CN108444980B - L IBS quantitative analysis method based on algebraic reconstruction relevance vector solution - Google Patents

L IBS quantitative analysis method based on algebraic reconstruction relevance vector solution Download PDF

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CN108444980B
CN108444980B CN201810089431.8A CN201810089431A CN108444980B CN 108444980 B CN108444980 B CN 108444980B CN 201810089431 A CN201810089431 A CN 201810089431A CN 108444980 B CN108444980 B CN 108444980B
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万雄
袁汝俊
王泓鹏
何强
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Abstract

The invention discloses an L IBS quantitative analysis method based on algebraic reconstruction relevance vector solving, which is characterized in that a high-precision optical chromatography reconstruction model is used for modeling a multivariate L IBS quantitative solving problem, a relation among a normalized spectral intensity matrix W of a standard sample, an association matrix F and a standard sample atomic fraction matrix P is established, each column vector of the association matrix F is solved by combining a high-precision ART iterative algorithm in optical chromatography with matrix column vector decomposition in a column-by-column iteration mode, so that the association matrix F reflecting the mutual connection between W and P is obtained, and then the high-precision solving and analysis of each element in an unknown sample is realized according to the operation of an actually measured normalized spectral intensity vector of the unknown sample and the association matrix.

Description

L IBS quantitative analysis method based on algebraic reconstruction relevance vector solution
Technical Field
The invention relates to a laser spectrum detection method, in particular to a quantitative laser-induced breakdown spectroscopy analysis method, which is suitable for simultaneous quantitative solution and analysis of multiple elements of a detection target and belongs to the field of photoelectric detection.
Background
Laser-induced breakdown spectroscopy (L ase-induced breakdown spectroscopy, L IBS for short) focuses on a target surface with short pulse laser, the target is ablated by instantaneous high temperature to generate plasma, atoms and ions fall back from a high energy state to a low energy state in a cooling process to emit spectral lines containing element information, and the spectral lines can be used for detecting substance constituent elements.
L IBS quantitative analysis is divided into unit analysis and multiple analysis methods, the unit analysis method calibrates a certain element to be measured to obtain a calibration curve, and calculates the content of the element according to the calibration curve and the spectral line intensity of the element of the target to be measured, the method has the advantages of simplicity and practicality, the defects are influenced by chemical matrix effect, and the precision of quantitative analysis is limited, the multiple analysis method can calibrate multiple spectral lines of multiple elements of the target to be measured simultaneously, and the content of multiple elements of the target to be measured is obtained by solving a multivariate mathematical matrix equation.
In order to solve the problem of the accuracy of multivariate quantification, a high-accuracy optical chromatography reconstruction model is used for solving the problem of multivariate L IBS quantification, high-accuracy algebraic reconstruction (ART for short) iterative algorithm in optical chromatography is combined with matrix column vector decomposition, column-by-column iterative solution is adopted, and a correlation matrix between a normalized spectral intensity matrix of a standard sample and the content of each element is obtained.
Disclosure of Invention
The invention aims to provide a multivariate L IBS matrix multielement analysis solving method, which comprises the steps of firstly establishing a L IBS multivariate analysis matrix mathematical model similar to optical chromatography reconstruction, then solving a correlation vector by combining with matrix column vector decomposition based on an ART iterative algorithm in optical chromatography to obtain a correlation matrix, and obtaining atomic fractions of a plurality of elements of a target to be detected according to the operation of normalized spectral intensity of the target to be detected and the correlation matrix so as to realize high-precision quantitative L IBS detection.
The present invention is achieved in such a way that,
1. assume that the number of elements to be quantitatively analyzed (i.e., element dimensions) is M and are well-ordered. N standard samples were prepared for calibration (i.e. sample dimension N). The N samples are solid, have equal size and size, contain the M elements in different proportions, the atomic fraction (atomic number percentage) of each element is known, and the components in each sample are uniformly distributed;
2. constructing a multivariate L IBS quantitative analysis matrix equation, namely a projection matrix, a flow field physical quantity image matrix and a measurement matrix in optical tomography reconstruction
WF＝P
In the formula, W is a normalized spectral intensity matrix of a standard sample, which is equivalent to a projection matrix in optical tomography reconstruction; f is a correlation matrix, which is equivalent to a flow field physical quantity image matrix in optical tomography reconstruction, and P is a standard sample atomic fraction matrix, which is equivalent to a measurement matrix;
3. the normalized spectral intensity matrix W of the standard sample is constructed according to the following method:
l IBS detection is carried out on the N standard samples under the same test conditions and test parameters to obtain N L IBS spectrograms corresponding to the N standard samples, and the N L IBS spectrograms are normalized to obtain N normalized L IBS spectrograms, wherein k characteristic spectral lines are respectively taken for each element (the sample dimension N is required to be larger than the spectral dimension kM), and then a standard sample normalized spectral intensity matrix W with N rows and kM columns is constructed as follows:
Figure GDA0002468886050000031
the kM values in the first row in the normalized spectral intensity matrix represent the normalized spectral intensity values of the spectral lines represented by the M elements kM of the first standard sample; the kM values in the second row represent the normalized spectral intensity values of the M elements kM of the second standard sample representing the spectral lines; and so on …; kM values in the nth row represent the normalized spectral intensity values of the spectral lines for the nth standard sample M elements kM;
4. constructing a standard sample atomic fraction matrix P with N rows by M columns as follows:
Figure GDA0002468886050000032
the M values in the first row of the atomic fraction matrix represent the atomic fractions of the M elements of the first standard sample; the M values in the second row represent the atomic fractions of the M elements of the second standard sample; and so on …; the M values in the Nth row represent the atomic fractions of the M elements of the Nth standard sample;
5. the correlation matrix F, which reflects the correlation between W and P, can be expressed as:
Figure GDA0002468886050000041
the incidence matrix F is a matrix of kM rows by M columns, and the required solution kM2The F matrix can be obtained by the unit value. Column decomposing the incidence matrix F into M incidence vectors F1、F2、F3、...、FM(ii) a Performing column decomposition on the atomic fraction matrix P of the standard sample into M atomic fraction vectors P1、P2、P3、...、PM；
6. Converting the solution of the incidence matrix F into M incidence vectors F1、F2、F3、...、FMThe solution model is as follows:
Pi＝WFi+Ei
wherein i is 1,2,3iFor the error vector, the error is minimum based on a certain optimization criterion, and the optimal approximate solution under the optimization criterion is obtained;
7. adopting ART iterative algorithm to carry out correlation vector FiAnd (3) solving:
Figure GDA0002468886050000042
Figure GDA0002468886050000043
t＝q(mod N)+1
in the above formula, the superscript 0 represents the initial value; superscript T represents transposition; the superscript q represents the q-th iteration value; superscript q +1 represents the q +1 th iteration value; pi(t) represents PiThe t-th cell value of (1); wtRepresents the t-th row vector in W; mod is a modulo (i.e., remainder) operation, i.e., t is the remainder of q divided by N plus 1; lambda is a relaxation factor, the value of which is between 0 and 1, and the value of the lambda represents the degree of tightness of iterative constraint;
the termination conditions for the iteration are:
Figure GDA0002468886050000052
is a very small number; after the iteration has terminated, FiThe last iteration value is FiThe solution result of (2);
8. all M relevance vectors FiCarrying out L IBS detection on the target to be detected under the same test conditions as the N standard samples to obtain a L IBS spectrogram, carrying out normalization processing on the L IBS spectrogram to obtain a normalized L IBS spectrogram of the sample to be detected, and obtaining a normalized spectral intensity vector of M element kM representative spectral lines of the target to be detected from the normalized L IBS spectrogram:
D＝[d1,d2,d3,...,dkM]
calculating the atomic fractions of M elements of the target to be detected according to the following formula:
Figure GDA0002468886050000051
the method has the advantages that the similar optical chromatography model is adopted and solved in multivariate analysis and calibration so as to solve the influence of competitive emission of different elements in the chemical matrix effect, and the ART chromatography iterative algorithm is adopted for solving so as to obtain the correlation matrix array with the minimum absolute error, thereby improving L IBS quantitative analysis accuracy.
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The principle of the invention is schematically shown in figure 1.
Detailed Description
The invention aims to provide a multivariate L IBS quantitative analysis solving method, which is characterized in that a high-precision optical chromatography reconstruction model is used for modeling a multivariate L IBS quantitative solving problem, a relation among a normalized spectral intensity matrix W, an incidence matrix F and a standard sample atomic fraction matrix P is established, each column vector of the incidence matrix F is iteratively solved column by adopting a high-precision ART iterative algorithm in optical chromatography and matrix column vector decomposition, so that the incidence matrix F reflecting the mutual relation between W and P is obtained, then, the high-precision solving analysis of each element in an unknown sample is realized according to the operation of the actually measured normalized spectral intensity vector of the unknown sample and the incidence matrix, and the solving precision problem of L IBS multivariate analysis is solved.
The L IBS quantification method is illustrated by the following specific example:
1. assuming that the number of elements to be quantitatively analyzed is 12 (i.e. taking the element dimension M ═ 12), including sodium, magnesium, calcium, iron, manganese, copper, silicon, carbon, oxygen, sulfur, nitrogen and hydrogen, in the order from 1 to 12. 100 standard samples were prepared for calibration (i.e., sample dimension N is taken as 100). The 100 samples are solid, have equal size and size, contain the twelve elements in different proportions, the atomic fraction (atomic number percentage) of each element is known, and the components in each sample are uniformly distributed;
2. constructing a multivariate L IBS quantitative analysis matrix equation shown in figure 1 by referring to the relationship among a projection matrix, a flow field physical quantity image matrix and a measurement matrix in optical tomography reconstruction, namely the multivariate L IBS quantitative analysis matrix equation
WF＝P
In the formula, W is a normalized spectral intensity matrix of a standard sample, which is equivalent to a projection matrix in optical tomography reconstruction; f is a correlation matrix, which is equivalent to a flow field physical quantity image matrix in optical tomography reconstruction, and P is a standard sample atomic fraction matrix, which is equivalent to a measurement matrix;
3. the normalized spectral intensity matrix W of the standard sample is constructed according to the following method:
l IBS detection is performed on the 100 standard samples under the same test conditions and test parameters to obtain 100L IBS spectrograms corresponding to the 100 standard samples, and the 100L IBS spectrograms are normalized to obtain 100 normalized L IBS spectrograms, 3 characteristic spectral lines are respectively taken for each element (i.e., k is taken to be 3, the sample dimension N is required to be greater than the spectral dimension kM, the sample dimension N is 100 in this embodiment, and the spectral dimension kM is greater than 36 in this embodiment), so that the following normalized spectral intensity matrix W of the standard samples with N rows by kM columns (in this embodiment, 100 rows by 36 columns) is constructed:
Figure GDA0002468886050000071
the kM values in the first row in the normalized spectral intensity matrix represent the normalized spectral intensity values of the spectral lines represented by the M elements kM of the first standard sample; the kM values in the second row represent the normalized spectral intensity values of the M elements kM of the second standard sample representing the spectral lines; and so on …; kM values in the nth row represent the normalized spectral intensity values of the spectral lines for the nth standard sample M elements kM;
4. constructing a standard sample atomic fraction matrix P with N rows by M columns as follows:
Figure GDA0002468886050000072
in practice, M is 12 and N is 100. The M values in the first row of the atomic fraction matrix represent the atomic fractions of the M elements of the first standard sample; the M values in the second row represent the atomic fractions of the M elements of the second standard sample; and so on …; the M values in the Nth row represent the atomic fractions of the M elements of the Nth standard sample;
5. the correlation matrix F, which reflects the correlation between W and P, can be expressed as:
Figure GDA0002468886050000081
the correlation matrix F is kMA matrix of rows by M columns, in the specific embodiment a 36 row by 12 matrix. Need to solve for kM2432 cell values, the F matrix can be obtained. As shown in fig. 1, the correlation matrix F is column-decomposed into M correlation vectors F1、F2、F3、...、FM(ii) a Performing column decomposition on the atomic fraction matrix P of the standard sample into M atomic fraction vectors P1、P2、P3、...、PM；
6. Converting the solution of the incidence matrix F into M incidence vectors F1、F2、F3、...、FMThe solution model is as follows:
Pi＝WFi+Ei
wherein i is 1,2,3iFor the error vector, the error is minimum based on a certain optimization criterion, and the optimal approximate solution under the optimization criterion is obtained;
7. adopting ART iterative algorithm to carry out correlation vector FiAnd (3) solving:
Figure GDA0002468886050000082
Figure GDA0002468886050000083
t＝q(mod N)+1
in the above formula, the superscript 0 represents the initial value; superscript T represents transposition; the superscript q represents the q-th iteration value; superscript q +1 represents the q +1 th iteration value; pi(t) represents PiThe t-th cell value of (1); wtRepresents the t-th row vector in W; mod is a modulo (i.e., remainder) operation, i.e., t is the remainder of q divided by N plus 1; λ is a relaxation factor, the value of λ is between 0 and 1, the magnitude of the value represents the degree of tightness of the iterative constraint, and the value of λ is 0.1 in this embodiment;
the termination conditions for the iteration are:
Figure GDA0002468886050000092
a very small number, in this example 0.001; after the iteration has terminated, FiThe last iteration value is FiThe solution result of (2);
8. all M relevance vectors FiCarrying out L IBS detection on the target to be detected under the same test conditions as the N standard samples to obtain a L IBS spectrogram, carrying out normalization processing on the L IBS spectrogram to obtain a normalized L IBS spectrogram of the sample to be detected, and obtaining a normalized spectral intensity vector of M element kM representative spectral lines of the target to be detected from the normalized L IBS spectrogram:
D＝[d1,d2,d3,...,dkM]
calculating the atomic fractions of M elements of the target to be detected according to the following formula:
Figure GDA0002468886050000091
wherein M is 12.

