applied
sciences
Article
A Matrix Effect Correction Method for Portable X-ray
Fluorescence Data
Jilong Lu
, Jinke Guo, Qiaoqiao Wei, Xiaodan Tang, Tian Lan, Yaru Hou and Xinyun Zhao *
Department of Geochemistry, Jilin University, Changchun 130026, China; lujl@jlu.edu.cn (J.L.);
guojk20@mails.jlu.edu.cn (J.G.); Qiaoqiao@jlu.edu.cn (Q.W.); tangxiaodan@jlu.edu.cn (X.T.);
lantian19@mails.jlu.edu.cn (T.L.); houyr20@mails.jlu.edu.cn (Y.H.)
* Correspondence: zhaoxy15@jlu.edu.cn
Abstract: Portable X-ray fluorescence spectrometry (pXRF) is an analytical technique that can be
used for rapid and non-destructive analysis in the field. However, the testing accuracy and precision
for trace elements are significantly affected by the matrix effect, which comes mainly from major
elements that constitute most of the matrix of a sample. To solve this problem, many methods based
on linear regression models have been proposed, but when extreme values or outliers occur, the
application of these methods will be greatly affected. In this study, 16 certified reference materials
were collected for pXRF analysis, and the major elements most closely related to the elements to
be measured were employed as correction indicators to calibrate the analysis results through the
application of multiple linear regression analysis. Some statistical parameters were calculated to
evaluate the correction results. Compared with the calibration data obtained from simple linear
regression analysis without taking major elements into account, those corrected by the new method
were of higher quality, especially for elements of Co, Zn, Mo, Ta, Tl, Pb, Cd and Sn. The results show
that the new method can effectively suppress the influence of the matrix effect.
Keywords: portable X-ray fluorescence; matrix effect; simple linear regression analysis; multiple
linear regression analysis
Citation: Lu, J.; Guo, J.; Wei, Q.; Tang,
X.; Lan, T.; Hou, Y.; Zhao, X. A Matrix
Effect Correction Method for Portable
X-ray Fluorescence Data. Appl. Sci.
2022, 12, 568. https://doi.org/
10.3390/app12020568
Academic Editor: Vlasoula Bekiari
Received: 12 November 2021
Accepted: 30 December 2021
Published: 7 January 2022
Publisher’s Note: MDPI stays neutral
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Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1. Introduction
Portable X-ray fluorescence spectrometry (pXRF) offers some unique advantages in
chemical composition analysis, which arise from the multi-element capability, the nondestructive nature and the immediate availability to the researcher of information on the
chemical composition of a sample in the field [1]. In addition, this technique is characterized by decreased production of hazardous waste and low running costs [2]. At present,
pXRF analysis has been widely used in mineral resource exploration [3], environmental
monitoring and evaluation [4], archaeological research [5], agricultural survey [6] and many
other fields.
However, the matrix effect that occurs in pXRF analysis generally has a heavy influence
on the quality of the analysis results, especially for trace elements [7]. The interference of
the sample particle size, surface structure, chemical composition and mineral morphology
on the analysis are all matrix effects [8]. In fact, the matrix influence is essentially the
impact that the sample matrix has on the X-ray intensity emitted during analysis, which is
mainly reflected by the absorption and enhancement or overlap in the spectral peaks [9]. As
a result, matrix effect correction has been the focus of much attention, and many correction
methods have been put forward from different points of view [10].
Currently, there are two kinds of correction methods for the matrix effect. One suppresses the matrix effect by experimental manipulations, such as the internal standard
method, the standard addition method and the dilution method. The other eliminates the
matrix effect by mathematical means, such as the linear regression method and machine
learning [11]. The experimental method will, however, make the experimental process
Appl. Sci. 2022, 12, 568. https://doi.org/10.3390/app12020568
https://www.mdpi.com/journal/applsci
Appl. Sci. 2022, 12, 568
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more complicated and increase the experimental workload. At present, the most commonly
used correction methods are mathematical methods [12], of which most are based on the
linear regression model, such as the fundamental parameter method and the experience
coefficient method [13]. These methods, however, only use the element to be measured
to establish a regression equation and pay little attention to the role of other elements,
especially major elements that have a great influence on the determination of trace elements.
For example, the element Fe will cause a significant increase in test value of Co [14], and the
resulting outlier (an extremely high value) may lead to an unrealistic regression curve [15].
As for the machine learning algorithm, such as artificial neural networks [16], random
forests [17] and geographically weighted regression [18], they have become an important
statistical tool in dealing with pXRF data, but they are not stable and sometimes have the
local optimum problem.
In this study, pXRF analysis was performed on 16 certified reference materials (CRMs),
including 10 rock samples and 6 soil samples, and some major elements were selected as
correction indicators to correct the analysis results with the application of a new method
based on multiple linear regression analysis [19]. For comparison, the simple linear regression (SLR) method is also performed on the data. Some statistical parameters, such
as the coefficient of determination (R2 ), mean absolute error (MAE) and root mean square
error (RMSE), are calculated and discussed in detail to evaluate the performance of these
two methods.
2. Materials and Methods
2.1. Samples and Analyses
A total of 16 CRMs were selected, including 10 rock samples and 6 soil samples
(Table 1), for pXRF analysis, and 34 elements were detected, including Mg, Al, Si, P, S, K,
Ca, Ti, Mn, Fe V, Cr, Co, Ni, Cu, Zn, As, Se, Rb, Sr, Y, Zr, Nb, Mo, Cd, Sn, Sb, Ba, Ta, W,
Tl, Pb, Bi and U. The pressed powder pellet technique was used for sample preparation.
