has dedicated machine room space in
the Warnock Engineering Building, includes

\subsubsection*{Shared-memory, multi-core computers}

\begin{itemize}
    \item
        % 1)
        2 IBM 32 3.0~GHz Xeon core clusters with 192~GB of RAM running
        SUSE Enterprise 10 linux.  Each of these systems can also be
        reconfigured into two separate 16 core systems with 96~GB of
        RAM.
    \item
        % 2)
        A 16-processor, 16~GByte shared memory SGI Altix-Linux system
        with Intel Itanium II 1.4~GHz processors.
    \item
        % 3)
        An 8-processor, 24~GByte shared memory SGI Altix-Linux system
        with Intel Itanium II 1.4~GHz processors.
    \item
        % 4)
        A variety of multi-core, linux desktop workstations.
\end{itemize}

\subsubsection*{Graphics and distributed-memory, multi-core computers}

\begin{itemize}
    \item
        % 1)
        The SCI Institute was recently named an NVIDIA Center of
        Excellence.  As a result of this collaboration, the institute
        has installed a 128-node NVIDIA Tesla compute cluster.  The GPU
        cluster is supported by a 512-core HP Nehalem-based CPU cluster
        and InfiniBand networking.
    \item
        % 2)
        Sun Cluster:  256 Dual-Quad Core Nodes (2048 total cores), 2.8
        GHz Intel Xeon (Harpertown) processors, 16~Gbytes memory per
        node (2~Gbytes per processor core), Qlogic Infiniband DDR (InfiniPath
        QLE 7240) interconnect, Gigabit Ethernet interconnect.
    \item
        % 3)
        3, 24 core, 2.5 GHz, AMD Opteron with NVIDIA Quadro FX 5600
        graphics card) with a dual Gigabit Ethernet backbone and 96
        GBytes RAM.
    \item
        % 4)
        An 8 core, 2.0 GHz, AMD Opteron, with NVIDIA Quadro 2FX graphics
        card) cluster with a dual Gigabit Ethernet backbone and 16
        GBytes RAM.
    \item
        % 5)
        A 2-processor (3.0 GHz, Intel Pentium D, with NVIDIA Quadro FX
        graphics card) visualization head with a Gigabit Ethernet
        backbone and 2 GBytes RAM.
    \item
        % 6)
        A 2-processor (3.0 GHz, Intel Pentium D, with NVIDIA Quadro FX
        graphics card) Windows based visualization head with a Gigabit
        Ethernet backbone and 2 GBytes RAM.
    \item
        % 7)
        A 16 core, 3.0 GHz AMD Opteron with NVIDIA 8800GTX graphics card
        with a dual Gigabit Ethernet backbone and 64 GBytes RAM.
    \item
        % 8)
        A 16 core, 2.4 GHz AMD Opteron, with NVIDIA 7900GTX graphics
        card with a dual Gigabit Ethernet backbone and 64 GBytes RAM.
        The SCI Institute computing facility contains the following
        network, regression
\end{itemize}

\subsubsection*{Testing and storage systems}

\begin{itemize}
    \item
        % 1)
        A hierarchical SGI Infinitefile file system consisting of both
        FibreChannel and SATA disk and online tape storage and with an
        initial capacity of over 120 TBytes and significant expansion
        capacity.
    \item
        % 2)
        Six Sun quad-core AMD Opteron fileservers attached to a Hitachi
        AMS500 modular FC high-availability storage array with over five
        TBytes of high-end FibreChannel disk.
    \item
        % 3)
        Three LTO-2 tape libraries providing backup for SCI Institute
        researchers.
    \item
        % 4)
        A 250 slot LTO-3 StorageTek SL500 tape library.
    \item
        % 5)
        A fully redundant Foundry BigIron switching core that provides a
        Gigabit network backbone for all HPC computers, servers, and
        individual workstations connected via Foundry floor switches.
    \item
        % 6)
        Connections to the campus backbone via redundant 10 Gigabit
        Ethernet links---the first such attachments on campus.
    \item
        % 7)
        A variety of Intel and AMD based desktop workstations running
        Linux with the latest ATI or NVIDIA graphics cards.
    \item
        % 8)
        Numerous Windows 2000, Windows XP, and Vista workstations.
    \item
        % 9)
        Numerous Mac workstations running OSX.
    \item
        % 10)
        Regression cluster made of 8 workstations of various OS/hardware
        configurations used for nightly regression builds of CIBC
        software.
    \item
        % 11)
        Three conference rooms with teleconferencing/video conferencing
        capabilities.
    \item
        % 12)
        Multiple, high availability Linux servers providing core SCI IT
        services - website (www.sci.utah.edu), ftp, mail, and software
        distribution.
    \item
        % 13)
        100 Megapixel powerwall composed of 24 Dell high-resolution
        monitors powered by six servers with high-end graphics cards.
    \item
        % 14)
        UPS power, including 100 minutes of battery backup for critical
        SCI servers.
\end{itemize}

\subsubsection*{David Evans Visualization Center (EVC)}

The David Evans Visualization Center is an important collaboration and
demonstration space within the SCI Institute and serves as a focal point
for CIBC members to work collaboratively together, prepare multi-media
components of presentations, and share work with visitors to the Center.

\subsubsection*{Equipment and functionality within the EVC include}

\begin{itemize}
    \item
        % 1)
        Three high-end, Mac based video editing bays with flatbed
        scanners, A/D video converters, Adobe CS Master Suite and Final
        Cut Pro.
    \item
        % 2)
        One midrange iMac flatbed scanning station.
    \item
        % 3)
        Two midrange Windows XP systems with Adobe CS Master Suite.
    \item
        % 4)
        One high-end Mac for running videos and 3D demos.
    \item
        % 5)
        One high-end PC for running videos and 3D demos.
    \item
        % 6)
        One Highlite 8000Dsx stereo 3D capable rear projection system
        with 1400x1050 native resolution.
    \item
        % 7)
        A 10 foot $ \times $ 8 foot, high-resolution, rear projection
        system with stereo capability for collaborative group
        interaction and code development.
\end{itemize}

\subsubsection*{Office space}

The SCI Institute houses its faculty and staff in individual offices and
students have individual desk space equipped with a workstation in
large, open common areas that facilitate student collaboration and
communication.  All workstations are connected to the SCI local area
network via full-duplex Gigabit Ethernet.

The University of Utah is a member of the Internet2 advanced networking
consortium.  It is connected to the new Internet2 Network via Gigabit
Ethernet (1 Gbit per second) and the National LambdaRail (NLR) via a
dedicated link to the Front Range Gigapop.  The University also has the
ability to extend dedicated wavelengths into the campus from the
Internet2 and NLR nodes collocated in a Level 3 Communications facility
west of Salt Lake City.




\eject




\section{Current and Pending Projects and Proposals}


\vspace{2ex}

\noindent
{\bf Kenneth M. Golden, Ph.D.}

\vspace{3ex}

\noindent
{\bf Current:}


\vspace{3ex}


\noindent
Title:  EMSW21-VIGRE: Vertical Integration in Mathematics at the University of Utah \\
Summary: Provides leading-edge training and research opportunities 
in the mathematical sciences for students. \\
Source:  National Science Foundation, DMS-0602219 \\
Percent effort:  5\%   \\
Technical contact:  Kenneth M. Golden     \\
Administrative/business contact:  Shauna Peterson; 1471 East Federal Way; \\
  Tel: 801-585-0062; Fax: 801-581-3007; email:  shauna.peterson@osp.utah.edu  \\
Period of performance:  05/17/06 - 07/31/12   \\
Total award (direct and indirect costs): \$3,500,000  \\
Person months devoted to project:  0    \\
Overlap:  None   


\vspace{3ex}

\noindent
Title:  CMG COLLABORATIVE RESEARCH: Mathematics and Electromagnetics 
for Monitoring Transport Processes In Sea Ice    \\
Summary:  To develop methods of electromagnetically monitoring 
the internal state of sea ice, the thermal evolution of its microstructure, 
and the transport processes it controls.  \\
Source:  National Science Foundation, ARC-0934721   \\ 
Percent effort:  20\%    \\
Technical contact:  Kenneth M. Golden   \\
Administrative/business contact:  Shauna Peterson; 1471 East Federal Way; \\
  Tel: 801-585-0062; Fax: 801-581-3007; email:  shauna.peterson@osp.utah.edu  \\
Period of performance:  10/01/09-09/30/13    \\
Total award (direct and indirect costs):  \$541,271    \\
Person months devoted to project:  0.75 summer months \\ 
Overlap:  None


\vspace{3ex}


\noindent
Title:  Collaborative Research:  Mathematics and Climate Change Research Network  \\
Summary:  The scientific objectives of the project are:  1) climate process modeling;  
2) the processes of past and future climates; and 3) data analysis and the 
incorporation of data into Earth system and climate process models. \\
Source:  National Science Foundation, DMS-0940249  \\
Percent effort:  10\%  \\
Technical contact:  Kenneth M. Golden    \\
Administrative/business contact:  Shauna Peterson; 1471 East Federal Way; \\
  Tel: 801-585-0062; Fax: 801-581-3007; email:  shauna.peterson@osp.utah.edu \\
Period of performance:  10/01/10-09/30/15    \\
Total award (direct and indirect costs):  \$527,727  (There are 12 institutions 
involved in this project, for a grand total of \$5,000,000.)\\
Person months devoted to project:  0  \\
Overlap:  None 

\vspace{3ex}


\noindent
Title:  Phase Transitions in Composite Media  \\
Summary:  To investigate the transitional behavior of electrorheological fluids and the existence of a critical electric field.  \\
Source:  National Science Foundation, DMS-1009704 \\
Percent effort:  10\%  \\
Technical contact:  Kenneth M. Golden \\
Administrative/business contact:  Shauna Peterson; 1471 East Federal Way; \\
  Tel: 801-585-0062; Fax: 801-581-3007; email:  shauna.peterson@osp.utah.edu  \\
Period of performance:  09/15/10-08/31/13    \\
Total award (direct and indirect costs):  \$307,000    \\
Person months devoted to project:  1 summer month \\
Overlap:  None




\vspace{3ex}

\noindent
Title:  Spectral Theory of Advective Diffusion in the Ocean    \\
Summary:  We will develop a systematic spectral analysis of advection enhanced diffusion.  \\
Source:  Office of Naval Research, N000141210861  \\
Percent effort:  10\%    \\
Technical contact:  Kenneth M. Golden   \\
Administrative/business contact:  Shauna Peterson; 1471 East Federal Way; \\
  Tel: 801-585-0062; Fax: 801-581-3007; email:  shauna.peterson@osp.utah.edu  \\
Period of performance:  07/01/12 -- 06/30/13    \\
Total award (direct and indirect costs):  \$50,000    \\
Person months devoted to project: 0 month \\
Overlap:  None




\vspace{6ex}



\noindent
{\bf Pending:}


\vspace{3ex}


\noindent
Title:  Geometry of the Antarctic Marginal Ice Zone \\
Summary:  Determine controls on variability in the width of the
Antarctic marginal ice zone, and investigate its evolution using 
mathematical models. \\
Source:  National Science Foundation, Office of Polar Programs \\
Percent effort:  10\% \\
Technical contact:  Courtenay Strong \\
Administrative/business contact: Jesse Pugh, Officer, Office of
Sponsored Projects; 1471 East Federal Way; Tel: 801-581-3008; Fax: 801-581-3007; email: Jesse.Pugh@osp.utah.edu \\
Period of performance:  01/01/13-12/31/15  \\
Total award (direct and indirect costs):  \$498,544   \\
Person months devoted to project: 1 \\
Overlap:  N/A

\vspace{3ex}




\noindent
Title:  Multiscale Models of Melting Arctic Sea Ice (this application) \\
Summary:  We will investigate the formation and evolution of melt ponds on Arctic sea ice. This work will help to improve climate models by quantifying our understanding of ice pack albedo and key feedback mechanisms. \\
Source:  Office of Naval Research \\
Percent effort:  15\% \\
Technical contact:  Kenneth M. Golden \\
Administrative/business contact:  Shauna Peterson; 1471 East Federal Way; \\
  Tel: 801-585-0062; Fax: 801-581-3007; email:  shauna.peterson@osp.utah.edu  \\
Period of performance:  12/1/12 -- 11/30/15  \\
Total award (direct and indirect costs):  \$481,616  \\
Person months devoted to project: 1 summer month \\
Overlap:  None



\vspace{5ex}


\noindent
{\bf Courtenay Strong, Ph.D.}


\vspace{3ex}

\noindent
{\bf Current:}


\vspace{3ex}

\noindent
Title:  Think Globally, Learn Locally  \\
Summary:  Graduate student fellows supported by this program serve as
scientific role models and mentors to high school students.\\
Source:  National Science Foundation, Division of Graduate Education \\
Percent effort:  5\%  \\
Technical contact:  Courtenay Strong   \\
Administrative/business contact: Jesse Pugh, Officer, Office of
Sponsored Projects; 1471 East Federal Way
Salt Lake City, Utah 84102-1821; Tel: 801-581-3008; Fax: 801-581-3007; email: Jesse.Pugh@osp.utah.edu \\
Period of performance:  05/01/09-04/30/14 \\
Total award (direct and indirect costs): \$2,143,634 \\
Person months devoted to project:  0 \\
Overlap:  None

\vspace{3ex}

\noindent
Title:  Negative sea ice-circulation feedback along Arctic marginal ice zones \\
Summary:  Innovative modeling frameworks (e.g., 
hybrid dynamical-statistical coupling) are used to investigate
observed feedback process in Arctic marginal ice zone.\\
Source: National Science Foundation, Office of Polar Programs \\
Percent effort:  17\%  \\
Technical contact:  Courtenay Strong   \\
Administrative/business contact: Jesse Pugh, Officer, Office of
Sponsored Projects; 1471 East Federal Way; Tel: 801-581-3008; Fax: 801-581-3007; email: Jesse.Pugh@osp.utah.edu \\
Period of performance:  08/15/10-07/31/13 \\
Total award (direct and indirect costs): \$242,991 \\
Person months devoted to project:  2 \\
Overlap:  None



\vspace{3ex}


\noindent
Title:  Collaborative Research: CI-WATER, Cyberinfrastructure to Advance High Performance Water Resource Modeling  \\
Summary:  Acquire and develop software and hardware to support new large-scale, high-resolution computational models.  \\
Source:  National Science Foundation, EPSCoR Research Infrastructure Improvement Program, Track-2  \\
Percent effort:  8\% \\
Technical contact:  Courtenay Strong  \\
Administrative/business contact: Jesse Pugh, Officer, Office of
Sponsored Projects; 1471 East Federal Way; Tel: 801-581-3008; Fax: 801-581-3007; email: Jesse.Pugh@osp.utah.edu \\
Period of performance:  09/01/11-08/31/14 \\
Total award (direct and indirect costs):  \$3,435,873 \\
Person months devoted to project:  1 \\
Overlap:  None

