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56

ISSN 0742-0463, Journal of Volcanology and Seismology, 2019, Vol. 13, No. 1, pp. 56–69. © Pleiades Publishing, Ltd., 2019.

Russian Text © A.A. Baranov, S.V. Baranov, P.N. Shebalin, 2019, published in Vulkanologiya i Seismologiya, 2019, No. 1.

A Quantitative Estimate of the Effects of Sea Tides

on Aftershock Activity: Kamchatka

A. A. Baranova, b, *, S. V. Baranovc, and P. N. Shebalina, **

aInstitute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences,

ul. Profsoyuznaya, 84/32, Moscow, 117997 Russia

bInstitute of Physics of the Earth, Russian Academy of Sciences, B. Gruzinskaya, 10, str. 1, Moscow, 123242 Russia

cKola Branch, RAS Unified Geophysical Survey Federal Research Center, ul. Fersmana, 14, Apatity,

Murmansk Region, 184209 Russia

*e-mail: baranov@ifz.ru

**e-mail: p.n.shebalin@gmail.com

Received September 19, 2017; revised November 14, 2017; accepted November 21, 2017

Abstract⎯The issue of whether tidal forces really affect seismicity has been raised many times in the litera-

ture. Nevertheless, even though there seems to be a kind of consensus that such effects do exist, no quantita-

tive estimates are available to relate tide parameters to changes in the level of seismic activity. Such estimation

for aftershocks of large earthquakes near Kamchatka is the goal of the present study. We consider the influ-

ence on seismicity due to ocean tides only, because their effects are stronger than those of solid earth tides.

Accordingly, we only consider earthquakes that occurred in the ocean. One important feature that distin-

guishes the present study from most other such research consists in the fact that we study the height of ocean

tides and its derivative rather than tidal phases as the decisive factors. We considered 16 aftershock sequences

of earthquakes near Kamchatka with magnitudes of 6 or greater. We also examined shallow background

earthquakes along the coast of Kamchatka. Our basic model of aftershock rate was the Omori–Utsu law. The

background seismicity distribution was assumed to be uniform over time. In both of these cases we used the

actual distributions in space. The heights of sea tides were estimated using the FES 2004 model (Lyard et al.,

2006). The variation in activity from what the basic model assumes in relation to tidal wave height and its time

derivative was estimated by the method of differential probability gain. The main practical result of this study

consists in estimates of averaged differential probability gain functions for aftershock rate with respect to both

of the considered factors. These estimates can be used for earthquake hazard assessment from aftershocks

with ocean tides incorporated. The results of our analysis show a persistent tendency of aftershock rate

increasing during periods when the ocean tide decreased at a high rate. For the background events, we found

a typical tendency of events rate increasing when the ocean tide decreased with high tidal amplitudes. The

difference in the main factors that affect aftershocks and background seismicity suggest the inference that the

effects of tides on aftershocks are more likely to be direct dynamic initiation of events during high strain rates,

while the effects on the background events were static in character.

DOI: 10.1134/S0742046319 010020

INTRODUCTION

The gravitational interaction among Earth, Moon,

and the Sun induces ocean and solid-earth tides. The

tides in turn produce periodic changes in the state of

stress in rocks, i.e., tidal stresses add a small sign-vary-

ing term to the background stress field, thus affecting

the geodynamic process and seismicity. Depending on

the phase, tidal forces can aid or impede the geody-

namic process, thus producing either an initiating or a

suppressing effect. Tides can also produce periodic

changes in water saturation for rocks in fault zones,

thus increasing or diminishing effective friction due to

changed pore pressure.

A large amount of evidence is available on the rela-

tionship of ocean tides and solid-earth tides to seis-

micity. As well, there are many critical publications

that call into question the significance or universality

of the relationship. The difficulty in obtaining reliable

inference consists in the fact that the investigated rela-

tionships are mostly weak and their detection requires

the correct use of statistical methods on the one hand,

while on the other hand, large numbers of seismic

events are to be analyzed (Heaton, 1975, 1982; Burton,

1986; Rydelek et al., 1992; Vidale et al., 1998). The

typical stress variations due to solid-earth tides reach

1–4 kPa (Melchior, 1983; Vidale et al., 1998). These

values are insufficient to initiate earthquakes (Harde-

beck et al., 1998; Kilb et al., 2002; Stein, 1999); thus,

JOURNAL OF VOLCANOLOGY AND SEISMOLOGY Vol. 13 No. 1 2019

A QUANTITATIVE ESTIMATE 57

there is no direct trigger mechanism that can account

for the observable correlation between seismicity and

solid-earth tides (Beeler and Lockner, 2003; Lockner

and Beeler, 1999; Rydelek et al., 1992; Tsuruoka et al.,

1995; Vidale et al., 1998). In this connection it is fre-

quently assumed that the tidal effects on seismicity are

possible, when the respective fault system is in a sub-

critical state, so that even a minute change in stress is

sufficient to initiate seismic events (Saltykov, 1995;

Saltykov and Ivanov, 2003; Saltykov et al., 2004;

Tanaka, 2010, 2012; Rydelek, 1992; Zhang and

Zhuang, 2011; Crockett et al., 2006). In a similar man-

ner, one can appeal to the same arguments to account

for the frequent observation that tidal effects on seis-

micity are local in character (Nikolaev, 1994; Saltykov

and Kugaenko, 2007; Yurkov and Gitis, 2005; Klein,

1976b; Emter, 1997; Souriau et al., 1982).

