Methods for estimating rotational components of seismic ground motion and their numerical comparisons

Chang Chen; Yun Wang; Yongxiang Wei; Shuilong Li; Qisheng Zhang

doi:10.29382/eqs-2020-0201-04

Earthquake Science > 2020 > 33(4): 201-208. > DOI: 10.29382/eqs-2020-0201-04

Chang Chen, Yun Wang, Yongxiang Wei, Shuilong Li, Qisheng Zhang (2020). Methods for estimating rotational components of seismic ground motion and their numerical comparisons. Earthq Sci 33(4): 201-208. DOI: 10.29382/eqs-2020-0201-04

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Methods for estimating rotational components of seismic ground motion and their numerical comparisons

1.
MWMC Research Group, School of Geophysics and Information Technology, China University of Geoscience, Beijing 100083, China
2.
Fujian Earthquake Agency, Fuzhou 530001, China

More Information

Corresponding author:
Yun Wang, e-mail: wangyun@mail.gyig.ac.cn
Received Date: 17 Mar 2020
Revised Date: 08 Oct 2020
Available Online: 28 Dec 2020
Published Date: 07 Dec 2020

Graphical Abstract

Abstract

Abstract

Rotational components play an important role in natural earthquake research, engineering seismic investigation, building monitoring, seismic exploration and other fields. Traditional researches mainly focus on three translational components, but less on rotational ones. As the precision of rotational sensing techniques has increased, many scholars have paid more attention to the seismic rotational motions. Because the rotational observations are not very popular before and now, approximately converting the translational components into rotational components is utilized in rotation analysis. Based on numerical six-component seismic data with the finite difference method, we compare three different conversion methods, the travelling-wave, frequency-domain and the difference method, to analyze their characteristics and feasibilities when they are applied to estimate rotational components with translational observations.
- rotational components,
- travelling-wave method,
- frequency-domain method,
- difference method

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1. Introduction

Ground motion is multi-dimensional freedom during earthquakes (Lee et al., 2007). In recent years, a lot of progress has been made in seismic multi-component research at home and abroad, which mainly concentrated on the three translational components. In addition to the translational components in fact, there are three rotational components in the earthquake motion, as illustrated in Figure 1. The three translational and rotational components along three coordinate axes of Cartesian coordinate all together form the complete six-component tensors field (Wang et al., 2017).

Figure 1. The six components in earthquake motion. (a) v_x, v_y and v_z are the three translational components; (b) r_x, r_y and r_z are the three rotational components. Traditional three-component seismic research mainly focuses on translational components, while six-component seismic research adds three rotational components to the discussion. Physically speaking, the six components are independent of each other

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It is of great significance to study the rotational ground motions in the research of passive earthquake and active seismology (Lee et al., 2007; Lee et al., 2009). In recent years, with the development of electrochemical sensing, laser sensing, optical fiber and other instrument techniques, the precision of rotational sensors has been continuously improved. And many scholars have conducted more in-depth research on the rotational motions and made a lot of progress. Igel et al. (2011) presented observations of rotational motions induced by free oscillations of the Earth after the M_W 9.0 Tohoku‐Oki earthquake of 2011, showed that rotational components provide benefits for observations of Earth’s long period ground motions and enhance source and Earth structure imaging. Hadziioannou et al. (2012) used ring laser to observe translational motion, showed that the measurements of rotational and translational motions in ambient noises at a single location could be used to make observations consistent with traditional methods which require arrays of translational instruments. Barak et al. (2014) conducted two six-component field experiments, validated that the recording of rotational sensors is agreed with that calculated with closely spaced translational geophones. Through employing the rotational components in singular value decomposition (SVD), it was verified that the rotation data were helpful to identify different wave modes and filter out undesired wave modes without spatial sampling requirements. Bernauer et al. (2014) ever had performed inversions for two scenarios, one case only translational velocity recorded, and the other one with half of the velocity records randomly replaced by rotational records, and found that the rotational ground motions were helpful to improve predictions of the rise time and rupture velocity.