The sample powder was pressed into a pellet with a diameter of 5 cm and a thickness of
about 2 cm under 30 kPa for 30 s. The experimental instrument was a portable energy
dispersive X-ray fluorescence spectrometer (X-MET7000) from the Oxford Instruments
Group, founded in Oxford, England in 1959. The instrument had a rhodium (Rh) anode
target and a fourth-generation silicon drift detector (SDD). Elements between Mg (atomic
number of 12) and U (atomic number of 92) on the periodic table of elements could be
analyzed [8]. The samples were determined under the conditions of a voltage of 40 kV
and current of 60 mA, and each test lasted 60 s. In order to reduce the influence of sample
heterogeneity, the analysis was repeated 5 times in different positions homogeneously
distributed in the sample, and the average value was calculated as the final result [20].
Table 1. Certified reference samples for pXRF analysis.
Lithology
Number
Rock
10
Soil
6
Sample No.
GBW07104, GBW07105, GBW07106, GBW07107, GBW07122,
GBW07162, GBW07163, GBW07165, GBW07825, ZBK336
GBW07401, GBW07403, GBW07404, GBW07405, GBW07406, GBW07408
Note: These samples were provided by the Institute of Geophysical and Geochemical Exploration of the Chinese
Academy of Geological Science.
2.2. Correction Method for the Matrix Effect
According to the Sherman equation (Equation (1)), which describes the relationship
between the measured intensities emitted by a sample and its composition [21,22], the
pXRF analysis result of an element is affected by all the other elements in a sample:
Ij = f Cj , Ck , . . . , Cm
(1)
where Ij is the net intensity of element j, Cj is the concentration of element j and Ck –Cm are
the concentrations of other elements.
Appl. Sci. 2022, 12, 568
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However, not all elements can be determined by pXRF analysis due to the low detection limits of some elements [23]. In addition, this equation is not reversible for calculating
unknown sample compositions. At present, the commonly used correction method is
the regression method. Note that major components of a sample contribute most to the
matrix effect and therefore should be considered important indicators in matrix effect
correction, such as Si, Al, Fe, Ca, K, Mn and Ti. As a result, major elements were added as
an independent variable into the commonly used SLR model, and then a multiple linear
regression equation (Equation (2)) could be established [11]:
Ci0 = αi Ci + α j Cj + ui
(2)
where Ci ’ is the reference value of element i, Ci and Cj are the pXRF testing values of
element i and major element j that is most closely related to element i, respectively, αi is
the regression coefficient of element i, αj is the influence coefficient of major element j to
element i and ui is the regression intercept of element i.
The values of αi , αj and ui are affected by the instrument conditions and can be
calculated by testing the CRMs. In this process, the ordinary least squares approach [24,25],
which is one the most popular chemometric algorithms for calibration model creation, is
used. Similarly, this method requires minimizing the sum of the squares of the residuals.
As a result, αi , αj and ui can be obtained by solving Equations (3)–(5):
αi =
n
0 M
M
M
∑nk=1 M2jk − ∑nk=1 Mik
∑
jk
ik jk
k =1
2
2
∑nk=1 Cik
∑nk=1 C2jk − ∑nk=1 Cik Cjk
0 M
∑nk=1 Mik
ik
αj =
n
2 −
0 M
M
M
∑nk=1 Mik
∑nk=1 Mik
∑
ik
ik jk
k =1
2
2
∑nk=1 Cik
∑nk=1 C2jk − ∑nk=1 Cik Cjk
n
0 −α ( n C )−α
C
∑
∑
∑nk=1 Cik
i
j
k=1 ik
k =1 jk
0 M
∑nk=1 Mik
jk
ui =
(3)
(4)
(5)
k
where n is the number of samples and Cik and Cjk are the testing values of elements i and j
in sample k, respectively. The testing values of i and j are zero centered to Mik and M jk by
0 and C 0 are the transposes of M and C , respectively:
the following equation. Mik
ik
ik
ik
Mik = Cik −
∑nk=1 Cik
k
(6)
Mij = Cjk −
∑nk=1 Cjk
k
(7)
2.3. Parameters for Evaluation of the Correction Results
Some statistical parameters were employed to evaluate the correction results, including
the coefficient of determination (R2 ), relative error (RE), mean absolute error (MAE), mean
absolute percentage error (MAPE), root mean squared error (RMSE) and root mean square
percentage error (RMSPE). The correlation between the pXRF testing value and the reference
value of an element could be estimated by the R2 values. The larger the R2 value is, the
stronger the correlation is [26]. It is generally considered that high-quality data can be
obtained when the relative error (RE) does not exceed 20%. In this paper, the accuracy of
the pXRF analysis was estimated by PRE , which represents the percentage of data with an
RE not exceeding 20% [26]. When the PRE value approached 1, the testing accuracy was
high, indicating that the experimental results were accurate. The parameter MAE was used
to measure the average absolute error between the corrected value and the reference value
of the experimental data set, and the RMSE was used to measure the deviation between the
Appl. Sci. 2022, 12, 568
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observed value and the true value [27]. As for the MAPE and RMSPE, they could be used to
evaluate the relative errors and dispersion of the whole test values of a sample, respectively.
The smaller the four values were, the more accurate and the less discrete the correction
results were [28]. These parameters could be obtained by solving the following equations:
2
R =
RE =
0 −C
∑nk=1 Cik
i
2
0 −C
∑nk=1 Cik
i
2
0 − C0
Cik
ik
MAE =
0
Cik
× 100%
0 − C0
∑nk=1 Cik
ik
n
n C 0 − C0
1
× ∑ ik 0 ik × 100%
n k =1
Cik
s
0 − C 0 )2
∑nk=1 (Cik
ik
RMSE =
n
v
!
u
0 − C0 2
n
u1
Cik
t
ik
×
× 100%
RMSPE =
0
n k∑
Cik
=1
MAPE =
(8)
(9)
(10)
(11)
(12)
(13)
0 denotes the predicted value of element i in sample k by the new method or the
where Cik
0 is the reference value of element i in sample k and C
simple linear regression method, Cik
i
represents the average reference value of element i in all samples.
