A method for picking up microseismic arrival time based on fuzzy clustering U-type neural network

The micro-seismic at-time labels are automatically produced by fuzzy clustering U-type neural network (FC-Net) combined with the fuzzy C-mean clustering (FCM) algorithm, which solves the label accuracy and objectivity problems caused by manual annotation in the existing technology, and achieves a more efficient micro-seismic signal pickup effect.

CN115097518BActive Publication Date: 2025-04-29UNIV OF ELECTRONICS SCI & TECH OF CHINA
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Patent Information

Application Number
CN202210739445.6
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2022-06-28
Publication Date
2025-04-29
Estimated Expiration
2042-06-28

AI Technical Summary

Technical Problem

The existing micro-seismic pickup method relies on manual annotation, which leads to a decrease in label accuracy and objectivity, affecting the learning and prediction effects of neural networks, especially the recognition error of micro-seismic signals with low signal-to-noise ratio is large.

Method used

The fuzzy clustering U-type neural network (FC-Net) combined with the fuzzy C-mean clustering (FCM) algorithm is used to automatically create the time tag to reduce human intervention, and signal feature extraction and label production are performed through feature kurtosis K, energy P and standard deviation Δ.

Benefits of technology

The automation of label production is realized, the objectivity and accuracy of labels are improved, human interference is reduced, and the recognition rate and accuracy rate when micro-seismic signals are picked up are improved.

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Abstract

The present invention discloses a method for picking up microseismic arrival time based on fuzzy clustering U-type neural network, comprising the following steps: step 1, preprocessing microseismic data: slicing the original microseismic data, and normalizing to obtain a microseismic sequence; step 2, data amplification; step 3, extracting the characteristic kurtosis K, energy P and standard deviation Δ of the amplified data; step 4, using the three features of K, P and Δ to automatically make the arrival time label of the microseismic signal through the FCM clustering algorithm; step 5, inputting the amplified data into FC‑Net for supervised training; step 6, the trained FC‑Net network model performs the first arrival picking of the sample. The present invention uses the FCM clustering method to make the arrival time label, realizes the automation of label making, can effectively reduce the human interference factors, and can better fit the signal waveform, improve the objectivity and accuracy of label marking, and improve the recognition rate and accuracy rate of microseismic signal picking up time.
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Description

Technical Field

[0001] The invention belongs to the technical field of microseismic monitoring, and in particular relates to a microseismic arrival time picking method based on fuzzy clustering U-type neural network. Background Art

[0002] Microseismic monitoring technology uses ground-based geophones to receive microseismic signals generated by internal rock fractures and sliding. By analyzing the effective information extracted from the signal waveform, spectrum and other characteristics, the underground rock structure is inferred. Therefore, in terms of geological disaster early warning and prevention, coal mine safety production and other aspects, microseismic monitoring technology has important practical significance for protecting the personal safety of the people and reducing economic losses. In early seismological research, the arrival time of the earthquake phase was picked by observing the waveform of the earthquake signal with the naked eye and relying on experience to identify the arrival time of the earthquake. Although this method has high picking accuracy, it consumes a lot of energy, has low efficiency, and is easily affected by the subjective judgment of the operator. With the development of seismology and the continuous improvement of hardware and software equipment, the amount of recorded seismic data has increased exponentially. The emergence of automatic extraction methods such as the long-short window average ratio method (STA / LTA), the autoregressive method (AIC), and the skewness / peak method has been favored by scholars in the field of seismology. However, most of these methods are based on objective functions for calculation and extraction, and some methods require professionals to set thresholds based on experience, which are subject to much human intervention. For microseismic signals with low signal-to-noise ratio, small amplitude, short duration and small sample size, the recognition error is large.

[0003] With the rise of machine learning, the use of neural networks to analyze seismic data has gradually become a major research trend in seismology. Since the 1990s, a series of scholars have used artificial neural networks (ANN) to monitor earthquake events and pick up their first arrivals. Later, Ross et al. used convolutional neural networks (CNNs) with fewer training parameters and easier optimization to pick up seismic phases, and made effective progress. Zhu et al. first used the U-Net network applied to the field of medical image segmentation to pick up natural earthquake phases. U-Net omits the fully connected layer in CNN and adds multi-level feature fusion from high dimensions to low dimensions, which can accurately identify targets and accurately locate target boundaries. However, the arrival labels required for training the U-Net model still need to be manually labeled by professional operators. For microseismic signals with low signal-to-noise ratio, due to the lack of obvious amplitude jumps and the influence of subjective factors in manual labeling, compared with natural earthquake signals, the accuracy of labeling is relatively low and has a larger error, which is not conducive to the learning of neural networks and will directly affect its prediction effect.

