Accuracy of the moment-tensor inversion of far-field P waves

SUMMARY

The far-field assumption is widely used and suitable for the moment-tensor inversion in which the source–receiver distance is quite long. However, the description of far field is uncertain and an explicit far-field range is missing. In this study, the explicit far-field range is determined and the errors of moment-tensor solutions produced by the far-field approximation are analysed. The distance, for which the far-field assumption is satisfied and the effect of the near-field term can be ignored, is directionally dependent. For the shear dislocation, in the directions near the nodal lines of the far-field P waves, the far-field distance is heavily dependent on the displacement component used to invert moment tensors. The radial component of displacement, which is parallel to the wave propagation direction, is recommended for the inversion and the corresponding far-field distance is quite short. In the directions far from the nodal lines, the selection of displacement components has little influence on the far-field distance. The maximum far-field distance appears in the directions of the pressure and tensile axes of the source and the value is about 30 wavelengths of radiated waves. Using more receivers (>6) in the moment-tensor inversion can shorten the far-field distance. The effect of the near-field term on the moment-tensor inversion for tensile dislocations and isotropic sources (explosion or implosion) can be ignored. The conclusions obtained in this study are helpful for determining the positions of receivers and evaluating the accuracy of moment-tensor solutions, with far-field assumption being applied in the inversion.

Inverse theory, Body waves, Earthquake source observations, Wave propagation

1 INTRODUCTION

Interpreting the source mechanisms of seismic events is important for understanding the evolution of stress field (Eaton et al.2014; Der Baan et al.2016) and provides an insight into source properties. A common method of calculating source mechanisms is the moment-tensor inversion, which can provide the knowledge of fracture type and propagation (Vavryčuk 2001; Silený 2009). The moment-tensor inversion utilizes the radiation pattern of seismic waves to obtain a 3 × 3 symmetric matrix in which each component represents a force couple (Aki & Richards 2002). Compared with surface waves (Aki & Patton 1978; Kanamori & Given 1981; Kanamori & Given 1982), the usage of body waves (Ohtsu 1988; Ohtsu & Ono 1988) in the moment-tensor inversion can simplify the inversion and is common in the engineering practice. Moment tensors can be calculated using the amplitude of seismic waves (Vavryčuk et al.2008; Fojtíková et al.2010), amplitude ratios (Hardebeck & Shearer 2003; Jechumtalova & Silený 2005) or full wave forms (Dziewonski et al.1981; Sipkin 1986; Silený et al.1992). For physically interpreting source mechanisms, the decompositions and source-type plots of moment tensors were provided (Hudson et al.1989; Vavryčuk 2001; Tape & Tape 2012; Vavryčuk 2015).

In the moment-tensor inversion, the Green's function is extremely important and has been provided by Aki & Richards (2002) for simple types of media. Actually, the function used in the moment tensor inversion is the spatial derivatives of the Green's function, which is called the Green's function of second kind. According to distance, seismic waves consist of two parts, the near-field and far-field terms. For simplifying the moment-tensor inversion, the far-field term is supposed to be dominated and the effect of the near-field term on moment-tensor solutions is ignored. The far-field assumption has been widely applied in earthquake monitoring (Trifu 2000; Cesca et al.2006; Lizurek 2017) and rock damage experiments (Yu et al.2005; Graham et al.2010; Stierle et al.2016; Xu et al.2017) in which the source–receiver distances are always quite long. In some specific engineering practices, the source–receiver distances are obviously very short and the far-field assumption is not suitable for the moment-tensor inversion, such as the volcano seismology (Legrand et al.2000; Lokmer & Bean 2010; Van Driel et al.2015).

