ANALYSIS OF SOURCE LOCATION ALGORITHMS Part II: Iterative methods

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1 ANALYSIS OF SOURCE LOCATION ALGORITHMS Part II: Iteratve methods MAOCHEN GE Pennsylvana State Unversty, Unversty Park PA 1680 Abstract Iteratve algorthms are o partcular mportance n source locaton as they provde a much more leble means to solve nonlnear equatons, whch s essental n order to deal wth a wde range o practcal problems. The most mportant teratve algorthms are Geger s method and the Smple method. Ths artcle provdes an overvew o teratve algorthms as well as an n-depth analyss o several major methods. 1. Introducton In Part I, we dscussed the non-teratve locaton methods. A restrcton that severely lmts the applcaton o these methods s the assumpton o a sngle velocty or all channels. Ths assumpton s not arbtrary or just or convenence; t s necessary n order to keep the source locaton equatons n the smplest orm so that they can be solved non-teratvely. In other words, the assumpton s a relecton o an nherent dculty assocated wth non-teratve locaton methods. I a sngle velocty model s not sutable, as n most cases, we have to turn to an teratve method. Whle the assocated searchng strateges may vary sgncantly, teratve methods n general reer to those algorthms that start rom an ntal soluton, commonly called a guess or tral soluton. Ths tral soluton s tested by the gven condtons, and then updated by the predened schemes, whch subsequently orms a new tral soluton. Thereore, teratve methods, n essence, are a process o testng and updatng tral solutons. An teratve method s dstngushed by ts updatng scheme, whch determnes ts ecency as well as ts other major characters. Whle there are many derent schemes, the ones that are truly sgncant both theoretcally and practcally are ew or the purpose o source locaton. The object o ths artcle s to provde an n-depth analyss o these methods. But rst we wll gve a bre revew on the basc searchng approaches.. An Overvew on Basc Searchng Approaches The teratve methods used or source locaton all nto several basc categores, whch are dervatve, sequental, genetc and Smple. A bre dscusson o these approaches s gven as ollow. Dervatve approach Dervatve approach reers to those methods that use the dervatve normaton to update tral solutons. Dervatve approach s a classcal means n mathematcs or solvng nonlnear problems and s probably one o the most wdely used numercal approaches or ths purpose. Root ndng by dervatves, an elementary topc n calculus, s such an eample. J. Acoustc Emsson, 1 ( Acoustc Emsson Group

2 Dervatve methods update ther tral solutons based on the nonlnear behavor normaton at the tral soluton as gven by dervatves. Ths searchng mechansm makes dervatve methods ar more ecent than any other teratve methods. The dervatve algorthms used or source locaton nclude Geger's method (Geger, 1910, 191 and Thurber's method (Thurber, The derence between these two methods s that the Geger's method uses the rst order dervatves whle the Thurber's method uses both the rst and second order dervatves. Mathematcally, Geger s method s an applcaton o Gauss- Newton s method (Lee and Stewart, 1981 and Thurber s method an applcaton o Newton s method (Thurber, Geger s method s probably the most mportant source locaton method, and has been used almost eclusvely or local earthquake locatons. Understandng ths method s mportant or both theoretcal and practcal reasons and we wll gve a detaled dscusson on ths method. We wll also dscuss Thurber s method. In addton to the act that Thurber s method consttutes a sgncant dervatve approach, the dscusson o ths method wll help us understand the dervatve approach as a whole. Sequental searchng algorthms Sequental searchng algorthms here reer to those methods that partton the montorng volume nto smaller blocks and study these blocks sequentally. Wth these methods, each block, whch s represented by ether the center o the block or other eature locatons, s consdered as a potental AE source. The block may be urther rened n the searchng process. Whle the approach s etremely smple, easy to use, and easy to mody, the man problem s necency, whch essentally blocks ts applcaton potental or problems requrng the good locaton accuracy. For nstance, we would lke to acheve 1 mm locaton accuracy on a 1 cubc meter block, the ponts that to be searched are n the order o one bllon. In contrast, t may take only a ew teratons or Geger s method to acheve ths accuracy. Smple algorthm A notable problem wth hgh ecency dervatve algorthms s dvergence, whch could become very severe the assocated system s not stable. Sequental searchng algorthms, on the other hand, ehbt very stable soluton process, although they are just too slow. A method that comes n between s the Smple algorthm, whch s qute ecent whle showng very stable characterstcs. The Smple algorthm s a robust curve ttng technque developed by Nelder and Mead (1965. It was ntroduced or the source locaton purpose n late 1980s by Prugger and Gendzwll (Prugger and Gendzwll, 1988; Gendzwll and Prugger, The mathematcal procedures and related concepts n error estmaton or ths method were urther dscussed by Ge (1995. Because o the rare combnaton o ecency and stablty, the Smple algorthm s suted or a wde range o problems and has rapdly become a prmary source locaton method. We wll gve a detaled dscusson on ths method. Genetc Algorthm The genetc algorthm was developed by Holland (1975. It s an optmzaton technque that smulates natural selecton n that only the ttest solutons survve so that they can create even better answers n the process o reproducton. The algorthm was appled by a number o researchers or earthquake locatons (Kennett and Sambrdge, 199; Sambrdge and Gallagher, 30

