An updated estimate of radium 228 fluxes toward the ocean: how well does it constrain the submarine groundwater discharge?

Abstract

Radium 228 (²²⁸Ra), an almost conservative trace isotope of the ocean, supplied from the continental shelves and removed by a known radioactive decay (T1/2 = 5.75 yr), can be used as a proxy to constrain shelf fluxes of other trace elements, such as nutrients, iron, or rare earth elements. In this study, we perform inverse modeling of a global ²²⁸Ra dataset (including GEOSECS, TTO and GEOTRACES programs, and, for the first time, data from the Arctic and around the Kerguelen islands) to compute the total ²²⁸Ra fluxes toward the ocean, using the ocean circulation obtained from the NEMO 3.6 model with a 2° resolution. We optimized the inverse calculation (source regions, cost function) and find a global estimate of the ²²⁸Ra fluxes of 8.01–8.49 × 10²³ atoms yr−1, lower and more precise than previous estimates. The largest fluxes are in the western North Atlantic, the western Pacific and the Indian Ocean, with roughly two thirds in the Indo-Pacific basin. A first estimate in the Arctic Ocean is assessed (0.20–0.50 × 10²³ atoms yr−1). Local misfits between model and data in the Arctic, the Gulf Stream and the Kuroshio regions could result from flaws of the ocean circulation in these regions (resolution, atmospheric forcing). As radium is enriched in groundwater, a large part of the ²²⁸Ra shelf sources comes from submarine groundwater discharge (SGD), a major but poorly known pathway for terrestrial mineral elements, including nutrients, to the ocean. In contrast to the ²²⁸Ra budget, the global estimate of SGD is rather unconstrained, between 1.3 and 14.7 × 10¹³ m³ yr−1, due to high uncertainties on the other sources of ²²⁸Ra, especially diffusion from continental shelf sediments. Better precision on SGD cannot be reached by inverse modeling until a proper way to separate the contributions of SGD and diffusion at a global scale is found.

Full text

An updated estimate of radium 228 fluxes toward the ocean : how
well does it constrain the submarine groundwater discharge ?
Guillaume Le Gland1, Laurent Mémery1, Olivier Aumont2, and Laure Resplandy3
1LEMAR, Institut Universitaire Européen de la Mer, Plouzané, France
2LOCEAN, Institut Pierre Simon Laplace, Paris, France
3Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA, USA
Correspondence to: Guillaume Le Gland (guillaume.legland@univ-brest.fr)
Abstract. Radium 228 (228Ra), an almost conservative trace isotope of the ocean, supplied from the continental shelves and
removed by a known radioactive decay (T1/2= 5.75 yr), can be used as a proxy to constrain shelf fluxes of other trace elements,
such as nutrients, iron, or rare earth elements. In this study, we perform inverse modeling of a global 228Ra dataset (including
GEOSECS, TTO and GEOTRACES programs, and, for the first time, data from the Arctic and around the Kerguelen islands)
to compute the total 228Ra fluxes toward the ocean, using the ocean circulation obtained from the NEMO 3.6 model with a 2◦
5
resolution. We optimized the inverse calculation (source regions, cost function) and find a global estimate of the 228Ra fluxes
of 8.01 −8.49 ×1023 atoms yr−1, lower and more precise than previous estimates. The largest fluxes are in the western North
Atlantic, the western Pacific and the Indian Ocean, with roughly two thirds in the Indo-Pacific basin. A first estimate in the
Arctic Ocean is assessed (0.20−0.50×1023 atoms yr−1). Local misfits between model and data in the Arctic, the Gulf Stream
and the Kuroshio regions could result from flaws of the ocean circulation in these regions (resolution, atmospheric forcing). As10
radium is enriched in groundwater, a large part of the 228Ra shelf sources comes from submarine groundwater discharge (SGD),
a major but poorly known pathway for terrestrial mineral elements, including nutrients, to the ocean. In contrast to the 228Ra
budget, the global estimate of SGD is rather unconstrained, between 1.3and 14.7×1013 m3yr−1, due to high uncertainties on
the other sources of 228Ra, especially diffusion from continental shelf sediments. Better precision on SGD cannot be reached
by inverse modeling until a proper way to separate the contributions of SGD and diffusion at a global scale is found.15
1 Introduction
Trace Elements and Isotopes (TEI) 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 TEIs 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.20
Some of them are studied because they constitute micronutrients for living organisms, like iron (Fe), or 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 (Jeandel et al., 2007), and thorium 234 (234Th) is related to
the biological carbon pump (Clegg and Whitfield, 1991; Buesseler et al., 1992; Henson et al., 2011). Radium isotopes are of
1
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

particular interest as they are proxies of all TEI fluxes from sediments and continents toward the ocean. More specifically,
radium isotopes have been used to estimate a still poorly known pathway to the ocean : Submarine Groundwater Discharge
(SGD).
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 (Burnett et al., 2006; Moore, 2010b). In these aquifers, water5
gets enriched in nutrients and trace elements, released from soils and rocks or coming from land pollution, before flowing back
to the ocean. It has traditionally been considered that the main coastal sources of terrestrial mineral elements to the ocean are
rivers, but there is growing evidence that SGD is in fact a source of nutrients of the same order of magnitude, affecting the
biogeochemistry at all scales, from coastal regions (Hwang et al., 2005; Kim et al., 2005) to ocean basins (Moore et al., 2008;
Rodellas et al., 2015). SGD is a suspected cause of algal blooms, including harmful algal blooms (LaRoche et al., 1997), and a10
pathway for contamination. In spite of this, their contribution is still much less precisely known than the river inputs. Because
of the strong heterogeneity in their distribution and intensity, properly estimating SGD by direct methods requires an intense
sampling, which is far from being fulfilled (Burnett et al., 2006; Moore, 2010a). Indirect methods should then be used: Radium
(Ra) isotopes offer a great potential due to their relation to SGD and their simple chemistry.
All the four natural radium isotopes, 223Ra (T1/2= 11.4 d), 224 Ra (T1/2= 3.6 d), 226Ra (T1/2= 1602 yr) and 228Ra (T1/2=15
5.75 yr), are produced within the rocks by the radioactive decay of thorium. Since radium is far more soluble in water than
thorium, its main source is not the decay of dissolved thorium in the ocean, but dissolution from lithogenic material. Therefore
it is used as a tracer of boundary fluxes. This element is released into the ocean by three main sources located on the continental
shelf: dissolution from riverine particles, diffusion from seabed sediments, and SGD, which are highly enriched. Dust inputs
account for less than 1% of all inputs (Moore et al., 2008). In the ocean, Ra is almost conservative. It is removed by radioactive20
decay and scavenging. Scavenging is associated with a residence time of approximately 500 yr (Moore and Dymond, 1991),
making it negligible for all Ra isotopes except 226Ra. Then the distributions of the other three isotopes, whose radioactive sinks
are known, depend only on the source distribution and transport by the ocean circulation. Since their half-life time scales are
small relative to the time scale of basin-wide horizontal mixing, typically a few years to a few decades, 223Ra and 224 Ra are
unsuitable for large scale studies. 228Ra, whose half-life of 5.75 yr is short enough to neglect scavenging but long enough to25
consider the global ocean, is suitable for global scale analyses and is thus considered here.
A simple way to use the information provided by this isotope is to make an observation-based inventory of the ocean 228Ra.
At steady state, the supply of 228Ra must balance the loss from disintegration, i.e. 12% every year. According to Charette et al.
(2016), these total 228Ra fluxes can be used to estimate fluxes of nutrients, iron and rare earth elements. 228Ra fluxes are also a
way to estimate the SGD fluxes by subtracting the contribution from rivers, diffusion and bioturbation, and dividing the remain-30
ing flux by the mean 228Ra concentration in groundwater. By this method SGD has been estimated at 0.03−0.48×1013 m3yr−1
in the Mediterranean Sea (Rodellas et al., 2015) and 2−4×1013 m3yr−1in the Atlantic Ocean (Moore et al., 2008). These
direct approaches suffer from strong potential biases associated with the relative sparsity of observations: raw assumptions
have to be made in order to estimate regional averages (Rodellas et al., 2015) or to interpolate scattered observations in space
(Moore et al., 2008). Therefore, it is suitable only in regions with dense sampling, such as the Atlantic basin.35
2
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

