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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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