Journal of Spectroscopy

Volume 2018, Article ID 5837214, 13 pages

https://doi.org/10.1155/2018/5837214

## Statistically Coherent Calibration of X-Ray Fluorescence Spectrometry for Major Elements in Rocks and Minerals

^{1}Instituto de Energías Renovables, Universidad Nacional Autónoma de México, Temixco, Mor 62580, Mexico^{2}División de Geociencias, Instituto Potosino de Investigación en Ciencia y Tecnología, Camino a la Presa San José # 2055, Col. Lomas 4a Sec., San Luis Potosí, SLP 78216, Mexico^{3}Instituto de Ciencias del Mar y Limnología, Unidad de Procesos Oceánicos y Costeros, Universidad Nacional Autónoma de México, Circuito Exterior s/n, 04510 CDMX, Mexico^{4}Posgrado en Geociencias Aplicadas, Instituto Potosino de Investigación en Ciencia y Tecnología, Camino a la Presa San José # 2055, Col. Lomas 4a Sec., San Luis Potosí, SLP 78216, Mexico^{5}Centro de Investigación en Ciencias, Instituto de Investigación en Ciencias Básicas y Aplicadas, Universidad Autónoma del Estado de Morelos, Cuernavaca, Mor 62209, Mexico^{6}Posgrado en Ingeniería, Instituto de Energías Renovables, Universidad Nacional Autónoma de México, Temixco, Mor 62580, Mexico^{7}División de Materiales Avanzados, Instituto Potosino de Investigación en Ciencia y Tecnología, Camino a la Presa San José # 2055, Col. Lomas 4a Sec., San Luis Potosí, SLP 78216, Mexico^{8}Doctorado en Ciencias, Instituto de Investigación en Ciencias Básicas y Aplicadas, Universidad Autónoma del Estado de Morelos, Cuernavaca, Mor 62209, Mexico^{9}Universidad Autónoma de Nuevo León, Facultad de Ciencias de la Tierra, Ex–Hacienda de Guadalupe, Carretera Linares–Cerro Prieto km 8, Linares, N.L. 67700, Mexico

Correspondence should be addressed to Surendra P. Verma; xm.manu.rei@vps

Received 8 August 2018; Revised 8 October 2018; Accepted 19 October 2018; Published 11 December 2018

Academic Editor: Rafal Sitko

Copyright © 2018 Surendra P. Verma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We applied both the ordinary linear regression (OLR) and the new uncertainty weighted linear regression (UWLR) models for the calibration and comparison of a XRF machine through 59 geochemical reference materials (GRMs) and a procedure blank sample. The mean concentration and uncertainty data for the GRMs used for the calibrations (Supplementary Materials) (available here) filewere achieved from an up-to-date compilation of chemical data and their processing from well-known discordancy and significance tests. The drift-corrected XRF intensity and its uncertainty were determined from mostly duplicate pressed powder pellets. The comparison of the OLR (linear correlation coefficient ∼0.9523–0.9964 and 0.9771–0.9999, respectively, for before and after matrix correction) and UWLR models (∼0.9772–0.9976 and 0.9970–0.9999, respectively) clearly showed that the latter with generally higher values of is preferable for routine calibrations of analytical procedures. Both calibrations were successfully applied to rock matrices, and the results were generally consistent with those obtained in other laboratories although the UWLR model showed mostly narrower confidence limits of the mean (slope and intercept) or lower uncertainties than the OLR. Similar sensitivity (∼2.69–46.17 kc·s^{−1}·%^{−1} for the OLR and ∼2.78–59.69 kc·s^{−1}·%^{−1} for the UWLR) also indicated that the UWLR could advantageously replace the OLR model. Another novel aspect is that the total uncertainty can be reported for individual chemical data. If the analytical instruments were routinely calibrated from the UWLR model, this action would make the science of geochemistry more quantitative than at present.

