Geostatistical Approach for Assessing Soil Volumes Requiring

Finally, to our knowledge, geostatistical modeling of polluted industrial sites has never ... At best, it is characterized through a highly dense, but...
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Environ. Sci. Technol. 2004, 38, 5120-5126

Geostatistical Approach for Assessing Soil Volumes Requiring Remediation: Validation Using Lead-Polluted Soils underlying a Former Smelting Works HELENE DEMOUGEOT-RENARD* AND CHANTAL DE FOUQUET† Eidegeno¨ssische Technische Hochschule Zu ¨ rich, Institut fu ¨r Raumplanung und Landwirtschaftsentwicklung, Ho¨nggerberg, 8093 Zu ¨ rich, Switzerland

Assessing the volume of soil requiring remediation and the accuracy of this assessment constitutes an essential step in polluted site management. If this remediation volume is not properly assessed, misclassification may lead both to environmental risks (polluted soils may not be remediated) and financial risks (unexpected discovery of polluted soils may generate additional remediation costs). To minimize such risks, this paper proposes a geostatistical methodology based on stochastic simulations that allows the remediation volume and the uncertainty to be assessed using investigation data. The methodology thoroughly reproduces the conditions in which the soils are classified and extracted at the remediation stage. The validity of the approach is tested by applying it on the data collected during the investigation phase of a former lead smelting works and by comparing the results with the volume that has actually been remediated. This real remediated volume was composed of all the remediation units that were classified as polluted after systematic sampling and analysis during cleanup stage. The volume estimated from the 75 samples collected during site investigation slightly overestimates (5.3% relative error) the remediated volume deduced from 212 remediation units. Furthermore, the real volume falls within the range of uncertainty predicted using the proposed methodology.

Introduction In most industrial countries, managing a polluted site requires two major stages (1). In a first stage, data are collected to identify the nature, the intensity, and the location of the contamination and to model the risks to human health and the environment. This investigation and evaluation stage is required to define site-specific remediation goals (i.e., maximal acceptable pollutant concentration levels). The remediation volume is then defined as the volume of soil * Corresponding author present address: Universite´ de Neuchaˆtel, Centre d’Hydroge´ologie de Neuchaˆtel, 11 Rue Emile Argand, 2007 Neuchaˆtel, Switzerland; phone: +41 (0)32-718-26-90; fax: +41 (0)32-718-26-03; e-mail: [email protected]. † Present address: Ecole National Supe ´ rieure des Mines de Paris, Centre de Ge´ostatistique, 35 Rue Saint Honore´, 77305 Fontainebleau, France. 5120

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exceeding the acceptable concentration (S). The site is remediated in a second stage. Assessing the remediation volume and the accuracy of the assessment between the two stages is of major importance for selecting appropriate remediation techniques, while taking into account the restoration costs, the environmental benefits, and the potential site use following restoration. In practice, remediation volumes are often estimated either empirically or using classical interpolation methods (e.g., inverse distance) without quantifying uncertainty. These practices lead to soil misclassifications and subsequent environmental risks: soils may remain untreated even though their pollutant concentrations exceed the acceptable concentration S. Soil misclassifications may also generate unexpected supplementary costs: polluted soils may be discovered unexpectedly during cleanup or, conversely, soils may be unnecessarily treated as their pollutant concentrations fall below S. In these two situations, soil misclassifications prevent correct assessment of remediation costs. Geostatistics provides a set of tools that allow uncertainty to be assessed, using a variographic model fitted to the spatial structure of the pollution phenomenon (2, 3). It has been applied to polluted industrial sites for over 20 years (4). However, the wide variety of the methods used (5-7) demonstrates the persistent difficulty in choosing a particular approach for a specific problem. Moreover, existing geostatistical studies of polluted soils are seldom dedicated to the assessment of remediation volumes (8-14). Finally, to our knowledge, geostatistical modeling of polluted industrial sites has never been validated with real data. Validating a model means that the adequation between the model and the physical reality is demonstrated. But in earth sciences, the major difficulty is that the physical reality is never perfectly known. At best, it is characterized through a highly dense, but still discrete (or indirect) and imperfect, sampling. A complete validation is thus only possible with a perfectly known “synthetic” reality, based for example on a stochastic nonconditional simulation. This type of validation has largely been applied within different fields of applications (nuclear wastes, petroleum engineering, environment; see, for example, ref 15), but the problem of testing the model against the physical reality is completely overlooked. This is nevertheless the most challenging question. A possibility is to test the remediation volume assesment model, based on scarce and irregular data available at the end of the investigation and evaluation stage (investigation data), against the actually remediated volume, resulting from a systematic sampling campaign executed during the remediation stage (remediation data). The difficulty is that both types of data are discrete and affected by different phenomena (sampling practice, analysis practice, soil classification mode, and soil extraction mode). In this paper, we propose a geostatistical approach that enables remediation volumes to be assessed along with the uncertainties. A key aspect of the approach is that it accounts for the specific conditions in which soils are sampled, classified, and extracted in a remediation process requiring soil excavation (i.e., ex situ remediation). The objective is to be close enough to the remediation reality to be able to test the geostatistical model built with the investigation data against the actual remediated volume deduced from the remediation data. A former smelting works underlain by soils polluted with lead has been selected for validation, as it has already been restored and the volume of remediated soils is known. 10.1021/es0351084 CCC: $27.50