\vspace{3ex}


\noindent
Title:  iUTAH-innovative Urban Transitions and Aridregion Hydro-sustainability  \\
Summary:  Observe and model fundamental interactions and
dynamic feedbacks among hydroclimate and the ecological and human
aspects of urban and montane landscapes.  \\
Source:  National Science Foundation, EPSCoR Research Infrastructure Improvement Program, Track-1  \\
Percent effort:  8\% \\
Technical contact:  Courtenay Strong  \\
Administrative/business contact: Jesse Pugh, Officer, Office of
Sponsored Projects; 1471 East Federal Way; Tel: 801-581-3008; Fax: 801-581-3007; email: Jesse.Pugh@osp.utah.edu \\
Period of performance:  08/01/12-07/31/17 \\
Total award (direct and indirect costs):  \$4,000,000 renewable
annually for up to five years\\
Person months devoted to project:  1 \\
Overlap:  None

\vspace{3ex}


\noindent
{\bf Pending:}


\vspace{3ex}

\noindent
Title:  Geometry of the Antarctic Marginal Ice Zone \\
Summary:  Determine controls on variability in the width of the
Antarctic marginal ice zone, and investigate its evolution using 
mathematical models. \\
Source:  National Science Foundation, Office of Polar Programs \\
Percent effort:  17\% \\
Technical contact:  Courtenay Strong \\
Administrative/business contact: Jesse Pugh, Officer, Office of
Sponsored Projects; 1471 East Federal Way; Tel: 801-581-3008; Fax: 801-581-3007; email: Jesse.Pugh@osp.utah.edu \\
Period of performance:  01/01/13-12/31/15  \\
Total award (direct and indirect costs):  \$498,544   \\
Person months devoted to project: 2 \\
Overlap:  N/A

\vspace{3ex}

\noindent
Title:  Multiscale Models of Melting Arctic Sea Ice (this application) \\
Summary:  We will investigate the formation and evolution of melt ponds on Arctic sea ice. This work will help to improve climate models by quantifying our understanding of ice pack albedo and key feedback mechanisms. \\
Source:  Office of Naval Research \\
Percent effort: 6\% \\
Technical contact:  Kenneth M. Golden \\
Administrative/business contact:  Shauna Peterson; 1471 East Federal Way; \\
  Tel: 801-585-0062; Fax: 801-581-3007; email:  shauna.peterson@osp.utah.edu  \\
Period of performance:  12/1/12 -- 11/30/15  \\
Total award (direct and indirect costs):  \$481,616  \\
Person months devoted to project: 0.75 summer month \\
Overlap:  None





\vspace{5ex}


\noindent
{\bf Bacim Alali, Ph.D.}


\vspace{3ex}


\noindent
{\bf Pending:}


\vspace{3ex}

\noindent
Title:  Multiscale Models of Melting Arctic Sea Ice (this application) \\
Summary:  We will investigate the formation and evolution of melt ponds on Arctic sea ice. This work will help to improve climate models by quantifying our understanding of ice pack albedo and key feedback mechanisms. \\
Source:  Office of Naval Research \\
Percent effort: 20\% \\
Technical contact:  Kenneth M. Golden \\
Administrative/business contact:  Shauna Peterson; 1471 East Federal Way; \\
  Tel: 801-585-0062; Fax: 801-581-3007; email:  shauna.peterson@osp.utah.edu  \\
Period of performance:  12/1/12 -- 11/30/15  \\
Total award (direct and indirect costs):  \$481,616  \\
Person months devoted to project: 1 summer month \\
Overlap:  None












\vspace{5ex}


\noindent
{\bf Tolga Tasdizen, Ph.D.}


\vspace{3ex}

\noindent
{\bf Current:}


\vspace{3ex}

\noindent
Title:  A Computational Framework for Mapping Long Range Genetic Circuits \\
Summary:  This project addresses neurobiologic technologies and software tools needed for gene-specific 3D reconstructions of long range axon projections in the limbic system, at the axon level and on the scale of the primate brain using light microscopy. \\
Source:  National Institutes of Health \\
Percent effort:  5\%  \\
Technical contact:  Tolga Tasdizen   \\
Administrative/business contact: Edward Black, Manager, Grants and Accounting, SCI; 72 S Central Campus Dr.; Tel: 801-587-7857; Fax: 801-585-6513; email: ed@sci.utah.edu \\
Period of performance:  09/30/09-08/31/11 \\
Total award (direct and indirect costs):  \$996,734 \\
Person months devoted to project:  0 \\
Overlap:  None


\vspace{3ex}


\noindent
Title:  MSPA-MCS:  High-Dimensional, Nonparametric Density Estimation for the Analysis of Images and Shapes \\
Summary:  This project addresses several computational challenges associated with estimating and representing probability densities in high-dimensional spaces and the application of this technology to several problems in image analysis. \\
Source:  National Science Foundation \\
Percent effort:  5\% \\
Technical contact:  Tolga Tasdizen    \\
Administrative/business contact: Edward Black, Manager, Grants and Accounting, SCI; 72 S Central Campus Dr.; Tel: 801-587-7857; Fax: 801-585-6513; email: ed@sci.utah.edu \\
Period of performance:  06/01/07-12/31/11 \\
Total award (direct and indirect costs):  \$474,000 \\
Person months devoted to project:  .50 summer months \\
Overlap:  None







\vspace{3ex}


\noindent
Title:  Scalable Large Analytic Segmentation Hybrid (SLASH) \\
Summary:  This project addresses scalable large analytic segmentation hybrid.\\
Source:  National Institutes of Health \\
Percent effort:  10\% \\
Technical contact:  Tolga Tasdizen  \\
Administrative/business contact: Edward Black, Manager, Grants and Accounting, SCI; 72 S Central Campus Dr.; Tel: 801-587-7857; Fax: 801-585-6513; email: ed@sci.utah.edu \\
Period of performance:  06/01/11-05/31/16 \\
Total award (direct and indirect costs):  \$635,903 \\
Person months devoted to project:  1.2 \\
Overlap:  None










\vspace{3ex}


\noindent
Title:  Fluorender: An Imaging tool for visualization and analysis of confocal data \\
Summary:  This project addresses visualization and analysis of confocal data specifically for Zebrafish developmental research . \\
Source:  National Science Foundation \\
Percent effort:  10\% \\
Technical contact:  Tolga Tasdizen    \\
Administrative/business contact: Edward Black, Manager, Grants and Accounting, SCI; 72 S Central Campus Dr.; Tel: 801-587-7857; Fax: 801-585-6513; email: ed@sci.utah.edu \\
Period of performance:  06/01/11-05/31/16 \\
Total award (direct and indirect costs):  \$1,253,900 \\
Person months devoted to project:  1.2 \\
Overlap:  None



\vspace{3ex}




\noindent
{\bf Pending:}









\vspace{3ex}


\noindent
Title:  CAREER: Deep sparse dictionary context models and their application to
image parsing and neuron tracking for connectomics \\
Summary:  This project addresses vnovel techniques for analysis of large scale electron microscopy data . \\
Source:  National Science Foundation \\
Percent effort:  8\% \\
Technical contact:  Tolga Tasdizen    \\
Administrative/business contact: Edward Black, Manager, Grants and Accounting, SCI; 72 S Central Campus Dr.; Tel: 801-587-7857; Fax: 801-585-6513; email: ed@sci.utah.edu \\
Period of performance:  01/01/12-12/31/16 \\
Total award (direct and indirect costs):  \$409,406 \\
Person months devoted to project:  1.0 \\
Overlap:  None









\vspace{3ex}



\noindent
Title:  Multiscale Models of Melting Arctic Sea Ice (this application) \\
Summary:  We will investigate the formation and evolution of melt ponds on Arctic sea ice. This work will help to improve climate models by quantifying our understanding of ice pack albedo and key feedback mechanisms. \\
Source:  Office of Naval Research \\
Percent effort: 10\% \\
Technical contact:  Kenneth M. Golden \\
Administrative/business contact:  Shauna Peterson; 1471 East Federal Way; \\
  Tel: 801-585-0062; Fax: 801-581-3007; email:  shauna.peterson@osp.utah.edu  \\
Period of performance:  12/1/12 -- 11/30/15  \\
Total award (direct and indirect costs):  \$481,616  \\
Person months devoted to project: 0.75 summer month \\
Overlap:  None





\eject



\section{Qualifications}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\vspace{2ex}





\begin{center}
 {\large {\bf Kenneth Morgan Golden \hspace{2ex} www.math.utah.edu/$\sim$golden}}
%{{\bf Kenneth M. Golden}} \\
 %\vspace{1ex}
%{ {\bf Kenneth Morgan Golden}} \\
\end{center}
%\vspace{.2ex}
%\begin{tabbing}
%{\bf Kenneth Morgan Golden} \hspace{20em}     \= University of Utah\\
%golden@math.utah.edu \>                          Department of Mathematics\\
%(801) 581 - 6176 (office)    \>                  155 S 1400 E RM 233 \\       
%(801) 581 - 4148 (fax)   \>                     Salt Lake City, UT 84112-0090\\
%\end{tabbing}
\vspace{-1ex}
\noindent {\bf Education}
\vspace{-1ex}
\begin{tabbing}
\hspace{2em}\=B.A.\hspace{1.7em}\=1980\hspace{2em}\=Dartmouth College, Mathematics and Physics\\
\>M.S.\>1983\>New York University, Mathematics\\                         
\>Ph.D.\>1984\>New York University, Mathematics\\                          
\end{tabbing}
\vspace{-3ex}
\noindent {\bf Employment}
\vspace{-1ex}
\begin{tabbing}
\hspace{2em}\=1984-87\hspace{4.5em}\=NSF Mathematical Sciences Postdoctoral Fellow, Rutgers University\\
\>1987-91\>Assistant Professor of Mathematics, Princeton University\\                         
\>1991-96 \>Associate Professor of Mathematics, University of Utah\\                          
\>1996- \>Professor of Mathematics, University of Utah\\

\>2007- \>Adjunct Professor of Bioengineering, University of Utah\\   

\end{tabbing}
\vspace{-2.5ex}
\noindent {\bf Professional Summary}
\vspace{-1ex}
\begin{enumerate}


\item[$\bullet$] Conducted field research on Antarctic sea ice 
                 in 1980, 1994, 1998, 1999, 2007, and 2010, 
                with SIPEX II expedition scheduled for September - November 2012; 
                 field research on Arctic sea ice in 
                 2000, 2001, 2002, 2003, 2004, 2007, and 2011,
                  with scheduled field work in May 2012 and May 2013.
      
\vspace{-1.3ex}
\item[$\bullet$] Served as Modeling Coordinator
                 for ONR Accelerated Research Initiative on Sea Ice 
                 Electromagnetics 1992-98, involving over 60 
                 researchers at 20 institutions. 
    

\vspace{-1.3ex}
\item[$\bullet$]
Published 56 papers in mathematics, physics, geophysics, ocean sciences,
electrical engineering, mechanical engineering, and biomechanics journals.
Given over 300 invited lectures on 6 continents.
Since 2008, invited to give over 70 plenary and keynote addresses, conference 
and university lectures.












\vspace{-1.3ex}
\item[$\bullet$] Member, Ed. Board of {\it SIAM J. on 
          Appl. Math.} 1996-99; {\it Applic. Analysis} 2004--2008 
       
 %     Served as reviewer for journals in mathematics,
 %                physics, engineering, and geophysics; Reviewed proposals
 %                for NSF and other agencies in US and abroad; Served on 
 %                panels and site reviews for NSF.

%\vspace{-1ex}
%\item[$\bullet$] Member at Large, Committee on Science Policy, 




\vspace{-1.3ex}
\item[$\bullet$]
Chair, Committee for Math Awareness Month April 2009 on 
``Mathematics and Climate,'' Joint Policy Board for 
Mathematics (AMS, SIAM, ASA, MAA)
%www.mathaware.org/mam/09.




\vspace{-1.3ex}
\item[$\bullet$] Chair, American Mathematical Society
                 Committee on Science Policy, 2012$-$2013;
                 Member, 2010$-$2013; 
               Member, AMS
                 Committee on Education, 2012$-$2013



\vspace{-1.3ex}
\item[$\bullet$] 
Organized or co-organized 19 conferences and workshops
on sea ice, polar climate, composite materials, 
%percolation, 
and inverse problems. Examples include:
Invited Minisymposium on {\it Polar Climate Modeling}, Joint Math Meetings 
(AMS, SIAM), Wash. D.C., 1/09;
Invited Symposium on {\it Sea Ice in the Changing Climate},
%Modeling a Multiscale Nonlinear System}, 
AAAS Natl. Meeting, San Diego, 2/10;
Invited Workshop on {\it Polar Climate}, 
IPAM, UCLA, 5/10;
Ocean Ecologies % and their Physical Habitats 
in a
Changing Climate, MBI, Ohio St., 7/11;
Scientific Program Committee, 2012
Ocean Sciences Meeting, Salt Lake City, 2/12.         

\vspace{-1.3ex}
\item[$\bullet$]
Mentored 27 REU students, 2 high school students, 
9 graduate students (4 current Ph.D. students), 
and 5 post-docs. Brought 8 undergrads (from 5 departments), 4 graduate students,
and 2 postdocs on polar expeditions.

\vspace{-1.3ex}
\item[$\bullet$]
Service to the U. of Utah Math Dept.:
Director of Undergraduate Studies (2002-08),
REU Program Coordinator (2003-07),
Chair of the Undergraduate Curriculum Committee (2002-08),
Executive Committee (1992-96, 2002-04),
Engineering Math Coordinator (1996-98),
many hiring committees and other positions.
%and course coord. positions;
%College of Science Dean's ``Kitchen Cabinet'' (2007-).






%\vspace{-1.3ex}
%\item[$\bullet$]
%Service to U. of Utah Math Dept. $-$
%Dir. Undergrad. Studies (2002-08),
%Research Experiences for Undergraduates (REU)
%REU Coordinator (2003-07),
%Chair, Undergrad. Curriculum Comm. (2002-08),
%Exec. Comm. (1992-96, 2002-04),
%numerous hiring committees
%and course coord. positions;
%College of Science Dean's ``Kitchen Cabinet'' (2007-10).

\vspace{-1.3ex}
\item[$\bullet$]
Since 1992, PI on 14 grants to U. of Utah from 
NSF (Division of Mathematical Sciences, Office of Polar
Programs, and Arctic Natural Sciences) 
and ONR, totalling \$3.7 M in funding. 
Co-PI or Senior Personnel on 2 NSF research and training 
grants, totalling \$8.5 M in funding.




\vspace{-1.3ex}
\item[$\bullet$]
Accounts of my research on sea ice and climate change,
and related polar expeditions
have been featured in newspapers, magazines, and web articles,
and on radio and television.
They include 4 TV appearances, several radio interviews, 
and over 30 articles, with
profiles in {\it Science} (2009) and {\it Science News} (2000, 2010).




\end{enumerate}




\noindent 
{\bf Honors and Awards}

\vspace{-0.3ex}

\begin{enumerate}


\item[$\bullet$]
2012 Distinguished Scholarly \& Creative Research Award,
University of Utah,
\$10,000 prize;
first math professor to receive the award since 1992.

