The apparent focal depth, emergence angle, and take-off angle of seismic wave measured by YRY-4-type borehole strainmeter as one kind of strain seismograph

Introduction: In theory, the observation objects and principles of strain seismograph and traditional pendulum seismograph are different, and the characteristics of observed signals should also be dissimilar. The observation results of pendulum seismograph show that seismic waves in inhomogeneous media will undergo refraction, reflection, and attenuation. Then, what signal characteristics can be detected by strain seismograph is great significance for understanding and explaining the observation results. Methods: Using YRY-4 type four-gauge borehole strainmeter as one kind of strain seismograph to detect the strain tensor change of the plane seismic wave emitted from the surface, a five-site strain seismograph observation network was built in Shanxi Province, with continuous observation for 2 years at a sampling rate of 100 Hz. In this paper, two local events occurring in the area covered by the strain seismograph observation network are taken as examples. We systematically studied the characteristics of seismic wave signals recorded by strain seismographs at five sites, inverted for the focal depth of the two local earthquakes and the relationship between the wave velocity and the wave velocity gradient of the focal depth, and calculated the apparent focal depth, the emergence angle and the take-off angle of seismic waves. Results: These results show stable uniqueness and apparent regularity, especially since the inverted focal depths are basically consistent with the seismic solutions based on those traditional pendulum seismographs. The observations from this study show that the strain seismograph can be used as an effective supplement to the pendulum seismograph. Discussion: In the future, we will continue to study the rupture process and focal mechanism of moderate-strong earthquakes and teleseismic earthquakes by combining two kinds of observations.


