| Citation: |
Songqin Nong, Chao Huang, Liangguo Dong (2018). Source-independent wave-equation based microseismic source location using traveltime inversion. Earthq Sci 31(2): 100-109. DOI: 10.29382/eqs-2018-0100-7
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Microseismic monitoring is now the primary technique for checking the effectiveness of hydraulic stimulation of petroleum reservoirs (Maxwell et al., 2010). Mostly, the main task for microseismic monitoring is to determine accurate locations of microseismic events (Maxwell et al., 2001). For hydraulic fracturing, the distribution of microseismic events is used to estimate the fracturing volume which helps to optimize the hydraulic fracturing operation. Also, with such information, engineers can start their standard workflows to do the interpretation (Maxwell, 2014).
Most traditional methods are traveltime inversion with phase identification and arrival-time picking (Geiger, 1912; Gibowicz and Kijko, 2013). But the strong background noise in surface microseismic data makes it difficult and challenging to pick arrival-time. Most traveltime inversion methods use ray tracing to compute both the traveltime and Fréchet derivative. However, it is subject to a high-frequency assumption about the seismic wave propagation and can therefore fail when the velocity variations are characterized by the same wavelength as the source wavelet (Luo and Schuster, 1991). Meanwhile, most of the traveltime inversion methods are based on ray theory, which ignores scattering and finite-frequency effects (Dahlen et al., 2000; Liu et al., 2009; Tape et al., 2007). Numerical solution of the wave equation overcomes the high-frequency approximation in traveltime inversion and it is a promising way to take the influence of finite-frequency effects and wave scattering into account (Font et al., 2004; Zhang et al., 2003).
Alternative methods without travel-time picking and taking into account the influence of scattering and finite frequency effects are migration-based and full wave inversion (FWI) based source location methods. In migration-based methods, the source-receiver geometry highly affects location results (McMechan, 1982; Gajewski and Tessmer, 2005; Kochnev et al., 2007; Artman et al., 2010; Eisner et al., 2010; Feroz and Baan, 2013; Eyre and Baan, 2015). And it remains a challenging task when the signal-noise ratio (SNR) is low. In FWI-based methods, they can examine the influence of the finite-frequency effects and the interactions between the seismic wave and complex velocity environments (Font et al., 2004; Liu et al., 2004; Brossier et al., 2009; Tong et al., 2016). But for waveform based inversion, the misfit function is highly nonlinear with respect to source location perturbations, and the inversion may easily suffer from cycle-skipping problem, the source signature and origin time are still required during inversion, which is challenging to be estimated when dealing with field data.
To inherit the best characteristics of traveltime inversion (quasi-linear misfit function and robust convergence properties), but take finite-frequency effects into consideration, Zhao (2016) and Nong et al. (2017) proposed wave-equation based traveltime inversion method for microseismic source location. However, this method is still affected by the accuracy of the estimated source signature and origin time. To overcome this problem, we further modify this method by introducing a source-independent strategy to eliminate the effects of the source signature and the origin time and finally present a wave-equation based source-independent travel time inversion method to locate the microseismic source.
In this paper, a source signature and origin time independent source location method is presented and new Fréchet derivatives of the traveltime-residuals based misfit function with respect to location parameters are derived, then an improved scattering-integral (SI) approach (Liu et al., 2015; Schumacher et al., 2016) is applied to perform the inversion to increase the computational efficiency. Finally, some numerical tests are presented to demonstrate the validity of the proposed method.
