A large earthquake of 8.0 magnitude occurred on 12 May 2008, 14:28 UTC, with the epicentre in Wenchuan. To investigate the pre-earthquake anomalous strain changes, negentropy is introduced to borehole strain data for three locations, approximated by skewness and kurtosis, revealing the non-Gaussianity of recorded fluctuations. We separate the negentropy anomalies from the background by Otsu's method and accumulate the anomaly frequency on different scales. The results show that the long-term cumulative frequency of negentropy anomalies follows a sigmoid behaviour, while the inflection point of the fitting curve is close to the occurrence of the earthquake. For the short-term analysis before the earthquake, there are two cumulative acceleration phases. To further verify the correlation with the earthquake, we compare our findings for different time periods and stations and rule out the possible influence of meteorological factors. We consider the negentropy analysis to exhibit potential for studying pre-earthquake anomalies.

Changes in crustal deformation fields over time have been recorded at least for some large earthquakes

The probability density function (PDF) of observation data is also an informative way of extracting potential anomalies contained in earthquake generation processes.

Rather than the whole PDF, its moments are often utilized; moments may be estimated quite reliably from relatively small amounts of data

Thereby, it is possible that precursor anomalies lead to an increase in disordered components in observation data.

Entropy can serve as a measure of the unknown external energy flow into the seismic system

Negentropy's definition is based on the entropy, and it is also widely used to detect non-Gaussian features.

YRY-4 borehole strainmeters, which are designed to record continuous deformation occurring over periods of minutes to years, have been deployed at depths of more than 40 m at more than 40 terrain-sensitive locations within China. These strainmeters are capable of resolving strain changes of less than one-billionth. The data sampling rate is once per minute.

The study period is from 1 January 2007 to 30 June 2009. The study area is shown in Fig. 1. We find that the Guza station stands on the southwestern end of the Longmenshan Fault zone. Besides this, the epicentre is about 150 km away from the station, which is within the monitoring capability of the borehole strainmeters

Location map showing the epicentre and three stations. The epicentre was located at 31.01

Because the four gauges of the YRY-4 borehole strainmeter are arranged at 45

In practical application, the higher the correlation between both sides of the Eq. (1), the more reliable the data. In that case, we can use areal-strain

The borehole strain of Guza station is highly consistent

Self-consistency of the borehole strain at Guza from 1 January 2007 to 30 June 2009.

It can be seen that when

Step 2 removes the periodic terms in the signal. We think that the period terms mainly include the periods related to the solid tide and also include the periodic effects of air pressure. The residual high-frequency signals are shown in Fig. 3. In particular, small changes in the curve are amplified by the processing.

High-frequency areal strain at Guza from 1 January 2007 to 30 June 2009.

The entropy-based negentropy is a statistically justified measure of non-Gaussianity

However, the theoretical calculation of negentropy also depends on the prior probability density of random variables and other information which is difficult to determine accurately. In practical applications, higher-order statistics (HOS) and density polynomial expansion are usually used to approximate one-dimensional negentropy

This definition suggests that any deviation from a Gaussian distribution will increase the negentropy

Moreover, the relation between the skewness and kurtosis is universal, and they approximately align along a quadratic curve (Sattin et al., 2009):

Here we calculate the normalized skewness and kurtosis in the study period, so Eq. (9) can be derived into

This relation is trivial in a Gaussian fluctuating system; it reduces to a fixed mass around zero (

To solve the negentropy anomaly detection problem, we designed a simple thresholding hypothesis test using the Otsu method

Using these classes, the weighted average value

In this test, our initial assumption is that the sliding window is composed of a Gaussian signal of non-seismic-related activities. When our test negentropy exceeds the critical value

According to the empirical hypothesis that geophysical signals deviate from the Gaussian distribution when they record abnormal activities, and based on the results of previous studies, we perform the following investigation.

As the negentropy is calculated using a 2 h sliding window, we assume that it reaches the maximum values when the time window contains anomalies from seismic-related activities. The negentropy during the study period is shown in Fig. 4.