Although many investigations (Toyokuni & Takenaka 2006; Nissen-Meyer et al.2007; Van Driel & Nissenmeyer 2014) analysed the waveforms caused by seismic sources, the far-field term is never studied separately and the boundary of far field is quite uncertain. Aki & Richards (2002) provided a simple definition of far field that distance of the receiver should be larger than a few wavelengths away from the source. But no more details about the number of wavelengths are specified. For example, Liu et al. (2014) assumed the number of wavelengths as 4. By contrast, Vidale et al. (1995) pointed out that the near-field term can be observable at great distance from a very large deep earthquake in some cases. It can be concluded that the boundary between the near and far field is uncertain, and one must assess the relative magnitude of each term (Aki & Richards 2002). For some earthquakes with shallow depth, it is quite uncertain whether the source–receiver distance is long enough and the error caused by the far-field assumption can be ignored. In addition, the traditional description of far-field range is based on the relative magnitude of the two terms (Aki & Richards 2002), which cannot indicate the errors of moment-tensor solutions directly. Consequently, it is significant to clarify the far-field range and analyse the errors of moment-tensor solutions caused by the far-field assumption. In the study, the moment-tensor inversion is carried out based on synthetic data and the explicit far-field range is determined by the errors of moment-tensor solutions. The sensitivity of far-field range to several factors, which are the wave velocity, frequency, positions of receivers and source mechanism, is analysed.

2 THEORETICAL ANALYSIS

2.1 Formulations

The moment tensor of seismic sources in isotropic media is expressed as follows:

$$\begin{eqnarray*} {m_{pq}} = (\lambda {l_k}{n_k}{\delta _{pq}} + \mu {l_p}{n_q} + \mu {l_q}{n_p})\mathit{ S_f}, \end{eqnarray*}$$

(1)

where p = 1, 2, 3 and q = 1, 2, 3 represent X, Y, and Z directions, λ and μ are the Lame constants, δ_pq is 1 while p = q, otherwise is 0, l_q are the components of the slip vector at the fault, and n_p are the components of the normal vector to the fault. S_f is the fault size.

In a homogenous and isotropic media, the wave produced by a seismic source contains three terms:

$$\begin{eqnarray*} {{\bf u}}\left( {{{\bf x}},t} \right) = {{{\bf u}}_{\mathrm{ near}}} + {{{\bf u}}_{\mathrm{ intermediate}}} + {{{\bf u}}_{\mathrm{ far}}}, \end{eqnarray*}$$

(2)

where, x is the position vector of receivers and t is time. As indicated by eq. (2), three parts, which are the near-field term |${{{\bf u}}_{\mathrm{ near}}}$|⁠, the intermediate-field term |${{{\bf u}}_{\mathrm{ intermediate}}}$| and the far-field term |${{{\bf u}}_{\mathrm{ far}}}$|⁠, are contained in the seismic wave field u(x, t), where the three terms are distinguished by the decrease of amplitudes with distance. The near- and intermediate-field terms decrease as r⁻² and the far-field term as r⁻¹, where r is the source–receiver distance. Consequently, there is no need to distinguish between the near- and intermediate-field terms and both terms are classified as the near-field terms in this study. In the moment-tensor inversion, the source–receiver distance is commonly quite large and the far-field term is supposed to be the predominant part in seismic waves. Because the compressional wave (P-wave) velocity is higher than the shear wave (S-wave) velocity, the P wave is commonly used to invert moment tensors independently (Ohtsu 2008). In the homogeneous and isotropic media, the P-wave amplitude excited by a point source and recorded by a receiver can be written as

$$\begin{eqnarray*} {A_n}\left( {{{\bf x}},t} \right) = \frac{{{r_n}}}{{4\pi \rho {\alpha ^3}R}} \cdot \left( {\begin{array}{*{20}{c}} {{r_1}}&{{r_2}}&{{r_3}} \end{array}} \right)\left[ {\begin{array}{@{}*{3}{c}@{}} {{m_{11}}}&{{m_{12}}}&{{m_{13}}}\\ {{m_{12}}}&{{m_{22}}}&{{m_{23}}}\\ {{m_{13}}}&{{m_{23}}}&{{m_{33}}} \end{array}} \right]\left[ {\begin{array}{@{}*{1}{c}@{}} {{r_1}}\\ {{r_2}}\\ {{r_3}} \end{array}} \right] \cdot \dot{S}\left( {t - \frac{R}{\alpha }} \right), \end{eqnarray*}$$

(3)

where, A_n(x,t) is the P wave, |$\rho $| is the density, α is the P-wave velocity, r_n are the direction cosine from sources to receivers, R is the source–receiver distance, m_pq are the moment-tensor components, and S(t) is the source-time function. The far-field range may be dependent on the wave velocity, frequency, positions of receivers, number of receivers, source mechanism and source-time function, and all these factors will be analysed in Section 3.

