PEAT8002 - SEISMOLOGY Lecture 11: Earthquake …rses.anu.edu.au/~nick/teachdoc/lecture11.pdf ·...

PEAT8002 - SEISMOLOGYLecture 11: Earthquake source mechanisms

and radiation patterns I

Nick Rawlinson

Research School of Earth SciencesAustralian National University

Earthquake sourcesIntroduction

In the next two lectures, we will study the generation ofseismic waves, and how they are used to studyearthquakes.Earthquakes almost always occur on faults, whichrepresent the boundary between two rigid media that arecapable of relative motion.Earthquakes that occur on land and close enough to thesurface often leave visible evidence in the form of grounddislocation e.g. the San Andreas fault.One of the first major breakthroughs in understandingearthquake mechanisms was the proposal of the so-calledelastic rebound theory of earthquakes on a fault.


Illustration of the elasticrebound theory. Materialon either side of the faultdeforms in response tostress, but friction on thefault locks it in place.Eventually, the strainaccumulation is more thanthe rocks on the fault canwithstand, and the faultslips, resulting in anearthquake.


The elastic rebound idea was a major conceptualbreakthrough, because the faulting seen at the surface hadpreviously been regarded as an incidental side-effect of anearthquake, rather than its cause.The largest earthquakes typically occur at plateboundaries. From the perspective of elastic reboundtheory, they represent the most dramatic part of theseismic cycle.During the interseismic stage, steady motion occurs awayfrom the fault, but the fault itself is “locked".Immediately prior to rupture, there is the preseismic stagethat can be associated with small earthquakes(foreshocks) or other possible precursory effects.


The earthquake itself marks the coseismic phase, duringwhich rapid motion on the fault generates seismic waves.Finally, a postseismic phase occurs following the rupture,and aftershocks and transient slip may occur for up toseveral years.Seismic waves radiated from an earthquake reflect thegeometry of the fault and the motion it experiences duringrupture. As such, they can be used to obtain a clear pictureof the fault kinematics.Geodetic measurements and geological observations canbe used to complement seismic studies of rupturemechanisms, but are of limited benefit in oceanic regionsor for deep focus events.

Earthquake sourcesFocal mechanisms

In the following examination of focal mechanisms, weassume that seismic energy is radiated from a planarsurface across which relative motion occurred during anearthquake.Geological observations of faults that reach the surfacedemonstrate that this is a reasonable approximation inmany cases, and is generally (but not always) consistentwith seismic observations.The geometry of a fault can be described in terms of theorientation of the fault plane and the direction of slip alongthe plane.The fault plane is characterised by n, its normal vector, andits direction of motion is given by d, the slip vector in thefault plane.


The slip vector indicates the direction in which the hangingwall (upper side) moves with respect to the foot wall (lowerside). The slip vector always lies in the fault plane, and istherefore perpendicular to n.


Several different coordinate systems are useful forstudying faults. One is aligned such that the x1 axis is inthe strike direction of the fault, x3 is vertical, and x2 isperpendicular to x1 and x3 (see previous figure).Alternatively, the orientation of the fault and slip can use ageographic coordinate system, where x1, x2 and x3correspond to north, west and vertical respectively.If δ represents the dip angle, φf the strike of the fault, and λthe slip angle of the fault, then the vectors n and d can bedescribed in terms of the geographic coordinate system.


The unit normal vector to the plane in this coordinatessystem is:

n =

− sin δ sin φf− sin δ cos φf

cos δ

The unit slip vector in the plane is given by:

d =

cos λ cos φf + sin λ cos δ sin φf− cos λ sin φf + sin λ cos δ cos φf

sin λ sin δ

The two different coordinate systems (φf , δ, λ) and (n, d)are useful for different purposes.


Although the slip angle can vary between 0 to 2π, severalbasic fault geometries can be defined.


If a fault is constrained to a rectangular shape, then thedistance along strike is the fault length, and the dimensionin the dip direction is the fault width.In reality, a fault may be a complicated surface, and anearthquake generated by movement along this surfacemay ultimately be composed of several sub-events atdifferent places.Complicated seismic events of this type can be treated asa superposition of simpler events, however, so it is usefulto understand how seismic waves can be generated by asimple fault with a rectangular geometry.

