*Article* **Method of Studying Modulation Effects of Wind and Swell Waves on Tidal and Seiche Oscillations**

**Grigory Ivanovich Dolgikh 1,2,\* and Sergey Sergeevich Budrin 1,2,\***


**Abstract:** This paper describes a method for identifying modulation effects caused by the interaction of waves with different frequencies based on regression analysis. We present examples of its application on experimental data obtained using high-precision laser interference instruments. Using this method, we illustrate and describe the nonlinearity of the change in the period of wind waves that are associated with wave processes of lower frequencies—12- and 24-h tides and seiches. Based on data analysis, we present several basic types of modulation that are characteristic of the interaction of wind and swell waves on seiche oscillations, with the help of which we can explain some peculiarities of change in the process spectrum of these waves.

**Keywords:** wind waves; swell; tides; seiches; remote probing; space monitoring; nonlinearity; modulation

#### **1. Introduction**

The phenomenon of modulation of short-period waves on long waves is currently widely used in the field of non-contact methods for sea surface monitoring. These processes are mainly investigated during space monitoring by means of analyzing optical [1,2] and radar images [3,4] received from the satellites to restore the structure of the rough sea surface. A two-scale model of the sea surface was used in analysis of radar images. The effect of short waves was taken into account in the framework of the Bragg scattering mechanism, and the effect of the large-scale component is taken into account by changing the slope of the surface. As a result, the small-scale wave component turned out to be responsible for the backscattering of radar signals, and the large-scale component was responsible for spatial modulation of the scattered signals [5,6].

Of particular interest during remote probing of seas and oceans in these types of research is the study of internal waves (IW) [7], current fields [8,9], and anthropogenic impacts on the aquatic environment [10]. Internal waves propagating in the ocean appear on the sea surface due to horizontal components of orbital velocities near the surface, which lead to variations in the characteristics of short wind waves. Thus, on the sea surface, IWs appear in the form of stripes and spots with increased (tidal rip) and decreased (slick) intensity of short gravity waves [11–13]. Modulation effects can occur in a wide range of wind waves. The modulation of wind waves can be described and explained within the kinematic mechanism framework [14], but there are also other theories [15,16]. For example, the authors of [17] propose a model where an increment modulation effect caused by the variation of the wind speed field over the water surface on which there is a field of currents generated by internal waves (IW) acts as the mechanism of this model. The modulation effect of wind increments is being actively discussed nowadays [18].

There are many works and studies devoted to modulation of infragravity (IG) waves. For example, [19] presents the results of studying short IG length changes, while longer waves move on the surface. In [20], the changes in wavelengths and amplitudes of a short-period wave process are carefully calculated with regard to the nonlinear interactions

**Citation:** Dolgikh, G.I.; Budrin, S.S. Method of Studying Modulation Effects of Wind and Swell Waves on Tidal and Seiche Oscillations. *J. Mar. Sci. Eng.* **2021**, *9*, 926. https:// doi.org/10.3390/jmse9090926

Academic Editor: Lev Shemer

Received: 19 June 2021 Accepted: 22 August 2021 Published: 26 August 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

between two wave processes. The change in the energy of short waves, in this case, corresponds to the effect of longer waves against the radiation stress of short waves, which was previously neglected. The radiation wave stress should be understood as an overflow of the impulse caused by wind waves, or, more exactly, by their nonlinearity. In [21], on the basis of field data and numerical models of sea surface gravity waves, it is shown that in the coastal zone during nonlinear wave-wave interaction, energy is transferred from low-frequency long waves back to higher-frequency motions. This explains the tidal modulation of the energy of IG waves, observed in the records of near-bottom pressure on the southern shelf of Sakhalin Island. Similar results were obtained in [22], where the strong tidal modulation of infragravity (200 to 20 s period) waves observed on the southern California shelf is shown to be the result of nonlinear transfers of energy from these low-frequency long waves to higher-frequency motions.

What is interesting is question of the evolution of the wind wave spectrum, which has many mechanisms. For example, [23,24] describes the mechanism of change in the wave spectrum, while waves are going out to a shallow; in this case, due to nonlinear interactions with the bottom, the frequency of the wave process decreases. In [25], numerical and experimental studies of the process of nonlinear energy transfer between two main spectral maxima conditioned by the mechanism of dispersive focusing indicate that nonlinear energy transfer plays a greater role than linear superposition. There is also a large number of works devoted to the evolution of the wave spectrum associated with the modulation effect caused by the interaction of wind waves and swell waves [26,27]. In this paper, we will try to show how the effects of the modulation of wind waves at long waves, such as tides and seiches, can affect changes in the wave spectrum.

We should also note the participation of modulation effects in the emergence of rogue waves. For example, in [28], it is stated that rogue waves (extreme waves) naturally originate as a result of evolution of spectrally narrow packets of gravity waves. We can say that rogue waves are a nonlinear stage of modulation instability. In [29], the Euler equation was solved for liquid with a free surface in deep water. Periodic and boundary conditions were created in the form of a Stokes wave, which was slightly modulated by a low frequency (10−5). At the same time, such a wave is unstable and modulation should increase with time, thereby generating an extremely high wave.

This paper presents a method for studying the effects of modulation of wind and swell waves on tidal and seiche oscillations. This method was developed and tested on hydrophysical data obtained from high-precision, modern instruments based on laser interference methods [30]. The paper also presents the results of using this method for processing and analyzing experimental data, the results of which several main types of modulation of wind waves on seiche oscillations were identified and described. The possibility of introducing this method for processing hydrophysical data received in real time mode is being considered.

All of the experimental data presented in this work were obtained using laser meters of hydrosphere pressure variations [31]. These devices were installed on the bottom for a period from several days to several months, transmitting data in real time to laboratory rooms located at Shultz Cape, the Sea of Japan, in the Primorsky Territory of the Russian Federation. There are also mobile versions of these devices, with which we conduct measurements of tidal and seiche oscillations in various harbors of the Posiet Bay and the Peter the Great Gulf of the Sea of Japan. Although most of the data presented were obtained in Vityaz Bay, the Sea of Japan, we also analyzed and compared the results obtained in other bays. Thus, the presented results will be valid for other closed water areas of the World Ocean.

#### **2. Wave Modulation Research Method**

As we know, the main change in the period of swell waves during propagation from the point of generation occurs due to dispersion during propagation; this change in the period is linear and has a decreasing character. However, if we consider large time-scale phenomena, such as typhoons, which originate hundreds of kilometers from the place of registration, the process of swell wave propagation can take several days. Additionally, in addition to dispersion, waves can be influenced by both large-scale phenomena, such as tides, and local phenomena, such as seiche oscillations that occur in closed sea areas and, as a result of these phenomena, the process of changing the period becomes nonlinear. At the same time, in order to study the emerging nonlinear processes, it is first necessary to separate the process of changing the period associated with dispersion from other processes that affect the variations in the period of wind waves and swell waves.

