CN114563826B - Microseismic sparse table network positioning method based on deep learning fusion drive - Google Patents
Microseismic sparse table network positioning method based on deep learning fusion drive Download PDFInfo
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- G01—MEASURING; TESTING
- G01V—GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
- G01V1/00—Seismology; Seismic or acoustic prospecting or detecting
- G01V1/40—Seismology; Seismic or acoustic prospecting or detecting specially adapted for well-logging
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Abstract
A microseismic sparse table network positioning method based on deep learning fusion driving is characterized by constructing a positioning waveform data set, utilizing collected long-term microseismic waveform data recorded by an underground station at this stage to divide the underground into a plurality of possible microseismic regions, constructing the microseismic positioning data set, researching a region positioning model based on deep learning under the sparse condition of the table network based on the constructed microseismic positioning data set, outputting a region with the maximum probability of microseismic occurrence position to be predicted by processing the waveform data by the model, utilizing all monitored microseismic positions and seismic wave data thereof in the region, calculating the distance from the station by the speed of seismic waves and p-wave s-wave arrival difference, calculating the position between each pair of microseisms, and finally obtaining the position of a target microseismic by constructing a microseismic coordinate system and a matrix calculation method. The method can accurately position the mine micro-seismic sources under the condition that the number of the underground stations is less than four.
Description
Technical Field
The invention relates to a microseismic positioning method, in particular to a microseismic sparse table network positioning method based on deep learning fusion driving, and belongs to the technical field of underground microseismic positioning.
Background
With the shift of the center of gravity of coal production to deep in China, the mining conditions are increasingly complex, and the rock burst disaster is increasingly serious, which becomes one of the major potential safety hazards in coal mining. In order to reduce the damage caused by rock burst to the maximum extent, a plurality of monitoring and early warning technologies, such as a drilling cutting method, a ground sound, electromagnetic radiation, stress on line and the like, are widely applied to monitoring and early warning of rock burst, wherein a positioning, monitoring and analyzing technology based on a microseismic system is the most basic and widely used rock burst monitoring and early warning technology. The accurate positioning of the micro-seismic event is the basis for developing subsequent monitoring and early warning analysis, but is influenced by complex mining environment of a coal mine, insufficient arrangement of a table screen and the like, and the current micro-seismic positioning method has certain limitations.
The key to locating microseismic events is to improve the accuracy of the location of the source center. The positioning of microseismic events can now be achieved more reliably with a sufficient number of stations. The mainstream microseismic positioning method comprises a longitudinal wave first entry time method and a relative positioning method. The positioning equation in the longitudinal wave first-time entering time method has four unknown parameters, the positioning can be performed only by data of at least four stations, and the positioning cannot be performed under the condition of less than four stations, so that the station is regarded as missing; however, the traditional relative positioning method, such as the double-difference positioning method, has a high requirement on the integrity of the monitoring data, greatly limits the positioning accuracy under the condition that the station is lack, and has poor positioning accuracy under the condition of a single station. Therefore, under the condition that the underground platform network is not laid perfectly, the traditional method often has information loss, so that the positioning performance is seriously reduced.
Disclosure of Invention
The invention aims to provide a microseismic sparse table network positioning method based on deep learning fusion driving, which can accurately position underground microseismic positions of less than four stations.