3. Results and Discussion
3.1. The pXRF Analysis Results
The pXRF analysis results and reference values of the samples are shown in
Tables S1 and S2, respectively. The data were imported into OriginPro Learning Edition,
and scatter plots were drawn with the testing values as the X-axis coordinate and reference
values as the Y-axis coordinate (Figure 1). It can be seen that the different elements had
different distribution patterns. For example, the plots of some elements show that there was
almost no difference between the testing and reference values, which indicates that these
elements had high testing accuracy, including Al, Si, P, S, K, Ca, Ti, Mn, Fe, Cu, Zn, As, Rb,
Sr, Zr, Pb and Bi. If not strictly required, the analysis results of these elements could be used
directly without correction. The plots of the other elements show that there were varying
degrees of difference between their testing values and reference values, indicating that
the analysis results of these elements needed to be corrected. Note that the plots of some
of these elements (e.g., Mg, Ni, Y, Nb and Sb) exhibited a significant linear relationship
between the testing values and reference values. Obviously, the SLR correction method
could be used for these elements. However, the elements V, Cr, Co, Se, Mo, Cd, Sn, Ba, Ta,
W, Tl and U did not have a linear relationship between their testing and reference values,
and therefore, it made little sense to perform the SLR correction.
3.2. The Correction of the Matrix Effect
The correction indicators should be determined before using the new method for data
calibration. The major elements that had the highest correlation with the element to be
corrected and are not or just slightly influenced by other elements could be selected as
correction indicators [10]. The results of the correlation analysis showed the major elements
with the highest correlation coefficients with the elements to be corrected (Table 2). The
number of trace elements mostly related to Al in the pXRF analysis was the largest, reaching
11 and accounting for almost half of all the elements. The following major elements were
Appl. Sci. 2022, 12, 568
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Ti and Fe, which affected six and four elements, respectively. The elements Mn, Si, K and
Ca had little effect on the other elements, and the number of elements affected by any
one
Appl. Sci. 2022, 12, x FOR PEER REVIEW
5 of 12
of them was generally less than three. The significant correlation was either reflected in
similar chemical properties or in similar positions in the periodic table.
Figure1.1.Scatter
Scatter plots
plots of
detected
elements.
The
unit
of Mg,
Figure
of testing
testing values
valuesvs.
vs.reference
referencevalues
valuesofofthe
the
detected
elements.
The
unit
of Mg,
−2, and that of V, Cr, Co, Ni, Se, Rb, Sr, Y, Zr,
Al,Si,
Si,P,P,S,S,K,
K,Ca,
Ca,Ti,
Ti,Mn,
Mn, Fe,
Fe, Cu,
Cu, Zn,
2 , and that of V, Cr, Co, Ni, Se, Rb, Sr, Y, Zr,
Al,
Zn, As
As and
andPb
Pbwas
was10
10−
Nb, Mo, Cd, Sn, Sb, Ba, Ta, W, Tl, Bi and U was 10−6.
Nb, Mo, Cd, Sn, Sb, Ba, Ta, W, Tl, Bi and U was 10−6 .
3.2. The
Correctionindicators
of the Matrix
Effect to be tested by pXRF analysis.
Table
2. Correction
for elements
The correction indicators should be determined before using the new method for
Correction
Trace correlation
Elements towith
Be Tested
data calibration.
TheIndicator
major elements that had the highest
the element to
Al not or just slightly influenced
Cu,by
Se,other
Rb, Mo,
Sb, Sn, could
W, Tl, be
As,selected
Bi, U as
be corrected and are
elements
Ti [10]. The results of the correlation analysis
P, Cr, Y, Zr,
Nb, Ta the major elecorrection indicators
showed
Fe correlation coefficients with the elements
Mg, V,toCo,
ments with the highest
be Ni
corrected (Table 2).
Mnelements mostly related to Al in the pXRF
Cd,
Zn
The number of trace
analysis
was the largest,
Si
S, Pb
reaching 11 and accounting for almost half of all the elements. The following major eleK
Ba
ments were Ti andCaFe, which affected six and four elements, respectively.
The elements
Sr
Mn, Si, K and Ca had little effect on the other elements, and the number of elements affected by any one of them was generally less than three. The significant correlation was
The
testingin
values
and
reference
values were
into IBM
SPSS
Statistics
25 to
either
reflected
similar
chemical
properties
or in imported
similar positions
in the
periodic
table.
calculate the coefficients of αi , αj and ui for each detected element using Equations (3)–(5).
Then,
correction
indicators
an independent
were substituted, along with the
Tablethe
2. Correction
indicators
for as
elements
to be tested variable
by pXRF analysis.
coefficients of αi , αj and ui , into Equation (2) to form multiple linear regression equations
Indicator
Tracethat
Elements
to Be Tested
for Correction
each element,
except the indicator elements
were slightly
influenced by the matrix
Al
Cu,
Se,
Rb,
Mo,
Sb,
Sn,
W,
Tl,
As, Bi,
effect (Table 3). The reference value and the testing value were used
as U
an independent
Ti
P,
Cr,
Y,
Zr,
Nb,
Ta
variable and dependent variable, respectively. For comparison, the SLR method was also
Fe
Mg, V, Co, Ni
Mn
Cd, Zn
Si
S, Pb
Appl. Sci. 2022, 12, 568
6 of 11
performed on the pXRF analysis results of the CRMs, and the testing and reference values of
each element were used to establish regression equations (Table 3). Subsequently, both the
multiple and simple linear regression equations were used to calculate the regression values
for each element. The corresponding scatter plots of the regression values vs. reference
values are shown in Figure 2.
Table 3. Correction equations for elements detected by pXRF analysis.