A standard centroid moment tensor (CMT) solution comprises moment tensor M, centroid location xs, source origin time t0, and source signature
For acoustic wave-equation, the synthetic date u and the observed data d can be expressed in time domain as
| u(xr,t)=scal(t+tu)∗Gcal(xr,t;xs), | (1) |
and
| d(xr,t)=sobs(t+td)∗Gobs(xr,t;xs). | (2) |
where
To eliminate the effects of the source signature, we construct new observed and synthetic data by convolving observed data and modeled data with the respective reference trace data
| u′(xr,t)=u(xr,t)∗d(xref,t)=scal(t+tu)∗Gcal(xr,t;xs)∗sobs(t+td)∗Gobs(xref,t;xs), | (3) |
and
| d′(xr,t)=d(xr,t)∗u(xref,t)=sobs(t+td)∗Gobs(xr,t;xs)∗scal(t+tu)∗Gcal(xref,t;xs). | (4) |
Obviously, the term
| u′(xr,ω)=u(xr,ω)d(xref,ω)=Gcal(ω)Gobs(xref,ω;xs)s(ω)eiω(tu+td), | (5) |
and
| d′(xr,ω)=d(xr,ω)u(xref,ω)=Gobs(ω)Gcal(xref,ω;xs)s(ω)eiω(tu+td), | (6) |
where
After modification, cross-correlation function between the new observed and synthetic data is defined as
| f(xr,τ,xs)=∫dtd′(xr,t+τ;xs)u′(xr,t;xs), | (7) |
where τ is the shift time between the modified synthetic and real seismograms. For each seismic trace, we seek a
| ∙fΔτ=[∂f(xr,τ;xs)∂τ]τ=Δτ=∫dt∙d′(xr,t+Δτ;xs)u′(xr,t;xs)=0, | (8) |
unless its maximum is at an end point
Equation (8) is the connective function which will be used to compute the Fréchet derivative. In frequency domain, it comes to
| F(xr,τ,xs)=∫dωd′(xr,ω;xs)¯u′(xr,ω;xs)eiωτ, | (9) |
where
| ∙FΔτ=[∂F(xr,τ;xs)∂τ]τ=Δτ=∫dωd′(xr,ω;xs)¯u′(xr,ω;xs)eiωΔτiω=0. | (10) |
The presented method attempts to determine accurate source location by minimizing the shift time for all the trace and define the objective function as
| S=12∑s∑rΔτ(xr,xs)2. | (11) |
The inverse problem is usually solved by a local optimization method. The increment
| xi+1s=xis+ai+1pi+1(i=0,1,2,...,n), | (12) |
where n is the number of iterations. The direction p is calculated from the misfit function gradient with respect to the source location parameters,
| p=−∂S∂xs=−∑s∑r∂(Δτ)∂xsΔτ(xr,xs). | (13) |
Using connective function and the rule for an implicit function derivative, the Fréchet derivative in frequency domain can be expressed as
| ∂(Δτ)∂xs=−[∂(∙fΔτ)∂(xs)][∂(∙fΔτ)∂(Δτ)]=−∫dωeiωΔτiω[∂d′(xr,ω;xs)∂xs¯u′(xr,ω;xs)+d′(xr,ω;xs)¯∂u′(xr,ω;xs)∂xs]∫dωd′(xr,ω;xs)¯u′(xr,ω;xs)eiωΔτ(iω)2, | (14) |
where
| u′(xr,ω;xs)=u(xr,ω;xs)d(xref,ω;xs), | (15) |
and
| d′(xr,ω;xs)=d(xr,ω;xs)u(xref,ω;xs). | (16) |
Substitution of equations (15) and (16) into equation (14) gives
| ∂(Δτ)∂xs=∫dωd(xr,ω;xs)¯d(xref,ω;xs)eiωΔτiω[∂u(xref,ω;xs)∂xs¯u(xr,ω;xs)+u(xref,ω;xs)∂¯u(xr,ω;xs)∂xs]∫dωd(xr,ω;xs)u(xref,ω;xs)¯u(xr,ω;xs)d(xref,ω;xs)ω2eiωΔτ. | (17) |
The derivative of the pressure field
| ∂u(xr,ω;xs)∂xs=[∇∇2u(x,xs,ω)+ω2v2(x)∇u(x,xs,ω)]G(x,xr,ω)−2ω2∇v(x)v3(x)u(x,xs,ω)G(x,xr,ω). | (18) |
Substitution of equation (18) into equation (17) gives
| k(x,ω|xr,xs)=∂(Δτ)∂xs=k1(x,ω|xr,xs)+k2(x,ω|xr,xs), | (19) |
where
| k1(x,ω|xr,xs)=∫dωd(xr,ω;xs)¯d(xref,ω;xs)eiωΔτiω(A1+A2)∫dωd(xr,ω;xs)u(xref,ω;xs)¯u(xr,ω;xs)d(xref,ω;xs)ω2eiωΔτ, | (20) |
where
| A1=¯u(xr,ω;xs)[∇∇2u(x,xs,ω)+ω2v2(x)∇u(x,xs,ω)]G(x,xref,ω),A2=u(xref,ω;xs)¯[∇∇2u(x,xs,ω)+ω2v2(x)∇u(x,xs,ω)]G(x,xr,ω), | (21) |
and
| k2(x,ω|xr,xs)=−∫dωdωd(xr,ω;xs)¯d(xref,ω;xs)eiωΔτiω(B1+B2)∫dωd(xr,ω;xs)u(xref,ω;xs)¯u(xr,ω;xs)d(xref,ω;xs)ω2eiωΔτ, | (22) |
where
| B1=¯u(xr,ω;xs)2ω2∇v(x)v3(x)u(x,xs,ω)G(x,xref,ω),B2=u(xref,ω;xs)¯2ω2∇v(x)v3(x)u(x,xs,ω)G(x,xr,ω). | (23) |
In the expression of the Fréchetderivative, k1 contains the spatial variation of the wavefield, which mainly accounts for the influence of the geometrical perturbation of the source position; and k2 mainly accounts for the effects of the spatial variation of velocity around the source. If the velocity model is complex, it is necessary to take k2 into consideration for source location (Huang et al., 2017).