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: tracer (Ra) observations, a forward
prognostic model which simulates the tracer distribution as a function of the fluxes to be assessed (here the spatial distribution
and intensity of the shelf fluxes), and an algorithm optimizing the fluxes of the model in order to minimize the misfit between
the observations and the model results. The advantage of this method is that the ocean model is used as an interpolator, based5
on physical considerations, which is expected to be robust and consistent. Kwon et al. (2014) used such an inverse modeling
approach to produce a global estimate of 228Ra fluxes of 9.1−10.1×1023 atoms yr−1between 60◦S and 70◦N, corresponding
to 9−15 ×1013 m3yr−1of SGD. Their study is based on a data-constrained global ocean circulation model (DeVries and
Primeau, 2011), considering 50 source regions on the continental shelf, and minimizing an ordinary least-squares cost function.
In this study, we estimate the radium fluxes from all continental shelves around the world and localize the most intense sources,10
using an inverse modeling technique with more data than previous studies (Kwon et al., 2014). The data set has been augmented
with data from two recent GEOTRACES cruises and from the Southern Ocean, the North Pacific, the Mediterranean Sea and
the Indonesian Seas. It also contains data from the Arctic, a basin absent from Kwon’s study. The forward model is built on
the Ocean General Circulation Model (OGCM) NEMO. Our main improvement is a careful analysis of sensitivity and errors,
which reveals that the result depends on the model mathematical parameters, such as the cost function and the number of15
regional sources that are considered. We have performed several inversions and analyzed the residuals and uncertainties, to de-
termine the most appropriate mathematical parameters and evaluate the precision of the flux estimates. This paper is organized
as follows. In section 2, we describe the different aspects of the inversion technique, e.g. the global dataset, the forward model
based on the NEMO OGCM, the different cost functions, the choice of the source regions related to SGDs, and the inverse
method. Section 3 presents the main results, e.g. the global and regional estimates of 228Ra supply, and the sensitivity of these20
estimates to several parameters of our approach, such as the cost function or the number of coastal 228Ra sources. Section 4
compares our results with results obtained in previous studies, and discusses issues associated with such an approach, with an
emphasis on SGD.
2 Methods
2.1 228Ra dataset25
Since the late 1960’s (Moore, 1969; Kaufman et al., 1973) tens of oceanographic cruises have carried out measurements of
228Ra (e.g. articles listed in Table S1). The dataset used in this study includes, among others, observations from three interna-
tional programs sampling trace elements. Data from the Indian Ocean cruise of GEOSECS, in 1977-1978, are included. From
1981 to 1989, the TTO (Transient Tracers in the Oceans) program produced a considerable number of 228Ra measurements in
the Atlantic, from 80◦N to 60◦S, at all depths, making of the Atlantic Ocean the best sampled ocean by far. Currently, new data30
from all oceans are being produced by GEOTRACES. 6059 data from all basins are used in total, of which 2789 are shallower
than 10 m, 1107 are located between 10 and 200 m deep, 606 between 200 and 600 m deep, and 1557 deeper than 600 m. Our
data set comprises 1359 more measurements than in Kwon et al. [2014]: Two GEOTRACES sections, GA03 (United States
3
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

– Cape Verde – Portugal) and GP16 (Ecuador – French Polynesia), and data from the Arctic, the Southern Ocean, the North
Pacific, the Mediterranean Sea and the Indonesian Seas have been added in the present study (See Table S1). These supple-
mentary observations extend the data coverage to regions north of 70◦N and around the Kerguelen islands. In a near future,
other sections from the GEOTRACES program will complete the global covering and may help in studying deeper sources.
For the purpose of our study, data have been averaged in each model grid cell (see section 2.2 for more details on the model5
configuration), leading to the 3076 cell averages shown on Fig. 1. The density of the measurement is noticeably uneven. The
Atlantic Ocean, north of 20◦S, and the Arctic, are the most densely covered basins and the only regions with a significant
number of data at depth deeper than 10 m. Other regions are sparsely sampled, leaving wide areas with no or very few mea-
surements, like the western Indian Ocean, the equatorial Pacific or the Pacific sector of the Southern Ocean.
Data are expressed in concentration or activity units, with the following conversion factor: 1 dpm m−3= 4.36×106atoms m−3.10
They range from 0.04 to 724.5 dpm m−3. The highest concentrations are found in the Bay of Bengal and the coastal seas of
eastern Asia, the lowest values are located in the Southern Ocean. Concentrations are generally higher than 10 dpm m−3in the
Indian Ocean, the Atlantic and the Pacific north of 30◦N, lower in the rest of the Pacific. In the Atlantic, west of a line running
from the Amazon delta to Newfoundland, most concentrations are higher than 30 dpm m−3.
2.2 Forward tracer model15
The second requirement of the inversion technique is a 228Ra transport model, allowing to link in situ observations to the
boundary conditions or shelf sources of 228Ra. The transport equation of tracer Ai(originating from the source region i) is
given by:
∂Ai
∂t =−U.∇Ai+∇.(K∇Ai)−λAi+Si(1)
Uand Kare the velocity field and the eddy diffusivity coefficient respectively. The two first terms on the right together20
constitute transport. They are derived from NEMO 3.6 model (Nucleus for European Modeling of the Ocean) using OPA
(Madec, 2015) as a general circulation component, coupled with the sea-ice model LIM3 (Vancoppenolle et al., 2009), with
an ORCA2 global configuration. The model has an horizontal resolution of 2◦×2◦cosφ(where φis the latitude) enhanced
to 0.5◦near the equator. The mesh is tripolar in order to overcome singularities, the North Pole being replaced by two inland
poles in the Northern Hemisphere. It has 31 vertical levels, ranging from the surface to 6000 m deep, the upper layer covering25
the first 10 m. The simulation is forced by a seasonal climatological dataset, based on NCEP/NCAR reanalysis and satellite
data.
λis the radioactive decay constant, 0.12 yr−1, given by the half-life of 228Ra which is 5.75 yr. Decay is the sole sink. It is
known and does not depend on environmental parameters, leaving the source term as the only unknown to be determined by
the inversion technique.30
Siis the source term specific to the ith region, representing riverine inputs, sedimentary diffusive fluxes and groundwater
discharge fluxes of 228Ra from region i. In this study, sources are assumed to be only on the continental shelf, defined as the
seabed shallower than 200 m. This depth range, spanning 16 model levels, is chosen because it is where most groundwater
4
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

discharge outflows. However, diffusion from sediments also occurs at higher depths (Hammond et al., 1990).
Equation (1) shows that the Aifields depend linearly on Si. That means that any 228Ra distribution can be written as a linear
combination of the Aifields. It is important to emphasize that the circulation is supposed to be "perfect", e.g. no correction
of the Ufield is looked for. Nevertheless, the simulated circulation obviously suffers from deficiencies, and that point has to
be kept in mind when interpreting the results. From now, what we refer to as "model concentration", [228Ra]mod , is a linear5
combination of the tracer final concentrations Ai, the coefficients being the source intensities xi:
[228Ra]mod =
n
X
i=1
Aixi(2)
A non-optimal model concentration was computed by assuming a uniform constant flux per unit of surface everywhere, with
a global fit using the average concentration estimated from the observations: This defines the first guess estimate before the
inversion. The inversion undertaken in this study aims at optimizing the parameters xiin order to minimize the total difference10
between the observations and the model 228Ra distribution.
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 over-
come that deficiency, sub-model grid scale bathymetric variations are accounted for by comparing the model grid to a global
20resolution bathymetry ETOPO2 of the National Geophysical Data Center (NGDC). The algorithm is detailed in Aumont15
and Bopp (2006). According to this method, the total surface of continental shelf is 2.73 ×1013 m2, 73% higher than the
1.58 ×1013 m2obtained with the coarser bathymetry.
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 Ai, which should be large enough to give independent information.20
Delimitation is done by trials and errors, using the posterior covariance matrix of the inversion (see section 2.3): The number of
sources is minimized by merging regions associated with negligible fluxes with close highly correlated regions. Most islands
are ignored, because the areas of their continental shelf and thus their expected contributions to the 228Ra balance are small.
In the inversion process, islands can give rise to spurious fluxes to accommodate for other types of errors. The only islands
considered in this study are the Kerguelen and Crozet Islands, in the Southern Ocean, because many samples have been taken25
in their surroundings which make it possible to constrain their contributions. Because of the lack of measurements and the
coarse model resolution, the Persian Gulf, the Red Sea, the Baltic Sea, the North Sea and the Hudson Bay are not taken into
account. The flux per unit of surface is assumed to be constant on each of the 38 emitting regions. Model simulations last for
the equivalent of 100 years in order to reach a quasi steady state. It is more than 17 times larger than the half-life of 228Ra,
so that the total amount of 228Ra does vary by less than 0.001%. As there are not enough data to study global inter-annual or30
seasonal variations, we do not take seasonal variations of 228Ra concentrations into account. We implicitly assume that radium
concentration is constant over time, and work with average concentrations over the 100th year of simulation.
5
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

