A Primer on Focal Mechanism Solutions for Geologists Vince Cronin, Baylor University

Transcription

1 Prefae A Primer on Foal Mehanism Solutions for Geologists Vine Cronin, Baylor University (Vine_Cronin@baylor.edu) The original ontents of this doument are 2010 by Vinent S. Cronin, and may be reprodued with attribution for use in non-ommerial eduational settings. Revised Marh 1, The goal of this primer is to help students develop the skill to reognize and interpret the beahball diagram that graphially represents the geometry of a moment tensor derived by seismologists. I leave most of the quantitative details of the proess of obtaining the moment tensor to ourses in seismology or to the literature; however, I find that some knowledge of how foal mehanism solutions (FMS) have been obtained manually is useful to students, so that the entire proess is not so muh of a blak-box dark art. A partially annotated bibliography is inluded near the end of this doument for the benefit of those who would like to delve more deeply into the theory and methodology used in omputing FMSs. Thanks to all who have improved this doument through their omments based on their experienes with earlier versions. Please send suggestions and onstrutive ritiisms to Vine_Cronin@baylor.edu. What is a foal-mehanism solution and why should strutural geologists know about them? A foal mehanism solution (FMS) is the result of an analysis of wave forms generated by an earthquake and reorded by a number of seismographs. It usually takes at least 10 reords to produe a reasonable foal mehanism diagram, and then only if the seismograph stations are well distributed geographially around the epienter. The omplete haraterization of an earthquake s foal mehanism provides important information, inluding the origin time, epienter loation, foal depth, seismi moment (a diret measure of the energy radiated by an earthquake), and the magnitude and spatial orientation of the 9 omponents of the moment tensor. And from the moment tensor, we an ultimately resolve the orientation and sense of slip of the fault. This wealth of information about an earthquake is plainly of interest to a strutural geologist working on ative strutures. Detailed modeling of the generation of an earthquake by fritional slip or non-linear flow along a fault surfae is, to say the least, a non-trivial matter. The fault itself is part of the problem, beause it onstitutes a signifiant disontinuity within the system. (It is muh easier to develop mehanial models of ontinuous media that have relatively simple rheologies than to work with disontinuous media.) As luk would have it, the radiation pattern of energy emitted by a rustal earthquake an often be modeled to the first order using a simpler proess that does not expliitly involve a fault disontinuity: the double fore ouple or simply the double ouple. The double ouple is mathematially desribed in 3 dimensions by a symmetrial tensor with 9 omponents, known as the moment tensor. Like the stress and strain tensors, the moment tensor an be desribed in terms of three orthogonal axes: P (for pressure; a ompressive axis), T (for tension) and (for null). The orientation and magnitude of these axes for a given earthquake an be resolved using data reorded by several seismographs that are distributed around the epienter. The orientation of the axes of a moment tensor is of interest to a strutural geologist beause the fault surfae along whih the earthquake was generated is 45 from the P and T axes, and ontains the axis. That s the good news. The bad news is that for any moment tensor, there are two possible planes that meet those riteria. The two planes are alled the nodal planes, and they are at right angles to one another and interset along the axis. One of the planes is the fault surfae, and the other is alled the auxiliary plane and has no strutural signifiane. All a seismologist an say with referene to the moment tensor alone is that the earthquake was generated on one or the other of the nodal planes. It takes geologial input to differentiate between the two possible fault-plane solutions. Assuming that the fault plane an be differentiated from the auxiliary plane, the FMS provides the strutural geologist with the orientation of the fault plane, the diretion of hanging-wall slip, and hene the type of fault involved in the earthquake: strike-slip, reverse, normal, oblique. And all of this information an be obtained within seonds by a strutural geologist who knows how to interpret a graphi depition of the FMS, known as a beahball diagram. Analysis of several FMS in a main shok-aftershok sequene allows us to map the path along the fault that slipped, and evaluate whether more than one fault generated earthquakes during the sequene. More sophistiated quantitative analysis of the earthquake soure mehanis an sometimes identify the diretion of fault propagation. Page 1

2 Quik-and-dirty eyeball interpretation of FMS beahball diagrams Strike-slip faulting results in FMS beahballs with a distintive ross pattern. The beahball at right (a) has vertial auxiliary planes that strike northsouth and east-west. At the instant the earthquake ourred, motion of the P wave through the material around the fous aused partiles in the blak quadrants to move away from the fous, while partiles in the white quadrants were drawn toward the fous (b). This ould have resulted either from right-lateral slip along an east-west fault (), or it ould have resulted from left-lateral slip along a north-south fault (d). Field data is needed to differentiate between these two options. FMS beahball diagrams are lower-hemisphere stereographi projetions that show two blak quadrants and two white quadrants separated by greatirle ars oriented 90 from eah other. The great-irle ars are the nodal planes, one of whih oinides with the fault surfae that generated the earthquake. The strike of the fault is indiated by a line onneting the two points at whih the great irle orresponding to the fault intersets the outer edge of the beahball diagram (i.e., where the fault great irle intersets the primitive irle -- the dashed line in the illustration at left). The dip diretion is 90 from strike, in the diretion indiated by the bold arrow from the enter of the plot to the middle of the great irle ar. So how an a mere mortal remember how to interpret the sense of slip along a partiular fault surfae represented in a beahball diagram? Using the strike-slip mehanism at left, assume that we know that the ative fault strikes toward the east-northeast (along the gray line). In my mind s eye, I plae myself on one of the white quadrants, and look parallel to the fault at the dark quadrant that beons just beyond the auxiliary plane. And I hear the immortal words of Darth Vader, Come to the dark side. The diretion I would move if I heeded his nefarious all indiates the sense of slip along the fault, from white quadrant to blak quadrant. Pure dip-slip faults inlude normal faults and thrust faults. Only three of the four quadrants are observable in the beahball diagram for pure dip-slip faults, as shown at right. The enter of the beahball plot is white for normalfault mehanisms (a), and the enter is blak for reverse-fault mehanisms (b). Reverse-fault FMS diagrams look like my at s eyes. a b a b Pratie with basi interpretation Oblique-slip faults have both strike-slip and dip-slip omponents. All four quadrants of a FMS beahball diagram are inluded for oblique slip earthquakes, as shown at left. If the enter of the beahball plot is in a white quadrant (a), the fault has a normal omponent of slip, regardless of whih of the two nodal planes is the fault; if the enter is in a blak quadrant (b), the fault has a reverse omponent of slip. Some FMS beahball diagrams are presented below. For eah one, identify the general type of faults depited (strike-slip, reverse, normal, oblique normal, oblique reverse) and sketh a map of eah of the two possible faults, representing their proper strike and indiating their dip diretion if any and sense of displaement. The first one is done for you as an example. a b d Type of faults: reverse oblique Map of one possible fault d u Map of other possible fault u d Page 2

