Environ. Sci. Technol. 1993, 27, 2035-2044
Sampling of Contaminated Soil: Sampling Error in Relation to Sample Size and Segregation Frank P. J. Lam6'vt and Peter R. Defize*
Institute of Environmental Sciences IMW-TNO and Institute of Applied Physics, Centre of Applied Statistics, The Netherlands Organization for Applied Scientific Research TNO, P.O. Box 601 1, 2600 JA Delft, The Netherlands When sampling a mixture of particles of different nature like soil, there will always be a minimal amount of variation between the samples due to the fact that the particle composition of each sample will differ. This minimal variation in sampling a particulate material is known as the fundamental error. It was previously unknown to what extent this fundamental error could influence the variance in a series of soil samples taken for analysis. On the basis of experiments with soil contaminated with cyanide, we conclude that the fundamental error only influences the variance when samples smaller than 10g are taken. When using larger samples, the segregation error determines the variance. However, there is no practically applicable method available for estimating the segregation error before sampling and analysis. Therefore, we developed a method for estimating the segregation error using a sampling board. This resulted in segregation plots which can be compared to the lot to be sampled.
Introduction In quality control of cleaned soil, both the mean concentration and the variance in this concentration are important testing criteria (1). In order to check on both criteria, samples of the cleaned soil must be taken and analyzed. Publications on the sampling of particulate materials, however, indicate that due to the fact that the samples consist of individual particles with different characteristics the variance is influenced by the size of the sample. As the variance is one of the quality control criteria for cleaned soil, it is essential to know to what extent the sample size affects the variance in concentration between samples. For this reason, we studied the effect of sample size on sampling error. In a strongly simplified model of the sampling of a particulate material, namely, the sampling of a mixture of red and white balls, the effect of sample size on sampling error can be calculated using simple binomial statistics. In the case of the sampling of a complex (and even partially unknown) mixture of different kinds of particles, like soil, the effect of sample size on sampling error cannot be predicted with simple statistics. More precise theories are needed to describe the relationship between sample size and sampling error. In a review of literature in Geophysical and Chemical Abstracts data bases, as well as in literature on this subject which we already possessed, we found only a relatively small number of publications in which the effects of sample size on sampling error have been described. The most relevant publications on this subject are shown in the reference list to this paper (214).
When comparing these publications with the problem of sampling contaminated soil, two major differences were observed: (a) The concentrations are very high (% level) compared with concentrations in contaminated soil (ppm level) and (b) for most materials, the particles are much larger than soil particles. Due to these two major differences, it was unknown if these theories could be used to predict sampling error in relation to sample size in contaminated soil. Therefore, two experiments were performed to investigate to what extent the existing theories are applicable to contaminated soil. In this paper we will give a short overview of the sampling theories, focusing on the two basic causes of sampling error when handling a particulate material: the fundamental error and the segregation error. The experimental results will be discussed and compared with the existing sampling theories. Based on these results, a supplementary approach will be presented to tackle the problem of sampling a particulate material, like contaminated soil, using an appropriate sample size. Theory The theory of sampling particulate materials is vast and complicated. It has been widely accepted that the most complete theory has been developed by Gy (2). Other references are Harnby (9), Ingamells (10,11),and Visman (3). As our main point of interest was to investigate the relation between sample size and sampling error, we compared experimental results with theoretical ones. The theories of Gy and Visman appeared to be the most useful. Gy's Theory. As mentioned by Pitard (12,131, the publications by Gy are hard to digest and difficult to read. Gy introduces a whole set of new terminology and concepts which are the building blocks of his sound and thorough theory. Here we shall only discuss in short those parts of his theory which are important for our study. Cy defines two forms of heterogeneity: constitution heterogeneity (CHL)and distribution heterogeneity (DHL). CHL is the heterogeneity that is inherent to the composition of each particle making up the lot. The greater the difference in compositionbetween each particle, the greater the constitution heterogeneity. The DHL of a lot is the heterogeneity that is inherent to the manner in which the individual fragments are distributed over the lot. Distribution heterogeneity is directly connected with segregation (a lot is segregated when the critical component is concentrated in certain parts of the lot). The distribution heterogeneity can be reduced by reducing the segregation, e.g., by mixing the lot. The mathematical definition of CHL is as follows:
* Author to whom correspondence should be addressed. t Institute t Institute
of Environmental Sciences IMW-TNO. of Applied Physics, Centre of Applied Statistics.
