An Algorithm of Acoustic Emission Location for Complex Composite Structure

Abstract

Acoustic emission (AE) is widely used in engineering and rock mechanics. The algorithm of AE location based on homogeneous medium or single velocity structure is confronted with lower accuracy when it is applied to the actual working conditions that are prevailing complicated and heterogeneous. In this paper, an AE location algorithm based on complex composite structure (CCS) is proposed via carrying out the following studies: (1) A new travel time calculation scheme suitable for CCS with step-like velocity change is proposed based on an optimized shortest path algorithm. By doing this, a reasonable ray path that is only deflected at the interface is obtained to improve the travel time accuracy. The time complexity of the new algorithm is $O\left(n{\log }_{2}n\right)$. (2) The availability of the new algorithm is verified via a theoretical analysis under a one-dimensional velocity structure as well as an AE experiment using a complex structure under artificial excitation. (3) The AE location during the failure of a simulated CCS in the laboratory indicates that the new algorithm can effectively calculate the travel time and ray path of the sample.

1. Introduction

Acoustic emission (AE) is widely used in engineering and rock mechanics [1,2,3,4,5,6,7,8,9,10,11,12]. The AE location, the algorithm of which mainly derives from seismology, is important to understand the stress state or stability of rock structures [13]. The one-dimensional (1D) velocity model, i.e., the horizontal or spherical layered model, is widely used in earthquake location. This simplification of the crustal structure can greatly reduce the difficulty of calculating seismic ray and travel time. However, the 1D velocity model cannot be well applied to locating fracture events within complex crustal structures. As a result, the earthquake location methods based on three-dimensional (3D) velocity model have been carried out to solve this problem [14,15,16,17,18,19,20,21,22]. However, 3D velocity models are still rarely used in earthquake location for the following reasons: (1) It is difficult to obtain the accurate 3D crustal structure; (2) The complex crustal structure is usually in the shallow crust, which has little influence on the location of most natural earthquakes, especially on the teleseismic location; (3) The algorithm of earthquake location based on 3D velocity model is too complex to implement. In recent years, the frequent occurrence of shallow earthquakes induced by human activities, such as mine-induced earthquakes, reservoir-triggered earthquakes, fluid injection-induced earthquakes and engineering-induced earthquakes [23,24,25,26,27,28,29,30], etc., has caused increasing security problems and subsequently requires accurate location of these events. On the other hand, the complex structures of the shallow crust, such as syncline, anticline, fault zone, tunnel, goaf, or terrain, where most induced earthquakes occur, have great impacts on the location of the induced earthquakes. Therefore, we should consider the location algorithm under 3D velocity model to improve the accuracy of the location of shallow induced earthquakes.

The main difficulty of seismic location under 3D velocity model is how to obtain the path and travel time of seismic rays. The studies of seismic tomography have greatly improved the techniques of seismic ray tracing and travel time calculation. As a result, a variety of algorithms have been proposed, such as trial shooting, bending, finite difference, and the shortest path method, etc. [31,32,33,34,35]. Among these algorithms, the shortest path method and the finite differential method have been widely used since they are easy to conduct 3D modeling and the subsequent seismic location calculation.

The shortest path method is based on Fermat’s principle. It determines the path and travel time of seismic rays by finding the path of the shortest time. The algorithm is stable and avoids the calculations of the normal direction, incidence angle, or refraction angle at the interface. Moreover, the problems of non-convergence or convergence to the local minimum point are also avoided by using the method. Thus, it has obvious advantages compared with other travel time calculation methods. Accordingly, various algorithms based on the shortest path method have been applied to the calculation of seismic ray and travel time [15,32,33,34]. For example, Moser’s algorithm [32] has a great impact, although the error of calculation will be large when this algorithm is applied to the velocity structures with step-like change. It was suggested to subdivide the grid to reduce the calculation error [32], but this will also increase the amount of calculation.