3 Type of faults: Map of one possible fault Map of other possible fault Type of faults: Map of one possible fault Map of other possible fault Type of faults: Map of one possible fault Map of other possible fault Type of faults: Map of one possible fault Map of other possible fault Type of faults: Map of one possible fault Map of other possible fault Type of faults: Map of one possible fault Map of other possible fault How is a foal mehanism solution derived? In urrent pratie, a FMS is based on the derivation of a moment tensor from analysis of sets of waveform data. There are several methods for finding the omponents of the moment tensor that best fits the available data for a given earthquake. The fault-plane solution or beahball diagram is a produt of the moment tensor inversion. The beahball diagram an also be obtained by graphi tehniques from a study of P-wave first motions, as desribed below; however, the graphi tehnique does not provide enough information to define the moment tensor for the earthquake. The basi tehniques for using P-wave first motions reorded by an array of seismographs to define a FMS beahball diagram were developed before the advent of the miroomputer, and it is still useful to have at least a rude understanding of how to make the diagram manually (desription after Kasahara, 1981, p ). From the preliminary determination of the epienter, we know the loation and origin time of the earthquake. Let us say that 14 geographially separate seismograph stations reorded the event. From the well-known loations of those stations, we ompute the distane between eah station and the epienter. Then we use a standard seismi-veloity model of Earth to define the exat time (t) that the P-wave of the earthquake should have arrived at eah station. Consulting the seismogram reord- Page 3

4 ing the vertial omponent at eah station, we evaluate whether the first motion deteted at that station was an up motion, a down motion, or no apparent signal at the expeted time. t first motion is up first motion is down no apparent signal at t t t The portion of the energy reeived by eah seismograph, that left the earthquake fous in the form of a ompressional P-wave, an be thought of as having traveled along a ray path from the fous to the seismograph. We need to learn two things about that ray for eah station: the azimuth along whih it traveled from the earthquake fous to the station, and its take-off angle (also known as angle of emergene). The take-off angle is the angle between the ray as it just emerges from the fous and an imaginary vertial line extending through the fous. In routine work, the take-off angle is interpreted from a table that relates take-off angle to distane between the fous and the station (e.g., Hodgson and Storey, 1953; Hodgson and Allen, 1954). We are now ready to plot the data on a lower-hemisphere stereographi projetion -- we hoose to use an equal-area stereonet (after Sykes, 1967; a stereonet is reprodued at the end of this doument). We will represent the data from eah seismograph station with one of three symbols: a irle ( ) if the P-wave first motion was down, a blak dot ( ) if the first motion was up, or an x ( ) if the first motion was too weak to differentiate. For eah station, the symbol is plaed along a line extending from the enter of the plot toward the azimuth of the station relative to the earthquake fous, and the take-off angle defines the angular distane from the enter of the plot to the symbol. In the example below, the symbol assoiated with station A is 60 from the enter of the plot along a line direted toward azimuth 50. a B 50 C Stn P wave symbol Stn P wave symbol Stn P wave symbol A A F K D B G L E 60 C H M F M D I G L E J H K I J After all of the first-motion data are plotted (illustration a ), we identify two great-irle ars on the stereonet representing two planes that are at b right angles to one another, that separate the irles from the blak dots, and that pass near or through the x symbols (b). These are the nodal planes, one of whih is oinident with the fault that produed the earthquake. Finally, we fill-in the quadrants aording to onvention (). Clearly, the solution is non-unique, but it is still useful in providing information about the type and orientation of the fault that produed the earthquake. The following data are adapted from two of the foal mehanism solutions published by Lynn Sykes in his lassi paper on the mehanisms of earthquakes along transform faults (Sykes, 1967, Figs. 11 and 14). The symbols used are the same as in the previous example. Sketh two admissible nodal planes, and shade the appropriate quadrants blak. What kind of fault produed these two earthquakes? Page 4

