Geophysical Journal International

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1 Geophyscal Journal Internatonal Geophys. J. Int. (2012) 190, do: /j X x Subevent locaton and rupture magng usng teratve backprojecton for the 2011 Tohoku M w 9.0 earthquake Huajan Yao, Peter M. Shearer and Peter Gerstoft Scrpps Insttuton of Oceanography, Unversty of Calforna, San Dego, 9500 Glman Drve, La Jolla, CA 92093, USA. E-mal: huyao@ucsd.edu Accepted 2012 May 7. Receved 2012 February 21; n orgnal form 2011 November 16 GJI Sesmology SUMMARY Knowledge of the rupture speed and spatal temporal dstrbuton of energy radaton of earthquakes s mportant for earthquake physcs. Backprojecton of telesesmc waves s commonly used to mage the rupture process of large events. The conventonal backprojecton method typcally performs temporal and spatal averagng to obtan relable rupture features. We present an teratve backprojecton method wth subevent sgnal strppng to determne the dstrbuton of subevents (large energy bursts) durng the earthquake rupture. We also relocate the subevents ntally determned by teratve backprojecton usng the traveltme shfts from subevent waveform cross-correlaton, whch provdes more accurate subevent locatons and source tmes. A bootstrap approach s used to assess the relablty of the dentfed subevents. We apply ths method to the M w 9.0 Tohoku earthquake n Japan usng array data n the Unted States. We dentfy 16 relable subevents n the frequency band Hz, whch mostly occurred around or west of the hypocentre n the downdp regon. Analyss of Tohoku aftershocks shows that depth phases can often produce artefacts n the backprojecton mage, but the poston and tmng of our man shock subevents are nconsstent wth depth-phase artefacts. Our results suggest a complcated rupture wth a component of blateral rupture along strke. The domnant energy radaton (between 0.2 and 1 Hz) s confned to a regon close to the hypocentre durng the frst 90 s. A number of subevents occurred around the hypocentre n the frst 90 s, suggestng a low ntal rupture speed and repeated rupture or slp near the hypocentre. The rupture reached the coastal regon about 106 km northwest of the hypocentre at 43 s and the regon about 110 km north of the hypocentre at 105 s wth a northward rupture speed 2.0 km s 1 at s. After 110 s, a seres of subevents occurred about km southwest of the hypocentre, consstent wtha3kms 1 along-strke rupture speed. The abundant hgh-frequency radaton n the downdp regon close to the coast suggests ntermttent rupture probably n the brttle ductle transton zone. The lack of hgh-frequency radaton n the updp regon suggests the rupture near the trench was more contnuous, probably due to more homogeneous frctonal propertes of the shallow slab nterface. The lack of early aftershocks n the updp regon ndcates that most of the accumulated slp n the updp regon durng the ntersesmc perod was probably released durng the man shock. Key words: Earthquake dynamcs; Earthquake source observatons; Subducton zone processes; Asa. 1 INTRODUCTION An mportant sesmc problem s understandng the rupture processes of large earthquakes. Conventonal (tme-doman) backprojecton of telesesmc P waves, startng wth Ish et al. (2005), s a popular and effcent method to determne the rupture extent, duraton and speed of large earthquakes (Walker et al. 2005; Ish et al. 2007; Walker & Shearer 2009; Xu et al. 2009; D Amco et al. 2010; Kser & Ish 2011; Zhang & Ge 2010). Backprojecton methods are smlar to mgraton technques n reflecton sesmology n that they use smple ray theory to compute tme-shfts for stackng sesmograms at canddate source locatons. They are related to tme-reversal (or reverse-tme) magng of sources or reflectors (e.g. Fnk 1992; Ekström et al. 2003; Larmat et al. 2006), but are faster to compute because they do not need to solve for the complete Green s functons for the sesmograms. Instead, the backprojecton approach approxmates the Green s functon smply as a tme delay that does not change the ampltude or shape of the waveforms. As such, backprojecton works best for uncomplcated portons of body-wave traveltme curves that do not suffer from trplcatons, 1152 C 2012 The Authors Geophyscal Journal Internatonal C 2012 RAS

2 Iteratve backprojecton of Tohoku earthquake 1153 large ampltude changes or sgnfcant dsperson. Fortunately, mantle P waves at epcentral dstances between 30 and 95 meet these crtera and nclude a large fracton of observed sesmograms. Emprcal correctons for the small tme-shfts caused by 3-D mantle structure along the source-to-recever ray paths can be obtaned usng waveform cross-correlaton on the frst arrvng waveforms, whch orgnate from the hypocentre (Ish et al. 2005). These advantages make backprojecton a valuable near-real-tme technque to analyse large earthquake rupture processes. Backprojecton provdes nformaton on hgh-frequency radaton, whch s a useful complement to other approaches to study earthquakes, for example, sesmc slp nversons (J et al. 2002; Uchde & Ide 2007), whch use local or telesesmc low-frequency waveforms to nvert for the detaled slp dstrbuton, rupture speed and moment rate functons at one or multple fault planes. The resoluton of the conventonal backprojecton method s lmted by the frequency range of the data and the array geometry used. Low-frequency P waveforms (f < 0.1 Hz) are less affected by scatterng and small-scale heterogenety and therefore are easy to stack coherently. However, ther relatvely long wavelengths result n poor lateral resoluton for backprojecton source magng. On the other hand, hgh-frequency data (f > 1 Hz) should be able to acheve good resoluton consderng ther short wavelengths, but ther waveforms are more easly dstorted by small-scale heterogenety and scatterng, whch results n less coherent waveform stacks and poorer source magng. So far, the most successful frequency band for backprojecton studes usng telesesmc P waves s Hz (e.g. Ish et al. 2005; Walker & Shearer 2009; Xu et al. 2009). In ths band the lateral resoluton and waveform coherence are well balanced to produce optmal source mages. Array geometry s another mportant factor that controls the lateral resoluton of backprojecton mages. In theory, global statons should gve better resoluton due to ther superor azmuthal coverage. However, n practce regonal arrays seem to work better. Ths s probably because waveforms from regonal arrays are less affected by 3-D heterogenety, waveform complextes (e.g. depth phases), and source effects (e.g. due to focal mechansm) and therefore more coherent for stackng than the more dstrbuted global statons. Results from a sngle regonal array sometmes show smearng of source mages n certan drectons, whch depends on the geometry of the array (Ish et al. 2005; Xu et al. 2009). Thus, combnng dfferent regonal arrays can often acheve better resoluton and less smearng of the source mages (Xu et al. 2009; Kser & Ish 2011). The depth resoluton of backprojecton