An Inverted Heterogeneous Velocity Model for Microseismic Source Location in Deep Buried Tunnels

Abstract

Microseismic (MS) monitoring is an effective and widely applied technology for early warning and prevention of rockburst disasters in deeply buried tunnels. The location of the microseismic source is the top priority of MS monitoring. However, an unknown complex and consistently changing velocity distribution of surrounding rocks is a major challenge for accurate MS source location. A real-time velocity inversion method of monitoring areas during tunnel drilling is urgently needed to provide a newly updated heterogenous velocity model. This paper develops a velocity inversion strategy for surrounding rocks using drilling and blasting seismic data. Inverted heterogeneous velocity models can be obtained at multiple construction stages with drilling. An L2 norm double-difference location method based on the simplex optimization method is provided for MS location using heterogeneous velocity models. Numerical studies test the accuracy of the proposed velocity inversion and MS location method. Noticeable improvement in location accuracy can be observed using updating inverted heterogeneous velocity models. Finally, we conducted a field experiment at two new tunnels with a burial depth of 2400 m in the China Jinping Underground Laboratory Phase II. Multiple blasting sources with known coordinates verified the performance of the proposed MS location method.

1 Introduction

A growing number of deep buried tunnels have been constructed worldwide for transportation, hydropower, and mining. However, rockburst, one of the most dangerous disasters in deep projects, occurs more frequently as the buried depth of the tunnel increases (Dai et al. 2017). For example, an extremely intense rockburst on November 28, 2009 in a 2525 m deep buried drainage tunnel of the Jinping II hydropower station in China featured several sudden and violent collapses of rock mass under high-stress conditions, causing severe casualties, destroying the tunnel boring machine, and delaying the construction schedule (Li et al. 2012; Feng et al. 2019). Previous studies have revealed a strong relationship between the evolution of microseismicity and the occurrence of high-strength rockburst (Li et al. 2017; Dai et al. 2018). Thus, microseismic (MS) monitoring techniques have been widely introduced in deep tunnel engineering to trace rock fractures and provide early warning of rockburst disasters (Xiao et al. 2016; Ma et al. 2019).

MS source location is the priority of MS monitoring and subsequent rockburst warning (Li et al. 2011; Ma et al. 2022). The accurate MS source location and the evaluation of MS parameters during construction provide essential information to reveal the distribution of the surrounding rock fracture network and the potential failure pattern of the rock mass (Pei et al. 2009; Hu and Dong 2019; Pu et al. 2022). The source location method of multiple scales from earthquakes and MS to acoustic emission (AS) sources has been studied for over a century (Geiger 1912). With the rapid improvement in calculation methods and hardware, researchers have explored many classic location methods, including the Geiger method (Geiger 1912), the US Bureau of Mines (USBM) method (Leighton and Blake 1970), Thueber’s method (Thueber 1985), the simplex method (Prugger and Gendzwill 1988), and the double residue difference method (Waldhauser and Ellsworth 2000). Note that the location accuracy relies strongly on a velocity model of the monitoring area. A single-velocity model can be applied in a simple and homogeneous lithology project but yields significant location errors in more complex geological settings. Feng et al. (2015a) proposed a sectional velocity model method for MS location and reported several successful application cases in tunnel projects. Castellanos and Van der Baan (2015) proposed a cross-correlation location method in mines. Collins et al. (2014) established a velocity model considering different infills of voids. Jiang et al. (2021) introduced a fast-marching method with a second-order difference approach (FMM2) as a forward modeling method employing the traveling time, making MS location possible using a velocity model containing a cavern. Dong et al. (2019, 2020) provided a velocity-free MS location method to mitigate the influence of wave velocity. The method locates the MS source in the mine without premeasured velocity, which is a beneficial complement to the traditional locating theory. However, there are still some challenges of MS source location in a deep tunnel, especially when the sensor network is placed near the drilling face. (1) Complex velocity models, such as the excavation damaged zone (EDZ) (Li et al. 2013), substantially change the ray path of the seismic wave traveling along the sidewall of the deep tunnel. To accurately calculate the travel time, a velocity model representing the characteristics of the EDZ is needed during the location. (2) The acquisition geometry of the MS monitoring network is ill-posed. Velocity analysis and inversion methods in mining and earthquake engineering are not applicable to linear observation acquisition in tunnels. (3) The condition of the surrounding rock changes dramatically over time during excavation. Continually updating velocity must be inverted while tunnelling to provide better location resolution.

The blasting source of tunnel excavation provides meaningful waveform information that can enable active seismic exploration without an accurate trigger time. Experiments and numerical studies (Leparoux et al. 2012) suggest that the EDZ underneath a concrete layer can be investigated with surface waves traveling along tunnel sidewalls. The S-wave velocity model can be estimated by inversion of the fundamental mode of the Rayleigh wave (Czarny et al. 2021). Then, the trigger time of blasting can be estimated by the inverted S-wave velocity. In that case, multiple blasting sources can be used for P-wave first-arrival traveltime tomography, which can continually invert the heterogeneous P-wave velocity during the whole tunnelling construction. Previous works (Dong et al. 2022) suggest a significant application of abnormal region identification in rock-mass using traveltime tomography, which provided insight into potential hazards detection in underground geotechnical engineering. However, first-arrival traveltime tomography is still challenging using an ill-posed acquisition system and limited blasting source. First-arrival traveltime tomography can be classified as traditional ray tracing-based tomography and the adjoint-state method (Sei and Symes 1994) depending on whether the ray path calculations are explicit or implicit. Ray tracing-based methods employ the straight-line method, curve method, or shortest path ray tracing method as a forward modeling operator. The least-square conjugate-gradient method (LSQR) (Yao et al. 1999) and simultaneous iterative reconstruction (SIRT) (Gregor and Benson 2008) are applied in the inversion of ray-tracing-based tomography. The velocity model updates are obtained by solving the tomographic equations. The definition and theory of traditional ray tracing-based inversion are clear, but the solution of tomographic equations is highly ill-conditioned and unstable. The adjoint-state method of first-arrival traveltime tomography, incorporated in the implicit ray tracing calculation, is provided by Taillandier et al. (2009) for surface seismic surveys. The gradient is calculated by dual modeling based on solving the eikonal equation using the fast-marching method (FMM) (Kroon 2021) or fast sweeping method (FSM) (Zhao 2005). Compared to the ray tracing method, the adjoint-state method has the advantage of imaging complex geological environments with limited pairs of sources and receivers.

In this work, we fully use drilling and blasting source data and developed an MS source location method in deep buried tunnels using an updated inverted heterogeneous velocity model. An improved perturbational dispersion inversion method is developed by combining a modified thin layer method (TLM) forward simulation and perturbational inversion method, which can estimate the S-wave velocity model and the depth of the EDZ. The starting time t₀ of blasting sources is calculated using S-wave back propagation tracing by the FMM on the inverted S-wave model. An adjoint-state first arrival tomography method is provided to continually invert the heterogeneous P-wave velocity model. A double-difference simplex-based method is introduced for MS location source. A realistic numerical study based on the finite difference in the time domain (FDTD) is used to visualize the blasting source wavefield in the surrounding rock. Sixty-four MS sources around the tunnel drilling face are used to test the accuracy of the proposed location method. A field experiment is conducted at two new tunnels with a buried depth of 2400 m in China Jinping Underground Laboratory Phase II. Blasting sources with known locations are used to test the performance of the proposed location method in practice.