Complex structures composed of blocks with different wave velocities are common in 3D velocity structures. Such a complex structure is referred to as a complex composite structure (CCS) in this paper. A CCS is characterized by homogeneous wave velocity inside a block but step-like change in wave velocity at block boundaries. Accordingly, the seismic ray is straight inside a block and only deflects at block boundaries. In addition, the path of the seismic ray at block boundaries is non-differentiable.

In this paper, a new algorithm of AE location for the CCS is proposed, which is still based on the 3D grid model and the shortest path algorithm. Furthermore, the grid node division, computing efficiency, and path correction are improved in the new algorithm to ensure that the ray path is straight inside a block and deflects only at block boundaries. Secondly, the travel time from all nodes of a five-layer 1D velocity structure to a designated position is calculated by using our algorithm and compared with the theoretical travel time. By doing this, the travel time error and influencing factors are confirmed. Thirdly, the location of artificial excited AE events in a simulated CCS is obtained via our algorithm combined with the time-reversal imaging method [36,37,38], which verifies the validity of our algorithm. Finally, as an application example of our algorithm, the micro fracture events during the failure of a simulated CCS with a single hole in laboratory are located.

2. Algorithms

2.1. Location Algorithm

As a kind of elastic wave, AE has the same wave equation as seismic wave. However, AE is an ultrasonic wave that is much higher in frequency than seismic wave. The first arrival phase is usually used to locate AE source since the first arrival phase is easy to identify while other seismic phases are often polluted by earlier seismic phases. For isotropic and homogeneous materials with known wave velocity, the equation for the location using the first arrival time can be written as:

$$\sqrt{{\left(x-x_i\right)}^2+{\left(y-y_i\right)}^2+{\left(z-z_i\right)}^2}=V\left(t_i-t\right)$$

(1)

where ($x_i$,$y_i$,$z_i$) is the 3D coordinate of the ith sensor, $t_i$ is the first arrival time detected by the ith sensor, ($x$,$y$,$z$) is the 3D coordinate of the source, $V$ is the wave velocity, and $t$ is the onset of the source.

Equation (1) has four unknowns. Therefore, at least four sensors are required to achieve the 3D location of the source.

For 3D wave velocity structure, Equation (1) can be rewritten as Equation (2) when the travel time of each node in the structure is known.

$$T\left(x_i, y_i, z_i,x,y,z\right)=t_i-t$$

(2)

where, $T\left(x_i,y_i,z_i,x,y,z\right)$ is the travel time from ($x_i, y_i, z_i$) to ($x$,$y$,$z$).

Equation (2) still has 4 unknowns and requires at least 4 arrival times to calculate the source location. However, the location error will be large when only four equations are used because of the uncertainties in arrival time and wave velocity. Thus, more than four sensors are usually used to obtain more travel time data and build overdetermined equations. By doing this, the optimal solution of the overdetermined equations can be solved with the least square method and subsequently the location error is reduced. The least square method is the optimal method when the errors of the arrival time obey the normal distribution. In addition, the location accuracy will increase with an increasing amount of arrival times. However, the least square method will produce large location errors under the existence of individual data (outliers) with large errors.

The time-reversal imaging method is another location algorithm commonly used in seismology [37,38,39]. The method assumes that the rays emit from the detectors to the source in the reverse time axis. In this case, the common intersection of more than four rays in the reverse time axis can be determined as the location and onset of the source. It is not necessary to determine which rays belong to the same event by using the method. Instead, the method focuses on how many rays intersect at one point in the reverse time axis, which means that individual outliers have little impact on the location. As a result, the time-reversal imaging method is more robust. Moreover, not only the arrival times but also the waveforms can be used in the method. Thus, the method is used in this paper to locate AE sources. However, this does not mean that other location algorithms cannot be used. In fact, all the seismic location algorithms based on travel time can be used to locate the sources of 3D complex structures after the 3D travel time table is obtained. Therefore, the key to AE location in 3D complex structure is how to obtain the 3D travel time table. In this paper, we focus on the algorithm of 3D travel time.

