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\journal{Can. J. Fish. Aquat. Sci.}
\received{Received (\emph{by robin}) September 1998}

\volyear{99}[1998]{1999}
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\specialtitle{Preprint: Can. J. Fish. Aquat. Sci.}

\title{Partitioning 
the influence of physical processes on the estimation of 
ichthyoplankton mortality rates.\\
II. Application to simulated and field data.}
\author{J.A. Helbig}
\address{Fisheries and Oceans,
  P.O. Box 5667,
  St. John's, Newfoundland,
  Canada  A1C 5X1}
\author{P. Pepin}
\shortauthor{Helbig and Pepin}


%\maketitle
%
%\newpage
%Helbig, J.A., Pepin, P. 1998.
%Partitioning 
%the influence of physical processes on the estimation of 
%ichthyoplankton mortality rates.  II. Application to 
%simulated and field data.
%Can. J. Fish. Aquat. Sci. 00: 000-000.

\begin{abstract}
Helbig and Pepin \cite{helbig98} developed a theory 
partitioning  the bias and variance of a number of population and mortality
measures  in terms of 
the space and time scales of natural variability of the underlying current and
plankton fields.
This study verifies that theory  
through a series of numerical simulations and by comparing predicted variances
in estimated fluxes with those observed in an earlier field study.
In the simulations, errors in estimated mortality rates were clearly dominated by 
errors in estimated fluxes due to transport, and 
estimates of variance and bias were all in good agreement with theoretical  
predictions.
For the field data,  the theory predicted a  
distribution of values slightly lower than that observed, but of the correct 
order.  That the model performed well for both
simulated and field data suggests it can be applied in practice with reasonable
confidence.
We also briefly explore the role of the model in survey design and analysis
for ichthyoplankton studies in which transport might be important.
\end{abstract}

\begin{resume}
  Tous cela, en fran\c cais
\end{resume}
\maketitle

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

\section{Introduction}

To assign a causal mechanism to perceived (i.e., estimated) changes in
abundance of planktonic organisms, it is essential that we determine the
losses attributable to transport. Success requires knowledge of the accuracy
(bias) and precision (variance) of a field study in order to be confident of
inferences. This is of particular importance in dealing with
pre-recruitment mortality as consensus concerning the major factors which
influence survival during the pelagic egg and larval stages is lacking
\cite{leggett94}.

Helbig and Pepin \cite{helbig98} 
presented a theoretical framework 
showing how error free measures of plankton population, transport, and mortality
vary with naturally occurring  environmental fluctuations.
Field based estimates of these quantities
vary to an even  greater degree, as experiments cannot 
adequately sample all the variability in the plankton and velocity fields.
In essence, the problem addressed by \cite{helbig98} is 
contained in the expression
%~~\\ % this would be catastrophic to the layout
\begin{equation}
  \rZn = Z + \rZf + \rZd \label{eq:0}
\end{equation}
%~~\\
relating the {\em apparent} mortality $\rZn$ estimated from the difference in 
plankton concentration between surveys to the 
the true biological mortality $Z$, the loss due to transport $\rZf$, 
and the diffusive loss  $\rZd$.
\cite{helbig98} showed how the bias 
and variance of these terms was partitioned according to
the space and time scales of variability of the underlying current and
plankton fields. As well, \cite{helbig98} indicated that their
theory could be used to
% ( <--- this is for the benefit of my editor's parenthesis-matching,
%         and should be ignored...
\begin{enumerate}[1)]
\item evaluate sampling plans to determine the length and times 
  scales on which estimates
  are vulnerable to
  degradation by environmental variations, 
\item estimate the bias and 
  variance of the 
  estimates for a given sampling plan and some knowledge of the current 
  and plankton fields, and
\item ultimately provide 
ichthyoplankton ecologists a toola tool  for the ``optimum'' design of
field programs prior to field experiments.
\end{enumerate}
The primary purpose of this paper is to verify the \cite{helbig98} framework through 
a series of numerical simulations
and by comparing predicted variances  of $\rZf$ to those 
observed in the field. 
In addition, a brief evaluation of alternative
sampling plans is made to illustrate how the framework might be used for 
array design. 

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

\section{Numerical Simulations}


A series of mortality estimation experiments were simulated by stepping the
larval balance equation% \\ these forced line-ends before equations
                       % are equally dangerous: there should be no
                       % blank space between an equation and its
                       % surrounding text, though comments (possibly
                       % with leading blank space, as here) are ok
\begin{equation}
\frac{\partial c}{\partial t}=-Zc-\nabla \cdot \left( \mathbf{u} \;c\right)
+K_H\nabla ^2c \label{eq:2}
\end{equation}
through time and sampling the resultant
larval and current fields appropriately. Here
 $c(x,y,t)$ is the plankton concentration ($\#\;\mathrm{m}^{-3}$), 
$\mathbf{u}=(u,v)$ is the horizontal velocity and $K_H$ is the horizontal
diffusivity.
Numerical experiments were run for a series of 
basins (widths of $L=5,10,20,50,100$, and $200$ km)
with a single boundary open to the deep ocean
using a fixed grid spacing $\Delta x = 1$ km (Fig. 1). 
\marginpar{Fig. 1\\near here}
Each experiment was run 100
times with current fields randomly drawn from an appropriately defined
ensemble.

We avoided
specifying open boundary conditions by modelling the plankton
distribution within a rectangular, closed basin of size $[L\;\times
\;(L+10)]\;\mathrm{km}$ and artificially inscribing
an open boundary across which the advective and diffusive fluxes were
measured (Fig. 1). These simulations were intended to be as realistic as
possible. A time- and space-varying current field was prescribed that
transported an initially specified larval field for periods up to 15 days.
Larval concentration was estimated at the beginning and end of each experiment 
by
sampling the plankton field non-synoptically at a fixed set of stations; the
measurements were corrupted to simulate experimental error. In the period 
separating the full
basin surveys, the flux of larvae across the open boundary was measured
non-synoptically in similar fashion at a specified set of stations.
Variability was introduced into both the full-basin and flux surveys by the
natural variability in the current and larval fields, which were modelled
stochastically.

In any given experiment, the velocity field was  
drawn from an ensemble of fields constrained only by a very
limited knowledge of the true field. Because the larvae were assumed to be
passive drifters, the plankton field was also stochastic although not 
independent 
of the velocity field. The current and larval fields were decomposed into 
mean and fluctuating parts as
\begin{align}
\mathbf{u}(\mathbf{x},t) &=\mathbf{0} +\mathbf{u}^{\prime }(\mathbf{x},t) \label{eq:6} \\
c(\mathbf{x},t) &=m(t) c_o+m(t) c^{\prime }(\mathbf{x},t) \label{eq:5}
\end{align}
with
\begin{equation}
\mathcal{E}(c_o)=c_o,\;\;\mathcal{E} (c^{\prime })=0,
\;\;\mathcal{E}(\mathbf{u}^{\prime })=\mathbf{0} \label{eq:7}
\end{equation}
where $\mathcal{E}$ denotes an ensemble average, and the non-dimensional, 
deterministic function
\begin{equation}
m(t)=e^{-Z t} 
\end{equation}
expresses the population decline due to mortality. (Note that 
true mortality  appears here, not 
the effective mortality defined in \cite{helbig98}. This is one benefit of working 
within a closed system.)
The underlying natural mortality $Z$ was set at $0.1~d^{-1}$.
Our choice includes a zero mean velocity field and a spatially constant 
mean plankton field
and thus 
focuses attention the variability in the velocity and plankton fields.



