A specially designed four-component strainmeter, named SKZ strainmeter, was invented in China in the 1970s (Su, 1977 [9] ) and continuously improved since (Chi et al. 2009 [21] ; Qiu et al. 2013 [22] ). It has four gauges set at 45 degrees intervals to measure extension and contraction of the cylindrical housing in four different azimuths (Figure 1).

Three parameters to define the dynamical regime of crust under the condition of plain strain (i.e. the maximum principal stress, the minimum principal stress, and the direction of maximum principal stress) can be derived using any three of the four measured strains. The fourth set of measurement can be utilized for redundancy check. The consistency parameters have been used to evaluate quality of the data and the self-checking of the instrument for the strainmeter network (e.g., Chen et al., 2015 [19] ). There is a consistency relation to characterize the regime of the crustal stress from the point of the horizontal strain.

The SKZ strainmeter is a long cylinder for radial displacement detection. Four sensing elements are installed in the middle of the cylinder to detect North-South (NS), North-West (NW), West-East (WE), and South-West (SW) alignments and deformations (Figure 1). Difference between the angles of two adjacent sensors is 45 degrees, which are equally distributed in accordance with the asterisk configuration.

Displacement transferring mechanism of the SKZ strainmeter was designed to increase sensitivity and accuracy. Eight narrow ribs were set precisely in the position where sensing elements were installed (see Figure 2). Metal cylinder casing with the ribs is cut from a single-piece, thick-walled metal tube using computer-controlled machines. The two polar plates of the capacitive sensors are arranged on the symmetrical fine ribs to constitute a simple non-contact measurement sensor. The SKZ units are shown in Figure 3, and specifications are summarized in Table 1.

The cylinder probe is installed into a borehole formation. The probe and formation

are bonded using the coupling medium. The first SKZ strainmeter was installed in Yiliang Station, Yunnan Province in March 2007, and followed by nine sites in several provinces in China, as shown in Table 2 and Figure 4. The depths of the installation range from 50 meters at the Yiliang Station to 135 meters at Dagan Station. The typical installation depth is several tens meters to reduce meteorological noises as far as possible (Qiu et al., 2013 [22] ).

Figure 5 illustrates 24-hour data record sampled at every half a minute rate at Datong Station in the Shanxi Province. The noise level at the site is around 5 × 10⁻⁹, much less than the solid earth tide influences.

When a strainmeter is installed close to the surface, the measured strain field can be generally considered to be horizontal, because vertical normal stress is

negligible. The sensor is expected to provide independent observations on strain variations in line with other geodetic observations such as GPS and InSAR.

Under this situation, the displacements (U₁ ~ U₄) of the four sensors are described by principal strains ε₁, ε₂ as follows (Su, 1977 [9] ).

U 1 = A ( ε 1 + ε 2 ) + B ( ε 1 − ε 2 ) cos 2 ( Ψ − φ ) (1)

U 2 = A ( ε 1 + ε 2 ) − B ( ε 1 − ε 2 ) sin 2 ( Ψ − φ ) (2)

U 3 = A ( ε 1 + ε 2 ) − B ( ε 1 − ε 2 ) cos 2 ( Ψ − φ ) (3)

U 4 = A ( ε 1 + ε 2 ) + B ( ε 1 − ε 2 ) sin 2 ( Ψ − φ ) (4)

where Ψ is azimuth of the principal stress, φ is the direction of the first sensing element, A and B are constants depending on parameters of the borehole (the radial displacement of the cylinder inner) and the crustal strain. A is the sensitivity coefficient to the areal strain, B is the sensitivity coefficient to the shear strain. These two coefficients are related with the mechanical parameters of cement and rock, and also related the cement layer depth, the inner and outer diameter of the cylinder. Normally A was significantly less than 1, and B was significantly greater than 1.

The following equations can be derived (Su, 1977 [9] ),

U o = ( U 1 + U 3 ) (5a)

U e = ( U 2 + U 4 ) (5b)

U o = U e = U c (5c)

Because the sum of odd-numbered displacements (U_o) equals to that of the even-numbered (U_e), Equation 5c actually is a self-consistent equation (Su, 1977 [9] ), which provides a check relation among the four parameters.

