Modelling the bimodal distribution of indoor gamma-ray dose-rates in Great Britain

Abstract

Gamma radiation from naturally occurring sources (including directly ionizing cosmic-rays) is a major component of background radiation. An understanding of the magnitude and variation of doses from these sources is important, and the ability to predict them is required for epidemiological studies. In the present paper, indoor measurements of naturally occurring gamma-rays at representative locations in Great Britain are summarized. It is shown that, although the individual measurement data appear unimodal, the distribution of gamma-ray dose-rates when averaged over relatively small areas, which probably better represents the underlying distribution with inter-house variation reduced, appears bimodal. The dose-rate distributions predicted by three empirical and geostatistical models are also bimodal and compatible with the distributions of the areally averaged dose-rates. The distribution of indoor gamma-ray dose-rates in the UK is compared with those in other countries, which also tend to appear bimodal (or possibly multimodal). The variation of indoor gamma-ray dose-rates with geology, socio-economic status of the area, building type, and period of construction are explored. The factors affecting indoor dose-rates from background gamma radiation are complex and frequently intertwined, but geology, period of construction, and socio-economic status are influential; the first is potentially most influential, perhaps, because it can be used as a general proxy for local building materials. Various statistical models are tested for predicting indoor gamma-ray dose-rates at unmeasured locations. Significant improvements over previous modelling are reported. The dose-rate estimates generated by these models reflect the imputed underlying distribution of dose-rates and provide acceptable predictions at geographical locations without measurements.

Appendices

Appendix A: the influence of period of construction and dwelling type on indoor gamma-ray dose-rates

This appendix discusses the information on period of construction and building type on indoor gamma-ray dose-rates. Two sources of information are available.

• The National Survey of the exposure of the UK population to naturally occurring radiation indoors (Wrixon et al. 1988) provides reasonably detailed information, but only for a subset of the measurement locations. The corresponding information for general locations, in particular the birth locations of study subjects in epidemiological studies, is not available.

• An approximate division into dwellings probably constructed before and after 1940 can be deduced from the Dudley Stamp land-use maps developed in the 1930s. This is outlined in “Materials and methods” and more details are given here.

Information from the National Survey

Relevant data were collected as part of the National Survey of Natural Radiation Exposures in UK Dwellings (Kendall et al. 2016a, b; Wrixon et al. 1988). Data were reported by Wrixon et al. (1988) for 2283 dwellings in Great Britain for which an indoor gamma-ray dose-rate measurement was available. For most of these, a radon measurement was also available. Mean indoor gamma radiation dose-rates and mean indoor radon concentrations for these 2048 dwellings are summarized in Table 4. On average, radon levels are higher in larger houses, while gamma-ray dose-rates are somewhat lower. Reasons for this are not known with certainty, but the lower mean radon concentration in flats (apartments) may be a consequence of many flats being above the ground floor and thus being less liable to radon ingress from the ground.

Table 4 Number of measurements with mean indoor radon level and gamma-ray dose-rate by house type based on the data of the UK National Survey

For 280 of the 2283 dwellings with a gamma-ray measurement, dwelling type and/or period of construction (generally the latter) were not known. The breakdown of the remaining 2003 dwellings by dwelling type and period of construction is given in Table 5. Table 6 shows the variation in mean indoor gamma-ray dose-rate with dwelling type and period of construction.

Figure 5 gives the variation with period of construction of the indoor gamma-ray dose-rate for each type of dwelling (i.e., the data of Table 6 plotted as a figure). It may be noted that mean indoor gamma-ray dose-rates differ between dwelling types and that the proportion of dwellings of each dwelling type varies significantly from period to period. The variation with time of the mean indoor gamma-ray dose-rate for all dwellings taken together is thus hard to interpret.

Table 5 Numbers of dwellings by dwelling type and period of construction (with percentages)

Table 6 Mean indoor gamma-ray dose-rate (nGy/h) by dwelling type and period of construction

Fig. 5

Breakdown of mean indoor gamma-ray dose-rates by dwelling type and year of construction

For all dwelling types, indoor gamma-ray dose-rates in homes constructed during 1945–1964 were lower than for those homes constructed in the previous period. This decline continued during 1965–1976. For semi-detached/end-terrace houses and for flats, this decline was reversed in the next period (homes constructed after 1976); for detached and mid-terrace houses, the decline continued.

Table 7 gives the variation in indoor radon concentration with building type and age. Radon levels in most dwellings tend to decrease over time, while, for flats, they increase.

Table 7 Mean indoor radon concentrations (Bq m^{− 3}) by dwelling type and period of construction

Approximate period of construction of dwellings from land-use maps

Methods for constructing dwellings have changed with time. Originally walls consisted of a single layer of masonry. Then, cavity walls with two layers, usually of brick, were introduced. Later, the inner layer of brick was replaced by one of blocks. These changes will have affected the gamma-ray dose-rate within the dwelling. In particular, the change from cavity walls consisting of outer facing bricks with an inner layer of common bricks (which became frequent between the First and Second World War) to walls consisting of an outer layer of facing bricks with an inner layer of blocks (which was introduced after the Second World War) (National House Building Council (NHBC) Foundation 2015) broadly coincided with a reduction in indoor gamma radiation dose-rates.

