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\begin{document}
\bc
\LARGE
{\bf Objective Analysis for Coastal Regimes}
\\
$ ~ $ \\
\normalsize
\Large
Daniel R. Lynch and Dennis J. McGillicuddy, Jr.
\\
$ $\\
\normalsize
Dartmouth College, Hanover NH USA 03755\\

Woods Hole Oceanographic Institution, Woods Hole, MA USA 02543
\\
$ $\\
\revdate
\\
$ $\\
Prepared for: Continental Shelf Research

\ec

\normalsize
\vskip1in

\section{Abstract}

Statistical interpolation of coastal oceanic fields is 
addressed within the general
framework of Gauss-Markov estimation.  Computation
of the prior covariance of the field to be estimated is posed as a
stochastically-forced 
differential equation subject to coastal boundary conditions with inhomogeneous,
anisotropic parameters.  Solution is readily implemented in standard
finite element methodology.  Examples illustrate the method for idealized 1-D
situations.  Analysis of real biological data from Massachusetts Bay 
is shown, using a covariance function derived from velocity and diffusivity
fields computed with a hydrodynamic model. 
The procedure defaults to standard OA methods using distance-based covariance
functions, far from boundaries and inhomogeneities.




\clearpage
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\section{Introduction}

Objective Analysis (OA) is a  
contemporary synonym for statistical estimation based on
the Gauss-Markov theorem.  This theorem provides a sound basis for
interpolation of irregularly-spaced data.  It was introduced to
meteorology by
Gandin \cite{Gandin65} and is widely used operationally.  
Bretherton et al \cite{Bretherton76} introduced the
theory to oceanography in the context of the Mid-Ocean Dynamics
Experiment.  Good theoretical descriptions are provided by 
Daley \cite{Daley91} and Wunsch \cite{Wunsch96}

The general theory requires a prior statistical description
of the field being estimated, and of the observational 
error or noise.  Knowledge of the field statistics 
does not come easy in oceanography, owing to the relative paucity of
observations.  
Practical procedures try to fit simple analytic covariance functions 
to data.  
Almost universally, these are spatially
homogeneous, isotropic functions with a small (1 or 2) number of degrees
of freedom.  It is widely recognized that this is a potentially serious
limitation, especially  in coastal regions and 
shelf seas.  
In these areas, topography (bathymetry and
coastline), local dynamics and transport processes exert 
strong local influences on oceanographic fields; 
and the assumptions of
homogeneity and isotropy are basically inappropriate.  Under these
conditions, the prior knowledge is defective and 
the Analysis is compromised.

In this paper we develop a general approach to OA which relieves some of
these deficiencies.  The central idea is the numerical construction of
the field covariance in a way which recognizes
local variations and the coastal constraint.  The approach involves
posing these influences via a discretized partial differential
equation subject to stochastic forcing. 

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\section{General Theory}

We wish to map an 
irregular array of observations $\{d\}$ to an unstructured computational 
grid of values $\{u\}$ representing realistic coastal geometries.
The base theory is Gauss-Markov estimation.  This theory treats the
general linear mapping 

\be 
\left\lbrace u \right\rbrace =\left [ B \right ] \left\lbrace d \right\rbrace
\ee
which is optimal (i.e. $u=\tilde{u}$) when the estimator $\left [B \right ]$ is given by

\be
\left [B \right ]= \left [C_{ud}\right ] \left [C_{dd}\right ]^{-1}
\ee
with $\left [C_{ab}\right ]$ the covariance matrix among arrays $\{a\}$ 
and $\{b\}$.  $\left [B \right ]$ is a
minimum variance estimator.  If $\{u\}$ and $\{d\}$ have zero means, 
$\left [B \right ]$ is unbiased (``Best Linear Unbiased
Estimator'', Wunsch \cite{Wunsch96} p. 183; Bennett
\cite{Bennett92} p. 48.)  The precision of the
optimal estimate $\tilde{u}$ is given by its covariance
${C}_{\tilde{u}\tilde{u}}$:

\be
[{C}_{\tilde{u}\tilde{u}}]=[C_{uu}]-[B][C_{ud}^*] 
= [C_{uu}]-[C_{ud}][C_{dd}]^{-1}[C_{ud}^*]
\ee
(Here and throughout, $C^*$ indicates the transpose of $C$.)

We are especially interested in mapping $\{d\}$ onto a continuous 
finite element representation $u(x,y)$ with finite number of degrees of
freedom $u_i$:

\be
u(x,y)=\sum\limits_i u_i \phi_i(x,y)
\ee
where $\phi_i(x,y)$ are the finite element basis functions providing a unique
interpolant among the nodal values $u_i$ of a FE grid.  Sampling this
function at the observation points gives

\be
\left [E \right ] \left\lbrace u\right\rbrace  
+ \left\lbrace n\right\rbrace = \left\lbrace d\right\rbrace
\label{eqn:constr1}\ee
where $\left [E \right ]$ is a measurement operator 
incorporating interpolation of the
$\phi$ to the observation points and local averaging appropriate to the
data; and  
$\{n\}$ is observational noise.\footnote{More precisely, $n$ is the model-data misfit,
which will be a priori ascribed to observational noise.}  
Assuming signal $\{u\}$ and noise $\{n\}$ 
to be uncorrelated, we have   