Solid-earth tides exert a complex effect on the

Earth’s crust that can be described by a tensor. The

maximum excitation is to be expected when the tidal

stresses act in a direction to enhance the regional tec-

tonic stresses (Tanaka et al., 2004; Stein, 2004). In

a similar manner, the degree of effect exerted by tides

is different for earthquakes with different types of slip.

Many authors have noted that the correlation between

tides and strike-slip earthquakes is the lowest com-

pared with that for thrust earthquakes, but especially

with normal-slip earthquakes (Tsuruoka et al., 1995;

Tanaka et al., 2002; Cochran, 2004; Stein, 2004).

At the same time, Métivier et al. (2009) detected a

strong effect of solid-earth tides on strike-slip earth-

quakes, although they note a low statistical signifi-

cance of that inference.

Ocean tides, as distinguished from earth tides, rep-

resent a cyclically varying load on the seafloor, which in

turn affects crustal stresses. The periodic variations in the

pressure due to sea tides is on the order of 10 to 100 kPa,

depending on the tide height at a site of interest (1–

10 m). The stress variation due to earth tides does not

exceed 2–3 kPa; thus, the effect of ocean tides at loca-

tions where they are high is considerably above that

due to solid-earth tides, as remarked upon by many

workers (Cochran, 2004; Tsuruoka et al., 1995;

Tanaka et al., 2002, 2004; Ide et al., 2014, 2016; Varga

and Grafarend, 2017). The variation in stress due to

movement of oceanic masses can be seen, not only in

variations of seismicity in the sea, but also in continen-

tal seismicity (Tsuruoka et al., 1995; Ide et al., 2014;

Stroup et al., 2007; Souriau et al., 1982; Cochran

et al., 2004; Tanaka et al., 2002; Tanaka, 2012; Wil-

cock, 2001; Crockett et al., 2006; Varga and Grafar-

end, 2017).

Ocean tides and earth tides produce substantially

different effects on the stress f ield. The vertical com-

ponent of tidal stresses for earth tides at depths shal-

lower than 200 km is negligibly small, with the domi-

nant component being horizontal normal stresses

(Melchior, 1983; Varga and Grafarend, 2017). In con-

trast to this, local stress variations due to ocean tides is

dominated by the vertical component, while the hori-

zontal components are no more than 30% of the verti-

cal (Varga and Grafarend, 2017).

There are several mechanisms to provide for a rela-

tionship between seismicity and ocean tides in use

today. The most obvious ones include the variation in

tangential stresses τ along faults, in normal stresses σn,

and in pore pressure P, and a combination of these.

Increased tangential stresses as an addition to the

available tectonic stresses can under definite condi-

tions lead both to the direct trigger effect (“dynamic

triggering”) and to an increase in the likelihood of

seismic events (“static triggering”) (Klein, 1976a;

Tanaka et al., 2002, 2004; Stein, 2004; Varga and Gra-

farend, 2017). In accordance with the Coulomb crite-

rion τc = τ + μ(σn – P), where μ is the frictional con-

stant, the friction threshold can be exceeded either

when the stresses that are normal to the fault surface

are lowered or when pore pressure is increased (Klein,

1976a, 1976b; Cochran et al., 2004; Stroup et al.,

2007; Wilcock, 2001; Métivier et al., 2009). When the

static triggering effect occurs, the action of tides can

lag in time (Lockner and Beeler, 1999; Beeler and

Lockner, 2003) owing to the fatigue strength effect

(Scholz, 1968; Atkinson, 1984; Lockner, 1998; Nar-

teau et al., 2002). This considerably complicates the

analysis of the relationships between seismicity and

tides. One important factor for the direct triggering

effect may also consist in the high strain rate (Vidale

et al., 1998). The tidal stress due to earth tides does not

exceed 4 kPa, which is a few orders of magnitude below

the stress released by a large earthquake. However, the

rate of tectonic stress buildup in the rupture zone of a

future large event is approximately 0.01 kPa/h, while the

rate of tidal stress variation is approximately 2 kPa/h,

which is two orders of magnitude greater.

Different workers have proposed other mecha-

nisms as well. Heaton (1975, 1982) sought to explain

the effect, as seen by this researcher, of tides on larger

earthquakes only by invoking a nonlinear dilatant dif-

fusion model. The periodicity of tides is the basis for a

resonant mechanism to explain the effect on seismic-

ity by A.V. Nikolaev (1996) as a viable solution.

Recently Saltykov (2014) proposed an alternative

mechanism, different from triggering, that incorpo-

rates tidal variations in physical properties of the earth

using a model of amplitude-dependent dissipation.

The effect of tides on seismicity rate according to this

model can be observed in biased magnitude estimates

and the associated change in the rate of seismic events

in a fixed magnitude range following the Gutenberg–

Richter magnitude–frequency relationship. In recent

years there is an increasing understanding of f luid

migration in earthquake generation (Shebalin and

Narteau, 2017). The part played by pore pressure

changes due to tides was noted by Klein (1976а,

1976b).

58

JOURNAL OF VOLCANOLOGY AND SEISMOLOGY Vol. 13 No. 1 2019

BARANOV et al.

The statistical tests that were conducted in order to

verify the relationship between tides and seismicity

were usually based on mainshock catalogs with after-

shocks eliminated using one of the known algorithms.