Negentropy at Guza from 1 January 2007 to 30 June 2009. The red dotted horizontal line is the optimal threshold

The within-class variance

Within-class variance

In the skewness–kurtosis domain, the statistical relationship of the borehole areal strain is consistent with parabolic behaviour as described in Eq. (10) (Fig. 6a), verifying that the turbulent system of borehole strain is significantly non-Gaussian. Besides this, the extracted negentropy anomalies are clustered strongly on the left side of the parabola, which exhibits similar characteristics different from the normal Gaussian distribution. Here, there are four points on the right side; one occurred in early 2007, and the others occurred after the earthquake. Therefore, we will not discuss them in the following.

Negentropy distributions in the skewness–kurtosis domain in

In addition, as shown in Fig. 6b, at times far from the earthquake, the negentropy distribution is basically Gaussian in the skewness–kurtosis domain. However, at times closer to the earthquake, the relatively stable state was broken due to the non-Gaussian mechanism, with more negentropy anomalies appearing on the left side of the parabola, while in 2008, almost all of the negentropy present was skewed to the left.

The phenomena prompt us to study its possible correspondence with the seismogenic process.

The transition of negentropy anomalies in the skewness–kurtosis domain is quantified as the change of the anomaly frequency per unit time through a logarithmic-linear model. Logarithmic-linear models of interest are often used to estimate the expected frequency of the response variable at the original scale for a new set of covariate values, such as the Gutenberg–Richer law, in which a linear relationship exists between the logarithm of the cumulative number of seismic events of magnitude

The logarithmic-linear regression model is proposed as

We use the logarithmic-linear model to solve the relationship between the negentropy anomaly frequency and different thresholds each day using the ordinary least-squares (OLS) method. Afterwards, an optimal threshold

The goodness of fit for each logarithmic-linear model was evaluated using analysis of

The

Estimated expected frequency

We calculate the negentropy cumulative frequency of the study period, as shown in Fig. 8. There is not only a long-term analysis of the whole period, but also a short-term analysis of the pre-earthquake process. In general, accumulated value of a typical random process usually has a linear increase. In particular, in case of critical phenomena, we would expect more frequent anomalies when they approach the critical point and less frequent anomalies after

For the entire earthquake process, a 2-month sliding window is selected for accumulation. In Fig. 8a, after July 2007, the negentropy anomalies gradually accumulated. Qiu (2009) and Chi (2014) also observed anomalies of this period at the Guza station, and they speculated that abnormal strain may reflect small-scale rock formation rupture before the earthquake. In particular, we find more frequent negentropy anomalies in 2008 as the earthquake approaches and less frequent anomalies after the earthquake; thus a sigmoid function is used to fit the acceleration before the earthquake and to fit the deceleration after the earthquake. The sigmoid function is expressed as

When we narrowed the accumulated window to 1 d, we observed two negentropy anomalies before the earthquake, as shown in Fig. 8b. The first anomaly frequency increase occurred from August to October 2007. In March 2008, there was a second phase of anomaly increase, and the cumulative frequency then slowly increased to a plateau period near the time of the earthquake. This, probably due to the stress, is in a deadlocked phase. This is because before the Wenchuan earthquake, the elastic deformation of the crust reaches its limit and the deformation is resisted in the hypocentral region, which is measured by GPS data

These two phases prior to the earthquake are also approximated with sigmoid functions. In order to further compare the anomalies of the two phases, we use linear regression to fit the central part of the two sigmoid curves. We find that the second acceleration is greater than the first acceleration.

Fault zones contain relatively weak and relatively strong parts. The former is the area where strain release begins, while the latter is the stress-locking part and the beginning of rapid instability

We randomly selected the strain data for 200 d before and after 20 March 2011 and 24 March 2014 at the Guza station. The selected data for the two periods are required to be in the absence of strong earthquakes and have higher quality. We performed negentropy analysis on these two observations and compared them with the results of negentropy analysis associated with the Wenchuan earthquake, as shown in Fig. 9.

The comparative analysis of cumulative frequency of negentropy anomalies between earthquake period and random time periods. The zero point of green dots is 20 March 2011, and the zero point of blue dots is 24 March 2014.

As we can see in Fig. 9, the cumulative frequency of negentropy anomalies of random periods has a linear increase. However, in the Wenchuan-earthquake periods, as the earthquake approaches, the cumulative frequency of negentropy anomalies increases rapidly and recovered to a slow growth after the earthquake.