Earlier, from the fragments of the record, which contained swell waves created by passing typhoons, we derived the general function of period change [32,33]. With high accuracy, this function describes the dispersion of waves as they propagate from the source to the receiving point; the general view of the function is presented below.

$$
\overline{T}(t) = K\_{10} \cdot \frac{\Delta \ T}{\Delta t} \cdot t + T\_0 \tag{1}
$$

where *<sup>K</sup>*<sup>10</sup> = −2753·10−4, <sup>Δ</sup> *<sup>T</sup>* is the total change in the period in the investigated section, Δ*t* is the total duration of the section, and *T*<sup>0</sup> is the initial period of wave for *t* = 0.

Thus, as applied to the above problem, we only need to identify the nonlinear part of the period change process. To do this, we will use regression analysis.

Based on the selected values of the spectral maxima, the surface wave signal, the coefficients of the polynomial regression are calculated followed by its construction. The coefficients are calculated from the system of equations presented below.

$$\begin{cases} b\_0 + b\_1 t + b\_2 t^2 + \dots + b\_k t^k = T \\ b\_0 t + b\_1 t^2 + b\_2 t^3 + \dots + b\_k t^{k+1} = Tt \\ b\_0 t^2 + b\_1 t^3 + b\_2 t^4 + \dots + b\_k t^{k+2} = Tt^2 \\ \vdots \\ b\_0 t^k + b\_1 t^{k+1} + b\_2 t^{k+2} + \dots + b\_k t^{2k} = Tt^k \end{cases} \tag{2}$$

where *T*—values of spectral maxima at time *t*, *b*<sup>0</sup> ... *bk*—regression coefficients.

This system can be represented in matrix form as *AB = C*, where

$$A = \begin{pmatrix} 1 & t & t^2 & \cdots & t^k \\ t & t^2 & t^3 & \cdots & t^{k+1} \\ t^2 & t^3 & t^4 & \cdots & t^{k+2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ t^k & t^{k+1} & t^{k+2} & \cdots & t^{2k} \end{pmatrix}, B = \begin{pmatrix} b\_0 \\ b\_1 \\ b\_2 \\ \vdots \\ b\_k \end{pmatrix}, C = \begin{pmatrix} T \\ Tt \\ Tt^2 \\ \vdots \\ Tt^k \end{pmatrix} \tag{3}$$

We can write the general regression equation as follows:

$$T(t) = b\_0 + tb\_1 + t^2 b\_2 + \dots + t^{2k} b\_k \tag{4}$$

To estimate the polynomial regression and the general function of the period change, we will use two criteria for evaluating the regression analysis: the coefficient of determination R<sup>2</sup> (equal to 1 in the ideal case) and the standard deviation S (equal to 0 in the ideal case). However, as practice and the theory of regression analysis show, to describe processes with the n number of extrema, it is sufficient to use a polynomial of degree *n +* 1. Thus, to describe the effects of modulation of wind waves on tidal oscillations, which can have 4–5 extrema in a few days, polynomials of 5–6 degrees will be sufficient for the description. As the practice of the conducted research shows, this approach to the choice of the degree of polynomial regression is also the best for parameters of the coefficient of determination and standard deviation. Thus, the use of polynomial degrees greater than *n +* 1 is not practical.

Subtracting the period change function from the obtained expression, we eliminate the constant component of the graph, which is the dispersion that occurs during propagation of wind and swell waves. Thus, we obtain the absolute values of nonlinearities arising during propagation. The general expression for the function describing the absolute values of nonlinearities can be written as:

$$T(t) = \left(b\_0 + tb\_1 + t^2 b\_2 + \dots + t^{2k} b\_k\right) - \left(K\_{10} \cdot \frac{\Delta \ T}{\Delta t} \cdot t + T\_0\right) \tag{5}$$

#### **3. Modulation of Swell Waves on Tides**

Figure 1 shows an example of processing experimental data using this processing method, highlighting the nonlinearity of the period change.

**Figure 1.** (**a**) Spectrogram of the wind wave signal obtained in the period from 3 to 5 July 2013 in Vityaz Bay, the Sea of Japan; (**b**) spectral maxima and the regression graph constructed according to (4); (**c**) absolute values of the change in the period of the swell waves identified according to (5).

As we can see from Figure 1c, the graph contains three pronounced maxima: the time interval between them is 12 h, which may indicate the modulation of swell waves by tidal fluctuations, which is also proved by other analyzed fragments. Thus, we can conclude that the nonlinearity of the period change is associated with the effect of the modulation of swell waves by low-frequency wave processes, in this case, tidal oscillations. Let us consider one more case of the modulation of wind waves by tidal oscillations of 24 h, shown in Figure 2.

**Figure 2.** (**a**) Spectrogram of the wind wave signal obtained in the period from 8 to 11 July 2013 in Vityaz Bay, the Sea of Japan; (**b**) spectral maxima and regression graphs; (**c**) absolute changes of period and amplitude of wind waves on tidal oscillations.

In Figure 2c, there is an explicit modulation of both the wind wave period and the amplitude. As a result of modulation at the points of maximum tide values, waves with a large period and amplitude are concentrated, which corresponds to the general idea of this process. However, not everything is as clear at first glance as it seems. When processing the data, we found a number of fragments that do not correspond to generally accepted concepts. One of these fragments is shown in Figure 3.

As we can see in Figure 3a, the record contains two pronounced tides within a period of 12 h, but in Figure 3b, the modulations of the period and the amplitude of the waves are in antiphase, i.e., despite the fact that the waves within aa large period are at the maximum tide point, waves within a smaller period have a greater amplitude. At the same time, all similar cases have one common feature: in all of the fragments, there are seiche oscillations; in this case, these are oscillations are within period of 18 min, which is characteristic of the place where the measurements were completed (Figure 4).

In connection with the above, we can assume that these ambiguous cases may arise due to complex modulation processes associated with the "submodulation" of several wave phenomena.

**Figure 3.** (**a**) Record of tidal fluctuations with a period of 12 h obtained in the period from 11 to 12 July 2013 in Vityaz Bay, the Sea of Japan; (**b**) change in the period and amplitude of waves associated with modulation on tidal oscillations and calculated by the following section of the record.

**Figure 4.** Seiche oscillations within a period of 18 min against the background of a 12 h tide.

#### **4. Influence of Atmosphere on Modulation of Wind Waves by Tidal Oscillations**

An additionally important issue in this topic is the influence of variations in atmospheric pressure and wind regime on the process of the modulation of wind waves by tidal oscillations. Let us consider this issue on the example of a processed recording fragment with tidal oscillations with a period of 12 h and provide meteorological data received from the meteorological station for the same period of time (Figure 5).

**Figure 5.** (**a**) Absolute values of wave periods and amplitude variations calculated from the site of the wind wave records obtained during the period from 3 July to 5 July 2013 in Vityaz Bay, the Sea of Japan; (**b**) data on atmospheric pressure and wind speed and regression graphs built on this data for the same period of time.