Otherwise, when the start of a large aftershock

sequence occurred in a tidal phase, the effect would be

overwhelming. An incomplete elimination of after-

shocks can also distort the analysis of how tides affect

the rate of earthquakes. One alternative could consist

in a study of tidal forces on aftershock activity, but

there are very few studies dealing with this problem.

Souriau et al. (1982) fitted an aftershock sequence

using the Weibull distribution and examined the cor-

relation between tidal phases and the departures from

the average model. Nikolaev and Nikolaev (1993) ana-

lyzed the relationship between tides and the after-

shocks of the Spitak and Racha earthquakes (Cauca-

sus). Chen et al. (2012) carried out a joint study of tides

and the aftershock sequence of the 2011 Christchurch,

New Zealand earthquake.

The overwhelming number of publications dealing

with the relationships between tides and earthquakes

since Schuster (1897) were concerned with tidal

phases, mostly the semi-diurnal phase. A recent pub-

lication (Ide et al., 2016) suggests that it was this factor

which was responsible for the lack of or low correlation

between seismicity and tides found in nearly all publi-

cations. The amplitude–frequency properties of tides

are very complex and one needs to incorporate the fact

that the semi-diurnal component has a varying ampli-

tude.

The main practical goal of this study is to find

quantitative estimates of how ocean tides affect the

activity of aftershocks following large earthquakes that

occur near the coast of Kamchatka. The problem was

to estimate the number of times the probability of

aftershock occurrence is increased or decreased for

definite parameters of sea tides. Another goal was to

achieve some progress in the understanding of the

mechanisms that are responsible for the effects of tides

on seismicity. Can the direct triggering effect occur or

do the more likely mechanism consists merely in a

changed probability, hence, a changed rate of seismic

events? Can tidal effects occur only in zones or during

periods in which the fault system is in a special condi-

tion, or can the background seismicity be affected as

well by tides? Is the part played by liquids essential for

a fault system to be affected by tides? In our investiga-

tion of these and other problems we relied on results of

previous research, while focusing on the results and

inferences that we considered to be the most import-

ant for dealing with the main problem. Based on the

inferences reached by Ide et al. (2016), and unlike the

overwhelming majority of previous studies, we com-

pare seismicity, not with tidal phases, but with abso-

lute values using the FES 2004 model (Lyard et al.,

2006). One example of our calculation is shown in Fig.

1. Following the idea from Vidale et al. (1998), we also

consider the rate of tidal strain as a possible factor for

the direct triggering effect. We compare time-depen-

dent tidal parameters and aftershock rate by fitting the

Omori–Utsu law to the averaged (without tides taken

into account) rate of aftershocks. A similar approach

was first proposed by Souriau et al. (1982), but later

practically not used. We note that this approach pro-

vides much greater amounts of seismological informa-

tion compared with the analysis of background seis-

micity for similar time spans, while using the same

catalog of earthquakes, not cleared from aftershocks,

is incorrect. The present study also compares the

results from a joint analysis of aftershock sequences

and ocean tides with similar results for background

seismicity, with the background seismicity being sepa-

rated by the method that was proposed by Zaliapin

and Ben-Zion (2013), which is now widely popular.

The quantitative estimation of tidal effects on the rate

of seismic events is here made using the method of dif-

ferential probability gain (DPG) method due to (She-

balin et al., 2012, 2014). The tide height or its first

derivative is used as the control parameter of the

method. The differential probability gain is found as

the ratio of the actual seismicity rate at a specified

value of the control parameter to the rate as estimated

by the model. We note that a similar approach was

proposed by Vidale et al. (1998) who, however, did not

detect any relationship between tides and the varying

seismicity rate.

It was not necessary for the purposes of the present

study to model changes in the stress field due to tides,

since we only considered ocean tides whose effect in

areas near Kamchatka is at least 5 times stronger than

the effect of earth tides (see our estimation in what fol-

lows). A joint use of ocean tides and solid earth tides

would necessarily require the construction or use of

ready-made complex models. This would make the

results largely dependent on the model, which would

reduce their value.

Fig. 1. A sample calculation of ocean tide heights using the

FES 2004 program for the area of the magnitude 6.6

March 2, 1992 earthquake off Kamchatka. Vertical bars

mark the times of M ≥ 4.5 aftershocks.

‒2.0

2.0

1.5

1.0

0.5

0

‒0.5

‒1.0

‒1.5

0 400300200100 500 600 700

Tide height, m

Time since main shock, hours

JOURNAL OF VOLCANOLOGY AND SEISMOLOGY Vol. 13 No. 1 2019

A QUANTITATIVE ESTIMATE 59

THE OBJECTS OF STUDY

AND THE RAW DATA

Most of the great earthquakes that occurred in the

Kamchatka region were caused by the subduction of

the Pacific plate (Ermakov, 1993; Fedotov et al.,

1985). The earthquakes in this subduction zone occur

at depths as large as a few hundred kilometers, with

many being shallow. The epicenters of the shallow

earthquakes are generally in the Pacific Ocean, not

very far from the shoreline. It is these shallow events

that are the subject of our study, since they are thought

to be more seriously affected by ocean tides.

The database of this study is the earthquake catalog

produced by the Kamchatka Branch (KB) of the Uni-

fied Geophysical Survey (UGS) of the Russian Acad-

emy of Sciences (RAS) (Catalog; Chebrov et al., 2016)

for the period between 1962 and 2016. The catalog can

be found at http://www.emsd.ru/sdis/earth-

quake/catalogue/catalogue.php

We use two types of seismic data, aftershock

sequences and “background” earthquakes, that is,

earthquakes that are not aftershocks of other earth-

quakes.

