Geophysical inversion with a neighbourhood algorithm I. Searching a parameter space

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1 Geophys. J. Int. (1999) 138, Geophysical invesion with a neighbouhood algoithm I. Seaching a paamete space Malcolm Sambidge Reseach School of Eath Sciences, Institute of Advanced studies, Austalian National Univesity, Canbea, ACT 0200, Austalia. malcolm@ses.anu.edu.au Accepted 1999 Mach 12. Received 1999 Mach 5; in oiginal fom 1998 Decembe 3 SUMMARY This pape pesents a new deivative-fee seach method fo finding models of acceptable data fit in a multidimensional paamete space. It falls into the same class of method as simulated annealing and genetic algoithms, which ae commonly used fo global optimization poblems. The objective hee is to find an ensemble of models that pefeentially sample the good data-fitting egions of paamete space, athe than seeking a single optimal model. (A elated pape deals with the quantitative appaisal of the ensemble.) The new seach algoithm makes use of the geometical constucts known as Voonoi cells to deive the seach in paamete space. These ae neaest neighbou egions defined unde a suitable distance nom. The algoithm is conceptually simple, equies just two tuning paametes, and makes use of only the ank of a data fit citeion athe than the numeical value. In this way all difficulties associated with the scaling of a data misfit function ae avoided, and any combination of data fit citeia can be used. It is also shown how Voonoi cells can be used to enhance any existing diect seach algoithm, by intemittently eplacing the fowad modelling calculations with neaest neighbou calculations. The new diect seach algoithm is illustated with an application to a synthetic poblem involving the invesion of eceive functions fo custal seismic stuctue. This is known to be a non-linea poblem, whee lineaized invesion techniques suffe fom a stong dependence on the stating solution. It is shown that the new algoithm poduces a sophisticated type of self-adaptive seach behaviou, which to ou knowledge has not been demonstated in any pevious technique of this kind. Key wods: numeical techniques, eceive functions, wavefom invesion. 1 INTRODUCTION The incease in computational powe coincided with the ealization that, fo many inteesting poblems, it was inconvenient, In the past decade and a half, Monte Calo (MC) methods o simply not possible, to use a lineaized appoximation. (Hammesley & Handscomb 1964) have enjoyed a esugence Lineaization involves the calculation of patial deivatives of in populaity amongst geophysicists, paticulaly in application data with espect to model paametes (o functionals), and to invese poblems. The ealiest use of MC methods was fo fo some non-linea poblems these can eithe be difficult to pobabilistic, o andomized, seaching of a (finite dimensional) calculate, o have a vey limited ange of applicability. [Fo paamete space. Notable papes wee by Keilis-Book & a ecent discussion on the ole of non-lineaity in invese Yanovskaya (1967), Pess (1968), Wiggins (1969), Andessen poblems see Sniede (1998).] Deivative-fee diect seach (1970), and Andessen & Senata (1971). These studies wee methods wee seen as an attactive altenative. This was well known to geophysicists, but it was only when the necessay especially tue in seismic poblems, whee high-fequency body computational powe became widely accessible (aound the wavefoms wee being used to constain eath stuctue fom beginning of the 1980s) that inteest in Monte Calo methods length-scales of geophysical exploation to those of egional became widespead. Pio to that (and continuing to this day), tectonics. many of the invese poblems posed by geophysicists wee The ealy diect seach methods wee based on unifom addessed using linea invese theoy (see Menke 1989; Taantola pseudo-andom sampling of a paamete space, sometimes 1987; Pake 1994 fo compehensive summaies). with constaints imposed (as in Wiggins 1969). The inefficiency 1999 RAS 479

2 480 M. Sambidge of this type of appoach makes it impactical once the dimen- space epesents a set of paametes descibing some physical sionality of the paamete space inceases. Rothman (1985, 1986) popety of the eath, fo example seismic wavespeed, and the intoduced the method of simulated annealing (SA) (Kikpatick objective function usually depends on a measue of data misfit et al. 1983) into geophysics, which is a stochastic diect seach and some function of the model. The data misfit measues the method designed pimaily fo global optimization poblems. discepancy between obsevations and theoetical pedictions Fo a suvey of papes see Sen & Stoffa (1995), and fo detailed fom a model (detemined fom the solution of a fowad desciptions of the method see Aats & Kost (1989), poblem). This fomulation applies to nealy all geophysical Moe ecently, genetic algoithms have made the jump fom invesion. Fo example, this is the case in the statistical view thei oigins in the compute science liteatue (Holland 1975; of invese theoy (e.g. Taantola & Valette 1982), whee one Goldbeg 1989) to geophysical poblems (Stoffa & Sen 1991; is often inteested in locating a model that maximizes the Sambidge & Dijkoningen 1992). Again the objective is often posteio pobability density function, and also in the extemal stated as seeking a model giving a globally optimal data misfit model view, which advocates optimizing some popety of the value (within a pe-defined finite-dimensional paamete space). model unde the constaint that it fits the data to a satisfactoy This type of appoach has found many applications in ecent level (e.g. Pake 1977). yeas (fo suveys see Gallaghe & Sambidge 1994; Sen & In many poblems (even linea ones), the data alone do Stoffa 1995). As with simulated annealing, many vaiants of not detemine the model uniquely. Thee may exist no o an the basic method have been developed. These often esult infinite numbe of models that satisfy the data. Optimal models in the intoduction of exta contol paametes (in addition to ae one way of chaacteizing the ensemble of acceptable data- those inheent in the basic method) which must be empiically fitting models. When a poblem is both non-unique and nonlinea, tuned fo each application. The goal of the tuning pocess is it may still be infomative to chaacteize the ensemble usually to achieve computational efficiency, and some level of of accptable solutions by seeking models with exteme values obustness against entapment in local minima. in appopiately chosen model popeties (Pake 1977). In this pape we pesent an entiely new class of