Evaluation of the seismic moment of the April 20, 2013 Lushan earthquake

Abstract

The April 20, 2013 Lushan earthquake which occurred in Sichuan, China had only moderate thrust. However, the computed seismic moments (M₀) for the Lushan earthquake calculated by several institutions differ significantly from 0.4 × 10¹⁹ to 1.69 × 10¹⁹ Nm, up to four times difference. We evaluate ten computed M₀s by using normal mode observations from superconducting gravimeters in Mainland China. We compute synthetic normal modes on the basis of moment tensor solutions and fit them to the observed normal modes. Comparison of our results indicates that M₀ is the main cause for some large differences between observations and synthetics. We suggest that a moment magnitude of M_w6.6, corresponding to a M₀ of 0.97–1.08 × 10¹⁹ Nm, characterizes the size and strength of the seismic source of the Lushan earthquake.

1 Introduction

A moderate earthquake occurred on April 20, 2013 at Lushan, Sichuan, China. This is in close proximity to the May 12, 2008 Wenchuan earthquake of magnitude M_w7.9. Shortly after the earthquake, several academic institutions including the Chinese Academy of Sciences, the Lamont-Doherty Earth Observatory of Columbia University, and the United States Geological Survey (USGS) reported the moment tensor solutions of the earthquake and concluded that it was a moderate shallow thrust event. Table 1 lists the fault motion parameters and scalar seismic moment (M₀) from ten solutions. The values of M₀ of solutions 8, 9, and 10 are derived from teleseismic inversion rupture processes of the Lushan earthquake. Their results indicated that the source time was about 10 s and rupture size was about 20 km (Wang et al. 2013; Liu et al. 2013a; Zhang et al. 2013). The results in Table 1 show some obvious differences, especially, M₀ differ significantly from 0.4 × 10¹⁹ to 1.69 × 10¹⁹ Nm, up to 4 times difference, corresponding to moment magnitudes (M_w) varying from 6.37 to 6.8. Such a large discrepancy in the values of predicted M₀ poses the question: which value is the most reliable?

Table 1 Ten focal mechanism solutions for the April 20, 2013 Lushan earthquake

Amplitudes of normal modes are dependent on the location of the epicenter, the seismic station, earthquake depth, focal mechanism, and seismic moment (Stein and Geller 1977). The normal mode is the only seismological signal that can be used to evaluate focal mechanism solutions and constrain the upper limit of the scalar seismic moment of large earthquakes. Observations of oscillatory modes below 1 mHz suggest that the moment magnitude of the 2004 Sumatra megathrust earthquake should be M_w9.3 (Stein and Okal 2005), or M_w9.15 (Park et al. 2005), instead of 9.0 predicted by Global CMT solutions. Note that M₀ of an M_w9.3 earthquake is about 2.5 times larger than the M₀ of an M_w9.0 earthquake. For the great Tohuku earthquake on March 11, 2011, observations of mode₀S₀ suggest that the dip angle provided by the USGS Centroid Moment Tensor Solution might be excessive (Xue et al. 2012). As a typical moderate earthquake, the Lushan event only generated observable signals with spheroidal modes in the range 2.5–5.5 mHz. According to Aki’s theory (Aki 1967), normal modes below 5.5 mHz should be sufficient to evaluate the seismic moment M₀ of moderate events. This study shows that the M₀ of the Lushan earthquake can be evaluated by normal mode observations made by superconducting gravimeters (SGs) in Mainland China.

2 Methods

Displacement at a seismic station caused by Earth’s free oscillations can be modeled using the moment tensor solution of the earthquake and the Green’s function, which represents the impulse response of the medium between source and receiver and thus contains the seismic wave propagation effects through the medium from source to receiver. These include energy losses through reflection and transmission at seismic discontinuities, anelastic absorption, and geometrical spreading.

By using a seismic signal that has much longer wavelengths than the dimensions of the source (point source approximation), the observed displacement within seismic source time $\overline{t}$ on the Earth’s surface at a station at time t can be expressed as a linear combination of the second-order seismic moment tensor elements convolved with the derivative of the Green’s functions (Backus and Mulcahy 1976; Stump and Johnson 1977; Aki and Richards 1980).

$$u_n(\stackrel{\rightharpoonup }{x},t)=[G_{\text{nk},j}(\stackrel{\rightharpoonup }{x},\stackrel{\rightharpoonup }{\mathit{\xi }},t,\overline{t})\ast M_{\text{kj}}(\stackrel{\rightharpoonup }{\mathit{\xi }},\overline{t})]$$

(1)

where * denotes the temporal convolution, and

• $\stackrel{\rightharpoonup }{x}$: position vector of station with coordinates x₁, x₂, x₃ for north, east, and down.