#### 1. Introduction

All modern analytical instruments require some kind of calibration of the instrumental response (-variable) as a function of the concentration (-variable) [1–3]. This calibration is generally achieved through an ordinary least-squares linear regression (OLR) model. However, such a procedure is not strictly valid because all requirements for the statistical validity of the OLR model are not fulfilled. Usually, the assumptions “independent concentration variable is error-free or less than one-tenth of the error in the dependent response variable ” and “error in is homoscedastic” (i.e., equal errors for all values) are not satisfied and, therefore, more sophisticated and statistically coherent regression procedures, such as weighted least-squares linear regression (WLR) models, should be used [4–18].

X-ray fluorescence (XRF) spectrometry is among the most popular analytical techniques for the determination of all major and some trace elements in rocks [4, 19–27]. Natural geochemical reference materials (GRMs) are commonly used for XRF calibrations and posterior characterization of those and other GRMs as well as of similar rock and mineral matrices [4, 19, 28–30]. As for most other analytical instruments, XRF spectrometers are also calibrated under the statistically incoherent OLR model.

To apply the WLR and compare it with the OLR, both central tendency (e.g., mean) and dispersion (e.g., confidence limits of the mean) estimates on both -axis (concentration, generally expressed in the unit of % m/m, i.e., mass/mass unit expressed in percent) and -axis (response, in this case XRF intensity, generally reported in the unit of kc·s^{−1}, i.e., kilo counts per second) variables are required. More precise (and accurate) estimates of the central tendency will also be useful for both types of regressions. Therefore, precise concentrations of GRMs with the respective lowest possible “confidence limits of the mean” (referred hereafter as the “uncertainty” of the measured variable) [2, 17, 18] are required to apply the regression procedure. Sometimes, we had to use also the term “error” (instead of the uncertainty) because the use of the error is widespread in the literature.

We report the following five aspects: (a) evaluation of 59 GRMs to achieve the least possible uncertainties in the mean concentrations of all major elements (SiO_{2} to P_{2}O_{5}); (b) the comparison of regression models (OLR and WLR) applied to net drift-corrected XRF intensities before the correction of matrix effects; (c) the second (or final) comparison of both models after achieving the matrix correction as well as for the estimation of sensitivities of the regression models; (d) application of the entire procedure to four GRMs treated as “unknown” samples and their comparison with the previous literature compilations; and (e) development of a computer program to achieve the abovementioned objectives. Thus, the regression equations (intercept and its uncertainty, slope and its uncertainty, and linear correlation coefficient values) for each constituent from SiO_{2} to P_{2}O_{5} and their application to similarly complex rock matrices are presented in this work.

#### 2. Evaluation of Major Element Data for GRMs

A total of 59 GRMs (listed in alphabetical order in Table S1; this and four other tables are provided in Supplementary Materials), along with a procedure blank, were used in this study. The procedure blank was a pellet prepared in duplicate with only pure N,N′-Ethylene bis(stearamide) beads without any sample (Section 3). The individual data reported in earlier compilations [31–47] were first compiled in new databases.

The statistical parameters obtained in these early compilations could not be directly used for instrumental calibrations due to the following reasons: (i) the statistical methods used to achieve the statistical estimates were outdated (see [17, 18, 48, 49] for possible reasons), and the inferred statistical values were of low quality (high values of dispersion); (ii) there are still determinations reported during about 30 or more years (postcompilation years) that were not obviously available to those compilers; (iii) the precision of more recent determinations is likely to have improved due to the availability of online computers on most modern instruments; (iv) newer more reliable statistical techniques are now available for improving both precision and accuracy of the statistical inferences, e.g., the use of discordancy tests with the highest power and lowest swamping and masking effects [18, 48, 50–52]; and (v) importantly, new computer programs have been developed by our group [52–54], available at http://tlaloc.ier.unam.mx for download or online processing of data (after previous registration onto our server), which can be advantageously used for efficient processing of experimental databases.

