3.1 Inversion of Epicenter Location Based on High-Rate GNSS
The inversion of the epicenter location is an urgent issue that needs to be addressed after an earthquake. Traditional seismology usually employs the Geiger method, a classical linear localization technique(Geiger 1912). Literature (Fang et al.2014) assumes that the propagation speed of seismic waves is equal in different directions. The inversion of the epicenter location using high-rate GNSS displacement waveforms primarily combines the earthquake arrival times and station coordinates of multiple high-rate GNSS stations. Using the least squares adjustment method, it is possible to simultaneously invert for the epicenter location (B, L), the seismic wave propagation speed v, and the origin time T0. Using the method proposed in literature (Fang et al.2014), assuming the epicenter location is (B, L), the epicentral distance for each high-rate GNSS and strong motion station can be calculated as follows:
$${D_{\text{i}}}=\arccos (\sin {B_i}\sin B+\cos {B_i}\cos B\cos ({L_i} - L))R$$
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In the equation, R represents the average radius of the Earth, set at 6371 km; Di is the epicentral distance for each high-rate GNSS and strong motion station to the epicenter location; Bi and Li are the latitude and longitude coordinates of each high-rate GNSS and strong motion station.
Based on the displacement waveforms from 22 high-rate GNSS and strong motion stations, the short-time average/long-time average (STA/LTA) method (Allen and Ziv 2011) is used to determine the arrival times of the seismic waves ti. Assuming the seismic waves propagate at the same speed v in all directions, the arrival times of the seismic waves at each observation station can be determined by the following equation:
$$\left\{ \begin{gathered} {D_2} - {D_1} - v({t_2} - {t_1})=0 \hfill \\ {D_3} - {D_1} - v({t_3} - {t_1})=0 \hfill \\ ...... \hfill \\ {D_{14}} - {D_1} - v({t_{14}} - {t_1})=0 \hfill \\ \end{gathered} \right.$$
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After linearization, the least squares adjustment method can be used to invert for the epicenter location (B, L) and the seismic wave propagation speed v. Based on these, the origin time T0 can be further calculated. The calculation formula is as follows:
$${T_0}=\frac{{\sum\limits_{{i=1}}^{n} {({t_i} - \frac{{{D_i}}}{v})} }}{n}$$
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In the formula, n is the number of high-rate GNSS and strong motion stations.
3.2 Magnitude Inversion Based on High-Rate GNSS Data
The earthquake magnitude is a measure that characterizes the strength of an earthquake. There are four commonly used magnitude scales: local magnitude ML, body wave magnitude MB, surface wave magnitude MS, and moment magnitude Mw. Among them, ML, also known as the Richter magnitude, is determined based on the maximum amplitude recorded by the Wood-Anderson standard seismograph. Because the Wood-Anderson standard seismograph is not widely used, ML is not commonly employed. MB and MS were both proposed by Gutenberg and are frequently used in rapid earthquake reporting. Mw is a physical quantity that represents the absolute size of an earthquake and is directly related to the physical processes at the earthquake source(Hanks 1979). It is the magnitude scale recommended for priority use by the international seismological community today.
The methods commonly used to determine Mw using high-rate GNSS include the finite fault inversion method and the empirical relationship method.
The finite fault inversion method starts from the physical process of the earthquake source. It uses GNSS data to obtain co-seismic displacement, which is then inverted to derive the slip distribution on the fault. From the slip amount, the seismic moment M0 is calculated using the following formula:
$${M_0}=\mu LW\overline {m}$$
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In the formula, L and W are the length and width of the finite fault (unit: km), \(\mu\)is the rigidity coefficient of the medium, usually 30 GPa, and \(\overline {m}\) is the average slip (unit: km). Then, using the relationship between M0 and Mw, the moment magnitude Mw can be calculated:
$${M_w}=\frac{2}{3}\lg {M_0} - 6.033$$
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This method yields the Mw result directly during the slip distribution inversion, serving as an effective validation tool for rapid magnitude assessment using empirical relationships.
The empirical relationship method uses displacement or velocity waveform data obtained from high-rate GNSS, combined with empirical relationships between certain seismic phase amplitudes and magnitude, to obtain Mw. Common methods include the empirical relationship of the horizontal maximum peak displacement Pd within 5 seconds after the P-wave arrival, the peak ground displacement (PGD) empirical relationship method, and the peak ground velocity (PGV) empirical relationship method. According to existing research results, when using the same high-rate GNSS station data for strong earthquake magnitude estimation, the PGD and PGV methods provide more source information over time compared to the Pd method, leading to more accurate magnitude estimation results(Gao et al. 2021).
In this paper, the PGD method is used for magnitude estimation. The PGD values can be extracted based on the three-direction components of the dynamic displacements obtained from high-rate GNSS and strong motion data. The calculation formula is as follows:
$$PGD=\hbox{max} (\sqrt {N_{d}^{2}(t)+E_{d}^{2}(t)+U_{d}^{2}(t)} )$$
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In the formula, \(N_{{\text{d}}}^{{}}(t)\),\(E_{{\text{d}}}^{{}}(t)\) and \(U_{{\text{d}}}^{{}}(t)\) represent the north-south, east-west, and vertical displacement components, respectively. Literature (Crowell et al. 2013) utilized high-rate GNSS data from five earthquakes in the United States and Japan to derive the empirical relationship between PGD and Mw, with the regression model being:
$$\log (PGD)=A+B \times {M_{\text{w}}}+C \times {M_{\text{w}}} \times \log (R)$$
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$${M_{\text{w}}}=\frac{{\log (PGD) - A}}{{B+C \times \log (R)}}$$
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In the formula, R represents the epicentral distance (unit: km), and A, B and C are regression coefficients. The coefficients are shown in Table 1. Additionally, literature summarized a series of regression models for magnitude estimation based on more earthquake cases, with the coefficients also listed in Table 1. To avoid the effects of earthquake rupture directionality and station location amplification, and to ensure the reliability of the magnitude estimation results, the average value of all station magnitude estimates is taken as the final magnitude estimate.
Table 1
PGD magnitude estimation model coefficients
Number | A | B | C | Source | Note |
1 | -5.013 | 1.219 | -0.178 | Literature(Crowell et al. 2013) | PGD in cm |
2 | -4.434 | 1.047 | -0.138 | Literature(Melgar et al. 2015) |
3 | -6.687 | 1.500 | -0.214 | Literature(Crowell et al. 2016) |
4 | -5.919 | 1.009 | -0.145 | Literature(Ruhl et al. 2019) | PGD in m |