2.2. Dijkstra’s Algorithm

The problem of the shortest path between nodes in complex networks is extensively studied in graph theory. Dijkstra proposed an algorithm [40] to solve the problem under a single source. In Dijkstra’s algorithm, the network nodes and their adjacent nodes need to be first determined, and then the shortest paths from a node to other nodes can be calculated. Here, two nodes are referred to as adjacent nodes if there is a direct path between them.

It is convenient to use discrete 3D grid nodes to describe complex structures, which is suitable for Dijkstra’s algorithm. The travel timetable is usually calculated in advance to avoid the double calculation of travel time during AE location since the calculation of travel time is time-consuming. We used Dijkstra’s algorithm to calculate the travel time from the node where each detector stays to all nodes via the following steps:

(1)

A travel time array and a ray source array are defined for each detector. The size of each array is equal to the total number of nodes. The elements in each array and the nodes are in one-to-one correspondence. Each element of the travel time array stores the travel time from the node, where the detector is located, to the corresponding node of the element. Consider that a ray emits from Node A to Node B, and the element of the ray source array corresponding to Node B is used to store the identifier number of Node A. Furthermore, a candidate queue, which is initially empty, is defined to store the identifier numbers of candidate nodes.

(2)

Define P as a source node where the detector locates. The element in the travel time array corresponding to Node P is set to zero. Define Q as the current secondary source node. The initial value of the element in the travel time array corresponding to Node Q is equal to TP-Q which is the travel time from Node P to Node Q.

(3)

Calculate the travel time from Node P to all the adjacent nodes of Node Q. Let $A_{Qi}$ be the ith adjacent node of Node Q, $T_{P-A_{Qi}}$ be the travel time from Node P to Node $A_{Qi}$, and $T_{Q-A_{Qi}}$be the travel time from Node Q to $A_{Qi}$, then we can obtain $T_{P-A_{Qi}}$ = TP-Q + $T_{Q-A_{Qi}}$. If $T_{P-A_{Qi}}$ is less than the value that has been previously stored in the travel time array, the previously stored value is replaced by $T_{P-A_{Qi}}$. Accordingly, the path source array is also modified, and the identifier numbers of $A_{Qi}$ are added to the candidate queue if they are not in the queue.

(4)

The calculation ends when the candidate queue is empty. Otherwise, let Q to be the node with the minimum travel time in the candidate queue, then Node Q should be removed from the candidate queue since the path and travel time of Node Q has been determined. Then, go back to the step (3) and continue to calculate.

The distribution range of the nodes is similar to the wave front, which gradually expands as the calculation process continues. After the calculation, the minimum travel time from the detector to each node has to be stored in the travel time array, and the ray path to each node from its source node can be obtained in the ray source array.

Dijkstra’s algorithm is a classic algorithm that is discussed in the textbooks of graph theory and data structure. Please refer to [41] for further details of the algorithm.

2.3. Optimization of Dijkstra’s Algorithm

To improve computing efficiency, we optimize the original Dijkstra’s algorithm in two following aspects.

(1)

The use of heap sort. It is very popular to add heap sort to Dijkstra’s algorithm [41]. In the original Dijkstra’s algorithm, the candidate queue is in a disordered state, the time complexity for seeking the node of minimum travel time from the candidate queue is $O\left(m\right)$, where $m$ is the total number of nodes in the candidate queue. If heap sort is used to manage the candidate queue, the time consumption for seeking the node of minimum travel time from the candidate queue includes k times of inserting or adjusting nodes into the heap, and one time of heap reconstruction after removing the minimum value. Specifically, k, which depends on the adjacency radius, is the number of nodes that need to be inserted or adjusted in the candidate queue. Thus, the time complexity is $O\left(k{\log }_{2}m\right)$ to insert or adjust nodes, and is $O\left({\log }_{2}m\right)$to rebuild the heap after removing the minimum value. On average, $m$ is a big number, but $k$ is relatively small, then $k{\log }_{2}m$ is much less than $m$. Therefore, we add heap sort into the original Dijkstra’s algorithm to improve the calculation efficiency for seeking nodes of minimum travel time from the candidate queue.

