Trans. Nonferrous Met. Soc. China 25(2015) 293-302

Three-dimensional analytical solution of acoustic emission source location for cuboid monitoring network without pre-measured wave velocity

Long-jun DONG^1,2, Xi-bing LI¹, Zi-long ZHOU¹, Guang-hui CHEN¹, Ju MA¹

1. School of Resources and Safety Engineering, Central South University, Changsha 410083, China;

2. Australian Centre for Geomechanics, The University of Western Australia, Perth 6009, Australia

Received 27 August 2013; accepted 25 November 2014

Abstract: To find analytical solutions of nonlinear systems for locating the acoustic emission/microseismic(AE/MS) source without knowing the wave velocity of structures, the sensor location coordinates were simplified as a cuboid monitoring network. Different locations of sensors on upper and lower surfaces were considered and used to establish nonlinear equations. Based on the proposed functions of time difference of arrivals, the analytical solutions were obtained using five sensors under three networks. The proposed analytical solutions were validated using authentic data of numerical tests and experiments. The results show that located results are consistent with authentic data, and the outstanding characteristics of the new solution are that the solved process is not influenced by the wave velocity knowledge and iterated algorithms.

Key words: acoustic emission; seismic source; sensor; location; analytical solution

1 Introduction

It is generally accepted that most solids emit low-level seismic signals when they are stressed or deformed. The solution of the problem of locating a signal source using time difference of arrival (TDOA) measurements has numerous applications in aerospace, surveillance, structural health, navigation, industrial process, speaker location, machine condition, monitoring of nuclear explosions, and mining induced areal seismology [1,2]. In the geotechnical field, this phenomenon is generally referred to as acoustic emission/microseismic (AE/MS) activities. When rock fractures, it produces AE/MS signals that transmit through the rock as elastic waves [3-5]. The application of the AE/MS system, which monitors self-generated acoustic signals occurring within the ground, has now rapidly increased for the monitoring of underground structures such as mines [6-8], tunnels [9-12], natural gas, nuclear engineering, and petroleum storage caverns, as well as surface structures such as foundations, rock, and soil slopes [13-16].

The location of a seismic event (earthquake, MS or AE) has been the first and most basic step in any study of seismicity on any scale since 1910 [17]. In general, earthquakes are predicted for their source locations, defined by the coordinates and the origin time, assuming a seismic velocity model and minimizing the difference between the observed and the calculated travel times. Source location is one of the classic problems in seismic areas [18,19]. Considerable number of studies published in the past more than 100 years on seismic source location have proved the importance and, at the same time, the complexity of this problem.

Many researchers have developed different AE/MS/seismic source location techniques, and some of which have been mature technologies and widely used in the positioning of AE/MS currently[20-23], for example the joint hypocenter determination method, the double- difference method, and topographic inverse [24-26]. Nevertheless, the problem of determining the four source parameters (geocentric: x, y, z and origin time) has still not be definitively solved. The iterative analytic procedures, which are nowadays most often used for the calculation (cf. Geiger’s method), are not infrequently divergent, or at any rate do not give very reliable results, which considerably reduces the number of well-located events. This can lead to negative consequences when interpreting the activity itself. It is well-known that the correct location of the source is dramatically hindered by the following factors: 1) insufficient knowledge of the seismic wave velocity; 2) inadequate distribution of the stations; 3) intrinsic limitations of the iteration algorithm applied. Generally, those factors do not act independently each other. Taking Geiger’s algorithm which is actually the widely used and known one as an example, the use of initial evaluated hypocenter coordinates is presupposed, and an iterative least-square technique is used. These conditions significantly influence the location accuracy.

