\documentstyle[epsf,psfig,12pt,titlepage]{article}
\topmargin=-.5in
\oddsidemargin=0in
\evensidemargin=0in
\textwidth=6.5in
\textheight=9.0in
\setcounter{totalnumber}{1}

\begin{document}

\noindent
{\bf Example 2, continued.}

To demonstrate the utility of this methodology, we applied it to a
complex geometry in a coastal domain off Massachusetts.  Velocity and
diffusivity fields were specified from a simulation of the March-April
climatological flow (Figure \ref{flow}).  [An alternative representation
of the hydrodynamic field in terms of streamfunction is shown in
Figure \ref{flow2}; we need to choose one of these two figures.]  
The basis of these simulations
is a fully nonlinear, three-dimensional primitive equation model with
turbulence closure (Lynch {\it et al.}, 1996).  This model has been
used extensively to study coastal currents in this region, particularly 
in the area north of Cape Cod (Lynch {\it et al.}, 1997).  Archived
solutions (see {\tt http://www-nml.dartmouth.edu/CircMods/}) were
vertically averaged to obtain $D$ and ${\bf V}$ fields suitable for 
use in Equation (21).  The circulation is dominated by a southward
flowing coastal current, which partially bifurcates south of Cape
Ann.  During this time period, a small portion of the current
splits off and flows through Massachusetts and Cape Cod Bays.  This
branch subsequently rejoins the bulk of the coastal current off
the northern tip of Cape Cod, where the southward flow intensifies
(note that the fine mesh on the steep topography east
of Cape Cod makes the current appear more intensified in Figure
\ref{flow} than it actually is).  The highest diffusivities 
(in excess of 500 ${\rm m^2 s^{-1}}$
are associated with Martha's Vineyard and Nantucket islands south
of Cape Cod; another local maximum (on the order of 400 ${\rm m^2 s^{-1}}$
is located west-northwest of the northern tip of Cape Cod.  Elsewhere,
diffusivity is on the order of 100 ${\rm m^2 s^{-1}}$.  Flow in the
deeper areas offshore is generally sluggish compared to the coastal
current.

Covariance structures $C_{uu}$ for various nodes off the Cape Cod shoreline
illustrate the impact of the coastal geometry and transport fields
(Figure \ref{nodecov}).  Information on nodes along the northeast
coast of Cape Cod (upper two panels) spreads southward with the
coastal current.  In addition, there is significant propagation of
information in the ``upstream'' direction owing to the very
high diffusivity present along the northern tip of Cape Cod, with
significant covariance penetrating all the way into Cape Cod Bay.
In contrast, the covariance structure associated with a node in
the southeast corner of Cape Cod Bay is confined to the bay itself,
due to the weak currents and low diffusivity present in that region.
Similarly, covariance with a node off the southern shore of Cape
Cod is limited to Vineyard and Nantucket Sounds.  These latter two
examples highlight the fact that the coastal geometry is built in to
the covariance structure, as information clearly does not propagate
across land boundaries.

An example objective analysis is shown in Figure \ref{rmrp}a.  The
data are observed concentrations of the toxic dinoflagellate
{\it Alexandrium} spp. from a survey conducted in 1993 as part
of the Regional Marine Research Program in the Gulf of Maine.
Although the distribution is quite patchy, abundance is generally
high in Massachusetts Bay and low elsewhere.  The objectively
analyzed field captures the essence of this pattern, spreading
information according to the covariance structure determined by
the hydrodynamics and assumptions about the underlying process.
Note that the analyzed field does not match the observations exactly;
smoothing is implied by the first-order decay specified in Equation
(21), and there is noise associated with both the observations
and the process.  The accompanying error map (Figure \ref{rmrp}b)
is a normalized presentation of the diagonal of $C_{\tilde u \tilde u}$.
The effect of the hydrodynamic field on the expected error is dramatic;
the coastal current propagates the impact of observations off the
northeast tip of Cape Cod all the way down its eastern shore
to Nantucket island where high diffusivity spreads the information
in all directions.