(2)

The use of parallel computing. This can largely improve the utilization of multi-cored CPU. Specifically, the nodes in the sample space are assigned to different threads, and a sub-heap is established for each thread. The travel time calculation of adjacent nodes as well as the addition and adjustment of the nodes in candidate queues are implemented by parallel computing. However, the removal of the nodes of minimum travel time from the candidate queue still uses the serial processing.

2.4. Construction of 3D Structure Model

We propose a new meshing method to calculate the travel time in CCS. Firstly, a CCS is divided into several uniform blocks with unique identifiers. Then, each block is divided into small regular cuboid cells that each cell only belongs to one block. Each vertex of a cell represents a node. By doing this, the block boundaries, ray paths, detector positions, and source positions are determined by the nodes. The nodes on block boundaries can belong to multiple blocks because they are shared by the adjacent cells.

Two nodes are referred to as adjacent nodes if there is a direct path between them. AE rays can directly propagate between the adjacent nodes. Two nodes in the same block are treated as adjacent nodes if the spacing in all directions of $x$, $y$ and $z$ between the two nodes is less than or equal to the adjacency radius. Here, the adjacency radius must be preset. The calculation accuracy as well as the amount of computation will increase with increasing adjacency radius.

The node adjacency relationship is schematically shown in Figure 1. The CCS is composed of two blocks in pink and blue. The red node in Figure 1a is the internal node of the pink block. According to the definition of the adjacent nodes, the adjacent nodes of the red node, are all internal nodes of the pink block. In addition, the spacing in the direction of $x$, $y$ and $z$ between the red node and the black solid nodes is less than or equal to the adjacency radius, Ra. However, the red node in Figure 1b is the boundary node between the pink and blue blocks. In such situation, the adjacent nodes of the red node, denoted by the black solid nodes in Figure 1b, could belong to different blocks as long as the spacing in the direction of $x$, $y$ and $z$ between the red node and the black solid nodes is less than or equal to $R_a$. The travel time between a node and its adjacent node is the linear distance between the two nodes divided by the wave velocity of the block, to which the two nodes belong. When the two nodes belong to multiple blocks, the highest wave velocity among the blocks is selected.

The attributes of a node are stored in 8 bytes (64 bits) in our program, whereas a block can be represented by 1 bit. This means that the number of blocks can be up to 64 in one model. As such, more bytes are required to store node attributes if larger complex structures with more than 64 blocks are used. In our program, two nodes are determined as adjacent nodes if the spacing in all directions of $x$, $y$ and $z$ between the two nodes is less than or equal to the adjacency radius and non-zero is obtained after “and” operation is applied to corresponding elements of the attributes between the two nodes. By doing this, all rays passing through the block boundary will pass through one of the boundary nodes.

2.5. Correction of Travel Time Error for Straight Path Inside a Block

The discretization of the continuum will inevitably cause errors although it brings convenience to describe complex structures. The error caused by spatial discretization is related to grid density. Specifically, the calculation accuracy as well as the calculation amount increase with increasing grid density. In addition, the shortest path algorithm also brings errors. For a CCS, AE rays are straight inside a block and deflected only at the block boundaries. However, the ray path inside the block is often not a straight line, but a polyline according to the above method due to the limitation of adjacency radius and the node discretization. Moreover, the length of each segment of the polyline is less than or equal to the adjacency radius. As a result, the travel time is calculated along the polyline path with error. This error will increase with the decrease in adjacency radius.

Table 1. Maximum relative error of straight-path travel time for various adjacency radii.

Adjacency Radius (Cell Number)	Maximum Relative Error
4	1.46%
5	1.15%
6	0.74%
7	0.50%
8	0.40%
9	0.34%
10	0.27%

lists the maximum relative error of straight path travel time for various adjacency radii.

