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Keywords:

  • Analytical precision;
  • differentiation power;
  • laser diffraction;
  • log-ratio analysis;
  • particle-size distribution

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. References
  9. Supporting Information

This paper presents a methodological framework for inter-instrument comparison of different particle-size analysers. The framework consists of: (i) quantifying the difference between complete particle-size distributions; (ii) identifying the best regression model for homogenizing data sets of particle-size distributions measured by different instruments; (iii) quantifying the precision of a range of particle-size analysers; and (iv) identifying the most appropriate instrument for analysing a given set of samples. The log-ratio transform is applied to particle-size distributions throughout this study to avoid the pitfalls of analysing percentage-frequency data in ‘closed-space’. A Normalized Distance statistic is used to quantify the difference between particle-size distributions and assess the performance of log-ratio regression models. Forty-six different regression models are applied to sediment samples measured by both sieve-pipette and laser analysis. Interactive quadratic regression models offer the best means of homogenizing data sets of particle-size distributions measured by different instruments into a comparable format. However, quadratic interactive log-ratio regression models require a large number of training samples (n > 80) to achieve optimal performance compared to linear regression models (n = 50). The precision of ten particle-size analysis instruments was assessed using a data set of ten replicate measurements made of four previously published silty sediment samples. Instrument precision is quantified as the median Normalized Difference measured between the ten replicate measurements made for each sediment sample. The Differentiation Power statistic is introduced to assess the ability of each instrument to detect differences between the four sediment samples. Differentiation Power scores show that instruments based on laser diffraction principles are able to differentiate most effectively between the samples of silty sediment at a 95% confidence level. Instruments applying the principles of sedimentation offer the next most precise approach.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. References
  9. Supporting Information

This paper presents a methodological framework for the inter-comparison of particle-size analysis instruments. The aims of this paper are to provide solutions to the following interconnected issues:

  1. Quantifying the difference between complete particle-size distributions.
  2. Adjusting for differences measured between particle-size distributions analysed by different instruments.
  3. Quantifying the precision of different particle-size analysers.
  4. Objectively identifying the most appropriate instrument for a given set of sediment samples.

The methods used by this study explicitly acknowledge the limitations of analysing percentage-frequency data using traditional multivariate statistical techniques. An alternative approach is offered by a suite of techniques known collectively as ‘compositional data analysis’. Compositional data analysis has been applied to particle-size data for over 30 years, yet remains a comparatively unknown approach within the field of sedimentology (Aitchison, 1982, 1986, 1999, 2003; von Eynatten, 2004; Weltje & Prins, 2003; Jonkers et al., 2009; Tolosana-Delgado & von Eynatten, 2009, 2010; Weltje & Roberson, 2012). Many other examples of compositional data analysis exist in geoscience, including: mineral compositions of rocks (Thomas & Aitchison, 2006; Weltje, 2002, 2006), pollutant profiles (Howel, 2007), pollen populations (Jackson, 1997) and trace element compositions (von Eynatten et al., 2003). Initially, it is helpful to provide a brief introduction to compositional data analysis and its relevance to inter-instrument comparison of particle-size analysers using a short example.

The mathematical properties of particle-size distributions

To appreciate the best way to analyse particle-size distributions, expressed as discrete particle-size percentage values, it is necessary to highlight some of their fundamental (often obvious) mathematical properties.

  1. Particle-size distributions contain relative information about the proportions of different particle sizes in a sediment sample.
  2. Particle-size categories are expressed as percentages, so are greater or equal to zero and less than or equal to 100%.
  3. Particle-size distributions sum to 100%.

These properties are very useful in that they normalize measurements of particle mass or particle volume, allowing for widespread data comparison; however, they also represent serious obstacles to multivariate statistical analysis and subsequent data interpretation. A data set of 100 simulated size-frequency distributions is presented as: (i) a percentage-frequency plot (Fig. 1); and (ii) ternary diagrams (Fig. 2A and C). These data are used below to illustrate that these mathematical properties act as constraints, limiting the extent to which standard statistical tests can be applied to particle-size distributions.

image

Figure 1. A size-frequency biplot of 100 simulated particle-size distributions (grey). The mean and upper and lower 95% confidence levels are plotted respectively as solid, dot-dashed and dashed lines. These levels have been calculated using percentage-frequency data (red) and log-ratio data (black). The upper and lower 95% confidence levels calculated from percentage data violate the constant-sum constraint, summing respectively to 197·5% and 2·5%. The lower confidence intervals calculated from percentage data are negative, falling outside the parameter space. Confidence intervals calculated from log-ratio values obey both constant-sum and parameter space constraints. Both confidence intervals are predictions of feasible particle-size distributions.

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image

Figure 2. Simulated particle-size distributions (n = 100) summarized by their relative proportions of sand silt and clay, plotted as: (A) A ternary diagram with hexagons of variation delineating confidence regions of the population at 90%, 95% and 99% confidence levels. The red cross indicates the population mean. (B) A three-dimensional scatterplot of centred log-ratio transformed values showing confidence regions of the population at 90%, 95% and 99% confidence levels. Confidence regions for three orthogonal axes are calculated as a volume, but are shown here as two-dimensional areas for graphic simplicity. (C) A ternary diagram with log-ratio confidence regions of the population at 90%, 95% and 99% confidence levels [plotted in (B)] transformed back into percentage-frequency values. The red cross indicates the population mean. In contrast to (A), all areas of the confidence region fall with the parameter space (0 ≤ x ≤ 100).