\vspace{-1.3ex}

\item[$\bullet$]
2012 Myriad Faculty Award for Research Excellence, 
U. of Utah, College of Science,
\$20,000 prize. 

\vspace{-1.3ex}


\item[$\bullet$] Selected as a Fellow of the Society for Industrial
                 and Applied Mathematics (SIAM), March 2011,
                 for ``extraordinary interdisciplinary
                 work on mathematics of sea ice.''

\vspace{-1.3ex}

\item[$\bullet$] Presented my research on sea ice and climate change
                 in the US Congress three times:
                 for the
                 American Mathematical Society in 2003 and 2007
                 and for SIAM in 2011.


\vspace{-1.3ex}

\item[$\bullet$] 2007 University Distinguished Teaching Award,
                 University of Utah; first math professor to receive
                the highest teaching award since 1987, fourth since 1965.

\vspace{-1.3ex}

\item[$\bullet$]
Excellence in Teaching Award, Princeton Engineering Council,
Princeton, 1989

\vspace{-1.3ex}

\item[$\bullet$]  Member, Electromagnetics Academy, 1996$-$; Member,
                Electrical Transport and Optical Properties
           of Inhomogeneous Media (ETOPIM) Association Committee, 2006$-$.


%\item[$\bullet$] Faculty Fellow Award, University of Utah, 1994

%\item[$\bullet$] Hertz Foundation Fellow, NYU, 1981$-$84


\end{enumerate}















\vspace{-0.2ex}


\noindent{\bf Related Publications}
%\vspace{-0.5ex}
\begin{enumerate}

\vspace{-0.3ex}


\item K. Golden and G. Papanicolaou,  Bounds for effective parameters of
      heterogeneous media by analytic continuation,
     {\it Communications in Mathematical Physics},
      90, pp. 473--491, 1983.


\vspace{-1ex}

\item K. Golden, S. Goldstein and J. L. Lebowitz,  Nash estimates and
      the asymptotic behavior of diffusions, {\it Annals of  Probability},
      16, pp. 1127--1146, 1988.


\vspace{-1ex}

\item K. Golden, Convexity and exponent inequalities for conduction near
      percolation, {\it Physical Review Letters}, 65, pp. 2923--2926, 1990.


\vspace{-1ex}

\item K. M. Golden, Statistical mechanics of conducting phase transitions,
      {\it Journal of Mathematical Physics}, 36, pp. 5627-5642, 1995.


\vspace{-1ex}

\item K. M. Golden, Critical behavior of transport in lattice and
       continuum percolation models,
      {\it Physical Review Letters}, 78,  pp. 3935-3938, 1997.



\vspace{-1ex}

\item K. M. Golden, D. Borup, M. Cheney, E. Cherkaeva,
      M. S. Dawson, K. H. Ding, A. K. Fung, D. Isaacson,
      S. A. Johnson, A. K. Jordan, J. A. Kong, R. Kwok,
      S. V. Nghiem, R. G. Onstott, J. Sylvester,
      D. P. Winebrenner and I. H. H. Zabel,
      Inverse electromagnetic scattering models for sea ice,
      {\it  IEEE Trans. on Geosci.
      and Remote Sensing}, 36(5), pp. 1675-1704, 1998.


\vspace{-1ex}

\item K. M. Golden, S. F. Ackley and V. I. Lytle, The percolation phase
      transition in sea ice, {\it Science}, 282, pp. 2238-2241, 1998.



\vspace{-1ex}

\item K. M. Golden, H. Eicken, A. L. Heaton, J. Miner,  D. Pringle,
      and J. Zhu,
      Thermal evolution  of permeability and
      microstructure in sea ice,
      {\it Geophysical Research Letters}, 34,
      L16501 %doi:10.1029/2007GL030447
      (6 pages and issue cover), 2007.



\vspace{-1ex}

\item  K. M. Golden, Climate change and the mathematics
       of transport in sea ice, invited feature article for the
       {\it Notices of the American Mathematical Society},
       Volume 56, Number 5, pages 562-584,
       (including issue cover), May 2009.




\vspace{-1ex}

\item
K. M. Golden, N. B. Murphy, and E. Cherkaev, Spectral analysis
and connectivity of porous microstructures in bone,
{\it Journal of Biomechanics}, Vol. 44, pp. 337-344, 2011.



\vspace{-1ex}

\item L. B. Simeonova, D. C. Dobson, O. Eso, and K. M. Golden,
Spatial bounds on the effective complex
permittivity for time-harmonic waves
in random media,
{\it Multiscale Modeling and Simulation},
Vol. 9, No. 3, pp. 1113-1143, 2011.


\vspace{-1ex}

\item C. Orum, E. Cherkaev and K. M. Golden,
Recovery of inclusion separations in strongly heterogeneous
composites from effective property measurements,
{\it Proceedings of the Royal Society A:
Mathematical, Physical \& Engineering Sciences},
Vol. 468, No. 2139, pp. 784-809, 2012.


\vspace{-1ex}

\item
N. B. Murphy and K. M. Golden, The Ising model and
critical behavior of transport in binary
composite media, {\it Journal of Mathematical Physics},
53, 063506, pp. 1-25, doi: 10.1063/1.4725964, 2012.






\vspace{-1ex}

\item
C. Hohenegger, B. Alali, K. R. Steffen,
D. K. Perovich, and K. M. Golden, Transition in the
fractal geometry of Arctic melt ponds, 
{\it The Cryosphere Discussions}, 6, 2161–-2177, 
doi:10.5194/tcd-6-2161-2012;
in submission to {\it The Cryosphere}, 2012.


\end{enumerate}



\noindent
{\bf Recent Selected Invited Lectures} 

\vspace{1ex} 


\begin{enumerate}

\vspace{-1.6ex}

\item[2008] Conf. on Frontiers of Climate Science, 
          Kavli Inst. for Theor. Phys., Santa Barbara 

\vspace{-1.6ex}

\item[2008] Invited Guest Lecture at the Climate Change Summer School, MSRI, Berkeley, CA

\vspace{-1.6ex}

\item[2008] Keynote Address, Intl. Polar Year Forum, 
           SACNAS Natl. Conf., Salt Lake City, UT

\vspace{-1.6ex}

\item[2009] The SIAM Invited Address at the Joint Math Meetings 
            (AMS, MAA, SIAM), D.C.



\vspace{-1.6ex}

\item[2009]
Houghton Lectures on {\it Sea Ice, Climate, and Multiscale Composites}, 
Department of Earth, Atmospheric and Planetary Sciences, MIT

\vspace{-1.6ex}

\item[2009] SIAM Conference on Dynamical Systems, Minisymposium on {\it Mathematical 
            Challenges in Climate Change: Some Examples}, Snowbird, UT

%\vspace{-1.6ex}

%\item[2009] Eighth International Conference on Electrical Transport and 
%            Optical Properties of Inhomogeneous Media (ETOPIM8), Crete

\vspace{-1.6ex}

\item[2009] Mathematics of Climate Change, First Pacific Rim Math. 
            Assoc. Congress, Sydney

\vspace{-1.6ex}

\item[2009]
Kieval Lecture, Humboldt State University, Arcata, CA
 


\vspace{-1.6ex}

\item[2010]
Random Matrix Theory and Appl., 
AMS Spring Western Sect. Meeting, Albuquerque


\vspace{-1.6ex}

\item[2010]
Mini-Workshop on {\it Mathematics of Sea Ice in the 
Climate System} (two lectures), Institute of Pure and 
Applied Mathematics (IPAM), UCLA, Los Angeles

\vspace{-1.6ex}

\item[2010]
Intl. Symp. on Sea Ice in the Phys. and Biogeochemical System,
Tromso, Norway

\vspace{-1.6ex}

\item[2010] 
Plenary Science Lecture, %on Mathematics and Climate, 
Conference on Emerging Topics in Dynamical Systems and 
Partial Differential Equations, DSPDE's 10, 
Barcelona

\vspace{-1.6ex}

\item[2010] 
MSRI-NCAR Summer School on Mathematics of Climate Change, NCAR, Boulder

\vspace{-1.6ex}

\item[2010] 
Workshop on The Multiphase Physics of Sea Ice, Santa Fe

\vspace{-1.6ex}

\item[2011] Public Lecture, 
Conf. on Prob. Theory \& Stat. Phys.,
NYU Abu Dhabi, UAE

\vspace{-1.6ex}

\item[2011]
Simon Fraser University Seminar Series on Global Warming, 
a Scientific Perspective, Invited Lecture on
{\it Evidence for Warming in the Arctic and Antarctic},
Burnaby




\vspace{-1.6ex}

\item[2011]
Invited Plenary Address, Spring Eastern Sectional Meeting of the American Mathematical Society, College of the Holy Cross, Worcester, MA



\vspace{-1.6ex}

\item[2011]
Ocean Ecologies and their Physical Habitats in a 
Changing Climate (two lectures), Mathematical Biosciences
Institute (MBI), Ohio State University, Columbus



\vspace{-1.6ex}

\item[2011]
7$^{th}$ Intl. Congress on Industrial and Applied Math. (ICIAM 2011), 
Minisymposium on {\it Composites and Inversion: Asymptotic and Computational Methods}, 
Vancouver

\vspace{-1.6ex}

\item[2011]
NSF-SIAM Workshop on Collaborations in Mathematical Geoscience, Washington DC

\vspace{-1.6ex}

\item[2011]
AMS Fall Western Section Meeting, Minisymposium on
{\it Electromagnetic Wave Propagation in Complex and Random Environments},
Salt Lake City

\vspace{-1.6ex}

\item[2011] Keynote Address, Ann. Conf.
Amer. Math. Assoc. for Two-Year Colleges, Austin

\vspace{-1.6ex}

\item[2011] Neal Thorpe Memorial Lecture,
20$^{th}$ Conference Undergraduate Research,
Murdock Charitable Trust, Seattle

\vspace{-1.6ex}

\item[2012]
Public Lecture, Pacific Institute for the Mathematical Sciences,
Vancouver

\vspace{-1.6ex}

\item[2012]
European Geosciences Union General Assembly 2012,
Session on {\it Sea ice physical and biological 
processes and interactions with climate},
Vienna


%\vspace{-1.6ex}

%\item[2012]
%Arctic Ocean Ecosystem Workshop, Barrow, AK





\vspace{-1.6ex}

\item[2012]
IUGG Conference on Mathematical Geophysics, Edinburgh



\vspace{-1.6ex}

\item[2012]
9$^{th}$ AIMS Conf. on Dynamical Systems, Diff. Eqs. and Applications,
Session on {\it Mathematical Modeling of Upwelling Ocean Currents and Related Phenomena},
Orlando




\vspace{-1.6ex}

\item[2012]
White Nights Workshop: Exotic Struct. and Homogenization,
St. Petersburg, Russia

\vspace{-1.6ex}

\item[2012] Plenary Address, Ninth International Conference on 
           Electrical Transport and
            Optical Properties of Inhomogeneous Media (ETOPIM9),
             Marseille



\vspace{-1.6ex}

\item[2012]
Second Reunion Conference for the UCLA IPAM Climate Modeling Program,
Lake Arrowhead, CA




\vspace{-1.6ex}

\item[2013]
MAA-AMS-SIAM Gerald and Judith Porter Public Lecture, 2013 Joint Math
Meetings, San Diego


\vspace{-1.6ex}

\item[2013]
Session on {\it Recent Advances and New Challenges in Applied Analysis}, 2013 Joint Math
Meetings, San Diego




\vspace{-1.6ex}

\item[2013] Inaugural Bernoulli Society Public Lecture, 
      36$^{th}$ Conference on Stochastic Processes and their Applications,
      Boulder




\end{enumerate}




\eject










\eject






\begin{center}
 {\large {\bf Courtenay Strong \hspace{2ex} http://www.inscc.utah.edu/$\sim$strong}}
%{{\bf Kenneth M. Golden}} \\
 %\vspace{1ex}
%{ {\bf Kenneth Morgan Golden}} \\
\end{center}
%\vspace{.2ex}
%\begin{tabbing}
%{\bf Kenneth Morgan Golden} \hspace{20em}     \= University of Utah\\
%golden@math.utah.edu \>                          Department of Mathematics\\
%(801) 581 - 6176 (office)    \>                  155 S 1400 E RM 233 \\       
%(801) 581 - 4148 (fax)   \>                     Salt Lake City, UT 84112-0090\\
%\end{tabbing}
\vspace{-1ex}
\noindent {\bf Education}
\vspace{-1ex}
\begin{tabbing}
\hspace{2em}\=B.A.\hspace{1.7em}\=1993\hspace{3em}\=University of
Texas at Austin, Journalism\\
\>B.S.\>1999\>Mississippi State University, Meteorology\\                         
\>M.S.\>2003\>University of Virginia, Environmental Sciences\\                         
\>Ph.D.\>2005\>University of Virginia, Environmental Sciences\\                           
\>Postdoc\>2007-2009\>University of California, Irvine, Climate dynamics\\                          
\end{tabbing}
\vspace{-3ex}
\noindent {\bf Employment}
\vspace{-1ex}
\begin{tabbing}
\hspace{2em}\=1994-2000\hspace{4.5em}\=Broadcast Meteorologist,
television network affiliates\\
\>2001-2002\>Research and Teaching Assistant, University of Virginia\\                         
\>2003-2005\>NSF Graduate Research Fellow, University of Virginia\\                         
\>2005-2006\>Visiting Scientist / Instructor, University of Missouri\\                         
\>2007-2009 \>Postdoctoral Scholar, University of California, Irvine\\
\>2009- \>Assistant Professor of Atmospheric Sciences, University of Utah\\

\end{tabbing}
\vspace{-2.5ex}
\noindent {\bf Professional Summary}
\vspace{-1ex}
\begin{enumerate}


\item[$\bullet$] PI on NSF grant investigating the role of sea ice in
  atmosphere-cryosphere feedback \$0.25 M; Co-PI or senior personnel
  on two NSF Experimental Program to Stimulate Competitive Research (EPSCoR)
  grants (Track-1 and Track-2) totaling approximately \$7.5 M in
  awarded funding to date. Co-PI on NSF
  Think Globally, Learn Locally grant (Division of Graduate Education)
  \$2.1 M. 
      
\vspace{-1.3ex}
\item[$\bullet$] Lead author on 20 manuscripts in atmospheric,
  oceanic, and climate dynamics journals. Given 15 invited lectures
  and more than 20 contributed presentations in North America and
  Europe. 
    
\vspace{-1.3ex}
\item[$\bullet$] Currently advising three graduate students (two Ph.D.) and one
  postdoctoral researcher. Member of advising committee for
  15 additional students in three departments (Atmospheric Sciences,
  Biology, and Civil and Environmental Engineering)

\vspace{-1.3ex}
\item[$\bullet$] Developed and published the first method for
  objective and automated measure of marginal ice zone width using remote
  sensing data (method also provides marginal ice zone position and area) 

\vspace{-1.3ex}
\item[$\bullet$] Expert reviewer for Southwest Technical Input to the
  2013 National Climate Assessment. Proposal reviewer for Natural Sciences and
  Engineering Research Council (Canada) and National Science
  Foundation. 
       
