A geostatistical approach for mapping and uncertainty ... - Springer Link

Environ Earth Sci (2010) 61:491–505 DOI 10.1007/s12665-009-0360-6

ORIGINAL ARTICLE

A geostatistical approach for mapping and uncertainty assessment of geogenic radon gas in soil in an area of southern Italy Gabriele Buttafuoco • Adalisa Tallarico Giovanni Falcone • Ilaria Guagliardi

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Received: 28 April 2009 / Accepted: 3 November 2009 / Published online: 20 November 2009  Springer-Verlag 2009

Abstract Spatial distribution of concentrations of radon gas in the soil is important for defining high risk areas because geogenic radon is the major potential source of indoor radon concentrations regardless of the construction features of buildings. An area of southern Italy (CatanzaroLamezia plain) was surveyed to study the relationship between radon gas concentrations in the soil, geology and structural patterns. Moreover, the uncertainty associated with the mapping of geogenic radon in soil gas was assessed. Multi-Gaussian kriging was used to map the geogenic soil gas radon concentration, while conditional sequential Gaussian simulation was used to yield a series of stochastic images representing equally probable spatial distributions of soil radon across the study area. The stochastic images generated by the sequential Gaussian simulation were used to assess the uncertainty associated with the mapping of geogenic radon in the soil and they were combined to calculate the probability of exceeding a specified critical threshold that might cause concern for human health. The study showed that emanation of radon gas radon was also dependent on geological structure and

G. Buttafuoco (&) CNR, Institute for Agricultural and Forest Systems in the Mediterranean (ISAFOM), Rende (CS), Italy e-mail: [email protected] A. Tallarico  G. Falcone Department of Physics, University of Calabria, Ponte Bucci, 31-C, 87036 Rende (CS), Italy I. Guagliardi Department of Geological and Environmental Studies, University of Sannio, Via dei Mulini, 59-A, 82100 Benevento, Italy

lithology. The results have provided insight into the influence of basement geochemistry on the spatial distribution of radon levels at the soil/atmosphere interface and suggested that knowledge of the geology of the area may be helpful in understanding the distribution pattern of radon near the earth’s surface. Keywords Radon mapping  Uncertainty  Stochastic simulation  Radon gas in soil  Faults

Introduction Radon generates the main natural source of radiation exposure for human beings (Nazaroff 1992; Man and Yeung 1998; Ielsch et al. 2001) and health risks that are associated with inhalation and ingestion of radon’s decay products in buildings has become an important issue (Nazaroff 1992; Steinbuch et al. 1999). Radon-222 (222 Rn) is the radioactive decay product of radium-226 (226 Ra) and is a colorless, odorless, and almost chemically inert radioactive gas. Radium-226 (226 Ra) is in turn one of the products of the natural decay of uranium and is ubiquitous in rocks and soils, albeit in variable concentrations depending on the specific mineralogy (Tanner 1964; Gundersen 1992; Gundersen et al. 1992; Choubey and Ramola 1997; Choubey et al. 1997, 1999; Clamp and Pritchard 1998; Minda et al. 2009). Radon-222 has a short half-life (3.82 days) allowing it time to escape through the soil and into the air before decaying by emission of an a particle into a series of short lived radioactive progeny (Darby et al. 2005). Most radon in buildings is generated by radium decay in nearby soil and rock which then migrates through soil pores by gas-phase diffusion and advection to the interfacial region between the soil and the

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building (Nazaroff 1992). Moreover, radon gas, originating deep within the earth’s crust, can be captured in the rising gas column close to the surface and migrate to the surface by means of two mechanisms (convection or advection) (Wilkening 1980; Durrance and Gregory 1990; Ciotoli et al. 1999, 1998; Ershaidat et al. 2008). Concerning advection, radon must be carried upward by the flow of another gas (probably carbon dioxide) which has sufficient mass, because the amount of radon in the underground environment (orders of 10-10 kg m-3) is too low to form a macroscopic quantity of gas which can flow adjectively through the geological formations (Kristiansson and Malmqvist 1982; Gold and Soter 1985; Pinault and Baubron 1997; Etiope and Lombardi 1995). Since the advection mechanism is mainly associated with fault and fracture systems, surface distribution of deep or mixed gases may act as tracers of such tectonic discontinuities, such as fractures and faults of various dimensions (Ciotoli et al. 1993; Klusman 1993; King et al. 1996; Sugisaki et al. 1980; Gold and Soter 1985; Torgersen and O’Donnell 1991; Guerra and Lombardi 2001; Inceoz et al. 2006; Toutain and Baubron 1999; Walia et al. 2005; Erees et al. 2006; Choubey et al. 2007; Fu et al. 2005). Among others, He and Rn have proved to be the most reliable fault tracers (Reimer 1990; Ciotoli and Lombardi 1999; Baubron et al. 2002; Borchiellini et al. 1991). The geogenic radon gas in soil is the main source of indoor radon concentrations independent of the construction features of buildings (Kemski et al. 2001; Adepelumi et al. 2005). Mapping the geogenic soil gas radon potential would be valuable in understanding and interpreting its spatial variations because radon maps can be both a predictive tool for radon concentration when planning housing developments (Kemski et al. 2001) and for identifying areas where houses are likely to have high radon risk that should be considered as ‘radon prone’ (ICRP 1993; Council of the European Union 1996; Jo¨nsson 1997; Kemski et al. 2009). Radon entry depends on the specific building characteristics of houses, therefore, geogenic radon potential maps cannot be used for the prediction of indoor radon concentrations of individual houses (Kemski et al. 2001). Radon concentration information is obtained at more or less sparse sampling points and geostatistical methods (Matheron 1971) provide us a valuable tool to study the spatial structure of radon concentration. They take into account spatial autocorrelation of data to create mathematical models of spatial correlation structures commonly expressed by semivariograms. The interpolation technique of the variable at unsampled locations, known as kriging, provides the ‘best’, unbiased, linear estimate of a regionalized variable in an unsampled location, where ‘best’ is defined in a least-square sense (Chile`s and

