Anisotropic Extensions of Space-time Point Process Models for Earthquake Occurrences (1)

7/25/2019 Anisotropic Extensions of Space-time Point Process Models for Earthquake Occurrences (1)

1/85

University of California

Los Angeles

Anisotropic Extensions

of Space-time Point Process Models

for Earthquake Occurrences

A dissertation submitted in partial satisfaction

of the requirements for the degree

Doctor of Philosophy in Statistics

by

Ka Leung Wong

2009


2/85

cCopyright byKa Leung Wong

2009


3/85

The dissertation of Ka Leung Wong is approved.

Qing Zhou

Jan de Leeuw

David Jackson

Frederic Paik Schoenberg, Committee Chair

University of California, Los Angeles

2009

ii


4/85

To Ivy

iii


5/85

Table of Contents

1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Focal Mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Analysis of Aftershock Spatial Distribution . . . . . . . . . . . . 7

3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3 Stochastic Aftershock Assignment . . . . . . . . . . . . . . . . . . 11

3.4 Relative Location of Aftershocks with respect to Mainshock Focal

Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.5 Tapered Pareto-Wrapped Exponential . . . . . . . . . . . . . . . . 13

3.6 Alternative Aftershock Spatial Distributions . . . . . . . . . . . . 17

3.6.1 The Normal Modell . . . . . . . . . . . . . . . . . . . . . . 17

3.6.2 The Kagan-Jackson Model . . . . . . . . . . . . . . . . . . 18

3.7 Residual Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.7.1 Quadrat Residuals . . . . . . . . . . . . . . . . . . . . . . 19

3.7.2 Weighted K-function . . . . . . . . . . . . . . . . . . . . . 19

3.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.8.1 Fit of proposed models . . . . . . . . . . . . . . . . . . . . 22

3.8.2 Diagnostics and Model Comparison . . . . . . . . . . . . . 23

3.9 Additional Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.9.1 Scaled Distance . . . . . . . . . . . . . . . . . . . . . . . . 25

iv


6/85

3.9.2 Relocation Catalog . . . . . . . . . . . . . . . . . . . . . . 26

3.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Focal Mechanism-dependent Anisotropic Spatial Kernel for Space-

Time Earthquake Point Process Models . . . . . . . . . . . . . . . . 40

4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Epidemic Type Aftershock Sequence Models . . . . . . . . . . . . 42

4.2.1 Anisotropic Clustering . . . . . . . . . . . . . . . . . . . . 44

4.3 Fault Plane Strike Angle and Relative Aftershock Angle . . . . . . 46

4.4 Anisotropic Extensions of ETAS Models . . . . . . . . . . . . . . 47

4.5 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . 49

4.6 Goodness-of-fit and Diagnostic Methods for Spatial and Spatial-

temporal Point Process Models . . . . . . . . . . . . . . . . . . . 50

4.7 Deviance Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.8.1 Pearson residuals . . . . . . . . . . . . . . . . . . . . . . . 53

4.8.2 Comparison of Isotropic Pareto and Isotropic Tapered Pareto

Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.8.3 Impact of the Wrapped Exponential Distribution of Rela-

tive Angles Between Mainshocks and Aftershocks . . . . . 55

4.8.4 Models with only Strike-slip Mainshocks . . . . . . . . . . 56

4.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 66

v


7/85

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

vi


8/85

List of Figures

2.1 Beach ball diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Ternary diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1 Definition of relative aftershock location. . . . . . . . . . . . . . . 29

3.2 Scatterplot of aftershocks . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Survival function ofr . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Conditional histograms of . . . . . . . . . . . . . . . . . . . . . 33

3.5 Estimates of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6 Density plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.7 Quadrat residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.8 WeightedK-functions . . . . . . . . . . . . . . . . . . . . . . . . 36

3.9 Survival function ofr/L . . . . . . . . . . . . . . . . . . . . . . . 37

3.10 Estimates of, with respect to r/L . . . . . . . . . . . . . . . . . 38

3.11 Estimates of, from a relocation catalog . . . . . . . . . . . . . . 39

4.1 Fault plane strike angle and relative aftershock angle . . . . . . . 61

4.2 Pearson residuals for model (4.5), . . . . . . . . . . . . . . . . . . 62

4.3 Deviance residuals of (4.5) against (4.8). . . . . . . . . . . . . . . 63

4.4 Deviance residuals of (4.5) against (4.10). . . . . . . . . . . . . . . 64

4.5 Deviance residuals of (4.13) against (4.14), with only strike-slip

mainshocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

vii


9/85

List of Tables

3.1 ETAS parameters used in stochastic aftershock assignment . . . . 30

4.1 MLE and AIC for ETAS models. . . . . . . . . . . . . . . . . . . 59

4.2 MLE and AIC for ETAS models, with only strike-slip mainshocks. 60

viii


10/85

Acknowledgments

I owe my deepest gratitude to my advisor Rick Schoenberg. His kindness, men-

torship, and guidance are vital to the development and completion of this work.I am also grateful to the members of my committee: Qing Zhou, Jan de Leeuw,

and David Jackson. Special appreciation goes to David Jackson who has shared

with me his invaluable expertise in seismology.

I would like to extend my gratitude to Yan Kagan and Alex Veen for their

stimulating discussions, Mark Hansen for taking me under his wings early in my

career, and Glenda Jones for her briliant and dedicated service. Many of my

colleagues also deserve praises. I like to thank Chris Barr, David Diez, and Gong

Chen for their humor, support, and encouragement.

Lastly, I like to acknowledge my dear parents for their unwavering support,

and my lovely Ivy, to whom I owe everything.

ix


11/85

Vita

1981 Born, Hong Kong, China.

19992004 B.A. Architecture, University of California, Berkeley.

20042009 Ph.D. Statistics, University of California, Los Angeles.

20062008 Teaching Assistant, Statistics Department, University of Cali-

fornia, Los Angeles.

2008 Ph.D. Candidate in Statistics, University of California, Los An-

geles.

Publications and Presentations

Wong, Ka and Schoenberg, Frederic. On Mainshock Focal Mechanisms and

the Spatial Distribution of Aftershocks, Bulletin of the Seismological Society of

America, in press.

Intl Workshop on Statistical Seismology 2009 (Lake Tahoe, CA)

Invited paper: A focal mechanism-dependent anisotropic spatial kernel for ETAS

models.

Annual SCEC Meeting 2008 (Palm Springs, CA)

Contributed poster: Estimation of ETAS models for earthquake occurrences.

x


12/85

Abstract of the Dissertation

Anisotropic Extensions

of Space-time Point Process Models

for Earthquake Occurrences

by

Ka Leung Wong

Doctor of Philosophy in Statistics

University of California, Los Angeles, 2009

Professor Frederic Paik Schoenberg, Chair

Focal mechanism provides a reasonable approximation to an earthquakes rupture

mechanics in terms of its fault plane orientation and direction of slip. The first

part of this dissertation explores focal mechanism as a means to describe the

anisotropic spatial distribution of aftershocks. Based on empirical analysis of

aftershock patterns in Southern California seismicity, a spatial distribution for

the relative location of aftershocks with respect to mainshock focal mechanism

is proposed. When compared to alternative models for aftershock and seismicity

patterns, the proposed model appears to offer superior fit to Southern California

seismicity.

The second part proposes a general framework for extending space-time earth-

quake point process models to incorporate focal mechanism data via an anisotropicspatial kernel. Using the proposed model for relative aftershock locations as an

example, the effectiveness of using focal mechanism in modeling earthquake oc-

currences is assessed. In addition, a new residual method is proposed for assessing

xi


13/85

the relative performance of models to spatial and spatial-temporal point process

data. This graphical tool is used to illustrate the advantages and disadvantages

of extended ETAS models compared to alternative models and appears quite

effective.

xii


14/85

CHAPTER 1

Introduction

Focal mechanism provides a reasonable approximation to an earthquakes rupture

mechanics in terms of its fault plane orientation and direction of slip. This dis-

sertation explores focal mechanism as a means to describe the anisotropic spatial

distribution of aftershocks and extends a state-of-the-art earthquake point pro-

cess model to incorporate focal mechanism via an anisotropic spatial kernel. In

addition, new methods are proposed for the assessment of goodness-of-fit of such

spatial-temporal point process models, and these methods are applied to the com-

parison of branching point process models for earthquakes and their aftershocks

using recent seismological data from Southern California.