\vspace{-1.3ex}
\item[$\bullet$] Manuscript reviewer for 14 leading journals in my discipline: \emph{Atmospheric Chemistry and Physics, Climate Dynamics, Climate Research, Geophysical Research Letters, International Journal of Climatology, Journal of the Atmospheric Sciences, Journal of Climate, Monthly Weather Review, Nature, Nature Climate Change, Nature Geoscience, Pure and Applied Geophysics, Quarterly Journal of the Royal Meteorological Society}

\vspace{-1.3ex}
\item[$\bullet$]
Teach mathematically rigorous graduate course in climate
dynamics. Developed advanced teaching materials for my co-taught graduate course in
statistical methods, and introduced a new quantitative climate course for
undergraduate majors and graduate students from outside atmospheric sciences. 

\end{enumerate}




\noindent 
{\bf Honors and Awards}

\vspace{-0.3ex}
\begin{enumerate}
\item[$\bullet$] 
2010 Invited speaker to NCAR Climate and Global Dynamics Division 
\item[$\bullet$]
2008 James R. Holton Junior Scientist Award, American
Geophysical Union
\item[$\bullet$]
2008 Invited postdoctoral participant, Leverhulme Climate
Symposium, Cambridge University and The Royal Society of London
\item[$\bullet$]
2003-05	National Science Foundation Graduate Research Fellowship
\item[$\bullet$]
2002 American Meteorological Society First Prize for Student Oral Presentation:   
		      25th Conference on Agricultural and Forest
                      Meteorology, Norfolk, VA 
\item[$\bullet$]
2001-05	Consecutive Merit-based Departmental Dupont Fellowships
\item[$\bullet$]
2001 Outstanding First Year Graduate Student in Atmospheric Science
(University of Virginia)
\item[$\bullet$]
1999 American Meteorological Society Seal of Approval for Broadcast
Meteorology 

\end{enumerate}

\vspace{-0.2ex}


\noindent{\bf Select Related Publications}
%\vspace{-0.5ex}
\begin{enumerate}

\vspace{-0.3ex}
\item J. Liptak and C. Strong, Propagating atmospheric patterns
  associated with sea ice motion through the Fram Strait, {\it
    Geophysical Research Letters}, submitted, 2012.

\vspace{-0.3ex}
\item C. Strong, Atmospheric influence on Arctic marginal ice zone
  position and width in the Atlantic sector, February-April
  1979-2010, {\it Climate Dynamics}, in press,
  doi:10.1007/s00382-012-1356-6, 2012. 

\vspace{-0.3ex}
\item C. Strong and S. C. Maberly, The influence of atmospheric wave
  dynamics on interannual variation in the surface temperature of
  lakes in the English Lake District. {\it Global Change Biology}, 17,
  pp. 2013--2022, 2011. 

\vspace{-0.3ex}
\item C. Strong and G. Magnusdottir, Dependence of NAO variability on
  coupling with sea ice, {\it Climate Dynamics}, 36, pp. 1681--1689, 2011.

\vspace{-0.3ex}
\item C. Strong and G. Magnusdottir, The role of Rossby wave breaking
  in shaping the equilibrium atmospheric circulation response to North
  Atlantic boundary forcing, {\it Journal of Climate}, 23,
  pp. 1269--1276, 2010. 

\vspace{-0.3ex}
\item C. Strong and G. Magnusdottir, Modeled boreal winter sea ice
  variability and the North Atlantic Oscillation: a multi-century
  perspective, {\it Climate Dynamics}, 34, pp. 515--525, 2010. 

\vspace{-0.3ex}
\item C. Strong, G. Magnusdottir and H. Stern, Observed feedback
  between winter sea ice and the North Atlantic Oscillation, {\it Journal
  of Climate}, 22, pp. 6021-6032, 2009. 

\vspace{-0.3ex}
\item C. Strong, J. D. Fuentes, R. E. Davis, and J. W. Bottenheim,
  Thermodynamic attributes of Arctic boundary layer ozone depletion
  {\it Atmospheric Environment}, 36: pp. 2641--2652, 2002. 
\end{enumerate}




\eject






\begin{center}
 {\large {\bf Bacim Alali\hspace{4ex} www.math.utah.edu/$\sim$alali}}
\end{center}
\noindent {\bf Education}
\begin{tabbing}
\hspace{2.5em}\=B.S.\hspace{2em}\=1998\hspace{2em}\= Mathematics and Physics, Yarmouk University, Jordan\\
\>M.S.\>2005\>Mathematics, Louisiana State University\\                         
\>Ph.D.\>2008\>Mathematics, Louisiana State University\\                          
\end{tabbing}

\vspace{-2ex}

\noindent {\bf Employment}
\begin{tabbing}
\hspace{2.5em}\=2011- \hspace{1.9em}\=Ed Lorenz Postdoctoral Fellow in the Mathematics of Climate, U. of Utah\\
\hspace{2.5em}\=2008-2011 \=Wylie Assistant Professor, University of Utah\\
\>1998-2002\>Research Mathematician, Estarta Solutions\\                         

\end{tabbing}

\vspace{-3ex}


\noindent{\bf Publications}
\begin{enumerate}
\item C. Hohenegger, B. Alali, K. R. Steffen,
D. K. Perovich, and K. M. Golden, Transition in the
fractal geometry of Arctic melt ponds, 
{\it The Cryosphere Discussions}, 6, 2161–-2177, 
doi:10.5194/tcd-6-2161-2012;
in submission to {\it The Cryosphere}, 2012.


\item B. Alali and G. W. Milton, Corrections to the effective conductivity in composites in the presence of imperfect interfaces, in submission.

\item B. Alali and R. Lipton, New bounds on local strain fields inside random
  heterogeneous materials, {\it Mechanics of Materials}, Volume 53, pages 111-122, 2012.

\item B. Alali and R. Lipton,  Multiscale dynamics of heterogeneous media in the peridynamic formulation. {\it Journal of Elasticity} (Online First), Dec. 8, 2010, pp. 1-33, DOI 10.1007/s10659-010-9291-4.
 
\item B. Alali and R. Lipton, Optimal lower bounds on local stress inside random media, 
 {\it  SIAM Journal on Applied Mathematics}, Volume 70, Issue 4, pages 1260-1282, 2009.

\item B. Alali and R. Lipton, Multiscale analysis of fiber-reinforced composites in the peridynamic formulation,
 {\it IMA Preprint Series 2241}, Institute for Mathematics and its Applications,
University of Minnesota, 2009.


\end{enumerate}


\noindent {\bf Academic Awards}
\begin{itemize}
\item[] 2006-2008 \hspace{1em} Research assistantship supp. by The Boeing Company, LSU.
\item[] 2004-2006 \hspace{1em} Research assistantship supp. by grants from AFOSR, NSF, and LSU.
\item[] 2002-2006 \hspace{1em} Graduate School Enhancement Award, LSU.
\item[] 1995-1998 \hspace{1em} Jordan's Ministry of Higher Ed. Math Fellowship,
    Yarmouk University.
\end{itemize}

\vspace{1ex}


\noindent
{\bf Invited Talks and Recent Workshops}
%\vspace{-3ex} 
\noindent
\begin{enumerate}
\item[2012] Poster Presentation,  2012 Ocean Sciences Meeting, Salt Lake City.
   \item[2012] Large Scale Multimedia Search, Institute for Pure and Applied Mathematics (IPAM), Los Angeles.
   \item[2011] Poster Presentation, Mathematics and Climate Research Network Annual Meeting, Chapel Hill, North Carolina.

   \item[2011] Poster Presentation, New Kind of Science Summer School, Boston.

   \item[2011] Minisymposium Speaker, 2011 Joint Mathematics Meetings, American Mathematical Society (AMS), New Orleans.


   \item[2010] Modeling and Mathematics of Sea Ice, Institute for Pure and Applied Mathematics (IPAM), University of California, Los Angeles. 

   \item[2009] Stochastic Multiscale Methods: Mathematical Analysis and Algorithms, University of Southern California, Los Angeles. 

   \item[2009] Minisymposium Speaker, 10TH US National Congress On Computational Mechanics (USNCCM10),
   Columbus, Ohio.
   
   \item[2008] Minisymposium Speaker, SIAM Conference on Mathematical Aspects of Materials Science (MS08),
   Philadelphia.

   \item[2007] Minisymposium Speaker, SIAM Conference on Analysis of Partial Differential Equations (PD07), 
   Mesa, Arizona.
    \item[2007] Contributed Poster, Workshop on Modeling, Analysis and Simulation of Multiscale Nonlinear Systems,
    Oregon State University, Corvallis, Oregon.

    \item[2007] DOE Summer School in Multiscale Mathematics and High Performance Computing,
Oregon State University, Corvallis, Oregon.
 
   \item[2007] Contributed Poster, Workshop on Computational and Mathematical Aspects of Materials and Fluids (CAM2007),
     Department of Mathematics and Ames Laboratory, Iowa State University, Ames, Iowa.
 \end{enumerate}


\vspace{3ex}

\noindent {\bf Teaching Experience}
%\vspace{1ex}
%\begin{enumerate}
\begin{itemize}
\item[] Mathematics Department, University of Utah, 2008-present\\
 Taught Calculus I, Calculus II, Business Algebra, ODE and Linear Algebra, and PDE's for Engineering Students.

 \item[] Department of Mathematics, Louisiana State University, 2002-2005\\
  Taught College Algebra and Calculus I.
\end{itemize}



\vspace{1ex}

\noindent
{\bf Ph.D. Advisor:} Robert P. Lipton, Louisiana State University.

%\vspace{3ex}

%\noindent {\bf Recent Collaborators}
%\begin{itemize}
%\item[] Tadele Mengesha, Pennsylvania state university.
%\item[] Tolga Tasdizen, University of Utah.
%\item[] Greame Milton, University of Utah.
%\item[] Stewart Silling, Sandia National Laboratories.
%\end{itemize}










\eject

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\begin{center}
 {\large {\bf Tolga Tasdizen \hspace{2ex} www.sci.utah.edu/$\sim$tolga}}
\end{center}

\vspace{-1ex}


\noindent{\bf Education}
\vspace{-1ex}
\begin{tabbing}
\hspace{1em}\=B.S. \hspace{1em}\=1995\hspace{2em}\= Bogazici University, Istanbul,  Electrical Engineering\\
\> M.S. \>1997 \>Brown University, Engineering\\
\> Ph.D\> 2001 \> Brown University, Engineering\\
\end{tabbing}
\vspace{-3ex}
\noindent{\bf Employment}
\vspace{-1ex}
\begin{tabbing}
\hspace{1em}\=2001-04\hspace{3em}\=Postdoctoral Research Scientist, Scientific Computing and Imaging\\\>\> Institute, University of Utah.\\
\>2004-08\>Research Assistant Professor, School of Computing, University of Utah.\\
\>2006-\>Adjunct Assistant Professor, Department of Neurology, Center for \\\>\>Alzheimer's Care, Imaging and Research, University of Utah.\\
\>2008-\>Adjunct Assistant Professor, School of Computing, University of Utah.\\
\>2008-12\> Assistant Professor, Electrical and Computer Engineering, U. of Utah.\\
\>20012-\> Associate Professor, Electrical and Computer Engineering, U. of Utah.\\

\end{tabbing}

\vspace{-2.5ex}
\noindent {\bf Professional Summary}
\vspace{-1ex}
\begin{enumerate}


\item[$\bullet$] PI on one grant from NIH (NIBIB National Institute of Biomedical Imaging and Bioengineering) \$1.14 M in funding. 
Co-PI or Senior Personnel on 5 NIH (NINDS National Institute of Neurological Disorders and Stroke,  NIBIB) and NSF, totalling \$5 M in funding.

\vspace{-1.3ex}
\item[$\bullet$]Inventors on three patents on pattern recognition, image segmentation and characterization of datasets.

\vspace{-1.3ex}
\item[$\bullet$]Associate Editor, IEEE Signal Processing Letters and BMC Bioinformatics

\vspace{-1.3ex}
\item[$\bullet$]Review Editor, Frontiers in Computational Physiology and Medicine, 2011-present. Reviewer for prominent IEEE journals, Medical Image Computing and Computer-Assisted Intervention (MICCAI), Journal of Neuroscience Methods, Journal of Mathematical Imaging and Vision, Pattern Analysis and Applications, ACM Solid Modeling, SIAM Journal of Scientific Computing, VisSym, Eurographics, Eurographics Workshop on Visual Computing for Biology and Medicine, Journal of Electronic Imaging, The Visual Computer, SIGGRAPH Asia, Elsevier Methods.

\vspace{-1.3ex}
\item[$\bullet$]  Panelist and reviewer for joint NSF/NIH Collaborative Research in Computational Neuroscience program, 2006, 2008, 2009 and 2010.

\vspace{-1.3ex}
\item[$\bullet$] Organizer MIAAB (Microscopic Image Analysis with Applications in Biology), 2008, 2009.

\vspace{-1.3ex}
\item[$\bullet$] IEEE Signal Processing Society, Bio imaging and Signal Processing (BISP) Technical Committee Associate Member, 2009-2012, Regular Member 2012-.

\vspace{-1.3ex}
\item[$\bullet$] Senior Member IEEE, IEEE Signal Processing Society and IEEE Computer Society.

\vspace{-1.3ex}
\item[$\bullet$] Advisor to eight students (five Ph.D.) 

\vspace{-1.3ex}
\item[$\bullet$]  Best paper award MICCAI 2010 MedIA special Issue

\vspace{-1.3ex}
\item[$\bullet$]  Best Student Paper Award Honorable Mention, 15th IEEE Computer Society International Conf. on Pattern Recognition, 2000.

\vspace{-1.3ex}
\item[$\bullet$]  Imaging and Computer in the Loop breakout session speaker, Opportunities in Biology at the Extreme Scale of Computing, Chicago, 2009.

\vspace{-1.3ex}
\item[$\bullet$]  $6^{th}$ IEEE International Symposium on Biomedical Imaging (ISBI): From Nano to Macro; {\em Electron Microscopy} session chair, 2009. 

\vspace{-1.3ex}
\item[$\bullet$]  Program committee, International Symposium CompIMAGE 2010, CompIMAGE 2012.

\vspace{-1.3ex}
\item[$\bullet$]  Program committee, IMAGAPP 2010


\vspace{-1.3ex}
\item[$\bullet$]  Dean's letter for top instructors in the College of Engineering, Fall 2009, Spring 2010, Fall 2010

\end{enumerate}

\vspace{-0.2ex}


\noindent{\bf Related Publications}
\vspace{-0.5ex}
\begin{enumerate} 

\item E. Jurrus and A. R. C. Paiva and S. Watanabe, J. R. Anderson, B. W. Jones, R. T. Whitaker, E. M. Jorgensen, R. E. Marc and T. Tasdizen, Detection of Neuron Membranes in Electron Microscopy Images using Auto-context, \textit{Medical Image Analysis}, 2010.

\vspace{-0.7ex}

\item T. Tasdizen, Principal Neighborhood Dictionaries for Non-local Means Image Denoising, \textit{IEEE Trans. on Image Processing}, 2009. 