[0004] The U-Net network was proposed by Ronneberger and was first used in the field of medical image segmentation. The U-Net network has the following advantages: (1) For training sample sets with small amounts of data, the U-Net network can rely on data enhancement to make more effective use of data; (2) The U-Net network realizes multi-scale feature recognition of the target; (3) The U-Net network realizes multi-scale feature fusion. In the field of seismology, most of the existing U-Net network structures are based on Zhu's research. The PhaseNet network model structure proposed by Zhu is as follows: Figure 1 As shown in the figure, there are five layers in total, and the left and right stages are the downsampling stage and the upsampling stage respectively. There are four submodules in the downsampling stage, each of which contains a convolution layer and a pooling layer for extracting data features. There are also four submodules in the upsampling stage, each of which contains a convolution layer and a deconvolution layer for restoring features, and each submodule will be connected with the output features of the submodule of the same scale in the upsampling to achieve feature fusion and improve the accuracy of image edge recognition. Zhu used the three-component seismic data of the Northern California Seismic Data Center as input data, and the output is the relevant probability distribution of the arrival time of P waves, S waves and noise, so as to realize the picking of earthquake phases.

[0005] Compared with natural earthquake signals, microseismic signals have the characteristics of low signal-to-noise ratio, small amplitude, short duration and small sample size, so the method of picking up microseismic signals has more stringent requirements. Although traditional methods such as STA / LTA and AIC have high computational efficiency, they are less sensitive to microseismic signals with low signal-to-noise ratio, are prone to mispickup or miss, and are greatly affected by parameter settings. In order to achieve more intelligent recognition and reduce human intervention, artificial intelligence has been applied to seismic phase picking by more and more scholars in the field of seismology. Among them, the U-Net network based on convolutional neural network, which was used in medical image segmentation in the early days, has been applied to seismic phase picking in recent years, giving full play to its advantage of being more sensitive to image edge detection and achieving good results. However, microseismic signals with low signal-to-noise ratio are still a challenge it faces. The arrival labels required for its training still need to be manually labeled. Since the effective signal amplitude of the microseismic signal itself is small and the duration is short, it is greatly affected by noise, and the objectivity and accuracy of manual labeling are reduced, resulting in interference in the accuracy of feature extraction of the neural network during the learning process, resulting in large result errors.

[0006] In the process of picking the first arrival of seismic signals, the labels of training samples were mainly made based on Gaussian distribution, that is, the time point of the first arrival of the earthquake event was judged by humans as the center, and the Gaussian distribution was formed by setting the mean value and standard deviation, which was used as the probability distribution of the arrival label. However, the waveform distribution of microseismic signals does not completely follow the Gaussian distribution, and this label distribution is only a manifestation of the first arrival of the earthquake event, but cannot fully show a complete and effective earthquake event, which has limitations in the processing and analysis of subsequent seismic data. This method of judging the first arrival time and setting the distribution parameters has the problem of human subjective factors. Summary of the invention

[0007] The purpose of the present invention is to overcome the shortcomings of the prior art and provide a method for picking up microseismic arrival time based on fuzzy clustering U-type neural network, which utilizes the FCM clustering method to make arrival time labels, realizes the automation of label production, can effectively reduce human interference factors, and can better fit the signal waveform, thereby improving the objectivity and accuracy of label annotation, and improving the recognition rate and accuracy of microseismic signal picking up time.