In order to achieve the purpose, the invention provides a microseismic sparse table network positioning method based on deep learning fusion driving, which comprises the following steps:
step 1: installing the station in an underground mining roadway, collecting long-term micro-seismic waveform data, and recording micro-seismic waveform data generated in a target detection area;
step 2: in the target detection area, labeling the microseismic waveform data according to the position of the microseismic occurrence, and dividing the microseismic occurrence into different geographical clusters; clustering the microseisms by using a K-means algorithm to obtain U clustering centers, defining a clustering region according to the U clustering centers, wherein the clustering center is the centroid of the clustering region, dividing U polygons, and the methodology for dividing the polygon region is called voronoi polygon, namely the distance from each point in the divided polygon to the centroid of the polygon is less than the distance to the centroids of other polygons, all microseismic generating positions on a map are represented by points and are distributed to the nearest clustering region, so that the waveform data of the microseisms are classified into N classes: category 0 corresponds to downhole noise without any microseisms, and categories 1 through N correspond to microseismic clusters from the corresponding geographic region;
and step 3: generating an additional new waveform window by using a method for generating a countermeasure network, and generating an additional data window by using a tiny data set of which the data in the classes 1 to N in the step 2 is obviously less than that of other classes of data so as to enlarge the data set;
and 4, step 4: dividing the data in the step 2 and the data generated in the step 3 into two independent test sets and training sets, wherein the proportion of the test sets to the training sets is 2:8;
and 5: adopting microseismic waveform data as input, cutting the microseismic waveform data with the frequency of 100hz into windows of 10s, generating 1000 waveform sampling samples in the data window of 10s, inputting a waveform diagram of each 1hz waveform data window, and designing a microseismic region detection model, wherein the microseismic region detection model comprises the following steps: the method comprises the following steps of waveform data input, data regularization, convolution operation, activation, pooling, full connection and Softmax classification operation, initialization weight and bias are carried out on a microseismic region detection model after training, an optimization model is obtained through a back propagation algorithm, the microseismic region detection model outputs an N-dimensional vector after operation, the Nth bit of the N-dimensional vector is the probability that the corresponding microseismic belongs to a position U, and the method for calculating probability distribution data by the microseismic region detection model comprises the following steps:
in the formula: w is the set of all weights;
b is the set of all deviations;
Z c calculating a score for each class;
p c calculating the probability for each class;
c is a class label;
Z 0 is waveform data;
Z k calculating a score for each class in the summation function;
n is a class;
k is an internal parameter of the summation function;
and 6: under the condition of a single station, the regional positioning can be carried out on microseismic events in the same horizontal plane, under the condition of double stations and three stations, the accurate spatial positioning can be carried out, and firstly, the station d calculates the spacing distance | r between the microseismic a and the microseismic b ab Acquiring microseismic signals generated nearby an underground mining roadway by using a station d to acquire microseismic velocity V of p waves and s waves reaching the station d p And V s Will V p And V s Calculating to obtain a speed coefficient k v ；
k v ＝V p V s /(V p -V s ) (1)
And 7: obtaining the time of p-wave and s-wave reaching station d, determining the microseismic position according to the different time of p-wave and s-wave to station d, when p-wave and s-wave are propagated to station d, triggering detector in station d to record the time of p-wave and s-wave reaching detector, according to different time-of-arrival theories obtaining the distance | r of microseismic to station d respectively ed |：
And step 8: if r ab If is much less than the distance between these microseisms and station d, then the microseismic separation distance is approximated by the equation below, where
Is an approximation of the distance between the microseisms a, b, | r ad I and | r bd And | is the distance from the microseismic a and the microseismic b to the single station d respectively:
in the formula:
is the time of arrival of the s-wave at microseismic a;
and step 9: if there are two stations d 1 And d 2 And also satisfies the characteristic of being far away from two seismic sources, respectively for two stations d 1 And d 2 Is calculated and then the pythagorean theorem is used to get a better approximation:
step 10: if there are three stations d 1 、d 2 And d 3 Then according to three stations d 1 、d 2 And d 3 Comparing the positions of the seismic sources, and selecting two stations with farther positions to execute the step 9;