2 Methodology

2.1 Motivation

Obtaining a reliable velocity model of the monitoring area is the priority of MS source location. The velocity model determines the wave propagation path and arrival time, which significantly impacts the accuracy of the location. The presence of an EDZ, the discontinuity of the rock mass and different intactness and stress conditions make the wave propagation path complex. Figure 1 presents the influence of the EDZ on wave propagation as an example. The wave propagation is represented by straight lines connecting sources and sensors if the velocity model is homogeneous (Fig. 1a). However, a zone that features lower velocity can be observed near the existing tunnel when the EDZ is considered, as shown in Fig. 1b. The propagation paths in the heterogenous velocity model are calculated by the FMM, and the paths follow Fermat’s principle of least time. The wave transmitted at the source travels toward the high-velocity area and backward to the sensors. The propagation paths in Fig. 1b are longer than the straight lines shown in Fig. 1a. The assumption of the homogeneous velocity model can cause location errors of MS events.

Fig. 1

Influence of the heterogeneous velocity on wave propagation (taking the EDZ of the tunnel as an example): wave propagation paths in a homogeneous velocity model and b heterogeneous velocity model

In the case shown in Fig. 1, a new tunnel is planned at an existing tunnel constructed for several years. The development of the EDZ and its effect on actual wave propagation paths cannot be ignored during MS location. However, it is possible to invert the velocity model by the drilling blast source of the new proposal tunnel. A drilling and blasting source can be used for active seismic exploration with an unknown accurate starting time of the source. Dispersion analysis of surface waves traveling along the tunnel sidewall and first-arrival traveltime tomography are two potential technologies in seismic exploration that can invert the S-wave and P-wave velocities. Extraction of the dispersion curve using limited sensors and determination of the starting timepoint t₀ during tomography are two critical steps during the inversion of velocity. We develop an inversion velocity model method and a double-difference MS location method in this study. The proposed method aims to visualize the heterogenous velocity field using ongoing drilling and blasting signals and enhance the accuracy of MS location using the updated inverted heterogenous velocity.

2.2 Overall Workflow

A workflow of heterogeneous velocity inversion and MS location is shown in Fig. 2. The workflow consists of four steps: drilling and blasting signal analysis, surface wave dispersion inversion, adjoint-state traveling time tomography, and L2 norm double-difference location. The absolute arrival time of the P-wave and S-wave can be extracted from the processed drilling and blasting source signals. The dispersion inversion provides an estimated layered S-wave velocity of the monitoring area, indicating the development of the EDZ and geological setting in the depth direction (y-axis in Fig. 1). The blasting starting time t₀ can be determined by wave backward propagation analysis using the known location of the source and sensors, arrival time of the S-wave and inverted S-wave velocity model. Then, the absolute traveling time of the P-wave can be calculated, because the t₀ values of the S-wave and P-wave are consistent. Traveling time tomography using multiple drilling and blasting source images is used to create a heterogenous P-wave velocity model between sources and sensors. More sources can be taken into account with deeper drilling. Double-difference location is conducted using the P-wave arrival time and the last inverted P-wave heterogeneous velocity model. The four steps of the workflow are described as follows:

• a. Drilling and blasting signal analysis

  Fig. 2

  

  Overall workflow of MS source location in a deep tunnel using heterogeneous velocity updating

  The bandpass filter is deployed to remove the background noise and enhance the signal-to-noise ratio. The drilling blast wave profile is shown in Fig. 2. The P-wave and S-wave signals can be identified based on the slope corresponding to the events. The drilling blast source in the actual project is a long signal of several seconds due to multiple blasting events. We analyze the signal of the first blast only because of its clean first arrival time without any overlap. The accurate pick of the P-wave is more straightforward than that of the S-wave. It is difficult to identify the S-wave arrival due to the interference of the P-wave (e.g., the left trace in the scheme figure of step 1). However, the S-wave and P-wave gradually separate as the offset increases. The arrival time of the S-wave at the most significant offset is used to estimate t₀ in the next step.

  • b. Surface wave inversion analysis

  Dispersion inversion of the surface wave is applied to reconstruct the S-wave velocity model, so that t₀ can be calculated by ray backward tracing, as shown in step 2 in Fig. 2. The dispersion curve is extracted from the processed drilling blast wave profiles by the phase-shift method (Park et al. 1998). The initial model is a linearly increasing velocity S-wave model. The 1D S-wave velocity profile can be obtained by iteration to fit the observed and simulated dispersion curves. The S-wave velocity profile in step 2 describes the EDZ range. Note that t₀ is not needed during surface wave dispersion inversion. Thus, it is possible to calculate t₀ by an inverted S-wave model and the S-wave arrival time and location of the source and sensors. Ray tracing can be conducted on the inverted S-wave by solving the eikonal equation using the FMM. The contour of traveling time from the sensor (red triangle at the right in step 2 scheme) is also shown in the figure. Thus, the backpropagating traveling time can be obtained and used to calculate t₀ as Eq. (1):

  $$t_{0} = t_{{\text{s}}} - t_{{\text{b}}} ,$$

  (1)

  where t_s is the arrival time S-wave recorded by the sensor with the largest offset and t_b is the corresponding backpropagating traveling time from the sensor to the blasting source. The parameter t₀ of the blasting source at different depths can be calculated similarly.

  • c. Adjoint-state traveling time tomography

    After obtaining t₀ of the blasting source, the absolute traveling time of the P-wave can be calculated. First-arrival traveltime tomography using the adjoint-state method is developed to invert the P-wave velocity model. The high-density ray path gradually covers the monitoring area when multiple blasting sources are included in the tomography (scheme of step 3 in Fig. 2). The P-wave velocity within the ray path coverage is updated during the tomography iteration. The initial P-wave velocity model can be estimated based on the S-wave profile. The P-wave velocity model is updated until the misfit between the observed and simulated traveltimes converges. An adjoint-state parameter is introduced during the inversion to speed up the iteration. Tomography can be conducted after every drilling and blasting, which updates the P-wave velocity model using the newly added blasting source signal. Thus, a constantly updated heterogenous P-wave velocity can be obtained during tunneling.d. Double-difference MS location using updating the heterogenous velocity model

A heterogeneous P-wave velocity model is used to locate the source of MS during tunneling. During the location of the MS, an arrival time difference between two sensors of one event is measured as Eq. (2):

$$r_{ij}^{k} = \left( {t_{i}^{k} - t_{j}^{k} } \right)^{{{\text{obs}}}} - \left( {t_{i}^{k} - t_{j}^{k} } \right)^{{{\text{cal}}}} ,$$

(2)

where $r_{ij}^{k}$ is the arrival time difference between sensors i and j (i, j = 1,2 … n, n is the total number of sensors) for MS event k. Obs and Cal are the observed and calculated values of the arrival time, respectively. The FMM (Kroon 2021) is applied to model the travel time using an updating heterogenous velocity model. Thus, the objective function of location for MS event k can be established as an L2 norm of the arrival time difference:

$$F\left( {x,y} \right) = \mathop \sum \limits_{i = 1,j = 2}^{n} \left( {\left( {t_{i}^{k} - t_{j}^{k} } \right)^{{{\text{obs}}}} - \left( {t_{i}^{k} - t_{j}^{k} } \right)^{{{\text{cal}}}} } \right)^{2} ,$$

(3)

where (x, y) is the location of the MS source. The simplex method (Nelder and Mead 1965; Li et al. 2014) is used to minimize the objective function [Eq. (3)]. The simplex method searches the MS source's location by moving and distorting the simplex shape to converge to the lowest misfit.

