Keith Priestley Leture 14 The Earthquake soure

Physis of the Earth as a Planet

EARTHQUAKE SOURCE

EARTHQUAKE LOCATION

Figure 107: Diagram of the rupture surfae of a fault. The rupture spreads from a point of initial rupture,

the hypoenter, over the fault surfae. The point on the surfae diretly above the hypoenter is the

epienter. All moving points on the fault radiate P and Swave energy. The average loation of the

energy release is alled the entroid loation.

The hypoenter of an earthquake is given in terms of four values: the (1) time, (2) latitude, (3) longitude

and (4) depth of initiation of the seismi energy. The hypoenter has to be determined from the arrival

times of phases reorded at seismographs around the earthquake. At least four phases are required to loate

an earthquake. The loation problem is then, given the arrival times of various seismi waves at a set of

seismographs, to dedue the origin time and spatial oordinates of the hypoenter, the initiation point of

the radiated seismi energy. There are two problems: (1) the identiation of the seismi phases whose

arrival times are measured, and (2) the hoie of an earth model to alulate the propagation time of the

various seismi phases.

In many ases only the rst arriving P-wave is used in the loation making the phase identiation less of a

problem but seondary phases greatly improve the a

uray of a loation, espeially the depth. The hoie

of an earth model is more diult. Ideally we should onsider the 3D struture of the Earth or at least use

regional veloity struture in the region of the soure. A pratial rst step is to determine a preliminary

loation in a 1D or spherially symmetri earth model, then rene the loation using a veloity model

spei to a region. The a

uray of the loation depends on a number of fators: (1) the a

uray of the

arrival time readings, (2) the validity of the veloity model, and (3) the azimuthal distribution of stations

about the event.

88

GRAPHICAL LOCATION USING SP TIMES

Pg

Pn

Sg

Distance

Tim

e

S P time

Epicentraldistance

Pg

Pg

Pn Pg

Sg

Sg

Sg

Figure 108: Determination of epientral distane from S - P times

Arrival travel time: ta(p) = tt(p) + to = x/vp + to

S minus P time : (ta(s) to) (ta(p) to) = tt(s) tt(p) = ta(s) ta(p) S P TIME

Distance : (S P) = tt(s) tt(p) =x

vs

x

vp= x(

1

vs

1

vp) = x = (S P)

vpvsvp vs

One the epienter is determined, the origin time an be determined from the distane and veloity. Beause

the travel time urves are only approximate and beause of measurement errors, the irles will rarely

interset at a point. The epienter is taken as the average of where the pairs of ars interset. If the

veloity model (travel time urves) are a good the average veloity for the region and the ars are all

onave outward from the epienter, then the origin time is too late. If the ars are all onvex outward

from the epienter, then the origin time may be too early or the event may be deep.

distance fromS P time

N

10 km

Cross section

Map view

sta 2

sta 1 sta 2

sta 3

epicentrehypocentre

sta 1

Figure 109: Graphial solution for the hypoentral loation

89

LEASTSQUARES HYPOCENTER DETERMINATION

h

V(r)P

Figure 110: Earthquake hypoenter seismograph geometry

Four parameters are required to dene the hypoenter: the latitude, the longitude, the depth, and the

origin time:

h = (x, y, z, to)

If we have n travel time observations To,i at a network of seismographs loated at (xi, yi, zi), we an invertthese for the loation vetor h. The observed arrival time at the ith station is

To,i = ti (h, xi, yi, zi, v(r)) (71)

where ti is the travel time to the ith

seismograph and v(r) is the Earth veloity struture. We rst hoosea trial hypoenter ho and a referene Earth model vr(r) and alulate the predited arrival times, Tp,ifor eah seismograph. The hypoenter an then be found by minimizing the travel time residual ri

ri = To,i Tp,i

Tp,i is a nonlinear funtion of both the Earth model and individual station loations and this ompliateshypoenter determination. This nonlinearity is lear even for a 2D loation within a plane uniform veloity

struture. The travel time in this ase for a station with oordinates (xi, yi) from a point (x, y) is given by

Tp,i =(x xi)2 + (y yi)2 / v

where v is the veloity. Sine Tp,i does not sale linearly with either x or y, standard leastsquares methods

annot be used and an iterative method is used. Geiger (1912) linearized (71) giving

To,i Tp,i =tix

x +tiy

y +tiz

z +tit

t0

[tit

= 1

](72)

where ti/x, . . . are the partial derivatives of the travel time to the ith

reeiver. If the travel times and

partial derivatives an be alulated for the veloity model v(r) then (72) is a linear system of n equationswith four unknowns and of the form

d = G h (73)

whih an be solved by standard leastsquares. The iterative proedure then involves hoosing a trial

hypoenter ho and solving (73) for the orretion vetor h = (x,y,z,to), then using this to denean improved trial hypoenter ho(new) = ho(old) + h and repeating the proedure until the hypoenter

onverges.