mages usng only the telesesmc P phase s very poor due to nearly vertcal take-off rays n the source regon (Xu et al. 2009). However, Kser et al. (2011) recently showed that combnng depth phases (.e. pp or sp) wth the drect P phase n the backprojecton method can provde good depth resoluton for deep sources n whch the depth phases are vsble. Although backprojecton has been successful n magng the gross rupture propertes of bg earthquakes, ts ablty to resolve small-scale detals of the rupture process s lmted by the relatvely coarse resoluton of the backprojecton operator, the rapd oscllatons n the raw backprojecton tme-seres and the presence of sweepng artefacts n backprojecton mages, whch propagate towards the statons at hgh speed (.e. the P-wave phase velocty at the recordng statons). The conventonal backprojecton method frst obtans the waveform stack, whch s a tmeseres, at each grd locaton n the source regon. To obtan stable estmates of energy radaton, the raw waveform stack s averaged over a smoothng nterval that reduces the temporal resoluton but removes some mage artefacts. The duraton of ths tme wndow s typcally about s (Ish et al. 2005; Xu et al. 2009) and sometmes both space and tme averagng are performed to suppress the sweepng artefacts (Walker & Shearer 2009). Rupture velocty s then typcally estmated by selectng locatons and tmes of peak energy radaton along the fault (e.g. Ish et al. 2005). However, ths tme averagng may lmt the subevent detals that can be resolved, such as ther number, duraton, locaton and tme. Our approach here s to consder more explctly the possblty that the rupture can be modelled as a seres of subevents (large energy bursts), rather than a smooth contnuous rupture. Our method apples a waveform cross-correlaton and sgnal strppng technque to teratvely dentfy subevents from the backprojecton results. The tme-shfts from the waveform cross-correlaton can then be used to locate the subevents more precsely than can be obtaned from the backprojecton mages. Once a subevent s found, ts prncpal waveforms are determned from prncpal component analyss and then subtracted from the sesmograms. The resdual sesmograms are then used to dentfy the next largest subevent. In ths way, the waveforms are modelled as a seres of subevents (defned by ther tmes, locatons and ampltudes), whch can be used to characterze the rupture propertes and dynamcs of the earthquake. The dea of decomposng a bg earthquake nto a number of subevents has a long hstory n sesmology. For example, Wyss & Brune (1967) used sx subevents to explan the pulses n telesesmc P records of the 1964 Alaska earthquake. Recently, Tsa et al. (2005) appled an teratve multple-source centrod-moment-tensor (CMT) analyss (nverson) to the 2004 Sumatra earthquake and dentfed fve subevents to ft the long-perod waveforms. Our tmedoman sgnal strppng technque shares some smlar features to the mode-branch strppng technque for measurng surface wave overtone phase veloctes (Van Hejst & Woodhouse 1997). It s also analogous to the CLEAN algorthm used n the frequency doman n rado astronomy (Högbom 1974) and for multmode surface wave dsperson analyss (Nolet & Panza 1975). Because our method s largely emprcal, t s dffcult to address questons of unqueness and resoluton n our results formally. As a substtute, we perform both bootstrap resamplng tests of our data and synthetc tests to evaluate how well we can recover varous scenaros of subevent dstrbutons and tme hstores. Ideally, wth good data coverage n azmuth and dstance, our method works f the subevents are separated n ether space and tme. However, n practce usng a regonal array wth a lmted aperture, our method works best when the subevent waveforms are well separated n tme, n whch case n prncple we obtan better tme and space resoluton than conventonal backprojecton. Our method has trouble when the subevent waveforms overlap substantally n tme at most statons. In ths case, other methods, such as the frequency-doman approach of MUSIC (Schmdt 1986; Meng et al. 2011) and compressve sensng (Yao et al. 2011) work better n smultaneously resolvng multple subevents, but at the cost of poorer tme resoluton than our sgnal strppng approach. To llustrate the method, we use telesesmc P-wave data recorded n the Unted States from the 2011 Tohoku M w 9.0 earthquake n Japan. In Secton 2, we provde the basc theory and mathematcal formulaton of the backprojecton method for a large earthquake consstng of multple subevents. The conventonal backprojecton method and the proposed teratve backprojecton method are llustrated n Sectons 3 and 4, respectvely. Secton 5 gves a method to relocate the subevents obtaned from teratve backprojecton. Secton 6 shows the results of the spatal and temporal dstrbuton of subevents of the Tohoku earthquake, whch are used to estmate Geophyscal Journal Internatonal C 2012 RAS

3 1154 H. Yao, P. M. Shearer and P. Gerstoft the rupture speed of ths earthquake. In Secton 7 we show some results of synthetc tests, the effects on subevents from depth phases, the relablty of the dentfed subevents from bootstrap approaches and compare our results wth other backprojecton results and slp nverson results. 2 BASIC THEORY OF BACKPROJECTION Durng the rupture of a large earthquake, sesmc energy s radated from the source regon and recorded by sesmometers around the globe. If we treat large energy bursts as subevents durng the man shock, the record at each staton can be vewed as a sum of waveforms generated by subevents. The receved sgnal V (t) (usually a velocty sesmogram) at the th staton due to N s subevents can be modelled as N s V (t) = a k (t) G(x k, x, t) + n (t), (1) where t s the recordng tme, a k (t) sthekth subevent source tme functon, G(x k, x, t) s the Green s functon from the source locaton x k to the recever locaton x and n (t) represents random nose. To determne the subevents, we assume that the Green s functons are manly due to one wave type (e.g. drect P waves), therefore, G(x k, x, t) = G k δ(t t k ) + g(x k, x, t), (2) where G k s the ampltude of the domnant wave type, t k = t(x k, x ) gves ts traveltme from x k to x,andg(x k, x, t) s the Green s functon of other mnor phases (ncludng scatterng) wth ampltude much smaller than G k. Eq. (1) s then modfed to [ N s Ns ] V (t) = G k a k (t t k ) + a k (t) g(x k, x, t) + n (t) N s = G k a k (t t k ) + V R (t), (3) where V R (t) s called the resdual waveform. We assume that all subevents are close enough relatve to the propagaton dstance (e.g. at telesesmc dstances) so the ampltudes of the subevent Green s functon may be approxmated as constant (G k = G ) and therefore N s V (t) = G a k (t t k ) + V R (t). (4) We assume that each subevent has startng tme t (k), duraton δt (k) and ampltude C k, { Ck a k a k (t) = (t) for t (k) < t < t (k) + δt (k), (5) 0 otherwse where a k (t) s a normalzed tme hstory (max{a k (t)} =1). Hence, eq. (4) becomes N s V (t) = G C k a k (t t k) + V R (t). (6) Each trace s then normalzed wth ts maxmum ampltude C max = max {V (t)} so that N s v (t) = C k a k (t t N s k) + v R (t) = v (k) (t) + v R (t), (7) where v (t) = V (t)/c max s the normalzed waveform, C k = G C k /C max, v R (t) = V R (t)/c max s the normalzed resdual waveform that ncludes waveforms of mnor phases and nose and v (k) (t) = C k a k (t t k) s the normalzed kth subevent waveform. The backprojecton mage s smply a slant stack of traces along some traveltme curve, whch approxmates the source tme hstory at each model pont as the weghted sum of the data ponts t affects. In a general case wth consderaton of other mnor phases and nose (eq. 7), the source tme hstory at grd locaton x m s approxmated by backprojecton (slant stack) of all N normalzed sesmograms: s(x m, t) = 1 N N v (t + t m ) = 1 N =1 + 1 N N N s C k a k (t + t m t k ) =1 N =1 v R (t + t m ), (8) where t gves the source tme. When the potental source locaton x m s at a subevent locaton x k,thats,t m = t k,thewaveforms due to the domnant phase wll stack coherently only at the source locaton x k (frst term n eq. 8) and the other mnor phases (wth dfferent traveltme moveout) and nose (second term n eq. 8) wll stack ncoherently, so s(x k, t) C k a k (t) (9) approxmates the source tme hstory at source locaton x k.these methods thus allow for nvestgaton of multple subevents n both space and tme from backprojecton. 3 CONVENTIONAL BACKPROJECTION In ths secton, we demonstrate the conventonal backprojecton method usng the frst 170 s of vertcal component waveforms of the M w 9.0 Tohoku earthquake, whch contan manly telesesmc P-wave data, as recorded by about 500 statons n the western and central Unted States (ncludng 370 USArray statons). The statons are at dstances of from the epcentre. Waveform data are frst fltered to a relatvely broad band ( Hz) and the ampltude of each trace s normalzed to unt peak. To obtan the coherent stack at the subevent locaton (eq. 8), we need to know the traveltme t k from the source locaton x k to each recever at x. The traveltmes are obtaned from a 1-D Earth model, wth emprcal correctons for 3-D structure computed from waveform cross-correlaton of the ntal part of the P wave. The waveforms are frst algned after correctng for the predcted P-wave propagaton tme t 0 from the hypocentre locaton [lat 38.19, lon , depth 21 km; from Chu et al. (2011)] to the staton usng the IASP91 1-D earth reference model (Kennett & Engdahl 1991). However, due to 3-D heterogenety of the Earth structure, the P waves are not perfectly algned, and addtonal tme correctons must be appled to ensure a coherent stack. Ths s usually acheved by cross-correlatng the frst few seconds of the P waves to determne the addtonal tme-shfts for each sesmogram. Here we apply a multchannel cross-correlaton method (VanDecar & Crosson 1990) and clusterng analyss (Romsburg 1984) for the frst 8 s of the P waves. Sesmograms from the largest cluster and wth correlaton coeffcent above 0.6 are lnearly stacked to generate a reference stack (the black trace n Fg. 2). The reference stack s then correlated aganst each sesmogram (stll for the frst 8 s of the P waves) to obtan the correlaton coeffcent and polarty wth respect to the reference stack. Then sesmograms wth correlaton coeffcents above 0.6 and postve polarty are stacked to generate Geophyscal Journal Internatonal C 2012 RAS

4 Iteratve backprojecton of Tohoku earthquake 1155 Fgure 1. Epcentre of the 2011 Tohoku earthquake (red star) and the statons (blue trangles) n the western and central Unted States used for the teratve backprojecton. Fgure 2. (a) Top panel: frst 170 s of the vertcal-component waveform (0.2 1 Hz bandpass-fltered and normalzed) recorded by statons n the western and central Unted States (Fg. 1). Bottom panel: the stack s(x, y, t) (eq. 11, black) of all the traces and the tme-averaged stack power P(x, y, t) (eq. 12, red) at the hypocentre (x, y) = (0, 0). The two vertcal lnes ndcate the frst 8 s of the P-wave wndow for the ntal waveform cross-correlaton and algnment usng a broader frequency band ( Hz). The traces are sorted n an azmuth-ascendng order. (b) Frst 15 s of the P-wave wndow after algnment. the next-generaton reference stack (Ish et al. 2007). The process s repeated a few tmes to obtan the stable addtonal tme-shfts t 0 and polarty nformaton (p =±1, due to focal mechansm) wth respect to the reference stack for each sesmogram. For all subsequent analyss we use only data from statons that have ntal P waves wth postve polarty (relatve to the reference stack). The dstrbuton of statons s shown n Fg. 1. The selected trace s then Hz bandpass fltered and normalzed, denoted as v (t) atthe th staton ( = 1,..., N = 476). The fnal algned P waveforms u (t) (Fg. 2) wth respect to the hypocentre event are defned as ( ) u (t) = v t + t 0 + t 0, (10) where t s the source tme. The traveltme perturbaton t 0 accounts for the heterogenety along the ray path from the hypocentre to the staton. If we assume the source regon (earthquake rupture area) s small compared to the epcentral dstances and ts structural varaton s not large, t 0 should also approxmate the traveltme shft wth respect to the 1-D reference model from other grd locatons n the source regon to the staton due to very smlar paths. Therefore, the traveltme from each source locaton x to staton at x s gven by t (x) = t P (x)+ t 0 where t P (x) s the reference traveltme from x to x usng the 1-D reference model. For conventonal backprojecton usng a lnear stack, the sesmograms are summed to make the stack s(x, t) as a functon of the, source tme t and the source locaton x = (x, y, z): s(x, t) = s(x, y, z, t) = = N =1 N p w v (t + t (x)) =1 p w u ( t + t P (x) ), (11) where p and w are the polarty and ampltude weght of the th sesmogram, and t P (x) = t P (x) t 0. The ampltude weghts are normalzed such as N =1 w = 1. The weght w can be determned usng a staton-weghtng scheme from the spatal densty of statons (Walker et al. 2005; Walker & Shearer 2009). In ths study we smply set w = 1/N consderng the relatvely even dstrbuton of the array statons. Here we defne the hypocentre locaton at x = (x, y, z) = (0, 0, H), where H = 21 km s the focal depth. For ths study the coordnates of all other locatons n the source regon for backprojecton are wth respect to the hypocentre locaton, wth x and y beng postve to the east and north of the hypocentre. For telesesmc waveform backprojecton only usng P waves, the depth resoluton s poor (Xu et al. 2009), thus we only backproject waveforms on the 2-D plane at the focal depth. From here on, we drop z n all expressons for smplcty. Snce s(x, y, t) s a stacked waveform, t cannot be drectly used to represent the source power. We often compute a runnng-wndow tme-averagng of s 2 (x, y, t) to obtan the tme-averaged stack (or Geophyscal Journal Internatonal C 2012 RAS