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The first constraint is that each part of a particle-size distribution must be considered in relation to all its other parts; this is because a change in one part of a distribution automatically results in an inverse change in all the other parts of the distribution. For example, if there is an absolute increase in the mass of silt in one sample compared to another sample, the relative proportions of sand and clay will decrease, even if the mass of both of these size fractions remains constant. The lengthy description of the relative proportions of each part of a particle-size distribution can be avoided if log-normal distribution coefficients are used instead (Folk & Ward, 1957; Inman, 1952; Evans & Benn, 2004). Comparisons between particle-size distributions are quantified most commonly using mean, standard deviation, skewness and kurtosis statistics. The limitations of this approach have been documented by several authors (Bagnold & Barndorff-Nielsen, 1980; Fredlund et al., 2000; Fieller et al., 1992; Beierle et al., 2002; Friedman, 1962), the most fundamental of which being that particle-size distributions are frequently not log-normal. Figure 3 illustrates how a series of markedly different multimodal distributions can have identical log-normal distribution coefficients (mean and standard deviation). Quantifying the analytical precision of an instrument is clearly problematic if the statistics used are unable to differentiate between particle-size distributions that are evidently dissimilar. Alternative probability distribution functions (for example, log-hyperbolic and skew-Laplace) have been suggested to circumvent this issue (Bagnold & Barndorff-Nielsen, 1980; Fieller et al., 1992), but the limitations of non-uniqueness are still applicable. Moreover, all distribution function statistics mask potentially important variations in empirical data. One means of avoiding the pitfalls of distribution function statistics has been to compare complete particle-size distributions using factor analysis (Syvitski, 1991; Stauble & Cialone, 1996). Unfortunately, in most cases this approach is not valid because percentage values in particle-size distributions are subject to bias (Falco et al., 2003). The source of this bias is detailed by the second constraint acting on particle-size distributions and is described below.

image

Figure 3. Semi-log plots of randomly simulated size-frequency data. Each subplot contains a unimodal, bimodal, trimodal and quadramodal frequency distribution with identical log-normal mean and standard deviation values. These plots illustrate one of the potential problems involved with using log-normal distribution coefficients to describe complex multimodal particle-size distributions.

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The second constraint is that percentage-frequency data occupy a mathematically limited space, 0 < x < 100. This has important implications for applying regression models to particle-size distributions, used for example when making adjustments to homogenize data sets analysed by a range of different instruments (Konert & Vandenberghe, 1997; Beuselinck et al., 1998; Buurman et al., 2001; Eshel et al., 2004). Figure 4A, B and C show the simulated data set (‘Data A’) of percentage-frequency values plotted against another synthetic data set (‘Data B’). Data set ‘B’ has been simulated to replicate the underestimation of clay particles by a laser diffraction instrument. A least-squares linear regression model has been calculated for each size fraction (solid red line) with model coefficients and R2 correlation coefficients given for each. R2 statistics are close to one for both sand and silt, indicating a good agreement between the two data sets. Close inspection of the regression models reveal that for the silt fraction (Fig. 4B) percentage values greater than 100 are predicted for the upper range of data, indicated by the horizontal dashed line. The inverse case is also true for the clay fraction (Fig. 4C), where the regression model predicts negative percentage values for data set A.

image

Figure 4. Biplots of the simulated data set (n = 100) summarized as the relative proportions of sand, silt and clay plotted against another simulated data set, replicating the underestimation of clay particles by a laser diffraction based instruments. (A), (B) and (C) Percentage data with ordinary least-squares linear regression model (red line), model coefficients (β0, β1) and root mean squared error (R). Regression models constructed using percentage data are capable of predicting values outside the mathematical parameter space. These predictions, if corrected post hoc, have implications for the constant-sum constraint. (D), (E) and (F) Log-ratio data with ordinary least-squares linear regression model (red line), model coefficients (β0, β1) and root mean squared error (R). Log-ratio data are free to range between −∞ and ∞, avoiding the prediction of invalid values. The constant-sum constraint is ensured by the back transformation of predicted values (Eq. (3)). Note the bias inherent in calculating correlation coefficients (R) of percentage-frequency data, which are consistently higher than for log-ratio data because all the data are positive.

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Calculating confidence regions around populations of particle-size distributions is important if differences between populations are to be determined reliably. Confidence regions for ternary diagrams have traditionally been calculated using hexagonal fields of variation (Stevens et al., 1956; Weltje, 2002). These are plotted for the simulated data set in Fig. 2 as confidence regions of the population at 90%, 95% and 99% confidence limits. Confidence intervals calculated using percentage-frequency values have also been plotted in Fig. 1. The red-dashed line in Fig. 1 indicates the lower 95th confidence limit of the population. In both the figures, some of the values within the confidence regions fall outside the range of zero and 100, which is clearly impossible.

The third constraint operating on particle-size distributions is that they must sum to 100. This restriction applies as equally to confidence intervals as it does to particle-size distributions modelled using regression functions. In the former case, the upper and lower 95% confidence levels of the synthetic data calculated using percentage-frequency values plotted in Fig. 1 sum respectively to 197·5% and 2·5%. Were these estimates taken as correct, they would violate principles of conservation of mass, implying that mass has the potential to both leave and enter the system. In the latter case, there is no way to guarantee that distributions predicted from percentage data will sum to 100. Post hoc adjustments performed to ensure constant sum, unless done with extreme care, are liable to violate the first constraint by changing the relative proportions of the distribution. These three mathematical constraints impose serious limitations on how well particle-size data can be described and compared and, consequently, the reliability with which the performance of particle-size analysers can be defined and compared. The simulated data sets ‘A’ and ‘B’ are available online in Appendix S3.