Element
Multiple Linear Regression Equations
Simple Linear Regression Equations
Mg
Al
Si
P
S
K
Ca
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
As
Se
Rb
Sr
Y
Zr
Nb
Mo
Cd
Sn
Sb
Ba
Ta
W
Tl
Pb
Bi
U
y = 1.312x − 0.019106Fe − 0.035
y = 0.882x + 0.011691Ti − 0.002
y = 1.136x + 0.088753Si − 1.965
y = 0.066x + 17.255Fe + 43.219
y = 0.827x + 27.540Ti − 11.885
y = −0.172x + 5.366Fe + 5.390
y = 0.914x + 3.297Fe − 0.558
y = 0.928x − 10.245Al + 147.296
y = 1.171x − 0.000Mn − 888.827
y = 1.018x − 25.559Al + 422.940
y = 1.692x − 0.406Al − 13.035
y = 0.841x + 1.068Al − 2.205
y = 0.787x + 0.690Ca ± 0.001
y = 0.719x − 0.357Ti + 1.495
y = 0.814x + 0.000Ti + 14.385
y = 0.604x + 10.429Ti − 0.092
y = 0.340x − 0.172Al − 1.664
y = 1.879x + 331.155Mn − 13.213
y = 0.635x + 4.047Al − 6.683
y = 1.110x − 5.948Al − 106.578
y = 0.069x + 596.818K − 566.860
y = 0.033x + 1.880Ti + 4.516
y = 0.286x + 4.809Al − 9.415
y = 1.199x + 23.460Al − 59.340
y = 1.314x − 9284.000Si + 3365.584
y = 4.199x + 0.428Al − 5.591
y = 0.7463x − 0.006Al − 0.641
y = 1.299x − 0.610
y = 0.846x − 0.037
y = 1.052x − 0.058
y = 0.911x + 0.019
y = 1.075x − 0.893
y = 0.888x + 0.005
y = 0.879x + 0.150
y = 0.828x − 0.003
y = 1.107x − 47.244
y = 0.878x + 4.045
y = 0.853x − 0.004
y = 0.935x + 1.353
y = 0.028x + 17.870
y = 0.965x − 1.632
y = 0.928x + 54.855
y = 1.168 x− 810.710
y = 1.021x + 46.514
y = 2.483x − 37.067
y = 0.866x + 2.394
y = 0.790x + 3.256
y = 0.710x + 4.660
y = 0.844x + 15.891
y = 0.776x − 8.625
y = 0.333x − 10.07
y = 1.737x − 113.910
y = 0.494x − 1.312
y = 1.088x − 59.504
y = 0.076x + 270.120
y = 0.038x − 0.070
y = 0.360x + 3.3911
y = 0.011x + 0.554
y = 1.381x − 540.560
y = 2.358x − 13.724
y = 0.739x − 7.474
Note: x and y represent the testing and regression values of the element to be tested, respectively. The unit is the
same as in Figure 1.
3.3. Evaluation of the Correction Results
3.3.1. Different Correction Effects
The regression results show that the matrix effect correction (MEC) and SLR methods
had different correction effects on V, Cr, Co, Mo, Cd, Sn, Ta and W. It can be seen that their
regression values calculated by the SLR method were quite different from their reference
values, while the results of the MEC method were much closer (Figure 2), indicating that
the MEC method was better than the SLR method. The statistical parameters of R2 , PRE ,
MAE, MAPE, RMSE and RMSPE were calculated for the detected elements to evaluate
the correction effect of these two methods. Not surprisingly, the MEC method had large
R2 values, PRE values close to one and small MAE, MAPE, RMSE and RMSPE values
(Figures 3–5), while the results for the SLR method were the opposite.
Appl. Sci. 2022, 12, x FOR PEER REVIEW
7 of 12
U
y = 0.7463x − 0.006Al − 0.641
y = 0.739x − 7.474 8 of 12
7 of 11
Note: x and y represent the testing and regression values of the element to be tested, respectively.
The unit is the same as in Figure 1.
, 12, x FOR PEER REVIEW
Appl. Sci. 2022, 12, 568
P
K influence
Si fluorescence
0.4intensities,
and weak Al
characteristic X-ray
whichSmay exacerbate3the
10
30
20
of the matrix effect.
2
2
0.2
5
10
15
Generally, the influence of the matrix effect can give rise to some extremely
large or
1
Mg
Al
K
Si
P
S
0
small
testing
values
called
outliers
during
the
pXRF
analysis,
which
will
deviate
signifi0
1
2
3
0
5
10
0
10
20
0
15
30
0.0
0.2
0.4
0
2
4
Ca
Fe
Ti the changing trend
V
Cr
Mn example, the testing
cantly from
of normal points
(Figure 2). For
values
6
0.30
40
300
1.0
200
of Co were much higher than its reference values, which may have been caused by the
4
0.15
0.5
100
enhancement
of the characteristic
X-ray fluorescence
intensity
by Fe (Figure202). The testing
150
2
Ca
V
Fe
Ti
Cr
Mn
values
of0.0V and
Cr,
however,
were
lower than
their
reference
values,
which
may
have
0
0
20
40
0.00
0.15
0.30
0
2
4
6
150
300
0.5
1.0
100
200
Cu
As by Fe and Ti, Se
Co
Ni by the absorption
been caused
of secondaryZnX-ray fluorescence
respectively
4
150
40
4
(Figure100
2). If the outliers10are not omitted 10
from the analysis results before SLR
regression
2
5
5
20
analysis,50 a realistic regression curve will not
be established, and therefore,2the resulting
Ni 0
Cu
As eliminated be-Se
Co
Zn if 0the outliers are
0
regression
values
will also0 be unreliable.
