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\title{Evaluating the Synopticity of the U.S. Globec \\
       Georges Bank Broad-scale Sampling Pattern With \\
       Observational System Simulation Experiments}

\author {D.J. McGillicuddy, Jr. \
           \thanks{Department of Applied Ocean Physics and Engineering,
                   Woods Hole Oceanographic Institution Woods Hole, MA 02543,
                   USA.  Tel: 508-289-2683 Fax: 508-457-2194
                   Email: dmcgillicuddy@whoi.edu (Corresponding Author).}
         D.R. Lynch \
           \thanks{Dartmouth College,
                   Hanover, NH 03755, USA.}
         P. Wiebe \
            \thanks{Biology Department,
                   Woods Hole Oceanographic Institution Woods Hole, MA 02543,
                   USA.}
         J. Runge \
            \thanks{Department of Fisheries and Oceans, 
                    Maurice-Lamontagne Institute, 
                    Division of Ocean Sciences,
                    850 Route de la Mer,
                    Mont Joli, QC G5H 3Z4, CANADA.} \\
         E.G. Durbin \
            \thanks{Graduate School of Oceanography,
                   University of Rhode Island,
                   Narragansett, RI 02882, USA.}
         W.C. Gentleman$^2$ \
         and C.S. Davis$^3$}

\date {Revised and re-submitted to {\it Deep-Sea Research} \\
       Special Issue on Globec - Georges Bank \\
       October 29, 1999}

\maketitle

\begin{abstract}

A set of observational system simulation experiments (OSSEs) have been
designed to quantitatively assess the synopticity of the broad-scale
surveys of Georges Bank carried out as part of the U.S. Globec
program.   The approach makes use of model simulations which contain
realistic spatial and temporal fluctuations of adult females of the
planktonic copepod
{\it Calanus finmarchicus} observed during February and March 1995.
Simulations are constructed with two types of
assimilation procedures (nudging and the adjoint method) which are 
used to dynamically interpolate
between the two broad-scale surveys taken one month apart.
Using these simulations as representations of the real ocean, the model
fields are subsampled in space and time along a typical
cruise track. These simulated data are then objectively analyzed and the
resulting maps compared with ``reality'' as represented in the original
simulation.  Results indicate a total error of approximately 50\%,
which is comprised mostly of simple mapping error (incomplete spatial
sampling) and a smaller contribution
from space/time smearing.  Adjustment of the station positions for
displacement by the mean flow reduces the latter error
by about half.

\vskip .5in
\noindent
Key words: 
biological sampling, mathematical modeling, ocean currents, 
oceanographic surveys, simulation.


\end{abstract}

\noindent
{\bf Introduction}

Biological
oceanography is fraught with a particularly difficult sampling problem 
because there are two distinct sources of space/time fluctuations in 
organism distributions: heterogeneity in biological processes and
stirring by ocean currents.  In principle, these two effects would be
separable if it were possible to collect a sequence of instantaneous
snapshots of both organism abundance and the circulation.  Such a
situation would lend itself to direct calculation of advective transport,
and the biological contribution could be inferred by difference.
Of course, practical considerations preclude such an approach; 
observations are time-consuming and difficult to make at sea, and the
speed with which ships can transit between stations is relatively slow.
These constraints generally give way to compromise between geographic 
coverage and resolution in space and time. 


The effectiveness of any sampling strategy is ultimately determined
by the accuracy with which the observations can be used to reconstruct
reality-- the state of the natural system being measured.  In this 
context, reality is an elusive metric, for biological distributions in the
ocean have been rarely (if ever) oversampled.  Given the dearth of
opportunity for testing sampling strategies against objective criteria
with purely observational means, numerical models offer an attractive
framework for investigation of these issues.  The approach begins with
the construction of a simulation which is characteristic of the 
natural system.   The model run serves as a space/time continuous
representation of reality, which is then subsampled in a specified
fashion to produce a simulated data set.  The simulated data are then fed
into an analysis scheme in which they are synthesized into a
reconstruction of reality.  Direct comparison of the reconstructed field
with the ``truth'' as defined by the original simulation thus provides a
quantitative evaluation of that particular sampling strategy.  This
general procedure, sometimes called Observational System Simulation
Experiments (OSSEs), has its origins in dynamic meteorology ({\it e.g.}
Charney {\it et al.}, 1969) and is recognized as an important tool for
the development of oceanographic sampling systems (Smith, 1993;
Robinson {\it et al.}, 1998).

\nocite{charneyetal69}
\nocite{smith93}
\nocite{robinsonetal98}

One aspect of sampling which is particularly amenable to assessment with
the OSSE approach is the synopticity of spatial surveys.  
A set of measurements is synoptic if it is collected in
a time interval which is short enough that the underlying distribution
does not change appreciably.  OSSEs provide a means to quantify the
extent to which dynamics in the underlying field can compromise
the fidelity of a map generated from data collected over a finite time
interval.  The procedure consists of the following three steps:
(1) subsampling model output along a realistic
cruise track over a time interval which reflects the actual time
required to occupy those stations at sea, 
(2) objectively mapping the simulated data, and
(3) comparing the analysis with an instantaneous snapshot from the
original model calculation.
Of course, this estimate of the space/time smearing associated with
the sampling strategy is 
robust only to the extent to which the model simulation used as the
basis for the OSSE is representative of the real ocean.