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Delfiner 1999; Webster and Oliver 2007). Geostatistical techniques are commonly used to generate soil maps and have been described in a wide number of texts (Journel and Huijbregts 1978; Isaaks and Srivastava 1989; Goovaerts 1997; Chile`s and Delfiner 1999; Wackernagel 2003; Webster and Oliver 2007). Examples of such geostatistical applications in studies on radon concentration are found in Badr et al. (1993), 1996; Oliver and Badr (1995); ; Zhu et al. (1996); Durrani et al. (1997); Ciotoli et al. (1999); Oliver and Khayrat (1999, 2001); Khayrat et al. (2001); Zhu et al. (2001); Chaouch et al. (2003); Buttafuoco et al. (2007). The choice of a kriging algorithm should be primarily guided by the characteristics of the data under study (Buttafuoco et al. 2007). Asymmetry is the most common form of departure from normality (Webster and Oliver 2007) and positively skewed data are encountered in many fields. The variogram is sensitive to outliers of a few very large or small values and in such cases a more flexible approach is Gaussian anamorphosis (Chile`s and Delfiner 1999; Wackernagel 2003). This uses Hermite polynomials and allows one to transform a variable with a skewed distribution into a Gaussian variable regardless of the shape of the sample histogram. The kriging of such normalized data is referred to as multi-Gaussian kriging (Verly 1983; Goovaerts 1997). However, every kriging algorithm essentially leaves the job of spatial pattern reproduction unfinished because the kriging map is unique and smooth (Caers 2003). In order to visualize heterogeneity and assess uncertainty of radon gas concentrations in soil at unsampled locations, the single kriging estimate should be replaced by a set of alternative maps of radon gas concentrations in soil that honor sample information and try to reproduce the true spatial variability of the radon gas. Stochastic simulation is a development of geostatistics (Journel and Alabert 1989; Gomez-Hernandez and Srivastava 1990; Deutsch and Journel 1998; Goovaerts 2000; 2001), which represents an alternative modeling technique, particularly suited to applications where global statistics are more important than local accuracy. Simulation reproduces the essential statistical characteristics of data distribution, such as histogram and spatial continuity, by computing a set of alternative stochastic images of the random process and then carrying out an uncertainty analysis. Furthermore, these techniques were developed largely in direct response to the inadequate measures of spatial uncertainty associated with the classical methods and are explicitly linked to the assessment of uncertainty (Castrignano` et al. 2002). Each simulation provides a map of radon gas concentration in soil and is consistent with the known radon gas concentration, the sample histogram and the spatial continuity patterns displayed by the data. Therefore, each map is an equally probable description of

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the unknown reality. The statistical quantities, derived from post-processing of a large set of simulated images, allow uncertainty to be assessed and then the consequences of data uncertainty on decision-making to be evaluated. Error variance associated with simulation is twice as high as kriging variance (Heuvelink and Webster 2001), but optimal prediction is not the main aim of the simulation, which is uncertainty assessment. Under strict assumptions of multi-Gaussian spatial behavior, the pixel-by-pixel variance of a large number of simulations converges to the kriging variance; however, when the actual data depart from ideal multi-Gaussian assumptions, the statistics derived from simulation post-processing will be more robust in representing reality. On the other hand kriging, being an estimator, even an optimal one, is inevitably smoother than the ‘‘true’’ value, i.e. the kriged values are generally less dispersed than the actual values (Journel and Huijbregts 1978; Delhomme 1979). The main aim of this paper was to study the relationship between radon concentrations, geology and structural pattern in an area of southern Italy. To do that, the spatial structure of radon concentration in soil was explored and the geogenic radon was mapped by using a multi-Gaussian approach. Another objective of this paper was to assess uncertainty associated with the mapping of geogenic radon in soil gas by using stochastic simulation. Moreover, this paper presents one application of stochastic simulation for assessing the probability that soil radon gas concentration exceeds a critical threshold at which it may become a concern for human health.