Chapter 2 provides an introduction to earthquake focal mechanism. Chapter 3

explores focal mechanism as a means to describe the anisotropic spatial distribu-

tion of aftershocks using seismological data from Southern California. A spatial

distribution is proposed for the relative location of aftershocks with respect to

mainshock focal mechanism. This model is compared to previously proposed

models based on the normal distribution and the squared cosine function. Using

residual analysis and weighted K-function as diagnostic measures, the normal and

squared cosine models are found to suffer from several serious problems, and that

the proposed distribution has features similar to both alternative models but fits

much better to Southern California earthquake data.

Chapter 4 proposes a general framework for extending space-time point pro-

1


15/85

cess earthquake models to incorporate focal mechanism via an anisotropic spatial

kernel. The proposed model for relative aftershock locations is as an example of

such a spatial kernel, and the effectiveness of using focal mechanism in mod-

eling earthquake occurrences is assessed. Some methods for assessing purely

spatial and spatial-temporal point process models are briefly summarized. Build-

ing upon some of these methods, a new residual method that involves inspecting

differences between competing models in contributions to the log-likelihood over

pixels is proposed. This graphical tool appears to be quite effective at portraying

the relative fit of models to space-time point process data, and is used to illus-

trate the advantages and disadvantages of extended ETAS models compared to

alternative models.

Chapter 5 concludes this dissertation and suggests important topics for future

research in this area.

2


16/85

CHAPTER 2

Focal Mechanism

This chapter provides a brief summary of earthquake focal mechanism; for further

details see Bullen and Bolt (1985) or Bolt (2006).

Earthquakes generally result from seismic failures and can be highly non-linear

and fractal-like processes (Kagan 1997, Turcotte 1989). The observed seismic

wave patterns of most earthquakes can be effectively explained by a double couple

as if the event were equivalent to a single nonelastic slippage on a single fault

plane (Bullen and Bolt, 1985). The focal mechanism of an earthquake, which

includes the direction of slip and the orientation of the fault on which it occurs,

provides a reasonable approximation to an events rupture mechanics (Bullen and

Bolt, 1985).

Focal mechanisms are derived from a solution of the moment tensor of an

earthquake. A seismic moment tensor is a 3 3 symmetric matrix estimatedby an analysis of observed seismic waveforms. Its determinant represents the

moment, or size, of an earthquake, and its eigenvectors give the directions of the

earthquakes N- (neutral), T- (tension), and P- (compression) axes relative to its

hypocenter. Inferred from the moment tensor are two ambiguous nodal planes,

one of which is the fault plane and the other is its perpendicular auxiliary plane.Differentiating the two planes requires knowledge of the events lateral orientation

(left or right), and such information can be inferred from local geological evidence

and/or the events aftershock pattern. The task of resolving fault plane ambiguity

3


17/85

in Southern California is aided by the large presence of two major right-lateral

strike-sip fault zones, the San Andreas and San Jacinto fault zones (Sanders 1989,

1993).

Depending on an earthquakes faulting style, each earthquake can be classified

into three major categories: strike-slip, normal, and reverse. The majority of

earthquakes in Southern California are called strike-slip events, which are events

with nearly vertical fault planes and nearly horizontal slippage. Focal mechanisms

are typically displayed graphically using so-calledbeach balldiagrams. The beach

ball diagram in Figure 2.1 corresponds to the focal mechanism of a strike-slip

event whose nodal planes are exactly vertical. The black and white quadrants

represent the tension and compression zones, respectively, containing the T- and

P- axes. If the fault is right-lateral, then the fault plane is represented by nodal

plane 1.

When many focal mechanisms are displayed collectively, a ternary diagram

(Frohlich 1992, Frohlich 2001) may be a useful graphical device. A ternary di-

agram projects each focal mechanism as a point onto a triangle in which each

corner represents a type of earthquake, slip-slip, normal, or reverse. Figure 2.2illustrates a ternary diagram for focal mechanisms in a typical Southern Califor-

nia focal mechanism catalog. Such a technique therefore allows one to summarize

a large set of focal mechanisms in a compact fashion. Minor area distortion is

introduced in the projection step.

Estimating focal mechanisms requires an extensive network of seismic stations

and data on focal mechanism have not been historically available in large quan-

tities. Such data recently become more widely available due to advances in in-

strumentation technology and inversion algorithms. For events located in or near

Southern California, focal mechanism estimates are provided by the Southern

4


18/85

California Earthquake Data Center (SCEDC) which has implemented an auto-

matic inversion process since 1999 (Clinton et. al. 2006). Owing to the difficulty

in the inversion process, focal mechanism solutions are often subject to quality

issues. Factors that affect the quality of a solution include an events epicentral

location relative to the monitoring stations, magnitude, and depth (Clinton et.

al. 2006). For instance, a small-magnitude event on the edge of a network may

not have a successful inversion. Published solutions on SCEDC are labeled with

a quality grade that reflects the precision of the inversion process. The three

quality grades are A, B, and C, in decreasing order of their precision.

Taxis

Paxis

nodal plane 1

nodal plane 2

Figure 2.1: A beach ball diagram corresponding to the focal mechanism of a

strike-slip earthquake.

5


19/85

Figure 2.2: A ternary diagram showing the distribution of earthquake types in a

typical Southern California data set. The dotted lines demarcate the definitions

of strike-slip, normal, and thrust, used in Frohlich (1992) and Frohlich (2001).

6


20/85

CHAPTER 3

Analysis of Aftershock Spatial Distribution

3.1 Background

Since Utsu (1969) first famously noted that aftershock regions tend to be ellipti-

cal, it has been widely observed that aftershock activities are often non-circular.

A convenient model used to describe aftershocks anisotropic spatial distribution

is the bivariate normal distribution. Ogata (1998) uses the normal model to

approximate the ellipsoidal contours of aftershock spatial distribution. Kagan

(2002) uses a similar method to measure the mainshock focal zone size. While

the normal distribution may serve as an acceptable first-order approximation for

the distribution of aftershock locations, there is scant evidence of its optimality.

Kagan (2002) lists a few reasons for which that the normal model cannot be

exact. For instance, aftershocks may happen at large distances from the main-

shock where no other traces of earthquake rupture can be found. In addition,

aftershocks exhibit the feature of secondary clustering and are not mutually in-

dependent in space.

Another shorting coming of the normal model is that it does not incorporate

information on mainshock focal mechanisms, which may have value in forecasting

aftershock patterns since it has been widely observed that aftershocks generally

occur on or near the fault planes of their associated mainshocks. For instance,

Willemann and Frohlich (1987) used the Anderson-Darling statistic to test the

7


21/85

distribution of aftershock hypocenters on the focal sphere of mainshocks against

a uniform distribution to show that for deep focus earthquakes, aftershocks that

were greater than 20 km from the mainshock exhibit significant clustering in the

plane of the Wadati-Benioff zone. Michael (1989) applied the same method to six

shallow focus aftershock sequences in California and found significant clustering

on the fault plane. Kagan (1992) adopted a more exploratory approach by using

equal-area projection rather than the Anderson-Darling statistic. He too reached

a similar conclusion regarding earthquake clustering along fault planes. Based

on this result, Kagan and Jackson (1994) introduced an anisotropic function for

their spatial smoothing kernel in their long-term earthquake forecast.

The aforementioned studies focused exclusively on the clustering of hypocen-

ters in certain directions on the focal sphere. I argue that such analyses essen-

tially ignore the potentially important relationship between the distancefrom an

earthquakes hypocenter or epicenter to those of its aftershocks and the angular

separation between an earthquake and its aftershocks. In this chapter, I attempt

to model both the angle and distance between earthquakes and their aftershocks

in concert.

This analysis focuses on strike-slip earthquakes with local magnitude ML >

3.0 in Southern California between 1999 and 2006 as mainshocks. Due to the

difficulty and subjectivity associated with the assessment of the precise branching

structure in earthquake catalogs, we follow Zhuang et. al. (2002) in using a

model-based method to identify aftershock sequences stochastically. I propose a

semi-parametric model for the spatial distribution of aftershocks that is composed

of a marginal distribution for distance and a conditional distribution for the

relative angle from the mainshock. This model is compared to two alternatives

that have been used to describe aftershock and seismic patterns respectively: the

8


22/85

normal model and the spatial smoothing kernel of Kagan and Jackson (1994).

