Microseismic signals are generally considered to follow the
Gauss distribution. A comparison of the dynamic characteristics of sample
variance and the symmetry of microseismic signals with the signals which
follow

Microseismic monitoring technology has been widely applied to mine rock burst monitoring, oil and gas field fracturing monitoring, reservoir seismic monitoring, slope stability evaluation and so on. Seismic source location is one of the key technologies used (Zhao et al., 2017). The conventional seismic source localization method usually first needs to pick up the P arrival time of multi-channel seismic signals and then calculate the time difference of arrival (TDOA) of the signals to solve the equation to obtain the source location (Schwarz et al., 2016). As a result, the accuracy of the calculated TDOA directly affects the accuracy of the seismic source location. However, in the process of actual operation, the first arrival time of the microseismic signals is not obvious, and there is much external noise (Jia et al., 2015). Therefore, it becomes very difficult to determine the time difference between waves from the same seismic source.

The basic problem the TDOA solves is to measure and estimate the TDOA between waves from the same seismic source accurately and rapidly. Since the classic article on TDOA written by Knapp and Carter was published in 1976, this problem has perpetually been a research focus in the field of international signal processing (Knapp et al., 1976). The common method of TDOA includes the generalized cross correlation method (Knapp et al., 1976; Souden et al., 2010; Jin et al., 2013), the phase spectrum estimation method (Youn et al., 1982; Qiu et al., 2012), the generalized bi-spectral estimation method (Hinich et al., 1992; Hou et al., 2013), the adaptive estimation method (Gedalyahu et al., 2010; Salvati and Canazza, 2013), the energy method based on Hilbert–Huang transform (Sun et al., 2016) and so on. These methods have been widely used in many fields. However, the vast majority of these methods assume that the signals and noises follow the Gaussian distribution. In the case of non-Gaussian distributions, their algorithm shows serious degradation in spatial resolution and does not function anymore (Ma and Nikias, 1996; Cornelis et al., 2010; Park et al., 2011).

Noisy microseismic signals have conspicuous non-stationary characteristics,
such as impulsiveness and randomness; therefore, they belong to the category
of non-Gaussian signals. If the microseismic signal is simulated by Gaussian
signal, it is inevitable that the TDOA algorithm will have serious
performance degradation. To solve the problem in theory, we intend to
introduce the

In order to facilitate data processing, the three component time travel curve
of microseismic is first transformed into a set of energy gradient time
travel curve (He et al., 2016). The basic model of TDOA is shown in Fig. 1.
The original microseismic signal is represented by

Two-sensor model of TDOA estimation.

If the microseismic acquisition system is discrete, the signals received by
the geophones

Correlation analysis is commonly used to calculate the TDOA estimation of
two signals. In the case that the substance of the problem is not affected
and the calculation is simplified, we take

In the process of microseismic monitoring, external noises are composed of
man-made noises, mechanical vibration, etc. The common characteristics of
these noises are that their time domain waveforms have conspicuous pulse
characteristics, the energy diminishes from low to high frequencies and
their corresponding probability density functions have a thicker tail than
that of Gaussian signals. In the field of signal processing, this type of
non-Gaussian noise is usually described by the

The

We can infer from Eq. (

The difference in determining a signal between the Gaussian distribution and the

If

Gaussian signal,

A comparison of the waveform characteristic of the signals (Fig. 2a–c) shows
that with the gradual decrease of the characteristic exponent

We select a measured microseismic wave and calculate its dynamic sample
variance according to Eq. (10) (Fig. 3). It shows that the microseismic
signal's dynamic sample variance jumps stepwise and also does not converge to a
stable value. Thus, one can conclude that the microseismic signal
follows the fractional lower-order

Before the parameter estimation of the

Draw the probability density curve of the sample sequence and observe the symmetry

Count the number of positive and negative values in the sample sequence. If the number of positive and negative values are approximately same, the signal is symmetric.

Figure 4a shows that when the skew parameter

For further validation of the symmetry of microseismic signal, we randomly select 30 signals from the microseismic records in different places, truncate the continuous 3000 sampling points of each signal and then count the number of positive and negative values. The absolute value for the difference between the numbers of positive and negative is shown in Fig. 5.

The absolute value for the difference between the positive and negative counts in each microseismic record.