In this study, the moment tensors are inverted using the P-wave amplitudes. In addition, we use the standard moment tensor decomposition (Vavryčuk 2015), which decomposes moment tensors into isotropic (ISO), double-couple (DC) and compensated linear vector dipole (CLVD). The corresponding orthonormal form of a moment tensor can be written as

$$\begin{eqnarray*} {{\bf M }} = {M_1}{{{\bf e}}_1}{{{\bf e}}_2} + {M_2}{{{\bf e}}_2}{{{\bf e}}_2} + {M_3}{{{\bf e}}_3}{{{\bf e}}_3}, \end{eqnarray*}$$

(4)

where |${M_1} \ge {M_2} \ge {M_3}$| and vectors e₁, e₂ and e₃ define the tensile, neutral and pressure axes, respectively. Then the moment tensor M can be decomposed into the three components and written as

$$\begin{eqnarray*} {{\bf M}} = {M_{\mathrm{ ISO}}}{{{\bf E}}_{{\mathop{\rm ISO}\nolimits} }} + {M_{\mathrm{ DC}}}{{{\bf E}}_{\mathrm{ DC}}} + {M_{{\mathop{\rm CLVD}\nolimits} }}{{{\bf E}}_{\mathrm{ CLVD}}}, \end{eqnarray*}$$

(5)

where E_ISO, E_DC and E_CLVD are the ISO, DC and CLVD elementary tensors. Then the relative scale factors C_ISO, C_DC and C_CLVD are defined as:

$$\begin{eqnarray*} \left[ {\begin{array}{@{}*{1}{c}@{}} {{C_{\mathrm{ ISO}}}}\\ {{C_{{\mathop{\rm CLVD}\nolimits} }}}\\ {{C_{{\mathop{\rm DC}\nolimits} }}} \end{array}} \right] = \frac{1}{M}\left[ {\begin{array}{@{}*{1}{c}@{}} {{M_{\mathrm{ ISO}}}}\\ {{M_{{\mathop{\rm CLVD}\nolimits} }}}\\ {{M_{{\mathop{\rm DC}\nolimits} }}} \end{array}} \right], \end{eqnarray*}$$

(6)

where |$M = | {{M_{\mathrm{ ISO}}}} | + | {{M_{\mathrm{ CLVD}}}} | + | {{M_{\mathrm{ DC}}}} |$|⁠.

2.2 Source-time function and source mechanisms

The source-time function is extremely important, because the relative magnitude of different terms is dependent on the source-time function. In many studies, the source-time function is supposed to be a Heaviside step function (Ohtsu 2008), because its derivative is the impulse function, which can simplify the inversion equation for moment tensors. However, the step function is not suitable for calculating synthetic seismic waves because it is too idealized and different from the observations. Here, the source-time function is expressed according to Ohtsu (1995) as

$$\begin{eqnarray*} S(t) = \left\{ {\begin{array}{@{}*{2}{c}@{}} {\frac{t}{{{T_\mathrm{ r}}}} - \frac{2}{{3\pi }}\sin \left( {\frac{{2\pi t}}{{{T_\mathrm{ r}}}}} \right) + \frac{1}{{12\pi }}\sin \left( {\frac{{4\pi t}}{{{T_\mathrm{ r}}}}} \right)}&{t < {T_{\mathop{\rm r}\nolimits} }}\\ 1&{t \ge {T_\mathrm{ r}}} \end{array}} \right., \end{eqnarray*}$$

(7)

where T_r is the rise time and t is the time. The source-time function in eq. (7) is broadband with its maximum at the zero Hz. The source-time function, far-field waveform and amplitude spectra of the far-field waves are plotted in Fig. 1.

Figure 1.

Source-time function, far-field waveform and amplitude spectra of the far-field waves.

The length of wave affected area is α × T_r. Then the predominant wavelength can be approximately equal to α × T_r and the predominant frequency can be approximately equal to 1/T_r. Actually, 1/T_r is also close to the central frequency of the spectrum of the far-field waves.