 2004 American Chemical Society Published on Web 09/03/2004

(iii) Calculation of block conditional simulations of pollutant concentrations, modeling sampling and analytical errors. (iv) Calculation of probabilities of exceeding S for block pollutant concentrations. One RU probability is equal to the ratio of the number of block-simulated values exceeding S to the total number of block-simulated values in the RU. (v) Assessment of the remediation volume Ve and the remaining uncertainty outside Ve on the basis of both block simulations and block probabilities, after defining two probability thresholds R and β. Assessment of the volumes Vr and Vs is carried out by calculating the distributions of volumes Vr(i) and Vs(i) on the basis of the block simulations within the spatial limits of Ve.

FIGURE 1. Common remediation works include (1) soil classification within the envelope (Ve), according to pollutant concentrations measured in RUs; (2) excavation and transport of nonpolluted soils ([pollutant] < S, volume (Vs)) to a storage area; (3) excavation, transport, and cleanup of polluted soils ([pollutant] > S, volume (Vr)) to a remediation unit ([pollutant]: pollutant concentration, S: regulatory cutoff).

Materials and Methods Nonlinear geostatistics is required to calculate the remediation volume since it is defined as a sum of blocks or remediation units (RUs) where the measured pollutant concentrations exceed S. Among the existing nonlinear geostatistical methods, stochastic conditional simulations have been selected for their flexibility. They allow (i) the difference of sample size between the investigation data and the remediation data (change of support; see, for example, ref 16) on the volumetric assessment to be taken into account; (ii) the effect of the difference between the data collected for soil classification in the RUs and the true RU pollutant concentrations, always unknown (information effect; see, for example, ref 16), to be modeled; (iii) the effect of sampling and analytical errors affecting any data collected in the field to be modeled; and (iv) the usual soil extraction conditions to be taken into account. As illustrated in Figure 1, polluted soils are commonly classified and extracted for remediation in the following manner. Within the envelope Ve of the volume that is first delineated on the basis of investigation data, soils are classified as polluted or nonpolluted according to the remediation data collected in the RUs. The set of RUs with pollutant concentration exceeding S constitutes the volume of soil Vr that will be really remediated. However, due to residual uncertainty within the envelope, some of the RUs may still show pollutant concentrations falling below S (constituting the volume Vs). Both polluted and nonpolluted RUs are subsequently excavated from the superficial layer to the deepest layer. The polluted RUs are excavated to be treated, while the nonpolluted RUs are excavated to access the polluted RUs. Calculations Steps. The geostatistical approach adopted for estimating soil remediation volumes and the uncertainties consists of the following five major calculation steps: (i) Generation of point conditional simulations of pollutant concentrations on a fine grid, using site investigation data as conditional data. (ii) Calculation of block conditional simulations of pollutant concentrations on a coarse grid that reproduces the grid used for selecting RUs of soils requiring remediation. One block-simulated value is the average of the sum of pointsimulated values included in the RU.