Here we evaluate the synopticity of the U.S. Globec Georges Bank
broadscale sampling plan using OSSEs based on data-driven simulations
constructed with the Dartmouth finite element circulation model
(Lynch {\it et al.}, 1996).  The approach is to initialize the model
with a set of observations and run forward for a period of one month,
using a second set of observations as an end point.
Two different data assimilation procedures are used to construct 
these simulations: nudging and the adjoint method.
The forward simulations are
then subsampled without error\footnote{Incorporation of
measurement error would enhance the realism of this exercise; we have
chosen not to do so in order to focus attention on the space/time
smearing issue.} along the broad-scale
survey track over the ten-day time period which is required to occupy 
the sampling grid.  The simulated data are
then objectively analyzed and compared with truth as defined by an
instantaneous snapshot of the model solution at the central time of
the survey (day 5.0).  The utility of an ``advective correction'' to
the objective analysis reconstruction is investigated by adjusting the 
locations of the observations from the original stations to their positions
at the central time assuming transport by the mean flow.

\vskip .1in
\noindent
{\bf Methods}

The U.S. Globec Georges Bank Broadscale Sampling Program was designed to
measure seasonal variations in the spatial abundance patterns of the target
species (U.S. Globec, 1992; Wiebe {\it et al.}, 1996a).  
A typical sampling plan is shown in Figure \ref{bss}, which
consists of approximately 40 stations spread fairly 
uniformly over the bank.  Such a grid
takes ten days to complete.  These surveys were occupied
monthly from January through June during the years 1995-1999.
Station data include CTD profiles of water column
properties and abundance estimates for zooplankton and larval fish derived 
from 1m$^2$ and 10m$^2$ Multiple Opening and Closing Net and Environmental
Sensing Systems (MOCNESS - Wiebe {\it et al.}, 1985) and Bongo net tows 
as well as pump profiling for microzooplankton.  
The OSSEs described herein are based on simulations created using 
abundance data on adult female {\it Calanus finmarchicus} derived from
MOCNESS tows
on two sequential broadscale surveys: R/V {\it Endeavor} cruises
EN261 (10-20 February
1995) and EN263 (13-23 March 1995).  Animal concentrations were
computed by dividing the vertically integrated abundance estimates
by the depth interval that was sampled.

\nocite{usglobec92}
\nocite{wiebeetal96a}
\nocite{wiebeetal85b}

\vskip .1in
\noindent
{\it Objective Analysis}

Construction of maps suitable for input into models requires an
algorithm by which irregularly spaced observations can be interpolated
onto an arbitrary grid.  Objective analysis is a technique for doing so
that minimizes the expected error (in a least squares sense) at each
analysis point via a linear combination of the neighboring observations.
Each data point is weighted according to its space/time distance from
the analysis point and a four-dimensional correlation function derived
from the ensemble of prior data.  Thus statistical information about
the measured field is
used to optimize the resulting map.  This procedure has been used routinely
for nearly three decades in open-ocean applications where the statistics
are fairly homogeneous and well-behaved (Bretherton {\it et al.}, 1976;
Freeland and Gould, 1976).

\nocite{brethertonetal76}
\nocite{freelandgould76}

Objective analysis in coastal domains is a much more difficult problem
for essentially two reasons.  First, the presence of a coastline
imposes geometrical constraints which are not easily handled analytically.
Second, coastal processes are in general strongly impacted by topography
which results in highly non-uniform statistics.  For example, Petrie and
Dean-Moore (1996) demonstrated that correlation scales of physical 
properties on the Scotian shelf were generally larger in the along-isobath
direction than across isobaths.  He {\it et al.} (1997) have implemented
these aspects into an objective analysis procedure based on the original
algorithm developed by Bretherton {\it et al.} (1976).  The basic idea
is that a set of domain-wide correlation scales are modulated by the
local bathymetric gradients, such that the anisotropy with respect to
along versus across isobath correlations is explicit.  The domain-wide
covariance model used here is of the form

$$ {\rm Cov}(C_i,C_j) = \sigma^2 e^{-r} \bigl( 1 + r + {r^2 \over 3} \bigr) $$

\noindent
where $\sigma^2$ is the variance estimated from the data and
$r$ is the pseudo-distance between points $i$ and $j$ defined as

$$ r = \sqrt{ \Biggl( {\Delta x' \over S_{x'}} \Biggr)^2
             +\Biggl( {\Delta y' \over S_{y'}} \Biggr)^2
             +\Biggl( {\Delta t  \over S_t}    \Biggr)^2 } $$

\noindent
in which the coordinates $x'$ and $y'$
refer to the local cross-isobath and along-isobath directions,
respectively.
The scale factors are oriented with respect to the local topography
in the following manner,

$$ S_{x'} = S_0 \bigl( 1 - C_x \sqrt{s} \bigr) \qquad 
   S_{y'} = S_0 \bigl( 1 + C_y \sqrt{s} \bigr) $$

\noindent
where $s$ is the magnitude of the bathymetric gradient.
For the purposes
of this study, the same correlation scales were used to map zooplankton
distributions as were used in the objective analysis of hydrographic
measurements used to drive the circulation model (see below):
$S_0 = 30$km, $C_x = 3$, $C_y = 10$ and a temporal scale $S_t = 30$ days.
Sensitivity
analysis showed that the resulting maps of zooplankton distributions on
Georges Bank were generally robust over a range of choices for the global
correlation scales.  However, rigorous assessment of the spatial and
temporal covariance statistics of the target species is clearly needed
in future research.