Introduction
Since the 1960s, relative stress observations have been carried out in the United States, China, Japan and other countries (Sacks et al., 1971;Gladwin, 1984;Ishii, 2001); borehole strain observation is a vital observation method gradually developed from relative stress observation. Scholars in mainland China have developed RZB-type four-component borehole strainmeter (Ouyang, 1977), TJ-type borehole volume strainmeter (Su, 1982), and YRY-type four-component borehole strainmeter (Chi et al., 2009), and continuously improved the observation technology. It has developed to more than 130 stations equipped with borehole strainmeters. These borehole strain observations can record clear solid tides and play an essential role in the field of earthquake monitoring and prediction in mainland China (Huang et al., 2017).
Unlike GTSM strainmeter such as the United States, which can produce high frequency data of 20 Hz, the data sampling rate of strainmeter in mainland China is minute sampling due to data storage and transmission technology. Using these minute data, scholars in mainland China have carried out a lot of research, including the tidal Variation and calibration technology (Qiu et al., 2015), abnormal changes of different earthquakes (Qiu et al., 2010;Liu et al., 2014), co-seismic strain steps (Gong et al., 2019;Li et al., 2020), free oscillation of the earth excited by large earthquakes (Tang et al., 2007;Tang et al., 2008;Qiu, 2017), and also studied and analyzed the influence of water level, air pressure, rainfall and other factors on borehole strain, and studied the data processing methods to identify and eliminate these influencing factors (Zhou et al., 2008;Zhang and Huang, 2011;Zhang et al., 2015). However, these studies are limited to longer period signals because of the low sampling rate.
Numerous studies have shown that the borehole strainmeter can be used for seismic wave observation (Byerly, 1926;Johnston et al., 1986;Borcherdt and Glassmoyer, 1989;Borcherdt et al., 2006;Johnston et al., 2006;Blum et al., 2010;Barbour and Agnew, 2012;Qiu et al., 2015;Barbour and Crowell, 2017;Canitano et al., 2017;Cao et al., 2018;Farghal et al., 2020;Barbour et al., 2021). This initially realized seismologists' expectations of strain seismographs (Benioff et al., 1961;Aki and Richards, 2002). Although the current horizontal component borehole strainmeter can only detect the two-dimensional strain variation in the horizontal plane, the seismic wave is usually approximated as a plane wave, and both have only three independent components, which can convert to each other. Therefore, this instrument can still be used to observe and analyze the strain variation of seismic waves.
Notably, strain seismographs are different from traditional pendulum seismographs. The traditional pendulum seismograph measures a vector (displacement), while the strain seismograph detects a tensor (displacement gradient). The strain seismograph can provide new information that can be used to carry out further research. One of the most critical research directions is the inversion of focal mechanism with strain seismic observation. Theoretically, the source moment tensor  can be converted from the data of at least two strain observation points in different directions. Solving the focal mechanisms of local earthquakes by strain seismic observation can be used as supplementary information for pendulum seismograph observation.
The determination of the focal mechanism by seismic strain observation is theoretically based on the assumption of an ideal homogeneous medium, but the actual crustal strata are inhomogeneous. Figure 1 shows the geometrical interpretation of seismic wave propagation in homogeneous and inhomogeneous media. In a homogeneous medium, the angle (take-off angle) i e between the seismic wave ray from the hypocenter and the normal direction of the ground remains unchanged when the seismic wave propagates linearly, which is equal to the emergence angle i 0 when reaching the ground. Seismic waves propagating in non-uniform strata can reflect and refract, and the rays are not straight lines. When the focal depth cannot be ignored, the emergence angle i is generally not equal to the take-off angle, i e . This makes it difficult to solve the focal mechanism by seismic strain observation. The crust is separated from the mantle by the Moho surface, and it can be considered that there is no similar interface within the crust (Wyllie PJ, 1963). However, crustal rock is not a homogeneous medium. Basically, the wave velocity generally linearly increases with depth. As a first-order approximation, it can be considered that the gradient of wave velocity of crustal rocks with depth is a constant; this is defined as a constant velocity gradient model (Stein and Wysession, 2003;Wan, 2016). Under this condition, for local earthquakes, when the observation site is very close to the epicenter (within tens of kilometers), because the seismic waves only propagate in the crust, the initial motion is a direct wave, so the situation becomes relatively simple.
As the technical management and planning team of the borehole strain network in Mainland China, we have upgraded and updated the high-sampling data collectors of 10 four-component borehole strainmeters which are producing high frequency data of 100 Hz since 2017, only one site could produce high-frequency data before. Using these data, the ability of four-component borehole strainmeters to record strain seismic waves of earthquakes of different magnitudes has been studied (Qiu et al., 2015;Tang et al., 2022), the study of determining the strain magnitude of seismic surface wave based on borehole strain seismic wave is carried out , Shi et al. (2021) studied the direct observation of co-seismic static stress deviation changes consistent with the theoretical prediction. These sites can record the seismic waves of earthquakes with M6.0 or above in the world (Tang et al., 2022), and can also record the local earthquakes in the area where the site is located. In this article, two earthquakes are experimentally examined through seismic strain observation using a high-density borehole strainmeter network in Xinzhou, Shanxi Province, China. The results show that there is an obvious rule that Frontiers in Earth Science frontiersin.org 02 the emergence angle of seismic waves reaching the observation sites varies with the epicentral distance, which can be simulated by the crustal wave velocity structure model with constant velocity gradient. On this basis, not only the focal depth can be calculated, the take-off angle of seismic wave rays arriving at each observation site can be determined.

