Using rainfall radar data to improve interpolated maps of dose rate in the Netherlands

Using rainfall radar data to improve interpolated maps of dose rate in the Netherlands Paul H. Hiemstra a,, Edzer J. Pebesma b, Gerard B.M. Heuvelink c, Chris J.W. Twenhöfel d a University of Utrecht, Department of Physical Geography, P.O. Box 8.115, 358 TC Utrecht, The Netherlands b University of Münster, Institute for Geoinformatics, Weseler Straße 253, 48151 Münster, Germany c Environmental Sciences Group, Wageningen University, P.O. Box 47, 67 AA Wageningen, The Netherlands d National Institute for Public Health and the Environment (RIVM), Antonie van Leeuwenhoeklaan 9, 3721 MA Bilthoven, The Netherlands Abstract The radiation monitoring network in the Netherlands is designed to detect and track increased radiation levels, dose rate more specifically, in 1 minute intervals. The network consists of 153 monitoring stations. Washout of radon progeny by rainfall is the most important cause of natural variations in dose rate. The increase in dose rate at a given time is a function of the amount of progeny decaying, which in turn is a balance between deposition of progeny by rainfall and radioactive decay. The increase in progeny is closely related to average rainfall intensity over the last 2.5 hours. We included decay of progeny by using weighted averaged rainfall intensity, where the weight decreases back in time. The decrease in weight is related to the half-life of radon progeny. In this paper we show for a rain storm on the 2 th of July 27 that weighted averaged rainfall intensity estimated from rainfall radar images, collected every 5 minutes, performs much better as a predictor of increases in dose rate than using the non-averaged rainfall intensity. In addition, we show through cross-validation that including weighted averaged rainfall intensity in an interpolated map using universal kriging (UK) does not necessarily lead to a more accurate map. This might be attributed to the high density of monitoring stations in comparison to the spatial extent of a typical rain event. Reducing the network density improved the accuracy of the map when universal kriging was used instead of ordinary kriging (no trend). Consequently, in a less dense network the positive influence of including a trend is likely to increase. Furthermore, we suspect that UK better reproduces the sharp boundaries present in rainfall maps, but that the lack of short-distance monitoring station pairs prevents cross-validation from revealing this effect. Keywords: density ordinary kriging, universal kriging, interpolation, dose rate, rainfall intensity, rainfall radar, trend, network 1. Introduction In case of releases of radioactive material into the atmosphere, a fast and accurate estimate of the spatial distribution of radiation levels is needed to estimate health effects on the population. In the Netherlands, radiation levels are measured by 153 monitoring stations of the National Radioactivity Monitoring network (NRM), see figure 1. The NRM provides point information on radiation level, dose rate more specifically, no data are available in between the stations. Interpolated maps provide estimated dose rate in between the monitoring stations and provide an estimate of the spatial distribution of dose rate. EUR 21595 EN (25), Dubois et al. (27) and Hiemstra et al. (29) explored the mapping of dose rate using geostatistics. Geostatistical mapping, i.e. kriging, has an advantage over more simple interpolation methods in that it can take in Corresponding author. Email addresses: p.hiemstra@geo.uu.nl (Paul H. Hiemstra), edzer.pebesma@uni-muenster.de (Edzer J. Pebesma) account trends and provides an estimate of the predicton error. Hiemstra et al. (29) focused on the interpolation of dose rate in non-emergency, background situations, providing a first step towards an interpolation system suitable for emergency situations. In addition, Hiemstra et al. (29) suggested to use trend information in a universal kriging (UK) approach (Chilès and Delfiner, 1999; Christensen, 1996), using soil type to improve the interpolated map. Many other studies (Knotters et al., 1995; Bishop and McBratney, 21; Bourennane and King, 23; Lloyd, 25; Yemefack et al., 25; Hengl et al., 27) showed that accounting for a trend can improve an interpolated map. Rainfall intensity is a major factor determining the spatial distribution of dose rate on a short time scale (Smetsers and Blaauboer, 1997b; Horng and Jiang, 24). Therefore, we hypothesized that accounting for rainfall intensity as predictor would improve the accuracy of the interpolated maps of dose rate. Rainfall intensity influences dose rate because radon daughter products or progeny, primarily bismuth ( 214 Bi) and lead ( 214 Pb), are washed out Preprint submitted to Science of the Total Environment August 27, 21

