Based on the statistical analysis of spatiotemporal distribution of
earthquake epicenters and perennial geodetic observation series, new evidence
is obtained for the existence of slow strain waves in the Earth. The results
of our investigation allow us to identify the dynamics of seismicity along
the northern boundary of the Amurian plate as a wave process. Migration of
epicenters of weak earthquakes (2

The inhomogeneous blocky structure of the crust and the lithosphere considerably affects the deformation, seismic, filtration and other processes. The effect of the blocky structure on the distribution of earthquakes can be especially clearly traced. It is exactly the blocky structure of the geological medium which results in the generation of waves of different types, including slow strain waves (Bykov, 2008). Clarification of the link between movements of tectonic structures and slow strain wave processes is of fundamental importance for expanding our understanding of the physics of earthquakes.

The most important problem of recent geodynamics is to clarify the mechanisms
responsible for the propagation of the energy of deformation processes and
tectonic stress transfer at the boundaries between the blocks and the
lithospheric plates, and to explore the causes of migration of earthquake
epicenters. The problem has been argued for more than 45 years since
Elsasser's publication (1969), suggesting the equation of local stress
transfer in the rigid elastic lithosphere underlain by the viscous
asthenosphere. The possibility of using Elsasser's model to describe
migration of seismicity was further discussed in papers published by other
researchers. Bott and Dean (1973) introduced the term “stress or strain
waves” and obtained the expression for the velocity of the wave propagating
along the lithospheric plate. According to their calculation, the stress wave
velocity attains 0.1–100 km yr

The advances in theoretical studies of slow strain waves in the Earth
initiated the search for the possibilities of experimentally detecting the
propagation effects of the waves of this type, and, especially, the intense study of
earthquake migration. By now, the deformographic, geodetic and hydrological
measurements performed worldwide have revealed the migration of deformations
at velocities of about 10–100 km yr

In the present study, we have obtained new evidence of the existence of strain waves in the Earth resting upon a comprehensive statistical analysis of the dynamics of seismicity along the northern boundary of the Amurian plate and the data derived from in situ GPS experimental observations performed near this boundary.

Slow strain wave transmittance through the fault-blocky geological medium is accompanied by various seismic, hydrogeological, electrokinetic, geochemical and other effects. The methods for strain wave detection are divided into indirect methods that display the wave-shaped variations in the geophysical fields due to temporal variations of the stress state of the medium, and direct ones immediately recording the migration of deformations.

The seismic, geoelectric and geochemical methods of strain wave recording are referred to as the indirect methods. Indirect evidence of the existence of strain waves is manifested in the targeted migration of large earthquakes (Stein et al., 1997), the occurrence of seismic velocity anomalies (Lukk and Nersesov, 1982; Nevskii et al., 1987), a cyclic wandering of aseismic strips in the Earth's mantle (Malamud and Nikolaevsky, 1983, 1985), oscillatory movements of the seismic reflection sites (Bazavluk and Yudakhin, 1993; Bormotov and Bykov, 1999) and the migration of geophysical field anomalies (radon, electrokinetic signals) in proximity to faults (Nikolaevskiy, 1998).

The direct indications of strain waves are displayed in wave fluctuations of the ground water level and the migration of slopes and surface deformations. The direct methods exploring temporal variations of crustal deformation comprise the deformographic (Kasahara, 1979; Ishii et al., 1983; Nevskii et al., 1987; Bella et al., 1990; Harada et al., 2003), hydrogeodynamic (Barabanov et al., 1988; Kissin, 2008) and geodetic measurements (Kuz'min, 2012), including the methods of deformation measurements using laser ranging (Milyukov et al., 2013) and GPS observations (Reuveni et al., 2014; Yoshioka et al., 2015).

To detect the main physical mechanisms of seismicity migration and the generation of signals of a different nature that accompany strain waves, we need to perform further observations and improve GPS and seismological data processing techniques, and conduct theoretically prepared and purposeful experiments.

The answer to the question “where to search for slow strain waves?” is directly linked with the detection of the main types of tectonic structures generating these waves.