Stochastic Conditional Simulations. We assume that the soil pollutant concentrations at an industrial site are described by a regionalized variable z(x), where x is a vector of spatial coordinates of R n. The variable z(x) is known at n sampling locations. z(x) can be considered as one realization of the random function {Z(x): x ∈ R n}. A simulation T(x) of the random function Z(x) is defined as one realization of Z(x), chosen in the set of all possible realizations with the same second-order moments as Z(x) (1, 4). T(x) follows the same expectation, covariance, and histogram as Z(x) but does not minimize the estimation variance: it is a nonbiased, but not an optimal estimator of Z(x). Because T(x) reproduces the spatial distribution of Z(x), the application of a cutoff S to T(x): {T(x) g S} provides a nonbiased estimator of {Z(x) g S}. Moreover, a conditional simulation T(x|n) honors the experimental data at the n sampling points. Probabilities of Exceeding S. The cumulative distribution function F(x;S) of Z(x) defines the probability that Z(x) is greater than S: F(x;S) ) P[Z(x) > S|n]. This probability can be estimated using a set of K conditional simulations. At each location x, the probability estimate is calculated as the ratio of the number of simulated values Ti (x|n) that exceed S, to the total number K of simulated values:

P[T(x) > S|n] )

1

K

∑[I(T (x|n);S)]

K i)1

(1)

i

where I(Ti(x|n);S) ) 1 when Ti(x|n) > S, otherwise I(Ti(x|n);S) ) 0. Modeling the Change of Support. The variance of the distribution of the variable z, representing pollutant concentrations measured on samples of a given size, decreases as the sample size increases (1, 2). Applying a cutoff to the distribution of z will thus provide different probabilities, depending on the sample support. In a common site restoration process, remediation volumes are first estimated using site investigation data collected from boreholes or trenches. The sample size is then small enough (order of magnitude: dm3) to consider the samples as points. Soils are subsequently classified for remediation using pollutant concentrations measured systematically in RUs, made of blocks of a regular grid. The sample support during this stage is rather large (order of magnitude: m3). The change of support between investigation stage and remediation stage is modeled as follows (Figure 2). (a) A set of conditional simulations of point pollutant concentrations TP(x|n) is generated on a fine grid, using the variogram model fitted to the site investigation data. (b) A coarse grid, consisting of blocks of same shape, orientation, and dimension as the RUs, is superimposed on the fine grid. For each simulation, point-simulated values included in blocks are averaged. The resulting averages VOL. 38, NO. 19, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 2. Change of support and information effect are modeled while assessing remediation volume, by simulating pollutant concentrations on different supports. Site investigation data are modeled by point simulations at the nodes of a fine grid. Real remediation data, collected for soil classification, are modeled by block-simulated values, averaging l point-simulated values included in the RU (e.g., l ) 5). True RU pollutant concentrations are also modeled, averaging all the L point-simulated values included in the RU (e.g., L ) 49). represent block-simulated values. In this way, each point simulation provides one block simulation, TV(x|n), such that

TV(x|n) )

1

l

∑T

l j)1

Pj(x|n)

1

K

∑[I(T

K i)1

Vi(x|n);S)]

(3)

where I(TVi(x|n);S) ) 1 when TVi(x|n) > S, otherwise I(TVi(x|n);S) ) 0. TVi(x|n) are the block-simulated values in each RU and K is the total number of simulations. Modeling Sampling and Analytical Errors. Blocksimulated values cannot be regarded as realistic without accounting for the presence of sampling and analytical errors that may affect the data. Sampling errors may be modeled by a variable (x), which is added as follows:

HV(x|n) ) TV(x|n)[1 + (x)]

(4)

The variable (x) is a relative error chosen as centered, between -1 and +1, and independent of TV(x|n). Ultimately, the probabilities of exceeding S while taking the change of support, the information effect, and the sampling and analysis errors into account are calculated according to

P[HV(x) > S|n] )

1

K

∑[I(H

K i)1

Vi(x|n);S)]