We can see from the graphs of the modulation of the amplitude and the period of the waves (Figure 5a) that they practically coincide in phase. This means that, as in the previously considered cases, at the maximum tidal points, there will be waves with a longer period and greater amplitude. The atmospheric pressure graph (Figure 5b) shows three peaks corresponding to atmospheric tides; the maxima of these tides are shifted several hours to the right of the maximum values of the swell modulation, which indicates that these oscillations occur after the sea tide. On the same graph, you can see that the graph of the wind speed variations is practically in antiphase with pressure variations and has a minor delay of several hours. Thus, we can conclude that diurnal pressure variations occur after sea tide. Variations in wind speed occur due to fluctuations in atmospheric pressure since air masses begin to move from the area of increasing pressure to areas with lower pressure, which results in a change in wind direction. An abrupt decrease in the wave amplitude at the second maximum (Figure 2a) at the max wind speed may be due to the fact that the wind direction was opposite to the direction of the swell wave propagation. At the same time, at the third maximum of the period change graph, we see an increase in the modulation effect, which may indicate that waves with periods of 4 s and below are more susceptible to the modulation effect than waves with higher periods. When processing and analyzing more than five more data fragments with modulations of wind waves and swell waves on tidal oscillations present on them and the meteorological data received over the same period of time, there were no impacts on the modulation effect on the part of the variations in the atmospheric pressure and wind speed.

#### **5. Modulation of Wind Waves on Seiche Oscillations**

Analyzing more than 30 fragments of the records on which seiche oscillations were explicitly presented simultaneously with strong wind waves, several characteristic types of modulations, "two-tone" and "four-tone", were identified. These types of modulations are shown in Figure 6.

**Figure 6.** (**a**) A recording fragment with two seiche oscillations of 18 min and a change in the period of wind waves ("two-tone" modulation); (**b**) a recording fragment with one seiche vibration of 20 min and a change in the period of wind waves ("four-tone" modulation).

As we can see in Figure 6a, in one seiche oscillation with period of 18 min, the wind wave period changes two times, which means that on opposite seiche fronts, the processes of compression (red arrows) and extension (green arrows) occur, while in Figure 6b, in one oscillation with period of 20 min, the wind waves at the front have one compression– extension cycle, which, in turn, generates modulation.

We can describe this type of modulation using a simple equation.

$$
\Delta T(t) = \sin\left[\left(\frac{2\pi t}{T\_{\text{ww}}} + \alpha\_1\right) + m \cdot \sin\left(\frac{2\pi t}{T\_{mo}} + \alpha\_2\right)\right] \tag{6}
$$

where *T*ww is the period of the wind waves, *Tmo* is the period of modulating oscillation, *m* is the modulation index.

Figure 7 shows the spectrum of wind waves within a period of 4.8 s modulated by seiche oscillations within a period of 18 min constructed on experimental data. Figure 6 also shows the spectrum calculated using Expression (5), with the same wave parameters and modulation index *m* = 4.

**Figure 7.** (**a**) Wave spectrum constructed on experimental data; (**b**) wave spectrum built on Expression (6).

**Figure 8.** (**a**) Example of a combined modulation on one seiche oscillation; (**b**) example of a combined modulation on two seiche oscillations.

In the spectra shown in Figure 7, as in other similar cases, there are two characteristic maxima. The first is the main period of wind waves, while the second is responsible for the modulation process of wind waves on seiche oscillations. In this case, the main spectral maximum always remains in its place, and the position of the second (modulation) maximum can vary depending on the modulation index. Thus, when analyzing the spectra of experimental data based on two characteristic maxima, we can confidently speak about the presence of seiche oscillations and their modulation of wind waves.

We have now looked at the most common modulation ideal cases. However, we understand that in nature, not everything is so unambiguous and, of course, there are cases of combined modulations. As such, for example, on one (Figure 8a) and on two seiche oscillations, both "two-tone" and "four-tone" modulation can occur (Figure 8b). Such combined modulations, as we can assume, arise in cases of different steepness of the front of the modulating oscillation, which is caused by the seiche asymmetry. As a result, on the flatter part, wind waves and swell waves can be modulated several times.

These types of modulations can also be described using Expression (6); however, the modulation index, in this case, will be equal to the number of extrema per the number of studied modulating oscillations. To show this, let us calculate the spectrum of the signal using Expression (6) in Figure 8a and compare it with the spectrum constructed based on the experimental data. The initial data are as follows: *Tww* = 8.8 s; *Tmo* = 24 s; *m* = 6. The calculation results are shown in Figure 9.

**Figure 9.** (**a**) Wave spectrum constructed on experimental data; (**b**) wave spectrum built on Expression (6).

As we mentioned above, the width of the modulated oscillations spectrum depends directly on the value of the modulation index, which is well demonstrated in Figure 9. The form of the spectra is almost identical although it has minor deviations in the numerical values of the maxima.

However, as we all perfectly understand, the change in the period is not only associated with the processes of modulation of wind waves but are also associated with

48

dispersion during wave propagation. In order to account for this variance, we substitute period change Function (1) into the expression describing Modulation (6). As a result, we obtain an equation describing both the modulation process and the dispersion of wind waves during propagation.

$$T(t) = \sin\left[\left(\frac{2\pi \cdot t \cdot \Delta t}{K\_{10} \cdot t \cdot \Delta t + T\_0 \cdot \Delta T}\right) + m \cdot \sin\left(\frac{2\pi \cdot t}{T\_{mo}} + \alpha\right)\right] \tag{7}$$

Figure 10 shows the spectrum of wind waves within a period of 5.2 s obtained based on experimental data; in the studied fragment, there is a pronounced dispersion of waves Δ*T* = 0.4 s, and seiches within a period of 18 min are present. The figure also shows the spectrum calculated based on Expression (7), according to the initial data indicated above.

**Figure 10.** (**a**) Wave spectrum constructed on experimental data; (**b**) wave spectrum built on Expression (7).

As you can see in Figure 10, there are now four characteristic maxima in the spectra. The first two are responsible for the modulation of wind waves on seiches, and the other two are responsible for wave dispersion. In this case, the spectrum width does not depend on the modulation index. The modulation index is responsible for the amplitude of spectrum maxima, and the change in the wave period due to dispersion during propagation is responsible for the spectrum width.

#### **6. Conclusions**

The method of studying modulation effects presented in this work, which is based on regression analysis and the general functions of the period change, has shown good results when applied to studies of the modulation of wind and swell waves on tidal oscillations. Using this method, it was shown that, in general, when wind waves are modulated by tides, waves with large a period and amplitude are concentrated in the upper points of the tide. However, when extraneous wave processes, such as seiches, occur, the modulation of wave amplitude can have an extremum in the lower tide point, i.e., the modulation of wave period and its amplitude will be in antiphase.

Studying the modulation of wind waves on seiches by the above method, we identified several main types of modulation: "two-tone" and "four-tone". These types of modulation are well described using the common frequency modulation, Equation (6). When comparing the spectra of the experimental data and the spectrum calculated using the frequency modulation formula, we identified two spectral maxima, the first of which is responsible for the main wave period, and the second of which is responsible for the modulation process. From these two characteristic maxima, we can speak with good confidence about the presence of seiche oscillations and their modulation of wind waves, while the width of the spectrum depends on the modulation index, i.e., the number of wave modulations per one period of the seiche oscillation.