Some are needs to be taken in numerial implementation beause there an be signiant dierenes in

the sizes of the derivatives of the arrival time with respet to the hypoentral parameters. Note that

tci/ts = 1, while the spatial derivatives are often muh smaller. Convergene to a suitable hypoenter

an generally be obtained if the errors in the arrival times are small and the assumed earth model gives

a good representation of the travel times for the paths employed. However, there is a strong trade-o

between the depth estimate zh(est) and the estimated origin time th(est) that is best resolved if observationsof dierent wave types P and S or a wide range of ray parameter an be made.

90

Calulating travel time and partial derivatives for earthquake loation

We an alulate the travel times using ray theory where the travel time is given by the integral

T =

Rab(v)

dl

v(r)(74)

where Rab is the ray path between a soure at a, and a reeiver at b. Using the stationarity of the traveltime in (74) it an be shown that for any heterogeneous veloity eld v(r) the partial derivatives in (72)are given by

tix

= cos

v(r),

tiy

= cos

v(r),

tiz

= cos

v(r), (75)

where , , and are the take-o angles or diretion osines of the rays at the soure.

Examples:

i) Constant veloity model, v = vo, rays are straight lines.

T =1

v

[x2 + z2

]1/2

a

b

z

x

Figure 111: Ray path in a uniform half-spae.

ii) 1-D ontinuous model, v = v(z) = v0 + gz where g is the veloity gradient, ray paths beome ars of

irles.

T =

ba

d

g sin=

1

gln

[tan b/2

tan a/2

]

ba

a

b

z

x

Figure 112: Ray path in a half-spae with a uniform veloity gradient.

91

iii) Multi-layered models (1-D)

Diret Td =[x2 + z2

]1/2/v (same as i)

Refrated Tr = x/v2 + h1 cos a/v1 + ha cos a/v1

h1

ha V1V2a

a

b

z

x

Figure 113: Ray paths in a model onsisting of a uniform layer over a uniform half-spae.

NON-LINEAR LOCATION ALGORITHMS

Now that muh faster omputers are available, it is feasible to alulate the travel times for eah postu-

lated soure loation rather than rely on the linearized proedure just outlined. This formulation is more

exible and allows the introdution of better representations of the expeted mist distribution between

the observed and alulated arrival times for the dierent phases.

Grid search earthquake location

Figure 114: Grid searh earthquake loation.

92

UNCERTAINTIES IN THE HYPOCENTER

The a

uray of the hypoentral loation has to be estimated from the information available in the inversion.

The implied preision of the estimates is therefore model based and an be quite dierent from the true

errors of the hypoenter.

Soures of error in earthquake loation inlude: (1) errors assoiated with the piking proedure whih an

be simulated by Monte-Carlo analysis; (2) errors assoiated with the hoie of Earth models whih an

generally be redued by using a full range of available phase data, inluding later phases; and (3) errors

assoiated with the magnitude of the event.

The inuene of the magnitude an be quite subtle sine it is primarily related to the available network

geometry for the loation. As the magnitude of the event inreases, it reords more widely and the in-

uene of partiular propagation paths is minimized. But for small events, ertain propagation paths are

emphasized by the pattern of reording, with a onsequent bias in the loation estimate. For example,

small events in the Helleni ar south of the Aegean are well reorded by a large number of stations in

Europe but relatively few stations to the south in Afria. Paths to the north are fast paths with respet

to spherially symmetri earth models so the hypoenters are pulled to the north, resulting in a systemati

misloation of 15-20 km. For large events there are more reordings from Afrian stations to the south and

the eet of the fast paths to the north is minimized.

Una

ounted for lateral variations in the veloity struture:

Figure 115: Distribution of earthquakes epienters in California (left) and the San Franiso bay area (right)

for 1969 1971.

The mist in earthquake loation problems is usually larger than is expeted from timing errors alone

and this is the result of unmodeled lateral heterogeneity in the referene veloity model. For example,

earthquakes o

ur in faults suh as the San Andreas fault in California and lateral motion on the fault

an result in high veloity roks on one side of the fault and low veloity roks on the other side. Events

o

urring on the fault will tend to be misloated o the fault onto the faster veloity blok due to a

systemati bias in the travel times.