[0008] The object of the present invention is achieved through the following technical solution: A microseismic arrival time picking method based on fuzzy clustering U-type neural network comprises the following steps:

[0009] Step 1: Preprocess the microseismic data: slice the original microseismic data and perform normalization using the min-max method to obtain a microseismic sequence;

[0010] Step 2, data augmentation: randomly generate a moving scale, and shift the data of the microseismic sequence to the left or right as a whole. In order to ensure that the data after the shift is still the original size, the part of the data shifted to the left or right is truncated, and the right or left side of the data is filled with 0 to the same length as the stage data;

[0011] Step 3, feature extraction: extract the characteristic kurtosis K, energy P and standard deviation Δ of the amplified data;

[0012] Step 4: Create labels: Using the three features K, P, and Δ extracted in step 3, set the time window length parameter, and automatically create arrival labels for microseismic signals using the FCM clustering algorithm;

[0013] Step 5, training model: input the amplified data into FC-Net for supervised training, with epoch set to 50 and batch set to 20;

[0014] Step 6: The trained FC-Net network model performs first arrival picking on samples and analyzes its performance. Microseismic events with a picking error within 20 ms are considered valid picking events and the evaluation results are output.

[0015] Furthermore, in step 3, the specific calculation method of characteristic kurtosis K, energy P and standard deviation Δ is:

[0016]

[0017]

[0018]

[0019] in is the mean of the sample set X, x i is the i-th sample in X, ω is half the length of the sliding window, δ is the standard deviation of X, and A i is the average amplitude within the sliding window.

[0020] Furthermore, the FC-Net network structure has a total of six layers from the input layer to the output layer, which is divided into two stages, the left half is the downsampling stage, and the right half is the upsampling stage; the downsampling stage includes five submodules, each of which includes a convolution layer and a maximum pooling layer to extract data features and reduce the resolution of feature maps and reduce the number of model parameters; the upsampling stage, which is symmetrical to the downsampling stage, also includes five submodules, each of which includes a convolution layer and a deconvolution layer, and is fused with the features of the same resolution in the downsampling stage to supplement the recovery features; all convolution layers and deconvolution layers use Dropout layers to prevent overfitting, and use ReLU nonlinear activation functions; features are extracted in the downsampling stage, and then the feature maps are restored to their original size in the upsampling stage, and the output is converted into a probability distribution using a Softmax layer to obtain the output probability distribution of the arrival of the microseismic event.

[0021] The beneficial effects of the present invention are as follows: the present invention proposes a U-shaped neural network based on the FCM clustering method, namely FC-Net, to pick up microseismic events in time, and uses the FCM clustering method to make arrival labels, thereby realizing the automation of label production, effectively reducing human interference factors, and being able to better fit the signal waveform, thereby improving the objectivity and accuracy of label annotation, being conducive to the learning of the neural network, and improving the recognition rate and accuracy of microseismic signal picking in time.

[0022] Microseismic monitoring technology has a wide range of applications. It can be used in any field of activities that can produce tiny vibrations through geological rock strata. It is not limited to the natural geological disasters mentioned in this article. Microseismic monitoring technology also plays a very important role in the fields of coal mining monitoring and gas storage safety and health operation monitoring. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 PhaseNet structure diagram;

[0024] Figure 2 It is a flow chart of the method for picking up microseismic arrival time of the present invention;

[0025] Figure 3 is the characteristic graph of K, P and Δ;

[0026] Figure 4 This is the result graph of label distribution for each type;

[0027] Figure 5 This is the FC-Net network structure diagram;

[0028] Figure 6 The prediction results of FC-Net for microseismic events and noise;

[0029] Figure 7 The following are the prediction results of two networks for the same low signal-to-noise ratio microseismic event. DETAILED DESCRIPTION

[0030] The technical solution of the present invention is further described below in conjunction with the accompanying drawings.

[0031] like Figure 2 As shown, a microseismic arrival time picking method based on fuzzy clustering U-type neural network of the present invention comprises the following steps:

[0032] Step 1, preprocessing the microseismic data: slicing the original microseismic data, each slice length is 1000 sampling points in this embodiment, and normalizing by using the min-max method to obtain a microseismic sequence;

[0033] Step 2, data augmentation: randomly generate a moving scale, and shift the data of the microseismic sequence to the left or right as a whole. In order to ensure that the data after the shift is still the original size, the part of the data shifted to the left or right is truncated, and the right or left side of the data is filled with 0 to the same length as the stage data;

[0034] Step 3, extract features: extract the characteristic kurtosis K, energy P and standard deviation Δ of the amplified data; the specific calculation method of the characteristic kurtosis K, energy P and standard deviation Δ is:

[0035]

[0036]

[0037]

[0038] in is the mean of the sample set X, x i is the i-th sample in X, ω is half the length of the sliding window, δ is the standard deviation of X, and A i is the average amplitude within the sliding window.