There are two main kinds of methods to obtain seismic rotational components: recording the rotation directly by the rotational sensors (Nigbor, 1994; Takeo, 1998; Liu et al., 2010), and indirectly converting translational components into rotational components. Because the rotational observations are not very popular before and now in China, observing three translational components and then indirectly estimating rotational components becomes more general, significant strategy and alternative. Through numerical simulations of the six-component seismic field with finite difference method, this paper compares and analyzes the differences and feasibility conditions of several approximate conversion methods.

2. Theories and methods

2.1 Rotation tensor

Based on the theory of elastic wave, Green tensor

${E}_{ij}=\frac{1}{2}\left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}+\frac{\partial {u}_{k}}{\partial {x}_{i}}\frac{\partial {u}_{k}}{\partial {x}_{j}}\right) \text{,}$

(1)

can be defined as combinations of the strain tensor ${e}_{ij}$ and the rotational tensor ${r}_{ij}$ (Hu, 1989)

${E}_{ij}={e}_{ij}+\frac{1}{2}{e}_{ij}^{2}+\frac{1}{2}\left({e}_{ij}{{r}_{ij}{-r}_{ij}e}_{ij}\right)-\frac{1}{2}{r}_{ij}^{2}\text{.}$

(2)

When defining

${e}_{ij}=\frac{1}{2}\left(\frac{\partial {u}_{j}}{\partial {x}_{i}}+\frac{\partial {u}_{i}}{\partial {x}_{j}}\right)\text{,}$

(3)

${r}_{ij}=\frac{1}{2}\left(\frac{\partial {u}_{j}}{\partial {x}_{i}}-\frac{\partial {u}_{i}}{\partial {x}_{j}}\right)\text{,}$

(4)

where the strain tensor ${e}_{ij}$ and the rotation tensor ${r}_{ij}$ are symmetrical and antisymmetric, respectively. Under the conditions of smaller and linear deformation assumed in the linear theory of elastic dynamics, Green tensor can be approximately equal to the strain tensor, i.e.

${E}_{ij}={e}_{ij}\text{,}$

(5)

which is one of the basic assumptions supporting the modern linear elastics. So in the three-dimensional Cartesian coordinate system, equation (4) can be rewritten as

$\left\{ {\begin{array}{*{20}{c}} {{r_x} = \dfrac{1}{2}\left( {\dfrac{{\partial {u_z}}}{{\partial y}} - \dfrac{{\partial {u_y}}}{{\partial z}}} \right)}\\ {{r_y} = \dfrac{1}{2}\left( {\dfrac{{\partial {u_x}}}{{\partial z}} - \dfrac{{\partial {u_z}}}{{\partial x}}} \right)}\\ {{r_z} = \dfrac{1}{2}\left( {\dfrac{{\partial {u_y}}}{{\partial x}} - \dfrac{{\partial {u_x}}}{{\partial y}}} \right)} \end{array}} \right. \text{,}$

(6)

where ${u}_{x}$ , ${u}_{y}$ , ${u}_{z}$ are displacements along X-, Y-, and Z-axis respectively, and ${r}_{x}$ , ${r}_{y}$ , ${r}_{z}$ are three rotational components.

2.2 Finite difference simulation

Based on the basic theory of the first-order velocity-stress equation (7) and elastic wave equation (8), we can use the staggered grid finite difference method to deduce the discrete formula of elastic wave velocity and stress under the two-order time difference precision, then the simulation of six-component seismic wave field can be approximately estimated by equations (6)−(8) (Sun, 2018).

