In the present work, we aim to analyse the regularity of a seismic process based on its spatial, temporal, and energetic characteristics. Increments of cumulative times, increments of cumulative distances, and increments of cumulative seismic energies are calculated from an earthquake catalogue for southern California from 1975 to 2017.

As the method of analysis, we use the multivariate Mahalanobis distance calculation, combined with a surrogate data testing procedure that is often used for the testing of non-linear structures in complex data sets. Before analysing the dynamical features of the seismic process, we tested the used approach for two different 3-D models in which the dynamical features were changed from more regular to more randomised conditions by adding a certain degree of noise.

An analysis of the variability in the extent of regularity of the seismic process was carried out for different completeness magnitude thresholds.

The results of our analysis show that in about a third of all the 50-data windows the original seismic process was indistinguishable from a random process based on its features of temporal, spatial, and energetic variability. It was shown that prior to the occurrence of strong earthquakes, mostly in periods of generation of relatively small earthquakes, the percentage of windows in which the seismic process is indistinguishable from a random process increases (to 60 %–80 %). During periods of aftershock activity, the process of small earthquake generation became regular in all of the windows considered, and thus was markedly different from the randomised catalogues.

In some periods within the catalogue, the seismic process appeared to be closer to randomness, while in other cases it became closer to a regular behaviour. More specifically, in periods of relatively decreased earthquake generation activity (with low energy release), the seismic process appears to be random, while during periods of occurrence of strong events, followed by series of aftershocks, significant deviation from randomness is shown, i.e. the extent of regularity markedly increases. The period for which such deviation from random behaviour lasts depends on the amount of seismic energy released by the strong earthquake.

The process of earthquake generation remains a focus of diverse interdisciplinary investigations by Earth science researchers worldwide. The practical and scientific reasons for this interest are well known and easily explainable. However, despite this strong interest and the enormous research efforts that have already been applied, many important aspects of the complex seismic process characterised by space and time clustering are still not clear (Bowman and Sammis, 2004; Godano and Tramelli, 2016; Kossobokov and Nekrasova, 2017; Matcharashvili et al., 2018; Pasten et al., 2018).

One of the fundamental questions of modern Earth science concerns the dynamics of the seismic process. As a logical compromise between the different approaches that have been proposed for this problem, it has been suggested that the dynamical features of the seismic process may vary, ranging from periodic (primarily for large events) to the totally random occurrence of earthquakes (Matcharashvili et al., 2000; Corral, 2004; Davidsen and Goltz, 2004). The same, in terms of the concept of intermittent criticality of earthquake generation, can be expressed as the ability of a tectonic system to approach and/or retreat from the critical state, i.e. the state of the system in which strong earthquakes occur (see e.g. Sornette and Sammis, 1995; Bowman et al., 1998; Bowman and Sammis, 2004; Corral, 2004).

Current knowledge of the scaling and memory characteristics of the overall seismic process indeed supports this proposed diversity in the dynamics of earthquake generation (Sornette and Sammis, 1995; Bowman et al., 1998; Abe und Suzuki, 2004; Chelidze and Matcharashvili, 2007; Czechowski, 2001, 2003; Białecki and Czechowski, 2010; Kossobokov and Nekrasova, 2017). Moreover, the results of analyses carried out to assess the dynamical features of the seismic process in terms of its separate domains (time, space, and energy) also indicate differences in behaviour (see e.g. Goltz, 1998; Matcharashvili et al., 2000, 2002; Abe and Suzuki, 2004; Chelidze and Matcharashvili, 2007; Iliopoulos et al., 2012). More specifically, it has been shown that the seismic process in the temporal and spatial domains may reveal features that are close to so-called low-dimensional dynamical structures, although the features of the behaviour in the energy domain appear close to randomness, i.e. representing high-dimensional dynamical processes (Goltz, 1998; Matcharashvili et al., 2000; Iliopoulos et al., 2012). This has been shown for whole catalogues as well as for their parts and for different time periods.

Coming back to the concept of a critical state, it should be emphasised that intermittent criticality implies time-dependent variations in the activity during a seismic cycle. Thus, since the critical state is usually described as the state of the system when it is at the boundary between order and disorder (Bowman et al., 1998), we can describe the time variability of the seismic process in terms of the contemporary concept of geocomplexity (Rundle et al., 2000).

