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 c(z,t), flux j(z,t) as well as source Q(z,t) or sink strength S(z,t) in the vertical direction z (with the z coordinate defined as positive downwards and z=0 at the soil–atmosphere interface).

In this case the equation of continuity in the soil air can be reduced to one spatial dimension, namely dc(z,t)dt+∂j(z,t)∂z=Q(z,t)+S(z,t).

We further assume that, at any depth in the soil, the only sink process is radioactive decay, which is described by S(z,t)=-λc(z,t), with the decay constant λ(222Rn) = 2.0974 × 10-6 s-1.

The source term Q, i.e. the production rate of 222Rn gas in the soil, is calculated according to Schüßler (1996) from Q(z)=λρb(z)cRa(z)ε(z), with ρb the dry bulk density of the soil (kg m-3), cRa the 226Ra activity concentration in the soil material (Bq kg-1), and ε the 222Rn emanation coefficient, which is defined as the probability that a 222Rn atom produced in a soil grain can actually escape into the soil air.

If we consider steady state conditions, i.e. no explicit dependence on time, Eq. (1) simplifies to 0=-∂j(z)∂z+Q(z)+S(z)=-∂j(z)∂z+Q(z)-λc(z)or∂j(z)∂z=Q(z)-λc(z).

Taking into account only molecular diffusion of the trace gas in the soil air, we can apply Fick's first law j(z)=-De∂c(z)∂z, where De is the effective diffusion coefficient of the trace gas in the soil air (hereafter also named effective diffusivity or simply diffusivity). De is assumed to be constant with depth. Note that in Eq. (5), j(z) is the flux per unit area of the bulk soil. This is not immediately obvious. In the respective Eq. (1) in Griffiths et al. (2010) and also in Sun et al. (2004), j(z) is multiplied with θP, the porosity of the soil to yield the flux density per bulk unit area. However, they also use a different expression to calculate the source strength Q in the soil (Eq. 3 in Griffiths et al., 2010) where they divide our Eq. (3) by θP.

Combining Eqs. (4) and (5) yields ∂2c(z)∂z2-λDec(z)=-Q(z)De.

If we further assume that the 226Ra activity concentration in the soil particles, the 222Rn emanation coefficient, and the soil bulk density are constant with depth, we obtain a depth-independent source strength, i.e. Q(z)=Q, and Eq. (6) becomes De∂2c(z)∂z2-λc(z)+Q=0.

The general solution of the inhomogeneous differential equation Eq. (6a) is c(z)=c∞(1-e-zz¯), where c∞ is the asymptotic concentration at large depths and z¯ the relaxation depth.

With the boundary conditions of (1) the 222Rn concentration approaching zero at the soil–air interface and (2) zero concentration gradient, i.e. equilibrium between 222Rn production and decay, at great depths c(z=0)=0anddcdz(z=∞)=0 the solution of Eq. (6a) takes the following form: c(z)=c∞(1-e-zz¯)=Qλ(1-e-λDez) i.e.c∞=Qλ and z¯=Deλ.

Introducing solution (7) into the diffusion Eq. (5), we can calculate the 222Rn flux at the soil surface j(z=0)=-De∂c(z)∂zz=0=-Dec∞z¯=-z¯c∞λ=-QDeλ=-ρbcRaεDeλ.

Note that the last term in Eq. (8), which allows calculating the 222Rn flux density per unit bulk surface of the soil from “bottom-up” parameters and the effective diffusivity in the soil, is now identical to Eq. (4) in Griffiths et al. (2010).

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 222Rn production and exhalation into the atmosphere (if we consider that the molecular diffusion coefficient of 222Rn in water is lower by 2–3 orders of magnitude compared to air, and that there is only negligible 222Rn flux from ground water into the unsaturated soil zone). With the boundary conditions of zero 222Rn activity concentration at the soil–air interface and zero 222Rn flux, i.e. zero concentration gradient, at water-table depth zG c(z=0)=0 and dcdz(z=zG)=0, the solution of the differential Eq. (6) gives the modified flux at the surface according to j(z=0)=-QDeλtanh⁡zGz¯.

The solution Eq. (8a) has the same form as Eq. (8) and for zG≫z¯ it yields Eq. (8).

The role of snow cover on the 222Rn exhalation rate is not yet fully understood. Robertson (2004) found in her measurements that a layer of snow had no significant influence on the 222Rn exhalation rate. However, when the top layer of the snow melted and froze again, a smaller 222Rn exhalation rate was measured. This finding suggests that the physical properties of the snow, such as a thin ice layer on its top, determine the magnitude of the 222Rn flux. However, most of the studies cited in Robertson (2004) found no or merely a small effect of snow cover on 222Rn exhalation rate. Thus, although a shielding effect of snow cover has been included in the López-Coto et al. (2013) flux map, this effect is not taken into account in our 222Rn flux estimates.