In the engineering practice, the source dimension and frequency of radiated waves vary greatly because the frequency depends on the size of the source area. However, if the far-field range is expressed in the number of wavelengths, the dependence on the source dimension and frequency of radiated waves is eliminated. The moment tensors for the shear dislocation, tensile dislocation and isotropic source are expressed as follows:

$$\begin{eqnarray*} \begin{array}{@{}*{3}{c}@{}} {{{{\bf M}}_{\mathrm{ shear}}} = \left[ {\begin{array}{@{}*{3}{c}@{}} 0&1&0\\ 1&0&0\\ 0&0&0 \end{array}} \right],}&{{{{\bf M}}_{\mathrm{ tensile}}} = \left[ {\begin{array}{@{}*{3}{c}@{}} 1&0&0\\ 0&{\frac{\lambda }{{\lambda + 2\mu }}}&0\\ 0&0&{\frac{\lambda }{{\lambda + 2\mu }}} \end{array}} \right],}&{{{{\bf M}}_{\mathrm{ isotropic}}} = \left[ {\begin{array}{@{}*{3}{c}@{}} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right]} \end{array}, \end{eqnarray*}$$

(8)

where λ and μ are the Lame constants.

2.3 Positions of receivers

In general, six one-component receivers are needed to invert for a moment tensor. In this study, the positions of receivers are defined as

$$\begin{eqnarray*} {{\bf r}}_{{\mathop{\rm rec}\nolimits} }^n = R{{{\bf e}}_{{\mathop{\rm azi}\nolimits} }} + {R_r}{{\bf e}}_{{\mathop{\rm rel}\nolimits} }^n, \end{eqnarray*}$$

(9)

where |${{\bf r}}_{{\mathop{\rm rec}\nolimits} }^n$| is the direction vector pointing from the source to the nth receiver, R is the source–receiver distance, |${{{\bf e}}_{\mathrm{ azi}}}$| is the unit vector pointing from the source to the central point of the distribution region of receivers, R_r is the radius of the circle (as shown in Fig. 1) and indicates the size of the distribution region of receivers, |${{\bf e}}_{{\mathop{\rm rel}\nolimits} }^n$| is the unit vector pointing from the centre of the distribution region of receivers to the nth receiver. For simplicity, the distribution of receivers is defined as a regular-pentagonal configuration, which one receiver locates at the centre of the region and the other five locate around. This configuration has been proved superior to suppress the effect of noise on the moment-tensor inversion (Kong et al.2019).

In Fig. 2, the angle γ defines the direction of receivers and the distribution of receivers with varying azimuthal gap is investigated in the next. In real engineering applications, receivers are typically arranged as Fig. 2(a). However, when angle γ is close to 0° or 180°, the source is close to the plane with receivers and the inversion equation for moment tensors is ill conditioned. Consequently, the plane with receivers is set perpendicular to the line from the source to the centre of the plane (as shown in Fig. 2b). Because R_r is much smaller than R, this setting has little effect on the far-field distance. For simplicity, R_r is supposed to be proportional to R and written as R_r = ηR. The sensitivity of far-field range to parameter η is investigated in Section 3.3.

Figure 2.

Positions of the six receivers, which are represented by the five-pointed stars (a). In the calculation, a setting is made that the plane with receivers is perpendicular to the line from the source to the centre of the plane (b).

3 FAR-FIELD RANGE

In this section, the far-field range is calculated and expressed in wavelengths of radiated waves. The far-field range is related to the relative magnitude of the two field terms, which vary greatly with the direction, as shown in Fig. 3. The circumferential coordinate represents the direction of receivers and the radial coordinate represents the amplitude of waves.

Figure 3.

For the shear dislocation, the relative magnitude of the two field terms at different source–receiver distances. The circumferential coordinate represents the direction of receivers and the radial coordinate represents the amplitude of waves.