Unfortunately, information on period of construction is not generally available for British dwellings. However, an approximate classification of those built before and after about 1940 can be obtained by discovering whether the house in question was in an area categorized as “Urban” or “Suburban” at this time. Those that were in such (Sub)Urban areas may be taken as probably already in existence when the map was compiled. Those built on areas not categorized as (Sub)Urban may be taken to have been built later (Appleton and Cave 2018).

The maps available for this categorization were compiled under the direction of Dudley Stamp in the 1930s (Southall et al. 2007). Fourteen categories of land use were assigned. Maps for England and Wales (and a small part of Southern Scotland) were converted for use in a GIS in a project of the Environment Agency about 10 years ago (Southall et al. 2007; Environment Agency 2010). This was a substantial task, partly because the maps had been printed over a long period by different printers and colour codings were inconsistent between sheets, but also because details such as text and contour lines were printed in colours also used for land-use categories. For the present study, these GIS maps for England and Wales were used.

No GIS-ready versions of the Dudley Stamp land-use maps for Scotland were available. Nine sheets covering areas of the highest population density were selected (see Table 8) and polygonised for GIS use. While much of the Scottish land area was not included in this exercise, a total of about 78% of the Scottish population were included and, in total, 98% of the population of Great Britain. Overall, 56 and 44% of the GB dwellings were classified as constructed pre- and post-1940, respectively. In Scotland and Wales, the proportions were more nearly equal than in GB as a whole (50 and 47% constructed pre-1940 for Wales and Scotland, respectively; these figures exclude the 22% of Scottish dwellings that were unclassified).

Table 8 Sheets of the Dudley Stamp land-use maps for Scotland that were converted for GIS use

It is clear that this Pre/Post-1940 construction classification can only be approximate. Modern buildings which were built in pre-1940 areas during infill or redevelopment will be wrongly classified as “pre-1940”. Testing the predictions is difficult, because they relate to a spectrum of years from 1962 until 2010. The acid test in the present context is how useful this parameter proves as a predictive tool. Nevertheless, approximate testing may be of interest.

The Department for Communities and Local Government published an English Housing Survey Housing stock report for 2008 (Department for Communities and Local Government, DCLG 2010) in which it was reported (their Table 1.1) that 43% of English housing was built before 1945; the total for Great Britain will be similar. These figures exclude the “Post-1990” category which covers only the end of the period used in the present work. As described above, the UK National Survey of Natural Radiation Exposure in UK Dwellings (Wrixon et al. 1988) collected data on period of construction. The percentages of the sample built before and after 1944 were 47 and 53%, respectively. Both the DCLG and National Survey data thus have a slight majority of dwellings constructed after 1944, while our Dudley Stamp data indicate a similar-sized majority pre-1940. It is unlikely that the difference between 1944 and 1940 as the cutoff is significant. As noted above, the Dudley Stamp classification will fail to pick up redevelopment or infill, and this probably accounts for the difference. Despite its approximate nature, the Dudley Stamp pre- and post-1940 construction classification is found to have predictive value for indoor gamma-ray dose-rate modelling. Figure 6 compares the median indoor gamma-ray dose-rates for dwellings constructed pre- and post-1940 for different underlying geologies. In each case, the dose-rate in the newer houses is lower than that in older ones, indicating that the age of construction effect is not a consequence of changes in location.

Fig. 6

Median indoor gamma-ray dose-rates (nGy/hr) for dwellings constructed pre-1940 (“OLD”) compared to those constructed post-1940 (“NEW”) by underlying geology (Carboniferous, Cretaceous, Jurassic, Permian–Triassic, and Tertiary). (Box = median 95% confidence limits; Horizontal line in box = median; Pre-1940 data derived from © L. Dudley Stamp/Geographical Publications Ltd., Audrey N. Clark, Environment Agency/DEFRA, and Great Britain Historical GIS)

Appendix B: revised ordinary least-squares linear-regression analysis

Indoor measurement data and other covariates used

The indoor gamma-ray dose-rate measurement data are as reported previously (Kendall et al. 2016b). There have been a few small changes made to coding of some of the other data, in particular the 50 k-BEDSUP surface codes, county district, Carstairs score, and population density (see OLR-1). Here, the candidate models were refitted using a number of candidate dose-interpolation measures. These comprise jackknife estimates (Davison and Hinkley 1997) of dose-rate based on interpolated “nearby” dose-rate measurements. Three basic types of interpolation were used: (a) weighting of neighbouring dose-rates by an inverse power of distance, (b) an average over some administratively defined areas (e.g., county district), or (c) an average over some geologically defined areas.

There are small differences in the present analysis from what was done in the previous paper (Kendall et al. 2016b) relating to the simple areal-average estimates, types (b) and (c) above. Previously, the jackknife area means, e.g., for 50 k-BEDSUP, were computed for each candidate point $i$ by finding the nearest point (using Euclidean distance) in the remaining $n=10,198$ data set, ${j_{\hbox{min} }}(i)$, then using the mean dose-rate of all points having the same, e.g., 50 k-BEDSUP, value as ${j_{\hbox{min} }}(i)$ in the $n=10,198$ data and assigning that to the candidate point. In the present paper, area means for the relevant area are simply averaged, e.g., given by 50 k-BEDSUP code, corresponding to the candidate point $i$ using the mean dose-rate of all points having the same, e.g., 50 k-BEDSUP, value as $i$ in the $n=10,198$ data.