The same kinds of objections are applicable even today for the originator’s websites, such as https://gbank.gsj.jp/geostandards/welcome.html for Japanese GRMs or https://crustal.usgs.gov/geochemical_reference_standards for United States GRMs. The statistical information at these websites is based on early compilations (around 30 or more years ago). Furthermore, we were unable to use the recent work [55] because this paper reported significantly larger uncertainty values as compared to those achievable from our new validated statistical procedure [51–54]; besides, updated statistical information on the mean and its uncertainty was not available in [55] for many GRMs used in our work.

The initial databases were complemented by individual data from a large number of posterior publications (∼480; Table S1), whose complete listing is available at our server http://tlaloc.ier.unam.mx under the heading of “Quality Control.” These major element data were classified according to the analytical method groupings [56]. Data from each method group were considered as a univariate statistical sample. Appropriate discordancy and significance tests were applied from thoroughly automatized software UDASys2 [52] and UDASys3 (unpublished), which, in their “recommended procedure,” apply the most powerful five (two new and three conventional) recursive tests with prior application of respective single-outlier tests having nil swamping and low masking effects [48, 57–60]. Although the application of discordancy tests is identical for both UDASys2 and UDASys3, the difference lies in that the latter applies the significance (ANOVA, F and *t*) tests in order to provide the final results automatically.

The resulting statistical information after the application of well-known discordancy tests at the strict 99% confidence level (mean and uncertainty values rounded according to the flexible rules [18]) is listed in Table S2. These GRM compositional data showed by far the lowest 99% uncertainty (Table S2), much lower than any existing compilation [31–47, 55]. We may also stress once again that this was achieved through an objective combination of discordancy tests having the highest performance and lowest swamping and masking effects [17, 18, 48, 53], i.e., from the methodology having the lowest type I and type II errors and the highest power.

Therefore, the population mean of these GRMs is now known within the narrowest possible 99% confidence limits of the mean to best represent the concentration (*x*) axis in the instrumental calibrations as suggested [2, 5, 7, 10, 17, 18, 53]. These data (in units of % m/m; Table S2) will also be useful for those who wish to achieve instrumental calibrations or simply use them for quality control of their results for rock and mineral matrices.

#### 3. XRF Instrumentation and Intensity Measurements

A wavelength dispersive X-ray fluorescence (WDXRF) spectrometer Rigaku ZSX Primus II model (rhodium X-ray tube; 4 kW maximum power) was used for this work. We made the effort to best represent the response () axis (-ray intensity in the units of kilo counts per second, kc·s^{−1}) for the calibrations. For each GRM, duplicate (41 samples) or even triplicate (8 samples) pressed powder pellets were prepared. First, an appropriate amount of each GRM was dried overnight in an oven at about 105°C. For each pellet, accurately weighed 3.5 g of moisture-free GRM was thoroughly mixed with accurately weighed 3.0 g pure N,N′-ethylene bis(stearamide) beads, <840 *μ*m as wax (Sigma-Aldrich), and stored in a desiccator. Pressed powder pellets were prepared at 20 tons·inch^{−2} pressure (about 310 MPa). However, for 10 GRMs, sufficient material was not available; therefore, only a single pellet could be prepared but the measured intensity uncertainty () at the 99% confidence level was increased by a factor of 2 to take into account the sample preparation variance. Similarly, accurately weighed 6.5 g of pure N,N′-ethylene bis(stearamide) beads, <840 *μ*m as wax, was pressed to prepare a procedure blank sample. This was done in duplicate.

For the intensity measurements, the optimum instrumental conditions were first established through preliminary experiments prior to the routine measurements (Table S2). Each pellet was run at least 8 to 10 times in a random sequence, along with two drift monitors prepared from two volcanic rocks (basalt and rhyolite) from the San Luis Potosí Volcanic Field, San Luis Potosí (central Mexico).