\vspace{-0.7ex}

\item A. R. C. Paiva, E. Jurrus and T. Tasdizen, Using Sequential Context for Image Analysis, \textit{In} Int. Conf. on Pattern Recognition, 2010.

\vspace{-0.7ex}
\item S. M. Seyedhosseini, A. R. C. Paiva and T. Tasdizen, Image Parsing with a Three-State Series Neural Network Classifier, {\it Int. Conf. on Pattern Recognition}, 2010.

\vspace{-0.7ex}
\item S. Gerber, T. Tasdizen and R. T. Whitaker, Dimensionality Reduction and Principal Surfaces via Kernel Map Manifolds, {\it Int. Conf. on Computer Vision}, 2009.

\vspace{-0.7ex}
\item E. Jurrus, T. Tasdizen, P. Koshevoy, P. T. Fletcher, M. Hardy,
C. Chien, W. Denk, and R. Whitaker, Axon Tracking in Serial Block-Face Scanning Electron Microscopy, {\it Medical Image Analysis} 2009.

\vspace{-0.7ex}
\item S. Gerber, T. Tasdizen, P. T. Fletcher, S. Joshi, R. Whitaker and the Alzheimers Disease Neuroimaging Initiative (ADNI), Manifold modeling for brain population analysis, {\it Medical Image Analysis}, 2010.

\vspace{-0.7ex}
\item T. Tasdizen, R. T. Whitaker, P. Burchard and S. Osher, Geometric Surface Processing via Normal Maps, {\it ACM Transactions on Graphics}, 2003.

\vspace{-0.7ex}
\item T. Tasdizen, R. T. Whitaker, P. Burchard and S. Osher, Geometric Surface Smoothing via Anisotropic Diffusion of Normals, {\it IEEE Visualization}, 2002. 

\vspace{-0.7ex}
\item T. Tasdizen, J.-P. Tarel and D. B. Cooper,  Algebraic Curves that Work Better, {\it IEEE Computer Society Conf. on Computer Vision and Pattern Recognition}, 1999. 
\end{enumerate}





\eject







\end{document}



Budget Justification:
The bulk of the funds being requested are to support the
participants in the project. In particular, the University
of Utah Math Department will generously contribute 1/3 of 
the $50,000 salary for a post-doc in years 2-5, in exchange 
for teaching one course each year in Mathematics. We have
requested a modest sum to help support a project coordinator.
The Math Department is not properly staffed to support the
polar field work that the PI will be conducting. The project
coordinator will oversee procedural matters, ordering supplies 
and equipment, arranging travel and meetings, and assisting 
in preparation of reports. Requests for supplies and computer 
and field equipment are to support the theoretical modeling 
efforts and our field research.   

extend info obtained from heli data on ship to 
probe larger scales with remote sensing and satellite data









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




INTRO



The precipitous decline of the summer Arctic sea ice 
pack is a strong signal that 
Earth's climate is changing.
Most current climate models, however, have significantly underestimated this retreat 
\cite{Stroeve:GRL-2007,Boe:2009:NG}.
We propose to investigate key processes and feedback
mechanisms to improve 
our understanding of the role of sea ice in the climate
system, as well as 
projections of the fate of Arctic sea ice.
In particular, our focus will be on the evolution 
of melt pond and ice pack configurations, 
and ice-albedo feedback.


As the Arctic warms in the spring, small melt ponds on the 
surface of sea ice grow and coalesce to form much larger 
connected structures with complex boundaries.   
Melt pond area fraction can exhibit critical behavior
with a rapid rise from 0\% to over 60\%
in just a few days
with a corresponding critical drop in albedo.
Analysis of surface images suggests that 
melt pond formation resembles a percolation process, with 
the onset of large scale connectivity %and complex geometry
beyond a critical threshold. In fact, 
area {\it vs.} perimeter data for melt ponds
indicate a transition from simple smooth shapes
to highly ramified regions whose boundaries are similar
to space filling curves \cite{Hohenegger:2011}.
At a critical length scale, the fractal dimension 
of the boundaries appears to transition 
from 1 (Euclidean) to about 2. 


The rich behavior we observe for melt pond evolution 
is similar in many respects to phase transitions in
statistical mechanics \cite{Christensen:2005,Thompson-88}
and composite materials \cite{Stauffer-92,Torquato:RHM-02}. 
Such systems often exhibit universal critical behavior,
where order parameters such as the effective conductivity
of a composite or the magnetization of an Ising ferromagnet
are described near their threshold
by critical exponents depending only on 
dimension and not on the details of the system. 
Developing such a theory for melt ponds 
could provide fundamental new insights into the 
evolution of ice pack albedo and Arctic melting, as
well as a rigorous framework for finding 
large scale properties from local information. 
For example, the scale of connected
fluid pathways on the surface influences melt pond drainage
through cracks and seal holes, as well as through the 
porous microstructure of sea ice.
Given the important role of
ice$-$albedo feedback in summer
Arctic sea ice losses \cite{Perovich:GRL-2008}, 
an analysis of critical behavior 
and related phenomena in dynamical systems
will advance our understanding of the 
Arctic sea ice pack,
and our ability to model its future trajectory. 

\vspace{-1ex}

\begin{enumerate}
\item [$\bullet$] We will investigate in detail the development of
the space filling boundaries
of melt ponds, and construct mathematical models of their
evolution. 

\vspace{-1.2ex}


\item [$\bullet$] Models of phase transitions in 
statistical physics will be used 
to explore critical behavior in the evolution of melt pond 
configurations. 

\vspace{-1.2ex}


\item [$\bullet$] Homogenization theory and spectral 
measures will be used to 
analyze melt pond and ice pack geometry.
%considered as composite structures of ice and water.
Effective properties 
relevant to climate models
such as albedo 
will be calculated in terms of the 
{\it composite geometry} and its multiscale features.



\vspace{-1.2ex}


\item [$\bullet$] 
A dynamical systems model of ice$-$albedo feedback 
for a composite of ice floes and open water 
will be developed. 



\vspace{-1.2ex}


\item [$\bullet$] 
We will conduct related field research on melt pond evolution
with aerial photos, and investigations of critical behavior
in the porous
micro- and macro-structure of the ice pack, 
and its transport properties relevant to 
fluid flow and melt pond drainage. We will work in 
close collaboration with our
colleagues such as Don Perovich.


\end{enumerate}






















\subsection{Critical behavior in the Ising model and random resistor network}


In order to provide the basis for our statistical mechanics
framework for composites, 
we briefly review the 
nearest neighbor Ising model of a ferromagnet in a field $H$ and 
at temperature $T$ \cite{Thompson-88,Christensen:2005,Golden:JMP-5627}. 
Our treatment will also expose the connection
to the analytic continuation method.
We consider
a finite box $\Lambda \subset \mathbb{Z}^d$ containing $N$ sites.
At each site there is a spin
variable $s_i$ which can take the values $+1$ or $-1$. 
We consider a Hamiltonian with ferromagnetic pair interaction 
$J \geq 0$ between
nearest neighbor pairs $<i,j>$,
\begin{equation}                                                                            
{\cal{H}}_\omega=-J\sum_{<i,j>}{s_is_j}\;-\;H\sum_i{s_i}\; ,                 
\end{equation} 
for any configuration $\omega \in \Omega=\{-1,1\}^N$ of the spins. 
The canonical partition function
$Z_N$ is given by
\begin{equation}                                                        
     Z_N(T,H)=\sum_{\omega \in \Omega}\exp(-\beta {\cal{H}}_\omega) = 
         \exp(-\beta N f_N)  ,      
\end{equation}
where  $\beta=1/kT$, $k$ is Boltzmann's constant, 
and $f_N(T,H)=\frac{-1}{\beta N}\log{Z_N(T,H)}$ 
is the free energy per site.
We are interested in the infinite volume limit 
$f(T,H) = \lim_{N\->\infty} f_N(T,H)$.  
Then the average magnetization                                      
$
M(T,H) =\lim_{N\->\infty}     
\frac{1}{N}\langle \sum_{i=1}^{N}{s_i} \rangle,
$
where $\langle \cdot \rangle$ in this context
denotes averaging over $\omega \in \Omega$ with
Gibbs weights, can be expressed in terms of the
free energy as 
\begin{equation}     \label{mag_free}
M(T,H)=-\frac{\partial f}{\partial H}\;.
\end{equation}
The magnetic susceptibility $\chi(T,H)$ is given by
$\chi(T,H)=\frac{\D M}{\D H}=-\frac{{\D}^2f}{\D H^2}\geq0.$  
When $H=0$, $M(T) \sim(T_c-T)^{\beta}$ as $T\->{T_c}^-$, and 
$\chi \sim (T-T_c)^{-\gamma}$ as $T\->{T_c}^+$. 
Along the critical isotherm $T=T_c$, 
$M(H)\sim H^{1/ {\delta}}$ as $H\->0^+$.
%where $\delta$ and the other exponents have different numerical 
%values  from their analogues in transport or the classical
%percolation model.









The correspondence  between transport in composite media
and statistical mechanics is based on the following 
remarkable property of the partition function.
In 1952, Lee and Yang \cite{Lee:PR-411,Ruelle}
found that the zeros of the partition function of the Ising
ferromagnet (or lattice gas) lie on the unit circle in the $z-$plane, 
where $z=exp(-2 \beta H)$ is the ``activity.''
%$\beta=1/kT$, and $k$ is Boltzmann's constant.
Then %$Z_N$ can be expressed as
\begin{equation}     \label{partition_function}
 Z_N(z) = a_N\prod_{n=1}^N{(z-z_n)}, \;\;\;\; |z_n|=1.
\end{equation}
Equivalently, the Lee-Yang zeros 
lie on the imaginary axis in the $H-$plane. For $T>T_c\;$,
there is a gap $\theta_H$ in these zeros 
around $H=0$ \cite{Ruelle:1971:PRL}, 
which
collapses as $T\->{T_c}^+$, with $\theta_H \sim (T-T_c)^{\Delta}$.
In the $T-$plane
the situation is more complicated,
although in $d=2$ for $H=0$, ``Fisher's zeros'' 
lie on two circles in the complex $v=\tanh (2 \beta J)$ plane
\cite{Fisher-65},  where $J$ is the interaction strength.










The Lee-Yang Theorem leads to an elegant analytic structure for 
the Gibbs free energy per site $f(T,H)$. For finite volume,
%$f_N=-\frac{1}{\beta N}\log{Z_N}$, or
%\begin{equation}        \label{free_energy_N}
$f_N = -\frac{1}{\beta} \sum_{n=1}^N \frac{1}{N} \log{(z-z_n)} -
       \frac{1}{\beta N} \log{a_N}.$
%\end{equation}
In the thermodynamic limit as $N \rightarrow \infty$, 
\begin{equation}        \label{free_energy}
 f(T,H) = -\frac{1}{\beta} \int_0^{2\pi} \log{(z-e^{i\theta})}d\nu(\theta) -
                      2d\beta J,
\end{equation}
where $\nu$ is a measure on the circle describing the limiting distribution
of Lee-Yang zeros, and $d$ is dimension. 
The integral is one half the electrostatic potential of
the corresponding distribution of point charges on the unit circle.  
The minimization of closely related sums and integrals on manifolds,
and their relation to many problems in mathematics and applications is 
surveyed in \cite{Hardin:NAMS-2004}.
Baker \cite{Baker:PRL-990} used this logarithmic potential representation
to show that $M$ has the following
analytic structure in $\tau = \tanh(\beta H)$,
\begin{equation}              \label{baker_magnetization}
  M(\tau) = \tau + \tau(1-\tau^2)G(\tau^2), \;\;\; \qquad 
   G({\t}^2)=\int_0^{\infty}\frac{d\psi(y)}{1+{\t}^2 y}\;,
\end{equation}
where $G$ is a Stieltjes (or Herglotz) function of $\tau^2$,
and $\psi$ is a positive measure which for $T>T_c$ is supported 
only in $[0,S(T)]$, where $S(T) \sim (T-~T_c)^{-2\Delta}, T\->{T_c}^+.$
This integral representation was used to obtain the 
classical (two-parameter) scaling relations of statistical  mechanics,
$\beta = \Delta - \gamma$ and $\delta = \Delta / (\Delta-\gamma)$ 
\cite{Gaunt:PRB-1184,Baker-1990}, 
%(which are satisfied by
%the mean field exponents 
%$\beta=1/2, \gamma = 1, \delta = 3, \Delta = 3/2$, 
%and the exact exponents for $d=2, 
%\beta=1/8,  \gamma = 1 \frac{3}{4},  \delta = 15, 
%\Delta=1 \frac{7}{8}$),
as well as {\it Baker's inequalities}
$\gamma_{n+1} - 2\gamma_n + \gamma_{n-1} \geq 0$ for the critical
exponents $\gamma_n$ of the higher field derivatives of the free energy $f$,
or equivalently, of the moments $\psi_n$ of $\psi$,
$\psi_n\sim(T-T_c)^{-\gamma_n}, T\->{T_c}^+$
with $\gamma_0 = \gamma$ \cite{Baker:PRL-990}. 
%The sequence $\gamma_n$
%is actually linear in $n$, $\gamma_n=\gamma+2n\Delta , n\geq0$, 
%with constant gap 
%$\gamma_i-\gamma_{i-1}=2\Delta$ \cite{Fisher:RPP-615,Baker:PRL-990}.



It was observed in \cite{Golden:PRL-3935} that the
Lee-Yang-Baker critical theory outlined above applies to classical transport
problems such as electrical conductivitiy and
permittivity, and magnetic permeability.
In particular, we obtained a new Stieltjes
integral representation 
\begin{equation}
  m(h) = 1 + (h-1) g(h), \qquad \;\;\; 
  g(h)=\int_0^{\infty}\frac{d\eta(y)}{1+h y}\;,
\end{equation}
where $\eta$ is a positive measure depending only on the geometry
of the composite and $y=\lambda/(1-\lambda)$, which is the
direct analogue of Baker's formula (\ref{baker_magnetization}), 
and is a different form of 
formula (\ref{gold1.eq.3}).
For the percolation models such as the
random resistor network, we assume that 
for $p>p_c$, there is a gap 
${\theta_h}\sim(p-p_c)^\Delta$ in the support of $\mu$ around 
$h=0$ or $s=1$, (or that any spectrum in 
this region does not affect power law behavior, which is discussed
in detail in \cite{Golden:PRL-3935} and is supported by 
\cite{Clerc:AP-191,Day:JPCM-96,Bruno:PRSLA-353}), 
so that $\eta$ is supported 
only in $[0,S(p)]$, where $S(p) \sim (p-~p_c)^{-\Delta}, \; p\->{p_c}^+.$
Then, the methods developed by Baker 
\cite{Baker:PRL-990,Gaunt:PRB-1184,Baker-1990}
to analyze the critical behavior of the Ising model can be
modified to obtain scaling relations for the
critical exponents for transport,
$ t = \Delta - \gamma,\;
\delta =\Delta/(\Delta-\gamma).$
The relations hold for lattice and continuum models, even though
continuum models can exhibit non-universal behavior
\cite{Halperin:PRL-2391,Stauffer-92,Torquato:RHM-02}.
The critical
exponents $\gamma_n$ of the higher derivatives of $m(h)$ at $h=0$,
or the moments $\eta_n$ of $\eta$,
with $\eta_n(p) \sim (p-p_c)^{-\gamma_n}$,
also satisfy Baker's inequalities.
%Furthermore, like the Ising model,
%$\gamma_n$ for transport is linear in $n$, $\gamma_n=\gamma+(n-1)\Delta , n\geq1$, 
%with constant gap $\gamma_i-\gamma_{i-1}=\Delta$, consistent with
%the absence of multifractal behavior for the bulk conductivity \cite{Stauffer-92}. 