diect Howeve, thee may be cases whee we do not know what seach methods, which we show has some distinct advantages model popeties might be appopiate to optimize, o the data ove pevious methods. The method is conceptually simple misfit function may be vey complex, containing many local with at most two contol paametes, but is able to exhibit minima, theeby making the optimal data-fitting model difficult complex self-adaptive behaviou in seaching a paamete space. to find and potentially of little use. Unlike pevious methods, the new appoach is not designed An altenative to seeking single optimal models is to chaacspecifically to pefom global optimization; howeve, we show teize the entie ensemble of acceptable solutions diectly that in ou test poblem it pefoms just as well as a pevious by fist tying to geneate as many membes of it as possible, appoach in this espect. The objective hee is to sample the and then analysing them. This two-stage appoach has been egion of paamete space that contains models of acceptable adopted many times befoe; indeed, it was the oiginal pupose data fit (o any othe objective function), and then to extact of applying Monte Calo methods in geophysics (Pess 1968; obust infomation fom the ensemble of models obtained. Andessen & Senata 1971). Since then, seveal authos have This pape deals lagely with the fist stage of this poblem, stated simila views, and poposed methods of appaising the i.e. geneating an acceptable ensemble, and only in a qualitative ensemble (Kennett & Nolet 1978; Kennett 1978; Dosso & manne with the second state, i.e. appaising the ensemble. Oldenbug 1991; Vasco et al. 1993; Lomax & Sniede 1995; A elated pape, Sambidge (1999; heeafte efeed to as Sniede 1998). Hee we adopt a simila appoach, although Pape II), shows how a simila appoach to the one pesented we pefe to make infeences based on popeties of all models hee can be used to extact quantitative obust infomation in the ensemble, not just the acceptable ones, because in fom an ensemble of models. In Pape II, no assumption is pinciple bad data-fitting models may also tell us something made on how the ensemble of models is geneated, and in useful. The focus of this pape is on the sampling poblem. paticula it need not necessaily be geneated with the method pesented in this pape. We compae the new appoach to existing methods on the 3 SEARCHING A PARAMETER SPACE invesion of seismic eceive functions fo S-wave velocity Stochastic (Monte Calo) seach methods such as unifom stuctue in the cust. This is a complex non-linea wavefom Monte Calo seach (UMC), simulated annealing (SA) and fitting poblem, whee lineaized techniques eithe fail o have genetic algoithms (GA) ae becoming inceasingly popula fo a stong dependence on the stating model (Ammon et al. 1990). optimizing a misfit function in a multidimensional paamete We also pefom some detailed analysis of the new method space. The wok pesented hee was motivated by a question and show how its computation time scales with quantities that aose in the autho s own use of these methods. How can such as dimension and numbe of samples. The new class of a seach fo new models be best guided by all pevious models methods makes use of some simple geometical concepts which fo which the fowad poblem has been solved (and hence the we show can also be used to enhance existing seach methods data-misfit value evaluated)? It is instuctive to conside how such as genetic algoithms and simulated annealing. each of the thee methods above addesses this issue. UMC by definition makes no use of pevious samples in paamete space, since each new sample is independent of the pevious 2 ENSEMBLE INVERSION RATHER THAN samples. Both SA and GA make use of pevious samples, OPTIMIZATION but in athe complex ways. The infomation inheited fom It is common to fomulate a geophysical invese poblem as one of optimization in a finite-dimensional paamete (model) space (possibly unde constaints). Each point in model pevious samples may also be highly dependent on contol paametes (GA) o tempeatue pofiles (SA), each of which must be tuned fo each poblem individually.

3 Neighbouhood algoithm: seaching a paamete space 481 In this pape we ague that a simple way of making use of appoximation to the misfit suface (NA-suface) is geneated pevious model space samples is to use them to appoximate by simply setting the misfit to a constant inside each cell. (i.e. intepolate) the misfit (objective) function eveywhee Theefoe to evaluate the appoximate misfit at any new point in model space. Assuming that this can be done, a simple we need only find which of the pevious samples it is closest to. genealized algoithm fo seaching a paamete space would The NA-suface povides a simple way of pefoming nonsmooth take the following fom. intepolation of an iegula distibution of points in d-dimensions and has some vey useful popeties. Fo any (1) Constuct the appoximate misfit suface fom the distibution and density of samples, the Voonoi cells ae always n pevious models fo which the fowad poblem has unique, space filling and have size invesely popotional to the p been solved; sampling density (see Fig. 1). This means that the NA-suface (2) use this appoximation (instead of actual fowad will contain shot-scale vaiations in misfit only whee they modelling) togethe with a chosen seach algoithm to ae pesent in the oiginal samples, and longe-scale vaiations geneate the next n samples; whee sampling is spase. We can egad the NA-suface as a s (3) add n to n and go back to (1). minimal -featue appoximation of the misfit suface influenced s p equally by all available samples. The inteesting featue is that IDEALIZED ALGORITHM the size and shape of the neighbouhoods about each sample (and hence the egions of bette and wose data fit) ae com- We will efe to this as ou idealized seach algoithm pletely detemined by the samples themselves. In paticula, containing one fee paamete n (initially n =n ). The two s p s no spatial scalelengths o diectionality ae imposed extenally. missing details ae how to constuct the appoximate misfit It is theefoe a useful candidate to dive ou idealized sampling suface, and how to geneate new samples with it. In the next algoithm above. Note that if the NA-suface can be used to section we pesent a simple but (we ague) poweful way of geneate new samples (geaed towads egions of lowe misfit) constucting an appoximate misfit suface, and in the subthen these will only change the Voonoi cells locally, and once sequent sections we see how this can be used with existing thei tue misfit values ae geneated