The proposed MS source location algorithm to invert updating heterogeneous velocity models using blasting seismic data. The surface wave inversion is used to consider the effect of the EDZ. The FMM is used to calculate the traveling time and ray tracing and estimate the t0 of the blasting source. Traveling time tomography is then conducted to obtain a heterogeneous P-wave velocity model. More blasting sources are added to the tomography with drilling construction going on. Finally, the MS location can be determined based on the double-difference objective function and the simplex method. The details of surface wave inversion analysis, adjoint-state traveling time tomography and double-difference MS location algorithms are described as follows:

2.3 Surface Wave Inversion

In the surface wave inversion, the phase-shift method (Park et al. 1998) is applied to extract the dispersion curve from the drilling blast wave profile. Then, we introduce an improved perturbational inversion of the fundamental mode of the Rayleigh wave dispersion curve (Liu and Wang 2021) to invert a layered S-wave velocity model. The improved perturbational inversion consists of two parts: the modified TLM forward simulation and perturbational inversion.

The modified TLM forward simulation first builds a matric function (Kausel and Roësset 1981):

$${\varvec{K}} = {\varvec{k}}^{2} {\varvec{B}}_{2} + {\varvec{kB}}_{1} + {\varvec{B}}_{0} - {\varvec{\omega}}^{2} {\varvec{M}}$$

(4)

where k is the wavenumber and ${\varvec{\omega}}$ is the angular frequency of excitation. The stiffness matrices ${\varvec{B}}_{0}$, ${\varvec{B}}_{1}$, and ${\varvec{B}}_{2}$ are matrix functions of Lame’s first parameter and shear modulus, whereas the mass matrix M depends only on the density. The problem of ${\varvec{Det}}\left( {\varvec{K}} \right) = 0$ becomes a typical generalized quadratic eigenvalue problem in terms of wavenumber k:

$$(k^{2} B_{2} + kB_{1} + B_{0} )v = \omega^{2} Mv$$

(5)

which can be solved using the root-searching method (HE et al. 2006) to obtain the eigenvalue k and the eigenvector $v$. An eigenvalue k and its corresponding eigenvector $v$ are together called an eigenpair. For a fixed frequency $\omega$, multiple eigenpairs can often be obtained. The eigenvector $v$ represents a mode shape, and the eigenvalue k (together with $\omega$) represents the Rayleigh wave phase velocity $V_{{\text{R}}} = \frac{\omega }{k}$. In the modified TLM, we use inverse iteration (Moler and Stewart 1973) to compute the eigenvector $v_{j}$ corresponding to $\left( {\omega_{j} ,V_{{{\text{R}}j}} } \right)$ for $j = 1,2, \ldots ,m$ and complete the forward simulation.

The perturbational inversion method updates the model $m = \left( {V_{s} ,V_{p} , \rho } \right)$ by solving the optimization problem:

$$J\left( m \right){\Delta }m = {\Delta }b,$$

(6)

where $\Delta b$, a function of the frequency $\omega$, is the difference between the observed and simulated dispersion curves. ${\varvec{J}}\left( {\varvec{m}} \right)$ is a Jacobian matrix calculated based on the Haney and Tsai (2017) formula:

$$J\left( m \right) = \frac{{\delta V_{{\text{R}}} }}{{\delta V_{{\text{s}}} }}.$$

(7)

To solve the optimization problem, we define a weighting matrix $W$ depending on the absolute difference in the Rayleigh wave velocity with respect to frequency as

$$W = L^{T} L, L = {\text{Diag}}\left( {\frac{{\left| {{\Delta }b} \right|}}{{{\Delta }b_{{{\text{inf}}}} }}} \right) ,$$

(8)

where $\left| {{\Delta }b} \right|$ is the absolute of column vector ${\Delta }b$, $\left| {{\Delta }b} \right|_{{{\text{inf}}}}$ is the maximum element of $\left| {{\Delta }b} \right|$, and $L$ is a diagonal matrix whose diagonal terms consist of $\frac{{\left| {{\Delta }b} \right|}}{{\left| {{\Delta }b} \right|_{inf} }}$. We apply this weighting matrix to Eq. (5) to obtain

$$A\left( m \right){\Delta }m = d,$$

(9)

where $A\left( m \right) = L*J\left( m \right)$ and $d = L*{\Delta }b$. During an iterative optimization process $A_{k} \left( {m_{k} } \right)\Delta x_{k} = d_{k} , k = 1,2, \ldots ,{\text{iter\_max}}$, we calculate the least-square solution ${\Delta }x_{k}$ for the kth iteration and update the model by

$$m_{k + 1} = m_{k} + {\Delta }m.$$

(10)

Upon obtaining the S-wave velocity model by surface wave inversion, the FMM (Kroon 2021) is introduced to calculate the S-wave travel time from the sensors to the blasting source and estimate t₀ of the blasting source by Eq. (1).

2.4 Adjoint-State Eikonal Equation Tomography in Deep Tunnels

The P-wave traveling time can be calculated based on the t₀ of the blasting source and the arrival time of the P-wave. The objective misfit function of P-wave traveltime tomography in a deep tunnel can be established as

$$E\left( c \right) = \frac{1}{2}\mathop \int \limits_{\partial \Omega }^{ } \left[ {T\left( {c,r} \right) - T^{{{\text{obs}}}} \left( r \right)} \right]^{2} {\text{d}}r$$

(11)

in which observed traveltimes T^obs(r) are picked from the test data and r represents the sensors at the designed monitoring acquisition surface $\partial$Ω. T(c,r) is the modeled traveltime obtained by solving the eikonal equation for model velocity c:

$$F\left( {t,c} \right) = \left[ {\nabla t\left( x \right)} \right] ^{2} - \frac{1}{{c^{2} \left( x \right)}} = 0,$$

(12)

where x is any point between the blasting sources and sensors Ω and where t ($x_{{{\text{soure}}}}$) is equal to zero. The eikonal equation can be solved by the FMM (Kroon 2021). In the classic approach, the objective function can be expanded at c₀ by employing the Gaussian expansion method:

$$E\left( c \right) \approx E\left( {c_{0} } \right) + \left[ {\frac{\partial E\left( c \right)}{{\partial c}}|_{{c = c_{0} }} } \right]^{T} \delta c + \frac{1}{2}\delta c^{T} \frac{{\partial^{2} E\left( c \right)}}{{\partial c\partial c^{T} }}|_{{c = c_{0} }} \delta c.$$

(13)

The gradient can be defined as the first derivative of the objective function relative to the velocity of medium c. The Hessian matrix is the second derivative, as shown in Eq. (13):

$$g = \frac{\partial E\left( c \right)}{{\partial c}},$$

(14)

$$H = \frac{{\partial^{2} E\left( c \right)}}{{\partial c\partial c^{T} }}.$$

(15)

To obtain the minimum of the objective function, we set g = 0 to obtain

$$H|_{c0} \delta c = - g|_{c0} ,$$

(16)

$$\delta c = - \left( {H|_{c0} } \right)^{ - 1} g|_{c0} .$$

(17)

The updated model can be written as

$$c_{n + 1} = c_{n} - \left( {H|_{cn} } \right)^{ - 1} g|_{cn} .$$

(18)

The updated model parameter c_n+1 is used as the initial model in the next iteration. However, it is difficult to compute the inverse of Hessian matrix $\user2{H }$ and Fréchet matrix g. The adjoint-state method is introduced to compute the gradient of the misfit function without introducing Fréchet derivatives. The Lagrangian formulation is added to the objective misfit function [Eq. (10)]. The extended objective function is

$$L\left( {c,t,\lambda } \right) = \frac{1}{2}\mathop \int \limits_{\partial \Omega }^{ } \left[ {t\left( r \right) - T^{obs} \left( r \right)} \right]^{2} {\text{d}}r - \frac{1}{2}\mathop \int \limits_{\Omega }^{ } \lambda \left( x \right)\left[ {\nabla t\left( x \right)} \right] ^{2} - \frac{1}{{c^{2} \left( x \right)}}{\text{d}}\Omega ,$$