A number of supplementary geological measures are also included in the present analysis, specifically a 23-level 50 k-BEDSUP surface code, a 23-level 50 k-BEDSUP-surface-bedrock code and a 3-level (pre-1940, post-1940, unknown) house construction period code that were not previously employed. The variables used are listed in Table 9. The optimal model and associated parameter values are given in Table 10. As can be seen, the model is somewhat different in form from the previously fitted optimal model (Table 17 in (Kendall et al. 2016b)).

Table 9 Variables used in ordinary least-squares (OLS) regressions as part of the forward–backward stepwise Akaike information criterion (AIC) variable selection

Table 10 Parameter values and 95% CIs for AIC-optimal model fitted by ordinary least squares

Empirical ordinary least-squares model selection and fitting, and Gaussian process maximum-likelihood fitting

As previously, a highly parameterized empirical model was constructed based on linear combinations of the interpolation measures described in the previous section. To construct an empirical model that satisfactorily explains the spatial variation of mean dose-rate, linear regression with ordinary least squares (OLS) (Rao 2002) was used. This model assumes a standard Normal error, and attempts to model the spatial correlation in dose-rate using combinations of explanatory variables. To avoid over-parameterized models, the Akaike information criterion (AIC) (Akaike 1973, 1981) was minimized to select the optimal set of interpolation variables (the calculation of which is outlined above). AIC penalises against overfitting by adding 2 × [number of fitted parameters] to the model deviance (residual sum of squares). An iterative mixed-forward–backward stepwise procedure was used to minimize AIC for OLS using R (R Project version 3.4.4 2018). There is literature, e.g., Hurvich and Tsai 1989, suggesting that the AIC may lead in some cases (in a class of autoregressive models) to over-parameterized models where the data sets are small, but not for problems such as the present ones.

The fit of all these models was tested using a standard cross-validation process. The models were fitted to a randomly selected 70% of the data, and the indicated models were then used to predict gamma-ray dose-rates in the remaining 30% of the data. The randomly chosen 70% (and 30%) samples were identical to those previously used (Kendall et al. 2016b). The resulting mean square error was 355.71 (Table 2); a substantial decrease on the previously reported estimate, 377.64 (Kendall et al. 2016b). It is possible that another random 70–30 partition of the data would result in a different optimal model with a different MSE. Here, a different partition was not explored, because: (1) there are currently no guidelines as to how many such random partitions which one must explore; (2) a reliable method to combine predictions across multiple random partitions does not exist in a frequentist setting. In a Bayesian setting, one could employ Bayesian model averaging, but this falls outside the scope of the current paper.

Calculating uncertainties in E-OLS estimates

A possible way of estimating the errors in estimated dose-rates taking into account the indicated non-normality of model residuals (see Fig. 7) is as follows:

• 1. Take N bootstrap samples from the M = 10,199 measurement data and estimate the projected dose-rates for the K = 124,454 measurement data points for each, resulting in a K × N bootstrap array.

• 2. For each of n = 1, …, K, estimate percentile confidence intervals (CIs) in the usual way from this K × N array, sampling for each k = 1, ..., K the percentile-based CI based on the N dose-rate estimates.

Bias corrected adjusted bootstrap CIs (Efron 1987) could, in principle, be calculated in step (2), but the computational overheads might be very onerous.

Fig. 7

Normal-quantile plot of optimal model residuals (observed–model expected). The two label numbers are the sequence numbers of these points in the file

Appendix C: Gaussian–Matérn maximum-likelihood fitting

Here, it is outlined how a standard geostatistical model was fitted to the indoor gamma-ray dose-rate data by maximum likelihood (Diggle and Ribeiro 2007). Specifically, it was assumed that the ln[gamma-ray dose-rate] ${Y_i}$ at spatial location ${x_i}$ was given by:

$${Y_i}=\mu ({x_i})+S({x_i})+{Z_i},$$

(2)

where $\mu ({x_i})=E[{Y_i}]$ is the mean of ${Y_i}$, S is a stationary Gaussian process with mean 0, variance ${\sigma ^2}$, and spatial correlation $\rho (u)={\text{Corr}}\left[ {S(x),S(x - u)} \right]$, and ${Z_i}$ are independent Gaussian random variables with mean 0 and nugget variance ${\tau ^2}$. The Matérn model (Matérn 1960) was employed, which assumes that:

$$\rho (u|\phi ,\kappa )={2^{\kappa - 1}}{(u/\phi )^\kappa }{K_\kappa }(u/\phi ),$$

(3)

where ${K_\kappa }(.)$ is a modified Bessel function of order $\kappa$ (Diggle and Ribeiro 2007). The Matérn model was chosen because of its flexibility. The integer part of the parameter $\kappa$ determines the mean-square differentiability of the process S—the process is mean-square differentiable when $\kappa>1$, and mean-square twice differentiable when $\kappa>2$. Formal estimation of the parameter $\kappa$ was difficult and was not attempted here. Preliminary curve fitting and examination of the empirical variogram, by eye, suggested that $\kappa =2.5$ gave a reasonable fit; this value was used in all the analyses, as in the previous such modelling (Kendall et al. 2016b). Various models of the mean process, $\mu (x)$, were constructed as outlined in Table 11. All associated model parameters ($\mu (x)$, $\phi$, ${\tau ^2}$) were estimated by maximum likelihood.