3 1993; Bllngs et al., 1994, Xe et al., Whle the algorthm seems very leble or ncorporatng varous source locaton condtons, t s less conclusve on ts ecency and accuracy. The vablty o the algorthm or source locaton wll largely depend on how these questons are answered. 3 Geger s Method Geger s method, developed at the begnnng o the last century (Geger, 1910, 191, s the classcal source locaton method by all accounts. In addton to ts long hstory, the method s the best known and most wdely used source locaton method. In sesmology, t s used almost unversally or local earthquake locatons. 3.1 Algorthm Geger's method (Geger, 1910, 191 s an eample o the Gauss-Newton s method (Lee and Stewart, 1981, a classcal algorthm or solvng nonlnear problems. The method s dscussed here n terms o the rst-degree Taylor polynomals and the least-squares soluton to an nconsstent lnear system. Let ( represent the arrval tme uncton assocated wth the th sensor, where denotes the hypocenter parameters: T = (, y, z, t. (1 The unknowns,, y and z, are the coordnates o an event and t s the orgn tme o ths event. Epand ( at a nearby locaton, 0, and epress the epanson by the rst-degree Taylor polynomal: where and ( = ( 0 + δ = ( 0 + δ + y δ y + z δz + δt ( t = 0 + δ, T =, y, z, t, 0 ( δ = (δ,δ y,δz,δt T. (3 Eq. ( may also be epressed n vector notaton: ( = ( 0 + δ = ( 0 + g T δ (4 where g s the transpose o the gradent vector g and s dened by T g T = ( = (,,, (5 y z t In source locaton, the nearby locaton, 0, s conventonally called guess or tral soluton. Snce the tral soluton s ether assgned by users or generated rom the prevous teraton, t s always known at the begnnng o each teraton. As such, ( 0 s also a known quantty and s called calculated arrval tme. The term calculated arrval tme relects the act that ths quantty s obtaned by calculaton, assumng the tral soluton, 0, as the hypocenter. 31

4 The term on the let hand sde o Eq. (, ( 0 + δ, represents the arrval tme recorded at the th sensor, whch s conventonally termed observed arrval tme. As such, the physcal meanng o Eq. ( s that an observed arrval tme s epressed by the arrval tme calculated rom a nearby locaton, and by δ + δy + δz + δt, y z t a correcton actor, whch s a uncton o the partal dervatves o the hypocenter parameters. All the partal dervatves o the arrval tme uncton are known quanttes here as they can be numercally evaluated based on the tral soluton. In solvng a system o the equatons dened by Eq. (, our goal s to nd an 0, such that the calculated arrval tmes wll best match the observed arrval tmes so that 0 can be consdered as the hypocenter o the event. Ths s done n a sel correcton process: the tral soluton s updated at the begnnng o each teraton by addng δ, known as the correcton vector, obtaned rom the prevous teraton. For the convenence o the soluton or δ, we rearrange Eq. ( nto the orm: δ + δy + δz + δt = γ (6 y z t where γ = t t, (7 o t o = (, and t =. c ( 0 c Here, γ s known as channel resdual. In matr notaton, a system dened by Eq. (6 can be wrtten: Aδ = γ (8 where δ y z t δ y A = δ = m δz m m m y z t δt γ 1 γ = γ m The least squares soluton to the system dened by Eq. (8 satses (Strang, 1980 A T Aδ = A T γ (9 or δ = (A T A -1 A T γ The total eect o the msmatch between the observed and calculated arrval tmes s called the event resdual or smply resdual. The event resdual that s dened by the least-squares soluton s (Ge, 1985: 3

5 γ T γ Res = (10 m q where m s the number o equatons and q s the degree o reedom. For the number o hypocenter parameters dened by Eq. (1, the degree o reedom s 4. Now the correcton vector, δ, has been ound and t can be added to the prevous tral soluton to orm a new tral soluton. Ths process s repeated untl the gven error crteron s ullled, and the nal tral soluton s then regarded as the true source. 3. Implementaton The algorthm o Geger s method dscussed n the prevous secton was developed or general arrval tme unctons, that s, we can use ths algorthm or any arrval tme unctons as ar as the unctons and ther partal dervatves can be evaluated. To urther enhance our understandng o ths algorthm, we now dscuss the mplementaton o Geger s method through eamples. The mplementaton o Geger s method s a three-step process: establshng arrval tme unctons, data preparaton, and solvng a system o smultaneous equatons. Establshng arrval tme unctons The rst task n mplementng the Geger s method s establshng arrval tme unctons. Arrval tmes are aected by many actors. Categorcally, there are three major ones: structure and composton o meda where stress waves propagate, source mechansm and relatve orentaton o the source and sensors, and the shape and geometry o the structure under study. Whle real travel tme models are complcated n nature, the arrval tme unctons that are used to descrbe a model have to be smpled or ether theoretcal and/or practcal reasons. As an eample, the ollowng s an arrval tme uncton or a homogeneous velocty model: 1 ( z = (, y, z, t = t + ( + ( y y + ( z (11 v where the unknowns,, y and z, are the coordnates o an AE event; t s the orgn tme o ths event;, y and z, are the coordnates o the th sensor, and v s the velocty o the stress wave. We note the derence o the velocty model used here and the velocty model assumed or those non-teratve methods dscussed n the precedng paper. For those non-teratve methods, we have to assume a sngle velocty or all arrval tme data. Wth the homogeneous velocty model used n ths eample, each equaton can have ts own velocty. Ths allows us to assgn the velocty based on the arrval type, whch s crtcal or accurate source locaton. The arrval tme unctons used or Geger s method can be much more complcated than the one used n ths eample. In act, Geger s method posts no restrctons on the velocty model to be utlzed as ar as arrval tme unctons can be establshed and ther rst-order dervatves can be evaluated. 33