According to the data in Fig. 5, we can use estimate maximum likelihood
estimator for parameters

In conclusion, the microseismic signal follows the symmetric

According to the study of Sect. 2.4, the microseismic signals and noises are
more consistent with the

In the case that the noise follows the

For the random variable

If the random variable

Comparison of the TDOA estimation results of GCC, FLOC and PHAT-GCC.

If

for a given sequence of discrete signal

assigning

add the Hanning window to

detect the peak of the function

The signals Ricker1 and Ricker2 used in the simulation are two Ricker
wavelets. Their spectral peak frequency is 25 Hz. The sampling frequency is
1 kHz, and the number of sampling points is 1000. The time delay between the
two Ricker wavelets is set to 70 ms (Fig. 6a). The generalized
signal-to-noise ratio (GSNR) is defined in Eq. (

The spatial resolution on TDOA estimation of the generalized cross correlation (GCC), PHAT-GCC (phase transfer–generalized correlation)
method based
on the Gaussian distribution and the FLOC method based on the non-Gaussian
distribution are compared and verified.

It is evident from Fig. 6d, f, h and j that the GCC and PHAT-GCC method shows serious performance degradation when GSNR

The influence of different

The influence of different

It is evident from Fig. 7 that a smaller

To verify the effectiveness of the FLOC method for TDOA estimation of real microseismic signals, we select eight microseismic signals from the same seismic source to do the experiment. The eight signals come from the ISS microseismic monitoring system of a coal mine in central China. Seismic geophones are laid along the mining roadway every 50 m in the system. The frequency bandwidth of the seismic geophones is between 3 and 2000 Hz. The data acquisition frequency is 1 kHz. For convenience of comparing and analyzing the experiment results, the first 2000 sampling points of each waveform are picked as the data object. The P arrival time of each microseismic signal is recorded manually, and the time delay between any two of the microseismic signals as a reference of the experimental result is calculated.

As an example, the microseismic signals in roadway nos. 2 and 7 are selected
to explain the result. The waveforms of the microseismic signals after
interception are shown in Fig. 8a and b. The time delay between the two signals
obtained by manual method is 19 ms. First, when the microseismic signal
follows the Gaussian distribution, the PHAT-GCC method, which, of the GCC
methods, performs best, is chosen for the TDOA estimation of microseismic
signals from the same seismic source. The result is shown in Fig. 8c. Second,
when the microseismic signal follows the

The comparison of TDOA estimation results based on PHAT-GCC and FLOC.

Figure 8c and d show that the two methods both obtain the correct result, 19 ms, but the peak of the FLOC method is sharper than the GCC method. This implies that the FLOC method performs better.

Each of the eight microseismic signals is considered to be a set of data
following the

The characteristic exponent

We can obtain 28 pairs of microseismic signals by the pair combination of the eight signals in Table 1. The comparison of TDOA estimations obtained by the PHAT-GCC, FLOC and manual method is shown in the table (Table 2).

The comparison of TDOA estimation results of microseismic signal.

Comparison of the characteristic exponent

An analysis of tables (Tables 1, 2) indicates that the pulse of actual
microseismic signal is stronger than the one following the Gaussian
distribution. Because the characteristic exponent of the actual microseismic
signal is less than 2, it is considered to be a signal following the

Through the analysis of the convergence of dynamic sample variance, the
microseismic signal with noises is shown to follow the

Because of the absence of second-order statistics of

Microseismic monitoring data obtained from a coal mine in central China are
used for TDOA estimation based on the GCC method and the FLOC method to
study cases when the microseismic signals follow the Gaussian distribution
and the

The microseismic data we use in this paper are derived from a coal mine. These microseismic data are not published online and are not intended to be published, because these data contain technical secrets of the coal mine.

The authors declare that they have no conflict of interest.

This work is funded by the State Key Research Development Program of China (2016YFC0801406), the Key Research and Development Program of Shandong Province (2017GSF20115), the Natural Science Foundation of Shandong Provice (ZR2018MEE008), China Postdoctoral Science Foundation (2015M582117), Qingdao Postdoctoral Applied Research Project and Special Project Fund of Taishan Scholars of Shandong Province. Edited by: Shaun Lovejoy Reviewed by: two anonymous referees