5 USGS seismologists routinely generate foal mehansm solutions using a semiautomated method developed by Stuart Sipkin, whih he has desribed in detail in the literature (Sipkin, 1982, 1984, 1994). A similar (but not idential) semiautomated method for omputing FMSs has been developed and used by Dziewonski and others (1981) at Harvard, providing very useful alternative interpretations. Solutions are heked or even manually produed using a wider array of methods for earthquakes in whih there is a partiular researh interest, inluding large earthquakes that ause damage or asualties. As an example of how the USGS routinely omputes solutions, the EIC and the IRIS Data Management Center (IRIS DMC) have developed a program to produe rapid estimates of the seismi moment tensor for most earthquakes with a body-wave magnitude or surfae-wave magnitude of 5.5 or greater. In many ases an estimate of the moment tensor an be produed within 20 minutes of the arrival of the gophered broadband P-waveform data from the IRIS DMC. The resulting determination of earthquake depth, geometry, and size an be very useful in the planning of post-earthquake studies, suh as damage surveys and the deployment of temporary seismi arrays. In some ases suh information an be ruial in the deision whether or not to issue a tsunami warning. Beause of the manner in whih the data and the syntheti seismograms are aligned, the method is insensitive to the effets of timing errors, epientral misloation, and lateral heterogeneity. The flow of data is as follows. An automati proess at the EIC, using real-time data from the U.S. ational Seismograph etwork (USS), identifies phase arrivals, assoiates the data, and omputes an origin time and hypoenter. This information is sent by to the IRIS DMC where it triggers the Gopher system. Four minutes of broadband data ontaining the P-arrival are retrieved from the ring buffers of the open IRIS stations and transmitted to the EIC by Internet. These data are merged with the USS dataset and used to refine the loation and ompute a body-wave moment tensor using a tehnique based on the theory of optimal filter design. This solution is then broadast by to a list of subsribers. Beause of the growing number of broadband seismi stations with either real-time or dial-up apability, these preliminary solutions are usually very similar to those produed later, using data from the entire network (nei.usgs.gov/neis/fm/fast_moment.html). Where an FMSs be obtained? Preliminary FMSs are routinely omputed for earthquakes with reported magnitudes of 5.5 or greater, and for smaller earthquakes with a large number of reported arrivals. Fast moment tensor solutions for reent earthquakes are posted by the USGS ational Earthquake Information Center (EIC) at EIC also maintains a webbased searhable database of EIC first-motion, EIC moment tensor, EIC radiated energy, Harvard entroid moment tensor, salar moment from the University of California at Berkeley for regional earthquakes, and teleseismi moment from the Laboratoire de Geophysique, Papeete (PPT) at nei.usgs.gov/neis/sopar/. Foal mehanism solutions are also produed by partiipants in The Global CMT Projet (formerly Harvard s Centroid-Moment Tensor (CMT) Projet), whose work is desribed at The CMT atalog an be searhed online via Many more FMS are embedded in researh papers and not ontained within searhable earthquake atalogs. Geometri regularity of an ideal FMS beahball diagram Beause of the symmetry of the fore double ouple and moment tensor on whih it is based, the FMS beahball diagram has a rystal-like regularity to it. The two nodal planes are perpendiular to eah other. The pole of the auxiliary plane is olinear with the slip vetor on the fault plane. The axis is oinident with the intersetion of the two nodal planes, and so is ontained within both of the nodal planes. The, T and P axes have an orthogonal relationship to one another: eah is 90 from the other. The T and P axes biset the dihedral angles between the nodal planes; that is, the T and P axes are 45 from the nodal planes. The plane defined by the T and P axes also ontains the vetors normal to the nodal planes, one of whih is the slip vetor. The P axis is in the middle of the white quadrant, and the T axis is in the middle of the blak quadrant. Page 5

6 Minimum data to speify a foal mehanism solution s geometry Just three data elements are needed to ompletely speify the orientation of a FMS beahball diagram (e.g., two elements to speify the orientation of the fault plane and one element to speify the orientation of the slip vetor on that plane). Seismologists ommonly report the dip azimuth and dip angle of the inferred fault plane, and the rake of the hanging-wall slip vetor in that plane. Cronin and Sverdrup (1998) reported the dip azimuth and dip angle of the inferred fault plane and the azimuth of the hanging-wall slip vetor, ognizant of the fat that geologists ommonly use the word rake in a far less preise manner than seismologists use the same term. The rake of the hanging-wall slip vetor is measured in the plane of the fault, relative to the referene strike of the fault plane. The referene strike is defined using the right-hand method. An angle measured through an antilokwise rotation from the referene strike is onsidered a positive angle; an angle measured lokwise from the referene strike is a negative angle. A slip vetor that is direted up relative to strike has a positive rake, and a slip vetor that is direted down the plane is negative. The range of permissible rake values is +180 to 180. a b dip vetor rake of slip vetor a: +30 rake of slip vetor b: 115 Hanging-wall slip vetors that have a positive rake have at least some omponent of reverse slip. A rake of 90 indiates slip that is entirely reverse with no strike-slip omponent. Similarly, hanging-wall slip vetors with a negative rake have at least some omponent of normal slip, with a rake of 90 indiating slip that is entirely normal with no strike-slip omponent. Hauksson (1990) defined normal-slip vetors as having rakes of 45 to 135 and reverse-slip vetors having rakes of 45 to 135, with strike-slip vetors having rakes of 44 to 44 or 136 to 224 ( 136 ). A geologist might be more onfortable with a differentiation of fault types along the lines outlined in the following table: rake of slip fault type 0 or 180 pure strike-slip 90 pure dip-slip reverse 90 pure dip-slip normal 20 to 20 left-lateral strike-slip 20 to 70 reverse left-lateral oblique 70 to 110 reverse 110 to 160 reverse right-lateral oblique 160 to 160 right-lateral strike-slip 110 to -160 normal right-lateral oblique 70 to 110 normal 20 to 70 normal left-lateral oblique Construting a FMS beahball from 3 input data Tables of earthquake foal mehanism data do not do us muh good if we are not able to reonstrut an aurate FMS beahball diagram from them. Suh tables usually (but not always) list the orientation of the inferred fault and its orresponding slip vetor. Otherwise if the fault has not been interpreted, the table might simply list orientation data for one of the two nodal planes and the orresponding potential slip vetor. Page 6

