Bulletin
of
the
Seismological
Society
of
America,
Vol.
72,
No.
6,
pp.
2181-2200,
December
1982
TIDAL
TRIGGERING
OF
EARTHQUAKES
BY
THOMAS
H.
HEATON
ABSTRACT
Analysis
of
the
tidal
stress
tensor
at
the
time
of
moderate
to
large
earthquakes
fails
to
confirm
an
earlier
hypothesis
that
the
origin
times
of
shallow
dip-slip
earthquakes
correlate
with
solid-earth
tidal
shear
stress,
Furthermore,
no
cor-
relation
is
seen
for
either
tidal
shear
stress
or
tidal
normal-to-the-fault
compres-
sive
stress
with
shallow
strike-slip
earthquakes
or
with
deep
earthquakes.
INTRODUCTION
In
an
earlier
study
(Heaton,
1975),
I
reported
on
the
results
of
an
investigation
that
suggested
a
correlation
between
the
origin
times
of
moderate
to
large
shallow
dip-slip
earthquakes
and
tidal
shear
stress
as
resolved
into
a
coordinate
frame
defined
by
the
slip
vectors
of
individual
earthquakes.
In
that
investigation,
I
computed
the
solid
earth
tidal
shear
stress
for
107
earthquakes
for
which
the
focal
mechanisms
were
known.
Although
no
significant
tidal
correlation
was
found
for
the
entire
data
set,
which
contained
earthquakes
of
all
depths
and
mechanisms,
a
fairly
striking
correlation
was
seen
for
those
34
earthquakes
that
were
classified
as
shallow
dip-slip
earthquakes.
Because
of
that
result,
I
concluded
that
there
is
good
reason
to
believe
that
larger
shallow
dip-slip
earthquakes
are
tidally
triggered.
In
order
to
test
the
validity
of
that
hypothesis,
in
this
study
I
calculate
tidal
stress
histories
for
222
earthquakes
that
were
not
considered
in
the
previous
study.
Of
these,
68
are
classified
as
shallow
dip-slip.
If
the
previous
hypothesis
is
to
be
considered
physically
meaningful,
then
a
similar
correlation
between
tidal
shear
stress
history
and
the
origin
times
of
earthquakes
should
be
seen
for
this
new
data
set.
The
purpose
of
this
study,
then,
is
to
confirm
or
reject
the
hypothesis
that
the
origin
times
of
shallow
dip-slip
earthquakes
correlate
with
tidal
shear
stress.
As
geophysical
problems
go,
the
problem
solved
in
this
paper
is
very
well
posed.
That
is,
there
are
very
few
subjective
judgments
which
must
be
made.
Both
the
way
in
which
the
data
are
chosen
and
the
method
of
statistical
analysis
are
defined
by
the
earlier
study.
Having
well-defined
rules
is
essential
if
statistics
are
to
have
meaning.
Furthermore,
the
statistics
are
invalid
if
results
are
examined
before
deciding
whether
to
play
the
game.
This
is
a
common
problem
with
many
earthquake
prediction
statistical
studies
and
one
which
I
seem
to
have
poorly
understood
in
my
earlier
study
(Heaton,
1975).
However,
it
seems
clear
that
professional
casinos
would
not
allow
their
patrons
to
make
this
same
mistake.
EARTHQUAKES
AND
TIDES
In
this
study,
a
very
simple
earthquake
tidal
triggering
mechanism
for
earthquakes
is
tested.
That
is,
do
earthquakes
occur
preferentially
at
times
when
solid
earth
tides
increase
the
shear
stress
on
faults?
In
order
to
answer
this
question,
the
solid
body
tidal
stress
is
computed
as
a
function
of
time
and
then
rotated
into
the
coordinate
frame
which
is
defined
by
the
earthquake
fault
plane
and
the
slip
vector.
In
this
way,
the
tidal
shear
stress
that
can
be
considered
to
be
sympathetic
to
failure
can
be
plotted
as
a
function
of
time.
The
coordinate
frame
that
is
used
in
this
study
is
illustrated
in
Figure
1.
The
procedure
used
for
calculating
the
solid
earth
tidal
stress
is
described
by
Heaton
(1975)
and
identical
computer
codes
are
used
in
both
2181
2182
THOMAS
H.
HEATON
these
studies.
In
addition
to
calculating
the
shear
stress,
the
normal-to-the-fault
compressive
tidal
stress,
and
the
hydrostatic
stress
(Tll
+
z22
+
T33)/3
are
also
computed
for
each
earthquake.
The
fault
plane
and
slip
vector
of
each
earthquake
are
defined
from
published
source
studies.
