Earthquake nucleation on dip-slip faults

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109,, doi: /2003jb002894, 2004 Erthquke nucletion on dip-slip fults Chunli Zhng College of Mechnicl Engineering, Yngtze University, Jingzhou, Chin Dvid D. Oglesy Deprtment of Erth Sciences, University of Cliforni, Riverside, Cliforni, USA Gunshui Xu Deprtment of Mechnicl Engineering, University of Cliforni, Riverside, Cliforni, USA Received 14 Novemer 2003; revised 13 April 2004; ccepted 26 July 2004; pulished 9 Novemer [1] The nucletion of unstle slip on fult is of key importnce in our understnding of the seismic cycle. We investigte how the symmetric geometry of dip-slip fults ffects the nucletion of unstle slip on such fults. Previous reserchers hve devoted much effort to understnding this nucletion process on geometriclly simple fults, using vriety of frictionl prmeteriztions. However, there re mny resons to elieve tht erthquke nucletion my e ffected y fult complexity. The rekdown of symmetry on nonverticl dip-slip fult mens tht norml stress is not constnt during the slip process nd tht the hnging wll nd footwll my not necessrily move eqully. Using slip-strengthening nd -wekening friction lw in two-dimensionl qusi-sttic model sed on vritionl oundry integrl method, we show tht nucletion on dip-slip fults is ffected y oth the dip ngle nd the direction of slip (norml versus thrust/ reverse). Under otherwise identicl conditions in homogeneous hlf-spce (i.e., neglecting depth-dependent frictionl or mteril properties), thrust fults nuclete closer to the Erth s surfce thn norml fults nd tke less time to do so. These differences decrese s the dip ngle of the fult increses to 90. The mount of preseismic surfce slip is much more complicted function of dip ngle. The results show tht fult geometry my hve n importnt role in the nucletion process s well s the process of dynmic rupture. Further reserch is needed to comine rigorous nucletion models with full rupture dynmics. INDEX TERMS: 7209 Seismology: Erthquke dynmics nd mechnics; 7260 Seismology: Theory nd modeling; 3210 Mthemticl Geophysics: Modeling; 3220 Mthemticl Geophysics: Nonliner dynmics; KEYWORDS: thrust fult, erthquke nucletion, qusi-sttic modeling Cittion: Zhng, C., D. D. Oglesy, nd G. Xu (2004), Erthquke nucletion on dip-slip fults, J. Geophys. Res., 109,, doi: /2003jb Introduction [2] One of the chief gols of seismology is to understnd the processes tht led to unstle fult slip (n erthquke). A thorough understnding of the erthquke nucletion processes rises the tntlizing prospect of eing le to predict erthqukes efore they rech instility, nd thus provide erly wrnings. While such deterministic forecsts pper still to e out of rech, experimentl nd numericl studies hve mde much progress on the scientific issues surrounding erthquke nucletion. Experimentl studies rnging from those of Dieterich [1978] to Ohnk nd Shen [1999] hve helped to shed light on the process y which slip loclizes to smll region on fult nd then propgtes dynmiclly over lrger re. In prticulr, the ltter study indictes erthquke nucletion strts y time Copyright 2004 y the Americn Geophysicl Union /04/2003JB002894$09.00 of slow, qusi-sttic slip, followed y short time period of ccelerting slip, nd ending with the onset of unstle, rpidly propgting rupture. [3] A numer of numericl studies hve lso investigted the process of seismic nucletion. These studies typiclly use rte-nd-stte friction lws [Dieterich, 1979, 1979; Ruin, 1983] motivted y rock friction experiments, or slip-wekening friction lws [Id, 1972; Andrews, 1976] motivted y either theoreticl rguments or simplifiction of the ove rte-nd-stte lws. Qusi-sttic nd dynmic models hve exmined the onset of instility s well s multiple slip events on fults [Tse nd Rice, 1986; Okuo, 1989; Rice, 1993; Cochrd nd Mdrig, 1996; Lnger et l., 1996; Ben-Zion nd Rice, 1997; Lpust et l., 2000; Shizki, 2002; Lpust nd Rice, 2003]. These studies typiclly investigte the effects of depth-dependent frictionl properties nd other fult heterogeneities on the loction nd onset time of erthquke nucletion. These studies lso find tht nucletion loctions re often ssocited with 1of18