The inversion for source location may suffer from a cycle skipping if the initial source location is too far from its true position. Migration-based or ray-based methods maybe used for the initial estimate of the source location, but application of these methods is not always efficient and flexible because of the time-consuming procedure or data preprocessing. In this paper, at the first iteration of source location procedure an arbitrary point is selected as the simulated source location in the synthetic record, then correlation is applied to find the space and time shifts that maximize the correlation function in equations (24) and (25). These shifts are used to move the estimated source from its current position to a new position, which then becomes the starting source position for the subsequent method.
| f(Δh)=max | (24) |
| f(\Delta \tau) = \max \left({\sum\limits_{\rm r} {\int {u({x_{\rm r}}, t + \Delta \tau, x_{\rm s}^0} })d({x_{\rm r}}, t, x_{\rm s}^{}){\rm d}t|\Delta \tau \in [ - T, T]} \right), | (25) |
where u and d are the synthetic and observed data,
After defining the misfit function and calculating the Fréchet derivative, we can perform the inversion using either the adjoint-state or the improved SI method. However, a self-adjoint operator is required in the adjoint-state method, which is not always satisfied. On the contrary, the improved SI method can calculate the approximate Hessian matrix efficiently. Besides, the inversion does not update the velocity, so the Green’s functions at the same receiver are identical in every iteration, which can be stored in advance, avoiding repeating calculations. It will reduce the computation cost of inversion by almost half of that the adjoint-state method. Therefore, the improved SI method is adopted in this study.
To calculate the direction p in equation (13) without storing the whole Fréchet kernel k in equation (19), a vector operator can be applied as Liu et al. (2015) did. In equation (19), once the
| {p} = \left( {\begin{array}{*{20}{c}}{{p_1}}\\ \vdots \\ {{p_j}}\\ \vdots \\ {{p_n}}\end{array}} \right) = {{K}^{\dagger}}\delta {d} = \left( {\begin{array}{*{20}{c}}{{\overline k_{11}}}& \cdots & {{\overline k_{i1}}} & \cdots & {{\overline k_{M1}}}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\{{\overline k_{1j}}}& \cdots & {{\overline k_{ij}}}& \cdots & {{\overline k_{Mj}}}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\{{\overline k_{1N}}}& \cdots &{{\overline k_{iN}}}& \cdots & {{\overline k_{MN}}}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{\delta {d_1}}\\ \vdots \\ {\delta {d_i}}\\ \vdots \\ {\delta {d_M}}\end{array}} \right) = \sum\limits_{i = 1}^M {\left( {\begin{array}{*{20}{c}}{{\overline k_{i1}}}\\ \vdots \\ {{\overline k_{ij}}}\\ \vdots \\ {{\overline k_{iN}}}\end{array}} \right)\delta {d_i}} , | (26) |
where
| {{ H}_a} = {{{K}}^{\rm T}}{{K}}. | (27) |
During inversion, forward modeling is conducted in time domain using the high-order finite-difference scheme while the inversion is performed in the frequency domain. Time domain forward simulation has a balance performance in both simulation accuracy and efficiency, and the inversion performed in frequency domain allows a flexible choice of frequency bands, further reducing computation cost.
Some synthetic examples are used to test the validity of the proposed method. In the following tests, the true Ricker wavelet is used to generate the observed data, while the theoretical data are synthesized using Gauss or Dirac wavelet. In this paper we select a single trace as the reference. If the S/N is not extremely low (Ao et al., 2015; Huang et al., 2017), it is more efficient to locate sources by selecting a single trace as the reference. Of course, it is better to use the mean value of the stacked trace to reduce the noise affection, but it requires additional computational cost to do the NMO before stacking. Therefore in the numerical tests of this paper we select the central trace as the reference trace for simplicity.
