CN110376290B - Acoustic emission source positioning method based on multi-dimensional nuclear density estimation - Google Patents

Acoustic emission source positioning method based on multi-dimensional nuclear density estimation Download PDF

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CN110376290B
CN110376290B CN201910656557.3A CN201910656557A CN110376290B CN 110376290 B CN110376290 B CN 110376290B CN 201910656557 A CN201910656557 A CN 201910656557A CN 110376290 B CN110376290 B CN 110376290B
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周子龙
芮艺超
周静
杜雪明
臧海智
林成龙
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Central South University
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Abstract

The invention discloses an acoustic emission source based on multi-dimensional nuclear density estimationBit method. Firstly, combining arrival time data of an acoustic emission sensor to obtain a plurality of groups of arrival time data; and constructing a corresponding set of defined equations according to different combinations of arrival time data. Solving each equation set to obtain multiple groups of closed-form solutions, and screening out the closed-form solutions containing the virtual root to obtain r0And (5) obtaining a preliminary positioning result. And secondly, excluding abnormal positioning results in the primary positioning results, and finally obtaining r primary positioning results. Thirdly, constructing a multi-dimensional kernel density estimation function of the sound emission source coordinate theta by utilizing r primary positioning results
Figure DDA0002137034090000011
Finally, a multi-dimensional kernel density estimation function is solved
Figure DDA0002137034090000012
The maximum point is the optimal acoustic emission source positioning result. The method has high positioning precision.

Description

Acoustic emission source positioning method based on multi-dimensional nuclear density estimation
Technical Field
The invention relates to an acoustic emission source positioning method based on multi-dimensional nuclear density estimation.
Technical Field
The invention with the application number of CN201510973875.4 provides a signal source positioning method for a uniform velocity field, which can obtain a more ideal positioning result when the number of abnormal values is very small, but the probability of positioning failure of the method is greatly increased when the number of abnormal values is more large, the invention with the application number of CN201610571666.1 provides a microseism or abnormal arrival time data identification method based on a minimum distance, which can obtain better abnormal arrival time data when the number of abnormal values is very small, but the probability of positioning failure of the method is greatly increased when the number of abnormal values is more, the invention with the application number of CN201610571666.1 provides a preliminary calculation method for determining microseism or abnormal arrival time data based on a minimum distance, which is difficult to determine the initial distribution of abnormal values when the acoustic emission data is more accurate, so that the initial calculation method for determining the abnormal arrival time distribution of the acoustic emission data is more difficult to obtain, and the initial calculation method for determining the initial distribution of the abnormal emission parameters is not more difficult to determine the initial distribution of the abnormal arrival time, which is more difficult to determine the initial distribution of the abnormal arrival time of the acoustic emission data, and the initial distribution of the abnormal emission parameters are more difficult to determine if the initial distribution of the abnormal arrival time distribution of the abnormal emission parameters.
Disclosure of Invention
The invention aims to solve the technical problem that the existing acoustic emission positioning technology is easily influenced by abnormal arrival time data, and provides an acoustic emission source positioning method based on multi-dimensional nuclear density estimation.
The technical scheme provided by the invention is as follows:
an acoustic emission source positioning method based on multi-dimensional nuclear density estimation comprises the following steps:
step 1, combining arrival time data of each acoustic emission sensor in an acoustic emission detection system to obtain multiple groups of arrival time data; obtaining a plurality of preliminary positioning results based on the plurality of groups of arrival time data;
step 2, constructing a multi-dimensional kernel density estimation function of the acoustic emission source coordinate theta by using the obtained multiple primary positioning results
Figure BDA0002137034070000021
Step 3, solving a multidimensional kernel density estimation function
Figure BDA0002137034070000022
The maximum point is the optimal acoustic emission source positioning result.