Compositional data

To overcome the constraints detailed above, the principles of compositional data and the log-ratio transformation must be introduced. Data characterized mathematically as positive constant-sum vectors are known as compositional data (Aitchison, 1986). A single compositional data point is known as a composition, for example a particle-size distribution. Aitchison (1986) recognized that because each part of a composition must be considered relative to all its other parts, they were best treated as ratios. Ratios are mathematically awkward, so Aitchison (1986) logically extended the transformation to derive the logarithm of the ratios, log-ratios. There are a number of different ways to calculate the log-ratio of compositions. For particle-size distributions composed of more than three parts, the centred-log-ratio transform (clr) is generally the most useful:

  • display math(1)

where q is the log-ratio transform of p, a particle-size distribution with D-particle-size categories, i is the ith category and g(p) is the geometric mean of the particle-size distribution p. For a three-part particle-size distribution [0·4 0·35 0·25] the log-ratio of the first category is:

  • display math(2)

The log-ratio transformation of compositional data moves it from closed space to real space, also referred to as co-ordinate space. The transformation into co-ordinate space is best visualized by comparing Fig. 2A to Fig. 2B. In the latter three-dimensional scatterplot, the log-ratios of the sand, silt and clay fractions are a dimension in co-ordinate space, analogous to (x, y, z) Cartesian co-ordinates. In co-ordinate space where there are no parameter limits (data values range from −∞ to ∞), these data can be analysed using standard multivariate statistics without any of the restrictions described above.

The application of linear regression models to particle-size distributions in co-ordinate space is illustrated by Fig. 4D to F. In co-ordinate space, linear regression models are incapable of predicting ‘impossible’ values. Equation (1) also removes the influence of bias on root mean squared error (RMSE) calculations because the data are both positive and negative. With the removal of this bias, correlation coefficients calculated using log-ratio data are notably lower than for the equivalent percentage-frequency data in Fig. 4A to C. Standard goodness-of-fit statistics can also be applied to data in co-ordinate space without the need to apply to moment measure statistics. The root mean squared error statistic applied in this case is identical to the Euclidian distance measurement normalized by degrees of freedom. This can be applied to compare the output of predictive models and instrument precision using replicate sample measurements.

Working with log-ratio data allows population confidence regions to be predicted in a straightforward manner (van den Boogaart & Tolosana-Delgado, 2008). The red ellipses plotted in Fig. 2B delineate confidence regions of the synthetic data population at 90%, 95% and 99% confidence limits. These are actually calculated as volumes using a trivariate probability model, but are plotted as orthogonal ellipses for graphical simplicity. To turn these confidence regions, along with modelled distributions, into meaningful information they must be converted back to percentage-frequency values. The transformation of log-ratio data back into constrained space is performed using the inverse log-ratio transform (clr−1):

  • display math(3)

where qi is the ith part of a log-ratio particle-size distribution q. To arrive at percentage-frequency values p* must be further adjusted using the closure operation C so that its component parts sum to 100%:

  • display math(4)

The inverted confidence intervals are plotted as a ternary diagram in Figs 2C and 1. All of the values within these confidence regions are conveniently within the parameter space and all sum to 100%. Modelled particle-size distributions predicted using log-ratio regression models also obey closed-data restrictions following back transformation by Eq. (3). The example given above demonstrates how the log-ratio transformation (Eq. (1)) can be used to overcome the restrictions of closed space and apply regression analysis and concepts of statistical confidence to comparing particle-size analysers.

Methods

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. References
  9. Supporting Information

The data sets

This study makes use of an extensive database of high quality particle-size distributions (4837) collected and maintained by the Geological Survey of the Netherlands. Particle-size distributions were measured by Fritsch A22 XL (Fritsch GmbH, Idar-Oberstein, Germany) laser particle sizers at the University of Amsterdam (see Appendix S1 for details). A subset of 138 sediment samples from the database were re-analysed using traditional sieve-pipette methods (see Appendix S2 for details). The samples analysed were selected to be representative of the entire range of sedimentary environments in the Netherlands, i.e. marine, aeolian, fluvial and glacial (these data are available in Appendix S3).

The issue of particle-size category mismatch between laser and sieve-pipette data was circumvented by amalgamating both data formats to a single uniform format composed of 16 categories following Weltje & Roberson (2012). The amalgamated particle-size categories are at half-phi intervals for particles between −1 to 4 φ and at φ intervals for particles between 4 and 9 φ. These are the standard category intervals for sieve-pipette analysis. Included in the amalgamation process is a multiplicative zero-replacement strategy following Martín-Fernández et al. (2003). This step is important when working with log-ratio particle-size distributions because zeros cannot be log transformed. The method used here ensures that zeros in the data are replaced by statistically realistic values (equivalent to machine accuracy), while also preserving the relative information contained in each distribution.

Particle-size distributions published in a comparison study by Goossens (2008) are re-analysed by the present study to quantify the relative precision and differentiation power of ten different instruments (these data are available in Appendix S3). The data set published by Goossens (2008) consists of four clayey silt sediment samples taken from Korbeek-Dijle, Belgium. Each sediment was measured ten times by ten different instruments, giving a total of 400 particle-size distributions. The data were extracted from cumulative-frequency particle-size distributions given in Goossens (2008, fig. 11) by a digital image analysis algorithm. Particle-size categories from 3 to 14 φ at quarter φ intervals were used for these data. A high digitization accuracy was achieved during this process, primarily because the graphs were available in a vector format. To extract the data, each individual cumulative-frequency curve was converted to a raster image (946 by 510 pixels). The images were registered using standard digital techniques, substituting the axis limits as (x,y) co-ordinates. Horizontal resolution was 0·11 μm for samples A and B and 0·08 μm for samples C and D. Vertical resolution was 0·2% for all samples.