Figure 1 shows a three-layer model with a surface monitoring survey carried out by 149 receivers equally spaced at 0.04 km. The true source is located at the position (3 km, 1.5 km) using a Ricker wavelet. Three tests, which use different modified Gauss wavelet and Dirac wavelet (Figure 2) to simulate the synthetic data, are performed with the same initial location at (0.1 km, 0.1 km). Source locations are iteratively updated using the proposed method and conventional method (Figure 3). It can be seen from the lines of the misfit value versus iterations that the proposed method is much superior to the conventional method (Figure 4). Table 1 shows the initial and final coordinates of tested locations. It is clear that the final inversion results are very close to the true source position in terms of the proposed method. However, the conventional wave-equation based source location method cannot converge to the correct position because of its strong dependency on the known source signature.
| X(km) | Z(km) | ||
| Gauss wavelet | Start | 0.1 | 0.1 |
| Final | 3.003 | 1.507 | |
| Dirac wavelet | Start | 0.1 | 0.1 |
| Final | 3.013 | 1.516 | |
| Target location | 3 | 1.5 |
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If the velocity model is complicated, as in real microseismic monitoring, the following tests show that the proposed method still behaves very well on locating microseismic source.
We use Ricker wavelet to generate the observed data with dominant frequency 10 Hz. A modified Gauss wavelet is used for simulating the theoretical data with dominant frequency 8 Hz. The Marmousi model is used for these tests. 149 receivers are equally distributed on the surface with intervals of 0.04 km. The true source is located at the position (2.8 km, 1.5 km). Tests are performed with the same initial location at (0.1 km, 0.1 km). The velocity of the Marmousi model (shown in Figure 5) varies from 1500 m/s to 5500 m/s. The origin time of Ricker wavelet is the same as Gauss wavelet (shown in Figure 6). With the proposed method, start point converges to the correct position gradually, while the conventional method locates far from the true position (Figure 7). Referring to the curves in Figure 8, the normalized misfit value of the proposed method almost reduces to zero by 6 iterations. Table 2 shows the initial and final coordinates of each tested location. The same as previous test, the final inversion results are very accurate by using the proposed method.
| X(km) | Z(km) | ||
| Gauss | Start | 0.1 | 0.1 |
| Final | 2.821 | 1.492 | |
| Target location | 2.8 | 1.5 |
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In the case of low SNR and unknown origin time of the source, as in most of microseismic monitoring, following tests show that the proposed method still behaves well.
The true source locates at position (2.8 km, 1.5 km). Ricker wavelet is used to generate observed data. Then we added Gaussian noise to observed data in each trace as
| \begin{split} {\rm signal_{new}^{obs}} & = {\rm signa{l^{obs}} }+ {\rm scale} \times {\rm noise} \hfill \\ {\rm scale} & = \frac{1}{{\sqrt 2 {\rm SNR}}} \times \frac{{\left| {\max ({\rm signa{l^{obs}}})} \right|}}{{\sqrt E }} \hfill \\ \end{split}, | (28) |
where
We use Ricker wavelet to generate the observed data with dominant frequency 10 Hz and origin time is set to 120 ms, and adding noise to the observed at the same time. A modified Gauss wavelet is used to simulate the synthetic data with dominant frequency 8 Hz and the origin time is set to 620 ms. All other parameters are as described in the previous test. Different signature and origin time used in the tests are shown in Figure 9. The observed data is contaminated with noise (SNR=1) as represented in Figure 10. With the proposed method, start point converges to the correct position gradually from position (0.1 km, 0.1 km) to the final position (2.812 km, 1.484 km), which is very close to the correct position (2.8 km, 1.5 km) (shown in Figure 11). After 6 iterations with our proposed source-independent location method the normalized misfit value almost declines to zero (Figure 12). Table 3 shows the initial and final coordinates of each tested location. We can see that the final inversion results are very accurate even when the origin time is incorrect and the observed data is contaminated with noise.