For a finite sample of a resistor network (or continuum model)
with $N$ resonances $\lambda_n\in[0,1]$, or poles of $F(s)$,
we can construct an analog of (\ref{partition_function}) 
for transport,
%\begin{equation}
${\Z}_N(s) =\prod_{n=1}^N{(s-\lambda_n)}$, 
%\end{equation}
whose zeros are distributed according to a
normalized counting measure by
$\nu_N=\frac{1}{N}\sum_{n=1}^N \delta_{\lambda_n}$.
Then we define a
finite volume ``Gibbs free energy'' \cite{Golden:JMP-5627} via
$
\Phi_N(s)=-\frac{1}{N}\log{\Z}_N(s)=-\int_0^1 \log(s-\lambda)d\nu_N(\lambda).
$
In the infinite volume limit \cite{Assche},  
\begin{equation}        \label{free_energy}
\Phi(s)=-\int_0^1 {\log(s-\lambda)d\nu(\lambda)}.
\end{equation}
It is a standard result of logarithmic potential theory that $\nu$ is the unique equlibrium measure of the ``electrostatic energy'' given by $\mathcal{E}[\nu]=\inf_m\mathcal{E}[m]=\inf_m\left( -\int_{\sigma\times\sigma} \log|\lambda^\prime-\lambda|dm(\lambda)dm(\lambda^\prime) \right)$ varied over all probability measures $m$ supported on $\sigma=supp(\nu)$ 
\cite{Pastur,Assche}.
We expect this minimum principle to aid in the characterization
of ground states and equilibrium configurations of composite systems.


%It is a standard result of logarithmic potential theory that $\Phi(s)=\inf_m\left[ -\int_\sigma \log|s-\lambda|dm(\lambda) \right]$ varied over all probability measures $m$ supported on $\sigma=supp(\nu)$ \cite{Pastur,Assche}.

%******Logarithmic capacity - Ben********







In \cite{Golden:JMP-5627,Golden:PRL-3935} 
the zeros of the transport partition function ${\Z}_N(s,p)$ in the complex
$p-$plane were considered. They are the analogs of Fisher's zeros 
\cite{Fisher-65} in the complex temperature plane.
For the $d=2$ random resistor network we found that the
percolation threshold $p_c=\frac{1}{2}$ can be characterized as 
an accumulation point of
its zeros.  
%It was also proven \cite{Golden:JMP-5627} that 
%for {\it infinitely interchangeable media}, (see, e.g. \cite{Bruno:JSP-365}), 
%which include lattice and cell materials in the 
%continuum, such as the random checkerboard,
Also, $m(p,h)$ is analytic in $p$ for $|h-1|<1$ in a 
domain which includes [0,1],
rigorously establishing the existence of a gap $\theta_p$ in the zeros 
in the $p-$plane around $p_c$. With $\theta_p \sim (p-p_c)^{\Delta_p}$,
${\Delta_p} \leq 1$.






**********************************



\section{Mathematical Methods}


\subsection{Analytic continuation and the spectral measure}

{\it Homogenization} denotes a field of applied mathematics
where the goal is to find a homogeneous medium which behaves
macroscopically just like a given inhomogeneous medium.
We briefly describe the analytic continuation method
for studying the effective properties of composite materials
\cite{Bergman:PRL-1285,Milton:APL-300,Golden:CMP-473,Golden:IMA-97},
or for homogenizing inhomogeneous media.
Let $\epsilon(x,\o)$ be a spatially stationary random field
in $x\in \mathbb{R}^d$, where $d$ is spatial dimension,
and $\o \in \O$, where $\O$ is the
set of all realizations of the random medium,
which represents the local values 
of the complex permittivity. This key parameter is defined in 
basic electromagnetism and determines the propagation
properties of an electromagnetic wave in a medium.
The real part represents the polarizability of the medium,
and the imaginary part the losses. 
We assume  $\epsilon(x)$ (where we supress the $\o$ notation)
takes the values 
$\epsilon_1$ in bone marrow
and $\epsilon_2$ in bone, where both of these 
parameters depend on frequency,
and write
%\begin{equation}                                 \label{eq2.1}
$\epsilon(x) = \epsilon_1 (1-\chi (x)) + \epsilon_2 \chi(x),$
%\end{equation}
where $\chi$ is the characteristic function of the bone, 
which equals one for all realizations $\o \in \O$
having bone at $x$, and equals zero otherwise.
Let $E (x)$ and $D(x)$ be the stationary
random electric and displacement fields satisfying
the constitutive law
%\begin{equation}                                 \label{eq2.2}
$    D (x)= \epsilon (x) E (x) $
%\end{equation}
and the equations
\begin{equation}                                 \label{eq2.3}
\nabla \cdot D  = 0, \;\;\;\;\;
\nabla \times E  = 0,
\end{equation}
with
%\begin{equation}                                   \label{eq2.5}
$\langle E (x) \rangle = e_k,$
%\end{equation}
where $e_k$ is a unit vector in the $k^{th}$ direction for
some $k = 1, \ldots, d$, and $\langle \cdot \rangle$ means
an ensemble average over $\O$ or spatial average over all of
$\mathbb{R}^d$.

The effective complex permittivity tensor ${\beps}^*$
is defined by
\begin{equation}                                    \label{eq2.6}
    \langle D \rangle= {\beps}^* \langle E \rangle.
\end{equation}
Without loss of generality, 
we focus on one diagonal coefficient
$\e^*~=~\e^*_{kk}$. Due to homogeneity,
$\e^*(a \epsilon_1, a \epsilon_2)= a
\e^*(\epsilon_1,\epsilon_2)$, for any complex number $a$,
$\e^*$ depends only on the ratio
$h = \epsilon_2 / \epsilon_1$, and we define
$m(h) =\e^* / \epsilon_1$.
The two main properties of $m(h)$ are that it is analytic
off $(- \infty, 0]$ in the $h$--plane, and that it
maps the
upper half plane to the upper half plane,
%\citep{Bergman:PRC-377,Golden:CMP-473},
so that it is an example of a Herglotz, or Stieltjes function.
The key step in the
analytic continuation method is obtaining
an integral representation for $\e^*$.

It is more convenient to work with 
the function %\citep{Bergman:PRC-377}
%\begin{equation}                                    \label{eq2.7}
$F(s) = 1 - m(h)$, where
$s = 1/(1-h)$,
%\end{equation}
which is analytic off $[0,1]$ in the $s$--plane.  It
was proven \cite{Golden:CMP-473,Bergman:PRC-377} 
that $F(s)$ has the 
representation
\begin{equation}                                     	\label{eq2.8}
    F(s)=\int_0^1 {\frac{d\mu(\lambda)}{s-\lambda}}\;, 
\qquad  s\notin[0,1]\;,
\end{equation}
where $\mu$ is a positive measure on $[0,1]$. 
This formula arises from the resolvent
representation  of the electric field 
$E = s(s + \Gamma \chi)^{-1}e_k$,
where $\Gamma  = \nabla (- \Delta )^{-1} \nabla \cdot$ 
and $\Delta=\nabla^2$ is the Laplacian, 
yielding
\begin{equation}                              	\label{gold1.eq.3a}
F(s) = \langle \chi [(s + \Gamma \chi)^{-1} e_k] \cdot e_k \rangle.
\end{equation}
%The operator $\Gamma$ acts as a
%projection onto curl-free fields, and is based on
%convolution with the 
%free-space Green's function associated with $-\Delta$. 
In the Hilbert space $L^2 (\Omega)$ with weight $\chi$ in
the inner product, $\Gamma \chi$ is a bounded self adjoint
operator. % with $\parallel \Gamma \chi_1 \parallel \leq 1$.
Formula (\ref{eq2.8})
is the spectral representation of the resolvent, 
and $\mu$ is a {\it spectral measure} of $\Gamma \chi$, in the 
$e_k$ state.
The integral representation
(\ref{eq2.8}) separates the
parameter information in $s$
from information about the mixture geometry 
contained in $\mu$.
Statistical assumptions about the 
geometry are incorporated into $\mu$ via its moments
%\begin{equation}                                   \label{eq2.9}
$\mu_n = \int^1_0 \lambda^n d \mu (\lambda)$,
which can be calculated from the
correlation functions of the random medium, with 
\begin{equation}
\mu_n = (-1)^n \langle \chi 
[( \Gamma \chi )^n e_k ] \cdot e_k \rangle.
\end{equation}
The mass $\mu_0$ is the 
bone volume fraction $\phi$ (with $1-\phi$ the porosity),
\begin{equation}                                   \label{BVF}
\mu_0 = \int^1_0  d \mu (z) = 
\langle \chi \rangle = \phi.
\end{equation}
%In the notation we use here, the porosity of the
%bone composite, or volume fraction of the marrow phase,
%is $1-\phi$.


A principal application of the analytic continuation method
is to derive {\it forward bounds} on $\e^*$ given partial
information on the microgeometry
\cite{Bergman:PRL-1285,Milton:APL-300,%
Golden:IMA-97,Golden:TGRS-1655,Gully:2007:Physica}.
The objective of {\it inverse bounds}
is to use data about 
the electromagnetic response of a composite to bound 
its structural parameters such as $\phi$ 
\cite{Cherkaeva:WRM-437,Golden:TGRS-1675,%
McPhedran:1982,McPhedran:1990,Cherkaev:1996,Tripp:1998}.
Below we will use a more sophisticated inversion
scheme over frequency
to reconstruct the spectral measure, and obtain
accurate estimates of bone volume
fraction $\phi$.
A powerful application of this inverse 
homogenization approach \cite{Cherkaev:2001}   
is that once the spectral measure has been 
reconstructed, it can be used to calculate other effective 
properties such as DC electrical conductivity or thermal 
conductivity, which have
%These other transport properties have 
analogous integral representations. 
%but involving the 
%ratio of the relevant component parameters. 
Thus, through this spectral coupling, other transport properties can 
be estimated indirectly from EM monitoring 
\cite{Cherkaev:2001,Cherkaev:2003,Cherkaev_Zhang:2003}.

 





















 




\subsection{Reconstruction of the spectral measure}



The problem of extracting microstructural information
from bulk property measurments %using analytic continuation
was introduced previously by 
\cite{McPhedran:1982} and \cite{McPhedran:1990}.  
They estimated the
volume fraction of one component in a two phase mixture from
measurements of the effective complex permittivity of the composite
material. An analytical approach
to estimating the volume fractions of materials in a composite was
developed \cite{Cherkaev:1996,Tripp:1998},
and extended to the problem of finding bounds on the
microstructural parameters for isotropic composite materials
in \cite{Cherkaeva:WRM-437}. 
The analytical method is based on the integral
representation in (\ref{eq2.8}) and gives explicit formulas 
for bounds on the volume fraction of one 
of the constituents \cite{Cherkaeva:WRM-437}.












The problem of characterizing
microstructural information from effective property measurements was
formulated as an inverse problem for the spectral measure $\mu$ in
the Stieltjes analytic representation \cite{Cherkaev:2001}. 
Uniqueness of the reconstruction
of the spectral measure \cite{Cherkaev:2001} gives a
basis for the theory of inverse homogenization. 
The spectral measure $\mu$ in the
integral representation (or spectral function) 
can be uniquely reconstructed if
measurements of the effective properties of the composite are
available along an arc in the complex $s$--plane 
\cite{Cherkaev:2001}.
Such data can be
obtained from measurements in an interval of frequency provided that
at least one of the constituents is frequency dependent.









The numerical problem of reconstructing the spectral function is
ill-posed, that is, the solution is highly sensitive to changes in 
the data,
and requires regularization \cite{Cherkaev:2001}. Several
regularized algorithms have been developed
\cite{Cherkaev:2001,Cherkaev:2003,%
Cherkaev_Zhang:2003}. To obtain a stable
reconstruction of the spectral measure, an inversion method based on
constrained rational approximation of the spectral function was
developed in \cite{Zhang:2008,Zhang:2009}.




The inverse homogenization method was extended to the viscoelastic problem in
\cite{Bonifasi:2006,Bonifasi:2008} for torsion
of a cylinder whose microstructure does not change in the axial direction. With this
simplified model of bone, this method was used successfully to recover porosity
from measurements of the effective shear modulus simulated using micro-CT images
of cancellous bone \cite{Bonifasi:2008, Bonifasi:2009a}.
An electrical impedance spectroscopy method developed in
\cite{Bonifasi:2009} shows potential for reconstructing bone
porosity from effective electrical measurements.
The method is based on reconstruction of the spectral 
function from electrical measurements and calculating the
zeroth moment, which gives the volume fraction of one of 
the bone components \cite{Bonifasi:2009}. 
The method was applied to the
effective electrical properties of cancellous bone, and
numerically calculated using micro-CT images of human 
vertebrae to find the bone porosity. 
It was shown that bone porosity can be accurately 
reconstructed even in the presence of errors in the 
measured data and uncertainty in the properties of bone 
and bone marrow.











 The problem of reconstructing the
spectral measure $\mu$  can be reduced to an inverse potential
problem. It was  shown in \cite{Cherkaev:2001} that 
$F(s)$ admits a representation
as  a logarithmic potential of the measure $\mu$:
\begin{equation}                               \label{ln}
F (  s  )   =
   \,  \frac{\D \Phi} {\D s }\,  , \qquad
 \Phi(s) = \int_0^1 \ln\,  | s - \lambda | \, \,   d \mu(\lambda) \;,
\end{equation}
where $\frac{\D}{\D  s} =   \frac{\D}{\D x} - i \frac{\D}{\D y}\;$.
The potential $\Phi$ solves the Poisson equation
$ - \Delta \Phi = \rho$,  
where  $\rho(\lambda)$  is  a density on
$[0,1]$. A solution to the forward problem is given by the
Newtonian potential with $ \mu(d\lambda) = \rho(\lambda)d\lambda$. 
The inverse problem is to find $\rho(\lambda)$ (or $\mu$) given
values of $\Phi, \partial \Phi \, /
\partial n $,  or  $\nabla \Phi$.



%It is known that the reconstruction problem for the logarithmic
%potential is extremely ill-posed  and  requires  regularization to
%develop a stable numerical algorithm. 
To construct the solution to the inverse problem we
formulate the related minimization problem:
\begin{equation}                               \label{minimization}
\mbox{min}_{\mu} \; || A \mu - F || \,,
\end{equation}
where $A\mu = 
\frac{\D}{\D s}\int_0^1 \ln\, | s - \lambda | \, \,d \mu(\lambda) $, 
%is the integral operator in  (\ref{ln}), 
the norm  is
the $L^2-$norm, and $F$ is the given function of the measured data, $F(s)
= 1-\e^*(s)/\e_1, ~ s \in {\mathbb{C}} $.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The solution of the problem does not depend continuously on the
data. Unboundedness of the operator $A^{-1}$  leads to arbitrarily
large variations in the solution, and the problem requires a
regularization technique.

