Radium-228 (

Trace elements and isotopes (TEIs) are low-concentration components of the
ocean, but they contain decisive information for our understanding of its
dynamics. The international program GEOTRACES has been designed to improve
our knowledge on the TEI concentrations and the oceanic processes
controlling their distribution, by means of observations, modeling and
laboratory experiments. Since 2006, GEOTRACES cruises have been mapping the
global distribution of tens of these TEIs. Some of them are studied because
they constitute micronutrients for living organisms, like iron (Fe), or are
pollutants, like lead (Pb) and cadmium (Cd). Others are proxies of ocean
dynamics or of biogeochemical processes. For instance, neodymium (Nd) is a
proxy of the exchanges between seabed and seawater

SGD is defined as the flux of water from coastal aquifers to the ocean,
regardless of its composition and origin. Part of it is meteoric freshwater,
but the largest part is infiltrated seawater

All the four natural radium isotopes,

A simple way to use the information provided by this isotope is to make an
observation-based inventory of the ocean

Inverse modeling techniques represent an alternative and powerful approach to
estimate the fluxes, providing that the ocean circulation is known with
sufficient accuracy. Inverse modeling is based on three elements:
observations of the physical quantity of interest (here oceanic

In this study, we estimate the radium fluxes from all continental shelves
around the world and localize the most intense sources, using an inverse
modeling technique with more data than previous studies

Observed concentrations of

Since the late 1960s

For the purpose of our study, data have been averaged in each model grid cell
(see Sect. 2.2 for more details on the model configuration), leading to the
3076 cell averages shown in Fig. 1. The density of the measurement is
noticeably uneven. The Atlantic Ocean, north of 20

Data are expressed in concentration or activity units, with the following
conversion factor: 1

The second requirement of the inversion technique is a

Equation (1) shows that the

As a consequence of its coarse horizontal resolution, continental shelves are
only poorly resolved by the ORCA2 grid. The emitting surface is
underestimated and some regions with narrow continental shelves would be
completely omitted. To overcome that deficiency, sub-model grid-scale
bathymetric variations are accounted for by comparing the model grid to a
global

The ocean–continent interface, including the Arctic and the Antarctic, is
divided into 38 regions (Fig. 2). This first guess takes into account the
sampling coverage (very low in the Antarctic for instance, and higher in the
North Atlantic Basin or Bay of Bengal) and differences in the tracer
distributions

Model

Cost functions.

The last requirement of the inversion technique is to define a cost function
measuring the misfit between the data and the model. This cost is then
minimized by a method already used to assess air–sea gas fluxes

The distance between model concentrations and data is summed up in a scalar,
the cost function

The residuals after inversion indicate what the inverse model cannot fit. In
a “perfect” inversion, these residuals should be assimilated to noise – e.g.,
small and without structure, due for instance to coarse resolution. In most
inversions, that is not the case, and the distribution of residuals
emphasizes biases or errors either in the chosen hypotheses (such as a
perfect circulation) or in the setting of the inversion (number and choice of
the regions). The posterior uncertainties on radium fluxes and correlations
between regions are computed following the method described in Appendix A. A
regional flux has a large uncertainty when it is constrained by few data or
correlated to other regions

The

Model surface

The global

Although having the same order of magnitude, uncertainties are generally lower than fluxes. They are highest in the western Pacific and Indian oceans (regions 22 to 34), because of data sparsity. It is lower in better sampled oceans: the Arctic (35 to 37) and the Atlantic (2 to 16) oceans, except for region 13 (Cape Hatteras to Newfoundland). The eastern Pacific (17 to 21) also has low uncertainties in absolute values, probably due to the low concentrations and prior errors there.

The two other inversions produce roughly similar results, although fluxes are
generally lower when derived from

Correlation between data and model concentration fields.

The residuals – i.e., the differences between model concentrations and observations – determine how well the model reproduces the observations and quantify the improvements in the tracer distribution provided by the inversion. They are also a basic tool to identify biases in the model and to assess the quality of the assumptions.

Surface

Radium fluxes obtained by the inverse method largely improve the model match
to observations compared to the prior radium flux (Figs. 4 and 5). The
improvement is quantified by the increase in the model–data correlations
(Table 2) and the decrease in the root mean square of the residuals
(Table 3), a proxy of the cost function. The correlation coefficient is
increased from 0.383 to 0.813 on a linear scale and from 0.809 to 0.902 on a
logarithmic scale. The correlation is higher on a logarithmic scale because
it is less sensitive to the few very high residuals associated with the
highest concentrations (see Fig. 6). On average, the inversion is able to
reduce the ordinary residuals (

Root mean square of residuals before and after inversion

In spite of being smaller, the order of magnitude of the residuals remains
comparable to the data (Fig. 5). On the one hand, in all oceans, positive and
negative residuals are observed with no clear patterns at the scale of a few
grid cells. Because of the rather low model resolution (2

Having assumed specific prior error statistics when choosing the cost functions, we need to check that there is no a posteriori contradiction. Figure 6 displays the residuals and model concentrations as a function of the observations. If the residuals depend on the observed concentrations, it means some observations are more precise than others, contain more information, and should be given a higher weight in order to obtain the best linear unbiased estimate. Figure 7 shows the probability density functions (PDFs) of residuals after all three inversions and compares them with a Gaussian curve representing the expected distribution given the root mean square of residuals. Each PDF should look like a Gaussian curve for the computed posterior uncertainties to be relevant descriptors of errors.