We identified 16 aftershock sequences for the

period of interest, with most epicenters in the ocean

(Table 1, Fig. 2). We note that the aftershock processes

in the area of study were discussed in detail by

S.V. Baranov and V.N. Chebrov (2012), I.A. Lutikov

and S.N. Rodina (2013), S.V. Baranov and P.N. She-

balin (2016), Shebalin and Baranov (2017a).

Table 1 lists the main parameters of the aftershock

sequences under consideration here.

METHODS USED FOR IDENTIFICATION

AND MODELING

OF AFTERSHOCK SEQUENCES

AND BACKGROUND SEISMICITY

Aftershocks were identified using the algorithm of

G.M. Molchan and O.E. Dmitrieva (Molchan and

Dmitrieva, 1991, 1992). This method is currently in

wide use in Russia thanks to the computer implemen-

tation due to V.B. Smirnov (2009). Many researchers

have noted that the method provides a more accurate

identification of aftershocks compared with the other

techniques. However, when the method is applied to

background seismicity, the resulting catalog still

retains some clustered events. We made our own cata-

log of background events using the recent “nearest

neighbor” method (Zaliapin and Ben-Zion, 2013),

which is based on a proximity function in the time–

space–magnitude coordinates (Baiesi and Paczuski,

2004). The proximity function involves the b-value

and the fractal dimension of seismicity df over space.

The method has become widely popular owing to its

relative simplicity and the high degree of visibility in

separating clustered and background events.

Table 1. The main parameters of the aftershock sequences considered here

H denotes the mainshock depth of focus; M is the mainshock magnitude (magnitude ML as converted from energy class is shown (Cata log));

Ndenotes the number of M ≥ Mc aftershocks in the interval (tstart, 720 hours) measured from the mainshock time; the maximum ampli-

tude of ocean tide in the interval (tstart, 720 hours) measured from mainshock time was converted to pressure, kPa.

no. Mainshock

date H, km MMctstart, hours N

Maximum

amplitude of

ocean tide, kPa

1 15 D ec 1971 20 7.0 3.8 12 287 19

2 28 Feb 1973 59 6.8 4.0 1 53 13

3 28 Dec 1984 19 6.3 3.8 5 55 16

4 10 Jul 1987 49 6.3 3.8 0.1 34 19

5 02 Mar 1992 20 6.6 4.0 0.1 55 14

6 08 Jun 1993 40 6.8 3.5 0.1 173 15

7 13 Nov 1993 40 6.6 3.5 0.1 48 16

8 05 Dec 1997 10 7.0 3.6 25 605 16

9 05 Dec 1997 24 6.4 3.5 22 218 16

10 08 Mar 1999 7 6.5 3.5 2 47 10

11 08 Oct 2001 24 6.3 3.8 1 106 13

12 15 Mar 2003 5 6.3 4.0 0.1 76 14

13 05 Dec 2003 29 6.7 4.0 0.1 59 17

14 30 Jul 2010 38 6.3 3.5 0.1 44 13

15 28 Feb 2013 61 6.9 3.5 20 58 13

16 24 Mar 2013 48 6.5 3.8 1 123 12

60

JOURNAL OF VOLCANOLOGY AND SEISMOLOGY Vol. 13 No. 1 2019

BARANOV et al.

Although, in our opinion, the method is less suitable

for aftershock identification compared with the Mol-

chan–Dmitrieva algorithm, the identification of

background events leaves practically no clustered

events. In this study we used the values b = 1.0 and

df=1.6 for the Kamchatka region (depths of focus

have been disregarded for the purposes of the present

study). These estimates were obtained for the catalog

from 1985 onward. The b-value was estimated using

the lowest magnitude of complete reporting Mc = 3.5.

We note that the estimate b = 1.0 is in agreement with

the regional estimate γ = 0.5 (Saltykov and

Kravchenko, 2009). Background events were defined

to be those whose “nearest neighbors” are situated

farther (in the sense of the proximity function) beyond

the threshold value that separates clustered and

unclustered events. Our estimate of the threshold

value η0 = 0.00001 turned out to be equal to that in

(Zaliapin and Ben-Zion, 2013). The resulting catalog

of background earthquakes covers the period between

1985 and 2016 and includes 1637 events with magni-

tudes M ≥ Mc = 3.5.

The time-dependent aftershocks λ(t) in each

sequence were fitted by the Omori–Utsu law (Utsu,

1961) for M ≥ M

c events in the time interval (tstart,

720 hours):

(1)

() .

()

p

K

t

tc

λ=+

The background seismicity was assumed to be con-

stant over time. The lowest level of complete reporting

Mc for background events was estimated using the

Maximum Curvature (MAXC) method (Wiemer and

Wyss, 2000; Woessner and Wiemer, 2005). However,

since we are dealing with aftershocks, we need to note

that the catalog is less complete at the start of an after-

shock sequence (Helmstetter et al., 2006; Hainzle,

2016). This was the reason we introduced the parame-

ter tstart (the initial time of data analysis as measured

from the mainshock time). The lowest magnitude Mc

decreases when that time increases, and conversely,

the value of tstart is to be increased as Mc decreases. In

this way the completeness of an aftershock catalog is

defined by the pair Mc and tstart. For the parameter tstart

we fixed the value of 0.1 hours as the minimum in all

cases, since the catalog is not complete for main

shocks of magnitude 6.0 or greater with any Mc

(Holschneider et al., 2012). Having estimated Mc by

the MAXC method from the data in (tstart, 720 hours),

we proceeded to refine tstart(Mc) by availing ourselves

of the idea of the MBS method (Method of b-value

stability) (Cao and Gao, 2002) in which the lowest

magnitude of complete reporting is based on the sta-

bility of the b-value in the Gutenberg–Richter law.