Table 11 Log likelihood, Akaike information criterion (AIC) and number of fitted parameters in fits of Gaussian–Matérn model to ln[gamma dose-rate]

$({Y_i})_{{i=1}}^{N}$ is multivariate Gaussian with mean $\mu ({x_i})$ and variance ${\sigma ^2}V={\sigma ^2}[R+{\upsilon ^2}I]={\sigma ^2}R+{\tau ^2}I$ where $I$ is the $N{\text{x}}N$ identity matrix, and $R=({r_{ij}})_{{i,j=1}}^{N}=\left( {\rho (||{x_i} - {x_j}||)} \right)_{{i,j=1}}^{N}$. If now one has an arbitrary point $x$, with associated “true” (unobserved) ln[gamma-ray dose-rate] $T(x)=\mu (x)+S(x)$, then $(T(x),(Y({x_i}))_{{i=1}}^{N})$ is multivariate Gaussian with mean $(\mu (x),(\mu ({x_i}))_{{i=1}}^{N})$ and variance $\left[ {\begin{array}{*{20}{c}} {{\sigma ^2}}&{{\sigma ^2}{r^T}} \\ {{\sigma ^2}r}&{{\sigma ^2}V} \end{array}} \right]$ where the vector $r$ is given by $r=({r_i})_{{i=1}}^{N}=\left( {\rho (||x - {x_i}||)} \right)_{{i=1}}^{N}$. Then, it is easily shown [e.g., p. 136 in (Diggle and Ribeiro 2007)] that the conditional distribution of $T(x)$ given $(Y({x_i}))_{{i=1}}^{N}$ is also multivariate Gaussian with mean:

$$m(x)=\mu (x)+{r^T}{V^{ - 1}}\left[ {Y({x_i}) - \mu ({x_i})} \right]_{{i=1}}^{N},$$

(4)

and variance:

$$v(x)={\sigma ^2}\left[ {1 - {r^T}{V^{ - 1}}r} \right].$$

(5)

Finally, one can transform back to the natural dose-rate scale, so that the mean dose-rate at $x$ is given by:

$$M(x)=\exp [m(x)+v(x)/2].$$

(6)

In estimating these, the maximum-likelihood estimates of $\mu (),V,{\sigma ^2}$, namely $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\mu } (),\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V} ,{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\sigma } ^2}$, were substituted.

The optimal model includes Easting and Northing and also the Dudley Stamp Pre/Post 1940 construction category, the urban–rural (6-level) classification, the external gamma dose-rate, and the 50K-BEDSUP Surface Geological classification (23-level). The fit of this model was tested using a standard cross-validation process, by fitting to the randomly selected 70% of the data, and the indicated model was then used to predict gamma-ray dose-rate in the remaining 30% of the data. Table 2 of the main text shows the mean-square error estimated in the 30% test sample.

Table 12 compares quantiles of the distributions of measured dose-rates with those predicted by the optimal E-OLS and Gaussian–Matérn models. It can be seen that both models have a narrower spread than the actual data. While Gaussian–Matérn and E-OLS results are broadly similar, the E-OLS predicts a slightly longer upper tail, which accounts for the lower SD of the Gaussian–Matérn model. Figure 8 compares the distribution of residuals (i.e., of differences observed–predicted values at the measurement points) for the E-OLS and for the Gaussian–Matérn models. It can be seen that, while the former are symmetrically distributed about zero, the latter are slightly displaced towards positive values, indicating that the Gaussian–Matérn predictions are somewhat lower, as can be seen more directly from Table 2.

Table 12 Quantiles of distribution of actual dose-rate and model-fitted dose-rate for optimal E-OLS and Gaussian–Matérn model

Fig. 8

Distribution of residuals for E-OLS and GM models

Appendix D: decomposition of the observed indoor gamma-ray dose-rate distribution as a superposition of Normal random variables

A more formal decomposition of the measured indoor gamma-ray dose-rate distribution as a superposition of Normal distributions may be made as follows. It was assumed that the dose-rates were given by a weighted superposition of $N$ normally distributed random variables, comprising ${N_1}=3$ distributions with means, ${\mu _i}$, at 47.5, 52.5, 57.5 nGy/h, ${N_2}=60$ with means at 60.5, 61.5, 62.5, …, 119.5 nGy/h, then a further ${N_3}=17$ with means at 122.5, 127.5, 132.5, …, 197.5, 202.5 nGy/h with probabilities ${p_i}$. The distribution at each of the $N={N_1}+{N_2}+{N_3}=80$ points is ${X_i}\sim N({\mu _i},{\sigma ^2})$, with common standard deviation $\sigma$, so that the overall distribution is a superposition of these. [Note: this ensemble of random variables is not independent, since a given individual can be assumed to drawn from one and only one of the $N$ distributions with probability ${p_i}$, so that the probability of the individual being drawn from distribution $i$ and distribution $j \ne i$ is $0 \ne {p_i}{p_j}$.] This implies that the overall cumulative density function is:

$$F(d|\left( {{p_i}} \right)_{{i=1}}^{N},\left( {{\mu _i}} \right)_{{i=1}}^{N},\sigma )=P\left[ {X \leqslant d|({\mu _i})} \right]=\sum\limits_{{i=1}}^{N} {{p_i}P\left[ {N({\mu _i},{\sigma ^2}) \leqslant d} \right]} .$$