they will impove the seach methods such as SA and GA, and also as the basis of local esolution of the next NA-suface. a new seach algoithm. Sambidge (1998) fist suggested this appoximation as a way of sampling a posteio pobability density function fo 3.1 The neighbouhood appoximation to a misfit evaluating Bayesian integals. Hee we use it to appoximate function any misfit function and show how it leads to a new class of seach algoithms fo high-dimensional paamete spaces. Given a set of n samples in model space fo which the misfit p function has been detemined we use the geometical constuct known as the Voonoi diagam ( Voonoi 1908) (see Okabe 3.2 Incopoating the neighbouhood appoximation et al. 1992; Watson 1992; Sambidge et al. 1995, fo full details). into existing methods This is a unique way of dividing the d-dimensional model Ou idealized seach algoithm above may be summaized space into n egions (convex polyheda), which we call Voonoi p as intemittently eplace the eal fowad modelling calcucells. Each cell is simply the neaest neighbou egion about lation with the appoximation povided by the NA-suface. one of the pevious samples, as measued by a paticula Obviously, then, it is independent of the seach algoithm, and distance measue. Hee we use the L -nom, and so the distance 2 it may be incopoated into any existing diect seach method. beween models m and m is given by a b (Note hee that we could not use the NA-suface with any d(m m )d=[(m m )TC 1 gadient-based method since all deivatives ae effectively zeo a b a b M (m a m b )]1/2, (1) o infinite). Hee we biefly discuss how the NA-suface can whee C is a matix that non-dimensionalizes the paamete M be used with seveal popula diect seach methods, namely space. (Fo example, a pio model covaiance matix.) Note genetic algoithms and two foms of simulated annealing. In that the size and shape of the Voonoi cells will depend on the each case we inset eithe SA o GA at step (2) of the idealized choice of C. Its ole is to put each paamete on an equal M algoithm descibed above. footing. A simple choice would be a diagonal matix with elements 1/s2. Hee s can be intepeted as a scale facto fo i i the ith paamete. In this case we can effectively educe C to A genetic algoithm M the identity by escaling each paamete axis by the set of scale Thee ae many vaiants of genetic algoithms and these have factos, s (i=1,, d). In what follows we will assume that i been descibed thooughly in the efeences cited. At each this is the case and dop the C matix. M iteation a new set of n samples is geneated using stochastic A fomal definition of the Voonoi cell follows: Let GA opeatos. This population will often contain new models and P={m,,m } be a set of points of d-space, whee 1 np some which ae identical to pevious models. The use of the 2 n 2, and let m m fo i j. The Voonoi cell about p i j NA-suface hee is quite tivial. One could simply eplace point m is given by i the misfit calculation (equiing fowad modelling) with an V(m )={x dx m d dx m d fo j i, (i, j=1,, n )}. evaluation of the NA-suface fo a fixed numbe of iteations, i i j p say n. Afte this, one would calculate the eal misfit values fo (2) f the latest population, and use these to update the NA-suface. Fig. 1 shows a set of Voonoi cells about 10, 100 and 1000 Afte I iteations one would have solved the fowad poblem GA iegulaly distibuted points in a plane. Since the data misfit I n /(n +1) times, as compaed to I n times if the GA GA f GA GA function is known at all pevious samples, the neighbouhood NA-suface wee not used.

4 482 M. Sambidge Figue 1. (a) 10 quasi-unifom andom points and thei Voonoi cells. (b) The Voonoi cells about the fist 100 samples geneated by a Gibbs sample using the neighbouhood appoximation. (c) Simila to ( b), but fo 1000 samples. (d) Contous of the test objective function. As samples ae geneated, the appoximation to the multi-modal suface becomes moe accuate. Note that all fou maxima of the objective function ae well sampled but the lowe valleys ae pooly sampled. B, (3) Clealy, then, thee is a cost saving. The pice one pays fo to look at how the samples ae geneated. Two methods ae this is the isk that the seaching ability of the algoithm with common: the Metopolis Hastings method (Metopolis et al. the appoximate misfit is less effective than fo that with the 1953; Hastings 1970); and the Gibbs sample (Geman & tue misfit. The degee to which this inhibits the effectiveness Geman 1984). of the GA depends on the natue of the fowad poblem and pobably vaies between applications. On the positive side, T he Metopolis Hastings method: Metopolis Hastings howeve, with the NA-suface one could un a GA fo many (M H) pefoms a Makov-chain andom walk in model space moe iteations [a facto of (n +1) times] fo the same oveall (see Gelfand & Smith 1990; Smith 1991; Smith & Robets f computation time. 1993; Mosegaad & Taantola 1995, fo details). At each iteation the cuent model, m, is petubed andomly, usually A along one of its d paamete axes, accoding to a pobability Simulated annealing distibution, q(m m ). Fo example, in Fig. 2 the petubation B A A simulated annealing algoithm woks by geneating samples, is fom point A to point B. A second (unifom) andom deviate, m, whose distibution in model space is asymptotically equal, is then geneated between zeo and one, and the walk moves to a pescibed sampling density function, S(m): fom A to B if S(m)=exp A w(m) T <min S(x C1, B )q(m B m A ) S(x )q(m m A A B )D, (4) whee T is a tempeatue paamete used to gadually change whee q(m A m ) is the pobability that a walk at B would be B the sampling density function as the algoithm poceeds. petubed to A. Usually q is symmetic in its aguments and The method is commonly used in a Bayesian invesion one obtains the moe familia Metopolis condition on : (see Taantola 1987; Duijndam 1988a,b, fo a discussion), and fo T =1 the function w(m) is often chosen so that sampling density, S(m), is equal to the posteio pobability density <min S(x C1, B ) S(x A )D. (5) (PPD). In pinciple, w(m) can be any data misfit function (fo example see Scales et al. 1992; Sen & Stoffa 1995). To examine how the NA-suface can be used in this case we need If this condition is not met then the walk stays at A and a new petubation is made (along the next axis). This accept/ eject pocedue equies the misfit to be evaluated at the