5 1156 H. Yao, P. M. Shearer and P. Gerstoft source) power P(x, y, t) and ampltude A(x, y, t) as τ=t+th /2 P(x, y, t) = A 2 τ=t t (x, y, t) = h /2 s2 (x, y,τ)h(τ)dτ τ=th /2 h(τ)dτ, (12) τ= t h /2 where h(t) s the wndow functon and t h gves the wndow duraton. Typcally h(t) s set to a boxcar functon (Ish et al. 2005), that s, h(t) = 1fort [ t h /2, t h /2], and therefore P(x, y, t) s the mean squared ampltude of the wndowed stack waveform. In ths study we set h(t) as the Hann wndow (symmetrc cosne taper functon) wth h(t) = 1 [1 + cos(2πt/t 2 h)] for t [ t h /2, t h /2]. The use of the Hann wndow nstead of the boxcar wndow may suppress some edge effects of tme averagng. An example of the stacked waveform and the stack power (wth t h = 10 s) s shown n Fg. 2(a). The tme-ntegrated stack power ˆP(x, y) and ampltude Â(x, y) from the backprojecton s defned as τ=te ˆP(x, y) = Â 2 (x, y) = s 2 (x, y,τ)dτ, (13) τ=t s where t s and t e are the startng and endng source tmes for backprojecton. It s possble that strong 3-D local heterogenetes exst wthn the source regon. Therefore, for subevents that are far away from the hypocentre, the traveltme perturbatons due to regonal 3-D structural heterogenetes are dfferent from t 0 n eq. (10). Ths may result n less coherent stacks of the waveforms at some magng locatons. Ish et al. (2007) ntroduced a tme calbraton method usng aftershocks to account for ths effect, but found that the changes were relatvely small for the backprojecton mage of the 2004 Sumatra earthquake. However, ths effect lkely causes at least some underestmaton of subevent ampltudes due to less coherent waveform stacks at ncreasng dstances from the hypocentre. For the backprojecton waveform stack (eq. 11), we only use waveforms whch have correlaton coeffcents above 0.6 and postve polarty wth respect to the reference stack. For those traces wth negatve polartes and smaller correlaton coeffcents we can smply set the weght w = 0 n eq. (11). The tme-ntegrated stack power ˆP(x, y) of the Tohoku earthquake s around the hypocentral area f we use data fltered between 0.2 and 1 Hz (Fg. 3a). However, the peak stack power shfts about 70 km towards the Japan coast wth respect to the hypocentre for hgh-frequency data (1 2 Hz; Fg. 3b), whch suggests frequency-dependent energy radaton of ths earthquake (Yao et al. 2011; Koper et al. 2011b). 4 METHODOLOGY OF ITERATIVE BACKPROJECTION In prncple we can decompose the orgnal sesmograms v (t) as the sum of subevent waveforms and resdual waveforms (eq. 7). Smlarly for the algned waveforms u (t), we decompose t as the sum of waveforms from N s subevents, u (k) (t)(k = 1, 2,...,N s ), and the resdual waveforms u R(t)as N s u (t) = u R (t) + u (k) (t). (14) We search for all local maxma of the tme-averaged stack ampltude [A(x, y, t) n eq. 12] of the conventonal backprojecton (eq. 11; see Fg. 4a for an example). A few local maxma wll cluster n space and tme and tend to have smlar arrval tmes to statons and many of them have small ampltudes. We perform a local maxma declusterng process to remove nsgnfcant maxma. Frst local maxma wth ampltude less than 0.05 of the global maxmum are removed. Secondly, startng from the largest local maxmum, we remove any other smaller local maxma that wll arrve at a smlar tme (wthn 5 s) wth respect to the arrval tme of the selected local maxmum to a reference staton n the centre of the array. Ths largest local maxmum s consdered as the sgnfcant maxmum and then removed from the lst of the local maxma. Ths process s repeated for the remanng local maxma untl there s only one local maxmum left n the lst. The obtaned sgnfcant maxma (locaton and tme) are shown n Fg. 4(b) and wll be consdered as possble subevents for the teratve backprojecton analyss. The teratve backprojecton approach combnes the conventonal waveform backprojecton method wth subevent sgnal strppng to determne the locaton, tme, duraton and waveforms of each ndvdual subevent n an teratve way. At each teraton, parameters for the next subevent (prelmnary locaton and source tme) are estmated from the local maxma (Fg. 4b) n the temporally smoothed stacked mage. We then wndow subevent waveforms and re-crosscorrelate them wth ther stack to compute correlaton coeffcents Fgure 3. Tme-ntegrated stack power n the frequency band Hz (left-hand sde) and 1 2 Hz (rght-hand sde) from conventonal backprojecton wth the epcentre locaton determned by Chu et al. (2011) and the strke orentaton of the Tohoku earthquake from USGS W Phase moment tensor soluton (dashed) ( earthquakes/eqnthenews/2011/usc0001xgp/nec_c0001xgp_wmt.php; last accessed on 2012 February 15). Fgure 4. (a) Locatons and tmes of all local maxma (coloured crcles) and (b) the sgnfcant maxma (coloured crcles) of the tme-averaged stack ampltude A(x, y, t). The subevent waveform ampltude s proportonal to the radus of crcle. The colour bar shows the source tmes of maxma. The plus symbol (+) n (b) shows the maxmum. Geophyscal Journal Internatonal C 2012 RAS

6 Iteratve backprojecton of Tohoku earthquake 1157 and addtonal traveltme shfts. The duraton of each subevent s determned by the length over whch the sgnals are well correlated wth the stack usng a short-tme runnng-wndow cross-correlaton method. Prncpal component analyss s then used to extract the subevent prncpal waveforms wthn the determned tme wndow. The next-generaton resdual waveform s then obtaned by subtractng the subevent prncpal waveforms from the current resdual waveforms, whch we term subevent sgnal strppng. Ths process s repeated untl no more qualfed subevents can be dentfed from the resdual waveforms. After larger subevents have been dentfed and ther waveforms are strpped out, smaller subevents can be dentfed one after another. The fnal resdual waveforms and the waveforms from each subevent are ndvdually backprojected and summed together to form the fnal backprojecton stack. Usng the addtonal tme-shfts derved from the subevent waveform re-cross-correlaton, we can locate the subevents more accurately than can be estmated from the conventonal backprojecton mage, whch should translate nto mproved estmates of rupture speed. The flow chart n Fg. 5 shows the steps of ths teratve method, whch are lsted as follows: (1) Bandpass-flter and algn (usng waveform cross-correlaton) the sesmograms wth respect to the ntal hypocentre subevent usng the frst few seconds of the P waves (see Secton 3 for the detal). (2) Perform backprojecton of the resdual waveforms u R (t) and obtan the stack s(x, y, t) at each grd locaton n the source regon. The resdual waveforms are ntally the orgnal sesmograms u (t) (see eq. 10) and k = 1. (3) Determne the prelmnary subevent locatons and source tmes. Here we frst determne the sgnfcant maxma of the tmeaveraged stack ampltude (Fg. 4b). These maxma are potental subevents to be further analysed. The frst subevent s the sgnfcant local maxmum at the hypocentre (x (1), y (1) ) = (0, 0) wth the source tme T s (1) corresponds to the maxmum of the tme-averaged stack ampltude A(x, y, t), usngthefrstfewseconds