Obviously, the far-field range is dependent on the positions of receivers relative to sources. With the increase of the source–receiver distance, the effect of the near-field term on moment-tensor solutions decreases and the inversion results are more close to the true value. For the shear dislocation, the actual proportion of the DC component in the moment tensor is 100 per cent. Because of the effect of the near-field term, the inversion results become approximate and the DC component is distorted. As shown in Fig. 3, in the directions of the nodal lines of the far-field P waves, the amplitude of the far-field term is zero and the near-field term becomes significant (Vavrycuk 1992), which indicates that the corresponding far-field distance may be quite different from those in the directions far from the nodal lines. In this study, we use different displacement components in different directions to invert moment tensors. Specifically, in the directions of −45° (315°) to 45° and 135°–225°, the X displacement component is used. In the directions of 45°–135° and 225°–315°, the Y displacement component is used. Thus, in the inversion, the amplitude of the corresponding displacement component of the far-field term is relatively large and has a better signal-to-noise ratio.

The change of the proportion of the DC component in the moment tensors is plotted in Fig. 4. Several directions, which are γ = 15°, 25°, 35° and 45°, are considered in Fig. 4. The source–receiver distance in the horizontal axis is expressed in wavelengths of radiated waves.

Figure 4.

For the shear dislocation, the change of the proportion of the DC component in moment tensors with the increase of source–receiver distance in different directions, γ = 15°, 25°, 35° and 45°.

Fig. 4 suggests that the effect of the near-field term on the retrieved DC component changes with directions and is extremely difficult to be eliminated. For the purpose of this study, we define the far-field range as a source–receiver distance, beyond which the error of the proportion of the DC component is less than 5 per cent, and call as the far-field distance. In the next sections, the far-field distance is calculated and its sensitivity to the wave velocity, frequency, receivers and source mechanism is analysed.

3.1 Wave velocity

According to Schön (2016), the wave velocity in common rock is among the range of 2000–9000 m s⁻¹. The far-field distance is plotted in Fig. 5 in which the wave velocity is plotted on the X-axis, the direction of receivers γ is plotted on the Y-axis and the corresponding far-field distance expressed in wavelengths is given by the Z-axis. The projection of the surface in the directions of γ = 10°, 15°, 20°, 25°, 30°, 35°, 40° and 45° is plotted on the X–Z plane. The colour bar represents the number of wavelengths. The directions above are representative, because the radiation patterns of the two field terms are all symmetrical (as shown in Fig. 3).

Figure 5.

Far-field distances for different wave velocities in different directions.

As shown in Fig. 5, the far-field distance is not affected by the wave velocity, even though the velocity is present in the formula for the Green's function. Because the far-field distance is independent of the velocity, the definition of far-field range obtained in the homogeneous and isotropic media can be applied to the moment-tensor inversion in a more complex media. As long as the propagation path of seismic waves is known, the propagation distance expressed in the number of wavelengths can be calculated easily and compared with the far-field distance directly.

3.2 Wave frequency

The wave frequency used in this study indicates the predominant frequency of seismic waves. For the generality of the conclusions in this study, the wave frequency range considered in this study is 10⁻¹–10⁷ Hz, for the application of the moment-tensor inversion is not limited to earthquakes but also to hydrofractures and acoustic emissions in which the frequency of waves is different (Cai et al.2007). The far-field distances for different frequencies are plotted in Fig. 6 in which the wave frequency is plotted on the X-axis, the direction of receivers γ is plotted on the Y-axis and the corresponding far-field distance expressed in wavelengths is given by the Z-axis. The projection of the surface in the directions of γ = 10°, 15°, 20°, 25°, 30°, 35°, 40° and 45° is plotted on the X–Z plane.

Figure 6.

Far-field distances for different wave frequencies in different directions.

Fig. 6 suggests that the far-field distance is independent of the wave frequency, which indicates that the conclusions obtained in this study are suitable for the moment-tensor inversion in various applications.

3.3 Receivers

In the traditional guideline for the moment-tensor inversion, it is desirable that the number of receivers should be as many as possible and the distribution region of receivers should be as large as possible to cover the focal sphere adequately (Eyre & Der Baan 2015). Consequently, in this section, we study the far-field distance with different distances between receivers and numbers of receivers.