Since the rays should be straight lines inside a block, we can correct the path and travel time of the rays according to the straight-line path. In our algorithm, it is easy to determine whether the ray reaches the block boundary since block boundaries are all located at nodes. Thus, the ray path and travel time in a block can be corrected when a ray reaches block boundary but does not enter another block. Namely, a path segment should be corrected as a straight line if the nodes, except the starting and ending nodes, of the path segment are intra-block nodes (Figure 2). Accordingly, the corresponding travel time is the straight-line distance between the starting and ending nodes of the path segment divided by the wave velocity of the block. By doing this, the ray path obtained by our algorithm will be straight inside a block and only deflected at the interface between blocks. As a result, the accuracy of the travel time calculation is improved.

2.6. Efficiency Test of Travel Time Calculation

Dijkstra’s algorithm has high efficiency because it only needs to search once to complete the travel time calculation from one source to all other nodes.

The computation amount of travel time is affected by the number of nodes and adjacency radius size. The spatial resolution depends on node density, whereas the angular resolution of a ray path is determined by the adjacency radius. Namely, both the calculation accuracy and the amount of computation increase with increasing node density or adjacency radius. Therefore, we need to achieve a balance between accuracy and computation consumption.

Our algorithm includes the calculation of the shortest path and the correction of ray path and travel time, which is difficult to estimate the time complexity. Thus, we evaluate the efficiency of the algorithm by using a laptop computer (CPU: Intel (R) Xeon (R) W-11,855M, 6 cores, basic frequency of 3.20 GHz) to calculate the travel time from one node (where one detector is located) to all other nodes. The adjacency radius is equal to 7 cells in all calculations. The time consumption and memory usage under different amounts of nodes are listed in

Table 2. Time consumption and memory usage under various number of nodes.

Total Nodes n	Time Consumption t (s)	Memory Usage (MB)
1,000,000	6	94
8,000,000	65	385
27,000,000	292	1169
64,000,000	786	2647
125,000,000	1745	5126
216,000,000	3139	8826
343,000,000	5231	13,863
512,000,000	7377	20,699

. Memory usage is nearly proportional to the number of nodes. Figure 3 shows the approximate linear relationship between the time consumption $t$ and $n{\log }_{2}n$, where $n$ is the number of nodes.

Therefore, the time and space complexity of our algorithm is $O\left(n{\log }_{2}n\right)$ and $O\left(n\right)$, respectively.

3. The Algorithm Testing

3.1. Theoretical Verification

In an ideal condition, a 3D model should be built to test the algorithm, where a detector is fixed on the model surface. Then, the travel time from the detector to all nodes is calculated and compared with the theoretical travel time to obtain the maximum relative error among all travel times. However, it is difficult to obtain the theoretical travel time of each node for complex structures. In this paper, we carry out this part of the work under a 1D wave velocity structure as a special case of the 3D case. This is because the 1D theoretical time is easy to calculate and can also satisfy the purpose of testing our algorithm.

We build a test model, which is divided into five layers (five blocks). the thickness of each layer is 20 mm and the P-wave velocity from top to bottom layers is 3000 m/s, 3500 m/s, 4000 m/s, 4500 m/s, and 5000 m/s, respectively. The sensor is set in the center of the upper model surface (Figure 4).

Errors of arrival time are mainly caused by discrete meshing and the limitation of adjacency radius. Reducing the mesh spacing can reduce the errors caused by discrete meshing, but cannot reduce the ones produced by the adjacency radius. To reduce the error caused by adjacency radius, we need to increase adjacency radius or to perform a straight path correction.

Table 3. The maximum relative errors obtained by calculating travel time under different options.