%
%\bea
%\left [C_{ud}\right ] &=& \left [C_{uu}\right ] \left [E^* \right ]
%\\
%\left [C_{dd}\right ] &=& \left [ E \right ]\left [C_{uu}\right ] \left [ E^* \right ]
%+ \left [C_{nn}\right ]
%\eea
%

\bea
\left [C_{ud}\right ] = \left [C_{uu}\right ] \left [E^* \right ]
& ~~~~~ and ~~~~~~ &
\left [C_{dd}\right ] = \left [ E \right ]\left [C_{uu}\right ] \left
[ E^* \right ]
+ \left [C_{nn}\right ]
\eea

\ni and the optimal estimate is

\be
\left\lbrace \tilde{u}\right\rbrace  =[C_{uu}][E^*]\biggl[ [E ][C_{uu}][E^*] 
+ [C_{nn}]   \biggr]^{-1} \left\lbrace d\right\rbrace  
\label{eqn:oa1}\ee
or, equivalently (Wunsch \cite{Wunsch96}, p. 184)


\be
\biggl[ [C_{uu}]^{-1} +    [E^*][C_{nn}]^{-1}[E]  
\biggr]\left\lbrace \tilde{u} \right\rbrace  
= [E^*][C_{nn}]^{-1}\left\lbrace d\right\rbrace 
\label{eqn:oa2}\ee



\ni The precision of the estimate $\tilde{u}$ is given by 
\footnote{Typically, $u\approx \tilde{u} \pm S$, where $S^2$ is the 
diagonal of $[{C}_{\tilde{u}\tilde{u}}]$ .}


\be
[{C}_{\tilde{u}\tilde{u}}]=[C_{uu}]-[B][C_{ud}^*] = [C_{uu}]-[B][E][C_{uu}^*] 
\label{eqn:oa3}
\ee


\ni The key to Objective Analysis, then, is knowledge of 
$[C_{uu}]$ and $[C_{nn}]$, the signal and 
noise covariances.  Computing $[C_{uu}]$ is our main interest.  

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%\clearpage
%\section{Review}



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\section{Covariance and Stochastically-Forced Differential Operators}

Classically, the covariance of the underlying field $[C_{uu}]$ has been 
expressed in simple analytical forms with a few degrees of freedom, 
leading to practical procedures as in Bretherton et al
\cite{Bretherton76}.  Typical are covariances of the form $C(r)$, where 
$r$ is scaled distance separating any two points.

Freeland and Gould \cite{FreelandGould76} estimated streamfunction from
velocity measurements in the Mid-Ocean Dynamics Experiment.   
Isotropic, homogeneous 
covariance functions $C_{\psi\psi}(r)$, $C_{uu}(r)$,  
$C_{vv}(r)$ describe the streamfunction and the 
longitudinal and transverse\footnote{The directions are
relative to the separation vector ${\bf r}$.} 
velocity covariances, as functions of the
spatial separation distance $r$.  Constraining these
functions are the physical relations defining them  (non-divergent,
geostrophic flow) and several constraints representing statistical
realizability.  The forms chosen were
\bea
C_{uu}(r)&=&(1+br)e^{-br}
\\
C_{vv}(r)&=&(1+br-b^2r^2)e^{-br}
\\
C_{\psi\psi}(r)&=&(1+br+b^2r^2/3)e^{-br}
\eea
Thus we have one-parameter, homogeneous isotropic 
covariances.  The MODE data were used to estimate $b$.   
McWilliams \cite{McWilliams76} made similar estimates using MODE data,
assuming geostrophic, non-divergent flow and homogeneous, isotropic
covariance.   
Two-parameter functions were fitted to the data:
\bea
C_{\psi\psi}(r)&=&(1-\gamma^2r^2)e^{-\delta^2r^2/2}
\\
C_{uu}(r)&=&(1-b^2r^2)e^{-\delta^2r^2/2}
\\
C_{vv}(r)&=&(1-[5+\delta^2/\gamma^2]b^2r^2 +
\delta^2b^2r^4)e^{-\delta^2r^2/2}
\eea
with the constraint
$b^2= \gamma^2 \delta^2 /(\gamma^2 + \delta^2)$.  


Denman and Freeland \cite{DenmanFreeland85} estimated 2-parameter 
covariance functions from observations over the continental shelf 
west of
Vancouver Island.  
Fitted functions are reported for geopotential
height and velocity under similar assumptions as Freeland and
Gould \cite{FreelandGould76}, plus temperature, salinity, and 
chlorophyll.  Functional forms used included

\bea
C(r)&=&(1-r^2/b^2)e^{-r^2/a^2}
\\
C(r)&=&\cos(r/b)e^{-r^2/a^2}
\\
C(r)&=&J_0(r/b)e^{-r^2/a^2}
\eea
($J_0$ is the Bessel function of the first kind.)  Good fits to several of these 2-parameter forms were found; but   
the data were insufficient to support the estimation of 
anisotropic covariance functions suitable for the shelf.   