93

t = 3.54 s21t = 2.36 s

10 km 10 kmsta #1 sta #2

average velocity = 5 km/s

6 km/s 4 km/s10

km

14.14 km 17.7

km

14.14

km11.8 km

apparent epicenter

actual epicenter4.3 km shift

Figure 116: Veloity model for the upper rust whih a

ounts for the shift in epienters o of the San

Franiso Bay area faults.

Inadequate station distribution:

Stationsequidistant

Error ellipse elongatedaway from array

with origin timedepth tradeoff

Seismic network

distant stations

Figure 117: Examples of poor station distributions whih result in the reording of a small ray parameter

range.

The distribution of rays leaving the soure determines the onstraints that may be plaed on it. Usually

the depth is the least well onstrained parameter along with origin time. If all rays leave the soure in

suh a manner that the partial derivatives with respet to the depth are roughly equal, then the olumn

in the G matrix orresponding to the depth will be nearly proportional to the vetor [1, 1, ..., 1]T whih isequivalent to origin time olumn. Therefore two olumns of G are linearly dependent and the problem isnon-unique. The situation an arise in a regional ase where the earthquake lies outside a network, or in

teleseismi studies where a limited distribution of stations and phases ause a similar problem.

94

MASTER EVENT METHODS

In many ases, the relative loation between events in a loalized region an be determined with muh

greater a

uray than the absolute loation of any of the events beause the lateral veloity variations

outside the loal region aet all of the events in nearly the same way. If one event is designated the master

event then the relative arrival times an be omputed relative to the master event

Trel,i = To,i Tmaster,i

and setting the master event loation to trial solution ho (72) and solving for the orretions to this loation

Trel,i =tihj

hj

where hj is the orretion to the master event loation. The solution is valid as long as h is small.

Figure 118: Comparison of ISC aftershok loation for four large earthquakes in the Kurile ar with relo-

ations of the aftershoks using the master event tehnique. The map views (right) show the epienters

for the aftershok zones and the average foal mehanism for the main shoks. The rosssetions of the

ISC foal depths show a great deal of satter whereas the reloated hypoenters dene the dipping plane

of subdution (from Shwartz it et al, 1989).

95

dept

h (km

)de

pth

(km)

distance (km)distance (km)

distance (km) distance (km)

LSHSCSN

LSHSCSN

SCSN LSH

115.8 115.4115.6 115.8 115.6 115.4

32.8

33.2

33.0

Figure 119: Comparison of earthquake loations in the Imperial Valley of southern California. SCSN

loations are the standard atalog of 408,105 events from 1981 to 2005 loated using a layered 1-D veloity

model for southern California. LSH loations inlude 399,521 events from 1981 to 2005 loated using

luster analysis, waveform ross-orrelation data and a dierential time loation method based on 3-D

starting loations. From Lin et al, 2008.

96

ELASTIC REBOUND THEORY (REID, 1910)

Figure 120: Results of the triangulation measurements analysis in the San Franiso region prior to (1861,

1885) and following (1906) the San Franiso earthquake of 1906 (Reid, 1910).

Mt. Moche

Mt. Diablo

Farallon Lighthouse

Mt. Tamelpias

Figure 121: Map of the San Andras fault in the San Franiso region whih moved in the 1906 earthquake.

Solid triangles denote triangulation points loations (Reid, 1910).

97

Figure 122: Slip on the San Andras fault during the 1906 earthquake, showing triangulation points loations

(Reid, 1910).

Following lastEarthquake

EarthquakeStrain buildup

time = ? 1861 1885 1906

Fau

lt

Figure 123: The elasti rebound model of an earthquake assumes that between earthquakes strain aross

the fault ause stress to a

umulate beause frition has loked the fault surfae. At some point in time

the stress exeed the strength of the rok and slip on the fault the earthquake o

urs relieving the

a

umulated stress.

98

EARTHQUAKE FAULTS

Figure 124: Fault geometry The oordinate system is oriented with respet to the fault suh that x1 isalong the fault strike, x2 is perpendiular to fault, and x3 is vertial. The fault plane separates the footwall blok from the hanging wall blok (Kanimori and Cipar, 1974).

n normal to the fault plane

d slip vetor whih indiates the motion of the hanging wall blokwith respet to the foot wall blok

f fault strike measured lokwise from north 0 360

fault dip whih is less than 90 measured from the x2 axis 0 90

rake or slip angle between the x1 axis and d in the fault plane 0 360

Figure 125: Basi types of faults The slip angle an vary between 0 and 360 but faults are grouped intothree basi types depending on their slip angle. Strike-slip fault two sides of the fault slide horizontally

by eah other; when = 0 the upper wall moves to the right and the motion is alled left-lateral beauseif we stand on one side of the fault, the other side moves to the left; if = 180 right-lateral motiono

urs. Normal or dip slip fault upper wall slides downwards, = 270 Thrust or reverse fault upper wall moves upwards, = 90 (Eakins, 1987).