Grid Spacing (mm)	Adjacency Radius (Cell Number)	Path Correction	The Maximum Relative Errors
2	5	No	1.43%
2	5	Yes	1.28%
1	5	No	1.16%
1	5	Yes	0.64%
0.5	5	No	1.16%
0.5	5	Yes	0.61%
0.5	7	No	0.50%
0.5	7	Yes	0.33%

shows that the errors are mainly caused by the influence of the grid spacing when the grid spacing is 2 mm, while the errors are mainly caused by adjacency radius when the grid spacing is less than or equal to 1 mm. Straight-line path correction at block interfaces can significantly reduce errors stemmed from adjacency radius.

3.2. Experimental Verification

The travel time algorithm and its maximum relative error are compared with theoretical calculation in the 1D wave velocity structure above. However, for complex structures, theoretical calculations are too difficult. Therefore, we carry out an experiment to verify the algorithm.

The sample is composed of three materials, including copper, aluminum, and steel (Figure 5). The P-wave velocity of copper, aluminum, and steel is 4394 m/s, 6305 m/s and 5918 m/s, respectively. The dimension of the sample is 100 mm × 100 mm × 100 mm and is divided into 1,000,000 regular cubic cells. By doing this, the side length of each cell is 1 mm. A piezoelectric ceramic can generate an electric signal after receiving the vibration of the sample surface, which can be used as a receiving sensor. On the other hand, the piezoelectric ceramic can also be used as an excitation source by applying an electric pulse to generate vibration. In testing, we arranged 16 piezoelectric ceramic sensors on the sample surface, one of them was used as the excitation source, whereas the other 15 sensors were used for receiving AE waveforms. The diameter of each piezoelectric ceramic sensor was 8 mm. A total of 25 AE events were excited at the same location during the experiment.

We first calculate the ray paths and travel times, and then use the time-reversal imaging method to locate the excited signals. Figure 5 shows the ray paths (white lines) obtained by correcting the travel time of ray paths from the excitation source to each sensor, which is straight inside the same block and only deflected at the block interfaces. Among them, several ray paths in the copper block are similar to the primary wave in seismology. Because the P-wave velocity in aluminum is faster than that in copper, and the first wave runs along the interface between copper and aluminum.

The results of the 3D AE location (red dots in Figure 5) show that 25 AE sources are concentrated near the excitation point with the absolute error less than 4 mm (it is equivalent to the sensor radius). This indicates that an algorithm is effective in complex layered structures. The calculation results also show that the travel time error is acceptable for the experimental sample meshing with the cell size of 1 mm and the adjacency radius of 7 cells.

4. Application: An Experiment on a Simulated CCS with a Hole

Here, we provide a case highlighting the algorithm’s application, based on an experiment on a simulated CCS with a hole in the laboratory. The sample used in the experiment was composed of three layers of rocks, namely, granodiorite, sandstone, and gabbro were assembled together to form a simulated CCS with the dimension of 298 mm × 300 mm × 48 mm (Figure 6). The thickness of the granodiorite, sandstone, and gabbro layer is 80 mm, 140 mm and 80 mm, respectively. A circular cavern with diameter of 83 mm was excavated in the center of the sandstone layer (Figure 6).

A roughly horizontal non-penetrating crack in the middle of the sandstone layer was produced accidently during the preparation of the sample prior to the experiment. Pressure was applied to the sample in a vertical direction (Figure 6). We progressively raised the pressure until the sample fractured. The goal of the experiment was to simulate the failure process of the layered rock cavern by detecting the fractures.

Cubic cell with side length of 1 mm was used to mesh the sample for computation. The adjacency radius was set as 7 times the cell size. Figure 7 shows the locations of the AE sources in four stages during the experiment. The results show that the AE events initiated along the pre-existing crack in the sandstone layer (subplot of 25 s in Figure 7). Then, the tensile cracks, which emerged at the top and bottom of the hole, gradually expanded through the granodiorite and gabbro layers, indicating that the confining hard rocks at the top and bottom cannot obviously hinder the propagation of the tensile cracks (subplots of 600 s, 840 s, and 1200 s in Figure 7).