6 Data preparaton Once arrval tme unctons are establshed, the net step s preparng data. It s know rom Eq. ( that there are our types o data that we have to prepare, whch are: tral soluton, observed arrval tme, calculated arrval tme, and partal dervatves. Tral soluton At the begnnng o the teraton process, a tral soluton has to be assgned manually by users or generated automatcally by the locaton code. Ater ths, t s updated automatcally by addng the new correcton vector. A queston that s requently asked s: s t necessary or the tral soluton to be very close to the true event locaton? Whle t would never hurt to have a close guess, t may not be achevable n many cases, especally n the stuaton where source locaton s carred out automatcally. Fortunately, the answer to ths queston s no. In general, t wll be good enough a tral soluton s wthn the montorng area. A practce that s requently adopted by the author s to use the locaton o the rst trggered senor as the tral soluton we do not have any pror knowledge on event locatons. There s a percepton, however, that the choce o the tral soluton s mportant. Whle t s possble that one has to play the ntal tral solutons n order to get the rght answer, t usually ndcates that the assocated system s unstable, a ar more serous problem than the choce o the tral soluton. When ths s the case, the condence that one can put on the nal soluton s sgncantly dmnshed. Observed arrval tme The observed arrval tmes are the data provded eternally. Snce the physcs o source locaton s to nd a locaton that ts assocated arrval tmes best match the observed arrval tmes, the accuracy o the observed arrval tmes has to be compatble wth the requred accuracy. For nstance, the requred locaton accuracy s 1 mm and the travel velocty o the stress wave under study s 1 km/sec, then the allowed tmng error s 1/ = 1 µs. Calculated arrval tme The calculated arrval tmes are obtaned by substtutng the tral soluton nto the arrval tme unctons, such as Eq. (11. Partal dervatves The partal dervatves dened by A n Eq. (8 have to be ullled. Ths s a two-step procedure: dervng the general epressons o the partal dervatves and numercal evaluaton o these partal dervatves n terms o the tral soluton. As a demonstraton, the ollowng are the general epressons o the partal dervatves o the arrval tme uncton gven by Eq. (11: 0 = v R y y = y v R 0 z z = z v R t R = 0 = 1 ( z 0 + ( y y0 + ( z 0 34

7 Solvng a system o smultaneous equatons The least squares soluton to an nconsstent system s gven by Eq. (9. Usually, the sze o correcton vector, δ, wll decrease rapdly and reach a prescrbed accuracy wthn a ew teratons. However, t s possble that δ wll not converge: t may oscllate or even ncrease beyond control. The problem o the dvergence s a sgn o the nstablty o the assocated mathematcal system, whch s usually the result o poor array geometry. 3.3 Mechancs o teraton by rst order dervatves Although Geger s method s relatvely straght orward rom a computatonal pont o vew, conceptually, the method s stll qute conusng despte ts enormous popularty and long hstory. For nstance, t s a generally accepted percepton n sesmology that Geger s method s a lnear appromaton o nonlnear source locaton problems (Thurber, The mplcaton o ths percepton s that the method s unable to take nto account o the nonlnear behavor o arrval tme unctons. Ths s a serous mstake. It aects not only our theoretcal understandng o the method, but also ts applcatons. Whle the conusons that surround Geger s method may be attrbuted to many causes, undamentally, t s the lack o the correct understandng o the mechancs o dervatve methods. In order to solve ths problem, there are two ssues we have to dscuss urther: Taylor s theorem and the uncton o rst-order dervatves Taylor s theorem and ormulaton prncple The key element n developng Geger s method s the epanson o arrval tme unctons nto the rst-degree Taylor polynomals, and we begn our dscusson wth Taylor s theorem. The ocus o ths dscusson s whether the epanson o a uncton by the Taylor polynomals s an appromaton o that uncton. The Taylor's theorem states that a uncton at a pont may be evaluated by the Taylor polynomal o the uncton at ts neghborng pont and the error or ths appromaton can be evaluated by the assocated error uncton. The key here s that, when the Taylor polynomals are used or the uncton evaluaton at the locaton by ts neghbors, the accuracy o ths appromaton s the uncton o the sze o ths neghborhood and, unless demonstrated otherwse, t has to be assumed very small. Thereore, the Taylor polynomal n general s a hghly localzed uncton n that t changes wth each neghborng pont that s beng chosen and there s no unque Taylor polynomal that can represent a uncton or ts entre doman. Furthermore, the Taylor polynomals used or the purpose o numercal computatons are mostly assocated wth a very low degree, typcally rst or second. Under ths condton, t s vrtually mpossble to appromate a uncton by polynomals. I we consder the act that Geger s method only uses the rst order dervatves, t s mpossble to represent arrval tme unctons by these polynomals. When an arrval tme uncton s epanded n the orm o the Taylor polynomal, the resultng equaton, such as Eq. (, no longer represents the orgnal arrval tme uncton. The orgnal arrval tme uncton s the uncton o hypocenter parameters and the new equaton s the uncton o a correcton vector. Wth ths new equaton, the observed arrval tme s represented by the arrval tme calculated or a nearby pont and a correcton actor. As t has been dscussed earler, the calculated arrval tme s the result o the evaluaton o the orgnal arrval tme uncton n terms o the tral soluton, whch eventually represents the hypocenter. The correcton actor 35

8 determnes how the tral soluton s to be changed n the net teraton. As such, none o these terms can be regarded as the lnear porton o the orgnal arrval tme uncton. It s understood rom the above analyss that the rst-degree Taylor polynomals used n Geger s method are not the appromaton o orgnal arrval tme unctons and, thereore, ths epanson process cannot be characterzed by lnearzaton. The analyss o the components o the epanded uncton also shows that there s no physcal evdence to characterze ths process by lnearzaton. So ar we have demonstrated that the Taylor polynomals used n Geger s method are not the appromatons o arrval tme unctons, and, thereore, lnearzaton s not an approprate term to characterze the ormulaton o Geger s method. We now dscuss the queston whether the searchng process can be termed as lnear because Geger s method uses only rst order dervatves. The answer to ths queston s actually qute smple: any dervatve method s a nonlnear searchng method. Ths s because dervatves, regardless o ther orders, are used to catch up wth the nonlnear behavor o unctons at tral locatons and correcton vectors are determned by ths normaton. Thereore, the queston wth a dervatve method s not whether t s a nonlnear searchng method; the queston s the type o the nonlnear behavor that s utlzed. We now demonstrate the geometrc meanng o the searchng process assocated wth Geger s method. where Consder the general orm o a nonlnear system: and = F( = 0, F ( = ( 1(, (,, m (, ( 0, ( 0,, ( = 0, 1 = ( = T, n 1,,. m The soluton o ths system s to nd the common ntersecton o the unctons o F( at F( = 0. Because o the nherent dculty to solve a nonlnear system analytcally, t s usually done numercally, and the correspondng process s commonly known as root ndng. One o the best known methods or ths purpose s the Gauss-Newton s method, and Geger s method s an applcaton o ths method (Lee and Stewart, The Gauss-Newton s method uses the rstorder dervatve normaton to determne the correcton vector. Gradent vectors, such as the one gven by Eq. (5, represent the drectons o the steepest slopes. The key to understand the Gauss-Newton s method, and thereore Geger s method, s how these rst order dervatves are used to determne correcton vectors. Although t s mpossble to demonstrate the geometrcal meanng o gradent vectors or problems wth more than two ndependent varables, t s ortunate that the mechancs remans the same or all dmensons. As such, we wll use the Newton-Raphson method, the Gauss-Newton s method n one varable, to llustrate how rst order dervatves are used to determne correcton vectors. 36