We selected the Xiaomiao station and Renhe station to find out if their observations received strain changes. Their locations are shown in Fig. 1. Compared with the Guza station, we did the negentropy analysis of these two stations, as shown in Fig. 10.

Cumulative frequency of negentropy anomalies of Xiaomiao station and Renhe station from 16 September 2007 to 30 June 2009. The negentropy analysis of Guza station is from 1 January 2007 to 30 June 2009 because of the different installation time of the instruments. The red vertical line is the inflection point of the fitting curve of Guza station. The blue vertical line is the inflection point of the fitting curve of Xiaomiao station. The black dotted line is the earthquake day.

As we can see in the Fig. 10, the cumulative frequency of negentropy anomalies of the Xiaomiao station is also well fitted by the sigmoid function. The accumulation curve grows rapidly before the earthquake and is concave downward after the earthquake, which is similar to the Guza station, although the inflection point of the Xiaomiao station precedes the earthquake moment by about 2 months. However, since the curve is approximately linear before and after the inflection point, we consider that the inflection point value is reasonable in the range from January to June 2008. Cumulative anomalies of the Renhe station are basically linear, indicating that the Renhe station may not detect pre-earthquake anomalies.

The Renhe station is far from the end of the Wenchuan-earthquake fault, so it is reasonable that no abnormal changes are observed. However, the Xiaomiao station is located between the Guza station and Renhe station, and the fitting result shows that there is a similar trend to the Guza station, with a weaker curvature. So, for the nearest station to the epicentre, the Guza station may be able to record more pre-earthquake anomalies.

Furthermore,

The strain signals are sensitive to a few meteorological factors; therefore, we display the pressure variations, temperature variations recorded at the Guza station and the daily rainfall measured by Tropical Rainfall Measuring Mission (TRMM) satellite, which can be downloaded through NASA Giovanni-4 for the same period and the same area (

Then we calculated the differential data of the strain for negentropy analysis. We also make differential calculations for all three influencing factors, as shown in Fig. 12.

Borehole stain, air pressure, temperature and rainfall variations during study period at Guza station.

Differential borehole stain, air pressure, temperature and rainfall variations during study period at Guza station.

We observed that the air pressure, temperature and rainfall did not change abnormally during the period when the extracted anomalies increase whether we do differential calculation or not. Therefore, we consider that the abnormal variations on the processed strain signals are not caused by these factors.

In our work, the extracted negentropy anomalies of borehole strain associated with the Wenchuan earthquake are analysed. The cumulative frequency of negentropy anomalies are studied in both the long and short term. In the Comparison discussion section, we compare the cumulative anomalies of different time periods and different stations with those at the Guza station during the study period and preliminarily exclude meteorological factors. We suspect that the negentropy anomalies at the Guza station may have recorded abnormal changes related to the Wenchuan earthquake.

Since the tectonic dynamics of earthquakes during seismogenic and seismic processes are very complex, the mechanism of such abnormal changes undoubtedly needs to be discussed. In particular, borehole strain signals are sensitive to external influence. Besides this, because of the different characteristics and accuracy of different types of observations, a joint analysis has not been carried out yet. Further research is needed to decipher a potential precursory phase. However, we may be able to ensure that the negentropy analysis has great potential in the study of earthquake precursors.

The borehole strain data, air pressure and temperature data are confidential information and therefore cannot be made publicly accessible. The rainfall data are downloaded through NASA Giovanni-4 (

The authors contributed in accordance with their competence in the research subject. The first author, KZ, was responsible for the key technical guidance and ideas. ZY was responsible for method improvement, data analysis and paper preparation. CC helped to ensure the graph quality of the paper. MF and KL contributed through active participation in the paper preparation.

The authors declare that they have no conflict of interest.

The authors would like to thank the China Earthquake Networks Center for providing the borehole strain data and the NASA Giovanni team for rainfall data. Moreover, the authors are grateful to Zehua Qiu for his guidance and helpful suggestions.

This research was supported by the National Natural Science Fund (grant no. 41974084) and the Institute of Crustal Dynamics, China Earthquake Administration (grant no. 3R216N620537).

This paper was edited by Luciano Telesca and reviewed by two anonymous referees.