Currents were drawn from an ensemble defined by
the wavenumber-frequency
spectra $S_u(\mathbf{k},\omega )=S_v(\mathbf{k},\omega ) = S(k,\omega )$
where
\begin{equation}
S(k,\omega )=
\begin{cases}
Ak^{-2}\omega e^{-\omega/\omega_m} & k_l \leq k \leq k_h \\
                                   & \omega_l \leq \omega \leq \omega_h \\ 
+ Bk^{-2} & k_0 \leq k \leq k_h \\
          & \omega_0 \leq \omega \leq \omega_h \\
\end{cases}
% stretched out those cases vertically to make them fit within a
% line's width; removed the eccentric \text in the conditional
% expressions
\end{equation}
represents a 
horizontally isotropic process broadly peaked in frequency about a period of 
5 days
($\omega_m/2\pi = .2 \;\daym$) together 
with an additional high frequency, high wavenumber plateau.
Frequencies extend over the range of $\omega_l/2\pi =1/(25\:\mathrm{day})$ to $ \omega_h/2\pi =
1/(2\:\mathrm{hr})$. The upper wavenumber limit of $k_h/2\pi = 1/(2\;\mathrm{km})$
was chosen to represent the smallest possible length scale resolvable by the
model (model Nyquist wavenumber), while the lower limit $k_l$ varied with basin
size and was chosen to admit fluctuations no larger than the basin width 
$2\pi /k_l=\min {\ (L,\;100)}\;\mathrm{km}$.
% that last expression _was_ written as display maths, for some
% reason; also removed second \; before the `km'

The frequency dependent portion of the spectrum was meant to represent
directly wind driven currents and eddies spawned by inertial instabilities
that are common in coastal systems (e.g., \cite{smith91}). 
Currents in such systems
are most energetic in the ``weather'' band extending over a range of about
3-10 days. 
The wavenumber part of the spectrum is 
consistent with the relatively few measures of the 
spatial variability available. For example, Helbig and Pepin (unpublished data)
indicate that currents measured in the top meter of a $20\times 50$ km bay
with high frequency groundwave radar had a spectrum proportional to $k^{-2}$ on
scales extending from 2 to at least 20 km ( $k=[k_x^2+k_y^2]^{1/2}$ is the
wavenumber magnitude). Similarly, Firing \cite{firing93} reported upper layer
currents measured with an acoustic Doppler current profiler at mid-latitudes
in the North Pacific that exhibited a $k^{-2}$ dependence over the range of
10 to greater than 100 km.
The
additional small scale plateau  was chosen to
mimic small scale ``noise'' present in realistic systems that is
characteristic of more transient wind induced fluctuations and larger scale
turbulence (at scales near and below the internal Rossby radius of
$\sim\text{--}10\,\mathrm{km}$.
% was $\sim 5 - 10 $ km
We selected the cutoff wavenumber and frequency as $k_o/2 \pi =
1/(10 \;\mathrm{km}) $ and $\omega_o /2\pi = 1/(18 \;\mathrm{hr})$,
respectively.

$A$ and $B$ were fixed to achieve 
a maximum variance of $(0.10\;%
\mathrm{ms}^{-1})^2$ from the major component of the spectrum and an additional
variance of $(0.05\;\mathrm{ms}^{-1})^2$ from the small scale plateau.
The maximum variance  was realised for $L=100$ and $%
200$~km, while the simulated variance for the smaller basins was
correspondingly lower  (Table 1). 
\marginpar{Table 1\\near here}
The small scale variance was constant in all runs. We emphasize that the
simulated currents were modest, with typical mean speeds of $0.10\;\mathrm{ms}%
^{-1} $ and peak values  up to four time larger. The actual method of
simulating currents from the spectrum is described in the appendix.

Although the choice of $S_u=S_v$ is the most expedient to achieve the desired
distribution of variance in the velocity field, it is not the
most general one as it effectively treats $u$ and $v$ as independent scalars
\cite[Chapter 6]{monin75}. This has implications for the
form of the eddies that comprise the fluctuations, as the cross-spectrum $%
S_{uv}$ vanishes and the spectrum of the divergence is relatively energetic.
These considerations re-emphasize the fact that the simulated current field
was not constrained to conform to a simple choice of dynamics 
(cf.\ Bennett and Denman \cite{bennett85}) who study plankton
patchiness with a
quasi-geostrophic velocity field with vanishing horizontal  divergence). 
An alternative approach 
would be to specify forms for the the divergence and kinetic energy
spectra, and to deduce the appropriate forms for $S_u$ and $S_v$. That
exercise lay beyond the objectives of the simulations here, the most 
important of which was
to verify the forms for the variance and bias given by \cite{helbig98}.
We note however, that our divergence field may be overly
energetic at large length scales, where the geostrophic constraint is 
likely to hold. 
Maximum model values of divergence  were about $10^{-4} s^{-1}$---%
large but not unrealistic.


The initial plankton field
for each simulation was drawn from an ensemble defined by a space
independent mean $c_o=0.2 \;\mathrm{m}^{-3}$
and a two-dimensional horizontally isotropic
wavenumber spectrum proportional to $k^{-2}$ with a variance of ($0.08 \;%
\mathrm{m}^{-3})^2$. In practice, however, the initial form of the plankton
field  proved unimportant, as it  adjusted very quickly to the
imposed velocity field.
Each experiment was run for 24~hr without mortality prior to
sampling to allow relaxation to equilibrium.

The only boundary conditions imposed
were that the normal component of
both velocity and larval concentration gradient vanish at all
solid boundaries. The model used standard second-order, centred-in-space,
centred-in-time finite differences \cite{roache76}, and was a scaled down and
modified version of the SPEM model \cite{haidvogel91}. 
A timestep of 10 minutes was used in all simulations, but model
stability appeared to be determined by the value of the horizontal P\'eclet
number $Pe=V\Delta x/K_H$ where $V$ 
is a characteristic speed.
As shown
by \cite{weaver90}, stability in most centred difference schemes
requires that $Pe\leq 2 $. For $V=0.10\;\mathrm{m}\;\mathrm{s}^{-1}$,
 this demands that $K_H\geq 50\,\;\mathrm{m}%
^2\;\mathrm{s}^{-2}$, near the upper end of values estimated in the field
for eddy diffusivities for small systems from drifter data (e.g., \cite{deyoung95}) and probably an order of magnitude larger than the small
scale eddy diffusivity for tracers \cite{ledwell93}.
We used a value of $100 \;\mathrm{m}\;\mathrm{s}^{-1}$ is all simulations.

The choice of geometry, in which the open boundary was embedded within the
basin, and the fact that the dynamics were uniform throughout the full basin
dictated that the average concentration on both sides of the open boundary must
be the same. The model achieved this balance through the joint action of
advection and diffusion, and the size constraint on $K_H$ coupled with the
absence of a mean flow (small advection) led to roughly 
equal magnitudes for both.
Diffusive effects were monitored during the
simulations, and some of the results presented in the next section have been
corrected for them as indicated.
We denote this correction as $\Zd$ (without caret) to indicate that it is measured without error.

Table 1 summarizes the 76 sampling plans that were explored in the
simulations in terms of basin size, the number of basin and open boundary
stations, the number of flux surveys, the overall duration of the
experiment, and the times required to complete basin and flux surveys. In
each case, the basin and flux stations were symmetrically arrayed (Fig. 1).
Basin stations were occupied consecutively along cross-basin transects. Flux
surveys were spaced symmetrically in time between the two basin surveys, and
were centred on noon of any given day. Between-station times were estimated
assuming 30 min on station and a steaming speed of 10 knots between
stations. Each realistic estimate of plankton concentration and velocity was
corrupted by randomly drawing from normal distributions with zero means and
respective standard deviations of $0.02\;\mathrm{m}^{-3}$ and $0.001\;%
\mathrm{ms}^{-1}$. Any corrupted plankton estimates with negative values
were set equal to zero, but these were very few.

These sampling plans range from very good to rather poor. The worst 
are obviously bad while some of the best are probably  not logistically 
feasible,
either in the field or later in terms of sample analysis. The ``best'' 50~km
basin survey, for example, requires 64~hrs to complete. The results of the
next section should be interpreted with these comments in mind.

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

\section{Simulation Results}


Experimental results are summarized in terms of the
normalised means, biases and variances of $\rZn$, $\rZf$, and 
the corrected mortality $\rZc=\rZn -\rZf -\Zd$ (Fig.~2 and Table~1).
\marginpar{Fig. 2\\near here}
Note  from (\ref{eq:0}) that $\rZc$ coincides with $Z$ if the flux and
diffusive corrections are accurately determined . Since the ensemble means
of $\rZn$, $\rZf $, and $\Zd $ are
unknown analytically (they depend on the unknown correlation between $%
v^{\prime }$ and $c^{\prime }$ on the open boundary) and are needed to 
compute bias and variance, we computed them
numerically during the simulations. Results are shown here for the case of
no measurement error, i.e. for $\epsilon _c=\epsilon _v=0$; the relative
effect of measurement error is discussed later.

Errors in $\rZc $ were clearly dominated by variations in 
$\rZf $ (Fig. 2) which  exceeded  100\% 
for some plans for small basins. 
Even with the $L^{-2} $ scaling indicated in \cite{helbig98}, they ranged upwards to $50\%$ for $L= 200$~km.
By contrast, the corresponding
fractional standard deviation in $\rZn $ 
varied from $5-20\%$ for most sampling plans, although it too was especially large
for small systems for some sampling plans. Interestingly, the variances in $\rZn$ 
and $\rZf$ 
attained minima at
$L=100 $~km 
and thereafter increased.
The mean value of $\rZn$  was near the true value $Z$ 
and differences diminished with increasing $L$, while  
fractional biases in $\rZf$ 
exceeded $50\%$ at small scales and decreased to 
$ 10-20\%$ for larger basins. 
These biases were 
positive for all systems meaning that the advective contribution was
routinely overestimated and that $Z $ would be underestimated if corrected for advection. 

