Email: ralph@crustal.ucsb.edu
ERI Notes: RJ Archuleta

http://www.geol.ucsb.edu/directory/ralph.html

1Outline for the Lecture 1
1. Strong ground motion—a summation of waves
from a finite fault
2. P and S waves
3. Finite fault geometry
4. Static field
5. Hypocenter, epicenter, slip areas, rupture
velocity
6. Seismic Moment, Magnitude
7. Point Source
8. Attenuation

ERI Notes: RJ Archuleta

2

Denali Fault Earthquake M7.9
November 3, 2002

ERI Notes: RJ Archuleta

3

Denali Acceleration

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4

FP velocity (m/s)

Denali Fault Station 10:
Particle Velocity
1.5
10.500FN velocity (m/s)

A51015120

B

2530353035C0.50-0.5-10ERI Notes: RJ Archuleta

D5101520Time (s)255Small Event Ground Motion

ERI Notes: RJ Archuleta

6

P and S waves
Displacement (mm)3020
100-10-20-3004080120
Time (s)λ+2µVP = α =ρ160;200240280µVS = β =ρPoisson solid: λ =µ; VP = 3 VSERI Notes: RJ Archuleta

7Wave Propagation in an Elastic Solid

ERI Notes: RJ Archuleta

8Wave Propagation in an Elastic Solid

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9P-Wave Propagation in an Elastic Solid

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S-Wave Propagation in an Elastic Solid

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11Near Source Ground Motion

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12
Oglesby and Archuleta, 1997

Faulting: Finite Fault Area

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13

Dipping Fault PlaneERI Notes: RJ Archuleta

14

Elastic Rebound (Reid, 1910)

ERI Notes: RJ Archuleta

15Vertical Strike-Slip Fault

Two blocks move relative to each other at a relative velocity of V/2.
The fault is locked from the free surface to a depth D. Below D theTwo blocks are slipping freely past each other.
ERI Notes: RJ Archuleta

16Strain and Displacement
for a Vertical
Strike-Slip Fault

Fault extends from a depth of 0 km to12 km. Slip is 1 m.

(u(x) = ΔS 1 2 − 1 π tan −1 (x / D)

⎛
⎞
⎛
⎞
ΔS1
⎜
⎟
ε (x, y) = ∂u ∂x = ⎜2⎟⎝ ( 2π D ) ⎠ ⎜⎝ 1 + ( x D ) ⎟⎠(ERI Notes: RJ Archuleta)17

Normalized Strain andDisplacement Attenuation
with Distance from a
VerticalStrike-Slip Fault• Amplitude is 0.5 of maximum at x = D• x>>D, strain decreases ~1/x2 whiledisplacement ~ 1/x• Fault is infinitely long; for a real fault
the decay will be more rapid.

ERI Notes: RJ Archuleta

Slip is normalized by ΔS; strain isnormalized by (ΔS/πD). Where ΔS isthe relative motion of one side withrespect to the other. Note (ΔS/2πD) isthe maximum strain at x/D = 0.
18

Earthquake Source Models❶ Point Source {Maruyama, 1963; Burridgeand Knopoff, 1964}❷ Centroid Moment Tensor {point source}❸ Finite Fault--Kinematic {Haskell, 1964}❹ Finite Fault--Dynamic {Kostrov, 1964}

ERI Notes: RJ Archuleta

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Point Source: Double Couple without Moment
Geometry specified by the strike, dip and rake angles
Size of earthquake determined by seismic moment = µAΔsERI Notes: RJ Archuleta

20Centroid Moment Tensor
The point source can be shifted from the hypocenter to provide
the best fit between synthetics and seismograms
CentroidLengthDip AngleDowndip
WidthHypocenter
ERI Notes: RJ Archuleta

21

Example: Oct 6, 2000 Tottori, JapanERI Notes: RJ Archuleta

22

Seismic Moment and Magnitude
Magnitude: Measures the relative size of an earthquake (Charles Richter, 1935)
Seismic Moment: Measures the absolute size of an earthquake (Keiiti Aki, 1966)

Richter’s Magnitude Formula
M = Log A - LogA0
Where A is the maximum amplitude (measured in mm) on a WoodAnderson seismogram with amplification ~2800.
Log A0 is the constant from Richter’s curve; it is the amplitude
created by a M=0 earthquake.
Aki’s Seismic Moment Formula
Mo = μ s A
Where:
μ : is the shear modulus.
s : is the average slip on the faulted area.
A : is the area of the fault plane over which slip has occurred.

