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Forecasting directivity in large earthquakes in terms of the conditional hypocenter distribution

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Forecasting directivity in large earthquakes in terms of the conditional hypocenter distribution

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Forecasting Directivity in Large Earthquakes in Terms of
the Conditional Hypocenter Distribution
by
Jessica Donovan

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(GEOLOGICAL SCIENCES)
August 2015

ii

Acknowledgements
Thank you to God first and foremost for direction, correction, strength, and always
providing for me. Thank you to my family for all their encouragement and support: to my
parents for instilling in me a strong work ethic and the belief that I could do anything I wanted
and providing me with the love, support, and encouragement to do just that, and to my siblings
for helping to keep me sane in insane times (Chris, a special thank you for the phone calls and
helping set me up in LA). Thank you Dad for introducing me to all the hiking in LA that helped
me escape, blow off steam in difficult times, and appreciate life here in LA. Thank you Mom for
taking care of me and being there for me for the move out to LA and during my qualification
exam. Thank you Mel, for all your support (including all the food you made me during this
writing process), for always being there and being my place to come home to: one step at a time.
Thank you to all my friends I’ve made here in LA, in particular to Feng and Yaman (Fight On!).
Thank you to my PhD advisor, Dr. Tom Jordan, for your support and countless hours of
drawings and equations on the whiteboard in your office, and for providing me with this
opportunity to pursue my passion.
I would also like to thank Scott Callaghan and Kevin Milner for their help with
CyberShake, Gavin Hayes for providing his finite source inversions, and Keith Richards-Dinger
for providing the RSQSim results. Thank you for your help and insightful discussions over the
years. This research has been funded through the Southern California Earthquake Center (NSF
Cooperative Agreement EAR-1033462 and USGS Cooperative Agreement G12AC20038).
This research contributed to and benefitted from funding provided by the National
Science Foundation-support project FESD Type I: Earthquake Fault System Dynamics (NSF
Grant EAR-1135455). This research used resources of the Blue Waters sustained-petascale

iii

computing project, which is supported by the National Science Foundation (awards OCI-
0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the
University of Illinois at Urbana-Champaign and its National Center for Supercomputing
Applications. This work is also part of the "Extending the Spatiotemporal Scales of Physics-
based Seismic Hazard Analysis" PRAC (award OCI-1440085) allocation support by the National
Science Foundation. This research used resources of the Oak Ridge Leadership Computing
Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of
the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.

iv

Table of Contents
0 Acknowledgments ……………………………………………………………………………. ii
0 List of Tables ………………………………………………………………………………... vii
0 List of Figures ……………………………………………………………………………… viii
1 Chapter 1 Introduction ............................................................................................................. 1
1.1 Overview ........................................................................................................................... 1
1.2 Probabilistic Seismic Hazard Analysis .......................................................................... 4
1.3 Rupture Directivity and the Finite Moment Tensor (FMT) ........................................ 6
1.4 Mapping from Directivity Parameter to Apparent CHD ............................................ 7
1.5 Data Sets ........................................................................................................................... 8
1.5.1 Published Finite Fault Inversions ................................................................................ 8
1.5.2 CyberShake ............................................................................................................... 13
1.5.3 RSQSim ..................................................................................................................... 16
2 Chapter 2 The Finite Moment Tensor and Directivity Vector ............................................ 17
2.1 Introduction .................................................................................................................... 17
2.2 Finite Moment Tensor (FMT) ...................................................................................... 17
2.3 Rake Variations and the FMT ...................................................................................... 21
2.4 1D Finite Moment Tensor ............................................................................................. 24
2.5 2D Finite Moment Tensor ............................................................................................. 26
2.6 Summary ........................................................................................................................ 31

v

3 Chapter 3 Displacement Along Strike .................................................................................... 33
3.1 Introduction .................................................................................................................... 33
3.2 Data ................................................................................................................................. 34
3.3 Creating 1D Displacement Profiles .............................................................................. 34
3.4 The Beta Distribution .................................................................................................... 40
3.5 Finite Source Inversion Displacement Profiles ........................................................... 43
3.6 Rupture Simulator Displacement Profiles ................................................................... 57
3.6.1 CyberShake Displacement Profiles ........................................................................... 58
3.6.2 RSQSim Displacement Profiles ................................................................................ 73
3.7 Discussion ....................................................................................................................... 85
3.7.1 Finite Source Inversion Profiles ................................................................................ 85
3.7.2 Rupture Simulator Profiles ........................................................................................ 88
3.8 Conclusion ...................................................................................................................... 89
4 Chapter 4 Directivity Mapping in Simulations ..................................................................... 91
4.1 Introduction .................................................................................................................... 91
4.2 Mapping Directivity Parameter to Apparent Hypocenter ......................................... 93
4.3 Simulation Sets ............................................................................................................... 95
4.3.1 CyberShake ............................................................................................................... 96
4.3.2 RSQSim ..................................................................................................................... 99
4.4 Results ........................................................................................................................... 102
4.5 Discussion ..................................................................................................................... 109
4.6 Conclusion .................................................................................................................... 111

vi

5 Chapter 5 Apparent Conditional Hypocenter for Observed Earthquakes ...................... 112
5.1 Introduction .................................................................................................................. 112
5.2 Finite Source Inversions .............................................................................................. 114
5.3 Hypocenter Distribution ............................................................................................. 117
5.3.1 Inferred from Aftershock Locations ........................................................................ 118
5.3.2 Determined from Finite Source Inversions ............................................................. 121
5.4 Directivity distribution ................................................................................................ 122
5.5 Apparent hypocenter distribution .............................................................................. 127
5.6 Discussion ..................................................................................................................... 131
5.6.1 Centroid-Bias of the CHD ....................................................................................... 131
5.6.2 The CHD and Hazard Estimates in CyberShake ..................................................... 138
5.7 Conclusion .................................................................................................................... 143
6 References ............................................................................................................................... 145

vii

List of Tables

Table 1.1: Finite Source Inversions .............................................................................................. 10
Table 1.2: CyberShake Rupture Section Abbreviations ............................................................... 14
Table 1.3: CyberShake Rupture Set .............................................................................................. 15
Table 3.4: Parameters for Best-Fitting Beta Distributions ........................................................... 54
Table 3.5: Parameters for Best-Fitting Beta Distributions (CyberShake) ................................... 68
Table 3.6: Parameters for Best-Fitting Beta Distributions (RSQSim) ......................................... 83

viii

List of Figures

Figure
1.1:
Map
of
Global
CMT
solutions
(Dziewonski
et
al.,
1981;
Ekström
et
al.,
2012)
for

the
published
finite
source
inversions
listed
in
Table
1.1.
...........................................................
9

Figure
2.1:
Analytic
solution
(Equation
2.4)
for
the
relationship
between
nucleation
and

directivity
parameter
for
a
simplified
1D
source
in
a
homogeneous
medium
with

constant
rupture
velocity
and
an
instantaneous,
uniform
slip
rate
(zero
rise
time
so

that
d(0)
=
1
from
Equation
2.6).
..........................................................................................................
26

Figure
2.2:
Circular
effects
on
the
directivity
components
for
nucleation
in
the
corner
of
the

rupture
area.
The
rupture
initially
propagates
as
a
circle
(a),
which
has
a
different

expected
FMT
than
a
rectangle
(e.g.,
see
the
comparison
of
equations
at
the
beginning

of
this
section),
until
it
reaches
the
edge
of
the
smallest
dimension.
For
small
aspect

ratio
faults
(b),
this
circular
effect
can
dominate
the
FMT
components
(note
the

behavior
of
the
directivity
components
before
the
red
line
in
d).
The
evolution
of
the

directivity
components
is
shown
in
(d)
with
the
red
line
corresponding
to
the
red
lines

in
(a-‐c),
before
which
circular
effect
dominate.
The
solid
black
line
corresponds
to
the

solid
line
in
(c),
which
represents
the
point
at
which
the
circular
effects
fall
off.
The

dashed
lines
mark
the
bounds
of
the
expected
behavior
of
|d2|
for
a
rectangular

rupture,
where
a
is
the
aspect
ratio.
Note
that
these
calculations
are
done
using
zero

rise
time
so
that
|d1|
=
1
for
a
perfectly
unilateral
rupture.
......................................................
29

Figure
2.3:
Circular
effects
on
the
directivity
components
for
nucleation
in
the
middle

center
of
the
rupture
area.
This
figure
is
directly
analogous
to
Figure
2.2
and
shows
the

aspect
ratio
necessary
to
dissipate
the
circular
effects
depends
on
where
the
rupture

nucleates
(note
that
for
this
nucleation
point,
|d2|
falls
within
the
expected
range
at

W/v
instead
of
1.5W/v
as
in
Figure
2.2).
Note
that
these
calculations
are
done
using

zero
rise
time
so
that
|d1|
=
1
for
a
perfectly
unilateral
rupture.
............................................
30

Figure
2.4:
Behavior
of
the
directivity
components
with
increasing
aspect
ratio.
The
left

column
shows
the
magnitude
of
the
directivity
components
(|d1|
on
top
and
|d2|
on

bottom)
for
each
hypocenter
location
on
the
fault
for
a
rupture
with
aspect
ratios
a
=
2

(a)
and
a
=
4
(c).
The
|d2|
colorbar
is
limited
to
a.
The
right
column
shows
the
relative

size
of
the
directivity
components
for
the
same
two
aspect
ratios
a
=
2
(b)
and
a
=
4

(d).
Note
that
these
calculations
are
done
using
zero
rise
time
so
that
d1
=
±1
for
a

perfectly
unilateral
rupture.
...................................................................................................................
31

Figure
3.1:
The
2011
MW
7.6
Kermadec
Islands
region
finite
source
inversion
slip
(Hayes,

USGS
website)
overestimates
the
length
of
the
rupture,
creating
a
profile
with
an

artificially
long
displacement
profile
(a).
..........................................................................................
36

Figure
3.2:
The
same
finite
source
inversion
shown
in
Figure
3.1
trimmed
according
to
the

methodology
employed
in
the
Next
Generation
Attenuation
models
by
Chiou
et
al.

(2008)
(b).
The
trimmed
slip
profile
(a)
eliminates
the
artificially
long
slip
profile
in

ix

Figure
3.1a,
providing
a
better
estimate
of
the
slip
profile
along
strike
for
this
rupture.

.............................................................................................................................................................................
37

Figure
3.3:
(a)
CyberShake
slip
distribution
for
a
MW
8.2
southern
San
Andreas
rupture

(UCERF2
Source
ID:
89,
Rupture
ID:
4)
with
strike
307°,
dip
84°,
and
rake
177°.
Grey

dot
marks
the
hypocenter,
red
triangle
marks
the
spatial
centroid,
red
arrow
is
the

directivity
vector,
black
ellipse
outlines
the
characteristic
dimensions
from
the
FMT,

and
rupture
initiation
times
are
contoured
in
white.
(b)
1D
slip
profile
computed
from

(a)
by
summing
up
dip.
(c)
1D
slip
profile
resampled
at
5%
resolution
from
the

original
slip
profile.
.....................................................................................................................................
39

Figure
3.4:
Examples
of
different
Beta
distributions
B(x;
α,
β)
with
various
α
and
β

parameters.
The
Beta
distribution
is
able
to
represent
the
uniform
distribution
(i.e.,

α=β=1)
as
well
as
concave,
convex,
asymptotic,
symmetric,
and
asymmetric
functions.

Note:
Figures
that
show
only
the
Beta
distribution
are
plotted
on
the
range
from
0
to
1

to
be
consistent
with
the
definition;
however,
the
Beta
distribution
is
shifted
to
range

between
-‐0.5
and
0.5
when
plotted
with
displacement
profiles.
............................................
41

Figure
3.5:
Comparison
of
the
Beta
distribution
least
squares
fit
(blue
dashed
line)
to
the

displacement
function
of
Biasi
and
Weldon
(2006),
(black
solid
line)
and
a
plot
of
the

residuals
where
e
=
Disp(x)
–
B(x;
α,
β).
Note:
Figures
that
show
only
the
Beta

distribution
are
plotted
on
the
range
from
0
to
1
to
be
consistent
with
the
definition;

however,
the
Beta
distribution
is
shifted
to
range
between
-‐0.5
and
0.5
when
plotted

with
displacement
profiles.
.....................................................................................................................
42

Figure
3.6:
An
example
of
the
symmetric
displacement
profile
for
28
finite
source
inversion

strike-‐slip
events
to
visually
define
the
parameter
ε.
The
trimming
process
defines
the

ends
of
the
slip
profiles
as
non-‐zero;
however,
we
know
that
slip
must
go
to
zero
at
the

ends
of
the
rupture;
therefore,
when
we
find
the
best-‐fitting
Beta
distribution,
we
add

an
additional
parameter
ε,
which
is
the
distance
from
the
end
of
the
slip
profile
to

where
slip
would
go
to
zero
given
a
Beta
distribution
with
parameters
α
and
β.
The

average
slip
profile
is
shown
as
the
bold
black
line
with
the
0.1
and
0.9
quantiles
as
the

shaded
region
between
the
dashed
black
lines,
and
the
best-‐fitting
Beta
distribution
to

this
average
displacement
profile
is
shown
as
the
dashed
blue
lines.
The
red
dashed

lines
show
the
distances
ε1
and
ε2
from
the
trimmed
ends
of
the
fault
to
where
the

best-‐fitting
Beta
distribution
goes
to
zero.
.......................................................................................
43

Figure
3.7:
Symmetric
normalized
displacement
profiles
for
all
faulting
types
(86
finite

source
inversions).
The
top
shows
each
slip
profile
as
colored
symbols
with
the
mean

profile
shows
as
a
solid
black
line,
the
10%
and
90%
percentiles
are
shown
as
the
light

gray
area
bounded
by
a
black
dashed
line.
The
best-‐fitting
Beta
distribution
is
shown

as
the
dashed
blue
line
with
its
parameters
in
the
legend.
The
blue
histograms
show

the
centroid
distribution,
red
histograms
show
the
hypocenter
distribution,
and
the

black
histogram
show
the
normalized
distance
between
the
centroid
and
hypocenters

(i.e.,
events
on
the
tails
of
this
distribution
are
expected
to
have
higher
directivity

parameters).
..................................................................................................................................................
45

x

Figure
3.8:
Symmetric
normalized
displacement
profiles
for
dip-‐slip
ruptures
(58
finite

source
inversions).
Directly
analogous
to
Figure
3.7.
..................................................................
46

Figure
3.9:
Symmetric
normalized
displacement
profiles
for
strike-‐slip
ruptures
(28
finite

source
inversions).
Directly
analogous
to
Figure
3.7.
..................................................................
47

Figure
3.10:
Asymmetric
normalized
displacement
profiles
for
all
faulting
types
(86
finite

source
inversions).
Analogous
to
Figure
3.7.
..................................................................................
48

Figure
3.11:
Asymmetric
normalized
displacement
profiles
for
dip-‐slip
ruptures
(58
finite

source
inversions).
Analogous
to
Figure
3.7.
..................................................................................
49

Figure
3.12:
Asymmetric
normalized
displacement
profiles
for
strike-‐slip
ruptures
(28

finite
source
inversions).
Analogous
to
Figure
3.7.
......................................................................
50

Figure
3.13:
Normalized
displacement
profiles
for
all
faulting
types
oriented
so
that
each

profile’s
directivity
vector
points
toward
the
right
(86
finite
source
inversions).

Analogous
to
Figure
3.7.
...........................................................................................................................
51

Figure
3.14:
Normalized
displacement
profiles
for
dip-‐slip
ruptures
oriented
so
that
each

profile’s
directivity
vector
points
toward
the
right
(58
finite
source
inversions).

Analogous
to
Figure
3.7.
...........................................................................................................................
52

Figure
3.15:
Normalized
displacement
profiles
for
strike-‐slip
ruptures
oriented
so
that
each

profile’s
directivity
vector
points
toward
the
right.
(28
finite
source
inversions)

Analogous
to
Figure
3.7.
...........................................................................................................................
53

Figure
3.16:
An
example
of
the
Bootstrap
distribution
of
α
(left)
and
ε
(right)
based
on

generating
1000
bootstrapped
datasets
from
the
original
dataset
of
28
symmetric

strike-‐slip
profiles.
The
mean
and
standard
deviations
are
shown
as
red
solid
and

dashed
lines
respectively.
........................................................................................................................
55

Figure
3.17:
Distribution
of
the
number
of
asperities
(defined
to
be
local
maxima
in
the
slip

profile
whose
height
is
at
least
66%
of
the
global
maximum,
analogous
to
the

definition
of
a
“very
large
slip”
asperity
in
Mai,
2005)
for
dip-‐slip
(top)
and
strike-‐slip

(bottom)
ruptures.
......................................................................................................................................
56

Figure
3.18:
CyberShake
symmetric
normalized
displacement
profiles
for
all
faulting
types

(5010
rupture
variations).
The
top
shows
each
slip
profile
as
colored
symbols
with
the

mean
profile
shows
as
a
solid
black
line,
the
10%
and
90%
percentiles
are
shown
as

the
light
gray
area
bounded
by
a
black
dashed
line.
The
best-‐fitting
Beta
distribution
is

shown
as
the
dashed
blue
line
with
its
parameters
in
the
legend.
The
blue
histograms

show
the
centroid
distribution,
red
histograms
show
the
hypocenter
distribution,
and

the
black
histogram
show
the
normalized
distance
between
the
centroid
and

hypocenters
(i.e.,
events
on
the
tails
of
this
distribution
are
expected
to
have
higher

directivity
parameters).
............................................................................................................................
59

xi

Figure
3.19:
CyberShake
symmetric
normalized
displacement
profiles
for
strike-‐slip

ruptures
(4917
rupture
variations).
Directly
analogous
to
Figure
3.18.
............................
60

Figure
3.20:
CyberShake
symmetric
normalized
displacement
profiles
for
dip-‐slip
ruptures

(93
rupture
variations).
Directly
analogous
to
Figure
3.18.
.....................................................
61

Figure
3.21:
CyberShake
asymmetric
normalized
displacement
profiles
for
all
faulting
types

(5010
rupture
variations).
Directly
analogous
to
Figure
3.18.
................................................
62

Figure
3.22:
CyberShake
asymmetric
normalized
displacement
profiles
for
strike-‐slip

ruptures
(4917
rupture
variations).
Directly
analogous
to
Figure
3.18.
............................
63

Figure
3.23:
CyberShake
asymmetric
normalized
displacement
profiles
for
dip-‐slip

ruptures
(93
rupture
variations).
Directly
analogous
to
Figure
3.18.
..................................
64

Figure
3.24:
CyberShake
normalized
displacement
profiles
for
all
faulting
types
oriented
so

that
each
profile’s
directivity
vector
points
toward
the
right
(5010
rupture
variations).

Analogous
to
Figure
3.18..........................................................................................................................
65

Figure
3.25:
CyberShake
normalized
displacement
profiles
for
strike-‐slip
ruptures
oriented

so
that
each
profile’s
directivity
vector
points
toward
the
right
(4917
rupture

variations).
Analogous
to
Figure
3.18.
................................................................................................
66

Figure
3.26:
CyberShake
normalized
displacement
profiles
for
dip-‐slip
ruptures
oriented
so

that
each
profile’s
directivity
vector
points
toward
the
right
(93
rupture
variations).

Analogous
to
Figure
3.18..........................................................................................................................
67

Figure
3.27:
Comparison
between
CyberShake
and
the
finite
source
inversions
for
the

frequency
of
occurrence
of
normalized
displacement
values
from
0
to
5.5.
The

normalized
displacements
are
binned
at
0.5
resolution
for
the
symmeterized
profiles

for
both
CyberShake
(dark
grey)
and
the
finite
source
inversions
(light
grey).
..............
69

Figure
3.28:
The
asperity
distribution
shown
for
finite
source
inversions
(FSI)
in
Figure

3.17
plotted
with
the
asperity
distribution
for
CyberShake
(CS)
ruptures.
.......................
70

Figure
3.29:
Schematic
of
two
example
rupture
lengths
of
55
and
75
km
and
how

CyberShake
would
assign
hypocenters
(red
dots).
The
CyberShake
symmetric

hypocenter
distribution
for
all
faulting
types
(5010
ruptures)
is
shown
in
the
bottom

histogram.
The
solid
and
dashed
black
lines
in
the
histogram
are
the
mean
and
0.1
and

0.9
quantiles
respectively
for
a
random
uniform
distribution
from
Monte
Carlo

simulation.
......................................................................................................................................................
72

Figure
3.30:
RSQSim
symmetric
normalized
displacement
profiles
for
all
faulting
types

(940
ruptures).
The
top
shows
each
slip
profile
as
colored
symbols
with
the
mean

profile
shows
as
a
solid
black
line,
the
10%
and
90%
percentiles
are
shown
as
the
light

gray
area
bounded
by
a
black
dashed
line.
The
best-‐fitting
Beta
distribution
is
shown

as
the
dashed
blue
line
with
its
parameters
in
the
legend.
The
blue
histograms
show

xii

the
centroid
distribution,
red
histograms
show
the
hypocenter
distribution,
and
the

black
histogram
show
the
normalized
distance
between
the
centroid
and
hypocenters

(i.e.,
events
on
the
tails
of
this
distribution
are
expected
to
have
higher
directivity

parameters).
..................................................................................................................................................
74

Figure
3.31:
RSQSim
symmetric
normalized
displacement
profiles
for
strike-‐slip
ruptures

(907
ruptures).
Directly
analogous
to
Figure
3.30.
......................................................................
75

Figure
3.32:
RSQSim
symmetric
normalized
displacement
profiles
for
dip-‐slip
ruptures
(33

ruptures).
Directly
analogous
to
Figure
3.30.
.................................................................................
76

Figure
3.33:
RSQSim
asymmetric
normalized
displacement
profiles
for
all
faulting
types

(940
ruptures).
Directly
analogous
to
Figure
3.30.
......................................................................
77

Figure
3.34:
RSQSim
asymmetric
normalized
displacement
profiles
for
strike-‐slip
ruptures

(907
ruptures).
Directly
analogous
to
Figure
3.30.
......................................................................
78

Figure
3.35:
RSQSim
asymmetric
normalized
displacement
profiles
for
dip-‐slip
ruptures

(33
ruptures).
Directly
analogous
to
Figure
3.30.
.........................................................................
79

Figure
3.36:
RSQSim
normalized
displacement
profiles
for
all
faulting
types
oriented
so
that

each
profile’s
directivity
vector
points
toward
the
right
(940
ruptures).
Analogous
to

Figure
3.30.
.....................................................................................................................................................
80

Figure
3.37:
RSQSim
normalized
displacement
profiles
for
strike-‐slip
ruptures
oriented
so

that
each
profile’s
directivity
vector
points
toward
the
right
(907
ruptures).
Analogous

to
Figure
3.30.
...............................................................................................................................................
81

Figure
3.38:
RSQSim
normalized
displacement
profiles
for
dip-‐slip
ruptures
oriented
so

that
each
profile’s
directivity
vector
points
toward
the
right
(33
ruptures).
Analogous

to
Figure
3.30.
...............................................................................................................................................
82

Figure
3.39:
Comparison
of
the
normalized
displacement
frequency
distributions
for
all

three
datasets
binned
by
0.5
normalized
displacement.
............................................................
84

Figure
3.40:
Best-‐fitting
triangles
to
the
finite
source
inversions
asymmetric
displacement

profiles
for
comparison
with
Manighetti
et
al.
(2005).
...............................................................
86

Figure
3.41:
Two
example
random
walk
normalized
displacement
profiles
generated
to
test

whether
the
finite
source
inversion
profiles
match
what
is
expected
if
each
individual

profile
is
fractal.
Each
fractal
profile
is
then
processed
in
the
same
way
as
the
finite

source
inversion
profiles
(including
resampling
at
5%,
indicated
by
the
red
dots)
and

their
average
is
fit
with
a
Beta
distribution
(Figure
3.42).
........................................................
88

Figure
3.42:
Symmetric
normalized
displacement
profiles
for
a
set
of
100
random
walk

fractal
distributions
resampled
to
5%.
The
mean
is
shown
as
the
thick
black
line
and

its
corresponding
best-‐fitting
Beta
distribution
is
shown
as
the
dashed
blue
line.
.......
88

xiii

Figure
4.1:
An
example
of
a
MW
7.2
RSQSim
rupture
on
the
Hayward
fault
in
which
rupture

complexity
makes
the
hypocenter
(red
star)
and
apparent
hypocenter
(grey
dot)
very

different.
The
dashed
black
lines
show
the
locations
of
the
spatial
and
temporal

centroids
for
this
rupture.
The
hypocenter
location
is
0.5
and
the
apparent
hypocenter

location
is
0.1.
................................................................................................................................................
92

Figure
4.2:
Influence
of
slip
distribution
and
rise
time
on
the
apparent
hypocenter
mapping.