The distribution of earthquake epicenters in the zone of interaction
between the Amurian, Eurasian and Okhotsk lithospheric plates; 1 –
lithospheric plate boundaries: EU – Eurasian, PA – Pacific, PH –
Philippine, OK – Okhotsk; 2 – epicenters of earthquakes with magnitude

From the published results it follows that subduction, collision, active riftogenesis and transform fault zones are the most probable types of tectonic structures generating strain waves. These intensive sources of different tectonic natures possess a common property – they are the interaction zones between crustal blocks and the lithospheric plates.

Migration of shear deformation in subduction zones is directed from the ocean
toward the coast. This general tendency was first revealed in the area of the
Japan island arc where migration is oriented east–west, and in the opposite
Pacific coastal area – in the western Cordilleras, where deformations
migrated from south to north (Kasahara, 1979). Migration of the maximum of
the vertical crustal deformation from the subduction zone toward the
continent at a velocity of about 10 km yr

The seismicity pattern observed in the south of central Asia can also be explained by strain waves excited under the oscillating regime of the Eurasian and Indian lithospheric plate collision in the Pamir and Tien Shan junction zone (Nersesov et al., 1990). The compression at the Indostan and Eurasian lithospheric plate boundary in the Himalayan collision zone is the source of “fast” and “slow” waves of plastic deformation that trigger earthquakes in central and eastern Asia (Wang and Zhang, 2005).

In the Baikal rift system, four main groups of strain waves with different
velocities (7–95 km yr

Based on continuous long-term seismic and laser ranging observation data, the
effect of propagation of slow waves of tectonic deformations traveling along
transform faults at velocities of 40–50 km yr

The rotational block movements in the fault zones due to tectonic processes or earthquakes are considered one of the main physical mechanisms of strain wave generation (Nikolaevskiy, 1996; Lee et al., 2009; Teisseyre et al., 2006).

In order to specifically investigate the relationship between strain waves and the dynamics and seismicity pattern observed in fault-blocky geological media, we have selected the study area on the northern margin of the Amurian plate – the most seismically active area of the interaction zone between the Amurian and Eurasian plates.

The analysis of the spatiotemporal seismicity pattern observed in vast regions is commonly performed based on statistical processing of earthquake catalogs. The directions of earthquake epicenter (or groups of epicenters) displacements are defined and their displacement rates are determined. As opposed to the standard regional approach, we here applied a comprehensive analysis including both conventional statistical methods and those of cluster analysis adapted by the authors for the geodynamic zone gradation. The details of developed clustering techniques and statistical analysis of background seismicity can be found in Trofimenko et al. (2015). In the paper by Trofimenko et al. (2016a), the statistical validity of the applied method is shown and the correctness of the models is evaluated.

To study the dynamics of seismicity in different zones, the area along the northern boundary of the Amurian plate was divided into separate clusters (Fig. 1). When clustering, we applied the criterion of earthquake grouping near active faults, and the geomorphological and tectonic features of active structures, as well as the presence of meridional (submeridional) first-rank faults within the distinguished zones, were taken into consideration.

When developing space–time models of seismicity, the spatial relationship between separate seismic clusters during a year was revealed and taken into account. Based on statistical distributions of earthquakes, the analysis of seismicity maxima passage over easterly–westerly arranged clusters has been performed.

The basic data were derived from the Earthquakes of Russia
catalog
(

As a result of the calculation, the average period of seismicity maximum passage in days from the beginning of the year has been determined for each cluster, which is assigned to the average value of the cluster longitude. These values were used for the calculation of the displacement rate of seismicity maxima. We calculated the velocities and wavelengths of slow strain waves from the maxima of the spatial correlation of seismicity.

The spatiotemporal distributions of earthquake epicenters reflect synchronization of seismicity maxima in the annual cycles over a certain spatial interval (migration period). The statistical calculations performed for each cluster allowed the identification of six similar spatiotemporal cycles of seismicity maxima migration A, B, C, D, E and F (Fig. 1), for which the spatial periods of migration and displacement rates of seismicity maxima have been calculated.