We use point process residuals and the weighted K-function to assess the fit of

the various models and show that the proposed semi-parametric model appears

to offer superior fit to Southern California seismicity.

The outline of this chapter is as follows. Sections 3.2 and 3.3 describe the

catalog and aftershock assignment procedure used in this analysis. Section 3.4

defines the relative location of aftershocks with respect to mainshock focal mech-

anism. A new model is proposed in Section 3.5 for the spatial distribution of

relative aftershock locations. Two alternative models used to describe aftershock

and seismic patterns are given in Section 3.6. Section 3.7 discusses two diagnostic

methods to be used to compare the competing models. Section 3.8 is the results

section. Section 3.9 investigate two additional topics in seismology that are rel-

evant to understanding the spatial distribution of aftershocks. A discussion and

suggestions for future work are given in Section 3.10.

3.2 Data

We focus here on earthquakes in the Southern California Earthquake Data Center

(SCEDC) catalog occurring between September 18, 1999, and Dec 31, 2005, with

epicenters in Southern California and a moment magnitude of M3.0 or above (see

Data and Resources Section). The published hypocenters in the SCEDC moment

tensor catalog are used for the locations of both mainshocks and aftershocks.

The hypocenters are based on first motion triggers and therefore correspond to

the initial points of rupture. Based on comparison of the frequency-magnitudedistribution of the catalog to the Gutenberg-Richter distribution, the catalog

is believed to be complete for earthquakes above M3.0, though some events on

the edge of the network can be missing due to the lack of a qualified moment

9


23/85

tensor solution (Clinton et al. 2006). As mentioned in Chapter 2, moment tensor

solutions provided by SCEDC are labeled with a quality factor, A, B, or C, in

decreasing order of their precision. Only focal mechanisms of quality A or B are

considered because solutions of worse qualities are deemed unstable and are often

discarded (Clinton et. al. 2006).

Because different types of earthquakes are likely to have very different pat-

terns of aftershock activity, in this paper we consider only the aftershock activity

surrounding strike-slip earthquakes of quality A or B, with the notion that the

aftershock activity surrounding other types of events may be analyzed using sim-

ilar methods in future work. As pointed out by Kagan and Jackson (1998), the

simplicity of the geometry of strike-slip faulting facilitates the description and

interpretation of aftershock patterns: in a strike-slip event, the fault plane inter-

sects the surface of the earth almost vertically and thus line of intersection is a

fairly accurate representation of the fault plane itself. In seismology, this line of

intersection is termed the strikeof the fault plane. Since the strike is a reasonable

proxy for the fault plane, we will refer to the two terms interchangeably.

Southern California is populated by a large number of right-lateral strike-slipfaults. As a result, strike-slip events make up the majority of earthquakes in

Southern California. In this paper, we categorize an earthquake mechanism as

strike-slip if its neutral axis of the moment tensor (B-axis) is within 20 of the

vertical (some previous authors have used a cutoff of 30 instead of 20) (Frohlich

1992, 2001). A strike-slip will thus have a nearly vertical fault plane and a nearly

horizontal rupture motion. Roughly 1/3 of earthquakes in the SCEDC catalog

fall into this category.

The Southern California Earthquake Data Center (SCEDC) data were ob-

tained via SCEDCs searchable data archive at

10


24/85

http://www.data.scec.org/catalog_search/CMTsearch.php

3.3 Stochastic Aftershock Assignment

In determining the mainshock-afershock assignments, we adopt a model-based

approach taken by Zhuang et. al. (2002). In this approach, one assumes an

aftershock sequence model and assigns the aftershock branching structure proba-

bilistically according to the model. For example, one may take a spatial-temporal

epidemic type aftershock sequence (ETAS) model(Ogata 1998) and assume the

conditional intensity at point (t,x,y) is the sum of the background and triggering

intensities,

(t,x,y|Ht) =(x, y) +i:ti


25/85

decisions that can arise in determining mainshock-aftershock assignments. Fur-

thermore, I repeated this probabilistic branching assignment multiple times and

verified that the main conclusions of this paper were not affected. We henceforth

refer to a single realization of this Zhuang et al. (2002) assignment procedure.

While this analysis is not concerned with discerning whether particular events

are foreshocks, mainshocks, aftershocks, or swarms, for purposes of explanation

in this paper, let us refer to the strike-slip earthquakes in the SCEDC catalog of

quality A or B as mainshocks. According to these definitions, the catalog con-

tains 190 distinct mainshocks, which collectively have a total of 1224 emulated

aftershocks.

3.4 Relative Location of Aftershocks with respect to Main-

shock Focal Mechanism

We consider an aftershocks relative location to any mainshock with respect to

the focal mechanism of the mainshock, as illustrated in a beach ball diagram in

Figure 3.1 representing the focal mechanism of a right-lateral strike-slip main-

shock. The location of an aftershock relative to a mainshock is measured by rand

, wherer is the epicentral distance between the two events, and is the angular

separation between the aftershock and the mainshocks fault plane. The fault

plane ambiguity is resolved by assuming all strike-slip faults in Southern Cali-

fornia are right-lateral unless individual aftershock sequences clearly delineate a

left-lateral fault upon examination. Gomberg (2003) observed strong directiv-

ity effects among a number of unilaterally-rupturing strike-slips with magnitudeMs> 5.4 in different tectonic environments. While events with directive ruptures

may have asymmetrically triggered aftershocks, such events and their propagation

directions are difficult to quantify in a large catalog of various-sized earthquakes.

12


26/85

Therefore we will ignore possible directivity effects and treat fault plane as an

axis without sense of direction. To tentatively differentiate between the compres-

sion zone and dilatation zone of the mainshock focal mechanism, one may define

as the angle measured clockwise from the aftershocks epicenter to the nearest

mainshock fault plane for a right-lateral strike-slip, and counter-clockwise for a

left-lateral event. Thus spans from 0 to , and (0, /2) and (/2, )represent the compression zone (containing the P-axis) and dilatation zone (con-

taining the T-axis) respectively. While the Coulomb stress changes caused by the

slip of the mainshock may be different in both zones (compression and dilata-

tion), the numbers of aftershocks in both zones are found to be similar: 618 in

dilatation zone and 606 in compression zone. To test whether the differences in

the observed aftershock patterns in the two zones are statistically significant, the

dilation zone is reflected along the y-axis into the first quadrant and a chi-square

(X2) test is conducted on whether the two distributions are significantly different.Each zone is first confined to a [0, 20] [0, 20] window and then partitioned intoa 3 3 rectilinear grid. With only cells containing 5 points or more entering intothe test, the

X2 test yields a test statistic of 3.98 with 5 degrees of freedom, which

corresponds to a p-value of 0.55. In light of such evidence, we do not distinguish

the two zones in the remainder of this paper and restrict to [0, /2] by defining

it as the absolute angular separation from the nearest fault plane.

3.5 Tapered Pareto-Wrapped Exponential

I propose to model the distribution of the relative locations of aftershocks, in

polar coordinates, as a product of two distributions: 1) a marginal tapered Pareto

distribution for the distancer between mainshocks and their aftershocks, and 2)

a wrapped exponential distribution for the relative angle between mainshocks

13


27/85

and their aftershocks, given the distance r. The distribution may be written

f(r, ) = 1r fr(r) f|r(|r), wherefr andf|r are each one-dimensional densities

to be estimated, so that f(r, )r dr d = 1. We refer to this model as theTapered Pareto Wrapped Exponential (TPWE) model in what follows.

The distribution of distance between mainshocks and aftershocks has been

investigated in previous studies using various methods. Utsu (1969) noted that

aftershock regions tend to be elliptical, and Ogata (1998) built upon this work,

questioning whether the distance decay function is short range (i.e. normal)

or long range (i.e. inverse power law) and proposing several moment-weighted

models for alternatives. The inverse power law was shown by Felzer et. al.

(2006) to be a good description of aftershock distances between 0.2 km and 50

km. In a time-independent framework, Kagan and Jackson (1994) used a density

proportional to 1/r to describe distances between mainshocks and aftershocks,

in producing a long-term seismic hazard map.