Note
that
even
0
5
10
0
20
40
0
2
4
5
10
0
2
4
0
50 100 150
Nb
Rb
Sr
Y a realistic regression
Zr
Mo
fore
SLR
regression
analysis,
curve
cannot
be
obtained
because
the
1,000
200
60
30 the regression 400
testing values involved in
analysis are incomplete, which will20affect the use
500
100
30
200 is not a fundamental
15 Obviously, this
of the regression
equation.
solution; in10other words,
Sr
Nb
Mo
Rb
Zr
Y
the influence of the matrix
0 effect cannot be eliminated by the SLR method.
0
100
200
0
30
60
0
200
400
15
30
0
500
1,000
0
10
20
Cd
Sn
Fortunately,
the MECSbmethod can useBathe data of the major
elements80toWmodify the
Ta
500
60
400
300
data of the outliers and, to a certain extent, weaken the influence
of the matrix effect. For
3
250
150
30
example, the testing values of Co, V and Cr were significantly corrected by40the participa300
0
Ba significantly
Ta
0
tion Cd
of Fe and Ti. TheSn quality
of theseSbelement data was
improved
by theW
0
0
300
400
0
150 300
250
500
0
3
0
30
60
0
40
80
simple linear
regression paMEC method.
Compared with
the SLR method,
the new methodData
hadwithbetter
statistical
Pb
Tl
U
Bi
200
6
with matrix effect correction
1.0
rameters4 for all of these elements (Figures 3–5). For example, allData
statistical
parameters of
Trendline with simple linear regression
100greatly improved.
3 As for Cr, W, V and
Co, Mo, 2Sn and Ta were
Cd, only
a part
of the
Trendline
with matrix
effect correction
X-axis
The
regression
values
of
the
element
0.5
2
Tl
Pb
U of Cr and W increased signifi0
parameters
was enhanced; that is, theBiR and PRE values
Y-axis The reference values of the element
0 100 200 300
0
3
6
0
2
4
0.5
1.0
cantly, and the MAE, MAPE, RMSE and RMSPE values of V and Cd decreased signifiFigurefor
2. Scatter
of the new method and the
simple
regression
method. The
unit
is the
cantly. The reason
the plots
incomplete
be linear
that
elements
also
Figure 2. Scatter
plots of the newcorrection
method and may
the simple
linearthese
regression
method. are
The unit
is the
same as in Figure 1.
as in Figure
1.
greatly affectedsame
by other
elements
aside from the selected major elements.
4
Mg
3.3. Evaluation of the Correction Results
1.0
R2
simple linear
The regression results
0.8 show that the matrix effect correction (MEC) and SLR methods
regression method
had different correction effects on V, Cr, Co, Mo, Cd, Sn, Ta and W. It can be seen that
their regression values0.6
calculated by the SLR method were quite different
from their refR2 (Figure 2), indicaterence values, while the results of the MEC method were much closer
matrix effect
ing that the MEC method
0.4 was better than the SLR method. The statistical parameters of
correction method
R2, PRE, MAE, MAPE, RMSE and RMSPE were calculated for the detected elements to evalElement
Element
uate the correction effect
0.2 of these two methods. Not surprisingly, the MEC method had
P
Ta
V
U
Cd
Y
Mg
Mo
Ni
Se
Cr
Nb
2
large R values, PRE values close to one and small MAE, MAPE,
RMSEREand RMSPE values
simple linear
(Figures 3–5), while the1.0results for the SLR method were the opposite.
regression method
Compared with the
0.8 other detected elements, these elements were more closely related to the major elements, including Fe, Mn, Ti and Al, mainly due to their adjacent
0.6
PRE
positions in the periodic table and similar chemical properties. As a result,
the determination of these elements0.4
may inevitably have been affected bymatrix
majoreffect
components during
method
pXRF analysis. In fact, 0.2
the influence of the matrix effect mainlycorrection
comes from
the absorption
or enhancement of the characteristic X-ray fluorescence intensity of the element to be
0.0
tested by adjacent
elements,
especially major elements.Element
These elements have low contents
Element
3.3.1. Different Correction Effects
0.8
0.6
0.4
0.2
0.0
Co
Sn
W
Sb
Zr
Rb
1.0
0.8
0.6
0.4
0.2
P
Pb
Ba
Zn
Sr
S
Bi
Cu
As
Tl
Figure 3. The R2 Figure
and PRE
for P
the correction
results
of the
newofand
SLRand
methods.
3. values
The R2 and
values for the
correction
results
the new
SLR methods.
RE
Appl.
Sci. REVIEW
2022, 12, 568
12, x FOR
PEER
2, x FOR PEER REVIEW
8 of 11
9 of 12
9 of 12
200
200
8
150
150
6
100
100
4
50
50
2
0
0
Cd
Cd
Tl
Tl
Sb
Sb
Pb
Pb
Element
Element
Zn
Zn
0
MAPE
MAPE
simple linear
simple linear
regression method
regression method
8
6
4
MAPE
MAPE
matrix effect
matrix effect
correction method
correction method
2
0
Element
Element
Mo W
As
Sn
Cu Co Ta
U
Mo W
As
Sn
Cu Co Ta
U
RMSPE
RMSPE
simple linear
simple linear
regression method
regression method
0.20
0.20
0.6
0.6
RMSPE
RMSPE
matrix effect
matrix effect
correction method
correction method
0.15
0.15
0.4
0.4
0.10
0.10
0.05
0.05
0.2
0.2
Mg
V
Mg
V
Ni
Ni
Element 0.00
Element 0.00
Bi
Nb
Se
Cr
P
Bi
Nb
Se
Cr
P
Y
Y
Ba
Ba
Zr
Zr
S
S
Element
Element
Rb
Sr
Rb
Sr
Figure 4. The MAPE
and The
RMSPE
values
for the correction
results ofresults
the newthe
and
SLR methods.
Figure
MAPE
and RMSPE
for theresults
correction
new
SLR methods.
Figure 4. The MAPE
and4.RMSPE
values
for the values
correction
of the newofand
SLRand
methods.
