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% Title     : Model Descriptions
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% Author    : Sara Mikaloff Fletcher et al.
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\documentclass[ gbc]{agu2001}


\authorrunninghead{MIKALOFF FLETCHER ET AL.}

\titlerunninghead{AIR-SEA EXCHANGE OF ANTHROPOGENIC CARBON}


\begin{document}

\title{Extended Methods}

\authors{S. E. Mikaloff Fletcher, \altaffilmark{1}
 N. Gruber, \altaffilmark{1}
 A. R. Jacobson, \altaffilmark{2}
 S. C. Doney, \altaffilmark{3}
 S. Dutkiewicz, \altaffilmark{4}
 M. Gerber, \altaffilmark{5}
 M. Follows, \altaffilmark{4}
 F. Joos, \altaffilmark{5}
 K. Lindsay, \altaffilmark{6}
 D. Menemenlis, \altaffilmark{7}
 A. Mouchet \altaffilmark{8}
 S. A. M\"uller \altaffilmark{5}
 and J. L. Sarmiento \altaffilmark{2}}

\altaffiltext{1}{Department of Atmospheric and Oceanic Sciences and
the Institute for Geophysics and Planetary Physics, University of
California, Los Angeles, California, USA.}

\altaffiltext{2}{Atmospheric and Oceanic Sciences Department,
Princeton, New Jersey, USA.}

\altaffiltext{3}{Marine Chemistry and Geochemistry,
Woods Hole Oceanographic Institution, Woods Hole, MA, USA.}

\altaffiltext{4}{Department of Earth, Atmosphere, and Planetary Sciences,
Massachusetts Institute of Technology, Cambridge, Massachusetts, USA.}

\altaffiltext{5}{Climate and Environmental Physics, Physics Institute,
University of Bern, Bern, Switzerland.}

\altaffiltext{6}{Climate and Global Dynamics Division,
National Center for Atmospheric Research, Boulder, Colorado, USA.}

\altaffiltext{7}{Estimating the Circulation and Climate of the Ocean,
Jet Propulsion Laboratory, Pasadena, California, USA.}

\altaffiltext{8}{Astrophysics and Geophysics Institute,
University of Liege, Belgium.}





\begin{article}


\section{Basis function simulations}

For a model grid box at position $i,j$ and time, $t$, the surface flux in units
of mass or moles per unit time, $F(i,j,t)_{dye}$, is equal to

\begin{equation}
F(i,j,t)_{dye}=F_{tot} \cdot P_{flux}(i,j,t_{seas}) \cdot \phi(t). \label{eq_1}
\end{equation}

The arbitrary unit flux, $F_{tot}$, is distributed within the model
region by a dimensionless, seasonal  spatial pattern, $P(i,j,t_{seas})$,
and scaled by the anthropogenic perturbation to atmospheric CO$_2$, $\phi(t)$.

The spatial pattern is determined from the seasonal climatology of
Takahashi et al. [2002] extrapolated to areas not covered by the
observations without using a sea-ice mask.  Previous ocean inversion
studies have found that the inverse estimates are relatively insensitive
to the choice of the spatial pattern (Gloor et al., 2001).

\section{Region aggregation and discussion of the covariance matrix}

Basis functions were originally generated for 30 surface regions
(Mikaloff Fletcher et al., 2003), and later aggregated to 24
regions. The regions that were aggregated were combined prior to the
inversion by weighting their corresponding basis functions by their
relative fluxes in the Takahashi et al. [2002] CO$_2$ climatology.
The inversion has been tested for biases associated with the
aggregation to 24 regions by comparing the 24 region inversion with
a 30 region inversion where the inverse estimates are then summed to
the 24 regions using three of the ten contributing models, NCAR,
PRINCE-LL, and PRINCE-HH.  The differences between the pre- and
post- aggregated inverse estimates are less than 0.10 Pg C yr$^-1$
regionally and are also less than the reported error except in a few
cases where both the reported error and aggregation differences are
less than 0.03 Pg C yr$^-1$.

These aggregations were selected to  minimize the covariance between
the modeled response to surface fluxes into each pair of regions, while
maintaining low residuals between the observationally-based and inverse
anthropogenic carbon concentration estimates. A matrix of
regional co-variances was calculated from the matrices produced by the
singular value decomposition, where the diagonal elements represent the square
of the random error for each flux estimate and the off-diagonal elements
represent the co-variances between model regions.

The covariance matrix for the 24 region aggregation generally has
small off-diagonal elements for all of the models relative to the
variances (shown in Figure 4 of the online supplement), indicating
that the errors due to The covariance matrix for the 24 region
aggregation generally has small off-diagonal elements for all of the
models relative to the variance, indicating that the errors due to
correlations between regions are smaller than the random errors.
Some models have high covariances between the Arctic Ocean (Region
1) and the North Atlantic High Latitudes (Region 2).  We chose not
to aggregate these regions because the partitioning of anthropogenic
carbon uptake between these two regions is of particular scientific
interest. The diagonal elements of the covariance matrix indicate
that the random errors are on the order of 0.01 Pg C yr$^-1$.  This
low random error estimate is due to the very large number of
observations constraining the flux estimates.  However, this does
not take into account systematic errors, which are addressed in
Section 4 of the manuscript.