• $\stackrel{\rightharpoonup }{\mathit{\xi }}$: position vector of point source with coordinates ξ₁, ξ₂, ξ₃ for north, east, and down.

• $u_n(\stackrel{\rightharpoonup }{x},t)$: component of displacement at position $\stackrel{\rightharpoonup }{x}$ and time t. The subscripts n = 1, 2, 3 indicate the components for north, east, and down, respectively.

• $G_{\text{nk},j}(\stackrel{\rightharpoonup }{x},\stackrel{\rightharpoonup }{\mathit{\xi }},t,\overline{t})$: derivative of Green’s function components with regard to source coordinate ξ _j .

• $M_{\text{kj}}(\stackrel{\rightharpoonup }{\mathit{\xi }},\overline{t})$: components of the second-order moment tensor released at source time $\overline{t}.$

If all the components of seismic moment tensor have the same time dependence $s(\overline{t})$(synchronous source, Silver and Jordan 1982), the equation (1) can be written as (Stump and Johnson 1977):

$$u_n(\stackrel{\rightharpoonup }{x},t)=[G_{\text{nk},j}(\stackrel{\rightharpoonup }{x},\stackrel{\rightharpoonup }{\mathit{\xi }},t,\overline{t})\ast s(\overline{t})]M_{\text{kj}}(\stackrel{\rightharpoonup }{\mathit{\xi }})$$

(2)

If the time function $s(\overline{t})$ is a delta function $\mathit{\delta }(\overline{t})$, the equation (2) can be simplified as (e.g., Jost and Herrmann 1989, Lay and Wallace T 1995; Udías 1999):

$$u_n(\stackrel{\rightharpoonup }{x},t)=\sum _{k=1}^3\sum _{j=1}^3\frac{\mathit{\partial }{G}_{\text{nk}}(\stackrel{\rightharpoonup }{x},\stackrel{\rightharpoonup }{\mathit{\xi }},t)}{\mathit{\partial }{\mathit{\xi }}_j}{M}_{\text{kj}}(\stackrel{\rightharpoonup }{\mathit{\xi }})$$

(3)

In this case, the nine components of derivative of Green’s function describe nine generalized couples. The component with respect to the source coordinate ξ _j is equivalent to a single couple with arm in the ξ _j direction. The Cartesian components of the symmetrical moment tensor in (3) in an isotropic medium for a double couple source can be expressed in terms of strike φ, dip δ, and slip λ of the source and the scalar seismic moment M₀ (Aki and Richards 1980; Jost and Herrmann 1989):

$$M_{11}=-M_0\left(sin\mathit{\delta }cos\mathit{\lambda }\text{sin2}\mathit{\phi }+\text{sin2}\mathit{\delta }sin\mathit{\lambda }{sin}^2\mathit{\phi }\right)$$

$$M_{22}=M_0\left(sin\mathit{\delta }cos\mathit{\lambda }\text{sin2}\mathit{\phi }-\text{sin2}\mathit{\delta }sin\mathit{\lambda }{cos}^2\mathit{\phi }\right)$$

$$M_{33}=M_0\text{sin2}\mathit{\delta }sin\mathit{\lambda }$$

(4)

$$M_{12}=M_0\left(sin\mathit{\delta }cos\mathit{\lambda }\text{cos2}\mathit{\phi }+0.5\text{sin2}\mathit{\delta }sin\mathit{\lambda }\text{sin2}\mathit{\phi }\right)=M_{21}$$

$$M_{13}=-M_0\left(cos\mathit{\delta }cos\mathit{\lambda }cos\mathit{\phi }+\text{cos2}\mathit{\delta }sin\mathit{\lambda }sin\mathit{\phi }\right)=M_{31}$$

$$M_{23}=-M_0\left(cos\mathit{\delta }cos\mathit{\lambda }sin\mathit{\phi }-\text{cos2}\mathit{\delta }sin\mathit{\lambda }cos\mathit{\phi }\right)=M_{32}$$

The synthetic normal modes can be computed according to focal mechanism solutions, Green’s function and equation (3) and (4). As normal modes are very long-period seismic waves, it is sufficient to calculate normal mode displacement caused at a station based on the Green’s function derived from 1-D Earth’s model, such as PREM (Dziewonski and Anderson 1981). By comparing the synthetic normal modes with the observed modes, we can evaluate focal mechanism solutions in Table 1 for the Lushan earthquake.