9 Newton-Raphson method The Newton-Raphson method s one o the most powerul numercal methods or ndng a root o ( = 0, and, yet, both the concept and the procedure are etremely smple. The method begns wth the rst-degree Taylor polynomal or (, epanded about 0, ( = ( 0 + ( 0 '( 0 Snce we are ndng the root o, so that ( = 0 and the above equaton becomes: 0 = ( 0 + ( 0 '( 0 Solvng or n ths equaton gves: ( 0 = 0 '( 0 where should be a better appromaton o the root. Ths sets the stage or the Newton-Raphson method, whch nvolves generatng the sequence { n } n an teraton process: ( n n+ 1 = n. '( n The geometry o the Newton-Raphson method s shown n Fg. 1. At each teraton stage, we determne the rst-order dervatve o the uncton at the tral soluton. Geometrcally, t represents the tangent lne o the uncton at ths locaton. The ntersecton o ths lne wth the -as denes the new tral soluton, n+1. The correspondng correcton vector s δ = n n+1. From the gure, t s easy to very the relaton: ' ( n ( n =. n n+ 1 The above equaton s another orm o the precedng equaton, but wth much clearer geometrcal meanng: the slope o the tangent lne represented by '( n on the let-hand sde o the equaton s the rato o the uncton value at the tral soluton and the correcton vector. In summary, the Newton-Raphson method can be vewed as a procedure that we use the ntersecton o the tangent lne at the -as to appromate the root o the uncton and the drecton o the tangent lne s dened by the rst order dervatve o the uncton at the tral soluton. 3.4 Stoppng crtera When an teratve method s used, the calculaton has to be stopped at a certan pont. There are three commonly used stoppng crtera or dervatve methods, and ther applcablty or the problem o source locaton s dscussed net. Resdual as stoppng crteron When resdual s utlzed as the stoppng crteron, we stop the calculaton soon ater the resdual s smaller than a pre-dened tolerance, such that: Res < ε, or ε > 0, where Res s the event resdual and ε s the tolerance. The event resdual s a measurement o the locaton error n terms o the total eect o the msmatch between the observed and calculated arrval tmes. Mathematcally, t s dened by the regresson method beng used. For the leastsquares method, the event resdual s (Ge, 1996 gven by Eq. (10: 37

10 Fg. 1 Geometry o Newton-Raphson method Res = as dened prevously. γ T γ m q (10 Although the approach appears qute natural or many applcatons as the resdual s a mathematcal measurement o the computatonal error, the problem n the case o source locaton s that t may vary n a very wde range or events under smlar condtons, say, covered by the same array. I the tolerance s set too low, many solutons may never be able to reach that level. I t s set too hgh, we lose the accuracy. Thereore, t s dcult to mplement ths approach or source locaton problems. Sze o correcton vector as stoppng crteron The second crteron s the sze o correcton vectors, δ. In the case o source locaton, correcton vectors nclude both spatal and tme components. Ths creates a problem or calculatng the sze o δ because o the derence n unts. A soluton to ths problem s to represent the sze o correcton vectors by δ d, the sze o the spatal components only, such as δd = δ + δ y + δz and the correspondng stoppng crteron s: δ d < ε When the sze o correcton vectors s used as a stoppng crteron, the underlyng assumpton s that t s a sgn o the convergency when δ d 0 as the number o teratons ncreases to nnty. 38

11 Relatve sze o correcton vector as stoppng crteron The thrd crteron s the relatve sze o correcton vectors. Agan, we only consder the spatal components, ths thrd crteron can be epressed: δd + y + z < ε where, y and z are the coordnates o the event locaton determned at the current teraton. The relatve error s a better measurement or many applcatons, although t has two serous drawbacks n the case o source locaton. Frst, the relatve error would be the uncton o the coordnate system. For nstance, the relatve error or an array descrbed by three dgts would be drastcally larger than the same array that s descrbed by ve dgts. Secondly, the approach would make the error as the uncton o event locatons. I we consder the act that the sze o an array s normally much larger than the locaton error, typcally o the order o or hgher, the event locaton wll sgncantly aect the outcome o ths approach. For these reasons, relatve sze s not a sutable stoppng crteron. In comparson o these three crtera, the sze o correcton vectors appears most sutable or the purpose o source locaton. In any case another stoppng crteron s always necessary: the number o teratons. Ths s especally true or dervatve methods, wth whch dvergence may occur. 3.5 Problem o dvergence The major problem assocated wth the Geger s method s dvergence. Dvergence s a common problem assocated wth many teratve algorthms. The cause o dvergence s comple and a detaled dscusson o the problem s beyond the scope o ths artcle. However, there are two ponts we wsh to make. Frst, although dvergence can be caused by many derent techncal reasons, t s undamentally governed by the stablty o the assocated mathematcal system. In the case o source locaton, ths stablty s determned by the sensor array geometry relatve to the actual event locaton. For nstance, because o the nherent dculty to spread the sesmographs n the vertcal drecton, the locaton accuracy o local earthquakes n the depth drecton s n general poorer than n the other drectons. Thereore, the best means to allevate the dvergence problem s to optmze the sensor array geometry, t s possble. The second pont s that dvergence s a phenomenon closely related to the convergence rate. For those methods wth a very hgh convergence rate, such as the Geger s method, the chances to develop the dvergence problem s also consderably hgher. Many research studes were carred out n sesmology on the mprovement o the convergence character o the source locaton algorthm, and some remedal measures were proposed (Smth, 1976; Buland, 1976; Lee and Stewart, 1981; Anderson, 198; Lenert and Frazer, 1983; Thurber, In general, the ecency o these remedal measures s very lmted as none o them could undamentally address the two nherent dcultes dscussed earler. 4. Thurber s Method Geger s method uses the rst order dervatves to catch up the nonlnear behavor o arrval tme unctons at the tral solutons. We now dscuss another sgncant dervatve approach, Thurber s method (Thurber, 1985, whch uses the normaton o both the rst and second dervatve to determne correcton vectors. 39