A regularization algorithm developed in \cite{Cherkaev:2001} is
based on constrained minimization.  It  introduces   a stabilization
functional   which  constrains the set of  minimizers. As a result,
the  solution depends continuously on the input data. Instead of
minimizing (\ref{minimization}) over all functions, the minimization is
performed over a convex subset of functions, which results in a
stable reconstruction algorithm. The stabilization functional 
added to the minimization functional in (\ref{minimization}), 
with regularization parameter  $\alpha$, was
chosen \cite{Cherkaev:2001,Bonifasi:2009} as a quadratic functional
corresponding to Tikhonov regularization. We use this approach here
as well.






\subsection{Percolation theory}





Lattice and continuum percolation theories 
\cite{Stauffer-92,Torquato:RHM-02,Hornung:1997}
have been used to model a broad range of disordered
materials where the connectedness of one phase
dominates effective behavior. 
Consider the square ($d=2$) or cubic ($d=3$) network of bonds 
joining nearest neighbor sites on the integer lattice 
%${\mathbb{Z}}^d$.
${\mathbb{Z}}^d$.
%In the random pipe network,
We consider the problem of electrical transport 
through the network. 
The bonds are assigned electrical conductivities
$\sigma_0 > 0$ (open) or 0 (closed) with probabilities $p$ 
and $1-p$. 
Groups of connected open bonds 
are called open clusters, and the average cluster size
grows as $p$ increases.
The striking feature of this model is that 
there is a critical probability $p_c$, $0<p_c<1$,
called the {\it percolation threshold}, 
where an infinite cluster of open bonds first appears.
In $d=2$, $p_c = \frac{1}{2}$, and in $d=3$,
$p_c \approx 0.25$. 
Typical configurations for the $d=2$ square lattice
above and below the threshold
are shown in Figure \ref{percolation_theory}. 
Near the threshold, or transition point, the order parameters
of the system, such as the infinite cluster density,
the correlation length, and the effective conductivity
are characterized by universal critical exponents
which depend only on dimension, and not on the 
details of the lattice.





An important feature of phase transitions in statistical mechanics,
such as in the Ising model of a ferromagnet
\cite{Thompson-88},
is the collapse of a spectral gap as the critical point is approached
\cite{Baker-1990,Baker:PRL-990}.
Similarly, there is a gap in the support of the spectral measure
for $p$ below $p_c$
\cite{Clerc:AP-191,Day:JPCM-96,Golden:PRL-3935,%
Golden:JMP-5627,Jonckheere_Luck_JPA_1998},
which collapses as the threshold is approached. 










\section{Results}



\subsection{Numerical calculation of the spectral measure for bone}
%

In Section 2 
we discussed how the effective
properties of a composite such as bone may be expressed in terms of the
spectral measure $\mu$ of a self adjoint operator $\Gamma\chi$, where
$\chi$ is the indicator function for the bone and
$\Gamma=\nabla(-\Delta)^{-1}\nabla\cdot$ acts as a 
projector onto curl free fields. 
We observed that $\Gamma \chi$ is a 
self-adjoint operator with respect to the inner product 
with weight $\chi$. Here it is more convenient to use the 
standard $L^2$ inner product and consider the 
self-adjoint operator $\chi \Gamma \chi$ (the two approaches 
are easily seen to be equivalent).
In order to identify bone microstructures with spectral measures, 
we have set threshold levels and then discretized two 
SEM images of trabecular bone
from http://hansmalab.physics.ucsb.edu. 
The first image, shown in 
Figure \ref{bone_pics_spectral_measures}a, is from a
young (22-year old) healthy adult man. The second, shown in
Figure \ref{bone_pics_spectral_measures}b, is from 
an elderly, osteoporotic woman in her eighties.
Discretizations of these images are shown in 
Figure \ref{bone_pics_spectral_measures}c and 
\ref{bone_pics_spectral_measures}d.
We remark that the images in 
Figure \ref{bone_pics_spectral_measures}a and 
\ref{bone_pics_spectral_measures}b
are three dimensional, but our discretizations
are two dimensional representations (surface slices)
of the underlying images. Our subsequent calculations
of the spectral measure are then two dimensional
as well.


The discretizations are associated with networks 
of 1's (bone) and 0's (marrow), as in the 2-D square 
lattice percolation model, 
in order to calculate the corresponding spectral
measures. On this square lattice
the action of the differential operators $\nabla$ and $\nabla\cdot$ are 
defined in terms of the
forward and backward difference operators
\cite{Golden:CMP-467}. Furthermore, the action of $\chi$ is that of a
square diagonal matrix with ones and zeros along the diagonal
corresponding to type two bonds (bone) and type one bonds (marrow),
respectively. Therefore the action of the self adjoint operator
$\chi\Gamma\chi$ on this lattice is given by that of a real symmetric matrix,
and the spectral measure may be calculated directly from its
eigenvalues  $\{\lambda_i\}$ and eigenvectors $\{v_i\}$ via the formula
%\citep{Golden:CMP-473}  
%
\begin{equation}                    \label{Spectral_Measure}
  d\mu(\lambda)=\sum_{i=1}^{n} m_i \delta(\lambda-\lambda_i) d\lambda,\quad
  m_i=\langle  e_0^T v_i v_i^T e_0  \rangle, 
\end{equation}
%
where $\delta(\lambda)$ is the Dirac delta function and $e_0$ is a vector of
ones. The effective complex permittivity is given by 
%
\begin{equation}                    \label{Lattice_permittivity}
F(s)=\sum_i\frac{m_i}{s-\lambda_i}\;, \qquad
\epsilon^*=\epsilon_1(1-F(s)),
\end{equation}
%
%(see equation \eqref{eq:Fs}). 
where $s=1/(1-\epsilon_2/ \epsilon_1)$.


%The central object of the theory is the spectral function $\mu(\lambda)$
%which, for matrices, is defined pointwise at the eigenvalues with
%$m_i=\mu(\lambda_i)$. 




As the system size $N$ increases, the size of the matrix
$n$ increases proportionally by $N^2$, and the eigenvalues become
increasingly dense in the spectral interval [0,1]. 
For a random system averaged over many realizations 
(or using a large-enough spatial sample of a fixed realization), 
a high resolution histogram of the spectral
measure begins to resemble a smooth curve,
as in Figure \ref{RRN_realizations_spectral_measures}c and 
\ref{RRN_realizations_spectral_measures}d. 
The presence or absence of spectral
gaps at the endpoints of the spectral interval
and the details of how large the gap is or how large the spectral
values $m_i$ are at the collapse give important information pertaining to
the connectivity and electromagnetic transport properties of the system. 









We have approximated the 2-D geometric structure in pictures of sliced
healthy and osteoporotic trabecular bone by a $540\times540$ square
lattice bond network. The CPU time for solving the spectral problem for a
given square sample grows like $N^6$. We have optimized the
use of our algorithm by averaging over six coarser discretizations of
the original $540\times540$ discretizations. 
Each of the $90\times90$ coarser representations sample
every 6th data point of the original $540\times540$ system with a shifted
starting point, sampling every data point of the fine mesh $540\times540$
discretization. 
As in \cite{Jonckheere_Luck_JPA_1998}, where the spectral
measure was obtained by an indirect method, in our method
of calculating the spectral measure, we separate the calculation
of the delta function at the left endpoint $\lambda =0$
from the calculation of the spectral measure in the rest of the 
interval. In order to see the details of the spectral measure
throughout the interval, we avoid plotting the delta function
at $\lambda =0$.





The spectral measures for healthy and osteoporotic bone
are shown  
in Figure \ref{bone_pics_spectral_measures}e and 
\ref{bone_pics_spectral_measures}f, respectively.
The actual bone volume fractions in the discretized images are
0.5386 and 0.2431, respectively. For comparison,  
in Figure \ref{RRN_realizations_spectral_measures}a and 
\ref{RRN_realizations_spectral_measures}b 
we show realizations of the 
$30\times30$
percolation model with $p=0.5$
and $p=0.25$, respectively, along with direct calculations of the
spectral meaures (averaged over 5000 statistical samples)
in c and d.
The graphs show linearly connected peaks
of histograms with bin sizes on the order of $10^{-2}$ and $10^{-1}$ for
$p=0.5$ and $p=0.25$, respectively. The function constructed
in this way from the weights $m_i$ in the spectral measure
is called the {\it spectral function} $\mu(\lambda)$, and provides
a graphical representation of the measure.  
The apparent smoothness of
the spectral function graphs corresponding to the random bond networks
is due to the large number ($\sim10^6$) of eigenvalues and eigenvectors
calculated. In contrast, the jaggedness of the spectral function
graphs corresponding to the discretizations of trabecular bone is due
to the relatively small number ($\sim10^4$) of eigenvalues and
eigenvectors calculated. In order to reveal the detail of the spectral
functions with $p=0.5$ for the lattice model and for the young bone,
we have limited the plot domain to
$(\eta,1-\eta)$ with $\eta\sim10^{-15}$, eliminating some of the divergent
behavior in the graph. No such restriction was necessary for
the spectral functions with $p=0.25$ for either the lattice model
or the old bone. 

In Figure \ref{brine_connectivity} we show how the connectedness 
of the brine microstructure in sea ice changes dramatically 
with small changes in temperature. If trabecular connectedness
behaved similarly, an individual could develop osteoporosis
from a fever!





columnar 
sea ice is effectively impermeable to fluid flow through sea ice.
When the brine volume fraction exceeds 5\%, the ice is increasingly
permeable with increasing temperature. For a typical 

 can be viewed 


and composite materials. 

Rather than focusing initially on the specific details of 


Our approach is to take a step back from detailed, 
mechanistic models of processes like melt pond 
evolution, and instead
explore more general features like
critical behavior, which could potentially transform
our understanding of this important process. 
Investigating 
critical behavior as a natural aspect of the polar 
marine environment and its role in the climate system
will shed light on key issues such as the rapidity
of the sea ice retreat and whether a 
so-called {\it tipping point} or critical transition
has been passed in the decline
\cite{Eisenman:PNAS-2009}. 






subsequent melt pond evolution;
quite complicated evolution influenced by a range of factors;
difficult to understand evolution and formation
here we begin a new approach.... view as a phase transition
look for universal behavior, percolation thresholds, etc.
\cite{Perovich:GRL-2008}
is thus affected by this decrease in albedo. 
\cite{Perovich:2003:JGR}



******************************


Given the
high contrast in the albedo of these two principal constituents
of the sea ice pack, 
It is not surprising, then, that the formation and evolution
of melt ponds is one of the physical processes which is not 
well represented in sea ice and climate models. 
\cite{Eicken:JGR-C08007}
\cite{Perovich:2002:JGR}
\cite{Perovich:2003:JGR}



The melting Arctic sea ice pack has as 
The albedos of the main constituents of of the
Arctic sea ice pack, namely snow, ice and  are highly contrasting. 
Developing a better understanding 
spin off of homogenization is the propagation of 
waves in the marginal
ice zone is effective propagation 
\cite{Flocco:2010:JGR}
\cite{Polashenski:JGR:2012}




******  ONR EM ARI 
let theoretical efforts
took the role of lead author on the principal modeling
papers
Prof Hohehegger is a very talented 
specialist in using partial differential equations
and numerical computation to model the behavior of
complex systems involving fluid motions



*************************



\textbf{Fractal behavior:} At the beginning of the Arctic Summer, small, shallow and mostly circular ponds form as the surface of the ice melts (see Fig. \ref{fig:meltponds}(a)). As the melting rate increases, the surface area and the depth of the melt ponds increase until a sharp percolation threshold. Past this transition, melt ponds become connected on larger length scales and start to fill the entire image space (see \ref{fig:meltponds}(b), (c)). We remark that smaller disconnected melt ponds remain during the entire season.  By mid-August (see  \ref{fig:meltponds}(f)), very large connected ponds spanning the entire image have formed. Finally, in September, the ponds freeze over and disappear. Figure \ref{fig:meltponds} illustrates the three distinct stages of melt ponds formation together with their connected water components. In Fig. \ref{fig:meltponds}(c)-(f), each color corresponds to a single connected component and these color coded figures give us an easy way to visualize the size and length scales of melt ponds.


****************************







Since melt ponds can extend beyond a single image, and 
since identifying and tracking melt ponds, ice floes, 
and their boundaries is
a significant image processing problem, 
we have begun a collaboration with Tolga Tasdizen 
and the Scientific Computing and Imaging Institute (SCI) 
at the U. of Utah. We plan to further develop this collaboration
through the DRI. We will also use remote sensing
and satellite images in our analysis 
and upscaling theories.




The co-Pi Hohenegger has taught a week long class on
the shape of the universe, and plan to stay involved with the program,
in particular in the ongoing development of classes like the one
entitled "scaling and fractal". 







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




One way of geometrically capturing the transition from small, 
circular ponds to fully connected large ponds is the fractal exponent $D$, 
defined as $P\sim \sqrt{A}^{\; D}$ where $A$ and $P$ are the area and 
perimeter, respectively. For regular objects like circles and polygons, 
the perimeter scales like the square root of the area and thus $D=1$. 
On the other hand, as the boundary becomes more complex and starts 
filling more of the available space, $D\sim A$ and thus $D\approx 2$. 
We remark that $D=2$ is the upper bound for the two dimensional 
fractal exponent, since it corresponds to curves which 
fill two dimensional space, such as the famous Peano curve or Brownian motion. 





Motivated by early work of Lovejoy
\cite{Lovejoy:S:1982} on the fractal dimension of clouds,
a couple year


whose albedo is another significant source of uncertainty in climate models $-$
we propose to investigate the complex boundaries of melt ponds as 
they evolve over the spring and summer. Our initial discovery 
of this transition, shown in Figure \ref{fig:areaperimeter},
occurred while preparing the planning letter for the MIZ DRI. 



One of the most exciting investigations we propose is 
to explore the
transition in fractal geometry at a critical length scale
as melt ponds grow. 


One way of geometrically capturing the transition from small, 
circular ponds to fully connected large ponds is the fractal exponent $D$, 
defined as $P\sim \sqrt{A}^{\; D}$ where $A$ and $P$ are the area and 
perimeter, respectively. For regular objects like circles and polygons, 
the perimeter scales like the square root of the area and thus $D=1$. 
On the other hand, as the boundary becomes more complex and starts 
filling more of the available space, $D\sim A$ and thus $D\approx 2$. 
We remark that $D=2$ is the upper bound for the two dimensional 
fractal exponent, since it corresponds to curves which 
fill two dimensional space, such as the famous Peano curve or Brownian motion. 
Figure \ref{fig:areaperimeter} displays area-perimeter data 
for a subset of segmented images for August 22nd (HOTRAX). 
We observe a transition between circular ponds with $D=1$ 
and ramified ponds with 
$D \approx 2$. The transition from
Euclidean to space filling fractal 
behavior occurs for ponds of area around a hundred square meters.
One of the key issues is to explain why the critical transition 
occurs there, and to explore its relation to the percolation threshold
discussed above.