Data
YRY-4 type of four-gauge borehole strainmeter (FGBS) is an important instrument for crustal deformation observation in china; the overall composition of the borehole observation  was designed by us as the drafters for the construction of the seismic industry standard borehole strain stations in mainland China and the four sensors of the YRY-4-type FGBS (Qiu et al., 2013;Li et al., 2021) are shown in Figure 2. The four gauges are arranged at 45°intervals, Si (i =1, 2, 3, 4) is the measurement obtained from each of the four gauges; it directly measure the change in diameter in the corresponding azimuth that results from changes in strain state. Since 2012, the YRY-4 -type FGBS has been used in Xinzhou area of Shanxi Province, and the sites such as Shenchi (SC), Fanshi (FS), Yuanping (YP), Daixian (DX), and Ningwu (NW) have been successively established, forming a high-density local observation network, as shown in Figure 3, and the measurement information of 5 sites are in Table 1. In 2018, we conducted an earthquake monitoring experiment using this observation network with a sampling rate of 100 Hz. Presently, more than 2 years of data have been accumulated. We have been very fortunate to capture two small local seismic events (YP M2.8 and DX M3.0) within the observation network (Figure 3), Table 2 gives the parameters of the two local events calculated by traditional pendulum seismograph, which are provided by the China Seismic Network.
YRY-4-type FGBS has good observation performance and has achieved great success in the study of geodynamic problems (Chi et al., 2009;Qiu et al., 2013;Liu et al., 2014). Some scholars have also studied the theoretical issues and feasibility of using the observed seismic strain waves (Zhang et al., 2019;Zhang et al., 2020;Zhang et al., 2021), but there has been no actual observation to verify the results. For the two earthquakes, the P and S waves of seismic strain waves recorded by the YRY-4-type FGBS at all the sites are apparent in Figure 4.
Based on NW in Figure 4, Figure 5 shows the more apparent seismic wave curve of YP M2.8 earthquake recorded at the NW site. For such local earthquakes, the first pulse (direct wave P g and S g ) reaching the observation site is the most direct reflection of the earthquake hypocenter, which is extremely valuable for examining the focal mechanism. In this article, the extreme value (wave peak or trough) of this pulse is called the initial motion. The arrow in Figure 5 indicates the selection time of direct strain wave observation data.
It should be noted that the observation curve in Figure 5 does not show the real strain variation. The borehole strainmeter directly measures the relative diameter change of the probe sleeve, which requires calibration and conversion to obtain the strain variation in the rock layer (Qiu et al., 2013). The long-period signals, such as hydrological and tidal effects have no effect on high-frequency seismic wave signals, so it is reasonable to remove the linear trend of the observed record only by fitting. Table 3 and Table 4 show the observed strains of the initial motions of P and S waves of the two earthquakes in Table 2. ε 11 , ε 12 , and ε 22 are three strain FIGURE 2 (A)The overall composition diagram of a standard borehole strain observation; (B) Schematic diagram of a standard borehole strainmeter, S1, S2, S3, S4 sensors in 4 directions,ε 1 and ε 2 represent the maximum and minimum principal strains, respectively.

FIGURE 3
Map of the observation sites and the two local events.

Frontiers in Earth Science
frontiersin.org components in the horizontal plane of the geographic coordinate system obtained by calibration and conversion.

Methods and models
3.1 Strain conversion of emergent wave

Hypothesis
To discuss the evaluation of actual seismic wave strain using the observed horizontal strain, there are some hypotheses: Firstly, it is assumed that the epicenter location is known. Since Shanxi Province has a relatively dense pendulum seismograph observation network, it can be considered that this epicenter location is reliable. Secondly, the seismic wave near the earthquake hypocenter must not be an ideal plane wave, but the seismic wave reaching the observation site can be treated as a plane wave. Finally, the local earthquake events are discussed here, so it can be assumed that the path deviations of the P wave and S wave from the source to the observation site are small, which can be regarded as approximate values . Practical calculations also confirm these hypotheses.

Fundamental formula
In the geographic coordinate system shown in Figure 6, the strain tensor of the seismic waves arriving at the observation site is denoted as However, the current borehole strainmeter can only observe symmetrical horizontal strain variation, i.e., only the three independent strain components (ε 11 , ε 12 , and ε 22 ) shown in the dashed frame of Eq. 1, the vertical strain components (ε 13 , ε 23 , and ε 33 ) are not considered.
According to the above assumptions, the seismic waves reaching the observation site can be regarded as plane waves. In the seismic ray coordinate system shown in Figure 6, where the seismic waves propagate along the x' 1 axis, the strain tensor should be written as (Qiu, 2017) where ε P R is the P wave strain, ε SH R is the SH wave strain, and ε SV R is the SV wave strain. It should be noted that ε 12 ′ is parallel to the ground, while ε 13 ′ is perpendicular to the ground. It is also noteworthy that ε R is asymmetric.
The relationship between the horizontal strain detected at the observation site and the actual plane seismic wave strain should conform to the following coordinate transformation equation: where the direction chord matrix is defined as follows: l GR l 11 l 12 l 13 l 21 l 22 l 23 l 31 l 32 l 33 The column vectors of l GR are the direction vectors of the axes of the seismic wave ray coordinate system (x' 1 , x' 2 , x' 3 ) in the geographic coordinate system (x 1 , x 2 , x 3 ).
In fact, P and S waves propagate separately, while SH and SV waves propagate together. For P waves, for the strain component that can be observed by the borehole strainmeter, Eq. 3 can be written as follows: The ε 11 ′ of the observation site can be calculated by Eq. 5. For S wave, Eq. 3 can be written as ε S 11 2ε 12 ′ l 11 l 12 + 2ε 13 ′ l 13 l 11 ε S 22 2ε 12 ′ l 21 l 22 + 2ε 13 ′ l 23 l 21 ε S 12 ε 12 ′ l 11 l 22 + l 12 l 21 ( ) + ε 13 ′ l 13 l 21 + l 11 l 23 ( ) ε 12 ′ and ε 13 ′ of the observation site can be calculated from Eq. 6.   Frontiers in Earth Science frontiersin.org 05 As shown in Figure 6, the emergence angle is expressed as i π − θ. The components of l GR can be expressed by azimuth angle φ and emergence angle i: l 11 sin i cos φ l 21 sin i sin φ l 31 −cos i l 12 −sin φ l 22 cos φ l 32 0 l 13 −cos i cos φ l 23 −cos i sin φ l 33 sin i