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Materials and methods The study area: geological and structural setting The study area covers about 1,105 km2 and is located in southern Italy (Fig. 1). It consists in a plain (Fig. 2) among the thick igneous-metamorphic Paleozoic massifs of the systems Chain Coastal-Sila to North and Serre-Aspromonte to South (Antronico et al. 2002). This area, named Stretta of Catanzaro, reverts in the regional geologic context of the Calabrian Arc, an accretionary wedge caused by the Africa–Europe collision (Amodio-Morelli et al. 1976; Tortorici 1982). Considering the geological setting (Fig. 2) in the study area, the following main stratigraphic units crop out (Antronico et al. 2002): •

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The Stilo Unit (Amodio-Morelli et al. 1976) represented by Paleozoic phyllite and metalimestone (Catanzaro me´lange Unit) and by granite and granodiorite (Decollatura granites Unit) (Tansi et al. 2007). Orthogneiss Unit (Castagna Unit of Amodio-Morelli et al. 1976) made of mylonitic augen-gneiss, micaschist, and subordinately marbles (Paleozoic) (Tansi et al. 2007). The Polı`a-Copanello Unit (Amodio-Morelli et al. 1976) made of pre-Alpine basement continental slices, represented by kinzigitic gneiss and metaperidotite, arranged in reverse order (with highest-grade rocks above the lowest-grade ones) (Tansi et al. 2007).

Fig. 1 Study area and sample data set (filled circle) locations

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Fig. 2 Lithologic map showing the distribution of major rock types in the study area (after Antronico et al. 2002, modified)

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The Bagni Unit (Amodio-Morelli et al. 1976) represented by gray phyllades. The unit is involved by a metamorphism in green schists facies (Gulla` et al. 2005). The Gimigliano Unit (Amodio-Morelli et al. 1976): a ophiolite unit with a HP-LT metamorphic serpentinitemetabasite-polychrome schist-Calpionella limestone sequence (Tithonian-Neocomian). The Frido Unit (Amodio-Morelli et al. 1976) made of metamorphic rocks from middle to low degree represented by schists and gray phyllades (Gulla` et al. 2005). Above these units, the following are to be found:

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Tortoniano-Pliocene deposits formed by silty clays, sandy clays, from fine to coarse sands and by conglomerate–calcarenite–clay-evaporite succession (Tansi et al. 2007). Pleistocene continental and marine terrace deposits made of thick conglomerate successions: they represent the basin-fill deposits of the main tectonic depression (Catanzaro Trough). Holocene deposits represented by stream sediments, dunes and windy sands, deposits of landslide and by alluvial fans (Gulla` et al. 2005).

The eastern border of the surfaces to the roof of these deposits follows the coastline (Fig. 2) and is characterized by the presence of a dense stream network with river lines running perpendicular to the coast. Structural analysis reveals the study area to be a tectonic structure of first order in the geologic context of the Central Mediterranean. Such a complex is better structurally defined as tectonic depression known as the Graben of Catanzaro. This structure is delimited by important tectonic

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segments which separate the Calabrian Arc into two sectors: northern and southern (Fig. 2). On the whole, the Stretta of Catanzaro is made of a transcurrent left fault with direction N110E: the ‘‘LameziaCatanzaro Fault’’. It forms a tectonic depression which is delimited, to the north, by Amantea-Gimigliano Fault including a system of faults with average direction EW (Tansi et al. 2007). The ‘‘Lamezia-Catanzaro Fault’’ and the ‘‘AmanteaGimigliano Fault’’ are two left-lateral strike-slip faults, arranged in a right-hand en echelon pattern (Tansi et al. 2007). The ‘‘Lamezia-Catanzaro Fault’’, clearly recognizable morphologically, is characterized by well-developed escarpments, with triangular and/or trapezoidal facets, that control the drainage network (Tansi et al. 2007). The ‘‘Graben’’ finishes, to the south, with another important system of faults with orientation WNW-ESE, named ‘‘Curinga-Girifalco Line’’ (Langone et al. 2006). Finally, important systems of NE-SW faults displace both the crystalline and the neogenic units, reducing toward the sea, the Sila and Serre slopes. During the Quaternary, huge regional uplifting affected the ‘‘Graben of Catanzaro’’ (Catalano et al. 2003): since Lower Pliocene an alternation of lifting and lowering and from Upper Pliocene a general tendency to uplifting has been recorded. These events controlled the erosional-sedimentary processes during the Quaternary, giving rise to different orders of marine and alluvial terraces (Monaco and Tortorici 2000). The occurrence, in this area and in the whole Calabrian Arc, of both this intense Quaternary faulting and active crustal seismicity suggests that these phenomena might be related to each other (Tortorici et al. 1995).