The obtained expression for frequency modulation accounting for Dispersion (7) well describes these phenomena. When comparing the spectra of the experimental data and the spectra calculated using this expression, four characteristic spectral maxima were identified. The first two are responsible for modulation on seiches; the rest are responsible for dispersion during propagation. In this case, the modulation index affects the amplitude of the maxima, and the change in period is responsible for the width of the spectra. In the presence of these maxima in the spectrum, we can speak not only about the presence of seiche oscillations, but also about the fact that the wave process modulated by them is not local in nature but most likely came from another point of the water area since its period varies linearly due to dispersion during propagation.

The efficiency of the considered method for studying the interaction of wind waves with wave processes of lower frequency—twelve-hour and round-the-clock tides and seiches—can be increased by combining it with space monitoring methods based on the analysis of optical and radar images, especially in closed water areas and in the shelf zone of the World Ocean.

**Author Contributions:** G.I.D. problem statement, discussion, and writing of the article. S.S.B. data processing, discussion, and writing of the article. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work was conducted with the financial support for the project by the Russian Federation represented by the Ministry of Science and Higher Education of the Russian Federation, Agreement No. 075-15-2020-776.

**Acknowledgments:** We would like to express our deep gratitude to all of the employees of the Physics of Geospheres laboratory.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


### *Article* **Deformation Anomalies Accompanying Tsunami Origination**

**Grigory Dolgikh 1,2,\* and Stanislav Dolgikh 1,2,\***


**Abstract:** Basing on the analysis of data on variations of deformations in the Earth's crust, which were obtained with a laser strainmeter, we found that deformation anomalies (deformation jumps) occurred at the time of tsunami generation. Deformation jumps recorded by the laser strainmeter were apparently caused by bottom displacements, leading to tsunami formation. According to the data for the many recorded tsunamigenic earthquakes, we calculated the damping ratios of the identified deformation anomalies for three regions of the planet. We proved the obtained experimental results by applying the sine-Gordon equation, the one-kink and two-kink solutions of which allowed us to describe the observed deformation anomalies. We also formulated the direction of a theoretical deformation jump occurrence—a kink (bore)—during an underwater landslide causing a tsunami.

**Keywords:** earthquake; tsunami; laser strainmeter; deformation jump; sine-Gordon equation; kink; anti-kink; underwater landslide

#### **1. Introduction**

We know that tsunamis are some of the most dangerous and catastrophic phenomena on Earth, which cause significant damage to humanity. A typical example is the tsunami that hit the Indian Ocean on 26 December 2004, killing more than 283,000 people. This was caused by a powerful earthquake with a maximum magnitude of about 9.3 [1]. Tsunamis affect various regions of the planet, although this is especially true for Japan. Considering the extent to which Japan suffers from the impacts of earthquakes and tsunamis and its high level of scientific and technical development, we can expect that more advanced scientific and technical ideas aimed at predicting the occurrence and development of earthquakes and tsunamis will be concentrated in this region of the planet. While the short-term forecasting of earthquakes is far from being solved, the detection of the moments of tsunami origination seems to be quite solvable. The Japanese Islands and the adjoining water areas are "crammed" with various seismic stations, GPS receivers, bottom seismic stations, and high-precision sea and ocean level meters. Nevertheless, the events of 2011 "exposed" the short-term tsunami forecasting problems even more.

Presently, the traditional method of short-term tsunami forecasting is based on seismological information (earthquake magnitude, main shock time, and epicenter location) [2]. An earthquake magnitude that exceeds a predetermined threshold, which will be different for different tsunamigenic zones, usually results in a tsunami warning. This approach, based on the "magnitude and geographical principle", is simple; it helps reduce the number of missed tsunamis, but also gives false alarms. Most current early tsunami warning systems are based on seismic data.

For example, the Pacific Tsunami Warning Center (PTWC) uses seismic data in conjunction with long-period wave (W-phase) data for global tsunami warnings in the Pacific [3]. Another tsunami warning center is the Japan Meteorological Agency (JMA), which provides local tsunami warnings within up to 3 min after near-field earthquakes by analyzing seismic data [4], then updates the warnings using the forms of seismic wave and tsunami

**Citation:** Dolgikh, G.; Dolgikh, S. Deformation Anomalies Accompanying Tsunami Origination. *J. Mar. Sci. Eng.* **2021**, *9*, 1144. https://doi.org/10.3390/jmse9101144

Academic Editor: Dong-Sheng Jeng

Received: 20 September 2021 Accepted: 12 October 2021 Published: 18 October 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

data [5]. The W-phase appears in seismic recordings between P-waves and S-waves and can be used to estimate the seismic moment, epicenter location, and fault mechanism. The efficiency of W-phase inversion has already been demonstrated in many studies and is actively used by tsunami warning centers [6,7].

Recently, two independent approaches have been proposed to determine the energy of a tsunami source: one is based on Deep-Ocean Assessment and Reporting of Tsunamis (DART) data during tsunami propagation, and the other on ground-based coastal global positioning system (GPS) data during tsunami generation. The GPS approach takes into account the dynamic earthquake process, while the DART inversion approach provides an actual estimate of the tsunami energy and propagating tsunami waves. Both approaches lead to coordinated energy scales for the earlier-studied tsunamis. Inspired by these promising results, the authors [8] researched an approach to determine the tsunami source energy in real time by combining the two methods. At the first stage, the tsunami source is determined immediately after the earthquake using the global GPS network for early warnings in the near-field zone. Then, the tsunami energy is defined more precisely based on the nearest DART measurements to improve the accuracy of forecasts or cancel the alarm. The combination of these two real-time networks could offer an attractive opportunity for the early detection of tsunami threats to save more lives and for the early cancellation of tsunami warnings to avoid unnecessary false alarms. In the past decade, the number of open ocean sensors capable of analyzing information about a passing tsunami has steadily increased, especially thanks to the national cable networks and international efforts such as DART systems. The received information is analyzed in order to warn people about a tsunami. Most of the current warnings, which include tsunamis, are aimed at the mid-to-far zone regions. In [8], the main directions of research using DART and GPS systems were formulated; however, the failure of this system in the Indian Ocean, when a powerful tsunami was missed, plunged once optimistic scientists into despondency. For early tsunami warnings, various methods can be used, including methods based on space monitoring of earthquake-endangered areas [9–11].

We should note that tsunamis can be caused not only by earthquakes, but also by underwater landslides, volcanic activity, and simply by the collapse of mountain massifs into the sea [12].

For the first time period, a deformation anomaly was recorded during the registration of a tsunamigenic earthquake, the laser strainmeter of which showed the form of a deformation jump that occurred after the earthquake started [13]. Later, this result was generalized in [14] and the deformation methods used for determining the tsunamigenic nature of earthquakes were based on it. The potential of this method is associated with the fact that its development will allow the nature of the movement of the Earth's crust to be determined remotely at planetary distances, which sets in motion huge masses of water that degenerate into tsunamis during their development. It is clear that oscillations originating in the source of an earthquake do not cause a tsunami. These oscillations are associated with the parameters of continuity breaches, i.e., with their geometrical dimensions and elastic plate deformations. As a rule, these oscillations occur in the time range from the first minutes to the first ten seconds. These oscillations will never cause a tsunami. Only quick displacements of huge masses of the Earth's crust, which unfortunately are not recorded by any broadband seismographs, lead to a tsunami. In this paper, we will consider the peculiarities of the appearance of the deformation anomalies accompanying (attending) tsunamis, using several examples and with descriptions of the physical mechanisms of their occurrence and development. The development prospects in this direction are associated with the fact that the speed of such deformation anomalies is more than an order of magnitude higher than the speed of tsunami propagation, which is extremely important for warning services.