step 11: and (5) acquiring the positions of the known sources of the non-coplanar microseisms in the region obtained in the step (5), calculating the distance between the positions of the known sources and the source to be measured through the station, and establishing a distance equation set to obtain the target position. Firstly, constructing a microseismic coordinate system according to the existing microseismic cluster positions in the region, and forming a microseismic cluster consisting of 1 +1 microseisms, wherein x 1 ,x 2 ,…x l+1 Is the center coordinate of the seismic source when each microseismic occurs in three dimensionsCoordinate system R 3 Middle definition, obtaining the Euclidean distance between each pair of microseisms i and j, | r ij L is the distance between the microseisms i and j, | x i I and I x j I is the distance of the microseisms i and j, respectively, from the origin of the reference system (i, j =1,2, \ 8230;, l + 1), one can obtain:
step 12: reconstructing the coordinate system by using the characteristic that the internal structure of the microseismic cluster does not change along with translation and rotation (namely, the microseismic cluster can be translated and rotated, but the internal structure of the microseismic cluster does not change), and therefore setting the (l + 1) th microseismic generating position as the origin, namely x l+1 = 0, in which case | r il+1 |＝|x i |，|r jl+1 |＝|x j L, thus step 11 is converted to:
step 13, deriving X by formula (7), where G = V (1 k,1 A The microseismic source coordinates for placing the new coordinate system of step 12, where x l+1 = 0, matrix X A X, the xyz three-dimensional coordinates of each seismic source, first using the translation vector
Mixing X A Conversion to X, conversion of X A Wherein each coordinate is subjected to an offset conversion, wherein
The positions of the geometric centers of the microseismic clusters in the original reference frame and the new reference frame respectively,
X＝x i i＝1,2,…,l (8)
step 14: defining a distance matrix R with the size of l multiplied by l, storing the distance between each pair of the micro-shocks in the step 11, wherein the ith row and the jth column are the distances between the micro-shocks i and j, and the calculation formula of each bit is below;
step 15: the matrix R can be obtained from the following equations (7) and (9)
R＝XX T R l×l (10)
Then the singular value decomposition of R is now available, where V is an orthogonal matrix and S (l x l) is the singular value diagonal matrix of R
Step 16: the parameters in X are derived by this formula, where G = V (1 l,1,
X＝G∑ 1/2 (12)
compared with the prior art, the method is improved by matching the traditional relative positioning method with a machine learning algorithm, and provides the microseismic sparse table network positioning method based on deep learning fusion driving, so that microseismic positions can be accurately positioned under the condition of insufficient station number; firstly, researching a microseismic positioning data labeling method based on a clustering algorithm, constructing a positioning waveform data set, utilizing collected long-term microseismic waveform data recorded by an underground station at the stage, calculating waveform data through the clustering algorithm, dividing the underground into a plurality of possible microseismic regions, and further constructing the microseismic positioning data set, researching a region positioning model based on deep learning under the condition of sparse platform network based on the constructed microseismic positioning data set, designing a machine learning model for prediction at the stage, and outputting the region with the maximum probability of the microseismic occurrence position to be predicted by the model through processing waveform data. Finally, on the basis of regional positioning, the invention further designs an accurate positioning method of mine microseismic events, utilizes all the monitored microseismic positions and seismic wave data thereof in the region, obtains the distance from a station through the speed and time-to-time calculation of seismic waves, calculates the position between each pair of microseisms, and finally obtains the position of a target microseismic through establishing a coordinate system and a matrix calculation method; the invention can accurately position under the condition that the number of underground stations is less than four.
Drawings
FIG. 1 is a flow chart of microseismic zone determination in a microseismic location system of the present invention;
FIG. 2 is a flow chart of the precise location of the microseismic zone microseismic in the microseismic location system of the present invention;
FIG. 3 is a schematic diagram of microseismic zone partitioning and downhole station design according to the present invention;
FIG. 4 is a schematic diagram of the single station spacing measurement of the present invention;
FIG. 5 is a schematic diagram of a dual station spacing measurement according to the present invention;
FIG. 6 is a schematic diagram of the three station spacing measurement of the present invention.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
As shown in fig. 1 to 3, a microseismic sparse table network positioning method based on deep learning fusion driving includes the following steps:
step 1: installing the station in an underground mining roadway, collecting waveform data of long-term micro-vibration, and recording the waveform data of the micro-vibration generated in a target detection area;