Earthquake sourcesFirst motions

Seismograms recorded at various locations can be used tostudy the geometry of faulting during an earthquake (thefocal mechanism).This can be done because the pattern of radiated seismicwaves depends on the fault geometry.One simple way of doing this is to study the first motion orpolarity of body waves.Focal mechanisms derived from first motions use the ideathat the polarity of the first P-wave arrival will vary betweenseismic stations at different directions from an earthquake.This occurs because the first motion is eithercompressional (material initially displaced towards thestation) or dilatational (material initially displaced awayfrom the station).


Example of first motions from a pure strike-slip earthquake.The P-waves impinge on a station from below.


The first motions define four quadrants, two compressionaland two dilatational. The division between quadrantsoccurs along the fault plane and a plane perpendicular to it.Seismograms exhibit small or zero first motions in thesedirections, because the first motion changes fromdilatational to compressional.The perpendicular planes are called nodal planes, and iftheir orientations can be found, then the fault geometry isknown.However, first motions alone cannot distinguish betweenthe fault plane and the perpendicular auxiliary plane.


Fault plane solutions for first motion studies are oftenrepresented using a focal sphere

First motion down(dilatation)

First motion up(compression)

The focal sphererepresents an imaginarysphere surrounding theearthquake location.The pattern ofcompressions anddilatations can be mappedonto this sphere accordingto where source-receiverpaths intersect the lowerhemisphere.


This can be done relatively easily (assuming a sphericallysymmetric Earth) provided the source-receiver azimuth isknown, in addition to the ray take-off angle at thehypocentre.Once the earthquake has been located, then the azimuth(A) is simple to calculate from spherical geometry. Forexample, one could use the equations:

cos ∆ = sin λr sin λs + cos λr cos λs cos(φr − φs)

cos A =cos λs sin λr − cos λr sin λs cos(φr − φs)

sin ∆

where (λs, φs) and (λr , φr ) are the latitude and longitudecoordinates of the source and receiver respectively, and ∆is the angular distance between source and receiver.


The inclination angle i of the ray at the source can becomputed from standard traveltime tables (e.g. assumingiasp91 or ak135), which provides an estimate of dT/d∆.The ray parameter for a spherically symmetric mediumwas derived earlier as:

p =r sin i

v=

dTd∆

Thus, the ray take-off angle can be determined from theradius and velocity at the earthquake location.On the lower hemisphere of the focal sphere, the firstmotions can then be plotted at the locations defined by(A, i) for each station.


The lower hemisphere of the focal sphere can be plottedonto a plane via a stereographic projection, referred to as astereonet. On a stereonet, lines and planes which passthrough the centre of the focal sphere plot as points andlines respectively.

Projected positionof station

FOCUS

RAY

PROJECTION PLANE

Lowerhemisphere

Focal Sphere


Note that it is only intersections between lines or planesand the focal sphere that are projected.For example, a vertically dipping N-S plane plots as astraight line through the centre. If the plane dips at anangle, then it will plot along one of the meridians of thestereonet.Planes striking at different azimuths can be plotted in asimilar way by rotating the stereonet.It is also straight forward to plot planes perpendicular to agiven plane; simply rotate the stereonet so that the planelies on a meridian, and find the point on the equator 90◦

from the intersection of the plane with the equator. Thispoint is the pole of the plane.


The pole to a plane represents where the normal to theplane intersects the sphere. Any plane perpendicular tothe original plane must contain the pole.


The plot on the leftdemonstrates howto represent a planedipping at 60◦E witha strike of N45◦W.The plot on the rightdemonstrates howto represent a raypath with azimuth40◦ and take-offangle 60◦.


Focal mechanisms forearthquakes with various faultgeometries. Compressionalquadrants are black.If the seismic source was anexplosion instead of anearthquake, the “beachball"diagram would be completelyblackIn many cases, earthquakesource mechanisms involvesome combination of strike-stripfaulting and normal or thrustfaulting.


Cascadia events

The example on theleft shows aselection of focalmechanisms forsome of the largestearthquakes inCascadia.This complex regionof the Earth isdominated by theeastward subductionof the Juan de Fucaplate.

Earthquake sourcesPressure and Tension axes

Earthquake focal mechanisms can be used to infer stressorientations in the Earth. One of the simplest fault modelspredicts that faulting occurs on planes 45◦ from themaximum and minimum compressive stresses.Equivalently, these stress directions are halfway betweenthe nodal planes.Thus, the maximum compressive (P) and minimumcompressive (T) axes can be found by bisecting thedilatational and compressional quadrants, respectively.To bisect the angle between two nodal planes on thestereonet, we find the poles for the two planes, draw thegreat circle connecting them, and mark the point on it halfway between the poles.