#### **2. Recording Complex**

The deformation anomaly, described in [13] and to a greater extent corresponding to the concept of a "deformation jump", showed a small amplitude equal to about 60 μm and was recorded by a laser strainmeter at a distance of about 5600 km from its location of origin. It is clear that such deformation anomalies cannot be recorded by any broadband seismograph (velocimeter, accelerometer, etc.), since these instruments are unable to register such disturbances. We could talk about the prospects of using GPS receivers to register such anomalies; however, at such distances from the locations of deformation anomalies, their registration by GPS receivers is impossible because the main samples of GPS receivers can only provide displacement registration accuracy of about 1 mm, which is much larger than the values of the registered anomalies. In the Conclusions section of this paper, we will discuss where these GPS receivers can be used. Currently, to register deformation anomalies associated with the process of tsunami generation, the most efficient instruments are laser strainmeters, which are capable of measuring microdisplacements of the Earth's crust in the frequency range of 0 (conditionally) to 1000 Hz with high accuracy (up to 1 pm) [15]. In our study, we will use the data on variations in the microdisplacements of the Earth's crust obtained from the horizontal laser strainmeter, which were created on the base of a Michelson interferometer with an unequal measuring arm length of 52.5 m and a "north–south" orientation. This instrument uses a frequency-stabilized helium–neon laser manufactured by Melles–Griott with long-term stability of 10−<sup>9</sup> as a light source [16]. Recently, we modernized this laser strainmeter, providing it with a frequency-stabilized laser, stabilized along the iodine lines in the eleventh digit, and an improved recording system. After the modernization process, it could detect variations of microdisplacements in the Earth's crust in the frequency range of 0 (conditionally) to 1000 Hz, with accuracy of 0.03 nm. The laser strainmeter was installed in thermally insulated underground rooms at depths of about 3–5 m under the Earth's surface at 42◦34.798 N, 131◦9.400 E, at an altitude of about 60 m above sea level. Figure 1 shows a general view of the underground part of the central interference unit (left) and the underground beam guide (right). Experimental data are transmitted via a cable line to the recording computer, where after pre-processing the data files are formed, with a sampling frequency of 1 kHz and duration of 1 h. The laser strainmeter is a part of the seismoacoustic and hydrophysical complex, located in the south of the Primorsky Territory of Russia at the sea hydrophysical study site of POI FEB RAS "Shultz Cape" [17]. The main purpose of the complex is studying the nature of variations in microdeformations of the Earth's crust; fluctuations in atmospheric and hydrospheric pressure over wide-frequency and dynamic ranges; and the regularities of the emergence, development, and transformation of oscillations and waves of the sonic and infrasonic ranges. The measurements at the complex are carried out in continuous mode and all obtained data are input into the database (approximately 10 terabyte capacity), which is being steadily complemented. A precise time clock based on the Trimble 5700 GPS instrument is used for synchronization.

**Figure 1.** Modernized 52.5 m laser strainmeter. The central interference unit with a frequencystabilized helium–neon laser, which is stabilized along the iodine lines (frequency stability 10−11, (**left**)), as well as an underground beam guide (**right**).

#### **3. Registration of Tsunamigenic Earthquakes Deformation Anomalies**

The laser strainmeter, which has been operating since 2000, has recorded many tsunamigenic and non-tsunamigenic earthquakes. Let us consider the specific differences in the behaviors of tsunamigenic and non-tsunamigenic earthquakes using the two example earthquakes described below. The first powerful tsunamigenic earthquake occurred on 26 December 2004, while the second non-tsunamigenic earthquake occurred on 4 August 2000. The epicenter of the first tsunamigenic earthquake was located at 3.30◦ N, 95.87◦ E, about 160 km west of Sumatra, at a depth of 30 km below the sea level. The distance from the earthquake epicenter to the laser strainmeter location was approximately 5600 km. The laser strainmeter recording of a peculiar signal from a tsunamigenic earthquake is shown in Figure 2. The recording shows a powerful deformation anomaly that appeared a short time after the earthquake started, with an amplitude of about 59.3 μm. The amplitude of this anomaly was much greater than the amplitude of the daily tide, as observed at the instrument location. In Figure 2, the earthquake onset is marked with an arrow. The laser strainmeter recorded the tsunamigenic earthquake signal at 19 min 54 s after the earthquake started.

**Figure 2.** Fragment of the 52.5 m laser strainmeter recording from December 2004 (**a**), enlarged fragment of the laser strainmeter recording (**b**), and dynamic spectrogram of the laser strainmeter recording (**c**). Universal time is shown on the abscissa axis.

Analyzing the dynamic spectrogram shown in Figure 2c, we found that the periods of the main oscillations caused by the earthquake gradually decreased from 30 to 14 s. If we know the relation connecting the propagation speed of the elastic waves with the period of oscillations, the magnitude of the change in the period of the main oscillations, and the time during which this change occurred, we can determine the distance to the earthquake site. The dynamic spectrogram also shows a strong disturbance in the low-frequency range.

We must note that the results from processing the space monitoring data, which was carried out after this catastrophic tsunamigenic earthquake, showed that ionospheric anomalies were observed 4–5 days before it, which were registered by way of analyzing data from the GPS satellite navigation system. These anomalies manifested themselves as changes in the electron density profiles [11] and in changes in the total electron content (TEC) in the ionosphere in the Sumatra area [10]. At the same time, the use of a geomechanical model [18] for this region showed that the recorded increase in atmospheric pressure led to an increase in the stress–strain state of the Earth's crust and brought it closer to the strength limit before the Sumatra earthquake [10]. As an example of a non-tsunamigenic earthquake, let us consider the 52.5 m laser strainmeter recording from August 2000 (Figure 3, top). At that time there was a recording of the earthquake, which occurred on 4 August 2000 at 21:13:05 (hereinafter—universal time) at N48.85◦ and E142.42◦, at a depth of 33 km, with a magnitude of 7.1. The recording showed oscillations of about 16 s, which were specific to an earthquake, although there was no deformation jump. The dynamic spectrogram in Figure 3 (bottom) shows that the amplitudes of oscillations in the period range of about 16 s are much greater than the amplitudes of oscillations in the low-frequency range.

**Figure 3.** Fragment of the 52.5 m laser strainmeter recording from August 2000 (**top**) and a dynamic spectrogram of the laser strainmeter recording (**bottom**).

Analysis of the laser strainmeter records of tsunamigenic and non-tsunamigenic earthquakes showed that the deformation jump recording was specific to a tsunamigenic earthquake only.