Curves
with
the
same
color
have
the
same
α
and
β
parameters,
and
curves
of
the
same

style
(i.e.,
solid
or
dashed)
have
the
same
rise
time
ratio
ρ,
ranging
from
0
rise
time
to

0.25
of
the
total
rupture
duration.
Increasing
the
rise
time
lowers
the
directivity

parameter
as
expected.
For
zero
rise
time,
increasing
α
and
β
parameters
push

apparent
hypocenters
toward
the
center
of
the
rupture
everywhere;
however,
for

finite
rise
time,
increasing
α
and
β
parameters
push
only
apparent
hypocenters
toward

the
center
of
the
rupture
only
within
30%
of
the
rupture
length.
At
the
edges,
an

increase
in
the
Beta
distribution
parameters
results
in
moving
the
apparent

hypocenter
toward
the
end.
....................................................................................................................
94

Figure
4.3:
An
example
rupture
variation
(slip
variation:
3,
hypocenter
variation:
3)
for
a

MW
7.8
CyberShake
rupture
on
the
Elsinore
fault
between
the
Glen
Ivy
and
Coyote

Mountain
sections
(UCERF2
Source
ID:
10,
Rupture
ID:
3)
with
strike
309°,
dip
85°,

and
rake
-‐178°.
Rupture
initiation
time
is
contoured
every
10
seconds
in
white
and

labeled
accordingly,
the
hypocenter
is
shown
as
the
grey
circle,
and
the
directivity

vector
from
the
FMT
is
shown
as
the
red
arrow.
...........................................................................
97

Figure
4.4:
The
CyberShake
symmetrized
hypocenter
distribution
along
strike
for
the
5010

rupture
variations
used
in
this
study
(4917
strike-‐slip
and
93
dip-‐slip).
The
solid
black

line
is
the
expected
number
in
each
bin
and
the
dashed
lines
are
the
0.1
and
0.9

quantiles
for
1000
Monte
Carlo
simulations
of
a
uniform
distribution.
..............................
98

Figure
4.5:
CyberShake
true
rupture
lengths
and
widths
for
the
17
ruptures
listed
in
Table

1.3
(“+”
is
used
for
strike-‐slip
and
“x”
for
dip-‐slip).
The
solid
lines
correspond
to

different
aspect
ratios
and
are
labeled
accordingly.
....................................................................
99

Figure
4.6:
An
example
RSQSim
slip
distribution
for
a
MW
7.3
rupture
on
the
Elsinore
fault

between
the
Glen
Ivy
and
Coyote
Mountain
sections
(Event
ID:
11548282)
with
strike

309°,
dip
90°,
and
rake
180°.
Rupture
initiation
time
is
contoured
every
10
seconds
in

white
and
labeled
accordingly,
the
hypocenter
is
shown
as
the
grey
circle,
and
the

directivity
vector
from
the
FMT
is
shown
as
the
red
arrow.
..................................................
100

Figure
4.7:
The
RSQSim
symmetrized
hypocenter
distribution
along
strike
for
the
940

ruptures
used
in
this
study
(907
strike-‐slip
and
33
dip-‐slip).
The
solid
black
line
is
the

expected
number
in
each
bin
and
the
dashed
lines
are
the
0.1
and
0.9
quantiles
for

1000
Monte
Carlo
simulations
of
a
uniform
distribution.
.......................................................
101

Figure
4.8:
RSQSim
true
rupture
lengths
and
widths
for
the
940
ruptures
in
this
study
(“+”

is
used
for
strike-‐slip
and
“x”
for
dip-‐slip).
The
solid
lines
correspond
to
different

aspect
ratios
and
are
labeled
accordingly.
......................................................................................
102

xiv

Figure
4.9:
The
CyberShake
apparent
CHD
as
mapped
using
the
directivity
parameter
D
for

the
5010
rupture
variations
in
this
study
(4917
strike-‐slip
and
93
dip-‐slip).
The
mean

(solid
red
line)
and
0.1
and
0.9
quantiles
(dashed
red
lines)
are
computed
using
Monte

Carlo
simulation
for
a
uniform
distribution
of
hypocenters
using
the
best-‐fitting
Beta

distribution
to
the
symmetric
profiles
for
all
faulting
types
(α
=
β
=
1.21,
Figure
3.18)

and
the
average
rise
time
ratio
ρ
=
0.03.
Note
that
events
missing
from
the
last
bin
are

expected
due
to
numerical
inaccuracy
as
well
as
deviations
of
the
ruptures
from
the

assumed
model.
For
example,
events
that
fall
within
the
last
bin
must
have
directivity

parameters
>
0.9987,
a
very
high
directivity
parameter.
Variations
in
rupture
velocity,

rise
time,
regions
of
zero
slip,
and
down-‐dip
nucleation
point
can
also
affect
the

directivity
parameter
in
third
decimal
place.
................................................................................
104

Figure
4.10:
The
RSQSim
apparent
CHD
as
mapped
using
the
directivity
parameter
D
for

the
940
ruptures
in
this
study
(907
strike-‐slip
and
33
dip-‐slip).
Directly
analogous
to

Figure
4.9,
using
the
best-‐fitting
Beta
distribution
of
slip
(α
=
β
=
1.37,
Figure
3.30)
and

the
average
rise
time
ratio
0.11.
Note
that
the
same
resolution
problem
arises
with
the

RSQSim
data
as
with
the
CyberShake
data
(see
Figure
4.9
caption).
The
last
bin
for

RSQSim
requires
events
to
have
a
D
>
0.9904.
.............................................................................
105

Figure
4.11:
The
relationship
between
true
and
apparent
hypocenters
(to
the
nearest
0.05)

mapped
using
D
for
(a)
CyberShake
and
(b)
RSQSim.
Red
circles
are
ruptures
in
Figure

4.13.
.................................................................................................................................................................
106

Figure
4.12:
A
MW
7.7
RSQSim
rupture
on
the
San
Andreas
Fault
from
the
San
Bernardino
to

the
Cholame
section
demonstrating
the
temporal
complexity
in
a
unilateral
RSQSim

rupture.
(Top)
Space-‐time
plot
colored
by
seismic
moment.
The
grey
dot
and
red
star

mark
the
hypocenter
and
apparent
hypocenters
respectively.
The
dashed
black
lines

mark
the
spatial
and
temporal
centroids.
(Bottom)
The
moment
distribution
with

rupture
initiation
times
contoured
every
1
second
in
black
lines.
The
grey
dot
marks

the
hypocenter
location.
Note
that
there
is
vertical
exaggeration
in
the
fault
plane
to

make
the
rupture
contours
more
visible.
........................................................................................
108

Figure
4.13:
The
spatiotemporal
histories
for
example
ruptures
from
(left)
a
CyberShake

MW
7.8
right-‐lateral
strike-‐slip
rupture
on
the
Elsinore
Fault,
and
(right)
a
RSQSim
MW

7.2
right-‐lateral
strike-‐slip
rupture
on
the
Panamint
Valley
Fault.
The
hypocenter
is

shown
as
a
grey
dot,
the
apparent
hypocenter
is
shown
as
the
red
star,
and
the
black

dashed
lines
mark
the
locations
of
the
spatial
and
temporal
centroids.
These
ruptures

are
shown
as
the
red
circles
in
Figure
4.11.
...................................................................................
110

Figure
5.1:
An
example
geometric
configuration
used
in
calculating
the
directivity

correction
factor
of
Spudich
and
Chiou
(2008).
The
red
star
xh
is
the
hypocenter

location
and
the
grey
dot
xclose
is
the
point
on
the
fault
closest
to
the
station
for
which

the
directivity
correction
is
being
calculated.
Figure
modeled
after
Figure
1
of
Spudich

and
Chiou
(2008).
......................................................................................................................................
113

xv

Figure
5.2:
An
example
of
the
trimming
procedure
used
in
this
study
for
the
2011
MW
7.6

Kermadec
Islands
region
earthquake
with
a
strike
of
170°,
dip
of
52°,
and
rake
of
-‐97°

(Hayes,
USGS
website;
also
shown
in
Figure
3.1
and
Figure
3.2).
The
slip
distribution
is

plotted
along
with
rupture
times
contoured
in
white
dashed
lines
and
labeled
in

seconds.
This
inversion
result
has
The
solid,
bold
box
outlines
the
rupture
area
of
the

trimmed
version
of
this
finite
source
inversion.
The
centroid
is
plotted
as
the
red

triangle,
the
hypocenter
is
shown
as
the
grey
circle,
the
characteristic
dimensions
are

shown
as
the
ellipse
(grey
dotted
ellipse
is
the
untrimmed
model),
and
the
directivity

vector
is
plotted
as
the
red
arrows
at
the
hypocenter
(smaller,
dashed
red
arrow
is
for

the
untrimmed
model).
...........................................................................................................................
114

Figure
5.3:
Comparison
of
the
low-‐order
components
of
the
FMT
and
the
hypocenter

distribution
between
untrimmed
and
trimmed
finite
source
inversions.
Each
point
is

scaled
by
its
characteristic
length
and
colored
by
magnitude.
The
black
lines
indicate
a

1:1
relationship.
The
spatial
centroid
is
a
fixed
location
in
3D
space;
therefore
we
plot

the
distance
between
the
untrimmed
and
trimmed
models
(the
x-‐axis
is
simply
the

model
number
from
1
to
86).
...............................................................................................................
116

Figure
5.4:
Comparison
of
the
higher-‐order
components
of
the
FMT
and
the
apparent

hypocenter
distribution
between
untrimmed
and
trimmed
finite
source
inversions.
As

in
Figure
5.3,
each
point
is
scaled
by
its
characteristic
length,
colored
by
magnitude,

and
the
black
lines
indicate
a
1:1
relationship.
............................................................................
117

Figure
5.5:
The
Henry
and
Das
(2001)
symmetric
hypocenter
distribution
for
a
set
of
105

MW
>
6.3
earthquakes
(38
strike-‐slip
and
67
dip-‐slip),
where
the
rupture
end
points

were
determined
from
the
first
24
hours
of
aftershocks.
The
solid
and
dashed
black

lines
are
the
mean
and
0.1
and
0.9
quantiles
from
1000
Monte
Carlo
simulations
of

randomly
draw
samples
from
a
uniform
distribution.
.............................................................
119

Figure
5.6:
An
example
for
the
1999
MW
7.2
Duzce,
Turkey
earthquake
of
the
procedure

used
to
estimate
rupture
dimensions
and
relative
hypocenter
location.
The
dots
are

the
relocated
positions
for
24
hours
of
aftershocks
with
their
error
ellipses
and
thin,

black
lines
point
to
their
starting
positions.
The
bold
red
line
is
the
rupture
plane

estimated
from
the
aftershock
distribution
and
the
bold
black
lines
are
surface
traces

for
two
finite
fault
inversions
(for
comparison).
The
hypocenter
and
projected

hypocenter
are
the
red
and
grey
stars,
respectively.
.................................................................
120

Figure
5.7:
The
combined
aftershock
inferred
hypocenter
distribution
for
the
45
strike-‐slip

events
from
Henry
and
Das
(2001)
and
this
study.
....................................................................
121

Figure
5.8:
The
finite
source
inversion
symmetric
hypocenter
distributions
for
the
set
of
86

finite
source
inversions
for
64
unique
events
(21
strike-‐slip
and
43
dip-‐slip)
used
in

this
study
with
their
associated
mean
and
0.1
and
0.9
quantiles
plotted
as
the
solid
and

dashed
black
lines
respectively
determined
from
Monte
Carlo
simulation.
...................
122

Figure
5.9:
The
distribution
of
41
directivity
parameters
computed
by
the
direct
inversion

of
seismic
data
for
35
unique
MW
≥
7
events
collected
from
literature
(McGuire
et
al.,

xvi

2002;
Clévédé
et
al.,
2004;
Llenos
and
McGuire,
2007).
In
cases
where
there
is
more

than
one
result
for
a
single
event,
each
result
is
given
1/n
weight
in
the
histogram

above,
where
n
is
the
number
of
results
for
a
given
event.
The
solid
and
dashed
red

lines
are
the
mean
and
0.1
and
0.9
quantiles,
respectively,
from
a
set
of
1000
Monte

Carlo
simulations
for
a
random
uniform
hypocenter
distribution
with
a
Beta

distribution
of
slip
assigned
according
to
the
results
in
Table
3.4
for
the
symmetric

distributions.
...............................................................................................................................................
124

Figure
5.10:
The
finite
source
inversion
directivity
distributions
for
the
set
of
86
trimmed

finite
source
inversions
of
65
unique
events
(21
strike-‐slip
and
43
dip-‐slip).
In
cases

where
there
are
more
than
one
inversion
for
a
unique
event,
each
inversion
is
given

equal
weight
in
the
histograms.
The
solid
and
dashed
red
lines
represent
the
mean
and

0.1
and
0.9
quantiles
respectively
from
1000
Monte
Carlo
simulations
for
a
simplified

1D
rupture
with
a
random
uniform
distribution
of
hypocenters,
a
Beta
distribution
of

slip
assigned
according
to
the
results
in
Table
3.4
for
the
symmetric
distributions
(α
=

β
=
1.92,
Figure
3.7),
and
the
average
rise
time
ratio
ρ
=
0.13.
..............................................
126

Figure
5.11:
The
apparent
hypocenter
distribution
calculated
from
the
86
trimmed
finite

source
inversions
for
65
unique
MW
≥
7
events
(21
strike-‐slip
and
43
dip-‐slip).
The

mean
and
0.1
and
0.9
quantiles
are
calculated
from
Monte
Carlo
simulations
for
a

random
uniform
distribution
of
hypocenters
and
a
Beta
distribution
of
slip
assigned

according
to
the
results
in
Table
3.4
for
the
symmetric
distributions
(α
=
β
=
1.92,

Figure
3.7),
and
the
average
rise
time
ratio
ρ
=
0.13.
................................................................
128

Figure
5.12:
Rupture
propagation
of
the
2001
MW
7.8
Kunlun
earthquake
in
southern

Qinghai,
China
(Hayes,
personal
communication).
The
top
is
the
final
moment

distribution
with
rupture
initiation
times
contoured
in
white
and
labeled
accordingly.

The
grey
dot
is
the
hypocenter,
red
triangle
is
the
spatial
centroid,
the
ellipse

represents
the
characteristic
dimensions,
and
the
red
arrow
is
the
directivity
vector.

Each
time
slice
is
incremented
by
12
seconds
and
is
on
the
same
scale
as
the
final

moment
distribution
at
the
top.
..........................................................................................................
130

Figure
5.13:
The
apparent
CHD
for
all
faulting
types
for
the
originally-‐trimmed
models

which
removes
regions
along
the
edges
of
the
rupture
with
>
10%
of
the
peak
slip,
but

retains
any
regions
with
≥
50
cm
slip
(top)
versus
the
apparent
CHD
the
modified

trimming
which
removes
regions
along
the
edges
of
the
rupture
with
<
10%
of
the

peak
slip,
regardless
of
slip
magnitude
(middle).
The
bottom
plot
shows
the
difference

between
the
two
distributions
(minimum-‐constrained
trimming
–
non-‐constrained

trimming).
Red
solid
and
dashed
lines
are
the
mean
and
0.1
and
0.9
quantiles
for
a
set

of
Monte
Carlo
simulations
assuming
a
random
uniform
distribution
of
hypocenters.

...........................................................................................................................................................................
132

Figure
5.14:
The
apparent
CHD
for
a
simulated
set
of
ruptures
with
a
uniform
distribution

of
hypocenters
both
along
strike
and
down
dip
for
a
rupture
with
an
aspect
ratio
of
3.

The
red
bold
and
dashed
lines
are
the
mean
and
0.1
and
0.9
quantiles
calculated
from

xvii

Monte
Carlo
simulations
for
a
random
uniform
distribution
of
hypocenters
on
a
1D

rupture.
..........................................................................................................................................................
133

Figure
5.15:
True
(top)
and
apparent
(bottom)
CHDs
for
all
faulting
types
with
aspect
ratios

>
3.5.
Solid
and
dashed
lines
mark
the
mean
and
0.1
and
0.9
quantiles
of
a
Monte
Carlo

distribution
for
a
uniform
random
hypocenter
distribution.
.................................................
134

Figure
5.16:
The
apparent
CHD
for
all
faulting
types
mapped
using
the
best-‐fitting
Beta

distribution
to
the
average
slip
profile
(top)
versus
the
apparent
CHD
mapped
using
a

uniform
slip
distribution
(middle).
The
bottom
plot
shows
the
difference
between
the

two
distributions
(Beta
mapped
CHD
–
uniform
mapped
CHD).
Red
solid
and
dashed

lines
are
the
mean
and
0.1
and
0.9
quantiles
for
a
set
of
Monte
Carlo
simulations

assuming
a
random
uniform
distribution
of
hypocenters.
......................................................
135

Figure
5.17:
The
apparent
CHD
for
all
faulting
types
mapped
using
the
average
rise
time

ratio
(0.13)
from
the
finite
source
inversions
(top)
versus
the
apparent
CHD
mapped

using
a
large
rise
time
ratio
of
0.25
(middle).
The
bottom
plot
shows
the
difference

between
the
two
distributions
(average
rise
time
ratio
CHD
–
large
rise
time
ratio

CHD).
Red
solid
and
dashed
lines
are
the
mean
and
0.1
and
0.9
quantiles
for
a
set
of

Monte
Carlo
simulations
assuming
a
random
uniform
distribution
of
hypocenters.
..
137

Figure
5.18:
The
apparent
CHD
for
finite
source
inversions
for
all
faulting
types
mapped
to

an
equivalent
CHD
for
CyberShake
is
shown
as
black
circles.
The
best-‐fitting
Beta

distribution
(α
=
β
=
10.03)
is
shown
as
the
blue
dashed
line
and
is
used
as
the
input

CHD
for
CyberShake.
................................................................................................................................
139

Figure
5.19:
CyberShake
results
for
the
ratio
of
the
probability
of
exceeding
0.1
g
between

the
modified
CHD
calculated
from
the
finite
source
inversions
(Figure
5.18)
and
the

uniform
CHD.
...............................................................................................................................................
141

Figure
5.20:
CyberShake
results
for
the
ratio
of
the
probability
of
exceeding
0.2
g
between

the
modified
CHD
calculated
from
the
finite
source
inversions
(Figure
5.18)
and
the

uniform
CHD.
...............................................................................................................................................
142

Figure
5.21:
CyberShake
results
for
the
ratio
of
the
probability
of
exceeding
0.3
g
between

the
modified
CHD
calculated
from
the
finite
source
inversions
(Figure
5.18)
and
the

uniform
CHD.
...............................................................................................................................................
143

1

1Chapter 1 Introduction
1.1 OVERVIEW
Forecasting the rupture directivity of large earthquakes is an important problem in
probabilistic seismic hazard analysis (PSHA) because directivity strongly influences ground
motion radiation patterns (e.g., Ben-Menahem, 1961). Whether a rupture propagates solely in
one direction versus in both directions will determine how energy is focused and can have a
significant impact on the amplitude, duration, and frequency content of the ground motion.
Ahead of a rupture, the early ground motions are higher amplitude, shorter duration, and
higher frequency. Conversely, in the opposite direction of rupture propagation, the arriving
waves are of lower frequency, higher duration, and lower in amplitude. As the rupture velocity
approaches that of the shear wave velocity, there are also significant differences between fault-
parallel and fault-perpendicular ground motions. Somerville et al. (1997) captured all three of
these features in an empirical correction factor, y, to ground motion prediction equations
(GMPEs) of the form
𝑦=
𝐶
!
+𝐶
!
𝑋cos𝜃 for strike-‐slip with 𝑀> 6.5
𝐶
!
+𝐶
!
𝑌cos𝜙 for dip-‐slip with 𝑀> 6.5,

where C
1
and C
2
are regression coefficients, X is the fraction of the fault that ruptures toward the
site, θ is the angle between the strike of the fault and the line between the epicenter and the site.
Y and ϕ are similar quantities to X and θ, respectively, for dip-slip faults. Spudich and Chiou
(2008) developed a more complex correction for directivity effects, but which still includes terms
related to the fraction of the fault that ruptures toward the site, the angle from the fault, and an
additional term for the ratio of the rupture and shear wave velocities. Like the groundwork laid
by Somerville et al. (1997), many of the GMPE directivity correction factors can be found using

2

simply the hypocenter location and the fault geometry. The choice of hypocenter distribution in
the earthquake rupture forecast (ERF), therefore, can have a significant effect on the hazard
(Wang, 2013; Wang and Jordan, 2014).
Traditionally, directivity is considered a site-specific effect and the GMPE correction
factors (e.g., Somerville et al., 1997; Bazzurro et al., 2006; Spudich and Chiou, 2008;
Rowshandel, 2010; Spagnuolo et al., 2012) treat it as such. These GMPE correction factors
require an input conditional hypocenter distribution (CHD; conditional on the rupture occurring),
and often assume a uniform CHD, claiming no a priori information about the distribution of
hypocenters (e.g., Abrahamson, 2000; Shahi and Baker, 2011). Wang (2013) showed how
different choices for the CHD can significantly influence the hazard calculations in CyberShake
(Graves et al., 2011), a physics-based earthquake simulator. Our goal is to develop a systematic
method of characterizing observed source directivity in the form of a CHD. This methodology
will have the ability to take into account complex rupture propagation to better characterize the
observed distribution of source directivity. That is, complexity in the rupture propagation can
make the true hypocenter location with respect to the ends of the rupture a poor proxy for
directivity. In such cases, if the true hypocenter location is used, this could result in incorrect
prediction of directivity effects. The resulting “apparent” CHD we determine with our
methodology better characterizes the observed distribution of source directivity and can then be
fed into existing machinery such as the GMPEs or simulations (for example, using the
averaging-based factorization methodology of Wang and Jordan 2014) to get out a better hazard
estimate. In this study, we use the existing framework of CyberShake (Graves et al., 2011) to
evaluate the difference our apparent CHD makes in the hazard calculations.

3

Rather than a site-specific measurement to quantify the directivity of a particular source,
there exists a low frequency, dimensionless “directivity parameter” contained in the finite
moment tensor (FMT) which measures the overall directivity of a source in terms of a second
polynomial moment of the source space-time function. This directivity parameter is dependent
on both a CHD and a conditional slip distribution (CSD). In this study, we use the distribution of
directivity parameters to determine an apparent CHD that best characterizes the distribution of
directivity observed in a global set of M
W
> 7 finite fault inversions for real events.
In addition to simply determining a CHD for use in PSHA calculations, there are several
questions about the CHD for large earthquakes that we will attempt to answer in this study:
• Is earthquake nucleation seemingly random (i.e., irrespective of the slip
distribution or geometrical complexities), producing a uniform CHD (McGuire et
al., 2002)?
• Do earthquakes tend to nucleate near geometric complexities where stress is
concentrated, and because of this barrier, rupture unilaterally, creating a
“periphery-biased” CHD in which hypocenters are preferentially located at the
ends of rupture areas (Oglesby, 2005)?
• Do earthquakes tend to nucleate in one spot, near or at an asperity for instance,
and propagate out in both directions, creating a “centroid-biased” CHD (Mai,
2005)?
• Is the CHD symmetric, or is there some asymmetry that can be attributed to a
physical process such as preferred rupture propagation direction (as with
bimaterial faults, Andrews and Ben-Zion, 1997)?

4

• How does the CHD change for different faulting types and in different tectonic
environments?
In this study, we use two earthquake simulators as well as a suite of finite-fault models
for observed earthquakes to investigate how our methodology for determining the apparent CHD
captures the characteristics of directivity.
1.2 PROBABILISTIC SEISMIC HAZARD ANALYSIS
Probabilistic seismic hazard analysis (PSHA) determines the probability of exceeding a
given ground motion by combining an earthquake rupture forecast (ERF) with predictions of
ground motions. Each earthquake specified in an ERF contains at least location and magnitude
information. An extended-ERF can also specify more detailed source parameters, such as
faulting type, rupture dimensions, and even source space-time functions. Potentially, each n
th

source in an extended-ERF can include a suite of rupture variations, representing the aleatory
variability in the source space-time functions.
In PSHA, following Field et al. (2005) and Wang and Jordan (2014), the exceedance
probability for an intensity measure im at site r in time t is given by

𝑃 𝐼𝑀>𝑖𝑚 𝑟,𝑡 =

1− 1−𝑝 𝑘 𝑡 𝑝 𝑥 𝑘 𝑝 𝑓 𝑘,𝑥
!∈!
!,!
𝑃 𝐼𝑀>𝑖𝑚 𝑘,𝑟,𝑥,𝑓
!∈!
!
!∈!

where K, X
k
and F
k,x
are the complete sets of, respectively, source and rupture variations,
hypocenters for the k
th
source-rupture variation, and source space-time functions for the k
th

rupture and x
th
hypocenter location, respectively.
Notice that in this formulation, the hypocenter distribution plays a role in three separate
probabilities: the probability of the hypocenter location given the k
th
rupture (what we call the

5

“conditional hypocenter distribution” or CHD), the probability of the source space-time function
given the k
th
rupture, and in the conditional probability of exceedance (determined by the
GMPEs). The choice of CHD; therefore, is non-trivial. Often, the CHD is assumed to be
uniform. Depending on the true shape of the CHD, this could either over-predict or under-predict
the directivity correction factor for the GMPEs. Wang (2013) showed that changes to the CHD
could alter the exceedance probabilities up to a factor of 2 compared to the uniform assumption.
The CHD can be found by examining the location of the hypocenter with respect to the
ends of the rupture (e.g., Mai, 2005 and qualitatively in Henry and Das, 2001). For ruptures with
complex temporal histories, this may not be an adequate descriptor of the directivity. For
example, a rupture that begins at one end, ruptures to an asperity or geometric complexity, and
then propagates back toward the hypocenter will radiate energy more like a bilateral rupture. In
this case, its hypocenter is a poor proxy for its directivity effect. We therefore, calculate an
“apparent hypocenter” from the directivity parameter obtained from the finite moment tensor
(FMT), as defined by Chen et al. (2005).
For rupture models in which the rupture velocity and rise time depend only on the local
slip, the distribution of the directivity parameter D, defined in terms of the degree-two
polynomial moments of the source space-time function, completely specifies the CHD. For more
complex rupture models, the directivity parameter D specifies an “apparent hypocenter” 𝑥
distribution that better characterizes the directivity (details of this are reserved for a later section,
Chapter 4). The GMPEs cannot account for complex rupture behavior and therefore may
incorrectly predict ground motions for complex sources. We therefore use the apparent CHD,
which better accounts for propagation complexities. For a simple rupture with uniform slip and

6

constant rupture velocity, 𝑝(𝑥|𝑘)=𝑝(𝑥|𝑘). We then amend the equation above, substituting the
apparent CHD as follows:

𝑃 𝐼𝑀>𝑖𝑚 𝑟,𝑡 =

1− 1−𝑝 𝑘 𝑡 𝑝 𝑥 𝑘 𝑝 𝑓 𝑘,𝑥
!∈!
(!,!)
𝑃 𝐼𝑀>𝑖𝑚 𝑟,𝑘,𝑥,𝑓
!∈!
!
!∈!