\nocite{heetal97}
\nocite{petriemoore96}
\nocite{heetal97}

\vskip .1in
\noindent
{\it Simulation Model}

The circulation in this region has been the subject of intense modeling 
activity by the Dartmouth Numerical Methods Laboratory 
(Lynch {\it et al.}, 1996). A finite element approach has been utilized
which facilitates realistic representation of the complex geometry
in this area.  Horizontal grid spacing in regions of steep topography
is as fine as 500m, and considerably coarser where such high resolution is not
required.
Perhaps the most
relevant circulation element in the present problem is the clockwise flow
around Georges Bank, caused by both tidal rectification and baroclinic
effects.  This aspect of the physical solutions has a scale which is
of the same order as the biological sampling.  Thus, the physical
and biological scales resolved in this coupled problem are reasonably well
matched.

This three-dimensional model
is hydrostatic, non-linear, and incorporates
advanced turbulence closure.  Published
solutions for the climatological mean circulation, broken down
into six bi-monthly periods, 
are demonstrably consistent with available observations
(Naimie 1996; Naimie {\it et al.}, this issue).  
Archived solutions
are stored in a form which is available for use in an off-line transport code
``Acadia'' which solves a depth-integrated form of the
advection-diffusion-reaction equation on the
same grid using the archived hydrodynamic information as input.

\nocite{lynchetal96}
\nocite{naimie96}
\nocite{naimieetal98}

Within this framework, the coupled physical--biological model for
vertically averaged zooplankton concentration $C$ is written

$${{\partial C}\over {\partial t}} +
{\bf v}\cdot\nabla C - {1 \over H} \nabla\cdot(H K \nabla C) = R \eqno{(1)}$$

\noindent
where ${\bf v}$ is the velocity, $K$ the diffusivity, $H$ the 
bottom depth, and $R$ the biological ``reaction'' term.
Positive $R$ implies net growth of the population, while negative $R$
implies net mortality.  Climatological mean velocity and diffusivity
fields are specified from the bi-monthly archive described above.
Boundary conditions consist of (1) no flux
through solid boundaries, (2) specification of concentration at inflow,
and (3) computation of concentration at outflow assuming no diffusive flux.

% In approaching coupled problems of this type, it is important that the
% physical and biological scales represented in the solutions are 
% consistent.  For example, it would not make sense to incorporate fine
% resolution biological observations directly into a coarse resolution
% circulation model.  Using coarsely sampled biology in high resolution
% physics could be equally problematic.  In some ways, the present case
% may seem to have attributes of the latter, with biological resolution
% of 20-30km and physical resolution as fine as 0.5km.  However, it is
% important to point out that the salient features of the physical
% solutions are very much larger than this lower bound of the node
% spacing.  Mesh this fine is deployed only in regions of very steep
% topography, where it is required for computational stability in the
% presence of very strong (order 1 m s$^{-1}$) tides.  
% Perhaps the most
% relevant circulation element in the present problem is the clockwise flow
% around Georges Bank, caused by both tidal rectification and baroclinic
% effects.  This aspect of the physical solutions has a scale which is
% of the same order as the biological sampling.  Thus, the physical
% and biological scales in this coupled problem are reasonably well
% matched.

\vskip .1in
\noindent
{\it Construction of realistic simulations}

% Two different approaches are used herein to create realistic
% simulations for the OSSEs conducted in this study: nudging and the
% adjoint method.

The OSSEs described herein are based on simulations which match observed 
{\em Calanus finmarchicus} concentrations at two points in time.  
Distribution maps at the two time points are derived from
objectively-analyzed broadscale surveys.  The matching is achieved by 
adjustment of the biological reaction term $R$ in equation (1).  
Two different approaches are used: nudging and the adjoint method.
These two assimilation procedures are discussed in turn below.

Nudging is a technique of data assimilation in which a model state
variable is relaxed to a particular value via a first order process.
For the present purposes, the simulation is initialized with the
concentration field derived from an objective analysis of the first survey
(EN261), 
and then nudged toward an objective analysis of 
the subsequent observations (EN263).  In equation (1) above, this amounts
to $R = -\alpha (C - C_{obs})$ where $\alpha$ is the nudging parameter
and $C_{obs}$ is the concentration observed in the second survey.
The rate constant of the relaxation thus specifies the e-folding time
scale at which the model solution converges.  At steady state, the
nudging term balances the model's local tendency to drift away from
the observed value.  Thus the stiffness of the nudging controls not only the
time scale for convergence, but also the magnitude of the steady state
difference between the numerical solution and the observations.