Estimates of the variance in $\rZn$ and $\rZf$   were in good agreement those predicted  
by \cite{helbig98} (Fig. 3), as
\marginpar{Fig. 3\\near here}
80\% and 65\% of the variance in $\var (\rZn)$ and $\var(\rZf)$ was explained by the 
leading term
in the respective formula derived in \cite{helbig98} (HPI-42 and 43 with $S-C$ approximated by
$\var(c)/\var(v) S_v$).
In particular, the  fall and subsequent rise in both $\var(\rZn)$  and $\var(\rZf)$
with basin size was predicted.

Measurement error had little effect on the estimations of the biases of $%
\rZn $ and $\rZf $ as values
were generally within 5\% of the non-corrupted estimates. Similarly, the
standard deviations of $\rZf $
were only a few percent higher than those plotted. Standard deviations of $%
\rZn  $ were, in general, 5-25\%
larger than those in its  noncorrupted counterpart. 


Histograms of the 100 values of $\rZn $, $\rZf $, 
and $\rZc  $ are
shown for two sampling plans for a 5 day survey of a 20 km basin (Fig. 4).
\marginpar{Fig. 4\\near here}
The first plan utilised 36 symmetrically arrayed stations 
(run 17, Table 1) coupled with 3 flux surveys of 10 stations each, whereas
the second used 9 basin stations aligned in 3 rows of 3 (run 26) together
with 3 flux surveys of 5 stations each.  The upper row of panels illustrate the natural
variability in $\rZn$ and $\rZf$, whereas the
remaining panels  demonstrate  how sampling
survey design induced variability overwhelms the natural variability, at least
in the systems simulated.
The superiority of the 36 station plan over the 9 station plan is evident.

The dependence on sampling plan is easily understood in terms of 
the \cite{helbig98} theory.
For example, the 
 variability in $\rZn$, as given by the first term in (HPI-42), is proportional to
\[
 \int S_c(\mathbf{k},\omega) [1 - \cos( \omega \DT)] 
     \left| \Hhat(\mathbf{k},\omega) \right|^2 d \mathbf{k} d \omega   .
\]
where $\Hhat(\mathbf{k},\omega )$ is the complex-valued, 
deterministic function
\begin{equation}
\Hhat(\mathbf{k},\omega )\approx \frac 1J\sum_{j=1}^Je^{i(\mathbf{k}\cdot
\boldsymbol{\xi }_j-\omega \tau _j)}.  \label{eq:H}
\end{equation}
defined in terms of the location $\boldsymbol{\xi }_j$ of each of $J$ stations relative to the basin centre
and the time lag $\tau_j$ the station occupation time relative to the central time $T$ of the survey.  
Contour plots of  $S_c(\mathbf{k},\omega)$ (dashed) and $|\Hhat(\mathbf{k},\omega )|^2$ (solid)
 are shown in Fig.~5 for the two 
sampling plans (Panels A and C), whereas
contour plots of the full integrand, $S_c (1 - \cos \omega \DT)
   |\Hhat|^2 $ are given in Panels B and D. These figures dramatically illustrate that the 
9 station plan 
is vulnerable to natural environmental fluctuations over a much wider range of space and time scales
than is the 36-station plan.
In fact, the ratio, 6.21,  of the integrals 
(i.e. the areas shown in Fig. 5B and D) is 
is very close to the ratio, 5.49,  of the variances in $\rZn$ observed in the 
simulations (Table 1).

These results paint a rather pessimistic picture of our ability to accurately
remove the physical contribution from mortality estimates,
especially for systems less than $50\;\mathrm{km}$ in extent (Fig. 2).
Many existing mortality estimates thus would appear to be severely
contaminated by physical effects. Ultimately, however, verification of 
a theory with
simulated data is unsatisfying, and we turn, therefore, in the next
section to a comparison made with real data.

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

\section{Application to Field Data}

In this section, we apply the \cite{helbig98} theory  to 
field data to see if its 
predictions are valid for an uncontrolled setting
  and to demonstrate how the theory can be applied 
in practice, even with 
incomplete data. We reconsider an earlier study of 
ichthyoplankton mortality (Pepin et al. 1995)
and focus attention directly on the errors associated in estimating transport, 
because these were 
clearly  dominant  in the simulations, and because fluxes were 
not discussed in
any detail in \cite{helbig98}.


The leading term in the expression for the variance of $\rZf$ (\cite{helbig98}-43) is
{\allowdisplaybreaks
\begin{xalignat}{2} 
\var&(\rZf) \approx \left(\frac{L_x}{A \wh{C}_o}\right)^2  \var(\rP )  && \notag\\
&= \left(\frac{L_x}{A \wh{C}_o}\right)^2 \left\{ 
       \int  S_v(k_x,\omega ) \left| \FC(k_x,\omega) \M(\omega) \right|^2 
                    \: dk_x\:d\omega \right. && [{\bf T_1}] \notag\\
%
&+ \int S_c(k_x,\omega ) \left| \FV(k_x,\omega) \M(\omega) \right|^2
                    \: dk_x\:d\omega&& [{\bf T_2}] \notag \\
&+2  \int  \:   S_{vc}(k_x,\omega ) \FC^{\ast}(k_x,\omega)
      \FV(k_x,\omega) \left|\M(\omega)\right|^2 \: dk_x\:d\omega && [{\bf T_3}] \notag\\ 
%
&+ \left( \sum_{m=1}^M(\frac{\Delta l_m}{L_x})^2 m^2(\delta t_m) v^2_{om}\right)
 \left( \sum_{n=1}^N (\frac{\Delta t_n}{\DT})^2 m^2(T_{fn}) \right) \sigma_c^2 && [{\bf T_4}] \notag\\
%
& +\left( \sum_{m=1}^M(\frac{\Delta l_m}{L_x}) m(\delta t_m) c_{om}\right)^2 
 \M^2(0) \; \sigma_u^2&&  [{\bf T_5}] \label{eq:100}\\
%
&= {\bf T_1 + T_2 +T_3 +T_4 +T_5} && \notag   
\end{xalignat}
}
where
{\allowdisplaybreaks
\begin{align}
&\wh{F}_C(k_x,\omega ) =\sum_{m=1}^M(\frac{\Delta l_m}{L_x})m(\delta t_m) 
           c_{om}e^{ik_xx_m}e^{-i\omega \delta t_m} \label{eq:FC}\\
&\wh{F}_V(k_x,\omega ) =\sum_{m=1}^M(\frac{\Delta l_m}{L_x}) m(\delta t_m) v_{om}e^{ik_xx_m}  
           e^{-i\omega \delta t_m} \label{eq:FV}\\
& \wh{M}(\omega ) =\sum_{n=1}^N(\frac{\Delta t_n}{\DT})m(T_{fn})e^{-i\omega T_{fn}} \label{eq:M} 
\end{align}
}
All of the notation coincides with that
used in \cite{helbig98}, but  we note that  
$L_x$ and $L_y$ are the bay width and length, respectively; 
$A=L_x L_y$ is the surface area;
$\DT$ is the time between full bay surveys; 
$\rP$ is the estimated flux and $\rP_0$ is its mean value;  
$\wh{C}_o$ is the estimated basin 
mean plankton concentration averaged over $\DT$;
$S_v$ and $S_c$ are the velocity and plankton concentration autospectra 
and $S_{vc}$ is the cross spectrum; 
$\FC $ and $\FV$ are the respective
 discrete Fourier transforms of the mean larval concentration $c_o$ and 
the mean out-bay current $v_o$; 
$\M$ is the discrete Fourier transform of the mortality function 
    $m(t) = e^{-Z_e t} $ defined in terms of the effective mortality; 
$M$ is the number of stations occupied during in a flux transect; 
$N$ is the number of flux transects executed during the experiment; 
$\Delta l_m$ is the length interval along the flux transect line encompassing the mth station
at $x=x_m$;
$\delta t_m$ is
the time interval between adjacent stations along the flux transect line; 
$T_{fn}$ is the central time of the nth flux transect and 
$\Delta t_n$ is the time interval over which that transect applies;  
and
$\sigma_c^2$ and $\sigma_u^2$ measure the variance in larval concentration 
and current speed due to experimental error.