7 An example Problem: Given a table of FMS data listing a fault with dip azimuth = 115, dip angle = 35, and azimuth of hanging-wall slip vetor = 50, reonstrut the orresponding foal mehanism solution beahball diagram and determine the approximate azimuth and plunge angle of the, P and T axes. Desribe the type of fault that produed the earthquake. Based on the foal mehanism solution, onstrut a map of what the fault might look like. Solution: a s b f sv d ) Use the azimuth (115 ) and the plunge (35 ) of the dip vetor (d) to plot the trae of the fault plane (f). The fault has a strike of = 25, and a dip of 35 (same as the plunge of the dip vetor). 2) Plot the pole of the fault plane (s ). 3) Given that the azimuth of the slip vetor is 50, onstrut a line from the enter of the plot () toward azimuth 50, and loate the slip vetor (sv) where that line intersets the trae of the fault (f). In the final presentation of the beahball diagram, we are going to leave a small portion of that line extending out from the irumferene of the diagram to indiate the diretion of hanging wall slip. 4) Determine the plunge of the slip vetor (sv): it plunges ~27. ap s n sv 5) Plot the trae of the auxiliary plane (ap) by using the slip vetor (sv) as the pole to that plane. ote that s is loated on the auxiliary plane. 6) The axis (n) is loated at the intersetion of the fault plane and the auxiliary plane. Determine the trend and plunge of the axis: it trends 150 and plunges ~ s 45 t p n sv 7) Plot the trae of the plane whose pole is the axis (dashed great irle). 8) Plot the P axis (p) 45 from s toward the slip vetor along the dashed great irle (i.e., half way between s and sv). 9) Plot the T axis (t) 45 from s and 135 from the slip vetor along the dashed great irle. d 12 p 253 t n 10) Determine the trend and plunge of the P axis (p): it trends ~12 and plunges ~52. 11) Determine the trend and plunge of the T axis (t): it trends ~253 and plunges ~ e map ) Fill the quadrants through whih the T axis passes with blak, and the quadrants through whih the P axis passes with white. Leave the short line indiating the diretion of slip on the outside of the beahball diagram. 13) The map is a depition of the orresponding fault as it exists near the fous of this earthquake. It is an oblique fault with omponents of normal and left-lateral slip. Page 7

8 Another example Problem: Given a table of FMS data listing a fault with dip azimuth = 205, dip angle = 25, and rake of hanging-wall slip vetor = +70, reonstrut the orresponding foal mehanism solution beahball diagram and determine the approximate azimuth and plunge angle of the, P and T axes. Desribe the type of fault that produed the earthquake. Based on the foal mehanism solution, onstrut a map of what the fault might look like. Solution: a b sva sva sva d ap 45 t s s 45 f s n p n ~133 1) Use the azimuth (205 ) and the plunge (25 ) of the dip vetor (d) to plot the trae of the fault plane (f). The fault has a strike of = 115, and a dip of 25 (same as the plunge of the dip vetor). 2) Plot the pole of the fault plane (s ). 3) The referene strike is toward azimuth 115. Given that the rake of the slip vetor is +70, it will not plot on a lower-hemisphere stereonet; however, the antipode of the slip vetor (sva) does plot on the stereonet -- the antipode is 180 from the slip vetor. The antipode is along the trae of the fault, 70 from the primitive irle as illustrated at left. Projet a line from the slip vetor antipode (sva) through the enter of the plot () to just beyond the irumferene of the stereo plot (i.e., beyond the primitive irle). That line, trending 47, defines the trend of the slip vetor. In the final presentation of the beahball diagram, we are going to leave a small portion of that line extending from the edge of the diagram to indiate the diretion of hanging-wall slip. 4) Determine the plunge of the slip vetor, whih is idential in magnitude to the plunge of the slip vetor antipode (sva): the slip vetor plunges ~23 upward, with a trend of 47. 5) Plot the trae of the auxiliary plane (ap) by using the slip vetor antipode (sva) as the pole to that plane. ote that s is on the auxiliary plane. 6) The axis (n) is loated at the intersetion of the fault plane and the auxiliary plane. Determine the trend and plunge of the axis: it trends 133 and plunges ~8. 7) Plot the trae of the plane whose pole is the axis (dashed great irle). 8) Plot the P axis (p) 45 from s and 135 from the slip vetor antipode (sva) along the dashed great irle. 9) Plot the T axis (t) 45 from s toward the slip vetor antipode (sva) along the dashed great irle. d 243 t 37 p n ) Determine the trend and plunge of the P axis (p): it trends ~37 and plunges ~21. 11) Determine the trend and plunge of the T axis (t): it trends ~243 and plunges ~78 e map ) Fill the quadrants through whih the T axis passes with blak, and the quadrants through whih the P axis passes with white. Leave the short line indiating the diretion of hanging-wall slip on the outside of the beahball diagram. 13) The map is a depition of the orresponding fault as it exists near the fous of this earthquake. It is a reverse or reverse-oblique fault with a small omponent of left-lateral slip. Page 8