12 Algorthm Smlar to our dscusson on Geger s method, let ( be the general orm o the arrval tme uncton assocated wth the th sensor, where denotes the hypocenter parameters dened by Eq. (1. Epand ( at a nearby locaton, 0, and epress the epanson by the second-degree Taylor polynomal: ( = ( 0 + δ = ( 0 + g T δ + 1 δ T H δ (1 where g T s the transpose o the gradent vector g dened by Eq. (5 and H s the Hessan matr: = 1 t t z t y t z t z z y z y t y z y y t z y H (13 The physcal meanngs o ( 0 + δ and ( 0 n Eq. (1 remans the same as n Eq. (, whch are the observed and calculated arrval tmes, respectvely. Eq. (1 s a quadratc uncton o the correcton vector. Wth the partal derentaton o the equaton and settng the resultng equaton to zero, we have g + H δ = 0. (14 For convenence, the hypocenter notaton gven by Eq. (1 may also be epressed as = T,,, ( (15 Wth the hypocenter notaton gven by Eq. (15, an alternatve orm o Eq. (14 s = = ( ( j k j j k δ, k = 1,, 3, 4, (16 whch gves us a detaled vew o the content o Eq. (14. The least squares soluton or the system dened by Eq. (14 s (Thurber, 1985: Δ = (A T A ( A T r 1 A T r (17 where both A and r are dened n Eq. (8. I we use N to represent r A ( T n Eq. (17, such that r A N ( T =, (18 and the entry N j o ths 4 4 matr s gven by

13 N j = m k k = 1 j γ k, j = 1,, 3 (19 N = 0, j = 4 j The step that holds the key to understand Newton s method, and thereore Thurber s method, s the transton rom Eq. (1 to Eq. (14. Eq. (1 s a quadratc uncton. By the partal derentaton o ths equaton and settng the resultng equaton to zero, Eq. (14 denes the etreme o ths quadratc uncton. Ths etreme n Newton s method s regarded as the optmum correcton vector or the tral soluton, and thereore, the soluton or Eq. (1. 4. Mechancs o teraton by rst and second order dervatves Followng the approach that the Newton-Raphson's method was used to llustrate the geometrc meanng o the rst order dervatves, we now use Newton s method n one varable to demonstrate geometrcally how the rst and second order dervatves are used to determne the correcton vector. Fndng the root wth rst and second dervatves Consder the second-degree Taylor polynomal or (, epanded about 0, 1 = ( 0 + ( 0 '( 0 + ( 0 "( (0 ( 0 Our goal s to determne so that ( = 0. Note that ( n ths case s epressed by a quadratc uncton o and the best appromaton o ( = 0 s to nd the mnmum o the uncton and ths can be done by takng the rst dervatve o the uncton wth respect to, d ( d ( 0 d(( 0 '( 0 1 d(( 0 "( 0 = + + d d d d = '( 0 + ( 0 "( 0 and let the resultng equaton equal to zero, '( 0 + ( 0 "( 0 = 0 Solvng the equaton or, = 0 '( 0 "( 0 the nal soluton can be approached teratvely by the sequence { } dened by = '( "( (1 The geometry o ths soluton process s llustrated n Fg.. The second-degree Taylor polynomal, denoted by p( 0 n the gure, s used to represent the uncton at the neghborhood o 0. The etreme o ths polynomal, 1, s consdered by Newton s method as the optmum soluton whch becomes the new tral soluton. Eq. (1 s a mathematcal denton o ths etreme. 41

14 Fg. Geometry o Newton s method n one varable. A very nterestng pont shown by Fg. s the derence between (, the orgnal uncton, and p( 0, the second-degree Taylor polynomal, whch clearly demonstrates that the Taylor polynomal s not the appromaton o that uncton. 4.3 Dscusson The essence o dervatve methods s usng nonlnear characters o unctons at the tral soluton, descrbed by dervatves, to determne correcton vectors. The derence among dervatve methods s hence the type o nonlnear characters that are used. The smplest dervatve method s Geger s method. The nonlnear character used by ths method s gradent, or the steepest slope at the tral soluton, whch are represented by rst order dervatves. Thurber s method uses both the rst and second order dervatves and ts soluton s geometrcally represented by the etreme o a quadratc uncton. Thurber s method s a more sophstcated dervatve method than Geger s method n that t utlzes the quadratc model or optmzaton. Although the method was demonstrated wth the mproved stablty n several cases (Thurber, 1985, t s not clear whether t can be regarded as a general soluton to the problem. A major uncertanty assocated wth the quadratc model based methods s whether the model s postve dente. Whle t s not a requrement, an underlyng assumpton or these methods s that the model s postve dente. I the assumpton s not ullled, the perormance o these methods s much more dcult to predct. 4