Figure 5. The MAE and5.RMSE
values
for thevalues
correction
results
of results
the new
and
SLR methods.
The MAE
andfor
RMSE
for the
correction
the
new
SLR methods.
Figure 5. The MAEFigure
and RMSE
values
the correction
results
of the
newofand
SLRand
methods.
Compared
with the other detected elements, these elements were more closely related
3.3.2.Similar
SimilarCorrection
Correction
Effect
3.3.2.
Effect
to the major elements, including Fe, Mn, Ti and Al, mainly due to their adjacent positions
Asfor
forthe
theother
other
elements,
these
two
methods
hadsimilar
similarcorrection
correction
effects.
However, of
in theelements,
periodic table
and
similar
chemical
properties.
As a result,
the determination
As
these
two
methods
had
effects.
However,
some
of
these
elements
were
well
corrected,
including
Al,
Si,
P,
S,
K,
Ca,
Ti,
Mn,
Fe,Cu,
Cu,
these elements
may corrected,
inevitably have
been affected
major
components
during
pXRF
some of these elements
were well
including
Al, Si, by
P, S,
K, Ca,
Ti, Mn, Fe,
analysis.
In fact,
the
influence
of the
matrix
effect were
mainlynot,
comes
from theSe,
absorption
Zn,
As,
Rb,
Sr,
Zr,
Pb,
Mg,
Ni,
Y,
Nb,
Sb
and
Bi,
and
some
including
Ba,
Tl
Zn, As, Rb, Sr, Zr, Pb, Mg, Ni, Y, Nb, Sb and Bi, and some were not, including Se, Ba, Tl or
of thewas
characteristic
X-ray fluorescence
intensity
ofthe
the SLR
element
be tested
andU.U.ItItcan
canbebeenhancement
seenthat
thatthere
there
nosignificant
significant
differencebetween
betweenthe
andto
MEC
and
seen
wasespecially
no
difference
SLRlow
and
MEC
by
adjacent
elements,
major
elements.
These
elements
have
contents
regressionvalues
valuesofofthe
theformer
formergroup
groupofofelements,
elements,allallofofwhich
whichwere
were closetototheir
theirreferrefer-and
regression
weak characteristic
X-ray fluorescence intensities,
which mayclose
exacerbate the influence
of
encevalues
values(Figure
(Figure2),2),indicating
indicating that these two regression methods both had a good corence
the matrix effect. that these two regression methods both had a good correctioneffect.
effect.
rection
However,
shouldalso
alsobebenoted
notedthat
thatthe
theMEC
MECmethod
methoddid
didnot
notplay
playa asignificant
significantrole
role
However, ititshould
eliminatingthe
thematrix
matrixeffect
effectfor
forthese
theseelements,
elements,which
whichmay
maybebebecause
becausethey
theywere
werejust
just
inineliminating
slightlyaffected
affectedby
bythe
thematrix
matrixeffect.
effect.For
Forexample,
example,although
althoughsome
sometrace
traceelements
elementshad
had
slightly
much
lower
contents
and
weaker
characteristic
X-ray
fluorescence
intensities
than
the
mamuch lower contents and weaker characteristic X-ray fluorescence intensities than the ma-
Appl. Sci. 2022, 12, 568
9 of 11
Generally, the influence of the matrix effect can give rise to some extremely large or
small testing values called outliers during the pXRF analysis, which will deviate significantly from the changing trend of normal points (Figure 2). For example, the testing values
of Co were much higher than its reference values, which may have been caused by the
enhancement of the characteristic X-ray fluorescence intensity by Fe (Figure 2). The testing
values of V and Cr, however, were lower than their reference values, which may have
been caused by the absorption of secondary X-ray fluorescence by Fe and Ti, respectively
(Figure 2). If the outliers are not omitted from the analysis results before SLR regression
analysis, a realistic regression curve will not be established, and therefore, the resulting
regression values will also be unreliable. Note that even if the outliers are eliminated before
SLR regression analysis, a realistic regression curve cannot be obtained because the testing
values involved in the regression analysis are incomplete, which will affect the use of the
regression equation. Obviously, this is not a fundamental solution; in other words, the
influence of the matrix effect cannot be eliminated by the SLR method.
Fortunately, the MEC method can use the data of the major elements to modify the
data of the outliers and, to a certain extent, weaken the influence of the matrix effect. For
example, the testing values of Co, V and Cr were significantly corrected by the participation
of Fe and Ti. The quality of these element data was significantly improved by the MEC
method. Compared with the SLR method, the new method had better statistical parameters
for all of these elements (Figures 3–5). For example, all statistical parameters of Co, Mo,
Sn and Ta were greatly improved. As for Cr, W, V and Cd, only a part of the parameters
was enhanced; that is, the R2 and PRE values of Cr and W increased significantly, and the
MAE, MAPE, RMSE and RMSPE values of V and Cd decreased significantly. The reason
for the incomplete correction may be that these elements are also greatly affected by other
elements aside from the selected major elements.
3.3.2. Similar Correction Effect
As for the other elements, these two methods had similar correction effects. However,
some of these elements were well corrected, including Al, Si, P, S, K, Ca, Ti, Mn, Fe, Cu, Zn,
As, Rb, Sr, Zr, Pb, Mg, Ni, Y, Nb, Sb and Bi, and some were not, including Se, Ba, Tl and U.
It can be seen that there was no significant difference between the SLR and MEC regression
values of the former group of elements, all of which were close to their reference values
(Figure 2), indicating that these two regression methods both had a good correction effect.