DONG and LI [27] proposed a set of analytical solutions for the AE/MS source location under a cuboid monitoring network of sensor location. A location method with P-wave velocity by analytical solutions (P-VAS) was obtained with the established equations. Virtual location tests show that the relocation results of analytical method are fully consistent with the actual coordinates for events both inside and outside the monitoring network; whereas the location error of traditional time difference method is between 0.01 and 0.03 m for events inside the sensor array, and the location errors are large, which are up to 1080986 m for events outside the sensor arrays. The broken pencil location tests were carried out in a granite rock specimen with 350 mm in length and the cross section of 100 mm×98 mm, using five AE sensors. Five AE sources were relocated with the conventional method and the P-VAS method. For the four events outside monitoring network, the positioning accuracy by the P-VAS method is higher than that by the traditional method, and the location accuracy of the larger one can be increased by 17.61 mm. The results of both virtual and broken pencil location tests show that their resolved analytical solutions are effective to improve the positioning accuracy. However, the problem of the P-VAS is that the wave velocity should be given in advance. It is difficult to apply in the conditions without pre-measured velocity or pre-given velocity knowledge.

In this work, the analytical solutions of the AE/MS source location coordinates without the knowledge of wave velocity were developed. Different locations of sensors on upper and lower surfaces were considered and used to establish nonlinear equations. Based on the proposed functions of time difference of arrivals, the analytical solutions were obtained using five sensors under several networks without need of wave velocity. The method highlights three outstanding advantages: 1) without using iterative solution; 2) without initial evaluated hypocenter coordinates; 3) without pre- measured velocity or pre-given velocity knowledge.

2 Statement of problem and solution method

The AE/MS/seismic source location method using P-wave arrival time is widely used to calculate source coordinates for two reasons: the fastest propagation velocity, and the easy identification of first arrival time. The AE/MS/seismic source location coordinate is (x, y, z); T_i(i=1, 2, …, n) is the ith monitoring station, and its coordinate is (x_i, y_i, z_i) (i=1, 2, …, n); l_i(i=1, 2, …, n) is the distance from the AE/MS/seismic source to the station T_i; t_i(i=1, 2, …, n) is the arrival time recorded by sensor in the station T_i; t₀ is the origin time of AE/MS source. Then, t_i can be expressed as

                                   (1)

where  is the P-wave velocity.

By the spatial distance formula between two points (the source location and the monitoring station location), it can be obtained

             (2)

By taking Eq. (2) into Eq. (1), we have

        (3)

In Eq. (3), t_i(i=1, 2, …, n), v, and (x_i, y_i, z_i) (i=1, 2, …, n) are known;  the seismic or AE source (x, y, z)  and  origin time  are unknown, which are needed to be solved. By taking each station data to Eq. (3), an equation can be obtained. Four stations correspond to four equations, and they can constitute a set of nonlinear equations. Generally, the greater the number of station is, the higher the positioning accuracy is. In the past more than 100 years, most researches were focused on the nonlinear optimization or iteration methods to locate the AE/MS/seismic source. The location precision was greatly influenced by the error of the wave velocity and the intrinsic limitations of the iteration algorithm applied. In this work, in order to find out the analytical solution of the AE/MS/seismic source location coordinates, the sensor location coordinates were optimized and simplified.

A cuboid monitoring network of sensor locations was selected, and the AE/MS/seismic source localization equations were established. The sensors are required to install at the vertices of the cuboid monitoring network. There are two cases including four sensors installed on one surface and additional one sensor on another surface (Fig. 1) as well as three sensors installed on one surface and additional two sensors on another surface (Fig. 2).

For every surface of the first case (Fig. 1), there are four types of monitoring network including Figs. 1(a), (b), (c) and (d). The first type (Fig. 1(a)) is analyzed in this work, and the others are similar. Five sensors are installed at vertices A, B, C, D and E of the cuboid monitoring network. The center of the cuboid is taken as the coordinate origin, and the coordinate direction is shown in Fig. 1.

Fig. 1 Three-dimensional location schematic of cuboid monitoring network

Fig. 2 Three-dimensional location schematic of cuboid monitoring network

The lengths of three sides of the monitoring network cuboid are 2a, 2b and 2c, respectively. The first sensor A is taken as a reference. The travel time of the sensor A from an AE/MS/seismic event is expressed as t₁₀, and the arrival time of sensors B, C, D and E is t₁₀+△t₂, t₁₀+△t₃, t₁₀+△t₄, and t₁₀+△t₅, respectively. According to Eq. (3), it can be obtained

              (4)

       (5)

        (6)

       (7)

        (8)

Taking subtraction of Eqs.(4) and (5), Eqs. (4) and (6), Eqs. (4) and (7), as well as Eqs. (4) and (8), we have