2.3 Inverse method
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 (Gloor et al., 2001; Mikaloff Fletcher et al.,
2006; Jacobson et al., 2007) and oceanic heat fluxes (Resplandy et al., 2016). If ηrepresents the residuals, e.g. the discrepancy
between model results and observations:5
Ax=[228Ra]obs +η(3)
Both sides are vectors, their size being the number of data. A is the matrix of footprints, representing the circulation model and
composed of the Aiat each data point. xis the vector of unknowns, the flux of 228Ra per unit of surface of each region. As
only shelf sources are modeled and as data coverage below 200 m is sparse, only data shallower than 200 m are considered.
The distance between model concentrations and data is summed up in a scalar, the cost function C(x). We look for the optimal10
flux vector xopt minimizing the cost function. Different choices for Care possible, depending on the assumed probability
distribution for the prior error on data and model footprints A. Errors due to biased sampling are not considered here. All
errors are supposed to be uncorrelated. In this study, three different cost functions have been tested and minimized. They all
correspond to the sum of squares of a specific type of residuals. Their respective equations are listed in Table 1. The first one
is an ordinary least-squares cost function, Cols. According to the Gauss-Markov theorem, its minimization produces the best15
linear unbiased estimator when prior errors have no correlation, zero expectation, and the same variance. It is the simplest
least-squares method, chosen in the study by Kwon et al. (2014). This function gives the same weight to all observations.
However, the hypothesis of the homogeneity of the variance is questionable: Far offshore, where concentrations are lower
and less sensitive to small changes in coastal sources, observations and model errors can be expected to be lower. Neglecting
this fact means these data are not fully exploited, as their contribution to the cost function is relatively small. Two other cost20
functions with a higher weight for smaller values are then considered for comparison. They are assuming heteroscedastic
data, with higher variances for higher concentrations. In the proportional least-squares cost function, Cprp, the error standard
deviation is supposed to be proportional to the observed concentrations. The logarithmic least-squares cost function, Clog,
works differently and assumes the logarithms of concentrations have the same error variance. It is less sensitive than Cprp
to model overestimations and more to underestimations. It is the only cost function which is not a quadratic function of the25
sources.
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 fluxes30
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 (Gloor et al., 2001), and two regions are strongly
correlated when the 228Ra emitted by each is transported to the same places and are then harder to differentiate. The computed
posterior uncertainties are precise only if all the preceding assumptions on prior errors are correct. The coherence of error
6
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

assumptions with the results has to be checked (see 3.2). Along with the main inversion considering 38 regions, we performed
4 other inversions with a higher or lower number of regions in order to estimate the sensitivity to this parameter (see 3.3).
3 Results
3.1 228Ra fluxes
The 228Ra fluxes from each of the 38 regions, deduced by minimizing each of the three cost functions, are shown on Fig. 35
with their confidence intervals, and compared with the prior estimates. The global fluxes for each method are also shown. As
they are sums of local fluxes, their standard deviations are proportionally lower.
The global 228Ra flux within one standard deviation is 8.01 −8.49 ×1023 atoms yr−1according to the Clog inversion. As we
will explain in section 3.2., this estimate is the most accurate of the three. Fluxes are found to be comparatively high in the
North Atlantic (regions 5 to 16), in the western Pacific (22 to 27) and in the Indian Ocean (28 to 34), together accounting for10
62.6% of the continental shelf and 82.8% of the global flux of 228Ra. Highest fluxes are located on the east coast of North
America (10 and 13), in the China Seas (23 and 25) and in the eastern Indian Ocean (29 to 32), where the inversion process
produces the largest increase compared to prior estimates. Conversely, inversion significantly reduces the prior estimates in the
Arctic Ocean (35 to 37), in the Bering Sea (21) and in the eastern Pacific Ocean (17 to 20). Fluxes are also quite low in the
Southern Ocean (1 and 38) and in the South Atlantic (2 to 4). The newly estimated Arctic and Antarctic sources are in the15
range of 0.43 to 0.50 and 0.31 to 0.37 ×1023 atoms yr−1respectively, accounting for 5% to 6.2% and 3.6% to 4.6% of the
total sources. In total, roughly two thirds of the 228Ra flows into the Indian and Pacific basins, contrasting with the 60% of the
global river discharge flowing into the Atlantic and Arctic basins (Milliman, 2001).
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 to20
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 Cprp and gen-
erally have higher uncertainties when derived from Cols. The global 228 Ra flux is estimated to 7.16 −8.14 ×1023 atoms yr−1
with Cols and 4.96 −5.28 ×1023 atoms yr−1with Cprp. The three inversions agree on which basins and continents have the25
largest and smallest sources. Yet, local disagreements occur. Regions 5 (Amazon delta), 21 (Alaska and Bering Sea) and 33
(Arabian Sea) have higher fluxes with Cols than with Clog, with non-overlapping confidence intervals, whereas the contrary is
true for regions 12 (Mediterranean), 26 (Indonesian Seas) and 34 (East Africa). Cprp fluxes are generally lower, and in eight
regions their confidence intervals overlap with none of the other inversions. Discrepancies happen because the fitting of the
model to each observation implies different and possibly opposite effects on the source intensities. Each inversion uses differ-30
ent weights, which translates into different flux corrections. Fluxes from each of the three inversions are most dissimilar for
regions where observations impose most dissimilar constraints, where the model fails to reconstruct the pattern of the data and
has to choose between fitting some data or others in priority. In such cases, all the results should be considered carefully. When
7
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

the confidence intervals between the different inversion techniques fail to overlap, it is likely that one or several estimates are
incorrect. As algorithms are built by assuming a prior error statistics, it is likely that some rely on wrong assumptions.
3.2 Model concentrations and residuals
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 tool5
to identify biases in the model and to assess the quality of the assumptions.
Radium fluxes obtained by the inverse method largely improve the model match to observations compared to the prior radium
flux (Fig. 4 and Fig. 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 scale10
because it is less sensitive to the few very high residuals associated with the highest concentrations (see Fig. 7). On average,
the inversion is able to reduce the ordinary residuals (Cols) by a factor 2, logarithmic residuals (Clog) by 1.4 and proportional
residuals (Cprp) by 3.5.
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 of15
the rather low model resolution (2◦, which is not sufficient to reproduce medium and small scale processes) and of issues
associated with temporal (seasonal or higher frequency) variability of the data, this kind of "noise" is expected. It is consistent
with the assumption of independent errors used when computing the error variances on fluxes. On the other hand, in several
regions, residuals can display coherent large scale patterns, which cannot be attributed to noise. These areas may suffer from
systematic overestimations, like in the Gulf Stream region, the western Pacific between 20◦N and 40◦N, and off Eastern Siberia,20
or underestimations, such as in the center of the North Atlantic Gyre. These residuals point out to possible flaws in the model
circulation. For instance, 228Ra is quite homogeneously distributed in the western North Atlantic according to data, but in the
model, the gradients are stronger, and no combination of sources manages to reduce these gradients. This is probably caused by
a bias in the Atlantic circulation, with a too low exchange rate between inshore and offshore waters. In a 2◦resolution model,
mesoscale eddies are not represented and cannot transport 228Ra south and east of the Gulf Stream or north of the North Brazil25
Current. Inversions minimize the misfit by increasing the fluxes from regions 5 (North Brazil), 8 (Caribbean) and 10 (southern
East Coast of the US), making the model concentration too high close to the coast while still too low in the gyre. Such large
scale biases are not consistent with the assumption of no prior error correlation, which may lead to underestimation of flux
uncertainties around these basins.
Having assumed specific prior error statistics when choosing the cost functions, we need to check that there is no a posteriori30
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
(PDF) of residuals after all three inversions and compares them with a Gaussian curve representing the expected distribution
8
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