As indicated by \cite{helbig98}, estimation of the various terms in (\ref{eq:100})  
requires knowledge of:
% ( [ignore that]
\begin{enumerate}[1)]
\item the basin geometry and sampling 
  plan---$L_x$, $L_y$, $\DT$, $N$, $M$,   $x_m$, $\delta t_m$, $T_{fn}$;
\item a first order guess at the effective mortality---$\Ze$;
\item the mean plankton and along-channel velocity along the open 
  boundary---$c_o(x,L_y)$ and $v_o(x,L_y)$; 
\item the auto- and cross-spectra---$S_c(k_x,\omega)$,
  $S_v(k_x,\omega)$,
% ` - ' -> `---'
  $S_{vc}(k_x,\omega)$;  and 
\item the experimental errors---$\sigma_c$ and $\sigma_u$.
\end{enumerate}
Before showing how each of these variables was calculated, we indicate how 
the various integrals
in (\ref{eq:100}) can be computed.  
Consider ${\bf T_1}$ as an  example. The integral can be approximated 
as a double summation over wavenumber and frequency if we specify an 
appropriate set for each.
Given a wavenumber range of interest of $k_l \leq k_x \leq k_h$, 
we define the set:
$k_i = k_l + (i-1)\Delta k \;\; i=1,N_k$ with $\Delta k = (k_h-k_l)/(N_k-1)$. 
A similar choice is made for $\omega$
so that
\begin{equation}
{\bf T_1} \approx \Delta k \Delta {\omega} \sum_{i=1}^{N_k} \sum_{j=1}^{N_{\omega}}
      S_v(k_i,\omega_j ) \left| \FC(k_i,\omega_j) \M(\omega_j) \right|^2  \label{eq:T1}.
\end{equation}
In order to evaluate (\ref{eq:T1}), one must know $S_v$, $\FC$ and $\M$ at 
the various $k_i, \omega_j$
points. It is thus advantageous to have simple analytical forms for these 
functions so that they can be
computed for arbitrary $k_x$ and $\omega$. We therefore fit simple 
polynomial curves to the
observations.


The field data come from multispecies ichthyoplankton surveys of
Conception Bay  carried out  over 2--3
% number range converted to en-dash (`2-3' -> `2--3')
week periods in 1990 and 1991 (Pepin et al. 1995). 
All the basin and sampling plan variables are listed in Table 2, which
\marginpar{Table 2\\near here}
reveals significant differences in execution between the two years.  
In 1990, biological
and physical measurements were made from a single vessel. Within-basin 
ichthyoplankton surveys were conducted every 6-7 days 
($\DT$ was chosen here  here  as the time
between the first and last surveys). On intervening days, transects 
were completed across the
$L_x = 20$ km bay mouth every 5-6 days to monitor fluxes 
(the relatively long time gaps were due
to weather  and a ship failure). Ichthyoplankton tows were made 
at $M=7$ stations, and currents were
measured continuously along the transect with a 
153 kHz boom mounted ADCP.   
In 1991, the weather was significantly calmer and two ships were used, one for 
ichthyoplankton tows and the other for physical measurements. 
Within-bay and flux surveys were made every 6 and 1-2 days, respectively. 
Currents were
again measured with an ADCP but using an improved, hydrodynamically smooth 
boom \cite{colbourne93}.
Further survey details are contained  in \cite{pepin95}.

The mean component of the along-bay current across the mouth and the spatial part of the along-bay 
current spectrum were estimated from 16 ADCP transects across the mouth.  The mean was reasonably fit 
by a quadratic (Fig. 6A). 
\marginpar{Fig. 6\\near here}
A raw wavenumber spectrum was computed from the discrete Fourier 
transforms of the residuals for each transect, and the set of raw spectra
were well fit by the curve $A k^{-1.9}$ (Fig. 6B).
However, the transects did not 
provide enough samples in time to permit computation of 
the temporal part of the spectrum. Consequently, we used 
the average  spectrum computed for three current meters moored at 20 m depth for 
6 months in Conception Bay 
(B. deYoung, Memorial University, pers. comm.), which shows a quadratic dependence on
$\omega$ (Fig. 6C) and is broadly peaked in the weather band
about a period of approximately 7 days. We then assumed that the full frequency-wavenumber spectrum 
was adequately represented by the product of the separately estimated parts.

There were six species present in sufficient numbers in each of the two years
to allow estimation of their transport (one more than reported for 1990 by
\cite{pepin95} ({\it Mallotus villosus})). 
According to our previous findings (Pepin et al. 1995), each species/cohort
combination could be treated as an independent realisation of the effect of transport
on population dynamics.
Mean larval fields
across the mouth showed no evidence of any consistent spatial patterns for any of the
species/cohorts, and
the mean field for each was thus considered to be spatially invariant.
Mean concentrations  ranged from 0.48 to 86 larvae per 1000 $\mathrm{m}^3$. 
The spatial part of the larval spectrum was computed as the Fourier transform of the
autocorrelation function of the 12 cohorts considered in this analysis augmented with
data on 20 other species/cohorts sampled in 1990, 1991 and 1994 (P. Pepin unpublished data)
over station separations of 3-30 km.  The analysis was not limited to the bay mouth, but
used data from within the bay as well, and thus effectively assumed the larval 
distribution to be isotropic.  We have no evidence to the contrary. 
On a log-log scale, $S_c(k_x)$ exhibits two slopes, $-1.2$ % converted
                                                           % this to maths
on scales between 7 and 30 km and 
$-3.2$ on smaller scales (Fig. 6B) and thus differs significantly from the velocity spectrum.
Again, appropriate data did not exist to determine the temporal part of the spectrum, and
we used that estimated for the current meters.


We approached the cross-spectral estimation in a similar manner by first estimating the 
cross correlation between larval concentration and currents measured on the flux surveys for
the 12 species considered here plus 4 others.
The resulting correlations were normally distributed about 0, but only
4 of the 82 estimates were significantly different from 0 ($p<0.05$) (Fig. 6D), so we
concluded there was no  consistent association between larvae
and currents and, hence,  that ${\bf T_3} \approx 0$.

We set $\Ze$  equal to the values reported by \cite{pepin95},
$0.10 \;\daym$
in 1990 and $0.05 \;\daym$ in 1991. 
Sensitivity to this choice is explored later.
The variance in currents due to measurement error, $\sigma_u^2$,
was set at $(0.007\; m s^{-1})^2$
for
1990 and $(0.006\; m s^{-1})^2$ for 1991   \cite{colbourne93}.
The corresponding measurement error in the plankton hauls was estimated
using the relationship given by \cite{pepin97} $ \sigma_c^2 = 1.6 c_o^{3/2}$
for concentrations measured in units of number $\mathrm{m}^{-3}$.

Each term in (\ref{eq:100}) is species specific in that it depends on 
either the mean concentration or $S_c$.
If we account for this dependence explicitly, for example by putting 
$S_c(k_x,\omega) = \var(c) S_c^{\prime}(k_x,\omega)$ 
and ${\bf T_1} = \var(c) {\bf T_1}^{\prime}$, then (\ref{eq:100})
can be rewritten as
\begin{equation}
{\bf T} = \left(\frac{L_x}{A}\right)^2 ( {\bf T_1}^{\prime} +V_c^2 {\bf T_2}^{\prime} + V_c {\bf T_3}^{\prime} 
       +c_o^{-1/2} {\bf T_4}^{\prime} +{\bf T_5}^{\prime} +c_o^{-1/2} {\bf T_6}^{\prime} )
\end{equation}
where
$V_c = \sqrt{\var(c)}/c_o $ is the coefficient of variation. The primed terms are all species independent
and need be computed only once. The $V_c$  were computed from the transect data alone
after first correcting the observed variance for sampling 
error, i.e.\ variance = observed variance---$\sigma_c^2$.
The results are summarized in Table 3. Both ${\bf T_1}$ and ${\bf T_2}$ contribute a
similar proportion of the average overall error in fluxes and together
\marginpar{Table 3\\near here}
account for more than 65\% of the predicted variance.
Measurement error (${\bf T_4}$ and ${\bf T_5}$) accounted for only about 30\% of the overall variability. This
is considerably higher than the value found in the simulations, but is consistent with the
larger values of $\sigma_c$ and $\sigma_u$.

Equation (\ref{eq:100}) shows that in order for ${\bf T_1}$ to be significant, the space and time scales
selected by  $|\wh{F}_C\wh{M}|^2$ must overlap the energetic part of the velocity spectrum $S_v$.
Similarly, if  ${\bf T_2}$ is to be important, then $|\wh{F}_V\wh{M}|^2$ and $S_c$ must coincide.
In the present example, 
$c_o$ is essentially white, and thus where one samples along the transect should 
not matter.   By contrast, $v_o$ is parabolic and and how it is sampled should be important.
These expectations are confirmed in the 
perspective plots of Fig.~7 which show that $|\wh{F}_C\wh{M}|^2$ is generally much
\marginpar{Fig. 7\\near here}
smaller than $|\wh{F}_V\wh{M}|^2$.
One would expect, therefore,  ${\bf T_2}$ to be larger than ${\bf T_1}$. That
the two terms are nearly the same size is due to fact that the velocity field is relatively
more energetic than the larval field. (Specifically,
${\bf T_1}/{\bf T_2}$ is proportional to the ratio of the coefficients of
variation for currents and concentration, and the former is about twice the latter.)  
The location and periodicity of the peaks, as explained in \cite{helbig98}, is determined
by the sample spacing; the widths of the peaks depends on the spatial structure of
$v_o$  and $c_o$.  