                       (9)

                  (10)

                      (11)

                     (12)

From Eqs. (9) , (10) , and (11) one can easily obtain

     (13)

Resolving Eq. (13) yields

                      (14)

Taking the ratio of Eq. (9) to Eq. (11), we have

                     (15a)

Supposing , Eq. (15a) can be rewritten as

                                 (15b)

From Eqs. (11) and (12) one can obtain

                      (16)

Supposing, Eq. (16) can be rewritten as

                                (17)

Eq. (4) divided by Eq. (9), we have

      (18)

Supposing , and substituting  Eqs. (15a) and (17) into Eq. (18), we have

   (19)

Eq. (19) can be rewritten as

                            (20)

where

and C=a²+b²+c²+.

Then, x, y, and z can be obtained by resolving Eqs. (20), (15b) and (17). The solutions can be defined as analytical solution I (ASI).

For every surface of the second case in Fig. 2, there are five types of monitoring network including Figs. 2(a), (b), (c), (d) and (e). The first type (Fig. 2(a)) is analyzed in this work, and the others are similar. Five sensors are installed at vertices A, B, D, E, and F of the cuboid monitoring network. The center of the cuboid is taken as the coordinate origin, and the coordinate direction is shown in Fig. 2. The lengths of three sides of the monitoring network cuboid are 2a_s, 2b_s and 2c_s, respectively. The first sensor A is taken as a reference. The travel time from an AE/MS/seismic source (x_s, y_s, z_s) to sensor A is expressed as t_s10, and the arrival time of sensors B, D, E and F is t_s10+△t_s2, t_s10+△t_s3, t_s10+△t_s4 and t_s10+△t_s5, respectively. The P-wave velocity is expressed as v_s. According to Eq. (3), it can be obtained

       (21)

  (22)

 (23)

  (24)

 (25)

Taking subtraction of Eqs. (21) and (22), Eqs. (21) and (23), Eqs. (21) and (24), as well as Eqs. (21) and (25), we have

                 (26)

                (27)

                 (28)

           (29)

From Eqs. (26), (28), and (29) one can easily obtain

    (30)

                    (31)

From Eqs. (26) and (27) one can obtain

                (32)

Supposing , Eq. (32) can be rewritten as

                              (33)

From Eqs. (28) and (27) one can obtain

                 (34)

Supposing , Eq. (34) can be rewritten as

                               (35)

From Eqs. (21) and (26) one can obtain

    (36)

Supposing and substituting Eqs. (33) and (35) in Eq. (36), we have

          (37)

Equation (37) can be rewritten as

                         (38)

where

and . x_s, y_s, and z_s can be obtained by resolving Eqs. (38), (33) and (35), respectively. The solutions can be defined as analytical solution II (ASII).

It is noted that the above two conditions are different networks which considered both upper and lower surfaces of the cuboid with five sensors. The first case is four sensors on upper surface and one sensor on the lower surface, while the second case is three sensors on upper surface and two sensors on the lower surface. The first case has four types of networks and the second one has five types of networks. It is easy to find the different and significant characteristics between the two types of networks. If we consider six surfaces of the cuboid networks, we can see that the two selected conditions have the same characteristic, four and one sensors on two different surfaces. It is noted that the condition that two and three sensors on two different surfaces is not considered. To fix the problem systematically, the third condition, two sensors on one surface and three sensors on another surface (Fig. 2(b)), is analyzed and the analytical solution is also obtained.

The lengths of three sides of the monitoring network cuboid are 2a_p, 2b_p and 2c_p, respectively. The first sensor A is taken as a reference. The travel time from the AE/MS/seismic source (x_p, y_p, z_p) to the sensor A is expressed as t_p10, and the arrival time of sensors A, B, D, F and H is t_p10+△t_p2, t_p10+△t_p3, t_p10+△t_p4 and t_p10+△t_p5, respectively. The P-wave velocity is expressed as v_p. According to Eq. (3), it can be obtained

         (39)

    (40)

  (41)

  (42)

  (43)

Taking subtraction of Eqs. (39) and (40), Eqs. (39) and (41), Eqs. (39) and (42), as well as Eqs. (39) and (43), we have