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Sampling radon gas in soil Radon gas in soil was sampled at 4,420 points following an irregular scheme (Fig. 1) with a regional survey (4 samples km-2) in 2004. Soil gas samples were collected with a probe driven into the ground to a depth of 0.5–1.0 m (depending on the soil thickness) to avoid measurements being affected by atmospheric variations. Surface features (variation of ground water table level, climatic changes, etc.) affect soil gas distribution and provide a truly random component that makes soil gas interpretation a tricky issue. However, the standardization of sampling conditions, the collection of a large number of samples and an appropriate statistical treatment of data can ensure that these methods can be powerful tools for geological investigation (Klusman 1993). A soil gas survey was performed during summer in a short and dry period characterized by stable temperature, moisture and rainfall, thus minimizing the effect of climate on soil gas distribution (Hinkle 1994). Alpha activity of radon samples in soil was measured in the laboratory using the alpha-scintillation properties of silver activated zinc sulfide (Lucas 1957; Semkow et al. 1994), using the Pylon AB-5 instrument. Geostatistical approach In Geostatistics each measured value, zðxa Þ, at location xa (x is the location coordinates vector and a ¼ 1; . . .; n is the sampling points) is interpreted as a particular realization, or outcome, of a random variable Zðxa Þ. The set of dependent random variables fZðxa Þ; a ¼ 1; . . .; ng constitutes a random function ZðxÞ. For a detailed presentation of the theory of random functions, interested readers should refer to texts such as Journel and Huijbregts (1978); Isaaks and Srivastava (1989); Goovaerts (1997); Chile`s and Delfiner (1999); Webster and Oliver (2007); Wackernagel (2003). Variogram estimation and modeling An important tool in geostatistics is the experimental variogram, which is a quantitative measure of spatial correlation of the regionalized variable zðxa Þ. The experimental variogram cðhÞ is a function of the lag h, a vector in distance and direction, of data pair values ½zðxa Þ; zðxa þ hÞ; it refers to the expected value of the squared differences; one way of calculating this is reported in Eq. 1: 1 X ½zðxa Þ  zðxa þ hÞ2 2NðhÞ a¼1 NðhÞ

cðhÞ ¼

ð1Þ

where NðhÞ is the number of data pairs for the specified lag vector h. A theoretical function, known as the variogram model, is fitted to the experimental variogram to allow one to

estimate the variogram analytically for any distance h. The function used to model the experimental variogram must be conditionally negative definite to ensure that the kriging variances are positive (see later Eq. 9). The aim is to build a model that describes the major spatial features of the attribute under study. The models used can represent bounded or unbounded variation. In the former models the variance has a maximum (known as the sill variance) at a finite lag distance (range) over which pairs of values are spatially correlated. The best fitting function can be chosen by cross-validation, which checks the compatibility between the data and the model. It takes each data point in turn, removing it temporarily from the data set and using neighboring information to predict the value of the variable at its location. The estimate is compared with the measured value by calculating the experimental error, i.e. the difference between estimate and measurement, which can also be standardized by estimating the standard deviation. The goodness of fit was assessed by the mean error and the mean squared deviation ratio (MSDR). The mean error (ME) proves the lack of bias of the estimate if its value is close to 0: n 1X ME ¼ ½z ðxa Þ  zðxa Þ ð2Þ n a¼1 where n is the number of observation points, z ðxa Þ is the predicted value at location xa , and zðxa Þ is the observed value at location xa . The MSDR is the ratio between the squared errors and the kriging variance r2 ðxa Þ: MSDR ¼

n 1X ½z ðxa Þ  zðxa Þ2 n a¼1 r2 ðxa Þ

ð3Þ

If the model for the variogram is accurate, the mean squared error should equal the kriging variance and the MSDR value should be 1. After selecting an appropriate variogram model, the parameters can be used with the data to predict radon gas concentrations in soil at unsampled locations using kriging. Multi-Gaussian approach The multi-Gaussian approach allows prediction at unsampled locations regardless of the shape of the sample histogram (Verly 1983; Goovaerts 1997; Wackernagel 2003). It is based on a multi-Gaussian model and requires a prior Gaussian transformation of the initial attribute fZðxÞ;  x 2 R2 g into a Gaussian-shaped variable YðxÞ; x 2 R2 with zero mean and unit variance, such that: ZðxÞ ¼ /½YðxÞ

ð4Þ

Such a procedure is known as Gaussian anamorphosis (Chile`s and Delfiner 1999; Wackernagel 2003) and it is a mathematical function which transforms a variable Y with a

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Gaussian distribution in a new variable with any distribution. To transform the raw variable into a Gaussian one, we have to invert this function: 1

YðxÞ ¼ / ½ZðxÞ

ð5Þ

The Gaussian anamorphosis can be achieved using an expansion into Hermite polynomials Hi ðYÞ (Wackernagel 2003; Bleines et al. 2008) restricted to a finite number of terms: n X wi Hi ðYÞ ð6Þ /ðYÞ ¼ i¼0