Hendry and He \cite{Hendry00} implemented OA for shelf 
estimations using
the covariance function as in Freeland and Gould
\cite{FreelandGould76}
\be
C_{uu}(\rho)= \left(1+\rho+\rho^2/3 \right)e^{-\rho}
\label{eqn:hendry}
\ee
In that implementation, $\rho$ is an anisotropic, space-time
pseudo-distance, with
principal axes and
scaling 
defined locally, prior to the data.  Several
studies of the Gulf of Maine/Scotian
Shelf region, 
including 
\cite{Naimie94}, \cite{LINW96}, \cite{Hannah96}, \cite{Loder97}, and
\cite{McGillic98} have
used this procedure to estimate
hydrographic and biological fields.  In
those cases the local correlation axes were keyed to the local
bathymetric gradient with longer 
correlation scales along isobaths than across them.      


Zhou \cite{Zhou98} studied space-time interpolation of plankton data,
using the covariance function 
\be
C(\rho)=(1-\rho)e^{-\rho}
\label{eq:zhou}
\ee
where again $\rho$ is a scaled pseudo-distance, in this case 
involving an 
isotropic 2-D spatial
scale and a temporal scale. 
This form was fit to the observed data.  A key idea is the Lagrangian
correction of the spatial
distribution.  The observations were passively 
advected either forward or backward to a common time, in an attempt to  
offset the spatial distortion associated with the 
non-synoptic sampling.  Following this advective adjustment, the
correlation function (\ref{eq:zhou}) was spatially 
isotropic and homogeneous.  
The advective field was itself based on geostrophic diagnosis of
objectively-analyzed hydrography.   





These types
of covariance functions have served well in the studies cited.  
But in the coastal regime, there are serious complications:
\bi
\item coastline constraints: there is no direct distance relationship
across landforms (islands, peninsulas)
\item anisotropy: advection creates locally-anisotropic covariance, 
oriented to streamlines; other dynamical processes may be oriented
by the local bathymetric gradient 
\item inhomogeneity: correlation scales and principal axes may vary 
locally, reflecting for example differential mixing or advective
regimes.
\ei
\ni Practical incorporation of these effects into $[C_{uu}]$ is our goal.

Our basic idea is to represent the underlying variability as the
outcome of a stochastic process; specifically, 
the result of a stochastically-forced differential equation (SDE).  
For example, the simple 
equation
\be
{\partial^2 u \over \partial x^2} - k^2 u = \epsilon(x)
\label{eq:pde}
\ee
with the {\em process noise} $\epsilon$ a $(0, 1)$ 
random disturbance with no spatial correlation, has covariance 
\be
C_{uu}(r) =  \left(1+kr\right)e^{-kr}
\label{eq:matpde}
\ee
when posed in an unbounded domain.  Thus (\ref{eq:pde}) and (\ref{eq:matpde}) are equivalent 
statements of the same problem.  The 2-D pair  
\be
\del^2 u  -  k^2 u= \epsilon(x,y)
\label{eq:pde2}\ee
\be
C_{uu}(r)=kr K_1(kr) \simeq \left({\pi\over 2} kr \right)^{1\over 2}\left(1+
{3\over 8 kr}\right)
e^{-kr} ~~~~~~~ kr\rightarrow \infty
\ee
are likewise equivalent statements (\cite{Balgovind83}).   
($K_1$ is the Bessel function of the second kind.)  


The importance of the SDE approach is that $C_{uu}$ can be computed numerically
for realistic conditions -- real topography, variable coefficients and resolution,
boundary conditions --  which make analytic solutions impossible.  And, the
parameters and process noise model $\epsilon$ can be chosen to represent real
processes affecting the field of interest.  And, the limiting case, far from
boundaries and inhomogeneities, defaults to an equivalent $C(r)$ structure.  
Balgovind et al.(\cite{Balgovind83}) used the SDE approach (specifically, the Helmholtz 
equation
(\ref{eq:pde2}) with $k^2$ varying with latitude) to compute a
realistic spatially-varying covariance for meteorological forecast error.  Here
we use it to analyze coastal oceanic fields.  


Discretization of (\ref{eq:pde}),  (\ref{eq:pde2}) or any other differential 
operator on a finite element
grid leads to the matrix form
\be
\left [A \right ]\left\lbrace u\right\rbrace=\left\lbrace e\right\rbrace
\label{eqn:constr2}
\ee
in which all coastal constraints (no transport across land boundaries) 
are incorporated automatically.  The covariance of the FE nodal
variables is obtained directly:
\be
\left [C_{uu}\right ]=\left [A\right ]^{-1}\left [C_{ee}\right ]\left [A\right ]^{-1*} 
\label{eqn:oa4}
\ee
\be
\left [C_{uu}\right ]^{-1}=\left [A^{*}\right ] \left [C_{ee}\right ]^{-1*}\left [ A \right ]
\label{eqn:oa4a}
\ee


\ni Immediately we have accounted, formally, for 
a) realistic oceanic transport processes and BCs in $[A]$, and b) 
realistic process
errors in $C_{ee}$, representing processes not modeled in $[A]$ 
or in the prior estimate.  
Knowledge of these two effects allows us to compute $[C_{uu}]$ using standard
finite element solution techniques.  
Coupling 
(\ref{eqn:oa4}) 
with (\ref{eqn:oa1}-\ref{eqn:oa3}), 
we have Objective Analysis.  
This is the central idea of this paper.  
Two examples follow.     