Another point concerning the 222Rn exhalation rate in winter months is the influence of frozen soils on 222Rn exhalation rates. While different authors, e.g. cited by Robertson (2004) report a reduction in 222Rn flux when the soil was frozen, Robertson (2004) found no evidence for a strong influence of frozen soils on 222Rn emissions. However, particularly when soil moisture is high or when an ice layer forms on the ground, this might cause a substantial decrease in 222Rn exhalation rates. Because no systematic analysis of the influence of soil freezing on the 222Rn flux is available, our standard 222Rn flux maps do not take into account any positive or negative effect of frozen soil on the exhalation rate. However, we will show one hypothetical scenario of the potential influence of frost on the exhalation rate with reduced fluxes, based on the number of ice days during winter months (Sect. 4.3).

From Eq. (8) we see that the 222Rn flux at the soil surface not only depends on the production rate Q in the soil (see Eq. 3), but also on the effective diffusivity De of 222Rn in the soil air. Estimating De in soil air is, however, not a trivial task. This parameter depends mostly on the percentage of soil air volume available for gas diffusion, but also on the grain size distribution of the soil, i.e. its texture. The unit volume of soil consists of the soil material fraction θm, the fraction that is filled with water θw, and the air-filled fraction θa so that θm+θw+θa=1.

The porosity θp of the soil is defined as θp=1-θm=θa+θw.

Different models were developed in the past to estimate De depending on soil properties and soil moisture. While the more recent models by Moldrup et al. (1996, 1999) require as input detailed parameters of the soil texture, i.e. percentages of clay, coarse sand, and fine sand, the earlier models by Millington and Quirk (1960, 1961) and also the parameterization reported by Rogers and Nielson (1991) only require information on soil porosity and soil moisture. The latter parameterization by Rogers and Nielson (1991) has been used by Zhuo et al. (2008), Griffiths et al. (2010) and López-Coto et al. (2013) in their 222Rn flux estimates. However, Jin and Jury (1996) could show that the original estimate of the effective diffusivity according to Millington and Quirk (1960), i.e. De=Daθa2θp23=Daθp-θw2θp23, where Da=1.1×10-5 m2 s-1 is the diffusion coefficient of radon in air, yields excellent agreement with a large set of available observational data of the effective diffusivity De for soils with different texture obtained from different studies in the literature (Jin and Jury, 1996, and references therein). Moreover, when comparing diffusivity calculated from the Millington and Quirk (1960) model with that of Moldrup et al. (1996), both agree very well (and for a hydraulic parameter b=6, which corresponds to a typical soil with about 20 % clay, they yield identical values of De (see Fig. S1 of the Supplement)). More importantly, when comparing measured 222Rn profile-based diffusivity values calculated from Eq. (7a) (see Sect. 3, Table 1) with the model-estimated results, we find the best agreement with these two models (Millington and Quirk, 1960; Moldrup et al., 1996). The Rogers and Nielson (1991) model seems to overestimate diffusivity, particularly during dry conditions, and the Moldrup et al. (1999) model largely underestimates the measured diffusivity. Therefore, we decided to use the Millington and Quirk (1960) model (Eq. 11), which is solely based on soil porosity and soil moisture, to estimate effective diffusivity.

The temperature dependence of the diffusivity for the 222Rn flux map has been estimated according to Schery and Wasiolek (1998): DeT=De0T273K32, with T the mean soil temperature in Kelvin and De0 the effective diffusivity at the reference temperature 273 K.

Schmithüsen (2012) measured the 222Rn exhalation rate and corresponding vertical concentration profiles of 222Rn in a loamy soil close to the Institut für Umweltphysik (IUP) in Heidelberg, Germany. These measurements provide a first validation of the theoretical concept described in Sect. 2, which is used for estimating the 222Rn exhalation rates in Europe from bottom-up data. Measured concentration profiles were binned into mean profiles for dry (nominal θw=0.124, actual range 0.098–0.145), medium dry (nominal θw=0.199, actual range 0.160–0.239), and wet (nominal θw=0.311, actual range 0.264–0.345) soil moisture conditions (Fig. 1). The ranges of the soil moisture classes resulted from a roughly equal distribution of all measured soil moistures (in the upper 20 cm of the soil) during the course of 1 year. By fitting a curve according to Eq. (7) to the mean profile data, one obtains the parameters z¯ and c∞ as well as values for the 222Rn source strength Q and the effective diffusivity De, exp (Table 1). The values for Q, which should be the same for all three moisture situations (wet, medium, dry), indeed agree rather well (i.e. to within ±25 %). The 222Rn exhalation rate at the soil surface (jprofile) calculated according to Eq. (8) from the parameters fitted to the measured profiles as well as the mean exhalation rates jchamber independently measured using accumulation chambers are also listed in Table 1. They agree within a factor of 2 for all three soil moisture regimes and within 15 % for the annual mean flux.