According to present knowledge, and in complete accordance with the concept of intermittent criticality, it is accepted that the extent of regularity (order) of the seismic process may vary in all its domains (temporal, spatial, and energetic) (Goltz, 1998; Abe and Suzuki, 2004; Chelidze and Matcharashvili, 2007; Iliopoulos et al., 2012; Matcharashvili et al., 2000, 2002, 2018). At the same time, despite the large number of recent publications demonstrating the diversity of these changes in the dynamics of the seismic process, interest in this issue continues to grow. In this context, it should be emphasised that it is important to assess these dynamical changes on the basis of multivariate analysis, taking into account all the temporal, spatial, and energetic constituents of the seismic process. Thus, one important research task is to understand the character of these changes in the entire seismic process.

Based on the state-of-the-art studies mentioned above, we aim in the present work to investigate the dynamical features of the seismic process based on all its temporal, spatial, and energetic characteristics. We carry out a multivariate comparison of the seismic process using an original earthquake catalogue for southern California and a set of randomised catalogues in which unique (temporal, spatial, and energetic) dynamical structures have been intentionally distorted by a shuffling procedure. This multivariate comparison of an original catalogue with randomised catalogues may help us to gain new knowledge about the character of the changes that occur in the extent of order/disorder of the seismic process. In addition, we will have stronger arguments regarding where and how the dynamics of the original seismic process in the analysed catalogue was close to disorder (irregularity) or order (regularity). We also aim to determine whether such changes are related to the process of preparation for strong earthquakes.

The results obtained in our research show that the extent of regularity in the analysed seismic process changes and is closer to randomness in the periods prior to strong earthquakes. After strong earthquakes, the regularity of the original seismic process assessed based on its temporal, spatial, and energetic characteristics is clearly increased.

We based our analysis on the southern California (SC) earthquake catalogue,
which is available from

Map of the area covered by the southern California earthquake catalogue (1975–2017).

As pointed out above, we aimed to carry out a multivariate analysis of the dynamical features of the seismic process. Thus, in order to preserve the original character of the temporal, spatial, and energetic characteristics of this process, we intentionally avoided any cleaning or filtering of the earthquake catalogue used here. This approach was based on a widely accepted practice (see e.g. Bak et al., 2002; Christensen et al., 2002; Corral, 2004; Davidsen and Goltz, 2004; Matcharashvili et al., 2018) in which all events are assumed to be on the same footing and the catalogue is considered as a whole. In other words, we did not pay attention to the details of tectonic features, the locations of the earthquakes, or their classification as mainshock or aftershock (Bak et al., 2002; Christensen et al., 2002; Corral, 2004).

In view of our research goal, i.e. a multivariate assessment of the extent of the regularity of the original seismic process, we need to analyse the seismic process in terms of the simultaneous variability in all three of its domains: temporal, spatial, and energetic. From this point of view, we consider cumulative sums of the characteristics of earthquakes in the temporal, spatial, and energetic domains (Fig. 2). The cumulative sum representation in the time domain is trivial, since time is already a cumulative characteristic, representing the cumulative sum of inter-earthquake times. Cumulative representation in the spatial domain is also quite feasible, and we consider cumulative sums of distances between consecutive earthquakes in the seismic catalogue. The cumulative sum of seismic energies released by consecutive earthquakes is also often used in the context of the different aspects of earthquake generation (e.g. Bowman et al, 1998; Bowman and Sammis, 2004; Nakamichi et al., 2018). Here, we add that despite some controversies (see e.g. Corral, 2004, 2008) over the question of the reliable energetic measurement of earthquake size, its relation to the magnitude of an earthquake is generally accepted. Thus, from the earthquake magnitudes in the SC catalogue, we can calculate the amount of seismic energy released, according to Kanamori (1977).

We start from the first earthquake in the catalogue (for the time period of
interest, from 1975 to 2017), which we consider a starting point, and
then follow the time sequence. Thus, ICT(

In order to have the same standard deviation for the three groups of data,
the standard deviations were calculated for each of the ICT(

In order to characterise the seismic process from a multivariate point of
view, we used a well-known statistical test, the Mahalanobis distance (MD)
calculation. Calculation of the MD is an effective multivariate method for
different classification purposes and is often used for data sets of
different origins. Thus, the objective of our analysis can be regarded as a
classification task of the features of a seismic process, assessed using the
variability in ICT(