0013-936X/93/0927-2035$04.00/0
0 1993 American Chemlcal Society
Envlron. Scl. Technol., Vol. 27, No. 10, 1993 2095
with
bottlenecks in applications): when when
where N is the number of particles in the lot, ai is the concentration of the critical component in particle i, U L is the concentration of the critical component in the lot, Mi is the weight of the particle i, and is the mean weight of the particles; hi is called the standardized deviation of particle i. The definition of DHL is similar to that of CHL.Instead of dividing the lot into particles, the lot is divided into groups of particles. The variance of the standardized deviation of the groups is defined as DHL.The magnitude of DHL for a lot is not a fixed value but depends on the grouping that has been chosen. The fundamental sampling error plays an important role in the theory of Gy. This is the error that ‘remains’ under ideal sampling conditions due to the different characteristics of the particles. These conditions are met if the sampling procedure can be thought of as a Bernoulli experiment where each particle of the lot has an equal probability to be part of the sample. Gy showed that the relative variance a 2 of~ the fundamental error is related to CHL as follows: (3)
where P is the selection probability of a particle, and a, is the concentration of the critical component in the sample. This theoretical formula cannot be applied in practice. The formula, however, is a basis for the derivation of formulas that can be used. In fact, Gy did this and derived the following formula under certain assumptions: Cd3
C = clfg (4) M.9 where M, is the weight of the sample, c is the composition factor, 1 is the liberation factor, f is the shape factor, g is the size distribution factor, and d is the maximum diameter of particles. These parameters and the maximum diameter can be estimated from the characteristics of the particles. The quantity C is called the sampling constant and is the product of 4 parameters: c, 1,f, andg. Foreachparameter, we give a short description., C o m p o s i t i o n Factor c. The following formula holds for this factor: ~ ~ ~ (=a-with , )
d 5 dl d > dl
then then
1=1 1 = (dl/d)l/2
S h a p e Factor f. This factor is related to the shape of the particles. It is defined as the ratio of the volume of a particle passing a certain sieve to the volume of a cube passing the same sieve [Minkkinen (14)l. Gy stated that a value of 0.5 for the shape factor is a good estimate for most materials. For particles with a coin-shape, however, avalue in the range of 0.1-0.2 should be taken. Pitard (ref 12, p 160) described a method to estimate f by means of sieve experiments. S i z e Distribution Factorg. Gy (2)gave a rule of thumb on how to calculate g. Let d be the maximum diameter and d’ give the minimum diameter of the particles: if if if if
did‘ > 4 2 < did’ 5 4 1 < did’ 5 2 d/d‘ = 1
then then then then
g = 0.25 g = 0.50 g = 0.75 g = 1.00
Thus, as the variation in size diameter increases, the size distribution factor decreases. From eq 4 it follows that the relative variance of the fundamental error is inversely proportional to the sample weight. It has been found by others [Ingamells (IO), Harnby (91,Visman (311 that such a relationship holds in many practical situations. It was Gy’s merit that he gave a method to estimate the constant C , purely on the basis of material properties and an initial estimate of the concentration of the critical component. Segregation. When sampling in the field or from a lot, segregation is an important factor which affects the sampling error. Usually, it cannot be assumed that lots are perfectly mixed. This section is therefore concerned with quantifying the effect of segregation on sampling error in relation to sample size. Consider a lot of N F particles to be divided into NG subsamples. A sample is obtained by a Bernoulli experiment with selection probability P for each subsample. The selected subsamples are combined in one final sample. Then the relative variance of the final sample can be shown to be equal to
This relative variance can be compared with the relative variance of the fundamental error as follows. Define the grouping factor as
(5)
(7)
where U L is the concentration of the critical component in the lot, A, is the density of the critical component, and A, is the density of the matrix. Liberation Factor 1. The liberation factor has a value between 0 and 1 and defines to what extent the critical component is ‘liberated’ from the matrix (complete liberation: 1 = 1). Gy (2) gave a rule of thumb on how 1 can be calculated from the values of the maximum diameter d and the liberation diameter di. The latter parameter is defined as ‘the particle size of a comminution that would ensure the complete liberation of the constituent of interest’ (this definition is not very precise and one of the
This factor is almost equal to the average number of particles in a subsample when the sample size is small relative to the size of the lot. Gy showed that the following inequality between CHL and DHL holds:
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By introducing a segregation factor [ with 0 5 5 5 1,one can write
From eqs 3 and 6 it follows that azo