Claims (1)

1. An L IBS quantitative analysis method based on algebraic reconstruction relevance vector solving is characterized by comprising the following steps:
1) assuming the number of elements needing quantitative analysis, namely the element dimension is M, and sequencing; preparing N standard samples for calibration, namely, the dimension of the sample is N, the N samples are solid, the size and the size are equal, the N samples contain the M elements in different proportions, the atomic fraction, namely the atomic number percentage, of each element is known, and the components in each sample are uniformly distributed;
2) constructing a multivariate L IBS quantitative analysis matrix equation, namely a projection matrix, a flow field physical quantity image matrix and a measurement matrix in optical tomography reconstruction
WF＝P
In the formula, W is a normalized spectral intensity matrix of a standard sample, which is equivalent to a projection matrix in optical tomography reconstruction; f is a correlation matrix, which is equivalent to a flow field physical quantity image matrix in optical tomography reconstruction, and P is a standard sample atomic fraction matrix, which is equivalent to a measurement matrix;
3) the normalized spectral intensity matrix W of the standard sample is constructed according to the following method:
l IBS detection is carried out on the N standard samples under the same test conditions and test parameters to obtain N L IBS spectrograms corresponding to the N standard samples, normalization processing is carried out on the N L IBS spectrograms to obtain N normalized L IBS spectrograms, k characteristic spectral lines are respectively taken for each element, and the normalized spectral intensity matrix W of the standard samples with N rows and K columns is constructed when the sample dimension N is larger than the spectral dimension kM:
Figure FDA0002468886040000021
the kM values in the first row in the normalized spectral intensity matrix represent the normalized spectral intensity values of the spectral lines represented by the M elements kM of the first standard sample; the kM values in the second row represent the normalized spectral intensity values of the M elements kM of the second standard sample representing the spectral lines; and so on …; kM values in the nth row represent the normalized spectral intensity values of the spectral lines for the nth standard sample M elements kM;
4) constructing a standard sample atomic fraction matrix P with N rows by M columns as follows:
Figure FDA0002468886040000022
the M values in the first row of the atomic fraction matrix represent the atomic fractions of the M elements of the first standard sample; the M values in the second row represent the atomic fractions of the M elements of the second standard sample; and so on …; the M values in the Nth row represent the atomic fractions of the M elements of the Nth standard sample;
5) the correlation matrix F, which reflects the correlation between W and P, can be expressed as:
Figure FDA0002468886040000031
the incidence matrix F is a matrix of kM rows by M columns, and the required solution kM2The F matrix can be obtained only by the unit value; column decomposing the incidence matrix F into M incidence vectors F1、F2、F3、...、FM(ii) a Performing column decomposition on the atomic fraction matrix P of the standard sample into M atomic fraction vectors P1、P2、P3、...、PM；
6) Converting the solution of the incidence matrix F into M incidence vectors F1、F2、F3、...、FMThe solution model is as follows:
Pi＝WFi+Ei
wherein i is 1,2,3iFor the error vector, the error is minimum based on a certain optimization criterion, and the optimal approximate solution under the optimization criterion is obtained;
7) adopting ART iterative algorithm to carry out correlation vector FiAnd (3) solving:
Figure FDA0002468886040000032
Figure FDA0002468886040000033
t＝q(mod N)+1
in the above formula, the superscript 0 represents the initial value; superscript T represents transposition; the superscript q represents the q-th iteration value; superscript q +1 represents the q +1 th iteration value; pi(t) represents PiThe t-th cell value of (1); wtRepresents the t-th row vector in W; mod is a modulo (i.e., remainder) operation, i.e., t is the remainder of q divided by N plus 1; lambda is a relaxation factor, the value of which is between 0 and 1, and the value of the lambda represents the degree of tightness of iterative constraint;
the termination conditions for the iteration are:
Figure FDA0002468886040000041
is a very small number; after the iteration has terminated, FiThe last iteration value is FiThe solution result of (2);
8) all M relevance vectors FiObtaining a correlation matrix F after solving, carrying out L IBS detection on the target to be detected under the same test conditions with the N standard samples to obtain a L IBS spectrogram, carrying out normalization processing on the L IBS spectrogram to obtain a normalized L IBS spectrogram of the sample to be detected, and obtaining a normalized spectral intensity vector of M element kM representative spectral lines of the target to be detected from the normalized spectral intensity vector:
D＝[d1,d2,d3,...,dkM]
calculating the atomic fractions of M elements of the target to be detected according to the following formula:
Figure FDA0002468886040000042
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