of clouds and the atmosphere and are deposited on the ground. Radioactive decay of the radon progeny increases dose rate at that location. The increase in dose rate at any given time is proportional to the amount of radon progeny that is decaying at that time. In turn, the amount of radon progeny is a balance between the deposition of progeny by rainfall and the radioactive decay of those progeny. We hypothesized that we could model this balance between deposition and decay by taking the weighted average over the history of rainfall intensity. Taking the average over the history of rainfall captured the deposition of radon progeny. In addition, letting the weight drop in time captured the decay of already deposited radon progeny. The rate at which the weight dropped was related to the halflife of the radon progeny. The goal of this study was to make interpolated maps of increase in dose rate based on NRM data and rainfall intensity estimated by rainfall radar. We defined the following research questions: 1. How well does the weighted averaged rainfall intensity perform as a predictor for increase in dose rate in comparison to using non-averaged rainfall intensity, measured at an individual time step? 2. Can the relationship between rainfall intensity and increase in dose rate improve our interpolated map? 3. Is there a relationship between monitoring network density and the improvement mentioned in research question 2? We fitted a linear model to a rain storm travelling over the Netherlands from southwest to northeast on the 2th of July 27. We used the goodness of fit R 2 as a measure for how well both non-averaged and weighted rainfall intensity explained the variation in increase in dose rate. Consequently, we made maps of increase in dose rate with and without accounting for rainfall, using leave-one-out cross-validation and the mean kriging prediction variance. Furthermore, we reduced monitoring network density and repeated the cross-validation procedure. 2. Methods In the study we used rainfall and dose rate data from the 2th of July 27. On this day, a large rainstorm passed over the Netherlands and caused significant increases in dose rate. 2.1. Measuring dose rate Dose rate in the Netherlands is measured at 153 locations (figure 1) every 1 minutes by the NRM (Twenhöfel et al., 25). Dose rate is commonly expressed in ambient dose equivalent rate, H (1) (ICRU, 1993), abbreviated in this study to dose rate. The unit used for dose rate in this study was nano Sievert per hour (nsv/h). Deposition of radon daughter products by rainfall increases 2 the dose rate. Consequently, we were only interested in increase of dose rate, not in the absolute value. We determined the increase in dose rate by subtracting the mean dose rate of each NRM station for the 2th of July from the 1-min dose rate data of that station. Note that the mean dose rate was calculate based on times without rainfall. Using the increase in dose rate has the added advantage of eliminating variations between stations, for example caused by calibration differences or soil type (Smetsers and Blaauboer, 1996). 2.2. Estimating rainfall intensity In this study rainfall intensity maps were estimated every five minutes using two C-band Doppler radars operated by the Royal Netherlands Meteorological Institute (KNMI) (figure 1). The radar emits radio waves and registers reflectivity. Increased reflectivity indicates more water present in the air, and thus a higher rainfall intensity. The radar provides the spatial distribution of radar reflectivity (Z, mm 6 m 3 ) for 2.5 km 2.5 km grid cells. Battan (1973) describes how the radar reflectivity can be converted to the rainfall intensity at the surface (R u, mm h 1 ). Figure 2 shows the radar rainfall intensity maps for the 2th of July 27 from 8AM to 7PM in hourly time steps. From the rainfall intensity maps we derived the rainfall intensity at the monitoring stations of the NRM. The rainfall intensity at a particular NRM monitoring station was defined as the closest cell centre of the rainfall intensity map. 2.3. Weighted averaged rainfall intensity Rainfall intensity was averaged over time using a weighted average where the weight was determined by the half-life of the deposited radon progeny. We calibrated an overall half-life to that part of the data that clearly shows the effect of radon progeny, without disturbances. We used large scale bound constrained optimization (Zhu et al., 1997) to perform the calibration, which lead to an overall half-life of 25.8 minutes. Using weighted averaged rainfall intensity assumes that the deposition of radon progeny is only influenced by rainfall intensity and not by how long it has already rained. This assumption is supported by the work of Fujinami (1996). The weighted averaged rainfall intensity (R w ) at time t was determined by: m i= R w (t) = α ir u (t i ) m i= α (1) i where t is the time for which we calculate the average, m is the number of timesteps of five minutes over which we averaged, R u is rainfall intensity measured at an individual time step and α i is the weight at t i = t i t, where t is the size of the timestep. We chose m equal to 3 because at α i=3 the weight is very low. The weight α i is determined by: ( α i = exp ln2 ) i t (2) t 1/2