= (1 + y[)a$
(10)
Logically, 620 is always larger than U ~ F .How much larger depends on the value of 75. Gy stated that ‘when designing a sampling plant, it is always safe to assume that 75 = 1’ (ref 2, p 343). This was also confirmed by Pitard, who mentioned: ‘Taking into account the large amount of variographic experiments he performed between 1960 and 1975, Gy reached the pragmatic conclusion that the product of yt stays slightly smaller than 1’ (ref 12, p 188). In most cases, however, the sampling of contaminated soil will deviate strongly from the ideal sampling that is possible in a specially designed sampling plant. In the majority of cases, soil contamination will have a strongly heterogeneous nature, and therefore the amount of segregation within a lot will be high. This is illustrated by Figure 2. The sampling of contaminated soil, therefore, is one of the cases in which the assumption that 75 I 1 is not correct. A source of inspiration for our own research was the work of Visman (3). He is one of the few who did practical research into the effect of segregation on sampling error. Visman did hypothetical experiments with a ‘sampling board’. This sampling board consisted of 10 000 positions, 2500 of which were occupied by lead pellets. The sample taken from the board consisted of n neighboring positions. The sample result was the fraction of positions containing a pellet. Visman studied the variance of this fraction (a2) in relation to the distribution of the pellets over the sampling board. From these empirical experiments, he concluded that the following relationship between variance and sample size approximately holds:
= (A/M,) + B (11) In this formula, A and B are parameters and M , is the sample size. Visman called the quantity AIM, the random variance and B the segregation variance. The parameters a’
A andB should be estimated from available data. Visman’s formula was discussed by Duncan (41, who also suggested some modifications. Equations 10and 11are identical if the segregationfactor 5 is taken as a constant. The second term in eq 10 (75 a 2 ~ ) is then independent of sample size because y is directly proportional to the sample size. By taking the product of y and &, the influences cancel each other. The segregation factor is not a constant by definition, but simulation results of Duncan (4) and ourselves with a sampling board give some confidence that it is a fairly good approximation. Experimental Section Introduction. In view of the sampling theory by Gy (2), a first experiment was carried out with soil contam-
inated with cyanide. Cyanide, mainly present as ferric hexacyanoferrate(II), is a stable salt (15);it is present in the soil in the form of small particles (ferric hexacyanoferrate(II1, Fe4[Fe(CN)&, better known as Prussian Blue; it is a cyanide-iron complex which is formed during the dry cleaning of coal gas and one of the major contaminants found on former gasworks sites). It may therefore be assumed that this type of contamination behaves as a particulate material. Furthermore, the contaminated soil can be dried at low temperature and, subsequently, be
ground and divided with the aid of a rotating divider, without affecting the concentration. In this way it is possible to make an analytical sample of the normal size (50g) without allowing alarge error during sample division. Of course, this assumption must be checked by duplicate subsampling and analysis. To confirm the results of the first experiment, it was followed by a second one. In this second experiment, a soil which was very heterogeneously contaminated with cyanide was sampled and analyzed. The results of these two experiments were compared with each other as well as with the theoretical relation between sample size and sampling error. First Experiment. During the thermal cleaning process of soil originating from a former gasworks site and contaminated with cyanide and PAHs, 60 bulk samples were taken from the conveyor belt which entered the plant. It was preferred to sample the contaminated soil because sampling of the cleaned soil would lead to a relatively large number of samples with concentrations below the detection limit, which could make it impossible to study the effect of the sample size on sampling error. From each bulk sample, subsamples of 0.1, 1, 50,400, and 6500 g were taken without any further homogenizing of the bulk samples. In this way, the subsamples taken were comparable to samples of these sizes taken directly from the conveyor belt. The subsamples were dried at 40 “C and, for samples larger than the normal analytical sample size of 50 g, ground over a 1-mm sieve and divided with a rotating divider. After the original sampling from bulk samples, duplicate samples were taken during pretreatment and analysis. As a result, it was possible to estimate the different variance components due to sampling (a2(,)), due to sample pretreatment ( Q ~ ( ~ ) )and , due to analysis (a2(*)), together resulting in the total variance (a2(t)= a2(s)+ a2(p) (the number of duplicate samples and analyses was based upon earlier studies with this type of contamination). By means of an unbalanced hierarchical analysis of variance (161,we were able