[0039] Step 4: Create labels: Using the three features K, P, and Δ extracted in step 3, set the time window length parameter, and automatically create arrival labels for microseismic signals using the FCM clustering algorithm;

[0040] The Hard C-means clustering algorithm (HCM) is the most classic and most commonly used clustering algorithm. Its main principle is that, given c initial cluster center points, each sample is assigned to the cluster represented by the cluster center point closest to it. After the assignment, each cluster center point is recalculated based on all samples in each cluster, and then the above sample assignment and cluster center point update steps are repeated until the iteration termination condition is met.

[0041] Fuzzy C-means clustering algorithm (FCM) uses the concept of fuzzy partitioning to generalize and improve HCM. In the HCM algorithm, each sample can only belong to one cluster, while in the FCM algorithm, the concept of membership is introduced for each sample to describe the degree of membership between the sample and the cluster center, so that each sample can belong to a different cluster. For a sample set X = {x1, x2..., x n}, there are c cluster centers with a central point set v = {v1...,v c}, and the membership matrix μ, the objective function of fuzzy clustering is calculated as follows:

[0042]

[0043] Among them, m>1 is the fuzzy weighted index, c>1 is the number of clusters, which is also the number of center points, η j is the size of each cluster, d ij is x i and v j The distance between them. Using the Lagrange multiplier method, the following update formula can be obtained to achieve the objective function J m Minimization of:

[0044]

[0045]

[0046] Since it is difficult to predict the size of the cluster, η j It is estimated and updated by the following formula:

[0047]

[0048] The steps of the FCM clustering algorithm are as follows:

[0049] 1) Initialize the cluster center set v;

[0050] 2) Initialize each cluster size η = 1;

[0051] 3) Update the membership matrix μ using formula (3);

[0052] 4) Estimate the size of each cluster using formula (4), and update the cluster center set v using formula (2);

[0053] 5) Repeat steps 3) and 4) until |J is satisfied m (R)-J m (R-1)|<εorR>R max , where ε is a preset minimum value greater than zero, R is the number of iterations, R max It is the preset upper limit of the number of iterations.

[0054] For labeling, we extracted five different signal features, namely, characteristic kurtosis K, skewness S, standard deviation Δ, absolute amplitude ABS and energy P. Their definition formulas are as follows:

[0055]

[0056]

[0057]

[0058]

[0059]

[0060] The Pearson coefficient and coefficient of variation are used to evaluate the various characteristics of microseismic signals. The definitions of the two are as follows:

[0061]

[0062]

[0063] Among them, V1 and V2 represent two vectors of the same length, and E(·) represents expectation. The Pearson coefficient reflects the correlation between two vectors. The larger the Pearson coefficient, the higher the correlation between the two vectors. It represents the average value of the sequence x(n), and the coefficient of variation reflects the degree of discreteness of the sequence. The larger the coefficient of variation, the more dispersed the data is. The Pearson coefficient and coefficient of variation of each feature are calculated for preliminary screening, and then three of the remaining features are selected to be input into the FCM clustering algorithm to cluster the signals containing microseismic events. The mean ME, standard deviation SD and prediction accuracy are used to analyze the errors between the results obtained from different combinations and the manually labeled labels, and finally the K / P / Δ combination is determined as the input of the FCM clustering algorithm.

[0064] The above three indicators are also used to evaluate the selection of the sliding window length (here in units of sampling points) for calculating each feature. When the length is determined to be 61 sampling points, the three evaluation indicators perform best. At this time, the clustering result has the highest accuracy and the highest effective picking rate. In summary, we will select three features, K, P and Δ, and the sliding window length is 61 sampling points to label the microseismic events. The effects of the three features are as follows: Figure 3 As shown, Amplitude represents the amplitude of data and each feature; Time is time.