A series of experiments were conducted in order to determine an appropriate
value for the nudging parameter.  Our aim is to simulate a 10 day period
(the time it takes to complete a survey) in which the temporal changes
are realistic.  Figure \ref{nudging} shows time series of the
differences between model solutions
and the data toward which they being nudged for three different 
values of the nudging parameter.
The solid line indicates the ``ideal'' solution for these purposes
which is a monotonic trajectory between the two observational periods.
Clearly the nudging simulations produce exponential rather than monotonic
adjustment from the initial conditions\footnote{Note that non-zero asymptotes 
in this family of curves demonstrate the strong role of advection in
these solutions.}.  However, the relatively linear
change in the first 10 days of simulation (c) are actually quite close to
``ideal'' as this term has been defined here.  A value of $\alpha = 40$
days$^{-1}$ is therefore used in the nudging experiments described below.

Adjoint data assimilation provides a more sophisticated (and arguably more
realistic) approach to creating a simulation of the type needed here.
This technique facilitates inversion for the population dynamics
implied by the changes in abundance between the two surveys and the
circulation during the intervening period.  The method begins with
the definition of a cost function, which is a measure of the misfit
between predicted and observed concentrations.  The goal of the
procedure is to minimize the cost function subject to the constraints
that the forward model equations are obeyed.  Formulation of a
Lagrange function, which is the sum of the cost function and
the product of a set of Lagrange multipliers with the model equations,
facilitates derivation of the adjoint model.  For the case in which 
the biological reaction term is a spatially varying constant $R=R(x,y)$,
the adjoint of equation (1) is

$$-{{\partial\lambda}\over {\partial t}}
-\nabla\cdot (\lambda {\bf v}) - {1 \over H} \nabla \cdot (H K \nabla \lambda )
= -2 \delta_M (C-C_{obs}) \eqno{(2)} $$

\noindent
where $\lambda$ are the Lagrange multipliers and $\delta_M$ is a measurement 
functional which is equal to one where observations exist in space-time and 
zero elsewhere.  Dynamically analogous to the forward model, the adjoint
model is an advection-diffusion-reaction equation which describes the
propagation of information about model-data misfits.  It can be shown that 
integration of the adjoint equation provides a method of computing the
gradient of the cost function with respect to the unknown
variable $R(x,y)$\footnote{
In this particular application, all of the model-data misfits are
attributed to biological processes.  Thus, the assimilation procedure
will amalgamate any errors in the prescribed circulation into the
inversion for $R(x,y)$.  The implicit assumption here is that errors
in the physics are small relative to biological sources and sinks.}.  
This provides the basis for a iterative 
solution to the variational problem which consists of the following
steps: (1) starting with some initial guess of the unknown variable
({\it e.g.}, $R=0$), initialize with data from the first survey and
run forward in time to evaluate the cost function; (2) integrate the
adjoint model, forced by the misfit between observed and predicted
values to evaluate the gradient of the cost function with respect to
the unknown variable; (3) feed the cost function and its gradient
into a descent algorithm to produce an updated estimate of $R(x,y)$;
and (4) repeat as necessary until convergence is attained.  In practice,
the assimilation procedure produces solutions which are nearly
identical to the observations.  See
McGillicuddy {\it et al.} (1998) for more details on the method and
its application to {\it Pseudocalanus} spp. populations in the Georges
Bank - Gulf of Maine region.

\nocite{mcgetal98b}

For the present purposes, either of these simulations -- 
nudging or adjoint -- are 
taken as reasonable surrogates for the natural system.  This is
demonstrated below in the {\bf Results and Discussion} section.  The
two simulations represent two equally plausible population dynamics
scenarios, in which the animal distribution changes in a manner that
the broad-scale survey is intended to resolve.  An effective sampling plan
should at least be successful with these two versions of reality.

\vskip .1in
\noindent
{\it Experimental Design}

The OSSEs described below consist of three separate experiments which
are performed on each case.  First, the model simulation is sampled
on the central date (day 5.0) as if the ship were able to occupy all 39
stations instantaneously.  This serves as the control in which the
simulated observations are perfectly synoptic.  Differences between
an objective analysis of the simulated observations and the ``true''
field (day 5.0 of the model simulation) are thus a result of mapping
error only.  Second, the model simulation is sampled in space and time
during days 1-10 along the prescribed broadscale sampling pattern
(see Table 1 for sampling schedule).
In this case, differences between the objective analysis of the simulated
observations and day 5.0 of the original simulation are a combination
of mapping error and the spatial/temporal smearing inherent in the
sampling plan.  The magnitude of the difference between this
and the control experiment is thus a measure of the error associated
with the lack of synopticity.  Finally, the space/time experiment is
repeated with the station positions adjusted (relative to the central time)
for the fluid displacement by the mean current.
That is, the locations of stations 
occupied prior to day 5 are advected forward, those on day 5 are 
unaffected, and those after day 5 are advected backward.  Comparison
of the objective analysis using the adjusted data
with the prior experiment thus facilitates evaluation of an advective
correction aimed at reducing spatial/temporal smearing.