(X)

where the apparent CHD, 𝑝 𝑥 𝑘 , is derived from the observed directivity distribution as
described in Section 4.2.
1.3 RUPTURE DIRECTIVITY AND THE FINITE MOMENT TENSOR (FMT)
There are multiple ways in which directivity can be manifested; therefore, there are
several methods for studying directivity. For example, high directivity events are expected to
have strong asymmetry in their radiation patterns, both in terms of amplitude and duration (Ben-
Menahem, 1961). Directivity can therefore be studied by examining the azimuthal variation in
ground motion about the hypocenter (e.g., Boatwright and Boore, 1982; Ammon et al., 1993;
Miyake et al., 2001; Calderoni et al., in review). A unilateral rupture sends a strong pulse of
energy in one direction, perturbing the stress field to a higher degree in the forward direction;
therefore, some studies have examined the asymmetry of aftershock patterns (e.g., Gomberg et
al., 2003; Zaliapin and Ben-Zion, 2011). If a rupture has a preferred propagation direction, as
expected in the case of bi-material faults (Ben-Zion and Andrews, 1998), after repeated events
on the same fault, one might expect to observe an asymmetry in the damage pattern across the
fault (Dor et al., 2006a; Dor et al., 2006b; Dor et al., 2008).
In this study, we are concerned with the overall character of directivity of a rupture, and
follow McGuire et al. (2002) in using the directivity parameter. The directivity parameter D is
the magnitude of a vector that tracks the spatiotemporal moment release during a rupture,

7

defined by the degree-two polynomial moments of the source space-time function 𝑓 𝒓,𝑡 . The
formulation and behavior of the FMT is presented in Chapter 2.
1.4 MAPPING FROM DIRECTIVITY PARAMETER TO APPARENT CHD
To map the directivity parameter D into an apparent hypocenter location 𝑥, we must
know the relationship between hypocenter and directivity. For example, for a simple 1D rupture
in a homogeneous medium with uniform slip rate and constant rupture velocity, we know that
this relationship depends on the rupture velocity, rise time, and slip distribution. With this
relationship, we can then ask the following question to obtain the apparent hypocenter:
Given a directivity parameter D obtained from an arbitrarily complex source
space-time function, what is the hypocenter location for a simple one-dimensional
rupture that would produce the same directivity parameter D?
The remaining chapters will follow the mapping process that relates the directivity
parameter D to an apparent hypocenter location in the following way:
Chapter 2: In the mapping process, we have taken a 2D fault and approximated it as a
simplified 1D source with constant rupture velocity, uniform slip rate, and
constant rise time. Chapter 2 will present the formulation of the FMT and
the behavior of the directivity vector in both 1 and 2 dimensions to show
the possible effect aspect ratio may have on the mapping process.
Chapter 3: In order to determine the relationship between hypocenter location and
directivity for a simple rupture, we must assume a rupture velocity and
slip distribution. Chapter 2 will step through the process for determining
these parameters and examine the properties of the along-strike
displacement profiles.

8

Chapter 4: Any new procedure needs to be investigated using known models. Chapter
4 will use two earthquake rupture simulators with different underlying
assumptions to explore the mapping process and serve as proof-of-concept
studies.
Chapter 5: The concepts set up in Chapters 1-4 are culminated into the final results
from a set of finite fault inversion for observed earthquakes. Chapter 5
will present the apparent CHD for a global set of large, observed
earthquakes and the influence of this CHD on hazard in the CyberShake
(Graves et al., 2011) region of Southern California.
1.5 DATA SETS
There are three basic datasets we take advantage of in this study: (1) Published finite fault
inversions for large, observed earthquakes, and two sets of simulation data from (2) CyberShake
(Graves et al., 2011), and (3) RSQSim (Dieterich and Richards-Dinger, 2010; Richards-Dinger
and Dieterich, 2012). These data will be referred to throughout the dissertation and so I will
outline the basic information about these datasets prior to the proceeding chapters.
1.5.1 Published Finite Fault Inversions
This study uses a collection of 86 finite-fault inversion models for global set of 64 M
W
≥
7 earthquakes gathered from the SRCMOD database (Mai, 2004; Mai and Thingbaijam, 2014),
the USGS website, personal communication with Dr. Gavin Hayes of the USGS Fast Finite-Fault
Project, and personal communication with several other authors (Figure 1.1). Within the set of all
possible published finite fault inversions, there exists a high level of variation. For example, the
studies can use different data, solution constraints, parameter spaces, assumptions, and
resolutions. In our selection criteria, we chose only events that use seismic data in their

9

inversions so that they contain complete time histories (so that directivity can be considered in
the analysis). The details of the chosen finite-fault inversions are given in Table 1.1. The dataset
consists of 21 strike-slip, 32 reverse, 7 normal, and 4 oblique-slip events. Of the 21 strike-slip
events, 10 are continental events (marked “S-CT” in Table 1.1), which are of particular interest
as we look to apply the results of this and further analysis to southern California.

Figure 1.1: Map of Global CMT solutions (Dziewonski et al., 1981; Ekström et al., 2012) for the
published finite source inversions listed in Table 1.1.
0˚ 45˚ 90˚ 135˚ 180˚ -135˚ -90˚ -45˚ 0˚
-80˚
-70˚
-60˚
-50˚
-40˚
-30˚
-20˚
-10˚
0˚
10˚
20˚
30˚
40˚
50˚
60˚
70˚
80˚

10

Table 1.1: Finite Source Inversions
Location YYYY-MM-DD M
W
Mechanism Reference/Source
near the west coast of Honshu, Japan 1948-06-28 7.0 S-CM Ichinose et al. (2005) - SRCMOD
near the south coast of Honshu, Japan 1974-05-08 7.0 S-CM Takeo (1990) - SRCMOD
Southern California (Landers) 1992-06-28 7.3 S-CT
Cotton and Campillo (1995) - SRCMOD
Hernandez et al. (1999) - SRCMOD
near the coast of central Peru 1996-11-12 7.7 O-R Salichon et al. (2003) - SRCMOD
western Turkey (Izmit) 1999-08-17 7.6 S-CT
Bouchon et al. (2002) - SRCMOD
Delouis et al. (2002) - SRCMOD
Sekiguchi and Iwata (2002) - SRCMOD
Taiwan (ChiChi) 1999-09-20 7.7 R
Iwata et al. (2000) - SRCMOD
Wu et al. (2001) - SRCMOD
Oaxaca, Mexico 1999-09-30 7.5 N Hernandez et al. (2001) - SRCMOD
Southern California (Hector Mine) 1999-10-16 7.1 S-CT
Ji et al. (2002) - SRCMOD
Salichon et al. (2004) - SRCMOD
western Turkey (Duzce) 1999-11-12 7.2 S-CT Delouis et al. (2004) - SRCMOD
Gujarat, India 2001-01-26 7.7 R Yagi (2003) - SRCMOD
near the coast of southern Peru 2001-06-23 8.4 R Hayes (USGS) - personal communication
southern Qinghai, China (Kunlun) 2001-11-14 7.8 S-CT
Wen et al. (2009) - personal communication
Hayes (USGS) - personal communication
Central Alaska (Denali) 2002-11-03 7.9 S-CT Hayes (USGS) - personal communication
offshore Colima, Mexico 2003-01-22 7.6 R Yagi et al. (2004) - SRCMOD
Carlsberg Ridge 2003-07-15 7.5 S-OT Wei (Caltech, Carlsberg Ridge 2003) - SRCMOD
Hokkaido, Japan region 2003-09-25 8.3 R Hayes (USGS) - personal communication
near the south coast of Papua, Indonesia 2004-02-07 7.3 S-CT Wei (Caltech, Irian Jaya, Indonesia 2004)
off the west coast of northern Sumatra 2004-12-26 9.1 R
Ammon et al. (2005) - SRCMOD
Rhie et al. (2007) - SRCMOD
Rhie et al. (2007) - SRCMOD
northern Sumatra, Indonesia 2005-03-28 8.6 R Hayes (USGS) - personal communication
south of Java, Indonesia 2006-07-17 7.7 R Yagi and Fukahata (2011) - SRCMOD
Kuril Islands 2006-11-15 8.3 R Hayes (USGS) - personal communication
Solomon Islands 2007-04-01 8.1 R Hayes (USGS) - personal communication
near the coast of central Peru 2007-08-15 8.0 R Hayes (USGS) - personal communication
southern Sumatra, Indonesia 2007-09-12 8.5 R Hayes (USGS) - personal communication

11

(continued) Location YYYY-MM-DD M
W
Mechanism Reference/Source
Kepulauan Mentawai region, Indonesia 2007-09-12 7.9 R Hayes (USGS) - personal communication
eastern Sichuan, China (Wenchuan) 2008-05-12 7.9 O-R Yagi et al. (2012) - SRCMOD
offshore Bio-Bio, Chile (Bio-Bio) 2010-02-27 8.8 R
Delouis at al. (2010) - SRCMOD
Shao et al. (UCSB, Maule 2010) - SRCMOD
Hayes (USGS) - personal communication
Baja California, Mexico (El Mayor-Cucapah) 2010-04-04 7.2 S-CT Wei et al. (2011) - SRCMOD
South Island of New Zealand 2010-09-03 7.0 S-CT Hayes (USGS, ID: us2010atbj) - USGS website
Kepulauan Mentawai region, Indonesia 2010-10-25 7.8 R Hayes (USGS, ID: usa00043nx) - USGS website
Bonin Islands, Japan region 2010-12-21 7.4 O-N
Hayes (USGS, ID: usc0000rxc)a - USGS website
Hayes (USGS, ID: usc0000rxc)b - USGS website
Vanuatu region 2010-12-25 7.3 N Hayes (USGS, ID: usc0000usf) - USGS website
southwestern Pakistan 2011-01-18 7.2 N
Hayes (USGS, ID: us2011ggbx)a - USGS website
Hayes (USGS, ID: us2011ggbx)b - USGS website
near the east coast of Honshu, Japan 2011-03-09 7.3 R Hayes (USGS, ID: usb0001r57) - USGS website
near the east coast of Honshu, Japan (Tohoku) 2011-03-11 9.0 R
Yagi and Fukahata (2011)a - SRCMOD
Wei et al. (2012) - SRCMOD
Hayes (USGS, ID: usc0001xgp) - USGS website
Hayes (USGS) - personal communication
Ammon et al. (2011) - personal communication
Ide and Beroza (2011) - personal communication
Shao et al. (UCSB, Honshu 2011)
Kermadec Islands region 2011-07-06 7.6 N Hayes (USGS, ID: usc0004pbm) - USGS website
Vanuatu 2011-08-20 7.1 R Hayes (USGS, ID: usc0005h9f) - USGS website
eastern Turkey 2011-10-23 7.1 O-R Hayes (USGS, ID: usb0006bqc) - USGS website
off the west coast of northern Sumatra 2012-01-10 7.2 S-OT Hayes (USGS, ID: usc0007ir5) - USGS website
Oaxaca, Mexico 2012-03-20 7.4 R Hayes (USGS, ID: usc0008m6h) - USGS website
off the west coast of northern Sumatra 2012-04-11 8.6 S-OT
Wei (Caltech, Sumatra 2012) - SRCMOD
Hayes (USGS, ID: usc000905e) - USGS website
off the coast of El Salvador 2012-08-27 7.3 R Hayes (USGS, ID: usc000c7yw) - USGS website
Philippine Islands region 2012-08-31 7.6 R Hayes (USGS, ID: usc000cc5m) - USGS website
Costa Rica 2012-09-05 7.6 R Hayes (USGS, ID: usc000cfsd) - USGS website
Haida Gwaii, Canada 2012-10-28 7.8 R
Wei (Caltech, Masset, Canada 2012) - SRCMOD
Hayes (USGS) - personal communication

12

(continued) Location YYYY-MM-DD M
W
Mechanism Reference/Source
offshore Guatemala 2012-11-07 7.4 R Hayes (USGS, ID: usb000dlwm) - USGS website
Southeastern Alaska 2013-01-05 7.5 S-CO Hayes (USGS, ID: usc000ejqv) - USGS website
76km W of Lata, Solomon Islands 2013-02-06 8.0 R Hayes (USGS, ID: usc000f1s0) - USGS website
31km SE of Lata, Solomon Islands 2013-02-08 7.1 S-OT Hayes (USGS, ID: usc000f40j) - USGS website
83km E of Khash, Iran 2013-04-16 7.7 N
Wei (Caltech, Khash, Iran 2013) - SRCMOD
Hayes (USGS, ID: usb000g7x7) - USGS website
218km SSE of Bristol Island, South Sandwich Islands 2013-07-15 7.3 S-OT Hayes (USGS, ID: usb000ief9) - USGS website
61km NNE of Awaran, Pakistan 2013-09-24 7.7 S-CT Hayes (USGS, ID: usb000jyiv) - USGS website
4km SE of Sagbayan, Philippines 2013-10-15 7.1 R Hayes (USGS, ID: usb000kdb4) - USGS website
Off the east coast of Honshu, Japan 2013-10-25 7.1 N Hayes (USGS, ID: usc000kn4n) - USGS website
Scotia Sea 2013-11-17 7.7 S-OT Hayes (USGS, ID: usb000l0gq) - USGS website
95km NW of Iquique, Chile 2014-04-01 8.2 R Hayes (USGS, ID: usc000nzvd) - USGS website
49km SW of Iquique, Chile 2014-04-03 7.7 R Hayes (USGS, ID: usc000p27i) - USGS website
93km SSE of Kirakira, Solomon Islands 2014-04-12 7.6 S-OT Hayes (USGS, ID: usc000phx5) - USGS website
112km S of Kirakira, Solomon Islands 2014-04-13 7.4 R Hayes (USGS, ID: usc000piqj) - USGS website
Guerrero, Mexico 2014-04-18 7.2 R Hayes (USGS, ID: usb000pq41) - USGS website
70km SW of Panguna, Papua New Guinea 2014-04-19 7.5 R Hayes (USGS, ID: usb000pr89) - USGS website
19km SE of Little Sitkin Island, Alaska 2014-06-23 7.9 S-CM Hayes (USGS, ID: usc000rki5) - USGS website
Southern East Pacific Rise 2014-10-09 7.1 R Hayes (USGS, ID: usb000sk6k) - USGS website
67km WSW of Jiquilillo, Nicaragua 2014-10-14 7.3 N
Hayes (USGS, ID: usb000slwn)a - USGS website
Hayes (USGS, ID: usb000slwn)b - USGS website
N – normal, O-N – Oblique Normal, O-R – Oblique Reverse, R – reverse, S-CM – Strike-Slip (Continental Margin), S-CT – Strike-Slip (Continental), S-
OT – Strike-Slip (Oceanic Transform).

13

1.5.2 CyberShake
The CyberShake (Graves et al., 2011) catalog used in this study is based on extending the
Uniform California Rupture Forecast, Version 2 (UCERF2, Field et al., 2009). CyberShake
begins with UCERF2 sources and their respective ruptures. CyberShake then creates rupture
variations with different slip distributions and hypocenter locations, representing the aleatory
uncertainty in these distributions. Spatial slip distributions are generated independently from the
hypocenter locations (which are placed every 20km along the fault, Graves et al., 2011)
according to the stochastic methodology of Graves and Pitarka (2010) where the slip is generated
to match the observed wavenumber fall off in the amplitude spectrum of Mai and Beroza (2002).
The associated time function is based on a prescribed hypocenter and perturbations to a
background rupture velocity so that the rupture velocity increases in areas of higher slip and
slows down in areas of lower slip (Graves and Pitarka, 2010). CyberShake is a pulse-like model
in that each grid is constrained to rupture only once. Each grid’s rise time is a function of its slip
and wavenumber.
The CyberShake simulated dataset consists of 17 southern California ruptures within the
magnitude range 7 to 8.2, of which 12 are strike-slip and 5 are reverse ruptures. We consider a
“rupture” to be unique section of a fault, and a “rupture variation” to be the different slip and
hypocenter distributions on that unique section of fault. For the 17 ruptures in this dataset, there
are a total of 5010 rupture variations (4917 of those are for the strike-slip ruptures, leaving 93
reverse rupture variations). The details of the selected ruptures are presented in Table 1.3. We
consider each rupture variation to be independent of one another.

14

Table 1.2: CyberShake Rupture Section Abbreviations
Fault Abbreviation Section
Elsinore CM Coyote Mountain
Elsinore GI Glen Ivy
Elsinore J Julian
Elsinore T Temecula
Elsinore W Whittier
S. San Andreas BB Big Bend
S. San Andreas BG Banning/Garnet Hill
S. San Andreas CC Carrizo
S. San Andreas CH Cholame
S. San Andreas CO Coachella
S. San Andreas NM North Mojave
S. San Andreas NSB North San Bernardino
S. San Andreas PK Parkfield
S. San Andreas SM South Mojave
S. San Andreas SSB South San Bernardino
San Jacinto A Anza-Clark
San Jacinto CC Coyote Creek
San Jacinto SBV San Bernardino Valley
San Jacinto SJV San Jacinto Valley

15

Table 1.3: CyberShake Rupture Set
Primary Fault (Source ID) Rupture Section(s)* (Rupture ID) M
W
Mechanism
Number of Rupture
Variations
Elsinore (10) GI+T+J+CM (3) 7.8 S-CT 200
Elsinore (15) T+J+CM (3) 7.7 S-CT 162
Elsinore (16) W (3) 7.0 S-CT 18
S. San Andreas (64) CH+CC+BB+NM+SM (4) 8.0 S-CT 512
S. San Andreas (85) PK+CH+CC+BB+NM+SM (1) 7.7 S-CT 648
S. San Andreas (87) PK+CH+CC+BB+NM+SM+NSB+SSB (3) 8.0 S-CT 882
S. San Andreas (89) PK+CH+CC+BB+NM+SM+NSB+SSB+BG+CO (4) 8.2 S-CT 1568
San Jacinto (112) SBV+SJV+A+CC (5) 8.0 S-CT 162
Cucamonga (149) (9) 7.0 R 8
Newport Inglewood Connected (219) Alt 2 (264) 7.5 S-CT 242
Palos Verdes (231) (85) 7.2 S-CT 50
Palos Verdes Connected (232) (456) 7.7 S-CT 450
San Cayetano (254) (27) 7.2 R 18
San Gabriel 0 (255) (61) 7.3 S-CT 32
Santa Susana (267) (12) 7.1 R 8
Sierra Madre (271) (41) 7.2 R 18
Sierra Madre Connected (273) (66) 7.3 R
32
*see Table
1.2 for rupture section abbreviations.

16

1.5.3 RSQSim
RSQSim (Dieterich and Richards-Dinger, 2010; Richards-Dinger and Dieterich, 2012) is
a quasi-dynamic rupture simulator based on the laws of rate-and-state friction (e.g., Dieterich,
1979) so that spatial and temporal slip histories are dictated by friction laws. An initial stress
state is established for each fault element by prescribing the frictional resistance, normal stress,
slip velocity, and the rate-and-state “state variable”. An element slips when the stress overcomes
the steady-state friction and slip velocity reaches a pre-defined seismic slip speed. During slip,
stress is transferred to surrounding elements and slip stops once the stress decreases back below
the steady state friction. In this way, elements trigger one another, and an element can rupture as
many times as the stress state for that element allows, which can produce considerably more
complex spatiotemporal histories than those of CyberShake. The details of RSQSim rupture
generation can be found in Dieterich and Richards-Dinger (2010, 2012).
The RSQSim simulated dataset consists of 1000 randomly-selected ruptures from a
million year synthetic catalog within the magnitude range 7 to 8. Of the 1000 RSQSim ruptures,
940 are geometrically suitable for this study. Of those remaining 940, 907 are strike-slip and 33
are reverse. The ruptures cover 100 segments of faults throughout California, the details of which
cannot be covered in a simple table, but are available upon request.

17

2Chapter 2 The Finite Moment Tensor and Directivity
Vector
2.1 INTRODUCTION
Traditionally, directivity is described as a site-specific term and the overall directivity of
the source is not quantified; rather, terms such as “unilateral” and “bilateral” are used to describe
the source. The degree to which a rupture is uni- or bi-lateral is usually based on the location of
the epicenter with respect to the ends of the rupture; however, as will be shown in Chapter 4, this
is a poor descriptor when the rupture is complex. The directivity vector of the Finite Moment
Tensor (FMT) is a much better quantitative description of the overall directivity of the source.
Properly characterizing the directivity of a source can improve our understanding of seismic
hazard, rupture dynamics, and applications such as aftershock forecasting (i.e., bilateral and
unilateral ruptures are expected to have different aftershock patterns, e.g., Rubin and Gillard,
2000; Gomberg et al., 2003). This chapter will cover the formulation of the FMT and the
behavior of the directivity vector in both one and two dimensions.
2.2 FINITE MOMENT TENSOR (FMT)
The low-order polynomial moments of the stress glut, which describe the non-elastic
deformation in the source volume, can be used to represent a seismic source (Backus and
Mulcahy, 1976). The stress glut field 𝚪 𝒓,𝑡 describes the inelastic deformation in the source
region. The integral of 𝚪 𝒓,𝑡 is the zeroth-order moment tensor:
𝐌
!
= 𝚪 𝒓,𝑡 dV 𝒓 dt.
For simplification, we neglect the limits of integration since the stress glut is defined to be zero
outside the source volume and rupture duration.

18

Following McGuire et al. (2001), the scalar source space-time function 𝑓 𝒓,𝑡 can be
defined as the projection of the stress glut field 𝚪 𝒓,𝑡 onto its unit zeroth-order tensor 𝐌
!

𝑓 𝒓,𝑡 =𝚪 𝒓,𝑡 :𝐌
!
where 𝐌
!
= 𝚪 𝒓,𝑡 dV 𝒓 dt 2𝑀
!
(2.1)
where M
0
is the scalar seismic moment (i.e., the magnitude of the zeroth-order moment tensor
where 𝑀
!
=
!
!
𝐌
𝟎
:𝐌
𝟎
) and the colon denotes the tensor (or “double”) dot product such that
𝐀:𝐁=𝐀
!"
𝐁
!"
.
The low-order polynomial moments of the source space-time function 𝑓 𝒓,𝑡 make up
the finite moment tensor (FMT; Chen et al., 2005), a compact finite-source representation which
expands on the more well-known centroid moment tensor (Dziewonski et al., 1981) by including
second-order polynomial moments of the source space-time function. These additional moments
describe non-point-source effects such as rupture extent, orientation, and directivity, and from
them we can infer stress drop.
The centroid moment tensor (Dziewonski et al., 1981) contains the zero- and first-order
polynomial moments 𝜇
(!,!)
where p and q are the degrees of the spatial and temporal moments
respectively, and are defined as
𝜇
(!,!)
= 𝑓 𝒓,𝑡 dV 𝒓 dt= 2𝑀
!

𝒓
!
=𝝁
(!,!)
𝜇
(!,!)
= 𝑓 𝒓,𝑡 𝒓dV 𝒓 dt 2𝑀
!

𝑡
!
= 𝜇
(!,!)
𝜇
(!,!)
= 𝑓 𝒓,𝑡 𝑡dV 𝒓 dt 2𝑀
!

where 𝒓
!
and 𝑡
!
are the spatial and temporal centroids respectively. The second-order centralized
polynomial moments are given by

19

𝜇
(!,!)
= 𝑓 𝒓,𝑡 (𝒓−𝒓
!
)
!
(𝑡− 𝑡
!
)
!
dV 𝒓 dt.

The directivity vector is specified by the second-order mixed space-time moment
𝝁
(!,!)
=𝝁
(!,!)
𝜇
(!,!)
, whose magnitude is the directivity parameter D, normalized to be unity for
a perfectly unilateral rupture with instantaneous slip and is 0 for a perfectly bilateral rupture. It is
given by
𝐷= 𝒅 =
𝝁
(!,!)
𝜇
(!,!)
𝐿
!
𝑇
!
0≤𝐷≤ 1 (2.2)
where L
c
and T
c
are the characteristic length and duration defined as
𝐿
!
=max 2 eig 𝛍
!,!

𝑇
!
= 2 𝜇
!,!
.
We define a 1D rupture model as a simple propagating dislocation whose width is
negligible in a homogeneous medium with constant rupture velocity, uniform slip, and a
constant, finite rise time τ (i.e., analogous to the Haskell, 1964 source). For this 1D simplified
rupture, 𝐿
!
= 𝐿 3 and 𝑇
!
= 𝑇
!
+𝜏
!
3, where L is the total length of the fault and T is the
total duration.
As defined above, the directivity vector d is a three-component vector oriented in the
geographic coordinate system chosen for the analysis. To make it more relevant, we rotate the
directivity vector into the eigensystem provided by the eigenvectors 𝒆
𝒊
of the second spatial
moment tensor 𝛍
!,!
so that the components of directivity are given in the principle axis
directions, and so that d
1
, d
2
, d
3
are ordered according to the eigenvalues 𝜆
!
≥ 𝜆
!
≥ 𝜆
!
.
𝑑
!
=𝒅 ∙𝒆
!

20

𝑑
!
=𝒅 ∙𝒆
!

𝑑
!
=𝒅 ∙𝒆
!

The third principle axis corresponds to the thickness of the source volume, which is zero
for a planar fault. For all ruptures used in this study, 𝑑
!
≈ 0. This study will therefore consider
only the first two components of the directivity vector. The magnitudes of the remaining two
directivity components can themselves be theoretically constrained.
Following (McGuire et al., 2002), we define the second moment tensor as
𝛍
(!,!)
𝝁
(!,!)
𝝁
(!,!)
!
𝜇
(!,!)
.

Rotating to the principle axis coordinate system and assuming the simplified 1D rupture defined
earlier, the second moment tensor becomes

1
2
𝐿
!
!
0 𝜇
!
(!,!)
0 𝑊
!
!
𝜇
!
(!,!)
𝜇
!
(!,!)
𝜇
!
(!,!)
𝑇
!
!
.

We then normalized by 1 2𝐿
!
!
and 1 2𝑇
!
!
to get
1 0 𝑑
!
0 𝑎
!!
𝑑
!
𝑑
!
𝑑
!
1

where a is the aspect ratio (i.e., a = L/W). Given that the second moment tensor is positive
definite, we place a positivity constraint on the determinant of its principle minors to get the
magnitude bounds on d
1
and d
2
. For the first and second principle minors, we obtain:

det
1 𝑑
!
𝑑
!
1
≥ 0 ⇒ 1−𝑑
!
!
≥ 0 ⇒ −1≤𝑑
!
≤ 1
(2.3)

det
𝑎
!!
𝑑
!
𝑑
!
1
≥ 0 ⇒ 𝑎
!!
−𝑑
!
!
≥ 0 ⇒ −𝑎
!!
≤𝑑
!
≤𝑎
!!
.