| X(km) | Z(km) | ||
| Gauss | Start | 0.1 | 0.1 |
| Final | 2.898 | 1.476 | |
| Target location | 2.8 | 1.5 |
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To be more realistic, the Marmousi model is smoothed (with smooth2 module in Seismic Unix software) to test the performance of the proposed method in the case of an inaccurate velocity model. The average velocity perturbation of every point is about 15% (as shown in Figure 13). All the other parameters are the same as the previous tests. After 5 iterations, the final locating point is very close to the true source position (as shown in Figures 14, 15 and Table 4). The normalized misfit value converges to 0.36 after 4 iterations, which is affected by the inaccurate velocity model. The sensitivity of the misfit to velocity error is also tested. We define smoothness as
| \delta = \frac{1}{N}\sum\limits_{i = 1}^N {\left|\frac{{V_i^\prime - {V_i}}}{{{V_i}}}\right|}, | (29) |
where
| X(km) | Z(km) | ||
| Gauss | Start | 0.1 | 0.1 |
| Final | 2.812 | 1.484 | |
| Target location | 2.8 | 1.5 |
DownLoad:
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Using a source-independent strategy, the proposed microseismic source location method can locate events accurately, even when the source signature is unknown. With the cross-correlation function, traveltime information is extracted from microseismic data without traveltime picking. By using acoustic wave equation, the proposed method takes account of the finite-frequency effect and the interactions between seismic waves and complex velocity structures, which helps to increase the location accuracy. The improved SI approach increases the inversion efficiency, which is important for real-time data processing. Numerical tests have demonstrated the validness and robustness of the proposed method. This method can also be used in earthquake location.
|
Ao RD, Dong LG, Chi BX (2015) Source-independent envelope-based FWI to build an initial model. Chin J Geophys 58(6): 1 998–2 000 (in Chinese with English abstract)
|
|
Artman B, Podladtchikov I, Witten B (2010) Source location using time-reverse imaging. Geophysical Prospecting 58(5): 861–873 doi: 10.1111/j.1365-2478.2010.00911.x
|
|
Brossier R, Operto S, Virieux J (2009) Robust elastic frequency-domain full-waveform inversion using the L1 norm. Geophysical Research Letters 36(20): 146–158
|
|
Dahlen FA, Nolet G, Hung SH (2000) Fréchet kernels for finite frequency traveltimes—I. Theory. Geophysical Journal International 141(1): 157–174 doi: 10.1046/j.1365-246X.2000.00070.x
|
|
Eisner L, Hulsey BJ, Duncan P, Jurick D, Werner H, Keller W (2010) Comparison of surface and borehole locations of induced seismicity. Geophysical Prospecting 58: 809–820 doi: 10.1111/j.1365-2478.2010.00867.x
|
|
Eyre TS and Baan MVD (2015) Overview of moment-tensor inversion of microseismic events. The Leading Edge 34(8): 882–888
|
|
Feroz A and Baan MVD (2013) Uncertainties in microseismic event locations for horizontal, vertical, and deviated boreholes. SEG Technical Program Expanded Abstracts: 592–596
|
|
Font Y, Kao H, Lallemand S, Liu CS, Chiao LY (2004) Hypeocentre determination offshore of eastern Taiwan using the Maximum Intersection method. Geophysical Journal International 158(2): 655–675 doi: 10.1111/gji.2004.158.issue-2
|
|
Gajewski D and Tessmer E (2005) Reverse modelling for seismic event characterization. Geophysical Journal International 163(1): 276–284 doi: 10.1111/gji.2005.163.issue-1
|
|
Geiger L (1912) Probability method for the determination of earthquake epicenters from the arrival time only. Bull St Louis Univ (8): 6
|
|
Gibowicz SJ and Kijko A (2013) An introduction to mining seismology (Vol. 55), Elsevier
|
|
Huang C, Dong L, Liu Y, Yang J (2017) Acoustic wave-equation based full-waveform microseismic source location using improved scattering-integral approach. Geophysical Journal Internaitonal 209(3): 1476–1488 doi: 10.1093/gji/ggx087
|
|
Huang C, Dong L, Liu Y, Yang J (2017) Waveform-based source location method using a source parameter isolation strategy. Geophysics 82(5): 85–97 doi: 10.1190/geo2017-0062.1
|
|
Kochnev VA, Goz IV, Polyakov VS, Murtayev IS, Savin VG, Zommer BK, Bryksin IV (2007) Imaging hydraulic fracture zones from surface passive microseismic data. First Break 25(10): 77–80