More recently, several authors have begun using the the tapered Pareto dis-

tribution as an alternative to the Pareto and truncated Pareto distributions. The

tapred Pareto was originally suggested by Vilfredo Pareto himself (Pareto 1897)and has been used to describe the distribution of phenomena which obey some

power-law type of behavior but are not quite as heavy-tailed as the Pareto, such

as seismic moments (Jackson and Kagan 1999; Vere-Jones et al. 2001) , the times

and distances between successive earthquakes in Southern California (Schoenberg

et al. 2008), and wildfire sizes (Schoenberg et al. 2003).

The tapered Pareto has cumulative distribution function:

Ftap(x) = 1 (a/x)exp

a x

, a x

where is a threshold after which frequency begins to decay especially rapidly.

Additional information concerning the density, characteristic function, moments,

14


28/85

and other properties of the tapered Pareto can be found in Kagan and Schoenberg

(2001).

While a variety of models for the density ofr have been proposed, the form of

f|r(|r) has been the subject of relatively scant scrutiny to date. As part of theirspatial smoothing kernel, Kagan and Jackson (1998) used a directivity function,

expressed as D = 1 + cos2(), where measures the concentration of earth-quake epicenters around the presumed fault plane. HereD does not serve exactly

the same purpose as f|r(|r) because f|r(|r) concerns aftershocks, i.e. seismicactivity at times after a mainshock, whereas D describes the time-independent

distribution of around any earthquake. Further, this investigations suggest

that, for the SCEDC data, the distribution of appears to depend on r, so that

the conditional distribution of givenr may be more meaningful than the over-

all marginal distribution of. Due to lack of theoretical support for a particular

functional form, I take a semi-parametric approach here. I propose a circular dis-

tribution called the wrapped exponential (WE), whose single parameter may

be estimated locally within selected bins, as described below.

Wrapped distributions are useful for modeling angular variables such as , therelative angle between mainshocks and aftershocks (Mardia and Jupp 2000). Such

a distribution is obtained by conceptually wrapping a distribution on the real line

around the circumference of a unit circle. That is, ifx is a real random variable

with an arbitrary probability density function f, then the wrapped analogue of

fhas density

fw(xw) =

k=

f(xw+ 2k), xw

[0, 2),

where

xw=x(mod 2)

15


29/85

This approach can be applied to any probability distribution to manufacture

a large class of circular distributions, among which the wrapped Gaussian and

wrapped Cauchy are examples. However, one shortcoming common to most such

wrapped distributions is the lack of a closed form expression, which renders pa-

rameter estimation difficult. One exception is the wrapped exponential (WE) in

which the infinite series converges and has a remarkably simple solution (Jam-

malamadaka and Kozubowski 2001). The WE has been used to model seismic

events triggered by periodic processes (Jupp et al. 2004), and is obtained by

applying the wrapping procedure to the exponential density, f(x) =ex , x >0.

Since in this analysis only spans [0, /2] by construction, it will cycle on a

quarter circle instead of a full circle. On a quarter circle, the WE has density

fwe() = e

1 e/2 , [0, /2].

Note that this is equivalent to the density of a truncated exponential random

variable on the line , i.e. f(X|X < /2), due to the memoryless property of theexponential distribution.

The WE has a single shape parameter . When= 0, the WE corresponds to

a uniform distribution on its support. As increases, the skewness of the distri-

bution increases, and asapproaches , the distribution degenerates to a pointmass at = 0. In the context of this analysis, can be thought as an aftershock

azimuthal concentration parameter. With only one parameter, the WE provides

a very good fit to the conditional distribution of given r. Evidence suggests,

however, that the conditional distribution of changes depending on the value

ofr. One way to model its dependency on r is to let the parameter vary as afunction ofr. I propose to estimate the value oflocally by maximum likelihood

(ML) successively on different bins, each containing n pairs of points, sorted ac-

cording to the distancer between mainshock and aftershock; in order to estimate

16


30/85

more accurately within each bin, I use all possible mainshock-aftershock pairs,

weighting each pair by its probabilityi,j of being an actual mainshock-aftershock

pairing. I experimented with several choices ofn and selected n= 200 in order

to achieve a satisfactory bias-variance tradeoff.

3.6 Alternative Aftershock Spatial Distributions

3.6.1 The Normal Modell

Two alternative models are considered in this paper for comparison. The first

model is the normal model and the second is the spatial smoothing kernel ofKagan and Jackson (1994). The normal model has been proposed to describe the

relative locations of aftershocks and thus is certainly applicable to the problem

in hand. Kagan and Jacksons model, by contrast, was suggested in a slightly

different context, as explained below, but may nevertheless constitute a relevant

model for comparison.

The normal model is often used as a conveniently simple spatial distribution

for aftershock sequences (Rhoades and Evison 1993, Kagan 2002, Ogata 1998).

A slight modification was made by Ogata (1998), who introduced an anisotropic

function based on the normal model as a spatial extension of his earlier ETAS

model (Ogata 1988), and proposed a separate normal model to be fit to each

aftershock sequence in the de-clustered catalogue, using an anisotropic metric

that replaces the Euclidean metric. Compelling evidence to support the normal

model is the commonly seen elliptical shape of aftershock zones (Rhoades and

Evison 1993).

17


31/85

3.6.2 The Kagan-Jackson Model

Kagan and Jackson (1994) estimated the long-term rate densities for earthquakes

as a weighted sum of smoothing kernels, each centered at the epicenter of a

previousith earthquake, using information on the earthquakes focal mechanisms.

Adopting current notation, the density at point (x, y) is estimated in Kagan and

Jackson (1994) as

f(x, y) =i

fi(ri, i, Mi) +s

where s = 0.02 is a small constant to allow for surprises far from past earth-

quakes. Of interest here is their proposed smoothing kernel

fi(ri, i, Mi) =A (Mi Mcut)

1 + cos2(j) 1

ri,

whereA is a normalization constant, Mi is the magnitude of earthquake, and

is a parameter controlling the degree of azimuthal concentration in a direction

relative to the earthquakes focal mechanism (Kagan and Jackson 1994). We refer

to this model as the KJ model in what follows. Kagan and Jackson used their own

expert knowledge to choose . In a region where the concentration of aftershocks

along the fault plane is high, should be assigned a high value, and should

be small if the earthquakes are dispersed relatively isotropically. The choice of

smoothing kernel was selected as a result of the analysis in Kagan (1992) of focal

mechanisms and the distribution of hypocenters on the focal sphere. Although

the model was motivated by analysis of subduction zone earthquakes, Kagan et

al. (2007) recently applied this model to seismicity in southern California using

= 100. In the current analysis, the value ofis selected via maximum likelihood.

18


32/85

3.7 Residual Analysis

3.7.1 Quadrat Residuals

I focus on a subset of relative mainshock-aftershock locations in [0, 20] [0, 20]within which I compare the goodness-of-fit for the different models. One method

of assessing the goodness-of-fit of a spatial density for point process is by exam-

ining the residuals over various quadrats, as suggested in Baddeley et al. (2005).

That is, one partitions the space into cells and calculates the residuals in each

cell, which may be standardized in various ways. For instance, the residualRi

corresponding to celli may be defined as

Ri =Ni Ei

Ei,

where Ni is the observed number of points in the cell, and Ei is the expected

number of points defined. Ei can be found by integrating the intensity f over

the cell i.

This is effectively Pearson residuals in the Poisson log-linear regression con-

text. By construction, the residuals are standardized to have mean 0 and standard

deviation approximately 1. Outliers and systematic patterns in the residuals may

indicate lack of fit.