The radius of the circle R_r, which represents the distance between receivers, is supposed to be proportional to the source–receiver distance R (written as R_r = ηR) and the ratio η is set to be 0.1, 0.2, 0.3, 0.4 and 0.5. The far-field distances for different ratios in different directions are plotted in Fig. 7. In Fig. 7, the directions near the nodal lines are excluded from the moment-tensor inversion, because the far-field distances in these directions are heavily dependent on the displacement component used in the inversion and need to be discussed separately.

Figure 7.

Far-field distances for different ratios in different directions. The directions close to the nodal lines are excluded from the moment-tensor inversion and represented by the light-coloured symbols.

According to Fig. 7, if only the directions far from the nodal lines are considered, the maximum far-field distance is along the directions of 45°, 225°, 135° and 315°, which are the tensile and pressure axes of the source, respectively. As the parameter η increases, the far-field distance increases, which indicates that large distribution region of receivers is harmful to the accuracy of moment-tensor solutions. This conclusion contradicts to the formal guideline of the distribution of receivers provided the Green's functions are calculated using the far-field approximation. The reason is that the proportions of the near-field term in the signals obtained by different receivers are different. According to Fig. 3, as the distance between receivers increases, the difference of the proportions of the near-field term in the signals obtained by different receivers increases. The principle of the moment-tensor inversion is to identify the radiation pattern of the seismic waves through the receivers, and then identify the corresponding dislocation type of the source. The difference of the proportions can distort the radiation pattern of the far-field term and its increase will lead to great errors.

In the directions (−10° to 10°, 80°–100°, 170°–190° and 260°–280°) close to the nodal lines of the far-field P waves, the far-field distances are heavily dependent on the displacement component used to invert moment tensors. The radiation patterns plotted by the displacement components are shown in Fig. 8.

Figure 8.

For the shear dislocation, the relative magnitude of the displacement components of the two field terms at the source–receiver distance of 1 wavelength. The circumferential coordinate represents the direction of receivers and the radial coordinate represents the amplitude of the displacement component.

In the directions near 0° and 180° (corresponding to the X-axis in the Cartesian coordinate system), the far-field distances calculated by the X displacement component is quite small because the X displacement component of the near-field term is close to zero (as shown in Fig. 8a). However, if we use the Y displacement component to invert moment tensors, we cannot determine the far-field distance because the Y displacement component of the far-field term is zero and that of the near-field term is relatively large (as shown in Fig. 8b). In the directions near 90° and 270° (corresponding to the Y-axis in the Cartesian coordinate system), things are the same, except that we should use the Y displacement component to invert moment tensors. In the directions far from the nodal lines, the difference of the far-field distances caused by different displacement components is quite small, because both the X and Y displacement components of the near-field term are relatively large. More generally, in the directions close to the nodal lines, the displacement component used in the inversion should be as parallel as possible to the wave propagation direction. Otherwise, great errors of moment-tensor solutions can be observed.

Fig. 8 also indicates that in the directions near the tensile and pressure axes (45°, 135°, 225° and 315°), the magnitude of the near-field term relative to the far-field term is very large and the change of the near-field term is very dramatic. The combination of these two causes seriously distorts the radiation pattern of the far-field term and results in long far-field distances.

In addition, increasing the number of receivers can suppress the effect of the near-field term on moment-tensor solutions. We use 10 receivers to invert moment tensors and the far-field distances are plotted in Fig. 9. The configuration with 10 receivers is that 4 receivers locate near the centre and the others locate around at equal intervals, which is similar to the configuration with 6 receivers in Fig. 2 and can achieve a relatively small condition number for the inversion equation.

Figure 9.

For the configuration with 10 receivers, the far-field distances for different ratios in different directions.

In Fig. 9, the directions near the nodal lines are also excluded from the moment-tensor inversion. If the number of receivers changes from 6 to 10, the maximum far-field distance changes from 30 to 26.6 wavelengths, which indicates that the usage of more receivers is helpful for the accuracy of moment-tensor solutions. In addition, the shape of the curve near the nodal lines in Fig. 9 is slightly different from that in Fig. 7. In the calculations of Figs 7 and 9, we use the X or Y displacement component instead of the radial component of displacement. In different configurations of receivers, the relative positions of receivers to the source are slightly different. This difference has minor effect on the far-field distance in the directions near the nodal lines.