15 5. Smple Method The Smple method s a relatvely new method developed by Nelder and Mead (1965. It searches the mnmums o mathematcal unctons through uncton comparson. The method was ntroduced or the source locaton purpose n late 1980s by Prugger and Gendzwll (Prugger and Gendzwll, 1988; Gendzwll and Prugger, The mathematcal procedures and related concepts n error estmaton or ths method were urther dscussed by Ge (1995. Readers should be aware that the Smple method dscussed here s derent rom the one used n lnear programmng where t reers an algorthm or a specal type o lnear problems. 5.1 Method concept The Smple method starts rom the dea that, or a gven set o arrval tmes assocated wth a set o sensors, an error can be calculated or any pont by comparng the observed and the calculated arrval tmes. An error space s thus the one n whch every pont s dened by an assocated error, and the pont wth the mnmal error s the event locaton. The process o searchng or the mnmal pont wth the Smple method s unque. The soluton s sad to be ound when a Smple gure alls nto the depresson o the error space. The Smple s a geometrc gure whch contans one more verte than the dmenson o the space n whch t s used. A smple or a two dmensonal space s a trangle, a smple or a three dmensonal space s a tetrahedron, etc. The search or the nal soluton wth the Smple source locaton method nvolves our general steps: settng an ntal Smple gure; calculatng errors or vertces; movng Smple gures; and v eamnng the status o convergency. At the begnnng o search, an ntal Smple gure has to be set, whch s then rollng through the error space, epandng, contractng, shrnkng and turnng towards the mnmal error pont o the space. The movement o the Smple gure s governed by the error dstrbuton at ts verte, whch s calculated each tme when the Smple gure s reshaped. 5. Smple gure and ts ntal settng It s understood rom the earler dscusson that the Smple s a geometrc gure that contans one more verte than the dmenson o the space n whch t s used. Because a source locaton problem s spanned by not only ts geometrc dmensons, but also tme dmenson, the Smple gure wll be tetrahedron and ve-vertces or two and three geometrcal problems, respectvely, we apply the Smple method drectly to our source locaton problems. A more convenent approach s to consder spatal coordnates and orgn tme separately n that error spaces are already optmzed n terms o orgn tme. Wth ths treatment, the dmenson o the Smple gures s solely determned by the geometrc dmenson o the source locaton problems: a trangle or two-dmensonal problems and a tetrahedron or three-dmensonal problems. We wll dscuss the mathematcal procedure o ths approach n the net secton. There s no rgd gudelne regardng how to set the ntal Smple gure. For the purpose o ecency, one may want to set t near the potental locaton wth a reasonable sze. Care must 43

16 Fg. 3 An eample o the Smple movng on the error space contour plot (ater Cacec and Cachers, be taken that the gure s not dmensonally reduced. A practce by the author s to set the ntal Smple gure near the rst trggered sensor wth the sze about one thrd to one hal o the montorng array. Because o the very robust nature o the Smple method, the sze, shape and locaton o the ntal gure can be qute arbtrary. 5.3 Error calculaton and orgn tme The locaton errors are dened by the assocated optmzaton method. Two most popularly employed such methods are the least squares method (L norm and the absolute value method (L1 norm method. The mplementaton o these methods or the Smple source locaton method s dscussed as ollows. Error estmaton by the Least squares method Assume that the observed arrval tme at the th staton s t o and the calculated arrval tme at ths staton s t c, the staton resdual as gven by Eq. (7 s γ = t t, (7 o c where γ s staton resdual. A calculated arrval tme conssts o two parts: orgn tme, t, and travel tme rom the verte under concern to the th staton, tt, whch are related by the ollowng equaton: t = tt t. ( c + It s noted that orgn tme s a source parameter, and hence an unknown here. Substtutng Eq. ( nto Eq. (7, we epress the staton resdual as a uncton o orgn tme: γ = t ( tt t, (3 o + Now the goal s to nd the best estmate o the orgn tme such that the total error wll be mnmzed. Wth the least squares method, the total error s dened by 44

17 γ and t s mnmzed when the orgn tme s determned by the equaton d( γ = 0 dt Solvng the above equaton, the best estmate o the orgn tme s t tt t = n n (4 By substtutng Eq. (4 nto Eq. (3, we epress the staton resdual n terms o observed arrval tme and calculated travel tme: The event resdual s t tt γ = ( t ( tt (5 n n γ Res = m q. (6 Eq. (6 s the equaton that s used or the error calculaton or each verte the least squares method s used. Note that the orgn tme wth ths approach s gven by Eq. (4. Error estmaton by the absolute value method For the absolute value method, the total error s dened by γ. The best estmate o the orgn tme has to satsy the objectve uncton Mnmze ( γ. Substtutng Eq. (3 nto the above epresson, we have Mnmze ( t tt + ( t. For a set o lnear equatons wth the orm o, = b where s the varable to be estmated and b s a constant, the analytcal soluton o the best t or dened by the L1 mst norm s the medan o b s (Chavatal, Accordngly, the best t o the orgn tme s the medan o all (t tt. Let us denote t as t m. The staton resdual dened by the absolute value method s thereore and the total error s γ = t tt t (7 m γ = t tt tm. (8 Eq. (8 s the equaton that s used or the error calculaton or each verte the absolute value method s used. Note that the orgn tme wth ths approach s the medan o all (t tt. 45