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.
Figure 7a emphasizes that the ordinary residuals do not follow a Gaussian distribution. On the contrary most residuals are
very close to zero and only a small number of them is 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 low5
concentrations should not be given the same weights, as in Cols, but the highest concentrations should be given the lowest
weights, as in Clog and Cprp. Flux estimates based on Cols are biased because the cost function puts more emphasis on
high concentrations, and this method then tries to fit more specifically the misfits at high concentrations. Cols also produces
very large error bars because the error variance is assumed to be constant and its computed value, influenced by a few very
large residuals, is larger than the actual error variances of the vast majority of data. Figure 6c and Fig. 7c show that the10
proportional residuals are not normally distributed either. They are not even symmetrical, as they cannot be smaller than -1 but
they do not have an upper limit. This asymmetry produces a bias in the flux estimate. The algorithm based on Cprp is more
sensitive to positive residuals because underestimations are never associated to proportional residuals lower than -1 whereas
overestimations can produce residuals higher than 1. As a consequence, this method tends to reduce the fluxes. The hypothesis
of constant variance is more realistic although the highest residuals occur at low concentrations. Finally, the distribution of the15
logarithmic residuals displayed on 6b) and 7b) is much closer to a Gaussian curve and much less dependent on concentrations,
which makes the logarithmic cost function more relevant for this study.
3.3 Sensitivity to the number of regions
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 Aifields). The global 228Ra should ideally not depend upon the number20
of regions. Therefore, alternative region geometries have been tested for comparison. The 228Ra fluxes are shown on Table
4 and the root mean square of their residuals is presented on Table 5. Case 1 inversion uses 52 regions: It was the original
distribution of regions before some of them were merged to define the 38 standard regions of this study. It includes more
regions in undersampled areas such as the Arctic, the South Atlantic, the western Indian or the equatorial Pacific. Case 5 has
just one emitting region for each of the following ocean basins: Southern, South Atlantic, North Atlantic, South Pacific, North25
Pacific, Indian and Arctic. Case 3 and Case 4 are intermediate cases with source regions built by merging regions from Case 2.
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 spurious30
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
9
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

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 Cols and zero with Clog and Cprp when there are 38 regions, and completely
disappears with 19 or fewer regions. Regions with fluxes within the error estimate are also very poorly constrained by the
observations and the circulation model: their number is also reduced from 26 out of 52 to 11 out of 38 with Cols, from 13 to 3
with Cprp and from 9 to 3 with Clog . All these fluxes have a low impact on the cost function, but make the global 228Ra flux5
less precise. This analysis shows that Case 1 (with 52 regions) is not constrained enough and that Case 3 to 5 display too large
residuals. Therefore Case 2 (38 regions) is considered to be the best choice.
The Cols-based global 228 Ra flux is varying in a non-monotonic way, with a difference of 23% between the highest and
the lowest. With 38 regions, it is lower than fluxes computed with both a higher and a lower number of regions. This high
sensitivity may be related to the high uncertainty associated with this inversion, certainly linked to the relatively poor data10
coverage. All the confidence intervals within one standard deviation from Case 1 to Case 4 overlap. The Cprp-based global
fluxes are always lower than the other fluxes, and they decrease as the number of regions decreases. This is consistent with our
previous hypothesis on Cprp. This cost function tends to fit the lowest data in priority. Larger regions suffer more from this bias
because they are constrained by more data, likely to be more dispersed. The confidence intervals within one standard deviation
based on Cprp fail to overlap. Only the logarithmic least-squares method produces very similar fluxes whatever the number of15
regions, with all confidence intervals overlapping. The global flux based on Clog again seems to be the most reliable.
3.4 Submarine groundwater discharge estimates
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 228Ra and transport 228Ra mainly with the sediments they carry (Moore and Shaw, 2008). A20
fraction of them is desorbed in the mixing zone, because of the salinity increase. According to various studies (Key et al.,
1985; Moore et al., 1995; Krest et al., 1999), the amount of 228Ra desorbed per gram of sediment lies in the range of 2.9to
8.7×106atoms. In this study, we follow Milliman (2001) who proposed a global river sediment flux of 1.8×1016 g.yr−1,
divided into fluxes from six basins: South Atlantic, North Atlantic, South Pacific, North Pacific, Indian Ocean and Arctic. This
leads to a global 228Ra river flux estimate of 0.53 −1.60 ×1023 atoms yr−1, significant but not dominant.25
228Ra is released from the sediments by diffusion, bioturbation and advection, the latter being associated with the SGD. Like
Moore et al. (2008), we assume that the 228Ra fluxes by diffusion and bioturbation from relict sands, composing 70% of the to-
tal continental shelf area, is weak, typically of the order of 109atoms m−2yr−1(Colbert, 2004; Hancock et al., 2006). Fluxes
from continental shelf muds, which correspond to the remaining 30%, have been estimated by several studies using different
methods, such as inventories (Moore et al., 1995), benthic chambers (Hancock et al., 2000), sediment profiles (Hancock et al.,30
2000) or modeling (Hancock et al., 2006). But some of them were done at a time when SGD were not considered to be an im-
portant source of radium and may have included them in their estimates. These estimates should then be considered as an upper
limit. Furthermore, their locations are often very close to the coast. Some recent studies addressing diffusive fluxes separately
in order to estimate local groundwater discharge (Crotwell and Moore, 2003; Kim et al., 2005) used the simplified equation of
10
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

diffusion from Krest et al. (1999) and pointed out to significantly lower values, but may not be representative of all continental
shelves. So far, it is not possible to be very precise and a wide range has to be considered: From the low recent values, close
to 5×109atoms.m−2.yr−1, to the higher values typically ranging from 25 to 75 ×109atoms m−2yr−1(Moore et al., 2008).
The full range is then 5−75 ×109atoms.m−2.yr−1. As the continental shelf area in our model is 2.73 ×1013 m2, this means
a total flux due to diffusion and bioturbation ranging from 0.43 to 6.51 ×1023 atoms yr−1. Then, using the logarithmic cost5
function, the SGD 228Ra flux estimate varies between 0.62 and 6.82 ×1023 atoms yr−1: these two fluxes are within the same
range.
Figure 8 shows the 228Ra fluxes for seven basins corresponding to the Southern Ocean and the six basins used by Milliman
(2001) to define sediment inputs by rivers. The largest 228Ra fluxes are found in the North Atlantic, the North Pacific and the
Indian Ocean, in roughly equal proportions, whereas those from the South Atlantic, the South Pacific and the Arctic Ocean10
are far smaller. Fluxes from the three latter regions are low enough in fact to be within the confidence intervals of riverine and
diffusive sediment 228Ra fluxes, so that the SGD confidence intervals include negative values, which are physically impossible.
SGD 228Ra end-member concentration varies considerably from one aquifer to another, ranging from 0.04 to 125×106atoms m−3
(Moore et al., 2008). Measurements so far have shown aquifer concentrations in the Atlantic Ocean higher than the average by
typically 30% (Kwon et al., 2014). Following Kwon et al. (2014), we assume that the aquifer concentrations are log-normally15
distributed with an average of 0.98 to 1.15 ×103dpm m3. Taking a geometric mean rather than an arithmetic mean is implic-
itly assuming that aquifers characterized by high 228Ra concentrations are emitting less water. It also suffers from wells being
concentrated in developed countries. Using the results from the inversion technique based on the logarithmic cost function, the
total SGD flux is estimated to 1.3−14.7×1013 m3yr−1, to be compared with the global river flow of 3.5×1013 m3yr−1
(Milliman, 2001).20
4 Discussion
4.1 Comparisons with previous studies
Our inversion estimates are in good agreement with previous regional studies of 228Ra based on inventories. The 228Ra inven-
tory of the Mediterranean Sea computed by Rodellas et al. (2015) led to a total flux due to rivers, sediments and groundwater
discharge of 1.86−2.48×1022 atoms yr−1. This is compatible with our estimate based on Clog ,1.96−2.28×1022 atoms yr−1.25
Other cost functions produce an underestimation in the model by a factor 2.5 (ols) or 3.8 (prp). In the Yellow Sea, Kim et al.
(2005) estimated a flux of 3.3×1015 dpm yr−1, or 1.4×1022 atoms yr−1. Expressed per unit of surface, it corresponds to
3.6×1010 atoms m−2yr−1. This is slightly lower than our flux for the larger region 23 (Sea of Japan, Yellow Sea, East China
Sea) of 4.2−5.6×1010 atoms m−2yr−1based on Clog, both these fluxes being larger than the global average. At larger scale,
Moore et al. (2008) estimated the total 228Ra flux over the Atlantic between 50◦S and 80◦N to be 2.8−4.2×1023 atoms yr−1.30
Restricting the inversion to the Atlantic with Clog yields 2.64−2.92 ×1023 atoms yr−1, in the lower range but compatible. At
least two reasons can explain the difference. Moore et al. (2008) have included data down to 1000 m, which allowed them to
estimate 228Ra release from sediments down to that depth. The authors have estimated the sources between 200 m and 1000 m
11
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