21

From these limitations, we can see that as aspect ratio increases, 𝐷≈ 𝑑
!
(shown in more detail
in Section 2.5).
2.3 RAKE VARIATIONS AND THE FMT
The formulation presented in the previous section relies on the assumption in Equation
2.1 that the rupture being represented is sufficiently planar, i.e., the unit stress glut tensor is
constant over the rupture so that 𝚪 𝒓,𝑡 =𝑀(𝒓,𝑡)𝐌
!
, where 𝑀(𝒓,𝑡) are the scalar seismic
moments. We investigate the effect of this assumption in finite source inversions on the
calculations by considering the projection of the stress glut onto its zeroth-order moment tensor
(as shown above) and its projection onto an orthogonal tensor.
The stress glut field can be expressed as a field of moment tensors:
𝚪 𝒓,𝑡 =𝑀 𝒓,𝑡 𝒔 𝒓,𝑡 𝒏 𝒓,𝑡
𝐓
+𝒏 𝒓,𝑡 𝒔 𝒓,𝑡
𝐓

𝐌
!
=𝑀
!
(𝒔
!
𝒏
!
!
+𝒏
!
𝒔
!
!
)

𝒔
!
= 𝒔(𝒓,𝑡)d𝑉 𝒓 d𝑡
𝒔
!
=
𝒔
!
𝒔
!

𝒏
!
= 𝒏(𝒓,𝑡)d𝑉 𝒓 d𝑡
𝒏
!
=
𝒏
!
𝒏
!

where s and n are the slip and normal vectors respectively, and superscript T denotes the
transpose.
All of the analysis in this study has been done using the projection onto 𝐌
!
, which is
exact for constant slip and normal vectors; however, for ruptures with changes in fault geometry
and/or rake, we can define a tensor orthogonal to 𝐌
!
which will capture deviations from the
constant moment tensor assumption. In our primary dataset, the finite source inversions, the fault
geometries tend to be rather simple and a constant normal vector is appropriate for most cases;
however, the rake variations can be rather extreme. For this reason, we focus our orthogonal

22

tensor on capturing variability associated with changes in rake. Here, we define the vector 𝒃
!
as
the vector orthogonal to both the slip and normal vectors and 𝐌
!
as the orthogonal tensor

𝒃
!
=𝒏
!
×𝒔
!
𝐌
!
=𝑀
!
(𝒃
!
𝒏
!
!
+𝒏
!
𝒃
!
!
)
(2.4)
𝐌
!
:𝐌
!
= 0
𝐌
!
:𝐌
!
=𝐌
!
:𝐌
!
= 2𝑀
!
!

where × denotes the cross product. The definition of 𝒃
!
as the cross product of the normal and
slip vectors determines that 𝒃
!
is in the same plane as 𝒔
!
and rotated counterclockwise by π 2.
Using a rotation matrix ρ, we can define 𝒃
!
as
𝒃
!
=𝛒
π
2
𝒔
!
=
cos π 2 −sin π 2 0
sin π 2 cos π 2 0
0 0 1
𝒔
!
=
0 −1 0
1 0 0
0 0 1
𝒔
!
.
As mentioned above, we concentrate on rake variations in the finite source inversions and
therefore assume a constant normal vector so that an angle 𝜃
!
is all that is necessary to relate
𝒔(𝒓,𝑡) to 𝒔
!
. Therefore, any slip vector 𝒔(𝒓,𝑡) can be defined as 𝒔(𝒓,𝑡)= 𝒔
!
cos 𝜃(𝒓,𝑡) +
𝒃
!
sin 𝜃(𝒓,𝑡) , where 𝜃(𝒓,𝑡) is the angle between 𝒔
!
and 𝒔(𝒓,𝑡) measured counterclockwise.
The total stress glut is given by
𝚪 𝒓,𝑡 =𝐌
!
𝑓 𝒓,𝑡 +𝐌
!
𝑔 𝒓,𝑡 .
Let 𝑚(𝒓,𝑡)=𝑀(𝒓,𝑡) 𝑀
!
so that the normalized functions 𝑓(𝒓,𝑡) and 𝑔(𝒓,𝑡) are
𝑓 𝒓,𝑡 =𝑚(𝒓,𝑡)cos 𝜃(𝒓,𝑡)
and
𝑔 𝒓,𝑡 =𝑚(𝒓,𝑡)sin 𝜃(𝒓,𝑡) .
Assuming no backslip is permitted (a common requirement in finite source inversions),
−π 2≤𝜃(𝒓,𝑡)≤π 2 and the zeroth-order equations for 𝑓(𝒓,𝑡) and 𝑔(𝒓,𝑡) are then as
follows:

23

𝜇
(!,!)
= 𝑚 𝒓,𝑡 cos 𝜃 𝒓,𝑡 d𝑉 𝒓 d𝑡= 1
𝜈
(!,!)
= 𝑚 𝒓,𝑡 sin 𝜃 𝒓,𝑡 d𝑉 𝒓 d𝑡= 0.
The spatial centroid is found by minimizing the tensor
𝓜
(!,!)
= 𝚪 𝒓,𝑡 𝒓−𝒓
!
dV 𝒓 d𝑡
=𝐌
!
𝑚 𝒓,𝑡 cos 𝜃 𝒓,𝑡 𝒓d𝑉 𝒓 d𝑡−𝒓
!
+𝐌
!
𝑚 𝒓,𝑡 sin 𝜃 𝒓,𝑡 𝒓d𝑉 𝒓 d𝑡
and because 𝐌
!
and 𝐌
!
are orthogonal, the tensor 𝓜
(!,!)
is minimized when
𝒓
!
= 𝑚 𝒓,𝑡 cos 𝜃 𝒓,𝑡 𝒓d𝑉 𝒓 d𝑡.
This shows that variations in rake (i.e., 𝜃 𝒓,𝑡 ≠ 0), as is common in finite source inversions, do
not affect the spatial centroid. The same will hold for the temporal centroid. The first-degree
polynomial moment minimizes a manageable tensor quantity; however, as the degree increases,
the rank of the tensor also increases. This quickly becomes computationally inefficient. We
instead use the definitions of f(r,t) (McGuire et al., 2001) and g(r,t) to define a geometry ratio
that quantifies the complexity of the rupture. This ratio can then be used to filter out complex
ruptures for which the 𝑓 𝒓,𝑡 =𝚪 𝒓,𝑡 :𝐌
!
approximation may not hold. This geometry ratio is
defined as

𝜂=
𝑔 𝒓,𝑡 d𝑉 𝒓 d𝑡
𝑓 𝒓,𝑡 d𝑉 𝒓 d𝑡

(2.5)
and is 0 for a perfectly planar fault with no rake variation.

24

2.4 1D FINITE MOMENT TENSOR
As defined in Section 2.2, for the one-dimensional case, we consider a simple rupture in a
homogeneous medium with constant shear modulus 𝜇, uniform slip rate 𝑠, constant rupture
velocity v, and a constant rise time τ, and where the width of the fault, W is negligible. The
stress-glut rate of this rupture can be written as

𝛤 𝑥,𝑡 = 𝜇𝑠H 𝑡−
𝑥−𝑥
∗
𝑣
H 𝜏−
𝑥−𝑥
∗
𝑣
+ 𝑡 𝑥< 𝑥
∗

𝛤 𝑥,𝑡 = 𝜇𝑠H 𝑡−
𝑥
∗
−𝑥
𝑣
H 𝜏−
𝑥
∗
−𝑥
𝑣
+ 𝑡 𝑥> 𝑥
∗

where 𝑥
∗
is the nucleation point (i.e., the 1D “hypocenter”) and H is the Heaviside function. To
normalize the stress glut, we divide by its zeroth-order moment M
0
.

𝑀
!
= 𝛤 𝑥,𝑡 d𝑡d𝑥
!
!
!
∗
!
+ 𝛤 𝑥,𝑡 d𝑡d𝑥
!
!
!
!
∗
= 𝜇𝑠H 𝑡−
𝑥−𝑥
∗
𝑣
H 𝜏−
𝑥−𝑥
∗
𝑣
+ 𝑡 d𝑡d𝑥
!
!
!
∗
!
+ 𝜇𝑠H 𝑡−
𝑥
∗
−𝑥
𝑣
H 𝜏−
𝑥
∗
−𝑥
𝑣
+ 𝑡 d𝑡d𝑥
!
!
!
!
∗
= 𝜇𝑠𝜏𝐿

𝛾 𝑥,𝑡 =
𝛤(𝑥,𝑡)
𝑀
!
=
1
𝐿𝜏
H 𝑡−
𝑥−𝑥
∗
𝑣
H 𝜏−
𝑥−𝑥
∗
𝑣
+ 𝑡 𝑥< 𝑥
∗
1
𝐿𝜏
H 𝑡−
𝑥
∗
−𝑥
𝑣
H 𝜏−
𝑥
∗
−𝑥
𝑣
+ 𝑡 𝑥> 𝑥
∗

Using the notation from the previous sections, the normalized polynomial moments up to
degree two are as follows:

25

𝜁
(!,!)
= 𝛾 𝑥,𝑡 d𝑡d𝑥
!
!
!
∗
!
+ 𝛾 𝑥,𝑡 d𝑡d𝑥
!
!
!
!
∗
=
1
𝐿𝜏
H 𝑡−
𝑥−𝑥
∗
𝑣
H 𝜏−
𝑥−𝑥
∗
𝑣
+ 𝑡 d𝑡d𝑥
!
!
!
∗
!
+
1
𝐿𝜏
H 𝑡−
𝑥
∗
−𝑥
𝑣
H 𝜏−
𝑥
∗
−𝑥
𝑣
+ 𝑡 d𝑡d𝑥
!
!
!
!
∗
= 1

𝜁
(!,!)
=
1
2
𝐿 𝜁
(!,!)
𝑥
∗
=
1
𝑣
1
2
𝐿+
𝑥
∗
!
𝐿
−𝑥
∗
+
1
2
𝜏𝑣

𝜁
(!,!)
=
1
12
𝐿
!
𝜁
(!,!)
𝑥
∗
=
1
𝑣
!
1
12
𝐿
!
−𝑥
∗
!
𝑥
∗
𝐿
−1
!
+
1
12
𝜏𝑣
!

𝐿
!
=
1
3
𝐿
𝑇
!
𝑥
∗
=
2
𝑣
1
12
𝐿
!
−𝑥
∗
!
𝑥
∗
𝐿
−1
!
+
1
12
𝜏𝑣
!

𝜁
(!,!)
𝑥
∗
=
1
𝑣
1
12
𝐿
!
+𝑥
∗
!
𝑥
∗
3𝐿
−
1
2

𝑑 𝑥
∗
=
𝜁
(!,!)
𝑥
∗
𝜁
(!,!)
𝑥
∗
𝐿
!
𝑇
!
𝑥
∗
=
2 3
1
12
𝐿
!
+𝑥
∗
!
𝑥
∗
3𝐿
−
1
2
𝐿
1
12
𝐿
!
−𝑥
∗
!
𝑥
∗
𝐿
−1
!
+
1
12
𝜏𝑣
!
.
(2.6)
In this case, the directivity is a scalar rather than a vector, so that in the previous notation
from Section 2.2 (Equation 2.2), d = D. As shown in Equation 2.6 and Figure 2.1, d is a non-
linear function of 𝑥
∗
; therefore, the characterization of McGuire et al. (2002) that D > 0.5 is
unilateral, is not a good definition. A D-value for this 1D fault of 0.5 can correspond to an 𝑥
∗

within 10% of the center of the rupture, and an 𝑥
∗
that begins at 25% of the rupture length can
correspond to D > 0.9. It is important to note that with non-zero rise time τ, the normalization
yields D < 1, even for a perfectly unilateral rupture (i.e., d(0) < 1).

26

Figure 2.1: Analytic solution (Equation 2.4) for the relationship between nucleation and
directivity parameter for a simplified 1D source in a homogeneous medium with constant rupture
velocity and an instantaneous, uniform slip rate (zero rise time so that d(0) = 1 from Equation
2.6).
2.5 2D FINITE MOMENT TENSOR
A one-dimensional approximation holds for ruptures with large aspect ratios (a = L/W),
such as we expect with large strike-slip ruptures, whose downdip widths are limited by steep dips
and the thickness of the brittle crust. For the two-dimensional case of a perfectly planar fault, we
consider the same simplified rupture as the previous section, but introduce the thickness W and
change the notation from the scalar locations x to the 2D vector r and the stress glut can now be
written as
𝜞 𝒓,𝑡 = 𝜇𝑠H 𝑡−
𝒓−𝒓
∗
𝑣
H 𝜏−
𝒓−𝒓
∗
𝑣
+ 𝑡 .
where 𝒓
∗
= 𝑥
∗
,𝑦
∗
is the nucleation point, 𝒓−𝒓
∗
is therefore the distance from a point r to the
nucleation point. We solved analytic expressions for the 2D polynomial moments much like in
0 0.1 0.5 1
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1
Distance Along Fault
Directivity Parameter
0.2 0.3 0.4 0.6 0.7 0.8 0.9

27

the section above; however, due to the complex nature of the mixed space-time moment, we
were unable to solve for the 2D directivity vector. We instead, use numerical simulations to
establish the relationship between nucleation point and directivity. We use the 2D analytic
equations to verify the code and establish its numerical accuracy, as well as place constraints on
the size and behavior of the components of the directivity vector.
The components of the 2D FMT for small aspect ratios behave differently from those
with large aspect ratios. This is due to circular propagation effects (Figure 2.2 and Figure 2.3). A
rupture propagates as a circle until it encounters the edge of the rupture area and the time it takes
for this to occur depends on where the rupture nucleates with respect to the rupture area (the red
contours in Figure 2.2 and Figure 2.3). After this point, the rupture behavior changes. For
example, for a circular rupture with radius L that nucleates in the corner of the rupture area (as in
Figure 2.2a), the spatial and temporal centroids are

𝝁
(!,!)
=
4
3π
𝐿
4
3π
𝐿

𝜇
(!,!)
=
2
3
𝐿
𝑣

compared to

𝝁
(!,!)
=
1
2
𝐿
1
2
𝑊

𝜇
(!,!)
=
1
𝑣
1
4
𝐿
!
+
1
4
𝑊
!

expected for a simple rectangular model with zero rise time (i.e., τ = 0). In this case, the spatial
centroids for L = W are separated by more than 10% of the length. The characteristic dimensions,
which are more sensitive, will have even more significant differences (for example, the
characteristic duration increases by more than 42% for a square compared to a circle).

28

Additionally, in the case of a square (i.e., L = W), the eigenvalues and eigenvectors are non-
unique.
From this analysis, we know that the aspect ratio of a rupture and where it nucleates with
respect to the overall rupture area affects when (and if) the FMT begins to behave as we expect
from our analytic solutions. We can see this in Figure 2.2d and Figure 2.3d as the circular effects
fall off and aspect ratio continues to increase, the |d
2
| values decrease from their upper bounds at
a
-1
to become more like a
-2
(as expected from Equation 2.3). For the example of nucleation in the
corner of the rupture area, we find from numeric simulations that the effects of the circular
rupture fall off once the aspect ratio (a = L/W) of the rupture gets to be ~1.5 (see the solid black
line in Figure 2.2c). The distance the rupture front has to travel to overcome the circular effects
depends on where rupture nucleates within the rupture zone. Figure 2.3 shows the example of a
rupture that nucleates at the top center of the rupture area, and in this case, the circular effects
fall off once the aspect ratio gets to be ~2. Figure 2.4 shows results from numeric simulations for
the magnitudes of d
1
and d
2
for different aspect ratios. From these numeric simulations, we find
that for ruptures with a > ~3.5, then 𝐷≈ 𝑑
!
.

29

Figure 2.2: Circular effects on the directivity components for nucleation in the corner of the
rupture area. The rupture initially propagates as a circle (a), which has a different expected FMT
than a rectangle (e.g., see the comparison of equations at the beginning of this section), until it
reaches the edge of the smallest dimension. For small aspect ratio faults (b), this circular effect
can dominate the FMT components (note the behavior of the directivity components before the
red line in d). The evolution of the directivity components is shown in (d) with the red line
corresponding to the red lines in (a-c), before which circular effect dominate. The solid black
line corresponds to the solid line in (c), which represents the point at which the circular effects
fall off. The dashed lines mark the bounds of the expected behavior of |d
2
| for a rectangular
rupture, where a is the aspect ratio. Note that these calculations are done using zero rise time so
that |d
1
| = 1 for a perfectly unilateral rupture.
0 W/v1.5W/v √(L
2
+W
2
)/v
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time

|d
1
|
|d
2
|
a
-1
a
-2
W/v
W/v
1.5W/v
a.)
b.)
c.)
d.)

30

Figure 2.3: Circular effects on the directivity components for nucleation in the middle center of
the rupture area. This figure is directly analogous to Figure 2.2 and shows the aspect ratio
necessary to dissipate the circular effects depends on where the rupture nucleates (note that for
this nucleation point, |d
2
| falls within the expected range at W/v instead of 1.5W/v as in Figure
2.2). Note that these calculations are done using zero rise time so that |d
1
| = 1 for a perfectly
unilateral rupture.
0
W/v
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time

|d
1
|
|d
2
|
√(L
2
+W
2
)/v
a
-1
a
-2
a.)
b.)
c.)
d.)
W/v W/v
0.5W/v

31

Figure 2.4: Behavior of the directivity components with increasing aspect ratio. The left column
shows the magnitude of the directivity components (|d
1
| on top and |d
2
| on bottom) for each
hypocenter location on the fault for a rupture with aspect ratios a = 2 (a) and a = 4 (c). The |d
2
|
colorbar is limited to a. The right column shows the relative size of the directivity components
for the same two aspect ratios a = 2 (b) and a = 4 (d). Note that these calculations are done using
zero rise time so that d
1
= ±1 for a perfectly unilateral rupture.
2.6 SUMMARY
The directivity parameter of the finite moment tensor (FMT) is a low frequency source
property that is derived from the spatiotemporal evolution of a rupture. The stress glut tensor
field gives way to a more manageable scalar source space-time function when projected onto its
zeroth-order moment tensor (Equation 2.1). This projection assumes the fault geometry and slip
vectors do not change significantly across the rupture. In the case of finite source inversions,
most have simple fault geometries; however, the rake can vary significantly. We therefore
construct a tensor orthogonal to the zeroth-order moment tensor that captures the deviations in
rake (Equation 2.4) and determine that variations in rake do not affect the zeroth and first-order
−1 −0.5 0 0.5 1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
d
1
d
2

a
-1
-a
-1
a
-2
-a
-2
Location Down Dip

0 0.5 1
0
0.1
0.2
d
1
0
0.5
1
Location Along Strike

0 0.5 1
0
0.1
0.2
d
2
0
0.1
0.2
−1 −0.5 0 0.5 1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
d
1
d
2

a
-1
-a
-1
a
-2
-a
-2
Location Down Dip

0 0.5 1
0
0.25
0.5
d
1
0
0.5
1
Location Along Strike

0 0.5 1
d
2
0
a.)
b.)
c.)
d.)
0
0.25
0.5
0.25
0.5

32

moments. For the second-order moments, we use the relative magnitudes of the two projections
to construct a geometry ratio (Equation 2.5), which we use to identify and remove more complex
ruptures for which the constant moment tensor assumption may not hold.
We solve analytically for the components of the FMT for the simplified 1D rupture
(defined in Section 2.4) and determine the 1D relationship between nucleation point and
directivity parameter (Figure 2.1). In 2D we show the limits on the components of the directivity
vector (Equation 2.3) and how they relate to the aspect ratio of the fault (Figure 2.4). We find
that the aspect ratio and where the rupture nucleates with respect to the overall rupture area
determine at what aspect ratio the components of the FMT begin to behave like our analytic
solutions (Figure 2.2 and Figure 2.3).
For the 2D simplified rupture, the d
2
component of directivity is bounded by ± a
-2
so that
as aspect ratio increases, the directivity parameter (i.e., the magnitude of the directivity vector) is
dominated by d
1
and the rupture can be approximated as 1D. We use this 1D approximation in
the process of mapping the directivity parameter to apparent hypocenter location in Chapters 4
and 5. In Chapter 5, Section 5.6, we show that for aspect ratios greater than ~3.5 (i.e., when the
upper bound on d
2
< ~0.08), this approximation holds.

33

3Chapter 3 Displacement Along Strike
3.1 INTRODUCTION
In order to determine the apparent hypocenter, we develop a mapping from the directivity
parameter defined in the finite moment tensor to nucleation point along a 1D rupture. Once we
have a catalog of directivity parameters, we use a forward modeling approach to determine the
relationship between directivity parameter and nucleation point for a simplified 1D rupture in a
homogeneous medium with constant rupture velocity, uniform slip rate, and constant rise time.
In this forward calculation, in order to focus strictly on the relationship between directivity and
nucleation point in the 1D simulations, we must assume a spatial distribution of slip.
Previous studies have investigated the character of the displacement profile along a fault,
with different motivations. Some studies have taken on the subject in order to devise magnitude
estimates for paleoearthquakes (Hemphill-Haley and Weldon, 1999; Biasi and Weldon, 2006),
others have used the displacement profiles to characterize fault displacement hazard (Moss and
Ross, 2011; Petersen et al., 2011), Manighetti et al. (2005) investigate it as possible insight into
the mechanics of earthquake rupture, and others investigate the nature of slip distributions for the
purpose of improving ground motion modeling (e.g., Somerville et al., 1999b; Mai and Beroza,
2002; Graves and Pitarka, 2010). Earlier studies have hypothesized about the nature of the
cumulative displacement profile along a fault based on the characteristic earthquake model
(Schwartz and Coppersmith, 1984) and determined that the cumulative displacement along a
fault over time becomes uniform (Sieh, 1996; Tapponnier et al., 2001). Even with more than a
dozen studies, including the ones mentioned here, there is no clear consensus on the along-strike
slip distribution, ranging from a triangle (Manighetti et al., 2005) to a function that looks like the
square root of sine (Biasi and Weldon, 2006). Several of the studies conclude that slip for any

34

single event is fractal in nature (e.g, Somerville et al., 1999a; Mai and Beroza, 2002; Milliner et
al., in press). Most of the previous studies use geological surface slip measurements to construct
a displacement profile. In this study, we use finite source inversion results to determine an
average rupture displacement profile and fit a Beta distribution to this profile. We also
investigate the relationships among slip, slip centroids, hypocenter locations, and directivity.
3.2 DATA
In this study, we focus on the global M
W
≥ 7 dataset of 86 finite source inversion models
(Figure 1.1) described in Chapter 1 as well as cataloged in Table 1.1. This dataset contains
source inversions which use, at least in part, seismic data and consists of 21 strike-slip, 32
reverse, 7 normal, and 4 oblique-slip events. Of the 21 strike-slip events, 10 are continental
events (marked “S-CT” in Table 1.1).
We present the results for the two additional synthetic datasets (i.e., CyberShake and
RSQSim) for reference purposes for discussion in Chapter 4. These datasets are also described in
the Chapter 1. The CyberShake dataset is cataloged in Table 1.3, and consists of 5010 rupture
variations from 17 southern California ruptures with 12 of those ruptures being strike-slip (4470
rupture variations) and 5 are reverse (540 rupture variations). The RSQSim dataset contains 967
synthetic events spanning 104 rupture segments in southern California, and of these 940 events,
907 are strike-slip and the remaining 33 are reverse.
3.3 CREATING 1D DISPLACEMENT PROFILES
Previous studies that use surface displacement data are already collected in 1D profiles
along strike (e.g., Hemphill-Haley and Weldon, 1999; Biasi and Weldon, 2006; Wesnousky,
2008); however, our dataset of rupture models is 2D. We therefore project the displacement onto

35

a line along the strike given by the finite moment tensor (FMT, see Chapter 2 for equations and
further explanations of the FMT).
Prior to creating the 1D displacement profiles, however, we must consider the dataset.
Finite source inversions tend to overestimate the fault dimensions (e.g., Somerville et al., 1999;
Mai and Beroza, 2000; Chiou et al., 2008) and so a 1D profile taken along the length of a finite
source inversion may yield artificially long profiles (e.g., Figure 3.1). In the inversions, the
dimensions are chosen to be at least large enough to resolve the slip distribution and the extra
fault space of zero slip is of little to no concern to the modelers, and they are therefore not often
concerned with trimming the dimensions. This typically leads to overestimates of the fault
dimensions with large areas of low or no slip, particularly along the edges of the fault. For this
reason, the finite source inversions should be trimmed to an effective rupture area (e.g., Mai and
Beroza, 2000; Chiou et al., 2008).
We choose the method used in the Next Generation Attenuation (NGA) project outlined
by Chiou et al. (2008) and implemented in more detail in Stewart et al. (2012) in which areas
along the edges of the rupture were trimmed back if their slip was less than 10-20% of the peak
slip (regardless of peak slip, areas with more than 50 cm of slip were not trimmed from the
models, following the recommendation of Chiou et al., 2008). For our dataset, we found no
significant difference between trimming using 10% versus 15%, and chose to use 10% to be
conservative.

36

Figure 3.1: The 2011 M
W
7.6 Kermadec Islands region finite source inversion slip (Hayes,
USGS website) overestimates the length of the rupture, creating a profile with an artificially long
displacement profile (a).

37

Figure 3.2: The same finite source inversion shown in Figure 3.1 trimmed according to the
methodology employed in the Next Generation Attenuation models by Chiou et al. (2008) (b).
The trimmed slip profile (a) eliminates the artificially long slip profile in Figure 3.1a, providing
a better estimate of the slip profile along strike for this rupture.
Once the finite source inversions are trimmed, the displacement profiles are stacked to
get an average displacement profile along strike. The inversions, however, vary widely in their
resolution. In order to stack the displacement profiles, we must normalize the lengths of the
profiles and resample them to a common increment. Previous studies have resampled the finite
source inversions themselves to a common resolution (typically 1 km by 1 km; i.e., Mai, 2005;
Manighetti et al., 2005); however, this approach is not helpful for stacking purposes once the
rupture lengths are normalized. Biasi and Weldon (2006) resampled the normalized distances to
1%. We found that this sampling rate for the finite source inversions may be too fine and
introduce too much additional slip in the interpolation (i.e., Manighetti et al., 2005). The finite
source inversions range in resolution from 0.65% (Bouchon et al., 2002) to 20% (Takeo, 1990),

38

and over 80% of the data fall within the resolution range 3% to 8% with mean and median values
of ~5.5% and ~5.3% respectively. We therefore resample the slip profiles at 5% increments of
the normalized lengths using cubic spline interpolation.
The RSQSim dataset shows similar resolution distribution with over 80% within the
resolution range 1.5% to 6% with mean and median values of ~3.7% and ~3.2% respectively.
We find that sampling at 5% for RSQSim ruptures is adequate. With its 1 km resolution and
large rupture areas (an order of magnitude larger than the RSQSim ruptures), the CyberShake
dataset is an order of magnitude finer in resolution than the other two datasets. Over 80% of the
CyberShake data have resolutions between 0.18% and 0.53% with a median value of ~0.29%.
CyberShake ruptures are very heterogeneous, showing a higher spatial frequency in their slip
distributions. In this study, we are interested in lower frequency properties of ruptures and a
resampling at 5% preserves the general shape of the profile while providing a smoothed, low
frequency view of CyberShake ruptures (Figure 3.3), more consistent with our low-order
moment analysis. When we downsample the higher resolution ruptures, we do not believe
aliasing to be a problem in the analysis as we are not interested in the spectral properties of the
slip distributions, nor do we need to retain information about the higher frequencies.