ofthep waves. Ths s because the ntatng subevent s free of contamnaton by later subevent waveforms. For later subevents, we choose the locaton and source tme correspondng to the largest sgnfcant maxmum (e.g. + n Fg. 4b) as the prelmnary kth subevent locaton (x (k), y (k) ) and source tme T s (k). Note that f ths s not a qualfed subevent (see step 5), we choose the next largest sgnfcant maxmum as the subevent. (4) Obtan the addtonal traveltme shfts of the selected subevent waveforms by re-cross-correlatng the wndowed subevent waveforms wth ther reference stack usng the adaptve stackng approach (Rawlnson & Kennett 2004). To acheve subsample accuracy of tme-shfts from cross-correlaton, we frst nterpolate the data and the reference stack to 50 Hz samplng rate usng cubc splne nterpolaton. Based on the prelmnary source tme T s (k), we wndow the waveforms for the th staton n the tme wndow T (k) s + t P (x (k), y (k) ) + [ t w /2, t w /2] to obtan the domnant subevent waveforms (e.g. wthn the two vertcal lnes n Fg. 6a), where t w s the wndow length. t w s set to 5 s, correspondng to the longest perod n the frequency band ([0.2, 1] Hz here). The (wndowed) reference stack s cross-correlated aganst each wndowed subevent waveform to obtan the cross-correlaton coeffcent c (k), polarty p (k) and addtonal tme-shfts t (k) (Fg. 6b). Sesmograms wth postve polartes and correlaton coeffcents above 0.6 are stacked to create the next-generaton reference stack. Ths step s repeated three tmes to obtan a stable reference stack and traveltme shfts t (k) for each trace. If the subevent locaton Fgure 5. Flow chart of the teratve backprojecton method. Geophyscal Journal Internatonal C 2012 RAS

7 1158 H. Yao, P. M. Shearer and P. Gerstoft Fgure 6. Subevent waveform algnment. (a) Intal 5-s-long wndowed subevent waveforms (background mage, wthn the two vertcal black lnes) and ther stack (sold). The subevent tme s 88.3 s and the locaton s (x, y) = ( 30, 20), asshownnfg. 4(b) (blue+). (b) Cross-correlaton coeffcents (background mage) between the tme-shfted subevent waveform and the stack [wthn the black lnes n (a)]. The maxmum absolute correlaton coeffcent s shown by the magenta (postve polarty) or yellow (negatve polarty) dot. The colour bar gves the correlaton coeffcent (b or d) or the ampltude of each (normalzed) trace (a or c). (c d): smlar as (a b), but after three teratons. s close to the hypocentre, t (k) s small, mostly less than 0.5 s for most traces from our experence (see Fg. 6d for an example). Therefore, we set the maxmum tme-shfts ( t max ) allowed n the re-cross-correlaton to be 1.0 s. Small t max may preclude some possble subevents from beng selected, whle large t max may produce cycle-skppng problems n the cross-correlaton and selecton of unqualfed subevents (see step 5 and eq. 15). (5) Assess whether the dentfed subevent s a qualfed subevent. Snce smaller cross-correlaton coeffcents (c (k) ) and larger addtonal traveltme shfts ( t (k) ) of the subevent waveforms are lkely to ndcate a less relable subevent, the qualty coeffcent r (k) of the selected subevent s emprcally defned as r (k) = N (k) j=1 c(k) j N (1) j=1 c(1) j { [ ] } 2 exp 2 σ (k) t / t max, (15) where N (k) s the number of qualfyng traces that have a postve polarty (p (k) =1) and correlaton coeffcents c (k) > 0.6 for the kth subevent, and σ (k) t s the standard devaton of the addtonal traveltme shfts ( t (k) )ofthen (k) qualfyng traces, and the summaton ndex j s only pcked from the lst of the N (k) qualfyng traces. For the frst subevent, N (1) = 451 and the qualty coeffcent s 1.0. Snce t max = 1s,exp{ 2[σ (k) t / t max ] 2 } s 0.92, 0.84 and 0.61 f σ (k) t s 0.1, 0.3 and 0.5 s, respectvely. From ths defnton, the subevent wth smaller N (k) and c (k) and larger σ (k) t wll naturally have a lower qualty coeffcent, whch s reasonable. The subevent we are usng as an example has a qualty coeffcent of 0.94 wth N (k) = 445 and σ (k) t = 0.1 s. If the qualty coeffcent of the selected subevent s less than 0.7, ths subevent s defned as an unqualfed subevent, and we go back to step (3) to choose the next sgnfcant local maxmum (Fg. 4b). If none of the subevents correspondng to the sgnfcant maxma meets the threshold of the qualty coeffcent (0.7 here), we stop searchng for subevents and jump to step (8). (6) Determne the approxmate duraton of the selected subevent by performng a short-tme runnng-wndow cross-correlaton between each algned qualfyng trace [.e. u R P (t + t (x, y) + t (k) ), wth p (k) = 1andc (k) > 0.6], and ther reference stack (black trace n Fg. 7a). The wndow s centred at tme t (for t [t s t w, t e 0.5 t w ]) wth duraton of t w (5 s here). For each effectve trace, we obtan a tme-dependent correlaton coeffcent functon (Fg. 7b), whch s then averaged to obtan the mean correlaton coeffcent (MCC), shown as the black curve n Fg. 7(c) (low-pass fltered below 0.5 Hz). We frst fnd the peak MCC wthn the subevent tme wndow n step (4) (wthn the two vertcal lnes n Fg. 7c). Typcally the MCC deceases quckly from the peak MCC, mplyng that the waveforms from other subevents (or nose) domnates when t s away from the selected subevent. However, n some cases, the MCC curve may become flat or decrease very slowly, whch probably ndcates other subevents occurred at a nearby tme and locaton. For determnng the approxmate duraton of the subevent, we choose the tme duraton that corresponds to the MCC larger than 0.75 tmes the peak MCC value (red dashed lne n Fg. 7c) wthn the tme wndow bounded by the two most nearby local MCC mnma. Ths approxmates the duraton of ths subevent (wth startng tme τ s and endng tme τ e ), over whch the sgnals are well correlated wth the reference stack. A wndow functon W(t), shown as the green curve n Fg. 7, s set to wndow the subevent waveforms. W(t) s1fort [τ s, τ e ] and has a cosne taper at each sde wth a tme duraton of 0.1 t w. (7) Extract the subevent prncpal waveforms usng prncpal component analyss for the wndowed subevent waveforms u w (t) = u R P (t + t (x, y) + t (k) )W (t) (only for t [τ s 0.1 t w, τ e t w ], where W(t) s non-zero). We then subtract the prncpal waveforms from the current resdual waveforms to obtan the nextgeneraton resdual waveforms. Ths step s termed subevent sgnal strppng. The wndowed subevent waveform u w (t) (Fg. 8b) can be expressed by an N M data matrx F, wth N number of traces and M number of ponts n each wndowed trace (M < N here). F can be decomposed usng sngular value decomposton (SVD) as F = U V T,whereU s a N N matrx, s a N M dagonal matrx wth non-negatve decreasng real numbers {λ 1,..., λ M } on the dagonal, V s a M M matrx and T denotes the transpose. The prncpal sgnal matrx s gven by F P = U P V T,where P s a N M dagonal matrx wth a few of the largest dagonal components {λ 1,..., λ L } (L M) of. Ifweonlykeepthesgnals correspondng to the largest component, the extracted prncpal sgnal of each trace s very smlar to that of the reference