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\footnotesize
\bc
\fbox{\fbox{
%\parbox[b]{\linewidth}{
\parbox[b]{12cm}{
\ni 
Aside, note that use of equations \ref{eqn:oa1}-\ref{eqn:oa3}, with
$[C_{uu}]$ from equation \ref{eqn:oa4} is equivalent to solving the
weighted least squares problem 

\be 
Minimize ~~~  \left \lbrace 
\left\lbrace n\right\rbrace^* [C_{nn}]^{-1} 
\left\lbrace n\right\rbrace 
+ \left\lbrace e\right\rbrace^* [C_{ee}]^{-1} 
\left\lbrace e \right\rbrace
\right \rbrace
\ee

\ni subject to the constraints \ref{eqn:constr1} and \ref{eqn:constr2}
which define the model-data mismatch $n$ and the process noise $e$ 
(Wunsch \cite{Wunsch96}).  
}}}
\ec
\normalsize
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\clearpage
\section{Example 1}

Consider the SDE

\be
{d~\over dx}D{du \over dx} + V{du \over dx} - k^2 u = \epsilon
\label{eq:adr1d}
\ee

\ni and its discretization (assuming constant $D$)

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%
%\be
%\left(\delta^2 +P_e ~\delta -K^2 \alpha \right)u_i = E_i
%\label{eq:diff}
%\ee
%
%
%\ni with $\delta^2$ and $\delta$ the first and second centered
%differences, and dimensionless quantities $P_e={Vh\over2D}$, 
%$K^2={(kh)^2\over D}$, and $E={\epsilon h^2\over D}$.
%
%
%$\alpha$ is a centered averaging operator
%
%\be
%\alpha u_i \equiv a~u_{i-1} + (1-2a)~u_{i}  + a~u_{i+1} 
%\ee
%
%which accounts for Galerkin FEM on linear elements ($a=1/6$) as well as 
%standard FD ($a=0$).  Expansion of equation \ref{eq:diff} gives the tridiagonal system
%
%
%\be
% [1-P_e -aK^2]~u_{i-1} 
% -[2+K^2( 1-2a)]~u_{i}  
% + [1+P_e -aK^2]~u_{i+1}
%=E_i
%\ee
%
%
%\ni and the matrix $[A^*][A]$ is pentadiagonal with columns
%
%\be
%\left\lbrace
%\begin{array}{c}
%\vdots\\
%0\\
%\vdots\\
%0\\
%(1-P_e^2) \\
%-2(2+K^2) \\
%2(1+P_e^2) + (2+K^2)^2 \\
%-2(2+K^2)   \\
% (1-P_e^2)   \\
% 0\\
%\vdots\\
%0\\
%\end{array}
%\right\rbrace
%\ee
%
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%
% This is the eqn with a.ne.0 i.e. the general fe/fd version.  a=1/6
% gives fe; a=0 gives fd.
%
%\be
% [1-P_e -aK^2]~u_{i-1} 
% -[2+K^2( 1-2a)]~u_{i}  
% + [1+P_e -aK^2]~u_{i+1}
%=E_i
%\label{eq:discreteADR}\ee
%
%\ni with dimensionless quantities 
%
%\be
%P_e={Vh\over 2D} ~~~~  
%K^2={k^2 h^2\over D}  ~~~~
%E={\epsilon h^2\over D}
%\ee
%The parameter 
%$a$ accounts for either Galerkin FEM on linear
%elements ($a=1/6$) or 
%standard FD ($a=0$).  $[A]$ is tridiagonal; 
%$[A^*][A]$ is pentadiagonal with columns
%
%\be
%\left\lbrace
%\begin{array}{c}
%0\\
%\vdots\\
%0\\
%(1-P_e^2) \\
%-2(2+K^2) \\
%2(1+P_e^2) + (2+K^2)^2 \\
%-2(2+K^2)   \\
% (1-P_e^2)   \\
% 0\\
%\vdots\\
%0\\
%\end{array}
%\right\rbrace
%\ee
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
\be
 [1-P_e ]~u_{i-1} 
 -[2+K^2]~u_{i}  
 + [1+P_e]~u_{i+1}
=E_i
\label{eq:discreteADR}
\ee

\ni with dimensionless quantities 

\be
P_e={Vh\over 2D} ~~~~  
K^2={k^2 h^2\over D}  ~~~~
E={\epsilon h^2\over D}
\ee

\ni $[A]$ is tridiagonal; 
$[A^*][A]$ is pentadiagonal with columns

\be
\left\lbrace
\begin{array}{c}
0\\
\vdots\\
0\\
(1-P_e^2) \\
-2(2+K^2) \\
2(1+P_e^2) + (2+K^2)^2 \\
-2(2+K^2)   \\
 (1-P_e^2)   \\
 0\\
\vdots\\
0\\
\end{array}
\right\rbrace
\ee

\ni  
The inverse of $[A^*][A]$, assuming $[C_{EE}]=[I]$, 
gives the covariance $C_{uu}$.  This will be a full matrix, with
the covariance structure centered on the diagonal.   
It is plotted in figures
\ref{fig:cov_k2}, \ref{fig:cov_k2_anal} and 
\ref{fig:cov_pe}, assuming periodic boundary conditions.  
Note that $K={1\over N}$
where $N$ is the number of grid cells per e-folding length for the
analytic solution; thus a reasonable discretization has $K=$O(1) or
less.  Similarly, $P_e=1$ is a threshold for poor resolution of
advection and accompanies loss of diagonal dominance in the case
$K^2=0$.   