[0065] The comparison results of FCM labels, Gaussian distribution labels and manually annotated 0-1 distribution labels are shown in the figure. Figure 4 As shown in the figure, Probability is the probability distribution of three types of labels. Figure 4 It can be seen intuitively that the Gaussian distribution label can only mark the time when the event starts, while the FCM label and the manual 0-1 label can not only mark the time when the event first arrives, but also reflect the continuous part of the entire event, which greatly facilitates the subsequent processing and analysis of related processing such as the energy and magnitude determination of microearthquakes.

[0066] Step 5, training model: the amplified data is input into FC-Net for supervised training, the epoch is set to 50, and the batch is set to 20; the FC-Net network structure has a total of six layers from the input layer to the output layer, which is divided into two stages, the left half is the downsampling stage, and the right half is the upsampling stage, as shown in Figure 5 As shown in the figure, Convolution represents the convolution operation, Max-pooling represents the maximum pooling operation, Deconvolution represents the deconvolution operation; Softmax is the activation function, which converts the output into a probability distribution between 0 and 1; Concatenate represents feature fusion;

[0067] The downsampling stage includes five submodules, each of which includes a convolution layer and a maximum pooling layer to extract data features, reduce the resolution of feature maps, and reduce the number of model parameters. The upsampling stage, which is symmetrical to the downsampling stage, also includes five submodules, each of which includes a convolution layer and a deconvolution layer, and is fused with the features of the same resolution in the downsampling stage to supplement the recovery features. All convolution layers and deconvolution layers use the Dropout layer to prevent overfitting, and the ReLU nonlinear activation function is used. After the downsampling stage, the features are extracted, and then the feature map is restored to its original size in the upsampling stage. The Softmax layer is used to convert the output into a probability distribution to obtain the output probability distribution of the arrival of the microseismic event.

[0068] The ReLU nonlinear activation function is:

[0069] Relu(x)=max(0,x)

[0070] The Softmax function is:

[0071]

[0072] Step 6: The trained FC-Net network model performs first arrival picking on samples and analyzes its performance, and regards microseismic events with a picking error (picking error refers to the error between the arrival time predicted by the network model and the arrival time judged manually) within 20 ms as valid picking events.

[0073] This embodiment selects accuracy, precision, recall, F1Score, mean error distribution (ME) and standard deviation (SD) of effective picking results and test samples for evaluation. The effect of network picking is judged according to the 20ms error range benchmark. The moment when the final result probability is higher than 0.6 for the first time is judged as the first arrival moment. The predicted first arrival error Δt less than 20ms is regarded as the effective picking time, and greater than 20ms is regarded as the invalid picking time.

[0074] This embodiment uses the microseismic records provided by Sichuan Jiayang Coal Mine, which has a sampling rate of 1000Hz and is a single-component microseismic signal. The microseismic records are manually segmented and intercepted, with a length of 1000 sampling points. Professional operators judge the microseismic events. There are 255 segmented records containing microseismic events. In addition, 255 noise segments that contain various waveforms are intercepted as much as possible, forming a data set of 510 segmented records in total. The StratifiedKFold function is used for hierarchical division, and the data is divided into a training set and a test set, with the test set accounting for 10% and the training set accounting for 90%. In the training set, 10% of the data is divided as a validation set for model training.

[0075] After the FC-Net model is trained, the prediction results for the test set are as follows: Figure 6 As shown in Figure 2, (a) is the microseismic event prediction result, and (b) is the noise prediction result. Figure 6According to the results, the FC-Net neural network based on FCM fuzzy clustering has a good effect on the prediction of microseismic events and can also suppress noise well. In order to better reflect the advantages of FC-Net, we selected five indicators, namely, accuracy, precision, recall, reconciliation index (F1Score), error distribution mean (ME) of effective picking results and test samples, and error distribution standard deviation (SD) to evaluate the network. The definitions of Accuracy, Precision, Recall and F1 Score are as follows:

[0076]

[0077]

[0078]

[0079]