to give unbiased estimates of the variance components, without analyzing a very large number of subsamples. An example of the resulting sampling schemes is shown in Figure 1. Table I shows estimates of the variance components and the total variance. An intensive study of the soil and its cyanide contamination was made in order to gain information about the distribution of the cyanide in the soil. This included the determination of the particle size distribution of the soil, the distribution of cyanide over the particle size classes, a microscopic study of the contaminated particles using an optical microscope as well as a scanning electron microscope (SEMI, astudyof the form in which the cyanide is present by means of infrared microscopicanalysis, X-ray microanalysis (XRMA), and specific analytical methods to determine the amount of complex cyanide (Prussian Blue), free cyanide (CN-), and thiocyanides (SCN-). As a result of this part of the study, it was possible to obtain estimates of the parameters which were used in the formula for the fundamental error (eq 4). A direct comparison of theoretical and practical results could therefore be made. Second Experiment. To ensure a strongly heterogeneous distribution of cyanide, we used a contaminated site for the second experiment. The top soil layer of a small part of the site was carefullyremoved before sampling in order to make the contamination visible. The exper-
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Environ. Sci. Technol., Vol. 27, No. 10, 1993 2037
Sampling
etc. (60samples)
Pretreatment
etc. (105 su bsamples)
Analysis
Flgure 1. Samplingscheme for the sample pretreatment of 400-g samples. The original number of samples (60) was based on previousexperlence with the analysls of cyanide and the variance components involved (sampling, sample pretreatment, analyses). Knowing the order of magnitude of the different variance components,a hierarchicalanalysis of variance could be designed which would give the deslred power of dlscrlmlnatlon. For each series of sample sizes a dlfferent pretreatment was necessary, resulting from the size of the original sample and the amount of soil used for the analysis (=50 9). Samples smaller than 50 g were dried and, excludlngthe 0.1- and 1-g samples, ground over a 1-mm sleve; samples larger than 50 g were also dlvlded In subsamples by means of rotating dividers. Therefore, each sample slze had its own pretreatment pattern and analysls of variance. For the 4 0 0 9 samples, this resulted in 105 subsamples and 120 analyses.
Table I. Mean Concentration, Total Variance, and Variance Contributions for First Experiment
sample size (9) no. of analysed mean concentration (mg/kg) total variance (&))
0.1
1
50
400
6500
60 1194 3.09 X 10’
72 501. 2.87 x 104
90 480 8.04 x 103
120 506 2.73 x 104
50 529 5.46 X 109
Variance Components absolute 2.72 x 104 7.93 x 103 2.48 x 104 5.16 x 103 sampling ( u 2 d pretreatment ( U Z ( ~ ) ) Ob 2.37 x 103 analysis 1.51 x 103 110 90 300 relative ( % ) sampling (&) 94.7 98.6 91.0 94.5 pretreatment ( u ~ ( ~ ) ) 8.7 Ob analysis (a!$*)) 5.3 1.6 0.3 5.5 The number of analyses depends upon the pretreatment scheme used; 0.1 g, no pretreatment, no duplicate analyses (no analysis of variance); 1 g, no pretreatment, some duplicate analyses; 50 g, drying and grinding, duplicates of pretreatment and analyses; 400 g, see Figure 1;6500 g, drying, grinding, subsampling, duplicates of pretreatment, and analyses. In the 6500-gseries, only half of the original bulk samples were used. Negative estimates, which are theoretically possible due to random variation, have been replaced by zero estimates.
imental area was then divided into 60 squares of 0.25 m2. Starting from the center of each square, samples were taken from a surface of 0.0001, 0.01,0.09, and 0.25 m2and a depth of approximately 1cm. With each sampling step, the center which had already been sampled was not sampled again. This resulted in 60 samples of approximately 1, 400, 2500, and 6000 g. The sampling of 0.1-g samples took place in the laboratory. Subsampling from both the 2500- and 6000-g samples resulted in two series of 0.1-g samples. Sample pretreatment and analysis comparable to the first experiment were carried out. Because of the heterogeneity within the lot, see Figure 2, the resulting distribution of the analytical data was strongly skewed, as shown in Table 11. The unbalanced hierarchical analysis of variance used in the first experiment resulted therefore in highly unreliable estimates of the variance components. Therefore, only an estimate of the totalvariance was made. By neglecting the variance caused by sample pretreatment and analysis (these components proved to be less than 10% of the total variance in the first experiment), the total variance slightly overestimates the sampling variance. In contrast to the first experiment, the distribution and particle characteristics of the cyanide contamination were not investigated in the second experiment. Therefore, it was not possible to give estimates for the parameters in the formula of the fundamental error (eq 4)and to compare the experimental results with theoretical estimates. Any2038
Envlron. Scl. Technol., Vol. 27, No.