For comparison with the measured profile-based diffusivity, we can calculate De with the Millington and Quirk (1960) model De,  MQ from measured porosity (θp=0.368) and measured mean soil moistures according to Eq. (11). The diffusivity was adjusted to the mean soil temperatures during the measurement dates for wet, medium and dry conditions according to Eq. (12). Likewise, we use the Rogers and Nielson (1991) model (their Eq. 19) to estimate De,  RN. The numbers of De,  MQ and De,  RN are given in the last two columns of Table 1. At our Heidelberg IUP sampling site the Millington and Quirk (1960) model underestimates diffusivity during wet and dry conditions by up to 25 %, while it overestimates diffusivity during medium dry conditions by about a factor of 2. However, the discrepancies between the diffusivity calculated with the Rogers and Nielson (1991) model and the experimental results are larger at wet and medium dry conditions, while they fit very well at dry conditions (θw < 0.15). Using an average Q=23.6 mBq m-3 s-1 from the measured profiles and the respective De,  i from Table 1, we also estimated Millington and Quirk- and Rogers and Nielson-based soil profiles according to Eqs. (7) and (7a). These profiles are plotted in Fig. 1 for comparison to the observations. Again the Millington and Quirk model fits the observations better than the Rogers and Nielson model. Hence, we favour the Millington and Quirk (1960) model (i.e. Eq. 11) for estimating moisture-dependent diffusivities for all European soils.

As mentioned in Sect. 2.3, water-table depth can be of huge importance limiting the 222Rn exhalation when it rises to levels that are of the same order as z¯. These situations are quite frequent in coastal areas, e.g. of northern Germany or the Netherlands, or in wetland regions. Measurements of co-located 222Rn exhalation rates and shallow water-table depths are available from a field site in Federovskoye, western Russia (Levin et al., 2002). Thus, we can test the validity of Eq. (8a) and compare the solution with the measurements from the Federovskoye transect measurements from Levin et al. (2002, their Fig. 3). The solid lines in Fig. 2 are estimates of the 222Rn exhalation rate for a soil with a mean source strength Q=12 mBq m-3 s-1 and relaxation depths z¯ of 0.2, 0.35 and 0.5 m (roughly corresponding to wet, medium and dry soil moisture conditions). The parameterization with the water table limitation reproduces the observed relation reasonably well and was thus applied to all areas with shallow water-table depth.

The 226Ra activity concentration in soils is the governing parameter for the 222Rn flux at the soil surface. It scales linearly with the exhalation rate. The Geochemical Atlas of Europe (Salminen, 2005) summarises results of a European-wide effort within the FOREGS (Forum of European Geological Surveys) Geochemical Baseline Mapping Programme to provide high quality environmental geochemical baseline data for European stream waters, sediments and soils. Besides many other elements and trace constituents, the uranium content was also measured in regularly distributed topsoil and subsoil samples from 26 European countries. Topsoil samples were collected at 0–25 cm depth (with a potential overlying humus layer being removed), while subsoil samples were collected from another 25 cm layer located between 50 and 200 cm depth. Uranium content was measured on residual soil samples (from the < 2 mm grain fraction, with total organic matter (TOC) being removed from these samples) and is reported in mg uranium per kg residual soil. As total uranium in soil material consists of ca. 99 % of 238U, the values given in the Geochemical Atlas (Salminen, 2005) can be directly transferred into 226Ra activity concentrations, when assuming secular equilibrium between 238U and its daughter 226Ra. The conversion factor from uranium concentration to 238U activity concentration was taken from IAEA (1989), i.e. 12.35 Bq kg-1 per mg kg-1 uranium.

The equally distributed 843 individual topsoil uranium measurements (median ± standard deviation: 2.03 ± 2.35 mg kg-1) and the 792 subsoil uranium measurements (median ± standard deviation: 2.00 ± 2.34 mg kg-1) were interpolated by ordinary kriging (e.g. Wackernagel, 2003) for both layers to the 0.083∘ × 0.083∘ grid of our map (see Fig. S2 of the Supplement). The resolution of our basic map is restricted in its spatial resolution by that of the global soil texture map of Reynolds et al. (2000), which we used to determine soil texture parameters (see Sect. 4.2). As the uranium content was measured on residual soil samples with total organic carbon being removed, we corrected the activity concentrations for “dilution” with organic carbon, using the TOC data that have also been reported in the Geochemical Atlas of Europe (Salminen, 2005). This correction is small with typical TOC values in topsoil between 0 and 6 % (median ± standard deviation: 1.73 ± 3.18 %) and in subsoil between 0 and 3 % (median ± standard deviation: 0.40 ± 2.86 %).

