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Bu l le t in o f th e S e i s mo lo g ic a l S o c ie ty o f Am e r ic a , Vo l. 7 2 , No . 4 , p p . 1 4 27 -1 4 32 , A u g u s t 1 9 8 2

COMMENTS ON "THE CORNER FREQUENCY SHIFT, EARTHQUAKESOURCE MODELS, AND Q," BY T. C. HANKS

BY CHARLES A. LANGSTON

INTRODUCTONHanks (1981) has recently offered a rather broad-based criticism of selected time-

domain methods for modelingearthquake sources. From he nature of and reasoningbehind Hanks' arguments, I perceive that there may be several fundamental mis-understandings regarding the objectives and techniques of using point source orvarious finite source models for earthquake data. Rather than reply directly to themany qualitative arguments he has presented concerning opinions of mode] fits, Iwould like to take this opportunity to examine what I feel is good scientificmethodology in the treatment of seismic sources and resulting interpretations. Inwhat follows, I will present a discussion of the various objectives that differentsource studies address and the assumptions involved. In doing so and in keepingwith the major themes presented by Hanks (1981), I would like to demonstrate thegenetic similarity between those studies which primarily employspectral techniquesfor interpretation purposes and those which use time-domain techniques. It isobvious that a digitally sampled seismogram is equally well represented by its timeseries or its complex spectrum; the differences are only a matter of form and notcontent. Similarly, the differences of opinion generated by those who work withthese different data forms should be easily reducible to a common quantitativeground by recognizing this fact. Toward the end of this commentary, I would like tosuggest ways whereby issues such as those brought up by Hanks (1981) may beincorporated in a more quantitative framework so that there can be some hope fortheir resolution.

S O U R C E M O D E L I N GLike in all other sciences, the data that are available to a seismologist do not

supply questions to be posed. It is entirely up to the scientist to construct thecontext in which questions may be asked. Th e context of questions in source studieshas been built up over the years through an interplay of theoretical results andobservational interpretation. Because of the unfortunate nature of trade-offs be-tween source parameters and earth s tructure parameters , much of the progress insource studies has come about only after a significant increase in knowledge aboutparticular aspects of wave propagation in the earth. Thus, there is a large body of"fact" outside the strict domain of source modeling which involves knowledge aboutearth structure. The quotation marks around "fact" is simply meant to emphasizethat occasionally some results are revised after further study of new data. Thisinformation derived from structural studies serves as outside constraints availablefor use in source modeling.

Context in source studies is also obtained by t he construction of reasonable andphysical hypotheses . All seismologists would agree that the basis for such hypot heseslie in application of Newton's laws and thermodynamics through continuum me-chanics. Howeve r, and thi s is where the trouble occurs, all seismologists do not agreeto what is reasonable. This is undou btabl y due to the situation that the earth is farmore complex than we let on and that all theories that are constructed to explain

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1428 LETTERS TO THE EDITORgeophysical phenomena are clearly approximations. There will always be somedeficiency (commonly attributed to "noise") in every theory. Agreement on aparticular theory or hypothesis in seismology usually occurs when an undefinednumber of people consider it most consistent with the available data in somestatistical sense. This, of course, does not mean theory is correct; at best it is onlyan approximat ion of the particular physical system.The development of context in trea tmen ts of the seismic data implicitly assumesthat the seismologist has some objective in mind. After all, there is always a reason,no matter how trivial, for doing a study. The objectives of a particular modelingexperiment can usually be plainly seen in the type of theory used or the hypothesesproposed to explain the data. For example, if source depth is the desired parame terof a particular earthquake study, then source depth must figure prominently as apara mete r in the t heory. Of course, there m ay be higher order objectives which arebased on interpretation of modeling parameters. These may even seem to beindependent of the modeling process because of their attached importance, anexample being the evaluation of earthquake risk. Unfortunately, higher orderobjectives, although more noble than the first order parameters of a source model,tend to magnify the effects of errors made at the lower level since more subjectiveor interp retive decisions are made.To perform the objectives of a source study, one ostensibly uses the scientificmethod; a hypothesis is proposed, an experimen t set up, and t he hypothesi s is testedby the experiment. If one is lucky or the experiment is exceptionally well designed,then the hypothesis is disproved or proved consistent with the data. If the result isambiguous th en t he process is recursively performed ad in f in i tum .

