Bulletin
of
the
Seismological
Society
of
America,
89,
4,
pp.
845-853,
August
1999
Estimates
of
the
Ground
Accelerations
at
Point
Reyes
Station
during
the
1906
San
Francisco
Earthquake
by
Abdolrasool
Anooshehpoor,
Thomas
H.
Heaton,
Baoping
Shi,
and
James
N.
Brune
Abstract
We
have
developed
an
analytical
solution
for
the
rocking
and
overturn-
ing
response
of
a two-dimensional,
symmetric
rigid
block
subject
to
a full
sine
wave
of
horizontal
ground
acceleration.
We
use
this
solution
to
provide
lower-bound
es-
timates
of
the
peak
ground
acceleration
at
Point
Reyes
Station,
California,
during
the
1906
San
Francisco
earthquake
that
toppled
the
San
Francisco-bound
train.
Our
results,
for
a 3%
damping
ratio,
indicate
that
for
a single
cycle
of
a sine
wave
the
minimum
toppling
accelerations
at
1,
1.5,
and
2 Hz
are
0.35g,
0.5g,
and
1.05g,
respectively.
For
more
realistic
accelerograms
the
toppling
accelerations
are
about
1.1g
(complex
synthetic)
and
0.76g
(Lucerne
record
of
the
1992
Landers
earthquake).
Introduction
It
has
long
been
recognized
that
most
of
the
slip
along
plate
boundaries
occurs
during
the
largest-magnitude
earth-
quakes.
Although
the
frequency
of
earthquakes
decreases
by
about
a factor
of
10
with
each
unit
of
magnitude,
the
seismic
moment
increases
by
a factor
of
32.
Thus,
great
earthquakes
on
the
San
Andreas
fault,
while
infrequent,
are
inevitable:
these
great
earthquakes
are
the
main
actors
in
California
plate
tectonics.
So
while
the
average
repeat
interval
of
these
great
earthquakes
may
be
hundreds
of
years,
it
is
clear
that
future
urbanized
areas
of
California
will
be
subjected
to
these
earthquakes.
What
is it like
to
be
close
to
a great
earthquake
on
the
San
Andreas
fault?
Unfortunately,
there
are
no
ground
mo-
tion
recordings
of
any
great
shallow
crustal
earthquakes.
Hall
and
others
(1995)
discuss
the
nature
of
near-source
ground
motions
from
large
crustal
earthquakes,
and
they
emphasize
the
importance
of
large-amplitude
displacement
pulses
that
occur
in
the
near-source
region.
In
particular,
when
rupture
is
propagating
toward
a site
located
near
a
fault,
the
pulses
are
largest
in
the
direction
perpendicular
to
the
rupture
surface,
and
they
have
displacement
amplitudes
that
are
roughly
comparable
to
the
fault
slip.
Haskell
(1969)
argued
persuasively
that
such
pulses
should
exist,
and
he
suggested
that
they
can
be
seen
in
ground
motions
recorded
during
the
1966
Parkfield
earthquake.
Numerous
other
ex-
amples
of
these
pulses
have
been
recorded
in
recent
mod-
erate-sized
earthquakes,
and
some
of
these
examples
are
shown
in
a report
by
Hallet
al.
(1995,
Fig.
2 therein).
While
the
existence
of
near-source
displacement
pulses
is now
well
accepted,
the
nature
of
such
pulses
in
great
earthquakes
is
still
a subject
of
discussion.
In
particular,
the
near-source
displacement
amplitudes
recorded
in
the
1994
Northridge
and
1995
Kobe
earthquakes
did
not
exceed
70
em
(Hallet
845
al.,
1995).
However,
much
larger
displacements
are
ex-
pected
in
great
earthquakes
that
have
larger
fault
slips.
Perhaps
the
best
example
of
ground
motion
recorded
close
to
a fault
with
large
slip
is the
Lucerne
accelerogram,
which
was
recorded
on
rock
about
1
km
from
the
trace
of
the
1992
Landers
earthquake.
Iwan
and
Chen
(1994)
pro-
cessed
these
accelerograms,
giving
particular
consideration
to
accurately
recovering
ground
displacement
in
an
inertial
coordinate
frame.