ERI Notes: RJ Archuleta

23

SeismicMoment

ERI Notes: RJ Archuleta

24

Seismic Moment and Magnitude
Magnitude: Measures the relative size of an earthquake (Charles Richter, 1935)
Seismic Moment: Measures the absolute size of an earthquake (Keiiti Aki, 1966)

M= (2/3)Log M0 – 6.033
Seismic Moment is measured in units of N-m (force x distance)Hanks and Kanamori (1979)

MW= (2/3)Log M0 – 6.067
Kanamori (1977)

ERI Notes: RJ Archuleta

25

Magnitude and Moment Magnitude

ERI Notes: RJ Archuleta

26
T. Heaton

Equivalent Energy and Magnitude

ERI Notes: RJ Archuleta

27
A. Johnston

Point Source For an Earthquake

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28

Body Force Equivalence

ERI Notes: RJ Archuleta

Kasahara, 1981
29

Geometrical Attenuation
Let E0 be the initial energy inside a sphere of radius R0
The sphere has an area 4π R 20 , thus the energy per unit area is E04π R 20
This energy will continue to flow outward. Imagine another sphereof radius R (with the same center as that of R0 ). The energy per unit
area through this surface is E04π R 2The ratio of the energy flux through R to the energy flux through R0 isE04 π R2

E04 π R 20

2R= 0R2

Amplitude is proportional to the square root of the energy, thus theamplitude A at a distance R relative to the initial amplitude A0 at R0 is2
RR0A ∝02 =A0RRERI Notes: RJ Archuleta

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Amplitude Attenuation with DistanceAttenuation with Distance, Q=100!
1.E+00!

Normalized Amplitude!

1/R1/2!

1.0 Hz!

1.E-01!

1/R2!

1.E-02!

1/R!

10 Hz!
1.E-03!

1.E-04!
1!

10!

100!

1000!

Distance (Km)!

ERI Notes: RJ Archuleta

31

Seismic Q

ERI Notes: RJ Archuleta

32

Amplitude Attenuation with Distance
Attenuation with Distance, Q=100!
1.E+00!

Normalized Amplitude!

1/R1/2!

1.0 Hz!

1.E-01!

1/R2!

1.E-02!

1/R!

10 Hz!
1.E-03!

1.E-04!
1!

10!

100!

1000!

Distance (Km)!

ERI Notes: RJ Archuleta

33

( )Qkappa = traveltime / Q = H V
Attenuation = exp ( −π f κ )π fκ = 1f = 1 / πκ → fmax

sKappaand Fmax

34
ERI
RJ Archuleta
ESG,Notes:
Tokyo Ins=tuteof Technology, March 23, 2010 Hanks, 1984, BSSA; Anderson and Hough, BSSA, 1984

We can diagonalize the moment matrix (equation 8) to find principal axes. In this case, theprincipal axes are at 45° to the original x1 and x2 axes, we get:!"#$!%&&&&'()*&"*&"%%+&&Double-Couple Radiation!"!" ! # """!""""$#"(10)

Principal axes become tension and pressure axis. The above matrix represents that 1x!"
coordinate is the tension axis, T , and !"# is the pressure axis, P. (see Figure 3)

The following figure shows the variation of the radiation patterns with the direction of thereceiver. It assumed that the radiation field is in spherical coordinates (will be explained in thissection), where ! is measured from the z axis and " is measured in the x-y plane.
(a)
x3x3
θ

x2

φx1

θReceiverx1x3
P waves

Auxiliary
planeF igure 3: The double-coupled forces and their rotation along the principal axes.(Source: Shearer,1999)

uFa

lt

ne-

P wavesx3+

pla

Nullaxis+-x2

x1
Auxiliary
plane+--+

x1

x3

Radiation Patterns

(b)

x3

x1

S waves(c)