5 Neighbouhood algoithm: seaching a paamete space 483 computation time of fowad modelling is epesented by T FM, and that of a neaest neighbou seach by T NN, then the time fo geneating each independent model with the NA-suface is T NA =T FM +T NN I d, (6) wheeas the equivalent using fowad modelling is T misfit =T FM I d. (7) This leads to a cost atio of T NA = 1 T I d + T NN. (8) T misfit FM In many poblems we would expect I d&1 and T FM &T NN, and hence one could geneate many moe samples using the NA-suface fo same oveall cost. By setting T NA =T misfit in the above equations, we find that this facto, F, is given by F= A 1 I d + T NN T FM B 1. (9) (Note that, if T =T, then F#1 and hence thee would NN FM be no eal benefit fom using the NA-suface; one might as Figue 2. A Makov-chain Monte Calo walk using the neighbouhood well use the tue misfit.) As in the case of the GA this appoximation to the misfit function. The Voonoi cells (polygons) ae impovement in efficiency comes at the cost of intemittently defined aound the set of pevious samples (black); the sampling density function is constant inside each cell. The cell in which the walk substituting the appoximate misfit fo the tue misfit. Again stats is shaded. The gey line shows the ange of values that can be the usefulness of the NA-suface will depend both on the taken in the fist step of the andom walk which is detemined natue of the fowad poblem and the sample size, n.asn s s pobabilistically accoding to a conditional PDF (plotted above the is inceased the NA-suface is used to geneate moe samples figue). Hee the walk may leave the cell in which it stats and is and so the infomation contained in the pevious models is attacted to the cells whee the data-fit function is high (see text). exploited fo longe. Convesely, as n is deceased the s NA-suface is updated moe apidly and less use is made of the global infomation contained in it. poposed model, B, and so one solution to the fowad poblem is equied fo each paamete axis in tun. A single iteation T he Gibbs sample: The Gibbs sample (also known is completed afte cycling though all d axes, and theefoe as a heat bath algoithm) is pesented in detail in Geman & equies the fowad poblem to be solved d times. Howeve, Geman (1984) and Rothman (1986). In contast to M H this many of the poposed petubation steps may be ejected, and is a one-step method in which a petubation to a model is so the model at the end of the iteation will have only some geneated and always accepted. Fig. 2 gives an example. Again of its components changed. The walk must poceed fo a the stating model is at A, but in this case the petubation numbe of iteations, say I, befoe the Makov-chain elaxes to B is poduced by dawing a andom deviate fom the 1-D and the cuent model becomes statistically independent of the conditional PDF shown above the figue. This conditional stating model, A. The whole pocess can then be epeated fo distibution is poduced by cutting S(m) along the x-axis the next independent model. It is impotant to note that it is though A. In Fig. 2, the NA-suface is used instead of the tue only the statistically independent samples that ae dawn fom misfit function, and so the conditional consists of six segments the desied density distibution S(m), and each one is at a cost fomed fom the intesection of the x-axis and the Voonoi of d I solutions to the fowad poblem. [In pactice, the cells. In each segment the PDF is constant, because the value of I is contolled by the accept/eject atio of the chain, NA-suface is constant inside each Voonoi cell. Afte cycling which is highly dependent on the chaacte of the misfit though all d axes, a new model is poduced, but in contast to function w(m) and the value of T : Rothman (1986); Gelfand M H it will have all components changed and be independent & Smith (1990); Smith (1991); Smith & Robets (1993).] of the pevious model A. As with a genetic algoithm, the M H method can be used To implement the Gibbs sample with the tue misfit function diectly in step (2) of the idealized algoithm. In this case, one usually appoximates the conditional PDF by evaluating howeve, we have a diect statistical intepetation of how it it at n points along the axis, each of which equies a solution woks. At each stage the M H method can be used to daw a to the fowad poblem (Rothman 1986). Theefoe the time n statistically independent samples fom the cuent NA-misfit s fo geneating each statistically independent sample with the suface; that is, the NA-suface is used to eplace w(m) ineq.(3). tue misfit is The impotant point is that this equies only neaest neighbou calculations and no solutions to the fowad poblem. If the T =T n d, misfit FM a (10)

6 484 M. Sambidge wheeas with the NA-suface we have T NA =T FM +T NN n a d. (11) An analysis simila to that fo the M H case leads to a cost atio of T NA = 1 T n d + T NN. (12) T misfit a FM As can be seen hee, the discetization of the axis, n, plays a a simila ole to the elaxation time, I, in the M H method. (In numeical expeiments with a Gibbs sample on the eceive function poblem discussed in Section 5 we had d=24, n =20, and found T /T >1/3000, indicating that fo this a NN FM case the NA-suface poduces consideable savings.) Oveall, the NA-suface with the Gibbs sample esults in a simila efficiency to that fo M H case, and comments and caveats cay ove. Sambidge (1998) pesented some esults of using the NA-suface in conjunction with the Gibbs sample to geneate samples fom successive neighbou appoximations to a misfit suface. It was found that the convegence of the Gibbs sample became excessively slow as the dimension inceased. Howeve, it was incoectly concluded that the NA-Gibbs sample would be impactical fo dimensions geate than 10. Late wok showed that the slow convegence was due to the lack of stuctue in the PDF used in that example. We see, then, that the neighbouhood appoximation can easily be used in combination with any existing diect seach method; howeve, its success will depend on the details of the application. Fig. 1 shows a simple example of seaching a 2-D paamete space using the NA-Gibbs sample fo a test function with T =1/15 and Figue 3. A unifom andom walk esticted to a chosen Voonoi cell. A Gibbs sample is used in a simila manne in Fig. 2; howeve, the conditional PDF outside the cell is set to zeo (plotted above). Asymptotically, the samples poduced by this walk will be unifomly distibuted inside the cell egadless of its shape. chosen Voonoi cells with a locally unifom density. The algoithm can be summaized in the following fou steps. (1) Geneate an initial set of n models unifomly s (o othewise) in paamete space; w(m)= f max f(m), (13) f f (2) Calculate the misfit function fo the most ecently max min geneated set of n models and