deep to 0.13 −0.37 ×1022 atoms yr−1. Furthermore, in order to compute the total 228Ra content of the Atlantic, they have
performed a linear interpolation of the data, potentially leading to errors, especially in areas where measurements are sparse.
At the global scale, Kwon et al. (2014) have used a method which is quite similar to ours and they have computed a total global
228Ra flux of 9.1−10.1×1023 atoms yr−1, of which between 4.2and 7×1023 atoms yr−1are released by SGD. This corre-
sponds to a global SGD flux of 9−15 ×1013 m3yr−1. The flux of 228Ra estimated by the present study is thus significantly5
lower than the estimates by Kwon et al. (2014), although our results include the Arctic and Antarctic sources ( 0.43 −0.50 and
0.28 −0.35 ×1023 atoms yr−1respectively). Kwon et al. (2014) minimized an ordinary least-squares cost function with 50
regions. We have shown here that both a high number of source regions and the use of the ordinary least-squares cost func-
tion concur to produce a higher estimate. However, this is at the expense of a higher uncertainty and of producing unrealistic
negative fluxes in some source regions. Additional differences in Kwon’s study may explain their higher estimates such as a10
different ocean circulation model, a coarser vertical resolution, and a bathymetry re-gridded onto the model domain. Finally,
they added dust deposition and removed data higher than 140 dpm m−3, but these two factors should rather tend to reduce
their fluxes.
As recently proposed by Charette et al. (2016), shelf 228Ra fluxes can be used as gauges of shelf fluxes of trace elements and
isotopes, including nutrients, iron, and rare earth elements. 228Ra is particularly relevant because it is chemically conservative15
and integrates information over annual to decadal timescales. At first approximation, the flux of TEIs is deduced from the
228Ra flux and the ratio of the nearshore gradients. Limited for now due to the lack of nearshore measurements, this method
could be more common in the future. As they are based on realistic assumptions on error statistics and have low uncertainties,
our radium fluxes are able to improve the current estimates of all elements originating from the continental shelf.
On the contrary, our uncertainties on groundwater discharge are large, even when compared to previous estimates. These larger20
uncertainties stem from the poor knowledge of the non SGD sources of radium that we subtract from the total flux. As diffu-
sion 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 Kwon et al. (2014) (2.73 ×1013 m2
compared to 1.5×1013 m2), our study also has a larger sedimentary flux, with an upper range close to the total 228Ra flux.
Thus, the lower range of SGD fluxes, 1.3×1013 m3yr−1, is very low, while the upper range, 14.7×1013 m3yr−1, is similar25
to other studies. The use of this model provides an upper estimate but cannot precisely compute the global SGD flux, and no
inverse model can if the surface diffusive flux and the area emitting radium are not clarified. Our expectation is that the lower
range of sedimentary flux is more likely, because it comes from studies where SGD is also considered and because it does
never produce negative regional fluxes. By contrast, the higher range is very similar to the total bottom flux, as expected when
no distinction is made between diffusion and SGD.30
Comparisons with local direct estimates of submarine groundwater discharge based on seepage meters and piezometers are
also possible but less conclusive because of the high spatial variability of SGD. Our average SGD fluxes are between 0.5and
5.5 m yr−1. Values from seepage meters and piezometers in the upper 200m reported by Taniguchi et al. (2002) and Knee
and Paytan (2011) range from 0.03 m yr−1in the Tokyo Bay (Taniguchi et al., 2002) to 1790 m yr−1near Mauritius (Burnett
et al., 2006). Most measurements have been performed in the upper 10 meters and range from 1to 50 m.yr−1. Thus, this range35
12
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

is very wide and many local studies span several orders of magnitude. At a global scale, Taniguchi et al. (2009) produced an
estimate of the global SGD flux based on seepage measurements of 6.1−12.8×1013 m3yr−1, in the upper range of our global
estimate. However, it suffers from most measurements being concentrated in developed countries.
4.2 Model biases
Our model of 228Ra is based on a circulation model and assumptions on the cycle of this isotope. Both are potential sources of5
errors.
Dust has not been included in the model. The model then replaces them with other sources, potentially leading to an overes-
timation. At global scale, it represents 1.7×1021 atoms yr−1(Kwon et al., 2014), less than 0.2% of the total flux and less
than most individual source regions, which cannot create large biases. Nevertheless, the largest dust deposition is associated to
Saharan dust transported to the North Atlantic, in the Canary Upwelling Region (Mahowald et al., 2005). In this region, dust10
may have some impact on the 228Ra distribution, since it brings this isotope directly to the open ocean. Its absence in our study
might explain the overestimation by our model near the coast of region 6 and underestimation offshore. Yet, the exclusion of
dust deposition in our analysis cannot explain the largest bias in the North Atlantic: The overestimation in the Gulf Stream
coupled with an underestimation just south east (see 3.2), because the area where the model cannot transport radium is located
west of the maximum of dust input and displays higher concentration.15
Scavenging is a neglected potential sink in this study. As the residence time related to scavenging is approximately 500 yr
(Moore and Dymond, 1991), it accounts for between 1% and 2% of all sinks. Thus, its inclusion in our computations would
increase the source intensity necessary to maintain global balance. Contrary to radioactivity, scavenging is highly heteroge-
neous, most intense where primary productivity and particle concentrations are highest. Fluxes from the high latitudes of the
North Atlantic are thus potentially more underestimated than fluxes in other regions, since they are areas of intense biological20
activity during spring blooms. As the actual total lifetime of 228Ra is overestimated when scavenging is not taken into account,
the gradient between coast and open ocean could be too low. However, as the horizontal mixing time scale of the ocean is a
few decades, the relative overestimation of open ocean concentrations is less than 10%.
The contribution of rivers to radium fluxes is considered when estimating the SGD, but only at a basin-wide scale. As most
riverine 228Ra travels attached to particles and is desorbed in the mixing zone, we have based our computation on the sediment25
loads of Milliman (2001), which are basin-wide. Although NEMO 3.6 takes rivers into account for their impact on salinity
(Madec, 2015), for now no information at the model grid scale is available on their sediment loads. If this information existed,
we would have been able to estimate the contribution of rivers, and consequently sediments and SGD, in the 228 Ra fluxes for
each of the 38 regions. Some local SGD fluxes close to large rivers (Amazon, Congo, Yellow River, etc.) could then appear to
be significantly lower than their shares of the total 228Ra fluxes suggest.30
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◦), isopycnal diffusion has
been used to parameterize sub grid processes and mixing (Redi, 1982) with a constant eddy diffusivity of 2000 m2s−1. Nev-
13
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

ertheless, in very energetic regions, such as the Western Boundary Currents (WBCs, e.g. the Gulf Stream and the Kuroshio),
higher eddy diffusivity might be needed to enhance the exchanges with the open ocean, possibly improving the fit with the
observations. Furthermore, it is well known that the mean currents are also dependent on this low resolution (impacting for
instance the position and intensity of the WBCs). Although it cannot solve all the flaws of the circulation, improvements could
thus be brought by an increase of the resolution towards eddy resolving models. However, as 100 years of simulation are5
needed at a global scale for each source region, this would be computationally expensive.
Other sources of errors are the four statistical assumptions on the errors: errors are assumed to have zero expectation, no corre-
lation, 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, model conserves mass: the quantity of radium present in the ocean from each model tracer is precisely known and if10
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 mea-
surements in this area does not multiply information proportionally. Where measurements are dense, with residuals far from15
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 dis-
tributed and independent from concentrations (see 3.2.), justifying the choice of Clog as a cost function. Cols leads to higher
uncertainties, especially for small sources, and Cprp to systematic underestimation, but both are useful for comparison, in order
to identify regions where physical assumptions are inaccurate (see 3.1.).20
4.3 What new data would be most useful ?
Observations are not evenly distributed. Some coastal regions cannot be constrained properly because 228 Ra data are lacking.
Improving the coverage would increase the quality of the inversion in two ways: It would reduce the uncertainties and make
it possible to divide wide regions into smaller regions as long as they have distinct footprints. For instance, the Philippines,
Papua, or the Gulf of Guinea, whose footprints are very different from the Indonesian Seas and the southwestern coast of25
Africa could be considered as independent regions. More samples in the Indian Ocean, the South Atlantic (south of 30◦S), the
Southern Ocean and the western equatorial Pacific are priorities. All these regions will be sampled by upcoming GEOTRACES
cruises shown on Fig. 8. At the same time, deep samples will be taken outside of the Atlantic, enabling a more comprehensive
global inversion with extra source regions at greater depths. This would improve our knowledge of the contribution of deeper
sediments.30
Most direct submarine groundwater discharge measurements have been performed in developed countries, with a focus on the
North Atlantic and the Mediterranean Sea (Taniguchi et al., 2002). At the same time, measurements are completely lacking
over large regions. More SGD studies in areas where they are potentially the highest, namely the Bay of Bengal, the Indonesian
Seas and the China Seas, would produce more representative estimates of the 228Ra content of SGD around the world and direct
14
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