$\left\{ {\begin{array}{*{20}{c}} {\dfrac{{\partial {\sigma _{xx}}}}{{\partial x}} + \dfrac{{\partial {\sigma _{xy}}}}{{\partial y}} + \dfrac{{\partial {\sigma _{xz}}}}{{\partial z}} = \rho \dfrac{{\partial {v_x}}}{{\partial t}}}\\ {\dfrac{{\partial {\sigma _{yx}}}}{{\partial x}} + \dfrac{{\partial {\sigma _{yy}}}}{{\partial y}} + \dfrac{{\partial {\sigma _{yz}}}}{{\partial z}} = \rho \dfrac{{\partial {v_y}}}{{\partial t}}}\\ {\dfrac{{\partial {\sigma _{zx}}}}{{\partial x}} + \dfrac{{\partial {\sigma _{zy}}}}{{\partial y}} + \dfrac{{\partial {\sigma _{zz}}}}{{\partial z}} = \rho \dfrac{{\partial {v_z}}}{{\partial t}}} \end{array}} \right. \text{,}$

(7)

$\left\{ {\begin{array}{*{20}{l}} {\dfrac{{\partial {\sigma _{xx}}}}{{\partial t}} = \left( {\lambda + 2\mu } \right)\dfrac{{\partial {v_x}}}{{\partial x}} + \lambda \dfrac{{\partial {v_y}}}{{\partial y}} + \lambda \dfrac{{\partial {v_z}}}{{\partial z}}}\\ {\dfrac{{\partial {\sigma _{yy}}}}{{\partial t}} = \lambda \dfrac{{\partial {v_x}}}{{\partial x}} + \left( {\lambda + 2\mu } \right)\dfrac{{\partial {v_y}}}{{\partial y}} + \lambda \dfrac{{\partial {v_z}}}{{\partial z}}}\\ {\dfrac{{\partial {\sigma _{xx}}}}{{\partial t}} = \lambda \dfrac{{\partial {v_x}}}{{\partial x}} + \lambda \dfrac{{\partial {v_y}}}{{\partial y}} + \left( {\lambda + 2\mu } \right)\dfrac{{\partial {v_z}}}{{\partial z}}}\\ {\dfrac{{\partial {\sigma _{xy}}}}{{\partial t}} = \mu \left(\dfrac{{\partial {v_y}}}{{\partial x}} + \dfrac{{\partial {v_x}}}{{\partial y}}\right)}\\ {\dfrac{{\partial {\sigma _{xz}}}}{{\partial t}} = \mu \left(\dfrac{{\partial {v_z}}}{{\partial x}} + \dfrac{{\partial {v_x}}}{{\partial z}}\right)}\\ {\dfrac{{\partial {\sigma _{yz}}}}{{\partial t}} = \mu \left(\dfrac{{\partial {v_z}}}{{\partial y}} + \dfrac{{\partial {v_y}}}{{\partial z}}\right)} \end{array}} \right. \text{,}$

(8)

where $\,\rho$ presents the density, $\sigma$ presents the stress, $\lambda$ and $\, \mu$ are Lame constants, ${v}_{x},{v}_{y},{v}_{z}$ are the velocities along three axes.

2.3 Difference method

There are mainly the following two kinds of methods to convert the translational components into rotational ones (Li et al., 2001):

(1) difference method with translational records of dense stations;

(2) travelling-wave method (TWM) developed from the elastic theory with one translational station.

The theoretical basis of the difference method is the smaller deformation dynamics theory in linear elastic medium. When the distances between observation stations are smaller enough, the partial differential term in equation (6) can be simplified as differences (Lin et al., 2009; Li et al., 2021). For example, assuming there are three adjoining stations numbered A, M and N, A is at the origin of coordinate system, M and N are on the X-axis and Y-axis respectively, with the Z-axis downward perpendicularly. Then the three rotational components can be converted as

$\left\{ {\begin{array}{*{20}{l}} {{r_x} = \dfrac{{u_z^N - u_z^A}}{{\Delta y}}}\\ {{r_y} = - \dfrac{{u_z^M - u_z^A}}{{\Delta x}}}\\ {{r_z} = \dfrac{1}{2}\left( {\dfrac{{u_x^M - u_x^A}}{{\Delta y}} - \dfrac{{u_y^N - u_y^A}}{{\Delta x}}} \right)} \end{array}} \right. \text{,}$

(9)

where ${u}_{x}$ , ${u}_{y}$ , ${u}_{z}$ are the translational components, the superscripts A, M, N are the number of stations, $\Delta x$ is the interval of stations A and N, $\Delta y$ is the interval of stations A and M.