\section{CFC skill score}

In order to consider which models are likely to have the most accurate
transport on the time scale of anthropogenic carbon perturbation,
we compare the GLODAP gridded CFC-11 data set with forward simulations of
CFC-11 from  OCMIP (Dutay et al., 2002).  Table 1 of the manuscript shows the
correlation between the  gridded CFC-11 data and the modeled CFC-11, $R$,
the standard deviation of the modeled CFC-11 normalized by the standard
deviation of the gridded CFC-11 data, $\hat{\sigma}$, and a CFC-11 model
skill score based on these two quantities, $S$ (Taylor, 2001).

\begin{equation}
S = \frac{4(1+R)^4}{(\hat{\sigma}+ 1/\hat{\sigma})^2(1+R_o)^4}   \label{eq_2}
\end{equation}

Where $R_o$ represents the maximum attainable correlation attainable.
This formula provides a skill score between one and zero, with 1 representing
a perfect model representation of the observations. $R_o$ was calculated based
on the correlation between the GLODAP gridded data set and the gridded data
set plus the estimated error due to griding the data.  The choice of $R_o$ is
critical in determining the absolute skill of a model; however, it is of
limited importance when evaluating the relative skill
of a variety of different models as in this study.

In the manuscript, the skill-weighted cross-model mean refers to a mean
of the quantity, $X$, of the $nmod$ models weighted according to their model
skill scores, $S_i$.

\begin{equation}
\bar {X}= \frac{\sum_{i=1,nmod}{S_i*X_i}}{\sum_{i=1,nmod}{S_i}}   \label{eq_3}
\end{equation}

Similarly, the weighted standard deviation refers to the standard deviation
of the models weighted by the skill score.

\begin{equation}
stdev_{wt}= \sqrt{\frac{\sum_{i=1,nmod}{S_i*(X_i-\bar{X})^2}}{(\sum_{i=1,nmod}{S_i})(nmod-1)}}   \label{eq_4}
\end{equation}


\section{Construction of scenarios that test possible biases in the data-based
anthropogenic carbon estimates}

Matsumoto and Gruber [2005] examined the accuracy of the anthropogenic carbon
estimates by applying the $\Delta$C$^*$ method to a forward model simulation
with known anthropogenic carbon concentrations. The authors found substantial
biases in the $\Delta$C$^*$ method due to the time evolution of the air-sea
disequilibrium, biases in the pCFC ventilation age and errors in identifying
water masses which contribute to a given water parcel.  As a result, they found
that the $\Delta$C$^*$ method tends overestimate the anthropogenic carbon
inventory in shallower waters by about 10\% and underestimate it in deeper
waters.  Globally, the $\Delta$C$^*$ inferred anthropogenic carbon inventory was
7\% larger than the true inventory.

It is not within the scope of this paper to re-assess the anthropogenic
carbon dataset based on the findings of Matsumoto and Gruber [2005]; however,
it is critical to address the effect these biases might have on the inverse
estimates.  To this end, we constructed a 'Matsumoto corrected' scenario, in
which a hypothetical correction factor was applied to the data-based
anthropogenic carbon estimates and these corrected anthropogenic carbon
estimates were used in the inversion.  In addition, two scenarios were
constructed to assess the impact of a globally uniform shift in the oxygen
to carbon remineralization ratio used to remove the effects of biology.

The hypothetical correction factor is  a function of potential density.
Below a given density threshold, $\sigma_\theta^0$, the anthropogenic carbon
concentration is decreased and above it the anthropogenic carbon concentration
is increased.

\begin{equation}
 C(\sigma_\theta) = A(ae^{(\sigma_\theta-\sigma_\theta^0)/k_1}-1) -
B(be^{-(\sigma_\theta-\sigma_\theta^0)/k_2}), \label{eq_5}
\end{equation}

The correction factor, $C(\sigma_\theta)$, is the fractional correction to the
data-based anthropogenic carbon estimates, $\sigma_\theta^0$ is 27.4, and the
constants $A, a, B, b, k_1$, and $k_2$ have been tuned based on the GLODAP
gridded anthropogenic carbon data set such that the global anthropogenic
carbon inventory is decreased by 7\% and the anthropogenic carbon inventory
at  densities less than $\sigma_\theta^0$ is decreased by 10\% (Figure 10
of the online supplement).

In addition, two scenarios were constructed to assess the impact of biases in
the stoichiometric ratio between carbon and oxygen,  $r_{C:O_2}$.
Gruber et al., [1996] estimated the data-based anthropogenic carbon for the
Atlantic Ocean using high and low values of $r_{C:O_2}$, which were 0.780 and
0.596 based on the $r_{C:O_2}$ and uncertainty estimates Anderson and Sarmiento
[1994].  In order to extrapolate these estimates to the entire ocean, we fit
the high and low $r_{C:O_2}$ estimates of Gruber et al. [1996] in Atlantic to
AOU and the anthropogenic carbon concentration using multiple linear
regression (coefficients shown in Table 3 of the online supplement).

\begin{equation}
\sigma= \alpha  AOU + \beta  C_{ant} + \gamma.    \label{eq_6}
\end{equation}

where $\alpha$, $\beta$, and $\gamma$ are optimized in the regression and
$\sigma$ is the difference between the standard anthropogenic carbon estimates
and the anthropogenic carbon based on high or low $r_{C:O_2}$ estimates.


\end{article}
\end{document}