In figure \ref{fig:cov_k2} we show $C_{uu}$ with periodic BCs, for $Pe=0$.  
From (\ref{eq:matpde}) we expect the decorrelation scale to be $1/K$ 
grid cells.  Figure \ref{fig:cov_k2} confirms that for large $K$ we have essentially
an unbounded domain.  Decreasing $K$ broadens the covariance.  
Figure \ref{fig:cov_k2_anal} confirms that for large $K$ we recover the analytical result;
while at small $K$ the BCs have effect and the free-space analytic solution becomes invalid. 

Figure \ref{fig:cov_pe} shows $C_{uu}$, again with periodic BCs, 
for various values of $Pe$. 
The effect of advection is to lengthen the correlation scale, increasing
covariance in the along-stream direction.  In the steady state, the 
upstream and downstream effect is symmetric.  
Creative use of these two parameters sets a baseline correlation scale via $K$ and
directional anisotropy via $Pe$, both of which can be locally variable.  Modulation near
shoreline boundaries is naturally incorporated from the outset.  

In figure \ref{fig:covstep} we show the effect of a spatial variation in
diffusion coefficient.  In this case there is a step change by a factor
of 10 in $D$ at node 19.  The result is asymmetry (right curve) and 
inhomogeneity (left curve versus right) in the covariance.  
Figure \ref{fig:covbc_k2} shows the result of an impermeable land
boundary between nodes 31 and 32.  Disturbances introduced near this
boundary must diffuse around it by a much longer path; thus
the covariance of geographic near-neighbors is greatly reduced.    


These results illustrate local control of the correlation length 
scales using  physical parameters and boundary conditions of the 
underlying differential equation.  

Objective analyses of 5 data sampled at intervals of $10 \Delta x$ 
are plotted in figure \ref{fig:interp}.   The three
cases illustrate the significant variations among the interpolants,
depending on the details of boundary conditions.  Figure
\ref{fig:interp_se} shows the estimated standard deviation of two
different interpolations of the same data, along with 
that of the data.  The discontinuity in the 
interpolation results in uncertainty growth near the boundary, where
data are effectively extrapolated from the left.  Another effect of the
internal boundary is the
reduction of the interpolation maximum at node 35, due to its isolation
from the data to the left.


\begin{figure} [ht]
%\begin{center}
%\epsfxsize = \linewidth
%\epsfxsize=8cm
\epsfysize=6cm
\epsfc{cov_k2.eps}
\caption{\label{fig:cov_k2} Covariance for the
advective-diffusive-reactive equation \ref{eq:discreteADR}, for three
different values of $K^2$.  Periodic BCs.  
The plots have been normalized by their
individual variances.} 
%\end{center}
\end{figure}

\begin{figure} [ht]
%\begin{center}
%\epsfxsize = \linewidth
%\epsfxsize=8cm
\epsfysize=6cm
\epsfc{cov_k2_anal.eps}
\caption{\label{fig:cov_k2_anal} Covariance for the
advective-diffusive-reactive equation as in figure \ref{fig:cov_k2} 
(blue, dots), compared with the
analytic free-space result (red, solid).  Periodic BCs.  
The plots have been normalized by their
individual variances.} 
%\end{center}
\end{figure}


\begin{figure} [hb]
%\begin{center}
%\epsfxsize = \linewidth
%\epsfxsize=8cm
\epsfysize=6cm
\epsfc{cov_pe.eps}
\caption{\label{fig:cov_pe} Covariance for the
advective-diffusive-reactive equation, for 5 values of $P_e$. Periodic BCs.} 
%\end{center}
\end{figure}



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\begin{figure} [ht]
\begin{center}
%\epsfxsize = \linewidth
%\epsfxsize=8cm
\epsfysize=7cm
\epsfc{covstep.eps}
\caption{\label{fig:covstep} Effect of variable diffusion coefficient
on covariance.   
To the left of node 19, $D=D_o/10$; to the right, $D=D_o$.   
The two curves are associated with  
nodes 10 (left curve) and 25
(right curve).  Periodic BCs; $P_e = 0.$; $K^2_o = .1$ 
} 
\end{center}
\end{figure}

\clearpage

\begin{figure} [hb]
\begin{center}
%\epsfxsize = \linewidth
%\epsfxsize=8cm
\epsfysize=7cm
\epsfc{covbc_k2.01.eps}\\
\epsfysize=7cm
\epsfc{covbc_k2.1.eps}
\caption{\label{fig:covbc_k2} Effect of no-flux boundary condition
between nodes 30 and 31 on covariance 
with nodes 10 (left curves) and 25
(right curves).  
Periodic BCs; $P_e = 0.$; $K^2 = .01$ (top)
and $0.1$ (bottom).  } 
\end{center}
\end{figure}