Since, according to Aki’s formula (Aki, 1965), the

b-value is in one-to-one correspondence with the

average magnitude, we found the lowest value of tstart at

which the average magnitude begins to be stable. We

Fig. 2. A map of M ≥ 6 main shocks off Kamchatka, 1971–2013, whose aftershocks were analyzed. Shades of grey show the max-

imum sea tide amplitudes at sites. The mainshock fault-plane solutions were taken from the Global Centroid-Moment-Tensor

catalog (Ekström et al., 2012); when the catalog did not contain data on an earthquake its epicenter is marked by an asterisk.

50°

56°

57°

55°

54°

53°

51°

52°

156°160°159°158°157 °161°162°163°166°165°164°

28 Dec 1984 6.3

7

7

6.3

6.7

05 Dec 2003

6.5

6.5

10 Jul 1987

28 Feb 2013

28 Feb 1973

6.9

6.8

6.8

24 Mar 2013

15 D ec 1971

05 Dec 1997

05 Dec 1997

15 Mar 2003

6.3

6.3

6.3

30 Jul 2010

6.6

6.6

02 Mar 1992

08 Oct 2001

08.06.1993

08 Mar 1999

13 Nov 1993

6.4

100

80

60

40

20

*

*

cm

JOURNAL OF VOLCANOLOGY AND SEISMOLOGY Vol. 13 No. 1 2019

A QUANTITATIVE ESTIMATE 61

used that approach in an earlier study (Shebalin and

Baranov, 2017b). Our criterion for the short-term

average value to be stable as obtained by averaging the

magnitudes of ten successive aftershocks was to reach

the long-term average based on the aftershock

sequence for the entire period of interest 0.1–720 h

(Fig. 3). When the data were insufficient for such an

estimation, we found the value of tstart(Mc) using the

relation Mc = Mm – 4.5–0.76 log10 (tstart), where Mm is

the mainshock magnitude (Helmstetter et al., 2006),

with tstart in days. In addition, when the resulting value

was below 0.1 h, we adopted the value tstart = 0.1 h.

The space–time model for the aftershock process

that was required for comparison with actually

observed distributions in our study of tidal effects was

constructed as follows. The aftershock area was subdi-

vided into rectangular 0.4° × 0.4° bins and the interval

(tstart, 720 hours) was subdivided into subintervals of

0.2 hours. Assuming a spatial homogeneity for the

parameters in the Omori–Utsu relation, we estimated

the overall parameters c and p by Bayesian methods

(Holschneider et al., 2012) for the entire aftershock

sequence. With this assumption, we found the

expected number of events in space–time bins as

(2)

where i is the index of the space bins, and j is the num-

ber of the appropriate time interval, while

start

start

0.2 0.2

0.2

1

,

()

tj

ij i p

tj

Kdt

tc

++

+

λ= +

∫

(3)

with being the actual number of aftershocks in

block i for the interval (tstart, 720 hours).

NUMERICAL MODELING OF OCEAN TIDES

Ocean tides are very important for coastal areas.

Sea level fluctuations and offshore marine currents are

complex processes. The modeling of pelagic and off-

shore tides provides a better understanding of the phe-

nomenon. Tides can be as high as 12–18 m in some

bays. The tidal amplitudes can reach 2 m near the

Pacific coast of Kamchatka, producing a pressure

contrast of approximately 20 kPa. At the same time,

tides can reach 10 m or greater heights in some loca-

tions along the western Kamchatka coast (the Pen-

zhina estuary). We need to know the height of tides at

a specified site at a specified time instant. There are

several programs that can provide tide heights to high

precision for any point over the globe at any given time

instant. One such program (FES) began to be devel-

oped during the 1990s at the Centre National d'Études

Spatiales (CNES) using Christine Le Provost’s work

in the 199 0s (Le Provost et al., 199 4, 1996).

The FES 2004 program (Lyard et al., 2006) was

developed at the CNES Department of Oceanography

(http://www.aviso.altimetry.fr/). The program is

based on the solution of tidal barotropic equations by

finite elements (triangles) on a global element grid

(~1 million elements). It uses numerical models of sea

bottom topography and shoreline. The program can

compute 15 main tidal components on a 1/8° grid

(amplitudes and phases), as well as 28 additional tidal

components. The presence of ice is incorporated for

polar regions. The accuracy is within a few centimeters

for open ocean and is within 10 cm for offshore areas.

The grid is not uniform, being denser near the shore

and less detailed in the open ocean, according to Le

Provost’s criterion (Le Provost and Vincent, 1986).

The program requires an input file that contains site

coordinates and time in hours as measured from Jan-

uary 1, 1985. The program uses the sites as specified in

the input file to yield tide heights at a required time

instant.