Practical algorithm
In addition to the known observations, the azimuth φ can be calculated from the epicenter position for an observation site. In other words, we are facing with an overdetermined problem; four unknowns need to be solved by Eqs 5, 6: ε 11 ′ , ε 12 ′ , ε 13 ′ , and the emergence angle i.
In a homogeneous medium, if the epicentral distance d and focal depth h are known, the emergence angle i can be easily calculated. However, on the one hand, although the epicentral distance d is generally reliable, the focal depth h given by the existing methods is not reliable; on the other hand, the crustal strata are not a homogeneous medium. Therefore, the emergence angle i cannot be given in advance.

Site
Epicentral distance (km) P wave strain (10 μs) S wave strain (10 μs Here, the apparent focal depth at each observation site is defined as We first calculate the apparent focal depth h i rather than the emergence angle i. Because the focal depth is not a high-precision quantity, it is generally accurate to the km level. Therefore, we use the enumeration method, where all the values from 1 km to 300 km are tried, and the optimal value is selected as the apparent focal depth; the error does not exceed 1 km. For an observation site, we proposed a specific algorithm to determine the focal depth as follows. Firstly, for the attempted h i , three ε 11 ′ are obtained by using Eq. 5, and the standard deviation std(ε 11 ′ ) is calculated. Secondly, for the attempted h i , three ε 12 ′ and three ε 13 ′ are obtained by using Eq. 6, and the corresponding standard deviations std(ε 12 ′ ) and std(ε 13 ′ ) are calculated. Thirdly, the objective function is established as follows: Finally, all h i are tested, and the minimum target function h i is selected as the apparent focal depth of the observation point. Further, the corresponding ε 11 ′ , ε 12 ′ , and ε 13 ′ are selected as the strain component value of the plane seismic wave reaching the observation site.
For the two earthquakes, Figure 7 shows the calculated apparent focal depth of the five observation sites. There are two notable points. First, the objective function can reach the obvious minimum extreme point; second, as shown in the sub-graphs at the lower right corner of Figures 7A, B, compared with Figure 3 and Table 3, and Table 4, the apparent focal depth of each observation site shows linear behavior: when the epicentral distance is larger, the apparent focal depth is also larger. These two points prove the reliability of the observation data as well as the accuracy of the calculation results. Further, the calculation results validate the feasibility of the above-mentioned assumptions.