3.7.2 Weighted K-function

TheK-function described by Ripley (1981) is commonly used to detect excessive

clustering or inhibition in a point process. The function K(h) is defined as

the average number of additional points within h of any given point, divided

by the overall rate. The null hypothesis is that the underlying point process is

homogeneous Poisson. In cases where the hypothesis is not uniform, each point

19


33/85

may be weighted according to the rate of the point-process in question, yielding

theweightedor inhomogeneousK-function (Baddeley et al. 2000). The weighted

K-function has been used by Veen and Schoenberg (2005) to assess the spatial

distributions in point process models for earthquakes in Southern California. To

test the null hypothesis that the spatial intensity of points (mainshock-aftershock

pairs) in region D is f0(x, y), the weighted K-function may be defined as

KW(h) = 1

f2N

i

wij=i

wj1(|pi pj | h),

whereNis the total number of observed pairs of points,f:= inf{f0(x, y); (x, y) D}

is the infimum of the density over the observed region, 1() is an indicator

function, and wr = f/f0(pr), where f0(pr) is the modeled density of pairs of

points at vector distance pr apart. Veen and Schoenberg (2005) verified that

for the Poisson case where f0 is locally constant on distinct subregions whose

areas are large relative to the interpoint distance hn, the weighted K-function

is approximately normal with mean h2 and variance 2h2A/[E(N)]2, where A

is the size of the area being studied and N is the number of points observed

in A. One common issue in applying the K-function or weighted K-function isthe problem of boundary correction. One method of edge correction is to make

mirror images of the points (where each point is a mainshock-aftershock pair)

along the boundaries over which these points are observed, as suggested e.g. in

Ripley (1981). Since all of observed points are restricted to the first quadrant on

the plane, I reflect each point along both the x- and y-axes.

Statistical inference is drawn from simulation-based confidence bounds. 4,000

samples are taken and a weighted K-function is estimated for each sample. Based

on these weighted K-functions, I form 95% pointwise confidence bounds to make

statistical inferences.

20


34/85

3.8 Results

Figure 3.2 shows a subset of mainshock-aftershock pairs as described in Section

3.3. Not surprisingly, it is immediately noticeable that the concentration of pointsis much higher near the x-axis (i.e. the fault plane) than elsewhere. One may

observe that aftershocks seem to be distributed nearly uniformly in all directions

when they are close to the respective mainshock, whereas when they are further

away, they tend to lie more predominantly along the azimuth of the fault plane

(i.e. along the x-axis). One may also observe the discreteness in the observed

distances r between mainshocks and their aftershocks, for pairs that are very

close together; this is a result of the resolution of measurements recorded in the

catalog. Note that rounding errors in the locations, compounded by estimation

errors in epicenter locations, may translate into large errors in the estimation of

, especially for small values ofr, where a tiny change in location will translate

into a large change in . Willemann and Frohlich (1987) and Michael (1989)

discarded aftershocks within 5 km of each mainshock, in order to avoid dealing

with these noisy observations.

Figure 3.3 shows the survival functions ofr, a fitted Pareto (i.e. inverse power

law), and a fitted tapered Pareto distribution, on log-log scale. The diagram

indicates the Pareto offers a good fit to the data for r


35/85

explanation for its uniform distribution in this range ofr is that it is dominated

by noise. In higher ranges ofr values, the skewness of the distribution increases

as tend to concentrate at low values. Overlaid on the histograms are fitted WE

densities. We can see that the WE is able to capture the shape of the conditional

histograms in different regions and seems to fit each of the densities rather well.

The local behavior of is sensitive to the value of r, as shown in Figure

3.5, which displays the local weighted maximum likelihood estimates ofplotted

against the mean distance in the corresponding bins. One sees that when r is

small, the estimates ofare unstable and seem again dominated by noise. As

r increases, climbs steadily until roughly r = 18 km before it declines slowly.

For the purposes of interpolation, extrapolation, or forecasting, one may seek

a parametrization of the estimates of . The F-distribution provides a good

approximation to the shape of the estimates of . Let fv1,v2(x) be the density

function of the F-distribution with v1 and v2 degrees of freedom, the function

2.7 f10,600(x/22) is found to be the best fit by least squares among functions ofcomparable form. The fitted function is superimposed on the estimates of in

Figure 3.5.

3.8.1 Fit of proposed models

Figure 3.6 displays the density surfaces on logarithmic scale for the fitted KJ,

normal, and TPWE models, to relative mainshock-aftershock locations. A com-

parison between these surfaces reveals characteristic differences and similarities

between the models. The KJ model has a rather sharp peak that reaches about

the same height as the normal model. The KJ density decays very slowly out-

ward and, as a result, retains a substantial density through the entire region.

Its shape mimics a 2-petalled rose that expands along the x-axis and tightens

22


36/85

along the y-axis. The normal model in contrast is comparatively smooth near

the origin and has relatively flat tails, obtaining densities that are very close to

zero outside the visible contours. The contour lines themselves, for the normal

model, all have an elliptical shape that seems to resemble aftershock zones. The

TPWE model can perhaps be viewed as a hybrid of the KJ and normal models.

On the one hand, it possess a sharp peak, like the KJ model (though the TPWEs

peak density is much higher than that of the KJ model). On the other hand,

its tail is quite thin and the visible contour lines cover roughly the same area as

the normal density. In addition, the bow-tie shape of the contours of the TPWE

model seems to have features that encompass characteristics of both the KJ and

normal models. While the TPWE model is not meant to be a compromise of

the other two models by construction, it nevertheless shares some similarity with

both of them.

3.8.2 Diagnostics and Model Comparison

The absolute values of the quadrat residuals of the KJ, normal, and TPWE

models are shown in Figure 3.7, on a logarithmic scale in order to facilitatevisualization. One sees immediately that the normal model has several outlying

residuals of very large size. This is due to the occurrence of mainshock-aftershock

pairs at relative distances where the normal model assigns a density very close to

zero. By contrast, the KJ model assigns a substantial density to these outliers.

However, it does so at the cost of having a substantial density throughout the

entire region. As a result, the model is over-predicting in most of the upper half

of the top-left panel of Fig. 8, at relative distances where very few mainshock-

aftershock pairs were observed. The proposed TPWE joint distribution seems

to achieve a balance between the other two. On the one hand, the outlying

23


37/85

residuals are several orders of magnitude smaller than those in the normal model;

on the other hand, the TPWE models residuals are much smaller than the KJ

model in relative locations where observations are rare. Near the origin, both the

normal and KJ models tend to under-predict the density. For instance, in the KJ

model, there is a vertical cluster of large residuals in the region above the origin,

indicating a systematic lack of fit. A similar problem is seen in the residuals of

the normal model. The peaks in these two densities are too low, relative to the

observed mainshock-aftershock pairs. The TPWE model, by contrast, has much

smaller residuals near the origin, indicating superiority of fit.

The weightedK-functions for the three models are shown in Figure 3.8. Also

plotted are the theoretical mean and simulation-based pointwise 95% confidence

bounds. There is serious departure from the confidence bounds in both the KJ

and normal models, indicating statistically significant lack-of-fit. The weighted

K-function for the KJ model is below the lower threshold of the 95% confidence

bounds for all values ofh, as a result of its wide-area over-estimation of aftershock

density. In the normal model, the weightedK-function is plotted on a logarithmic

scale because the estimates of the KWare orders of magnitude above the upper

bounds of the 95% confidence intervals. Such a dramatic departure is a result

of serious underestimation of the density at the origin as well as a few outlier

locations where aftershocks are observed. The TPWE does not seem to have

systematic over-estimation or under-estimation of the density of relative distances

between mainshocks and their aftershocks, nor is there any serious indication in

Figure 3.8 of clustering or inhibition of the mainshock-aftershock pairs relative

to this joint distribution.

24


38/85

3.9 Additional Topics

This section provides an exploration on two additional topics in seismology that

are relevant to understanding aftershock spatial distribution: scaled distance, andrelocation catalog. These topics provide alternative ways of analyzing aftershocks

and are subjects of on-going research. Although an in-depth examination is

beyond the scope of this dissertation, this section attempts to explore these topics

in relation to the analysis of aftershocks in this chapter and estimate the impact

they may have on the results.

3.9.1 Scaled Distance

It has long been a subject of debate whether the spatial distribution of aftershocks

is independent from mainshock magnitude. Previous authors have reached dif-

ferent conclusions using different criteria (Ogata 1998, Kagan 2002, Huc et. al.

2003, Davidsen et. al. 2005, Felzer et. al. 2006). Amidst such controversy,

this section repeats part of the analysis in this chapter using a scaled distance

in an attempt to capture possible scaling effects due to mainshock magnitude.

One way to relate aftershock distances to mainshock magnitudes is to measure

distances in terms of mainshock rupture lengths (Felzer et. al. 2006), which have

been shown intimately related to magnitude for large events (Wells et. al. 1994).