We used the FES 2004 program to find ocean tide

heights hij for each space–time block at a time step of

0.2 hours. The derivative of tide height was found from

the difference of values in each block at successive

time steps. Figure 4 shows plots of tide height as a

function of time for several spatial blocks using the

magnitude 7.0 December 5, 1997 Kronotsky earth-

quake as an example. Table 1 lists the maximum

amplitudes of ocean tides at the epicenters of main

shocks for the period of 30 days considered here; the

amplitudes were computed by the FES 2004 program

()

start

720

,

i

i

p

t

N

K

tc dt

−

=

+

∫

i

N

Fig. 3. The estimate of tstart for a specified value of Mc: the

aftershock sequence of the magnitude 7.0 December 5,

1997 earthquake. The parameter Mc = 3.6 is based on the

0.1–720-h data after the main shock; our estimate of tstart

was taken to be the time from the main shock to the point

of the first intersection between the average magnitude

curve based on ten consecutive events (solid curve) and the

straight line corresponding to the average magnitude based

on the entire 0.1–720-h sequence (4.2 for this sequence at

Mc = 3.6).

5

4

0 40302010 50 60 70 80 90 100

Magnitude

Time since main shock, hours

62

JOURNAL OF VOLCANOLOGY AND SEISMOLOGY Vol. 13 No. 1 2019

BARANOV et al.

and converted to pressure values. We note that these

amplitudes are 5–10 times greater than the maximum

stress changes due to solid earth tides.

ESTIMATING THE DEGREE

OF TIDE EFFECTS ON SEISMICITY

USING THE METHOD OF DIFFERENTIAL

PROBABILITY GAIN

Ocean tide height or its time derivative are here

considered as parameters that control the tide-

induced relative change in the frequency of seismic

events. The degree of the effect is estimated in a

space–time region where the parameter has a value A

by the ratio g = ω(

Α

)/λ(A) between the actual total

number of seismic events ω(A) in the region and the

number to be expected in the region according to the

total model λ(A); this is called the differential proba-

bility gain (Shebalin et al., 2012, 2014). The total num-

ber and the total model were found by summing over

those bins with space index i and the number of time

interval j in which the parameter has the specified

value A. The differential probability gain shows by how

much the rate of events is increased or decreased at a

fixed value of the control parameter. When no exci-

tation occurs, we have g ≅ 1. The control parameter

can also be treated as the “alarm function” of a predic-

tion algorithm (Zechar and Jordan, 2008). “Alarm”

means here that those space–time regions in which

the control parameter is above (or below) a certain

threshold value have the probability of earthquake

occurrence accordingly increased (or decreased). The

error diagram (Molchan, 1991) can be used to estimate

the efficiency of the control parameter relative to the

average model in retrospect. The differential probabil-

ity gain is found from the error diagram as local slope

(Shebalin et al., 2012, 2014). Both of the variants of

the control parameter, h, ocean tide height and h', the

first time derivative of tide height, are continuous vari-

ables. For this reason we estimate g(h) and g1(h') for

intervals of h and h', ensuring some smoothing in this

way:

Fig. 4. A map of epicenters and tide heights for the aftershocks of the magnitude 7.0 Kronotsky earthquake of December 5, 1997.

Crosses mark aftershock epicenters. The aftershocks that occurred on land were used to estimate the parameters of the Omori–

Utsu law, but were not used to calculate the effects of ocean tides on seismicity. The boundaries of the spatial bins considered here

are shown as white straight lines. The two insets demonstrate some difference among ocean tide heights for different bins.

–1.0

–0.5

0

0.5

1.0

048 96 144192240288336384432480528576624672720

53.0

54.0

55.0

56.0

54.5

55.5

53.5

160.0 161.0 162.0160.5 161.5 162.5

Tide height, m

Time since main shock, h

–1.0

–0.5

0

0.5

1.0

048 96 144192240288336384432480528576624672720

Tide height, m

Time since main shock, h

05 Dec 1997_M = 7.0

05 Dec 1997_M = 7.005 Dec 1997_M = 7.0

JOURNAL OF VOLCANOLOGY AND SEISMOLOGY Vol. 13 No. 1 2019

A QUANTITATIVE ESTIMATE 63

(4)

The functions g(h) and g1(h′) can then be used

(Shebalin et al., 2014) for fitting the time-dependent

aftershock rate with tides incorporated. When h and h′

are weakly correlated, the resulting model can be

derived from the relationship (Shebalin, 2017)

(5)

where hij and are the tide height and its first deriv-

ative at the ith space bin at time j as determined

according to the FES 2004 model.

RESULTS AND DISCUSSION

Aftershock sequences have been identif ied for

16 large (M ≥ 6) events with epicenters near Kam-

chatka (see Fig. 2, Table 1) using the methods outlined

above. For these sequences we estimated Mc and tstart;

using these parameters we estimated с and p in (1) for

the entire sequence during the first month after the

main shock. We note that the aftershock sequences of

the two December 5, 1997 earthquakes (magnitudes 7

and 6.4) that were separated by more than 7 hours and

whose epicenters were at a distance of over 300 km

were treated separately. The parameter K in (1) was

estimated for each spatial bin using (3). Relation (2)

was used to find the number of M ≥ Mc aftershocks as

expected in compliance with the Omori–Utsu model

in successive time intervals at a step of 0.2 hours and

the actually observed number of such aftershocks was

found from the catalog. The next stage was to con-

struct time series of tide height and its first derivative

at a step of 0.2 hours, as well for each spatial bin using

the FES 2004 algorithm. The derivative was found as

the difference of two consecutive values of tide height

divided by the length of the interval. Lastly, (4) gave

differential probability gain as a function of tide height

and its first derivative. The smoothing was carried out

in moving windows of 0.3 m for tide height and of 2 ×

10–5 m/s for the derivative.