Constant velocity gradient model
The two earthquakes recorded by our experimental network occurred in the crust. In a homogeneous medium, seismic waves should propagate linearly. If the crust is a homogeneous medium, the apparent focal depth calculated at all the observation sites should be equal to the actual focal depth when the seismic wave emitted by the hypocenter reaches different observation sites. However, actually, the apparent focal depths of each observation site are different, indicating that the crust may not be homogeneous medium.
The variation in the wave velocity of crustal rocks with depth is very complex, but the primary feature of the crustal structure is that the wave velocity increases with depth. As a first-order approximation, we can consider that the wave velocity steadily increases with depth, which is called the constant velocity gradient model (Stein and Wysession, 2003;Wan, 2016). When the epicentral distance is larger, the observed phenomenon that the focal depth is larger can be explained by this model.
The constant wave velocity gradient model is usually established on the premise that the hypocenter is located on the surface. To examine the seismic wave rays of local earthquakes, the constant velocity gradient model with a certain depth is considered, as shown in Figure 1. Assuming that the wave velocity gradient is k, the variation in the wave velocity with depth can be expressed as follows: v where v e is the wave velocity at the focal depth, and v 0 v e − kh is the wave velocity near the surface. In a homogeneous medium, the rays emitted from the take-off angle i e propagate in a straight line, and the emergence angle i 0 i e reaches the surface. In the medium, where the wave velocity steadily increases with depth, the emergence angle i ≠ i e reaches the surface. The characteristic of the rays emitted from the emergence angle i e is that the radius of curvature ρ remains constant (Wan, 2016), which can be written as where p is the ray parameter, which conforms to Snell's law as follows: According to the previous assumptions, the path of P and S waves is the same, i.e., p is the same. Substituting Eq. 12 into Eq. 11, we get where H ve k , H reflects the relationship between wave velocity gradient and wave velocity at the focal depth. Geographical coordinates (x 1 , x 2 , x 3 ) and ray coordinates (x' 1 , x' 2 , x' 3 ). "O" represents the seismic focus; "S" represents the seismographic site.
Frontiers in Earth Science frontiersin.org

Formula of take-off angle
As mentioned above, in a constant velocity gradient medium, the equation of the circle (Figure 1) of the seismic wave rays emitted from the source is as follows: where On the ground, y = 0. Substituting Eq. 15 into Eq. 14 and simplifying it, we get Then, inserting Eq. 13 with Eq. 16, we obtain Thus, the take-off angle can be expressed as where x is the epicentral distance, which can also be denoted by d.
Using Eqs 14, 15 we get Then, the actual seismic wave ray equation can be written as Using Eq. 20, seismic wave rays with different departure angles can be obtained. Differentiating the above equation to x and substituting Eq. 13 into it, we get dy dx Finally, using Eqs 13, 18, we get The emergence angle of seismic waves reaching the observation site is expressed as 4 Results and discussion

Fitting results
The feasibility of using the constant velocity gradient model to explain the actual observation results depends on whether the emergence angle of each observation site can be approximately fitted with Eq. 22. The observation Eq. 22 and related Eqs 21, 18 involve two quantities with clear physical significance: focal depth h and take-off angle, i e , which are the quantities to be determined through the fitting.
Equation 22 gives the emergence angle i corresponding to different epicentral distances x. To explain our experimental results, we must determine two parameters in Eq. 22: h and H, where the focal depth h has a more significant application value.
This problem is non-linear, and the enumeration method is still used for inversion. For the focal depth h, all the integer values from 1 km to 100 km are traversed, and the error does not exceed 1 km.
Since v e generally does not exceed 8 km/s in the crust, assuming that the wave velocity near the surface is 5 km/s and the crustal thickness is 30 km, then k is approximately 0.1/s, and the corresponding H is equal to 80 km. Based on this analysis, the variation range of H is selected as 1-200 km, and all the integer values are traversed with an The corresponding x-i relationship is considered as the criterion. Figure 8 shows the fitting results of two earthquakes. Here, the dash-dotted curve shows the variation in the emergence angle with the epicentral distance in the homogeneous medium under the same focal depth. The solid curve is the x-i relationship obtained by the constant velocity gradient model. Obviously, the observed x-i relationship does not conform to the homogeneous medium model, but it is in good agreement with the constant velocity gradient model. The inversion results of h and H are also shown in Figure 8, which shows that the focal depths h of the two earthquakes is 11km and 7 km respectively, which are consistent with the results of 8km and 5 km given by the pendulum seismograph (Table 2). It is noteworthy that the H of the two local events is 48 km and 49 km, respectively, and the results are largely consistent; this also explains that the medium structure should also be consistent results for two earthquakes in the same area.
Red dash-dotted line: homogeneous medium; solid line: constant velocity gradient medium.
Once h and H are determined, seismic wave rays can be plotted for different take-off angles, as shown in Figure 9 (solid lines). It is also necessary to give the take-off angle corresponding to each observation site to plot the seismic wave rays at each observation site, which are shown by red dash-dotted lines in Figure 9.
Eq. 21 can be rewritten as Squaring and expanding both sides of Eq. 24, we obtain Using trigonometric function relation, we get Substituting Eq. 14 into Eq. 25 (y = 0 on the ground), we have Then, Squaring both sides of Eq. 28 and substituting in Eq. 13, we get Using Eq. 29, the following expression of Snell's law can be obtained: When h and H are known, H−h H can be calculated, and then the take-off angle, i e is calculated from the emergence angle i according to Eq. 29. According to Figure 6, i e < 90 + corresponds to the usual Snell's law, where Eq. 30 takes a positive sign on the right. When i e > 90 + , the seismic waves emitted from the source are downward, and the negative sign should be taken. At the same time, the obtained negative take-off angle should be added to 180°. Eq. 18 shows that when i e 90 + , the epicentral distance is expressed as follows: Therefore, the epicentral distance is positive when x < x 90 , and it is negative when x > x 90 . In fact, the take-off angle can be directly calculated using Eq. 18. The two methods give the same results and substantiate each other. Table 5 shows the calculation results.
It can be seen from Figure 9 that all the seismic wave rays represented by the dash-dotted lines do not exceed the depth of 30 km, and they should not exceed the crust range and are propagated in the crust.