detemine the n models whee the test function ange is f =2200 and f = s max min with the lowest misfit of all models geneated so fa; independent samples ae geneated at each iteation (n =10, s (3) Geneate n new models by pefoming a unifom n =20). The initial 10 samples ae distibuted unifomly, s a andom walk in the Voonoi cell of each of the n chosen and Voonoi cells ae plotted fo 100 and 1000 samples. As models (i.e. n /n samples in each cell); the algoithm poceeds, the infomation in the NA-suface is s (4) Go to step 2. exploited to concentate sampling in the egions whee S(m) is high [shown in pat (d)]. Note that all fou of the local A NEIGHBOURHOOD ALGORITHM maxima (dakest egions) ae densely sampled, while the toughs in S(m) ( lighte) ae spasely sampled. An example of the The unifom walk within a chosen Voonoi cell can be NA-Gibbs sample in a 24-dimensional space is pesented in geneated using a Gibbs sample, in a manne simila to that Pape II, although thee it is applied to the appaisal athe descibed above. Fig. 3 shows an example. At each step the ith than the seach stage of the invese poblem. component of the cuent model, x, is eplaced with a unifom A andom petubation esticted to the boundaies of the cuent Voonoi cell (l, u ). This is essentially the same as in the i i pevious case (shown in Fig. 2), except that the conditional 4 A NEIGHBOURHOOD SAMPLING ALGORITHM PDF is set to zeo outside the cell. The esulting andom walk asymptotically geneates a spatially unifom distibution of samples with any d-dimensional convex polygon. The philosophy behind the algoithm is that the misfit of each of the pevious models is epesentative of the egion of space in its neighbouhood (defined by its Voonoi cell). Theefoe at each iteation new samples ae concentated in the neigh- bouhoods suounding the bette data-fitting models. In this way the algoithm exploits the infomation contained in the pevious models to adapt the sampling. The pevious section showed how the neighbouhood appoximation can be used to enhance any existing diect seach method. It was only used, howeve to eplace the fowad poblem. In this section we pesent a new diect seach method which uses the spatial popeties of Voonoi cells to diectly guide the sampling of paamete space. The algoithm is conceptually simple and summaized in Fig. 3. The key idea is to geneate new samples by esampling

7 Neighbouhood algoithm: seaching a paamete space 485 Thee ae two impotant featues of the new algoithm. The fist is that the size and shape of the neighbouhoods ae not A bute foce appoach imposed extenally but athe detemined automatically and The intesections can be appoximated using a simple appoach. uniquely by the pevious samples. Note that the boundaies We discetize the axis into n points, and fo each find its a of each Voonoi cell ae detemined by all pevious samples, neaest neighbou amongst the pevious n samples. The p and, egadless of how iegulaly distibuted the samples ae, appoximate position of the intesection is given by a change the neighbouhood will be a easonable one, in the sense of in the neaest neighbou node. If we calculate the distance to an L -nom. The second featue is that the algoithm only each of the n points, then the time fo finding a single neaest 2 p equies models to be assessed fo thei elative fit to the data, neighbou satisfies because it uses only the ank of a misfit/objective function. T 3n d. (14) This is vey useful because in many poblems (e.g. seismic NN p wavefom invesion) it is often much easie to answe the By using eq. (11), we see that the time fo geneating each question is model A a bette fit to the data than model B? independent sample becomes than to quantify the diffeence in a pecise way. With most T =T +l n n d2, (15) NA FM 1 a p seach methods one is foced to define an absolute measue of whee l is a constant of popotionality. In pactice this is data misfit (e.g. the chi-squae o the PPD), which equies 1 vey easy to implement, and, with moden vecto computation, accuate knowledge of the statistics of the data eos. With a may be pactical fo many applications. Howeve, since the ank-based appoach, the weight of each of the pevious intesection poblem is at the vey heat of the algoithm samples in diving the seach depends only on thei position (and will need to be solved many times), an efficient solution in the ank list and not on any analytical estimate of the noise is essential. Note that the atio of the two tems in eq. (15) is statistics. Of couse noise fluctuations still play a ole because, when lage enough, they can influence the anks of the misfit T = l 1 n a n p d2. (16) values. Ultimately, an absolute measue of data fit is always R T FM needed in ode to detemine if any models satisfy the data; Ideally we seek T <1, so that the oveall cost of the algoithm howeve, by using only misfit ank, the seach pocess becomes R is dominated by the time equied to solve the fowad poblem. independent of the absolute scale of the misfit function. Even if this is not the case howeve, the algoithm is still With diect seach methods that do not use ank, one is viable. Note also that, since the cost of geneating a single often foced to escale the misfit function to avoid stability sample depends on all n pevious samples (eq. 15), the cost poblems o loss of efficiency. Fo example, misfit escaling is p fo geneating all samples depends quadatically on n. often used with genetic algoithms to educe exhaustion p poblems (Goldbeg 1989). [We note that misfit ank has also been intoduced into GAs to deal with this poblem (Goldbeg A efined appoach & Deb 1991; Sambidge & Gallaghe 1993).] Similaly, the It tuns out that this simple bute foce appoach can be main ole of the tempeatue paamete (cucial to the pe- impoved upon consideably by using a moe complex method fomance of the simulated annealing algoithm) is to escale which avoids a discetization of the axis and also allows exact the misfit function (cf. eq. 3), theeby contolling the influence intesections to be calculated. If we define the kth Voonoi cell of lage changes in misfit on the seach pocess. A ank-based as the one about sample v, and the point whee the bounday k appoach can be applied to any misfit citeion that can discen between cells k and j intesects the axis as x (see Fig. 3), then j between completing models (even one based on a combination by definition we have of citeia o a set of heuistic ules). d(v x )d=d(v x )d. (17) In defining the neighbouhood algoithm a second contol k j j j paamete, n, has been intoduced. Again the influence of Taking C =I we have M this is quite