Prior to any analysis, it was established by using a Kolmogorov–Smirnov test for normality that none of the samples used in this study exhibited a log-normal distribution. This made comparison of distributions using a moment measures approach (Inman, 1952) unsuitable.

Regression models

The adjustment of sediment samples measured by both sieve-pipette and laser techniques were performed in co-ordinate space using the clr transformation (Eq. (1)). Continuing with the notation introduced above, q is given as the log-ratio of particle-size distributions, p is measured by sieve-pipette analysis and r is the log-ratio of particle-size distributions measured by laser analysis s. A range of multivariate regression models was used to model the relation between q and r: linear, polynomial, quadratic and quadratic with interaction terms. Regression models were applied iteratively to each particle-size category to model the complete distribution. Polynomial and quadratic regression models were run using a range of two to 16 terms, limited by the total number of particle-size categories available. Including the linear model, a total of 46 different regression models were applied to particle-size calibration. The straightforward linear regression model is given in the familiar form:

  • display math(5)

where inline image is the ith particle-size category of the modelled distribution r, β0 is the intercept and β1 is slope. The second-degree polynomial form is given as:

  • display math(6)

where β1 and β2 are the first and second-order model coefficients. The second-order quadratic model without interaction is given as:

  • display math(7)

where qi–1 is the particle-size category before qi and β4 and β3 are the coefficients for the squared terms. In the case where i = 1 the second term becomes q2. This rule was also applied to higher order models. The second-order quadratic model with interactive terms is given as:

  • display math(8)

where β5 is the coefficient of the interaction term inline image. It is impractical to write out the formulae for all regression models used, so readers are directed to Devore (2011) for an up to date introduction to multivariate regression models.

Goodness-of-fit statistics

A normalized Euclidean distance statistic (ND) is used here to quantify the difference between log-ratio particle-size distributions a and b:

  • display math(9)

where D is the number of particle-size categories. Having defined a robust means of quantifying the difference between two particle-size distributions, it is now possible to apply the ND statistic to measuring: (i) the performance of the regression models used for calibration; (ii) the precision of different analytical techniques; and (iii) the power with which an analytical technique can differentiate between two sediment samples.

Model performance

The performance of each regression model was assessed by calculating inline image, the median ND between distributions measured by laser granulometry, r, and distributions predicted from sieve-pipette measurements, inline image. In addition to calculating which model best fits the data using all the available samples, it is also pertinent to determine the influence of sample numbers on model performance. This was performed using a series of cross-validation tests. Cross-validation tests involve partitioning a data set into two subsets: a training set and a testing set. The training set is used to build the regression model, which is then tested by applying it to the testing set. The type of cross-validation applied here is known as k-fold cross-validation, where the data set is partitioned randomly into k subsamples and run k times. Random partitioning ensures that each iteration uses a different training set to create the model, which is then tested against all subsets, meaning that all data are eventually used for training and testing. To summarize model performance for ns/k training samples, the ND statistic is averaged over all the results, where ns is the total number of samples in the dataset.

A series of k-fold cross-validation tests were run for each of the 46 regression models (linear, polynomial, quadratic and interactive quadratic). The tests were run for a range of training set sizes ns/k, from eight to 138 at intervals of ten. To ensure that all combinations of training–testing portioning were used each cross-validation test was repeated 1000 times. The performance of each model for a given training set size was measured as the median ND statistic across the range of results. The results of these tests are a measure of model robustness and may be particularly important in determining which regression model should be used for any given set of samples.

Analytical precision

The precision of an analytical technique has been defined as the spread of repeat measurements around a central value, in this case precision is defined as the median ND score between all possible combinations of repeat measurements. The standard deviation of these ND statistics therefore quantifies how representative the median ND value is of technique precision, i.e. how well technique precision has been established.

It should also be pointed out that with samples of natural sediment, it is essentially impossible to determine the accuracy of a technique because the actual size of particles is difficult to measure when they are in large numbers. Determining machine accuracy is therefore best attempted using standardized glass spheres in the manner of Konert & Vandenberghe (1997).

Differentiation power

The ability to quantify instrument precision raises some interesting questions. For example, if the precision of an instrument is low, at what point is it still possible to reliably detect the difference between two populations of particle-size distributions? Furthermore, if there is a measurable difference between the distributions, at what point is that difference significant?

In order to answer these questions, the ND statistic is extended to calculate the power of an analytical technique to differentiate between sediment samples. The differentiation power inline image of analytical technique T quantifies the ability of that technique to distinguish between two populations of particle-size distributions a and b. The DP statistic takes the form of a standard Z-score statistic, stating that if the distance between two populations is less than the spread of those populations then it is not possible to distinguish between them. DP is therefore defined as:

  • display math(10)

where inline image is the median ND between all possible combinations of repeat measurements of sediment a and b, and inline image is the pooled standard deviation. inline image is defined as:

  • display math(11)

where inline image is the median ND score between all repeat measurements a. Equation (10) then is the log-ratio of the distance between two populations and the average precision of the technique used: n and m are, respectively, the sample size of populations a and b. This statistic has the same form as an independent two sample t-test (Davis, 2002), allowing significance to be assessed at a 95% confidence level for n + m − 2 degrees of freedom. Larger DP scores therefore correspond to a greater ability to distinguish between two groups of particle-size distributions. In such cases, DP scores may be considered analogous to a high signal-to-noise ratio. DP scores of one mean that two populations of particle-size distributions are indistinguishable from one another.

Note that the definition of technique precision given in the denominator of Eq. (10) is specific to each sediment, rather than the technique as a whole. This is an important distinction to make, given that particle shape and density in natural sediments may differ markedly between locations. Machine precision is then heavily dependent upon the particular shape and density of particles in the sediment measured.