Equation (9) indicates that the approximations of r_x and r_y require translational records of two stations, while r_z requires three. Therefore, the difference method used for r_x and r_y components is called the two-point difference method, and the difference method for r_z component is called the three-point difference method.

2.4 Travelling-wave method and frequency-domain method

At the end of 1960s, Newmark first put forward the basic idea of travelling-wave method for estimating rotational components based on travelling wave theory (Newmark, 1969; Newmark et al., 1972; Nathan et al., 1975). Assuming that there is an apparent wave velocity ${C}_{\rm{a}}$ , which presents the velocity of seismic wave propagating along the surface (Wang et al., 1991), as shown in Figure 2, and setting the azimuth $\theta$ denotes the angle between the wave propagation direction and the $X$ -axis, then the apparent wave velocities ${C}_{x}$ and ${C}_{y}$ in the two horizontal directions can be obtained through the apparent wave velocity ${C}_{\rm{a}}$ by

Figure 2. Schematic diagram of apparent wave velocity decomposition. The velocity of wave propagation

${C}_{\rm{a}}$ can be decomposed into the apparent wave velocity

${C}_{x}$ along the X-axis and

${C}_{y}$ along Y-axis

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${C}_{x}={C}_{\rm{a}}/{\cos}\theta , \;{C}_{y}={C}_{\rm{a}}/{\sin}\theta \text{.}$

(10)

Assuming that the earth medium is homogeneous and elastic isotropy, and the seismic waves propagate in a stable direction at a constant velocity, then the equation (6) can be indirectly converted as

$\left\{ {\begin{array}{*{20}{l}} {{r_x}\left( t \right) = \dfrac{1}{{{C_{\rm{a}}}}}{u_z}\left( t \right){\rm{sin}}\theta }\\ {{r_y}\left( t \right) = \dfrac{1}{{{C_{\rm{a}}}}}{u_z}\left( t \right){\rm{cos}}\theta }\\ {{r_z}\left( t \right) = \dfrac{1}{2} \dfrac{1}{{{C_{\rm{a}}}}}\left[ {{u_y}\left( t \right){\rm{cos}}\theta - {u_x}\left( t \right){\rm{sin}}\theta } \right]} \end{array}} \right. \text{,}$

(11)

where $t$ presents the seismic wave is a function of time.

Equation (11) is the basic formulas for converting rotational components by travelling-wave method in time domain. Considering the dispersion effect of seismic waves, the rotational components can also be converted by using travelling wave theory in the frequency domain (Trifunac, 1982), which is called frequency-domain method (FDM). Through Fourier transform of the translational components and inverse Fourier transform, the traveling-wave method in frequency-domain can be completed with following equation (Li, 1991; Li et al., 2001)

$\left\{ {\begin{array}{*{20}{l}} {{r_x}\left( t \right) = \dfrac{{{\rm{i}}\omega }}{{{C_{\rm{a}}}}}{u_z}\left( t \right)\sin \theta }\\ {{r_y}\left( t \right) = \dfrac{{{\rm{i}}\omega }}{{{C_{\rm{a}}}}}{u_z}\left( t \right)\cos \theta }\\ {{r_z}\left( t \right) = \dfrac{1}{2} \dfrac{{{\rm{i}}\omega }}{{{C_{\rm{a}}}}}\left[ {{u_y}\left( t \right)\cos \theta - {u_x}\left( t \right)\sin \theta } \right]} \end{array}} \right.,$

(12)

where $\omega$ presents the circle frequency, ${\rm{i}}$ presents the imaginary.

3. Conversion

In order to verify the feasibility of the conversion methods, we perform finite difference simulation on a model and compare the difference between the converted rotations and the simulated ones.