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\begin{figure} [p]
\begin{center}
%\epsfxsize = \linewidth
%\epsfxsize=8cm
\epsfysize=6cm
\epsfc{interp.eps}\\
\epsfysize=6cm
\epsfc{interp1.eps}\\
\epsfysize=6cm
\epsfc{interpbc.eps}
\caption{\label{fig:interp} Objective analysis  of 5 data (plotted as
asterisks) for $P_e = 0.$; $K^2 = .01$.  Top: periodic BCs; Middle:
Neumann BCs at left and right; Bottom: Neumann BCs in center of domain.
Standard deviations of observational noise $n$ ($Sn$) and 
process noise $e$ 
($Se$) are $0.5$ and $0.1$, respectively.  } 
\end{center}
\end{figure}


\clearpage

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%\begin{figure} [p]
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%%\epsfxsize=8cm
%\epsfysize6cm
%\epsfc{interp_se.eps}\\
%\epsfysize=6cm
%\epsfc{interpbc_se.eps}
%\caption{\label{fig:interp_se} Objective Analysis as in figure
%\ref{fig:interp}, with standard deviations of data (error bars) and the estimated
%interpolant (solid lines) shown. Top: periodic BCs; Bottom: Neumann BC 
%in center of domain.  Finite difference ($a=0$).} 
%\end{center}
%\end{figure}
%

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

\begin{figure}[htbp]
\centering
\mbox{

\subfigure
{\epsfig{figure=interp_se.eps,width=.49\textwidth}}\quad

\subfigure
{\epsfig{figure=interpbc_se.eps,width=.49\textwidth}}}


\caption{\label{fig:interp_se} Objective Analysis as in figure
\ref{fig:interp}, 
with standard deviations of data (red asterisks with blue error bars) 
and the
estimated
interpolant (black solid lines) shown.  The data are 
connected by a blue solid line representing linear interpolation.
The left panel is a standard OA.  
The right panel illustrates the
successful blocking of interpolation across a coastal boundary.  Both
panels have periodic conditions at left and right; the right panel has a
Neumann condition just the left of node 30.}
\end{figure}

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\clearpage

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\section{Example 2}

Next we consider the 2-D transport equation

\be
{1\over h} \div hD \grad u - {\bf V}\cdot \grad u -k^2 u 
= \epsilon_s + \epsilon_p
\label{eq:adr2d}
\ee

\ni with 


$~$\\
$u$=field anomaly\\
$h$=bathymetric depth\\
$D$=dispersion coefficient\\
${\bf V}$=fluid velocity\\
$k^2$=first-order decay rate\\
$\epsilon_s$= surface forcing\\
$\epsilon_p$= isolated inputs from point sources (e.g. river discharges)\\


\ni We imagine for example that climatological estimates of all transport 
parameters are available, and that 
a climatological prior estimate of the transported 
field has been subtracted from the data.  Standard Galerkin
FEM discretization leads to 

\be
\left [A \right ]\left\lbrace u\right\rbrace 
= \left\lbrace e_s\right\rbrace 
+ \left\lbrace e_p\right\rbrace  
+ \left\lbrace r\right\rbrace  
\ee

\be A_{ij} = <-hD\grad \phi_j \cdot \grad \phi_i 
- h{\bf V}\cdot \grad \phi_j~ \phi_i
- hk^2\phi_j \phi_i >
\ee
\be r_i=-\oint hD{\d u \over \d n} \phi_i
\ee
\be e_{si}= <h\epsilon_s\phi_i>
\ee
\be e_{pi}= <h\epsilon_p\phi_i>
\ee
\ni where $<~>$ indicates integration over the spatial domain, and $\oint$ integration over its
boundary.  The $\{e_s\}$ vector might be constituted as random noise plus a highly structured response
correlated with weather; $\{e_p\}$ might be correlated with
hydrological conditions or industrial activity.


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

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}).    
The basis of these simulations
is a fully nonlinear, three-dimensional primitive equation model with
turbulence closure (\cite{LINW96}).  This model has been
used extensively to study coastal currents in this region, particularly 
in the area north of Cape Cod (\cite{LHN97}).  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 (\ref{eq:adr2d}).  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.  Control
parameters for the objective analysis (Table \ref{oapars}) were chosen
as follows.  The observational noise $n$ was set to the precision
of the measurement, which is 10 cells l$^{-1}$.  The first-order
decay constant $k^2$ in Equation (\ref{eq:adr2d}) sets the horizontal scale of
patchiness
in the absence of advection and process noise.  For a purely
diffusive balance ($D \nabla^2 u \sim k^2 u$),
the non-dimensional ratio $\sqrt{D} \over k$ is the e-folding
scale for patches of $u$.  Assuming a typical patch size of
1 km and a diffusivity of 100 m$^2$ s$^{-1}$ yields $k^2=1.0 \cdot
10^{-4}$.
The process noise is scaled to meet the variance of
the observations ($d$) in the absence of hydrodynamic transport in 
Equation (\ref{eq:adr2d}):

$$ Var(\epsilon) \sim Var(k^2 d) $$

\noindent
This particular data set has a mean of 92 cells l$^{-1}$ and
a standard deviation of 142 cells l$^{-1}$, resulting in an estimated
process
noise size $\epsilon_s = 1.42 \cdot 10^{-2}$ cells l$^{-1}$ s$^{-1}$.  
The covariance of the process noise is assumed to take the form of 
equation (\ref{eqn:hendry}), with a 
2 km spatial scale.  The point source noise $\epsilon_p$ was zero.  