Each of the three experiments are performed on several simulations based on
the EN261/EN263 MOCNESS data.
Sensitivity experiments are performed with
respect to the flow field and the representation of population dynamics
used in the simulations.
Ascertaining the sensitivity of the results to the flow field
is of particular interest because the timing of the observations is such
that they straddle the boundary between periods for which the climatological
flow is archived (January-February and March-April).  Sensitivity
to the method of biological data assimilation is relevant in that the results of
neither technique have been verified with biological measurements
of associated source/sink terms; in the absence of such information
it is not possible to determine which technique is more appropriate.  
In all, four sets of the three experiments
each were performed (Table 2).

\vskip .1in
\noindent
{\bf Results and Discussion}

Objective analysis of the EN261 and EN263 data reveals a great
deal of structure in the distribution of adult female {\it C. finmarchicus}
(Figure \ref{data}a,b).  In the February survey, four distinct maxima in 
abundance are present: on the northeast peak, the northeast portion of the 
crest just inside the 60m isobath, the center of the crest, and just north
of the Great South Channel.  In addition, there is a broad region of
low concentration south of the crest.  The March survey is characterized
by a north-south band of very high abundance which cuts across the center
of the bank;  although it is quite coherent, local maxima are clearly
evident within the band.  It is flanked by large areas of relatively
low concentration to the east and west.  The peak in abundance present
just to the north of the Great South Channel in February appears to have
intensified.

Two different data assimilation techniques were used to create simulations
of the time period in between the two surveys.  Nudging is an inherently
local process, so the structure of the forcing term arising from this
procedure reflects the Eulerian difference between the evolving concentration
field and the terminal condition toward which the solution is being nudged.
Strong sources to the north and south of the peak at station
31\footnote{Note that in this discussion we use a chronological
numbering scheme which reflects the order in which the stations are
occupied at sea.  See Figure 1 for a conversion to Globec station
identifiers.}
build up the meridional band 
of high concentration (Figure \ref{data}c).
Sinks to the east and west maintain low concentrations in those regions,
while a weak source north of the Great South Channel is responsible for
amplification of the peak in that area.  In contrast, the adjoint method
explicitly represents the effects of the fluid transport in the construction
of the biological reaction term (Figure \ref{data}d).
Net sources occur upstream of areas in which abundance
increases, whereas net sinks are required where high concentrations are
advected into regions where the animals are not observed.  A prime example
of this occurs along the southern flank of the bank, where strong 
southwestward flow is present.  Buildup of the peak in the vicinity of
station 6 is brought about by net growth upstream (to the northeast),
and net mortality downstream (to the southwest).  In other areas where
the mean current is weak, changes in abundance are largely controlled
by local balances.  In such circumstances, the source term computed with
the adjoint method resembles the nudging solution.  Examples include
the net growth present near stations 11-12 and 39, and the net mortality
which straddles the 60m isobath northeast of the Great South Channel.

Biological rates implied by the distribution of sources and sinks
created by the two assimilation procedures are not unrealistic.
The magnitude of the reaction term varies from about -1 individual
m$^{-3}$ day$^{-1}$ to +1 individual m$^{-3}$ day$^{-1}$.  Note that
this is an absolute rate of population change, not a specific rate.
Interpreting positive $R$ (a source) as molting of C5 copepodites to
adults, the rate of 1 individual m$^{-3}$ day$^{-1}$ is plausible
based on the
abundance of C5s on Georges Bank at this time (data not shown; see
the U.S. Globec Georges Bank Zooplankton Home Page at
{\tt http://globec.gso.uri.edu/}) and
temperature dependent development (Belehradek, 1935; see also
Lynch {\it et al.}, 1998).  Negative $R$ implies net mortality
in the adult population; consumption of 1 individual m$^{-3}$ day$^{-1}$
is easily justifiable in terms of the abundance of predators and 
rates of ingestion measured on Georges Bank (Davis, 1984;
Sullivan and Meise, 1996; Madin {\it et al.}, 1996).
Thus, from the point of view of biological rates, both the nudging and 
adjoint solutions are 
reasonable candidates for realistic simulations.

\nocite{davis84c}
\nocite{sullivanmeise96}
\nocite{madinetal96}
\nocite{belehradek35}
\nocite{lynchetal98b}

Results of the first set of experiments (Table 2) are presented in
Figure \ref{nudge_JF}.  Animations of the simulated space/time
sampling are available on the accompanying CD-ROM and also at
{\tt http://science.whoi.edu/users/mcgillic/globec/OSSE\_paper/animations/}.
The base simulation was created with nudging
using the January/February flow field.  The top row of panels shows
three instantaneous snapshots of simulated abundance during the first
ten days of integration.  Note that the first panel (Sim Day 0) is
identical to the objective analysis of the EN261 survey (Figure \ref{data}a), 
as it was used for the initial condition.  In the control experiment,
simulated measurements are collected from an instantaneous occupation of 
stations 1-39 on day 5.  The resulting objective analysis and its
difference field with the ``reality'' as defined by the simulation on
day 5 reveal that the general features of the distribution have been
captured.  The most significant discrepancies are associated with
(1) smoothing of the peaks near stations 11, 21 and 39; and (2)
fine scale structure in the ``true'' field which is not resolved by
the sampling.  The most obvious example of the latter occurs at the
western end of the southern flank, where a local minimum develops in 
the simulation in between stations 2 and 3.  