The peak and background measuring conditions and time periods are also listed in Table S3. Appropriate mean drift corrections from two monitors were applied to all intensity measurements. Both monitors were run randomly 8 to 10 times each day. First, the expected monitor intensity was established as an average value of the first two days when the intensities were fairly stable and reproducible. Then, the average drift correction factors were calculated for each chemical element from the two monitors run in the XRF instrument periodically before and after a set of GRMs used for the calibration. These correction factors were then applied to the bracketed GRMs for the entire period of calibration, including the first two days and analysis of “unknown” samples.

Now, although the X-ray counts may obey a Poisson distribution, we are dealing with average values of count rates, which are likely to follow a normal distribution because of the central limit theorem. A normal distribution of measured intensities was also assured for each pellet from the application of discordancy tests as explained above for GRM concentrations. The intensity results for all pellets from a given GRM were then combined, the tests applied again to the combined data, and new mean and 99% uncertainty values were calculated for X-ray intensity of each GRM. This was done to take into account the variance of the sample preparation method, which was significantly higher than the instrumental variance of intensity measurements for individual pellets. The drift-corrected intensity values and their 99% uncertainties (kc·s^{−1}) for all GRMs, along with the concentration data and their 99% uncertainties (% m/m), are listed in Table S2.

#### 4. Regression Models

Two different regression models (OLR and UWLR) were used and compared in this work. The OLR model most frequently used for instrumental calibrations (-axis concentration and -axis response; GRM concentration and X-ray intensity, respectively, in XRF spectrometry) requires the following assumptions to be fulfilled [4, 7, 10, 12–18]: (i) all errors are in the -axis; (ii) -axis is either error-free or has at most 10% error of the -axis errors; (iii) errors in both axes are normally distributed; and (iv) errors in the -axis are homoscedastic. Some or all of these assumptions are violated in most instrumental calibrations through the OLR model.

Thus, from the literature on the GRMs, it has been demonstrated that the concentration axis is not error-free (see non-zero uncertainties for all GRM concentrations in Table S2) [31–47, 51–53]. One can also clearly see that the errors in the intensity axis are not homoscedastic (see unequal, i.e., heteroscedastic uncertainties for any element in different GRMs in Table S2). For a heteroscedastic linear regression system, even if each error or noise term is still Gaussian, the OLR model is no longer the maximum likelihood estimate and consequently, it is no longer efficient [10]. The main advantage that the WLR has over the OLR is the ability to handle regression situations in which the data points are of varying quality as is the case with most instruments including the XRF spectrometers.

However, the major disadvantage of the WLR is that the approach is based on the assumption that the weights are known exactly. They can be estimated using several different equations or algorithms, but when the weights are produced from small numbers of replicated observations, the regression parameters can be unpredictably affected [10]. In the example of the XRF calibration that we are presenting, the numbers of observations were relatively large for both the and axes (concentration and X-ray intensity parameters). Besides, instead of the sample variance, we used the uncertainty values (that take into account the number of observations in the formula for uncertainty or confidence limits of the mean calculations) [2, 18] for estimating the weight factors. The problem of the sensitivity to outliers in the regression equations [10] was also appropriately handled by discordancy tests programmed in the UDASys and BiDASys software [53, 54, 61].

Therefore, although frequently used, the OLR model is not statistically correct or coherent. The statistically coherent WLR, especially the uncertainty-based WLR (UWLR [17]) model, should be used. The confidence level, such as 95% or 99% (significance level of 5% or 1%, respectively, or of 0.05 and 0.01, respectively), can be explicitly expressed in the confidence limits of the mean or uncertainty used in the UWLR model as well as to estimate the weight factors [17]. We will deal with the 99% uncertainty to have the type I error small (about 1%). Unfortunately, software of most analytical instruments, including XRF spectrometers, allows only the OLR calibration. Therefore, any sophisticated regression model, such as the UWLR, will have to be applied outside the instrumental software. Thus, the probability concept (99% confidence level) can be explicitly used in the UWLR model for weight factors based on the inverse of the squared 99% uncertainty of the mean.