For calculating the 222Rn exhalation rates for each pixel, we used the mean values of topsoil and subsoil from the TOC-corrected interpolated 226Ra activity concentrations (i.e. cRa of Eq. 3). Assuming a depth-constant cRa seems to be well justified in view of the very good agreement between topsoil and subsoil uranium concentrations reported in the Geochemical Atlas of Europe (Salminen, 2005).

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 226Ra activity concentration based on geological information available from the high-resolution global lithological map “GLiM” (Hartmann and Moosdorf, 2012). First, a median 226Ra activity concentration was computed for each lithological class in GLiM using the measured uranium content at all sampling sites together with co-located GLiM data. The resulting relation was then used to extrapolate the 226Ra activity concentration map to the regions not covered by the Geochemical Atlas. Due to this very indirect approach, the resulting 222Rn exhalation rates will have a much higher uncertainty in these regions (hatched area in Fig. S2 of the Supplement).

Soil texture, i.e. the percentages of sand (0.5–2 mm), silt (0.002–0.5 mm) and clay (< 0.002 mm) for our 222Rn exhalation map have been taken from Reynolds et al. (2000), a soil database that is frequently used in modelling studies of similar problems. Porosity and soil bulk density were computed from soil texture according to Saxton et al. (1986). The data are given at a horizontal resolution of 0.083∘ × 0.083∘ and for two different depth intervals (from 0–30 cm and from 30–100 cm). Here we use weighted mean values for 0–100 cm depth for all parameters. As has been shown by Zhuo et al. (2006, 2008), the emanation coefficient ε, i.e. the likelihood of a newly formed 222Rn atom to escape the grain and reach the air-filled soil volume, depends on the soil type and on soil moisture. The soil moisture dependency is, however, only relevant at very small moisture content below 15 % water saturation (i.e. at θw < 0.06 for a typical porosity of θp = 0.4). Outside this range ε was shown to be largely constant (Zhuo et al., 2006). For simplicity and because water contents below 15 % saturation are very rare in European soils, we used constant (saturation) values for each texture class. We also neglected the temperature dependence of ε, as it changes by only a few percent within a temperature range of 0–20 ∘C (Iskandar et al., 2004). The numbers to calculate εsat for sand, silt, and clay are given in Zhuo et al. (2008) in their Table 2. The values must, however, be exchanged, as was noted by Griffiths et al. (2010) and confirmed by W. Zhuo (personal communication, 2013). From this we estimated εsat= 0.285 for sand, εsat= 0.382 for silt, and εsat= 0.455 for clay. These numbers are well in accordance with emanation coefficients determined by Schüßler (1996) from measured 222Rn profiles and known 226Ra contents in different soils of the surroundings of Heidelberg (M1–M5, see Sect. 5.3 and Table 2). We used weighted mean values for the different texture classes to estimate the emanation coefficients for each pixel of our map.

As in the case of soil moisture, systematic European-wide measurements of water-table depth that could be used as input for our 222Rn exhalation map are not existing. Hence, we use data from a hydrological model simulation by Miguez-Macho et al. (2008). Supplementary Fig. S4 shows the influence of low water-table depth on 222Rn fluxes for Europe. Large areas of the Netherlands, northern Italy, and Hungary with water table above 2 m are affected. For these areas, the 222Rn exhalation rate was reduced according to our estimation described in Sect. 2.3.

Figure 3 shows the maps and frequency distributions of European 222Rn fluxes as estimated with the model parameters described in Sect. 4, applying the two different soil moisture model estimates (GLDAS Noah (left panels) and ERA-Interim/Land (central panels)) for January (top panels) and July 2006 (middle panels). Both maps show some areas of very high 222Rn exhalations rates, most pronounced in July, which coincide with the areas in Europe where the 226Ra activity concentration in the upper soil layer is very high. These areas concern for example the Massif Central in southern France, the Iberian Peninsula and areas in central Italy (compare 226Ra distribution displayed in Fig. S2 of the Supplement).