The number of aftershocks that could be used for

analysis is not large for any individual sequence (see

Table 1). It is only for the 1971 sequence and for both

of the 1997 sequences that the number exceeds 200.

Figure 5 shows plots of g and g' for these three

sequences. As to the remaining 13 sequences, as well as

for all the 16 sequences and 14 sequences with the 1997

sequences eliminated (the number of aftershocks is the

highest for these sequences), we found the total (aver-

aged) functions g and g' using (4) as well, but summing

over the space–time bins of all sequences (Fig. 6).

,;

,;

'

,; ' ' '

1

'

,; ' ' ' '

() ;

(') .

ij

ij

ij

ij

ijhdhh h dh ij

ijhdhh h dh ij

ij

ijh dh h h dh

ij

ijh dh h h dh

gh

gh

−<≤+

−<≤+

−<≤+

−<≤+

Σω

=Σλ

Σω

=

Σλ

1'

()() ,

tide

ij ij ij ij

gh g h

λ= λ

'

ij

h

It can be seen from Figs. 5 and 6 that the most gen-

eral phenomenon consists in an increasing probability

of aftershock occurrence at a high rate of tide decay

(negative derivative); the higher the rate, the greater is

the value of g', which reaches approximately 2 at the

highest negative rate. We note that this phenomenon

occurs both for all the sequences considered here (see

Fig. 5) and for the totality of the 13 remaining

sequences (see Fig. 6d). A rapid increase in g for the

highest tide heights was also observed, as well as a

slight increase in g during deep ebbs and an increasing

g' at the highest rate of tide rise. However, these phe-

nomena are less obvious, since they are not observed

for all sequences, or because the numbers of events in

these sequences are too small to derive the estimates.

All the same, the functions g and g' for the 13 sequences

and the sequence following the second of the 1997

earthquakes are similar; the rapid increase in g during

high water for these sequences was based on a large

number of events. Nevertheless, the phenomenon did

not occur for the aftershock sequences of the two great

earthquakes of 1971 and 1997.

We sought to refine the possible mechanisms that

are responsible for the effects of sea tides on seismicity

by a similar joint analysis of background seismicity

and tides using (4). We selected ten nonintersecting

spatial 1° × 1° bins along the Kamchatka coast that

include the greatest numbers of earthquakes with

depths of focus less than 50 km (the centers have the

following coordinates, N, W: 49.5, 156; 50.5, 157;

51.5, 158; 52, 159; 52.8, 160; 53.8, 161; 54.8, 162; 56,

163; 56, 164; and 55.5, 166). The total selection in

these bins from the background seismicity catalog with

events shallower than 50 km was used to estimate the

lowest magnitude of complete reporting: Mc = 3.8.

The depth limit of 50 km was fixed in order to have an

approximate agreement between the depth ranges for

the background earthquakes considered here and the

aftershock sequences. The resulting catalog with epi-

centers in ten bins, M ≥ 3.8, and depths of focus less

than 50 km contains 1176 events for 1985–2016. Time

series h(t) and h′(t) were constructed at a step of

0.2 hours using the FES 2004 program. The results

from a joint analysis of these and the rate of back-

ground events using the differential probability gain

(4) are plotted in Fig. 7. The expected number of

events λij was estimated separately for each bin as the

summed number of events for the 1985–2016 period

divided by the number of 0.2-h intervals for that

period.

It is apparent from Fig. 7 that the effect of tides on

background seismicity only occurs at extremal values

of both tide heights and its rate of change. Extremal

values occur for large tide amplitudes. This result is

identical with the conclusions of Ide et al. (2016). The

increase in the rate of events at high ebb rates (large

negative values of the derivative h') is not as evident as

that for aftershocks. In fact, one even finds a decrease

64

JOURNAL OF VOLCANOLOGY AND SEISMOLOGY Vol. 13 No. 1 2019

BARANOV et al.

in the rate at intermediate values. Comparison

between the respective results for aftershocks and

background events suggests that the effect of ocean

tides on background seismicity is more likely to be

static excitation due to stress rearrangement. The most

definite effect for aftershocks is a relative increase in

their activity at high tide decay rates. This type of exci-

tation looks more definitely like a dynamic triggering

effect: a high rate of tide decay reduces the normal

stress component on fault sides, the friction is dimin-

ished, and slip occurs that ought to have occurred

eventually without tidal effects (Narteau et al., 2002),

though somewhat later. One possible distinguishing

feature in the influence of tides on background earth-

quakes consists in the circumstance that such events

occur under conditions where dynamic initiation of

events is unlikely (no clustering is seen for this reason,

i.e., some events being initiated by others).

The main goal of the present study is to determine

the degree of influence that tides exert on the proba-

bility of aftershock occurrence in quantitative terms.

Quantitative estimates will enable us to incorporate a

possible inf luence of tides when fitting the aftershock

process following large earthquakes. Since the avail-

able data do not permit estimating that influence sep-

arately for earthquakes with different focal mecha-

nisms, the estimates can be derived as averaged differ-

ential probability gain functions g (h) and g′ (η′) (see

Figs. 6a, 6b). The quantities h(t) and h′(t) are signa-

tures of the same periodic process, but they are weakly

correlated in formal terms (Fig. 8). It follows that one

can use (5) in order to make use of both g (h) and g′ (η′)

(Shebalin, 2017).