9 Pratie with onstruting a beahball diagram Problem: Given a table of FMS data listing a fault with dip azimuth = 130, dip angle = 60, and rake of hanging-wall slip vetor = 125, reonstrut the orresponding foal mehanism solution beahball diagram and determine the approximate azimuth and plunge angle of the, P and T axes. Desribe the type of fault that produed the earthquake. Based on the foal mehanism solution, onstrut a map of what the fault might look like. Representing unertainty in FMS Earthquakes with smaller magnitudes tend to be reorded by fewer seismographs, and so their foal loations are more poorly known. The parameter that is most poorly known for many earthquakes is the foal depth. For earthquakes deteted by seismographs that are not well distributed in azimuth around the fous, even the loation of the epienter may be poorly onstrained. For example, earthquakes along the Blano transform fault zone are routinely misloated to the northeast of their atual foi due largely to the uneven distribution of seismographs available to reord the moderateto-small earthquakes along the Blano (e.g., Cronin and Sverdrup, 2003a, 2003b). The unertainty assoiated with published foal mehanism solutions is generally unstated. Beause the onstrution of FMSs involves defining a best-fit of interpreted first-motion or wave-form data, the unertainties in a FMS tend to inrease as the number of different seismograph stations whose data reports were used to ompute the FMS dereases. Fewer reported arrivals, more unertain FMS. Some earthquakes are generated by a nearly ideal double-ouple mehanism, while others are generated in a more omplex manner, perhaps involving mutliple faults or other foal onditions that are not simply desribed by fritional slip on an individual surfae. An analysis of the radiation pattern of energy away from the fous allows seismologists to estimate the perentage of the earthquake that an be desribed by a double-ouple mehanism. A graphial approah to this problem has been taken by the Global CMT Projet (formerly Harvard s CMT Projet). In their beahball diagrams, the perentage of double-ouple in the CMT solution is suggested by the degree to whih the four quadrants are sharply defined as quadrants. A set of examples drawn from CMT solutions for earthquakes reorded along the Blano transform fault are shown below, with the perentage of double-ouple inreasing from left to right: Some problems 1. A published foal mehanism solution for a magntude (mb) 6.5 earthquake below northern Peru on Otober 28, 1997, lists the azimuth (trend) and plunge of the three prinipal axes as follows: T 247, 8; 337, 0; P 69, 82. What is the strike and dip of the two nodal planes, one of whih is parallel to the fault that generated the earthquake? What are the orientations (trend and plunge, or rake) of the two possible hanging-wall slip vetors? 2. In a published datafile listing the fault strike azimuth (right-hand rule), dip angle, and rake of hanging-wall slip vetor, the data for a given earthquake are 135, 60, 120. Make a FMS beahball diagram (lower-hemisphere stereographi plot) for this earthquake. 3. In a published datafile listing the fault dip diretion, dip angle, and rake of hanging-wall slip vetor, the data for a given earthquake are 216, 65, Make a FMS beahball diagram (lower-hemisphere stereographi plot) for this earthquake. 4. A field geologist measures the orientation of a fault surfae and the diretion of hanging-wall slip indiated by grooves and striations on that surfae. The fault s strike-and-dip in the azimuth method is 35, 63SE, and the rake of the hanging-wall slip vetor is 75 (as rake is defined by seismologists). For no partiularly good reason other than a persistent ase of seismologist envy, he wants to represent these data on his map as a pseudo-foal-mehanism solution beahball (whih, by the way, is not a good idea). For the sake of humoring this poor sod, make a FMS beahball diagram from the field data. 5. You will be provided with a topographi map on whih the surfae trae of some faults are indiated. Two earthquake our in the area. Given the loation of the fous of eah earthquake and their respetive foal mehanism solutions, projet likely fault planes to the surfae to assess if any of the urrently-mapped faults are ative, or if there might Page 9

