Shock and Vibration

Volume 2016, Article ID 4051701, 6 pages

http://dx.doi.org/10.1155/2016/4051701

## Algorithm Indicating Moment of P-Wave Arrival Based on Second-Moment Characteristic

^{1}KGHM CUPRUM Ltd., CBR, Sikorskiego 2-8, 53-659 Wrocław, Poland^{2}Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Janiszewskiego 14a, 50-370 Wrocław, Poland^{3}KGHM Polska Miedź S.A. O/ZG “Rudna”, Henryka Dąbrowskiego 50, 59-100 Polkowice, Poland

Received 22 April 2016; Accepted 1 August 2016

Academic Editor: Shimin Liu

Copyright © 2016 Jakub Sokolowski et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The moment of P-wave arrival can provide us with many information about the nature of a seismic event. Without adequate knowledge regarding the onset moment, many properties of the events related to location, polarization of P-wave, and so forth are impossible to receive. In order to save time required to indicate P-wave arrival moment manually, one can benefit from automatic picking algorithms. In this paper two algorithms based on a method finding a regime switch point are applied to seismic event data in order to find P-wave arrival time. The algorithms are based on signals transformed via a basic transform rather than on raw recordings. They involve partitioning the transformed signal into two separate series and fitting logarithm function to the first subset (which corresponds to pure noise and therefore it is considered stationary), exponent or power function to the second subset (which corresponds to nonstationary seismic event), and finding the point at which these functions best fit the statistic in terms of sum of squared errors. Effectiveness of the algorithms is tested on seismic data acquired from O/ZG “Rudna” underground copper ore mine with moments of P-wave arrival initially picked by broadly known STA/LTA algorithm and then corrected by seismic station specialists. The results of proposed algorithms are compared to those obtained using STA/LTA.

#### 1. Introduction

Obtaining accurate information about seismic phenomena induced by mining activity might be a difficult task. The recordings strongly depend on distance between source and measuring device, energy of the event, lithology of the rock mass, device parameters, noise induced by transmission line, and so forth. In order to acquire exact features of the event (like, e.g., 3-dimensional location) recordings from at least four different one-axial sensors are required.

When the seismic event occurs, its energy is transported via different types of seismic waves, which can be primarily classified as body waves (P-wave, S-wave) and surface waves (Rayleigh wave, Love wave, and Stoneley wave). P-waves possess the highest velocity among others; thus they indicate onset of the event. Therefore, in order to receive detailed information about particular phenomenon, the first step is to indicate its moment of P-wave arrival.

From mathematical point of view the problem is isometric with finding a moment in time series where it loses stationarity (as the background noise is considered to be stationary) or as a problem of finding structural break point.

The moment of P-wave arrival is commonly used in estimation of event location [1], energy [2], and focal mechanism [3]. Determining such a moment manually is time-consuming and requires considerable experience. However, under development of science and technology, many automatic P-wave picking algorithms were proposed. Implementation and use of such methods are a much faster solution but not 100% reliable, as the results frequently differ from indications given by seismic station specialists. Thus the algorithms are frequently used as an initial pick followed by experts’ manual correction.

So far, there are plenty of different algorithms which can be divided into 2 main groups: proceeding in time and proceeding in frequency domain [4]. Broadly known time domain methods include AR-AIC [5, 6], which fits autoregressive model to the data and determines the moment of P-wave arrival in a point where Akaike Information Criterion [7] is minimized, and STA/LTA algorithm [8, 9], which for fixed characteristic function (e.g., square of the signal) computes its average over short and long time window and indicates the onset time when the ratio of averages exceeds predefined value. The moment of P-wave arrival might be also determined with use of neural networks [10, 11], methods based on wavelet transform [12, 13], spectrogram [14, 15], and cross-correlation [16].

When dealing with the problem of P-wave arrival moment, one may investigate it as an element of a signal segmentation procedure [17, 18], as the indication of onset moment is basis for segmentation. Common methods are often used in both problems.

Recently, a method of finding a critical point which divides the time series into two stationary parts with different variances has been proposed [19]. The basis for this method is statistical property of the second central statistical moment; that is, the expected value of cumulative sum of squares for stationary time series increases linearly with time. Such property is independent of the underlying probability distribution, as long as the variance is finite. The method has been already utilized in structural break detection method [20]. It was decided to involve this idea for P-wave arrival point estimation. However, the entire seismic event does not possess stationarity property, nor it can be split into two stationary time series. Thus the method requires a modification. In this paper two similar methods are proposed and compared to a widely used STA/LTA algorithm. All of the investigated automatic P-wave picking methods are compared with arrivals indicated by specialists of O/ZG “Rudna” underground copper ore mine seismic station experts due to their extensive experience in analysis of mining-induced seismic events.

The rest of the paper is organized as follows: in Methodology the new method of structural break detection is presented. Moreover, we recall the STA/LTA algorithm (the classical method used to detection of P-wave arrival time). Next, in Section 3 of application to real data, the new methodology is applied to the real seismic signals. Obtained results are compared with the STA/LTA technique. The last section contains conclusions.