staightfowad. Fo lage values, the sampling d2+(v x )2=d2+(v x )2, (18) (at each iteation) is spead ove moe cells, and so we would k k,i j,i j j,i j,i whee d is the pependicula distance of sample k fom the expect the algoithm to be moe exploatoy in natue. k cuent axis, and a subscipt of i denotes the ith component Convesely, fo smalle values it is esticted to fewe cells and of the coesponding vecto. Solving fo the intesection point so the sampling should be moe localized. In Section 5 we we obtain examine the influence of this paamete in moe detail. x j,i = 1 2 C v k,i +v j,i + (d2 k d2 j ) (v k,i v j,i )D. (19) 4.1 Sampling Voonoi cells in a high-dimensional space If we ignoe the calculation of d2 and d2 fo the moment then k j It tuns out that, in ode to implement the neighbouhood this expession equies just six aithmetic opeations to algoithm, full details of the high-dimensional Voonoi diagam evaluate. To find the equied boundaies of the Voonoi cell, do not have to be detemined (which would be an impossible eq. (19) must be evaluated fo all n cells and the two closest p task). As can be seen fom Fig. 3, all that is equied is to find points eithe side of x etained. Moe fomally, we have the A the points whee the boundaies of the d-dimensional Voonoi lowe bounday given by cell intesect the ith axis passing though a given point x. A max[l,x ], (fo x x ;j=1,, n ), (20) The next step of the unifom andom walk is then esticted i j,i j,i A,i p and the uppe bounday by to lie between these two points on the axis (i.e. x and x j l in Fig. 3). min[u,x ], (fo x x ;j=1,, n ), (21) i j,i j,i A,i p

8 486 M. Sambidge whee l and u ae the lowe and uppe bounds of the i i paamete space in the ith dimension, espectively. Afte 5.1 Paametization and synthetic eceive functions cycling ove all dimensions a new sample in paamete space The custal stuctue is divided into six hoizontal layes, is geneated, and the time taken fo each satisfies which we name Sediment, Basement, uppe, middle and lowe Cust, and Mantle. The model compises of fou T =T +l n d, (22) NA FM 2 p paametes in each laye: the thickness of the laye (km), S velocity at the topmost point in the laye (km s 1), S velocity which gives a cost atio at the bottommost point in the laye (km s 1), and the atio of P and S velocity in the laye. A linea gadient in velocity T = l 2 n p d, (23) is assumed in each laye. The philosophy behind the paa- R T FM metization is to be as flexible as possible, allowing a vey wide ange of eath models to be epesented by a finitewhich in pactice will be a consideable impovement ove the dimensional (d=24) paamete space. Theefoe vey loose bute foce appoach since a facto of n d has been emoved a pio bounds ae placed on each paamete. These ae appaent (cf. eq. 16). in the figues below and also act as the scale factos equied To complete the desciption we need the set of squaed to non-dimensionalize the paamete space. (Numeical values pependicula distances, d2( j=1,, n ), available at each j p and all othe details can be found in Shibutani et al. 1996). step of the walk. These can be calculated fo the initial axis A synthetic eceive function (RF) was calculated fom [an O(dn ) calculation], and fo each new axis by a ecusive p the tue model using the Thomson Haskell matix method update pocedue. Fo example, afte the ith step has been (Thomson 1950; Haskell 1953), and unifom andom noise completed and the walk moves fom x to x, the cuent set A B was added in the fequency domain (with a noise-to-signal of d2 values can be calculated fo the (i+1)th axis using j atio of 0.25). This esulted in the obseved RF shown in (d2) =(d2) +(v x )2 j i+1 j i j,i B,i Figs 5(c),(d) etc. (gey line). A chi-squae misfit function was used to measue the discepancy between the tue, wobs(m), (v x )2 j,i+1 B,i+1 fo (j=1,, n ). p (24) and pedicted, wpe(m), wavefoms fom an eath model m: The cost of this pocedue fo each model also depends linealy on n d, and so only the multiplicative constant in (22) is p changed, and oveall the time taken will be linea in both the numbe of pevious points, n, and dimension, d. This com- p pletes the desciption of the method. In the next section we pesent a numeical example of its application to the poblem of inveting seismic eceive functions fo custal seismic stuctue. In this case we found that the total time taken fo solving the fowad poblem fo all models was a facto of 14 times geate than the total time of geneating all of the samples (i.e. T =1/14). R 5 EXAMPLES To illustate the neighbouhood algoithm we examine its pefomance in the invesion of eceive functions fo custal seismic stuctue. This is a highly non-linea wavefom-fitting poblem, exemplifying many of the difficulties of seismogam invesion. Typically, eceive functions contain both highamplitude evebeation phases, due to nea-suface stuctue, and much smalle-amplitude conveted phases fom the Moho. It is well known that when lineaized techniques ae applied to eceive function invesion they can have a stong dependence on the assumed stating model (Ammon et al. 1990). It is theefoe a suitable test poblem on which to illustate the algoithm. We use a synthetic data poblem so that the tue model is known in advance and can be compaed with the esults of the algoithm. To be as ealistic as possible we use exactly the same paametization of custal stuctue as was used by Shibutani et al. (1996). In that study, a genetic algoithm was applied to the invesion of eceive functions ecoded in easten Austalia. We also apply this genetic algoithm to ou synthetic eceive functions and compae the pefomance with that of the neighbouhood algoithm. x2(m)= 1 n n N d Awobs wpe i i s i=1 i B 2, (25) whee s is an estimate of the standad deviation of the noise i calculated fom wobs using the appoach descibed by Gouveia & Scales (1998), and n is the numbe of degees of feedom (n#n d). (Note that all off-diagonal tems of the data d covaiance matix ae ignoed, and so this misfit measue does not take into account any tempoal coelation in the noise.) The tace was sampled at a fequency of 25 Hz with 30 s duation, giving a total of N =876 data points. d 5.2 Seaching fo data-fitting models Fig. 4 shows the impovement in data fit (of the best model) in thee tials with a neighbouhood and a well-tuned genetic algoithm stating fom vaious initial (andom) populations. (The details of the genetic algoithm ae exactly those of Shibutani et al ) At each iteation the neighbouhood algoithm geneates 10 samples of a unifom andom walk inside each of the Voonoi cells of the cuent two best models (i.e. n =20, n =2). These values wee selected afte a few s tials with n in the ange and n in the ange No s exhaustive testing was done and we do not expect the values to be in any way optimal. The tuning of the genetic algoithm was accomplished with moe cae (and not by the autho) (Shibutani, pesonal communication). Ou pimay inteest in Fig. 4 is to compae the chaactes of the misfit eduction in the two cases, athe than the paticula values of data misfit achieved. (The latte will no doubt vay between applications and with the pecise details of the algoithms.) Notice how the GA cuves in Fig. 4(a) consist of a lage numbe of small changes and a few lage steps which gadually diminish. In contast, the thee NA cuves (Fig. 4b) all have a staicase -like featue eminiscent of the ealy stages of a unifom Monte Calo seach (shown dashed); howeve, unlike the case fo unifom sampling, the