Results and Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. References
  9. Supporting Information

Predictive models

This section presents the results of regression modelling applied to 138 Dutch particle-size distributions analysed by both sieve-pipette and laser diffraction. The specific results are therefore important only to geologists working with Dutch sediments that have been analysed by these two instruments. The wider aim here is to illustrate how the best regression model for any data set may be objectively identified using compositional data analysis.

Figure 5 shows the sieve-pipette data plotted against laser data in co-ordinate space for all of the 16 integrated particle-size categories. The basic correlation structure between the two data sets is quantified by a squared correlation coefficient, R2. The two data sets correspond most closely across the silt range (inline image) and least closely for very coarse sands at −1 φ (inline image) and coarse sands at 0·5 φ (inline image). The comparatively high correlation between particles in the silt range is consistent with previous observations (Konert & Vandenberghe, 1997; Beuselinck et al., 1998) and is probably attributable to a relatively high proportion of near-spherical particles. The data structure of coarse-sand size particles in Fig. 5 deviates considerably from the ideal 1 : 1 ratio (diagonal black line). Horizontal and vertical data clusters in these plots are caused by large proportions of particles being detected in a specific size range by one instrument which were not detected in the same size range by the other instrument. This discrepancy is likely to be caused by non-spherical particles in these size ranges (Beuselinck et al., 1998). Squared-mesh sieve stacks preferentially separate particles on the basis of the intermediate axis length, which can mask the presence of both elongated and bladed particles. Laser granulometry, in contrast, detects particles transported in a flowing medium using a diffraction algorithm, indicating that the long axes of elongated particles are more likely to be detected.

image

Figure 5. Plots of log-ratio transformed percentage-frequency data comparing laser granulometry data and sieve-pipette data in co-ordinate space for each of the 16 particle-size distributions used. Distributions measured by laser analysis have been down-sampled to the same format as the sieve-pipette data using a log-ratio interpolation algorithm (Weltje & Roberson, 2012). R2 correlation coefficients show that there is poor to moderate correlation between the two data sets, with the closest correspondence in the silt to clay size fractions. In contrast, there is relatively poor agreement between the data sets across the sand fraction, caused in this case by zero values being detected by one instrument and non-zero values being detected by the other.

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The results of predicting laser particle-size distributions from sieve-pipette data are shown in Fig. 6 for five randomly selected samples. The first row of plots in Fig. 6 compare cumulative-frequency distributions for both laser and sieve-pipette data. Plots in rows two to six compare laser cumulative-frequency distributions against modelled distributions for six of the 46 regression models applied: linear, second-degree polynomial, three-term quadratic and five-term, seven-term and nine-term quadratic interaction models. The different regression models are highly variable in terms of how closely they fit the original laser data for individual particle-size distributions. No single model guarantees a better prediction for any one sample. For example, predictions for samples 45 and 66 (Fig. 6) are best achieved using a nine-term interactive quadratic model. In contrast, the five-term quadratic model for sample 107 has a higher ND score than the sieve-pipette measurement, 0·351 and 0·282, respectively.

image

Figure 6. Cumulative-frequency distribution plots for five randomly selected sediment samples from the Dutch data set showing the results of six different regression models applied to predict equivalent laser measurements from sieve-pipette data. The first row shows measured sieve-pipette data (red-dashed line) plotted against laser analysis data (black solid line). Cumulative-frequency plots in rows two to seven compare laser analysis data (solid black line) with the results of a range of regression models: (i) linear; (ii) 2nd order polynomial; (iii) three-term quadratic; (iv) five-term quadratic with interaction; (v) seven-term quadratic with interaction; and (vi) nine-term quadratic with interaction. Model performance is quantified using the Normalized Difference (ND) statistic (Eq. (9)). Note that an increase in the number of model terms does not automatically guarantee a better prediction, and that not all regression models offer an improvement over the original sieve-pipette measurements.

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The tendency for regression models to not always generate an improved fit to the data reflects the fact that the difference between laser and sedimentation methods is not a linear problem, but a function of both three-dimensional particle shape and the density of individual sediment grains. Approaching this problem with a physically based model that would enable complete conversion between two instruments based on different physical principles requires knowledge of both particle density and shape. An underlying issue in particle-size analysis therefore remains: our knowledge of particle-size remains incomplete in all but the most simple of cases.

The performance of all regression models with respect to the number of samples used is summarized in Fig. 7. For each type of model, both an increase in the number of training samples and coefficients used resulted in an increase in model performance. While this is not surprising, the results quantify for the first time the added value of using quadratic regression models over linear models for adjusting particle-size distributions measured by different techniques. Importantly, the results of the cross-validation analysis indicate which model is most suitable for the number of samples available. Models with ND scores higher than 0·41 (green-yellow-brown colours) should not be used because they do not predict better fit distributions than the original sieve-pipette data. Quadratic interaction models naturally require the largest number of samples to function adequately owing to the interaction terms, resulting in the best performance scores. For models with seven or more coefficients, predictions made using all the available samples produced ND scores better than the precision of the sieve-pipette data (0·23). Given that model performance must be limited by the precision of the input data, this must therefore be the result of modelling noise in the data set. Any further improvements to modelling laser data may therefore only be achieved by acquiring higher precision sieve-pipette data. Taking into account the high levels of laboratory personnel experience, it is difficult to see how this could be achieved.