(5)

where I(HVi(x|n);S) ) 1 when HVi(x|n) > S, otherwise I(HVi(x|n);S) ) 0. Modeling the Information Effect. Samples used for soil classification at the remediation stage are assumed to be representative of the RUs. These samples are often composites of small size samples, considered as points. However, 5122

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CV(x|n) )

L

1

∑T L

Pj(x|n)

(6)

j)1

These RU simulations are used to calculate the probabilities, similarly to eq 3:

(2)

where TPj is a point simulation, l is the number of pointsimulated values included in the RU, and V indicates the RU (Figure 2, l ) 5). (c) The regulatory cutoff S is subsequently applied to the distribution of block-simulated values in each block to calculate the probabilities that block pollutant concentrations exceed S:

P[TV(x) > S|n] )

even when a careful sampling has been carried out, a difference always exists between the pollutant value measured in the composite samples and the true yet unknown value of the RUs (2, 16). Applying S to the distribution of pollutant concentrations measured within composites will therefore provide a different result than the same operation that would be performed, if possible, on the distribution of the true RU pollutant concentrations. This information effect is studied as follows (Figure 2): (i) Pollutant concentrations in composites are simulated using conditional simulations of point pollutant concentrations generated on the fine grid, as described in the prior paragraph (eq 2). A simulated value for a composite is calculated as the average of a selection of l point-simulated values included in the RU. Block simulations are used to calculate the probabilities (eq 3) and the remediation volumes (eqs 8, 10, 11, 14, and 15). (ii) The same set of point conditional simulations is subsequently used to calculate simulations of the true RU pollutant concentrations. One true RU pollutant concentration is simulated by averaging all the L point-simulated values included in a RU:

ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 38, NO. 19, 2004

P[CV(x) > S|n] )

1

K

∑[I(C K

Vi(x|n);S)]

(7)

i)1

where I(CVi(x|n);S) ) 1 when CVi(x|n) > S, otherwise I(CVi(x|n);S) ) 0. CVi(x|n) are the true block-simulated values in each RU, and K is the total number of simulations. The RU simulations and these probabilities are then used to calculate the remediation volumes, as described eqs 9, 12, 13, 16, and 17. (iii) Probabilities and volumes calculated with the two different models of blocks, sample composites, and true RUs, are compared. Assessing Volumes. Block conditional simulations and probabilities of exceeding S provide the basis for calculating the remediation volumes, while accounting for real cleanup conditions. Assessment is performed in two steps (Figure 1): (i) The envelope of polluted soils Ve is delineated at the end of risk assessment, using site investigation data. Probabilities that RU pollutant concentrations exceed S allow for such a delineation. The maximum uncertainty accepted by the decision makers in Ve is defined in terms of a probability threshold β. For the block simulations HV(x|n), Ve is subsequently defined as the sum of the unit volumes Vb of the RUs where the probabilities exceed β:

Pmax(S, β) ) P[HV(x) > S|n] g β

(8)

plus the unit volumes Vb of the RUs where the probabilities fall below β but that have to be excavated to access the first type of RUs. These last RUs are located above the first ones. For the true block simulations CV(x|n), Ve is defined similarly as the sum of the unit volumes Vb of the RUs where the probabilities exceed β: true Pmax (S, β) ) P[CV(x) > S|n] g β

(9)

plus the unit volumes Vb of the RUs where the probabilities fall below β but that have to be excavated to access the first type of RUs.

misclassification are modeled as probabilities of exceeding S, but they may also be translated in volumetric uncertainties as follows. (i) Outside Ve. A second probability threshold R, where R < β, is applied to the RU probabilities. RUs with a probability less than R:

Pmin(S,R) ) P[HV(x) > S|n] e R

(14)

are considered to pose no risk (i.e., risk of soil misclassification is considered as low enough to leave them on site without any additional investigation). These RUs constitute the lowrisk volume Vlr. RUs with probabilities greater than R and less than β: FIGURE 3. Different estimated volumes are defined in relation to three thresholds. The low-risk volume (Vlr), the uncertain volume (Vu), and the envelope of remediation volume (Ve) are defined according to the upper and lower probability thresholds r and β. Within Ve (i.e., where probabilities of exceeding S are greater than β), the remediated volume (Vr) and the volume of clean and stored soils (Vs) are defined according to the pollutant concentration threshold S. (ii) Within Ve, RUs are generally classified according to remediation data (Figure 3). Due to residual spatial uncertainty, the pollutant concentration of some of the RUs falls below S, while the pollutant concentration of others exceeds S. The first type of RU is excavated in order to access the polluted material, while the second type of RU is excavated for cleanup. The set of excavated but “clean” RUs is the volume of stored soil, Vs. The set of RUs to be treated constitutes the remediated volume, Vr. For each blocksimulation HVi(x|n), the volume Vs(i) is calculated as the sum of the unit volumes Vb of the t block-simulated values falling below S within Ve:

Vs(i) )

∑V [1 - I(H b

Vit(x|n);S)]

(10)

t

The calculation algorithm scans the RUs from the deepest layer to the superficial layer to take account for the deepest polluted RUs. Similarly, for each block simulation, the volume Vr is the sum of the unit volumes of the t block-simulated values exceeding S within Ve:

Vr(i) )

∑V I(H b

Vit(x|n);S)

(11)

t

These calculations lead to two distributions of volumes Vs(i) and Vr(i). Depending on the skewness of the distributions, the mean or the median provides an estimate of the actual volumes Vs and Vr excavated from Ve. In the same way, true distributions V true s (i) and V r (i) are calculated within the volume Ve estimated for the true block simulations CV(x|n):

Vtrue s (i) )

∑V [1 - I(C b

Vit(x|n);S)]

(12)

t

Vtrue r (i) )

∑V I(C b

Vit(x|n);S)

(13)

t

Volumetric Uncertainties. Soil misclassifications can be found both in zones outside and inside Ve. These misclassifications have associated environmental and financial risks. The major environmental risk consists of omitting soils with pollutant concentrations exceeding S outside Ve, which should have been excavated for cleanup. In contrast, the major financial risk involves unnecessarilly treating soils with pollutant concentrations below S within Ve. Risks of soil

Pu(S,R,β) ) R < P[HV(x) > S|n] < β

(15)

are considered to pose a risk (i.e., the risk of soil misclassification is insufficient to incorporate them into Ve, but probabilities are not low enough to permit the soils to be left without any additional characterization). This set of RUs defines the uncertainty remaining outside Ve, in the form of a volume Vu, called the uncertain volume. Similar equations are applied to estimate the low-risk volume and the remaining volumetric uncertainty for the true block simulations CV(x|n): true Pmin (S,R) ) P[CV(x) > S|n] e R

(16)

Ptrue u (S,R,β) ) R < P[CV(x) > S|n] < β

(17)

(ii) Within Ve. The uncertainty associated with the remediated volume Vr and the stored volume Vs is modeled using the dispersion of the distributions Vr(i) and Vs(i) true (respectively, V true r (i) and V s (i) for the true block simulations CV(x|n)). The higher the dispersion of the distributions, the higher the risk of soil misclassification within Ve. This may be quantified by the interquartile range [Q25% - Q75%] or by the coefficient of variation σ/µ, where σ is the standard deviation and µ is the mean of the distributions. Study Site. The approach for estimating the remediation volume and its accuracy was tested using data collected at a former smelting works. Lead was spread accross the site from a chimney without any filter. The soil polluted with lead has already been remediated. The site covers an area of 3 ha. The uppermost meters of soil consist of gravels and argillaceous sands, with the water table occurring approximately 2 m below ground. The groundwater flow direction is west/southwest, toward the stream that crosses the site. Prevailing wind directions at the site is west to north/ northeast. Using 75 homogeneous soil samples (Figure 4), an investigation and risk assessment study concluded that soils with Pb concentrations exceeding 300 mg/kg had to be remediated, due to risk of dust inhalation. The envelope of soil requiring remediation was empirically determined at the end of the investigation stage and corresponded to a volume V real of 7830 m3 of soil. e Within the V real envelope, soils were subsequently clase sified based on the results of chemical analysis of 212 composite samples, collected in each RU (10 m × 10 m × 0.30 m) of a grid including three layers. RUs with [Pb] > 300 mg/kg were excavated, layer by layer, and treated on site by soil washing. Whenever a soil Pb concentration was less than 300 mg/kg, RUs were not excavated, and the underlying RUs remained untouched. The polluted RUs discovered in these classification conditions represented V real ) 3902 m3 of soil. r It is apparent that these classification conditions were less VOL. 38, NO. 19, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 4. Location map of the investigation data obtained at the former smelting works (distance unit: m). The semivariograms in the horizontal (northeast and northwest) and vertical directions with the fitted model are displayed at the bottom of the figure. than optimal and that a risk that polluted blocks may remain unremediated existed, especially at the greater depths.

