Detailed

One of the limiting factors for applying atmospheric

In the present work, we use a similar approach as Griffith et al. (2010) for
Australia and López-Coto et al. (2013) for Europe. We estimate the

The derivations below essentially follow those presented in Dörr and
Münnich (1990), Born et al. (1990), Schüßler (1996), and
Griffiths et al. (2010). They are valid for an infinitely deep unsaturated
homogeneous soil, and we consider only changes of concentration

In this case the equation of continuity in the soil air can be reduced to
one spatial dimension, namely

We further assume that, at any depth in the soil, the only sink process is
radioactive decay, which is described by

The source term

If we consider steady state conditions, i.e. no explicit dependence on time,
Eq. (1) simplifies to

Taking into account only molecular diffusion of the trace gas in the soil
air, we can apply Fick's first law

Combining Eqs. (4) and (5) yields

If we further assume that the

The general solution of the inhomogeneous differential equation Eq. (6a) is

With the boundary conditions of (1) the

Introducing solution (7) into the diffusion Eq. (5), we can calculate
the

Note that the last term in Eq. (8), which allows calculating the

The solution of the differential Eq. (6a) given by Eqs. (7) and (7a) is
only valid if we can assume an infinitely deep unsaturated soil. This
assumption is not always fulfilled. Particularly in northern Europe or in
Siberian wetland areas, the water-table depth can be as close to the surface
as 10 or 20 cm. In that case there is only a very shallow soil depth
available for

The solution Eq. (8a) has the same form as Eq. (8) and for

The role of snow cover on the

Another point concerning the

From Eq. (8) we see that the

The porosity

Different models were developed in the past to estimate

Parameters of the fit curves plotted in Fig. 1, mean exhalation
rates estimated from the measured radon concentration profiles
(

The temperature dependence of the diffusivity for the

Schmithüsen (2012) measured the

Mean vertical profiles of the

For comparison with the measured profile-based diffusivity, we can calculate

Dependency of the

As mentioned in Sect. 2.3, water-table depth can be of huge importance
limiting the

Estimation of bottom-up

The

The equally distributed 843 individual topsoil uranium measurements (median

For calculating the

For those regions of our map, for which the uranium content was not
available in the Geochemical Atlas (e.g. Belarus, Ukraine), we estimated the

Soil texture, i.e. the percentages of sand (0.5–2 mm), silt (0.002–0.5 mm)
and clay (

Soil moisture has a strong impact on the effective diffusivity

Characteristics of the long-term

Both soil moisture data sets were compared to observations from the
International Soil Moisture Network (ISMN;

Soil moisture estimates are only available at lower spatial resolution than
the other (constant) soil parameters described above. In order to apply
internally consistent data sets for the flux estimates, based on the two
different soil moisture models, we use the porosities originally applied in
the respective land surface model to calculate effective diffusivity
according to Eq. (11). Consequently, the different flux maps shown in Figs. 3
and 4 have different spatial resolutions. For flux estimates at higher spatial resolution, i.e. 0.083

In Eq. (13)

Soil temperature estimates are available from both soil models that provide soil moisture for the different depths. For respective flux estimates, we thus used these values to calculate the temperature dependence of diffusivity according to Eq. (12).

While the reduction of the

As in the case of soil moisture, systematic European-wide measurements of
water-table depth that could be used as input for our

In this section, we first present results for a typical year (2006) of our
two

Figure 3 shows the maps and frequency distributions of European

For both soil moisture models, we find in many regions seasonal differences
of the fluxes that are as large as a factor of 2. As mentioned before,
these differences originate from the large changes of soil moisture and thus
soil diffusivity between the drier summer and the, in general, wetter
winter conditions. The frequency distribution of

As both maps use the same

Annual mean

Latitudinal gradient of annual mean

Comparison of the observed climatology of monthly

The mean

The huge differences between the estimates with different soil moisture
input data emphasize the importance of direct comparison of our
process-based