10, 1993
how, the results of this second experiment made a comparison irrelevant, as will be shown in the discussion. Discussion of the Results. By knowing the distribution and the nature of the cyanide contamination in the first experiment, we were able to make good estimates of the shape factor f (=0.5), the size distribution factor g (=0.25), the composition factor c (=3600 g/cm3), and the maximum particle size d (=0.8 cm). As already mentioned, the estimate of the liberation factor 1 is more difficult. Calculation of 1 from d and dl gave a very low value for 1. In line with Gy’s recommendation (ref 2, p 2621, a minimum value of 0.03 for 1 was chosen to calculate the fundamental error. In Figure 3, the coefficient of variation according to Gy’s formula (= CJF) is plotted against the sample size, together with the coefficient of variation due to the sampling calculated from the data of the first experiment. When the experimental variance for the 0.1-g samples is not taken into account, there seems to be no significant difference between the variances found for the different sample sizes. Furthermore, the shape of the experimental relationship does not support the theoretical relationship based upon Gy’s theory with respect to the fundamental error. The fundamental error is, for at least part of the investigated sample size range, larger than the experimental sampling error. As the fundamental error is the smallest possible sampling error, it is expected that the
4
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C 0 .J
4
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c
8 h
4
Q)
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8 u,
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Flgure 2. Cyanide distribution on the sampling location for the second experiment.
Table 11. Mean Concentration (mg/kg) and Total Variance for Second Experiment samde size (a) no. of analysesn mean total variance median lower quartile upper quartile minimum maximum
0.1
0.1
1
400
2500
6000
72 99.6 1.16x 105 19.5 6.5 92.0 2.0 1690
72 118 2.70 X lo6 21.5 3.0 57.0 2.0 6460
72 267 5.55 x 105 72.0 27.5 217 1.0 2770
84 116 0.64 x 104 28.8 13.8 88.5 2.0 3320
84 126 0.81 x 104 30.0 16.0 96.5 5.0 2430
84 200 3.01 X 106 36.0 19.0 147 7 2280
Variance is estimated on the basis of 60 samples. When more than one concentration was available due to duplicate sampling, the variance was calculated using the mean concentration for a sample which was analyzed in duplicate.
fundamental error is always smaller than the experimental sampling error. Therefore, neither the shape nor the position of the experimental relationship supports the relationship based upon the fundamental error. In the first experiment, the sampling error seemed to be more or less independent of the sample size for samples larger than approximately 1g. For samples smaller than
1g, the experimental variance increased with decreasing sample size. In the second experiment, with a much more strongly segregated lot, there did not seem to be a direct relationship between sample size and sampling error for the whole range of investigated sample sizes (0.1-6000 g). Therefore, it must be concluded that the fundamental error alone is of no use for predicting the sampling error. Envlron. Scl. Technol., Vol. 27, No. 10, 1993 2099
-
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Uirmmn
1000
i
‘+\ i ‘t
0.1
1
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rmmplm miz.
ice
ieee
ieeee
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Figure 3. Coefficient of variation in relation to sample size. Plotted are the experimentalresultsand the relationshipbetweenthe coefficient of variation and the sample size based on the formula of the fundamental error given by Gy. Experimental results: [(experimental variance)l/*/mean concentration]. According to fundamental error: fJF
8.1
1
ie
$00
ieee
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m r m p l m r i z m (grmrnm)
Figure 4. Coefficient of variation In relation to sample size. Plotted are the experimentalresultsand the relationship betweenthe coefficient of variation and the sample size based on Visman’s theory. Experimental results: [(experimental variance)”2/mean concentration], According to Visman’s theory: ( A M , i-
.