(19)

where $t\left( r \right)$ is the solution of the eikonal equation at receiver x and where $\lambda \left( x \right)$ is the Lagrangian multiplier and is named the adjoint-state parameter, which can ensure that $t\left( r \right)$ equals $T^{{{\text{obs}}}}$ at the optimum. When searching for the minimum of the objective, the traveltime $t\left( r \right)$, Lagrangian multiplier $\lambda \left( x \right)$, and velocity $c\left( x \right)$ are independent. When the objective function reaches the minimum value, the variable satisfies Eqs. (20), (21) and (22):

$$\frac{\partial L}{{\partial \lambda }} = 0,$$

(20)

$$\frac{\partial L}{{\partial t}} = 0,$$

(21)

$$\frac{\partial L}{{\partial c}} = - \mathop \int \limits_{\Omega }^{ } \frac{\lambda }{{c^{3} \left( x \right)}}{\text{d}}x = \frac{\partial E}{{\partial c}}.$$

(22)

According to Eq. (21), we have

$$\frac{\partial L}{{\partial t}} = \mathop \int \limits_{\partial \Omega }^{ } \left[ {t\left( r \right) - T^{{{\text{obs}}}} \left( r \right)} \right]{\text{d}}r - \mathop \int \limits_{\Omega }^{ } \lambda \left( x \right)\nabla t\left( x \right)\frac{\partial \nabla t}{{\partial t}}{\text{d}}x = 0,$$

(23)

We apply the Gauss formula to the second term of Eq. (23):

$$\frac{\partial L}{{\partial t}} = \mathop \int \limits_{\partial \Omega }^{ } \left[ {t\left( r \right) - T^{{{\text{obs}}}} \left( r \right) - \lambda n\nabla t} \right]{\text{d}}r + \mathop \int \limits_{\Omega }^{ } \nabla \lambda \left( x \right) \cdot \nabla T\left( x \right){\text{d}}x = 0.$$

(24)

Two integral terms are set to zero in Eq. (23). Thus, ${\varvec{\lambda}}$ is the solution of

$$\nabla \cdot\left( { - {\varvec{\lambda}}\nabla {\varvec{T}}} \right) = 0\quad {\text{between}}\, {\text{the}}\, {\text{sources}} \;{\text{and}}\; {\text{sensors}}$$

(25)

with boundary condition

$$\left( {n \cdot \nabla T} \right)\lambda = T\left( r \right) - T^{{{\text{obs}}}} \left( r \right)\quad {\text{in}}\; {\text{the}} \;{\text{tunnels}}.$$

(26)

The FSM (Taillandier et al. 2009) is used to calculate $\lambda$ in Eq. (25) over the whole model with the boundary condition of Eq. (26). Thus, the gradient $\frac{\partial E}{{\partial c}}$ can be obtained by Eq. (22). The model update is as follows:

$$c_{n + 1} = c_{n} + \alpha \frac{\partial E}{{\partial c}},$$

(27)

where ${\varvec{\alpha}}$ is the step length. The inversion iteration is stopped when the objective function is less than a given threshold.

2.5 MS Source Location Using the Heterogeneous Velocity

The objective function of the MS source location in deep tunnels is established as Eq. (2) using the L2 norm double traveling time difference. When searching the location, the forward modeled traveling time is calculated by the FMM using the inverted heterogeneous P-wave velocity model. The simplex method is introduced as a nonlinear optimization technique to speed up the search for the minimums of the objective function. The simplex method creates an N + 1 dimensional simplex in an objective function value space to search for a minimum in an N-dimensional space. The lower value point replaces the high-value point by reflection, expansion, contraction and shrinkage (Li et al. 2014) until the minimum is reached.

Note that inverted heterogenous velocity models are 2D due to the tunnel construction's linear monitoring acquisition geometry. The starting time of the source is not cancelled in the double-difference objective function. Thus, a simplex geometry for a 2D space is a triangle. The workflow of the double-difference source location method based on the simplex method can be concluded as:

• a. Establish a double-difference source location objective function as Eq. (2). Three points P₁, P₂ and P₃ are selected as an initial simplex triangle based on the MS monitoring area.

• b. Calculate the three objective function values corresponding to three points P₁, P₂ and P₃ as y₁, y₂ and y₃ using the FMM of the current inverted heterogeneous velocity model. Define y_min, y_mid and y_max as the minimum and maximum of y₁, y₂ and y₃ and the corresponding minimum and maximum points as P_min, P_mid and P_max, respectively. The reflection, expanding, and contracting points and corresponding values are defined as P*, P**, P*** and y*, y**, y***, respectively.

• c. Start the searching iteration. The reflection point P* is searched, and the corresponding objective function value y* is calculated and compared to y_s and y_m. Then, a new simplex can be determined based on the principle of four transformations (i.e., reflection, expansion, contraction and shrinkage), as shown in Fig. 3. The formula and details of transformation can be found in Li et al. (2014).

• d. Repeat steps (b)–(c) until the misfits are lower than the threshold and the best fit location has been found.

• e. Repeat steps (a)–(d) when more blasting sources are added to the travel tomography to obtain a new heterogeneous P-wave velocity model.

Fig. 3

Workflow of double-difference simplex-based MS source location

3 Numerical Study

The numerical simulation of a realistic deep tunnel model illustrates the proposed inversion method of the heterogenous velocity model. We present wave field snapshots to feature the characteristics of P-wave, S-wave and surface wave propagation in tunnels. Blasting seismic data are used to invert the velocity model considering the effect of the EDZ. Multiple MS events around the drilling tunnels are located to evaluate the accuracy of the proposed MS source location method.

3.1 Numerical Modeling of the Blasting Source

Figure 4 shows the numerical model, including an existing tunnel and two new tunnels in construction that are vertical to the existing tunnel. A convolutional perfectly matched layer (CPML) boundary is assigned at the boundary of the computing domain, which can absorb the reflection from the computed boundary. Thus, the wave propagation simulation can be conducted in a defined domain. The acoustic–elastic boundary approach (AEA) (Xu et al. 2007) is applied to address the free (air) boundary. We implement the high-order FDTD to solve the viscoelastic wave equation. The generalized standard linear solid (GSLS) method (Bohlen 2002) considers the attenuation and dispersion of viscoelastic waves. The stress–velocity formulation of the viscoelastic waves based on GSLS are shown as

$$\rho \frac{{\partial v_{i} }}{\partial t} = \frac{{\partial \sigma_{ij} }}{{\partial x_{j} }} + f_{i} ,$$

(28)

$$\frac{{\partial \sigma_{ij} }}{\partial t} = \frac{{\partial v_{k} }}{{\partial x_{k} }}\left\{ {\pi \left( {1 + \tau^{p} } \right) - 2\mu \left( {1 + \tau^{s} } \right)} \right\} + 2\frac{{\partial v_{i} }}{{\partial x_{j} }}\mu \left( {1 + \tau^{s} } \right) + \mathop \sum \limits_{l = 1}^{L} r_{ijl} \quad {\text{if}}\quad i = j,$$

(29)

$$\frac{{\partial \sigma_{ij} }}{\partial t} = \left( {\frac{{\partial v_{k} }}{{\partial x_{k} }} + \frac{{\partial v_{k} }}{{\partial x_{k} }}} \right)\mu \left( {1 + \tau^{s} } \right) + \mathop \sum \limits_{l = 1}^{L} r_{ijl} \quad {\text{if}}\quad i \ne j,$$

(30)

where $\sigma_{ij}$ denotes the ijth component of the stress tensor, $v_{i}$ are the components of the particle velocities. $f_{i}$ are the components of the external body force. $\tau^{{\text{p}}}$ and $\tau^{{\text{s}}}$ are the levels of attenuation for P- and S-waves, respectively, which are related to the seismic quality factors Q_p and Q_s, respectively.