(7)

The probabilities $({P_i})_{{i=1}}^{N}$ are constrained so as to be positive and $\sum\nolimits_{{i=1}}^{N} {{p_i}=1}$. This cumulative distribution function was fitted to the empirical data, consisting of counts of persons in each of the $M=80$ intervals defined by the cut-points ${d_0}=0,\,{d_1}=50,\,{d_2}=55,\,{d_3}=60,\,{d_4}=61,\,{d_5}=62, \ldots ,{d_62}=119,\,{d_{63}}=120,$ ${d_{64}}=125,{d_{65}}=130, \ldots ,{d_{79}}=200,{d_{80}}=\infty$ nGy/h via (multinomial) maximum likelihood, i.e., by maximizing in the weights $({p_i})_{{i=1}}^{N}$:

$$L\left[ {({p_i})_{{i=1}}^{N},\sigma } \right]=\sum\limits_{{j=1}}^{M} {{n_j}\,\,\ln } \left[ {F({d_j}|({p_i})_{{i=1}}^{N},({\mu _i})_{{i=1}}^{N},\sigma ) - F({d_{j - 1}}|({p_i})_{{i=1}}^{N},({\mu _i})_{{i=1}}^{N},\sigma )} \right].$$

(8)

The results of maximizing (8) with the ${p_i}$ corresponding to dose-rates above 160 nGy/h constrained equal for stability, which yields an estimate of the probabilities to be attached to each component Normal distribution, as given by the ${p_i}$, which are plotted in Fig. 9. As can be seen, the dose-rate distribution is largely given by a combination of three Normal distributions ${X_i}\sim N({\mu _i},{\sigma ^2})$ with means at 80.5, 97.5, and 117.5 nGy/h, with weights ${p_i}$ = 0.198, 0.706, and 0.052, respectively. The model provides a good fit to the observed dose-rate distribution. The common estimated SD was $\sigma =19.79$ nGy/h.

Fig. 9

a Empirical and model-fitted distributions and b probabilities $\left( {{p_i}} \right)_{{i=1}}^{N}$ associated with component Normal distributions

While the choice of fitting three Normal distributions was essentially arbitrary and for illustrative purposes, it is plausible that the two Normal distributions with lower means represent the underlying bimodality, while the third reflects a slight shoulder on the high dose side of the distribution.

Appendix E: distributions of indoor gamma-ray dose-rates in other countries

Population distributions of indoor gamma-ray dose-rates are discussed in Annex B of the 2000 Report of the United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR 2000); Table 12 of that report presents data for 11 countries. UNSCEAR (UNSCEAR 2000) notes that “The distribution in Italy is also wide and approximately bimodal. The distributions in the Russian Federation, Finland, and Lithuania are characterized by separate peaks in the distributions at decades 2 or 3 above the country mean.” In the case of Lithuania, the two peaks are presumably too close together to be resolved in the data presented, or else one may lie in the substantial tail above 100 nGy/h.

The indoor gamma-ray dose-rate distribution for Denmark, while apparently unimodal in the UNSCEAR (2000) tabulation, is bimodal when examined at a finer resolution (Ulbak et al. 1987, 1988). In fact, Ulbak et al. (1987, 1988) made measurements in 489 buildings and analyzed the data for one-family houses and apartment blocks separately.

The distribution of indoor gamma-ray dose-rates in Spain also exhibits bimodality, with distinct peaks at 60–69 and 80–89 nGy/h (44% of measurements were > 100 nGy/h). This bimodality would provisionally be assigned to geological factors (Marta García-Talavera, private communication).

The dose-rate distribution in Hungary is also bimodal, with a low dose-rate peak at 20–29 nGy/h, then a steady rise to 90–99 nGy/h, and a second and much larger peak somewhat greater than 100 nGy/h. The data for Bulgaria exhibit distinct peaks at 60–69 nGy/h and 80–89 nGy/h.

Thus, at least seven countries (Denmark, Finland, Hungary, Italy Lithuania, the Russian Federation, and Spain), of the eleven for which UNSCEAR (2000) presents distributions of indoor gamma-ray dose-rates, have bimodal distributions; it is not clear whether the other four do so. In particular, the data for Bulgaria, while rather sketchy, do hint at bimodality. The Belgian distribution, while unimodal, is based on only 100 measurements (Uyttenhove et al. 1984) and any structure may be difficult to see. The data for Romania appear to be based on theoretical calculations rather than direct measurements (Iacob and Botezatu 2004), and while this may well give a good estimate of the mean population dose, it is likely to overlook any finer structure that might be present.

