PEAT8002 - SEISMOLOGY Lecture 12: Earthquake source...

PEAT8002 - SEISMOLOGYLecture 12: Earthquake source mechanisms

and radiation patterns II

Nick Rawlinson

Research School of Earth SciencesAustralian National University

Earthquake sourcesWaveform modelling

P-wave first-motions often cannot constrain focalmechanisms very accurately.Additional information can be obtained by comparingobserved and synthetic waveforms and adjusting sourceparameters until a suitable fit is obtained. This can be doneusing either a forward or inverse approach.Waveform analysis also gives information aboutearthquake depths and rupture processes that cannot beobtained from first motions.The generation of synthetic seismograms is often done inthe frequency domain, which allows efficient manipulationof harmonic waves.A recorded seismic wave is a function of the earthquakesource, Earth structure sampled by the propagating wave,and the seismometer.


It is useful to think of the seismogram u(t) in terms of itsFourier transform U(ω), which represents the contributionof different frequencies. These two quantities are relatedby:

u(t) =1

2π

∫ +∞

−∞U(ω)eiωtdω, U(ω) =

12π

∫ +∞

−∞u(t)e−iωtdt

The above equations allow a seismogram or itscomponents to be transformed from the time domain intothe frequency domain or vice versa.Generating synthetic seismograms from earthquakes canbe done using a convolution approach, where wedeterministically combine contributions from known effects.


The convolution s(t) of two time series w(t) and r(t) in thetime domain can be written:

s(t) = w(t) ∗ r(t) =

∫ +∞

−∞w(t − τ)r(τ)dτ

where τ applies a relative time shift to the waveforms.Convolution is often carried out in the frequency domainbecause it simply equals the product of the Fouriertransforms i.e. S(ω) = W (ω)R(ω).A seismogram can therefore be written as:

u(t) = x(t) ∗ e(t) ∗ q(t) ∗ i(t)

where x(t) is the source-time function (i.e. the signalimparted by the earthquake), e(t) and q(t) represent theelastic and anelastic effects of Earth structure respectively,and i(t) represents the instrument response of theseismometer.


Equivalently, the seismogram in the frequency domain canbe written:

U(ω) = X (ω)E(ω)Q(ω)I(ω)

Response of an 1960sanalogue WWSSN longperiod seismometer.The impulse responseindicates how theseismometer wouldrespond to an input deltafunction.


The effects of earth structure are divided into two factors.e(t) is a function of the Earth’s elastic properties, whichgive rise to focusing, defocusing, reflection, refraction etc.of the propagating wave.q(t) describes anelastic attenuation, which is caused bythe conversion of wave energy to heat. The quality factor Qcharacterises this attenuation.The earthquake source time function x(t), can be long andcomplex, depending on the nature of the rupturemechanism.In the case of a short fault that slips instantaneously, theseismic moment function M(t) = µD(t)S(t) is a stepfunction. Since x(t) = M(t), the source time function issimply a delta function.


In order to synthesise greater complexity in the source timefunction, consider a simple case in which the rupture ateach point on a rectangular fault radiates an impulse. Thetotal radiated signal is not impulsive because the pointimpulses occur over a finite time period.If we consider the simple situation of a fault of length Loriented at an angle θ from the receiver, and rupturing atspeed vR, then the first arrival at the receiver will be att = rs/v . Here rs is the distance from the start of the faultand v is the propagation speed of either a P-wave (v = α)or an S-wave (v = β).The far end of the fault ruptures at a time L/vR later, givinga seismic arrival time of t = re/v + L/vR, where re is thedistance from the end of the fault to the receiver.


From the diagram below, we see that b = rs − L cos θ anda = L sin θ. Therefore:

re2 = a2 + b2 = rs

2 − 2rsL cos θ + L2

For points far from the fault (r >> L), re ≈ rs − L cos θ.

rs

Rv

TR

θ

re

LRupturestarts

Ruptureends

Station

t

A

Boxcar time pulsea

b


The time pulse TR due to the finite length of the fault isgiven by:

TR =LvR

+re

v− rs

v=

LvR

− L cos θ

v=

Lv

[vvR

− cos θ

]TR is known as the rupture time. vR is typically assumed tobe about 0.7-0.8 times the shear velocity β, so the ratiov/vR is about 1.2 for shear waves and 2.2 for P-waves.The maximum rupture duration occurs for θ = 180◦, andthe minimum rupture duration occurs for θ = 0◦.These expressions can be modified for different faultshapes and rupture propagation directions.