estimates of local SGD magnitude to be compared with regional inversion results. They have begun more recently (Knee and
Paytan, 2011) and are still sparse.
Information contained in 228Ra might be completed with 226 Ra concentrations, measured during the same campaigns and for
this reason available with a similar coverage. Associated with the same source as 228Ra, but with a much longer half-life,
1602 yr,226Ra would constrain deeper sources and would help in assessing the quality of the thermohaline circulation and5
deep ventilation of the circulation model. However, inverting 226Ra data would require a precise modeling of scavenging,
which is not negligible at these longer time scales, as well as a much longer integration duration in order to get steady state
distribution.
5 Conclusions
Based on inverse modeling, we have computed a global 228 Ra flux from continental shelves of 8.01 −8.49 ×1023 atoms yr−1,10
with the largest sources in the western Pacific, the western North Atlantic and the Indian Oceans and the smallest sources in
the eastern Pacific. The Arctic and Antarctic sources have been estimated for the first time, accounting for 0.43 −0.50 and
0.28−0.35 ×1023 atoms yr−1respectively. These precise estimates are obtained by minimizing the squares of the differences
of logarithmic concentrations between model and data, a cost function we think is more realistic than the other functions we
have tested because it is based on more realistic assumptions on error statistics. Given the number of available measurements,15
we were able to constrain 38 regional fluxes. The shelf fluxes produced using these optimal parameters are lower than previous
estimates. In a near future, they will enable to quantify continental shelf fluxes of trace elements and isotopes to the oceans at
any place where nearshore gradients are measured (Charette et al., 2016). The estimated global SGD flux is far less precise,
ranging between 1.3and 14.7×1013 m3yr−1, because of the very large uncertainty on the two other sources of 228Ra, i.e.
riverine particles and most of all diffusion from bottom sediments, also located on the continental shelf. Only the upper range20
is compatible with previous estimates. After inversion, we were able to reproduce the basin scale patterns of 228Ra distribution
with nevertheless systematic biases in several regions, especially in the Arctic, and west of the sub-tropical gyres. Shortcomings
in the circulation model are the most probable explanation of these biases (too weak exchanges between continental shelves
and open ocean). Therefore, besides estimating the sources and sinks of tracers, interesting information about the ocean model
can be obtained by Ra like tracer inversions. Extensive regions are lacking observations, mostly in the Southern Hemisphere25
(Pacific, Indian, and mostly Southern Ocean, as well as western equatorial Pacific Ocean), and better coverage in basins where
SGDs are known to be influential and to produce large horizontal gradients is also needed (such as in South Asia). But the
main impediment to achieve precise estimates of global Ra SGD fluxes comes from the very poor knowledge of diffusive
sedimentary fluxes: without a proper way to separate diffusion and SGD, inversions can compute the total bottom flux but are
not able to precisely evaluate these two components.30
15
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

6 Code and Data availability
The source code of NEMO is available on the NEMO website. Studies providing data are listed in Table S1 in the supplementary
materials. The inversion code and data used in this study can be obtained directly by contacting the authors.
Appendix A: Error Statistics
This section describes the way regional 228Ra flux estimates and their error bars are computed. The inverse problem is the5
following one : using p measurements of oceanic 228Ra concentrations, their n sources must be traced back, by means of a
circulation model. The model produces n 228Ra concentration fields, one from each of the source regions. It is run through
steady state, the ith field reaching concentration Ai. The model concentration is a linear combination of the latter:
[228Ra]mod =
n
X
i=1
Aixi
Inversion requires fitting the model concentration to the observations. As the expression of model concentration is linear in10
terms of the unknowns, the problem can be written:
Ax=b+η
where b∈ Mp,1is the vector of the p measured radium concentrations, x∈ Mn,1is the unknown and the vector of the n
source intensities, A∈ Mp,n is the footprint matrix, composed of the Aiat each data point, and η∈ Mp,1is the vector of
residuals, accounting for the misfits, which are inevitable in our over-constrained problem (p>n).15
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:
Cols(x)=(Ax−b)>(Ax−b)
The cost function has to be minimal at the optimal x, called xopt, implying, in the absence of inequality constraints:
∇Cols(xopt )=2A>(Ax−b) = 020
As the n 228Ra concentration fields are independent from each other, A>Ais invertible. Then:
xopt =Ainvb(A1)
Ainv = (A>A)−1A>b
Ainv is the pseudo-inverse of A, transforming the consequences into their most probable causes and justifying the word25
"inversion". Equation (A1) corresponds to (2.95) in Wunsch (2006). In practice, fluxes are constrained to be positive. This is
16
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

managed by just reducing the number of source regions if necessary, without changing the principle of the inversion.
Matrix Hsi defined by :
H=A(A>A)−1A>=AAinv
As H2=H>=H,His an orthogonal projector on a subspace of dimension n. As Axopt =Hb, the model concentration
is the projection of the observations on this subspace whereas the vector of residuals is the projection on the complementary5
subspace.
Simple algebraic transformations yield:
HA =A(A>A)−1A>A=A
AinvA= (A>A)−1A>A=In
AinvA>
inv = (A>A)−1A>A(A>A)−1= (A>A)−1(A2)10
Uncertainties on xresulting from uncertainties on data (b) and model (A) must now be evaluated. By principle of the ordinary
least-squares cost function, the error covariance of (Ax−b) is considered to be a constant diagonal matrix:
h(Ax−b)(Ax−b)>i=σ2Ip
The posterior covariance matrix of xis:
Vxx =h(x−xopt)(x−xopt )>i15
=h(AinvAx−Ainvb)(Ainv Ax−Ainvb)>i
=Ainvh(Ax−b)(Ax−b)>iA>
inv
=σ2AinvA>
inv (A3)
Equation (A3) corresponds to (2.102) in Wunsch (2006).
σ2is related to the root mean square of residuals vηthe following way:20
vη=h(Axopt −b)>(Axopt −b)i
=h(Hb−b)>(Hb−b)i
=h((Ip−H)(Ax−b) + (HA −A)x)>((Ip−H)(Ax−b) + (HA −A)x)i
=h(Ax−b)>(Ip−H)(Ax−b)i
=σ2p−n
p(A4)25
Then, using Eqs. (A2), (A3) and (A4), we can compute Vxx :
Vxx =p
p−nvη(A>A)−1(A5)
17
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

The diagonal terms of Vxx are the squared uncertainties on xwhereas the other terms are covariances, indicating source
regions with similar footprints.
The result can be extended to cases where the cost function is weighted, such as the proportional least-squares inversion, by
just normalizing matrix Aand vector b. Extension to logarithmic inversion is done by linearizing log(Ax)−log(b)to Fx−β
with β= log(b) + Fxopt −log(Axopt ), assuming a small error.5
Competing interests. The authors declare that they have no conflict of interest.
Acknowledgements. 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 Council10
of Brittany.
18
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