We now present a synthesis of the regression equations for instrumental calibrations [2, 10, 17, 18, 61].

##### 4.1. Ordinary Least-Squares Linear Regression (OLR) Model

Let us assume that we have a series of reference materials or standard calibrators having individual mean concentrations with respective uncertainties where varies from to . In order to calibrate an instrument, each of these calibrators were run several times, obtaining individual mean responses with respective uncertainties where varies from to . Thus, we have bivariate concentration-response data pairs or calibrators () with the respective uncertainties ().

We can apply the OLR model to these data for obtaining a calibration equation. The OLR fits a least-squares linear equation to the pairs () but does not take into account the respective uncertainties ().

The general regression equation for the OLR is as follows (the subscript is for the OLR model):where is the slope, is the resulting uncertainty in the slope, is the intercept, is the resulting uncertainty in the intercept, is the independent variable, is the dependent variable from the OLR model, and is the resulting uncertainty in . The following equations allow the calculations of these parameters:where and are, respectively, the mean values of the and variables:where is the value of for in equation (1) and is the Student’s *t* test value for degrees of freedom, and the superscript is the confidence level, generally 95% or 99%:

It is a general practice in most instrumental calibrations to ignore all uncertainties in equation (1) and use an OLR equation without any error (or uncertainty) as follows:

The resulting standard deviation values of repeat measurements of unknown samples are reported as the final errors. However, these are only partial errors because the errors in the calibration equation (1) are not taken into account. In this work, we will use equation (1) to report total errors (in fact, 99% uncertainties) for the OLR model.

##### 4.2. Uncertainty Weighted Least-Squares Linear Regression (UWLR) Model

For the UWLR model, the pairs () of calibrators as well as the respective uncertainties () are taken into account in order to achieve the best least-squares linear fit.

The uncertainties in the *x*-axis are first propagated to the *y*-axis, combined with the , and the total uncertainty values on the *y*-axis are used for the weighting factors [2, 10, 17, 18, 61]:

The weights are calculated from as follows:where values have the following property:

Thus, the UWLR fits a linear equation to the pairs () with the respective weighting factors as follows (the subscript is for the UWLR model):

Note that this regression line will pass closer to the data with lesser uncertainty . The intercept and slope variables and their uncertainties are calculated from the following equations:where and are, respectively, the weighted mean values of the and variables:where is the value of for in equation (9):

The best regression equation for a calibration curve should have the following characteristics (without distinguishing the subscripts and ): (i) intercept small approaching to zero; (ii) slope large; and (iii) both and small. Further, the quality of the regression, whether a calibration curve or any other bivariate relationship, is also expressed as the linear regression coefficient (; and , respectively, for the OLR and UWLR), which is ideally +1.00000 for a calibration curve [5, 18, 61].

#### 5. Application of Regression Models for XRF Calibration

##### 5.1. Original Drift-Corrected Net Intensities and GRM Concentrations: The First Set of Two Regression Equations for Each Element

The evaluations for both regression types on the drift-corrected net intensity-concentration (Int-Conc) relationships (Table S2) for all major elements from SiO_{2} to P_{2}O_{5} were performed (Table S4), for which the new online software BiDASys was used [61] at http://tlaloc.ier.unam.mx. BiDASys allows the application of the conventional OLR as well as the newly proposed UWLR model [17] and provides the output of all regression parameters in an Excel® file. Contrary to the common practice, we will refrain from showing the numerous - (variable is drift-corrected net intensity “Int” and variable is the GRM concentration “Conc”) plots. This is because Table S4 statistically quantifies the visual interpretation of such diagrams. The quality parameters (standard errors and , uncertainty and , and linear correlation coefficient and its squared value parameters) are reported in Table S4. Because we are using these several different quality parameters, the concern against the use of solely parameter [62] is not important for comparison purposes.