Normal modes in 2.5–5.5 mHz were clearly observed by two SGs in Mainland China after the Lushan earthquake. OSG057 and OSG053 are about 1,149 and 1,060 km away from the epicenter, locating at Lhasa and Wuhan stations (see Fig. 1). The seismic source of the Lushan earthquake can be regarded as a point source for observations made at the stations for which the distances to the source are much greater than the linear dimension of the source.

Fig. 1

The locations of Lhasa and Wuhan Stations and the epicenter of the Lushan earthquake on April 20, 2013

Since SGs have constant scale factor and stable, flat transfer functions in the amplitude and period bands from DC to 100 s, SGs theoretically have an advantage over most broadband seismometers in detailed analyses of the amplitude variations of normal modes and the Earth’s hum. In fact, at frequencies below 4 mHz the noise levels of OSG057 and OSG053 are much lower than those of broadband seismometers in Mainland China. For example, normal mode signals cannot be clearly identified in the broadband seismometer STS-2 at Lhasa after the Lushan earthquake.

We calculated theoretical tidal signals for Lhasa and Wuhan Stations and then substracted them from the SG data. We also removed noise signals caused by local air-pressure variations from the data, which may affect the amplitudes of oscillatory modes below 3 mHz (Zürn and Widmer 1995; Hu et al. 2005; Hu et al. 2006). Synthetic normal modes are calculated using the Mineos software package (http://www.geodynamics.org/cig/software/mineos). We perform discrete Fourier transforms on both the corrected data and the synthetic data to get the amplitude spectra.

3 Observations

Figure 2a–c shows ten comparisons among the amplitude spectra at Lhasa. The synthetic spectra, based on solutions 1, 2, 3, 5, and 9 (see Table 1), fits the observations well; however, there are significant differences between the remaining synthetic data and the observations. Table 2 lists these differences between the synthetic data and observations at Lhasa. Synthetic modes derived from solution 6 have the smallest average amplitude, only about 0.37 times the observations. Whereas synthetic modes from solution 10 have the largest average amplitudes, about 1.7 times the observations. The average amplitude of synthetic modes derived from solution 10 is about 5 times larger than that from solution 6. Similar results are observed at Wuhan Station (see Fig. 3a–c).

Fig. 2

a Comparisons among synthetic modes (blue) and SG observations (red) at Lhasa after the Lushan earthquake. Numbers 1–10 denote the synthetic modes derived from the solutions listed in Table 1. b The same as Fig. 1a, but the synthetic modes are derived from solutions 5–7. c The same as Fig. 1a, but the synthetic modes are derived from solutions 8–10

Table 2 Amplitude differences between synthetic modes and observations at Lhasa

Fig. 3

a Comparisons among synthetic modes (blue) and SG (red) observations at Wuhan after the Lushan earthquake. Numbers 1–10 denote the synthetic modes derived from the solutions listed in Table 1. b The same as Fig. 2a, but the synthetic modes are derived from solutions 5–7. c The same as Fig. 2a, but the synthetic modes are derived from solutions 8–10

Both the seismic moment M₀ and the fault motion parameters affect the amplitude of the synthetic data. To distinguish between their respective contributions, we assume that the solutions 6 and 10 in Table 1 have the same M₀ = 1.0 × 10¹⁹ Nm but fault motion parameters are kept unchanged, and then calculate their moment tensor components by the formula (4). We recompute synthetic normal modes for solutions 6 and 10 according to the newly calculated moment tensor components. The results are compared in Fig. 4, showing that the discrepancy between them becomes very small. This indicates that M₀ is the main cause for large discrepancies between synthetic modes and observations. Synthetic modes derived from solutions 1, 2, 3, and 4 in Table 1 fit the observed modes well, suggesting that M_w of 6.6, corresponding to M₀ = 0.97–1.08 × 10¹⁹ Nm, characterize the size and strength of the seismic source of the Lushan earthquake.