Their
processed
displacement
is
domi-
nated
by
a 4-sec
pulse
of
displacement
having
a maximum
amplitude
of
255
em.
W ald
and Heaton
( 1994)
demonstrated
that
the
observed
displacement
pulse
can
be
reproduced
by
a fault-slip
model
that
is
compatible
with
other
data
includ-
ing
strong
ground
motions,
teleseismic
body
waves,
geodetic
deformations,
and
surface
fault
offsets.
Hartzell
and
Heaton
(1995)
and
Archuleta
and
Seale
(1995)
both
produced
long-
period
ground
motions
with
striking
similarities
to
the
Lu-
cerne
record
in
a
1990
workshop
to predict
the
motions
from
a hypothesized
M
7.5
strike-slip
earthquake
(the
displace-
ment
records
from
the
Hartzell
and
Heaton
(1995)
simula-
tion
are
also
reproduced
in
the
paper
by
Hallet
al.
[1995].
A
remarkable
account
of
a long-period
displacement
pulse
that
overturned
a train
at
Point
Reyes
Station
during
the
18
April
1906
San
Francisco
earthquake
(M
7.8)
is
de-
scribed
by
Jordan
(1907).
The
overturned
train
is shown
in
Figure
1,
and
a sketch
map
showing
the
location
of
the
train
and
the
San
Andreas
fault
are
shown
in
Figure
2.
The
train
was
pointing
southeastward
approximately
parallel
to
the
fault
when
the
earthquake
struck.
The
following
passage
is
from
Jordan's
description
(p.
19).
At
Point
Reyes
Station
at
the
head
of
Tomales
Bay
the
5:15
train
for
San
Francisco
was
just
ready.
The
con-
846
A.
Anooshehpoo
r,
T.
H.
Heaton,
B.
Shi,
and
J.
N.
Brune
Figure
1.
The
San
Francisca-bound
train
at
Point
Reyes
Station
that
was
tipped
on
its
side
during
the
1906
San
Francisco
earthquake.
ductor
had
just
swung
himself
on
when
the
train
gave
a
great
lurch
to
the
east
(from
which
we
infer
that
the
ground
initially
moved
southwest),
followed
by
another
to
the west
(from
which
we
infer
that
the
ground
then
moved
back
to
the
northeast),
which
threw
the
whole
train
on
its
side.
The
astonished
conductor
dropped
off
as
it went
over,
and
at
the
sight
of
the
falling
chimneys
and
breaking
windows
of
the
station,
he
understood
that
it was
the
Temblor.
The
fireman
turned
to
jump
from
the
engine
to
the west
when
the
return
shock
came.
He
then
leaped
to
the
east
and
borrowing
a Kodak
he
took
the
picture
of
the
train
here
presented.
In
this
paper,
we
present
an
analysis
of
the
character-
istics
of
ground
motion
pulses
that
are
consistent
with
the
story
of
the
tipped
train
at
Point
Reyes
station.
The
account
given
above
suggests
a pulse
of
displacement
in
the
south-
west
direction.
This
is
consistent
with
the
polarization
ex-
pected
for
an
SH
wave
propagating
to
the
northwest
pro-
duced
by
rupture
on
the
San
Andreas
fault
that
is
approaching
Point
Reyes
Station
from
the
southeast.
If
the
rupture
was
approaching
from
the
northwest,
then
we
would
expect
a displacement
pulse
in
the
northeast
direction.
Thus
the
current
observation
seems
consistent
with
the
epicenter
located
approximately
40
km
to
the
southeast
near
the
Golden
Gate,
as
inferred
by
Bolt
(1968)
. We
begin
by
de-
veloping
an
analytical
solution
for
the
rocking
and
overturn-
ing
of
a two-dimensional
rigid
block
subject
to
a full
sine
wave
of
horizontal
ground
acceleration.
The
solution,
when
applied
to
the
toppled
train,
provides
a lower-bound
estimate
of
the
peak
ground
acceleration
at
Point
Reyes
Station
. (We
assume
a
full
sine-wave
shape
for
the
fault-normal
acceler-
ation
pulse.)