The predicted variance in estimated fluxes ranges
from $1.46 - 5.36 \times 10^{-14}~s^{-2}$ for the species considered in 
this analysis (Table 3). When
compared with the observed error in estimated fluxes,
\cite{helbig98}'s model gives a distribution of values which is
consistent with the observations although the average is somewhat
lower than the observations (Fig 8A). On a one-to-one species/cohort basis,
\marginpar{Fig. 8\\near here}
individual estimates display little relationship with one another (Fig.  8B). 
However, when normalised relative to the estimated mortality rates for each
species, there is a close correspondence in the distribution of
observed and predicted values (Fig. 8C). Probably the most important finding
is that, on average, the predicted and observed variances are remarkably similar: \cite{helbig98}
predicts the correct order of magnitude.

The comparison of observed and predicted values of the estimated error in
the fluxes raises important issues. First, we should not expect a close one-to-one relationship.
Each variance estimate made from the field data is, in effect, a single realisation of a 
stochastic process which need not be near the average value given by the theory. In this example,
predicted 
estimates of the flux variance  were strictly
based on the overall average means and variance. The only elements which
differed in our estimates of $\mathrm{var} (\widehat \Psi)$ among the
species/cohorts sampled in 1990 and 1991 were the coefficients of variation of the
larvae concentration  estimated from the surveys, whereas the field derived
values used the observed plankton and current measurements at each station.
Second, any individual estimate of mortality correction due to transport must be
interpreted in terms of its expected error.
Our results suggest this error is usually large and, that the only practical way
to reduce it is by obtaining a sufficient number of independent samples.
We were fortunate in Pepin et al. (1995) in that 
each survey sampled several species simultaneously thus allowing us to
obtain samples from several realisations of the same underlying physical
process. This yielded quasi-independent estimates of the underlying
imprecision of our estimates of the fluxes.

Of course, the theoretical predictions are of value only if they 
do not depend too strongly on the parameters describing the environmental variability.
To test the sensitivity of the predictions, we computed ${\bf T}$ for 
several values of the important parameters 
(Fig. 9). The greatest sensitivity 
\marginpar{Fig. 9\\near here}
was to the choice for the effective mortality, as ${\bf T}$ ranged between values 30\% less to 25\% more
than the values reported here for values of $\Ze$ one-half and twice that assumed. 
${\bf T}$ showed little sensitivity to variations of the slope of $S_v$ and 
moderate dependence on the slope of $S_c$.
Not shown are the dependencies on the variances of the current and plankton fields  or the
measurement errors.
Since ${\bf T_1}$ and ${\bf T_2}$ are directly proportional to $\var(v)$ and $\var(c)$, respectively,
variations in either would yield a similar sensitivity to that found for $\Ze$.
Sensitivity to measurement error is also strictly proportional.
The relatively small range for each parameter   demonstrates that 
the estimates are reasonably robust and can be applied with some confidence. 
Individual components of ${\bf T}$ vary more widely, but the overall effect on ${\bf T}$ is 
relatively uniform.

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

\section{Array Design}

We have referred several times here and in \cite{helbig98} to the potential use of the 
theory in {\em a priori} array design for ichthyoplankton experiments.
Array design is a difficult subject that is the focus of an on-going study that is attempting to 
link the present theory with other approaches. Our only purpose here is to demonstrate some
of the theory's potential.

To do so, we return to the example considered in \cite{helbig98} of the estimation of the single survey 
within-bay plankton concentration for a 20~km square basin using a 36 station, 24 hour 
sampling scheme. The variability in estimated concentration is given by (\cite{helbig98}-28)
%
\begin{equation}
\mathrm{var}(\wh{C}_T)=m^2(T) \int S_c(\mathbf{k},\omega )\left| \Hhat(%
\mathbf{k},\omega )\right| ^2\;d\mathbf{k}d\omega   \label{eq:36}
\end{equation}
%
where 
$T$ is the central time of the survey. 
\cite{helbig98} displayed two-dimensional slices of $|\Hhat|^2$ for an array in which the stations 
were occupied row-wise (Fig. 10A). 
\marginpar{Fig. 10\\near here}
A comparison of the dependence on the cross-bay
wavenumber $k_x$ against that on the along-bay wavenumber $k_y$ (Fig. 10B \& C) revealed a dramatic
asymmetry that originates in the row-wise strategy. For contrast, we show 2D slices here for a 
sampling plan that uses a spiral-like pattern (Fig. 10D, E, F). In terms of the total distance
and total time required, the two sampling plans are identical. Yet, $|\Hhat|^2$
is much more nearly isotropic for the spiral plan, and the area under the curve in Fig. 10F is only
about 60\% that of Fig. 10C (for length scales less than 20~km and time scales less than 25~days).
The implication is not necessarily 
that the spiral plan is a better one, but that fairly minor differences in
design can result in major differences in sensitivity to environmental variability. The
spiral plan is only better if $|\Hhat|^2$ is large where $S_c$ is not. Lacking some prior 
knowledge of basin dynamics, this information provided through $|\Hhat|^2$ would be difficult
to exploit. However, the necessary information might be inferred from factors as simple as
basin shape. 
In any case, the experimenter should possess some understanding of the characteristics
of the sampling plan prior to executing a field study.

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

\section{Discussion}

The theory developed  by \cite{helbig98} partitions the bias and variance of
mortality and flux estimates according to the space and time scales of
the underlying current and larval fields. We validated this  theory 
with an extensive set of numerical experiments as well
as with field data from a previous study. The model performed well in  
the simulations,
where all environmental variability is known exactly, as well as
for field conditions, where environmental conditions could only be estimated. 
This implies that the theory can be applied in practice with reasonable confidence.

Accurate measurement of mortality (and growth) rates is fundamental to
determining the factors that influence the dynamics of the early life stages
of fish which appear to be important to ultimately determining their
relative year-class strengths. For example, accurate knowledge of these
rates is vital in understanding the role of predator-prey interactions in
regulating egg and larval survival \cite{bailey89}. 
The present results imply that the expected error in the apparent
mortality, $\rZn$, is about 20\% for many systems, indicating that useful estimates of the 
effective mortality can be made in practice. 
Whether or not $Z_{\mathrm{e}}$ is close to $Z$ depends on transport.
Our comparison with field data showed  that  individual flux estimates were about
50\% of the overall mortality estimates, and we thus conclude that many mortality estimates 
should be corrected for transport effects.
However,  the large error in the flux estimates
suggests a limited ability to do so  
unless a sufficient number of independent flux 
estimates are be obtained. The most obvious solution is to measure the mortalities of several
independent species simultaneously. In fact, it was only by doing so that 
\cite{pepin95} were able to obtain meaningful results.

Our results thus imply that  many 
previous estimates of egg and larval mortality
rates are likely contaminated by  transport effects, especially for systems 
less than $50\,\mathrm{km}$ in extent.
Although this has been previously suggested
\cite{taggart90,pepin95,helbig98}, the support for
this hypothesis is now much stronger.
Ultimately, this raises questions about earlier inferences concerning
the role of biotic processes in regulating ichthyoplankton mortality
rates
\cite{peterson84,mcgurk86,houde89,pepin91}.



Ichthyoplankton survey designs often adopt either systematic (e.g.,
\cite{smith77}) or stratified approaches (e.g., \cite{sherman84}) and
generally implicitly assume that the populations
can be sampled synoptically within the scope of each survey. The present
study demonstrates the fragility of this assumption. 
Repeated estimates of larval fish mortality
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%to here%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
rates from \cite{zweifel81,lo85,taggart87} suggest that the relative
variance in effective mortality $Z_{\mathrm{e}}$ ranges from 20 to
100\% among cohorts and is greater in small systems
in consonance with the results presented here. The present study suggests that it
may be difficult, if not infeasible, to determine the sources of that
variation simply because of the difficulties in accurately sampling
populations which are highly dynamic in both space and time.