Other reports of indoor gamma-ray dose-rates are worthy of note. In particular, Schmier et al. (1982) summarized a large study in West Germany in which 30,000 measurements of exposure were made in dwellings (generally three measurements per building). Differences were found between regions (“Länder”) and between buildings constructed with different materials. However, the overall distribution of exposure-rates was reported in fairly broad categories and it is likely that any deviations from unimodality would not have been apparent.

Mjönes (1986) reported from a survey of 1300 Swedish dwellings that dose-rates in apartments were twice those in single-family houses. This no doubt accounts for the bimodality in the overall distribution, and there were also substantial regional differences. The Swedish dwellings with the highest indoor gamma-ray dose-rates tended to be those in which alum shale-aerated concrete had been used as a building material (Mjönes 1986; Axelson et al. 2002).

Storruste et al. (1965), in a study of 2026 Norwegian dwellings reported substantially lower indoor gamma-ray dose-rates in buildings made of wood as compared to concrete or brick. Alum shale-aerated concrete was not used in Norway.

It is clear that bimodality may be present, but not apparent, if insufficient measurements have been made or if the tabulated data are too coarsely stratified. To this, the situation which we report in Great Britain should be added where the underlying bimodality is obscured by inter-house variations and measurement errors.

The reasons for bimodality are likely to be characterized by specific features in each country. In the case of Italy, the broad bimodality is ascribed to two regions in Central Italy (Lazio and Campania, out of a total of 21) having conspicuously high dose-rates (Bochicchio et al. 1996). It appears that geology and the distribution of population by geology drive the differences, at least in part because local materials are used for building (Bochicchio et al. 1996). In Spain, the bimodality is believed to correspond to different geologies. In Denmark and Finland, it appears to correspond to two types of dwelling, but, in Finland, there are also strong geographical variations which are ascribed to varying radionuclide concentrations in soils and construction materials (Vesterbacka 2015). As noted in the main text, in Great Britain, a number of factors seem to be operating; none of which is dominant.

In the general situation, indoor gamma-ray dose-rates may vary depending on local geology, building materials, and house styles; there is thus no obvious reason why the indoor distributions should be bimodal rather than multimodal. The Hungarian distribution looks more complex than simply bimodal, and it may be that, in other countries, further complexity is concealed by limitations on the accuracy of measurements and the number of homes measured. It is also unclear why bimodality is shown by some original measurement data sets, but is only obvious in the UK data when measurements are averaged over comparatively small geographical areas. Of course, in some countries, the separation of the peaks is greater than in the UK data, which makes it easier to distinguish the separate modes. It is also likely that the bimodality (or greater complexity) may be more marked in some countries than others.

Appendix F: the influence of building materials on indoor gamma-ray dose-rates

The indoor gamma-ray dose-rate in a building may be regarded as the outdoor dose-rate before the house was constructed, reduced by the shielding provided by the building, but increased again by emissions from radionuclides within the building materials. These emissions are normally the largest contribution to the indoor dose-rate as noted by the UNSCEAR 2000 Report, Paragraph 57, (UNSCEAR 2000), at least for buildings constructed of the conventional masonry materials, as are a large majority of those in the UK (Department for Communities and Local Government 2010). Gamma-ray emissions from building materials come largely from ⁴⁰K and radionuclides in the ²³⁸U and ²³²Th decay chains, the three contributions being roughly equal, UNSCEAR 2000 Report, Annex B, Paragraph 43 (UNSCEAR 2000). These radionuclides are ubiquitous in the environment, but concentrations vary from one material to another. The extent to which building materials with high radionuclide concentrations might lead to high indoor gamma-ray dose-rates has been reviewed (European Commission 1997, 1999).

Building materials apart from timber are derived from “earth materials” such as clays or rocks. In some cases, building materials are made from by-products of other processes; for example, the use of power station coal ash in building blocks. The radioactivity in building materials will be determined by the radioactivity in the materials from which they are made, although the activity may be modified in the fabrication process. There will also be variation in the radioactive content of different material from the same broad geological strata. Relatively few data have been published on radioactivity in British building materials. A review by the European Commission (1997) cites three publications (Cliff et al. 1984; Hamilton 1971; O’Riordan and Hunt 1977) and an unpublished contract report. Models are available to calculate indoor gamma-ray dose-rates given the activity concentrations in the building materials (Markkanen 1995; Risica et al. 2001; de Jong and van Dijk 2008). However, the lack of information on the materials used in the construction of the dwellings of interest precludes their application here.

A detailed analysis has been published of the radiological consequences of using building materials (blocks, but possibly also concrete) made using coal ash (bottom or fly). Smith et al. (2001) state that power station ash amounts to about 16% by weight of the coal that is burned and that there is a considerable concentration of radionuclides in it. Smith et al. (2001) go on to estimate that “the total external dose to a resident of a building constructed using building materials containing ash, from all radionuclides in the material, is approximately 893 µSv year^− 1. The corresponding dose in a building constructed from similar materials that do not contain ash is approximately 758 µSv year^− 1.”