3.4 Tensile dislocation and isotropic source (explosion or implosion)

The conclusions above are for the shear dislocation, because the shear dislocation is of great interest in seismology. Actually, tensile dislocations and isotropic sources are also quite common in some engineering applications. For the consistency of the study, we are still concerned about the DC component in the moment tensors of tensile dislocations and isotropic sources.

For the pure tensile dislocation, the actual proportion of the DC component is 0. In moment-tensor solutions, the change of the proportion of the DC component with the increase of source–receiver distance is plotted in Fig. 10.

Figure 10.

For the tensile dislocation, the change of the proportion of the DC component in moment tensors with the increase of source–receiver distance in different directions, γ = 15°, 25°, 35° and 45°.

By comparing Fig. 10 with Fig. 4, the effect of the near-field term on the moment-tensor inversion for tensile dislocations is much smaller than that for shear dislocations, thus the far-field distance is much shorter for the tensile dislocation. According to the radiation patterns of the two terms of the tensile dislocation (as shown in Fig. 11), the amplitude of the near-field term decreases more quickly than that of the shear dislocation. In addition, the radiation pattern of the near-field term is similar to that of the far-field term, which can reduce the effect of the near-field term on moment-tensor solutions.

Figure 11.

For the tensile dislocation, the relative magnitude of the two field terms at different source–receiver distances.

For the isotropic source (explosion or implosion), there is no effect of the near-field term on the moment-tensor inversion because the radiation pattern of the far-field term is exactly the same as that of the near-field term (as shown in Fig. 12). It should be noted that the two curves have been normalized by the maximum value respectively in order to compare the shapes of the two curves. Actually, the value of the far-field term is much larger than that of the near-field term.

Figure 12.

Radiation patterns of the near- and far-field terms.

Based on the results above, it can be concluded that the far-field distance for the shear dislocation is higher than for other dislocation types. Consequently, for all moment-tensor inversions, keeping the source–receiver distance larger than the maximum value of the far-field distance for the shear dislocation is recommended for achieving a sufficient accuracy.

4 CONCLUSION

In the moment-tensor inversion, clarifying far-field range is extremely important for improving the accuracy of source mechanisms, when the far-field Green's function is applied. In this study, the explicit far-field range is determined based on the errors of moment-tensor solutions, and the sensitivity of far-field range to several factors is analysed. Based on the numerical experiments, we arrived at the following conclusions:

1. The far-field distance, which is expressed as the number of wavelengths of radiated waves, is not related to the wave velocity, thus the conclusions obtained in the homogeneous and isotropic media can be applied to more complex media, such as the layered media. In the layered media, the ratio of the propagation distance to the wavelength can be calculated in each layer along the wave propagation path and the sum of the ratios can be compared with the far-field distance directly.

2. The far-field distance is not related to the wave frequency, thus the explicit definition of far field in this study can be applied to many other applications, such as hydraulic fracturing and acoustic emissions with a broad range of radiated frequencies.

3. If the far-field assumption is applied, a near-source position of receivers is harmful to the accuracy of moment-tensor solutions. The common recommendation is to use a large coverage of receivers but of course with the Green's function calculated accurately, that is, including the near-field waves.

4. The far-field distance differs in different directions. For the shear dislocation, in the directions near the nodal lines of the far-field P waves, the far-field distances are heavily dependent on the displacement component used to invert moment tensors. The radial component of displacement, which is parallel to the wave propagation direction, is recommended for the inversion and the corresponding far-field distance is quite short. Otherwise great errors of the moment-tensor solutions will be caused and the far-field distances cannot be determined. In the directions far from the nodal lines, the difference of the far-field distances calculated by different displacement components is very small. The maximum far-field distance appears in the directions of the pressure and tensile axes of the source and the value is about 30 wavelengths.

5. Using more receivers (>6) in the moment-tensor inversion can suppress the effect of the near-field term on the solutions and shorten the far-field distance.

6. For the tensile dislocation and isotropic source (explosion or implosion), the effect of the near-field term on moment-tensor solutions is small and can be ignored in the moment-tensor inversion.

ACKNOWLEDGEMENTS

The authors of this paper would like to thank the financial supports provided by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA22000000).