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 222Rn fluxes, displayed in the lower part of Fig. 3, is most confined during winter (January 2006) and when calculated with the ERA-Interim/Land soil moisture data; these fluxes also show a low median value of only 5.83 mBq m-2 s-1 (interquartile range (IQR) = 5.38 mBq m-2 s-1). This is about half of the median flux estimated with the GLDAS Noah soil moisture data set for January 2006 (12.08 mBq m-2 s-1). During summer (July 2006), both frequency distributions of fluxes are broader than during winter (IQR: ERA-I/L = 8.39 mBq m-2 s-1 and GLDAS Noah = 11.47 mBq m-2 s-1). The median values are much larger than in January 2006, i.e. in the case of the ERA-I/L soil moisture being more than a factor of 2 larger, while the difference of the medians in July 2006 between the two maps is much smaller than in winter (only about 30 %).

As both maps use the same 226Ra distribution and also the same 222Rn emanation coefficient (i.e. the same 222Rn source term), differences of 222Rn flux of the two maps are solely due to the differences of diffusivity, which we calculate from modelled soil moisture using the individual soil porosity data from the two models (according to Eq. 11). The right panels in Fig. 3 show the flux differences between the two maps for January and July 2006. In fact, the differences of fluxes between the two maps are not homogeneous all over Europe, but they show a distinct north to south gradient. While fluxes estimated with ERA-I/L soil moisture for January 2006 are slightly higher than those estimated based on GLDAS Noah in Sweden, Denmark and some parts of northern Germany and Poland, they are much smaller than GLDAS-Noah-based fluxes in central and southern Europe. The differences in soil porosity in the two models are only small (i.e. ERA-I/L uses about 10 % smaller porosity in northern than in central Europe, while porosity is pretty homogeneous all over Europe in GLDAS Noah and similar to ERA-I/L in central Europe) but very distinct differences are found in the soil moisture distributions. Soil moisture is much lower in the GLDAS Noah model estimates for central and southern Europe than in ERA-I/L. Only in some areas of Scandinavia and the northern coasts of central Europe, ERA-I/L estimates lower soil moisture than GLDAS Noah. This directly translates into higher 222Rn fluxes in the mentioned regions of Scandinavia.

The mean 222Rn flux for the period 2006–2010 is estimated to be 10 mBq m-2 s-1 (ERA I/L soil moisture) or 15 mBq m-2 s-1 (GLDAS Noah soil moisture) with mean seasonal variations ranging from 7 mBq m-2 s-1 in February to 14 mBq m-2 s-1 in August (ERA I/L soil moisture) and from 11 mBq m-2 s-1 in March to 20 mBq m-2 s-1 in August (GLDAS Noah soil moisture).

The huge differences between the estimates with different soil moisture input data emphasize the importance of direct comparison of our process-based 222Rn flux estimates with measured fluxes, in order to find out, which soil moisture model would better fit real ambient conditions. This comparison is shown below in Sect. 5.4 and 5.5.

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∘ latitudinal bands are compared in Fig. 5. The general shape with higher 222Rn exhalation rates in regions of high 238U activity concentrations (e.g., on the Iberian Peninsula) is similar in all four maps. The difference between GLDAS-Noah- and ERA-I/L-based fluxes, with generally higher fluxes estimated based on the GLDAS Noah soil moisture model (except for some areas in northern Europe), was discussed before for January and July 2006 (Fig. 3) and is also visible in annual mean flux estimates. The annual median values for the 2006–2010 period differ by more than 50 % (Fig. 4, lower four panels). There is relatively good agreement in the spatial pattern, in the annual medians and IQRs between the ERA-I/L and the López-Coto et al. (2013) map. This is because the basis of the López-Coto et al. (2013) map is also the 238U distribution from the Geochemical Atlas of Europe (Salminen, 2005), and López-Coto et al. (2013) use a similar process-based soil transport model as described here, but the parameterization for diffusivity developed by Rogers and Nielson (1991). Soil moisture estimates in López-Coto et al. (2013) are from ERA-40 reanalyses, which are based on an earlier version of the land surface model than used in ERA-I/L. Soil moistures in ERA-40 show an overall smaller variability than the ERA-I/L model estimates (Balsamo et al., 2015) used in our study (compare also Sect. 5.4, which discusses time profiles in comparison to observations). The maps of differences between our study and the López-Coto et al. (2013) climatology are displayed in Fig. S5 in the Supplement. While our GLDAS-Noah-based estimates are higher than López-Coto et al. (2013) throughout Europe (with the exception of northern Ireland and a few areas in Italy) the higher fluxes of our ERA-I/L-based estimates compared to López-Coto et al. (2013) are most prominent in Scandinavia. Differences in annual mean 222Rn fluxes between these two maps are small in central Europe. The difference in annual fluxes in regions north of 60∘ N (Fig. 5) might, at least to some extent, be caused by the reduction of 222Rn fluxes in snow-covered regions, which is included in the flux map of López-Coto et al. (2013) but not in our standard estimates. Including a restriction during frozen soil conditions in our flux estimates (orange and cyan lines in Fig. 5) reduces the difference of the annual mean in this region, but they are still more than 50 % higher than López-Coto et al. (2013). However, it is important to keep in mind that López-Coto et al. (2013) use a ca. 40 % smaller emanation coefficient of 0.2 for all soils, compared to a median value of 0.35 in our study. This difference is responsible for a generally 40 % lower 222Rn exhalation rate in the López-Coto et al. (2013) map than estimated for our two maps.