39

Figure 3.3: (a) CyberShake slip distribution for a M
W
8.2 southern San Andreas rupture
(UCERF2 Source ID: 89, Rupture ID: 4) with strike 307°, dip 84°, and rake 177°. Grey dot
marks the hypocenter, red triangle marks the spatial centroid, red arrow is the directivity vector,
black ellipse outlines the characteristic dimensions from the FMT, and rupture initiation times
are contoured in white. (b) 1D slip profile computed from (a) by summing up dip. (c) 1D slip
profile resampled at 5% resolution from the original slip profile.
Following Hemphill-Haley and Weldon (1999), we then stack the resampled
displacement profiles along strike, normalized by the average displacement. Each profile 𝑓 𝑥
can be written as a sum of an even and odd function such that
𝑓 𝑥 = 𝑓
!"!#
𝑥 +𝑓
!""
(𝑥) (3.1)

𝑓
!"!#
𝑥 =
1
2
𝑓 𝑥 +𝑓(−𝑥) 𝑓
!""
𝑥 =
1
2
𝑓 𝑥 −𝑓(−𝑥) .
The orientation of 𝑓 𝑥 is arbitrary and there is no proposed mechanism to determine to
which direction any asymmetry should be; therefore, we study only the symmetric displacement
profile 𝑓
!"!!
𝑥 and fit the running average displacement along strike of 𝑓
!"!#
𝑥 with a
normalized, symmetric Beta distribution.

40

3.4 THE BETA DISTRIBUTION
The Beta distribution is versatile in that it can take on a variety of different shapes,
including symmetric, asymmetric, concave, and convex functions (Figure 3.4). It is also
convenient for normalized data because it is defined on the interval from 0 to 1. It is written in
the following form:

𝐵 𝑥;𝛼,𝛽 =
𝑥
!!!
1−𝑥
!!!
𝑥
!
!!!
1−𝑥
! !!!
!
!
𝑑𝑥
!

𝛼,𝛽> 0
where α and β are shape parameters, and for a symmetric distribution, α = β. Of the many
functions a Beta distribution can represent (Figure 3.4), two that are of particular interest for slip
distributions in particular are the uniform distribution (α = β = 1), which is assumed in Chapter 0,
and α = β = 1.56 (Figure 3.5), which describes the functional form of the average displacement
profile determined in Biasi and Weldon (2006):
𝐷𝑖𝑠𝑝 𝑥 = 1.311 sin (𝜋𝑥)
as printed in Appendix F of the Uniform California Earthquake Rupture Forecast, Version 2
(UCERF2) Report (Field et al., 2013), where x is the normalized distance along the fault. The
displacement profile Disp(x) makes use of surface slip measurements for the 13 historical
earthquakes in Hemphill-Haley and Weldon (1999), most of which are strike-slip events (Biasi
and Weldon, 2006).

41

Figure 3.4: Examples of different Beta distributions B(x; α, β) with various α and β parameters.
The Beta distribution is able to represent the uniform distribution (i.e., α=β=1) as well as
concave, convex, asymptotic, symmetric, and asymmetric functions. Note: Figures that show
only the Beta distribution are plotted on the range from 0 to 1 to be consistent with the definition;
however, the Beta distribution is shifted to range between -0.5 and 0.5 when plotted with
displacement profiles.
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
2.5
3
x
B

α=0.5, β=0.5
α=3, β=1
α=1, β=1
α=2, β=2
α=2, β=3

42

Figure 3.5: Comparison of the Beta distribution least squares fit (blue dashed line) to the
displacement function of Biasi and Weldon (2006), (black solid line) and a plot of the residuals
where e = Disp(x) – B(x; α, β). Note: Figures that show only the Beta distribution are plotted on
the range from 0 to 1 to be consistent with the definition; however, the Beta distribution is
shifted to range between -0.5 and 0.5 when plotted with displacement profiles.
In each dataset, the ruptures do not necessary go to zero at the ends. In fact, in RSQSim
ruptures, elements with zero slip are not reported in the slip maps. Additionally, due to trimming
of the finite-source inversions, the edges of these displacement profiles are defined to be non-
zero at the ends and will be no less than 10% of the maximum slip or ≥ 50 cm slip. We therefore
also solve for an additional parameter ε, which is the offset of the data from the true ends of the
fault (Figure 3.6). We assume that the truncated displacement profiles are not dramatically
different in behavior near the ends than a full profile that ends with zero slip, i.e., the behavior of
α=β=1.56
1.311√(sin(πx))
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
Distance Along Strike
Normalized Displacement

0 0.1 0.2 0. 3 0.4 0. 5 0.6 0.7 0. 8 0.9 1
−0.02
−0.01
0
0.01
0.02
Distance Along Strike
Residual, e
Σe
2
= 0.0587

43

ruptures near the ends is smooth and we can therefore extrapolate the slip to zero using the best-
fitting Beta distribution. We then use the full Beta distribution, ending in zero slip, to forward
simulate the relationship between nucleation point and directivity parameter. This relationship
provides a mapping between the computed directivity parameters and nucleation along strike.
Figure 3.6: An example of the symmetric displacement profile for 28 finite source inversion
strike-slip events to visually define the parameter ε. The trimming process defines the ends of the
slip profiles as non-zero; however, we know that slip must go to zero at the ends of the rupture;
therefore, when we find the best-fitting Beta distribution, we add an additional parameter ε,
which is the distance from the end of the slip profile to where slip would go to zero given a Beta
distribution with parameters α and β. The average slip profile is shown as the bold black line
with the 0.1 and 0.9 quantiles as the shaded region between the dashed black lines, and the best-
fitting Beta distribution to this average displacement profile is shown as the dashed blue lines.
The red dashed lines show the distances ε
1
and ε
2
from the trimmed ends of the fault to where the
best-fitting Beta distribution goes to zero.
3.5 FINITE SOURCE INVERSION DISPLACEMENT PROFILES
We find that for the global set of finite source inversions listed in Table 1.1, the
distributions for strike-slip and dip-slip faulting are different. The dip-slip events tend to have a
more concentrated displacement profile, coincide more with results from Manighetti (2005),
while the strike-slip events have a slightly more flat-top distribution, more consistent with
Hemphill-Haley and Weldon (1999) or Biasi and Weldon (2006). All three of the previously-
mentioned studies used surface slip measurements. Manighetti (2005) used both surface slip
−0.5 0 0.5
0
1
2
3
Distance Along Strike
Normalized Displacement

ε
1
ε
2
Average Displacement
α=1.76, β=1.76

44

measurements and finite source inversions. Manighetti (2005) found that the stacked slip profile
along strike could be described by an asymmetric triangle. Note, however, that in their stacking
algorithm, they chose to align the profiles so that the apex of an asymmetrical profile was always
on the same side. Initially, we found no compelling underlying mechanism that would prescribe
to which side the asymmetry should be biased; therefore, we did not initially orient our rupture
profiles with any known bias. The centroid of our full, unfolded average displacement profile,
oriented along strike, is located at 0.004, indicating a symmetric profile. Nevertheless, we
explore three additional schemes for orienting rupture profile directions and compare them with
the centroid and hypocenter distributions; they are as follows:
(1) Symmetric: Displacement profiles are folded in half and averaged, representing the
even portion in Equation 3.1.
(2) Asymmetric: Oriented in the direction of maximum asymmetry; i.e., all centroids are
located in the positive range 0 to 0.5.
(3) Directivity: Oriented so that the directivity vector points in the same direction.
The results of these different orientations are shown in Figure 3.7 through Figure 3.15
and Table 3.4. We fit (1) using a symmetric Beta distribution (i.e., we require that α = β), and fit
(2) and (3) with an asymmetric Beta distribution. The symmetric profiles (1) allow us to compare
to previous surface-slip studies and clearly show the centroid bias in all three distributions (i.e.,
centroid locations, hypocenter locations, and the distance between the centroid and hypocenter),
but are uninformative for studying the relationship among slip, hypocenter, centroid, and
directivity as that information is lost in the symmetrization. The asymmetric distributions of (2)
and (3) should provide more insight into these relationships.

45

Figure 3.7: Symmetric normalized displacement profiles for all faulting types (86 finite source
inversions). The top shows each slip profile as colored symbols with the mean profile shows as a
solid black line, the 10% and 90% percentiles are shown as the light gray area bounded by a
black dashed line. The best-fitting Beta distribution is shown as the dashed blue line with its
parameters in the legend. The blue histograms show the centroid distribution, red histograms
show the hypocenter distribution, and the black histogram show the normalized distance between
the centroid and hypocenters (i.e., events on the tails of this distribution are expected to have
higher directivity parameters).
−0.5 0 0.5
0
5
10
15
20
25
0
1
2
3
4
Distance Along Fault

Average Displacement
α=1.92, β=1.92
−0.5 0 0.5
0
5
10
15
20
25
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.92, β=1.92
−1 −0.5 0 0.5 1
0
5
10
15
Centroid − Hypocenter

46

Figure 3.8: Symmetric normalized displacement profiles for dip-slip ruptures (58 finite source
inversions). Directly analogous to Figure 3.7.
−0.5 0 0.5
0
5
10
15
20
0
1
2
3
4
Distance Along Fault

Average Displacement
α=2.01, β=2.01
−0.5 0 0.5
0
5
10
15
20
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=2.01, β=2.01
−1 −0.5 0 0.5 1
0
5
10
Centroid − Hypocenter

47

Figure 3.9: Symmetric normalized displacement profiles for strike-slip ruptures (28 finite source
inversions). Directly analogous to Figure 3.7.
−0.5 0 0.5
0
5
10
0
1
2
3
4
Distance Along Fault

Average Displacement
α=1.76, β=1.76
−0.5 0 0.5
0
5
10
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.76, β=1.76
−1 −0.5 0 0.5 1
0
1
2
3
4
5
Centroid − Hypocenter

48

Figure 3.10: Asymmetric normalized displacement profiles for all faulting types (86 finite
source inversions). Analogous to Figure 3.7.
−0.5 0 0.5
0
5
10
15
20
25
0
1
2
3
4
Distance Along Fault

Average Displacement
α=2.34, β=1.95
−0.5 0 0.5
0
5
10
15
20
25
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=2.34, β=1.95
−1 −0.5 0 0.5 1
0
5
10
15
Centroid − Hypocenter

49

Figure 3.11: Asymmetric normalized displacement profiles for dip-slip ruptures (58 finite source
inversions). Analogous to Figure 3.7.
−0.5 0 0.5
0
5
10
15
20
0
1
2
3
4
Distance Along Fault

Average Displacement
α=2.47, β=2.06
−0.5 0 0.5
0
5
10
15
20
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=2.47, β=2.06
−1 −0.5 0 0.5 1
0
5
10
Centroid − Hypocenter

50

Figure 3.12: Asymmetric normalized displacement profiles for strike-slip ruptures (28 finite
source inversions). Analogous to Figure 3.7.
−0.5 0 0.5
0
5
10
0
1
2
3
4
Distance Along Fault

Average Displacement
α=2.13, β=1.71
−0.5 0 0.5
0
5
10
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=2.13, β=1.71
−1 −0.5 0 0.5 1
0
1
2
3
4
5
Centroid − Hypocenter

51

Figure 3.13: Normalized displacement profiles for all faulting types oriented so that each
profile’s directivity vector points toward the right (86 finite source inversions). Analogous to
Figure 3.7.
−0.5 0 0.5
0
5
10
15
20
25
0
1
2
3
4
Oriented for Directivity →
Distance Along Fault

Average Displacement
α=2, β=1.89
−0.5 0 0.5
0
5
10
15
20
25
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=2, β=1.89
−1 −0.5 0 0.5 1
0
5
10
15
Centroid − Hypocenter

52

Figure 3.14: Normalized displacement profiles for dip-slip ruptures oriented so that each
profile’s directivity vector points toward the right (58 finite source inversions). Analogous to
Figure 3.7.
−0.5 0 0.5
0
5
10
15
20
0
1
2
3
4
Oriented for Directivity →
Distance Along Fault

Average Displacement
α=2.14, β=1.97
−0.5 0 0.5
0
5
10
15
20
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=2.14, β=1.97
−1 −0.5 0 0.5 1
0
5
10
Centroid − Hypocenter

53

Figure 3.15: Normalized displacement profiles for strike-slip ruptures oriented so that each
profile’s directivity vector points toward the right. (28 finite source inversions) Analogous to
Figure 3.7.

−0.5 0 0.5
0
5
10
0
1
2
3
4
Oriented for Directivity →
Distance Along Fault

Average Displacement
α=1.76, β=1.77
−0.5 0 0.5
0
5
10
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.76, β=1.77
−1 −0.5 0 0.5 1
0
1
2
3
4
5
Centroid − Hypocenter

54

Table 3.4: Parameters for Best-Fitting Beta Distributions
Faulting Type α β ε*
Symmetric
All 1.92±0.12 1.92±0.12 3±0.11 3±0.11
Dip-Slip 2.02±0.15 2.02±0.15 3±0.38 3±0.38
Strike-Slip 1.76±0.15 1.76±0.15 3±0.33 3±0.33
Asymmetric
All 2.34±0.11 1.95±0.13 0±0.81 4±0.65
Dip-Slip 2.47±0.11 2.06±0.13 0±0.63 4±0.74
Strike-Slip 2.13±0.16 1.71±0.18 2±1.91 3±1.01
Directivity
All 2.00±0.12 1.89±0.12 3±0.56 3±0.51
Dip-Slip 2.14±0.15 1.97±0.16 3±0.83 2±0.60
Strike-Slip 1.76±0.17 1.77±0.15 3±0.83 2±0.77
All values and their uncertainties are given as the result of bootstrapping analysis where the
estimated errors are given as one standard deviation.
*The parameter ε and its estimated errors are reported in terms of percent of fault length.
Due to the additional parameter ε, the analytic calculation of uncertainties in the Beta
distribution parameters is not straight-forward. We therefore estimate the uncertainties in the
Beta parameters using the bootstrap method (Efron, 1979). Following this methodology, we
randomly sample our original dataset with replacement to generate 1000 additional datasets, each
with the same number of samples as in our original dataset. The standard deviations from these
1000 bootstrapped datasets are reported in Table 3.4 as the errors and an example distribution of
these bootstrapped parameters is shown in Figure 3.16.

55

Figure 3.16: An example of the Bootstrap distribution of α (left) and ε (right) based on
generating 1000 bootstrapped datasets from the original dataset of 28 symmetric strike-slip
profiles. The mean and standard deviations are shown as red solid and dashed lines respectively.
In the distribution that includes all faulting types, 95% of the full slip profiles have
normalized displacements below 1.62. Events with high normalized displacements are generally
characterized by compact regions of high slip and large regions of low slip. For example, the
USGS inversion for the 2012 M
W
7.4 Oaxaca, Mexico earthquake has less than 50 cm of slip on
over 84% of its trimmed rupture plane (and nearly 88% of the untrimmed rupture plane). This
brings the average slip down to ~1 m, causing the normalized displacement to be larger (i.e., 4 m
maximum displacement yields a normalized displacement of 4).
The dip-slip best-fitting Beta distributions are more peaked than the strike-slip profiles
with α = β = 2.02±0.15 compared to α = β = 1.76±0.15. In the individual slip distributions, the
dip-slip inversions tend to show that the rupture broke one or two major asperities; whereas, the
strike-slip inversions tend to have a more complicated slip distribution, often with multiple
asperities. To verify this difference, we find the number of local maxima in each displacement
profile whose peak is at least 66% of the global maximum (corresponding to a “very-large-slip”
asperity analogous to the definition in Mai, 2005). Of the ruptures with the highest 90% number
of peaks (i.e., ruptures with 7 or more peaks), 60% are strike-slip ruptures; in contrast, of the
1.5 2 2.5
0
50
100
150

Number of Occurrences
0.02 0.03 0.04
0
500
1000

α = β
ε
1

56

ruptures with the lowest 10% number of peaks (i.e., ruptures with only one peak), only ~22% are
strike-slip.
Figure 3.17: Distribution of the number of asperities (defined to be local maxima in the slip
profile whose height is at least 66% of the global maximum, analogous to the definition of a
“very large slip” asperity in Mai, 2005) for dip-slip (top) and strike-slip (bottom) ruptures.
The asymmetric profiles show that within the centroid-bias of the hypocenter distribution,
hypocenters tend to occur on the side of the profile with lower slip. Even with maximal
asymmetry, the profiles are still quite centralized. For all faulting types, the centroids are within
18% of the center of the fault and the mean profile’s centroid is located at ~0.06. The asymmetry
in these profiles, however, cannot be ignored. When compared to the best-fitting symmetric Beta
distribution for this set of profiles, the residual increases by over 250%. The profiles oriented
with respect to their directivity vectors show strong asymmetry in their hypocenter distributions.
These figures show, as one might expect, that the directivity vector tends to point from the
0 1 2 3 4 5 6 7 8
0
20
40
Dip−Slip
Number of Aperities
Percentage of Events
0 1 2 3 4 5 6 7 8
0
20
40
Strike−Slip
Number of Aperities
Percentage of Events

57

hypocenter toward the centroid. Exceptions are primarily for events with low directivity. In
terms of their slip distributions, however, the profiles oriented in the direction of directivity are
symmetric within error (i.e., α = 2.00±0.12 and β = 1.89±0.12 for all faulting types). In all three
schemes, it is clear that hypocenters are centroid-biased and preferentially located near the
centroid of the slip profile.
3.6 RUPTURE SIMULATOR DISPLACEMENT PROFILES
The two rupture simulators used in this study have very different methods for creating
slip distributions. The CyberShake ruptures are generated according to the stochastic
methodology of Graves and Pitarka (2010) to create random slip distributions whose spectra
follow a prescribed wavenumber falloff. RSQSim slip distributions, on the other hand, are
generated quasi-dynamically. Once an element’s shear stress overcomes the normal and
frictional forces, slip can accelerate to seismic slip speeds and an element continues to rupture
(and redistribute stress) at a constant slip speed until the shear stress decreases again below
levels necessary to generate slip (Dieterich and Richards-Dinger, 2010). The slip distribution,
therefore, depends on the initial stress fields as well as the changes in stress during the rupture.
Additionally, the two rupture simulators have very different levels of complexity in their
spaciotemporal slip evolution. CyberShake ruptures are relatively simple in that each element of
a rupture is allowed to slip once (though slip, rake, and slip velocity are allowed to vary).
RSQSim ruptures can be considerably more complex with each element allowed to slip as many
times as the shear stress permits (though slip velocity and rake are held constant). We expect this
complexity to be expressed in the relationships among the hypocenter distributions and
directivity.

58

3.6.1 CyberShake Displacement Profiles
CyberShake slip distributions begin as uniform with tapering along the edges of the fault
(except the top), and are randomly perturbed to give a prescribed wavenumber spectral decay to
match observed events (Mai and Beroza, 2002; Graves and Pitarka, 2010). CyberShake
hypocenters are assigned independently from the slip distributions and, in this set of CyberShake
ruptures, are placed every 20km along strike and are symmetric about the center of the rupture
(Graves et al., 2011). We therefore expect the slip, hypocenter, and centroid distributions for
each of the three orientation schemes defined in Section 3.5 and presented here for CyberShake
ruptures in Figure 3.18 through Figure 3.26 and in Table 3.5 to reflect these assumptions. In this
way, CyberShake provides a verification of our processing and we expect to see the following
assumptions reflected in the distributions:
(1) a near uniform slip distribution with tapering at the ends and normalized displacements
similar to those of the finite source inversions,
(2) a uniform hypocenter distribution that does not correlate with slip or directivity, and
(3) a tightly-clustered centroid distribution, centered about 0 (i.e., the center of the rupture).

59

Figure 3.18: CyberShake symmetric normalized displacement profiles for all faulting types
(5010 rupture variations). The top shows each slip profile as colored symbols with the mean
profile shows as a solid black line, the 10% and 90% percentiles are shown as the light gray area
bounded by a black dashed line. The best-fitting Beta distribution is shown as the dashed blue
line with its parameters in the legend. The blue histograms show the centroid distribution, red
histograms show the hypocenter distribution, and the black histogram show the normalized
distance between the centroid and hypocenters (i.e., events on the tails of this distribution are
expected to have higher directivity parameters).
−0.5 0 0.5
0
500
1000
1500
0
2
4
Distance Along Fault

Average Displacement
α=1.21, β=1.21
−0.5 0 0.5
0
500
1000
1500
Number of Events
−0.5
0
2
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.21, β=1.21
−1 −0.5 0 0.5 1
0
50
100
Centroid − Hypocenter

60

Figure 3.19: CyberShake symmetric normalized displacement profiles for strike-slip ruptures
(4917 rupture variations). Directly analogous to Figure 3.18.
−0.5 0 0.5
0
500
1000
1500
0
2
4
Distance Along Fault

Average Displacement
α=1.03, β=1.03
−0.5 0 0.5
0
500
1000
1500
Number of Events
0
2
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.03, β=1.03
−1 −0.5 0 0.5 1
0
50
100
Centroid − Hypocenter

61

Figure 3.20: CyberShake symmetric normalized displacement profiles for dip-slip ruptures (93
rupture variations). Directly analogous to Figure 3.18.
−0.5 0 0.5
0
5
10
15
20
25
0
1
2
3
Distance Along Fault

Average Displacement
α=1.06, β=1.06
−0.5 0 0.5
0
5
10
15
20
25
Number of Events
0
1
2
3
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.06, β=1.06
−1 −0.5 0 0.5 1
0
5
10
Centroid − Hypocenter

62

Figure 3.21: CyberShake asymmetric normalized displacement profiles for all faulting types
(5010 rupture variations). Directly analogous to Figure 3.18.
−0.5 0 0.5
0
500
1000
1500
0
2
4
6
8
Distance Along Fault

Average Displacement
α=1.22, β=1.01
−0.5 0 0.5
0
500
1000
1500
Number of Events
0
2
4
6
8
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.22, β=1.01
−1 −0.5 0 0.5 1
0
50
100
Centroid − Hypocenter

63

Figure 3.22: CyberShake asymmetric normalized displacement profiles for strike-slip ruptures
(4917 rupture variations). Directly analogous to Figure 3.18.
−0.5 0 0.5
0
500
1000
1500
0
2
4
6
8
Distance Along Fault

Average Displacement
α=1.22, β=1.01
−0.5 0 0.5
0
500
1000
1500
Number of Events
0
2
4
6
8
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.22, β=1.01
−1 −0.5 0 0.5 1
0
50
100
Centroid − Hypocenter

64

Figure 3.23: CyberShake asymmetric normalized displacement profiles for dip-slip ruptures (93
rupture variations). Directly analogous to Figure 3.18.
−0.5 0 0.5
0
5
10
15
20
25
0
2
4
6
Distance Along Fault

Average Displacement
α=1.2, β=1
−0.5 0 0.5
0
5
10
15
20
25
Number of Events
0
2
4
6
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.2, β=1
−1 −0.5 0 0.5 1
0
5
10
Centroid − Hypocenter

65

Figure 3.24: CyberShake normalized displacement profiles for all faulting types oriented so that
each profile’s directivity vector points toward the right (5010 rupture variations). Analogous to
Figure 3.18.
−0.5 0 0.5
0
500
1000
1500
0
2
4
6
8
Oriented for Directivity →
Distance Along Fault

Average Displacement
α=1.21, β=1.21
−0.5 0 0.5
0
500
1000
1500
Number of Events
0
2
4
6
8
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.21, β=1.21
−1 −0.5 0 0.5 1
0
50
100
150
200
Centroid − Hypocenter

66

Figure 3.25: CyberShake normalized displacement profiles for strike-slip ruptures oriented so
that each profile’s directivity vector points toward the right (4917 rupture variations). Analogous
to Figure 3.18.
−0.5 0 0.5
0
500
1000
1500
0
2
4
6
8
Oriented for Directivity →
Distance Along Fault

Average Displacement
α=1.02, β=1.03
−0.5 0 0.5
0
500
1000
1500
Number of Events
0
2
4
6
8
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.02, β=1.03
−1 −0.5 0 0.5 1
0
50
100
150
200
Centroid − Hypocenter

67

Figure 3.26: CyberShake normalized displacement profiles for dip-slip ruptures oriented so that
each profile’s directivity vector points toward the right (93 rupture variations). Analogous to
Figure 3.18.

−0.5 0 0.5
0
5
10
15
20
25
0
2
4
6
Oriented for Directivity →
Distance Along Fault

Average Displacement
α=1.08, β=1.06
−0.5 0 0.5
0
5
10
15
20
25
Number of Events
0
2
4
6
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.08, β=1.06
−1 −0.5 0 0.5 1
0
5
10
Centroid − Hypocenter

68

Table 3.5: Parameters for Best-Fitting Beta Distributions (CyberShake)
Faulting Type α β ε*
Symmetric
All 1.21±0.09 1.21±0.09 1±0.49 1±0.49
Strike-Slip 1.03±0.01 1.03±0.01 0±0 0±0
Dip-Slip 1.06±0.03 1.06±0.03 4±2.37 4±2.37
Asymmetric
All 1.22±0 1.01±0 1±0 0±0
Strike -Slip 1.22±0 1.01±0 1±0 0±0
Dip-Slip 1.21±0.02 1.00±0.02 1±0 1±3.25
Directivity
All 1.21±0.09 1.21±0.09 1±0.49 1±0.49
Strike -Slip 1.02±0.01 1.03±0.01 0±0 0±0
Dip-Slip 1.08±0.03 1.08±0.03 6±4.00 3±3.51
All values and their uncertainties are given as the result of bootstrapping analysis where the
estimated errors are given as one standard deviation. Errors reported as 0 are less than 1e
-3
.
*The parameter ε and its estimated errors are reported in terms of percent of fault length.
The CyberShake distributions (i.e., slip, hypocenter, and centroid) provide verification of
our methodologies because they reflect their underlying assumptions. The profiles show features
of the slip generation process in their relatively uniform slip distributions (e.g., α=β=1.03±0.01
for the symmetric strike-slip profiles, Figure 3.19) with sharp fall-off at the ends of the rupture.
The CyberShake slip profiles are significantly different from the finite source inversions (i.e.,
α=β=1.21±0.09 compared to α=β=1.92±0.12 for symmetric profiles for all faulting types);
however, they show similar ranges in their normalized displacement, ranging from ~0 to ~5
compared to ~0 to ~4 for finite source inversions. CyberShake shows a similar frequency
distribution of normalized displacements as the finite source inversions (Figure 3.27). This result
reflects the prescribed wavenumber spectral decay in CyberShake, which is based on the
observations of Mai and Beroza (2002) of finite source.