Cushing \cite{cushing75}, Sissenwine \cite{sissenwine84} and Myers and
Cadigan \cite{myers93}
have suggested that variability in the survival of fish during the early
life history is the primary cause of variations in year class strength. In
general, few studies have been able to observe catastrophic events
\cite{leggett94}
suggesting that the length of time that eggs and larvae
spend in a stage vulnerable to high mortality has a direct impact on overall
survival \cite{mcgurk86,houde89,pepin91}. Under such circumstances,
small changes in either growth or mortality rates can generate substantial
variation in year class strength \cite{houde87,beyer89}. If this
hypothesis is correct, then the results of our study indicate that it is
unlikely that attempts at measuring larval mortality rates will provide the
accuracy and precision required to understand the processes regulating
survival. Under some circumstances, broad scale surveys of highly abundant
species may provide adequate representation of the magnitude in estimated
mortality rates with limited bias and variance (e.g., \cite{lo89}; recall that 
$\var (\rZn)$ is inversely proportional to $\rC_{oT}^2$).
However, even if it is possible to correct changes in abundance for losses
attributable to physical processes, variations in estimated mortality rates
caused by sampling variability alone (i.e., their coefficient of variation)
may be of the same order as the changes required to cause substantial
variations in year class strength. The only alternative may
be to develop studies using multispecies sampling strategies. One could then
ascertain if estimated vital rates differ significantly among species.
Failure to reject the null hypothesis may then allow us to use each species
as an independent sample of a single process \cite{pepin95}. Two
important points of interpretation must be seriously considered. First,
failure to measure significant differences among species may be due simply
to inappropriate survey design, a possibility which must be seriously
explored. Alternatively, similarity of vital rates may reflect vulnerability
to similar environmental conditions and would thus suggest a degree to
synchrony among species.

Oceanographers and biologists may be most capable of estimating the average
vital rates of a population (i.e., mean mortality and growth rates). As a
result, our principle research objectives may need to focus on the
underlying factors and processes which govern average rates of growth
and loss of populations of fish eggs and larvae rather than the fluctuations
about the mean. Ichthyoplankton surveys have been used to estimate adult
biomass, determine spawning locations, and to describe the basic life cycle
and environmental vulnerability of marine fishes.
For example, supply of larvae to suitable habitats (e.g., nursery areas) 
has been
proposed as the underlying process of greatest importance in determining
recruitment variability in a number of species
\cite{doherty88,roughgarden88,underwood89}. 
Understanding the factors that determine the general
variability in supply and distribution and post-settlement or aggregative
processes may prove to be fundamental requirements to understanding the
regulation of variations in year-class strength in marine fish populations.

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

\section*{\emph{Acknowledgements}}

We thank Drs. Gordon Mertz and Geoff Evans for their critical reviews of
earlier versions of this manuscript. We also acknowledge Dr. Evans for
several penetrating conversations. 
We are also greatful to Drs. C. Taggart and J. Power for their thoughtful
evaluation and suggestions for the presentation of this work.


\renewcommand{\arraystretch}{0.9}
%\setlength{\parskip}{6pt}
%\setlength{\parindent}{0pt}