18 5.3 Rules or movng Smple gures The movement o Smple gures s realzed by deormaton. There are our deormaton mechansms: relecton, epanson, contracton, and shrnkage, whch are eplaned n Fg. 4. Fg. 4 An llustraton o our deormaton mechansms: relecton, epanson, contracton and shrnkage. BWO = the Smple gure pror to deormaton, B = best verte, W = worst verte, E = epanded verte, C = contracted verte, and S = shrunken vertees (ater Cacec and Cachers, Trangle BWO n the gure represents a two-dmensonal Smple gure. Assumng that W and B represent the worst and best verte, respectvely. R s then called relected verte, a mrror mage o W about the mdpont o all the other vertees. E s epanded verte, whch doubles the mage dstance. C s contracted verte, located a halway rom W to the mdpont. S represents shrunken vertees, the mddle locatons between the best verte and the others. The logc to apply these mechansms s llustrated n the low dagram n Fg. 5. The ollowng s a summary o ths logc: calculatng the locaton error or each verte, determnng the vertces wth the mamum and mnmum errors, locatng the relected verte and calculatng ts locaton error, v ths error s less than the old mamum but more than the mnmum, a new Smple gure s ormed by replacng the old mamum by the relected verte, and go back to to restart the process. v the error at the relected verte s less than the mnmum, locatng the epanded verte and calculatng ts locaton error. I the error s less than the mnmum, a new Smple gure s ormed by replacng the old mamum by the epanded verte, and go back to to restart the process. I the error s larger than the mnmum, a new Smple gure s ormed by replacng the old mamum by the relected verte, and go back to to restart the process, v the error at the relected verte s larger than the old mamum, locatng the contracted verte and calculatng ts locaton error. I ths error s less than the old mamum, replacng the old mamum by the contracted verte. Otherwse, shrnkng the Smple gure by movng all vertces other than the old mnmum to the shrunken vertces. Go back to to restart the process. 46

19 Fg. 5. Flowchart o the Smple algorthm (ater Cacec and Cachers,

20 5.5 Eamnng the status o convergency In secton 3.4 we dscussed the stoppng crteron or dervatve methods and determned that the sze o correcton vectors would be most approprate or the purpose o source locaton. It s or the same reason that we use the sze o Smple gures as the measurement o the convergency status o Smple solutons. We accept the soluton the sze o the Smple gure s less than the tolerance. The sze may be dened derently. For nstance, the average dstance rom a verte to others would be a convenent and representatve measurement o the sze. 5.6 Dscusson The Smple method oers several dstnctve advantages over dervatve methods. The most mportant one s that dvergence s mpossble. The author manually eamned several thousands o locaton data by the Smple method and dd not observe a sngle dvergence case. In act, the same Smple code developed by the author has been nstalled at a numerous mne stes or the contnuous montorng or many years and there has been no dvergence problems ever reported. The reason or ths robust perormance s that the Smple gure wll not leave the lowest error pont whch has been ound unless a better one s located. Thereore, the Smple method wll always keep the best soluton that has been ound, whereas or others t may be lost n teraton processes. Ths character s especally mportant or the automated montorng. The other mportant advantage o the method s ts leblty. Unlke dervatve methods, wth whch arrval tme unctons have to be establshed pror to the analyss, arrval tme unctons used n the Smple method can be establshed durng the data processng, whch s a necessary condton n order to handle many sophstcated problems. A requently mentoned advantage o the Smple method s that we avod many computatonal problems assocated wth partal dervatves and matr nversons. It s, however, mportant to understand that ths s not equvalent to say that the underlyng problems are also gone. For nstance, an ll-condtoned matr n source locaton s a relecton o the nstablty o the assocated mathematcal system. It would be a serous mstake to epect the dsappearance o the problem by usng the Smple method. The truth s that the problem s physcal; ts estence s ndependent o the soluton methods. 6. Conclusons Iteratve methods are o prmary mportance n source locaton methods. Ths s because o ther leblty n dealng wth arrval tme unctons, whch s essental or realstcally approachng a great majorty o source locaton problems. Non-dervatve methods have to assume a sngle velocty or all channels, whch severely lmts ther applcaton. The best known group o teratve methods s dervatve algorthms. Wth dervatve methods, we approach the nal soluton through a contnuous updatng process o tral soluton, and ths s done by addng the correcton vector ound rom the prevous teraton at each stage. The correcton vector s determned by the assessment o nonlnear behavor o arrval tme unctons at the tral soluton. The derence among dervatve methods s thereore the type o the nonlnear behavor that s utlzed. The nonlnear behavor used by Geger s method s the gradent o arrval tme unctons, or the steepest drecton o these unctons, represented by the rst-order dervatves. The one utl 48

21 zed by Thurber s method s the etreme o the quadratc unctons, whch s the second-degree o Taylor polynomals, ncludng both the rst- and second-order dervatves.. Because the nonlnear behavors are used decsvely or the calculaton o correcton vectors, dervatve methods, such as Geger s method and Thurber s method, are nherently ecent, and oer superor perormance to other teratve methods. The nal solutons are usually approached wthn a ew teratons the assocated systems are stable. A shortcomng o dervatve methods s dvergence. Whle the cause o dvergence s comple and a detaled dscusson o the problem s beyond the scope o ths artcle, t s mportant to know that dvergence, n general, relects the problem o nstablty, whch, n turn, s largely governed by the sensor array geometry. The Smple method s the most mportant source locaton method emerged n recent years. Because o ts unque teraton approach, dvergence s mpossble. Ths has gven the method a huge advantage over dervatve methods. The other major advantage o the method s ts leblty to deal wth complcated velocty models. Unlke dervatve methods, wth whch arrval tme unctons have to be establshed pror to the analyss, arrval tme unctons used n the Smple method can be establshed durng the data processng, whch s a necessary condton n order to handle many sophstcated problems. Fnally, we would lke to emphasze that source locaton s a subject aected by many actors and the locaton algorthm s only one o them. In order to use a locaton algorthm ecently, we need not only a good understandng o the algorthm tsel, but also a perspectve vew on how the algorthm relates to other actors. The two most mportant actors are sensor array geometry and errors assocated wth nput data. Sensor array geometry and system stablty The mportance o the sensor array geometry n AE source locaton les on the act that we would never be able to elmnate nput errors completely and the mpact o these ntal errors on the nal locaton accuracy s controlled by the sensor array geometry (Ge, Good array geometry can eectvely lmt the mpact o ntal errors whle a poor one wll mamze the mpact. In other words, the stablty o the assocated mathematcal system s determned by the sensor array geometry. Thereore, the sensor array geometry s undamental or an accurate and relable AE source locaton. Understandng the relaton between sensor array geometry and system stablty s mportant rom two perspectves. Frst, the stablty o an event locaton s ndependent o the locaton algorthm beng used; that s, we can not change the degree o the senstvty o a soluton to ntal errors by swtchng the locaton algorthm. I we want to mprove the relablty o event locatons, the only means s to mprove the sensor array geometry. There s no other way around. Second, a phenomenon that s closely assocated wth an unstable system s dvergence. It s more dcult to approach the soluton numercally when the assocated system s unstable. It s, however, mportant to note that the convergence character does der rom method to method. We can reduce the rsk o dvergence by choosng an algorthm wth the better convergence character, and Smple method s such an eample. 49