                 (44)

                (45)

          (46)

         (47)

From Eqs. (44), (45), and (46) one can easily obtain

      (48)

              (49)

Taking the ratio of Eq. (44) and Eq. (45), we have

               (50)

Supposing , Eq. (50) can be rewritten as

                              (51)

Submitting Eq. (45) into Eq. (47) yields

    (52)

Taking the ratio of Eq. (52) and Eq. (45), we have

    (53)

Supposing , Eq. (53) can be rewritten as

                              (54)

Taking the ratio of Eq. (39) and Eq. (44) yields

    (55)

Supposingand substituting Eqs. (51) and (54) in Eq. (55), we have

            (56)

Eq. (56) can be rewritten as

                       (57)

where

And

.

x_p, y_p and z_p can be obtained by resolving Eqs. (57), (51) and (54), respectively. The solutions can be defined as analytical solution III (ASIII).

3 Validated examples and discussion

3.1 Numerical examples

In the first example, a positioning system includes five sensors at the five cuboid vertices, and the coordinates are A(-130, 165, 220), B(-130, -165, 220), C(130, -165, 220), D(130, 165, 220), E(-130, 165, -220). The average equivalent P-wave velocity in the medium is expressed as v, and v=5000 m/s. The AE/MS sources are O(110, 200, 180), P(210, 97, -89), Q(-77, -89, 190), R(-98, 22, 168), and S(99, -289, 190) (all coordinates have the length unit of m). The arrival time recorded by sensors is listed in Table 1, and the accuracy of time is 10^-6 s. By using the proposed analytical solution to calculate the AE/MS source coordinates, coordinate values of the five sensors and arrival time of five sensors for five events are taken into Eqs. (20), (15b) and (17), and the coordinate values (x, y, z) of five acoustic emission events can be resolved. The actual and calculated results are listed in Table 2. It can be seen from Table 2, one set of the location results of the proposed analytical solutions are fully consistent with the actual coordinates.

In the second example, a positioning system includes five sensors at the five cuboid vertices, and the coordinates are A(-130, 165, 220), B(-130, -165, 220), D(130, 165, 220), E (-130, 165, -220), and F(-130, -165, -220), and average equivalent P-wave velocity in the medium is expressed as v, and v=5000 m/s. Assume that AE/MS sources O, P, Q, R and S are as the same as the first example (all coordinates have the length unit of m). The arrival time recorded by sensors is listed in Table 3. By using the proposed analytical solution to calculate the AE/MS source coordinates, the coordinate values of the five sensors and arrival time of the five sensors for the five events are taken into Eqs. (38), (33) and (35), and the coordinate values (x_s, y_s, z_s) of five acoustic emission events can be resolved. The actual and calculated results are listed in Table 4. It can be seen from Table 4 that one set of the location results of the proposed analytical solutions are fully consistent with the authentic coordinates.

Table 1 Arrival time recorded by sensors O, P, Q, R, and S in the first example

Table 2 Comparison between actual and calculated coordinates and errors of absolute distance in the first example

In the third example, a positioning system includes five sensors at the five cuboid vertices, and the coordinates are A(-130, 165, 220), B(-130, -165, 220), D(130, 165, 220), F (-130, -165, -220) and H(130, 165, -220). The average equivalent P-wave velocity in the medium is expressed as v, and v=5000 m/s. Assume that AE/MS sources O, P, Q, R and S are as the same as the first example (all coordinates have the length unit of m). The arrival time recorded by sensors is listed in Table 5. By using the proposed analytical solution to calculate the AE/MS source coordinates, the coordinate values (x_p, y_p, z_p) of the five sensor and arrival time of five sensors for the five events are taken into equations (57), (51) and (54), and the coordinate values of the five acoustic emission events can be resolved. The actual and calculated results are listed in Table 6. It can be seen from Table 6 that one set of the location results of the proposed analytical solutions are fully consistent with the actual coordinates.