Although the observed distribution is quite patchy, abundance is 
generally high in Massachusetts Bay and low elsewhere 
(Figure \ref{rmrp}a).  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
(\ref{eq:adr2d}), and there is noise associated with both the observations
and the process.  The accompanying map of standard error (Figure
\ref{rmrp}b)
is simply the square root of the diagonal of $C_{\tilde u \tilde u}$.  
Expected error is reduced in the vicinity of the observations, toward
the level of the observational noise. Far from the data, 
the posterior
covariance $C_{\tilde u \tilde u}$ is no different from the prior
covariance $C_{uu}$. 
Note that there is
structure in the far field standard error.  
This structure comes as a direct result of the 
hydrodynamic field.  For example, the very high diffusivity near
Martha's Vineyard and Nantucket island would tend to smooth any
patchiness in that area, thereby reducing the expected variance.

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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Tables
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage

\begin{table}[b]
\bc
\begin{tabular}{|l|l|l|} \hline\hline

Parameter  &  Value               & Definition                     \\
\hline
$n$        &  10 cells l$^{-1}$   & observational noise            \\
\hline
$k^2$      &  $1.0  \cdot 10^{-4}$ s$^{-1}$ 
                                  & first-order decay coefficient  \\
                                  \hline
$\epsilon$ &  $1.42 \cdot 10^{-2}$ cells l$^{-1}$ s$^{-1}$
                                  & process noise                  \\
                                  \hline
$r$        &  2 km                & spatial scale of process noise \\
\hline
\hline

\end{tabular}
\ec
\caption{Objective analysis control parameters for Example 2.}
\label{oapars}
\end{table}



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% Figures
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% Figures
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\clearpage
%
\begin{figure}
\epsfxsize=10.0cm
\centerline{\epsfbox{dennis/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
%\epsfxsize=16.0cm
%\centerline{\epsfbox{dennis/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}
%
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%\clearpage

\begin{figure}
%\epsfxsize=14.0cm
\epsfxsize=16.0cm
\centerline{\epsfbox{dennis/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=12.0cm
%\epsfxsize=14.0cm
\epsfxsize=16.0cm
\centerline{\epsfbox{dennis/exp23.eps}}
\caption{(A) Objective analysis of {\it Alexandrium} spp.
concentration.
Observed values are shown as colored dots, with the same
color scale
used to map the objectively analyzed field. (B) Standard
error estimate,
${C_{\tilde u \tilde u}}^{1 \over 2}$.}
\label{rmrp}
\end{figure}

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

\begin{figure}
%\epsfxsize=12.0cm
%\epsfxsize=14.0cm
\epsfxsize=16.0cm
\centerline{\epsfbox{dennis/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}

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


%
%\begin{figure}[htbp]
%\centering
%\mbox{
%
%\subfigure
%{\epsfig{figure=Gaussian.eps,width=.49\textwidth}}\quad
%\subfigure
%{\epsfig{figure=NoGaussian.eps,width=.49\textwidth}}}
%
%
%\caption{\label{fig:OAFE2d} Coastal covariance maps based on 
%Finite Element transport model in vicinity of Cape Cod, Massachusetts. 
%Left: offshore covariance, nearly unconstrained by coastline.  Right: 
%the landward boundary traps the covariance in the concave
%portion of the Bay.  Parameters: ${\bf V}=0$; $D=500 m^2/s$; $k^2
%=2\times 10^{-6} s^{-1}$;
%Gulf of Maine bathymetry; uniform, uncorrelated process noise.}
%\end{figure}
%

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\clearpage



\section{Conclusion}



The observational data base for the coastal ocean is growing in sophistication and volume 
at a remarkable rate.  
In parallel there is a growing need for 
state estimation procedures for coastal fields.  Classical Gauss-Markov theory is
readily adapted; but knowledge of the prior covariance of the field to be
estimated is problematic and potentially limiting.  Conventional distance-based
covariance functions are not suited to the coastal ocean because of its
inhomogeneous nature; but
stochastically-forced differential equations representing coastal processes
provide a natural formulation, and numerical solution is readily obtained with standard
Finite Element procedures.  The combination of G-M estimation with SDE-based field covariance
provides a compact version of coastal OA.\footnote{
For an implementation, see the OAFE Users Guide
and Software \cite{oafe99}.}  It 
focuses attention on a) what {\em processes}
are relevant to the observed field variability; and b) the nature of the
stochastic forcing (the {\em process noise}).  

The use of the simple Helmholtz SDE operator provides a locally-variable,
isotropic correlation scale -- in terms of number of grid cells M, 
we have  $M=1/K\sim 
\left[{D\over k^2 A}\right]^{1/2}$, with $A$ the element area.  
This SDE defaults to conventional distance-based
covariance when far from boundaries and inhomogeneities.   Adding coastal boundary
conditions (no normal transport) blocks interpolation across land.  
Where a coastal boundary isolates an estimation point from the data, the local
estimation error increases.  
This Helmholtz operator alone is a significant advance over conventional practice.  
Variation in the diffusion coefficient modulates the correlation scale.  Inclusion of an
advective term elongates the covariance along streamlines.  In the Massachusetts
Bays example shown,  the climatological circulation and mixing provides a natural
specification for these parameters.  Even better would be a circulation hindcast
of ambient conditions during the cruise.   