Earthquake sourcesPressure and Tension axes

Illustration of therelationship between faultplanes and maximum (P)and minimum (T)compressive stresses.Using first motion studies,it can be difficult toaccurately constrain nodalplanes due to unevensampling of the wavefield(e.g. few stations in oceanbasins).

Earthquake sourcesBody wave radiation patterns

The radiation patterns of P and S waves due to motion ona fault plane is what would occur for a pair of force couples,which are pairs of forces in opposite directions but only asmall distance apart.However, due to the conservation of angular momentum,force couples do not exist in isolation. If one couple isoriented in the slip direction, then another must exist in theperpendicular auxiliary plane. This pair of force couplesare referred to as double couples.

It is worth noting that thisidea of equivalent forces isonly a simple model of apotentially complex faultingprocess.


The radiation patterns of double couples have a naturalsymmetry about the fault plane, and are therefore normallywritten using a coordinate system oriented along the fault.In such a system, the fault plane lies in the x1 − x2 plane,so its normal is the x3 axis. The slip vector lies in the faultplane, and is parallel to the x1 axis.The direction orthogonal to x1 and x3 is referred to as theNull axis (i.e. x2).To see how the radiation patterns vary with the direction ofthe receiver, consider the radiation field in sphericalcoordinates, where θ is measured from the x3 axis and φ ismeasured in the x1 − x2 plane.


It turns out from seismic source theory, that the far fielddisplacement due to compressional waves can be given by:

ur =1

4πρα3rM(t − r/α) sin 2θ cos φ (1)

noting that for a compressional wave moving away fromthe source, the displacement will only be in the radialdirection (for a homogeneous medium).The first term in Equation (1), 1/4πρα3r , is an amplitudeterm, which shows that the amplitude decays as 1/r .The second term, M(t − r/α), describes the energy pulseradiated from the fault, which propagates away at theP-wavespeed α, and arrives at a distance r at a timet − r/α.


M(t) is called the seismic moment rate function or sourcetime function. It is the time derivative of the seismicmoment function:

M(t) = µD(t)S(t)

which describes the faulting process in terms of the rigidityof the material (µ) and history of the slip D(t) and faultarea S(t).Later, we will describe earthquake size and energy releasein terms of the seismic moment M0 = µDS, where D is theaverage slip (or dislocation of the fault) with area S.The seismic moment function can be writtenM(t) = M0x(t), where x(t) is the source time function.


The final term in Equation(1), sin 2θ cos φ, describesthe P-wave radiationpattern. It is four-lobed,with two positivecompressional lobes andtwo negative dilatationalones.The displacement is zeroon the fault (θ = 90◦) andauxiliary (φ = 90◦) planes.The maximum amplitudestherefore occur midwaybetween these two nodalplanes.


For an S-wave, which has two components of displacement(uθ, uφ) oriented perpendicular to the propagationdirection, the radiation pattern is quite different.uθ and uφ are given by:

uθ =1

4πρβ3rM(t − r/β) cos 2θ cos φ

uφ =−1

4πρβ3rM(t − r/β) cos θ sin φ

where β is the S-wavespeed.In this case, the S-wave motion does not have nodalplanes. Displacement is perpendicular to the P-wave nodalplanes, and at the null axis (θ = φ = 90◦), displacement iszero.


Particle motion converges towards the centre of thecompressional quadrants, which is the location of theT-axis, and diverges from the centre of the dilatationalquadrants, which is the location of the P-axis.

The above equations for P-and S-wave displacementclearly show why S-waveson a seismogram usuallyhave a large amplitudethan P-waves. Theequations predict anaverage ratio of α3/β3 ≈ 5.


Radiation amplitude patters of P and S-waves in thex1 − x3 plane.

PEAT8002 - SEISMOLOGY Lecture 11: Earthquake …rses.anu.edu.au/~nick/teachdoc/lecture11.pdf ·...

Documents

Transcript of PEAT8002 - SEISMOLOGY Lecture 11: Earthquake …rses.anu.edu.au/~nick/teachdoc/lecture11.pdf ·...