In the step 1, based on the plurality of sets of arrival time data, a method disclosed in the prior art, such as an analytic positioning algorithm or a numerical positioning method disclosed in the invention patent with application number CN201610571666.1 (the method selects 6 arrival time numbers each time to obtain a positioning result), or a conjugate gradient method or a marquardt method disclosed in the invention patent with application number CN201610571029.4, may be used to obtain a plurality of preliminary positioning results.
Further, in the step 1, in order to reduce the amount of calculation and improve the calculation efficiency, the invention also provides a method for obtaining the preliminary positioning result, that is, a set of equations is constructed according to each set of arrival time data, and the unknowns in the set of equations are the coordinates (X, Y, Z) of the acoustic emission source, the average wave velocity v of the acoustic emission signal propagation medium and the triggering time t of the acoustic emission signal; respectively solving each set of the equations to obtain a plurality of closed-form solutions, screening out the closed-form solutions containing the virtual root, and using the rest r0Combined solution to obtain acoustic emission source r0And (5) obtaining a preliminary positioning result.
Further, in the step 1, each set of arrival-time data includes arrival-time data of 5 acoustic emission sensors; the set of equations constructed from a set of arrival time data is:
Figure BDA0002137034070000023
wherein (X, Y, Z) is the coordinate of the acoustic emission source, and v is the average wave velocity of the acoustic emission signal propagation mediumT is the triggering time of the acoustic emission signal, v and t are unknown numbers, and the rest are known; t is tiFor the ith arrival time data in the set of arrival time data, (x)i,yi,zi) Is tiCorresponding acoustic emission sensor SiThe coordinates of (a).
Further, the closed form solution of the set of equations is defined as:
Figure BDA0002137034070000024
wherein the content of the first and second substances,
p＝a3b4c5-a3b5c4-a4b3c5+a4b5c3+a5b3c4-a5b4c3
Figure BDA0002137034070000031
Figure BDA0002137034070000032
Figure BDA0002137034070000033
furthermore, the unknown parameter ω can be obtained by solving the following one-dimensional cubic equation:
Aω3+Bω2+Cω+D＝0
wherein, A, B, C and D are constants, and their expressions are:
Figure BDA0002137034070000034
Figure BDA0002137034070000035
Figure BDA0002137034070000036
Figure BDA0002137034070000037
and the number of the first and second electrodes,
Figure BDA0002137034070000038
Figure BDA0002137034070000039
wherein m isi(i＝1,2,3)、ni(i＝1,2,3)、w、p、ai(i＝3,4,5)、bi(i＝3,4,5)、ci(i＝3,4,5)、di(i＝3,4,5)、ei(i＝3,4,5)、xi,1(i＝2,3,4,5)、yi,1(i＝2,3,4,5)、zi,1(i＝2,3,4,5)、ti,1(i＝2,3,4,5)、Li,1(i＝2,3,4,5)、A、B、C、D、Q1、Q2Are all intermediate variables. The invention does not solve the triggering time t of the acoustic emission signal.
Further, in the step 3, firstly, each preliminary positioning result obtained in the step 2 is calculated, and the euclidean distance between the preliminary positioning result and the origin is calculated; then based on the calculated Euclidean distance, r is eliminated by combining a quartile method0And obtaining r initial positioning results finally according to the abnormal positioning results in the initial positioning results.
Further, r is excluded0And obtaining r initial positioning results finally according to the abnormal positioning results in the initial positioning results.
Further, the method for eliminating the abnormal positioning result comprises the following steps: first to r0Calculating the Euclidean distance from each initial positioning result to the origin point respectively; then based on the calculated Euclidean distance, r is eliminated by combining a quartile method0And obtaining r initial positioning results finally according to the abnormal positioning results in the initial positioning results.