Fig. 4

Scheme of the MS source location numerical simulation model

The two new tunnels are alternately drilled, as shown in Fig. 4. The model measures 100 m in width × 50 m in depth with a 0.2 m spatial interval in both x- and y-directions. The starting points (x, z) of tunnels A and B (the middle point of the drilling face) are (15,0) and (85, 0), respectively. The MS monitoring network is installed in the middle of the two new tunnels, consisting of 8 sensors located at (35, 1), (40, 1), (40, 4), (45, 1), (55, 1), (60, 1), (60, 4) and (65, 1). The drilling and blasting sources are set at the middle axis of tunnel drilling every 2 m. The blasting source wavelet is a Ricker wavelet with a dominant frequency of 120 Hz. The background (surrounding rock) S-wave, P-wave velocity and density are 3500 km/s, 5250 km/s and 2.5 g/cm³, respectively. The S-wave velocity varies from 3500 m/s to 2500 m/s in the zone of 10 m around the tunnel (i.e., EDZ). The ratio of the P-wave and S-wave velocities is 1.5 in the EDZ zone. The total simulation time length is 50 ms with a sampling interval of 0.02 ms.

Snapshots of the first blasting source are shown in Fig. 5. The Ricker wavelet is transmitted at the middle axis of tunnel A as a blasting source. The P-wave and S-wave gradually separate due to the velocity difference. The surface wave (Rayleigh wave) traveling along the sidewall of the tunnel can be observed after t₀ + 13.5 ms. The propagation of the surface wave is slightly slower than that of the S-wave, since the velocity of the surface wave is typically 0.9 times that of the S-wave. The P–S converted wave, which can be seen in the snapshot of t₀ + 21.0 ms, is caused by the P-wave reflected at the sidewall of the tunnel. The S-wave travels with a higher velocity in the deeper area in the y-direction, because the S-wave velocity of intact rock is higher than that in the EDZ zone. The dispersion of the surface wave can be observed at t₀ + 13.5 ms. The surface wave with different frequencies travels along the sidewall with different velocities.

Fig. 5

Snapshots of the first blasting source at tunnel A

Figure 6 presents the vertical velocity component of four sensors at (35, 1), (45, 1), (55, 1) and (65, 1) and their corresponding frequency spectrum. Note that t₀ is an unknown parameter in most drilling and blasting tunneling cases. Thus, t₀ in the time axis in Figs. 5 and 6a is to be determined. The P-wave, S-wave and surface wave events can be identified based on the different slopes of the events. The arrival time of the P-wave and S-wave can be extracted as the blue and red lines in Fig. 6a. The accurate arrival time of the P-wave can be easily determined for every trace. However, the single arrival time of the S-wave of the no. 4 sensor can be identified directly. The overlap influences the identification of the arrival time of the S-wave of sensor nos. 1, 2 and 3. The dispersion of the surface wave can be observed from the more extended events in the farther offset recordings. The dominant frequency of recordings varies from 150 to 170 Hz (Fig. 6b), consistent with the actual blasting source signal in tunnels.

Fig. 6

Simulated recordings of four sensors at the (35, 1), (45, 1), (55, 1) and (65, 1) a wave profiles and b corresponding frequency spectra

3.2 Determining t ₀ of the Blasting Source

The phase-shift method (Park et al. 1998) is used to calculate the four recordings' dispersion curves. The dispersion curve calculated by four recordings is shown in Fig. 7a. The limited number of sensors causes spatial aliasing, as shown in Fig. 7a. The dispersion curve is calculated by 11 assuming sensors located from (25, 1) to (75, 1) with the same spacing, as shown in Fig. 7b. The dispersion curves of the fundamental mode in Fig. 7a, b are the same, and the dispersion curve energy is consistent with the theoretical solutions (white stars in Fig. 7). The comparison demonstrates that the surface wave modeling is correct, and the dispersion curve of the fundamental mode can be extracted by the four linear monitoring sensors.

Fig. 7

Dispersion curve of blasting source recordings. a Dispersion curve calculated by four recordings. b Dispersion curve calculated by the recordings of simulated sensors along the red dashed line in Fig. 5. White stars are the theoretical dispersion curve solutions

We use the proposed improved perturbational inversion met hod to invert the dispersion curve extracted from Fig. 7a. The process of the inversion is shown in Fig. 8. The initial S-wave velocity model varies from 2500 to 2700 m/s (the blue dashed line in Fig. 8c). The corresponding dispersion of the initial model is shown in Fig. 8a. A significant difference between the observed and initial dispersion curves can be seen. The inversion aims to narrow the misfits of the dispersion curve by updating the S-wave velocity model. The ratio of the P- and S-wave velocities is 1.5, and the density is constant during the whole inversion. The misfits rapidly decrease, as shown in Fig. 8b. After the 120th iteration, the misfit curve converges to a 0.14 normalized misfit and stops updating. The inverted S-wave presents a low-velocity zone in the depth range from the surface to 10 m, which is consistent with the preset true S-wave velocity model. Meanwhile, the dispersion curve of the final results fits the observation well.

Fig. 8

Inversion process of the dispersion curve using the improved perturbational inversion method: a dispersion curve comparison at the 120th iteration; b misfit evolution; and c S-wave velocity comparison at the 120th iteration

The inverted S-wave layered model is established in Fig. 9a. The contour lines calculated by the FMM in Fig. 9a are the isochrones, which are curves of equal travel time. The ray tracing of backpropagation from the sensor to the source is calculated by searching the normal direction of every contour line. The ray tracing results show that the wave propagation path is significantly different between the inverted S-wave and homogenous models. The EDZ zone of low velocity deepens the curve of wave path travel, while the wave path in the homogeneous model is a line. The traveling time of the S-wave can be obtained from the contour lines, as shown in Table 1. The travel times from the four sensors to the source are calculated based on actual, inverted, and homogeneous models. The travel times t_b calculated by the inverted and actual models are similar. However, the traveling time t_b calculated by the homogeneous model is much shorter due to the lack of consideration of the EDZ. The parameter t₀ is calculated according to the S-wave arrival time and Eq. (2). As shown in Table 1, the error of t0 identified by the inverted S-wave model is approximately 0.2 ms, much less than the error (2–3 ms) calculated by the homogenous model. Thus, t₀ estimated by the inverted S-wave model can be regarded as the trigger time of the blasting source.

Fig. 9

Backpropagation of the S-wave from sensor no. 4 to the first blasting source at tunnel A: a ray tracing based on the inverted S-wave velocity model and b ray tracing based on the homogeneous S-wave velocity model

Table 1 Calculation of the drilling blast source t₀ of the numerical study

3.3 P-Wave Traveling Time Tomography While Drilling

The P-wave traveling time tomography can be conducted after the blasting source trigger time t₀. The absolute traveling time of the P-wave can be calculated by the blasting source t₀ and selected arrival time of the P-wave (e.g., the blue line in Fig. 6a). The simulated blasting source is set in the middle lines of the two tunnels with a 2 m interval. Eight sensors between the two tunnels are used to obtain the travel time and invert the P-wave velocity. The results of P-wave traveling time tomography during drilling are shown in Fig. 10. Figure 10a presents the true P-wave velocity model, which overlays the low-velocity zone (i.e., the EDZ) of the existing tunnel and two new tunnels. The surrounding rock away from the tunnel sidewall has a higher P-wave velocity. The initial model is estimated by the inverted S-wave velocity model. Four traveling time tomography are conducted by the proposed adjoint-state method when the tunnels are drilled to 5 m, 10 m, 14 m, and 18 m, respectively.