stack. However, the subevent waveforms recorded at dfferent statons may be more complcated than the reference stack due to source complextes, propagaton effects and ste effects. Therefore, we keep a few of the largest dagonal components (Fg. 8c). We requre λ L > 0.25λ 1. Therefore, less coherent sgnals or noses assocated wth smaller dagonal components of are removed. The subevent prncpal waveform of each trace u (k) (t + t P (x, y) + t (k) )s obtaned from each row of the prncpal sgnal matrx F P. The nextgeneraton resdual waveforms (Fg. 8d) are updated by subtractng the subevent prncpal waveforms (Fg. 8c) from the current resdual waveforms (Fg. 8a). Then we reterate from step (2) for fndng the next subevent and k = k + 1. (8) Perform the backprojecton for the fnal resdual waveforms u R (t) and each subevent prncpal waveforms u(k) (t)(k = 1,...,N s ) after N s subevents have been dentfed. The fnal resdual waveform stack S R (x, y, t) s obtaned usng eq. (11). We use the addtonal tme-shfts t (k) from waveform re-cross-correlaton (step 4) to Geophyscal Journal Internatonal C 2012 RAS

8 Iteratve backprojecton of Tohoku earthquake 1159 Fgure 7. (a) Algned waveforms (background mage) and ther stack (black trace) for the dentfed subevent. (b) Short-tme runnng-wndow correlaton coeffcents (background mage) between each trace and the stack (see text for the detal). (c) Mean correlaton coeffcent after 0.5 Hz low-pass flterng below (black curve) and the subevent tme wndow (green curve). The red dashed lne has a correlaton coeffcent equal to 0.75 tmes the peak correlaton coeffcent wthn the ntal subevent tme wndow (between the vertcal black lnes). Fgure 8. (a) Algned current resdual waveforms for the dentfed subevent. (b) Wndowed subevent waveforms (between the dashed lnes). The wndow functon s shown as the green curve, same as the wndow functon n Fg. 7 (the green curve). (c) The subevent prncpal waveforms obtaned from prncpal component analyss. (d) The next-generaton resdual waveforms obtaned by subtractng the subevent prncpal waveforms n (c) from the current resdual waveforms n (a). Geophyscal Journal Internatonal C 2012 RAS

9 1160 H. Yao, P. M. Shearer and P. Gerstoft Fgure 9. Stack peak ampltude A p (x, y) = max { A(x, y, t) } for t [t s, t e ] of (a) the current resdual waveforms, (b) current subevent waveforms and (c) the next-generaton resdual waveforms wth the locatons of the current (blue +) and next subevents (green ). obtan each subevent waveform stack S (k) (x, y, t)as S (k) (x, y, t) = N =1 p (k) w (k) u (k) ( ) t + t P (x, y) + t (k). (16) We set w (k) = 0 for non-qualfyng traces (wth negatve polarty (p (k) = 1) and correlaton coeffcent c (k) < 0.6) and w (k) = 1/N (k) for the N (k) qualfyng traces. The complete teratve backprojecton stack S(x, y, t) s thus obtaned by summng the fnal resdual waveform stack S R (x, y, t) and all the subevent prncpal waveform stacks S (k) (x, y, t), that s, N s S(x, y, t) = S R (x, y, t) + S (k) (x, y, t). (17) Fg. 9(a) shows an example of the stack peak ampltude (A p (x, y) = max { A(x, y, t) } for t [t s, t e ]) from the current resdual waveforms (Fg. 8a). The stack peak ampltude usng the determned current subevent waveforms (Fg. 8c) s shown as Fg. 9(b). After the waveforms of ths subevent are strpped out, the stack peak ampltude of the next-generaton resdual waveforms (Fg. 8d) s shown as Fg. 9(c), where the next largest subevent (green n Fg. 9c) appears more vsble. Here we defne the resdual waveform energy rato as N te { =1 t R E = s u R (t) } 2 dt N =1 te t s {u (t)} 2 dt. (18) Fg. 10(a) shows the evoluton of the resdual waveform energy rato versus number of dentfed subevents. After 16 subevents (wth qualty coeffcents above 0.7) are dentfed, the resdual waveform energy rato decreases to 0.25, mplyng that the 16 subevents contrbute 75 per cent of the energy n the waveforms. The fnal resdual waveforms are shown n Fg. 10(b). 5 RELOCATION OF SUBEVENTS The prelmnary subevent locatons are determned at the predefned (coarse) grd locatons correspondng to some sgnfcant maxma of the tme-average stack ampltude (see the step 3 n Secton 4). However, the subevent can be relocated usng the relatve traveltme shfts ( t (k) ) for each subevent determned from subevent waveform re-cross-correlaton. Here we use a grd search and l 1 -norm method (Shearer 1997) to mprove the subevent locatons and source tmes. Due to the poor depth resoluton of telesesmc data, we only solve for horzontal locatons at the plane of the hypocentre depth, just as for the backprojecton magng. Fgure 10. (a) Resdual waveform energy rato after each subevent waveforms have been strpped. (b) The fnal resdual waveforms after 16 subevent waveforms have been strpped. The predcted traveltme from a locaton (x, y) to a staton usng the 1-D model s t P (x, y)( = 1, 2,...,N). For the subevent at the locaton (x (k), y (k) ) wth the source tme T s (k), the measured relatve traveltme shfts (or traveltme resduals) are t (k) = t (k) t P (x (k), y (k) ), where t (k) s the actual traveltme through the real heterogenous Earth. If the source s moved from (x (k), y (k) )to(x, y), the traveltme shfts becomes δt (x, y) = t (k) + [ ( t P (x, y) t P x (k), y (k))]. (19) Our goal s fndng locaton (x, y) and a source tme perturbaton δt s such that the l 1 -norm traveltme msft functon χ(x, y,δt s ) = 1 N (k) δt N (k) (x, y) δt s (20) =1 Geophyscal Journal Internatonal C 2012 RAS

10 Iteratve backprojecton of Tohoku earthquake 1161 Fgure 11. Example of subevent relocaton. (a) The traveltme msft χ(s) (eq. 20) dstrbuton wth the prelmnary subevent locaton from backprojecton (black +) and the optmal subevent locaton after relocaton (yellow +). (b) and (c) show traveltme shfts δt (x, y) (eq. 19) before and after subevent relocaton, respectvely. The black lne n (c) gves the medan of all traveltme shfts, that s, the source tme perturbaton δt s by eq. (21). The trace number s sorted n an azmuth-ascendng order. s mnmzed. Snce the source tme perturbaton δt s smply gves the baselne shft of the traveltme shfts, we obtan t from the medan of the traveltme shfts (Shearer 1997), that s, δt s = medan{δt (x, y)} ( for [ 1, 2,...,N (k)]). (21) The use of the l 1 -norm (eq. 20) and medan (eq. 21) nstead of the l 2 -norm and mean for source locaton problems has the advantage of more robust response of the former to outlers n the data (Shearer 1997). We perform the grd search for the subevent relocaton on much fner grds (2 km spacng) than the sparser grds used for backprojecton (10 km spacng) to fnd the optmal subevent locaton (ˆx (k), ŷ (k) ) and source tme perturbaton δt s by mnmzng the traveltme msft functon (eq. 20). The new source tme at the optmal subevent locaton s updated by ˆT s (k) = T s (k) +δt s. We show one example of subevent relocaton n Fg. 11. The optmal subevent locaton s about 12 km east and 6 km south of the prelmnary subevent locaton (Fg. 11a) from the teratve backprojecton. The