Further, the euclidean distance from the jth preliminary positioning result to the origin is:
Figure BDA0002137034070000041
wherein (X)j,Yj,Zj) As the jth primary positioning result, j is 1,2, …, r0。
Further, based on the calculated Euclidean distance, r is excluded by combining a quartile method0The abnormal positioning result in the initial positioning result is specifically as follows: if sj＞q3+1.5(q3-q1) Or sj＜q1-1.5(q3-q1) If so, the jth preliminary positioning result is considered as an abnormal positioning result and is eliminated; wherein q is1Is r0The first quartile of Euclidean distance, q, from the initial positioning result to the origin3Is r0A third quartile of Euclidean distance from the initial positioning result to the origin, j being 1,2, …, r0. The existing quartile position determination methods are of several kinds, and any quartile position determination method can be adopted in the invention.
Further, in step 2, a multi-dimensional (multi-component) kernel density estimation function of the coordinates θ of the acoustic emission source
Figure BDA0002137034070000042
The specific expression form is as follows:
Figure BDA0002137034070000043
wherein θ ═ θ1,θ2,θ3) (X, Y, Z) is a multivariate random vector of the probability density function f (θ), i.e., the acoustic emission source coordinates; theta1、θ2And theta3X, Y and Z, respectively; thetaj,1、θj,2And thetaj,3Respectively represent Xj、YjAnd Zj；(θj,1,θj,2,θj,3)＝(Xj,Yj,Zj) Is a probability density function f (theta)) J is 1,2, …, r, r is the number of the preliminary positioning results finally obtained in step 1; d is an element position index in the variable theta, 1,2 and 3; k (-) is a kernel function.
Further, the kernel function adopts a probability density function of standard normal distribution (in the multidimensional kernel density estimation, the kernel function adopts the probability density function of standard normal distribution, that is, normal information diffusion is performed), and the expression is as follows:
Figure BDA0002137034070000051
wherein h isdFor bandwidth, the concrete expression is:
Figure BDA0002137034070000052
wherein sigmadThe scale parameter is expressed in the following specific form:
Figure BDA0002137034070000053
where med (-) represents the median.
Further, in step 3, the function is estimated by multidimensional kernel density
Figure BDA0002137034070000054
And (3) searching the maximum value of the target function by adopting an iteration method, taking the average value of r primary positioning results obtained in the step (3) as initial acoustic emission source coordinates (X, Y and Z) in the searching process, continuously correcting the coordinates (X, Y and Z) to find the optimal acoustic emission source coordinates, terminating iteration when an iteration termination condition is met, and obtaining the final corrected result, namely the optimal acoustic emission source coordinates.
Further, the iteration termination condition is: when the variable quantity of the objective function value obtained by two adjacent iterative computations is smaller than a preset value, the step length of X, Y and Z correction is smaller than the preset value or the iteration number exceeds the preset value.
Has the advantages that:
1) the method gives a closed-form solution of acoustic emission source parameters under a set equation set, does not invert the triggering time, uses five arrival time data for positioning each time, is one less than the traditional method, reduces the calculation amount, and can ensure the positioning precision each time;
2) according to the method, the abnormal positioning result in the primary positioning result is eliminated through the quartile method, so that the fitting of the multidimensional probability kernel density function is better, and the over-fitting phenomenon is avoided;
3) the multi-dimensional nuclear density estimation method does not utilize prior knowledge of data distribution of the primary positioning result by adopting multi-dimensional nuclear density estimation, does not need to carry out any additional assumption on the data distribution, starts from the data of the primary positioning result, obtains more accurate density estimation, and avoids errors caused by unreasonable assumed distribution. The method ensures the accuracy and robustness of the obtained positioning result from the statistical angle of non-parameter estimation, so that a more ideal positioning result can be obtained even if the time-out error is serious;
4) the method adopts a multi-dimensional nuclear density estimation model, considers the correlation between the acoustic emission parameters X, Y and Z, and has higher accuracy;
5) the method takes the average value of a plurality of preliminary positioning results with small deviation as an iteration initial value, and adopts an iteration method to search the maximum value of the multidimensional kernel density function, so that the efficiency of searching the extreme value of the function can be improved while the search is prevented from getting into local optimum;
6) compared with the traditional method, the method has very obvious stability, is more suitable for the practical engineering practice problem, and better solves the technical problems of unstable positioning result and low positioning precision caused by the fact that time data in the acoustic emission positioning field contains abnormal values.
