Earthquake faulting - largely reflects motion of thelithosphere

Elastic rebound theory

Friction on the fault locks itpreventing sides from slipping

Material at distance on oppositesides of the fault move relativeto each other

Eventually strain overcome whatthe fault can withstand and thefault slips -- EARTHQUAKE

Fault geometry

!

ˆ n =

"sin# sin$ f

"sin# cos$ f

cos#

%

&

' ' '

(

)

* * *

Normal vector to thefault plane

Unit vector in the slip direction

!

ˆ d =

cos" cos# f + sin" cos$ sin# f

%cos" cos# f + sin" cos$ cos# f

sin" cos$

&

'

( ( (

)

*

+ + +

Orientation of fault plane Right/left lateral strike-slip

Extensional normal fault Thrust fault

First motions of P-waves can be used to determinethe orientation of the fault plane

Vertical component of seismogramsrecord upward or downward first

motion corresponding to eithercompression or dilatation.

Four quadrants divided up into twocompressional and two dilatational

divided by the fault plane and aplane perpendicular to it.-- called

nodal planes - first motions can notresolve which plane is the actual

fault plane

!

ur

=1

4"#$ 3rM (t % r /$)sin2& cos'

Body wave radiation pattern for a double-couple source

Radial Displacement

!

M (t) = µD(t)S(t)

Seismic moment rate or source time function

product of rigidity, slip (D), and area (S)

!

M0

= µDSStatic moment Tensor:

M(t) ---> the pulse radiatedfrom the fault which propagates

away at speed α and arrives atdistance r at a time (t-r/α)

!

u" =1

4#$% 3rM (t & r /%)cos2" cos'

Shear-wave displacement

!

u" =1

4#$% 3rM (t & r /%)(&cos' sin")

!

p =r sin i

v=dT

d"

Take-off angle

Distance from earthquake toseismic stations can beconverted to take-off angles

So you want to make youown plot of fault planes todetermine a focalmechanism?

Maximum compressive(P) andminimum compressive (T) axes

Waveform modeling:seismogram u(t) can be written

!

u(t) = x(t)" e(t)"q(t)" i(t)

!

U(") = X(")E(")Q(")I (")

This expression can also be written in the frequency domainwhere convolutions in the time domain become multiplications.

x(t) is the signal imparted to the ground by the earthquake, bothe(t) and q(t) represent effects of earth structure and i(t) is theinstrument response of the seismometer.

Effects of directivityon the source timefunction seen atstations at differentazimuths

The resulting displacement on a seismometeru(t) is then a convolution of the responses ofthe source, structure and instrument. Thegoal is to solve for the source.

L=fault length

arrival timeL/VR+r/V

arrival timer0/V

Rupture fronts propagate at rupture velocity VR over atime TR

!

u(t,",#) = i(t)$q(t)$M

0

4%&h'h

3

g(")

aC(i

0)(

[RP(#,ih )x(t )*

P)+ R

P(#,% ) ih )+

PP(ih )x(t )*

PP)

+RSV(#,% ) jh )

'h cos ih

,h cos jh+SP

( jh )x(t )*sP)]

P-wave displacement at a distance of 30°-90° as a function oftime, distance, and azimuth:

!

g(")

a

# $ %

Geometric spreading term

!

"PP,"

SP} Relfection coefficients

!

RP,R

SV,R

SH} Radiation pattern

- P-wave incidence anglei0

-free surface amp. correction

C(i0)

-source time function laggedby travel times

x(t)

!

g(t) = e(t)"q(t)

To study the source time function of an EQ we can estimate theearth response using a Green’s Function:

where e(t) is the structure and q(t) is the attenuation

!

u(t) = x(t)" e(t)"q(t)

!

X(") =U(")

G(")I (")

Synthetic seismograms depend on the assumed focal depth, whichcontributes to the separation between arrivals, therebycontributing to the pulse shape:

Effects of different STF on synthetic seismograms at teleseismicdistances:

Instrument effects can make itdifficult to resolve somedifferences in the STF.

Effects of including a crustallayer when calculating P-wavesynthetic seismograms froman earthquake below theocean:The crustal layer can have asmaller effect than the waterlayer.

Modeling a single large eventsas a series of smaller events:

For a complex EQ the source is treated as a sum of STF withdifferent amplitudes cj at different times τj.

!

u(t) = cj x(t "# j )$ g(t)$ i(t)[ ]j=1

k

%