image

Figure 7. The results of cross-validation showing the performance of regression models as a function of the number of samples used to construct regression models (A) to (D) and the number of model coefficients used (B) to (D). (A) Performance of linear regression model in co-ordinate space shows that the best model performance is achieved with 118 samples (Normalized Difference: ND = 0·35), although model performance with only 38 training samples is almost as good (ND = 0·35). Dot-dashed line in (A) indicates goodness-of-fit between laser data and sieve-pipette data. Precision of sieve-pipette analysis is shown by a dotted line and the precision of laser analysis by a dashed line. (B) Performance of polynomial regression models in co-ordinate space show that low performance gains are to be had by increasing the number of training samples or model coefficients. White contour lines at 0·41 in plots (B) to (D) indicate the goodness-of-fit between laser data and sieve-pipette data. Models with scores lower than 0·41 should not be used. (C) Results of quadratic regression model shows that discernible improvements in model performance can be made by using both more training samples and model coefficients. (D) Quadratic interaction regression models provide the best method of homogenizing sieve-pipette and laser particle-size distributions. Contour line at 0·23 corresponds with the precision of sieve-pipette analysis and, hence, the limit of the models' predictive performance. Performance scores higher than this are the result of modelling noise in the data. The best model to use ultimately depends on the available number of samples, although the use of fewer coefficients will limit computational overhead.

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On the basis of the cross-validation results presented in Fig. 7, it is recommended that procedures adjusting for differences between laser and sieve-pipette data use a minimum of 30 replicate samples, and apply a two-term quadratic regression model. If more resources are available, using at least 90 replicate samples is recommended, allowing an eight-term interactive quadratic regression model to be applied. It should be noted that the maximum number of model terms is strictly limited by the number of particle-size categories measured. Interpolating particle-size distributions to increase the number of size categories will not lead to an increase in model performance because the underlying data structure still remains. This was the reason why the laser diffraction data analysed in this study were down-sampled to the sieve-pipette data format.

Analytical precision

The results presented in this and the following section illustrate how instrument precision and the related differentiation power statistic can be quantified using compositional data analysis. It is emphasized that while the results here refer specifically to the sediment samples published by Goossens (2008), a limited range of sediment types, the analytical framework presented is generic and may be applied to any given data set or sediment type where repeat sample analysis has been performed. These results are intended to illustrate how the performance of different particle-size analysers can be objectively quantified for a given set of samples in terms of precision and power to differentiate between populations of particle-size distributions. The objective is to allow future users to empirically identify the optimum instrument for any given set of samples.

The particle-size distributions analysed by Goossens (2008) are presented in Fig. 8. These consist of ten repeat measurements of four silty sediment samples A, B, C and D by ten different instruments, giving a total of 400 particle-size distributions. Individual distributions are given in red and the median distribution for each plot is given in black. The area of red visible is therefore a qualitative indicator of instrument precision: the more red visible the lower the precision.

image

Figure 8. Particle-size frequency distributions of the four silty sediments originally analysed and presented by Goossens (2008, fig. 11). Each sample was measured ten times by ten different particle-size analysers. Each of the original particle-size distributions is plotted in red, with a median particle-size distribution given in black. A qualitative impression of instrument precision can be gained by the amount of red lines showing.

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Analytical precision can also be illustrated using principal component analysis (Fig. 9). These plots show the first two principal components of each log-ratio particle-size distribution plotted against each other for each technique, which account for between 72% and 97% of the population variance. Principal component plots are equivalent to traditional mean-sorting plots, but are able to account for the influence of multimodal distributions on population variability. Principal component analysis of particle-size distributions is only feasible in co-ordinate space because the angles between particle-size categories are orthogonal. The amount of scatter within each sediment population (Sediments A, B, C or D) corresponds inversely to technique precision, noting that the scale of x-axis and y-axis are not equal and that the first principal component always describes more variability than the second and so on. For example, the Horiba Partica (Horiba Ltd, Kyoto, Japan) and the Fritsch Analysette 22C (Fritsch GmbH, Idar-Oberstein, Germany) both show very low total spread for each of the four samples. The principal component scores plot almost on top of one another, so it can be said that these techniques have a high precision. In contrast, the spread of principal component scores from the Histolab (Microvision Instruments, Evry, France) show that this technique is relatively imprecise.

image

Figure 9. Plots of the first two principal components (u1s1 and u2s2) of log-ratio transformed particle-size distributions grouped for each instrument. Sediment samples are indicated by marker type. Principal component plots are useful visualization tools as they show the spread of repeat measurements for each sediment sample (precision) and how much overlap there is between different sediment samples (differentiation power). Note the difference in scale between x-axes and y-axes and compare the spread of each population to Differentiation Power (DP) scores are given in Table 3. u1s1 and u2s2 are the product of unitary matrix U and singular matrix S, where q = USVT is the singular value decomposition (SVD) of log-ratio particle-size distributions q and VT is the transpose of the right unitary matrix V. r2 scores indicate the total data variance accounted for by the first two principal components.

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The precision of each analytical technique is shown in Table 1, defined as the median ND score between all distributions in a population (Eq. (9)). The most precise techniques are highlighted in bold for each sediment population. Median precision scores for each technique show that there is a marked difference between techniques that use laser diffraction [Coulter LS 200 (Beckman Coulter Inc., Fullerton, CA, USA), Fritsch Analysette 22C, Horiba Partica LA-950, Malvern Mastersizer S (Malvern Instruments Ltd, Malvern, UK)] and those that rely upon on other physical principles. The laser-based techniques all show precision scores lower than 0·029, the Fritsch Analysette 22C achieving the best overall score (inline image). In contrast, the next most precise technique is over an order of magnitude less precise. Of the other instruments, settling methods were the next most precise (Atterberg cylinder and Sedigraph 5100 (Micromeritics Instrument Corporation, Norcross, GA, USA)), followed by electro-resistance (Coulter Multisizer 3: Beckman Coulter Inc., Fullerton, CA, USA), dynamic image analysis (Eyetech: Ankersmid B.V., Oosterhout, The Netherlands), time-of-transition (CIS-100: Galai Production Ltd, now maintained by Ankersmid B.V.) and static image analysis (Histolab, Microvision Instruments).