Drawings
FIG. 1 is a flow chart of method steps according to an embodiment of the present invention.
FIG. 2 is a comparison of positioning results of the method of the present invention with other methods.
Detailed description of the invention
An acoustic emission source is preset with coordinates S (110, 160, 180) and is surrounded by 9 acoustic emission sensors with coordinates a (0,0,0), B (300,0,0), C (300, 0), D (0,300,0), E (0,0,300), F (300,0,300), G (300,300,300), H (0,300,300), I (300,150,150). The units are all mm. The wave velocity is unknown. In the test, a group of arrival time data is generated by a simulation method, the influence of environmental noise on positioning is simulated by adding an error with a variance of 0.2 mu s in the obtained arrival time data, and in addition, a large error of +/-5 mu s is randomly added to one arrival time data to simulate the interference of an abnormal value. The set of arrival time data generated by the above random process is: 38.38, 42.90, 40.89, 50.75, 53.10, 53.13, 55.27, 59.41, 61.55, in μ s.
The method is explained in detail by this example. For the sake of clarity, the method of the present invention is illustrated in the following five steps:
(1) closed form solution of the set of equations is defined:
the total number of the arrival time data is 9, 5 arrival time data are selected each time to form a set of equations, and different combinations can be used for obtaining the arrival time data
Figure BDA0002137034070000061
Each set of the defined equations is solved to obtain multiple closed-form solutions, and the closed-form solutions with virtual root are excluded from the rest r0(r083) to obtain r of acoustic emission source0(r083) preliminary positioning results, as shown in table 1. This example shows the calculation of the closed-form solution of only one set of equations for the construction of the selected arrival time data 38.38, 42.90, 40.89, 50.75 and 59.41 (in mus) as follows
Firstly, the formula
Figure BDA0002137034070000071
Is calculated to obtain
Figure BDA0002137034070000072
Second formula of
Figure BDA0002137034070000073
Is calculated to obtain
Figure BDA0002137034070000074
And can then calculate
p＝a3b4c5-a3b5c4-a4b3c5+a4b5c3+a5b3c4-a5b4c3＝1.18×1015
Figure BDA0002137034070000075
Figure BDA0002137034070000081
In addition, the unknown parameter ω can be obtained by solving the following cubic equation
Aω3+Bω2+Cω+D＝0
Wherein, the specific expressions of the coefficients A, B, C and D are respectively
Figure BDA0002137034070000082
Figure BDA0002137034070000083
Figure BDA0002137034070000084
Figure BDA0002137034070000085
And the number of the first and second electrodes,
Figure BDA0002137034070000086
Figure BDA0002137034070000087
calculated, the only real solution to ω excluding the imaginary root is 2.90 × 107。
Finally, a preliminary localization result (acoustic emission source coordinates) of the acoustic emission source can be obtained from the set of equations:
Figure BDA0002137034070000088
table 1: 83 groups of screened localization results
Figure BDA0002137034070000089
Figure BDA0002137034070000091
Figure BDA0002137034070000101
(2) Calculation of Euclidean distance of acoustic emission source to origin
According to the formula
Figure BDA0002137034070000102
Calculating the Euclidean distance s from the jth primary positioning result to the originjThe specific calculation results are shown in table 1.
(3) Method for eliminating abnormal group by quartile method
The specific formula of the quartile method for eliminating the abnormal positioning result in the preliminary positioning result is as follows:
Figure BDA0002137034070000103
wherein q is1Is the first quartile of data, q3Is the third quartile of data.