Fig. 4

Comparisons among synthetic modes (black dash) and SG (red) observations at Lhasa by assuming that synthetic modes have the same seismic moment M₀ = 1.0 × 10¹⁹. Synthetics 6 and 10 denote the moment motion parameters provided by the solutions 6 and 10 in Table 1

4 Discussions

Scalar seismic moments quantitatively measure the size and strength of a seismic source based on the final static displacement and the rupture area. It is clear that M₀ is associated with displacement at zero frequency. According to Aki (1967), a simple seismic shear source with linear rupture propagation should have a smooth far-field displacement and velocity spectra. The source spectra can be derived from the station spectra after correction of the effects of geometrical spreading and attenuation. According to the ω⁻² source model (Aki 1967), the spectral amplitude of displacement at a seismic source increases linearly with its seismic moment for frequencies f < f_c (corner frequency), and the increase with moment is reduced for f > f_c. The maximum seismic energy is released around f_c because the peak of the ground-velocity spectrum is at f_c. To evaluate M₀ of a great earthquake of M_w ≥ 9.0, the frequencies of seismic waves have to be less than 1.0 mHz, but for a moderate earthquake (about M_w 6.5), the frequencies need to be less than 20 mHz. Thus, normal modes within range of 2–5.5 mHz are sufficient to evaluate the seismic moment of the Lushan event.

The USGS Body-Wave Moment Tensor Solution (USGS BMT) according to Sipkin (1982, 1986) uses 15–55 s passband filtered body waveform data. Therefore, this procedure underestimates M₀ for a really great earthquake, mainly because the frequencies of body waveform are much higher than the corner frequency f_c of the great earthquake. The USGS significant earthquake archives used now do not provide the result of USGS BMT for earthquakes of M_w > 8.0. For a moderate earthquake (5.0 < M_w < 7.0), the M₀ from USGS BMT usually is slightly less than those from Global CMT Project Moment Tensor Solution (Global CMT). That the M₀ predicted by USGS BMT for the Lushan earthquake is about 70 % less than M₀ by Global CMT presumably reflects slow slip which was not detectable from the body waves.

The synthetic modes derived from the moment tensor solutions 1, 2, 3, and 5 in Table 1 fit the observed modes well in a wide range of 2.5–5.5 mHz, suggesting that synchronous processes played a key role in the rupture propagation of the Lushan event. Solutions 5, 6, and 7 used body wave data and similar inversion methods, CAP or modified CAP method (Zhao and Helmberger 1994; Zhu and Helmberger 1996), but they used different station observations, source time, and velocity model of seismic waves. The predicted M₀s from the three solutions differ significantly. This result confirms that the predicted M₀s not only depends on the inversion methodology itself, but also on the details of in the calculation, including data quality, azimuthal distribution of seismic stations, data pre-processing, velocity model, and waveforms used in the inversion. It needs a reliable way to evaluate the values of the predicted M₀s.

M₀s values of solutions 8, 9, and 10 are estimated according to average slip over rupture area, which are results of inversion for rupture process of the Lushan earthquake based on teleseismic body waves and finite-fault models (e.g., Hartzell and Helmberger 1982; Hartzell and Heaton 1983; Olson and Apsel 1982; Ji et al. 2002). Finite-fault rupture models represent the space–time distribution of slip on a parameterized fault surface (usually setting strike and dip of fault slip). Differential times and amplitudes of seismic waves arriving at stations distributed around the source provide the resolution of slip history. The M₀s values provided by the three solutions are directly cacluated by $M_0=\mathit{\mu }·A·\overline{D}$, where μ is the average shear modulus of the medium around the fault, A the surface area of the rupture, and $\overline{D}$ the average displacement after the rupture. The normal mode observations suggest that M₀s values are significantly overestimated in solutions 8 and 10.

5 Conclusions

Although measuring moment magnitude of an earthquake has become a routine work, there are sometimes significant discrepancies among the predicted M₀s values from different research institutions. The practice used in this paper suggests that the predicted M₀ for the moderate earthquake like Lushan event can be well constrained by normal mode observations in the frequency range of 2–5 mHz. It would be worthwhile to employ this method of M₀ evaluation to the general case of a moderate earthquake.