The modeling of the anamorphosis starts with the discrete version of the curve on the true data set; then a model expanded in terms of Hermite polynomials (Eq. 6) is fitted to the discretized anamorphosis. This model gives the correspondence between each item of the sorted raw data and the corresponding frequency quantile in the standardized Gaussian scale. For a detailed description of the Hermite polynomials, interested readers should refer to Chile`s and Delfiner (1999); Webster and Oliver (2007); Wackernagel (2003). Multi-Gaussian kriging In the multi-Gaussian approach we can choose between simple and ordinary kriging. Unlike simple multi-Gaussian kriging, like disjunctive kriging (Matheron 1976), ordinary multiGaussian kriging accounts for the local mean calculated after the data located in the kriging neighborhood (Emery 2005). The use of ordinary kriging in the multi-Gaussian approach allows a weakening of the stationary assumption when using non linear geostatistical methods and makes it robust to a departure from the ideal stationarity model. The transformed data are used for interpolation at all unsampled locations as a linear combination of nðx0 Þ Gaussian data surrounding the unsampled point ðx0 Þ using the following expression: yMK ðx0 Þ ¼

nðx 0Þ X

kMK ðxa Þ yðxa Þ a

ð7Þ

a¼1

The key issue is the determination of the weights kMK a ðxÞassigned to each sample. The kriging weights are chosen so as to minimize the estimation variance r2E ðxÞ ¼ Var½Y  ðxÞ  YðxÞ under the constraint of unbiasedness that E½Y  ðxÞ  YðxÞ ¼ 0. The kriging weights are calculated by solving the following system of linear equations: 8 nðxÞ P MK > > > > kb ðxÞcðxb  xa Þ  lðxÞ ¼ cðx  xa Þ a ¼ 1;. ..;nðxÞ > < b¼1 > nðxÞ > P MK > > > : kb ¼ 1 b¼1

ð8Þ

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and the kriging variance is: r2MK ¼ 1 

nðxÞ X

kMK a ðxÞcðx  xa Þ  lðxÞ

ð9Þ

a¼1

Unbiasedness of the estimator is ensured by constraining the weights to sum to one, which requires the definition of the Lagrange parameter lðxÞ. Finally, the values of the Gaussian variable were estimated at the nodes of a 50 m 9 50 m interpolation grid and back-transformed using the inverse Gaussian anamorphosis (Eq. 4). Stochastic simulation Radon gas concentrations in soil, zðxÞ, are treated as a realization of a stochastic process, ZðxÞ and its variation is characterized by the variogram. Stochastic simulation involves drawing a large number of equi-probable images (also called realizations) that honor the sample data and reproduce statistical characteristics and spatial features. The actual realization is only one and it is possible to simulate many equally probable realizations that have the same statistical characteristics (Webster and Oliver 2007). In this way it is possible to obtain dense fields of values from sparse data, but the original data is retained. In geostatistical simulation, the sample data, the sample histogram and the model variogram are used to determine the conditional probability distribution function (cpdf) of the variable of interest. In Gaussian simulation, the cpdf is assumed to be a random function of multivariate Gaussian form, i.e. any linear combination of its variables follows a Gaussian distribution. In the stationary case, spatial distribution of the multi-Gaussian random function is entirely characterized by its mean value and its covariance. As Gaussian simulation technique requires a multi-Gaussian framework, each variable must be transformed into a normal distribution beforehand and the simulation results must be back-transformed to the raw distribution afterwards. To transform the raw distribution into a normal one the Gaussian anamorphosis is used (Eq. 5). The transformed data will be later used in the simulation process to represent the sample values on the multi-Gaussian probability function and then to estimate this function at all unsampled locations. The simulations were generated by using the conditional sequential Gaussian simulation algorithm (Journel and Alabert 1989; Deutsch and Journel 1998), which involves defining a regularly spaced grid covering the region of interest and a random path through the grid, such that each node will be visited once and only once. At each unsampled location, nearly normal score transformed data values are used to estimate the expected value of the multi-Gaussian probability distribution function for that grid node using

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ordinary kriging. Through the Gaussian assumption of spatial behavior, the form of the probability distribution function becomes known after which the probability of obtaining any possible value can be computed. A random value is drawn from that probability distribution as one possible simulation of the variable at that location. This value is then added to the conditioning data set and the calculation proceeds to the next node along the whole random path through the grid. As simulated values are conditioned both to the measured data and any nearby previously simulated values, spatial continuity is introduced into the calculation as required by the spatial covariance function. Several simulations can be produced which differ only in the random-number seed used in their construction for initiating the simulation process. As this input seed affects the random drawing of a value from the conditional probability distribution function, which will be then used to calculate each simulated value, it is quite likely that the same location will be assigned different values in different simulations. However, these alternative simulations are virtually identical in their statistical character, as each simulation, by construction, reproduces the sample data at the original data locations and the univariate sample statistics. The number of realizations was fixed at 500 because a high level of accuracy is reached only when the number of runs is sufficiently large.

Such decisions, however, are very often made in the presence of uncertainty, because the estimates of soil radon are invariably affected by error, whichever interpolation technique used. It is, therefore, critical to assess the uncertainty when estimating soil radon. One approach consists in delineating all locations where soil radon is below a maximum level in the site. This approach requires the kriging estimation of soil radon at the unsampled locations but, because of estimation error, we could declare ‘‘safe’’ a hazardous location and, conversely, ‘‘hazardous’’ a safe location (Goovaerts 1997). These two misclassification risks can be assessed from the probability that the value of the variable z (soil radon concentration) at any unsampled location x (coordinates vector) is not greater than a given threshold zk , (Buttafuoco et al. 2000). Indicating with F the conditional cumulative distribution function of probability, it follows:

Probabilistic summary of the set of simulations

2.