The experimental results clearly showed that the sampling error is dominated by the heterogeneity within the lot. Only for very small samples is the fundamental error a substantial part of the sampling error. The results of the first experiment made clear that a minimum sample size of approximately 10g must be used for the investigated lot to obtain a good estimate of the heterogeneity within the lot. Using smaller samples, the particulate hetsrogeneity started to have an important effect on total variance, and the sampling error increased due to the fundamental error. No significant increase of the sampling error was found in the sample size range of the second experiment. Compared to the first experiment, the heterogeneity of the second lot was much greater, even for small samples, resulting in a relatively small contribution of the fundamental error to the total sampling error. Although Gy’s sampling theory (2) also introduced the segregation error, which describes the variance caused by heterogeneity within the lot, Gy himself gave no practical method for calculating this sampling error. In fact, due to the use of composite samples, Gy stated that the segregationerror is normally smaller than the fundamental error (ref 2, p 343). In environmental sampling and specifically in quality control, the use of composite samples is, however, limited, and mostly individual samples are analyzed. Therefore, the segregation error is one of the main sources of sampling error. When the importance of the segregation error is recognized, it is important to know that Visman (3) did take this sampling error into account. In order to apply eq 11, it is necessary to know the variances which occur when sampling and analyzing large and small samples. By knowing these variances, it is possible to calculate the values of A and B which are used in eq 11. Based on the 2040
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1-and 6500-g samples, this resulted in A = 2.2035 X 10-8 (M, in grams). The relationship is and B = 5.1536 X shown in Figure 4 With these estimates, and the assumption that f is constant, the product -yf in eq 10 can be estimated. For 1-g samples the product equals 0.23, and for 6500-g samples -yf = 1520. The last value is much greater than l! It must be stated that by using the 0.1-gsamples instead of the 1-g samples the ‘Visman curve’ would, within the investigated sample size range, resemble much more the ‘Gy curve’ which is shown in Figure 3. Only for samples larger than approximately 1000 g would the relationship flatten. We used the estimated variance of the 1-gsamples because this estimate is more accurate than the estimated variance of the 0.1-g samples (because the variance itself is larger). When comparing the sampling error in the first experiment with the theoretical prediction according to Visman, there seems to be a large amount of similarity between the two. The more or less constant variance found for the highly segregated second lot can be explained by the fact that the segregation error is independent of the sample size. Therefore, Visman’s formula could be useful in predicting the minimum sample size to be used. However, the practical applicability of this formula is limited because the variance between samples must be known for two sample sizes. Therefore, we used the basic ideas of Visman to develop an alternative method to predict the sampling error in relation to sample size and segregation patterns. Application of Geostatistics. Although based on a different concept, segregation can also be analyzed by using geostatistics. Geostatistics is based on the assumption that data are correlated in space and/or time. When this correlation exists, it can be used to predict data at locations
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that have not been sampled (17). The existence of segregation implies the existence of spatial correlation. The fundamental error, as described in this paper, has no relation with spatial or temporal correlation. So in principle geostatistics is a useful method to investigate whether the segregation error is predominant over the fundamental error.
Spatial and/or temporal correlation can be measured by means of a semivariogram. In a semivariogram, the variance of differences between samples is plotted against the distance (in time or space) between the sample locations. The semivariograms of the first experiment (measuremente in time) show no temporal correlation. Apparently, Envlron. Scl. Technol., Vol. 27, No. 10, lQQ3 2041
64.14
54.18
64.19
0:
54.21
64.2
SEGREGATION PLOTS
Coefficient of variation : given in upper right-hand corner Segregation factor Flgure 8. Segregation plots, N = 6561, n = 81, u y = 0.4.
the scale of segregation is much smaller than the scale on which the temporal correlation can be studied. This can be explained by the amount of soil taken for the exper2042
Environ. Scl. Technol.. Vol. 27, No. 10, 1993
iments (approximately 10-15 kg each), the sampling interval (between 10 and 60 min), and the amount of soil passing through the soil cleaning plant in this period (20-
25 tons per hour). For the second experiment, two semivariograms are given in Figure 5a,b. Figure 5a is the semivariogram of the first series of 0.1-g samples which were taken in the laboratory from the 2500-g samples. Figure 5b shows the semivariogram of the 2500-g samples. In both plots the semivariogram values drop for distances larger than approximately 2.5 m. This is caused by the fact that high values are concentrated in a relative small area of the investigation site, see Figure 2. Only Figure 5b does show some correlation for data at distances less than 1.5 m. Semivariograms of other sample sizes are comparable to the ones given, showing no spatial correlation for the 0.1-g samples and some correlation for larger sample sizes.These results are in line with our conclusion that the segregation error is of much more importance than the fundamental error in sampling (contaminated) soil.