69

Figure 3.27: Comparison between CyberShake and the finite source inversions for the frequency
of occurrence of normalized displacement values from 0 to 5.5. The normalized displacements
are binned at 0.5 resolution for the symmeterized profiles for both CyberShake (dark grey) and
the finite source inversions (light grey).
These displacements, however, are not located at similar distances along strike.
CyberShake distributes the slip randomly along strike; whereas finite source inversions tend to
concentrate slip in asperities, which localizes areas of high slip (Figure 3.28).
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Normalized Displacement
Frequency of Occurrence

Inversion
CyberShake

70

Figure 3.28: The asperity distribution shown for finite source inversions (FSI) in Figure 3.17
plotted with the asperity distribution for CyberShake (CS) ruptures.
As with the finite source inversions, the asymmetry of CyberShake rupture profiles is
significant; however, because hypocenters are assigned independently from slip distributions, the
CyberShake hypocenter distribution does not correlate well with the regions of higher slip,
unlike with the results of the finite source inversions (Figure 3.10). The unsymmetrized profiles
have significantly higher normalized displacements than both the symmeterized profiles and the
displacement profiles in the finite source inversions. For example, a M
W
7.8 rupture on the
Elsinore fault that has a normalized displacement near 10 has relatively low slip along the
majority of the rupture with one localized region (~7.5% the length of the rupture) of very high
slip, making it significantly higher than its average slip.
0 1 2 3 4 5 6 7 8 9 10
0
10
20
30
40
50
Percentage of Events
Strike−Slip

Inversion
CyberShake
0 1 2 3 4 5 6 7 8 9 10
0
10
20
30
40
Number of Aperities
Percentage of Events
Dip−Slip

Inversion
CyberShake

71

The CyberShake profiles oriented in the direction of the directivity vector also show the
effects of the assumed hypocenter distribution. For each slip distribution, there are multiple
hypocenter locations, symmetric about the center of the rupture. Hypocenters at opposite ends of
ruptures with the same slip distributions will cancel each other out and create a symmetric
profile. The directivity profiles are symmetric and within error match the symmeterized profiles.
All three orientation schemes for the displacement profiles reflect underlying
assumptions in CyberShake. In terms of the assumptions listed at the beginning of this section:
(1) Slip: The symmeterized profiles show the slip generation process in their nearly
uniform slip distributions with tapers at the end, and demonstrate the assumed wavenumber
spectrum by matching the displacement frequency distribution of finite source inversions (Figure
3.27).
(2) Hypocenters: The asymmetric profiles highlight the assumed hypocenter distribution
in their departure from the finite source inversion behavior in this orientation scheme. The
hypocenter distribution itself; however, is not truly uniform, but this is not an artifact of our
procedure (because it is not discernable in the figures presented showing the three distributions,
the hypocenter distribution is plotted in Figure 3.29 for reference). This is a product of the
algorithm used to assign hypocenter locations. Figure 3.29 shows how hypocenters are assigned
in CyberShake for two examples rupture lengths of 55 and 75 km. Hypocenters are located 20
km apart and are required to be both symmetric about the center of the rupture and must not be
within 2 km of the edge of the rupture area (Scott Callaghan, personal communication). The 55
km length has three hypocenter locations with one in the center and two on the ends. For this
rupture, the hypocenter distribution is “periphery-biased”, as it has twice as many hypocenters
located near the end of the fault than it does in the center of the fault. The 75 km length rupture

72

has four hypocenter locations, but none of them are located directly in the center. For this
rupture, the hypocenter distribution is uniform. With only 17 CyberShake ruptures, this leads to a
slight periphery bias in the hypocenter distribution.
Figure 3.29: Schematic of two example rupture lengths of 55 and 75 km and how CyberShake
would assign hypocenters (red dots). The CyberShake symmetric hypocenter distribution for all
faulting types (5010 ruptures) is shown in the bottom histogram. The solid and dashed black
lines in the histogram are the mean and 0.1 and 0.9 quantiles respectively for a random uniform
distribution from Monte Carlo simulation.
(3) Centroids: Each of the oriented profiles shows the tight clustering of centroid
locations within CyberShake. If each of the profiles was exactly the same, the centroids would
all be located in the center of the fault. Tight clustering suggests similarity in slip distributions.
0 -20 20
10 -10 -30 30
L = 55 km
L = 75 km
−0.5 0 0.5
0
50
100
150
200
250
300
350
Hypocenter Location
Number of Events
All

73

3.6.2 RSQSim Displacement Profiles
RSQSim slip distributions are generated quasi-dynamically based on rate-and-state
friction laws, allowing elements to slip when their shear stress overcomes the normal and
frictional forces, and they stop slipping once the shear stress decreases (Dieterich and Richards-
Dinger, 2010). In this way, RSQSim slip distributions are products of both the initial stress field
and the stress transfers during rupture. RSQSim hypocenters are not prescribed, but are also
subject to the nucleation requirements in rate-and-state friction (Dieterich, 1992). Following the
three orientation schemes outlined in Section 3.5, RSQSim results are presented here in Figure
3.30 through Figure 3.38 and in Table 3.6.

74

Figure 3.30: RSQSim symmetric normalized displacement profiles for all faulting types (940
ruptures). The top shows each slip profile as colored symbols with the mean profile shows as a
solid black line, the 10% and 90% percentiles are shown as the light gray area bounded by a
black dashed line. The best-fitting Beta distribution is shown as the dashed blue line with its
parameters in the legend. The blue histograms show the centroid distribution, red histograms
show the hypocenter distribution, and the black histogram show the normalized distance between
the centroid and hypocenters (i.e., events on the tails of this distribution are expected to have
higher directivity parameters).
−0.5 0 0.5
0
100
200
300
0
1
2
3
Distance Along Fault

Average Displacement
α=1.37, β=1.37
−0.5 0 0.5
0
100
200
300
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.37, β=1.37
−1 −0.5 0 0.5 1
0
20
40
60
Centroid − Hypocenter

75

Figure 3.31: RSQSim symmetric normalized displacement profiles for strike-slip ruptures (907
ruptures). Directly analogous to Figure 3.30.
−0.5 0 0.5
0
100
200
300
0
1
2
3
4
Distance Along Fault

Average Displacement
α=1.36, β=1.36
−0.5 0 0.5
0
100
200
300
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.36, β=1.36
−1 −0.5 0 0.5 1
0
20
40
Centroid − Hypocenter

76

Figure 3.32: RSQSim symmetric normalized displacement profiles for dip-slip ruptures (33
ruptures). Directly analogous to Figure 3.30.
−0.5 0 0.5
0
5
10
15
0
1
2
3
4
Distance Along Fault

Average Displacement
α=1.47, β=1.47
−0.5 0 0.5
0
5
10
15
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.47, β=1.47
−1 −0.5 0 0.5 1
0
1
2
3
4
5
Centroid − Hypocenter

77

Figure 3.33: RSQSim asymmetric normalized displacement profiles for all faulting types (940
ruptures). Directly analogous to Figure 3.30.
−0.5 0 0.5
0
100
200
300
0
1
2
3
4
Distance Along Fault

Average Displacement
α=1.56, β=1.31
−0.5 0 0.5
0
100
200
300
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.56, β=1.31
−1 −0.5 0 0.5 1
0
20
40
60
Centroid − Hypocenter

78

Figure 3.34: RSQSim asymmetric normalized displacement profiles for strike-slip ruptures (907
ruptures). Directly analogous to Figure 3.30.
−0.5 0 0.5
0
100
200
300
0
1
2
3
4
Distance Along Fault

Average Displacement
α=1.56, β=1.31
−0.5 0 0.5
0
100
200
300
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.56, β=1.31
−1 −0.5 0 0.5 1
0
20
40
Centroid − Hypocenter

79

Figure 3.35: RSQSim asymmetric normalized displacement profiles for dip-slip ruptures (33
ruptures). Directly analogous to Figure 3.30.
−0.5 0 0.5
0
5
10
15
0
1
2
3
4
Distance Along Fault

Average Displacement
α=1.61, β=1.4
−0.5 0 0.5
0
5
10
15
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.61, β=1.4
−1 −0.5 0 0.5 1
0
1
2
3
4
5
Centroid − Hypocenter

80

Figure 3.36: RSQSim normalized displacement profiles for all faulting types oriented so that
each profile’s directivity vector points toward the right (940 ruptures). Analogous to Figure
3.30.
−0.5 0 0.5
0
100
200
300
0
1
2
3
4
Oriented for Directivity →
Distance Along Fault

Average Displacement
α=1.38, β=1.35
−0.5 0 0.5
0
100
200
300
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.38, β=1.35
−1 −0.5 0 0.5 1
0
20
40
60
Centroid − Hypocenter

81

Figure 3.37: RSQSim normalized displacement profiles for strike-slip ruptures oriented so that
each profile’s directivity vector points toward the right (907 ruptures). Analogous to Figure
3.30.
−0.5 0 0.5
0
100
200
300
0
1
2
3
4
Oriented for Directivity →
Distance Along Fault

Average Displacement
α=1.38, β=1.35
−0.5 0 0.5
0
100
200
300
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.38, β=1.35
−1 −0.5 0 0.5 1
0
20
40
Centroid − Hypocenter

82

Figure 3.38: RSQSim normalized displacement profiles for dip-slip ruptures oriented so that
each profile’s directivity vector points toward the right (33 ruptures). Analogous to Figure 3.30.

−0.5 0 0.5
0
5
10
15
0
1
2
3
4
Oriented for Directivity →
Distance Along Fault

Average Displacement
α=1.52, β=1.44
−0.5 0 0.5
0
5
10
15
Number of Events
0
1
2
3
4
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.52, β=1.44
−1 −0.5 0 0.5 1
0
1
2
3
4
5
Centroid − Hypocenter

83

Table 3.6: Parameters for Best-Fitting Beta Distributions (RSQSim)
Faulting Type α β ε*
Symmetric
All 1.37±0.01 1.37±0.01 1±0 1±0
Strike-Slip 1.36 ±0.01 1.36±0.01 1±1.00 0±0
Dip-Slip 1.47±0.03 1.47±0.03 2±2.00 0±0
Asymmetric
All 1.56±0.01 1.31±0.01 2±1.00 1±0
Strike -Slip 1.56±0.01 1.31±0.01 2±1.00 0.13±0
Dip-Slip 1.61±0.05 1.40±0.04 2±2.00 0.45±0.40
Directivity
All 1.38±0.01 1.35±0.01 1±1.00 0±0
Strike -Slip 1.38±0.01 1.35±0.01 1±1.00 0±0
Dip-Slip 1.52±0.05 1.44±0.04 2±2.00 0.40±0.35
All values and their uncertainties are given as the result of bootstrapping analysis where the
estimated errors are given as one standard deviation. Errors reported as 0 are less than 1e
-3
.
*The parameter ε and its estimated errors are reported in terms of percent of fault length.
RSQSim symmetrized profiles do not show the range in normalized displacement values
that both the finite source inversions and CyberShake exhibit. Although they span a similar range
(i.e., ~0 to ~3 normalized displacement), toward the center of the ruptures, none of the profiles
have any zones that extend down into the low to no slip region. The displacement curves appear
much smoother with less scatter than the finite source inversions or CyberShake displacement
profiles. Figure 3.39 shows this difference among the three datasets with RSQSim ruptures
underrepresenting the lower slip values and significantly overrepresenting the displacements near
the average compared to the other two datasets. This is expected given the smooth slip
distributions compared to CyberShake ruptures (for example, see Figure 4.3 for an example
CyberShake slip distribution compared to Figure 4.6 for an example RSQSim slip distribution
for the same fault). The number of asperities in RSQSim strike-slip ruptures, however, is quite
similar to the distribution seen for finite source inversions. Despite the fact that RSQSim
hypocenters are not assigned, the distribution is remarkably uniform with the exception of
significantly higher numbers of hypocenters located at the very ends of ruptures (-0.5 to -0.45
and 0.45 to 0.5) and those located near ±0.1. The symmetrized RSQSim “Centroid-Hypocenter”

84

distributions are much more skewed toward ±0.5 than either of the previous datasets, showing
the periphery-bias of the hypocenter distribution.
Figure 3.39: Comparison of the normalized displacement frequency distributions for all three
datasets binned by 0.5 normalized displacement.
RSQSim, like the previous two datasets, have significant asymmetry in the slip profiles
with α = 1.56±0.01 and β = 1.31±0.01. Like the prescribed CyberShake hypocenter distribution,
RSQSim hypocenters do not exhibit the same behavior as the finite source inversions; i.e, the
RSQSim hypocenter distribution for the asymmetric orientations does not show a strong bias
toward the lower slip portion of the profiles. For finite source inversions, Mai (2005) noted that
hypocenters are preferentially located in regions of low slip, but near asperities, where stress
concentrations are high. This may be what is observed in the finite source inversion asymmetric
profiles where the hypocenters are located on the lower slip portion of the profile. RSQSim
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Normalized Displacement
Frequency of Occurrence

Inversion
CyberShake
RSQSim

85

ruptures, however, may be too smooth and the stress field not heterogeneous enough to
concentrate stress in specific locations near an asperity.
The hypocenter distribution in the case where profiles are oriented in the direction of the
directivity vector shows the same features as that observed in the finite source inversions: the
expected behavior that the directivity vector tends to point from the hypocenter toward the
centroid. The RSQSim directivity-oriented slip profiles show very slight asymmetry so that slip
is higher away from the hypocenters (α = 1.38±0.01 and β = 1.35±0.01 for all faulting types).
3.7 DISCUSSION
To better understand our results, we want to put them into the context of previous studies
(for the finite source inversions) and discuss how the profiles and distribution behaviors reflect
the underlying assumptions in the simulator datasets.
3.7.1 Finite Source Inversion Profiles
Previous studies have come up with different shapes for slip profiles along strike.
Manighetti et al. (2005) determined that the slip profiles are triangular when oriented in the
maximal asymmetric direction. We find that the finite source inversion profiles are well-
represented by triangular functions. For comparison, Figure 3.40 shows the asymmetric
displacement profiles with their best-fitting triangles. A triangle is however, a poor descriptor of
the displacement profiles of CyberShake and RSQSim ruptures. We therefore prefer the Beta
distribution for its versatility.

86

Figure 3.40: Best-fitting triangles to the finite source inversions asymmetric displacement
profiles for comparison with Manighetti et al. (2005).
Biasi and Weldon (2006) conclude that the symmetric average displacement profile for
strike-slip faults in southern California is well approximated by a function that looks like the
square root of sine. As shown in Figure 3.5, their square root of sine function can be
approximated by a Beta distribution with α = β = 1.56. For the finite source inversion symmetric
profiles, the strike-slip ruptures are best fit by a Beta distribution with α = β = 1.76±0.15. This is
−0.5 0 0.5
0
1
2
3
4

Average Displacement
Best−Fitting Triangle
−0.5 0 0.5
0
1
2
3
4
Normalized Displacement

Average Displacement
Best−Fitting Triangle
−0.5 0 0.5
0
1
2
3
4
Distance Along Fault

Average Displacement
Best−Fitting Triangle
All
Strike-Slip
Dip-Slip

87

consistent with the previous results of Biasi and Weldon (2006) for surface slip measurements
within the 95% confidence interval computed assuming a Gaussian distribution of Beta
parameters (Figure 3.16) as follows:
𝛼±1.96∗𝜎
where 𝜎 is the standard deviation/reported error.
Others have concluded that slip in individual ruptures is fractal (Somerville et al., 1999b;
Mai and Beroza, 2002; Milliner et al., 2015). We generate a set of fractal distributions using a
random walk and keep only those that start and end at zero (as we expect slip to go to zero at the
ends of ruptures, examples in Figure 3.41). From these 100 fractal distributions, we use the same
resampling, symmetrization, normalization, and Beta fitting procedures as used for the data. The
best-fitting Beta distribution for the symmetric average displacement profile for a stack of fractal
ruptures is α = β = 1.69±0.09 (Figure 3.42). This result is consistent with both the findings of
Biasi and Weldon (2006) and the Beta parameters for the symmetric distribution for strike-slip
events in the finite source inversion data within the 95% confidence interval as computed above.
Dip-slip profiles, however, are neither consistent with the random walk profiles nor the results of
Biasi and Weldon (2006). This may be due to the fewer asperities for dip-slip ruptures in that
they tend to rupture one or two major asperities that dominate the slip profile (Figure 3.17).

88

Figure 3.41: Two example random walk normalized displacement profiles generated to test
whether the finite source inversion profiles match what is expected if each individual profile is
fractal. Each fractal profile is then processed in the same way as the finite source inversion
profiles (including resampling at 5%, indicated by the red dots) and their average is fit with a
Beta distribution (Figure 3.42).
Figure 3.42: Symmetric normalized displacement profiles for a set of 100 random walk fractal
distributions resampled to 5%. The mean is shown as the thick black line and its corresponding
best-fitting Beta distribution is shown as the dashed blue line.
3.7.2 Rupture Simulator Profiles
Both the CyberShake and RSQSim displacement profiles are markedly different from
those of the finite source inversions. The simulation profiles reflect their underlying assumptions
and their departure from the distributions of the finite source inversions may contribute insights
into the underlying processes of observed earthquakes.
CyberShake slip distributions reproduce exactly what is put into them, verifying that our
procedure does not introduce significant artifacts or biases. The slip distributions in CyberShake
0 0.5 1
0
1
2
3
4
5
Distance Along Strike
Normalized Displacement
0 0.5 1
0
2
4
6
Distance Along Strike
−0.5 0 0.5
0
5
10
Distance Along Fault
Normalized Displacement

Average Displacement
α=1.69, β=1.69

89

are randomly generated, highly heterogeneous (for example, see Figure 3.3), and forced to taper
at the ends (Graves and Pitarka, 2010). Their average slip profile, then, is nearly uniform across
the majority of the fault and tapered at the ends, yielding significantly lower α and β values than
the finite source inversion profiles.
RSQSim slip distributions are generated according to rate-and-state friction laws and
reflect the scale and heterogeneity of the initial stress field. If the field was highly heterogeneous
on length scales much smaller than the rupture length, we may expect more variation in the slip
distributions. The resulting smooth displacement profiles for RSQSim therefore suggest that the
stress field is fairly homogenous on rupture length scales.
3.8 CONCLUSION
The simulator results of both CyberShake and RSQSim show the features of their
underlying assumptions, which we then use to make inferences about the features shown in the
finite source inversion distributions. From these rupture simulator results, we may infer that
observed earthquakes, insofar as they are represented by finite source inversions, have slip
distributions that are not random (as in CyberShake), but rather, are more concentrated in the
form of asperities on the fault. The discrepancy between both simulators and the finite source
inversions in terms of the relationship between slip profile asymmetry and hypocenter location
suggest that (1) hypocenters are not independent of the slip distributions as assumed in
CyberShake, and (2) the apparent homogeneity of the stress field in RSQSim may not allow
realistic earthquake nucleation. Hypocenters preferentially occur on the lower slip portion of an
asymmetric slip distribution, consistent with the findings of Mai (2005); likewise, the directivity
vector tends to point in the direction of higher slip.

90

The results obtained from finite source inversions for the distribution of slip along strike
for large strike-slip earthquakes are consistent with the distribution obtained by Biasi and
Weldon (2006) for surface offsets. Furthermore, the average strike-slip profiles from finite
source inversions are also consistent with the idea posed by several authors (Somerville et al.,
1999a; Mai and Beroza, 2002; Milliner et al., 2015) that individual slip profiles are fractal, in
that they are consistent with random walk fractals (Figure 3.42). Note that extension to any other
type of fractal would require further study.
We found that dip slip profiles are inconsistent with both the distribution of Biasi and
Weldon (2006) and the idea that individual slip profiles are fractal. We find in the source
inversion dataset that dip slip profiles tend to have fewer asperities than strike-slip ruptures
(Figure 3.17), concentrating slip in just one or two asperities that dominate the slip profile.

91

4Chapter 4 Directivity Mapping in Simulations
4.1 INTRODUCTION
There exist sets of empirically-derived ground motion prediction equations (GMPEs)
which predict the ground shaking at a specific site conditional on source and site parameters (e.g,
Abrahamson et al., 2008; Boore and Atkinson, 2008; Campbell et al., 2008; Spudich and Chiou,
2008). These GMPEs take into account factors such as magnitude, distance from the source,
faulting type, and site effects. Directivity is a strong control on ground motion and several
authors have developed empirically-derived directivity correction factors for GMPEs (e.g.,
Somerville et al., 1997; Spudich and Chiou, 2008; Rowshandel, 2010). Both the GMPEs
themselves and the directivity correction factors depend on the hypocenter location. Although
these equations are empirically-derived, even when they are used to investigate the effect
directivity has on probabilistic seismic hazard analysis (PSHA), a uniform hypocenter
distribution is assumed (e.g., Abrahamson, 2000; Shahi and Baker, 2011). GMPEs are
empirically-derived and therefore based solely in observations. It seems inconsistent, therefore,
to assume we have no a priori information about the distribution of hypocenters to feed into
these observation-based equations.
The CHD for finite source inversions was presented in Mai (2005); however, the CHD
itself is not sufficient for characterizing the directivity of complex ruptures using the simplified
equations and correction factors. For example, consider a rupture that begins at one end, ruptures
unilaterally to an asperity, and then large slip in the asperity region sends the rupture back
toward the hypocenter (Figure 4.1). Because energy is radiated in both directions, in terms of its
ground motion pattern, this rupture would not be well characterized by its initial hypocenter
location. If real earthquakes are able to exhibit appreciable complexity in their rupture

92

propagation, then an apparent CHD is necessary to better estimate the directivity effects in the
GMPEs.
Figure 4.1: An example of a M
W
7.2 RSQSim rupture on the Hayward fault in which rupture
complexity makes the hypocenter (red star) and apparent hypocenter (grey dot) very different.
The dashed black lines show the locations of the spatial and temporal centroids for this rupture.
The hypocenter location is 0.5 and the apparent hypocenter location is 0.1.
In the previous chapters, we outlined a methodology for mapping directivity parameters
from the finite moment tensor (FMT) to apparent hypocenters, which characterize the directivity
of the event. This apparent hypocenter distribution can then be used in the existing PSHA
framework to better estimate seismic hazard.
In this study, we test our mapping framework by using it on two different rupture
simulators: (1) CyberShake (Graves et al., 2011), which uses a kinematic rupture model (Graves
and Pitarka, 2010) and whose hypocenters are currently prescribed to be uniform, and (2)

93

RSQSim (Dieterich and Richards-Dinger, 2010), a quasi-dynamic rupture model whose
hypocenters are not pre-determined, but rather nucleate according to rate-and-state friction laws
(Dieterich, 1992). Rupture simulators allow us to investigate our procedure in an environment
where we know the inputs and underlying assumptions.
4.2 MAPPING DIRECTIVITY PARAMETER TO APPARENT HYPOCENTER
In Chapter 2, we presented the formulation and behavior of the directivity parameter D
from the FMT. For a simplified 1D rupture with constant rupture velocity, uniform moment
release rate, and constant, finite rise time, we have solved the analytic equations for each
component of the FMT (Section 2.4). With a simple relationship between nucleation point and
directivity parameter, we can map one to the other. For a non-uniform moment release, we can
use forward simulations to determine this relationship. In Chapter 3 we determined the best-
fitting Beta distribution for the average along-strike slip profiles. We use this slip distribution to
forward simulate the curve to relate nucleation point to directivity. The relationship between
nucleation point and directivity parameter is also dependent on the rise time. The effect of rise
time on the FMT depends on its size relative to the total rupture duration. That is, if τ is large
compared to L/v, it can considerably decrease the directivity (Equation 2.6 is reproduced below
for reference).
𝑑 𝑥
∗
=
𝜁
(!,!)
𝑥
∗
𝜁
(!,!)
𝑥
∗
𝐿
!
𝑇
!
𝑥
∗
=
2 3
1
12
𝐿
!
+𝑥
∗
!
𝑥
∗
3𝐿
−
1
2
𝐿
1
12
𝐿
!
−𝑥
∗
!
𝑥
∗
𝐿
−1
!
+
1
12
𝜏𝑣
!
.
Since we are working with normalized values, we use ρ, defined to be the ratio of the rise
time to the total rupture duration 𝜌= 𝜏 𝑇, where 𝑇= 𝐿 𝑣. Figure 4.2 shows how the mapping
changes for different α and β parameters for the Beta distribution of slip as well as for different
values of ρ. For the forward simulations, we use the average ρ determined from the ruptures.

94

Figure 4.2: Influence of slip distribution and rise time on the apparent hypocenter mapping.
Curves with the same color have the same α and β parameters, and curves of the same style (i.e.,
solid or dashed) have the same rise time ratio ρ, ranging from 0 rise time to 0.25 of the total
rupture duration. Increasing the rise time lowers the directivity parameter as expected. For zero
rise time, increasing α and β parameters push apparent hypocenters toward the center of the
rupture everywhere; however, for finite rise time, increasing α and β parameters push only
apparent hypocenters toward the center of the rupture only within 30% of the rupture length. At
the edges, an increase in the Beta distribution parameters results in moving the apparent
hypocenter toward the end.
An apparent CHD calculated using one set of values for the Beta parameters and rise time
can be mapped to an equivalent apparent CHD for another set of parameters by using the original
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Distance Along Strike
Directivity Parameter

α=β=1, ρ=0
α=β=2, ρ=0
α=β=1, ρ=0.25
α=β=2, ρ=0.25

95

set of directivity parameters and mapping them using the new set of parameters. This process
finds the CHD under the new set that would produce the same distribution of directivity
parameters. This mapping will be used in Section 5.6.2 to map the apparent CHD for finite
source inversions for an equivalent CHD to use in CyberShake to test the impact the change in
CHD has on a deterministic ground motion forecast.
4.3 SIMULATION SETS
Simulated rupture sets allow us to both verify and explore our mapping procedure. By
using the simulations as our “data”, we are able to see how well our methodology preforms in an
environment where there is no noise and where the underlying assumptions and rupture behavior
are known. For this study, we use both a kinematic (as part of CyberShake, Graves and Pitarka,
2010) and quasi-dynamic (RSQSim; Dieterich and Richards-Dinger, 2010) rupture model.
CyberShake serves as an important verification model because it contains key features
assumed in the equations derived in Chapter 2, namely: (1) a nearly uniform slip distribution
(Figure 3.18), (2) simple rupture propagation (each grid ruptures once and no large variations in
rupture velocity), and (3) large aspect ratio ruptures. With these features in mind, the apparent
CHD for CyberShake should retain the features of the true CHD.
RSQSim is a physics-based model and when compared to observations, both serves to
verify the RSQSim model itself, but also to provide physical insights into features we may
observe in real earthquakes. RSQSim offers the tools to examine the physical source of features
we observe in the differences between true and apparent CHDs and verify whether these seem
reasonable.