The increased dose is thus ~ 135 µSv year^− 1, approaching 20% of the total dose. However, there does not appear to be published information on the proportion of buildings constructed using coal ash-based materials, nor on where these materials were used. It is plausible that such building materials are becoming increasingly common with pressures to reduce the amount of material sent to landfill. However, it is likely that coal might be transported considerable distances from mine to power station (and these coal ash blocks are a relatively recent innovation), so that these building materials are not used particularly close to where the coal was mined. It is interesting to note that the use of coal ash in building materials is not an exclusively modern practice; in the London area, ‘town ash’ (residue from domestic coal-burning) was used for brick-making in the nineteenth and twentieth century (Bloodworth 2016).

Bricks, perhaps, the most common building material in Great Britain, were generally made close to the source of raw materials (clay or shale), though this became a less powerful tendency over the twentieth century as production became more centralised (Brunskill 1997). Brick works also tended to be positioned close to sources of coal to provide fuel. In the early days, bricks were used close to where they were made (indeed, initially, bricks were made at the construction site). With the introduction of canals, railways and road transport bricks were used further from the source of the raw material (Brunskill 1997).

In Great Britain, the main brick-making raw materials have been derived from material from Carboniferous Coal Measures, Triassic Mercia Mudstones, Jurassic Oxford clay and Cretaceous clay (Department for Communities and Local Government (DCLG) 2007); Scottish Executive 2007; Bloodworth 2016). The first two of these are older than the second pair. Dwellings built on these older strata have, on average, higher mean indoor gamma-ray dose-rates (104 and 101 vs 85 and 85 nGy/h see Table 1). There is, however, considerable variation within each bedrock class.

If houses tended until relatively recently to be built using bricks from the nearest source of clay, then dwellings in the English Midlands and Northwest England conurbations will tend to have indoor gamma-ray dose-rates similar to those in houses built on the Carboniferous Coal Measures and Triassic Mercia Mudstones, whilst dwellings in the Northeast conurbations of England and the Central Valley of Scotland will tend to have indoor gamma-ray dose-rates similar to those in houses built on the Carboniferous Coal Measures. Conversely, indoor gamma-ray dose-rates in dwellings in London and the South-east of England are likely to be more similar to those built on the Jurassic Oxford clay and the Cretaceous clays. This provides a possible explanation for at least part of the general geographical variation in indoor gamma-ray dose-rates seen in OL-2b and OLR-2c. However, information on the sources and radioactive content of British building materials is very limited and firm deductions are not possible.

Appendix G: other approaches to estimating indoor gamma-ray dose-rates in unmeasured homes

Introduction

In this paper and its predecessors (Chernyavskiy et al. 2016; Kendall et al. 2016b), methods for estimating indoor gamma-ray dose-rates in unmeasured dwellings have been developed and tested. A main aim of the present work was to predict dose quantities for use in epidemiological studies of the effect of natural background gamma radiation, but the work has scientific interest in its own right. The data available for developing models consisted of a set of 10,199 measurements in dwellings. The present task was greatly simplified by the fact that the individual measurements of the quantity in question were available (rather than just a summary) and by the large size of the measurement set. This resulted in a closely spaced grid of measurements, such that most points at which predictions were required were close to a measurement location. The mean separation of measurement locations from their nearest neighbour was 1.2 km and the mean separation of the birth location of study subjects from the nearest measurement location was 1.1 km. However, these means were inflated by occasional large separations. Table 13 shows that well over a third of locations had a measurement within 500 m, about 70% within 1000 m and over 85% within 2000 m.

Table 13 Separations from nearest measurement location of measurement locations (excluding that in question) and a 10% sample of birthplaces of study subjects

In many countries, including most of those considered here, the majority of people spend most of their time indoors (UNSCEAR 2000). In theory, a more accurate estimate of doses from gamma-rays would be obtained by combining the estimates of indoor and outdoor dose-rates with the respective occupancy factors. However, a reasonable approximation [e.g., (Kendall et al. 2006)] is that the fraction of time spent outdoors is small, the dose-rates indoors and outdoors are not greatly dissimilar, and that the mean overall dose-rate will not differ greatly from the indoor rate.

Another point to consider is whether it matters that the measurements on which the dose prediction method is based are taken during the period of epidemiological follow-up (in the present case, 1962–2010). It is true that indoor gamma-ray dose-rates are not completely constant in time. Thus, Minato (1980) reported that atmospheric radon daughter concentrations and rainfall play an important role in short- and middle-term variations in the background radiation flux, while changes in soil dryness contribute to longer term variations. However, such changes are largely temporary, and it is argued here that mean indoor gamma-ray dose-rates in the existing homes do not change much with time over the timescale of interest for epidemiology. It is true that dose-rates in an older house may change if, say, it is replastered with gypsum containing relatively high concentrations of uranium. However, generally, constant concentrations of radionuclides in surrounding geology and in the building materials mean that gamma radiation dose-rates in a house will be roughly constant, much as they are outdoors. This is not necessarily the case with radon, for example, where changes in heating and in ventilation can affect indoor radon concentrations significantly.

As described in the main paper, in Great Britain (GB), the best predictive method was found to be a linear combination of various simple models—averages over small areas, over geological units, or a weighted sum of the nearest measurements. Geology was not a very powerful predictor, though it may be the best single predictor available. Despite testing several detailed models, with the available data, substantial mean-square error was associated with the optimal fit. Geostatistical (“kriging”) methods performed slightly less well; possible reasons for this are explored in the main text.