where t 1/2 is the overall half-life. Figure 3 shows maps comparing R w to R u at 5PM. 2.4. Relating dose rate to rainfall intensity To determine how well R w described increase in dose rate (H) compared to R u, we investigated the temporal relationship between both R u and R w and H. For the temporal relationship we kept location constant and varied time. The temporal relationship was determined using linear regression. The assumption in linear regression is that the n observations of increase in dose rate, H, at a certain location can be described by the following linear model (Christensen, 1996): H = Xβ + e, E(e) =, Cov(e) = σ 2 I. (3) where X is the n 2 design matrix where the i-th row equals (1, R u (t i )) or (1, R w (t i )), β = (β, β 1 ), are unknown regression coefficients describing the temporal relationship between H and R u or R w, and e is the residual. We used the R 2 as a goodness of fit for the fitted regression coefficients: R 2 = 1 SS n e i=1 = 1 (H i Ĥi) 2 SS n tot i=1 (H i H) (4) 2 where SS e is the residual sums of squares, SS tot is the total sums of squares, H i is the observed H, Ĥ i is the H estimated by the linear regression and H is the mean of H. 2.5. Mapping dose rate with- and without trend We compared ordinary kriging (OK) to universal kriging (UK) (Chilès and Delfiner, 1999; Christensen, 1996) to determine whether including a spatial trend improved the accuracy of the interpolated map. Note that in contrast to section 2.4, the trend is fitted in space and not in time. An important step in kriging is fitting the variogram model to the residuals. For UK these are residuals to a trend, for OK these are residuals to a spatially constant mean. The variogram model was automatically fitted to the residuals as described in Hiemstra et al. (29). Based on the sample variogram, calculated based on the residuals, we made an initial guess of the variogram parameters, nugget, sill and range. After that, we used iterative reweighted least squares, or Gauss-Newton fitting (Cressie, 1993), to fit the variogram model to the sample variogram. We fitted a single isotropic variogram model to the entire study area. We used ordinary least squares (OLS) residuals (assuming uncorrelated residuals) instead of generalized least squares (GLS) residuals to fit the variogram model. More information on using OLS residuals to find the variogram model is found in Kitanidis (1993). We used the gstat package (Pebesma, 24) in the statistical computing environment R (R Development Core Team, 21) for all geostatistical calculations. 3 2.6. Quantifying the accuracy of the map We quantified the accuracy of the maps produced by OK and UK using three different measures. The first was the Root Mean Squared Error (RMSE) of the leave-oneout cross-validation residuals: RMSE = 1 n (Ĥcv,i H i ) n 2 (5) i=1 where n is the number of observations, Ĥcv,i is the increase in dose rate estimated by cross-validation and H i is the measured increase in dose rate. A smaller RMSE indicates a smaller error and thus a more accurate map. The second measure was the Mean Error (ME) and is defined as: ME = 1 n n (Ĥcv,i H i ) (6) i=1 ME provides an indication for the systematic error or bias in the cross-validation residuals. The third measure is the Mean Kriging Variance (MKV) defined as the mean of the kriging variance calculated for each prediction location (Christensen, 1996). 3. Results 3.1. Weighted averaged vs. non-averaged rainfall intensity To compare non-averaged rainfall intensity (R u ) and weighted averaged rainfall intensity (R w ) as a predictor for increase in dose rate (H) figures 4 and 5 show time series of these three variables for four monitoring stations. In addition, scatterplots of R u versus H and R w versus H are shown with the fitted regression line. The goodness of fit (R 2 ) is shown below the scatterplots in the x-axis caption. At all four monitoring stations the R 2 increased when we used R w instead of R u. To compare the R 2 between using R u and R w for all monitoring stations, figure 6 shows these R 2 s. Filled dots represent the R 2 value for R w, open dots the R 2 for R u. On average, R 2 increased from.17 for R u to.78 for R w. 3.2. Estimation of the spatial distribution of dose rate Figure 7 shows scatterplots of R w versus H (location varies, time is constant) between 8 AM and 7 PM in hourly timesteps. For all the hourly timesteps the fitted regression parameters are significant (p <.25). The goodness of fit of the fitted regression parameters varies between.42 and.71. Figure 8 shows variograms of R w and the correlation length in kilometers. The correlation length is determined by fitting a spherical variogram model to the sample variogram and using the range of the variogram model as the correlation length. The correlation lengths are quite large in comparison to the typical distance between the NRM stations, which is about 12 km. Figure 9 shows the fitted variogram models of H for both OK and UK between