Under the condition of smaller deformation, a horizontal layered model with isotropic and homogeneous assumptions is established, and then finite difference simulation is performed on it (Sun et al., 2018). The model is set with the first layer depth of 50 m and the second layer depth of 70 m, as illustrated in Figure 3. The physical parameters of each layer are illustrated in Table 1, where ${v}_{{\rm{P}}}$ and ${v}_{{\rm{S}}}$ present the velocity of P- and S-wave respectively, and $\rho$ presents the density.

Figure 3. Structure diagram of the model. Two receiving-lines (red) are set on the surface of the first layer; A, B, C are the three recording stations on the receiving-lines. Orange and red cuboids are two layers of the medium, and the blue part outside the medium is the absorbing boundary condition of splitting perfectly matched layer (SPML)

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Table 1. Physical parameters of the model

Layer	v_P(m/s)	v_S(m/s)	$\rho$ (kg/m³)
1	2 000	1 400	2 600
2	3 000	2 100	2 700

| Show Table

DownLoad: CSV

The model size is set as 40 m (length) ×40 m (width) ×120 m (height), and the discrete grid interval is 1 meter in each direction. A P-wave source is allocated at the depth $z=5 \;{\rm{m}}$ below the surface of the first layer with the plane coordinates: $x=22 \;{\rm{m}}$ and $y=22 \;{\rm{m}}$ . The dominant frequency of Richer wavelet is 120 Hz. The top boundary of the model is a free surface. At the bottom and side boundaries of the model, we use a splitting perfectly matched layer (SPML) (Sun, 2018) with a thickness of 30 m to eliminate the influence of boundary reflections.

On the surface of the first layer, a main receiving-line is set along the X-axis direction at the position of $y=32\; {\rm{m}}$ , and an auxiliary receiving-line is set along the Y-axis direction at the position of $x=22 \;{\rm{m}}$ . Station A (22 m, 32 m) is at the intersection point of the two receiving-lines, stations B (26 m, 32 m) and C (30 m, 32 m) are 4 meters and 8 meters away from station A on the main receiving-line respectively. The sampling interval of time d_t is set as 0.1 ms, recording length ${t}_{{\rm{max}}}$ is equal to 100 ms with 1 000 samples in each station.

After simulating the three-component translations, the three-component rotations could be estimated by travelling-wave method and frequency-domain method. Because the estimated rotations are weaker than simulated ones, and there is about 10 to 1 000 times amplitude difference, it is necessary to normalize rotational motions so that they can be compared directly and visually.

To quantitatively compare the conversion accuracy of each method, the first 50 ms seismic records of the stations A, B and C, i.e. the first arrivals, are normalized, and the correlation coefficients between simulated rotations and estimated rotations are illustrated in Table 2.

Table 2. Waveform correlation coefficients between the simulated and converted rotations

Station	Method	Correlation coefficient
Station	Method	r_x	r_y	r_z
A	TWM	0.875 794	0.875 795	0.668 701
A	FDM	0.947 949	0.973 025	0.657 353
B	TWM	0.796 342	0.796 342	0.759 599
B	FDM	0.959 054	0.957 479	0.962 875
C	TWM	0.524 416	0.524 416	0.798 854
C	FDM	0.880 858	0.872 079	0.951 143

| Show Table

DownLoad: CSV

Then, spectrum analysis is executed on the rotational components of station A and the correlation coefficients of the amplitude spectrum are calculated, as shown in Table 3. The time-domain waveforms and amplitude spectrum are illustrated in Figures 4 and 5.