[0080] We consider the moment when the probability of the final result is higher than 0.6 for the first time as the time. The predicted initial arrival error Δt less than 20ms is considered as the effective pickup time, and greater than 20ms is considered as the invalid pickup time. p Indicates the time information when the microseismic event is effectively picked up, F p Indicates the information when it is picked up by mistake, T n Indicates that the noise is correctly judged, F n Indicates that the microseismic event is misjudged as noise. ME and SD are calculated for the error Δt between the network judgment and the manual judgment of all test samples, where the first arrival time of the noise sample is set to -100 (to avoid affecting the label of the noise sample due to the Gaussian label width); Accuracy is calculated based on the error Δt of all test samples being less than 20ms; Precision, Recall and F1Score are all evaluated only for microseismic event records. Precision indicates how many real microseismic events can be effectively picked up among all predicted microseismic events, reflecting the accuracy of network prediction; Recall indicates how many microseismic events can be effectively picked up among all real microseismic events, reflecting the accuracy of network prediction; F1 Score is a comprehensive consideration of Precision and Recall. FC-Net is compared with PhaseNet, and the comparison results are shown in Table 1.

[0081] Table 1 Model evaluation results

[0082] Evaluation Metrics Precision Recall F1 Score ME(ms) SD(ms) Accuracy(%) PhaseNet 0.858 0.599 0.706 91.776 206.486 78.431 FC-Net 0.772 0.821 0.796 32.175 122.174 87.843

[0083] From the analysis of Table 1, it can be concluded that although PhaseNet has high accuracy in predicting samples as microseismic events, its accuracy in predicting microseismic events is low, and its recognition of microseismic events and noise is low. Overall, the prediction results of FC-Net are more stable and accurate than PhaseNet. The reason for this is that PhaseNet has stronger noise suppression than FC-Net. For microseismic events with low signal-to-noise ratio, it is easy to misjudge the event as noise. FC-Net is more sensitive to the event part and has moderate noise suppression ability, so it performs better. Figure 7 As shown, Figure 7 (a) is the prediction result of PhaseNet network. Figure 7 (b) is the prediction result of the present invention, Probability is the probability distribution between 0 and 1 of the network model output, amplitude is the amplitude of the data data, and pred indicates that the curve shown in the second sub-graph is the probability distribution curve of the network model prediction output.

[0084] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A microseismic arrival time picking method based on fuzzy clustering U-type neural network, characterized in that: The following steps are involved: Step 1: Preprocess the microseismic data: slice the original microseismic data and perform normalization using the min-max method to obtain a microseismic sequence; Step 2, data augmentation: randomly generate a moving scale, and shift the data of the microseismic sequence to the left or right as a whole. In order to ensure that the data after the shift is still the original size, the part of the data shifted to the left or right is truncated, and the right or left side of the data is filled with 0 to the same length as the stage data; Step 3, feature extraction: extract the characteristic kurtosis K, energy P and standard deviation Δ of the amplified data; Step 4: Create labels: Using the three features K, P, and Δ extracted in step 3, set the time window length parameter, and automatically create arrival time labels for microseismic signals through the FCM clustering algorithm; Step 5, training model: input the amplified data into FC-Net for supervised training, epoch is set to 50, batch is set to 20; the FC-Net network model has a total of six layers from the input layer to the output layer, which is divided into two stages, the left half is the downsampling stage, and the right half is the upsampling stage; the downsampling stage includes five submodules, each of which includes a convolution layer and a maximum pooling layer to extract data features and reduce the resolution of feature maps and reduce the number of model parameters; the upsampling stage, which is symmetrical to the downsampling stage, also includes five submodules, each of which includes a convolution layer and a deconvolution layer, and is fused with the same resolution features of the downsampling stage to supplement the recovery features; All convolutional and deconvolutional layers use the Dropout layer to prevent overfitting, and the ReLU nonlinear activation function is used. The features are extracted in the downsampling stage, and then the feature map is restored to the original size in the upsampling stage. The output is converted into a probability distribution using the Softmax layer to obtain the output probability distribution of the microseismic event. Step 6: The trained FC-Net network model performs first-arrival picking of samples and analyzes its performance. Microseismic events with a picking error within 20 ms are considered valid picking events and the evaluation results are output.

2. The microseismic arrival time picking method based on fuzzy clustering U-type neural network according to claim 1 is characterized in that: In step 3, the specific calculation method of characteristic kurtosis K, energy P and standard deviation Δ is: Where X is the mean of sample set X, x i is the i-th sample in X, ω is half the length of the sliding window, δ is the standard deviation of X, and A i is the average amplitude within the sliding window.

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