8AM and 7PM. The fitted variogram models show a drop in both sill and range for UK in comparison to OK. The semivariance in these plots is on the log-scale, to make the differences between the fitted models more clear. Figure 1 shows interpolated maps for OK and UK for three moments in time. Table 1 shows the mean increase in dose rate ( H), root mean squared error (RMSE) of the cross-validation residuals, the mean error (ME) of the cross-validation residuals and the mean kriging variance (MKV) for OK and UK. The differences between OK and UK in terms of RMSE are small compared to H, indicating that the results for OK and UK are comparable. In addition, ME and MKV for OK and UK are comparable in size. RMSE is large in comparison to the ME, suggesting that there is no bias in the cross-validation residuals. We reduced the network density to see how this could effect the RMSE. We randomly took out 2%, 4%, 6% and 8% of the stations. We repeated this procedure a number of times for each moment in time. From these randomly reduced networks we selected the one that had the change in RMSE mose favorable to UK. Note that we kept the variogram models and the regression coefficients for cross-validating the reduced network equal to those of the full network. Figure 11 shows the results of reducing the network with panels for different reduction percentages, time on the y-axis and the change in RMSE on the x-axis. The vertical lines represent the mean values for the full and reduced network respectively. These lines indicate that the change in RMSE shifts in favor of UK when the network is reduced. 4. Discussion 4.1. Weighted averaged vs. non-averaged rainfall intensity Weighted averaged rainfall intensity (R w ) performs much better as a predictor for the increase in dose rate (H) than non-averaged rainfall intensity (R u ). The R 2 increases for all monitoring stations when we use R w instead of R u (figure 6), on average from.17 to.78. This confirms our hypothesis that taking a weighted averaged rainfall intensity is a much better description of the radon washout and decay process. The high R 2 s for R w underline the fact that rain out of radon progeny is an important process in describing the variations in dose rate. Figure 5 shows that not all monitoring stations show a high R 2. We offer two possible explanations: firstly, the observed dose rate enhancements include contributions from sources other than rainfall. The spiky patterns in e.g. figure 5(a) may well be attributed to the transport of medical radioactive sources or to radiographic screening during welding activities in the vicinity of the monitoring station. Secondly, the correlation between rainfall at surface level and the rainfall radar images may fail occasionally (figure 5(b)). For example when rainfall detected by the radar system high up in the atmosphere does not 4 reach the ground surface. In addition, reflectance from buildings or large flocks of birds can produce false rainfall patterns. EUR 26 EN (23) provides a more thorough description of complications when using rainfall radar to estimate rainfall intensity. Our estimate of the rainfall intensity could be improved by combining rainfall radar with ground measurements of rainfall intensity (Schuurmans et al., 27). Including ground measurements combines the spatial coverage of the radar images with the accuracy of ground measurements. In this study we expressed the temporal structure of the relation between rainfall intensity and increase in dose rate by taking a weighted average. An alternative approach could be to use space-time kriging, see e.g. Jost et al. (25). Space-time kriging captures the temporal aspect in defining a variogram model not only in space, but also in time. 4.2. Estimation of the spatial distribution of dose rate The accuracy in terms of RMSE, ME and MKV is more or less the same for OK and UK (table 1). Consequently, there is no significant improvement in our estimate of the distribution of dose rate when we take into account the relationship between R w and H. This is surprising given the fact that UK takes into account a significant trend with a goodness of fit of up to.71 (figure 7). We discuss this fact for MKV and RMSE seperately in the next two sections. Because ME is very small in comparison to RMSE, i.e. there is no bias in the cross-validation residuals, and we will not further discuss ME. MKV We expected MKV to drop because the fitted linear model explains part of the variance in the data (sill becomes lower), decreasing the kriging variance. To illustrate why the MKV sometimes does not drop for UK, we discuss the way the kriging variance is calculated and the role of the variogram model in this calculation (see figure 9 for the fitted variogram models). The kriging variance at a prediction location is calculated as a weighted average of the semivariance of the surrounding observations, similar to the kriging prediction. The semivariance of the surrounding observations is obtained from the variogram model and the weights are equal to the kriging weights. In our case the kriging weight is mainly distributed over the points within a 4 km radius. Consequently, the kriging variance is mainly determined by the behaviour of the variogram model in this distance interval. When the variogram model shows greater semivariances in this distance interval for UK than for OK, the MKV for UK increases. A good example of the increase when using UK are the fitted variogram models for 9AM (see figure 9). The sill drops, but the decrease in range causes the semivariance values in the range upto 4 km to be greater for UK than for OK. In conclusion: the total variance in the dataset (the sill) drops, but because of the density of the monitoring network the kriging weights are mainly distributed