References
Aumont, O. and Bopp, L.: Globalizing results from ocean in situ iron fertilization studies, Global Biogeochemical Cycles, 20,
doi:10.1029/2005gb002591, http://dx.doi.org/10.1029/2005GB002591, 2006.
Buesseler, K. O., Bacon, M. P., Cochran, J. K., and Livingston, H. D.: Carbon and nitrogen export during the JGOFS North Atlantic Bloom
estimated from 234Th:238U disequilibria, Deep Sea Research Part I: Oceanographic Research Papers, 39, 1115–1137, doi:10.1016/0198-5
0149(92)90060-7, http://dx.doi.org/10.1016/0198-0149(92)90060-7, 1992.
Burnett, W., Aggarwal, P., Aureli, A., Bokuniewicz, H., Cable, J., Charette, M., Kontar, E., Krupa, S., Kulkarni, K., Loveless, A., and et al.:
Quantifying submarine groundwater discharge in the coastal zone via multiple methods, Science of The Total Environment, 367, 498–543,
doi:10.1016/j.scitotenv.2006.05.009, http://dx.doi.org/10.1016/j.scitotenv.2006.05.009, 2006.
Charette, M. A., Lam, P. J., Lohan, M. C., Kwon, E. Y., Hatje, V., Jeandel, C., Shiller, A. M., Cutter, G. A., Thomas, A., Boyd, P. W., and et al.:10
Coastal ocean and shelf-sea biogeochemical cycling of trace elements and isotopes: lessons learned from GEOTRACES, Philosophical
Transactions of the Royal Society A: Mathematical,Physical and Engineering Sciences, 374, 20160 076, doi:10.1098/rsta.2016.0076,
http://dx.doi.org/10.1098/rsta.2016.0076, 2016.
Clegg, S. L. and Whitfield, M.: A generalized model for the scavenging of trace metals in the open ocean—II. Thorium scavenging,
Deep Sea Research Part I: Oceanographic Research Papers, 38, 91–120, doi:10.1016/0198-0149(91)90056-L, http://dx.doi.org/10.1016/15
0198-0149(91)90056-L, 1991.
Colbert, S. L.: Ra Isotopes in San Pedro Bay, CA: Constraint on Inputs and Use of Nearshore Distribution to Compute Horizontal Eddy
Diffusion Rates., Ph.D. thesis, University of Southern California, 2004.
Crotwell, A. M. and Moore, W. S.: Nutrient and Radium and Fluxes from Submarine and Groundwater Discharge to Port Royal Sound, South
Carolina, Aquatic Geochemistry, 9, 191208, doi:10.1023/b:aqua.0000022954.89019.c9:10.1023/b:aqua.0000022954.89019.c9, http://dx.20
doi.org/10.1023/b:aqua.0000022954.89019.c9, 2003.
DeVries, T. and Primeau, F.: Dynamically and Observationally Constrained Estimates of Water-Mass Distributions and Ages in the Global
Ocean, Journal of Physical Oceanography, 41, 2381–2401, doi:10.1175/jpo-d-10-05011.1, http://dx.doi.org/10.1175/JPO-D-10-05011.1,
2011.
Gloor, M., Gruber, N., Hughes, T. M. C., and Sarmiento, J. L.: Estimating Net Air-sea Fluxes from Ocean Bulk Data, Global Biogeochemical25
Cycles, 15, 767–782, doi:10.1029/2000GB001301, http://dx.doi.org/10.1029/2000GB001301, 2001.
Hammond, D. E., Marton, R. A., Berelson, W. M., and Ku, T.-L.: Radium 228 Distribution and Mixing in San Nicolas and San Pedro
Basins,Southern California Borderland, Journal of Geophysical Research, 95, 3321–3335, doi:10.1029/jc095ic03p03321, http://dx.doi.
org/10.1029/jc095ic03p03321, 1990.
Hancock, G. J., Webster, I. T., Ford, P. W., and Moore, W. S.: Using Ra isotopes to examine transport processes controlling benthic fluxes30
into a shallow estuarine lagoon, Geochimica et Cosmochimica Acta, 64, 3685–3699, doi:10.1016/s0016-7037(00)00469-5, http://dx.doi.
org/10.1016/s0016-7037(00)00469-5, 2000.
Hancock, G. J., Webster, I. T., and Stieglitz, T. C.: Horizontal mixing of Great Barrier Reef waters: Offshore diffusivity determined
from radium isotope distribution, Journal of Geophysical Research, 111, 1–14, doi:10.1029/2006jc003608, http://dx.doi.org/10.1029/
2006JC003608, 2006.35
19
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

Henson, S. A., Sanders, R., Madsen, E., Morris, P. J., Le Moigne, F., and Quartly, G. D.: A reduced estimate of the strength of the oceans
biological carbon pump, Geophysical Research Letters, 38, 1–5, doi:10.1029/2011gl046735, http://dx.doi.org/10.1029/2011GL046735,
2011.
Hwang, D.-W., Kim, G., Lee, Y.-W., and Yang, H.-S.: Estimating submarine inputs of groundwater and nutrients to a coastal bay using radium
isotopes, Marine Chemistry, 96, 61–71, doi:10.1016/j.marchem.2004.11.002, http://dx.doi.org/10.1016/j.marchem.2004.11.002, 2005.5
Jacobson, A. R., Mikaloff Fletcher, S. E., Gruber, N., Sarmiento, J. L., and Gloor, M.: A joint atmosphere-ocean inversion for surface fluxes
of carbon dioxide: 1. Methods and global-scale fluxes, Global Biogeochemical Cycles, 21, doi:10.1029/2005gb002556, http://dx.doi.org/
10.1029/2005GB002556, 2007.
Jeandel, C., Arsouze, T., Lacan, F., Techine, P., and Dutay, J.: Isotopic Nd compositions and concentrations of the lithogenic inputs into
the ocean: A compilation, with an emphasis on the margins, Chemical Geology, 239, 156–164, doi:10.1016/j.chemgeo.2006.11.013,10
http://dx.doi.org/10.1016/j.chemgeo.2006.11.013, 2007.
Kaufman, A., Trier, R. M., and Broecker, W. S.: Distribution of 228 Ra in the World Ocean, Journal of Geophysical Research, 78, 8827–8848,
doi:10.1029/jc078i036p08827, http://dx.doi.org/10.1029/jc078i036p08827, 1973.
Key, R. M., Stallard, R. F., Moore, W. S., and Sarmiento, J. L.: Distribution and Flux of 226 Ra and 228Ra in the Amazon River Estuary,
Journal of Geophysical Research, 90, 6995–7004, doi:10.1029/jc090ic04p06995, http://dx.doi.org/10.1029/jc090ic04p06995, 1985.15
Kim, G., Ryu, J.-W., Yang, H.-S., and Yun, S.-T.: Submarine groundwater discharge (SGD) into the Yellow Sea revealed by 228 Ra and
226Ra isotopes: Implications for global silicate fluxes, Earth and Planetary Science Letters, 237, 156–166, doi:10.1016/j.epsl.2005.06.011,
http://dx.doi.org/10.1016/j.epsl.2005.06.011, 2005.
Knee, K. and Paytan, A.: Treatise and on Estuarine and Coastal and Science, chap. Submarine Groundwater Discharge: A Source of Nutrients,
Metals, and Pollutants to the Coastal Ocean, pp. 206–228, Elsevier, 2011.20
Krest, J. M., Moore, W. S., and Rama: 226Ra and 228 Ra and in the mixing zones of the Mississippi and Atchafalaya Rivers and indicators
of groundwater input, Marine Chemistry, 64, 129–152, doi:10.1016/s0304-4203(98)00070-x, http://dx.doi.org/10.1016/s0304- 4203(98)
00070-x, 1999.
Kwon, E. Y., Kim, G., Primeau, F., Moore, W. S., Cho, H.-M., DeVries, T., Sarmiento, J. L., Charette, M. A., and Cho, Y.-K.: Global estimate
of submarine groundwater discharge based on an observationally constrained radium isotope model, Geophysical Research Letters, 41,25
8438–8444, doi:10.1002/2014gl061574, http://dx.doi.org/10.1002/2014GL061574, 2014.
LaRoche, J., Nuzzi, R., Waters, R., Wyman, K., Falkowski, P. G., and Wallace, D. W. R.: Brown tide blooms in Long Island’s coastal
waters linked to inter-annual variability in groundwater flow, Global Change Biology, 3, 397–410, doi:10.1046/j.1365-2486.1997.00117.x,
http://dx.doi.org/10.1046/j.1365-2486.1997.00117.x, 1997.
Madec, G.: NEMO ocean engine, Institut Pierre-Simon Laplace, 2015.30
Mahowald, N. M., Baker, A. R., Bergametti, G., Brooks, N., Duce, R. A., Jickells, T. D., Kubilay, N., Prospero, J. M., and Tegen, I.:
Atmospheric global dust cycle and iron inputs to the ocean, Global Biogeochemical Cycles, 19, 1–15, doi:10.1029/2004GB002402.,
http://dx.doi.org/10.1029/2004GB002402., 2005.
Mikaloff Fletcher, S. E., Gruber, N., Jacobson, A. R., Doney, S. C., Dutkiewicz, S., Gerber, M., Follows, M., Joos, F., Lindsay, K., Mene-
menlis, D., and et al.: Inverse estimates of anthropogenic CO2 uptake, transport, and storage by the ocean, Global Biogeochemical Cycles,35
20, doi:10.1029/2005gb002530, http://dx.doi.org/10.1029/2005GB002530, 2006.
Milliman, J.: Encyclopedia of Ocean Sciences, chap. River Inputs, pp. 2419–2427, Elsevier BV, doi:10.1006/rwos.2001.0074, http://dx.doi.
org/10.1006/rwos.2001.0074, 2001.
20
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