Comparison with the
experiment in which the model is sampled during days 0-10 along the
cruise track reveals the additional error that arises from spatial and 
temporal smearing associated with the lack of synopticity.
The most prominent example occurs in the vicinity of stations 3, 5 and 6
where a local minimum present in the initial condition turns into a
local maximum by day 10.  Because this area is sampled early in the
survey (days 1-2), the resulting objective analysis depicts a local minimum
there.  The difference field in the region thus reflects an underestimate of 
the ``truth'' in model day 5 which is characterized by intermediate
concentrations in that area.  Another notable example of smearing is
evident in the further reduction of the peak concentration around
station 31.  During the simulation, the peak is advected southward
and decreases in magnitude.  Because those stations are occupied late
in the survey (day 8), the simulated measurements (and therefore the
resulting objective analysis) reflect the degraded peak.

Correcting the station positions for advection by the mean flow
helps reduce the impact of space/time smearing.  In particular, 
southwestward advection of stations 3 and 6 noticeably improves the
objective analysis along the western portion of the southern flank.
In the first few days of the simulation (during which this region is
sampled), the broad area of low concentration around stations
3,5,6 and 9 is neutralized by the nudging term seeking to create
the maximum that accumulates there one month later.  The advective
correction improves the analysis here in two ways.  First, the local
minimum in between stations 2 and 3 on day five of the simulation 
is recreated almost perfectly by the advective correction: because
station 3 does not lie within the region of positive nudging, the 
westernmost portion of the low concentration water breaks off and
is swept to the southwest.  Accordingly, repositioning the simulated observation
at station 3 places the subgridscale local minimum
right where it should be.  The second improvement relates to the
southwestward displacement of the simulated observation at station 
6, where the maximum in positive nudging is located.  Advection
of the increased concentration toward station 3 helps flatten out
the minimum there, resulting in improvement of the difference
field in a strip oriented northeast-southwest in between stations 3
and 5.

In an attempt to assess the sensitivity of these results to the
specified circulation, the experiments were repeated using the climatological
flow from the March/April period.  The resulting maps were qualitatively
similar to Figure \ref{nudge_JF} and are therefore not shown here.
Only minor differences are evident between the January/February and
March/April experiments in terms of their root mean square differences
with ``truth'' integrated over the bank (Figure \ref{results}a).

Simulations based on the adjoint method show more dramatic evolution
in the first ten days than the nudging runs.  The case using the
January/February flow field (Figure \ref{adjoint_JF}) reveals a higher
dynamic range in concentration values over the same time period, as
well as more fine structure.  In terms of the bank-wide structure, the
biggest differences with the nudging case occur along the southern
flank.  The strip of net growth from just east of station 6 all the way
to station 17 (Figure \ref{data}d) causes strong accumulation there,
which is most intense furthest downstream.  As it turns out, the local
maximum in that region is located in between stations 6, 7, 8, and 9
on day 5 (model ``truth'').  Because only the periphery of the peak is
sampled, the concentration in that area is underestimated even in the
perfectly synoptic control experiment.  That issue aside, the general
features of the bank-wide distribution are represented, including the
peaks in the vicinities of stations 11, 21, 31 and 39.  However,
there is in general more fine structure in between stations than in
the nudging cases, thereby increasing the overall magnitude of the
mapping error quantified by the control experiment (Figure \ref{results}b).

Spatial and temporal smearing associated with occupation of the ship track
further degrades the fidelity of the objective analysis.  Sampling of
stations 6-9 early in the survey (day 2) causes the concentration to be
underestimated there.  Another noticeable difference with the control
experiment occurs on the eastern side of the north flank, where a small
patch of low concentration water originating between stations 27, 28 and
29 is transported to the very edge of the northeast peak during the
ten days of simulation.  On day 5, the patch is centered on station 27.
However, station 27 is not occupied until day 7, just after the low has
passed; this results in an overestimate of the concentration there.

Correcting the station positions for the mean flow causes little
change in this case.  Some areas are improved slightly.
For example, southwestward displacement of the simulated observation
at station 3 helps partially compensate for the unresolved
minimum in between there and station 2.  However, averaged over the bank
the improvement is very minor (Figure \ref{results}b).  

As in the nudging cases, basing the experiments on the March/April
flow field causes little change in the qualitative behavior and
so the detailed maps are not presented here.  But, there is a
significant increase in the magnitude of the mapping error 
(Figure \ref{results}b), owing to the creation of more fine scale
structure in the March/April case.  The increased mapping error is
carried over into the space/time sampling and advective correction
experiments, but again the qualitative sense of the results is the
same.