In the northeastern segment, the maxima of statistical distributions are
located in the clusters arranged nearly equally apart from each other, at

The determined spatial period

The spatial distribution of seismicity in the annual cycles with respect to the strain wave fronts and meridional structures. Active tectonic faulting: Tanlu fault zone, Aldan–Stanovoy block (Al–St) and Baikal rift zone (BRZ). Figures in the circles denote the faults: 1 – Limurchan, 2 – Tyrkanda, 3 – Temulyakit meridional faults, 4 – meridional structures of the eastern flank of the Baikal rift zone, 5 – Gastakh, 6 – western Turanian, 7 – Levo-Minsky. 1 – submeridional interblock faults of the Aldan shield; 2 – strain wave fronts (Sherman, 2013); 3 – the direction of seismicity maxima migration in the annual cycles and movements of the strain wave fronts.

The displacement rate values for seismicity maxima are obtained from
regression equations using the linear approximation method and are equal to

For the entire northeastern segment, the average value of the velocity
modulus of the seismicity maxima displacement (with a relative determination
error of 7 %) is equal to

To explore the deformation processes in the geological medium with a discrete blocky structure and to perform special GPS experimental observations, we selected the South Yakutia geodynamic polygon located near the northern boundary of the Amurian plate, at the junction of two major tectonic structures – the Aldan Shield and the Stanovoy Range. Recently, a number of blocks of different sizes and configurations have been inferred here from geological data. These blocks experience the vertical and horizontal movements of different directions, velocities and amplitudes (Imaeva et al., 2012), which are responsible for the complicated character of tectonic movements.

We have analyzed a set of time series obtained at two collocated GPS sites
NRGR and NRG2 situated near the active fault intersection area in the central
part of the Stanovoy Range (Fig. 1). The NRGR site is located in the area of
the Chulman depression on a 15

It is necessary to emphasize that the meteorological factors in the annual cycles influence the shapes of the movement trajectories of the collocated sites equally (van Dam et al., 1994). Therefore, the detected paradox cannot be explained by the meteorological causes.

The dynamics of displacement components of NRGR and NRG2 station
daily positions in different directions:

The generalized model of block movement in the vertical plane along
differently oriented local faults of the hinge type due to variable vertical
loading.

This paradox can only be resolved in the case when the observation sites are adjacent to the boundaries of specific – “hinge” – type local faults (Fig. 4a). Really, for site NRGR, a local feathering fault of the Sunnangyn–Larba northeast-trending fault system is the “hinge”, whereas for site NRG2, the” hinge” is one of the branches of the Berkakit northwest-trending fault (Fig. 1). The physical model of this fault-blocky structure can be represented as a set of rods – physical pendulums (Fig. 4b) – whose lower parts are fixed, while the upper parts are disturbed from the equilibrium condition. In this case, the upper parts of the rods (blocks) are displaced with respect to some central line (the fault hinge).

The approximation curve fitting for the vertical component of block
displacement has led to one more unexpected result. The shape of the best fit
function approximating the experimental curve appeared to coincide with a
breather – the solution (2) of the sine–Gordon equation (see below). When
selecting the theoretical curve in the shape of a breather (2), this result
for the north–south component is obtained at

Seasonal variations of NRGR and NRG2 station positions.
Approximation of the observed displacement curves by the theoretical curves
for the N–S

The coincidence of the trajectory shape of measured vertical displacements with the shape of a breather, and the correspondence of the blocky structure in the area of GPS site locations to the model of coupled pendulums, served as a motivation for application of the sine–Gordon equation to describe the evolution of the vertical components of block movements.