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 γ-dose rate observations and an empirical correlation with measured 222Rn fluxes, their fluxes are significantly different in certain areas of Europe. In particular, the pronounced maximum in the French Massif Central, where high 238U concentrations are measured in the soils (Fig. S2), is only slightly visible in the Szegvary et al. (2009) map. A detailed picture of the differences between our maps and the Szegvary et al. (2009) estimate for 2006 is shown in the Supplement (Fig. S5). Largest differences are seen in central Europe, where our GLDAS-Noah-based estimates are in many places larger than the Szegvary et al. (2009) estimates by a factor of 2, while ERA-I/L-based estimates are often about 50 % smaller compared to the Szegvary et al. (2009) estimates. In northern Scandinavia our two estimates are higher than the Szegvary et al. (2009) map. The reason might be the shielding effect of snow cover on the observed γ-dose rate (Szegvary et al., 2007a). Including the frost restriction in our flux estimates reduces the difference of the annual mean in this region but leads to values lower than in the Szegvary et al. (2009) map in southern Scandinavia (Fig. 5).

A large number of systematic direct 222Rn flux measurements using the accumulation chamber technique were carried out in the 1980s and 1990s at five sampling sites south of Heidelberg, Germany. Dörr and Münnich (1990) started these measurements in 1984 at a sandy soil site (M1) as well as at a clay-loam soil site (M4). Schüßler (1996), who sampled additional sites close to the earlier plots from Dörr and Münnich (1990), continued measurements on these plots. The soil parameters of the five sampling sites M1–M5 are listed in Table 2. For these sites we estimated the percentages of clay, silt, and sand according to Cosby et al. (1984, Table 2) from the soil type descriptions given by Schüßler (1996). The soil properties of other IUP sampling sites studied by Schell (2004; Gebesee), Schmithüsen (2012; IUP) and Schwingshackl (2013; Gif-sur-Yvette (Gif)) at locations in Germany and France are also listed in Table 2. In addition, the soil parameter values of the 0.083∘ × 0.083∘ pixels from the high-resolution soil parameter map, in which the measurement sites are located, are listed. From comparison, we can assess the representativeness of the measurement sites for their corresponding pixel of the map. While the Sandhausen sites M1–M3 are not at all representative for the corresponding map pixel, the soil texture and 226Ra activity concentration of the loamy sites M4 and M5 as well as the IUP site, discussed already above (Sect. 3.1), are well comparable with the map pixel. The latter are thus suitable for validation of our maps and the transport model approach. For Gebesee in northern Germany, actual site parameters agree well with the soil parameters of the map. Only the 226Ra content is about 20 % lower in the map than measured by Schwingshackl (2013). Contrary, for Gif-sur-Yvette in France porosity, bulk density and 226Ra activity concentration are significantly different from the pixel values. This should be kept in mind when comparing our process-based maps with these observations.

Figure 6 shows the climatology of the monthly mean 222Rn exhalation rates measured at the Heidelberg M1–M5 stations over the periods of 1987–1995 (M1, M2, M4) and 1987–1998 (M3, M5). Jutzi (2001) calculated these averages from the individual data of regular 1–2-weekly flux measurements reported by Schüßler (1996). The strong dependency of the mean exhalation rate on soil type is clearly visible. The clay or loamy soils (M4 and M5) show the highest fluxes with significant seasonal variations of the exhalation rate with up to a factor of 2 larger values in July/August compared to January/February. In contrast, the seasonality at M1 and M3 is only very weak, and fluxes at the sandy sites (M1–M3) are about 3 times lower than at M4 and M5.