\begin{table*}[tp]
\caption{Simulations. List of numerical experiments run classified by basin
size L, number of basin survey stations J, number of flux survey stations M,
number of flux surveys N and simulation duration $\Delta T$, and indicating
the duration of basin survey $T_{\mathrm{basin}}$, duration of flux survey $%
T_{\mathrm{flux}}$, the root mean square speed $\sigma _{\mathbf{u}}$, and
the biases $ \rZn/Z$ and $ \rZf/Z$ and the 
standard deviations of $\rZn/Z$ and $\rZf/Z $.}
\label{tab:simulations}
\begin{tabular}{rcccccccccccc}
\hline
\hline
\mc{1}{r}{run} & L & \mc{1}{c}{$\Delta T$} & J & M & N & 
$T_{\mathrm{basin}}$ & $T_{\mathrm{flux}}$ & $\sigma_{\mathbf{u}}$ & bias & 
std dev & bias & std dev \\ 
\mc{1}{r}{} & (km) & \mc{1}{c}{(days)} & & & & (hours)
& (hours) & $(m\:s^{-1})$ & $\rZn $ & $ \rZn /Z $ & $\rZf/Z$ & $ \rZf/Z $ \\ 
\hline
\hline
\mc{1}{r}{} & & \mc{1}{c}{} & & &  & & & & & 
&  &  \\ 
\mc{1}{r}{1} & 5 & \mc{1}{c}{5} & 25 & 5 & 3 & 14 & 3 & 
.054 & 0.003 & 0.06 & 1.307 & 1.199 \\ 
\mc{1}{r}{2} & & \mc{1}{c}{} & & 3 & & & 2 & & & 
& 0.874 & 2.354 \\ 
\mc{1}{r}{3} & & \mc{1}{c}{} & 9 & 5 & & 5 & 3 & & 
0.011 & 0.135 & 1.303 & 1.194 \\ 
\mc{1}{r}{4} & & \mc{1}{c}{} & & 3 & & & 2 & & & 
& 0.858 & 2.373 \\ \hline
\mc{1}{r}{5} & 10 & \mc{1}{c}{5} & 25 & 10 & 3 & 15 & 6
& .075 & -0.012 & 0.116 & 0.860 & 0.767 \\ 
\mc{1}{r}{6} & & \mc{1}{c}{} & & & 2 & & & & & & 
0.895 & 0.858 \\ 
\mc{1}{r}{7} & & \mc{1}{c}{} & & & 1 & & & & & & 
0.754 & 1.309 \\ 
\mc{1}{r}{8} & & \mc{1}{c}{} & & 5 & 3 & & 3 & & & 
& 0.829 & 1.012 \\ 
\mc{1}{r}{9} & & \mc{1}{c}{} & & & 2 & & & & & & 
0.884 & 1.174 \\ 
\mc{1}{r}{10} & & \mc{1}{c}{} & & & 1 & & & & & 
& 0.666 & 1.763 \\ 
\mc{1}{r}{11} & & \mc{1}{c}{} & 9 & 10 & 3 & 6 & 6 & & 
-0.006 & 0.207 & 0.863 & 0.770 \\ 
\mc{1}{r}{12} & & \mc{1}{c}{} & & & 2 & & & & & 
& 0.899 & 0.861 \\ 
\mc{1}{r}{13} & & \mc{1}{c}{} & & & 1 & & & & & 
& 0.755 & 1.306 \\ 
\mc{1}{r}{14} & & \mc{1}{c}{} & & 5 & 3 & & 3 & & & 
& 0.833 & 1.020 \\ 
\mc{1}{r}{15} & & \mc{1}{c}{} & & & 2 & & & & & 
& 0.887 & 1.183 \\ 
\mc{1}{r}{16} & & \mc{1}{c}{} & & & 1 & & & & & 
& 0.673 & 1.760 \\ \hline
\mc{1}{r}{17} & 20 & \mc{1}{c}{5} & 36 & 10 & 3 & 24 & 6
& .088 & 0.017 & 0.149 & 0.476 & 0.507 \\ 
\mc{1}{r}{18} & & \mc{1}{c}{} & & & 2 & & & & & 
& 0.926 & 1.196 \\ 
\mc{1}{r}{19} & & \mc{1}{c}{} & & & 1 & & & & & 
& 1.026 & 1.684 \\ 
\mc{1}{r}{20} & & \mc{1}{c}{} & & 5 & 3 & & 3 & & & 
& 0.358 & 0.600 \\ 
\mc{1}{r}{21} & & \mc{1}{c}{} & & & 2 & & & & & 
& 0.370 & 0.709 \\ 
\mc{1}{r}{22} & & \mc{1}{c}{} & & & 1 & & & & & 
& 0.329 & 1.181 \\ 
\mc{1}{r}{23} & & \mc{1}{c}{} & 9 & 10 & 3 & 8 & 6 & & 
0.011 & 0.349 & 0.476 & 0.515 \\ 
\mc{1}{r}{24} & & \mc{1}{c}{} & & & 2 & & & & & 
& 0.934 & 1.209 \\ 
\mc{1}{r}{25} & & \mc{1}{c}{} & & & 1 & & & & & 
& 0.995 & 1.664 \\ 
\mc{1}{r}{26} & & \mc{1}{c}{} & & 5 & 3 & & 3 & & & 
& 0.359 & 0.602 \\ 
\mc{1}{r}{27} & & \mc{1}{c}{} & & & 2 & & & & & 
& 0.372 & 0.719 \\ 
\mc{1}{r}{28} & & \mc{1}{c}{} & & & 1 & & & & & 
& 0.331 & 1.155 \\ \hline
\mc{1}{r}{29} & 50 & \mc{1}{c}{15} & 81 & 20 & 9 & 64 & 
13 & .100 & 0.004 & 0.057 & 0.148 & 0.102 \\ 
\mc{1}{r}{30} & & \mc{1}{c}{} & & & 5 & & & & & 
& 0.161 & 0.117 \\ 
\mc{1}{r}{31} & & \mc{1}{c}{} & & & 3 & & & & & 
& 0.175 & 0.146 \\ 
\mc{1}{r}{32} & & \mc{1}{c}{} & & 10 & 9 & & 8 & & 
& & 0.120 & 0.151 \\ 
\mc{1}{r}{33} & & \mc{1}{c}{} & & & 5 & & & & & 
& 0.309 & 0.222 \\ 
\mc{1}{r}{34} & & \mc{1}{c}{} & & & 3 & & & & & 
& 0.339 & 0.290 \\ 
\mc{1}{r}{35} & & \mc{1}{c}{} & 49 & 20 & 11 & 43 & 13
& & 0.012 & 0.066 & 0.169 & 0.103 \\ 
\mc{1}{r}{36} & & \mc{1}{c}{} & & 10 & 11 & & 8 & & 
& & 0.140 & 0.143 \\ 
\mc{1}{r}{37} & & \mc{1}{c}{} & 25 & 20 & 13 & 26 & 13
& & 0.014 & 0.115 & 0.200 & 0.142 \\ 
\mc{1}{r}{38} & & \mc{1}{c}{} & & 10 & 13 & & 8 & & 
& & 0.178 & 0.191 \\ \cline{3-13}
\mc{1}{r}{39} & & \mc{1}{c}{10} & 49 & 20 & 6 & 43 & 13
& & -0.009 & 0.095 & 0.162 & 0.191 \\ 
\mc{1}{r}{40} & & \mc{1}{c}{} & & & 3 & & & & & 
& 0.169 & 0.192 \\ 
\mc{1}{r}{41} & & \mc{1}{c}{} & & & 2 & & & & & 
& 0.177 & 0.229 \\ 
\mc{1}{r}{42} & & \mc{1}{c}{} & & 10 & 6 & & 8 & & 
& & 0.126 & 0.296 \\ 
\mc{1}{r}{43} & & \mc{1}{c}{} & & & 3 & & & & & 
& 0.102 & 0.105 \\ 
\mc{1}{r}{44} & & \mc{1}{c}{} & & & 2 & & & & & 
& 0.104 & 0.158 \\ 
\mc{1}{r}{45} & & \mc{1}{c}{} & 25 & 20 & 8 & 26 & 13 & 
& 0.017 & 0.126 & 0.092 & 0.098 \\ 
\mc{1}{r}{46} & & \mc{1}{c}{} & & 10 & 8 & & 8 & & 
& & 0.094 & 0.145 \\ \cline{3-13}
\mc{1}{r}{47} & & \mc{1}{c}{5} & & 20 & 3 & & 13 & & 
0.039 & 0.350 & 0.080 & 0.089 \\ 
\mc{1}{r}{48} & & \mc{1}{c}{} & & 10 & 3 & & 8 & & 
& & 0.081 & 0.125 \\ \hline
\mc{1}{r}{49} & 100 & \mc{1}{c}{15} & 81 & 20 & 7 & 88 & 
15 & .111 & -0.008 & 0.085 & 0.073 & 0.069 \\ 
\mc{1}{r}{50} & & \mc{1}{c}{} & & & 4 & & & & & 
& 0.077 & 0.087 \\ 
\mc{1}{r}{51} & & \mc{1}{c}{} & & & 3 & & & & & 
& 0.076 & 0.088 \\ 
\mc{1}{r}{52} & & \mc{1}{c}{} & & 10 & 7 & & 10 & & 
& & 0.062 & 0.081 \\ 
\mc{1}{r}{53} & & \mc{1}{c}{} & & & 4 & & & & & 
& 0.157 & 0.166 \\ 
\mc{1}{r}{54} & & \mc{1}{c}{} & & & 3 & & & & & 
& 0.155 & 0.172 \\ 
\mc{1}{r}{55} & & \mc{1}{c}{} & 64 & 20 & 9 & 75 & 15 & 
& 0.006 & 0.089 & 0.076 & 0.062 \\ 
\mc{1}{r}{56} & & \mc{1}{c}{} & & 10 & 9 & & 10 & & 
& & 0.071 & 0.086 \\ 
\mc{1}{r}{57} & & \mc{1}{c}{} & 36 & 20 & 11 & 50 & 15
& & 0.003 & 0.100 & 0.079 & 0.054 \\ 
\mc{1}{r}{58} & & \mc{1}{c}{} & & & 6 & & & & & 
& 0.082 & 0.071 \\ 
\mc{1}{r}{59} & & \mc{1}{c}{} & & & 4 & & & & & 
& 0.071 & 0.087 \\ 
\mc{1}{r}{60} & & \mc{1}{c}{} & & 10 & 11 & & 10 & & 
& & 0.079 & 0.083 \\ 
\mc{1}{r}{61} & & \mc{1}{c}{} & & & 6 & & & & & 
& 0.079 & 0.103 \\ 
\mc{1}{r}{62} & & \mc{1}{c}{} & & & 4 & & & & & 
& 0.080 & 0.136 \\ \cline{3-13}
\mc{1}{r}{63} & & \mc{1}{c}{10} & 36 & 20 & 6 & & 15 & 
& 0.012 & 0.174 & 0.090 & 0.076 \\ 
\mc{1}{r}{64} & & \mc{1}{c}{} & & 10 & 6 & & 10 & & 
& & 0.096 & 0.116 \\ \hline
\mc{1}{r}{65} & 200 & \mc{1}{c}{15} & 49 & 20 & 7 & 99 & 
20 & .111 & 0.005 & 0.126 & 0.139 & 0.152 \\ 
\mc{1}{r}{66} & & \mc{1}{c}{} & & & 4 & & & & & 
& 0.143 & 0.178 \\ 
\mc{1}{r}{67} & & \mc{1}{c}{} & & & 3 & & & & & 
& 0.135 & 0.194 \\ 
\mc{1}{r}{68} & & \mc{1}{c}{} & & 10 & 7 & & 15 & & 
& & 0.142 & 0.212 \\ 
\mc{1}{r}{69} & & \mc{1}{c}{} & & & 4 & & & & & 
& 0.288 & 0.356 \\ 
\mc{1}{r}{70} & & \mc{1}{c}{} & & & 3 & & & & & 
& 0.271 & 0.391 \\ 
\mc{1}{r}{71} & & \mc{1}{c}{} & 25 & 20 & 9 & 65 & 20 & 
& 0.001 & 0.194 & 0.160 & 0.127 \\ 
\mc{1}{r}{72} & & \mc{1}{c}{} & & & 5 & & & & & 
& 0.162 & 0.148 \\ 
\mc{1}{r}{73} & & \mc{1}{c}{} & & & 3 & & & & & 
& 0.134 & 0.188 \\ 
\mc{1}{r}{74} & & \mc{1}{c}{} & & 10 & 9 & & 15 & & 
& & 0.122 & 0.194 \\ 
\mc{1}{r}{75} & & \mc{1}{c}{} & & & 5 & & & & & 
& 0.118 & 0.213 \\ 
\mc{1}{r}{76} & & \mc{1}{c}{} & & & 3 & & & & & 
& 0.161 & 0.276 \\ 
\mc{1}{r}{} & & \mc{1}{c}{} & & &  & & & & & 
&  &  \\ 
\hline
\hline
\end{tabular}
\end{table*}

%\setlength{\parskip}{6pt} \setlength{\parindent}{0pt}

\begin{table*}[tp]
  
  \caption{Basin and sampling plan variables for the study by Pepin et al. (1995).}
  \label{tab:variables}

\begin{tabular}{ccc} \hline\hline
 & & \\
      & 1990 & 1991 \\ 
 & & \\
\hline\hline
 & & \\
$L_x$ (km) &  20 &20 \\ 
$L_y$ (km) &  50 &50\\ 
$\DT$ (day) & 17.5   & 10   \\  
 & & \\
\hline
 & & \\
$N $    & 4        & 8        \\
$T_{fn}$  (day)& -1.5, 4.5, 13.5 16.5    & 0.5, 1.5, 2.5, 3.5, 6.5, 7.5, 8.5, 9.5         \\
$\Delta t_n$&1.5, 7.5, 6.,2.5           &1.0, 1.0, 1.0, 2.0, 2.0, 1.0, 1.0, 1.0 \\ 
 & & \\
\hline
 & & \\
$M$     & 7        & 7        \\ 
$\Delta l_m $(km)   &2.7     &2.7        \\
$\delta t_m $ (min) & 43  & 22 \\ 
 & & \\
\hline
 & & \\
$J$                 & 7   & 7 \\ 
 & & \\
\hline\hline
\end{tabular}
\end{table*}


\begin{table*}[tp]

  \caption{Variance components of total error in flux estimates for
      the study by Pepin et al. (1995).  Columns 2 and 4 report the
      overall mean for each variance component as well as the range in
      total variance.  Columns 3 and 5 present the mean percentage and
      range of each variance component based on the estimate for each
      species included in the analysis.}
  \label{tab:variances}

\begin{tabular}{ccccc} \hline\hline
 & & & & \\
Term & 1990 & percent & 1991 & percent \\
 & & & & \\
     & variance $(10^{-14} s^{-2}) $&         & variance $(10^{-14}  s^{-2})$&         \\
 & & & & \\
\hline \hline
 & & & & \\
${\bf T}$  &  2.95 (1.46~-~5.36) & -& 3.09 (2.32~-~4.20)& - \\
 & & & & \\
${\bf T_1}$&  0.837 & 33 (16-57) & 1.06& 36 (25-44)   \\
${\bf T_2}$&  1.33 & 37 (1-72) & 1.02 & 30 (10-51)   \\
${\bf T_3}$&  0. &0  & 0.& 0  \\
${\bf T_4}$&  0.356 & 13 (1-49) & 0.190 & 6 (2-12) \\
${\bf T_5}$&  0.428 & 17 (8-30) & 0.815 & 28 (19-35) \\ 
 & & & & \\
\hline\hline
\end{tabular}
\end{table*}