In seismology and, in particular, source studies, there are major problems withthe experimental aspect of the process. The information that is commonly desired,such as exact fault geometry and slip as a function of space and time, is only crudelycontained within the available data. Most seismic data, in fact, is the resu lt of poorlydesigned experiments if exact source quantities are desired. The seismologist is,therefore, forced to construct passive experiments based on the availability of anydata. This is usually a painstaking process since almost everything must be knownabout the data in order to obtain those few extra parameters contained in thehypothesis. For example, if one wants to apply a source theo ry involving P and Swaves, then the P and S waves must be unambiguously identified on a seismogramand their structural interactions completely known. Unfortunately because of theearth structure-source parame ter trade-offs and the underparameterization of geo-physical systems, there is no certa inty tha t assumptions involving the nature of dataare correct. Thus, these assumptions must be constantly reevaluated to be madeconsistent with outside constraints and the data.In light of these previous comments, let us now discuss the several themes thatHanks (1981) presented. First, consider the basic methods which were discussed.There is the frequency domain technique for determining the two parameters in thesource model suggested by Brune (1970). The techn iques for applying this procedureto teleseismic P and S dat a were given by Hanks an d Wyss (1972) and to regionalseismograms by That che r and Hanks (1973). The basic assumptions made in thesetechniques were th at1. the radiation pattern of the source can be represented by a point-shear

dislocation situated in homogeneous, isotropic, wholespace;2. geometric spreading is taken for the di rect P or S body wave in the appropriatevertically inhomogenous earth model; and

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L E T T E RS T O T HE E DI T OR 14293. t he da ta contains only the di rect P or S wave.

The technique is applied by fitting a low- and high-frequency asymptote to theamplitude spectrum of a windowed portion of the seismogram. The intersection ofthe asymp totes is the corner frequency and the zero frequency limit is related to theseismic mome nt of the point dislocation. What Hanks {1981) calls P and S wavesare modeled separately. Indeed, each individual windowed seismogram is modeledseparately, essentially giving a different time function spectrum for each "P" and"S" wave.

Langston and H elmber ger (1975) discuss a modeling meth od based on the follow-ing assumption

1. assume a point-shear dislocation in homogeneous, isotropic, plane-layeredmedia.An appr oximate meth od for computing the teleseismic response of P and S wavesfrom a point dislocation was presente d which assumed

1. the effects of struct ure ne ar the source and ne ar the receiver are separable; and2. geomet ric spreading is tak en for the direct P or S body wave in the appropriat evertically inhomogeneous ear th model.

To apply these methods, one has to determine the appropriat e structure model fromoutside constraints or, more dangerously, recursively. Synthetic seismograms arethen computed for a source model to be compared directly with the time-domaindata. Although a point source model was stressed in the paper for simplicity, it wasmenti oned t hat finite sources could be built up by summing concat enated arrays ofpoint source solutions.

Th e connection between the two meth ods is clear. Both use solutions for a point-shear dislocation at their core. Indeed, I am totally at a loss to understand Hanks'(1981) obvious dislike of point- source me thods even in the expedient of treat ing theP and S waves separately as he seems to prefer. The methods outlined by Langstonand Helmberge r (1975) can be considered to be more realistic and helpful in sourcemodeling since assumptions concerning the st ructure response have to be explicitlyincluded in calculations. Comparison of the synthetics with features of the datainsures that these assumptions are constantly evaluated. In contrast, spectralmethods, as commo nly applied, do not have any consistency checks on assumptionssince the phase is ignored. High- and low- frequency asym ptot es can be fit to mostany band-limi ted data regardless of wave type.

Hanks (1981) also seems to disagree with a basic ten et of modeling philosophy.He makes the rather rigid assertion that even if a point-source model fits theavailable data, it still has to be wrong. As implied above, all geophysical models areultimatel y incorrect because they are under parameterized . If a simply parameter izedmodel, such as a point source, fits the data just as well as a more complex model,then there are obvious problems with the parameter resolution of the complexmodel. Both mode ls are equally "cor rec t" but we are lead to disbelieve or ignore themeaning of the more complex of the two. In other words, the complex model~is notneeded to explain the data. It will not yield any new information that is not alreadycontained within the simple model. This is why point-source models are extensivelyused. I f an ea rthqu ake can be a pproxima ted sufficiently well by a point source, thereis no justification for adding new parameters. Often, however, it is hard to judgewhat "sufficiently well" is. If deviations from the point source are systematic, the sedeviat ions may indicate t he type of new parametr izat ion to try, e.g., going to a finitepropagating source. More often than not, the deviations in the data are morecomplex than simple models would allow, including the simple expansion of the S-

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1430 LETT ERS TO THE EDIT ORwave pulse rel ative to P as Hanks (1981) would like. Upon reaching this point, themodeler usually gives up, having to reconcile the deviations with his lack ofknowledge of other parameters.