Further, we will highlight some of the peculiarities of the appearance and development of deformation anomalies at the times of tsunami generation in the three tsunamiendangered regions, namely Indonesia, Chile, and the west coast of North America.

#### *3.1. Earthquakes in Indonesia*

The first powerful earthquake occurred on 11 April 2012 at 08:38:36 on the west coast of northern Sumatra, Indonesia, at 2.327◦ N, 93.063◦ E, at a depth of 20 km and a magnitude of 8.6. The maximum recorded tsunami wave height was 1.08 m. The distance from the earthquake epicenter to the location of the laser strainmeter was more than 5800 km. The laser strainmeter recorded this earthquake signal almost 18 min later at 08:55:39. The average propagation speed of the elastic wave was 5.66 km/s. In the dynamic spectrogram (Figure 4a), we can identify oscillations with periods ranging from 30 to 14 s that are peculiar to an earthquake. There is also a strong disturbance in the lower frequency range.

**Figure 4.** Dynamic spectrograms of 52.5 m laser strainmeter recordings from April 2012 (**a**), April 2014 (**b**), and September 2018 (**c**).

In the dynamic spectrograms of the laser strainmeter recording (Figure 4b), we can identify the signal of the earthquake, which occurred on 12 April 2014 at 20:14:39 at 11.270◦ S, 162.148◦ E, near the Solomon Islands, at a depth of 22.6 km and a magnitude of 7.6. In the coastal zone, the tsunami height reached 0.5 m. The laser strainmeter, located at a distance of over 6700 km, recorded the earthquake signal almost 20 min later at 20:33:58. For this earthquake, the average speed was 5.58 km/s. In the dynamic spectrogram of the laser strainmeter recording shown in Figure 4b, the earthquake signal amplitude is lower than in the previous case, but it also contains oscillations with periods ranging from 30 to 14 s.

The next earthquake under study occurred on 28 September 2018 at 10:02:45, with a magnitude of 7.5 and wave height of about 11 m. The earthquake epicenter was located at 0.256◦ S, 119.846◦ E, at a depth of 20 km and a distance of more than 4800 km from the laser strainmeter. The calculated average speed of the surface elastic wave was 5.49 km/s. On the dynamic spectrogram of the laser strainmeter record (Figure 4c), the earthquake signal was recorded 15 min later at 10:17:19. In the spectrogram, there were oscillations at periods of about 20 s peculiar to earthquakes of such magnitude. From the analysis of the dynamic spectrograms of the three earthquakes that occurred in Indonesia, it follows that along with the oscillations of the earthquake itself, which simply "shake" the Earth, there are disturbances in the lower frequency range.

Figure 5 shows the fragments of the laser strainmeter recordings at the times of registration of the three earthquakes in Indonesia. All figures show a deformation jump peculiar to tsunamigenic earthquakes. For example, in Figure 5a–c, the middle line of the direction of the laser strainmeter recording in the absence of a jump is marked in red, but at the time of the earthquake the recording deviated from its natural behavior (a deformation jump was observed), indicating the tsunamigenic nature of the earthquake.

**Figure 5.** Fragments of 52.5 m laser strainmeter recordings from April 2012 (**a**), April 2014 (**b**), and September 2018 (**c**).

#### *3.2. Earthquakes in Chile*

The recordings from the 52.5 m laser strainmeter showed that three strong earthquakes occurred off the coast of Chile from 2010 to 2018. The first earthquake occurred on 27 February 2010 at 06:34:11 on the northwest coast of Chile at 36.122◦ S, 72.898◦ W, at a depth of 22.9 km; the maximum height of the catastrophic tsunami was 29 m. The distance from the earthquake epicenter to the location of the laser strainmeter was more than 17,800 km. The 52.5 m laser strainmeter recorded the signal of this earthquake at 07:19:00. Let us calculate the average speed of propagation of the elastic wave, which is equal to 6.77 km/s. When analyzing the dynamic spectrogram (Figure 6a) of the recording during this earthquake, we identified not only oscillations typical for an earthquake, but also a disturbance in the lower frequency range.

**Figure 6.** Dynamic spectrograms of 52.5 m laser strainmeter recordings from February 2010 (**a**), April 2014 (**b**), and September 2015 (**c**).

Let us analyze the dynamic spectrograms of the fragments of the 52.5 m laser strainmeter records from April 2014 and September 2015. During this period, there were two strong tsunamigenic earthquakes off the northwest coast of Chile. On 1 April 2014, at 23:46:47, a strong earthquake occurred at 19.610◦ S, 70.769◦ W, at a depth of 25 km, with a wave height of 4.6 m near the coast. The signal for this earthquake was recorded by the laser strainmeter at a distance of more than 16,700 km on 2 April 2014 at 00:24:10. On 16 September 2015 at 22:54:32, a strong earthquake occurred, with its epicenter at 31.573◦ S, 71.674◦ W, at a depth of 22.4 km. As a result of the earthquake, a tsunami with a height

of 13.6 m was generated. The laser strainmeter located at a distance of about 17,650 km recorded the signal for this earthquake at 23:45:01. For these earthquakes, the average speeds of elastic wave propagation were 7.44 km/s and 6.47 km/s, respectively. In the dynamic spectrograms of the laser strainmeter recordings of these earthquakes (Figure 6b,c), oscillations occurred at periods of about 20 s, which are peculiar to earthquakes of this magnitude. Moreover, we noted disturbances in the lower frequency range.

When analyzing the recordings from the laser strainmeter at the times these earthquakes occurred, we identified deformation jumps. Figure 7a–c shows fragments of these earthquakes recordings, whereby the red line indicates the medium direction of the laser strainmeter recording and the deviation from this line at the time the seismic waves were registered indicates the tsunamigenic nature of the earthquakes (the presence of a deformation anomaly—a deformation jump).

**Figure 7.** Fragments of 52.5 m laser strainmeter recordings from February 2010 (**a**), April 2014 (**b**), and September 2015 (**c**).

#### *3.3. Earthquakes on the West Coast of North America*

The first powerful earthquake occurred on 28 October 2012 at 3:04:08 on the southwest coast of Canada at 52.788◦ N, 132.101◦ W, at a depth of 14 km, with a magnitude of 7.8 and a tsunami height of 12.98 m on the shelf. The distance from the earthquake epicenter to the laser strainmeter was almost 6800 km. The laser strainmeter recorded this earthquake signal almost 19 min later at 03:23:13. For this earthquake, the average propagation speed of the elastic wave was 5.94 km/s. In the dynamic spectrogram (Figure 8a), we can see the oscillations ranging from 30 to 14 s, which are peculiar to earthquakes of this magnitude, as well as a strong disturbance in the lower frequency range.

**Figure 8.** Dynamic spectrograms of 52.5 m laser strainmeter recordings from October 2012 (**a**), January 2013 (**b**), and September 2017 (**c**).