It is important to pay attention to the spatial correlation in the process noise
($\epsilon$ in equations \ref{eq:pde},\ref{eq:adr1d},\ref{eq:adr2d}).  Assuming white noise at the nodes of a finite mesh 
amounts to large spatial scales where
there are big elements -- a subtle side effect.  
In the field application here, we used a
distance-based autocorrelation for $\epsilon$ which was
successful.  Naturally, a large spatial scale for $\epsilon$ 
overwhelms diffusion and smooths everything.  A small
spatial scale for $\epsilon$ defaults to the mesh-dependent $[C_{ee}]$ implied by its 
finite element basis, and allows this to pass   
through into the posterior covariance  $[{C}_{\tilde{u}\tilde{u}}]$.  
Relative to size, the process noise ought to be scaled to match the observed variance of
the data: $Var(\epsilon) \sim k^2 Var(d)$. 
There is substance in the specification of this 
error model and it deserves careful attention
in any analysis. 

Boundary conditions for the SDE also warrant attention.  We have used homogeneous Neumann BCs
which are natural at the coast, with zero variance; 
these produce a buildup of $[{C}_{\tilde{u}\tilde{u}}]$,
essentially reflecting the covariance.  More generally, boundary
conditions need to have their own error models explicitly incorporated into the $e$ term in
equations \ref{eqn:constr2}, \ref{eqn:oa4} and \ref{eqn:oa4a}. 


Two other important OA issues remain 
unaddressed and should not be overlooked.  First, there is the need to 
describe the observational noise via a relevant error model $C_{nn}$.  We have not looked at
this herein, and look forward to the development of error models for the many oceanographic
data products becoming available.  Second, the data to be
analyzed should in general be reduced by removal of a structured prior 
estimate (spatially-variable mean), in the hope of achieving the zero-mean 
condition for an unbiased estimate.  Explicit recognition of this step is critical to 
success with
any OA procedure.


\section{Acknowledgments}


We are grateful to D. Anderson, B. Keafer and R. Signell for providing
the
{\it Alexandrium} spp. data used in Figures \ref{rmrp} and \ref{cubic}.  
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; and the Office of Naval Research.

\ni This is Contribution \# XXXX of the Woods Hole Oceanographic
Institution.

%\nocite{*}
%\bibliography{../../bibtex/oafe,../../bibtex/Model}
\bibliography{../../bibtex/oafe}
\bibliographystyle{plain}

\end{document}




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% Example table and figure


\begin{figure} [hp]
\begin{center}
%\epsfxsize = \linewidth
%\epsfxsize=15cm
%\epsfxsize=8cm
\epsfysize=20cm
\epsfc{osse1a/inverse5a.results/inverse5dbce.uvbar.ts1.ps}
\caption{\label{fig:mooring.inverse.ts} Inverse solution sampled at
mooring ST1 (interpolation) and NEP (extrapolation).  Four cases are
shown: Base case (5C); unconverged Base Case (5B); Base Case but with
12-hour $\Delta t_{\rho}$ (5D); and Base Case with $W2$ increased by a
factor of 10.  The standard is the ``truth - prior'' panel in figure
\ref{fig:mooring.error}} 
\end{center}
\end{figure}




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\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{Physical Parameters}   \\
\hline
  $L$ & $200 ~km$ \\
\hline 
  $T$ & $216 ~hours$ \\
\hline 
  $h_{min}$& $10 ~m$ \\
\hline 
  $h_{max}$& $150 ~m$ \\
\hline
  Latitude & $42.3^{\circ} N$ \\
\hline 
  $k$ & $0.002 ~m/s$ \\
\hline 
  $N_z$  & $0.020 ~m^2/s$ \\
\hline
\multicolumn{2}{c}{ } \\
\multicolumn{2}{c}{Numerical Parameters} \\
\hline
  $\Delta x$& $\sim 3 ~km$ \\
\hline 
 $\Delta t$& $200 ~sec$ \\
\hline 
  $NNV$& $11$ \\
\hline 
  $\Delta z_{bl}$& $1 ~m$ \\
\hline 
  $\Theta$ & $0.75$ \\
\hline 
  $\tau_0$ & $2.0\times 10^{-4}$ \\
\hline 
\multicolumn{2}{c}{ } \\
\multicolumn{2}{c}{Unexplained Velocity Data}  \\
\hline
   $\vert V \vert $ (Signal) & $0.0083 ~m/s $ RMS \\
\hline 
   $\vert V \vert $ (Noise)  & $0 ~m/s $ RMS \\
\hline  
\multicolumn{2}{c}{ } \\
\multicolumn{2}{c}{Inversion Parameters} \\
\hline
   $W_0 $   & $0.0$ \\
\hline 
   $W_1 $   & $10^{13}$ \\
\hline 
   $W_2 $   &  $5. \times 10^{13}$\\
\hline 
   $V_{rms} $   & $0.002 ~m/s$ \\
\hline 
   $\Delta t_{bc} $   & $3 ~hours$ \\
\hline 
\end{tabular}
\caption{\label{tab:GBParams} Georges Bank OSSE parameters. Base Case
(5C)} 
\end{center}
\end{table}


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