We estimate the surface rupture length (L) of a fault from empirical relationships

(Wells et. al. 1994) and express the scaled distance in terms of fault lengths,

r/L. The marginal distribution ofr/Land the conditional distribution of with

respect tor/L are investigated in the same manner as in the case for r.

The marginal distribution of scaled distancer/Land the corresponding con-

dition distribution of are shown in Figures 3.9 and 3.10 respectively. The fit of

25


39/85

the tapered Pareto tor/Lis slightly worse than is the case for r but still appears

reasonable. In comparison, the Pareto distribution systemically deviates from the

empirical survival function over the entire range of data, rendering it a misfit.

The conditional distribution of with respect to r/L shows strong resemblance

to its counterpart for unscaled distance. The estimates ofare low in the upper

and lower ranges ofr/L and are largest at roughly 40 fault lengths. The shape

of the estimates seems again well capture by an F-distribution. Although it is

difficult to infer from Figures 3.9 and 3.10 whether the distribution of aftershocks

is dependent on mainshock magnitude, both figures seem to suggest that TPWE

is applicable to both scaledandunscaleddistances, at least for the range of data

considered.

3.9.2 Relocation Catalog

The use of relocation catalogs for the studies of aftershock spatial distribution

is open to debate. Despite the drastically reduced location errors in relocated

events, some authors have argued they may not be well suited such studies.

Some reasons as suggested by Kagan (2002) include 1) increased effort needed toreinterpret the data, 2) bias and statistical dependence that may be introduced in

the location estimates, and 3) difficulty in communicating the relocation proce-

dure and in reproducing the experiment. In spite of this, due to the more precise

locations, events in a relocation catalog are often found to align in linear and/or

planar structures suggestive of faults. It is therefore interesting as a descriptive

exercise to investigate the impact of relocation on the estimation of aftershock

concentration along mainshock fault planes. If clustering is indeed strong, larger

estimates ofwill be expected.

I examine relocated events in Southern California (Shearer et. al. 2005)

26


40/85

between 1984 and 2002 withM >3.3 and their focal mechanisms (Hardebeck et.

al. 2003) with quality A or B. Both catalogs are available from SCEDC at

http://www.data.scec.org/research/altcatalogs.html.

New estimates ofare obtained using the relocation catalog in a similar fashion

as before and are compared to the previous estimates of using the SCEDC

catalog. While the relocation catalog covers a longer period than does the SCEDC

catalog and has a higher magnitude cut-off, it should still provide a meaningful

comparison on the clustering of aftershocks along the fault direction.

Estimates offrom a relative relocation catalog are shown in Figure 3.11. Of

interest is the magnitude of as compared to the SCEDC catalog, particularly in

the lower range ofr. Despite improved location estimates, the relative relocation

catalog does not substantially increase the clustering of near-by aftershocks along

the fault direction as reflected in . A possible explanation is that while relocation

decreases location errors slightly for nearby events, it does not eliminate them,

and when the distance from the mainshock is small, is still very susceptible

to noise. Only at larger distances, such as r > 30, does one observe stronger

clustering due to more precise location estimates.

3.10 Discussion

This chapter explores focal mechanism as a means to describe the anisotropy of

aftershock spatial distribution. Using the strike angle as a proxy for the fault

plane, aftershocks are found to lie preferentially along mainshock fault plane forsouthern California seismological data. The tapered Pareto / wrapped exponen-

tial (TPWE) model appears to adequately describe the locations of aftershocks

relative to mainshocks focal mechanism. Using residual analysis and weighted

27


41/85

K-function as diagnostic measures, both suggest that TPWE vastly outperforms

competing models such as the Kagan-Jackson and normal models.

It must be emphasized, however, that this analysis is performed using only

Southern California strike-slip earthquakes of quality A or B as candidates for

mainshocks. The effect of using only strike-slips as triggering events is that af-

tershocks of other types of mainshocks (e.g. dip-slip events) are now mostly

identified as background events, though some may be falsely identified as after-

shocks to some strike-slip event. In other seismic regions, especially in areas

where the faulting is more heterogeneous, the TPWE model might not fit well,

and an important direction for future work is the investigation of the fit of such

models in other seismically active zones.

It should also be noted that earthquakes are treated as point sources in this

analysis. An important topic for future research is the investigation of models

for aftershock distances based on estimating the actual segment of fault which

ruptured for each earthquake and calculating minimal distances between such

segments. Related to this is the sensitivity of aftershock spatial distribution

to mainshock magnitude. This analysis considers an additional scaled distancebased on empirical relations in an attempt to capture possible scaling effects

due to mainshock magnitude. Nevertheless, such relations are only based on ap-

proximation and may not be accurate for small- and medium-sized mainshocks.

A topic for future research is a more refined and thorough approach that con-

siders mainshocks of various sizes separately. Lastly, as depth measurements for

earthquakes become increasingly accurate and other features of earthquake faults

become discernible, including possible directivity of aftershocks, such effects may

also be taken into account to reflect a more realistic and accurate description of

aftershock spatial distribution.

28


42/85

Taxis

Paxis

Fault

plane

Auxillary plane

aftershock

r

Figure 3.1: A beach ball diagram illustrating the definition of relative aftershock

location with respect to mainshock focal mechanism. r represents the epicentral

distance between the mainshock and aftershock, and measures the aftershocks

angular separation from the mainshock fault plane.

29


43/85

Table 3.1: ETAS parameters used in stochastic aftershock assignment.

c d K 0 p q

(M1) (days) (km) (shocks/day/km2)

.255 .346 2.903 1.008 1.888107 1.324 1.305

30


44/85

0 5 10 15 20

0

5

10

15

20

strike distance (km)

Figure 3.2: Scatterplot of mainshock-aftershock relative locations, with respect

to the mainshocks fault plane. Only a subset is displayed in a 20 20 km2

window.

31


45/85

1 2 5 10 20 50 100 200 500

0.0

01

0.0

05

0.0

50

0.5

00

r (km) on log scale

S(r)on

log

scale

data

Pareto

tapered Pareto

Figure 3.3: Survival function (1 F{r}) for mainshock-aftershock distances, r.The tapered Pareto model appears fit much better to r than does the Pareto

model.

32


46/85

10 km < r < 15 km

(degrees)

Density

0 20 40 60 80

0.0

0

0.0

2

0.0

4

5 km < r < 10 km

(degrees)

Density

0 20 40 60 80

0.0

00

0.0

10

0.0

20

r < 5 km

(degrees)

Density

0 20 40 60 80

0.0

00

0.0

10

Figure 3.4: Histograms of relative angle () between mainshocks and aftershocks,

arranged according to distancerbetween mainshocks and aftershocks. Top panel:

10 km r


47/85

0 10 20 30 40 50 60

0

1

2

3

mean r (km)

^

Fdistribution

Figure 3.5: Estimates of the aftershock azimuthal concentration parameter , as

a function of distancer between mainshock and aftershock. The gray dots are es-

timates obtained by bins of 200 mainshock-aftershock pairs each; the black dotted

curve shows the parameterization using the density function of an F-distribution.

20 10 0 10 20

20

10

0

10

20

20 10 0 10 20

20

10

0

10

20

20 10 0 10 20

20

10

0

10

20

11 or less

6.9

2.9

Figure 3.6: Density plots on logarithmic scale corresponding to three models for

mainshock-aftershock relative locations. Left: KJ model; middle: normal model;

right: TPWE model.

34


48/85

0 5 10 15 20

0

5

10

15

20

0 5 10 15 20

0

5

10

15

20

0 5 10 15 20

0

5

10

15

20

7 or less

1.6

3.9

Figure 3.7: Quadrat residuals from each of the three models for mainshock-after-

shock relative locations. Left: KJ model; middle: normal model; right: TPWE

model.

35


49/85

0 1 2 3 4 5 6

0

20

40

60

80

100

h (km)

K^

(h)

empirical weighted Kfunction95% pointwise confidence boundstheoretical mean function

0 1 2 3 4 5 6

1e03

1e+00

1e+03

1e+06

1e+09

h (km)

K^

(h)

empirical weighted Kfunction

95% pointwise confidence boundstheoretical mean function

0 1 2 3 4 5 6

0

50

100

150

200

250

300

350

h (km)

K^

(h)

empirical weighted Kfunction95% pointwise confidence boundstheoretical mean function

Figure 3.8: WeightedK-functions corresponding to three models for mainshock-

-aftershock relative locations. Top: KJ model; middle: normal model; bottom:TPWE model.