\section*{Appendix. Specification of the velocity field.}

%\setcounter{equation}{0}

Each velocity component takes the form 
\[
u(x,y,t)=\sum_{k=-N_k}^{N_k}\sum_{l=-K_k}^{N_k}\sum_{m=-N_\omega }^{N_\omega
}a_{klm}\cos (k_kx+k_ly+\omega _mt+\phi _{klm})
\]
where $N_k$ and $N_\omega $ are the number of wavenumbers and frequencies
included in the model (30 and 40 respectively) and the $\phi _{klm}$ are
random phases drawn from a uniform distribution on $[0,2\pi ]$ at the
beginning of the simulation. The coefficient $a_{klm}$ is defined by 
\begin{equation}
a_{klm}=\left[ S_u(k_k,k_l,\omega _m)\Delta k_k\Delta k_l\Delta \omega
_m\right] ^{1/2}
\end{equation}
where $\Delta k$, and $\Delta \omega $ are spacings between adjacent
wavenumber and frequency components. Note that the sums run over negative
wavenumbers and frequencies as well as positive ones in order to admit
constituent waves of all possible propagation directions. Note also that $%
a_{klm}=0$ for $k$, $l$, or $m=0$ so there is no mean component.

For gridpoint $(i,j)$, the dependence on wavenumber can be separated from
that on frequency to give 
\begin{equation}
u_{ij}(t)=\sum_{m=1}^{N_\omega }A_{ijm}\cos (\omega _mt)-B_{ijm}\sin (\omega
_mt). \label{eq:85}
\end{equation}
where 
\begin{eqnarray*}
\lefteqn{A_{ijm}=\sum_{k=1}^{N_k}\sum_{l=1}^{N_k}a_{klm}\left\{ {}\right. }
\\
&& \cos (k_kx_i+k_ly_j)(c_{\scriptscriptstyle +++}+c_{\scriptscriptstyle %
++-}+c_{\scriptscriptstyle --+}+c_{\scriptscriptstyle ---}) \nonumber \\
&&-\sin (k_kx_i+k_ly_j)(s_{\scriptscriptstyle +++}+s_{\scriptscriptstyle %
++-}-s_{\scriptscriptstyle --+}-s_{\scriptscriptstyle ---}) \nonumber \\
&&+\cos (k_kx_i-k_ly_j)(c_{\scriptscriptstyle -++}+c_{\scriptscriptstyle %
-+-}+c_{\scriptscriptstyle +-+}+c_{\scriptscriptstyle +--}) \nonumber \\
&&-\sin (k_kx_i-k_ly_j)(s_{\scriptscriptstyle -++}+s_{\scriptscriptstyle %
-+-}-s_{\scriptscriptstyle +-+}-s_{\scriptscriptstyle +--})\left. {}\right\}
% \nonumber -- no ... numbering suppressed in eqnarray*
\end{eqnarray*}
The $c$ and $s$ terms are cosines and sines of the random phases; for
example, %\begin{displaymath}
$c_{+-+}=\cos (\phi _{k_x,-k_y,\omega })$ %\end{displaymath}
with appropriate choices of $k_x$, $k_y$ and $\omega $ dictated by the
subscripts $k$, $l$, and $m$. A similar expression holds for $B_{ijm}$.

$A_{ijm}$ and $B_{ijm}$ were generated at the beginning of each run, and $%
u_{ij}$ was updated at each timestep using (\ref{eq:85}). The vast majority of
computer time was spent in the former task ($\approx 95$\%), while the
remainder proceeded very quickly.

\newpage
\section*{Figure Legends}

\begin{description}

\item {Figure 1.}  Basin geometry for the numerical simulations of a $20 \times
20 \mathrm{km}$ study system.  Tick marks represent 1 km intervals.  Open circles
show the location of stations used in the basin surveys, closed circles show the
location of stations sampled in the flux surveys.  Solid lines represent ship tracks.

\item  {Figure 2.} Results of the numerical simulations showing  means or biases  and
standard deviations of $\rZn/Z,\;\rZf/Z$ and $\rZc/Z=(\rZn-\rZf-\Zd)/Z$
vs. basin size for the 76 sampling plans listed in Table 1. The values of $Z_{%
\mathrm{diff}}$ used to correct $\rZc$ were computed during the simulations without error.
Note that the ordinate scale
used for $\rZn$ differs from that used for $\rZf$ and $\rZc$.
differ.

\item  {Figure 3.} Scatterplots of predicted versus simulated variance and
and bias in $\rZn$ ,  and $\rZf$.
A) $\mathrm{var}( \rZn/Z$  and 
B) $\mathrm{var}(\rZf/Z)$;
The predicted values 
were computed with the \cite{helbig98} formulation; the simulated values were calculated from 
the 6 sets of 100 numerical experiments each.  

\item  {Figure 4.} Histograms of the true and estimated values of $\rZn$,
$\rZf$ and $\rZc= \rZn -\rZf -\Zd$ for numerical experiments 17 and 23 (20~km basin,  5~days).   
Plan 17 uses a symmetric  $6 \times 6$ station grid coupled with 3 flux surveys of 10 stations each.
Plan 26 uses a $3 \times 3$, grid coupled with 3 flux surveys of 5 stations each.
The value of $\Zd$ was computed during the simulations without error.

\item {Figure 5.} Comparison of a $6 \times 6$ and $3 \times 3$ sampling plan for a $20$~km
basin. Panels A) and C): Contours of $k_x\omega \int S_u(k_x,k_y,\omega )dk_y$
    (dashed) and $\int \left| H(k_x,k_y,\omega )\right|^2dk_y$ (solid);
Panels B) and D)  contours of $k_x\omega \int S_u(k_x,k_y,\omega ) (1 - \cos \omega \DT)
\left| H(k_x,k_y,\omega )\right| ^2dk_y$; The factor $k_x\omega$ corrects for the 
distortion induced in a log-log plot and ensures that each factor contributes
to its respective integral in accord with the area subtended by its graph.
 


\item {Figure 6.} (A) Scatterplot of the along-bay current across the mouth measured
on 16 ADCP transects together with the best fit quadratic representing the mean;
(B) Wavenumber spectrum for currents ($S_v$) (solid line)
and larval abundance ($S_c$) (dashed line);
(C) Average frequency spectrum computed for three
current meters moored in Conception Bay (B. deYoung, Memorial University, pers. comm.);
(D) Cumulative probability distribution of the correlation coefficient ($r$) of
larval concentration and currents measured for each occupation of the transect at the
mouth of the bay.  The dotted vertical lines show the values required to obtain a 
significant correlation ($P<0.05$) based on a sample of 7 stations.

\item {Figure 7.} Perspective plots of $|\FC(k_x, \omega) \M(\omega) |^2$ and
$|\FV(k_x, \omega) \M(\omega) |^2$.

\item {Figure 8.} (A) Cumulative proportion of predicted (dotted line) and observed
(solid line) standard errors in the estimated flux based on the characteristics of
Pepin et al.'s (1995) study. (B) Scatter plot of the observed and estimated errors
in estimated fluxes for the 12 individual species/cohorts that were part of the study.
The solid line shows the one-to-one relationship. (C) Cumulative proportion of predicted 
(dotted line) and observed (solid line) standard errors in the estimated flux standardized
relative to the estimated mortality rate for individual species/cohorts.

\item {Figure 9.} Sensitivity of ${\bf T}$  to (A) $\Ze$ ; (B) the slope of the velocity wavenumber 
spectrum; (C) the low wavenumber slope of the plankton spectrum; and
(D) the high wavenumber slope of the plankton spectrum
\item {Figure 10.} A comparison of two sampling arrays.
A) Row-wise sampling plan; 
B)  $|\Hhat(k_x,k_y=0,\omega) |^2 $ for the row-wise plan;
C) $|\Hhat(k_x=0,k_y,\omega) |^2 $ for the row-wise plan; 
D) the spiral plan;
E)  $|\Hhat(k_x,k_y=0,\omega) |^2 $ for the spiral plan; and 
F) $|\Hhat(k_x=0,k_y,\omega) |^2 $ for the spiral plan; 

\end{description}

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\end{document}