96

4.3.1 CyberShake
As outlined in Chapter 1 (Section 1.5.2), the CyberShake slip distributions used in this
study are generated using the stochastic methodology of Graves and Pitarka (2010), where the
spatial distribution of slip is generated independent of hypocenter location. Beginning with an
initially uniform slip distribution with tapers on three sides (i.e., sides and bottom), the slip is
randomly perturbed and required to have a spectrum that fits the wavenumber fall-off of Mai and
Beroza (2002). Time histories are then generated using the hypocenter location and adjustments
to the rupture velocity and rise time are based on slip amplitude. Each element in a rupture model
is allowed to slip only once.
CyberShake ruptures are based on the Uniform California Earthquake Forecast, Version 2
(UCERF2, Field et al., 2009). The ruptures identified in UCERF2 define unique sections of a
fault. CyberShake then creates multiple hypocenter locations and slip distributions for each
rupture, creating “rupture variations” which are meant to represent the aleatory variability in the
ruptures. Hypocenters are located every 20km along a rupture (as shown in Figure 4.4), and there
are twice as many slip distributions as hypocenters. An example of a CyberShake rupture
variation is shown in Figure 4.3.

97

Figure 4.3: An example rupture variation (slip variation: 3, hypocenter variation: 3) for a M
W

7.8 CyberShake rupture on the Elsinore fault between the Glen Ivy and Coyote Mountain
sections (UCERF2 Source ID: 10, Rupture ID: 3) with strike 309°, dip 85°, and rake -178°.
Rupture initiation time is contoured every 10 seconds in white and labeled accordingly, the
hypocenter is shown as the grey circle, and the directivity vector from the FMT is shown as the
red arrow.

98

Figure 4.4: The CyberShake symmetrized hypocenter distribution along strike for the 5010
rupture variations used in this study (4917 strike-slip and 93 dip-slip). The solid black line is the
expected number in each bin and the dashed lines are the 0.1 and 0.9 quantiles for 1000 Monte
Carlo simulations of a uniform distribution.
CyberShake ruptures range from lengths of ~ 25km to over 500km. Due to the limited
seismogenic thickness of southern California, vertical and near-vertical faults tend to have large
aspect ratios, ranging from 2 to 25 with over 95% of the aspect ratios for strike-slip rupture
variations in this study between 5.5 and 24. The aspect ratios for dip-slip CyberShake rupture
−0.5 0 0.5
0
100
200
300
400
All
−0.5 0 0.5
0
100
200
300
400
Strike−Slip
Number of Events
−0.5 0 0.5
0
5
10
15
20
Dip−Slip
Distance Along Strike

99

variations used in this study range between 0.94 and 2.6. Figure 4.5 shows the distribution of
lengths and aspect ratios for the 17 CyberShake ruptures in this study.
Figure 4.5: CyberShake true rupture lengths and widths for the 17 ruptures listed in Table 1.3
(“+” is used for strike-slip and “x” for dip-slip). The solid lines correspond to different aspect
ratios and are labeled accordingly.
4.3.2 RSQSim
As outlined in Chapter 1 (Section 1.5.3), RSQSim ruptures are generated according to
rate-and-state friction laws (e.g., Dieterich, 1979) so that an element is locked when the stress is
below the steady state friction and slips once the stress overcomes the frictional force and
accelerates to a pre-defined seismic slip speed. Each element that slips transfers stress to
neighboring elements. This allows for triggering as long as the conditions for seismic slip are
met. Elements may, therefore, rupture as many times as the stress state allows, and this can
0 5 10 15 20 25 30 35 40
0
50
100
150
200
250
300
350
400
450
500
550
Width [km]
Length [km]
1
5
10
15
20
25

100

create considerably more complex ruptures than in CyberShake. An element continues to slip
until its slip velocity decelerates below the prescribed threshold and so the spatial slip
distribution depends on the initial stress state and the history of slip in neighboring elements. An
example of an RSQSim rupture is shown in Figure 4.6. Hypocenters in RSQSim are not
prescribed as they are in CyberShake; rather, they are permitted to nucleate wherever the stress
state permits, as detailed in Dieterich (1992). The hypocenter distribution for RSQSim ruptures
is shown in Figure 4.7).
Figure 4.6: An example RSQSim slip distribution for a M
W
7.3 rupture on the Elsinore fault
between the Glen Ivy and Coyote Mountain sections (Event ID: 11548282) with strike 309°, dip
90°, and rake 180°. Rupture initiation time is contoured every 10 seconds in white and labeled
accordingly, the hypocenter is shown as the grey circle, and the directivity vector from the FMT
is shown as the red arrow.

101

Figure 4.7: The RSQSim symmetrized hypocenter distribution along strike for the 940 ruptures
used in this study (907 strike-slip and 33 dip-slip). The solid black line is the expected number in
each bin and the dashed lines are the 0.1 and 0.9 quantiles for 1000 Monte Carlo simulations of a
uniform distribution.
RSQSim ruptures range in length from about ~19km to ~400km. Like the CyberShake
ruptures, RSQSim vertical and near-vertical ruptures are limited by the seismogenic thickness
and tend to have large aspect ratios (Figure 4.8), ranging from ~2.8 to over 34. For all
mechanisms, 90% of the aspect ratios are between 4 and 21 and over 95% of the aspect ratios for
−0.5 0 0.5
0
20
40
60
80
All
−0.5 0 0.5
0
20
40
60
80
Strike−Slip
Number of Events
−0.5 0 0.5
0
5
10
Dip−Slip
Distance Along Strike

102

strike-slip ruptures are between 4 and 27. The aspect ratios for dip-slip RSQSim ruptures used in
this study range between 0.86 and 7.1. Figure 4.8 shows this distribution of lengths and aspect
ratios for the 940 RSQSim ruptures used in this study with “+” denoting a strike-slip rupture and
“x” denoting a dip-slip rupture.
Figure 4.8: RSQSim true rupture lengths and widths for the 940 ruptures in this study (“+” is
used for strike-slip and “x” for dip-slip). The solid lines correspond to different aspect ratios and
are labeled accordingly.
4.4 RESULTS
Although RSQSim hypocenters are not prescribed as they are in CyberShake, both
simulator hypocenter distributions are qualitatively similar (Figure 4.4 and Figure 4.7). As is
evident by the difference in their slip distributions (Figure 4.3 and Figure 4.6), their average
profiles along strike are significantly different, with CyberShake having a more uniform
0 5 10 15 20 25 30 35 40
0
50
100
150
200
250
300
350
400
Width [km]
Length [km]
5
1
10
15
20
25

103

distribution of slip along strike than RSQSim (Chapter 3, e.g., Figure 3.18 and Figure 3.30). We
use the best-fitting Beta distribution (as outlined in Chapter 3, Section 3.4) to the symmetrized
slip profiles for both CyberShake and RSQSim to forward model the relationship between
nucleation point and the directivity parameter (i.e., Figure 4.2) to determine the mapping to
apparent hypocenter. Figure 4.9 and Figure 4.10 show the distribution of apparent hypocenters
that result from this mapping procedure. From these figures, it’s clear that the distribution of
apparent hypocenters is significantly different from the distribution of true hypocenters (Figure
4.4 and Figure 4.7), with the difference being most notable for RSQSim ruptures.

104

Figure 4.9: The CyberShake apparent CHD as mapped using the directivity parameter D for the
5010 rupture variations in this study (4917 strike-slip and 93 dip-slip). The mean (solid red line)
and 0.1 and 0.9 quantiles (dashed red lines) are computed using Monte Carlo simulation for a
uniform distribution of hypocenters using the best-fitting Beta distribution to the symmetric
profiles for all faulting types (α = β = 1.21, Figure 3.18) and the average rise time ratio ρ = 0.03.
Note that events missing from the last bin are expected due to numerical inaccuracy as well as
deviations of the ruptures from the assumed model. For example, events that fall within the last
bin must have directivity parameters > 0.9987, a very high directivity parameter. Variations in
rupture velocity, rise time, regions of zero slip, and down-dip nucleation point can also affect the
directivity parameter in third decimal place.
−0.5 0 0.5
0
200
400
All
−0.5 0 0.5
0
200
400
Strike−Slip
Number of Events
−0.5 0 0.5
0
10
20
Dip−Slip
Distance Along Strike

105

Figure 4.10: The RSQSim apparent CHD as mapped using the directivity parameter D for the
940 ruptures in this study (907 strike-slip and 33 dip-slip). Directly analogous to Figure 4.9,
using the best-fitting Beta distribution of slip (α = β = 1.37, Figure 3.30) and the average rise
time ratio 0.11. Note that the same resolution problem arises with the RSQSim data as with the
CyberShake data (see Figure 4.9 caption). The last bin for RSQSim requires events to have a D
> 0.9904.
−0.5 0 0.5
0
20
40
60
80
All
−0.5 0 0.5
0
20
40
60
80
Strike−Slip
Number of Events
−0.5 0 0.5
0
5
10
Dip−Slip
Distance Along Strike

106

Figure 4.11: The relationship between true and apparent hypocenters (to the nearest 0.05)
mapped using D for (a) CyberShake and (b) RSQSim. Red circles are ruptures in Figure 4.13.
a.)
b.)
0 0.1 0.2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
0.5
Apparent Hypocenter (Distance From Center)
Hypocenter (Distance From Center)
CyberShake
0 0.1 0.2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
0.5
RSQSim
Hypocenter (Distance From Center)
Apparent Hypocenter (Distance From Center)

107

The CyberShake apparent CHD retains the features of the true CHD (Figure 4.4), i.e.,
uniform except the increase in events at the end of the rupture. As mentioned previously, the lack
of events in last bin and increased number of events in the second-to-last bin in the CHD is
expected in part due to insufficient resolution, as well as deviations in the models from the
simple rupture model (i.e, regions of zero slip, down-dip hypocenter location, as well as
variations in rupture velocity and rise time). The CyberShake apparent CHD shows significant
departure from a uniform distribution in the bin range between ±0.3-0.4, compared to the
departure in the true CHD in the ±0.35-0.4 and ±0.45-0.5 range. The events near the end of the
rupture whose apparent hypocenters are much lower than their true hypocenters are events with
aspect ratios < 3.5, which is the minimum determined from numerical simulation to be the aspect
ratio below which the 1D approximation breaks down. Considering that CyberShake ruptures
have rupture histories with varying rise times and rupture velocities (e.g., Figure 4.13a) and our
1D representation assumes a constant rupture velocity and rise time, this is a very encouraging
result and verifies our methodology.
The RSQSim apparent CHD is very different from its true CHD (Figure 4.7). Although
the basic shape remains the same, the apparent hypocenters are pushed toward the center of the
rupture. There are significantly fewer apparent hypocenters located near the ends of the rupture
area in the range ±0.35-0.5, the last 15% of the rupture area on both ends. RSQSim ruptures are
considerably more complex than our 1D representation used in the mapping procedure. The 1D
simplified rupture assumes constant rupture velocity, constant rise time, and each point ruptures
only once. RSQSim rupture velocities can change significantly during a rupture based on the
stress distribution on the fault (Richards-Dinger and Dieterich, 2012), and in a RSQSim rupture,
an element can slip multiple times during the rupture.

108

Figure 4.12: A M
W
7.7 RSQSim rupture on the San Andreas Fault from the San Bernardino to
the Cholame section demonstrating the temporal complexity in a unilateral RSQSim rupture.
(Top) Space-time plot colored by seismic moment. The grey dot and red star mark the
hypocenter and apparent hypocenters respectively. The dashed black lines mark the spatial and
temporal centroids. (Bottom) The moment distribution with rupture initiation times contoured
every 1 second in black lines. The grey dot marks the hypocenter location. Note that there is
vertical exaggeration in the fault plane to make the rupture contours more visible.
Figure 4.12 shows an example unilateral RSQSim rupture with the rupture fronts
contoured in black lines. Once the rupture encounters an asperity, it slows down significantly
(indicated by the tightly-grouped contour lines). This deceleration through the region with
highest moment release decreases the directivity, and pushes the apparent hypocenter ~30% of

109

the fault length toward the center. The shift observed for the RSQSim apparent CHD simplifies
the ruptures to those more appropriate for use in the GMPEs.
4.5 DISCUSSION
The rupture behavior between CyberShake and RSQSim differs greatly, which allows us
to investigate the performance of our methodology. We can see that the hypocenter distributions
(Figure 4.4 and Figure 4.7) are similar, despite the fact that hypocenter locations are not
prescribed in RSQSim; however, their apparent CHDs are very different. With a simple rupture
model such as CyberShake, the apparent CHD matches the true hypocenter distribution.
CyberShake ruptures are favorable for this analysis because they have large aspect ratios (i.e.,
are well-approximated by our 1D assumption in the mapping process, see Figure 4.5), nearly
uniform slip (e.g., Figure 3.18), and simple rupture propagation (Figure 4.13a), all of which are
elements of the simplified 1D source presented in Chapter 2 (Section 2.4) used to map the
directivity parameter to the apparent CHD. The resulting apparent CHD demonstrates that for
simple rupture behavior, the true hypocenter can be used to describe the directivity (also shown
in Figure 4.11a). The only portion of the apparent CHD that deviates from the true hypocenter
distribution in Figure 4.11a is the located near the end of the fault. This due in part to insufficient
numerical accuracy, as well as deviations of the ruptures from the simplified source model used
for mapping, including regions of zeros slip, variations in both rupture velocity and rise time, and
the location of the hypocenter down-dip (Figure 2.2-Figure 2.4). For example, in order for an
apparent hypocenter to fall in the range 0.45 to 0.5, its directivity parameter must be > 0.9987 for
the symmetric CyberShake slip distribution α = β = 1.21 and rise time ratio ρ = 0.03.

110

Figure 4.13: The spatiotemporal histories for example ruptures from (left) a CyberShake M
W
7.8
right-lateral strike-slip rupture on the Elsinore Fault, and (right) a RSQSim M
W
7.2 right-lateral
strike-slip rupture on the Panamint Valley Fault. The hypocenter is shown as a grey dot, the
apparent hypocenter is shown as the red star, and the black dashed lines mark the locations of the
spatial and temporal centroids. These ruptures are shown as the red circles in Figure 4.11.
Unlike CyberShake ruptures, RSQSim ruptures can have considerably more complex
spatiotemporal histories. This is evident in the extreme difference between the real CHD (Figure
4.7) and the apparent CHD (Figure 4.10). In both Figure 4.10 and Figure 4.11b, it’s clear that
there is a centroid-bias of the apparent CHD compared to the true hypocenter location. The
potential for complex spatiotemporal history of RSQSim ruptures (Figure 4.13b) does not permit
the true hypocenter location to adequately describe the directivity. For example, the rupture in
Figure 4.13b has its true hypocenter at the end of the fault, it ruptures in one direction until it
encounters some barrier (e.g., a geometrical complexity or asperity), and then ruptures back
unilaterally toward the hypocenter. Since the rupture propagates in both directions, in terms of its
hazard and ground motion patterns, it would be better characterized as a bilateral rupture. The

111

directivity parameter for this rupture (D = 0.45) captures this behavior and puts the apparent
hypocenter at -0.08.
4.6 CONCLUSION
The existing probabilistic seismic hazard analysis (PSHA) framework cannot account for
complex ruptures. In order to properly forecast directivity, then, it is necessary to find a
conditional hypocenter distribution (CHD) that is able to characterize some observed directivity.
In the previous chapters, we have developed a methodology for turning observations of
directivity into an apparent CHD. In this chapter, we present the results of this methodology for
two different rupture simulators.
The results obtained from CyberShake ruptures verify our methodology in that for
sufficiently simple ruptures, the true and apparent hypocenters are representative of one another
(Figure 4.11a). The RSQSim results allow us to explore how our methodology works when
ruptures are more complex, and we find that for complex ruptures, the true hypocenter is a poor
descriptor of the directivity. The apparent CHD determined using the directivity parameter better
characterizes the directivity in complex ruptures.
Using our mapping technique, the resulting apparent CHD is able to take a suite of
ruptures, with both simple and complex rupture histories and specify a distribution of
hypocenters that can be used in the existing PSHA framework to improve forecasts of directivity.

112

5Chapter 5 Apparent Conditional Hypocenter for
Observed Earthquakes
5.1 INTRODUCTION
The ground motion prediction equations (GMPEs) make use of a directivity correction
factor (e.g., Somerville et al., 1997; Spudich and Chiou, 2008; Rowshandel, 2010) which
depends on the assumed conditional hypocenter distribution (CHD). Often, this CHD is assumed
to be uniform (e.g., Abrahamson, 2000; Shahi and Baker, 2011), which could lead to either over-
or under-prediction of the directivity effects, depending on the true CHD for observed
earthquakes. Mai (2005) determined a CHD from finite source inversions and found that finite
source inversions have a centroid-biased CHD. If this is indeed the case, using a uniform CHD
rather than a centroid-biased CHD would likely cause the correction factors to overpredict the
directivity effects in a forecast. The correction factors typically simplify directivity to a function
of station-source geometry, assuming a hypocenter location. For example, the “isochrone
directivity predictor” determined by Spudich and Chiou (2008) relies on the geometry set up in
Figure 5.1, where x
h
is the hypocenter and x
close
is the point on the fault closest to the station. The
specific form of this directivity predictor will not be presented here (interested parties can review
the details in Spudich and Chiou, 2008), but notice that four (R
hypo
, R
D
, h, and s) of the five
distances labeled in Figure 5.1 depend on the hypocenter location, and more specifically, the
location of the hypocenter with respect to the ruptured area. Wang (2013) showed the influence
different CHDs can have on the hazard calculations.

113

Figure 5.1: An example geometric configuration used in calculating the directivity correction
factor of Spudich and Chiou (2008). The red star x
h
is the hypocenter location and the grey dot
x
close
is the point on the fault closest to the station for which the directivity correction is being
calculated. Figure modeled after Figure 1 of Spudich and Chiou (2008).
The location of the hypocenter with respect to the ends of the rupture area can serve as a
proxy for directivity when ruptures are sufficiently simple in their spatiotemporal history. As
seen with RSQSim ruptures in Chapter 4, complex spatiotemporal histories can render the true
hypocenter location irrelevant for use in the simplified models of the GMPEs. For example, the
RSQSim rupture shown in Figure 4.13b begins at one end of the rupture area, propagates
unilaterally to the other end, and then ruptured unilaterally back toward the hypocenter. In terms
of the ground motion patterns and hazard for this rupture, it would be better characterized as a
bilateral rupture. In the preceding chapters, we have developed a methodology for determining
an apparent CHD, which is able to capture rupture complexity and produce a CHD that would
better predict the directivity.
In this Chapter, we apply the mapping methodology we’ve established in the previous
chapters and verified in Chapter 4 with simulator results (for which the underlying assumptions
R
close
R
hypo
R
D
h
x
close
x
h
Rupture Area
s

114

are known) to the set global of 86 M
W
≥ 7 finite source inversions used in the previous chapters
and cataloged in Table 1.1.
5.2 FINITE SOURCE INVERSIONS
The parameters of interest in this study are the relative hypocenter locations (i.e., relative
to the ends of the rupture area), the components of the finite moment tensor (FMT, McGuire et
al., 2001; Chen et al., 2005), and the apparent hypocenter locations (which depend both on the
FMT and the slip distribution determined in Section 3.5).
Figure 5.2: An example of the trimming procedure used in this study for the 2011 M
W
7.6
Kermadec Islands region earthquake with a strike of 170°, dip of 52°, and rake of -97° (Hayes,
USGS website; also shown in Figure 3.1 and Figure 3.2). The slip distribution is plotted along
with rupture times contoured in white dashed lines and labeled in seconds. This inversion result
has The solid, bold box outlines the rupture area of the trimmed version of this finite source
inversion. The centroid is plotted as the red triangle, the hypocenter is shown as the grey circle,
the characteristic dimensions are shown as the ellipse (grey dotted ellipse is the untrimmed
model), and the directivity vector is plotted as the red arrows at the hypocenter (smaller, dashed
red arrow is for the untrimmed model).
As explained in Section 3.3, finite source inversions often have poorly-resolved areas
along the edges of the rupture with little to no slip. This can introduce bias in calculating the
parameters of interest. We therefore use the trimmed versions of the models for this study to
decrease bias due to noise in the tail of the source function. For example, Figure 5.2 shows the
difference between a trimmed and untrimmed finite source inversion for the 2011 M
W
7.6

115

Kermadec Islands region earthquake (Hayes, USGS website; also shown in Figure 3.1 and
Figure 3.2). The centroids remain unchanged; however, the second moment parameters of the
FMT are more sensitive to the outliers. For this reason, the untrimmed directivity vector is
smaller (the dashed red arrow at the hypocenter in Figure 5.2) than the trimmed version (the
solid red arrow at the hypocenter). Additionally, the location of the hypocenter with respect to
the ends of the rupture change when the fault is trimmed, which could decrease an artificial bias
in the true hypocenter distribution as well (i.e., the hypocenter in the untrimmed model is located
0.8% from the center of the rupture compared to 13.8% from the center for the trimmed model).
Figure 5.3 (low-order moments) and Figure 5.4 (higher-order moments) show the difference
trimming the finite source inversions makes in the FMT parameters. The spatial centroid is a
fixed location; therefore, we plot the distance between the untrimmed and trimmed spatial
centroids (the x-axis in this case is simply the model number from 1 to 86). In both figures,
hypocenter locations are plotted so that 0 is the center of the rupture and 0.5 is an end, and in
each plot, the circles are scaled by their characteristic length and colored by magnitude. The low-
order moments remain reasonably stable (85% of the spatial centroids are less than 1 km from
one another). When trimming effects parameters, it tends to change them in the following way:
(1) Push true hypocenters toward the ends of ruptures,
(2) Slightly decrease the characteristic length and duration,
(3) Increase the directivity vector and push apparent hypocenters toward the ends of ruptures.

116

Figure 5.3: Comparison of the low-order components of the FMT and the hypocenter
distribution between untrimmed and trimmed finite source inversions. Each point is scaled by its
characteristic length and colored by magnitude. The black lines indicate a 1:1 relationship. The
spatial centroid is a fixed location in 3D space; therefore we plot the distance between the
untrimmed and trimmed models (the x-axis is simply the model number from 1 to 86).
10
20
10
22
10
20
Seismic Moment
Untrimmed [N*m]
Trimmed [N*m]
0 0.1 0.2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
Hypocenter Location
Untrimmed
Trimmed
20 40 60 80
10
0
Distance Between Centroids
Distance [km]
10
2
10
2
Temporal Centroid
Untrimmed [s]
Trimmed [s]
0.5
Magnitude [M
W
]
7 7.5 8 8.5 9
10
22
10
-5

117

Figure 5.4: Comparison of the higher-order components of the FMT and the apparent
hypocenter distribution between untrimmed and trimmed finite source inversions. As in Figure
5.3, each point is scaled by its characteristic length, colored by magnitude, and the black lines
indicate a 1:1 relationship.
5.3 HYPOCENTER DISTRIBUTION
The location of the hypocenter with respect the ends of the rupture is a first-order
approximation of directivity, and as shown in Figure 5.1, the directivity correction factors
10
2
10
2
Characteristic Length
Untrimmed [km]
Trimmed [km]
10
2
10
2
Characteristic Duration
Untrimmed [s]
Trimmed [s]
0 0.2 0.4 0.6 0.8 1
0
0.6
1
Directivity Parameter
Untrimmed
Trimmed
0 0.1 0.2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
Apparent Hypocenter Location
Untrimmed
Trimmed
0.5
Magnitude [M
W
]
7 7.5 8 8.5 9
0.8
0.4
0.2

118

depend on the location of the hypocenter within the rupture area. In Chapter 3, we calculated the
hypocenter distributions from finite source inversions and simulation results. These are full
source representations; however, a complete source description may not always be necessary to
determine such a low-order feature as the location of the hypocenter with respect to its rupture
area. This can also be inferred from the location of the mainshock with respect to its spatial
distribution of early aftershocks. Because early aftershocks tend to occur along the perimeter of
the rupture zone and near structural complexities where stress concentrations may still be high
(Scholz, 1990), they can be used to estimate the extent of the rupture and then the mainshock can
be located with respect to this estimated rupture extent. Here we present both the hypocenter
distributions inferred from aftershock locations as well as the hypocenter distribution calculated
from the finite source inversions.
5.3.1 Inferred from Aftershock Locations
Henry and Das (2001) used a global catalog of 105 M
W
> 6.3 events, 67 of which are dip-
slip and 38 are strike-slip to study the properties of early aftershocks. In their study, Henry and
Das (2001) use the methodology of Pegler (1995) to determine rupture zones from relocated
aftershocks within the first 24 hours and first 30 days. We present their findings using the first 24
hours worth of aftershocks in Figure 5.5 (symmetrized to match the hypocenter distributions
presented throughout this paper). Through Monte Carlo simulation for randomly selected
hypocenter locations from a uniform distribution, we find the mean and 0.1 and 0.9 quantiles
plotted as the solid and dashed black lines (respectively) in Figure 5.5. In their study, they claim
the hypocenter distribution is random (Henry and Das, 2001). For dip-slip ruptures, however,
less than 4% of the Monte Carlo simulations have more events in the -0.05 to 0.05 range than the
distribution determined by Henry and Das (2001).