T axis
S wavesx3

x3

Fault planex2-./0&1234&52&6372&839273&/5&9./04&(1(/:&x1

x1
The body wave radiation pattern for a double couple source has symmetry
in the spherical coordinate system shown. θ is measured from the x3 axis,
the normal to the fault (x1-x2) plane, and φ is measured in the fault plane.
The P-wave radiation pattern has four lobes that go to zero at the nodalplanes, which are the fault and auxiliary (x2-x3) planes. The S-wave
radiation pattern describes a vector displacement that does not have nodal
planes but is perpendicular to the P-wave nodal planes. S-wave motion
converges toward the T axis, diverges from the P axis, and is zero on
the null axis.

F igure 4: Radial componet

Radiation amplitude patterns of P and S waves in the x1-x3plane. a: Fault geometry, showing the symmetry of the double
couple about the x2 axis. b: Radiation pattern for P waves,
showing the amplitude (left) and direction (right). c: Same as(b), but for S waves.

Figure by MIT OpenCourseWare.

,&&x1

F igure 4: The left graph shows body wave radiation patter for a double couple source. The rightgraph shows the radiation amplitude patterns of P and S waves in the x-z plane. (Source: Steinand Wysession, p221)
P-wave potential in spherical coordinate is given by !"#$%&'%()*%ERI Notes: RJ Archuleta%#$%! &' " ( )& * + , # -$& # + % ;. " P-wave speed,#/ " + & 0 & " ;

(11)
35

&Double-Couple Radiation
Affected by Rupture Speed

Note change in scale when Vr/Vs= 0.9
ERI Notes: RJ Archuleta36

!"#$!%&&&&'()*&"*&"%%+&&&&Direction of Motion
From a Double-Couple Point Source'&terms of time is given by equation (9). The final term describes the P-wave radiation patterndepedig on the two nodal planes. The first term describes P-wave radiation pattern depending on
the two angles !!" "#$%At ! & " & '() , the siplacement is zero on the two nodal planes. Themaximum amplitudes are between the two nodal planes. Figure 8 shows the far-field radiation
pattern for P-waves and S-waves for a double-couple source.

P-wave Motion

S-wave Motion

ERI Notes: RJ Archuleta37

Double Couple Radiation Pattern
The following 10 slides are taken from the notes of Dr. Martin Kaser, fromwww.geophysik.uni-muenchen.de/~kaeser/WS0910/Lecture3.pdf

They are very useful in understanding how the elements of the moment
tensor are related to the geometry of the fault.

ERI Notes: RJ Archuleta38

Radiation Patterns
Radiation Patterns of Moment Tensor Sources
The far field displacement vector u = (u1,u2,u3)T for a moment tensor source can be
expressed bywhereSeparation into different wave type contributions gives

The first term represents motion into the direction of Γ!P-wave term

The second term represents motion
into the direction perpendicular to Γ!S-wave term

spherical coordinate systemwith origin at the source location

Question: What type of S-waves
do we usually distinguish?
© MartinKäser
ERI Notes:
RJ Archuleta

Seismic Sources and Source Inversion

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Farfield Ground Displacement
Radiation Patterns of Moment Tensor Sources
The S-wave term can be split into:• SV-waves – polarized in the vertical plane between source and receiver• SH-waves – polarized in a horizonal plane! introduction of unit vectors, that form an ortho-normal basis:

spherical coordinate systemwith origin at the source location! displacement radiation pattern is given by
withwith(I)with! angle of polarization ε is given by© Martin
ERI Notes:
RJKäser
Archuleta

Seismic Sources and Source Inversion

40Principal Axes
Double Couple – Compressional and Tensional Dipole(Nakano, 1923)

The moment tensor of a double couple is given by

The moment tensor of two dipoles in the rotatedsystem is given by

Then the moment tensor of the dipoles in the non-rotated system
is given byusing
and we get:Question: What does that mean physically ?
Question: Does this remind you of something
concerning “beach balls” ?