However, it should also be noted that the MEC method did not play a significant role
in eliminating the matrix effect for these elements, which may be because they were just
slightly affected by the matrix effect. For example, although some trace elements had much
lower contents and weaker characteristic X-ray fluorescence intensities than the major
elements, as long as they were just slightly affected by the matrix effect, a good correction
effect could also be obtained, such as the elements Rb, Sr, Zr, Ni, Y, Nb and Sb. As for
the elements Al, Si, P, S, K, Ca, Ti, Mn, Fe, Cu, Zn, As, Pb and Mg, they could produce an
extremely high intensity of characteristic X-ray fluorescence under X-ray irradiation due to
their high contents in the certified reference samples, and therefore, other elements would
have little impact on them. The statistical parameter results show that these two methods
both had large R2 values close to one and small MAE, MAPE, RMSE and RMSPE values
(Figures 3–5), which also indicate good correction effects.
For the correction of Ba, Tl, Se and U, neither of these two methods worked, and
their regression values were quite different from their reference values (Figure 2). The
testing accuracy of these elements was low, and there was no linear relationship between
their testing values and reference values (Figure 1), which may be the reason for the poor
correction effect of the SLR method. As for the MEC method, the poor correction effect
may be attributed to the fact that these elements are influenced not only by the selected
major elements but also by other elements or factors. Nevertheless, compared with the SLR
method, the new method had better statistical parameters for Ba and Tl (Figures 3–5). For
Appl. Sci. 2022, 12, 568
10 of 11
example, the MAE, MAPE, RMSE and RMSPE values of Ba and Tl decreased significantly.
As for Se and U, there was almost no difference between the parameters.
Regardless of whether the element was affected by the matrix effect, the new method
could attain a considerable correction effect, indicating that a lot of the matrix effect was
stripped off. Compared with the SLR method, the regression values of the elements V, Cr,
Ta and W obtained from the new method were closer to their reference values, especially
for elements Co, Mo, Cd and Sn (Figure 2). As for the elements slightly or not affected
by the matrix effect, these two regression methods had considerable and almost the same
correction effects. Collectively, the SLR method tends to deal with the matrix effect at a
macro level without exploring the relationship between elements, while the new method
has a better correction result for the matrix effect with the aid of the major elements.
4. Conclusions
In this study, 16 certified reference materials were determined by pXRF analysis,
and major elements were used to calibrate the analysis results with the application of
multiple linear regression analysis. The results show that the major components of a sample
contributed the most to the matrix effect and could be used as important indicators in
matrix effect correction. The regression method based on the correction indicators of
major elements can significantly improve pXRF analysis results. Although pXRF analysis
is not a substitute for laboratory analysis methods, such as X-ray fluorescence analysis,
inductively coupled plasma mass spectrometry analysis and other forms of high-precision
analysis, the data can provide reliable material composition information after correction.
Supplementary Materials: The following are available online at https://www.mdpi.com/article/10
.3390/app12020568/s1, Table S1: The pXRF analysis result of the samples, Table S2: The reference
values of the samples.
Author Contributions: Conceptualization, J.L.; methodology, X.Z.; software, J.G.; validation, X.T.
and Y.H.; formal analysis, J.G.; investigation, T.L.; resources, Q.W.; data curation, J.G.; writing—
original draft preparation, J.G.; writing—review and editing, X.Z.; visualization, X.T.; supervision,
Q.W.; project administration, J.L.; funding acquisition, J.L. All authors have read and agreed to the
published version of the manuscript.
Funding: This research was funded by the State Key Research and Development Program, grant
number 2016YFC0600606.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data is contained within the Supplementary Material.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
2.
3.
4.
5.
Jinke, G.; Jilong, L.; Yechang, Y.; Yuyan, Z.; Xiaodan, T.; Yuchao, F.; Yan, L. Application of Portable XRF in Core Geochemical
Characteristics of Qujia Gold Deposit in Jiaodong Area. Spectrosc. Spectr. Anal. 2020, 40, 1461–1466. [CrossRef]
Rawal, A.; Chakraborty, S.; Li, B.; Lewis, K.; Godoy, M.; Paulette, L.; Weindorf, D.C. Determination of base saturation percentage
in agricultural soils via portable X-ray fluorescence spectrometer. Geoderma 2019, 338, 375–382. [CrossRef]
Zhang, W.; Lentz, D.R.; Charnley, B.E. Petrogeochemical assessment of rock units and identification of alteration/mineralization
indicators using portable X-ray fluorescence measurements: Applications to the Fire Tower Zone (W-Mo-Bi) and the North Zone
(Sn-Zn-In), Mount Pleasant deposit, New Brunswick, Canada. J. Geochem. Explor. 2017, 177, 61–72. [CrossRef]
Kazimoto, E.O.; Messo, C.; Magidanga, F.; Bundala, E. The use of portable X-ray spectrometer in monitoring anthropogenic toxic
metals pollution in soils and sediments of urban environment of Dar es Salaam Tanzania. J. Geochem. Explor. 2018, 186, 100–113.
[CrossRef]
Holcomb, J.A.; Karkanas, P. Elemental mapping of micromorphological block samples using portable X-ray fluorescence
spectrometry (pXRF): Integrating a geochemical line of evidence. Geoarchaeology 2019, 34, 613–624. [CrossRef]
Appl. Sci. 2022, 12, 568
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
11 of 11
Pessanha, S.; Marguí, E.; Carvalho, M.L.; Queralt, I. A simple and sustainable portable triaxial energy dispersive X-ray fluorescence
method for in situ multielemental analysis of mining water samples. Spectrochim. Acta Part B At. Spectrosc. 2019, 164, 105762.