Due
to
the
symmetry
of
the
stress
tensor,
either
of
the
two
complementary
fault
planes
and
slip
vectors
that
are
obtained
from
focal-mechanism
studies
can
be
used
to
specify
the
coordinate
frame
into
which
the
tidal
stress
tensor
is
rotated.
The
effects
of
oceanic
tides
on
crustal
stress
are
ignored
in
this
study.
This
is
not
to
imply
that
oceanic
tides
are
unimportant.
On
the
contrary,
oceanic
tidal
stress
can
cause
significant
pertubations
in
the
phase
and
amplitude
of
crustal
tidal
stress
and
in
some
instances
may
dominate
over
the
solid
earth
tidal
stress
(Beaumont
and
Berger,
1975).
Unfortunately,
computation
of
tidal
stress
due
to
oceanic
loading
is
quite
difficult
and
beyond
the
scope
of
this
study.
Because
of
the
statistical
nature
of
this
problem,
one
may
argue
that
inclusion
of
the
contribution
of
oceanic
tides
is
FIG.
1.
Coordinate
system
used
in
this
report.
Fault
strike
is
defined
as
clockwise
from
north.
Dip
is
positive
for
a
fault
striking
north
and
dipping
east.
Rake
is
0
°
for
left-lateral
and
positive
90
°
for
thrust.
el
is
alined
parallel
to
the
slip
vector
and
ea
is
perpendicular
to
the
fault.
not
crucial
for
this
test
of
tidal
triggering.
Nevertheless,
it
is
clear
that
modeling
the
contribution
of
oceanic
loading
is
desirable.
CHOICE
OF
DATA
SET
The
purpose
of
this
study
is
to
determine
whether
the
origin
times
of
earthquakes
depend
upon
tidal
stress.
Thus,
it
seems
clear
that
only
earthquakes
whose
origin
time
is
independent
of
other
obvious
factors
should
be
chosen.
Therefore,
an
attempt
was
made
to
include
only
earthquakes
that
were
main
shocks
and
that
were
not
preceded
by
obvious
large
foreshock
activity.
Furthermore,
the
fault
plane
and
slip
vector
for
each
earthquake
should
be
known
and
well
constrained.
These
parameters
were
obtained
from
either
observed
surface
rupture
or
seismically
determined
fault-
plane
solutions.
Because
of
the
free-surface
boundary
condition,
T31
(the
shear
stress
sympathetic
to
failure)
for
dip-slip
earthquakes
on
vertical
faults
is
always
nearly
zero.
The
polarity
of
the
computed
tidal
shear
stress
for
dip-slip
earthquakes
on
nearly
vertical
faults
is
thus
sensitive
to
minor
errors
in
the
fault-plane
solution.
Therefore,
earthquakes
with
dip-slip
motion
on
near-vertical
faults
are
excluded.
Earthquakes
that
appear
to
meet
the
specifications
listed
above
were
rather
ran-
TIDAL
TRIGGERING
OF
EARTHQUAKES
2183
domly
chosen
from
a
variety
of
sources.
The
earthquakes
were
then
classified
according
to
focal
depth
and
mechanism.
They
are
listed
in
chronological
order
in
Tables
1
to
3.
Entries
with
an
asterisk
represent
earthquakes
that
were
considered
in
my
earlier
study
(Heaton,
1975).
Earthquakes
with
slip
angles
of
less
than
30
°
from
horizontal
are
classified
as
strike-slip
and
earthquakes
with
depths
greater
than
30
km
are
considered
as
deep.
However,
since
a
focal
depth
of
33
km
is
often
used
as
a
default
value
in
published
catalogs,
some
of
the
earthquakes
that
are
called
deep
may
have
actually
been
significantly
shallower.
To
protect
against
systematic
error
due
to
my
own
predjudices
(i.e.,
a
desire
to
repeat
my
previous
results),
the
decisions
were
made
about
earthquake
parameters
before
any
tides
were
computed
and
the
decisions
were
final.
DATA
ANALYSIS
The
data
analysis
in
this
study
is
identical
to
that
in
my
earlier
study
(Heaton,
1975).
Tidal
stresses
are
plotted
as
a
function
of
time
for
each
earthquake.
A
phase
is
then
assigned
using
a
linear
scale
(with
time)
from
0
°
to
360
°,
where
0
°
and
360
°
are
defined
by
the
times
of
tidal
stress
maxima
immediately
before
and
after
the
earthquake,
respectively
(see
Figure
2).
These
phases
are
then
plotted
on
rose
diagrams.