From the concept that replicate stochastic images are consistent with the observed data and are equally probable, it follows that pixel-by-pixel histograms, summarizing a large number of simulations, approximate the probability distribution functions corresponding to the locations. The probability of exceeding a particular threshold value can be computed from a set of simulations by counting the number of stochastic images that exceed the stated threshold and converting the sum to a proportion. It is then possible to display the spatially varying empirical probability in map form. Moreover, additional probabilistic information can be extracted from a set of simulated images, something that can be very useful in making decisions. If, instead of simply counting the number of times that each pixel exceeds a threshold value, one averages the simulated values for each pixel, it is possible to present a map of the ‘expected’ value at any given location (E-type or Expectedvalue estimate) (Journel 1983) and the related standard deviation. Decision-making in the presence of uncertainty Many surveys of radon in soil are used to as a basis for making important decisions, such as classifying risk areas.

1.

the risk aðxÞ (false positive), i.e. the probability of wrongly declaring a location x ‘‘hazardous’’, is given by: aðxÞ ¼ Prob fZðxÞ  zk jz ðxÞ [ zk ðnÞg ¼ Fðx; zk jðnÞÞ ð10Þ for all locations x such that the kriging estimate z ðxÞ [ zk . The symbol (n) means: conditional to the n sample data. the risk bðxÞ (false negative), i.e. the probability of wrongly declaring a location x as ‘‘safe’’, is given by: bðxÞ ¼ Prob fZðxÞ [ zk jz ðxÞ  zk ðnÞg ¼ 1  Fðx; zk jðnÞÞ

ð11Þ

for all locations x such that the kriging estimate z ðxÞ  zk : A logical deduction defines the risk bðxÞ where the risk aðxÞ is zero and conversely. The main practical difficulty with this approach consists in choosing an appropriate probability threshold for each type of misclassification risk (Goovaerts 1997). All statistical and geostatistical analyses were carried out by using the software package ISATIS, release 8.0.5 (http://www.geovariances.fr).

Results and discussion The descriptive statistics of the radon concentration values are presented in Table 1. Radon concentration varied spatially from a minimum value of 1.08 kBq m-3 to a maximum of 68.19 kBq m-3. The distribution of the sample values was positively skewed (Fig. 3) with values of

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Table 1 Statistics of soil radon gas concentration data Count

4,420

Mean (kBq m-3)

17.54

Stand. Dev. (kBq m-3)

13.24

Variance (kBq m-3)2

175.20

Maximum (kBq m-3)

68.19

Upper quartile (kBq m-3)

22.94

Median (kBq m-3)

14.75 -3

Lower quartile (kBq m )

7.19

Minimum (kBq m-3)

1.08

Skewness (–)

1.23

Kurtosis (–)

4.33

Fig. 4 Experimental variogram (filled circle) and fitted model (solid line) of the anamorphosed soil gas radon concentration data. Experimental variance (horizontal dashed line) is also reported. Sph(.) is the spherical model

Fig. 3 Histogram of the soil gas radon concentration

skewness of 1.23 (Table 1) and a secondary spike of values near 22 kBq m-3. Moreover, the distribution of the radon values was more peaked than a normal one with a kurtosis of 4.33 (Table 1). Also the lognormal transformed radon concentration data departed from Gaussian distribution because the result of a v2 test for normality clearly showed that the hypothesis of normality cannot be assumed at both 5% (23.69) and 10% (21.06) levels of probability because the experimental v2 was equal to 597.03. Then the radon concentration values were transformed to normality using a Gaussian anamorphosis by an expansion of Hermite polynomials restricted to the first 30 terms (Wackernagel. 2003). Therefore, all calculations were performed on the Gaussian transformed variables. No anisotropy was evident in the maps of the 2-D variograms (not shown) to a maximum lag distance of 18,000–20,000 m. To fit the shape of the experimental variogram (Fig. 4) three basic structures were combined (nested) including a nugget effect (c0 ), a spherical model (Sph) with a range a1 of 3,800 m and a spherical model

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with a range a2 of 23,900 m. The nugget effect implies a discontinuity in ZðxÞ and is a positive intercept of the variogram. It arises from errors of measurement and spatial variation within the shortest sampling interval (Webster and Oliver 2007). The spherical model (Webster and Oliver 2007) is given by: 8 "  3 # > 3 h 1 h > > : c if h\a where c is the sill and a the range. The ratio of nugget effect to total semivariance is about 12% and indicates a strong spatial dependence (Cambardella et al. 1994). The percentage of variance explained by the structure at shorter range (3,800 m) was about 46% (c1 = 0.4647) approximately corresponding to the width of the radon anomalies, while the one explained by the structure at longer range (23,900 m) approximately corresponding to the width of tectonic structures, is about 43% (c2 = 0.4313). The two structures to some extent balance the variability causes. The goodness of fit was checked by cross-validation and the results were quite satisfactory, because the statistics used, i.e. mean of the raw estimation errors, were close to 0 (0.00005), while the variance of the standardized error was 0.89. In Fig. 5a, b there is the map of radon concentration computed by ordinary multi-Gaussian kriging (Eqs. 7 and 8) and the standard deviation of ordinary multi-Gaussian kriging (Eq. 9).