10 be a previously unmapped fault in the area. For an example of how the nodal planes an be projeted to the ground surfae as depited using a digital elevation model, see Cronin and others (2008). Some definitions auxiliary plane The axis is along the auxillary plane, and the fault is perpendiular to the auxillary plane. The auxillary plane is orthogonal to the two fore ouples (i.e., either perpendiular or parallel to the fore ouples). beahball diagram Lower-hemisphere equal-area stereographi projetion of a foal-mehanism solution, inluding the two nodal planes separating opposite quadrants that are generally filled either with blak (ompressional quadrants in whih p-wave first motion is up) or white (dilatational quadrants in whih p-wave first motion is down). fore ouple A non-zero omponent of a fore field ontaining the derivative of a Dira delta funtion, whih an be visualized as two parallel vetors of equal magnitude but different diretions ating at a short distane from one another. double ouple This is the simplest fore ondition that explains or reprodues the observed energy radiation pattern observed in earthquakes generated by fault slip. A double ouple inludes two fore ouples at right angles (90 ) to eah other ating at a short distane from one another. hanging wall The blok that is above an inlined fault surfae, as ontrasted with the foot wall that exists below the fault surfae. The hanging wall rests on the foot wall. By onvention, slip vetors in foal mehanism solutions reflet the motion of the hanging wall relative to the footwall. epienter The loation of the point on Earth s surfae that is diretly (i.e., vertially) above the fous of an earthquake. fault plane Surfae or zone aross whih there is shear displaement; in this appliation, the fault is an assumed-planar surfae ontaining the earthquake fous. The fault plane is a nodal plane. The axis is in/along the fault plane. The slip vetor is in the fault plane, perpendiular to the axis. foal sphere Imaginary small projetion sphere that is entered on the earthquake fous. fous The best mean solution for the point of origin of an earthquake, and inludes the earthquake depth and the longitude and latitude of the point vertially above the fous (i.e., the loation of the epienter).. -axis The so-alled null axis, the -axis is one of the prinipal axes of the moment tensor. The orientation of the - axis oinides with the orientation of one of the eigenvetors of the moment tensor the eigenvetor with the intermediate eigenvalue between that of the P and T axes. A seismograph that lies along a ray path through the axis will not reord a disernable signal from the earthquake at the expeted arrival time. The axis is along the intersetion of the two nodal planes, and is orthogonal to (i.e., 90 from) both the P and T axes. nodal plane Eah nodal plane is parallel to one of the two fore ouples entered on the earthquake fous, and perpendiular to the other fore ouple. One of the nodal planes is oinident with the fault surfae, and the other is oinident with the auxiliary plane. A seismograph that lies along a ray path along one of the nodal planes will not reord a disernable signal from the earthquake at the expeted arrival time. P-axis The so-alled pressure or ompressional axis, the P-axis is one of the prinipal axes of the moment tensor. The orientation of the P-axis oinides with the orientation of one of the eigenvetors of the moment tensor the eigenvetor with the smallest eigenvalue (i.e., a negative value). A P-wave traveling along a ray path through the P axis will produe the largest amplitude negative (down) first motion as reorded on a distant seismograph, normalized for travel distane. The P axis bisets the dihedral angle between the two nodal planes, and is orthogonal to (i.e., 90 from) both the and T axes. rake Angle measured in the plane of the fault between the referene strike (right-hand rule) and the slip vetor. Vetors with a positive rake point above the strike line; vetors with a negative rake are direted downward, below the strike line. referene strike The strike of a given inlined plane is the azimuth of a horizontal line ontained within that plane. But sine a horizontal line may be haraterized by either of two azimuths, eah 180 from the other, it is neessary to differentiate between these two azimuths to define one of the two diretions as the referene strike. Hene, the referene strike is identified through a 90 anti-lokwise (right-handed) rotation from the dip vetor. For example, the referene strike for an east-dipping plane is toward the north; for a northwest-dipping plane, the referene strike is toward the southwest. Put another way, if the trend of the dip vetor is θ, the azimuth of the referene strike is θ 90. slip vetor By onvention among those who use foal mehanism solutions, the slip vetor represents the diretion of motion of the hanging wall relative to the foot wall. The slip vetor is always ontained within the fault plane (i.e., the point that represents the slip vetor lies along the trae of the fault plane on a stereographi plot or beahball Page 10

11 diagram). T-axis The so-alled tension axis, the T-axis is one of the prinipal axes of the moment tensor. The orientation of the T-axis oinides with the orientation of one of the eigenvetors of the moment tensor the eigenvetor with the largest eigenvalue. A P-wave traveling along a ray path through the T axis will produe the largest amplitude positive (up) first motion as reorded on a distant seismograph, normalized for travel distane. The T axis bisets the dihedral angle between the two nodal planes, and is orthogonal to (i.e., 90 from) both the and P axes. take-off angle The angle, measured from a vertial line extending down from an earthquake fous, that a partiular ray path takes as it leaves the fous. Partially annotated list of soures and referenes Brumbaugh, D.S., 1999, Earthquakes siene and soiety: Upper Saddle River, ew Jersey, Prentie-Hall, 251 p. very lear elementary text whose Chapter 5 deals with FMS. Bullen, K.E., and Bolt, B.A., 1985, An introdution to the theory of seismology: Cambridge, Cambridge University Press, 499 p. A standard seismology textbook whose Chapter 16 provides a tehnial desription of matters related to modeling the earthquake soure. Cronin, V.S., and Sverdrup, K.A., 2003a, Multiple-event reloation of histori earthquakes along Blano Transform Fault Zone, E Paifi, Geophysial Researh Letters, v. 30(19), 2001, doi: /2003gl Example of use of reloated earthquake data and FMSs to address a problem in struture/tetonis. Cronin, V.S., and Sverdrup, K.A., 2003b, Defining stati orretion for jointly reloated earthquakes along the Blano Transform Fault Zone based on SOSUS hydrophone data: Oeans 2003 MTS/IEEE Conferene Proeedings (ISB ), p. P Example of use of reloated earthquake data and FMSs to address a problem in struture/tetonis. Cronin, V.S., and Sverdrup, K.A., 1998, Preliminary assessment of the seismiity of the Malibu Coast Fault Zone, southern California, and related issues of philosophy and pratie, in Welby, C.W., and Gowan, M.E. [editors], A Paradox of Power--Voies of Warning and Reason in the Geosienes: Geologial Soiety of Ameria, Reviews in Engineering Geology, p Contains an earthquake dataset for most of the Malibu area that inludes 638 events, with 107 FMSs. FMS data reported as fault dip azimuth, fault dip angle, and azimuth of inferred slip vetor. The FMS data indiate that the average hanging-wall slip vetor for Malibu earthquakes is nearly perpendiular to the San Andreas fault through the Transverse Ranges. Cronin, V.S., Millard, M.A., Seidman, L.E., and Bayliss, B.G., 2008, The Seismo-Lineament Analysis Method [SLAM] - - A reonnaissane tool to help find seismogeni faults: Environmental and Engineering Geosiene, v. 14, no. 3, p Desribes how the nodal planes from a foal mehanism solution an be projeted to the ground surfae, as represented by a hillshade map reated using a digital elevation model, so that the resulting seismo-lineaments an be ompared with the ground-surfae trae of faults. Dziewonski, A.M., and Woodhouse, J.H., 1983, An experiment in the systemati study of global seismiity entroidmoment tensor solutions for 201 moderate and large earthquakes of 1981: Journal of Geophysial Researh, v. 88, p Provides a tehnial explanation of CMT solutions with examples. Dziewonski, A.M., and Woodhouse, J.H., 1983, Studies of the seismi soure using normal-mode theory, in Kanamori, H., and Boshi, E., eds., Earthquakes observation, theory, and interpretation notes from the international shool of physis Enrio Fermi (Varenna, Italy, 1982): Amsterdam, orth-holland Publishing Company, p Provides the most omplete tehnial explanation of CMT solutions with examples. Dziewonski, A.M., Chou, T.-A., and Woodhouse, J.H., 1981, Determination of earthquake soure parameters from waveform data for studies of global and regional seismiity: Journal of Geophysial Researh, v. 86, p Provides a tehnial explanation of CMT solutions with examples. Ekström, G., 1994, Rapid earthquake analysis utilizes the internet: Computers in Physis, v. 8, p Friedman, M., 1964, Petrofabri tehniques for the determination of prinipal stress diretions in roks, in Judd, W.R., ed., State of stress in the Earth s rust: ew York, Amerian Elsevier Publishing Company, p This is Mel Friedman s lassi desription of basi petrofabri tehniques, inluding the use of lower-hemisphere stereographi projetions. This was the authority ited in Lynn Sykes paper on transform fault seismiity (Sykes, 1967). Gubbins, D., 1990, Seismology and plate tetonis: Cambridge, Cambridge University Press, 339 p. Hardebek, J. L., and Shearer, P.M., 2002, A new method for determining first-motion foal mehanisms: Bulletin of the A Page 11