Table 3 Arrival time recorded by sensors A, B, D, E, and F in the second example

Table 4 Comparison between actual and calculated coordinates and errors of absolute distance in the second example

Table 5 Arrival time recorded by sensors in the third example

Table 6 Comparison between actual and calculated coordinates and errors of absolute distance in the third example

3.2 Experimental validation

The AE tests were carried out in a cuboid of granite rock using five AE sensors. The sensor coordinates are A(-60, 80, 90), B(-60, -80, 90), C(60, -80, 90), D(60, 80, 90), E(-60, 80, -90). AE/MS sources are O(60, 30, 40), P(28, -80, 38) and Q (-29, 80, 58) (all coordinates have the length unit of cm). The arrival time recorded by sensors is listed in Table 7, and the accuracy of time is 10^-6 s. By using the proposed analytical solution to calculate the AE/MS source coordinates, coordinate values of the five sensor and arrival time of the five sensors for five events are taken into equations of ASI, ASII and ASIII. The coordinate values of acoustic emission events can be resolved. The actual and calculated results are listed in Table 8. It can be seen from Table 8 that one set of the location results of the proposed analytical solutions are fully consistent with the actual coordinates.

It can be seen from the above validated examples that there are two groups of solutions using the proposed analytical solutions. One is the real and correct solution, the other one is meaningless solution. The problem is how to select the real solution and cancel the meaningless solution. The checking calculation with arrival time of the sixth sensor is an efficient approach to select a reasonable solution. For example, the arrival time of six sensors (i.e. sensor G) is listed in Table 5. The solutions in Table 6 are obtained only using the arrival time of the first five sensors. We can use arrival time t_p6 and coordinates of the sixth sensor G to select the reasonable solution. The distance D between the solved source and sensor G can be calculated according to the distance formula between two points in space. v_p can be solved using Eq. (39). According to t_p10, the original time t_original of the event can be obtained by taking the subtraction of t_p1 and t_p10, then the reasonable solution should meet the following criterion:

                          (58)

Taking the values of sensor G into Eq. (58), we can easily get the reasonable solutions which are listed in Table 6.

Table 7 Arrival time recorded by sensors for three groups of AE tests

Table 8 Results and comparison of AE experiments

4 Conclusions

1) The sensor location coordinates were simplified as a cuboid monitoring network. Different locations of sensors on upper and lower surfaces were considered and used to establish nonlinear equations.

2) Based on the proposed functions of time difference of arrivals, the analytical solutions were obtained using five sensors under two networks. The proposed analytical solutions were validated using authentic data. The results show that the proposed analytical solution is reasonable and a set of the resolved solutions are consistent with the authentic results.

3) The sixth sensor is needed to determine the unique solution of the source location. Based on a cuboid monitoring network of sensor location, the method can locate the coordinates of AE/MS source only using simple four arithmetic operations. The method highlights three outstanding advantages of without using iterative solution, without initial evaluated hypocenter coordinates and without pre-measured velocity or pre-given velocity boundary conditions.

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无需预先测速的长方体监测网络声发射源三维解析定位方法

董陇军^1,2, 李夕兵¹, 周子龙¹, 陈光辉¹, 马 举¹

1. 中南大学 资源与安全工程学院，长沙 410083；

2. Australian Centre for Geomechanics, The University of Western Australia, Perth 6009, Australia

摘  要：为得到未知波速结构声发射源定位坐标求解的解析解，将传感器阵列简化为长方体，以求得最小监测网络下的声发射源定位解析解。考虑传感器在长方体表面不同位置的各种情形，建立对应的定位控制非线性方程组。根据建立的方程组和到时差，将监测网络分为3种情况，分别求得未知波速情况下采用5个传感器进行定位的声发射源定位解析解。将得到的声发射源定位解析解应用到数值和声发射试验中进行验证。结果显示，所求得的解析解避免了预先测定波速与迭代算法求解给定位带来的误差，定位结果与真实坐标一致。

关键词：声发射；震源；传感器；定位；解析解

(Edited by Wei-ping CHEN)

Foundation item: Projects (11447242, 41272304, 51209236, 51274254) supported by the National Natural Science Foundation of China; Project (2015CB060200) supported by the National Basic Research Program of China

Corresponding author: Long-jun DONG; Tel: +86-13973160861; E-mail: rydong001@csu.edu.cn

DOI: 10.1016/S1003-6326(15)63604-4