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499

Fig. 5 Maps of soil gas radon concentration (a) and kriging standard deviation (b) obtained after back transform of estimates produced using multiGaussian kriging. The faults orientations are also reported (lines)

By using the same data and variogram, 500 simulations of soil radon concentrations were generated on the 50 m 9 50 m grid, which are to be viewed as statistically indistinguishable alternative realizations of the underlying, but unknown, real world. The differences among several images can provide a measurement of spatial uncertainty, because critical high values in radon concentrations will be deemed certain if observed on most of the realizations, which means that their probability of occurrence will be high. On the other hand, site characterization is generally undertaken for a specific purpose and one of our objectives was to collect sufficient information data to delineate portions of the study area, where the radon concentrations were above a critical threshold as regards safety. Although it is clear from what has been said above that the exact location of the ‘anomaly zones’ is still somewhat uncertain,

nevertheless it is possible to make some quantitative statements from the equi-probable simulations by calculating that a critical threshold is jointly exceeded at a series of locations. Figure 6 shows a way to treat the simulated images jointly by calculating the mean (Fig. 6a) and the standard deviation (Fig. 6b) of the 500 simulations at each grid node and then mapping the results. The comparison of the mean map with the one obtained by kriging (Fig. 5a) reveals the differences between the two approaches: the smoothing effect (typical of kriging) is greatly reduced in the mean simulated map, where the spatial fluctuations of radon concentration in soil are emphasized. As regards the issue of uncertainty assessment through stochastic simulation, the standard deviation map (Fig. 6b) represents a way of assessing uncertainty when the original variable is non-

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500

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Fig. 6 Maps of mean soil gas radon concentration a and standard deviation b obtained through sequential Gaussian simulation. The faults orientations are also reported (lines)

Gaussian and moreover, it allows us to overcome the drawback of kriging variance of independence from the actual sample values (Fig. 5b). Figure 7 presents one such probabilistic summary for radon concentration of 22.94 kBq m-3, corresponding to the upper quartile of the experimental data distribution. In the figure, the colored scale values represent the probability of exceeding the specified threshold value, rather than indicating estimates of soil radon concentrations. Of course, uncertainty remains regarding the actual radon gas concentration at any point, but the probability that it is above 22.94 kBq m-3 is very high. Conversely, such an occurrence is actually impossible in the blue areas. Maps of this type could be used in land management to identify the areas that have a specified probability of health risk or high indoor radon concentrations.

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Finally, we wanted to illustrate the risks of a misclassification based only on kriging estimate. Figure 8a shows the a risk map of false positive in declaring hazardous a location on the basis of the kriging estimate exceeding the critical threshold of 22.94 kBq m-3. Even inside the region where the kriging estimate is more than 22.94 kBq m-3 there is a not negligible probability that the true soil radon concentration falls short of the critical value. Conversely, Fig. 8b shows the b risk map of declaring safe zones where the kriging estimate is less than 22.94 kBq m-3, with high probability that the true value exceeds the critical threshold. In many cases, the trade off of the two types of risk is a subjective and political decision, which falls well beyond the realm of geostatistics. However, the joint availability of maps such as those in Figs 6, 7 and 8a, b (risks a and b) can assist in decision-

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501

Fig. 7 Soil gas radon concentration probabilistic map of exceeding the threshold of 22.94 kBq m-3 obtained from post-elaboration of simulated realizations. The faults orientations are also reported (lines)

making by ranking the areas most prone to anomalously high radon concentrations. Radon gas concentrations in soil were also compared with lithology and location of faults. As regards lithology, there was some evidence of a correlation between type of outcropping terrain and radon anomalies. Regional radon gas measurements at the soil/atmosphere interface showed marked variations according to the lithology of the basement. Distinct radon gas concentration characterized the regolith soils overlying the different rock types. Those differences reflected the radon emanation pattern of the bedrock underlying the soil. An overlay of the known geological map (Fig. 2) of the area on the radon concentration map (Fig. 5a) showed a good correlation. In the northern part of the area radon levels were higher and more variable than in the southern part; this observation was consistent with the fact that the northern part is mainly constituted of granitoids. The soil overlying the granite/ gneiss showed the highest radon concentration; the reason for the increase in the radon values was probably attributable to uranium content of the granitic bed rocks and its mineralization associated with granitic gneisses, while alluvial deposits, superficial continental deposits, terrace deposits, and sand and clays had the least concentration because of the sedimentary lithologies (Choubey et al. 2007). These spring systems were related to higher porosity and transitivity that allowed neither accumulation nor emanation of radon gas. On the other hand, higher radon concentration values were, in general, found to be associated with springs that were controlled by fault lineaments. In fact, the most important part of the increase of radon gas concentrations was due at the presence of different thrust/ shear planes, which are mainly E-W oriented, but also influenced by the superposition of the N-S and NW-SE