The
analysis
is
extended
to
more
realistic
sit-
uations
using
a synthetic
accelerogram
generated
for a mag-
nitude
8 earthquake,
and
the
Lucerne
record
of
the
1992
Landers
earthquake.
Analytical
Solution
Following
Housner
(1963),
Shi
et
al.
(1996)
developed
an
analytical
solution
for
the
rocking
response
of
a two-
dimensional,
symmetric
rigid
block
(Fig.
3)
subject
to
a half
sine
wave
of
ground
acceleration.
Here,
we
consider
the
rocking
response
of
a rigid
block
subject
to
a full
sine
wave
of
horizontal
ground
acceleration
that
is defined
as
Estimates
of
the
Ground
Accelerations
at
Point
Reyes
Station
during
the
1906
San
Francisco
Earthquake
847
Ukiah
•
Pt.
Arena
0
50
123'
122'
!OOkm
N
1
Mt.Hamilton
•
Figure
2.
A
map
showing
the
San
Andreas
fault
and
Point
Reyes
Station
(from
Boore,
1977).
{
-A
sin(
wt
+
1.f1)
a(t)
=
0
l.fl
2n
-
1.f1
--::::;
t::::;
---'-
(()
w
otherwise
39'
38'
37'
(1)
where
w
is
the
angular
frequency,
A
is
the
acceleration am-
plitude,
and
1.f1
is defined
by
l.fl
=
sin-
1
(7).
(2)
to
ensure
that
at
time
t
=
0
the
inertial
force
due
to
ground
acceleration
is large
enough
to
overcome
the
restoring
force
(ag)
and
to
initiate
the
rocking
motion.
Referring
to
Figure
3,
the
overturning
problem
we
are
interested
in
can
be
described
as
follows.
Initially,
at
t
=
0
when
ground
acceleration
exceeds
ag,
the
block
starts
rock-
ing
clockwise
about
0
1
;
then,
after
rotating
through
an
angle
B,
it comes
to
rest;
at
this
point,
the
block
changes
direction
and
rotates
counterclockwise
until
its
impact
with
the
base.
At
the
time
of
impact
(t
=
ti),
0
2
comes
in
contact
with
the
base,
while
0
1
loses
contact.
After
impact,
the
block
contin-
ues
to
rotate
counterclockwise
about
0
2
.
Its
angular
velocity
decreases
gradually
until
it becomes
nearly
zero
just
when
the
angular
displacement
exceeds
a;
at
this
point,
because
of
its
weight,
the
block
overturns.
It
is
assumed
that
the
coefficient
of
friction
is
sufficiently
large
so
that
there
will
02
01
Figure
3.
Rocking
of a rigid
block
subject
to
a
full
sine
wave
of
ground
acceleration,
a(t).
a(t)
be
no
sliding
between
the
block
and
the
base
(Hausner,
1963).
Equation
of
Motion
For
a symmetric
rigid
block
subject
to
ground
acceler-
ations
a,
the
differential
equation
of
motion
about
an
axis
of
rotation
is
IiJ
=
-mgR
sin
(a
-
B)
+
mRa
cos(a
-
B),
(3)
where
g
is
the
acceleration
due
to
gravity,
m
is
the
mass
of
the
block,
R
is
the
distance
between
the
center
of
mass
to
the rocking
axis,
I
is the
moment
of
inertia
about
the
rocking
axis;
B
is
the
instantaneous
angular
displacement
of
the
block,
and
a
is
the
maximum
angular
displacement
of
the
block
from
its
equilibrium
position
without
being
over-
turned.
Assuming
that
a
is
small
(
<
15°),
the
equation
of
motion
reduces
to
where
mRa
I
Overturning
by
Sinusoidal
Acceleration
(4)
The
overturning
of
a rigid
block
(Fig.
3)
by
the
full
sine
wave
of
ground
acceleration
specified
by
equation
(1)
in-
volves
an
impact.
During
the
course
of
rocking,
depending
on
the
frequency
of
the
ground
acceleration,
w,
impact
can
occur
either
before
or
after
the
instant
that
the
ground
acceleration
has
completed
its
full
cycle,
namely,
at