Discussion
Most of the seismic sites and borehole strainmeter sites in mainland China are planned and constructed separately, so there are few sites where the two observations are co-located. Although Shanxi Province has a relatively dense pendulum seismograph observation network, there are only three seismic stations in the study area of this paper, and only the NW site is co-located. Seismic observation is a velocity observation optimized by frequency band; the strain observed by the four-component borehole strainmeter is a vector. The results of the two observations may be different. Therefore, the research in this paper did not carry out a detailed comparative analysis with seismic observations; only the hypocenter parameters given by the seismic network were compared. With continuous observation, the number of local earthquakes recorded has increased. In the future, we will use the sites where seismic and borehole strainmeters are co-located to carry out a comparative analysis of the two observations through a large number of earthquakes. This study is part of our research on solving the focal mechanism solution with strain seismic observation, which broadens the application field of four-component borehole strain observation. The results can provide a practical basis for further development of the four-component borehole strainmeter in mainland China in the future. Other than that, some recent studies show that strain seismographs could help provide unique constraints for local earthquakes, which then further strengthening the efficiency of the local earthquake warning system (Juhel et al., 2018;Canitano et al., 2021). Based on the research in this paper, how to apply these high sampling sites to the earthquake early warning system in mainland China will be a meaningful study.

Conclusion
The apparent focal depth, the emergence angle, and the take-off angles of seismic waves given in this study for five sites of two local events have stable uniqueness and apparent regularity. In particular, the inverted focal depths are basically consistent with the interpretation results of the traditional pendulum seismograph, which shows the reliability of the research results. This provides another important application direction of the YRY-4 type fourgauge borehole strainmeter. The accuracy of the YRY-4 type fourgauge borehole strainmeter was the prerequisite for this study. The H values of the two earthquakes (H is the reflection of the relationship between wave velocity gradient and wave velocity at focal depth) are consistent, which also verifies the applicability of the crustal model with constant gradient wave velocity. The observation results suggest that the constant velocity gradient model is more realistic than the homogeneous medium crust model.
Currently, the research on solving focal mechanisms by seismic strain observation is limited to local earthquakes. Especially, in the range of direct wave with initial motion, the seismic moment is generally less than 100 km. At present, there is no good way to solve the problem that the emergence of the first wave and mantle  reflection wave in a larger range can cause greater errors in the inversion. However, the research in this paper gives a method to determine the emergence angle, which clears a major obstacle for further solving the focal mechanism of local earthquakes by strain seismographs. It should be noted that this method of determining the emergence angle requires more observation sites. We will continue to study the rupture process and focal mechanism of moderate-strong earthquakes and teleseismic earthquakes by combining two kinds of observations. Strain seismographs and pendulum seismographs are different with distinct advantages and disadvantages. The research in this paper shows that the strain seismograph can be used as an effective supplement to the pendulum seismograph, especially in areas with fewer measuring sites of pendulum seismographs. Using strain seismographs and seismographs for co-site or co-hole observation will be one of the development directions of seismic observation in China.

Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.