|
|
Liu Q, Polet J, Komatitsch D, Tromp J (2004) Spectral-element moment tensor inversions for earthquakes in southern California. Bulletin of the Seismological Society of America 94(5): 1748–1761
|
|
Liu Y, Dong L, Wang Y, Zhu J, Ma Z (2009) Sensitivity kernels for seismic Fresnel volume tomography. Geophysics 74(5): U35–U46 doi: 10.1190/1.3169600
|
|
Liu Y, Yang J, Chi B, Dong L (2015) An improved scattering-integral approach for frequency-domain full waveform inversion. Geophysical Journal International 202(3): 1827–1842 doi: 10.1093/gji/ggv254
|
|
Luo G and Schuster GT (1991) Wave-equation traveltime inversion. Geophysics 56(5): 645–653 doi: 10.1190/1.1443081
|
|
Maxwell SC, Urbancic TI, Prince M, Falls SD and Zinno R (2001) Real-time microseismic mapping of hydraulic fractures in Carthage, Texas: 70th Annual International Meeting, SEG, Expanded Abstracts: 1449–1452
|
|
Maxwell S (2014) Microseismic Imaging of Hydraulic Fracturing. Society of Exploration Geophysicists, Distinguished Instructor Series (17)
|
|
Maxwell SC, Rutledge J, Jones R, Fehler M (2010) Petroleum reservoir characterization using downhole microseismic monitoring. Geophysics 75(5): 75A129–75A137 doi: 10.1190/1.3477966
|
|
McMechan GA (1982) Determination of source parameters by wavefield extrapolation. Geophysical Journal of the Royal Astronomical Society 71: 613–628 doi: 10.1111/j.1365-246X.1982.tb02788.x
|
|
Mei J and Tong Q (2015) A Practical Acoustic Full Waveform Inversion Workflow Applied to a 3D Land Dynamite Survey. SEG Technical Program Expanded Abstracts: 1220–1224
|
|
Nong S, Huang C, Dong L (2017) Acoustic wave-equation based traveltime inversion for microseismic source location. SEG Technical Program Expanded Abstracts: 2924–2928
|
|
Schiemenz A and Igel H (2013) Accelerated 3-D full-waveform inversion using simultaneously encoded sources in the time domain: Application to Valhall ocean-bottom cable data. Geophysical Journal International 195: 1970–1988 doi: 10.1093/gji/ggt362
|
|
Schumacher F, Friederich W, Lamara S (2016) A flexible, extendable, modular and computationally efficient approach to scattering-integral-based seismic full waveform inversion. Geophysical Journal International 204(2): 1100–1119 doi: 10.1093/gji/ggv505
|
|
Tape C, Liu Q, Tromp J (2007) Finite-frequency tomography using adjoint methods-Methodology and examples using membrane surface waves. Geophysical Journal International 168(3): 1105–1129 doi: 10.1111/gji.2007.168.issue-3
|
|
Tong P, Yang D, Liu Q, Yang X, Harris J (2016) Source location using time-reverse imaging. Geophysical Prospecting 205(1): 464–478
|
|
Vigh D, Starr B, Kapoor J, Li H (2010) 3D full waveform inversion on a Gulf of Mexico WAZ data set. SEG Technical Program Expanded Abstracts: 957–961
|
|
Zhang H, Thurber C, Rowe C (2003) Automatic P-wave arrival detection and picking with multiscale wavelet analysis for single-component recordings. Bulletin of the Seismological Society of America 93(5): 1904–1912
|
|
Zhao L (2016) Microseismic event location by wave equation travel time inversion. SEG Technical Program Expanded Abstracts: 2530–2534
|
| 1. | Weng, L., Wu, Z., Wang, Z. et al. Acoustic Emission Source Localization in Heterogeneous Rocks with Random Inclusions Using a PRM-Based Wave Velocity Model. Rock Mechanics and Rock Engineering, 2023. DOI:10.1007/s00603-023-03236-z |
Figures(16) / Tables(4)
震球期刊
EQS
Supported by: Beijing Renhe Information Technology Co., Ltd. 
| X(km) | Z(km) | ||
| Gauss wavelet | Start | 0.1 | 0.1 |
| Final | 3.003 | 1.507 | |
| Dirac wavelet | Start | 0.1 | 0.1 |
| Final | 3.013 | 1.516 | |
| Target location | 3 | 1.5 |
DownLoad:
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| X(km) | Z(km) | ||
| Gauss | Start | 0.1 | 0.1 |
| Final | 2.821 | 1.492 | |
| Target location | 2.8 | 1.5 |
DownLoad:
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| X(km) | Z(km) | ||
| Gauss | Start | 0.1 | 0.1 |
| Final | 2.898 | 1.476 | |
| Target location | 2.8 | 1.5 |
DownLoad:
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| X(km) | Z(km) | ||
| Gauss | Start | 0.1 | 0.1 |
| Final | 2.812 | 1.484 | |
| Target location | 2.8 | 1.5 |
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