traveltme shfts before and after relocaton are shown n Fgs 11(b) and (c). From the medan of the new traveltme shfts after relocaton (black lne n Fg. 11c), we obtan the source tme perturbaton δt s = 0.17 s. Although the traveltme msft χ decreases as the subevent locaton moves to the optmal locaton, some trends n the azmuth-dependent traveltme resduals may reman, whch are probably caused by local heterogenetes around the subevent locaton or contamnaton by waveforms of other subevents or nose. The locaton uncertantes usng traveltme shfts from waveform cross-correlaton can be estmated usng a bootstrap approach (Shearer 1997). Here, we randomly select N (k) traveltme shfts from the total set of N (k) observed shfts. Then we apply the relocaton algorthm descrbed earler usng the N (k) randomly pcked shfts to fnd the best locaton. Ths procedure s repeated 100 tmes. Fnally, the locaton error s estmated from the standard devaton of the obtaned 100 best locatons. In the example we show here, the locaton error s estmated to be about 2.6 km n the E W drecton and 2.0 km n the S N drecton. Snce the error of the subevent locaton s from the formal statstcal bootstrap analyss, the true locaton error s lkely larger because we are assumng a smple 1-D model and do not consder effects of contamnaton from other phases or coherent nose. 6 RESULTS FOR THE TOHOKU EARTHQUAKE The teratve backprojecton method s used here to mage the rupture of the 2011 Tohoku M w 9.0 earthquake usng telesesmc P-wave data (fltered to Hz; Fg. 2) recorded by the statons n the central and western Unted States (Fg. 1). Fgs 12(a) and (b) show the spato temporal dstrbuton of 16 subevents before and after relocaton, respectvely. The detaled nformaton about the 16 subevents after relocaton s shown n Table 1. These 16 subevents are verfed as relable from bootstrap analyss (Secton 7.3). The peak of the tme-ntegrated stack power s slghtly west of the hypocentre (Fg. 12a). Fg. 13 shows the rupture mages (tme-averaged stack power of the complete stack S(x, y, t)) at some representatve tmes. The locaton dfferences between the subevent locatons before and after relocaton are shown n Fg. 14(a) and the relocaton error s shown n Fg. 14(b), whch gves the lower bound of the locaton error. The tme versus dstance to hypocentre along the strke of the relocated subevents s shown n Fg. 15. From the dstrbuton of subevents (Fg. 12b) ths megathrust earthquake shows apparent blateral rupture features along the strke (N S) drecton and also downdp rupture towards the coast of Japan. The spatal dstrbuton of subevents s more smlar to that of the aftershocks (wth magntude above 4.0) wthn the frst 2 d after the man shock than to the major slp dstrbuton nferred from slp nverson (Fg. 12c). Our results show qute complcated rupture behavour durng the frst 90 s and domnant hgh-frequency (0.2 1 Hz) radaton near the hypocentre. In the frst 50 s and also at later tmes (e.g. near 75 s and 88 s) a group of subevents occurred close to the hypocentre (wthn 30 km), whch suggests repeatng rupture around the hypocentre area, where few bg aftershocks occurred n the frst several days (Fg. 12c). The ntal rupture speed appears slow (less than 1.5 km s 1 ) from the dstrbuton of the frst three Geophyscal Journal Internatonal C 2012 RAS

11 1162 H. Yao, P. M. Shearer and P. Gerstoft Fgure 12. (a) Normalzed tme-ntegrated stack power (red contours) and spato-temporal dstrbuton of 16 subevents (coloured crcles wth colour bar showng the subevent tme τ max n Table 1) from teratve backprojecton. (b) Locaton of 16 subevents (crcles) after subevent relocaton. (c) Locaton of aftershocks (M j > 4, from JMA catalogue; crcles) wthn the frst 2 d after the man shock, the relocated subevents (magenta x, subevent tme black number) n the man shock and the domnant slp regon [green shaded area (Chu et al. 2011)]. The subevent waveform peak ampltude s proportonal to the radus of crcle n (a) and (b) and to the sze of the magenta x n (c). Each panel shows: epcentre locaton (black +), the trench locaton (blue lne), the strke (dashed blue) and Japan coastlne (black lne). subevents at about 6, 18 and 23 s (Fg. 12c), whch has also been nferred n slp nverson results (e.g. Lee et al. 2011). A clear subevent s observed near the coast regon at about 43 s at a dstance about 106 km northwest of the hypocentre, whch suggests an average speed of 2.5 km s 1 for the northwestward downdp rupture. The largest subevent (.e. wth the largest subevent waveform ampltude) occurred about 30 km northwest of the hypocentre at about 88 s (Fg. 12c). At 65 s a subevent occurred about 43 km north of the hypocentre and at 105 s the second largest subevent occurred about 115 km north of the hypocentre and n a regon also wthout bg early aftershocks. Meng et al. (2011) suggest a supershear northward rupture speed of about 5.0 km s 1 usng a frequency-doman MUSIC backprojecton method. If the subevent at 105 s to the north were ntated by the largest subevent at 88 s close to the hypocentre (Fg. 12c), the northward rupture speed would reach 5.0 km s 1. However, t s very lkely that the northward rupture speed s only about 2 km s 1 as nferred from the northern subevents at 65 and 105 s (Fgs 12c and 15), whch s more consstent wth the northward rupture speed from slp nverson results (Leeet al. 2011).The energy radaton n the northern rupture regon appears to dmnsh after 110 s. The southward rupture both along strke and downdp became energetc after about 110 s at a dstance about 120 km away from the hypocentre. The deepest subevent n the southwestern rupture area occurred at 128 s, located beneath the coast regon. The southernmost subevent occurred at 147 s at a dstance about 230 km Geophyscal Journal Internatonal C 2012 RAS

12 Iteratve backprojecton of Tohoku earthquake 1163 Table 1. Relocated subevents of the 2011 M w 9.0 Tohoku earthquake. N τ max (s) τ s (s) τ e (s) Lon ( ) Lat ( ) Max amp Qualty coef N, subevent ndex number; τ max, tme correspondng to the subevent waveform peak ampltude; τ s and τ e (s), subevent startng and endng tme; Lon ( )andlat( ), subevent longtude and lattude; Max amp, relatve subevent waveform peak ampltude; Qualty coef., subevent qualfy coeffcent defned as eq. (15). southwest of the hypocentre. There are three subevents that occurred close to the coast where the early aftershocks dmnshed. From the spatal and temporal dstrbuton of subevents to the south of the hypocentre after 100 s, we estmate the average southward rupture speed along the strke s about 3.0 km s 1 between 100 and 150 s (Fg. 15). 7 DISCUSSION 7.1 Synthetc examples Because our waveform teratve backprojecton and subevents recovery method s essentally emprcal, t s mportant to perform synthetc tests to verfy that t correctly recovers multple subevent tmes and locatons. Here we smply consder the drect P waves from a seres of subevents and gnore the depth phases and other multple reflected and converted phases. In realty, depth phases and other reflected or scattered phases, whch depend on the depth and focal mechansm of the earthquake as well as on structural heterogenetes, wll also contrbute to the waveforms, and we wll dscuss the effects of depth phases n the next secton. Fgure 13. Tme-averaged power of the complete stack S(x, y, t) at representatve tmes. The colour bar and contours gve the normalzed power. In each plot: purple star subevents occurred around the gven tme wth ts sze proportonal to subevent waveform ampltude, blue cross hypocentre locaton. Geophyscal Journal Internatonal C 2012 RAS