The source time function is also influenced by the fact thateven at a single point on the fault surface, slip does notoccur instantaneously.The slip history is often represented as a ramp functionthat begins at time zero and ends at the rise time TD. Thederivative of this ramp function, which provides the sourcetime function for a single point, is therefore a boxcarfunction.Convolving this rise time effect with the finite duration ofthe faulting process yields a trapezoid with length TR + TD.More complicated source-time functions than these can bederived from observational data, particularly for largeearthquakes.


Illustration of theeffect of includingrise time in theestimate of thesource time functionx(t).Here it is assumedthat at each point,the fault moves witha constant velocity


The time duration of the radiated pulse has an azimuthaldependence due to the finite rupture length (i.e. it has adirectivity).Since the same amount of energy arrives at a givendistance, the amplitude of the ground motion, and hencethe seismic signal, may vary depending on its locationrelative to the source.The integral of the source time function is equal to theseismic moment M0, which is the same at all azimuths.

Fault rupturedirection

M0Area=M0Area=

M0Area=

M0Area=


The elastic structure operator e(t) primarily reflectsinteractions near the Earth’s surface, where the largestchanges in seismic properties occurs.For a deep earthquake, the surface reflections and otherrelated phases arrive much later than direct P, so theseeffects will not influence the first-arriving wavetrain.At distance ranges 30◦ < ∆ < 90◦, the effects of uppermantle triplications and core structure can also be ignoredfor the direct arrival. Therefore, e(t) can be dropped in theconvolution for u(t).


For shallow earthquakes, reflections (pP and sP) from theEarth’s surface may arrive shortly after the first break,making the direct P arrival difficult to separate out.The time delay of the the pP pulse relative to P for anearthquake at depth h is given by:

δtpP =2h cos i

α

where i and α are the incidence angle and velocity forP-waves.

h

2h

i

i

i

P

pP

h

i

P

i

sP

jj


The time delay of the sP pulse for a Poisson solid (λ = µ)relative to P is given by:

δtsP =hα

(cos i +

√3− sin2 i)

noting that sin i/α = sin j/β and α2 = 3β2.Example: for a source at 10 km depth in a medium withα = 6.8 km/s, the time delays at ∆ = 50◦ are δtpP = 2.7 sand δtsP = 3.8 s.Source parameters can be studied by generating syntheticseismograms for various values, and finding the best fiteither by trial and error forward modelling or inversion.Often, first motion, body wave and surface wave analysesare combined.


A useful way to estimate the source time function is basedon the so-called Green’s function:

g(t) = e(t) ∗ q(t)

which combines the elastic and anelastic effects ofpropagation from the source to the receiver.Essentially, the Green’s function describes the signal thatwould arrive at the seismometer if the source time functionwere a delta function.The source time function of an earthquake can be found bydeconvolving the Green’s function and seismometerresponse from the seismogram u(t):

X (ω) =U(ω)

G(ω)I(ω)

Earthquake sourcesMoment tensors

So far, we have assumed that earthquakes result frommovement along a simple fault.This approach is now generalised to include other types ofseismic sources by using seismic moment tensors.The seismic moment tensor M can be written in the form ofa 3× 3 matrix as:

M =

Mxx Mxy MxzMyx Myy MyzMzx Mzy Mzz

where each component represents one of the ninepossible force couples.


A force couple consists of two forces acting together. Forexample, Mxy consists of two forces of magnitude f ,separated by a distance d along the y − axis, that act inopposite directions (±x). The magnitude of Mxy = fd ,which has units of Nm.In the case of Mxx , two forces of magnitude f areseparated by a distance d along the x − axis, and act inopposite directions (±x). This type of couple is sometimesreferred to as a vector dipole.The difference between Mxy and Mxx is that no torque isexerted in the latter case.


In the case of a simple planar fault with a strike directionaligned to the y − axis, and undergoing left-lateral purestrike-slip motion, the equivalent body forces Mxy and Myxmake up the double couple source.The Mxy couple is required in this case to ensure thatangular momentum is conserved.The magnitude of the equivalent body force is M0, thescalar seismic moment of the earthquake

M yx

M xy

−My’y’

M x’x’Fault

Slip

Slip


Thus, the moment tensorof an earthquakerepresents both its faultgeometry, via the differentcomponents, and its size,via the scalar moment.The moment tensor is asimple mathematicaldescription of the seismicwaves produced by acomplex rupture involvingdisplacements varying inspace and time on anirregular fault.