Table 3. Amplitude spectrum correlation coefficients between the simulated and converted rotations in station A

Method	Correlation coefficient
Method	r_x	r_y	r_z
TWM	0.941 690	0.941 690	0.995 362
FDM	0.982 657	0.983 524	0.978 669

| Show Table

DownLoad: CSV

Figure 4. Waveform comparisons between the simulated and converted rotations of station A. (a)−(c) is the comparisons between simulated and TWM rotations; (d)−(f) is the comparisons between simulated and FDM rotations

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Figure 5. Amplitude spectrum comparisons between simulated and converted rotations of station A. (a)−(c) is the comparisons between simulated and TWM rotations; (d)−(f) is the comparisons between simulated and FDM rotations. The dominant frequency of simulated r_x is about 140 Hz, while dominant frequencies of converted r_x are about 190 Hz (TWM) and 170 Hz (FDM). The discrepancy is also obvious for the r_z of FDM

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Through comparisons, it’s obvious that, the waveform of converted rotations fits the simulated ones very well, and the correlation coefficients between the simulated and frequency-domain method rotations are generally higher than those with travelling-wave method. The amplitude spectrum comparisons indicate discrepancies of the dominant frequency between simulated and calculated rotations, which up to 40 Hz in r_x-component converted by TWM.

For further comparison with difference method, a pair of reference stations is set located on the two receiving-lines with a distance of 1 meter from station A, then the rotational components are estimated by difference method, and the spectrum distribution is illustrated in Figure 6. After that, the other five pairs of reference stations are set with distances of 2 meters to 6 meters, and the correlation coefficients are shown in Table 4 and Figure 7.

Figure 6. Comparisons between difference method (DM) converted and simulated rotations. (a)−(c) is the waveform comparisons; (d)−(f) is the amplitude spectrum comparisons. The discrepancies of dominant frequency between simulated and calculated rotations are more obvious for difference method, which up to 50 Hz

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Table 4. Waveform correlation coefficients between the simulated and converted rotations by DM in station A

Reference station distance (m)	Correlation coefficient
Reference station distance (m)	r_x	r_y	r_z
1	0.877 902	0.863 620	0.654 201
2	0.864 612	0.861 864	0.493 617
3	0.817 992	0.855 324	0.330 769
4	0.731 687	0.844 936	0.179 654
5	0.607 310	0.829 621	0.050 284
6	0.458 299	0.808 423	−0.050 765

| Show Table

DownLoad: CSV

Figure 7. Variation curve of correlation coefficients between the simulated and converted rotations by difference method (DM) with reference station distance. As the distance of reference station increases, the accuracy of the DM decreases obviously, and the r_z is more affected by the reference station distance than r_x and r_y components

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Comparison diagrams shown in Figure 6 indicate that rotational components converted by difference method can also accurately fit the simulated values. Under the condition of the minimum station interval of 1 meter in the model, the fitting degree of difference method is similar to that of travelling-wave method, which is slightly lower than frequency-domain method.

4. Conclusions

(1) Travelling-wave method and frequency-domain method can approximately convert translational components into rotational components to a certain extent. The converted value has high correlation with the simulated value in both the waveforms and frequency spectrum.

(2) The frequency-domain method is generally more accurate than the travelling-wave method and the difference method in the small idealized layered model. However, frequency-domain method needs longer computational time.

(3) The difference method requires dense stations as reference, and the distance of the reference stations greatly affects the accuracy of the rotational components, especially the r_z.

(4) The six components in earthquake motions are independent of each other, therefore, both the three conversion methods and the finite difference simulation method are approximate estimates under the assumption of smaller deformation, which cannot replace the actual observation of rotational components.

Under the condition of smaller deformation, this paper mainly focuses on the model data with idealized source and constant wave velocity. Considering different source types, epicenter distances and underground geological structures, more six-component observing data should be used to further study the conversion methods.

Acknowledgement

This research is supported by the National Natural Science Foundation of China (grant No. U1839208). We sincerely thank the two reviewers for their suggestions, which have substantially improved the manuscript. We are also very grateful to all editors for their kind help.

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Article Contents

Abstract

1. Introduction

2. Theories and methods

3. Conversion

4. Conclusions

Acknowledgement

References

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Methods for estimating rotational components of seismic ground motion and their numerical comparisons