over stations that are closer than the range of the variogram model. Consequently, we do not take advantage of the decrease in the sill and the MKV does not drop significantly. RMSE The correlation length of R w is large in comparison to the average distance between the monitoring stations, about 12 km. Figure 8 shows an average correlation length of 184 kilometers. So the NRM is dense compared to the correlation length of the rain storm. The density of the network allows OK to be succesful in interpolating the increases in dose rate caused by rainfall. The succes of OK is also apparent from the interpolated maps in figure 1. The interpolated maps by OK and UK broadly show the same pattern. So the density of the network causes OK and UK to perform equally well in reproducing the spatial pattern of H, and thus have a comparable RMSE in crossvalidation. In conclusion: when the monitoring network is dense in comparison to the phenomenon causing the trend in the data, the increase in accuracy when including the trend is likely to be small. This conclusion is in line with the work of Journel and Rossi (1989). In case the correlation length is smaller than the typical distance between the monitoring stations, we expect an improvement in the accuracy of the map. This is supported by the results shown in figure 11. The figure shows that for a less dense network UK has smaller RMSE values. This indicates that in a less dense network, the positive influence of adding a trend increases. In addition to a less dense network, we expect an improvement in RMSE for rain storms with a smaller correlation length. Smaller correlation lengths occur with more localized thunder storms or in mountainous regions where rain storms are restricted to the valleys. Although for the current network density UK does not perform much better than OK in terms of cross validation statistics, we consider the maps resulting from UK to be the more realistic ones. OK tends to create highly smooth surfaces, where UK shows much sharper boundaries, see for example the interpolated of 12: in figure 1. Rainfall time series show that boundaries are often sharp rather than smooth, see e.g. figure 4(b). That this more realistic pattern does not lead to better cross-validation result might be attributed to the even spread of monitoring stations over the country. Lacking monitoring station pairs at short distances prohibit the detection, and thereby validation, of sharp boundaries. Fujinami, 1996), show that wash out of radon progeny is a very important process in describing the variations in dose rate. The accuracy of maps produced by ordinary kriging (OK, no trend) and universal kriging (UK) is comparable. This is mainly caused by the density of the NRM in comparison to the scale of the rainfall radar data. When the monitoring network is dense in comparison to the phenomenon causing the trend in the data, the increase in accuracy when including the trend is likely to be small. In support of this conclusion, our results show that for networks with a decreased density the performance of UK in comparison to OK increases. In a less dense network the positive effect of including a trend increases. In addition, for rainfall patterns with a shorter correlation length, we expect to see an improved performance of UK in comparison to OK. Furthermore, we suspect that cross validating the evenly spread monitoring stations works in the advantage of OK, when the external variable (rainfall) exhibits sharp boundaries. Maps resulting from UK better follow the sharp boundaries present in rainfall, but the lack of short-distance monitoring station pairs prevents crossvalidation to reveal this effect. Acknowledgements The authors thank the Royal Dutch Meteorological Institute (KNMI) for supplying the rainfall radar data. We gratefully acknowledge financial support from the innovation programme Space for Geo-Information (RGI), project RGI-32. This work has been partially funded by the European Commission, under the Sixth Framework Programme, by the INTAMAP project Contract N. 33811 with the DG INFSO, action Line IST-25-2.5.12 ICT for Environmental Risk Management. The views expressed herein are those of the authors and are not necessarily those of the RGI or the European Commission. The authors would also like to thank Stephanie Melles and three anonymous reviewers for providing comments that improved the manuscript. 5. Conclusions Our results show that the weighted averaged rainfall intensity performs much better as a predictor for increase in dose rate than the non-averaged rainfall intensity. This conclusion, in combination with the results from literature (Smetsers and Blaauboer, 1997a; Horng and Jiang, 23; 5