13 1164 H. Yao, P. M. Shearer and P. Gerstoft Fgure 14. (a) Relatve locaton dfference (coloured crcles) between new (after relocaton) and old locatons of all subevents. Postve dx means the new locaton s more east (closer to the trench) and postve dy more north. The subevent waveform ampltude s proportonal to the radus of crcle and the colour bar shows the subevent source tme. (b) Subevent source locaton uncertantes n x and y drectons from bootstrap analyss of traveltme resduals. Our tests assume sources near the hypocentre and the same staton locatons as the real data. The synthetc P wave pulse for each subevent s from the frst few seconds of the lnear stack of the P waves (Fg. 2a). Assumng a locaton and source tme for each subevent, we calculate the predcted P-wave traveltme to each staton. The synthetc waveform at each staton s the sum of the synthetc P-wave pulses at the arrval tmes of the subevents. We add Gaussan nose wth standard devaton 20 per cent of the peak sgnal ampltude to the synthetc waveforms. We smulate a blateral earthquake rupture, whch shares some features as the Tohoku earthquake. The synthetc waveforms (Fg. 16a) are formed from 13 subevents (Fg. 16b) wth the same source ampltude. The P waves from fve of these subevents have some degree of nterference (between 30 and 55 s n Fg. 16a). In ths case the teratve backprojecton method exactly recovers all subevent locatons and tmes (Fg. 16c). Our most realstc test s of blateral rupture, usng 15 subevents wth dfferent source ampltudes (Fg. 17). Waveforms from some of the subevents severely nterfere wth each other (.e. between 10 and 70 s n Fg. 17a). The teratve backprojecton method does resolve most of subevents correctly, partcularly the larger ampltude subevents. Although several of the smaller subevents are not recovered, the general pattern of blateral rupture s stll apparent from the teratve backprojecton results. The qualty of the recovered subevents can be accessed from the defned qualty coeffcent (eq. 15) and the relablty of the subevents can be tested through a bootstrap approach (Secton 7.3). 7.2 Effects from depth phases In most waveform backprojecton studes, the effects on the mages from depth phases (e.g. pp) are gnored. The ampltude and tmng of the depth phases mostly depend on the focal mechansm and depth. For very shallow earthquakes (e.g. wth focal depths less than 10 km) the depth phases closely follow the drect arrvals and have nearly dentcal moveout, and thus they have lttle dstortng effect on the backprojecton mage. However, for deeper ruptures (e.g. focal depths of 30 km or more), the depth phases are separated enough from the drect P phase that they may ntroduce artefacts n the backprojecton mage, whch appear at later tmes and offset n locaton from the drect phase mage. For the Tohoku man shock, the hypocentre depth s 20 km (e.g. from Chu et al. 2011) and the deepest rupture area may reach 40 km depth beneath the Japan sland. Therefore, t s mportant to access how depth phases may affect the teratve backprojecton results (e.g. the number of subevents and ther locatons). Depth phases for the Tohoku earthquake wll nclude the pp phase (reflected back at the seafloor) and the water phase pwp [reflected back at the water surface; see examples n Chu et al. (2011)]. Fgure 15. Tme versus dstance to hypocentre along strke of the subevents. The radus of each crcle s proportonal to the subevent waveform ampltude and s centred at the tme of the maxmum ampltude of that subevent. The red bar shows the estmated tme duraton of each subevent. The numbers n red and black gve the tme (same as n Fg. 12) and the qualty coeffcent of the subevent, respectvely. Dstance s from south to north along the strke (blue dashed lne n Fg. 12). Lnes of constant rupture velocty are shown n dashed for reference. Geophyscal Journal Internatonal C 2012 RAS

14 Iteratve backprojecton of Tohoku earthquake 1165 Fgure 16. Synthetc example of the teratve backprojecton method. (a) Synthetc waveforms (wth 20 per cent level of Gaussan nose added) by the array statons n Fg. 1 from contrbuton of 13 subevents shown as the coloured dots n (b); (c) shows the recovered subevent locatons and source tmes from the teratve backprojecton method. The locaton at (0, 0) n (b) and (c) s the hypocentre at (lat 38.19, lon ). The synthetc waveforms are already algned usng the P waveforms from the frst subevent at the hypocentre. Fgure 17. Synthetc example llustratng the teratve backprojecton method, assumng subevents of varyng ampltude (proportonal to crcle radus) and severe nterference of some subevent waveforms. It s dffcult to dentfy ndvdual depth phases n the complcated man shock wave tran. Thus, to assess the effect of depth phases on our results, we analyse USArray data from selected Tohoku aftershocks wth short source-tme functons and varous focal depths usng the same method we apply to the man shock. Fg. 18 shows results of teratve backprojecton from sx aftershocks of M w 6 wthn the man shock rupture regon. For the two aftershocks wth focal depths less than 20 km, we mage only a sngle subevent at the epcentre. Ths s because the depth phases (pp and pwp) and the drect P phase are too close n tme to be resolved separately. For the two aftershocks wth focal depths at 27 and 28 km, we resolve two or three subevents. The frst and largest subevent comes from the drect P wave and s located at the epcentre. The second, and most sgnfcant subevent, occurs 15 s later and results from the depth phases pp and pwp, but ts locaton s close to the hypocentre. The most substantal backprojecton artefacts occur for the two aftershocks wth focal depths close to 40 km, n whch the frst depth-phase subevent has a slghtly larger ampltude than the drect P-phase subevent, and s about km northeast (towards the USArray drecton) of the hypocentre. Ths occurs because the depth-phase surface bounceponts are n the drecton of the staton array. We can use these results to assess the lkelhood that any of the subevents that we mage for the Tohoku man shock are lkely depth-phase artefacts by searchng for event pars n whch the second event occurs s after the frst event and s dsplaced to the northeast by km. No obvous canddates for such pars are seen n Fg. 12. In addton, the waveform ampltudes Geophyscal Journal Internatonal C 2012 RAS