Segregation Plots We used Visman’s sampling board to apply Gy’s theory. Suppose one takes a sample of n neighboring locations from a sampling board with N locations on which a fraction of p pellets are placed at random. The relative variance of the fraction locations filled with pellets ( & I ) in the sample, disregarding the final population correction, is
or
P=
1 (nUzv+ 1)
(13)
With the aid of this relationship, it is possible to visualize the constitution heterogeneity (CHL). Suppose N = 6561 (812),n = 81, and UV =.0.4. Then it follows from eq 13 that p = 0.0717. A picture of a sampling board with 6561 locations with a fraction of 0.0717 pellets randomly distributed over the sampling board is shown in the top left corner of Figure 6. The heterogeneity of the picture is completely due to the random distribution of the pellets. It cannot be reduced. It can be enlarged, however, by distributing the pellets not randomly over the board but in a segregated pattern, as shown in the other pictures of Figure 6. The pellets are concentrated around certain locations (hot spots). Tne sampling board can be divided into 81 nonoverlapping squares of 81 positions. Then, by calculating the relative variance of these squares and using eq 10 with y = 81, one can estimate the segregation factor 4. The estimates of 5 are printed at the top right corner of each picture. The value of 5 is 0.18 in the third picture of Figure 6. Since y = 81, it follows from eq 10 that UG = 3.95UF in this case. The numerical exercises as performed above can help to get a feeling for the sampling error in relation to sample size and degree of segregation. The idea is first to calculate the relative variance of the fundamental error with eq 4; to then simulate different kinds of segregation plots using Visman’s sampling board, with the first plot representing only constitution heterogeneity; and then to calculate the variance in relation to sample size. An important point of consideration is the choice of the sample size on the sampling board (we used 81 in our example) and the size of the sampling board itself (we used 6561). All these
values can be changed and will lead to different plots which in fact visualize the lot and samples at different scales. Appropriate scales should be chosen for specific applications. We attempted to perform such an exercise for contaminated soil. The sampling constant C in eq 4 was estimated from the first experiment: C = 13.5 g/cm3 cf = 0.5, g = 0.25, 1 = 0.03, c = 3600 g/cm3,aL = O.O005g/g). For samples of 400 g, this resulted in a coefficient of variation for the fundamentalerror of 13.1% Suppose that this is the ‘true’ value of UF in the second experiment. From the data of experiment 2, a coefficient of variation of 268% has been calculated. The size of the experimental area in experiment 2 was 15 m2. A sample of 400 g took about 0.01 m2. In terms of the sampling board, this gave a value of 15/0.01= 1500 for the ratio N / n . Substituting an arbitrary value of 0.1 for p in eq 13 gave n = 524 and consequently N = 786 000. From eq 10 with y = 524, UF = 0.131, and UG = 2.68, it follows that 4 = 0.80. Unfortunately such a large degree of segregation can only be visualized by means of the sampling board by simulating 78 600 pellets (10%)concentrated in a subarea of a size just over 10% of the size of the totalarea (say 12-15 % ). Suchapicture willstrongly deviate from a picture with 78 600 pellets distributed completely at random over the complete area. Exercising with data, as has been done above, does give a rough idea of how much the pattern of contamination in experiment 2 is due to segregation, compared with the pattern that would arise if the distribution was purely random and only heterogeneous because the material under study consists of particles.
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Conclusions In the experimental part of this study, two lots of contaminated soil have been sampled. The first lot was sampled from a conveyor belt, while the second lot was a highly segregated part of a former gasworks site. From both experiments it was difficult to reach quantitative conclusions due to the uncertain assumptions which had to be made. Nevertheless, we can still obviously conclude that the distributional pattern of experiment 2 is highly due to segregation and that the influence of the fundamental error can be ignored. This can also be concluded for the first experiment in spite of the fact that the sampled lot of the first experiment was much more homogeneous. Sample sizes for the analysis of cyanide in soil should not be smaller than approximately 10 g to avoid a significant contribution of the fundamental error on the sampling error. By making segregation plots, it is possible to get an idea of the segregation in a lot to be sampled. When the estimated segregation of a lot is compared to a segregation plot with the same kind of segregation, an estimation of the segregation factor is available prior to sampling. This can be used to estimate a minimum sample size, which results in a negligible contribution of the fundamental error on sampling error.