The mathematical model of quasi-periodical vertical components of
oscillations of rigidly coupled crustal blocks with the adjacent
“hinge”-type faults corresponds to the equation

One of the solutions of Eq. (1) is called a breather (dynamic soliton) and
represents a nonlinear function, which for the case of the soliton with the
immobile center of gravity can be written as

Like a soliton, the breather has the shape of an impulse; it is localized in
space and is pulsating in time. In the low-frequency range

The detected high correlation of the observed site displacement trajectories with the theoretical curve corresponding to a breather allows us to suggest that the mechanism of these oscillations can be associated with the occurrence of strain waves in the fault intersection system. In this case, these waves can be qualitatively treated as standing waves of compression–extension in the blocky geological medium.

The sine–Gordon equation solution in the shape of a breather has previously been applied for modeling the wave dynamics of faults and strain waves (Mikhailov and Nikolaevskiy, 2000; Gershenzon et al., 2009; Erickson et al., 2011). Mikhailov and Nikolaevskiy (2000) considered a scenario when collision of two tectonic waves (kink–antikink collision) resulted in the occurrence of a large earthquake. The solution in the shape of a breather has also been applied for the interpretation of the features of fault dynamics observed after the 1989 Loma Prieta earthquake (Gershenzon et al., 2009). Based on a modified Burridge–Knopoff model, a solution has been obtained that corresponds to a localized failure – a breather that propagates along a fault and is damping in the fault segment of the final length (Erickson et al., 2011). Wu and Chen (1998) earlier reduced a one-dimensional Burridge–Knopoff spring-block model to the sine–Gordon equation and applied its solution in the shape of a solitary wave (kink) to investigate earthquakes.

The accumulated facts indicate the propagation of slow wave-like movements
within the crust and the lithosphere at different velocities on global and
regional scales (Bykov, 2014). The results of our investigation (the
periodicity of the seismic components, spatial cycles with phase shift of
seismicity maxima, migration velocity of earthquake epicenters) and their
comparison with the known data allow us to identify the dynamics of
seismicity along the northern boundary of the Amurian plate as a wave
process. We have revealed synchronous quasi-periodic seismicity variations in
equally spaced clusters with spatial periods of 7.26 and 3.5

The calculated average displacement rate value of the maxima of weak
seismicity (2

The displacement of seismicity in the annual cycles occurs from east to west
and coincides with the direction of migration of large earthquakes, strain
wave fronts and crustal deformation detected from direct deformographic and
GPS measurements (Kasahara, 1979; Bella et al., 1990; Harada et al., 2003;
Yoshioka et al., 2015). The slow strain wave fronts are triggers of large
earthquakes (

The spatial correlation of migration of seismicity and deformations as well as the migration of deformations – two different manifestations of the geodynamic process – may mean that seismicity migration is associated with the propagation of tectonic stresses in the form of slow strain waves that cause a complementary load and subsequent earthquake occurrence. The numerous results of observations of seismicity migration are hard to explain by causes other than wave-like variations of the global and local stress fields.

The conclusions on the wave pattern of the deformation process are consistent with the results of special experimental observations performed to explore crustal block interaction. The seasonal course of displacements of GPS stations NRGR and NRG2, involved in the in situ experimental observations, or of the deformations of the blocky structure of the crust, exhibits a wave-like rather than linear pattern. The wave-like displacements can be explained by transmittance of slow strain waves.

Based upon the statistical modeling, we have established the in-phase and anti-phase changes of the components of the full displacement vector, the relative time delay of the maxima and minima for separate components, and dissimilarity of the displacement trajectory from a sinusoid. In order to describe the evolution of oscillations of the interacting blocks, a simple mathematical model is proposed from which the explanation of the observed specific behavior of these blocks follows.

Based on experimental observation data and the developed model of crustal block movement, we have shown that there is one possible interpretation for this fact that the trajectory of GPS station position disturbance is induced by migration of crustal deformation in the form of slow waves.

The basic data were derived from the Earthquakes of Russia catalog
(

We are deeply grateful to the editor Arcady Dyskin and to two reviewers, Semen Sherman and the anonymous reviewer, for constructive comments on the first version of the paper. The reported study was funded by the RFBR according to research project no. 16-05-00097 a.Edited by: A. Dyskin Reviewed by: S. Sherman and one anonymous referee