99

POINT FORCES AND THE SEISMIC MOMENT TENSOR

We will onsider a seismi soure small enough ompared to the wavelength of the radiated waves that

it an be thought of as a point soure. A single, internal point soure annot at at a point beause

momentum would not be onserved. Internal soures must be represented by fore ouples whih an

either be two fore vetors of magnitude f ating in opposite diretions at a point, or two fore vetorsseparated by a small distane in a diretion perpendiular to the diretion of the fore. In this ase angular

momentum is only onserved if there is a omplementary ouple balaning the fores and this ase is alled

a doubleouple.

Figure 126: Fore ouples are opposing point fores separated by a small distane d. A double ouple is apair of omplementary ouples that produe no net torque (Shearer, 1999).

If we dene a fore ouple Mij in a Cartesian oordinate system as a pair of opposing fores pointing inthe i diretion and separated in the j diretion, then the nine dierent fore ouples dene the momenttensor

M =

M11 M12 M13M21 M22 M23M31 M32 M33

(76)

The magnitude of Mij is fd and is assumed to be onstant as d goes to zero. Sine angular momentum is

onserved (76) must be symmetri so M has only six independent omponents. This is a general represen-

tation of the internally generated fores that an at at a point and is a good approximation for modeling

distane seismi waves for soures whih are small ompared to the seismi wavelength.

Figure 127: The nine fore ouples whih make up the elasti moment tensor (Shearer, 1999).

100

Seismi energy radiated from a fault an be represented by a doubleouple soure. For example, a vertial,

rightlateral, strikeslip fault oriented in the x1 diretion orresponds to the moment tensor

M =

0 M0 0M0 0 00 0 0

(77)

where M0 is the salar seismi moment

M0 = DA

where is the shear modulus, D is the fault displaement, and A is the area of the fault.

Beause Mij = Mji there are two fault planes whih orrespond to the same doubleouple, i.e., (77)

orresponds to the two faults shown below. This is a fundamental ambiguity whih o

urs beause both

faults produe exatly the same displaement in the fareld. The real fault plane is alled the primary

fault plane and the other is alled the auxiliary fault plane.

Figure 128: Sine the seismi moment tensor is symmetri, both the rightlateral and leftlateral faults

have the same moment tensor and radiation pattern (Shearer, 1999).

Sine the moment tensor is symmetri it an be diagonalized by omputing its eigenvalues and eigenvetors

and rotating to a new oordinate system. For example, (77) has prinipal axes at 45

to the original x1

and x2 axes and the rotated moment tensor beomes

M =

M0 0 00 M0 00 0 0

(78)

The x1 oordinate axis is termed the tension axis and the x

2 oordinate axis is alled the pressure axis

and these give the diretions of the maximum ompression and tension if the fault plane orresponds to a

plane of maximum shear.

Figure 129: The double ouple (left) is represented by the odiagonal elements of the moment tensor M12and M21. If the oordinate system is rotated to be aligned with the P and T axes the moment tensor isdiagonal with opposing terms M11 and M22 (Shearer, 1999).

The trae of the moment tensor is a measure of the volume hange and is zero for doubleouple soures

but nonzero for explosion soures.

101

The moment tensor an be written in terms of the unit normal vetor to the fault plane n, the unit slipvetor d, and the salar moment Mo:

Mij = Mo(nidj + njdi)

or

M = Mo

2nxdx nxdy + nydx nxdz + nzdxnydx + nxdy 2nydy nydz + nzdynzdx + nxdz nzdy + nydz 2nzdz

Things to note about the moment tensor:

(a) interhangeability of n and d show that slip on the fault plane and auxiliary plane yield the same seismiradiation.

(b) trae given by 2nidiij = 2nd = 0 sine the slip vetor lies in the fault plane and is thus perpendiularto the normal vetor, moment tensors representing doubleouple soures orresponding to slip on a fault

always have a zero trae. If the trae is nonzero, some net volume hange (i.e. explosion) is represented.

This is alled the isotropi omponent.

The normal vetor to the fault plane is

n =

sin sinf sin cosf

cos

where

f = strike = dip = slip

and the slip vetor is

d =

cos cosf + sin cos sinf cos sinf + sin cos cosf

sin sin

Figure 130: Examples of moment tensor fault representations and beahball fault plane solutions (Dahlen

and Tromp, 1998).

[Shearer 241251

102