Acknowledgments We performed part of this study for the Netherlands Integrated Soil Research Programme and were financially supported by the Netherlands Ministry of Housing, Physical Planning, and the Environment. We thank Envlron. Sci. Technol., Vol. 27. No. 10, 1993 2049
Afvalstoffen Terminal Moerdijk B.V. and t h e City of T h e Hague for their cooperation in t h e sampling, a n d Mr. M. M. Albert for carrying out t h e larger part of t h e practical work.
segregation factor (dimensionless) relative variance of the fundamental error (% ) relative variance of the final sample (% ) variance due to analysis variance due to sample pretreatment variance due to sampling total variance relative variance of the fraction locations (% )
Nomenclature concentration of the critical component in particle i (kg/kg) concentration of the critical component in the lot (kg/kg) concentration of the critical component in the sample (kg/kg) sampling constant determined by random variance sampling constant determined by segregation variance composition factor (g/cm3) sampling constant (g/cm3) constitution heterogeneity; the heterogeneity that is inherent to the composition of each particle making up the lot (dimensionless) maximum diameter of particles (m) liberation diameter (m) distribution heterogeneity; the heterogeneity that is inherent to the manner in which the individual fragments are distributed over the lot (dimensionless) shape factor (dimensionless) size distribution factor (dimensionless) standardized deviation of particle i (76) mean weight of the particles (kg) liberation factor (dimensionless) weight of particle i (kg) weight of sample (kg) number of neighboring locations in one sample (dimensionless) number of particles in the lot; number of locations on a sampling board (dimensionless) number of particles in a lot number of subsamples in a lot selection probability of a particle or subsample (dimensionless) grouping factor (dimensionless) density of the critical component (g/cm3) density of the matrix or gangue (g/cm3)
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Literature Cited (1) Bosman, R.; De Kwaadsteniet, J. W.; Lamb, F. P. J.; Versluijs, C. W. A quality control method for decontaminated soil; Soil Protection Series, Part 79; Sdu Uitgenerij: Den Haag, 1989; ISBN 90 120 6466 x (in Dutch). (2) Gy, P. M. Sampling of particulate materials, Theory and practice; Developmentsin Geomathematics 4; Amsterdam, 1982; ISBN 0 444 42079 7. (3) Visman, J. Mater. Res. Stands. 1969, 9, 8-66. (4) Duncan, A. J. Mater. Res. Stands. 1971, 11, 25. (5) Visman, J.; Lerner, M. Mater. Res. Stands. 1971,11,32-37. ( 6 ) Stange, K. Chem.-Ing.-Tech. 1954,26,150-155 (inGerman). (7) Stange, K. Chem-Ing.-Tech. 1954,26,331-337 (inGerman). (8) Stange, K. Chem.-Zng.-Tech. 1963,35,580-582 (inGerman). (9) Harnby, N.; Edwards, M. F.; Nienow, A. W. Mixing i n the
process industries; Butterworths Series in Chemical Engineering;Butterworths: London, 1985; ISBN 0 408 11574 2. (10) Ingamells, C. 0.;Switzer, P. Talanta 1973, 20, 547-568.
(11) Ingarnells, C. 0. Talanta 1978,25,731-732. (12) Pitard, F. F. Pierre Gy’s sampling theory and sampling
practice, volume Z Heterogeneity and sampling; CRC Press: Boca Raton, 1989; ISBN 0 8493 6658 5. (13) Pitard, F. F. Pierre Gy’s sampling theory and sampling practice, volume IZ Sampling Correctness and sampling practice; CRC Press: Boca Raton, 1989; ISBN 0 8493 6659 3. (14) Minkkinen, P. Anal. Chim. Acta 1987, 196, 237-245. (15) Meeussen,J. C. L.; Keizer,M. G.; de Haan, F. A.M. Environ. Sci. Technol. 1992, 26, 511-516. (16) Gower, J. C. Biometrics 1962, 18, 537. (17) Journell, A. G.; Huijbregts, Ch. J. Mining geostatistics; Academic Press: London, 1978.
Received for review September 15, 1992. Revised manuscript received June 2, 1993.’ ~~
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Abstract published in Advance ACS Abstracts, August 15,1993.