36


50/85

1e01 1e+00 1e+01 1e+02 1e+03

0.0

01

0.0

05

0.0

50

0.5

00

r/L on log scale

S(r)on

log

scale

data

Pareto

tapered Pareto

Figure 3.9: Survival function (1F{r/L}) for normalized mainshock-aftershockdistances,r/L.

37


51/85

0 20 40 60 80 100 120 140

0

1

2

3

mean r/L

^

Fdistribution

Figure 3.10: Estimates of the aftershock azimuthal concentration parameter,

as a function of scaled distance r/L. The gray dots are estimates obtained by

bins of 200 mainshock-aftershock pairs each; the black dotted curve shows the

parameterization using the density function of an F-distribution.

38


52/85

0 10 20 30 40 50 60

0

1

2

3

mean r (km)

^

Figure 3.11: Estimates of the aftershock azimuthal concentration parameter

from a relocation catalog. The gray dots are estimates obtained by bins of 200

mainshock-aftershock pairs each.

39


53/85

CHAPTER 4

Focal Mechanism-dependent Anisotropic Spatial

Kernel for Space-Time Earthquake Point

Process Models

4.1 Background

Spatial-temporal Epidemic-Type Aftershock Sequence (ETAS) models for earth-

quake occurrences were proposed by Ogata (1998) and have since been widely

used to characterize modern catalogs of seismicity. The initial, simple versions

of these models posit an isotropic spatial distribution of aftershocks around each

mainshock. However, such a model fails to account for the anisotropic spatial dis-

tribution of aftershocks that has been widely observed since Utsu (1969). Indeed,

Ogata (1998) acknowledges the need for an anisotropic spatial kernel and sug-

gests altering the spatial decay function in ETAS models with ellipsoidal contours

corresponding to a bivariate normal fitted to aftershock regions.

Although the normal model is a convenient choice and may serve as a first

order approximation to aftershock spatial distribution, it is shown in Chapter 3

to suffer from several serious issues. Recently, modern catalogs of earthquakes

containing seismic moment tensor estimates for many of the events have become

available. These estimates, especially the resulting estimates of focal mechanism,

appear to be quite effective at describing the anisotropic spatial distribution

40


54/85

of aftershocks (see Chapter 3 and the references therein). Nevertheless, such

information has yet to my knowledge been previously used in ETAS models. A

primary purpose of the present chapter is therefore to explore of ETAS models

that incorporate information regarding focal mechanism, and in particular the

orientation of earthquakes in forecasting subsequent seismicity.

In the previous chapter, TPWE was proposed as a spatial distribution for

the relative aftershock locations with respect to mainshock focal mechanism.

The present chapter proposes general anisotropic extensions of space-time point

processes such as ETAS to incorporate focal mechanism information. A some-

what similar procedure was explored by Kagan and Jackson (1994), who formed

long-term seismic hazard maps by smoothing past seismicity in Southern Cal-

ifornia using an anisotropic spatial smoothing kernel that depends on the past

earthquakes focal mechanism. Here, the TPWE is used as an example of an

anisotropic, focal mechanism-dependent spatial kernel. The effectiveness of using

focal mechanism in modeling earthquake occurrences is assessed by comparing

ETAS models with isotropic and anisotropic spatial kernels.

A secondary purpose of this paper is to explore a new technique for goodness-of-fit assessment of space-time point process models. The proposed procedure

involves inspecting differences between competing models in contributions to the

loglikelihood over pixels, and are thus called deviance residualsbecause of their

obvious connection with deviances in the context of generalized linear models.

The resulting residual plots are a small extension of those proposed by Baddeley et

al. (2005) for the purpose of assessing purely spatial point processes by examining

their behavior over pixels. The key idea of the method proposed here is to assess

the relative performance of competing models, i.e. to inspect the differences

between each models residuals. This graphical tool appears to be quite effective

41


55/85

at portraying the relative fit of models to space-time point process data, and is

used to illustrate the advantages and disadvantages of extended ETAS models

compared to alternative models.

The format of this chapter is as follows. Section 4.2 provides background

information on models for aftershock behavior such as ETAS models. Section

4.3 details the mathematical relation between mainshock fault plane strike angle

and relative aftershock angle as defined in this analysis. Extended ETAS models

are proposed in Section 4.4. Sections 4.5 discusses parameter estimation. Section

4.6 summarizes some diagnostic methods for purely spatial and spatial-temporal

point process models. A deviance-based residual method is proposed in Section

4.7 method and is used in Section 4.8 to assess the relative fit of ETAS models

described in Section 4.4. Section 4.9 is the concluding section for this chapter.

4.2 Epidemic Type Aftershock Sequence Models

Point process models have proved to be an efficient tool for modeling earthquake

occurrences. Their use in seismology has been pioneered by Vere-Jones (1970,

1975), Ogata (1999) gives a nice review. Today, the ETAS model (Ogata 1988,

1998) is considered to be the standard branching process model in seismology.

First introduced by Ogata (1988) to describe the intense temporal clustering

observed in earthquake occurrences, the ETAS model is a type of branching or

self-exciting point process. Early applications of self-exciting point processes to

earthquake occurrence models can be found in Hawkes and Adamopoulos (1973)

as well as in Lomnitz (1974, Chapter 7). Modeling earthquake occurrences usinga self-exciting point process implies the separation of the seismicity into a long-

term background componentand a short-term, time-dependent clustering com-

ponent

{i:ti


56/85

component describes seismic activity as a superposition of earthquake clusters,

where each earthquake is either a mainshock or an aftershock, and the aftershocks

can themselves trigger further aftershocks, resulting in branching behavior. Note

that since the precise branching behavior of observed earthquakes is often difficult

to determine and open to debate, we use the terms mainshock and aftershock

here purely in reference to the branching behavior described by the ETAS model.

Spatial-temporal versions of the ETAS model were described in Ogata (1998).

The conditional intensity function of the spatial-temporal model can be written

as

(t,x,y

|Ht) =(x, y) +

{j:tj


57/85

chapter. The following forms for h(r, ; Mj) were proposed by Ogata (1998):

h(r, ; Mj) = (r2 +d)q, (4.1)

h(r, ; Mj) = r2

e(MjMc) +dq

, (4.2)

h(r, ; Mj) = exp

1

2

r2

de(MjMc)

. (4.3)

The discriminating features among the above models can be summarized in two

major ways: (1) h can have short range decay (i.e. normal) or long range decay

(i.e. power law), (2) the spatial distribution of aftershocks around a given main-

shock can be parameterized in terms of the Euclidean distance between the two

events, or between this distance scaled as an exponential function of mainshock

magnitude. Common to these models is they all describe isotropic (i.e. rotation

invariant) spatial clustering.

4.2.1 Anisotropic Clustering

It has long been observed that, after a given mainshock, subsequent events tend to

occur within a roughly elliptically shaped region around the mainshock (see e.g.

Utsu, 1969). Ogata (1998) lists several possible reasons, based on the dip angle of

the slipped fault of an earthquake, the proportion of the slipped length of the fault

to its width, and the location errors of aftershock hypocenters. Ogata (1998) and

Ogata and Zhuang (2002) suggest extending the spatial kernel h(r, ; Mj) to ac-

commodate ellipsoidal aftershock zone contours by altering the Euclidean metric

inh(r, ; Mj) for certain aftershock sequences that are anisotropically distributed.

One of the methods amounts to first identifying large aftershock sequences in acatalog and fitting a bivariate normal distribution to the spatial coordinates of

the aftershocks within each sequence. For every fitted normal whose covariance

matrixSj is significantly different from the identity matrix, one replaces the term

44


58/85

r2 inh(r, ; Mj) by

(r cos(), r sin())Sj(r cos(), r sin())T

for the corresponding mainshock so that its aftershocks are distributed with el-

lipsoidal contours. Applying this to model (1), for instance, the altered spatial

kernel can be written as

h(r, ; Mj) = ((r cos(), r sin())Sj(r cos(), r sin())T +d)q.