Results and Discussion Spatial Distribution. As it is commonly observed with pollution phenomenon, the distribution of the 75 Pb concentrations collected at the investigation stage was highly skewed and did not show any evident spatial structure. However an anisotropic three-dimensional spatial structure could be shown on the Gaussian anamorphosed transformed data and modeled by the following semivariogram γ(h) (Figure 4):

γ(h) ) 0.1 + 0.5 Sph(50 mNE, 70 mNW, 0.7 mVert) + 0.65 Sph(140 mNE, 1000 mNW, 4 mVert) (12) where h is a vector of distance, Sph is an anisotropic spherical variogram model with different ranges according to the directions NE (northeast), NW (northwest), and Vert (vertical). This anisotropy is consistent with the prevailing wind directions, which may be responsible for the dispersion of dust from the chimney of the smelting works. Stochastic Conditional Simulations. Conditional simulations of the Pb concentrations were generated with the turning band method, applied within the framework of a multi-Gaussian model (16). A total of K ) 200 conditional simulations of point Pb concentrations were generated on a fine grid oriented according to the anisotropy axis, in a unique neighborhood. Each fine grid mesh had a 1.43 m side and 0.30 m height, so that each RU of the coarse grid, applied at the remediation stage, included 49 of these fine grid meshes (Figure 2). The composite samples used in the field to select the polluted RUs consisted of five small grab samples, considered as points: four of these samples were taken at the corners; the fifth was taken at the center of each RU. Each block-simulated value was subsequently calculated by 5124

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averaging the five point-simulated values within the RU. The value of an independent variable, , was subsequently added to each of these block-simulated values. The  variable was taken to have a uniform distribution over the interval [-1;+1], whose variance is large (0.33), thus permitting the presence of a high sampling and analytical error to be modeled. Validation. To test the validity of the geostastistical approach, the distribution of the block conditional simulations was used to calculate a distribution of volumes Vr(i) within the volume V real e that had been empirically delineated at the end of investigation and risk assessment stage (Figure 6). Soil classification conditions applied during cleanup (i.e. a RU was not analyzed or excavated if Pb concentration of an upper layer block was less than 300 mg/kg) were modeled. Details of the volumetric estimation per layer of the excavation zone (Figure 5) show that the remediation volume is estimated with different levels of precision and accuracy in the three layers. The precision and accuracy are lower in the deepest layer (0.60-0.90 m) than in the surface (0-0.30 m) and subsurface layers (0.30-0.60 m). This is largely a consequence of investigation data scarcity in the deepest layers, which is responsible for high uncertainty on the vertical semivariogram model at the greater depths (Figure 4). Let us consider now the overall estimation of the volume, including the three layers. Table 1 shows the elementary statistics of the distribution Vr(i). The median Vr* of the distribution Vr(i) can be taken as an estimate of the real volume V real r , as its distribution is only slightly skewed. This estimate is equal to Vr* ) 4118 m3, while the volume of soil actually determined to be polluted within Ve is equal to V real r ) 3902 m3. The difference between V real and Vr*, as mear sured by the ratio:

- Vr*|/Vreal |Vreal r r

(18)

is equal to 5.3%. This small difference between V real and Vr* r shows that the proposed geostatistical approach is a means of estimating the remediated volume Vr, with an acceptable degree of accuracy, within the volume of soils that had been really excavated for cleanup. Furthermore, the low coefficient of variation of the distribution Vr(i) shows that the precision of the estimation is acceptable and, thus, that the risk of soil misclassification is satisfactory. Sensitivity Analysis. To study the influence of the volumetric uncertainties and soil classification conditions on the volume estimates, the volumes Ve and Vu were calculated for different probability thresholds β (from 20% to 90%), RU sizes (10 × 10 × 0.3 m and 20 × 20 × 0.3 m) and sample composites representative of RU (models for true RU, 5-point samples composite, 1-point sample). Volume estimates are compared calculating the ratios: |Vel - Vew|/ Vel and |Vul - Vuw|/Vul, where l and w denote the number of the lines in Table 2 providing the volume estimates that need to be compared. Calculations were performed as described in the Materials and Methods section, so that improved soil classification conditions during cleanup were modeled. In that way, calculations were slightly different from those performed during the validation process, when real soil classification conditions were modeled. Table 2 gives the volumetric estimates Ve and Vu for different probability thresholds β, when the other parameters R, size of the RU, and sample composite are fixed. First, the high dependence between the estimated volumes and the maximum level of uncertainty, defined as a maximum probability threshold β, is demonstrated. For given parameters R, size of RU, and sample composite, Ve decreases while Vu increases as β increases. For example, for R ) 20%, 10 × 10 × 0.3 m RU, and a model of true RU concentrations, Ve decreases from 39379 to 3968 m3 (line 1, Table 2), while Vu

FIGURE 5. Distributions of the calculated remediation volumes Vr(i) within the real excavated zone, layer by layer. The black vertical lines figure the volumes of soil that have actually been remediated in each layer. 10 × 10 × 0.3 m (lines 1 and 2) to 20 × 20 × 0.3 m (lines 3 and 4), Ve is higher and Vu is nearly stable. The difference between the Ve estimates, measured by the ratio |Vel - Ve3|/ Ve1el, goes from 5 to 30% as β increases from 20% to 90%. If the 10 × 10 × 0.3 m RUs are modeled by sample composites of 5-point samples (lines 5 and 6) or by unique point samples collected in the RUs (lines 7 and 8), volume estimates Ve and Vu are smaller. The difference between the Ve estimates, measured by the ratio |Ve5 - Ve7|/Ve5 goes from 18 to 43% as β increases from 20% to 90%. The ratio |Vu6 - Vu8|/Vu6 goes from 0 to 16% as β increases from 20% to 90%. Third, the information effect is quantified, comparing the volumes estimated with a model of composite samples to the volumes estimated with a model of true RU. For R ) 20% and 10 × 10 × 0.3 m RUs, the volume Ve calculated for true RU pollutant concentrations (line 1) is higher than the volume Ve assessed for 5-point sample composites (line 5). The difference between the Ve estimates, measured by the ratio |Vel - Ve5|/Vel goes from 10 to 37% as β increases from 20% to 90%. Conversely, the volume Vu is smaller with the true RU values (line 2) than with sample composite RU values (line 6). The difference between the Vu estimates, measured by the ratio |Vu2 - Ve6|/Ve6 goes from 0 to 6% as β increases from 20% to 90%.

FIGURE 6. Map of the probabilities that RU Pb concentrations exceed 300 mg/kg, for the superficial layer (0-0.30 m depth) of the former smelting works. The envelope of the real excavated zone is superimposed (distance unit: m).

TABLE 1. Elementary Statistics of the Distribution of the Calculated Remediation Volume Vr(i) within the Real Excavated Zonea no.

min

max

mean

median

SD

coeff of variation

200

2074

5772

4181

4118

617

0.15

a

The geostatistical approach for assessing remediation volume presented in this paper could be validated with the data collected from the smelting works, provided that the different phenomena affecting the representativeness of the data were taken into account. For polluted site managers, this validation means that they can be confident using such a geostatistical approach to assess the remediation volume at the end of the investigation stage. The validation shown for the smelting works is nonetheless partial, since it is limited to the interior of the real zone where soils had been classified and excavated. The approach was not validated outside this

Including three layers of 0.30 m depth. Volume unit: m3.

increases from 0 to 33908 m3 (line 2, Table 2) as β increases from 20% to 90%. Second, the effect of soil classification conditions is quantified. If the size of the remediation RU is changed from