Cumulative sums of

In other words, we aimed to assess the changes that occurred in the seismic
process over the period covered by the SC catalogue (1975–2017). It is
well known that the correctness of a multivariate assessment and
classification of a system is strongly dependent on correct feature
extraction (McLachlan, 1992, 1999). In other words, the data sets
used should be specifically focused on the targeted features of the process
under investigation. Hence, in order to have data sets with a similar
physical sense, enabling us to assess the dynamical features of seismicity
in three domains, we used ICT(

The MD (Mahalanobis, 1930; McLachlan, 1992, 1999) is a widely accepted
method of measuring the separation of two groups of vectors (e.g. one group
A, consisting of

When dealing with analysis of complex seismic processes, it needs to be pointed out that the MD calculation is sensitive to inter-variable changes in a multivariate system (Mahalanobis, 1930; Lattin et al., 2003) and that it takes into account the correlations between several variables providing information on the similarity or dissimilarity between the compared groups (Taguchi and Jugulum, 2002; Kumar et al., 2012).

If we are primarily interested in analysing dynamical changes occurring on short scales (short data sets), it is useful to combine the advantages of multivariate analysis and surrogate testing (Matcharashvili et al., 2017, 2018). In this case, we can use the multivariate MD calculation to examine whether the original seismic process is similar to or dissimilar from a random process (randomised catalogues) by comparing them based on the three main characteristics listed above.

In summary, we aim to analyse the way in which the order in the seismic
process, as assessed using its derivative temporal, spatial, and energetic
characteristics (the quantities ICT(

In order to verify whether the approach used here, which combines MD calculation with surrogate testing, is indeed useful for discerning any changes occurring in the natural 3-D system (the seismic process in a tectonic system), with slightly or strongly different dynamical features, we used time series generated by two 3-D simulated systems with added noise. These were a 3-D Lorenz system and a crack fusion model with added Gaussian noise.

The well-known Lorenz model describes the motion of an incompressible fluid contained in a cell that has a higher temperature at the bottom and a lower temperature at the top. Despite the simple form of this set of equations, very complex behaviour can be exhibited. This approach has therefore been commonly used to present the interesting non-linear dynamics of 3-D systems.

The Lorenz model has the following form (see e.g. Hilborn, 1994):

In this work, we need time series that are close to stationary, and thus in
order to avoid periodic orbits we assume

The kinetic crack fusion model (Czechowski,
1991, 1993, 1995) describes the evolution of a system of numerous cracks
which can nucleate, propagate, and coalesce under applied stress. Here, we
use a simple version of the model (related to seismic processes) in which
only three crack populations (small cracks

Thus, in order to ensure that the multivariate method used here enables us to discriminate between different conditions of dynamical systems, we use 3-D models in which the dynamical features are changed from more regular to more randomised conditions by adding some extent of noises. We start with the Lorenz system (Fig. 3) and then proceed to the crack fusion model (Czechowski, 1991, 1993, 1995) (Fig. 4). As explained above, in both cases we add noises of different intensity to the original 3-D system, assuming that the more intense the added noise, the closer the model system is to randomness. Figures 3 and 4 clearly show that the number (or portion) of the 50-data windows in which the condition of the 3-D system is indistinguishable from the initial condition (the system with no added noise) gradually decreases when the intensity of the added noise is increased. This means that the method of analysis used here enables us to distinguish the conditions of systems even in cases when they are only slightly different (i.e. only a small amount of noise is added) (see the left-hand parts of the curves in Figs. 3 and 4, showing a smaller amount of added noise).

Percentage of the 50-data windows (shifted by 50 data steps) of the Lorenz system with added noise that are indistinguishable from the initial condition (system with no added noise).

The percentage of the 50-data windows (shifted by 50 data steps) of the crack fusion model with added noise that are indistinguishable from the initial condition (system with no added noise).

For clarity, we note here that in Figs. 3 and 4, we show results for the case of windows 50-data long, since in the analysis of the seismic catalogue below, we also use this size of window. At the same time, it should be emphasised that the result of the above analysis depends on the timescale used (the size of the windows). For larger windows (500- or 1000-data long, for example) distinguishability from the starting condition (i.e. without added noise) requires a larger amount of added noise, although the general conclusion remains the same: the method of analysis used here enables us to distinguish between the states of 3-D systems with different extents (or degrees) of dynamical regularity.