Fig. 10

Results of P-wave traveling time tomography during drilling: a true model; b initial model; c–f P-wave tomography when two tunnels are drilled to depths of 5 m, 10 m, 14 m and 18 m, respectively

The inverted P-wave velocity model gradually updates to the true model, as shown in Fig. 10c–f. In the first drilling step (2 m in depth, Fig. 10c), the major update of the velocity model is at the near field of the existing tunnel, which the ray path of the first three blasting waves mainly covers. The inverted P-wave models (Fig. 10c–f) are almost symmetric due to the symmetry monitoring setup and true velocity model. The velocity model in Fig. 10d has a similar distribution, but the velocity value is still visibly different from the true model. The inverted models of steps 3 and 4 (Fig. 10e, f) are highly consistent with the true model. The details of the velocity comparison are shown in Fig. 11. The velocity logs of the inverted, true and initial models at x = 20, 30, 40 and 50 m demonstrate that the P-wave traveling time tomography effectively obtains reasonable heterogeneous models while drilling. In general, the inverted P-wave velocity is closer to the true value with the addition of blasting sources. The middle of the monitoring area (around x = 50, y > 12 m) presents a more significant error due to the lack of ray path travel. However, those misfits do not affect the accuracy of the location for the MS source, since the wave of the MS event around the tunnel also does not travel through that area to the sensors.

Fig. 11

Inverted P-wave velocity comparison: a x = 20 m, b x = 30 m, c x = 40 m, and d x = 50 m

3.4 MS Source Location

Sixty-four (8 × 8) MS sources that are uniformly distributed in the red square in Fig. 10a (ranging from 5 to 19 m on the X-axis and 8–22 m on the Y-axis) are used to test the location accuracy using different velocity models. The forward-simulated arrival time is calculated using the eikonal equation based on the FMM. The location results for all the MS sources can be found in Appendix Tab le 4. The location errors using different homogenous velocity models and inverted velocity models of different steps are shown in Fig. 12. Figure 12a shows the location error using the true model, which means that there is no error in the location velocity model. The location error in Fig. 12 is small and consistent, which demonstrates the accuracy of the double-difference simplex location method. Thus, when we conduct location using the inverted and homogenous velocity models, the errors are generally caused by the velocity model difference.

Fig. 12

Location errors of the MS source within the red dashed square in Fig. 10a using different P-wave velocity models. a True model. b Homogeneous model of 4500 m/s. c Homogeneous model of 5000 m/s. d Homogeneous model of 5500 m/s. e Inverted model of step 1 (Fig. 10c). f Inverted model of step 2 (Fig. 10d). g Inverted model of step 3 (Fig. 10e). h Inverted model of step 4 (Fig. 10f)

The location results of the homogenous velocity (4500, 5000, 5500 m/s) models present significant errors (Fig. 12b, c, d). The location accuracy is primarily affected by the velocity difference between the homogeneous and true velocity models, especially the velocity difference in the EDZ zone. The location at a depth of 9 m using a model velocity of 5500 m/s presents a smaller error, because 5500 m/s is close to the average velocity at that depth. However, the location error dramatically increases at other depths due to the heterogeneity of the true model velocity.

Note that the location accuracy improves using updating inverted velocity models (Fig. 12e–h). The location presents a high accuracy around the depth of the current drilling face. For example, the tomography for the traveling time uses blasting sources with drilling depths of less than 5 m, 10 m, 14 m and 18 m. Thus, the location using four inverted velocities fits the true location of MS events near the drilling face well (referring to 5 m, 10 m, 14 m and 18 m in the Y-axis in Fig. 12e–h, respectively). Most location errors in those areas are less than 2 m. A larger error can be found, where the ray path density is lower in the traveling time tomography. Compared to the location results using the homogenous velocity model, the location accuracy is much higher and continually improves with more drilling blasting sources added to the P-wave velocity tomography.

4 Case Study

MS monitoring of two new tunnels with a buried depth of 2400 m in China Jinping Underground Laboratory Phase II (CJPL-II) was conducted to warn of rockburst disasters. We inverted the heterogeneous P-wave velocity model of the monitoring area based on drilling and blasting source data analysis. Then, the blasting sources with a known location were used to test the proposed location method.

4.1 Project Background and MS Monitoring Setup

CJPL-II is one of the deepest underground laboratories on the Yalong River in Sichuan Province, China (Li et al. 2021; Zheng et al. 2021). The CJPL-II has a maximum burial depth of 2525 m. The surrounding rock of the tunnel and laboratories mainly consists of marble with high strength and brittleness. The maximum principal stress of the surrounding rock is 69.20 MPa (Jiang et al. 2021). Based on several previous studies, rockbursts frequently occur during the excavation of tunnels and laboratories (Feng et al. 2015b). The layout of the 8 caverns of CJPL-II and their connection to the traffic tunnels are shown in Fig. 13a. Two new tunnels were proposed at the no. 2 auxiliary tunnel, as shown in the dashed square. The two tunnels were constructed at the same time using the drilling and blasting method. Thus, the monitoring network was installed in the middle of two tunnels, so that the MS events occurring around the drilling face of two tunnels could be monitored using one equipment system. A SinoSeism (SSS) MS source monitoring system (Chen et al. 2021) with a 32-bit A/D acquisition apparatus was deployed in this case. Eight sensor monitoring networks were designed, as shown in Fig. 13c. The nearest sensor was 20 m from the tunnel, where the blasting wave would not damage the sensors. The inversion of the velocity model and location of the MS source were simplified to a 2D issue due to the plane-like monitoring network and objective area.

Fig. 13

Project background of deep buried tunnels in Jinping China and the MS monitoring setup of two new tunnels: a layout of CJPL-II, b location of Jinping China, and c MS monitoring setup

4.2 Inversion of the Velocity Model

Blasting signals were collected to invert velocity models of the monitoring area. The locations of the 14 blasting sources (7 in tunnel A and 7 in tunnel B) are summarized in Table 2. The geological conditions of the drilling faces during construction are shown in Fig. 14. Most tunnel drilling faces were grayish-white fine crystal marble. The rock mass was blocky with general integrity. Part of the rock mass was rusty yellow due to ferromanganese filling. The rock fracture was dry, fresh and not weathered. There was no noticeable sound when taking photos. The MS monitoring results suggested a minor rockburst risk at tunnel B 0 + 4.5 and a low rockburst risk at other stages. The drilling face at tunnel B 0 + 4.5 was sprayed, anchored and grouted in the actual project. There was no rock burst during the whole construction process. In this study, four sensors with the same height (S1, S4, S5 and S8 in Fig. 13c) were used for surface wave dispersion analysis, and all eight sensors were considered during the P-wave velocity tomography.