Uc is related to surface strain (ε₁ + ε₂) as shown in Equation (6).

U c = 2 R ( ε 1 + ε 2 ) (6)

It is a parameter of the plane strain independent of directions Ψ and φ. The difference between the measured values for mutually vertical sensors Ua and Ub,

U a = U 1 − U 3 = 4 R ( ε 1 + ε 2 ) cos 2 ( Ψ − φ ) (7)

U b = U 2 − U 4 = − 4 R ( ε 1 − ε 2 ) sin 2 ( Ψ − φ ) (8)

are related only to the differential stresses, (ε₁ − ε₂). Ψ, the direction of the principal stress, can be determined by the ratio of Ub to Ua (Su, 1977 [9] ).

U b / U a = − tan 2 ( Ψ − φ ) (9)

Measurements of the four components can reflect the plane strain regime, if the consistency relation (Equation (5)) is generally fulfilled. Otherwise, the inconsistency may be a result of system malfunction and/or anomalous variations regime of the crust which induces non-planer strain. The redundant data can be used for self-checking providing better performance compared with other types of borehole strainmeters (e.g., Gladwin, 1984 [2] ; Sakata and Sato, 1986 [23] ; Hart et al., 1996 [24] ; Ishii, 2001 [25] ).

The consistency relation can be used as an indicator of data quality (Chen et al., 2015 [19] ). Moreover, the relation can also be used to detect anomalous variations as a result of the breakdown of the normal state of plain strain around the region of earthquake preparation.

Figure 6 is an example of monthly data sampled at one per minute at the Daguan Station in the Yunnan Province between May 1 and May 31, 2012 (Table 2 and Figure 4). Two dominant variations in each component are steady with diurnal and semi-diurnal changes induced by solid earth tides. The trends of NS and NW components are almost linear, but EW and NE components are superposed by small random disturbances lasting for several days. The trends of the components are different from these reported by Sakata and Sato (1986) [23] . However our present object is to focus on shorter-period variations from the viewpoint of short-term earthquake predictions.

Figure 7 shows the calculated surface strains, U_o (5a) and U_e (5b) at Datong Station in Shanxi Province from June 1, 2012 to July 31, 2012. They are nearly equal, proving the self -consistency relation except the small trends. The difference between U_o and U_e is about 2.0 × 10⁻⁷ except for the shorter frequency components. It proved that the instrument is very steady and the consistency has been conserved for a very long period of a month or longer.

The rate of the satisfaction of self-consistency relation is defined by a ratio between the difference and the mean of the U_o and U_e by Qiu et al. (2013) [22] , as shown below.

C 95 = 1 − ∑ N 95 | ( U o − U e ) | ∑ N 95 ( U o + U e ) (10)

After eliminating the 5% outliers, the remaining correlation coefficients between U_o and U_e, instead of the self-consistency relation itself, was used to evaluate performances at particular sites.

The alternative parameter to quantize qualities of strain data that we use is the Pearson’s correlation coefficient Cp, which utilizes 25 data sampled at a once-per hour rate shifting by an hour. The selection is based on the fact that dominant components are the diurnal and semi-diurnal lunar tidal components S₁ (25.82 hours) and M₂ (12.42 hours) solid tides. The correlation coefficient Cp is defined from the following:

C p = C o v ( U o , U e ) σ U o σ U e (11)

where Cov is the covariance, and σ is the standard deviation.

Yiliang earthquake (M 5.7) occurred on September 7, 2012, in the Luozehe Town (104.0E, 27.5N), 15 km south of Yiliang County center (Figure 8).

The Ludian M6.5 earthquake occurred on August 3, 2014 and the epicenter was located in Ludian County in Yunnan Province (Figure 10). Figures 11(a)-(d) show the strain changes and the correlation coefficient at Ludian, Yudon, Yiliang, and Daguan stations during Ludian earthquake, respectively.

It is shown that the anomalous strain decreased as distances from earthquake epicenter increased. The changes of the correlation coefficients (Equation (11)) between the two components of consistency relation U_o and U_e are proved to characterize the preparatory stage of the earthquake (Table 3). Correlation coefficients during normal stages were typically between 0.8 and 0.98, substantially decreased during the imminent stage of the earthquake, and return to 0.98 or higher after the earthquake.