Table 1. The median (inline image) and standard deviation (σ) of Normalized Difference (ND) scores between all 10 particle-size distribution measurements for each sediment sample analysed by Goossens (2008) grouped by analytical technique. ND scores are a measure of technique precision, where a score of zero indicates perfect precision. The best scores for each sediment are highlighted in bold.
 Sediment ASediment BSediment CSediment DMedian
inline image σ inline image σ inline image σ inline image σ inline image σ
Atterberg cylinder0·2630·0760·1830·0440·0450·0950·0250·0080·1140·060
Sedigraph 51000·2410·0730·1310·0570·1440·0680·1460·0640·1450·066
Coulter Multisizer 30·2080·0930·2040·0720·1760·0810·2130·0790·2060·080
Eyetech (DIA)0·1970·0740·2380·0750·1480·0710·3260·1040·2170·075
Histolab (SIA)0·2850·0690·3040·0890·2070·0970·1610·0670·2460·079
CIS-100 (TOT)0·2760·0810·3280·0820·2060·0660·2110·0770·2440·079
Coulter LS 2000·1360·0560·0160·0070·0220·1390·0110·0030·0190·032
Fritsch Analysette 22C0·0250·090 0·011 0·002 0·009 0·002 0·010 0·003 0·011 0·002
Horiba Partica LA-950 0·010 0·004 0·0290·0530·0280·0080·0470·0530·0280·030
Malvern Mastersizer S0·1860·0700·0180·0060·0140·0080·0150·0040·0160·007
Median0·2030·0740·1570·0550·0950·0700·0960·0590·1290·063

Differentiation power

The difference between sediment samples A, B, C and D is quantified as the median ND value between all possible combinations of particle-size distributions (Table 2). The results in Table 2 show that the measured difference between sediment populations varies with the technique used to analyse particle-size distributions. More importantly, these data show that the measured differences between sediments are non-identical. For example, the difference between sediment C and D is on average the lowest (inline image) and the difference between sediment B and C the third lowest (inline image). However, ND scores from the CIS-100 suggest that the difference between sediment B and C is the lowest, and the difference between sediment C and D only the third lowest. Without knowing the precision of the analytical techniques used, the meaning and reliability of these scores is unclear.

Table 2. Median Normalized Difference (ND) between the four different sediment samples analysed by Goossens (2008) grouped by analytical technique. Higher ND values indicate greater difference between samples. ND scores of zero indicate identical samples.
 A : BA : CA : DB : CB : DC : D
Atterberg cylinder0·380·510·530·330·330·06
Sedigraph 51000·260·540·560·470·460·16
Coulter Multisizer 30·220·500·530·510·540·20
Histolab (SIA)0·440·480·550·270·310·25
Eyetech (DIA)0·360·590·660·470·530·38
CIS-100 (TOT)0·500·610·840·410·590·54
Coulter LS 2000·170·400·410·330·340·03
Fritsch Analysette 22C0·180·410·430·330·340·03
Horiba Partica LA-9500·510·660·690·350·350·07
Malvern Mastersizer S0·180·420·440·340·350·04
Median0·310·500·540·350·350·11

To address this problem, the measured differences between samples were re-analysed taking into account technique precision (Table 1). The resulting Differentiation Power (DP) statistics are presented in Table 3. Values highlighted in bold are those that were found to be statistically significant at the 95% confidence level. Scores of one indicate that there is no detectable difference between two populations of particle-size distributions (sediment samples) when taking into account the precision of the technique with which they were analysed.

Table 3. Differentiation Power (DP) scores quantify the power of an analytical technique to differentiate between populations of particle-size distributions. The DP statistic is the log-ratio of the median normalized compositional distance between two sample populations and the average precision of the technique used for those samples, effectively a signal-to-noise ratio statistic. The scores here show the ability of each of the ten analytical techniques used by Goossens (2008) to differentiate between populations of the four silty sediment samples analysed. Scores of one indicate no detectable difference between populations of particle-size distributions. Larger scores indicate greater differentiation power. Scores in bold indicate significant differences between populations at a 95% confidence level.
 A : BA : CA : DB : CB : DC : DMedian
Atterberg cylinder1·64 2·57 2·70 2·65 2·73 1·74 2·61
Sedigraph 51001·35 2·75 2·80 3·39 3·35 1·132·77
Coulter Multisizer 31·07 2·57 2·52 2·66 2·59 1·022·54
Eyetech (DIA)1·65 3·33 2·52 2·40 1·88 1·502·14
Histolab (SIA)1·51 1·93 2·40 1·061·291·351·43
CIS-100 (TOT)1·65 2·53 3·43 1·51 2·13 2·62 2·33
Coulter LS 200 1·75 4·16 4·29 17·16 25·02 1·524·23
Fritsch Analysette 22C 9·28 21·61 21·27 31·98 31·70 3·03 21·44
Horiba Partica LA-950 22·20 30·50 19·46 12·35 9·10 1·70 15·91
Malvern Mastersizer S1·39 3·22 3·31 21·52 21·43 2·93 3·27
Median1·642·983·063·023·041·612·69

Technique precision and the impact that this has on the potential to differentiate between two sediment samples is effectively illustrated by Fig. 9. The Histolab has the lowest differentiation power scores on average and is only able to differentiate between samples A and C and samples A and D at a 95% confidence level. Figure 9 illustrates that this is caused by the high amount of spread and overlap between the principal component scores of the different sediment populations. The ND scores between samples given in Table 2 for the Histolab are therefore only valid for comparisons between those two pairs of samples. In contrast, the Fritsch Analysette 22C showed the highest median DP scores, and was able to distinguish between all sediment samples at a 95% confidence level. Figure 9 shows that principal component scores for the Fritsch Analysette are clustered together very tightly, with the exception of sediment A. With this technique, it is therefore possible to differentiate between sediment samples that are very similar at a high confidence level.