Figure 8b shows the dynamic spectrogram of the fragment of the laser strainmeter recording for 5 January 2013, in which we can identify the tsunamigenic earthquake that occurred at 8:58:14 off the coast of Alaska, USA. The earthquake, with a magnitude of 7.5, occurred at 55.228◦ N, 134.859◦ W, at a depth of 8.7 km, resulting in a tsunami with a maximum height of 1.5 m. The signal for this earthquake was detected in the laser strainmeter recordings at 09:16:31. The laser strainmeter was located 6500 km away from the epicenter. In the dynamic spectrogram of the laser strainmeter recording (Figure 8b), oscillations occurred from 30 to 14 s and disturbances occurred in the lower frequency region. Another earthquake occurred off the coast of Mexico on 9 August 2017 at 4:49:19, with a magnitude of 8.2. After this, a tsunami with a height of 2.7 m appeared. The earthquake occurred at 15.022◦ N, 93.899◦ W, at a depth of 47.4 km, at a distance of 12,150 km from the laser strainmeter. Let us calculate the average propagation speed of the elastic waves. For the 2013 earthquake the speed was equal to 5.92 km/s, while for the 2017 earthquake the speed was 5.48 km/s. In the dynamic spectrogram of the laser strainmeter record (Figure 8c), the earthquake signal was detected at 05:34:28. In the spectrogram, along with oscillations from the earthquake at periods ranging from 30 to 14 s, disturbances in the lower frequency range also occurred.

Figure 9a–c shows fragments of the laser strainmeter recordings at the times the three earthquakes occurred off the west coast of North America. All figures show deformation jumps peculiar to tsunamigenic earthquakes. In the figure, the middle line of the laser strainmeter recording direction in the absence of a jump is indicated in red, although at the times the earthquakes occurred, the recordings deviated from their natural behavior, showing the tsunamigenic nature of the earthquakes.

**Figure 9.** Fragments of 52.5 m laser strainmeter recordings from October 2012 (**a**), January 2013 (**b**), and September 2017 (**c**).

Analysis of the dynamic spectrograms of the laser strainmeter recordings of all earthquakes showed that along with the oscillations of the earthquakes themselves, with periods ranging from 30 to 14 s, there were also disturbances in the lower frequency range.

#### **4. Analysis of Certain Characteristics of the Registered Deformation Anomalies**

An earthquake that occurs near a geological fault provokes the displacement of geoblocks relative to each other. It may also break the connections between geoblocks and cause destruction of separate geoblocks, with powerful movement of the released individual geoblocks and their parts. It is this displacement, along with landslides, that is the cause of a tsunami. It is impossible to register this shift directly in the center; it can only

be determined remotely. At large distances, the amplitudes of these slow displacements are very small. No instruments used in tsunami warning services are capable of registering such displacement; therefore, we can estimate this instead with various models, using continuous geodesic survey data, data from GPS receivers with low sampling rates, and tsunami data produced by the DART network. Models of finite faults from the USGS NEIC have an advantage. They use a kinematic approach based on the method proposed by Ji [19]. For calculations, both the body waves P and S and the Rayleigh and Love surface waves are used. To estimate the dissipative characteristics of deformation anomalies recorded by the 52.5 m unequal-arm laser strainmeter, we will use the calculated displacements of geoblocks (plates, parts of geoblocks, etc.) in the earthquake center according to this model. Table 1 lists the calculated displacements in the earthquake center and deformation anomaly values that the laser strainmeter registered at the moments the earthquakes were recorded.

**Table 1.** Calculated displacements of geoblocks and deformation anomaly values recorded by the laser strainmeter.


Since the intensity is proportional to the square of the amplitude, damping of the oscillation amplitude will be expressed by the law of intensity damping, whereby only the damping ratio will be two times smaller. To calculate the damping ratios, we will use the formula used to calculate the oscillation amplitude at the distance under consideration [20]:

$$A = A\_0 \mathfrak{e}^{-\frac{1}{2}\mu \mathbf{x}} \tag{1}$$

where *A* is the amplitude at the registration site, *A*<sup>0</sup> is the initial amplitude, *μ* is the damping ratio, and *x* is the distance.

Let us calculate the damping ratios for each earthquake using the displacement data from Table 1 and the distance from the earthquake epicenter to the laser strainmeter location.

From the obtained results (Table 2), it follows that the damping ratios for all considered tsunamigenic earthquakes are approximately identical. The average damping ratio for all earthquakes was 0.03.

**Table 2.** Earthquake damping ratios.


#### **5. Discussion**

Let us note some of the peculiarities mentioned above: (1) tsunamigenic earthquakes are characterized by the presence of a deformation anomaly—a deformation jump—and disturbances in the dynamic spectrogram in the lower frequency range, in comparison with the range of oscillations, which is peculiar to the earthquake source zone; (2) the damping ratios of deformation anomalies for all regions of the Earth are the same, within the measurement and calculation error ranges; (3) the durations of deformation anomalies for different tsunamigenic earthquakes vary from 15 s to 17 min.

Let us analyze some of the peculiarities of the above-mentioned tsunamigenic earthquakes. Let us focus on the presence in the dynamic spectrograms of disturbances in the low-frequency and ultra-low-frequency ranges that are not associated with oscillations, which are excited in the source zones of earthquakes. These disturbances are associated not with natural processes, but with the processing of records containing deformation jumps. The appearance of disturbances indicates only one thing, namely the presence of a deformation anomaly—a deformation jump—in the record. There are no oscillations in these areas. Let us demonstrate this using the example of processing an instrument recording containing oscillations caused by sea wind waves and a jump. Figure 10a shows an instrument recording containing a jump. Figure 10b shows its spectrum. Figure 10c shows the same instrument recording but without a jump, while Figure 10d shows its spectrum. All scientists involved in signal processing understand this effect. The increase in intensity in the low-frequency range is associated not only with the Gibbs phenomenon, but mainly with the presence of a jump in the recording; thus, the presence of disturbances in the spectrograms of tsunamigenic earthquake recordings containing deformation anomalies indicates only one thing—the influence of this deformation jump on the increases in intensity in the low-frequency and ultra-low-frequency ranges due to the processing effect, which is also remarkable. After all, looking only at the spectrogram, which was obtained in real time, one can notice the tsunamigenic nature of the earthquakes.

**Figure 10.** Fragment of the laser strainmeter recording showing hydrosphere pressure variations [21] when registering surface wind waves (**a**), along with its spectrum (**b**). The same recording is shown with the jump removed (**c**), along with its spectrum (**d**).