For the above example, the moment tensor can be written:

M =

0 M0 0M0 0 00 0 0

= M0

0 1 01 0 00 0 0

The moment tensor can be written in any orthogonalcoordinate system. In general, the tensor will appear morecomplex than the above example for arbitrary orientationsof slip direction and fault plane.For a double-couple earthquake in an arbitrary coordinatesystem, the components of the moment tensor areprovided by the scalar moment and the components of n,the unit normal vector to the fault plane, and d, the unit slipvector.


In index notation,

Mij = M0(nidj + njdi).

which can be written in full as:

M = M0

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

In this case M is symmetric. Physically, this corresponds toslip on either side of the fault or auxiliary plane yielding thesame seismic radiation pattern.The trace of M is tr[M] = 2M0n · d = 0 because the slipvector lies in the fault plane and is therefore perpendicularto the normal vector.


A non-zero trace implies a volume change (explosion orimplosion); such an isotropic component does not exist fora pure double-couple source.

The scalar moment M0 =√∑

ij Mij2/√

2 is analogous tothe magnitude of a vector.Using the definitions of the normal and slip vectors interms of fault strike, dip and slip directions, it is straightforward to write the moment tensor for any fault.The reverse process of finding the fault geometrycorresponding to a moment tensor is more complicated.However, this is necessary for waveform inversions thatyield the moment tensor.


It turns out that the eigenvectors of the moment tensor areparallel to the T, P and null axes, which enables the faultgeometry to be reconstructed from the moment tensor.

A selection of momenttensors and theirassociated focalmechanisms. The top rowshows an explosion and animplosion.The next three rows are fordouble-couple sources.The bottom two rows showCLVD sources


As illustrated in the previous slide, if all three diagonalterms of the moment tensor are nonzero and equal, thepolarity of the first motions is the same in all directions.This triple vector dipole of three equal and orthogonal forcecouples is the equivalent body force system for anexplosion or an implosion.The moment tensor therefore has the form:

M = E

1 0 00 1 00 0 1

= E I

which has a trace of 3E . A moment tensor with a non-zeroisotropic component represents a volume change.


Most explosions are artificial, but they can also occurnaturally. For example, they may be associated with fluidand gas migration linked to magmatic processes, or withsudden phase transitions or metastable minerals.High-velocity impacts of meteorites could also be modelledwith explosive sources.The physical process of an explosion is very different fromthat of an earthquake. As the initial shock wave expandsand diminishes in amplitude, a point is reached when thedeformations are small enough to occur elastically, yieldinga spherical P-wave.This propagating wave interacts with interfaces within theEarth, including the free surface, and generates SV andRayleigh waves. SH-waves are also observed, which issomething that is not expected in a sphericallysymmetric isotropic Earth.


One possibility to explain the presence of SH waves is thetectonic release of deviatoric stress triggered by theexplosion.Another class of non-double-couple seismic sources arecompensated linear vector dipoles (CLVDs). These aresets of three force dipoles that are compensated, with onedipole twice the magnitude of the other two:

M =

−λ 0 00 λ/2 00 0 λ/2

The trace of the moment tensor is zero, so there is noisotropic component. In contrast to the beachball focalmechanisms of double-couples, the first motions for CLVDslook like baseballs or eyeballs.


Two primary explanations have been offered fro CLVDs.One is in regard to volcanic regions, where an inflatingmagma dike can be modelled as a crack opening undertension. The moment tensor for such a crack turns out tobe:

M =

λ 0 00 λ 00 0 λ + 2µ

This can be decomposed as follows:λ 0 0

0 λ 00 0 λ + 2µ

=

E 0 00 E 00 0 E

+

−2µ/3 0 00 −2µ/3 00 0 4µ/3

where E = λ + 2µ/3 and the second term is a CLVD.


An alternative explanation is that CLVDs are due tonear-simultaneous earthquakes on nearby faults ofdifferent geometries.For example, consider the sum of two double-couplesources of different orientation with moments M0 and 2M0:M0 0 0

0 0 00 0 −M0

+

0 0 00 −2M0 00 0 2M0

=

M0 0 00 −2M0 00 0 M0

Thus, adding together these two double couples yields aCLVD. In this case, the P, B and T axes of the first sourceare the T, P and B axes of the second


Thus, if the first earthquake were a strike slip on a verticalfault, the second would be normal faulting on a 45◦ dippingfault.The next example shows CLVD-type focal mechanisms forearthquakes near a volcano in Iceland. The mechanism isthought to reflect reverse faulting on cone-shaped ringfaults surrounding the magma chamber.In this model, deflation of the magma chamber increaseshorizontal compression, so the roof block above themagma chamber subsides with respect to the surroundingrock.

PEAT8002 - SEISMOLOGY Lecture 12: Earthquake source...

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