Based on average differentiation power scores for all the sample pairs (Table 3), the Fritsch Analysette 22C is the technique recommended for analysing samples of slightly clayey silt. Differentiation power scores further indicate that, of the instruments used, those based on laser diffraction-refraction are best at distinguishing between silty sediment samples. In addition, while some instruments are able to differentiate between specific types of sediment to a very high degree, they may not necessarily be able to differentiate between other sediment types. For example, the Malvern Mastersizer S, while it has a good overall DP score, cannot reliably differentiate between samples A and B.

Fundamental differences between instruments in terms of how particle-size is measured have been the source of much discussion within the research community. Several authors have suggested that sedimentation methods are the most appropriate analogies for deposition in marine environments (McCave et al., 2006; McIntyre & Howe, 2009; Bianchi et al., 1999). The principal reasons behind this argument have been: (i) particles in the 63 to 10 μm size range preserve the most information about sedimentation; and (ii) that laser diffraction instruments tend to underestimate silt to clay-sized platy particles relative to sedimentation techniques. It is clear from DP statistics (Table 3) that neither Horiba nor Fritsch instruments, which are based on laser diffraction, have any difficulty in distinguishing between sediment samples, due to their high precision. However, operators preferring to use sedimentation techniques are recommended to use the Atterberg cylinder in preference to the Sedigraph 5100. Extension of the present work to encompass a wider range of sedimentary environments would provide a much better understanding of the overall capabilities and limitations of these instruments. Such a study would require a large inter-laboratory collaboration, because many laboratories analyse only a limited range of sediments.

Any recommendations made about the appropriateness of a specific analytical technique for sedimentary analysis should be based on empirical evidence for a specific range of samples. Averaged values of analytical precision, while important, do not fully convey the ability of a technique to distinguish between specific sediments. Indeed, the results presented in Table 3 indicate that differentiation power scores should be considered for a range of confidence levels, so that the operator is fully aware of the significance of the results produced.

Conclusions

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. References
  9. Supporting Information

This study presents a methodological framework for the inter-comparison of different particle-size analysers. The framework consists of: (i) quantifying differences between complete particle-size distributions; (ii) adjusting for differences observed between particle-size distributions measured by different analytical techniques; (iii) quantifying the precision of a range of particle-size analysers; and (iv) identifying the most appropriate instrument for analysing a given set of samples.

The main conclusions from this study are presented below.

  1. The normalized distance statistic provides a means of quantifying differences between particle-size distributions, predictive model performance and instrument precision.
  2. Interactive quadratic regression models provide the best means of homogenizing data sets of particle-size distributions measured by different instruments. Quadratic interaction models allow adjacent particle size classes to be used implicitly to predict log-ratio transformed percentage values. Quadratic models require a large number of training samples (n > 80) compared to linear or polynomial models (n = 50) to achieve optimum results. The ideal model therefore depends on the number of samples analysed. The results of cross-validation demonstrated that the performance of any regression model is ultimately limited by the precision of the instrument used.
  3. Data presented by Goossens (2008) were used to define the analytical precision of ten different instruments. Those instruments based on laser diffraction (Fritsch Analysette 22C, Horiba Partica LA-950, Coulter LS and Malvern Mastersizer S) scored markedly higher (two to five times) than settling-based instruments (Atterberg cylinder and Sedigraph 5100). The Coulter Multisizer was the next most precise instrument, followed by the CIS-100, the Eyetech, with the Histolab instrument scoring the lowest precision value.
  4. The differentiation potential statistic is introduced to assess the significance of differences measured between populations of particle-size distributions. Laser diffraction-based instruments were best able to distinguish between different populations of particle-size distributions at a 95% confidence level. Sedimentation, electro-resistance and dynamic image analysis based instruments were also able to differentiate between the majority of silty sediment samples, but at a lower degree of precision. In contrast, the Histolab, which is based on static image analysis, performed poorly and was only able to differentiate between two populations of sediment samples. The results of this study indicate that this instrument is not suitable for the analysis of silty sediments.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. References
  9. Supporting Information

The authors would like to thank the Geological Survey of the Netherlands for database access, Ronald Harting for co-ordinating sample logistics and Martin Konert at the Vrije Universiteit Amsterdam for sieve and pipette analysis of samples. Simon Blott, Stephen Rice and three anonymous reviewers are thanked for their thorough and thought provoking comments on previous versions of this manuscript which have helped to markedly improve it. This work was completed when SR was in receipt of a post-doctoral research fellowship at Delft University of Technology, The Netherlands.

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  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. References
  9. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. References
  9. Supporting Information
FilenameFormatSizeDescription
sed12093-sup-0001-AppendixS1.texapplication/x-tex999Appendix S1. Pretreatment of samples for analysis with laser particle sizer.
sed12093-sup-0002-AppendixS2.texapplication/x-tex999Appendix S2. Determination of particle-size distribution by sieve and pipette method.
sed12093-sup-0003-AppendixS3.texapplication/x-tex999Appendix S3. Data sets.

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