Other approaches

A French study of natural background radiation and childhood cancer (Demoury et al. 2017) made use of dose estimates by Warnery et al. (2015). Like the GB study, Warnery and coworkers also had a large set of indoor measurements made in 17,404 dental surgeries and veterinary clinics. Two kinds of variogram-based geostatistical modelling were conducted to estimate indoor terrestrial gamma-ray dose-rates in France on a grid of 1 km squares: ordinary kriging (considering only the locations of the measurement points) and multi-collocated cokriging, (also considering the geogenic uranium potential of the measurement locations). Warnery and coworkers used cross-validation in which single observed measurements were successively excluded from the total measurement set and then predicted using the remaining 17,403 measurements (Warnery et al. 2015; Marquant et al. 2018). The mean-square errors of the two methods calculated in this way were 409 and 407 (nSv/h)², respectively. The arithmetic mean was 76 nSv/h.

In spatial statistics, the total variance can be described as the sum of the “sill” and the “nugget effect”. The former is the variation that is explained by distance between observations; the latter is random variation that is non-spatial. Outside the context of spatial statistics, a nugget effect is just “random error”. The variogram-based modelling approach of Warnery et al. yields a nugget effect of 35% of the total variance of the 17,404 available measurements. Marquant et al. (2018) considered it likely that the nugget effect could mainly be due to the influence of local factors that are not taken into account by the modelling (i.e., inter-house variation) rather than metrological inaccuracies.

Kendall et al. (2018) queried whether the distribution of dose-rates in French dental surgeries and veterinary practices was necessarily similar to that in French homes (Kendall et al. 2018). However, whatever the case may be, it is unlikely to affect conclusions drawn from the experience of Warnery et al. (2015) in fitting the data.

Two other European epidemiological studies, in Finland (Nikkilä et al. 2016) and in Switzerland (Spycher et al. 2015), were less well placed than France and GB in that they had available only pre-existing maps of outdoor dose-rates.

The Finnish outdoor gamma radiation results were based on measurements from vehicles covering 15,000 km during 1978–1980 (Akima 1978; Arvela et al. 1995). The coordinates of the midpoints of 410 representative, evenly distributed, sections were used as the coordinates for the measurements. The SAS G3GRID procedure was used for interpolating values from an irregularly spaced set of points, generating a rectangular grid of 8 × 8 km². G3GRID uses a modification of the bivariate triangular interpolation method of Akima (Akima 1978). However, the Finnish workers also had available measurements of indoor gamma-ray dose-rates in 346 dwellings and also access to a housing register which allowed the homes of study subjects to be classified as a house or flat. Dwelling-type specific conversion factors were then used to estimate indoor gamma-ray dose-rates from the outdoor values.

Spycher et al. (2015) made use of the Swiss outdoor natural terrestrial radiation survey (Rybach et al. 2002) which combined a variety of measurements: airborne gamma-ray spectrometry (about 10% of the country’s surface surveyed by helicopter); in situ gamma-ray spectrometry (166 sites); in situ dose-rate measurements using ionization chambers (837 sites); laboratory measurements of rock and soil samples from 612 sites. These measurements were made from the early 1960s to mid-1990s. In addition to airborne measurements, a total of 1615 ground data points were available, which corresponds to about one point per 25 km². Dose-rates for cells of a 2 × 2 km grid were interpolated from the available data points using the inverse distance method and a search radius of 12 km. In most countries, no housing register was available, and the Swiss researchers analyzed their epidemiological data in terms of these outdoor dose-rate estimates.

The studies described so far had to make use of such pre-existing data as were available. However, two epidemiological studies comparing areas of high natural background with control areas were able to undertake special surveys and measurements to assess exposures. These studies were set in Guangdong Province, China (Tao et al. 2012), and in Kerala, India (Nair et al. 2009). In both cases, extensive sets of indoor and outdoor gamma-ray dose-rate measurements were made, and habit surveys gave sex- and age-specific house occupancy factors. The researchers were thus able to make the estimates of doses based on nearby measurements. These were not necessarily based on measurements for the individual concerned, but sometimes on village-specific parameters. Nevertheless, they were based on direct measurements of the quantities in question at locations close to the homes of study subjects. Moreover, measurements using personal dosemeters worn by a sample of local inhabitants were undertaken and used to validate the estimated doses. Agreement was reported to be good, but, in China, comparisons appear to have been of group averages rather than individuals.

Summary

Different workers have used very different approaches, driven by the data and resources available. In the Introduction to this paper, the E-OLS model favoured in the current paper is contrasted with the likelihood-based estimation as in the MRGP models favoured by Chernyavskiy et al. (2016) and the variogram-based (co)kriging models by Warnery et al. (2015). It is likely that developments in methodology and in computer hardware will increase the role for geostatistical models in the future (Heaton et al. 2018).

Both the French and the British investigations found that even their optimal models left large non-spatial inter-house variations that could not be explained with the explanatory variables available. However, approaches like those of the French and British investigators are dependent on access to sets of individual measurements. Other methods must be adopted where such data are not available. Conversely, if it is possible to undertake a significant number of new measurements, then a much more detailed approach is possible.