step 2: in the target detection area, labeling the microseismic waveform data according to the position of the microseismic occurrence, and dividing the microseismic occurrence into different geographical clusters; clustering the microseisms by using a K-means algorithm to obtain U clustering centers, defining a clustering region according to the U clustering centers, wherein the clustering center is the centroid of the clustering region, dividing U polygons, and the methodology for dividing the polygon region is called voronoi polygon, namely the distance from each point in the divided polygon to the centroid of the polygon is less than the distance to the centroids of other polygons, all microseismic generating positions on a map are represented by points and are distributed to the nearest clustering region, so that the waveform data of the microseisms are classified into N classes: category 0 corresponds to downhole noise without any microseisms, and categories 1 through N correspond to microseismic clusters from the corresponding geographic region;
and step 3: generating an additional new waveform window by using a method for generating a countermeasure network, and generating an additional data window by using a tiny data set of which the data in the classes 1 to N in the step 2 is obviously less than that of other classes of data so as to enlarge the data set;
and 4, step 4: dividing the data in the step 2 and the data generated in the step 3 into two independent test sets and training sets, wherein the proportion of the test sets to the training sets is 2:8;
and 5: adopting microseismic waveform data as input, cutting the microseismic waveform data with the frequency of 100hz into windows of 10s, generating 1000 waveform sampling samples by the data window of 10s, inputting the waveform diagram of each 1hz waveform data window, and designing a microseismic region detection model, wherein the microseismic region detection model comprises the following steps: the method comprises the following steps of waveform data input, data regularization, convolution operation, activation, pooling, full connection and Softmax classification operation, initialization weight and bias are carried out on a microseismic region detection model after training, an optimization model is obtained through a back propagation algorithm, the microseismic region detection model outputs an N-dimensional vector after operation, the Nth bit of the N-dimensional vector is the probability that the corresponding microseismic belongs to a position U, and the method for calculating probability distribution data by the microseismic region detection model comprises the following steps:
in the formula: w is the set of all weights;
b is the set of all deviations;
Z c calculating a score for each class;
p c calculating the probability for each class;
c is a class label;
Z 0 is waveform data;
Z k calculating a score for each class in the summation function;
n is a group;
k is an internal parameter of the summation function;
and 6: under the condition of a single station, the regional positioning can be carried out on microseismic events in the same horizontal plane, under the condition of double stations and three stations, the accurate spatial positioning can be carried out, and firstly, the station d calculates the spacing distance | r between the microseismic a and the microseismic b ab Acquiring microseismic signals generated nearby an underground mining roadway by using a station d to acquire microseismic speed V of p-wave and s-wave reaching the station d p And V s A V is measured p And V s Calculating to obtain a speed coefficient k v ：
k v ＝V p V s /(V p -V s ) (1)
And 7: obtaining the time of p-wave and s-wave reaching station d, determining the microseismic position according to the different time of p-wave and s-wave to station d, when p-wave and s-wave are propagated to station d, triggering detector in station d to record the time of p-wave and s-wave reaching detector, according to different time-of-arrival theories obtaining the distance | r of microseismic to station d respectively ed |：
And 8: if r ab If | is much less than the distance between these microseisms and station d, then the microseismic separation distance is approximated by the equation below, where
Is an approximation of the distance between the microseisms a, b, | r ad I and | r bd And | is the distance from microseismic a and microseismic b to single station d, respectively, as shown in fig. 4:
in the formula:
is the time of arrival of the s-wave at microseismic a;
and step 9: if there are two stations d, as shown in FIG. 5 1 And d 2 And also satisfies the characteristic of being far away from two seismic sources, respectively for two stations d 1 And d 2 Is calculated and then the pythagorean theorem is used to get a better approximation:
step 10: if there are three stations d, as shown in FIG. 6 1 、d 2 And d 3 If yes, comparing the positions of the stations far away from the seismic source, and selecting two stations with farther positions to execute the step 9;
step 11: and (5) acquiring the positions of the known sources of the non-coplanar microseisms in the region obtained in the step (5), calculating the distance between the positions of the known sources and the source to be measured through the station, and establishing a distance equation set to obtain the target position. Firstly, a microseismic coordinate system is constructed according to the existing microseismic cluster positions in the region, and a microseismic cluster consisting of 1 +1 times of microseisms is formed, wherein x 1 ,x 2 ,…x l+1 Is the center coordinate of the seismic source when each microseismic occurs and is in a three-dimensional coordinate system R 3 Middle definition, obtaining the Euclidean distance between each pair of microseisms i and j, | r ij I is the distance between the microseisms i and j, | x i I and | x j I is the distance of the microseisms i and j, respectively, from the origin of the reference system (i, j =1,2, \ 8230;, l + 1), one can obtain:
step 12: reconstructing the coordinate system by using the characteristic that the internal structure of the microseismic cluster does not change along with translation and rotation (namely, the microseismic cluster can be translated and rotated, but the internal structure of the microseismic cluster does not change), and therefore setting the (l + 1) th microseismic generating position as the origin, namely x l+1 = 0, in which case | r il+1 |＝|x i |，|r jl+1 |＝|x j L, thus step 11 is converted to:
step 13, finding X by formula (7), where G = V (1 k,1 A The microseismic source coordinates for placing the new coordinate system of step 12, where x l+1 = 0, matrix X A X, the xyz three-dimensional coordinates of each seismic source, first using the translation vector
Mixing X A Conversion to X, conversion of X A Wherein each coordinate is subjected to an offset conversion, wherein
The positions of the geometric centers of the microseismic clusters in the original reference frame and the new reference frame respectively,
X＝x i i＝1,2,…,l (8)
step 14: defining a distance matrix R with the size of l multiplied by l, storing the distance between each pair of the microseisms in the step 11, wherein the ith row and the jth column are the distances between the microseisms i and j, and the following calculation formula of each bit is as follows:
step 15: the matrix R can be obtained from the following equations (7) and (9)
R＝XX T R l×l (10)
Then the singular value decomposition of R is now available, where V is an orthogonal matrix and S (l x l) is the singular value diagonal matrix of R
Step 16: the parameters in X are derived by this formula, where G = V (1 l,1,
X＝G∑ 1/2 (12)。
the method mainly uses a station which can output microseismic waveforms and is installed in an underground mining roadway with microseismic risks, as shown in figure 1 (step 1-5), firstly, a microseismic positioning data labeling method based on a clustering algorithm is researched to construct a positioning waveform data set. And at the stage, the collected long-term microseismic waveform data recorded by the underground station is utilized, the waveform data is calculated through a clustering algorithm, and the underground is divided into a plurality of possible microseismic areas, so that a microseismic positioning data set is constructed. Secondly, based on the constructed microseismic location data set, researching an area location model based on deep learning under the condition of sparse platform network, designing a machine learning model for prediction at this stage, wherein the model outputs the area with the maximum probability of the microseismic occurrence position to be predicted by processing waveform data, and finally, as shown in figure 2 (step 6-16), on the basis of area location, the invention further designs an accurate location method of the mine microseismic event. The invention can accurately position under the condition that the number of underground stations is less than four.
Claims (1)
1. A microseismic sparse table network positioning method based on deep learning fusion driving is characterized by comprising the following steps:
step 1: installing the station in an underground mining roadway, collecting waveform data of long-term micro-vibration, and recording the waveform data of the micro-vibration generated in a target detection area;
and 2, step: clustering the microearthquakes by using a K-means algorithm to obtain U clustering centers, defining a clustering area according to the U clustering centers, wherein the clustering centers are centroids of the clustering area, dividing U polygons, and the methodology for dividing the polygonal area is named as voronoi polygons, namely, the distance from each point in the divided polygons to the centroid of the polygons is less than the distance from each point to the centroid of other polygons, all microearthquake generating positions on a map are represented by points and are distributed to the closest clustering area, so that the waveform data of the microearthquakes are classified into N classes: category 0 corresponds to downhole noise without any microseisms, and categories 1 through N correspond to microseismic clusters from the corresponding geographic region;
and 3, step 3: generating an additional new waveform window by using a method for generating a countermeasure network, and generating an additional data window by using a tiny data set of which the data in the classes 1 to N in the step 2 is obviously less than that of other classes of data so as to enlarge the data set;
and 4, step 4: dividing the data in the step 2 and the data generated in the step 3 into two independent test sets and training sets, wherein the proportion of the test sets to the training sets is 2:8;
and 5: adopting microseismic waveform data as input, cutting the microseismic waveform data with the frequency of 100hz into windows of 10s, generating 1000 waveform sampling samples by the data window of 10s, inputting the waveform diagram of each 1hz waveform data window, and designing a microseismic region detection model, wherein the microseismic region detection model comprises the following steps: the method comprises the following steps of waveform data input, data regularization, convolution operation, activation, pooling, full connection and Softmax classification operation, initialization weight and bias are carried out on a microseismic region detection model after training, an optimization model is obtained through a back propagation algorithm, the microseismic region detection model outputs an N-dimensional vector after operation, the Nth bit of the N-dimensional vector is the probability that the corresponding microseismic belongs to a position U, and the method for calculating probability distribution data by the microseismic region detection model comprises the following steps:
in the formula: w is the set of all weights;
b is the set of all deviations;
Z c calculating a score for each class;
p c calculating the probability for each class;
c is a class label;
Z 0 is waveform data;
Z k calculating a score for each class in the summation function;
n is a class;
k is an internal parameter of the summation function;
step 6: under the condition of a single station, the regional positioning can be carried out on microseismic events in the same horizontal plane, the accurate spatial positioning can be carried out under the conditions of double stations and three stations, and firstly, the station d calculates the spacing distance | r between the microseismic a and the microseismic b ab Acquiring microseismic signals generated nearby an underground mining roadway by using a station d to acquire microseismic velocity V of p waves and s waves reaching the station d p And V s Will V p And V s Calculating to obtain a velocity coefficient k v ；
k v ＝V p V s /(V p -V s ) (1)
And 7: the method comprises the steps of obtaining the time of arrival of p-wave and s-wave at a station d, determining the microseism position according to the difference of the time of propagation of the p-wave and the time of propagation of the s-wave to the station d, triggering a detector in the station d to record the time of arrival of the p-wave and the time of arrival of the s-wave when the p-wave and the s-wave are propagated to the station d, and obtaining the distance | r of the microseism respectively arriving at the station d according to different arrival times theories ed |：
And step 8: if r ab If | is much less than the distance between these microseisms and station d, then the microseismic separation distance is approximated by the equation below, where
Is an approximation of the distance between the microseisms a, b, | r ad I and | r bd And | is the distance from the microseismic a and the microseismic b to the single station d respectively:
in the formula:
is the time of arrival of the s-wave at microseismic a;
and step 9: if there are two stations d 1 And d 2 And if the characteristic of being far away from the two seismic sources is also met, the two stations d are respectively provided with the seismic source 1 And d 2 The data of (a) are calculated and then a pythagorean theorem is used to obtain a better approximation;
step 10: if there are three stations d 1 、d 2 And d 3 Then according to three stations d 1 、d 2 And d 3 Comparing the positions of the seismic sources, and selecting two stations with farther positions to execute the step 9;
step 11: acquiring the positions of the known non-coplanar micro-seismic sources in the area obtained in the step 5, calculating the distance between the positions of the known seismic sources and the seismic source to be detected through the station and establishing a distance equation set to obtain a target position, firstly establishing a micro-seismic coordinate system according to the positions of the existing micro-seismic clusters in the area, and establishing a micro-seismic cluster consisting of (1 + 1) times of micro-seismic, wherein x is 1 ,x 2 ,…x l+1 Is the center coordinate of the seismic source when each microseismic occurs and is in a three-dimensional coordinate system R 3 Middle definition, obtaining the Euclidean distance between each pair of microseisms i and j, | r ij I is the distance between the microseisms i and j, | x i I and I x j I is the distance of the microseisms i and j, respectively, from the origin of the reference system (i, j =1,2, \ 8230;, l + 1), one can obtain:
step 12: reconstructing a coordinate system by using the characteristic that the internal structure of the microseismic cluster does not change along with translation and rotation, and thereforeThe location of the occurrence of the 1 + microseismic is designed as the origin, i.e. x l+1 = (0, 0), at this time | r il+1 |＝|x i |，|r jl+1 |＝|x j Step 11 is converted into:
step 13, defining an l X3 matrix X for placing the microseismic source coordinates of the original coordinate system designed in the step 11, and then defining an l X3 matrix X A The microseismic source coordinates for placing the new coordinate system of step 12, where x l+1 = 0, matrix X A X, the xyz three-dimensional coordinates of each seismic source, first using the translation vector
Will matrix X A Convert to X, convert matrix X A Wherein each coordinate is subjected to an offset conversion, wherein
The positions of the geometric centers of the microseismic clusters in the original reference frame and the new reference frame are respectively:
X＝x i i＝1,2,…,l (8)
step 14: defining a distance matrix R with the size of l multiplied by l, storing the distance between each pair of the microseisms in the step 11, wherein the ith row and the jth column are the distances between the microseisms i and j, and the following calculation formula of each bit is as follows:
step 15: the matrix R can be obtained from the following equations (7) and (9)
R＝XX T R l×l (10)
Then the singular value decomposition of R is now available, where V is an orthogonal matrix and S (l x l) is the singular value diagonal matrix of R
Step 16: from this formula, parameters in X are derived, where G = V (1, 1) l+1 Three parameters, which are l +1 rows of the matrix X and correspond to the coordinates of the xyz three directions thereof, have been derived in the matrix X
X＝G∑ 12 (12)。
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