We will explain the implications of the statistical results for the first element SiO_{2}; the statistics for other elements (Table S4) can be similarly understood. The OLR regression equation from the first row of statistical information in Table S4 is as follows (after the element , subscript is for the OLR and is for provisional concentration; note many decimal places are used for the regression variables in such equations, because these values are not final results, and we should not introduce rounding errors during the calculation stage):

Similarly, the UWLR equation from the second row of statistical information in Table S4 is as follows:

The implications of these regression equations can be understood from the comparison of the uncertainties of the intercept and slope, which are lower for the UWLR (equation (14)) than for the OLR (equation (13)). This means that the uncertainty of the calculated concentration will be lower for the UWLR than for the OLR. Correspondingly, the value for the UWLR (0.99004, ; ) is much higher than that for the OLR (0.95229, ; ; Table S4). Similar trend in the (and ) values was obtained for all other elements except MnO (Table S4).

##### 5.2. Matrix-Effect-Corrected Intensities and GRM Concentrations: The Second Set of Two Regression Equations for Each Element

Matrix correction is certainly required because the abovementioned least-squares linear regression fits are far from “perfect” ( ≠ +1.00000; in fact, < 1; ; = 0.95229–0.99638 for the OLR and = 0.97715–0.99760 for the UWLR; Table S4). There is a vast literature on the subject of matrix effects in XRF and their correction procedures [63–75]. In this study, the Lachance-Traill algorithm [73] was used for the matrix effect correction [63, 71]. This was done outside the XRF instrument software. In a review of the existing algorithms, Rousseau [63] showed that the Lachance-Traill algorithm could be considered as one of the most appropriate procedures for the matrix effect correction because other algorithms have limited application range or lack of accuracy. Thus, for each element from SiO_{2} to P_{2}O_{5}, a system of overdetermined equations was solved and the resulting alpha coefficients were used to correct all intensities for matrix effects.

From the alpha coefficients, matrix-corrected intensities and improved concentration values for the GRMs and their uncertainties were calculated iteratively under the condition that the convergence parameter (absolute relative difference of the GRM calculated and input concentrations) for each compositional constituent (SiO_{2} to P_{2}O_{5}) be minimized.

New regression equations for achieving the corrected concentrations were established from the relationship of the calculated GRM concentrations (ConcCalc) and the original GRM concentrations (Conc) given in Table S2, for which the online BiDASys software [61] was used at http://tlaloc.ier.unam.mx. These equations can be formulated from the regression coefficient values given in Table S4 (see ConcCalc-Conc rows corresponding to the OLR and UWLR). Again, we will highlight their significance for SiO_{2} only.

The OLR regression equation from the third row of statistical information in Table S4 is as follows:where the subscripts and stand for the OLR model and calculated concentration (ConcCalc), respectively.

Similarly, the UWLR equation from the fourth row of statistical information in Table S4 is as follows:where the subscripts and stand for the UWLR model and calculated concentration (ConcCalc), respectively.

Equations (15) and (16) show that the concentration values from the UWLR would be more reliable (lesser uncertainty values in both intercept and slope) than the OLR model. The value is higher for the UWLR (0.99704, ; ; Table S4) than the OLR (0.97710, ; ).

After the matrix correction, in fact most regression equations are better because all and values are higher for both OLR and UWLR than without the correction (Table S4; Figure 1 for only). For the OLR, the matrix correction increased the values () from 0.95229–0.99638 () to 0.97710–0.99992 (). Similarly, for the UWLR, this increase was from 0.97715–0.99760 () to 0.99704–0.99993 (). Thus, after matrix correction, all values increased for both OLR and UWLR. For the UWLR, the values approached the ideal value of +1.00000 (Figure 1). One has to keep in mind that when the values are closer to the maximum possible value of 1 (the “ideal” fit), the improvement expressed by the actual (absolute) value of will apparently be small. However, as long as the value increases for the UWLR as compared to the OLR (Figure 1; Table S4), we can objectively infer that the UWLR is a better regression model than the OLR.