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t 8: 9: 1: 11: 12: 13: 14: 15: 16: H.27 2.99 3.6 4.47 9.93 12.32 1.37 8. 5.31 RMSE OK 3.58 2.66 3.11 3.71 6.69 4.69 3.94 3.46 3.2 RMSE UK 2.85 3.18 3.4 4.18 6.19 4.51 4.4 3.29 3.5 RMSE -.73.53 -.8.47 -.5 -.18.46 -.17.3 MKV OK 9.2 9.62 11.34 12.39 26.43 15.75 13.76 11.37 8.37 MKV UK 6.95 2.7 11.8 13.6 28.48 15.29 27.83 1.43 11.5 MKV -2.26 1.45 -.26.67 2.4 -.46 14.7 -.94 2.67 ME OK..3. -.4.2.7.7.4.2 ME UK.3 -.3 -.2 -.7.4.1 -.2.3. Table 1: Mean increase in dose rate ( H), Root Mean Squared Error (RMSE) and Mean Error (ME) of the leave-one-out cross-validation residuals, the Mean Kriging Variance (MKV) for Ordinary and Universal kriging and the difference between them. We performed crossvalidation for the data between 8AM and 7PM. A negative difference for either MKV or RMSE means that UK was performing better than OK. 7

25 km Figure 1: Monitoring stations of the National Radioactivity Monitoring network ( ) and the location of the rainfall radar stations ( ). 8

Figure 2: Rainfall intensity in mm/h estimated from rainfall radar images. 9

Figure 3: Maps of rainfall intensity and weighted averaged rainfall intensity in mm/h at 5PM. 1

1 2 3 4 5 R u: Rainfall intensity (mm/h) R w: Weighted averaged rainfall intensity (mm/h) H (nsv/h) 3 2 1..5 1. 1.5 2. 2.5 1 2 3 H: Increase in dose rate (nsv/h) H (nsv/h) 3 2 1 1 2 3 4 5 R u (mm/h, R 2 =.19 ) 6: 9: 12: 15: 18: 21: time..5 1. 1.5 2. 2.5 R w (mm/h, R 2 =.9 ) (a) 5 1 15 R u: Rainfall intensity (mm/h) R w: Weighted averaged rainfall intensity (mm/h) H (nsv/h) 5 4 3 2 1 1 2 3 4 5 1 2 3 4 5 H: Increase in dose rate (nsv/h) H (nsv/h) 5 4 3 2 1 5 1 15 R u (mm/h, R 2 =.26 ) 6: 9: 12: 15: 18: 21: time 1 2 3 4 5 R w (mm/h, R 2 =.94 ) (b) Figure 4: Non-averaged rainfall intensity (R u), weighted averaged rainfall intensity (R w) and increase in dose rate (H) versus time (left), and scatterplots of R u and R w vs H (right) for two stations (a, b) that show a high correlation between R w and H. 11