12 Seismologial Soiety of Ameria, v. 92, p The title says it all. Hauksson, E., 1990, Earthquakes, faulting, and stress in the Los Angeles Basin: Journal of Geophysial Researh, v. 95, no. B10, p. 15,365-15,394. This is an exellent paper that sought to assoiate the regional seismiity of southern California with ative fault trends. A signifiant number of FMSs are utilized with first-motion data represented on beah-ball diagrams. FMSs are tabulated using the dip azimuth and dip angle of the inferred fault plane and the rake of the hanging-wall slip vetor. Hodgson, J.H., and Allen, J.F.J., 1954, Tables of extended distanes for PKP and PP: Pub. Dominion Obs. Ottawa, v. 16, p Hodgson, J.H., and Storey, R.S., 1953, Tables extending Byerly s fault-plane tehnique to earthquakes of any foal depth: Bulletin of the Seismologial Soiety of Ameria, v. 43, p Jost, M.L. and Herrmann, R.B., 1989, A student s guide to and review of moment tensors: Seismology Researh Letters, v. 60, p This is a tutorial onerning foal mehanism solutions for students. Kasahara, K., 1981, Earthquake mehanis: Cambridge, Cambridge University Press, 248 p. A partiularly good tehnial disussion of earthquake mehanis, apparently meant for student seismologists. Lay, T., and Wallae, T.C., 1995, Modern global seismology: San Diego, Aademi Press, 521 p. Chapter 8 onerns seismi soures, and ontains a lear exposition of material related to foal mehanism solutions. Lliboutry, L, 2000, Quantitative geophysis and geology: Chihester, UK, Praxis Publishing and Springer, 480 p. Marrett, R., and Allmendinger, R.W., 1990, Kinemati analysis of fault-slip data: Journal of Strutural Geology, v. 12, no. 8, p Menke, W., and Abbott, D., 1990, Geophysial theory, ew York, Columbia University Press, 458 p. geophysis textbook, but the treatment of FMSs is rather short. A good general Pujol, J., 2003, Elasti wave propagation and generation in seismology: Cambridge, Cambridge University Press, 444 p. Chapter 10 of this book onerns soure mehanis and presents some interesting information about radiation patterns and the instantaneous motion of partiles on the foal sphere. Reasenberg, P.A., and Oppenheimer, D., 1985, FPFIT, FPPLOT and FPPAGE -- Fortran omputer programs for alulating and displaying earthquake fault-plane solutions: U.S. Geologial Survey, Open-File Report o , 25 p.. These software appliations provide a standard way of omputing fault-plane solutions using P-wave first arrivals. Reiter, L., 1990, Earthquake hazard analysis, issues and insights: ew York, Columbia University Press, 254 p. Shearer, P.M., 1999, Introdution to seismology: Cambridge, Cambridge University Press, 260 p. A partiularly good reent textbook on introdutory seismology. Sipkin, S.A., 1994, Rapid determination of global moment-tensor solutions: Geophysial Researh Letters, v. 21, p This paper provides a desription of the method used by the USGS to rapidly define foal mehanism solutions. Sipkin, S.A., 1986, Interpretation of non-double-ouple earthquake mehanisms derived from moment tensor inversion: Journal of Geophysial Researh, v. 91, p This paper provides a desription of the method used by the USGS to rapidly define foal mehanism solutions. Sipkin, S.A., 1982, Estimation of earthquake soure parameters by the inversion of waveform data: Syntheti waveforms: Physis of Earth and Planetary Interiors, v. 30, p This paper introdues and provides the basi desription of an automated method for determining foal mehanism solutions using omparisons with syntheti waveforms. Stein, S. and Wysession, M, 2003, An introdution to seismology, earthquakes and Earth struture: Malden, Massahusetts, Blakwell Publishing, 498 p. Chapter 4 inludes a lear and rather omprehensive summary of foal mehanism solutions. Stüwe, K., 2002, Geodynamis of the lithosphere: Berlin, Springer, 449 p. Sverdrup, K.A., Shurter, G.J., and Cronin, V.S., 1994, Reloation analysis of earthquakes near anga Parbat-Haramosh Massif, northwest Himalaya, Pakistan: Geophysial Researh Letters, v. 21, p Sykes, L., 1967, Mehanism of earthquakes and nature of faulting on the mid-oean ridges: Journal of Geophysial Researh, v. 72, p The lassi paper that demonstrated that the transform fault model proposed by Tuzo Wilson is orret. This paper also provides a good desription of how foal mehanism solutions an be derived manually, with examples of beahball diagrams that inlude the first-motion data. U.S. Geologial Survey, Foal mehanisms: A very brief, qualitative desription of FMSs for the general publi. The full text and illustration is appended to this doument. Page 12