9 Neighbouhood algoithm: seaching a paamete space 487 small changes in the NA cuves suggests that it exhibits no local seach chaacte; athe, it is pefoming unifom seaches in eve-deceasing volumes of paamete space. The impotant point is that the size and shape of these sampling egions ( Voonoi cells) ae completely detemined by the pevious samples, and also evolve as the algoithm poceeds. 5.3 Self-adaptive sampling in paamete space Examining the eduction in misfit of the best model gives only limited infomation on the chaacte of the seach pocess. Figs 5 and 6 show the distibution of S-wave velocity models fo the two bette uns in the GA and NA, togethe with the eceive functions of the best model found in each case. The most stiking featue of these figues is the diffeence in the sampling density of the best 1000 models between the GA and NA esults. In the GA case, both density plots (Figs 5a and b) show a vey localized sampling with high concentation in some places and holes in othes. This implies localized iegula sampling in paamete space. The density of the 1000 NA models is much smoothe in both cases. Again this is moe eminiscent of unifom sampling; howeve, the seach has achieved levels of misfit eduction that ae compaable to those fo the GA seach. The anges of models geneated (shown as black outlines) ae effectively at the boundaies of the paamete space, and ae simila in all cases. A compaison of the best fit model in each case (white line) with the tue model (black line) shows that fo both GA and NA the depth and velocity jump acoss the Moho ae ecoveed easonably well. This esults in the P-to-S convesion at 5 s being fit quite well in all cases, although the NA model in Fig. 6(a) has a slightly deepe Moho esulting in a slightly late conveted phase (see Fig. 6c). The NA models seem to ecove the depth and jump acoss the basement bette than the GA, esulting in bette alignment of dominant phases in the evebeations (0 3 s). This may account fo the lowe oveall data fit of the GA models. Figs 6(e) and (f ) show esults fom a sepaate NA un with paametes n =2, n =1. This is included because it s (fotuitously) povides an example whee the best fit model Figue 4. (a) x2 data misfit function plotted against the numbe of found has exactly the same numeical value as one of the GA models (poduced by a genetic algoithm) fo which the fowad esults (see GA3 in Figs 5b and d). This is useful because poblem has been solved. The thee cuves ae poduced by diffeent it allows a diect compaison of sampling between the two stating andom numbe seeds. ( b) As (a), fo thee uns of the algoithms without the complication of diffeences in sampling neighbouhood algoithm. In both panels the coesponding cuve leading to diffeences in the fit of the best model. The NA fom a unifom Monte Calo seach is plotted fo efeence (solid). sampling is again much moe distibuted and unifom than Notice how the GA cuves consist of a lage numbe of small changes the GA case, which confims the pevious conclusion that the while the NA cuves look moe like the ealy stages of the unifom moe distibuted sampling of the NA has not been at the Monte Calo seach. Afte models have been sampled, two of sacifice of impoving the fit of the best model. A compaison the thee NA cuves have lowe data misfits than the best GA cuve. of the two best fit models shows that the NA model (Fig. 6e) is simila at the Moho but distinctly diffeent fom GA3 steps continue as the iteations poceed, and esult in easonable well fit best models (x2 values of 1.42, 2.04 and 1.44). pefect fit to the tue eceive function in the high-amplitude n in the evebeation layes nea the suface [giving a nea Afte models have been geneated, the unifom sampling (0 1 s) pat of the tace]. This also suggests the pesence of gives a x2 value of 3.75 (a poo data fit), and two of the multiple minima in the misfit suface (as one might expect in n thee NA cuves have lowe misfits than the best GA cuve a wavefom-fitting poblem). (x2 values of 1.79, 2.51 and 1.69), although we do not feel that One might ague that the appaently spase sampling of the n this is paticulaly significant. NA is a disadvantage because it will not concentate as much The chaacte of the misfit eduction is moe intiguing. In sampling in the egion of paamete space about the tue the GA case the small changes suggest that a local seach model, compaed with a GA pehaps. To examine this question mechanism is pesent (consistent with pevious speculation on we plot the entie ensemble of models poduced by NA1 and GAs; Sambidge & Dijkoningen 1992). The absence of these GA1 pojected onto fou pais of paamete axes (see Fig. 7).

10 488 M. Sambidge Figue 5. (a) Density plot of the best 1000 data-fitting S-velocity models geneated by the GA with the fist andom seed (GA1). The best datafitting model is plotted in white with a black outline, and the tue model is in black. The dake shading epesents bette fit to the data. The outline epesents the extemes of all models geneated. (b) As (a), fo un GA3. (c) and (d) Receive functions of the tue (gey) and best fit ( black) models fo GA1 and GA3, espectively. In both cases the density of the models is highly iegulaly distibuted, showing concentations and holes about the best model.

11 Neighbouhood algoithm: seaching a paamete space 489 Figue 6. (a) Model density plot fo models poduced by the neighbouhood algoithm with the fist andom seed (NA1) and paametes n s =20, n =2. Details ae the same as fo Fig. 5. (b) As (a), fo un NA3. (c) and (d) Receive functions of the tue (gey) and best fit (black) models fo NA1 and NA3, espectively. (e) and (f ) Show the esults using contol paametes n s =2, n =2. Note that the final data fit is exactly the same as fo GA3.