22 Errors assocated wth nput data There are a number o error sources or AE source locaton data. The obvous ones are tmng, velocty model and sensor coordnates. The one that s oten not recognzed but may cause most damages s msdentcaton o arrval types. An assumpton that s requently made n AE source locaton s P-wave trggerng. In realty, many arrvals are due to S-waves or even not related sgnals called outler. An outler may be caused by ether culture noses or the ntererence o other events. The analyss o AE data shows that S-wave arrvals may account or as much as more than 50% o total arrvals (Ge and Kaser, As S-wave velocty s typcally hal o the P-wave velocty, mng o P- and S- wave arrvals mmedately ntroduces a large and systematcal error to the locaton system. Ths has been the prmary reason responsble or the poor AE source locaton accuracy n the past. Although there are means to reduce the mpact o nput errors, such as optmzaton o the sensor array geometry and statstcal analyss, one should not epect that any o these methods would be able to deal wth large and systematcal errors. These errors have to be detected and elmnated beore the locaton process starts. Acknowledgments I am grateul to Dr. Kanj Ono or hs encouragement to wrte my research eperence n the area, and hs detaled revew o the manuscrpt and comments. I thank Dr. Hardy or hs thorough revew and the anonymous revewer or hs comments and suggestons to mprove the manuscrpt. Reerences Anderson, K. R., (198. Robust earthquake locaton usng M-estmates, Phys. Earth and Planet Int., 30, Bllngs, S. D., B. L. N. Kennett, and M. S. Sambrdge, (1994. Hypocentre locaton: genetc algorthms ncorporatng pro lem-specc normaton, Geophys. J. Int. 118, Buland, R., (1976. The mechancs o locatng earthquakes, Bull. Sesm. Soc. Am. 66, Burden, R. L., J. D. Fares and A. C. Beynolds (1981. Numercal analyss, Prndle, Weber & Schmdt, Boston, Massachusetts. Cacec, M. S. and W. P. Cachers (1984. Fttng curves to data (the Smple algorthms s the answer, Byte 9, Ge, M., (1988. Optmzaton o Transducer array geometry or acoustc emsson/mcrosesmc source locaton. Ph.D. Thess, The Pennsylvana State Unversty, Department o Mneral Engneerng, 37 p. Ge, M. and P. K. Kaser (1990. Interpretaton o physcal status o arrval pcks or mcrosesmc source locaton. Bull. Sesm. Soc. Am. 80, pp

23 Ge, M. (1995. Comment on "Mcroearthquake locaton: a non-lnear approach that makes use o a Smple steppng procedure" by A. Prugger and D. Gendzwell, Bull. Sesm. Soc. Am. 85, Geger, L. (1910. Herbsetmmung be Erdbeben aus den Ankunzeten, K. Gessell. Wll. Goett. 4, Geger, L. (191. Probablty method or the determnaton o earthquake epcentres rom the arrval tme only, Bull. St. Lous. Unv. 8, Gendzwll, D. and A. Prugger (1989. Algorthms or mcro-earthquake locaton, n Proc. 4 th Con. Acoustc emsson/mcrosesmc Actvty n Geologc Structures, Trans Tech. Publcatons, Clausthal-Zellereld, Germany, Holland, J. H. (1975. Adapton n Natural and Artcal Systems, The Unversty o Mchgan Press, Ann Arbor. Kennett, B. L. N. and M. S. Sambrdge, (199. Earthquake locaton genetc algorthms or telesesms, Phys. Earth and Planet Int., 75, Lee, W. H. K. and S. W. Stewart (1981. Prncples and applcatons o mcroearthquake networks, Adv. Geophys. Suppl.. Lenert, B. R. and L. N. Frazer (1983. An mproved earthquake locaton algorthm, EOS, Trans. Am. Geophys. Unon, 64, 67. Nelder, J. A. and R. Mead (1965. A smple method or uncton mnmzaton, Computer J. 7, Prugger, A. and D. Gendzwll (1989. Mcroearthquake locaton: a non-lnear approach that makes use o a Smple steppng procedure, Bull. Sesm. Soc. Am. 78, Sambrdge, M., K. Gallagher, (1993. Earthquake hypocenter locaton usng genetc algorthms, Bull. Sesm. Soc. Am. 83, Smth, E. G. C., (1976. Scalng the equatons o condton to mprove condtonng, Bull. Sesm. Soc. Am. 66, Strang, G. (1980. Lnear algerbra and ts applcatons, Academc Press Inc., New York, New York. Thurber, C. H. (1985. Nonlnear earthquake locaton: theory and eamples, Bull. Sesm. Soc. Am. 75, Xe, Z, T. W. Spencer, P. D. Rabnowtz, and D. A. Fahlqust, (1996. A new regonal hypocenter locaton method, Bull. Sesm. Soc. Am. 86,