13 Woodhouse, J.H., and Dziewonski, A.M., 1984, Mapping the upper mantle three dimensional modeling of Earth struture by inversion of seismi waveforms: Journal of Geophysial Researh, v. 89, p Provides a tehnial explanation of CMT solutions with examples. Yeats, R.S., Sieh, K., and Allen, C.R., 1997, The geology of earthquakes: ew York, Oxford University Press, 568 p. A textbook treatment offering a (generally Californian) perspetive on the geology of ative faulting. Their explanation of FMSs is brief (p ), qualitative and reasonably lear, with a good illustration of first-motion waveforms and a orresponding beahball diagram showing the data (Figure 4-6, after abelek and others, 1987). Software The USGS Earthquake Hazards Program maintains a very useful web site through whih you an download a variety of software appliations, inluding several that relate to foal mehanism solutions. The URL of that site (as of Marh 2010) is In partiular, look for the following: [1] ArGIS plugin software 3D foal mehanisms by Keith Labay and Peter Haeussler, to view earthquake foal mehanism symbols three dimensionally using ArSene 9.x (Windows) [2] The FORTRA odes FPFIT, FPPLOT and FPPAGE by Paul Reasenberg and David Oppenheimer, to alulate and plot fault-plane solutions from first-motion data (Unix) [3] The FORTRA77 ode HASH 1.2 by Jeanne Hardebek and Peter Shearer to alulate earthquake foal mehanisms (Unix) [4] RAGE by Jim Luetgert to alulate distane and azimuth between two points (MaOS9) [5] SATSI by Jeanne Hardebek and Andy Mihael to ompute spatially and/or temporally varying stress field from foal mehanisms (Unix or any platform with C) A brief desription of foal mehanism solutions by Dave Oppenheimer of the USGS Seismologists refer to the diretion of slip in an earthquake and the orientation of the fault on whih it ours as the foal mehanism. They use information from seismograms to alulate the foal mehanism and typially display it on maps as a beah ball symbol. This symbol is the projetion on a horizontal plane of the lower half of an imaginary, spherial shell (foal sphere) surrounding the earthquake soure (A). A line is sribed where the fault plane intersets the shell. The stress-field orientation at the time of rupture governs the diretion of slip on the fault plane, and the beah ball also depits this stress orientation. In this shemati, the gray quadrants ontain the tension axis (T), whih reflets the minimum ompressive stress diretion, and the white quadrants ontain the pressure axis (P), whih reflets the maximum ompressive stress diretion. The omputed foal mehanisms show only the P and T axes and do not use shading. These foal mehanisms are omputed using a method that attempts to find the best fit to the diretion of P-first motions observed at eah station. For a double-ouple soure mehanism (or only shear motion on the fault plane), the ompression firstmotions should lie only in the quadrant ontaining the tension axis, and the dilatation first-motions should lie only in the quadrant ontaining the pressure axis. However, first-motion observations will frequently be in the wrong quadrant. This ours beause a) the Page 13

14 algorithm assigned an inorret first-motion diretion beause the signal was not impulsive, b) the earthquake veloity model, and hene, the earthquake loation is inorret, so that the omputed position of the first-motion observation on the foal sphere (or ray azimuth and angle of inidene with respet to vertial) is inorret, or ) the seismometer is mis-wired, so that up is down. The latter explanation is not a ommon ourrene. For mehanisms omputed using only first-motion diretions, these inorret first-motion observations may greatly affet the omputed foal mehanism parameters. Depending on the distribution and quality of first-motion data, more than one foal mehanism solution may fit the data equally well. For mehanisms alulated from first-motion diretions as well as some methods that model waveforms, there is an ambiguity in identifying the fault plane on whih slip ourred from the orthogonal, mathematially equivalent, auxiliary plane. We illustrate this ambiguity with four examples (B). The blok diagrams adjaent to eah foal mehanism illustrate the two possible types of fault motion that the foal mehanism ould represent. ote that the view angle is 30- degrees to the left of and above eah diagram. The ambiguity may sometimes be resolved by omparing the two faultplane orientations to the alignment of small earthquakes and aftershoks. The first three examples desribe fault motion that is purely horizontal (strike slip) or vertial (normal or reverse). The oblique-reverse mehanism illustrates that slip may also have omponents of horizontal and vertial motion (quake.wr.usgs.gov/reenteqs/beahball.html). orth Equal Area Stereonet Page 14