Table 2 Calculation of t₀ of the different drilling blast sources in the field case

Fig. 14

Drilling face conditions after blasting of two new tunnels

4.2.1 Data Processing of the Blasting Signal

The signal processing of the first blasting source signal in tunnels A and B (A1 and B1 in Table 2) is shown in Fig. 15. The original signals in Fig. 15a, e show prominent low-frequency noise in traces S1, S4 and S8, which is caused by the electric interference of the sensors. An FIR bandpass frequency filter is applied to remove the low-frequency noise. Multiple pulses can be observed from the filtered waveform profile in Fig. 15b, f, which indicates that multiple blast sources were transmitted independently and can be analyzed separately. The time slice of the first pulse was extracted, as shown in Fig. 15c, g, to investigate waveform characteristics. The first arrival of the P-wave in all the traces is clearly identified. The P-wave and S-wave gradually separate as the offset increases. Thus, the first arrival of the S-wave is difficult to determine in several near offset traces. However, in the further offset trace (S1–S4 in the tunnel A data set, S5–S8 in the tunnel B data set), it is easy to identify the first break of the S-wave arrival. Note that the same horizontal position pairs of S2 and S3 and S6 and S7 have highly similar waveform and arrival time information, which verifies that the 2D assumption is applicable in the inversion and location.

Fig. 15

Blasting signal processing of the first source in tunnels A and B (A1 and B1 in Table 2). a Original signal of blasting source A1. b Processed signal of blasting source A1. c Time slice of the first pulse in a. d Time slice of channels S1, S4, S5, and S8. e Original signal of blasting source B1. f Processed signal of blasting source B1 in a. g Time slice of the first pulse in e. h Time slice of channels S1, S4, S5, and S8 in e

4.2.2 Dispersion Analysis

The dispersion curves are extracted from Fig. 15d, h using the tunnel A and B blasting source data. The two dispersion energy distributions in Fig. 16a, b present similar sharpness, because they reflect the same velocity–depth curve. The dispersion phenomenon is shown in Fig. 16, which indicates a velocity change with the depth of the surrounding rock. The two dispersion curves suggest a lower velocity at higher frequency and a high-velocity area at 175 Hz and 50 Hz.

Fig. 16

Dispersion curve calculated by recordings of S1, S4, S5 and S8. a Dispersion curve calculated from blasting tunnel data of tunnel A. b Dispersion curve calculated from blasting source data of tunnel B

The proposed improved perturbational inversion method applies the average of two dispersion curves to invert the S-wave velocity model. Gaussian smooth and cubic spline interpolation are applied to preprocess the dispersion curve. The initial model for inversion linearly varies from 3200 to 3500 m/s (the blue dashed line in Fig. 17c). The misfits between the observed and simulated dispersion curves decrease rapidly and converge at 0.11 normalized misfits after the 100th iteration (Fig. 17a, b). The inverted S-wave presents a low-velocity zone in the depth range from the surface to 10 m, which is consistent with the preset true S-wave velocity model. The inverted S-wave velocity–depth curve is shown in Fig. 17c. The inverted results suggest a 2.8 m EDZ zone of the surrounding rock. Intact rock of high S-wave velocity can be found at approximately 9 m. Note that a lower velocity area indicates broken rock or developed discontinuities.

Fig. 17

Inversion process of the dispersion curve of field data: a dispersion curve comparison at the 100th iteration, b misfit evolution, and c S-wave velocity comparison at the 100th iteration

Ray path tracing of wave backpropagation from the S1 sensor to the sources is used to estimate t₀ of the blasting sources in tunnel A. The FMM calculates the traveling time and their isochrones, as shown in Fig. 18. The wave propagation path on the heterogeneous velocity model can be traced by searching the normal direction of the isochrones. The arrival time of the S-wave and traveling time of the S1 and S8 sensors are used to calculate t₀ of the blasting sources in tunnels A and B by Eq. (1), respectively. The results are presented in Table 2.

Fig. 18

Determining t₀ of blasting source A1 by backpropagation of the S-wave from S1 sensors

4.2.3 P-Wave Traveling Time Tomography

The calculated t₀ of the blasting sources was used to obtain the P-wave traveling time. The P-wave traveling time of every sensor and source is summarized in Table 3. The initial model of P-wave traveling time tomography is shown in Fig. 19a, which linearly varies from 5000 to 5500 m/s in the depth range from 0 to 10 m. The inversion is divided into four steps using different blasting sources. Step one uses blasting sources A1, A2, B1, and B2. Step two uses blasting sources A1–A3 and B1–B3. Step three uses blasting sources A1–A5 and B1–B5. Step four uses blasting sources A1–A7 and B1–B7. The inverted P-wave velocity models of different steps are shown in Fig. 19b–e. All the inverted velocity models can identify the EDZ. A higher velocity area can be observed in the depth range from 5 to 10 m, which suggests an intact rock layer. The inverted velocity models of steps 3 and 4 indicate a lower velocity area at depths from 14 to 20 m. The velocity models in Fig. 19c–e consistently suggest that the velocity of the surrounding rock near tunnel A is lower than that near tunnel B.

Table 3 P-wave traveling time of 14 blasting sources in the field case

Fig. 19

Results of P-wave traveling time tomography during drilling: a initial model. b Tomography results from blasting sources A1, A2, B1, and B2. c Tomography results from blasting sources A1–A3 and B1–B3. d Tomography results from blasting sources A1–A5 and B1–B5. e Tomography results from blasting sources A1–A7 and B1–B7

4.3 Validation by the Blasting Source Location

No significant rockburst occurred during the tunnel’s excavation, although many MS events were monitored. Thus, the 14 blasting source data with known locations are applied to validate the proposed location method. Four velocity models (Fig. 19c–e) obtained at different drilling steps are used to locate the blasting source. The location results of 14 blasting sources using different velocity models can be found in Appendix Tab le 5. The location errors are concluded in Fig. 20. Most location errors are less than 5 m. The location errors of the deeper drilling sources are lower. The location result using inverted model 1 (Fig. 19c) is shown by the blue line in Fig. 20. The location errors are stable during the whole drilling and less than 4 m. The location using inverted model 2 (Fig. 19d) presents better location accuracy at a depth of 9 m, where the blasting source (9 m and 7 m in depth of tunnels A and B) is newly added to the tomography of model 2. A similar phenomenon can be found in the depth range from 15 to 18 m of inverted model 3 (gray line in Fig. 20) and depth range from 18 to 20 m of inverted model 4 (black line in Fig. 20). When a blasting source is newly added to the tomography process, locating the source and its adjacent space has better accuracy (error less than 2 m).

Fig. 20

Location accuracy test of known blasting sources using different inverted velocity models. a Location error of the blasting source in Tunnel A. b Location error of the blasting source in Tunnel B

5 Discussion

Both numerical and field experiments suggest that the location method based on updating the inverted heterogeneous velocity significantly improved the location accuracy. However, the basic assumption, the reason for location error, and the method's potential application need to be pointed out. The aspects of the proposed location method that merit further discussion include the following:

• 1. Location error analysis

  The factors that affect the location accuracy using the proposed method can be generally divided into two parts: velocity inversion and source location.

During the inversion of the velocity model, the estimation of t₀ is the priority of P-wave traveling tomography. The proposed method uses dispersion analysis to invert an S-wave model and then backtraces the S-wave propagation from the sensors to the source to estimate the t₀ of the blasting source. The basic idea of the proposed method is that both the S-wave and the P-wave have the same trigger time. Moreover, t₀ is not mandatory for dispersion inversion of the surface wave. The estimation error of t₀ is mainly caused by the error of the S-wave velocity inverted by dispersion analysis. The basic assumption of surface wave dispersion inversion is the horizontal layer assumption (Park et al. 1998). Although the S-wave velocity of the EDZ is layered, which is consistent with the assumption of surface wave inversion, the range of the EDZ fluctuates in the axial tunnel direction. If the depth of the EDZ zone changes dramatically, the error in the estimated t₀ increases. A trigger instrument is required to directly record the t₀ blasting source in complex geological conditions, which can completely remove the error in the estimated t_0. The other factor that affects the accuracy of the inverted velocity model is the ray path density of the P-wave in the tomography, which is determined by the limited number of sources and sensors. The gradient mainly updates the velocity model, where the ray path density is high. Due to the acquisition geometry, some areas with no ray path are not updated during the tomography. A smooth Gaussian filter can be applied to mitigate the unbalanced updates for velocity. A study on the weight function of ray path density is meaningful to improve the traveling time tomography algorithm. Meanwhile, significant anisotropy of seismic waves (King and Talebi 2007; Gajek and Malinowski 2021) caused by complicated lithology, structure and stress state also may affect the location error, which is not considered in the proposed velocity inversion and location method.