The various modeling techniques often have differing objectives. The primaryobjectives of the freque ncy domain technique are clear; there are only two param-eters to invert. Det ermin ed for each wavetrain, the corner frequency can be used tointerpret other quantities such as stress drop or fault radius and the point-sourcemoment can yield the average slip. On the other hand, the time domain techniqueis tota lly general. Indeed, because the met hod is general, it is just as easy to includethe Brune (1970) point source as any other approximate parameterization for thefar-field time function. Since source and structure are tre ated together, there is thepossibility that source and/or structure parameters can be investigated. In additionto estimates of the duration of the source-time function, the many teleseismicwave form studies crit iqued by Hanks (1981) were also performed to obtain es timatesof source orienta tion and source depth. Although there are always trade-offs involvedamong these parameters, it has usually been my experience that small variations intime function width does little to estimates of the other parameters. It has beenshown in several studies (e.g., Burdick and Mellman, 1976) that the precise choiceof source model (point or finite) does not affect these ot her parameters.

This brings us to the inter pretat ion of modeling results and how to discuss them.Hanks (1981) rat he r forcibly argues that the S-wave corner frequency shift (S waveshaving lower corner frequen cy than P waves) is a general observation of mostspectral studies and can easily be seen in the time domain as well. Upon analysis, Ifind that this tenet is generally ambiguous and strongly tied to the interpretivetechniqu e used to formula te the stat ement. First, we have to ask, what is meant by" P " and "S" waves. With out a doubt, the wave trains which are called "S" by Hanksand others in the many frequency domain interpretations are often clearly longerperiod than those called "P." Amplitude spectra computed from these wave trainsclearly show this. To obtain a uniformly good estimate of corne r frequency , assumingthe Brune (1970) model is sufficient, the s tructur e response for the P and S wavesmust either be composed of a single delta function or a statistically random set ofarrivals which yield a white spectrum. In the first case, the original modelingassumption of a homogeneous whole-space is correct and in the second, a rando mscatterer of a particular kind is invoked (the random scatterer must also preservethe same power level for the signal as if the arrival were the simple direct wave). Ithas been shown in several studies concerning structure effects for shallow earth-quakes t hat these basic assumptions can be severely violated yielding moment s andcorner frequencies unrelate d to the source model or yielding biases in the frequencycon ten t between P and S wave trains (Helmberger, 1974; Helmb erger and Malone,1975; Heat on and Helmberger, 1978; Langston, 1978a). The p roblem comes aboutbecause what are called "P " and "S" waves (thinking about the simple whole-spaceideal) are actually composed of a complex series of interferring arrivals. Thus, itwould seem advantageous to carefully consider the structure effects before modelingthe source. In those spectral studies where the source is deep enough or the receive ris close enough to clearly distinguish the uncontaminated far-field P- and S-wavearrivals, then the spectral method may be appropriate.Hanks (1981) has taken selected figures from several time-domain studies toseemingly prove his point that S waves have lower corner frequencies tha n P waves.I find it difficult to unde rstan d Hanks' meth od of determining corner frequency fora wave train in the time domain. I suspect he is confusing high-f requency con tent or

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LETTERS TO THE EDITOR 1431

phase effects with corner frequency which are not at all synonymous. It is also notclear how the concept of corner frequency, as derived from Brune's (1970) model,applies to complex geometries and rupture models as in Langston's (1978b) SanFernando model. His presentation does point out a major problem in communica-tion, however: how can results be shown and compared? In the many time-domainstudies, some of which were discussed by Hanks (1981), the usual technique is toconstruct a deterministic source model and compute synthetic seismograms tocompare with windowed data. The quality of fit is usually defined within the contextof the study. For example, a "good" fit may only consist of having wave polarities beconsistent or it may consist of a quanti tative objective function which is minimizedin an inversion. Likewise, deviations from the fit are also discussed within thecontext of the study. Hanks (1981) points out what he considers deviations in thesource models used in several time-domain studies. However, there is no room formeaningful discussion because of a lack of agreement on context. On one hand,Hanks (1981) shows wha t he considers are the deviations and suggests in qualitativeway that the problems must be in the source model. On the other hand, the time-domain modeling studies employ specific and quantitative source and structuremodels where source effects inte ract strongly with structure effects. It is not so clearwhere the deviations of fit lie. In other words, to determine whether a hypothesis isat least consistent with the data, one has to test it as rigorously as possible. Thecontext of a dialogue between proponents of one model over another lies in aquantitative comparison of the same data in the same form. Hanks (1981) may beentirely correct in his assessment of deficiencies in point-source models but he hasnot pre sented a ny useful quantitative evidence to demo nstrate his point.