© MartinKäser
ERI Notes:
RJ Archuleta

Seismic Sources and Source Inversion

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Principal Axes
Compression, Tension, and Null-Axes
The eigenvectors of the matrix M representing a double couple source form
an orthogonal system and are given by
P axisT axisand point into the directions of the• T-axis (center of compressional quadrant)• P-axis (center of dilatational quadrant)• b-axis (or Null axis, where the fault and auxiliary plane intersect)

N axis

The eigenvalues are 1, -1, and 0, respectively, and give thestrength (seismic moment) of the source.
Note: The matrix R formed by the eigenvectors [t,b,p] can beused to reduce M to diagonal form, with elements equal
to the eigenvalues, i.e.

principle axis transformation!
Note: Knowing the eigenvectors gives you theknowledge about the fault orientation!
© MartinKäser
ERI Notes:
RJ Archuleta

Seismic Sources and Source Inversion

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Radiation Patterns for P and S
Radiation Patterns of Double Couple M13 + M31
Question: How does the corresponding
moment tensor look like ?
Using equations (I) leads toQuestion:
What is surprising looking
at the P-waves ?
The absolute value of theradiation pattern of uS isNote: For radiation patterns of the total field, see Pujol 2003.

© Martin
ERI Notes:
RJ KäserArchuleta

Seismic Sources and Source Inversion

43

Arbitrary Orientation of Double-Couple
Moment Tensor for Fault Slip of Arbitrary Orientation
Definition of a geographic coordinate system:• x1-axis points to the 0°-meridian intersecting the equator• x2-axis points to the 90°-meridian (east) intersecting the equator• x3-axis points to the north-poleDefinition of a epicentral coordinate system (Pujol):• x1-axis (Θ) points to the south• x2-axis (Φ) points to the east
• x3-axis (Γ) points upward from the Earth’s surface

In the epicentral coordinate system
we the define the fault parameters
strike (ϕϕ), dip (δδ), and rake (λλ).

© MartinKäser
ERI Notes:
RJ Archuleta

Seismic Sources and Source Inversion

44

Local to Epicentral Coordinates
Moment Tensor for Fault Slip of Arbitrary Orientation
In a local, fault-aligned coordinate system,
we have defined our double couple momenttensor.

ν! now, we want to transform this moment tensor
from the local coordinate system to the amoment tensor in the epicentral coordinate
system, that we usually use for all other parameters(e.g. model coordinates, mesh coordinates, etc).
! we have to find the rotation operator (matrix) R
Note: The rotated moment tensor willkeep its properties, like symmetry,seismic moment, etc.
© Martin
Käser
ERI Notes:
RJ Archuleta

Seismic Sources and Source Inversion

45

Point Source Moment Tensor
Moment Tensor for Fault Slip of Arbitrary Orientation
The rotation matrix R can be expressed as a theproduct of three simple rotation matrices:whereWe can express the vectors
andand the moment tensor as:or:!
© Martin Käser

ERI Notes: RJ Archuleta

Seismic Sources and Source Inversion

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Coordinates Relative to Aki-Richards
Moment Tensor for Fault Slip of Arbitrary Orientation
Note that the moment tensor for fault slip of arbitrary orientation depends
on the definition of your coordinate axes of the epicentral system!
Another convention often used is the one suggested by Aki and Richards:(Aki and Richards, Quantitative Seismology, University Science Books, 2002)

Definition of the (primed) epicentral coordinate system (Aki and Richards):• x1-axis points to the north• x2-axis points to the east
• x3-axis points downward from the Earth’s surface
Question: What is the effect of this modification on the moment tensor ?
In this case the absolute values of the components of the moment tensor remain
but the signs change, such that we get the primed components by:

M’ for Aki & Richards’
definition
ERI Notes:
RJ Archuleta© Martin
KäserM for Pujol’s
definition
Seismic Sources and Source Inversion

47

Properties of Double-Couple Moment Tensor
Properties of the Double Couple Moment Tensor! we cannot distinguish between the fault and the auxiliary plane! the trace (=sum of diagonal elements) of the tensor is zeroNote: A non-zero trace implies volume change (explosion orimplosion) and produces an isotropic component.

! the scalar seismic moment M0 gives the magnitude of the moment tensor! the t, p, and b-axis are eigenvectors of the moment tensor© MartinKäserERI Notes:
RJ Archuleta

Seismic Sources and Source Inversion

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