R u: Rainfall intensity (mm/h) 2 4 6 8 1 R w: Weighted averaged rainfall intensity (mm/h) H (nsv/h) 6 4 2..5 1. 1.5 H: Increase in dose rate (nsv/h) 6 2 4 6 8 1 R u (mm/h, R 2 =.1 ) 2 4 6 H (nsv/h) 4 2 6: 9: 12: 15: 18: 21: time..5 1. 1.5 R w (mm/h, R 2 =.16 ) (a) R u: Rainfall intensity (mm/h) 1 2 3 R w: Weighted averaged rainfall intensity (mm/h) H (nsv/h) 15 1 5 1 2 3 4 5 H: Increase in dose rate (nsv/h) 15 1 2 3 R u (mm/h, R 2 = ) 5 1 15 H (nsv/h) 1 5 6: 9: 12: 15: 18: 21: time 1 2 3 4 5 R w (mm/h, R 2 =.14 ) (b) Figure 5: Non-averaged rainfall intensity (R u), weighted averaged rainfall intensity (R w) and increase in dose rate (H) versus time (left), and scatterplots of R u and R w vs H (right) for two stations (a, b) that show a low correlation between R w and H. 12

R 2 stations..2.4.6.8 1. Weighted averaged rainfall intensity Rainfall intensity Figure 6: Goodness of fit (R 2 ) between increase in dose rate and non-averaged rainfall intensity (open dots) and weighted averaged rainfall intensity (filled dots) per station. Note how the R 2 shifts in favor of weighted averaged rainfall intensity. 13

2 4 6 8 2 4 6 8 8 8 R 2 =.56 9 R 2 =.55 1 R 2 =.54 11 R 2 =.71 6 4 2 Increase in radioactivity level (nsv/h) 8 12 16 R 2 =.48 R 2 =.59 13 17 R 2 =.47 R 2 =.51 14 18 R 2 =.5 R 2 =.42 15 19 R 2 =.61 R 2 =.52 8 6 4 2 6 4 2 2 4 6 8 2 4 6 8 Weighted averaged rainfall intensity (mm/h) Figure 7: Weighted averaged rainfall intensity versus increase in dose rate from 8 AM to 7 PM. Line fitted using linear regression. numbers in the plots represent the goodness of fit (R 2 ). The 14

5 1 15 2 25 5 1 15 2 25 8 9 1 11.8.6.4.2. 12 range : 144 km 13 range : 187 km 14 range : 147 km 15 range : 175 km.8 Semivariance (nsv/h) range : 239 km 16 range : 263 km 17 range : 255 km 18 19 range : 221 km.6.4.2..8.6.4.2. range : 186 km range : 165 km range : 15 km range : 22 km 5 1 15 2 25 Distance (km) 5 1 15 2 25 Figure 8: Spherical variogram models fitted to weighted averaged rainfall intensity from 8 AM to 7 PM. The number in the lower right corner is the correlation length. 15

2 4 6 8 1 12 2 4 6 8 1 12 8 9 1 11 148.4 54.6 2.1 7.4 2.7 12 13 14 15 Semivariance (nsv/h) 148.4 54.6 2.1 7.4 2.7 16 17 18 19 148.4 54.6 2.1 7.4 2.7 2 4 6 8 1 12 2 4 6 8 1 12 Distance (km) OK UK Figure 9: Hourly sample variograms and fitted models for ordinary kriging (o) and universal kriging (+) from 8AM to 7PM. Note that semivariances are shown on the log-scale. 16

Figure 1: Interpolated maps of dose rate (nsv/h) of 8AM, 12AM and 5PM using ordinary kriging (OK) and universal kriging (UK). 17

19: 18: 17: 16: 15: 14: 13: 12: 11: 1: 9: 8: Time 19: 18: 17: 16: 15: 14: 13: 12: 11: 1: 9: 8: 19: 18: 17: 16: 15: 14: 13: 12: 11: 1: 9: 8: Normal Decreased density x + 19: 18: 17: 16: 15: 14: 13: 12: 11: 1: 9: 8: 2. 1.5 1..5..5 Change in Cross validation RMSE Figure 11: The effect of reducing the size of the network by 2%, 4%, 6% and 8% on the change in cross-validation RMSE between OK and UK. Time is on the y-axis, the change in RMSE on the x-axis. A negative change in RMSE means that UK is outperforming OK and vice versa. The plusses (+) show the best RMSE in favor of UK for the reduced network and the crosses ( ) show the RMSE for the full network. The vertical lines represent the mean values for RMSE for the reduced and the full network. 18