We note that Fig. 8 also demonstrates a complex

phase structure of tides, which makes a joint analysis

of h and h′ with seismicity preferable to the analysis of

tide phases.

CONCLUSIONS

This study reports a joint analysis of ocean tide

height as modeled by the FES 2004 program and its

Fig. 5. The differential probability gain functions g (a, c, e ) and g′ (b, d, f) for three aftershock sequences following the main

shocks of 1971, M = 7.0 (a, b); 1997, M = 7.0 (c, d); and 1997, M = 6.4 (e, f). (a, c, e) the solid line shows values of g and g′, the

light line shows the number of aftershocks in a specified range of tide height h or of its derivative h′. The straight line marks the

level of differential probability gain equal to 1.0. The functions g and g′ were found from formulas (4) with dh = 0.3 m and dh′ =

2 × 10–5 m/s. n denote s the numb ers o f events in th ose spa ce–time bins wher e tide he ight or its de rivat ive are i n the relevan t inter-

val (the numerators in (4)).

g'

n

h', m

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒0.0001 ‒5e‒5 0 5e‒5 0.0001

(f)

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒0.0001 ‒5e‒5 0 5e‒5 0.0001

(d)

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒0.0001 ‒5e‒5 0 5e‒5 0.0001

(b)

g

h, m

n

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒1.5 ‒0.5‒1.0 0 0.5 1.0 1.5

(e)

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒1.5 ‒0.5‒1.0 0 0.5 1.0 1.5

(с)

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒1.5 ‒0.5‒1.0 0 0.5 1.0 1.5

(a)

JOURNAL OF VOLCANOLOGY AND SEISMOLOGY Vol. 13 No. 1 2019

A QUANTITATIVE ESTIMATE 65

derivative with respect to time on the one hand and the

aftershock sequences of 16 earthquakes with magni-

tudes equal to 6 or greater off Kamchatka on the other.

The analysis was carried out using the differential

probability gain function; this indicates how many

times the rate of events changes for definite values of

each factor relative to the basic model. The Omori–

Utsu power-law model was adopted as the basic

model.

We obtained averaged differential probability gain

functions for the occurrence of aftershocks relative to

the basic Omori–Utsu model with respect to two fac-

tors: tide height and its time derivative. These func-

tions can be used to model the rate of aftershocks fol-

lowing large earthquakes off Kamchatka, in particular,

for assessment of the seismic hazard posed by after-

shocks. Due to a formally low correlation between tide

height and its derivative, one can make use of a prod-

Fig. 6. Total differential probability gain functions g (a, c, e) and g′ (b, d, f): (a) and (b) are for all 16 aftershock sequences, (b) and

(d) are for 13 sequences (the 1971 sequence and the two 1997 sequences have been eliminated), and (e) and (f) are for 14 sequences

(the two 1997 sequences have been eliminated). See explanation in the caption to Fig. 5.

h, m h', m

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒1.5 ‒0.5‒1.0 0 0.5 1.0 1.5

(e)

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒1.5 ‒0.5‒1.0 0 0.5 1.0 1.5

(c)

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒1.5 ‒0.5‒1.0 0 0.5 1.0 1.5

(a)

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒0.0001 ‒5e‒5 0 5e‒5 0.0001

(f)

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒0.0001 ‒5e‒5 0 5e‒5 0.0001

(d)

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒0.0001 ‒5e‒5 0 5e‒5 0.0001

(b)

g'

n

g

n

Fig. 7. The differential probability gain functions g (a) and g′ (b) for background seismicity. For the legend, consult Fig. 5.

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒1.5 ‒0.5‒1.0 0 0.5 1.0 1.5

(a)

g

n

h', m h', m

0

3.0

2.0

2.5

1.5

1.0

1000

100

10

1

0.5

‒0.0001 ‒5e‒5 0 5e‒5 0.0001

(b)

g'

n

66

JOURNAL OF VOLCANOLOGY AND SEISMOLOGY Vol. 13 No. 1 2019

BARANOV et al.

uct of the two differential gain functions, thereby

incorporating both of the factors.

The analysis of tides and the rate of aftershocks

demonstrated a persistent effect of considerable rela-

tive increase in the aftershock rate (by two times at the

most), while having large negative time derivatives of

tide height, that is, at high ebb rates, which corre-

sponds to lifting the load from the seafloor. For some

of the aftershock sequences we also observed increas-

ing aftershock rates at the greatest tide heights and tide

height derivative. We compared these results with a

similar joint analysis of tides and background seismic-

ity, finding that an increasing rate of events during low

water periods is the most persistent effect for back-

ground seismicity. The fact that background seismicity

is preferentially affected by ebb depth and aftershocks

by ebb rate can be interpreted in the sense that the

dominant mechanism for initiation of aftershocks

consists in dynamic influence due to high rates of

strain unloading, while the static mechanism of

increased probability of events during large-amplitude

ebbs occurs for background earthquakes.

The phase structure of tides is extremely complex.

In this connection we believe that it is more reasonable

to compare the rate of events, not with different tidal

phases, but with tide height and its derivative. The

results obtained here demonstrate a substantial influ-

ence of ocean tides, when their amplitudes are large.

This is in agreement with the conclusions reached by

Ide et al. (2016).

ACKNOWLEDGMENTS

We thank the authors of the FES 2004 program for

the program and comments on it, and the reviewers for

valuable remarks. This study was supported by the

Russian Science Foundation, project no. 16-17-

00093.

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