Model global

Probability density functions (PDFs) of

Figure 7a emphasizes that the ordinary residuals do not follow a Gaussian
distribution. On the contrary, most residuals are very close to zero, while a
small number of them are much higher than the standard deviation. Figure 6a
shows that these high residuals occur at high concentrations only, and that
error variance is not homogeneously distributed. Then high and low
concentrations should not be given the same weights, as in

Root mean square of residuals after inversions with different numbers of source regions. The standard case with 38 regions is in bold.

The choice of the regions (and their number) has been made rather
subjectively, although several criteria have been used (spatial distribution
of the observations, independence of the

The root mean square of residuals is a proxy of the cost function. On the one
hand, this parameter should be as low as possible. Increasing the number of
regions always decreases it because the number of degrees of freedom
increases, which tends to improve the fit to the observations. In this
inversion, the largest decrease is found between 7 and 12 source regions.
Further increases in the number of regions have smaller impacts. On the other
hand, too many source regions may produce spurious results. Some regional
fluxes, with too few observations nearby to constrain them, would be computed
using observations farther away, already used by other fluxes. Because of the
lower sensitivity of the concentrations at these farther-off locations, this
process can create extreme fluxes, positive or even negative. The presence of
physically impossible negative values, set to zero by the constraint of
positivity, necessarily means such poor constraints exist. When 52 fluxes are
computed, 5 to 7 of them, according to the cost function, are so poorly
constrained that their fluxes have been set to zero to prevent them from
being negative. This number is reduced to 1 with

The

The shelf fluxes after inversion combine groundwater discharge, riverine particles, diffusion from sediments, and bioturbation. Here we deduce the contribution from groundwater discharge by using existing estimates of the other sources of radium.

Rivers are poor in dissolved

Estimates of

Radium-228 is released from the sediments by diffusion, bioturbation, and
advection, the latter being associated with the SGD. Like

Figure 8 shows the

SGD

In total, 63 % of the

Our inversion estimates are in good agreement with previous regional studies
of

At the global scale,

As recently proposed by

In contrast, our uncertainties on groundwater discharge are large, even
when compared to previous estimates. These larger uncertainties stem from the
poor knowledge of the non-SGD sources of radium that we subtract from the
total flux. As diffusion and bioturbation are expressed in flux per area, the
mean SGD fluxes and their uncertainties depend to a large extent on the
radium-emitting area we consider. Based on a more realistic refined
bathymetry than

Comparisons with local direct estimates of SGD
based on seepage meters and piezometers are also possible but less conclusive
because of the high spatial variability of SGD. Our global average SGD flux
lies between

The spatial distribution of the fluxes in this study is consistent with

Our model of

Dust has not been included in the model. The model then replaces them with
other sources, potentially leading to an overestimation. At global scale, it
represents

Scavenging is a neglected potential sink in this study. As the residence time
related to scavenging is approximately 500

The contribution of rivers to radium fluxes is considered when estimating the
SGD, but only at a basin-wide scale. As most riverine

The other part of the model is the circulation model. The climatological
circulation of NEMO 3.6 was not optimized in this study. However, the
residuals after inversion show that some regions are associated with
spatially structured residuals. There are good reasons to incriminate the
ocean circulation. Because of the low resolution of the model (2

Other sources of errors are the four statistical assumptions on the errors:
errors are assumed to have zero expectation, no correlation, a normal
distribution, and variances depending on the concentrations in a way specific
to each cost function. Systematic biases on data or model are not corrected
by least-squares algorithms. They increase or decrease values without leaving
clues. However, the model conserves mass: the quantity of radium present in
the ocean from each model tracer is precisely known and if concentration is
too high at some place, it will be too low elsewhere. As for the
measurements, their uncertainty is generally around 10 % or lower, and
cruises take them independently from each other, making the assumption of
zero expectation on the observational errors reasonable. The second
assumption is the absence of prior correlation. If prior uncertainties of
neighboring data are correlated, it means that the errors are likely to be of
the same sign whatever the solution, and that multiplying measurements in
this area does not multiply information proportionally. Where measurements
are dense, with residuals far from the expected white noise, for instance in
the North Atlantic, there may be correlation and uncertainties may be
underestimated. The last assumptions are about the structure of variance. We
have shown that logarithmic residuals were almost normally distributed and
independent from concentrations (see Sect. 3.2), justifying the choice of

Observations are not evenly distributed. Some coastal regions cannot be
constrained properly because

Map of GEOTRACES cruises (from

Most direct submarine groundwater discharge measurements have been performed
in developed countries, with a focus on the North Atlantic and the
Mediterranean Sea

Information contained in

Based on inverse modeling, we have computed a global

The source code of NEMO is available on the NEMO
website (

This section describes the way regional

We want to minimize the distance between data and model concentrations,
summed up in a cost function. The ordinary least-squares cost function we
first used is given by

Matrix

Simple algebraic transformations yield

The result can be extended to cases where the cost function is weighted, such
as the proportional least-squares inversion, by just normalizing matrix

The authors declare that they have no conflict of interest.

The authors thank all the scientists who produced the data used in this article. We thank Matt Charette, Eun Young Kwon, Virginie Sanial and Pieter van-Beek for helping us putting the data together. We also thank Olivier Marchal for a discussion about algorithms. This work is part of the first author's PhD, supported by the “Laboratoire d'Excellence” LabexMER (ANR-10-LABX-19) and co-funded by a grant from the French government under the program “Investissements d'Avenir”, and by a grant from the Regional Council of Brittany. Edited by: Jack Middelburg Reviewed by: Isaac Santos and one anonymous referee