The AE location of multilayer samples has been a difficult task for a long time in the laboratory due to low accuracy. Our experimental results show that the travel time and ray path of such samples can be effectively calculated, and subsequently the accurate AE location can be obtained.

5. Discussion

5.1. Algorithm Optimization

Parallel computing is an important way to improve computing speed since the improvement of the basic frequency of CPU is limited in spite of the quick increase in CPU cores at the present. Therefore, parallel computing is used in part of our calculations. The calculation results show that the running speed is doubled but does not scale with the number of CPU cores. So the further optimization is still required.

Although the location accuracy is lower by using the time-reversal imaging method compared with the least square method when the arrival time error obeys the normal distribution, the advantage of the time-reversal imaging method is that it can adapt to the situation that there are some individual outliers with large errors in arrival time data. The probability of the occurrence of such situation is high when the arrival time is automatically recognized. This is why we used the time-reversal imaging method to locate the AE hypocenters in our experiments.

5.2. Limitations of the Algorithm

Some limitations of our algorithm were found and listed as follows when it was applied to calculate the ray path and travel time.

(1)

Only the first arrival wave can be used to obtain the path and travel time by using our algorithm. The first arrival wave may be P wave, head wave, or diffraction wave.

(2)

Our algorithm is only suitable for models with step-like change in wave velocity, such as layered model or block model, but not for models with gradual change in wave velocity.

(3)

Our algorithm stipulates that two nodes in the same block are adjacent nodes if the spacing in all directions of $x$, $y$ and $z$ between the two nodes is less than or equal to the adjacency radius, and that the waves can directly propagate between the adjacent nodes. As a result, an interlayer of the third material between two nodes, which is thinner than the adjacency radius, will be omitted during the calculation so as to guarantee calculation speed. To solve this problem, the material, which contains the thin interlayer, should be cut into two blocks along the two boundaries of the thin interlayer during 3D model meshing.

5.3. Comparison with Moser’s Algorithm

Moser’s algorithm has a great impact on the 3D velocity model. For the convenience of description, we compared our algorithm with Moser’s algorithm in a 2D case. Regular rectangular grids are used in Moser’s algorithm, where each node connects to 48 nearest nodes. In addition, slowness is sampled at nodes. The travel time between two connected nodes is defined as the distance between the two nodes multiplied by the average slowness of the two nodes, which is an approximate method to calculate the travel time between nodes. The block boundary in Moser’s algorithm is not located on the node, which leads to inaccurate block boundary. Namely, the change in velocity structure is gradual. Thus, Moser’s algorithm can be well applied to velocity structures with gradual change. However, the error of calculation will be large when this algorithm is applied to velocity structures with sudden change. Although Moser suggested subdividing grid to reduce the calculation error, this will increase the amount of calculation.

6. Conclusions

In this paper, an algorithm based on the shortest time path algorithm is proposed to calculate the travel time and ray path in CCSs, where seismic ray is straight in the same block and deflects only at block boundaries. Such 3D models have a step-like velocity structure and are consistent with the ray theory.

Our study shows that the accuracy in travel time calculation strongly depends on gridding. The grid spacing determines the spatial resolution of the algorithm. The calculation accuracy as well as the amount of calculation increase as the grid spacing decreases. In addition, the calculation accuracy of seismic ray and travel time also depends on the adjacency radius. Namely, the calculation accuracy as well as the amount of calculation increase with increasing adjacency radius. According to our method, the correction of straight line path and travel time can significantly reduce the error caused by small adjacency radius.

It takes about 6 s to calculate all the travel time from one sensor to one million nodes by using the computer (CPU: Intel (R) Xeon (R) W-11,855M, 6-core, 3.20 GHz basic frequency) in our study. The time complexity of the algorithm is O (nlog2n).

The theoretical calculation under 1D velocity model and the location of AE sources in laboratory experiments suggest that the algorithm proposed in this paper is feasible and can solve the problem of source location in CCS.