Let us pay some attention to the fact that for different regions of the Earth, the damping ratios of deformation anomalies are practically the same. This coefficient should consist of summands associated with divergence and absorption due to dissipative energy losses. If we consider only the cylindrical divergence, then the signal amplitude should decrease with distance in proportion to the square root of the distance. Regarding spherical divergence, the signal amplitude, with distance, should decrease in proportion to the distance. Solving the inverse problem, using Table 1, on the basis of the data from the laser strainmeter, for the spherical divergence we can obtain the following initial amplitudes of deformation

anomalies (deformation jumps), which arose at the source of tsunami generation. In the sequence of column 3 in Table 1 (calculated displacement), these values are 19.6 (10.5 m), 16.7 (8 m), 10.6 (3.2 m), 11.6 (5.4 m), 2.7 (0.8 m), 4.8 (1.8 m), 2.7 (1.5 m), 5.2 (3 m), and 6.0 m (4 m). Taking into account the fact that the calculated data almost coincide with the model data (given in parentheses), we can state that the deformation anomaly arising at the source of tsunami generation, in the case of spherical divergence, moves similarly to the motion of a soliton; however, all calculations are correct when the signal moves along the surface of the Earth, i.e., in an arc rather than a chord. We do not know the path of the signal, so we take the limiting case, whereby the distances from the place of generation to the place of registration are equal to the lengths of the arcs of the circles, as determined by the coordinates of the points. Let us discuss the probable physical mechanism of formation of the deformation anomalies (deformation jumps) arising during the movement of geoblocks (joints) of the Earth's crust at the source of a tsunami generation. Let us assume that when an earthquake occurs, geoblocks are displaced relative to each other or one of the geoblocks (geological plate, joint of the Earth's crust) becomes out of balance and moves relative to the other geoblocks. This movement leads to the movement of huge masses of water, which subsequently degenerate into a tsunami. We are not interested in the origin point of the tsunami. We are only interested in the movement of the geoblock or geoblocks relative to each other. We can describe these geoblocks movements using the following equation:

$$\frac{\partial^2 u}{\partial t^2} + \sin u - \frac{\partial^2 u}{\partial x^2} = 0 \tag{2}$$

where *u* is the displacement of a geoblock. This is the classic sine-Gordon equation. One of the solutions for this equation in the factorized form, which is peculiar to solitons, allows us to obtain the following geoblock displacement:

$$u = 4 \operatorname{arctg} \left[ \exp \left( \pm \frac{\mathbf{x} - Vt}{\sqrt{1 - V^2}} \right) \right] \tag{3}$$

where *V* = 0.5. A solution with "+" gives a kink, while a solution with "−" gives an anti-kink. Figure 11 shows the displacement of the geoblock in the form of an anti-kink.

**Figure 11.** Displacement *u* at *V*= 0.5, shown as a function of time (anti-kink).

The displacement as a function of the coordinate and time at *V*= 0.5, demonstrating plastic deformation, is shown in Figure 12.

**Figure 12.** Geoblock displacement in the case of plastic deformation.

These geoblock displacements, when interacting with the environment, are transmitted to the surrounding space and propagate in the Earth in the form of a soliton—a deformation step, corresponding to a kink or anti-kink.

In addition to the exponential function, hyperbolic functions also satisfy Equation (2). In this case, the solution of Equation (2) gives a two-soliton solution. For this case, the displacement of the geoblock (homogeneity of the Earth's crust, a plate, etc.) depending on the coordinate and time for *V*= 0.5 is shown in Figure 13. This corresponds to elastic deformation under stretching. Plastic deformation can also occur under stretching only in the vicinity of *x* = 0.

**Figure 13.** Geoblock displacement in the case of elastic deformation.

Under special conditions, such as for imaginary solutions, the solution to Equation (2) will give breathers. Figure 14 shows the geoblock displacement in the breather at *V*= 0.5.

**Figure 14.** Geoblock displacement in the breather at *V* = 0.5.

These displacements (several deformation jumps) were observed during the 2011 Great East Earthquake in Japan (Tohoku). It is clear that any movements of a geoblock (or geoblocks) are transmitted to the neighboring environment, in which deformation anomalies propagate in the form of kinks, anti-kinks, and other similar disturbances.

Next, let us consider the behavior of an underwater landslide. At the advanced stage of the process, an underwater landslide can be represented as consisting of two parts: the head part (front) is a soliton (kink), while the tail part is a periodic wave. The landslide front is a same kink (bore) and can be described by a single-kink solution of the sine-Gordon equation, with "+" on the right-hand side of Equation (3). The formed soliton moves without experiencing any resistance from the medium. In expanding areas, the landslide front begins to blur. In such cases, the height of the soliton begins to decrease due to the energy conservation law. This process can be schematically represented in the following sequence during formation of an underwater landslide: (1) in the initial stage, a soliton (kink) and a periodic part are formed; (2) leaving the source of origin, the soliton begins to move in its environment with conservation of energy; (3) the motion of the soliton in the environment obeys the law of motion, ranging from cylindrical to spherical divergences. Taking into account the above, we must register a soliton with an ever-decreasing height and with increases in distance from the place of its registration. Figure 5 shows a laser strainmeter recording containing a deformation disturbance (a deformation jump) kink caused by an underwater landslide during the earthquake in Indonesia. Both earthquakes and underwater landslides can form the periodic oscillations observed in the figure.

#### **6. Conclusions**

During the processing of the experimental data from the laser strainmeter, we found that all tsunamigenic earthquakes are characterized by the presence of deformation anomalies deformation jumps—in the instrument records. These deformation anomalies, leading to the formation of tsunamis in the vicinity of the earthquake source areas, occur during the relative movement of geoblocks (plates, joints) and underwater landslides. These geoblock movements can be described by the sine-Gordon equation, the one-kink and two-kink solutions of which explain the appearance of the observed deformation anomalies in laser strainmeter recordings. The behavior of deformation anomalies is the same as the behavior of solitons in non-linear medium. Considering that the signals from tsunamigenic earthquakes containing deformation anomalies propagate at speeds much higher than those of surface waves (from 5.48 to 7.44 km/s), we can assume that the signals do not propagate along the Earth's surface, with the divergences ranging from cylindrical to spherical. Further research should investigate the spatial behavior of the deformation anomalies from

tsunamigenic earthquakes. For this, it will be necessary to place several laser strainmeters far from each other, along the assumed direction of movement of deformation anomalies (deformation jumps), kinks, anti-kinks, and breathers. These experimental studies will allow us to study the main parameters of the observed disturbances. Of particular interest is the conservation of the soliton shape with decreasing value due to divergence in space during movement. The development of this area of research, along with the application of the classical "magnitude geographical principle" used for determining the tsunami hazard of underwater earthquakes will bring us closer to short-term tsunami forecasting. Taking into account the above, with spherical divergence in accordance with Table 1, we can calculate the applicability of GPS receivers capable of registering displacement with an accuracy level of 1 mm for recording displacements (column 3 of Table 1). In this way, with an average displacement of 4.2 m (according to Table 1), a GPS receiver will be able to register a displacement of 2 mm at a distance of 2100 m under conditions of spherical divergence. It is clear that there is no point in discussing the prospects of using GPS receivers for registering displacements of geoblocks (plates, joints) leading to the occurrence of tsunamis. The cylindrical divergence of the signal does not help in situations.

**Author Contributions:** G.D.—problem statement, discussion, and writing of the article. S.D.—data processing, discussion, and writing of the article. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work was carried out with financial support from the Russian Federation represented by the Ministry of Science and Higher Education of the Russian Federation, Agreement No. 075-15-2020-776.

**Data Availability Statement:** 3rd Party Data. Restrictions apply to the availability of these data.

**Acknowledgments:** We would like to express our deep gratitude to all employees of the Physics of Geospheres laboratory.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