\vskip .1in
\noindent
{\bf Conclusions}

The Observational System Simulation Experiments described herein present
a framework in which some of the error characteristics of the U.S. Globec
Georges Bank broadscale sampling plan can be investigated.  The results
indicate that the largest source of error is associated with the
mapping procedure itself: perfectly synoptic surveys of instantaneous
model-generated fields result in normalized root mean square errors that
range between 22\% (for the nudging cases) and 53\% (for the adjoint
cases).  In essence, the mapping error is a measure of the degree to
which the sampling grid resolves the ``true'' variability.  In these
experiments, the adjoint data assimilation procedure creates more subgridscale
structure in the model simulations used to represent reality, and the
mapping error is commensurately higher.  Space/time smearing associated
with the lack of synopticity increases the overall error to 30\% (nudging)
and 58\% (adjoint).  As a percentage of the mapping error, this amounts to
36\% and 9\% for the nudging and adjoint cases, respectively.  Thus the
relative error caused by space/time smearing is more pronounced in the
case where the ``biological'' forcing used to create the simulation
is inherently local (nudging).  In that instance, correcting the station
positions for advection by the mean flow is particularly effective, 
reducing the additional space/time smearing error by approximately 80\%.  The 
advective correction is less powerful (35\% reduction) when the dynamics 
underlying the production of variability are not local, as in the adjoint 
method.  Averaging all the cases together, there is a mapping error of
approximately 40\% in perfectly synoptic sampling; space/time smearing 
contributes an additional 20\% (as a percentage of the mapping error),
of which about half can be recovered through advective corrections.

There are several important caveats which must be considered in interpreting
these results.  First, the OSSEs are based on simulations constructed from
data which were collected using the same sampling pattern that is being 
tested for robustness.  Thus the ``true'' fields contain features which 
may have been smeared by the exact same sampling problems addressed here.
Furthermore, they may not adequately represent variations that are not resolved 
by the grid.  Skewness in the statistics of patchiness structure and its
scale dependence has significant implications with respect to resolving
larger scale population distributions (Pinel-Alloul and Pont, 1991;
Cyr {\it et al.}, 1992).  We plan to investigate the issues of sampling
error and subgridscale patchiness in future work, using high resolution
data based on acoustical techniques (Wiebe {\it et al.}, 1996b) and the
Video Plankton Recorder ({\it e.g.}, Davis {\it et al.}, 1996).
Suffice it to say that the present OSSEs are biased in favor of 
the sampling scheme, having already
favorably filtered the observational basis of the simulated ``truth.''
Therefore, these experiments are more a test
of the internal consistency of the sampling scheme than an 
absolute assessment of
its accuracy.

\nocite{pinelalloul91}
\nocite{cyretal92}
\nocite{davisetal96}
\nocite{wiebeetal96b}

Of course, these checks for internal consistency are based on representations
of reality that may have their own intrinsic physical and biological
shortcomings.  Although the sensitivity of the results to some aspects 
have been assessed ({\it i.e.} choice of bimonthly
flow field and data assimilation method), the possibility of systematic
deficiencies in the simulations cannot be discounted.  Examples include
storm-driven washout events in which departures from the climatological
flow can be quite large, and the role of vertical shear and biological 
behavior in controlling organism distributions.  
Perhaps of even more concern is the prospect that even subtle errors in the
circulation could give rise to unrealistic behavior of the coupled
system.  It is important to remember that the OSSEs described
herein are applicable
only to the extent to which the baseline simulations represent
reality.

Finally, these results are based on one particular stage of
one species during one time period.   While the basic aspects of the 
space/time mapping problem may be
somewhat general, their manifestation in other circumstances could be 
quite different.  
However, subject to the caveats discussed above, this study suggests
that the error contribution from the lack of synopticity in the 
broad-scale surveys is smaller than the simple
mapping error associated with under-resolution
of spatial patterns.  Taken together, these two sources
result in an average error of approximately 50\% in abundance
maps.  This estimate represents a lower bound in that there are
additional sources of error which were not included in this analysis.
% Perhaps the most important of these is subgridscale variability,
% which we plan to address in a follow-on study using Observational
% System Simulation Experiments.

\vskip .1in
\noindent
{\bf Acknowledgments}
Support of this work by the National Science Foundation is greatly
appreciated.  This is WHOI contribution 10018 and U.S. Globec contribution
124.

\nocite{lynchetal98b}

\bibliography{ocean}

\vfill\eject

\begin{table}
\begin{tabular}{|c|c|} \hline
Sampling Day & Stations Occupied \\ \hline\hline
 0 &  1           \\ \hline
 1 &  2-5         \\ \hline
 2 &  6-9         \\ \hline
 3 &  10-13       \\ \hline
 4 &  14-17       \\ \hline
 5 &  18-21       \\ \hline
 6 &  22-25       \\ \hline
 7 &  26-29       \\ \hline
 8 &  30-33       \\ \hline
 9 &  34-37       \\ \hline
10 &  38-39       \\ \hline\hline
\end{tabular}
\caption{Sampling schedule for the 1995 U.S. Globec Georges Bank broad-scale
surveys.  Note that the station numbering here (and in Figures 3,4 and 6)
is sequential; see Figure 1 for a conversion to Globec station identifiers.}
\end{table}

\begin{table}[h]
\begin{tabular}{|c|c|c|c|c|c|} \hline

\multicolumn{3}{|c|}{Base simulation} & \multicolumn{3}{c|}{OSSE Experiments} \\

\multicolumn{1}{|c}{No.} & \multicolumn{1}{c}{Method} &
\multicolumn{1}{c|}{Flow} & \multicolumn{1}{c}{Control} &
\multicolumn{1}{c}{Space/Time} & \multicolumn{1}{c|}{Adv. Corr.} \\ \hline