2 regions t the trnsition etween stick-slip nd stedy sliding ut re lso strongly ffected y the stress heterogeneities left y previous events. Other studies hve used slip-wekening friction lws nd pplied them to fults tht re everywhere t the point of filure. These studies then use stility nlyses in the spectrl domin to investigte how instilities rise on such fults [Cmpillo nd Ionescu, 1997; Fvreu et l., 1999; Ampuero et l., 2002; Voisin et l., 2002]. [4] Few studies hve focused on the role of fult geometry in the nucletion process. For exmple, Nielsen nd Knopoff [1998] studied the effect of repeted ruptures on nonplnr fult system using qusi-sttic model. They found tht extensionl stopovers were common plces for erthquke nucletion, s expected from the reduced yield stress in these loctions. However, fult geometry is known to hve lrge effect on the dynmics of erthquke rupture nd slip. Fult ends nd rnches cn hve strong effect on the propgtion of rupture nd slip nd cn cuse stress uildups nd reductions ner these fetures [Bouchon nd Streiff, 1997; Nielsen nd Knopoff, 1998; Mgistrle nd Dy, 1999; Aochi et l., 2000, 2000; Aochi nd Fukuym, 2002; Aochi et l., 2002; Hrris et l., 2002; Polikov et l., 2002; Kme et l., 2003; Oglesy nd Archulet, 2003; Oglesy et l., 2003, 2003]. Nonverticlly dipping fults cn cuse n symmetry etween hnging wll nd footwll motion, s well s mplifiction of thrust fult motion in comprison to norml fult motion [Brune, 1996; Nielsen, 1998; Oglesy et l., 1998; Shi et l., 1998; Brune nd Anooshehpoor, 1999; Bonfede nd Neri, 2000; Oglesy et l., 2000, 2000; Agrd et l., 2001; Oglesy nd Dy, 2001, 2001]. In oth these cses, the complexity in fult motion nd ground motion is cused y the symmetric fult geometry, leding to greter feedck etween the rdition nd frictionl processes. It is resonle to ssume tht geometricl fult complexity would hve effects not only on the propgtion of unstle rupture ut lso on the loding nd nucletion processes tht led to this instility. This ltter suject is the focus of the present work. [5] We investigte the effect of nonverticl fult dip on erthquke nucletion. As shown in the work cited ove, dip-slip fults differ in fundmentl wy from typicl strike-slip fults: they tend to e nonverticlly dipping, with lrge verticl component of fult motion. This roken symmetry leds to norml stress tht chnges when the fult slips, n interction tht is sent in the more symmetricl cse of verticl strike-slip fults in lterlly homogeneous mterils or fults in homogeneous whole spce. The sic effect is s follows: in the slipping region of thrust fult, interction of the fult stress field with the free surfce of the Erth leds to decrese in the compressive norml stress. Conversely, hed of the rupture front, in the locked prt of the fult, the compressive norml stress is incresed. In the cse of norml fults, the reversed direction of slip leds to the reverse effect on the norml stress increment: compressive norml stress is incresed in the slipping region nd decresed hed of the rupture front. Becuse chnging norml stress chnges the frictionl stresses on the fult, the rupture process cn e much more complicted thn in symmetric fulting configurtions. An importnt impliction of this interction is the symmetry etween thrust nd norml fulting mentioned ove: the chnges in norml stress due to fult slip led to n incresed stress drop for thrust fults reltive to norml fults when these fults pproch the Erth s surfce. Additionlly, initilly homogeneous stress fields ecome inhomogeneous fter slip on such fults [Oglesy et l., 1998, 2000, 2000]. [6] Models for the nucletion process cn produce mny different results for nlysis, ut for simplicity, we will focus on reltively mcroscopic quntities for discussion. In prticulr, we investigte how fult geometry nd direction of slip ffect the hypocentrl loction, instility onset time, nd mount of preseismic surfce slip. Hypocentrl loction (i.e., the down-dip loction of the onset of unstle slip) is importnt in tht it strongly ffects the directivity nd intensity of ground motion. The onset time for instility is lso quite importnt ecuse hstened or delyed nucletion hs n effect on the energy udget of n erthquke nd could ffect oth the size nd frequency of erthqukes over repeted erthquke cycles. The mount of preseismic surfce slip is perhps more difficult to interpret, ut it could e importnt in interpreting interseismic deformtion dt ner dip-slip fults. Using n exponentil slip-wekening friction lw ( simplifiction from rte-nd-sttedependent friction lw) nd qusi-sttic modeling method, we follow the preseismic loding nd nucletion processes. For simplicity, nd to isolte the effects of fult geometry, we use homogeneous (depth-independent) mterils nd frictionl properties. However, we cknowledge tht the effects of depth-dependent frictionl properties my e of equl or greter importnce thn geometricl effects on the nucletion process [e.g., Shizki, 2002]; these depthdependent effects will e the suject of future work. Also for simplicity, we use qusi-sttic modeling method tht reks down when the frictionl instility is reched; thus our models re only vlid in the slow deformtion time period leding up to the onset of dynmic rupture. Full dynmic models tht include oth nucletion nd unstle propgtion of rupture nd slip re likewise the suject of future work. 2. Frictionl Constitutive Reltions [7] Nucletion of erthqukes on crustl fults results from the onset of unstle fult slip. In the frmework tht fults re modeled s surfces of displcement discontinuities in elstic mterils, the occurrence of such slip instilities criticlly depends on frictionl constitutive reltions tht govern fult slip under tectonic loding. The reltionship of slip instility to frictionl constitutive reltions hs een elorted in previous studies [e.g., Rice, 1983]. The essentil condition for instility to occur on fult is tht the strength of the fult decreses in n pproprite mnner with ongoing slip (slip wekening). In generl, fult friction is complex microphysicl process tht depends on mny prmeters such s effective norml stress, slip velocity, temperture, nd fult gouge minerlogy. The phenomenon of generic slip wekening is commonly oserved in lortory experiments [e.g., Ohnk et l., 1987; Ohnk nd Ymshit, 1989]. Two types of frictionl constitutive models hve emerged to e widely dopted for the nlysis of fult instilities. One is the simple rte-independent slip-wekening model, which ssumes tht friction is function of slip displcement only, nd the strength of the fult degrdes from pek resistnce down to fixed residul frictionl level sustined t lrge slip. 2of18

3 This type of frictionl model is suitle for the one-time motion of fult segment, which is ssumed to e locked in the preseismic period [Id, 1972; Plmer nd Rice, 1973; Andrews, 1976, 1976; Sturt, 1979; Sturt nd Mrvko, 1979; Sturt et l., 1985; Okuo, 1989; Shizki, 2002]. The other generl type of friction lw is the rte-nd-stte-dependent model, which ssumes tht the friction is function of slip rte nd set of stte vriles. The stte vriles evolve with ongoing slip, nd represent memory dependence on previous slip episodes [Dieterich, 1979, 1979, 1986; Ruin, 1983]. Becuse this model implies fult strengthening fter dynmic rupture, it is prticulrly suitle for description of sequences of repeted slip instilities on the sme fult surfce. The richness of this more comprehensive friction model is lso ccompnied y complexity in oth selection of physicl prmeters nd in the implementtion of dynmic models. At present, this type of model is typiclly used on fults with reltively simple geometry [e.g., Tse nd Rice, 1986; Rice, 1993; Sturt nd Tullis, 1995; Kto nd Hirsw, 1997; Lpust nd Rice, 2003]. The implementtion of this model, nevertheless, is essentil for understnding the long-term evolution of fult dynmics s well s shortterm fult seismicity. High-speed frictionl sliding, ccompnied y the effect of fult melting under high pressure, offers nother exciting possiility for explining the complexity of fult dynmics. [8] In this pper, we dopt the first type of frictionl model: A rte-independent friction constitutive reltion is ssumed to operte on dip-slip fult. The model is similr to the slip-strengthening nd -wekening frictionl model used y Shizki [2002] in study of the effect of the seismic-seismic oundry on the nucletion of lrge erthqukes. Such frictionl model hs een proposed sed on creful stick-slip lortory experiments [Ohnk et l., 1987]. Insted of using single prmeter to represent the strengthening nd wekening process, we, however, use two seprte prmeters to llow more flexiility. Furthermore, we derive the slip-wekening reltion y considering it s specil cse of the rte-nd-stte-dependent reltion in which the velocity effect is neglected. This pproch illustrtes the consistency etween these two kinds of friction models nd therefore llows us to use common physicl prmeters to represent the wekening process. [9] To this end, we strt with the most common rte-ndstte-dependent frictionl reltion, which my e expressed y ¼ V ½ L t tss ðv ÞŠ; ð1þ where t is the sher stress on the slip fult, t is the time, A is constnt, V is the slip