If
earthquake
origin
times
and
tidal
stress
are
independent,
then
the
phases
will
appear
uniformly
distributed
about
the
rose
diagram.
Clustering
of
the
phases
on
one
side
of
a
rose
diagram
indicates
a
possible
relationship
between
tides
and
earthquakes.
The
statistical
significance
of
clustering
is
evaluated
by
a
clever
and
simple
method
developed
by
Rayleigh
(1919).
Consider
a
random
walk
in
two
dimensions
(see
Figure
3).
Let
each
earthquake
phase
~i
represent
a
unit
step
in
the
~i
direction.
If
the
magnitude
of
the
vector
sum
of
m
unit
two-dimensional
vectors
(1,
~i)
is
denoted
by
R,
then
the
probability
PR
that
a
random
set
of
m
phases
will
produce
a
vector
sum
whose
magnitude
exceeds
R
is
approximately
equal
to
exp(-R2/m).
This
approximation
is
sufficient
when
m
is
larger
than
10.
Thus,
the
smaller
is
PR,
the
greater
becomes
our
confidence
in
tidal
triggering.
RESULTS
The
results
of
this
study
are
summarized
in
Tables
1
to
3
and
Figures
4
and
5.
Phases
of
the
hydrostatic
stress,
normal-to-the-fault
compressive
stress,
and
shear
stress
are
given
for
each
earthquake.
Zero
degrees
phase
denotes
either
maximum
tensile
or
maximum
shear
stress.
Figure
5,
a
and
b,
shows
no
apparent
correlation
between
earthquake
origin
times
and
tidal
shear
stress
or
normal-to-the-fault
compressive
stress
for
the
entire
data
set
(328
earthquakes).
The
same
conclusion
is
reached
when
only
those
earthquakes
deeper
than
30
km
are
considered
(Figure
5,
c
and
d).
If
the
data
set
is
restricted
to
shallow
(depth
<
30
km)
strike-slip
(slip
vector
<
30
°
from
horizontal)
earthquakes,
then
no
apparent
correlation
can
be
seen
for
either
shear
stress
or
normal-to-the-fault
compressive
stress
(Figure
5,
a
and
b).
All
of
the
above
results
are
compatible
with
my
earlier
study
(Heaton,
1975).
In
that
earlier
study,
however,
there
was
a
rather
striking
correlation
between
the
origin
times
and
tidal
shear
stress
seen
for
shallow
dip-slip
earthquakes
(34
events).
An
additional
68
shallow
dip-slip
earthquakes
are
investigated
in
this
study.
The
phases
of
the
tidal
shear
stress
for
those
new
earthquakes
are
shown
in
Figure
5c;
no
correlation
can
be
seen.
The
phases
of
the
tidal
shear
stress
for
shallow
dip-slip
earthquakes,
and
the
combined
data
set
are
shown
in
Figure
5d;
once
again,
the
correlation
is
not
statistically
significant.
Therefore,
I
conclude
that
the
previously
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noted
correlation
between
tidal
shear
stress
and
the
origin
times
of
shallow
dip-slip
earthquakes
is
not
reproducible.
DISCUSSION
Is
there
a
simple
relationship
between
solid
earth
tidal
shear
stress
and
the
origin
times
of
shallow
dip-slip
earthquakes
as
I
suggested
in
my
previous
study?
This
is
the
primary
question
addressed
in
this
paper,
and
the
answer
clearly
seems
to
be
no.
This
conclusion
raises
two
other
questions
for
which
the
answers
are
more
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FIG.
2.
Example
of
how
earthquakes
are
assigned
a
phase
relative
to
tidal
stress
time
history.
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FIG.
3.
Geometric
interpretation
of
PR,
the
probability
that
a
random
walk
consisting
of
m
unit
steps
will
traverse
a
distance
of
R
or
greater.
ambiguous.
The
first
is:
how
did
my
previous
study
manage
to
be
so
misleading?
The
second
is:
are
the
origin
times
of
earthquakes
ever
affected
by
tidal
stresses?
With
regard
to
the
first
question,
I
believe
that
I
made
two
mistakes,
both
of
which
I
hope
that
this
paper
remedies.
The
first
mistake,
if
it
can
be
called
that,
was
a
lack
of
followthrough.
That
is,
after
calculating
tidal
phases
for
all
earthquakes,
I
noticed
that
a
pattern
was
present
if
the
data
set
was
grouped
in
a
particular
manner.
Although
this
grouping
formed
the
basis
for
a
hypothesis,
it
did
not
constitute
proof
of
its
reality.
The
Second
mistake
is
related
to
the
first.
In
my
earlier
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