R is calculated according to the quartile method0(r083) a first quartile and a third quartile of Euclidean distances of the preliminary positioning results to the origin, which are q respectively1＝260.21×103，q3＝266.49×103. The criteria from which the anomaly groups can be derived are:
Figure BDA0002137034070000104
according to the criterion, 13 groups with abnormity are excluded, and r (r is 70) primary positioning results are remained, and the specific results are shown in table 1.
(4) Calculation of scale parameters
Scale parameter σdThe specific expression form of (A) is as follows:
Figure BDA0002137034070000111
where r is 70, calculated to give:
σ1＝1.78×10-3,σ2＝1.49×10-3,σ3＝2.09×10-3。
(5) calculation of a bandwidth matrix
The bandwidth matrix H is a diagonal matrix, H1/2The vector elements of the main diagonal of (a) may be expressed as:
Figure BDA0002137034070000112
calculating a normal scale parameter sigma according to the step (4)dCan obtain hdAre respectively as follows:
h1＝9.39×10-4,h2＝7.86×10-4,h3＝11.04×10-4
(6) constructing a multi-dimensional kernel density estimation function
Multidimensional kernel density estimation function
Figure BDA0002137034070000113
Comprises the following steps:
Figure BDA0002137034070000114
wherein, the kernel function k (·) adopts a probability density function of standard normal distribution, and the expression is as follows:
Figure BDA0002137034070000115
finally, the expression of the multidimensional kernel density estimation function can be obtained as follows:
Figure BDA0002137034070000116
(7) calculation of optimal positioning results
Searching the maximum value of a multi-dimensional nuclear density estimation function by adopting an iteration method, selecting an average value (110.38,156.52,180.68) (unit: mm) of 70 groups of initial positioning results as initial acoustic emission source coordinates (X, Y, Z) in the iteration searching process, and continuously correcting the coordinates (X, Y, Z) to find an optimal solution, wherein the deviation between the initial value and the true value is large at the initial searching stage, so that the step length of X, Y and Z correction is large; the X, Y and Z correction steps become smaller and smaller as the real value is continuously approached, and when the variation of the objective function value obtained by two adjacent iterative computations is less than 10-6Or both acoustic source coordinates X, Y and the step size of the Z correction are less than 10-6m, or the number of iterations exceeds 25, the iteration terminates. The result of the last correction (110.29160.09179.89) (in mm) is the optimal acoustic source coordinates that fit well with the real coordinates S (110, 160, 180) (in mm) and locateThe precision is higher.
The method has the following three advantages that (1) a closed-type solution of the coordinates of the acoustic emission source under a set equation is given, the least arrival time data can be used each time during positioning, inversion does not need to be carried out on the triggering time, and the calculation efficiency is improved. (2) A multi-dimensional kernel density estimation model is adopted, and the correlation advantage among parameters is considered; (3) by adopting the parameter-free estimation, the prior knowledge of data is not needed, the prior distribution of data is not needed to be assumed, the applicability is wider, and the accuracy is higher.
Compared with a two-step weighted least square method (2W L S), a non-iterative unknown wave velocity system acoustic emission source analytic positioning method (NIUV) and a comprehensive analytic method (CAS), the New positioning method (New) disclosed by the invention is compared with the two-step weighted least square method (2W L S), and a positioning result is shown in figure 2.

Claims (9)

1. An acoustic emission source localization method based on multi-dimensional nuclear density estimation is characterized by comprising the following steps:
step 1, combining arrival time data of each acoustic emission sensor in an acoustic emission detection system to obtain multiple groups of arrival time data; obtaining a plurality of preliminary positioning results based on the plurality of groups of arrival time data; wherein each set of arrival time data comprises arrival time data of more than 5 acoustic emission sensors;
step 2, constructing a multi-dimensional kernel density estimation function of the acoustic emission source coordinate theta by using the obtained multiple primary positioning results
Figure FDA0002417888240000011
The specific expression form is as follows:
Figure FDA0002417888240000012
wherein θ ═ (X, Y, Z) is the acoustic emission source coordinates; theta1、θ2And theta3Respectively representX, Y and Z; thetaj,1、θj,2And thetaj,3Respectively represent Xj、YjAnd Zj；(Xj,Yj,Zj) J is 1,2, …, r is the number of the preliminary positioning results finally obtained in step 1; h isdFor bandwidth, subscript d is 1,2,3 is an element position index in variable θ; k (·) is a kernel function;
step 3, solving a multidimensional kernel density estimation function
Figure FDA0002417888240000013
The maximum point is the optimal acoustic emission source positioning result.