12 490 M. Sambidge Figue 6. (Continued.) Immediately one sees the effect of the discetized paamete space used by the GA (left panels). All GA samples fall on a elatively cude gid. (Note that the level of discetization used by the GA is typical of poblems of this kind, and had to be limited in ode to stop the GA sting becoming too long; Shibutani et al. 1996). The holes in the GA ensemble ae also appaent, even close to the best fit model (coss). It is well known that the GA can poduce an ensemble containing models that ae patial o full copies of each othe, which will mean that many models plot on top of each othe in the left panel. The NA woks with a continuum and poduces a divese cloud of samples with sampling concentated in the neighbouhood of the best fit model (which, in this case, is on aveage close to the tue model, maked by an ). It is inteesting to note that even though each cloud in Fig. 7 is a pojection fom 24 to two dimensions, thee is a fai amount of consistency between the data fit of the model and its position elative to the cente of each cloud (dake shades indicate bette data fit). Futhemoe, the sampling density seems to incease close to the best fit model, indicating that the algoithm has adapted the sampling to concentate in egions of bette data fit. We ague hee that the divesity of the NA ensemble is likely to be moe useful in the appaisal stage of the invese poblem than that poduced by the GA, since it might bette chaacteize the egion of acceptable models. (This issue is dealt with in detail in Pape II.) Fig. 8 shows a simila scatte plot whee all models have been plotted as a function of Moho depth and velocity jump acoss the Moho. (These vaiables ae linea combinations of the oiginal model paametes.) One can see that the tue values (cosses) ae epoduced well by the best fit model and the NA ensemble povides thoough sampling about the tue values. We conclude fom this that the data contain easonable infomation on both the depth and jump acoss the Moho. 5.4 Popeties of NA sampling In these examples only a small numbe of Voonoi cells wee esampled at each iteation (n =1 o 2). Since a small value of n suggests a moe estictive seach (i.e. moe exploitation than exploation), one might suspect that this would seveely inhibit the sampling of the NA, pehaps diecting it away fom a global minimum towads seconday minima. Fig. 9 was poduced in an attempt to examine this question, and, it tuns out, allows us to obseve a emakable popety of the NA sampling. Hee we plot, as a function of iteation, the aveage distance in model space between the cuent best fit model and the set of models most ecently geneated inside its Voonoi cell. Fo the NA this is an aveage ove n /n models, s and gives a measue of the size of the Voonoi cell about the cuent best fit model. We also plot the same measue fo the GA and MC esults fo efeence (although in these cases the cuve has no simple intepetation). As the iteations poceed we see (fom Fig. 9) that the size of the Voonoi cell about the best fit model gadually deceases and then inceases in a spike-like featue only to decay again. This patten is epeated seveal times. The eason fo this becomes clea when we compae the cuve to the coesponding misfit eduction cuve NA1 in Fig. 4(b) (this cuve is shown in Fig. 9 in light gey). The spikes in Fig. 9 occu at exactly the same points as whee a new best fit model is found. Theefoe the gadual shinking of the Voonoi cell is a esult of moe and moe samples being geneated within it. (We ecall that at each iteation the Voonoi cells ae updated, and so they must shink as space is taken up by new cells.) Each

13 Neighbouhood algoithm: seaching a paamete space 491 Figue models poduced by the NA pojected onto the axes epesenting the S-velocity jump acoss (x-axis), and depth of, the Moho ( y-axis). Symbols and shading as in Fig. 7. The plot ange is the complete paamete space (11 to 60 km fo Moho depth, 0.5 to 1.8 km s 1 fo velocity jump). The best fit and tue models ae elatively close. The density of the samples is inceased in the egion aound the tue solution whee data fit is high (dake shades). Figue 7. Compaison of ensembles poduced by NA (ight panels) and GA (left panels) pojected onto fou pais of paamete axes (labelled). The best of the models in each panel is shown as a coss and the tue model as a. The dots ae shaded by data fit, and a dake shade indicates a highe data fit. The GA left panel poduces many copies of each model on a cudely discetized axis and yet still has holes. In each case the NA poduces a continuum of samples with a highe concentation of bette-fitting models in the egion of the tue model. spike in Fig. 9 occus when the sampling switches to a new Voonoi cell about a new best fit model, and this cell is much lage than the pevious one. This illustates the emakable popety of the NA efeed to above; that is, the local sampling density (invese of Voonoi size) automatically inceases and deceases (in a pulsating-like fashion) as new cells ae sampled. Futhemoe, the centes of the sampling must also change, because they ae esticted to a new set of n Voonoi cells whose centes diffe fom the old ones. Note that each new best fit model is not necessaily poduced by sampling the pevious best fit Voonoi cell, but may have come fom any of the n cells. Hence the centes of sampling may jump fom place to place. Note also that as the best model is eplaced with a bette one, the old cell is not necessaily discaded, since it meely moves down the ank list. We see then that the NA is always sampling the cuently most pomising n egions in model space simultaneously. A futhe question aises fom these plots. How can each new Voonoi cell in paamete space be lage than the oiginal cell, if its defining point in paamete space is inside the oiginal cell? The answe can be illustated with a simple 2-D test case (see Fig. 10). Fig. 10(a) shows a Voonoi cell (shaded) and the set of new samples geneated within it (open cicles). Afte updating the Voonoi cells we see that all six of the new Voonoi cells extend beyond the oiginal (shaded) cell. Theefoe, even though the sample itself lies inside the oiginal cell, its Voonoi cell can extend beyond the oiginal Voonoi cell. It tuns out that the likelihood of this occuing damatically inceases with dimension, and also the new cell is moe likely to occupy a lage egion of paamete space. (This effect is discussed in moe detail by Sambidge 1998.) 5.5 Tuning the fee paametes As mentioned above, the two fee paametes (n, n ) used in s the examples wee obtained afte tials ove a ange of values. Robust conclusions ae not possible fom these limited tests, and no doubt details will vay between applications. Fom