Moore, W. S.: Oceanic concentrations of 228radium, Earth and Planetary Science Letters, 6, 437–446, doi:10.1016/0012-821x(69)90113-7,
http://dx.doi.org/10.1016/0012-821x(69)90113-7, 1969.
Moore, W. S.: The Effect of Submarine Groundwater Discharge on the Ocean, Annu. Rev. Marine. Sci., 2, 59–88, doi:10.1146/annurev-
marine-120308-081019, http://dx.doi.org/10.1146/annurev-marine-120308- 081019, 2010a.
Moore, W. S.: A reevaluation of submarine groundwater discharge along the southeastern coast of North America, Global Biogeochemical5
Cycles, 24, 1–9, doi:10.1029/2009GB003747, http://dx.doi.org/10.1029/2009GB003747, 2010b.
Moore, W. S. and Dymond, J.: Fluxes of 226Ra and barium in the Pacific Ocean: The importance of boundary processes, Earth and Planetary
Science Letters, 107, 55–68, doi:10.1016/0012-821x(91)90043-h, http://dx.doi.org/10.1016/0012-821x(91)90043-h, 1991.
Moore, W. S. and Shaw, T. J.: Fluxes and behavior of radium isotopes, barium, and uranium in seven Southeastern US rivers and estuaries,
Marine Chemistry, 108, 236–254, doi:10.1016/j.marchem.2007.03.004, http://dx.doi.org/10.1016/j.marchem.2007.03.004, 2008.10
Moore, W. S., Astwood, H., and Lindstrom, C.: Radium isotopes in coastal waters on the Amazon shelf, Geochimica et Cosmochimica Acta,
59, 4285–4298, doi:10.1016/0016-7037(95)00242-r, http://dx.doi.org/10.1016/0016-7037(95)00242-r, 1995.
Moore, W. S., Sarmiento, J. L., and Key, R. M.: Submarine groundwater discharge revealed by 228Ra distribution in the upper Atlantic Ocean,
Nature Geosci, 1, 309–311, doi:10.1038/ngeo183, http://dx.doi.org/10.1038/ngeo183, 2008.
Redi, M. H.: Oceanic isopycnal mixing by coordinate rotation, Journal of Physical Oceanography, 12, 1154–1158, doi:10.1175/1520-15
0485(1982)012<1154:oimbcr>2.0.co;2, http://journals.ametsoc.org/doi/pdf/10.1175/1520-0485%281982%29012%3C1154%
3AOIMBCR%3E2.0.CO%3B2, 1982.
Resplandy, L., Keeling, R. F., Stephens, B. B., Bent, J. D., Jacobson, A., Rödenbeck, C., and Khatiwala, S.: Constraints on oceanic meridional
heat transport from combined measurements of oxygen and carbon, Climate Dynamics, 47, 3335–3357, doi:10.1007/s00382-016-3029-3,
http://dx.doi.org/10.1007/s00382-016-3029-3, 2016.20
Rodellas, V., Garcia-Orellana, J., Masqu, P., Feldman, M., and Weinstein, Y.: Submarine groundwater discharge as a major source of nu-
trients to the Mediterranean Sea, Proceedings of the National Academy of Sciences of the United States of America, 112, 3926–3930,
doi:10.1073/pnas.1419049112, http://dx.doi.org/10.1073/pnas.1419049112, 2015.
Taniguchi, M., Burnett, W. C., Cable, J. E., and Turner, J. V.: Investigation of submarine groundwater discharge, Hydrological Processes, 16,
2115–2129, doi:10.1002/hyp.1145, http://dx.doi.org/10.1002/hyp.1145, 2002.25
Taniguchi, M., Ishitobi, T., and Burnett, W. C.: From Headwaters to the Ocean: Hydrological Change and Watershed Management, chap.
Global assessment of submarine groundwater discharge, pp. 613–617, CRC Press, 2009.
Vancoppenolle, M., Fichefet, T., Goosse, H., Bouillon, S., Madec, G., and Maqueda, M. A. M.: Simulating the mass balance and salinity
of Arctic and Antarctic sea ice. 1. Model description and validation, Ocean Modelling, 27, 33–53, doi:10.1016/j.ocemod.2008.10.005,
http://dx.doi.org/10.1016/j.ocemod.2008.10.005, 2009.30
Wunsch, C.: Discrete inverse and state estimation problems, Cambridge University Press, 2006.
21
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

Table 1. Cost functions
Cost function Formula Residuals
Ordinary least-squares (ols) Cols(x) =
p
P
j=1
([228Ra]mod
i−[228Ra]obs
i)2ηols(x) = [228 Ra]mod
−[228Ra]obs
Logarithmic least-squares (log) Clog(x) =
p
P
j=1
(log[228Ra]mod
i−log[228Ra]obs
i)2ηlog(x) = log[228Ra]mod
−log[228Ra]obs
Proportional least-squares (prp) Cprp(x) =
p
P
j=1
([228Ra]mod
i−[228Ra]obs
i
[228Ra]obs
i
)
2
ηprp(x) = [228Ra]mod
−[228Ra]obs
[228Ra]obs
Table 2. Correlation between data and model concentration fields
Model Linear correlation Logarithmic correlation
Prior 0.383 0.809
ols 0.813 0.883
log 0.797 0.902
prp 0.754 0.877
Table 3. Root mean square of residuals before and after inversion
Model Ordinary residuals Logarithmic residuals Proportional residuals
rms (dpm.m−3) rms (no unit) rms (no unit)
Prior 70.7 0.892 1.91
ols 36.6 0.703 1.10
log 37.9 0.636 1.02
prp 47.4 0.940 0.558
Table 4. Model global 228Ra fluxes (1023 atoms yr−1) with different numbers of source regions
Cost function Case 1: 52 regions Case 2: 38 regions Case 3: 19 regions Case 4: 12 regions Case 5: 7 regions
ols 7.81 −9.93 7.16 −8.14 7.51 −8.49 7.36 −8.16 8.98 −9.78
log 8.09 −8.61 8.01 −8.49 7.90 −8.34 7.85 −8.27 7.93 −8.37
prp 5.15 −5.47 4.96 −5.28 4.70 −4.98 4.25 −4.49 3.75 −4.01
22
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

Table 5. Root mean square of residuals after inversions with different numbers of source regions
Cost function Case 1: 52 regions Case 2: 38 regions Case 3: 19 regions Case 4: 12 regions Case 5: 7 regions
ols 35.836.637.6 38.2 42.9
log 0.623 0.636 0.656 0.671 0.781
prp 0.545 0.558 0.581 0.597 0.665
23
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

Figure 1. Observed concentrations of 228Ra in the global ocean, averaged when available on the ORCA2 cells used for inversion. Data
plotted on subfigures b), c) and d) are also vertically averaged in order to take all layers into account.
24
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

Figure 2. Model 228Ra source regions
25
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

Figure 3. 228Ra annual flux from each source within one standard deviation, after minimization of three cost functions. Prior estimates,
proportional to shelf surfaces, are shown for comparison.
26
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

Figure 4. Model surface 228Ra concentration after minimization of three cost functions. Prior estimate is shown for comparison.
27
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

Figure 5. Surface 228Ra logarithmic residuals after minimization of three cost functions. Prior estimate is shown for comparison.
28
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

Figure 6. (a) Residuals, (b) logarithmic residuals, (c) proportional residuals and (d) model 228Ra concentration as a function of observed
228Ra concentration. Lines of zero residuals are drawn in black.
29
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

Figure 7. Probability Density Functions (PDF) of 228Ra (a) ordinary residuals, (b) logarithmic residuals and (c) proportional residuals after
inversion, compared with a Gaussian PDF (black full line) based on their standard deviations.
30
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

Figure 8. 228Ra sources by basin, divided into riverine input, diffusion from sediment and SGD. Only logarithmic least-squares results are
shown. Fluxes are divided the following way: Regions 1 and 38: Southern; 2−4: South Atl; 5−16: North Atl; 17 −18 and 26 −27: South
Pac; 19 −25: North Pac; 28 −34: Indian; 35 −37: Arctic
31
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.

Figure 9. map of GEOTRACES cruises (from http://www.geotraces.org/cruises/cruise-summary). Planned sections are in red, completed
sections in yellow, International Polar Year sections in black. Names in rectangles correspond to cruises potentially bringing the largest extra
information on 228Ra fluxes because they are performed in places lacking measurements.
32
Biogeosciences Discuss., doi:10.5194/bg-2017-25, 2017
Manuscript under review for journal Biogeosciences
Published: 6 February 2017
c
Author(s) 2017. CC-BY 3.0 License.