1 & Nudging & JF    & 0.212   & 0.296      & 0.241 \\ \hline
2 & Nudging & MA    & 0.222   & 0.295      & 0.209 \\ \hline
3 & Adjoint & JF    & 0.508   & 0.561      & 0.553 \\ \hline
4 & Adjoint & MA    & 0.554   & 0.596      & 0.570 \\ \hline
\multicolumn{3}{|c|}{Mean: Exps. 1-4} & 0.374 & 0.437 & 0.393 \\ \hline

\end{tabular}
\caption{Run table and error summary for the Observational System Simulation Experiments.
Results are presented in terms of the normalized root mean square difference
% $ \sqrt{{\bigl({\hat \phi_i - \phi_i \over \phi_i}\bigr)}^2} $
between objective analyses of
simulated data 
% ($\hat \phi$) 
and ``truth'' 
% ($\phi$) 
defined as day 5.0 of
the model simulation on which each set of experiments are based.
The values shown here represent an areal average of the gridded
fields on Georges Bank,
for the waters inside the 100m isobath and east of the 69W meridian
which cuts through the Great South Channel.  JF refers to January-February,
and MA to March-April.}
\end{table}

\clearpage

\begin{figure}
\caption{The U.S. GLOBEC Georges Bank Program
broad-scale sampling station plan for 1998. The first broad-scale sampling
was conducted in June 1994 with a sampling design (with 48 stations) that was
subsequently modified into a  pattern that was used in 1995 and had 38
stations.  Station 39 was added during the 1996 field season, Station 40 was
added in 1997, and Station 41 was added in 1998.  Dashed lines indicate
where cruise tracks prior to 1998 differ due these changes in the
station plan.  The accompanying table
provides a conversion between Globec station identifiers and the
sequential numbering scheme used in the text and Figures 3,4 and 6.}
\label{bss}
\end{figure}

\begin{figure}
\caption{Time series of the differences between model solutions
and the data toward which they being nudged for three different
experiments in which the relaxation time scales are (a) 20, (b) 30 and 
(c) 40 days.  The trajectory of monotonic convergence to the second
set of observations is indicated by the solid line.}
\label{nudging}
\end{figure}

\begin{figure}
\caption{Objective analyses of adult female {\it C. finmarchicus} abundance
(number of animals m$^{-3}$) observed during cruises (a) EN261, and
(b) EN263.  Shown below are the source terms (number of animals m$^{-3}$
s$^{-1}$) used to create a simulation 
of the time period between the two cruises using nudging (c) and the
adjoint method (d).  
The nudging term in panel (c) is averaged
over the time period of the simulation.  
These two fields correspond to the flow field for
January-February; results using the March-April flow are quite similar
and therefore not shown here.  
Note that the locations of the observations
in the top two maps (indicated by circles) do not exactly match 
each other or the station
positions called for in the broadscale sampling plan (depicted by numerals
in the lower two panels; see also Figure \protect{\ref{bss}}); 
such discrepancies are unavoidable during normal
seagoing operations.  
The 60m and 100m isobaths are overlayed on all
panels.
Note that the station numbering
is sequential; see Figure 1 for a conversion to Globec station identifiers.
}
\label{data}
\end{figure}

\begin{figure}
\caption{Observational system simulation experiments using the EN261/EN263
data and the nudging technique with the January/February flow field.
The top row shows three snapshots of the simulation representing ``truth.''
These maps are a subregion of a much larger model domain extending from
the New England Shelf up through the Gulf of Maine to Nova Scotia (for
more information see Lynch {\it et al.} (1998).  The middle row shows
objective analyses of simulated data from each of the three experiments
described in the text.  Difference fields with ``truth'' (defined as
day 5 of the model simulation) for each experiment are displayed in the 
bottom row.  Station locations are indicated by numerals; the displacement
of the observations associated with the advective corrections are shown in
the lower two panels of the right hand column.  Units for all panels are
number of animals m$^{-3}$.
Note that the station numbering
is sequential; see Figure 1 for a conversion to Globec station identifiers.}
\label{nudge_JF}
\end{figure}

\begin{figure}
\caption{Normalized root mean square difference between objective analyses of
simulated data and ``truth'' averaged over Georges Bank (see Table 2
caption for definitions).
Panels (a) and (b) pertain to the experiments based on EN261/EN263
data using nudging and the adjoint method, respectively.  Circles
represent results using the January/February flow field, and squares
March/April.}
\label{results}
\end{figure}

\begin{figure}
\caption{Same as Figure \protect{\ref{nudge_JF}}, but the simulation
on which the experiments are conducted is based on the adjoint method.}
\label{adjoint_JF}
\end{figure}

\end{document}