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 θp= 0.434 in the map pixels while the ERA-Interim/Land model used θp= 0.439, i.e. both significantly different from measured porosities, in particular at the sites with sandy soils (M1 and M3, Table 2). For our calculations, we thus individually adjusted the soil moistures for all sites M1–M5 according to Eq. (13) to better approximate the pore volumes available for diffusion at the different sites. With these adjustments, the flux estimates based on GLDAS Noah soil moisture agree very well with observations for the sites M1–M3, but are about 30 % too high for the stations M4 and M5. When using modelled ERA-I/L soil moisture data, estimated mean seasonal 222Rn fluxes are always lower than observations, by up to a factor of 3 at M1 and M3 and by about a factor of 2 at the loamy and clay sites M2, M4, and M5. Without adjustment of modelled soil moisture to the site porosities, for all sites and both soil moisture estimates, modelled 222Rn fluxes would be up to a factor of 6 too low (results not shown in Fig. 6). From this comparison of process-based estimates with long-term observations, we can conclude that (1) the agreement between estimates and observations strongly depends on the validity of soil texture parameters used in the map; (2) modelled soil moisture values need to be adjusted to the local porosity according to Eq. (13), if reliable flux estimates shall be calculated; (3) in the Heidelberg pixels associated to M1–M5, GLDAS-Noah-based 222Rn flux estimates agree rather well with existing observations, while ERA-I/L-based estimates largely underestimate fluxes at all sites. This comparison also emphasizes that quantitative validation of our 222Rn exhalation map can be misleading, if information on soil properties is missing at the measurement sites.

As demonstrated in the previous section, proper validation of our 222Rn flux estimates requires comparison with direct measurements carried out on soils representative for the respective pixel of the map. However, systematic 222Rn flux measurements in Europe are very sparse so that we include in this section all sites (except for M1–M4 that have already been discussed before, see Sect. 5.3), which have observations available to us over the course of at least 4 months. Figure 7 compares estimates from our 222Rn flux map based on the two soil moisture models GLDAS Noah (red lines: standard, orange: with frost restriction), ERA-I/L (blue lines: standard, cyan: with frost restriction) with those from Szegvary et al. (2009: dark green lines), from López-Coto et al. (2013: light yellow-green lines) and with observations (black dots). Note that in case the observations do not fall into the modelled time span of 2006–2008 displayed here, the data points have been repeated as climatology for all years. If the dotted red and blue lines can be distinguished, they show the effect of shallow water-table depth. Fluxes that are not restricted by the water table, contrary to those that are restricted, are then visible as dotted (red and blue) lines (relevant at Lutjewad and Gebesee where the water table is less than 2 m below the soil surface); otherwise, the solid and dotted lines fall onto each other. Figure 7 also shows the soil moisture estimates calculated by the two land surface models as well as direct soil moisture measurements in different depths, if available.

For most sites shown here, the ERA-Interim/Land-based 222Rn fluxes (plotted in blue and cyan) are significantly lower (often by more than a factor of 2) than those estimated with the GLDAS Noah soil moisture data (plotted in red and orange). Accordingly, ERA-I/L soil moisture estimates are significantly higher than those estimated by GLDAS Noah at these sites; note that porosities do not differ very much in between models at these sites, with a maximum difference of 6 % at Gebesee. Only at Lutjewad the two flux estimates are similar despite the high soil moisture in ERA-I/L; here also the porosity in the ERA-I/L model is by almost a factor of 2 higher than in GLDAS Noah. At all sites except for Gif-sur-Yvette and Lutjewad, ERA-I/L-based fluxes are significantly lower than observed fluxes.

At Pallas station in northern Finland, no direct 222Rn flux measurements are available. For this reason, we use flux estimates derived from summer observations in the atmosphere and atmospheric transport modelling (Lallo et al., 2009). For this time of the year, the GLDAS-Noah-based 222Rn flux results compare best with the data. For the winter months, López-Coto et al. (2013) predict very low fluxes at Pallas, and here the effect of frost restriction on GLDAS-Noah- and ERA-I/L-based estimates becomes visible (difference between red and orange as well as blue and cyan lines, respectively, in Fig. 7a).

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 222Rn exhalation rate (dotted lines in Fig. 7b) would largely overestimate the flux in both approaches by more than a factor of 4. Here the Szegvary et al. (2009) and the López-Coto et al. (2013) models overestimate observed fluxes by more than a factor of 2–3. Taking into account the restriction due to the shallow water table brings the modelled 222Rn exhalation rate closer to the observations but also reduces the amplitude of the seasonal variations. Note that ERA-I/L-based and GLDAS-Noah-based fluxes are almost identical under water table restriction and therefore hardly distinguishable in Fig. 7b.

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 222Rn fluxes fit the observations very well. Here again, the water-table depth flux restriction turns out to be important. Estimated GLDAS-Noah-based fluxes not restricted by water-table depth are significantly higher in early summer than observed fluxes (dotted red line in Fig. 7c), but those restricted by water table agree, on average, well with observations. At the end of the summer, local water-table depth may be deeper than in winter and spring, which is why observations then seem to fall on the unrestricted GLDAS-Noah-based model estimates.