Note that isotropic clustering is a special case where Sj is the identity matrix.

Although the normal model may serve as a reasonable first-order approxi-

mation for the spatial patterns of aftershock zones, it was shown in Wong and

Schoenberg (2009) to suffer from several issues when used to describe aftershocks

across different sequences in Southern California. Even for individual clusters

as suggested by Ogata (1998), its application in a point process model may not

be optimal. For instance, the normal distribution is sensitivity to outliers and it

may not fit well to observation regions that are irregularly shaped and/or difficult

to delineate. On the other hand, it may fit too well to short sequences, causingthe model to overfit. Ogata (1998) tries to prevent this problem by using after-

shock clusters of no less than a minimum length. Most importantly, aftershock

identification is often a subjective process and rests heavily on human input.

As suggested by the analysis of Chapter 3, focal mechanism appears to be pre-

dictive of aftershock pattern and may provide an alternatively way to account for

the anisotropy of aftershock distribution in a point process model such as ETAS.

As compared to normal approximation, such an approach has the advantages of

more being objective and easily reproducible.

45


59/85

4.3 Fault Plane Strike Angle and Relative Aftershock An-

gle

In Chapter 3, the relative angle of an aftershock with respect to mainshock focal

mechanism was defined as its angular separation from the nearest mainshock

fault plane. I now give a more precise mathematical definition of in relation to

the fault plane strike angle and the aftershocks conventional polar angle.

When one expresses the epicenter of an aftershock in relation to a mainshock

in conventional polar coordinates (r, ), as was done in Section 4.2, one implic-

itly uses the mainshock epicenter and the positive x-axis as reference point and

reference vector respectively. Denoting the strike angle of the mainshock fault

plane asj, which is also measured counter-clockwise against the positive x-axis,

can be calculated from andj as follows

(, j) =

| j| if 0 | j | < /2 | j| if/2 | j | < | j| if | j|


60/85

4.4 Anisotropic Extensions of ETAS Models

In extending ETAS model to incorporate the strike angle of mainshock fault

planes, we seek a focal mechanism-dependent spatial kernel that can encompassexisting isotropic models. In other words, given an isotropich(r, ; Mj) such as

(4.1)-(4.3), we seek h(r, ; Mj, j) that satisfies 20

h(r, ; Mj , j) d=

20

h(r, ; Mj) d,

for allj, so that the two marginal distributions agree. One possible parameter-

ization ofh(r, ; Mj, j) is

h(r, ; Mj

, j

) =h(r, ; Mj

)f(; r, j

),

wheref(; r, j) is a density, for any r, j. The predictive power of focal mecha-

nism can be evaluated by comparing ETAS models with and without the anisotropic

term f.

For instance, one may take the Pareto model (4.1) for h(r, ; Mj) and gener-

alize it by letting f(; r, j) =fWE((, j); r). The corresponding spatial decay

functions with isotropic and anisotropic clustering can be written respectively as

h(r, ; Mj) = (r2 +d)q, (4.5)

h(r, ; Mj, j) = (r2 +d)q fWE((, j); r), (4.6)

Alternatively, one may use the tapered Pareto model as suggested by the analysis

in Chapter 3. An isotropic spatial kernel according to which r has a tapered

Pareto marginal distribution can be written as

hTP(r, ; Mj) =

r +

1

expa r

a

r+1 (a r < ). (4.7)

Similar to (4.5) and (4.6), one can compare hTPand its extend version:

h(r, ; Mj) = hTP(r, ; Mj), (4.8)

h(r, ; Mj , j) = hTP(r, ; Mj) fWE((, j); r). (4.9)

47


61/85

All of the above spatial kernels are magnitude-invariant. Instead, one may

consider variations that scale with mainshock magnitude, as in models (4.2) or

(4.3). The main focus of this chapter is not on magnitude scaling, however, but

on the incorporation of focal mechanism estimates; interested readers can refer

to Section 3.9.1 for a discussion of a magnitude dependent version offWE.

The analysis in Chapter 3 suggests that fWEis applicable to strike-slip main-

shocks, but the spatial distribution of aftershocks of other types of earthquakes

may be different. One may define an indicator variable

I=

1 mainshock is a strike-slip

0 otherwise

and consider a modified version of (4.6) and (4.9) where the aftershock zones of

strike-slip mainshocks have directional preferences according to fWE, while non-

strike-slip events have circular aftershock zones. The restricted spatial kernels

can be written respectively as

h(r, ; Mj, j) = (r2 +d)q

I fWE((, j); r) + (1 I)/2

, (4.10)

and

h(r, ; Mj , j) =hTP(r, ; Mj)

I fWE((, j); r) + (1 I)/2

. (4.11)

Alternatively, considering the relatively small presence of qualified strike-slip

mainshocks in the catalog, one may put more emphasis on the directional ex-

tension by considering a model with only strike-slips type mainshocks. The con-

ditional intensity function of a space-time ETAS model with only strike-slips

actting as triggering events can be written as

(t,x,y|Ht) =(x, y) +

{j:tj


62/85

A response function with isotropic spatial clustering such as (4.5),

g(t,x,y|tj , xj , yj ; Mj) = K0e(MjMc)

(t tj+c)p (r2 +d)q, (4.13)

can be compared to its focal mechanism-dependent invariant (4.6)

g(t,x,y|tj , xj , yj ; j, Mj) = K0e(MjMc)

(t tj+ c)p (r2 +d)q fWE((, j); r). (4.14)

Similar comparison can also be made for a tapered Pareto model with isotropic

and anisotropic clustering such as (4.8) and (4.9) respectively

g(t,x,y|tj, xj, yj; Mj) = K0e(MjMc)

(t tj+c)p hTP(r, ; Mj) (4.15)

g(t,x,y|tj, xj, yj; j , Mj) = K0e(MjMc)

(t tj+c)p hTP(r, ; Mj) fWE((, j); r). (4.16)

Such models consist of only the background seismicity and aftershock activ-

ities following strike-slip mainshocks. Nevertheless, by focusing on the branch-

ing structure of strike-slip mainshocks, one is able to accentuate the observed

anisotropic aftershock activities surrounding strike-slip mainshocks as suggested

by Chapter 3 and the impact of a focal mechanism-dependent spatial kernel in

ETAS models.

4.5 Parameter Estimation

Given a catalog of distinct estimated occurrence times, spatial coordinates, mag-

nitudes, and focal mechanisms of earthquakes,{(ti, xi, yi, i, Mi); Mi M0, i =1, . . . ,} during a time interval [0, T] and within an observation region A, the log

likelihood of a point process model such as (4.0) is given by (Daley and Vere-Jones, 1988)

log L() =ni=1

log (ti, xi, yi|Ht) T0

A

(t,x,y|Ht) dx dy dt,

49


63/85

where is the parameter vector. Parameter estimates can be obtained via maxi-

mum likelihood. For details on maximizing the log-likelihood for isotropic models

using numerical methods, see Ogata (1998). Veen and Schoenberg (2004) explore

a relatively robust and efficient Expectation Maximization-based alternative to

gradient-based approaches.

4.6 Goodness-of-fit and Diagnostic Methods for Spatial

and Spatial-temporal Point Process Models

The goodness-of-fit of multi-dimensional point process models are commonly in-

vestigated using likelihood statistics such Akaike Information Criterion (AIC,

Akaike 1974) and Bayesian Information Criterion (BIC; Schwarz 1978). For given

data ofNobservations, letL() be the likelihood of which hask unknown pa-

rameters, and MLE be its MLE, then the AIC value of the model is defined

by

AIC = 2log L(MLE) + 2k;

and the BIC value is

BIC = 2log L(MLE) +k log N.

Both AIC and BIC examine the maximized likelihood values plus a penalty term.

In addition to penalizing for the number of unknown parameters, BIC also takes

into account the sample size effect so that it would correct the bias toward the

more complex models. When used for model selection, lower values of AIC and

BIC indicate better fits.

While the AIC and BIC are useful for scoring the overall quality of fit, graphi-

cal summaries may also be useful, especially in order to identify where one model

may be fitting well or poorly and to suggest possible ways in which a model may

50


64/85

be improved. Schoenberg (2003) invest

Documents

Anisotropic Extensions of Space-time Point Process Models for Earthquake Occurrences (1)