fault systems, which provided potential pathways for the emanation of radon gas from the deeper part of the crust. Therefore, also in the same tectonic settings (type of faults), there were different radon gas emissions, due to differences in lithologies. The results showed that the highest radon values tended to occur along elongated zones similar to the most representative trends obtained by geomorphological and mesostructural analyses, i.e. E-W trends and, secondarily, NW-SE orientations. The enhanced gas permeability zones were connected to the structural pattern (i.e. fault and fracture which are along the border of the graben and in the sedimentary basin) of the area; in this manner tectonics controls the leakage and the distribution of terrestrial gas in soil pores. The results also showed that the gas-bearing properties of faults depended on the enhanced permeability of fracture systems; however, due to structural heterogeneity and self-sealing processes, such permeability was not continuous throughout the faults leading to a spotted distribution of soil gas anomalies. In particular, in the northern sector, high radon concentrations occurred along elongated zones which corresponded to the most representative faults on the northern border of the graben; instead in the southern sector, results showed a spotted distribution of radon concentration in soil along the fault direction of the southern border of the graben. The normal faults in the study area were subdivided taking into account their state of activity and cumulative throw into four groups (Fig. 2): (1) active normal faults with high throw; (2) active normal faults with low throw; (3) normal faults with high throw; (4) normal faults with low throw. We have noted that active faults with high cumulative throw had a general continuous linear radon anomaly along their direction.

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Fig. 8 Maps of risk a (a) and risk b (b). The faults orientations are also reported (lines)

In the northern sector of the study area two main E-W anomalous zones are found. The first had a liner trend and showed a symmetrical gradient; the second had an irregular shape due to the superposition of two faults systems (NWSE and NE-SW) and also showed a symmetrical gradient. Main gradients were to be found in the northern borders of the graben. In the northern sector, in fact, there was a major strike-slip fault: Lamezia-Catanzaro fault (fault number 1; Tansi et al. 2007). In the NW sector, there was also the intersection of E-W and NW-SE faults, so that there were high radon concentrations. In fact, radon distribution could be explained physically as: (1) the presence of the discontinuous gasbearing fault properties; (2) the interaction of two or more faults with different directions.

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In the northern sector of the area (faults number 2, 1, and 3), we noted high radon values, probably due to the intersection of three fault systems: E-W faults of the border of the graben, NW-SE faults and N-S buried faults. In the southern sector, there were three spotted radon anomalies: the first was due to faults number 4, 5, and 6; the second to fault 4; the third to faults 4 and 7. In the area, peaks of radon anomalies were commonly found along the downthrown side of the normal faults (Tansi et al. 2000). Moreover, the goal of this study was to map faults whose presence had only been suggested by indirect stratigraphic and morphological evidence. To better understand the influence of buried structures on gas migration and concentration in the subsurface, radon gas data was compared with geo-structural data obtained by

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photo-interpretation and field-based structural mapping. Analyzing radon spatial variability in soil gas, we noted that there were some buried faults in the sedimentary basin: faults numbers 8 and 9. In the sedimentary basin bordered by E-W faults of the graben there were a considerable number of radon anomalies: the first was at fault number 10; the second at fault number 11; the third at faults number 11 and 12. Near the Tyrrhenian Coast, there were two faults buried by Holocenic deposits.

Conclusions The paper has described a geostatistical approach to studying the relationship between radon concentrations, geology and structural pattern in an area of southern Italy. Based on the multi-Gaussian approach a geostatistical technique was used to explore and map radon gas in soil. In order to assess uncertainty associated with the mapping of geogenic radon in soil, conditional sequential Gaussian simulation was used to yield a series of stochastic images, which represent equally probable spatial distributions of the radon concentration in soil throughout the study area. These maps were then combined to calculate the probability of exceeding a specified critical safety threshold that could be used to delineate the areas where houses are likely to have high radon risk. The novelty of the approach consisted in the probabilistic assessment of radon concentration, so as to enable planners to recognize explicitly and incorporate uncertainty in site characterization. Uncertainty derives from our incomplete site knowledge, whatever the available information. Therefore, rather than focusing on the ‘best’ estimation of radon concentration in soil at unsampled locations, the emphasis is placed on the likelihood that, in some places, the concentration may be hazardous for human health. Moreover, the paper provides a warning of the risks that decision makers can run when they base their strategic choices only on kriging estimates without any assessment of the uncertainty. The study showed that emanation of radon gas is dependent on two factors: geological structure and lithology. The approach described in this study provided insight into the influence of the basement geochemistry on the spatial distribution of radon levels at the soil/atmosphere interface. The results suggest that the knowledge of geology of the area may be helpful in understanding the distribution pattern of radon gas near the earth’s surface. The radon source-term of the lithologies, their uranium content, is most likely to be one parameter which determines radon concentrations in the outdoor environment.

503 Acknowledgements The authors thank the reviewers of this paper for providing constructive comments which have contributed to the improvement of the published version.

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