TABLE 2. Ve and Vu Are Calculated for Different Values Given to Probability Threshold β, the Sample Support, and the Information Effecta β (%) line no.

r ) 20%

1 2 3 4 5 6 7 8

10 × 10 × 0.3 m RU true RU [Pb] 20 × 20 × 0.3 m RU true RU [Pb] 10 × 10 × 0.3-m RU [Pb] of a composite of 5 point samples 10 × 10 × 0.3-m RU [Pb] of one centered point sample

Ve (m3) Vu (m3) Ve (m3) Vu (m3) Ve (m3) Vu (m3) Ve (m3) Vu (m3)

20

30

40

50

60

70

80

90

39379 0 41483 0 35501 0 29098 0

29669 8898 31984 8537 26272 8507 20982 7365

23447 14669 25611 14910 19930 14369 15691 12595

19118 18757 20681 19479 15511 18727 10431 17856

15541 22335 17194 22966 10912 23327 6222 22064

10942 26934 13347 26814 6222 28016 3667 24629

6403 31473 9138 31022 3908 30331 2645 25641

3968 33908 5170 34990 2495 31743 1413 26814

a Compared to the initial situation described on lines 1 and 2, one (boldface) parameter is changed while the two others are fixed. In lines 5-6 and 7-8, volumes are calculated for two types of samples in RUs, modeling a high sampling error ((x) has been chosen as a uniform distribution on the interval [-1, +1]).

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zone, since no remediation data were available. Superposition of the real zone delineated for excavation, with the map of probabilities of exceeding S (Figure 6), shows that the extent of the central domain of highest probabilities spreads out of the zone. Moreover, one high probability (g70%) hot spot is located at the west side outside the zone. It may thus be possible that the extent of the domain requiring remediation is larger than the real excavated zone. Unfortunately, no field data allow this assumption to be confirmed. The fact that assessing remediation volume with the approach described above requires definition of a maximal probability threshold constitutes both an advantage and a complication. Disposing of a tool for choosing accepted risks of soil misclassification, and as a result, environmental and financial acceptable risks with regard to various restoration scenarios, while acounting for decision-makers’ profiles, economical constraints, and possible reuse of the remediated site, is an advantage. However application of the approach raises the following issue: what is a site-specific acceptable risk of error? Human health and environmental risk assessment can then be helpful, along with a sensitivity analysis of the volumetric estimations to the probability thresholds. In any case, the approach emphasizes the need for thorough discussions on uncertainty in the decision-making process. Furthermore, the strong influence of the probability thresholds on the volumetric estimations presented in this paper suggests that spatial uncertainty should also be taken into account in the regulatory remediation goals, along with the well-established regulatory site-specific cutoff S, as a means to guarantee an acceptable reduction of risk to human health and the environment after cleanup. Defining probability thresholds could thus provide a possible means of quantifying this spatial uncertainty. Similarly, the support effect and the information effect demonstrated in the lead smelter site show that the sole definition of the soil remediation volume does not guarantee an acceptable reduction of risk after cleanup. Two solutions could nevertheless be envisaged to overcome this difficulty. The estimation of the remediation volume could be accompanied by the definition of classification and excavation conditions to be applied. Alternatively, an estimation of the mass of pollutant that will be removed from the excavated volume could be specified, so that the remediation goals can be achieved, whatever the soil classification conditions encountered. Selectivity curves, such as those used in mining industry for ores (2), could be plotted for the excavated volume of soil as a function of the pollutant mass removed, taking into account the size of the excavated blocks. These curves could then be used to select the best earthwork conditions to be used to reach specified remediation goals.

Acknowledgments The research was performed with financial and technical supports from Gaz de France (GDF) and the Agence de l’Environnement et de la Maıˆtrise de l’Energie (ADEME). The

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data of the former smelting works were provided by the ADEME. We express our gratitude for constructive comments and support provided by Philippe Renard, Martine Louvrier, Philippe Be´gassat, and Raymond Flynn.

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Received for review October 7, 2003. Revised manuscript received July 20, 2004. Accepted July 21, 2004. ES0351084