119

Figure 5.5: The Henry and Das (2001) symmetric hypocenter distribution for a set of 105 M
W
>
6.3 earthquakes (38 strike-slip and 67 dip-slip), where the rupture end points were determined
from the first 24 hours of aftershocks. The solid and dashed black lines are the mean and 0.1 and
0.9 quantiles from 1000 Monte Carlo simulations of randomly draw samples from a uniform
distribution.
We examined an additional 8 M
W
> 7 strike-slip ruptures in a similar way. For events
within the first 24 hours, we either used relocated positions from Hauksson et al. (2012) or
relocated the aftershocks using Hypocentroidal Decomposition (Jordan and Sverdrup, 1981),
−0.5 0 0.5
0
5
10
All
−0.5 0 0.5
0
1
2
3
4
5
Strike−Slip
Number of Events
−0.5 0 0.5
0
5
10
Dip−Slip
Hypocenter Location

120

estimated a fault rupture length from the characteristic length of the aftershock locations, and
project the hypocenter on the principle axis of the second spatial moment tensor of the aftershock
locations. Figure 5.6 shows an example of this methodology and Figure 5.7 shows the resulting
combined hypocenter locations for strike-slip events from Henry and Das (2001) and this study.
Figure 5.6: An example for the 1999 M
W
7.2 Duzce, Turkey earthquake of the procedure used to
estimate rupture dimensions and relative hypocenter location. The dots are the relocated
positions for 24 hours of aftershocks with their error ellipses and thin, black lines point to their
starting positions. The bold red line is the rupture plane estimated from the aftershock
distribution and the bold black lines are surface traces for two finite fault inversions (for
comparison). The hypocenter and projected hypocenter are the red and grey stars, respectively.
1999 M
W
7.2 Duzce, Turkey
30.5˚ 31˚ 31.5˚
40.5˚
41˚

121

Figure 5.7: The combined aftershock inferred hypocenter distribution for the 45 strike-slip
events from Henry and Das (2001) and this study.
5.3.2 Determined from Finite Source Inversions
Mai (2005) calculated the hypocenter distribution from a global set of 80+ finite fault
inversions for 50+ earthquakes, many of which are included in this study. He determined that the
hypocenters in finite source inversions are biased toward the center of the rupture (which we
describe as a “centroid-bias”). Our dataset of finite source inversions heavily overlaps with the
events used in their study. We therefore see similar results. The hypocenter distribution in the
trimmed finite source inversions is significantly centroid-biased for both strike-slip and dip-slip
events, concentrating within 15% of the center of the rupture (Figure 5.8). This on its own should
motivate the rejection of the uniform distribution as an input into forecasting models;
furthermore, as discussed in Section 0, the hypocenter can be a poor proxy for directivity in cases
where there is complex rupture propagation, which tend to decrease directivity, effectively
increasing the centroid-bias of the apparent hypocenter distribution.
−0.5 0 0.5
0
1
2
3
4
5
Hypocenter Location
Number of Events

Henry and Das (2001)
This Study

122

Figure 5.8: The finite source inversion symmetric hypocenter distributions for the set of 86
finite source inversions for 64 unique events (21 strike-slip and 43 dip-slip) used in this study
with their associated mean and 0.1 and 0.9 quantiles plotted as the solid and dashed black lines
respectively determined from Monte Carlo simulation.
5.4 DIRECTIVITY DISTRIBUTION
The directivity parameter of the finite moment tensor (FMT, McGuire et al., 2002; Chen
et al., 2005) is determined from a mixed, second-order polynomial moment of the source space-
time function as outlined in Chapter 2. This polynomial moment essentially tracks the movement
−0.5 0 0.5
0
5
10
15
All
−0.5 0 0.5
0
5
10
Strike−Slip
Number of Events
−0.5 0 0.5
0
5
10
15
Dip−Slip
Distance Along Strike

123

of the spatial centroid with time. For a simplified 1D rupture with constant rupture velocity and
instantaneous, uniform slip, the centroid for a perfectly unilateral rupture will move at the same
velocity as the rupture velocity. For a perfectly bilateral rupture of the same type, the centroid
will not move at all as the rupture front propagates out. McGuire et al. (2002) therefore defined
the directivity parameter D to be a ratio of the instantaneous centroid velocity (calculated from
the mixed space-time moment of the FMT) and the characteristic rupture velocity (both of which
are defined in Chapter 2, Section 2.2), ranging from 0 for bilateral and 1 for unilateral for zero
rise time (the limit on the directivity parameter is < 1 when there is a finite rise time).
McGuire et al. (2002) used a global dataset of 25 M
W
≥ 7 earthquakes and inverted for
the FMT directly from seismic waves. They found that the majority of ruptures have large
directivity parameters (Figure 5.9) and hypothesized that a uniform CHD could lead to this
distribution of directivity parameters. Additional studies have calculated the directivity
parameter in the same way (Clévédé et al., 2004; Llenos and McGuire, 2007). We compiled 41
directivity parameters from the literature (McGuire et al., 2002; Clévédé et al., 2004; Llenos and
McGuire, 2007) for 35 unique M
W
≥ 7 events. Without a way to properly assess the quality of
competing results for the same event, each result for the same event is given 1/n weight, where n
is the number of directivity parameters for a given event. Figure 5.9 shows the directivity
distribution for these direct inversion results. We use Monte Carlo simulation to determine the
expected number of events in each bin for a random uniform distribution of hypocenters, as
suggesting in McGuire et al. (2002).

124

Figure 5.9: The distribution of 41 directivity parameters computed by the direct inversion of
seismic data for 35 unique M
W
≥ 7 events collected from literature (McGuire et al., 2002;
Clévédé et al., 2004; Llenos and McGuire, 2007). In cases where there is more than one result
for a single event, each result is given 1/n weight in the histogram above, where n is the number
of results for a given event. The solid and dashed red lines are the mean and 0.1 and 0.9
quantiles, respectively, from a set of 1000 Monte Carlo simulations for a random uniform
hypocenter distribution with a Beta distribution of slip assigned according to the results in Table
3.4 for the symmetric distributions.
The directivity distribution for events collected from the literature shows a lack of events
in the highest directivity bin compared what is expected from a uniform distribution of
hypocenters. Less than 10% of the Monte Carlo simulations for a random uniform distribution
have a fewer number of directivity parameters in the range 0.9-1 than those in Figure 5.9. If this
discrepancy can be attributed to a centroid-bias in the CHD for observed earthquakes, then using
a uniform CHD would overpredict the directivity.
McGuire et al. (2002) and Clévédé et al. (2004) calculated the FMT using the inversion
methodology of McGuire et al. (2001); however, both studies showed the source space-time
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
2
4
6
8
10
12
14
16
18
20
22
Directivity Parameter
Number of Events

125

functions from finite source inversions can also be used to find the FMT. McGuire et al. (2002)
used a set of 22 finite source inversions to supplement their dataset of directivity parameters, and
Clévédé et al. (2004) calculated the FMT for 3 finite source inversions of the 1999 M
W
7.6 Izmit,
Turkey earthquake in order to compare to their direct inversion results. We compiled a catalog of
86 finite source inversions for 64 unique M
W
≥ 7 events to use in this study (Table 1.1).
The resulting directivity distribution from the trimmed finite source inversions is shown
in Figure 5.10. As in Figure 5.9, for the cases in which there are multiple inversions for the same
event, each inversion is assigned equal weight in the histogram (i.e., 1/n, where n is the number
of inversions for a given event).

126

Figure 5.10: The finite source inversion directivity distributions for the set of 86 trimmed finite
source inversions of 65 unique events (21 strike-slip and 43 dip-slip). In cases where there are
more than one inversion for a unique event, each inversion is given equal weight in the
histograms. The solid and dashed red lines represent the mean and 0.1 and 0.9 quantiles
respectively from 1000 Monte Carlo simulations for a simplified 1D rupture with a random
uniform distribution of hypocenters, a Beta distribution of slip assigned according to the results
in Table 3.4 for the symmetric distributions (α = β = 1.92, Figure 3.7), and the average rise time
ratio ρ = 0.13.
The directivities for the finite source inversions are lower than those of the direct
inversions in Figure 5.9. Not only do the finite source inversions significantly lack directivity
0 0.2 0.4 0.6 0.8 1
0
20
40
All
0 0.2 0.4 0.6 0.8 1
0
5
10
15
Strike−Slip
Number of Events
0 0.2 0.4 0.6 0.8 1
0
10
20
30
Dip−Slip
Directivity Parameter

127

parameters in the 0.9-1 bin, but they also have a significantly higher number of very low
directivity parameters (i.e., D < 0.5). This directivity distribution would not be well represented
by a uniform CHD. Our goal is to determine which CHD (i.e., the “apparent” CHD) would be
required to better represent this observed directivity distribution.
5.5 APPARENT HYPOCENTER DISTRIBUTION
In Chapter 4, we presented and verified the methodology for mapping between directivity
parameter and hypocenter location using simulation results from CyberShake (Graves et al.,
2011) and RSQSim (Dieterich and Richards-Dinger, 2010; Richards-Dinger and Dieterich,
2012). We saw that for a simple time history, the apparent hypocenter distribution represents the
true hypocenter distribution, and that for complex ruptures, the apparent hypocenter distribution
deviates from the true hypocenter distribution, but is better able to characterize the directivity of
the complex rupture. In this way, the apparent CHD provides a better estimate of the directivity
than the true CHD. We apply this mapping to the set of directivity parameters for the finite
source inversions in Figure 5.10 using the parameters from the best-fitting Beta distribution to
the symmetric profiles presented in Chapter 3, Table 3.4. The results of this mapping procedure
are shown in Figure 5.11.

128

Figure 5.11: The apparent hypocenter distribution calculated from the 86 trimmed finite source
inversions for 65 unique M
W
≥ 7 events (21 strike-slip and 43 dip-slip). The mean and 0.1 and
0.9 quantiles are calculated from Monte Carlo simulations for a random uniform distribution of
hypocenters and a Beta distribution of slip assigned according to the results in Table 3.4 for the
symmetric distributions (α = β = 1.92, Figure 3.7), and the average rise time ratio ρ = 0.13.
The difference between the apparent CHD (Figure 5.11) and true CHD (Figure 5.8) is a
shift of the hypocenters toward the center of the rupture, creating a centroid-bias similar to, but
not as extreme as the shift observed in RSQSim ruptures (Figure 4.7 and Figure 4.10), which we
determined to be a product of the complexity in RSQSim rupture propagation. Some finite
−0.5 0 0.5
0
5
10
15
Apparent Hypocenter Distributions
All
−0.5 0 0.5
0
5
10
Strike−Slip
Number of Events
−0.5 0 0.5
0
5
10
15
Dip−Slip
Distance Along Strike

129

source inversions do indeed show spaciotemporal complexity (albeit, not on the order of the
complexity shown in RSQSim; e.g., Figure 4.13b). Figure 5.12 shows the rupture propagation
for the 2001 M
W
7.8 Kunlun earthquake in southern Qinghai, China (Hayes, personal
communication). In a 12 second window, the rupture covers up to ~145 km. Long rise times in
this rupture (up to 24 seconds) contribute to backward portions of the rupture continuing to
contribute to the FMT as it propagates forward, essentially slowing down the instantaneous
centroid velocity (i.e., 𝝁
(!,!)
𝜇
(!,!)
in equation 2.2) and thereby decreasing the directivity
parameter to 0.98. Consequently, its original hypocenter location of -0.45 is changed to an
apparent hypocenter location of -0.36.
Both the true and apparent CHDs for finite source inversions significantly deviate from a
uniform distribution. The apparent CHD broadens this region compared to the true CHD, and in
general shifts hypocenters toward the center. This is particularly noticeable for dip-slip ruptures.
In the next section we will explore possible sources of this increased centroid-bias in the
apparent CHD.

130

Figure 5.12: Rupture propagation of the 2001 M
W
7.8 Kunlun earthquake in southern Qinghai, China (Hayes, personal
communication). The top is the final moment distribution with rupture initiation times contoured in white and labeled accordingly.
The grey dot is the hypocenter, red triangle is the spatial centroid, the ellipse represents the characteristic dimensions, and the red
arrow is the directivity vector. Each time slice is incremented by 12 seconds and is on the same scale as the final moment distribution
at the top.

131

5.6 DISCUSSION
Both the true and apparent hypocenter distributions show a significant deviation from the
uniform distribution. We investigate whether the increased centroid biased in the apparent CHD
is an artifact of our assumptions, and show an example of the effect a centroid-biased CHD can
have on a ground motion forecast by using the apparent CHD determined from finite source
inversions as an input into the deterministic forecasting framework of CyberShake (Graves et al.,
2011).
5.6.1 Centroid-Bias of the CHD
Both the true and apparent hypocenter distributions show a centroid-biased distribution,
significantly deviating from uniform in the center of the rupture area. In untrimmed finite source
inversions, this could be caused by the initial model setup. Without prior knowledge of the
location of the hypocenter with respect to the ends of the rupture, many researchers initialize
their models to have hypocenters in the center of a fault area that is several times larger than a
rupture length expected from a chosen scaling relation. They can then adjust the size and location
of the rupture plane(s) iteratively based on their results. If the centroid-bias is due to this initial
parameterization in finite source inversions, then the trimming procedure should remove this
bias. We note, however, that the trimming procedure is prohibited from removing any areas with
more than 50 cm of slip, regardless of the magnitude of peak slip. If even areas of 50 cm
displacements are artifacts in large-slip events, then the trimming procedure could be prevented
from removing all the bias. We therefore re-trim the data to allow any areas along the edge of a
fault to be trimmed if they are less than 10% of the peak slip without the condition that areas of
50 cm slip or larger must be retained. Figure 5.13 shows the true and apparent CHDs for the
more aggressively trimmed models.

132

Figure 5.13: The apparent CHD for all faulting types for the originally-trimmed models which
removes regions along the edges of the rupture with > 10% of the peak slip, but retains any
regions with ≥ 50 cm slip (top) versus the apparent CHD the modified trimming which removes
regions along the edges of the rupture with < 10% of the peak slip, regardless of slip magnitude
(middle). The bottom plot shows the difference between the two distributions (minimum-
constrained trimming – non-constrained trimming). Red solid and dashed lines are the mean and
0.1 and 0.9 quantiles for a set of Monte Carlo simulations assuming a random uniform
distribution of hypocenters.
Another possible source of centroid-bias is for events with small aspect ratios. Our
mapping procedure approximates the rupture as one dimensional. Chapter 2, Section 2.5
−0.5 0 0.5
0
5
10
15
All (Original Trimming)
−0.5 0 0.5
0
5
10
15
Number of Events
−0.5 0 0.5
−2
0
2
Distance Along Strike
All (Modified Trimming)
All (Original - Modified)

133

discusses the influence aspect ratio has on the components of the directivity parameter. Figure
5.14 shows the mapping of a uniform distribution of hypocenters for a 2D rupture with and
aspect ratio of 3 using the 1D mapping procedure. As shown in this figure, aspect ratio causes
misassignment in the bins toward the ends of the rupture where the relationship between
nucleation point and directivity becomes highly non-linear. This effect, therefore, cannot account
for the increased number directly in the center of the fault.
Figure 5.14: The apparent CHD for a simulated set of ruptures with a uniform distribution of
hypocenters both along strike and down dip for a rupture with an aspect ratio of 3. The red bold
and dashed lines are the mean and 0.1 and 0.9 quantiles calculated from Monte Carlo simulations
for a random uniform distribution of hypocenters on a 1D rupture.
The region of the apparent CHD where there is significant deviation from uniform is
within ±0.1 and between -0.45 to -0.2 (and with symmetry, 0.2-0.45). Through numerical
simulation for a constant rupture velocity and rise time with a uniform slip rate, we find that the
1D approximation holds within the relevant range along strike for ruptures with aspect ratios >
~3.5, which corresponds to an upper bound on d
2
of < ~0.08. Figure 5.15 shows the true and
apparent hypocenter distributions for all faulting types with a > 3.5. The centroid-bias is
persistent. Figure 5.14 and Figure 5.15 discount aspect ratio as a cause of the centroid bias.
−0.5 0 0.5
0
20
40
60
80
Distance from Center
Number of Events

134

Figure 5.15: True (top) and apparent (bottom) CHDs for all faulting types with aspect ratios >
3.5. Solid and dashed lines mark the mean and 0.1 and 0.9 quantiles of a Monte Carlo
distribution for a uniform random hypocenter distribution.
As shown in Section 4.2, the mapping depends on the assumed slip distribution. To test
whether the centroid-bias could be attributed to an incorrect slip distribution, we map the
directivity parameters using a uniform distribution of slip. The resulting apparent CHD is shown
in Figure 5.16.
−0.5 0 0.5
0
1
2
3
4
5
All
−0.5 0 0.5
0
1
2
3
4
5
All
Number of Events
Distance Along Strike

135

Figure 5.16: The apparent CHD for all faulting types mapped using the best-fitting Beta
distribution to the average slip profile (top) versus the apparent CHD mapped using a uniform
slip distribution (middle). The bottom plot shows the difference between the two distributions
(Beta mapped CHD – uniform mapped CHD). Red solid and dashed lines are the mean and 0.1
and 0.9 quantiles for a set of Monte Carlo simulations assuming a random uniform distribution of
hypocenters.
Both the difference in slip distribution (either from trimming methodology or assuming
an incorrect slip distribution) and the effect of aspect ratio represent deviations of our simplified
1D rupture model with reality. Another way in which our simplified rupture model could
−0.5 0 0.5
0
5
10
15
All (Beta Slip)
−0.5 0 0.5
0
5
10
15
Number of Events
−0.5 0 0.5
−2
0
2
Distance Along Strike
All (Uniform Slip)
All (Beta - Uniform)

136

possibly introduce bias in the apparent CHD is through the assumed rise time. Rise time is a
poorly-constrained property of earthquake rupture because of the tradeoff between rupture
velocity, slip amplitude, and rise time in finite source inversions (e.g., Olson and Anderson,
1988; Hernandez et al., 2001). For example, the USGS finite source inversion for the 2001 M
W

7.8 Kunlun earthquake (Gavin Hayes, personal communication) has a maximum rise time of 24
seconds; whereas, the Wen et al. (2009) finite source inversion for the same event has a
maximum rise time of less than 3 seconds. Although researchers are unable to agree on the form
of the rise time function, there must be a finite rise time associated with coseismic slip. Figure
5.12 showed an example of how long rise times decreases the directivity parameter for the 2001
M
W
7.8 Kunlun earthquake, in which the average rise time ratio is 0.11. Figure 4.2 shows the
difference between instantaneous slip (ρ = 0) and a finite rise time (ρ = 0.25). The larger the rise
time ratio, the less centroid-biased the resulting CHD should be. We therefore assume a rise time
ratio of ρ = 0.25 the rupture duration (compared to the ρ = 0.13 average used in the mapping
process). Figure 5.17 shows the resulting apparent CHD for all faulting types compared to the
mapping using the average rise time ratio. Assuming a higher rise time ratio decreases the
centroid-bias of the apparent CHD; however, like the changes associated with the slip
distribution (Figure 5.16), these changes are not statistically significant.
The increased centroid-bias of the apparent CHD compared to the true CHD is a
consistent feature that cannot be explained as an artifact of the trimming process or of the
assumptions about slip or rise time. This difference between the true and apparent CHDs
therefore, reflects true complexity in the ruptures. For observed earthquakes, the true CHD is not
an adequate proxy for directivity and the apparent CHD is a better estimate of the directivity
associated with these ruptures.

137

Figure 5.17: The apparent CHD for all faulting types mapped using the average rise time ratio
(0.13) from the finite source inversions (top) versus the apparent CHD mapped using a large rise
time ratio of 0.25 (middle). The bottom plot shows the difference between the two distributions
(average rise time ratio CHD – large rise time ratio CHD). Red solid and dashed lines are the
mean and 0.1 and 0.9 quantiles for a set of Monte Carlo simulations assuming a random uniform
distribution of hypocenters.
−0.5 0 0.5
0
5
10
15
All (Average Rise Time Ratio)
−0.5 0 0.5
0
5
10
15
Number of Events
−0.5 0 0.5
−2
−1
0
1
2
Distance Along Strike
All (Modified Rise Time Ratio)
All (Average - Modified)

138

5.6.2 The CHD and Hazard Estimates in CyberShake
The persistence of the centroid-bias of the CHD leads us to then investigate the potential
effect this could have on hazard estimates. We use the forecasting model of CyberShake (Graves
et al., 2011), which is a deterministic ground motion model based on the Uniform California
Earthquake Rupture Forecast, Version 2 (UCERF2, Field et al., 2009), to see how the CHD
changes the forecast. CyberShake uses the rupture set and associated probabilities in UCERF2
and then builds on these ruptures by creating “rupture variations”, which are variations in slip
distributions and hypocenter locations to represent the aleatory variability observed in nature.
CyberShake slip distributions are generated by perturbing an initially uniform field with slight
tapers at the end to match a specified wavenumber spectrum seen in finite source models (Mai
and Beroza, 2002; Graves and Pitarka, 2010). Hypocenters are assigned for each rupture
independently from the slip distribution, are located every 20 km along strike, and are symmetric
about the center of the rupture (Graves and Pitarka, 2010; Graves et al., 2011). Ground motions
from these rupture variations are then calculated based on full 3D wave propagation using the
Southern California Earthquake Center’s Community Velocity Model, version 4.26 (Lee et al.,
2014).
The mapping process to convert from directivity parameter to apparent hypocenter
depends on the assumed slip distribution and rise time ratio. We therefore map the apparent CHD
for finite source inversions and to an equivalent CHD based on the average slip profile (α = β =
1.21, Figure 3.18) and average rise time ratio (ρ = 0.03) of CyberShake ruptures. We find the
best-fitting Beta distribution to the resulting apparent CHD (Figure 5.18). This CHD is used as
input into the CyberShake framework to modify the probabilities of rupture variations based on
their hypocenter locations.

139

Figure 5.18: The apparent CHD for finite source inversions for all faulting types mapped to an
equivalent CHD for CyberShake is shown as black circles. The best-fitting Beta distribution (α =
β = 10.03) is shown as the blue dashed line and is used as the input CHD for CyberShake.
The results of this modified CHD for CyberShake are shown in Figure 5.19 through
Figure 5.21, which show the ratio of the probability of exceeding a 0.1 to 0.3 g for our modified
CHD compared to the original, uniform CHD. These results for the CHD determined from finite
source inversions show that the probabilities can change by more than 30% in either direction.
Changes in the hazard of only ~10% for ruptures with high recurrence intervals (i.e., less than
~250 years) were shown to be important for building code applications in the Uniform California
Earthquake Rupture Forecast, Version 3 (Field et al., 2014; Field and Jordan, 2015).
The effect of the CHD on the hazard increases from approximately ±15% at 0.1 g to over
±30% at 0.3 g. As ground motion increases, the overall effect on the hazard tends toward
lowering the probability of exceedance with respect to a uniform CHD, with obvious increases in
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Distance Along Strike
Number of Events

Apparent CHD
Beta Fit (α=β=10.03)

140

the San Gabriel Mountains that concentrate around the Sierra Madre and San Gabriel fault zones.
The CyberShake hazard decreases most significantly in the coastal mountain areas (i.e., the Santa
Monica and Santa Margarita Mountains). Understanding the detailed features of this map will
require an extensive deaggregation analysis and examining the specific assumptions and 3D
structural features in CyberShake, which is beyond the scope of the current study where we seek
only to show an example of how the CHD affects a hazard model.

141

Figure 5.19: CyberShake results for the ratio of the probability of exceeding 0.1 g between the
modified CHD calculated from the finite source inversions (Figure 5.18) and the uniform CHD.
POE Ratio (0.1 g)

142

Figure 5.20: CyberShake results for the ratio of the probability of exceeding 0.2 g between the
modified CHD calculated from the finite source inversions (Figure 5.18) and the uniform CHD.
POE Ratio (0.2 g)

143

Figure 5.21: CyberShake results for the ratio of the probability of exceeding 0.3 g between the
modified CHD calculated from the finite source inversions (Figure 5.18) and the uniform CHD.
5.7 CONCLUSION
In this study, we used a global dataset of 86 finite source inversions for 64 M
W
≥ 7
events, consisting of 43 dip-slip and 21 strike-slip events. We trimmed the inversions to decrease
bias due to poorly-constrained regions along the edges of ruptures with low or no slip, and
calculated their hypocenter (Figure 5.8), directivity (Figure 5.10), and apparent hypocenter
POE Ratio (0.3 g)

144

(Figure 5.11) distributions. We found a centroid-bias in the hypocenter distribution for both the
finite source inversions (consistent with the findings of Mai, 2005) and the hypocenter locations
inferred from early aftershock locations. We find a difference between the distributions of
directivity parameters calculated from direct seismic inversion (Figure 5.9) and those calculated
from finite source inversions (Figure 5.10). The directivity parameters calculated from direct
inversion show higher directivity than those calculated from finite source inversions. We also see
a lack of high-directivity events (D > 0.9) compared to what is expected for a simple rupture with
constant rupture velocity, a constant rise time, uniform slip rate, and a uniform distribution of
hypocenters. The centroid-bias of the hypocenter distribution and the comparative low directivity
of the finite source inversions lead to a centroid-biased apparent CHD.
The centroid-biased nature of the apparent CHD is a persistent feature and reflects the
underlying true CHD as well as spatiotemporal complexity in observed earthquakes compared to
our simplified 1D rupture. The apparent CHD better represents the directivity than the true CHD
because it takes into account the slip distribution as well as temporal complexity. When this
centroid-biased CHD is used as an input into the existing CyberShake (Graves et al., 2011)
framework, the probabilities of exceedance for 0.3 g fluctuate by more than ±30%.

145

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Abstract (if available)

Abstract Forecasting directivity in large earthquakes in terms of the conditional hypocenter distribution. A dissertation presented to the faculty of the USC Graduate School of the University of Southern California in partial fulfillment of the requirements for the degree Doctor of Philosophy (Geological Sciences).

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Creator Donovan, Jessica R. (author)

Core Title Forecasting directivity in large earthquakes in terms of the conditional hypocenter distribution

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Degree Doctor of Philosophy

Degree Program Geological Sciences

Publication Date 07/22/2015

Defense Date 06/11/2015

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Tag directivity,Forecasting,OAI-PMH Harvest,probabilistic seismic hazard analysis,PSHA,seismic hazard,source directivity

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Advisor Jordan, Thomas H. (committee chair), Lynett, Patrick J. (committee member), Miller, Meghan S. (committee member)

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