2. The method for positioning an acoustic emission source based on multi-dimensional nuclear density estimation according to claim 1, wherein in the step 1, a set of predetermined equations is constructed according to each set of arrival time data, and the unknowns in the set of predetermined equations are coordinates of the acoustic emission source, an average wave speed of a propagation medium of the acoustic emission signal, and a trigger time of the acoustic emission signal; respectively solving each set of the equations to obtain a plurality of closed-form solutions, screening out the closed-form solutions containing the virtual root, and using the rest r0Combined solution to obtain r0And (5) obtaining a preliminary positioning result.
3. The method according to claim 2, wherein each set of arrival time data in step 1 comprises arrival time data of 5 acoustic emission sensors; the set of equations constructed from a set of arrival time data is:
Figure FDA0002417888240000014
wherein, (X, Y, Z) is the coordinate of the acoustic emission source, v is the average wave velocity of the acoustic emission signal propagation medium, t is the triggering time of the acoustic emission signal, v and t are unknown numbers, and the rest are known; t is tiFor the ith arrival time data in the set of arrival time data, (x)i,yi,zi) Is composed oftiCorresponding acoustic emission sensor SiThe coordinates of (a).
4. The method of claim 2, wherein r is excluded from the multi-dimensional nuclear density estimation-based acoustic emission source localization0And obtaining r initial positioning results finally according to the abnormal positioning results in the initial positioning results.
5. The method of claim 4, wherein the method of excluding outlier localization results comprises: first to r0Calculating the Euclidean distance from each initial positioning result to the origin point respectively; then based on the calculated Euclidean distance, r is eliminated by combining a quartile method0And obtaining r initial positioning results finally according to the abnormal positioning results in the initial positioning results.
6. The method of claim 5, wherein the Euclidean distance from the origin of the jth preliminary localization result to the origin is:
Figure FDA0002417888240000021
wherein (X)j,Yj,Zj) As the jth primary positioning result, j is 1,2, …, r0。
7. The method of claim 6, wherein r is excluded by using a quartile method based on the calculated Euclidean distance0The abnormal positioning result in the initial positioning result is specifically as follows: if sj＞q3+1.5(q3-q1) Or sj＜q1-1.5(q3-q1) If so, the jth preliminary positioning result is considered as an abnormal positioning result and is eliminated; wherein q is1Is r0Of Euclidean distances of the initial positioning result from the originFirst quartile, q3Is r0A third quartile of the Euclidean distance of the preliminary positioning result to the origin.
8. The method of claim 1, wherein the kernel function is a probability density function of a standard normal distribution, and the expression is as follows:
Figure FDA0002417888240000022
hdthe specific expression form of (A) is as follows:
Figure FDA0002417888240000023
d＝1,2,3
wherein sigmadThe scale parameter is expressed in the following specific form:
Figure FDA0002417888240000024
d＝1,2,3
where med (-) represents the median.
9. The method according to claim 8, wherein in step 3, the function is estimated using multi-dimensional kernel density
Figure FDA0002417888240000025
And (2) searching the maximum value of the target function by adopting an iteration method, taking the average value of r primary positioning results obtained in the step (1) as initial acoustic emission source coordinates (X, Y and Z) in the searching process, continuously correcting the coordinates (X, Y and Z) to find the optimal acoustic emission source coordinates, terminating iteration when an iteration termination condition is met, and obtaining the final corrected result, namely the optimal acoustic emission source coordinates.
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