Comparison of these objectively analyzed distributions with maps of the
same data made with straightforward interpolation techniques demonstrates
the utility of this approach.  For example, cubic spline interpolation
can be used to fill the gaps in between data points (Figure \ref{cubic}).
With such an algorithm, the resolution of the mesh determines the spatial
scale of the interpolated structures.  The resulting fields are typically
truncated beyond the areas of data coverage (Figure \ref{cubic}a).  Various 
techniques can be used to force the extrapolation problem, such as
the introduction of artificial data points in the far field 
({\it e.g.}, Figure \ref{cubic}b).
However, this {\it ad hoc} procedure has obvious shortcomings.  The
objective analysis technique described herein thus has clear advantages
in that it accounts for coastal geometry, a specified
hydrodynamic field, and an explicit mathematical representation of
the underlying processes controlling the distribution of the quantity
being mapped.

\vskip .2in
\noindent
{\bf Acknowledgments}

We are grateful to D. Anderson, B. Keafer and R. Signell for providing the
{\it Alexandrium} spp. data used in Figures XX and XX+1.  Those
data were collected as part of the Regional Marine Research Program
for the Gulf of Maine, Project GMR-20.  Additional support for
this research was provided by NOAA and NSF through the ECOHAB-GOM
and Globec programs.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Figures
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage

\begin{figure}
\epsfxsize=10.0cm
\centerline{\epsfbox{oafe_resahiZ0.eps}}
\vspace{.2in}
\caption{Climatological velocity and diffusivity fields for the March-April
time period.  The scale for the velocity vectors appears in the upper left.
Selected bathymetric contours are overlayed.}
\label{flow}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage

\begin{figure}
\epsfxsize=14.0cm
\centerline{\epsfbox{oafe_resahi_MApsi.eps}}
\vspace{.2in}
\caption{Climatological diffusivity and streamfunction
fields for the March-April time period.  Selected bathymetric contours
are overlayed.}
\label{flow2}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage

\begin{figure}
\epsfxsize=14.0cm
\centerline{\epsfbox{oafe_point_source_exp19a_1.eps}}
\caption{Normalized covariance structures of four nodes off the
Cape Cod shoreline.  In each case the covariance is normalized to
the value at the node in question (indicated by a white dot).
In this example, $k^2 = 2.0 \cdot 10^{-6}$ and
$\epsilon = 2.0 \cdot 10^{-6}$.  The spatial scale of the process
noise is specified to be $2.0 \cdot 10^3$m.}
\label{nodecov}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage

\begin{figure}
\epsfxsize=14.0cm
\centerline{\epsfbox{oafe_rmrp_data_exp21b_cor2.eps}}
\caption{(A) Objective analysis of {\it Alexandrium} spp. concentration.
Observed values are shown as colored dots.  In this example, 
$k^2 = 2.0 \cdot 10^{-7}$ and
$\epsilon = 4.0 \cdot 10^{-3}$ (the variance of the data is
$2.0 \cdot 10^4$).  The spatial scale of the process
noise is specified to be $2.0 \cdot 10^3$m.  (B) Normalized
standard error estimate,
reported as $\Bigl({C_{\tilde u \tilde u} \over Var (u)}\Bigr)^{1 \over 2}$.}
\label{rmrp}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage

\begin{figure}
\epsfxsize=14.0cm
\centerline{\epsfbox{rmrp_cubic_cor2.eps}}
\caption{(A) Cubic interpolation of the same {\it Alexandrium} spp.
concentration data shown in Figure \protect{\ref{rmrp}}a.  Observed
values are shown as colored dots.  The field is truncated in areas
where extrapolation is required.  (B)  Remapping of (A) in which
artificial data points (of zero concentration) are inserted at the
corners of the mapping domain in order to force the interpolator
to fill the entire field.}
\label{cubic}
\end{figure}

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

\end{document}