A prime example o f this d isparit y is Hanks' (1981) discussion of results concerningthe 1968 Borrego Mountain earthquake. In it, he basically attacks some underlyingassumptions made in the studies performed by Helmberger (1974) and Burdick andMel lman (1976). Challenging assumptions is always a good way to learn somethingnew, especially when it comes to the basic ambiguities i nher ent in source modeling,and every study should certainly be scrutinized carefully. However, in this case,Hanks (1981) has been somewhat less tha n careful in his pres enta tion of the data bycomparing P waveforms between the Borrego Mountain and E1 Golfo earthquak esat only two stations (his Figure 5). The two stations that he has chosen are locatedvery near to P-wave nodes for the Borrego Mountain event and do not demonstrat ethe effects of the P wave [as Burdick and Mellman (1976) interp ret it] changingpolarity and amplitude. In any case, his multiple-source idea is an easily testableone. I would prefer to disparage a model by first presenting results. These commentsare especially applicable to Hanks' (1981) attenuation discussion. Burdick (1982)discusses the se points in satisfying detail.

CONCLUSONS

I hope this commentary has helped toward the resolution of the issues Hanks(1981) has brought up. The treatment of the seismic source is difficult and fraughtwith many problems. Indeed, the comments of Hanks (1981) not withstanding, Ihave always found tha t the data never behaves perfectly and that there are alwaysunexplainable deviations from th eory in every study. However, I firmly believe thatprogress in the field will not come about by proposing qualitative argumentsconcerning one model over another. Ma ny of the importa nt assumptions made in allsource studies are testable and testable in a quantitative way.

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1432 LETTERS TO THE EDITORMost of the d isag reemen ts I seem to have wi th H ank s (1981) seem to l ie in the

bas ic ph i l osop hy one fo llows to mode l se i smic sources . Because i t is imposs ib le to beaware of eve ry fac tor o f pa r ame te r which m ay be i mpo r ta n t in expla in ing the source,i t seems reaso nable to take a conse rva t ive approach in model ing . I be l ieve tha t i t i sgood pract ice to process the se ismic data as l i t t le as possible and to constructde te r min i s t ic mode ls wh ich a re a s s imple as poss ib le , in an e f fort to f i t the da ta . Th eef fec t o f s t ruc ture a ssu mpt ion s wil l be p la in ly v is ib le so tha t dec is ions on sourcepa r a m e te r s c a n be ma de mor e un a mbiguous ly . I n pa rt i c u l ar , t he u se o f po in t sou rc e sin se ismic mode l ing se rves a s an exce l len t s ta r t ing po in t fo r the mode l ing of anyevent . Higher orde r e f fects such as f in i teness or d i rec t iv i ty a re much more d i f ficu ltto assess in the se ismic data . Usually, these effec ts also overpa rame ter i ze thep r ob l e m f o r t he a va i l a b l e da t a so t ha t un ique n e s s p r ob le ms be c ome mor e s ever e.

REFERENCESBrune, J. N. (1970). Tectonic stress and spectra of seismic waves from earthquakes, J. Geophys. Res. 75,

4997-5009.Burdick, L. J. (1982). Comments on "The corner frequency shift, earthquake source models, and Q," byT. C. Hanks, Bull. Seism. Soc. Am. 72, 1417-1424.Burdick, L. J. and G. R. Mellman (1976). Inversion of the body waves from the Borrego Mountainearthquake to the source mechanism, Bull. Seism. Soc. Am. 66, 1485-1499.Hanks, T. C. (1981). The corner frequency shift, earthquake source models, and Q, Bull. Seism. Soc. Am.71, 597-612.Hanks, T. C. and M. Wyss (1972). The use of body-wave spectra in the determination of seismic-sourceparameters, Bull. Seism. Soc. Am. 62, 561-589.Heaton, T. H. and D. V. Helmberger (1978). Predictability of strong ground motion in the ImperialValley; modeling the M4.9, November 4, 1976 Brawley earthquake, Bull. Seism. Soc. Am. 68, 31-48.Helmberger, D. V. {1974). Generalized ray theory for shear dislocations, Bull. Seism. Soc. Am. 64, 45-64.Helmberger, D. V. and S. D. Malone (1975). Modeling local earthquakes as shear dislocations in a layeredhalf space, J. Geophys. Res. 80, 4881-4888.Langston, C. A. (1978a). Moments, corner frequencies, and the free surface, J. Geophys. Res. 83, 3422-3426.Langston, C. A. (1978b). The February 9, 1971 San Fernando earthquake: a study of source finiteness inteleseismic body waves, Bull. Seism. Soc. Am. 68, 1-29.Langston, C. A. and D. V. Helmberger (1975). A procedure for modelling shallow dislocation sources,Geophys. J. 42, 117-130.Thatcher, W. and T. C. Hanks (1973). Source parameters of southern California earthquakes, J. Geophys.Res. 78, 8547-8576.DEPARTMENT OF GEOSCIENCESPENNSYLVANIA STATE UNIVERSITYUNIVERSITY PARK, PENNSYLVANIA 16802

Manuscript received September 14, 1981