[CrossRef]
Declercq, Y.; Delbecque, N.; De Grave, J.; De Smedt, P.; Finke, P.; Mouazen, A.M.; Nawar, S.; Vandenberghe, D.; Van Meirvenne,
M.; Verdoodt, A. A Comprehensive Study of Three Different Portable XRF Scanners to Assess the Soil Geochemistry of An
Extensive Sample Dataset. Remote Sens. 2019, 11, 2490. [CrossRef]
Durance, P.; Jowitt, S.M.; Bush, K. An assessment of portable X-ray fluorescence spectroscopy in mineral exploration, Kurnalpi
Terrane, Eastern Goldfields Superterrane, Western Australia. Appl. Earth Sci. 2014, 123, 150–163. [CrossRef]
West, M.; Ellis, A.T.; Potts, P.J.; Streli, C.; Vanhoof, C.; Wegrzynek, D.; Wobrauschek, P. Atomic spectrometry update—X-ray
fluorescence spectrometry. J. Anal. At. Spectrom. 2012, 27, 1603–1644. [CrossRef]
Ang, J. Development of X-ray Fluorescence Spectrometry in the 30 Years. Rock Miner. Anal. 2012, 31, 383–398. [CrossRef]
Haijun, Q.; Jianying, W.; Xuefeng, Z.; Yang, W. Matrix Effect of Fe and Ca on EDXRF Analysis of Ce Concentration in Bayan Obo
Ores. Spectrosc. Spectr. Anal. 2015, 35, 3510–3513. [CrossRef]
Song, L.; Bomin, S.; Qinghui, L.; Fuxi, G. Application of Calibration Curve Method and Partial Least Squares Regression Analysis
to Quantitative Analysis of Nephrite Samples Using XRF. Spectrosc. Spectr. Anal. 2015, 35, 245–251. [CrossRef]
Gallhofer, D.; Lottermoser, B. The Influence of Spectral Interferences on Critical Element Determination with Portable X-ray
Fluorescence (pXRF). Minerals 2018, 8, 320. [CrossRef]
Ziyue, N.; Jiwen, C.; Mingbo, L.; Xueliang, L.; Shaocheng, H. Interference correction of energy dispersive X-ray fluorescence
spectrometric determination of chromium and manganese in soil. Metall. Anal. 2016, 36, 10–14. [CrossRef]
Tingting, G.; Ye, T.; Qian, C.; Boyang, X.; Fuzhen, H.; Lintao, W.; Ying, L. Matrix Effect and Quantitative Analysis of Iron Filings
with Different Particle Size Based on LIBS. Spectrosc. Spectr. Anal. 2020, 40, 1207–1213. [CrossRef]
Andrade, R.; Silva, S.H.G.; Weindorf, D.C.; Chakraborty, S.; Faria, W.M.; Guilherme, L.R.G.; Curi, N. Tropical soil order and
suborder prediction combining optical and X-ray approaches. Geoderma Reg. 2020, 23, e00331. [CrossRef]
Benedet, L.; Nilsson, M.S.; Silva, S.H.G.; Pelegrino, M.H.P.; Mancini, M.; Menezes, M.D.; Guilherme, L.R.G.; Curi, N. X-ray
fluorescence spectrometry applied to digital mapping of soil fertility attributes in tropical region with elevated spatial variability.
An. Acad. Bras. Cienc. 2021, 93, 4. [CrossRef] [PubMed]
Qu, M.; Liu, H.; Guang, X.; Chen, J.; Zhao, Y.; Huang, B. Improving correction quality for in-situ portable X-ray fluorescence
(PXRF) using robust geographically weighted regression with categorical land-use types at a regional scale. Geoderma 2021,
409, 115615. [CrossRef]
Fragkaki, A.G.; Farmaki, E.; Thomaidis, N.; Tsantili-Kakoulidou, A.; Angelis, Y.S.; Koupparis, M.; Georgakopoulos, C. Comparison
of multiple linear regression, partial least squares and artificial neural networks for prediction of gas chromatographic relative
retention times of trimethylsilylated anabolic androgenic steroids. J Chromatogr. A 2012, 1256, 232–239. [CrossRef]
Xiaoyong, T.; Xiaofang, N.; Zhaocong, S. Effect and Correction of Iron in Soil on Accuracy of Chromium Determination by
Portable X-ray Fluorescence Spectrometry. Rock Miner. Anal. 2020, 39, 460–474. [CrossRef]
Rousseau, R.M. Some considerations on how to solve the Sherman equation in practice. Spectrochim. Acta Part B At. Spectrosc.
2004, 59, 1491–1502. [CrossRef]
Rousseau, R.M. Corrections for matrix effects in X-ray fluorescence analysis—A tutorial. Spectrochim. Acta Part B At. Spectrosc.
2006, 61, 759–777. [CrossRef]
Fen, G.; Liang, G.; Xiaoxu, P.; Jin, Y.; Yufang, Z.; Zheng, W. Matrix Effect of Soil Samples in Laser Ablation-Atmospheric Pressure
Glow Discharge Atomic Emission Spectrometry. Chin. J. Anal. Chem. 2020, 49, 1111–1119. [CrossRef]
Balabin, R.M.; Safieva, R.Z.; Lomakina, E.I. Comparison of linear and nonlinear calibration models based on near infrared (NIR)
spectroscopy data for gasoline properties prediction. Chemom. Intell. Lab. Syst. 2007, 88, 183–188. [CrossRef]
Subedi, S.; Fox, T.R. Predicting Loblolly Pine Site Index from Soil Properties Using Partial Least-Squares Regression. For. Sci.
2016, 62, 449–456. [CrossRef]
Rongxi, K.; Wenyou, H.; Yue, H.; Biao, H.; Yanqun, Z.; Yuan, L.; Fangdong, Z.; Xiaoleng, Z.; Bao, W. Application of Portable X-ray
Fluorescence (PXRF) for Rapid Analysis of Heavy Metals in Agricultural Soils Around Mining Area. Soils 2015, 47, 589–595.
[CrossRef]
Cort, J.W.; Matsuura, K. Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing
average model performance. Clim. Res. 2005, 30, 79–82. [CrossRef]
Shuang, L.; Tiankai, Z.; Linzi, F. Analysis of Urban Green Development and Influencing Factors in Yangtze River Economic Belt.
Stat. Decis. 2019, 35, 125–128. [CrossRef]