Having shown that the multivariate testing method selected for this research
enables us to discriminate between different conditions of dynamical
systems, we proceed to analyse data sets from the original seismic catalogue and the randomised catalogues mentioned above. We start
from the case where MD values are calculated for non-overlapping, 50-data,
windows shifted by 50 data steps, in the same way as for the 3-D model data
sets. Figure 5 presents the results of this calculation. Groups consisting of
the ICT(

Seismic energy released (upper curve) and average MD values (bottom
curve) calculated for consecutive non-overlapping 50-data windows shifted
by 50 data steps, for the southern California earthquake catalogue
(1975–2017). Averages of the MD values and the corresponding standard
deviations (given in the lower plot by white circles and grey error bars)
were calculated by comparing ICT(

For a more precise analysis, we calculate the MD values for 50 data windows shifted by one data step (Fig. 6).

Average MD values calculated by comparing ICT(

The results in Figs. 5 and 6 support the view that despite the generality of the background physics (Lombardi and Marzocchi, 2007; Di Toro et al., 2004; Davidsen and Goltz, 2004; Helmstetter, 2003; Helmstetter and Sornette, 2002; Corral, 2008), the processes taking place prior to and after main shocks are nevertheless different (Sornette and Knopoff, 1997; Davidsen and Goltz, 2004; Wang and Kuo, 1998). According to recent research, the latter is characterised by long- and short-range correlations and thus is more ordered, while the former is apparently more uncorrelated and random-like (Touati et al., 2009; Godano, 2015). Indeed, according to Bowman et al. (1998), the loss of energy (released also in the form of seismic energy) that is related to the occurrence of strong events introduces memory into the system (Bowman and Sammis, 2004).

We can see from Figs. 5 and 6 that in the SC earthquake catalogue considered
here, the seismic process after strong earthquakes is more regular than in
the periods prior to these events. Indeed, in all windows, the seismic
process, as assessed based on the variability in ICT(

In order to exclude the possibility that some of the characteristics
selected here (ICT(

For better visualisation of the above results (see Fig. 6), Fig. 7 presents
MD values calculated for 50 data windows for the period from 14 May 1990 (the
window started from event 12100 in the SC catalogue) to 28 June 1992 (the
window started from event 13797 in the SC catalogue). Within this period,
two strong earthquakes occurred,

Average MD values calculated for the period from 14 May 1990 (12100)
to 28 June 1992 (13797) in which two strong earthquakes occurred:

The next period selected for detailed analysis was from 24 August 1997 (the window
started from event 20760 in the SC catalogue) to 16 October 1999 (the window
started from event 21160 in the SC catalogue). Two large events occurred in
this period: a moderate

The results shown in Fig. 8 are mostly similar to those in Fig. 7. Strong
and relatively strong (for this selected short period) earthquakes are
preceded by a significant number of windows in which the seismic process in
the original catalogue is indistinguishable from that observed for
randomised catalogues. In contrast, in all 50-data windows following strong
(or relatively strong) earthquakes, we can observe a statistically
significant difference. A multivariate comparison of these windows based on
the variation in ICT(

Average MD values calculated for the period from 24 August 1997 (20760) to
16 October 1999 (21160) in which two strong earthquakes occurred:

Separate consideration of the period of the strong

As expected, the behaviour of the seismic process prior to and following all of the strong events considered here is similar. The only difference is the length of the period during which the post-earthquake seismic process remains significantly regular compared to the randomised catalogues. For strong earthquakes, this period is clearly longer (see Fig. 6). This appears to be connected with the generation of a series of aftershocks, in which the spatial, temporal, and energetic features are causally related to the mainshock. This is in agreement with the well-known productivity law that states that the larger the magnitude of the mainshock, the larger the total number of aftershocks (Helmstetter, 2003; Baiesi and Paczuski, 2004; Godano and Tramelli, 2016). Here, we emphasise that the question of the temporal length of the aftershock sequence following a strong earthquake is still not understood, as it is related to the timescale of background seismic activity (Godano and Tramelli, 2016).

Average MD values calculated for the period from 30 October 2008 (27300)
to 5 April 2010 (28300) in which three moderate and strong earthquakes
occurred:

From Figs. 7 to 9, we can see that the extent of order in the seismic
process (as assessed based on the temporal, spatial, and energetic
distributions of earthquakes) may change not only in the periods prior to
and following strong (

Here we point out that this article was submitted to

Magnitudes and MD values calculated for part of the SC catalogue
after

Since the above results suggest that, prior to strong earthquakes, a
comparatively calm seismic process of relatively small (with