Before comparing with observations at individual sites, we compare the
distribution of annual mean fluxes calculated here based on the two soil
moisture models for 2006–2010 with the other published European maps of
Szegvary et al. (2009) for 2006 and of López-Coto et al. (2013). The
latter is shown as climatology for the years 1957–2002. The maps and
normalized frequency distributions are displayed in Fig. 4. Zonal averages
of 1

The Szegvary et al. (2009) map has lower spatial resolution and less
pronounced hotspots of exhalation rates, but the median of its annual mean
exhalation rates lies between our GLDAS-Noah- and ERA-I/L-based estimates.
However, as Szegvary et al. (2009) used

A large number of systematic direct

Figure 6 shows the climatology of the monthly mean

Figure 6 also shows calculated exhalation rates (according to Eq. 8) based
on the measured soil parameters listed in Table 2 and the climatology of
soil moisture for the Heidelberg pixel as calculated from the two soil
moisture models for the years 2006–2010. Note that for these process-based
calculations the GLDAS Noah model used a porosity of

Upper panels of each row: Comparison of estimated

As demonstrated in the previous section, proper validation of our

For most sites shown here, the ERA-Interim/Land-based

At Pallas station in northern Finland, no direct

A station with very shallow water table is Lutjewad, located at the
Netherland's North Sea coast. Not taking into account ground water table
restriction in the modelled

At Gebesee, co-located soil moisture measurements are available. They agree
very well with the GLDAS-Noah-based model estimates (Fig. 7c); further,
GLDAS-Noah-based

As has been indicated already in Fig. 6, the GLDAS-Noah-based estimates for
M5-Nußloch are slightly higher than observations, while the ERA-I/L-based
estimates underestimate the observations by about a factor of 2 (Fig. 7d).
Note, however, that in the current comparison, contrary to the results shown
in Fig. 6, we use for both modelled fluxes all parameters, including

At Gif-sur-Yvette, all models except for GLDAS Noah seem to reproduce well
at least the annual mean observed fluxes (Fig. 7e). However, the seasonal
amplitude seems to be best captured by the ERA-I/L-based and the
GLDAS-Noah-based estimates, whereas the Szegvary et al. (2009) model for
2006, if also valid for other years, and the López-Coto et al. (2013)
model underestimate the seasonal amplitude. GLDAS-Noah-based fluxes are
larger than observations by about a factor of 2. This is very surprising because

At Binningen, Switzerland, which is the measurement station that Szegvary et
al. (2009) also used for the empirical

In summary, we conclude that at three out of six stations the (generally
higher) GLDAS Noah soil-moisture-based

Map of episodic

Since only very few systematic

Temporal and spatial variations of soil moisture have a huge influence on
the effective diffusivity in the soil and thus on the

Even if we had good estimates of soil moisture, we need to keep in mind that
the Millington-Quirk (1960) model used in this study does not necessarily
describe effective diffusivity in the unsaturated soil zone correctly.
Comparison of model-based diffusivity with diffusivity estimated from
observed

An important parameter determining the

Besides the

One basic assumption in our estimate is homogeneity of soil parameters with
depth up to 1m. While the differences of

For consistency, we use in our European

In our sensitivity test with a very simple parameterization of frost and/or
snow conditions,

The overall uncertainty of

Inspecting now our comparison between single measurements and modelled flux
for the respective pixel (Fig. 8), the IQR of the differences were on the
order of 11 mBq m

A high-resolution

Using two different sets of soil moisture reanalyses underlines the strong
dependence of

The spatial resolution of the soil moisture models used here restricts
spatial resolution of the two realizations of our European

Validation of our estimated

It would also be interesting to apply our approach to other areas of the
world, which would allow for comparison with maps developed for areas outside of Europe, using different methodologies, e.g. of diffusivity estimation. We
decided to leave this effort to future work, also because of
non-availability of a systematic

The presented

Although we currently favour the GLDAS-Noah-based

Digital versions of the maps are available at

The research leading to these results has received funding from the European
Community's Seventh Framework Programme (FP7/2007-2013) in the InGOS project
under grant agreement n