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 226Ra activity concentration and soil porosity, from our map and not from observations. Although absolute fluxes are not perfectly reproduced, both of our models seem to capture the seasonal amplitude of observations much better than estimates by Szegvary et al. (2009) and López-Coto et al. (2013) models. The good agreement between GLDAS-Noah-based and observed 222Rn fluxes at M5 is accompanied by good agreement of GLDAS-Noah-modelled soil moisture and respective observations. Soil moisture data plotted for M5 do not exactly stem from the M5 site but are taken from a soil monitoring station north of Heidelberg at Grenzhof (Wollschläger et al., 2009). Modelled soil moistures as well as soil properties in the grid cells corresponding to the location of M5 and Grenzhof are identical in GLDAS Noah and very similar in ERA-I/L.

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 226Ra activity concentration of the map pixel is a factor of 2 smaller than those measured by Schwingshackl (2013) (see Table 2). From this difference alone, we would expect an underestimation of Gif-sur-Yvette flux observations by both of our flux estimates. On the other hand, the shallow water table at the measurement site (Campoy et al., 2013) might restrict the 222Rn fluxes. This situation is not represented in our maps, where the water table is well below 10 m in this region.

At Binningen, Switzerland, which is the measurement station that Szegvary et al. (2009) also used for the empirical γ-dose rate-based estimates of their 222Rn flux map for 2006, their measured data fall in between our GLDAS-Noah- and ERA-I/L-based fluxes (Fig. 7f). Only in spring 2007 both of our estimates are higher than the observed fluxes. Soil moisture estimates in both reanalyses are most of the time lower than the observations but capture the temporal variation rather well. In 3 summer months of 2006, the Szegvary et al. (2009) model estimates are slightly lower than the observations, while the López-Coto et al. (2013) model results are considerably lower than all other model estimates and lower than the observations by at least a factor of 2.

In summary, we conclude that at three out of six stations the (generally higher) GLDAS Noah soil-moisture-based 222Rn exhalation rates are in good agreement with observations. At two of these sites, where we have data available, this correlates with good agreement of model-calculated and observed soil moisture. Flux estimates based on ERA-Interim/Land soil moistures have the tendency to underestimate observed fluxes and only fit well at one of our comparison sites (Gif-sur-Yvette). The two published maps, in particular that developed by López-Coto et al. (2013), generally underestimate measured fluxes with the exception of the coastal site Lutjewad. There the shallow water-table depth is not taken into account in these models, which leads to large over-estimation. Concerning the seasonal amplitude of fluxes, the GLDAS-Noah-based estimates as well as those based on ERA-I/L soil moisture are in most cases very well in line with observations. Contrary, Szegvary et al. (2009) flux estimates largely underestimate seasonal amplitudes, at least in 2006. The same is true for the López-Coto et al. (2013) model estimates. As mentioned before, a large part of the general underestimation by López-Coto et al. (2013) may be due to the use of an overly low emanation coefficient in their estimates. Based on available observations, the effect of frozen soils cannot be evaluated. However, restriction due to shallow water table turns out to be important, not only at the coastal site Lutjewad, but potentially also in river plains such as in the surroundings of the Gebesee site.

Since only very few systematic 222Rn flux measurements during different seasons are available in the literature, validation of our new 222Rn flux map is far from exhaustive. In order to better judge at least the reliability of the European-wide flux estimates, we have compiled here all published 222Rn flux measurements available to us, even if they are only based on episodic field campaigns (see Table S1 in the Supplement). Larger short-term data sets from a single site have been averaged to monthly values and included in our model–data comparison in Fig. 8. The map in Fig. 8 (upper panel) shows the geographical distribution of these episodic observations and their individual values colour-coded. The frequency distributions of the differences between modelled and measured 222Rn fluxes are shown in the lower part of Fig. 8. We find no geographical dependency of the differences (not shown) but a large mean bias between ERA-I/L-based and measured fluxes. While the GLDAS Noah differences yield a mean value close to zero (0.82 mBq m-2 s-1 with IQR = 11.29 mBq m-2 s-1), the ERA-I/L-based estimates are on average lower than observations by 5.73 Bq m-2 s-1 (IQR = 11.21 mBq m-2 s-1). This is in accordance with the earlier comparison based on the more systematic long-term results at fewer stations, discussed in Sect. 5.4 and gives a strong hint that the GLDAS-Noah-based estimates on average provide the more accurate flux estimates than those based on ERA-I/L-soil moisture, which seem to be systematically too low. However, since IQR values of radon flux differences are large for both soil moisture models, a definitive decision, which soil moisture model to use is not yet possible. The large IQR values are most probably caused by the non-representativeness of many of our observations for the entire pixel and due to very similar uncertainties in the bottom-up information used for both flux estimates (see also Sect. 5.6).