Magnitudes and MD values calculated for part of the SC catalogue
after

Magnitudes and MD values calculated for part of the SC catalogue
after

We then carried out a similar analysis for the sequences of relatively small
earthquakes that occurred in periods when no strong earthquakes were
registered. These small earthquakes apparently cannot be regarded as
aftershocks of strong events. In Fig. 13, we present the results of an
analysis of an almost 2-year period of small earthquake activity. This
period began 5 months later, after the

Figure 14 presents the results for the next part of the catalogue, which
contained relatively small earthquakes in the observation period, which was
far from the occurrence of strong events. A moderately strong earthquake of

Magnitudes and MD values calculated for the non-aftershock part of the SC catalogue from 7 March 1983 (sequential number in SC catalogue 5000) to 5 February 1985 (sequential number in SC catalogue 6253). Average MD values are calculated for 50-data windows shifted by one data step.

Magnitudes and MD values calculated for the non-aftershock part of the SC catalogue from 7 April 2011 (sequential number in SC catalogue 31823) to 14 February 2012 (sequential number in SC catalogue 32240). Average MD values are calculated for 50-data windows shifted by one data step.

In Fig. 15, we present the results for the third part of the catalogue,
which was selected to contain relatively small earthquakes,

Magnitudes and MD values calculated for the non-aftershock part of the SC catalogue from 24 May 2006 (sequential number in SC catalogue 26259) to 5 August 2007 (sequential number in SC catalogue 26717). Average MD values are calculated for 50-data windows shifted by one data step.

As can be seen from our results, as assessed based on the ICT(

Average MD values calculated by comparing ICT(

In Fig. 16, we give results for a completeness magnitude threshold of

Average MD values calculated by comparing ICT(

As can be seen from Fig. 17, in the case of a higher threshold of

This behaviour is apparently caused by the small number of events above the

We have investigated the variability in the regularity of the seismic process, based on its spatial, temporal, and energetic characteristics. For this purpose, we used an SC earthquake catalogue over the period 1975 to 2017. Our method of analysis was a combination of multivariate Mahalanobis distance calculation and surrogate data testing. We carried out a multivariate assessment of changes in the regularity of the seismic process, based on increments of cumulative times, increments of cumulative distances, and increments of cumulative seismic energies calculated from the SC earthquake catalogue.

In order to assess the ability of the multivariate approach used here to discriminate between different conditions of dynamical systems, we used two 3-D models in which the dynamical features were changed from a more regular form to more randomised conditions by adding a certain degree of noise.

It was shown that in about a third of the analysed 50-data windows, the
original seismic process is indistinguishable from a random process by the
features of its temporal, spatial, and energetic variability. Prior to the
occurrence of strong earthquakes, in periods in which there are events with
relatively small magnitudes (

Based on the results of our analysis, we conclude that the seismic process cannot in general be regarded either as completely random or as completely regular (deterministic). Instead, we can say that the dynamics of the seismic process undergoes strong time-dependent changes. In other words, the regularity of the seismic process, as assessed based on the temporal, spatial, and energetic distributions, changes over time.

It was also shown that in some periods, the seismic process appears to be closer to randomness, while in other cases it becomes closer to regular behaviour. More specifically, in periods of relatively low earthquake generation activity (i.e. with smaller energy release), the seismic process looks more random, while in periods of occurrence of strong events, followed by a series of aftershocks, it shows significant deviation from randomness (i.e. the extent of regularity essentially increases). The period for which this deviation from random behaviour lasts depends on the amount of seismic energy released by the strong earthquake. The results obtained here from a multivariable assessment of the dynamical features of the seismic process are in accordance with our previous findings on the dynamical changes in the temporal distribution of earthquakes (Matcharashvili et al., 2018).

It should be underlined that the occurrence in July 2019 (during the
editorial process of our manuscript in NPG) of two strong earthquakes,

Used in this research, seismic data are available from the catalogue which is accessible from the official site (

The authors contributed in accordance with their competence in the research subject. The first author TM was responsible for all aspects of research and manuscript preparation. An immense contribution by ZC helped to ensure a high mathematical and modelling level of research, and NZ contributed through programming, data analysis, and active participation in the manuscript preparation.

The authors declare that they have no conflict of interest.

The authors acknowledge the useful comments of the reviewers, Eleftheria Papadimitriou and Antonella Peresan.

This research was supported by the Shota Rustaveli National Science Foundation (SRNSF) (“Investigation of the dynamics of the temporal distribution of earthquakes” (grant no. 217838)).

This paper was edited by Ilya Zaliapin and reviewed by Eleftheria Papadimitriou and Antonella Peresan.