During the location of the MS source, both the ill-posed monitoring network and 2D simplification affect the location accuracy. The layout of sensors cannot cover the monitoring area due to the limitation of tunnel construction. The MS events occur outside the array of sensors. The double-difference traveling time objective function and the simplex method improve the location convergence speed. However, the linear-like layout of sensors is still sensitive to the MS location source on the same line of the sensor array (e.g., the location results of blasting sources A1 and B1 in Appendix Tab le 5 and Fig. 20). The location of the MS source is simplified as a 2D problem, because the heterogeneous P- and S-wave velocity models are inverted in 2D using current acquisition geometry. This 2D simplification might induce extra errors. The 3D layout of monitoring sensors can be deployed in the future to invert a 3D heterogeneous velocity model, so that the MS source can be located in a 3D coordinate system.

• 2. Identifying the first arrival of the P-wave and S-wave

  Accurate identification of the first-arrival time of the P-wave and S-wave is critical to traveling time tomography and MS source location. From both numerical and field experimental waveforms in Figs. 6 and 15, it is not difficult to identify the first break of the P-wave. However, the overlap of the P-wave and S-wave challenges clear identification of the arrival of the S-wave for the near-field sensor. Arrival identification based on the instantaneous phase difference between the P- and S-waves can be explored in further studies to mitigate the picking error for both P- and S-wave arrivals.

In this study, only one S-wave arrival time needs to be extracted to estimate the t₀ of the blasting time. Thus, we recommend using the S-wave recorded by the sensor with the farthest offset, where the S-wave and P-wave completely separate, as shown in Figs. 6a, 15c, g. All the P-wave arrival times are suggested to be applied to ensure a sufficient ray path during the P-wave tomography. We use the P-wave arrival of MS sources to calculate their location. Compared to the layered S-wave velocity model, the continuous updating and heterogeneous P-wave velocity model have a higher resolution of the geological structure and can visualize the change in velocity distribution during tunnel excavation.

• 3. Selection of inverted velocity models

  Inverted velocity models of multiple stages can be obtained during tunnel excavation. We test the location accuracy using different inverted velocity models in both numerical and field experimental studies. The detailed location results and their errors are shown in Appendix Tab les 4 and 5 and Figs. 12 and 19. Based on the previous knowledge, the location accuracy should be consistently improved using the velocity models inverted by the most blasting sources. However, the location results show that the best resolution is in the depth range, where the blasting source of the corresponding depth is newly added to the tomography. Taking the location using model 4 in the field case (black dot in Fig. 19) as an example, the location of the A7 and B7 blasting sources is the most accurate. One reason is that the monitoring network is more robust for the location of the deeper source. More critically, gradients are mainly updated to the velocity model between newly added sources and sensors during tomography, which mitigates the location error of adjacent areas of these sources. Thus, we suggest selecting the inverted velocity model depending on the monitoring area.

  • 4. Potential application in linear excavated tunnels

    The numerical and field experimental study is based on two new tunnels intersected by an existing tunnel. The EDZ is deeply developed, because the existing tunnel has been constructed for a long time. The EDZ primarily affects the velocity distribution of the surrounding rock in the field case. The proposed method shows its advantages of location accuracy and estimation of EDZ in the intersected tunnel construction. The proposed velocity inversion and location method is also applicable to regular linear excavated tunnels and is an ongoing research topic. The estimation of t₀ of the blasting time is difficult, since the array sensors are much farther from the sources in the linear excavated tunnel scenario. A single inverted S-wave velocity curve is not suitable to be extended to a layered velocity model in the monitoring area. However, a trigger instrument on the blasting source can be deployed to measure t₀ directly. For P-wave traveling time tomography, a moving array of sensors provides an even better acquisition geometry than the case reported in this study. The proposed location algorithm can be easily extended to a 3D version using 3D FMM and simplex code if the velocity model can be inverted using a 3D monitoring network.

6 Conclusion

In this paper, we developed an inverted heterogeneous velocity model for microseismic source location in deep buried tunnels. An improved perturbational dispersion inversion of surface wave and adjoint-state traveling time tomography method using blasting data was provided to obtain a heterogenous P-wave velocity model. A double-difference simplex-based method was introduced for MS source location. A numerical study and a field case were conducted to test the accuracy of the proposed location method. The following conclusions can be drawn:

• 1. The EDZ significantly changes the wave propagation path in deeply buried tunnels. It is necessary to invert the velocity model of the EDZ before MS location. The drilling and blasting source data can be used for seismic exploration. The dispersion of surface waves traveling along the sidewall of the tunnel makes it possible to invert the S-wave velocity of the EDZ. An improved perturbational dispersion inversion method is developed by combining modified TLM forward simulation and perturbational inversion. The inversion method can estimate the S-wave velocity model and the EDZ depth.

• 2. P-wave first-arrival tomography of the blasting source can be used to obtain a heterogeneous P-wave velocity model. The starting time t₀ of blasting sources is estimated using S-wave back propagation tracing by the FMM on the inverted S-wave model. Adjoint-state first-arrival tomography was provided in this paper. The FMM solves the eikonal equation and forward-simulates travel time. The adjoint-state method is introduced to compute the gradient of the misfit function without introducing Fréchet derivatives during inversion. Updating velocity models are inverted when new blasting source data are added to tomography.

• 3. An objective function of MS source location is provided using the L2 norm of the double difference of the traveling time. The FMM is deployed to forward-simulate the traveling time of the inverted heterogeneous P-wave velocity model and calculate the double-difference traveling time in the objective function. The double-difference function removes the parameter t₀ of the MS source and improves the stability of the location method. The simplex method is introduced as a nonlinear optimization technique to search for the minimum misfit of the objective function and the best-fit location.

• 4. A numerical study based on a realistic tunnel model was conducted to test the location accuracy of the proposed method. The surface wave dispersion inverts a layered S-wave model and successfully estimates t₀ of the blasting source by the FMM. P-wave first-arrival time tomography imaging is used to provide more detailed heterogeneous velocity models. Sixty-four MS sources around the tunnel drilling face are simulated to test the accuracy of the proposed location method. The results show that location by the inverted heterogenous velocity model significantly improves the location accuracy compared to that of the homogeneous velocity model.

• 5. A field experiment was conducted at two new tunnels with a buried depth of 2400 m in China Jinping Underground Laboratory Phase II, one of the deepest underground laboratories. We continually inverted the heterogeneous P-wave velocity model of the monitoring area during tunnel drilling based on the analysis of the blasting source data. Blasting sources with a known location were used to test the proposed location method. The results showed that most location errors were less than 5 m using inverted velocity models. The location error can be less than 2 m if the updated inverted velocity model is appropriately selected.

Appendix

See Tables

Table 4 Location results of the MS source using different velocity model in the numerical study

4 and

Table 5 Location of the blasting source in the field case

5.