The behavior we observe for melt ponds is in stark contrast 
to similar calculations done for clouds and rain areas where 
the fractal dimension is 1.35, and constant over the entire 
range of accessible length scales \cite{Lovejoy:S:1982}. 
For melt ponds however, the area-perimeter relation in 
Figure \ref{fig:areaperimeter} shows that small length scales 
can be separated from large scales features, which is relevant
for climate models. Interestingly, sea ice floe size
distributions display a similar separation of scales with 
two fractal dimensions \cite{Toyota:JGR:2006}, which 
is another line of inquiry we will pursue.
We note that the transitional length scale appears to be 
about the same across a broad range of data sets for both
SHEBA and HOTRAX. After our initial discovery, Don Perovich 
provided us with area vs. perimeter data sets for SHEBA 
which included sets with as many as 140,000 melt ponds.
They all show the transition occurring at a similar length scale.
It is important in 
this project that we develop software to be able to 
examine closely the area-perimeter relation across multiple 
images. We do not expect the fundamental behavior of 
Fig. \ref{fig:areaperimeter} to change, but with more 
temporal data we will be able to determine the time at 
which the transition occurs. Therefore, one 
key component of the proposed field work will be to 
collect data at the early stages of melt pond formation.


Even though large connected melt ponds have an 
area-perimeter fractal exponent of 2, they of course do not 
entirely fill the space, as do mathematical idealizations.
In particular, their boundary 
curves does not pass through every point of the image 
and thus they cannot be entirely space filling. 
In this project, we will further examine the 
self-similarity of melt ponds by computing the 
Hausdorff dimension of the boundary itself, which 
preliminary calculations show to be approximately 1.8. 
Furthermore, we will develop simple geometrical models,
such as one shown in Figure \ref{geometric_evolution_models}
(a) and (b) 
exhibiting a similar fractal behavior. These models 
are not meant to be physically accurate, but will 
be the starting point for the development of more 
evolved PDE models which will be included 
in the albedo feedback loop and ultimately, climate models.







The Hele-Shaw equations describe the Stokes flow between two parallel
flat plates separated by a thin gap. The resulting flow is purely two
dimensional and it is well known that the shape of the interface
exhibits a fingering phenomenon 
\cite{Saffman:PRSA:1958}. The Hele-Shaw equations describe the evolution
of an interface displaying symmetry breaking and self-similar
structure. Although the physics and length scales are
different, the growth process is similar to what we observe for
melt ponds. Recent numerical progress \cite{Fast:JCP:2004} 
has made it possible to evolve interfaces fast and
accurately. Using such techniques, we propose to develop a Hele-Shaw
type equation for the melt pond evolution. Physical parameters and
initial conditions will be inferred from the available images.
Moreover, the richness of the melt pond images will permit extensive model
validation.



We will also formulate melt pond evolution as a
Stefan problem for melting considered as a non-convex variational 
problem of optimal shape design. In this approach, the shape 
of the boundary between the two phases, water and ice, 
is constructed as an optimal solution to a variational functional. 
This part of the project will be led (informally) by 
Professor Elena Cherkaev in the Math Department,
who has expressed great interest in this project.
She pioneered a robust approach to non-convex structural 
design \cite{Cherkaev:2003,Cherkaev:2008} by formulating 
the variational problem as a min-max optimization problem.
Fractal behavior of the boundary between the different phases 
is typical for optimal composite structures due 
to the ill-posedness of the non-convex optimization problem.
Other PDE approaches will include the eikonal equation 
where the level set of the solution represents the melt pond
boundary.








\begin{figure}                \centering
\includegraphics[width=6.5in]{ONR_Figure_sea_ice_NEW.eps}
   \caption{{\it Spectral analysis of sea ice structures.} 
   Increasingly connected configurations
    of the percolation model (a)-(c),
    melt ponds (d)-(f), and leads in the ice pack
    (g)-(i), with corresponding spectral measures
    and eigenvalue spacing distributions
    below. The area under each spectral function
    in (d)-(f) is the melt pond area fraction $\phi$, and 
    the open water fraction $\phi$ in (g)-(i).
    For the network, the width of the
    gap in the spectrum near
    $\lambda = 1$ for $p=0.1$ shrinks to 0 
     as the percolation threshold $p=0.5$ is
    approached, with similar behavior for melt ponds
    and the ice pack.  }
     
    \label{spectral_measures_RRN_meltponds_icepack}
        
\end{figure}





\subsubsection{Field experiments}
We plan to take a
very active role in the field work $-$ we have found that 
such opportunities play a pivotal role in theoretical
developments, and shape the mathematical work.
For example, participation in the ANZFLUX expedition
to the Eastern Weddell Gyre in 1994 
led to the rule of fives for columnar ice. 
Sea ice electrical experiments during the 2007 SIPEX 
expedition led to finding the electrical signature of this
critical behavior of fluid transport
\cite{Golden:2011,Sampson:DSR:2011}, which will enable us
to investigate percolation horizons in the ice,
and associated drainage processes
during the field campaign. In 2007 we also found
that fine grained granular ice has a higher 
percolation threshold $-$ around 10\% $-$
for fluid permeability \cite{Golden:2011:FTASI}, 
which has significant implications 
for surface layers in the Arctic and melt pond evolution
processes. It will be important to thoroughly 
investigate this phenomenon directly in the Arctic.


We will obtain extensive image
sets of melt pond and ice pack evolution. 
For example, we want to capture the same region on 
multiple days before and after the transition 
to fully formed melt ponds, as well as drainage events. 
We would also like to correlate this evolution 
with changes in the porous microstructure of the ice
as well as larger scale drainage features. 
Using electrical methods to help understand melt pond evolution
would build on the foundation laid in an NSF 
CMG project where we 
(with Hajo Eicken) are developing EM methods to monitor 
such processes. In November-December of 2010 the PI led a team
of 4 from the University of Utah to collaborate with our
New Zealand partners at a field site in McMurdo Sound,
where we advanced our methods for measuring 
the fluid and electrical transport properties of sea ice.
For the DRI, a useful scenario for our work would be  
an aircraft-supported icebreaker survey cruise, combined with a drift station
to obtain time series data.
We will also rely on data from and closely collaborate with colleagues  
such as Don Perovich, to inform and guide the model development and
validation. 







Thanks to Don Perovich we have access to two image sets captured during two measurement campaigns: Healy Oden TRans Arctic EXpedition (HOTRAX, 2005) and Surface Heat Budget of the Arctic Ocean (SHEBA,1998). During the HOTRAX campaign, ten helicopter photographic survey flights following a modified rectangular pattern around the Healy were conducted during the transition from Arctic summer to fall (August-September) at an altitude between 500 and 2000 feet.  Images were recorded with a Nikon D70 mounted on the helicopter with the intervalometer set to ten seconds, resulting in almost no photo overlaps. During the SHEBA campaign, about a dozen helicopter survey flights following a box centered on the Des Groseilliers were conducted between May and October at an altitude of 6000 feet. Images were recorded with a Nikon 35 mm camera mounted on the back of the helicopter. Again there was minimal overlap between pictures. We will use both sets for validation of the algorithms and models, before collecting our own field data.

Recorded images show three distinct and geometrically complex features: open water (ocean), melt ponds and ice. The HOTRAX images are easier to analyze, as the histograms of the RGB pixel values are clearly bimodal. In this case, ice is characterized by high red pixel values, while water corresponds to lower red pixel values. A finer thresholding allows classification of the water as ocean (close to black), shallow water (light blue) and deep ponds (dark blue), as in Figure \ref{fig:segmentation} (a). In the ensuing image, black correspond to ocean, white to ice, green to shallow ponds and red to deep ponds, as in Figure \ref{fig:segmentation} (b). This segmentation introduces errors as some very deep ponds are mislabeled as ocean. To correct this, the directional derivative of the color intensity in each direction or the change in colors surrounding a deep water region is calculated. If the region is part of a melt pond, then the change in color in the original picture is smooth and the directional derivative is continuous in all directions. A discontinuity however reveals the transition from an ocean to ice or shallow ponds. In the last step the correctly labelled shallow and deep ponds are combined to produce a white, blue and black image for ice, melt ponds and ocean, as in  Figure \ref{fig:segmentation} (c). 








*********************************


It has been difficult so far to automatically segment SHEBA images as the distribution of RGB values is not bimodal. In this project, we will thus generalize the steps of the previous segmentation algorithms to the SHEBA data set. This step will require the use of the Mahalanobis distance instead of the Euclidean distance to classify a pixel as ice or water.

Tasdizen \emph{et al.} \cite{Tasdizen10a} have developed a suite of automatic and interactive software tools for mosaicking of microscopy image tiles. More specifically, {\em ir-tweak} is an interactive graphical user interface for image alignment. It allows users to drop and drag fiducial control points on either the moving or the stationary image. As the user moves the control points, the resulting warping of the moving image is shown in real time to the user allowing for a smooth interaction. The user utilizes information in overlapping areas of the moving and stationary images to bring them into alignment. By adding more control points, users can achieve highly deformable warps if necessary. When aligning more than two images, only adjacent (overlapping) image tiles are aligned. All images are then brought into the same coordinate frame by creating a cascade of transformations from the first (anchor) tile to all the other tiles. Figure~\ref{fig:align} shows a series of image tiles aligned into a strip using {\em ir-tweak}. 










Currently, {\em ir-tweak} does not support rigid rotation because it was designed to handle microscopy images which typically require deformable maps. In this project, we will add a rigid rotation capability into {\em ir-tweak} which is needed for better alignment of the ice images into strips. More importantly, strips corresponding to a series of passes will be aligned individually and then to each other using {\em ir-tweak}. This step cannot be done with the current set of transformations in {\em ir-tweak} and requires the rigid rotation transformation. 

Hogrebe \emph{et al.} \cite{Hogrebe11a} have  developed an automatic landmark based algorithm to align light microscopy sections of retina with deformable transformations. Using an established neurite tracing algorithm to identify where axons exist in a section, they were able to define point landmarks at the section boundaries. After calculating landmark similarities based on local landmark spatial configurations, temporary correspondences between two sections using minimum weighted bipartite matching were assigned. In conjunction with the RANSAC~\cite{Fischler81a} algorithm, the correspondences allow a suitable least squares solution for the transformation parameters to then be recovered. They iteratively update the correspondences and non-rigid transformations, with the intent of first correcting for large landmark  displacements and shifting towards making more local corrections.

 In this project, we will modify the Hogrebe algorithm to track melt ponds in time. After segmentation of individual melt ponds, we will use their properties such as color, size, location and other shape properties to match them across time using bipartite matching. This matching will drive a  deformable warping of the image taken at time $t+1$ to the image taken at time $t$. This calculated warp will then allow us to refine the landmark matching. The process will be iterated until convergence.



Finally, the helicopter path is not perfect and there are therefore gaps in the row of images. Small gaps could be closed by extrapolation and iteration, but big gaps in the existing data set will remain a problem. However for future data collection, a couple of vertical helicopter paths could be flown to give more overlap and reference points. 

The development of our specific algorithm is to be combined with existing toolboxes, in particular the Matlab Image Processing Toolbox. Given a black and white image, this toolbox can efficiently find connected components (either ice or water) as in Figure \ref{connected_components} and areas and perimeters of connected components.











%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Mapping Melt Ponds to Mathematical Graphs}
One key idea that gives a new perspective of the structure of melt ponds and allows for the quantitative analysis of its geometry is to recognize that a melt pond can be viewed as  a {\it graph} consisting of {\it vertices} and {\it edges}. In general, a melt pond consists  of simpler round ponds, which correspond to vertices of a graph, connected through thin elongated channels, which correspond to its edges. 
Viewing melt pond as graphs allows for the classification of melt ponds as complex, simple, and transitional. Also, this point of view is directly related to self-similarity of melt ponds, as described in Section \ref{sec:segmentation}. But the main applications of considering melt ponds as graphs are quantifying the horizontal (surface) percolation of melt ponds and understanding the evolution of melt ponds by studying their corresponding graphs. This directly leads to the following question. How to accurately and automatically map melt ponds from photographs to mathematical graphs?

We plan to achieve this by developing an image analysis algorithm and implementing it into a computer software. The main steps of the algorithm can be described as follows:
\begin{enumerate}
\item {\bf Segmentation}. Identify melt ponds from  the given photographs by using the melt ponds segmentation software described in Section \ref{sec:segmentation}. 
\item {\bf Separation}. For each melt pond, separate the thin elongated channels from the round convex parts  by applying appropriate mathematical morphological operations. This is achieved by finding the  binary image $X$ where each pixel belonging to the melt pond is assigned a value of $1$ and all other pixels are assigned a value of $0$. Next
apply a morphological opening to $X$ using an operator that is slightly larger than the maximum width of the  elongated channels. This operation removes these channels from $X$ and leaves the convex parts disconnected. Next apply connected component analysis to these disconnected parts to identify each individual part. 
\item {\bf Graph formation}. Each individual part identified from Step 2 is mapped to a vertex. Next form the appropriate edges between  vertices by matching  the individual vertices with their corresponding ones in $X$. 
The graph $(V_X,E_X)$ is now formed, where $V_X$ denotes the  set of vertices and $E_X$ denotes the set of edges.
\end{enumerate}









\noindent
{\bf 1. IceBridge.}
We have had preliminary discussions with Ken Jezek of the Byrd Polar Research Center,
who serves as co-Lead of the Science Team for the NASA IceBridge Program.
Upon hearing of our proposed research on critical behavior of melt pond evolution
in the Arctic, he indicated a strong willingness to begin planning to make 
available an unmanned aerial vehicle (predator drone) to obtain aerial photos
of melt ponds during the critical formation period in early June. 
By exploiting the capability of these aircraft to loiter for several
days over an area, or transit across a large region, we could dramatically 
increase the amount and quality of imagery available for our
analyses. Moreover, we would attempt to begin this very exciting opportunity
next year in May$-$June 2012, well before the field campaign in 2014. 
Jezek said that our project would advance the objectives 
of their program as well. 


\vspace{1ex}






\eject



\subsection{Outreach Activities}


 
The PI Golden has been 
been extensively involved in 
outreach to students and broader communities,
in undergraduate 
education, and in attracting students into science, engineering, and math. 
We anticipate that this project will fuel even more success in this 
direction, particularly since it combines math, geophysics, 
ocean science, and climate change. 

%We will be involving undergraduates directly in this 
%research program.  
%The co-PI Hohenegger has been very involved with the ACCESS program for
%women in science and mathematics at the University of Utah. During
%the summer of their freshman year, a group of forty women follows a
%specially designed course aimed at giving them a taste of each of the
%major sciences. The ACCESS program provides an
%excellent opportunity to expose smart and motivated women in 
%science to ongoing and exciting research. During their second
%semester in ACCESS, each student gets placed in a laboratory
%and we expect our proposed research on Arctic sea ice to fit well into
%this program component.