#### 2. Methodology

##### 2.1. STA/LTA Algorithm

One of the classical algorithms that are often used in the problem of P-wave arrival moment detection is based on the short-term-average and long-term-average (STA/LTA) trigger method. The underlying idea of this method is to evaluate in a continuous fashion the value of characteristic function (CF) of a seismic signal in two moving-time windows (one short and one long) in order to detect the seismic event. The characteristic function used for calculation purpose can be defined as energy, absolute amplitude, or envelope function of the microseismic trace. Irrespective of the definition of the characteristic function (CF), the short time window (STA) is supposed to measure the instantaneous amplitude of the seismic signal, whereas the long time window (LTA) provides information about the amplitude of seismic noise. When their ratio exceeds a predefined value (activation threshold), the following recorded samples are marked to be event-driven until the ratio falls below another predefined value called the deactivation threshold. In this algorithm, for a raw signal , the following statistic is being calculated:where and denote short and long time windows lengths (in samples), respectively. Moreover, in the above equation is a specific characteristic function defined in terms of signal energy. In the literature different characteristic functions can be found, such as absolute value of the signal or envelope of the microseismic trace. In this paper we consider

In the STA/LTA algorithm the inspection of the statistic is performed and on such basis one can detect the moment of P-wave arrival. This moment is the minimum for which the ratio STA/LTA exceeds the predefined value ; that is,In this paper we compare the classical approach based on the STA/LTA algorithm with the new algorithm based on the cumulative empirical second moment of given raw signal.

##### 2.2. Algorithm Based on the Empirical Second Moment

As it was mentioned, the proposed method is based on the empirical second moment of given raw signal First, we introduce the statistic which is a cumulative second moment of given sample:The statistic was used in [19] as a base of the method applied in the segmentation problem in case when in real data we observe that some characteristics change with respect to time. This statistic was also a main point of the testing procedure whether in the given sample a structural break point exists or not.

In this paper we extend the methodology presented in [19] and propose to analyze the following statistic: This choice is motivated by seismic recordings characteristics and discussion is carried out in further sections. As one can expect can tend to if at least the first reading is equal to 0. In order to avoid this problem we modify the raw signal and in the further analysis instead of we substitute the first reading with the first nonzero reading. This technical issue is related to a single sample at the very beginning of the recording; thus it does not influence the results. We denote corrected series as .

Until the moment of P-wave arrival, the seismic recordings consist of ambient noise which is considered stationary [21]; obviously they can be described by independent identically distributed Gaussian random variables. Moreover, we assume that the theoretical second moment of the distribution is finite. It can be shown that for data before the moment of P-wave arrival we have the following:Our methodology is therefore based on this observation. In the procedure, in contrast to [19], we fit the logarithm function to first points of statistic. After the moment of P-wave arrival the character of the statistic changes. It is not exactly known what kind of function we can observe after the moment of P-wave arrival; however it was noted that in general the statistic is concave with respect to . Here we decided to test two different concave functions: exponential and power . These functions are fitted with time shift; , . In order to reduce computational time we subtract or and then fit the exponential or power functions, respectively. Fitted functions coefficients are obtained by using of Levenberg-Marquardt algorithm (LMA) [22, 23] which is an iterative algorithm used to solve nonlinear least squares problems. It combines features of Gauss-Newton method and the method of gradient descent [24]. The LMA algorithm requires at least 3 points to fit considered functions. The next step is to calculate the squared errors between and fitted functions. The estimated point of P-wave arrival is for which the error is minimized.

Entire detection algorithm can be described as follows:(1)Set 3.(2)If go to (7).(3)Fit to , to , and to .(4)Calculate . Calculate .(5)Set .(6)Go to (2).(7), is “exponent” estimator, and the “power” one.

#### 3. Application to Real Seismic Data

In this paper the proposed algorithm was applied to a 188 single-event recordings from O/ZG “Rudna” underground copper ore mine. The signals were gathered by seismic system ELOGOR-C which is used to rock mass observation. The system consists of 2 sets of 32 seismometers Willmore MK-III type; each collects velocity data in the frequency band 0.5–150 Hz which is adequate frequency band containing mining-induced events. Such band is enough for localization, seismic energy estimation, and focal mechanism indication by analysis of first motion direction, which is the basic purpose of the monitoring system. The microseismic events in higher frequency are registered in this mine by a different system. The data is transmitted to seismic station using analog transmission (frequency modulation) and sampled with sampling frequency 500 Hz. Due to characteristics of the deposit, the seismic system network is relatively flat and a few additional sensors are located in shafts. Analyzed signals are dated from August 1, 2015, to August 19, 2015. The events length extent from 4.6 s to 33 s. Moments of P-wave arrival was indicated preliminarily using the STA/LTA algorithm and then manually corrected by seismic station experts.

In Figure 1 an exemplary seismic event is presented with moment of P-wave arrival marked by red cross. In Figure 1(b) zoom on the arrival time is shown. It is easy to spot stationarity of the background noise before the arrival of P-wave (red cross).