rte, L is the chrcteristic length, nd t ss is the stedy stte fult strength. The first term on the right-hnd side of eqution (1) represents the sher stress jump due to the slip rte jump. The second term represents the sher stress decrese to stedy vlue over the chrcteristic length L. By ssuming the fult ccelertion to e smll, the first term my e neglected. Replcing t ss y t 0 nd integrting ove the eqution results in t t 0 ¼ Dt m expð d r =LÞ; ð2þ where t 0 represents the residul frictionl strength, Dt m is the fult frictionl strength drop from the pek to the residul vlue, nd d r is the slip on the fult. Eqution (2) represents typicl slip-wekening reltion, with the sher strength grdully decresing from the pek vlue t 0 + Dt m to the residul vlue t 0 over the chrcteristic length L. Note tht in this cse, L is the distnce over which the stress drops to 1/e of its pek vlue; it is not strictly equl to the criticl slip-wekening prmeter d 0 of liner slip-wekening friction lws [e.g., Andrews, 1976]. It is lso not equl to roughly 1/15d 0, which is the rough reltionship found in the dynmic rupture models of Cocco nd Bizzrri [2002]. As Cocco nd Bizzrri note, in the full dynmic cse the reltionship etween L nd d 0 is complicted one tht incorportes the vrious constnts nd tht define the stedy stte frictionl stress [Rice, 1993]. In the lowvelocity limit such s in the present study, the reltionship of L to d 0 is somewht simpler: If we define the stress drop to e complete when the stress drops to 5% of its pek vlue, then d 0 is pproximtely three times L. [10] At the strt of our simultions we ssume tht the fult hs zero sher stress, such s would e the cse immeditely fter prior lrge event with complete stress drop. By nlogy with the rte-nd-stte formlism, we ssume tht the fult my slip smll mount (under the effects of the stedy loding t the fult se) t low stress efore it egins to lock up. To incorporte this strengthening prt of the friction lw, we follow previous work [Ohnk nd Ymshit, 1989; Bizzrri et l., 2001] y using n initil logrithmic increse of the friction with slip displcement nd express the frictionl reltion s t t 0 d r ¼ Dt m exp 1 d r ; d r < d rc d rc d rc t t 0 ¼ Dt m exp d r d rc L ; d r > d rc ð3þ where d rc is the criticl displcement t which the friction reches the pek vlue. A typicl frictionl stress vrition s function of slip (normlized y the chrcteristic strengthening length scle (d rc ) is illustrted in Figure 1. We note tht the friction is continuous in displcement ut the derivtive of the friction is discontinuous t d r = d rc.to ensure the convergence in the process of solving nonliner equtions, locl smoothing is dopted in the numericl itertion. We point out tht one dvntge of incorporting the strengthening prt is to void the sujective choice of initil vlue of the wekening rte [Bizzrri et l., 2001]. [11] As pointed out y S. B. Nielsen (personl communiction, 2004), one key difference etween the current simplified friction lw nd the full rte-nd-stte formultion is tht the ltter, due to its rte dependence, will regin strength s the fult slows down. Thus the rte-nd-stte lw hs mens y which the fult my restrengthen with time. Our simplified friction lw hs no such mens, nd for this reson it is not sufficient to model multiple erthqukes on our fult system. However, s will e seen in the results, s the fult pproches erthquke nucletion, the slip velocity monotoniclly increses t ll times for ll points on the fult. Thus the lck of restrengthening mechnism should not ffect our results. However, it does imply tht to model multiple erthquke cycles (including the full dynmics of 3of18

4 Figure 1. () Sher stress vrition s function of normlized slip. () Norml stress vrition s function of normlized norml displcement. Note tht the norml stress vrition is n pproximtion to the step function which descries the contct ehvior in the norml direction; it is used s convenient lterntive wy to ccurtely compute the norml contct stress numericlly. rupture nd slip), more complete friction lw tht includes some sort of rte-sensitive effects should e used. [12] To incorporte vriility of norml stress due to the symmetric geometry of the dip-slip fult [Brune, 1996; Oglesy et l., 1998; Shi et l., 1998], we rewrite eqution (3) s t ¼ ms ð n Ds n Þ; ð4þ where m is the friction coefficient, s n is the in situ norml stress on the fult, nd Ds n is the vrition of the norml stress cused y slip motion in dip-slip fults. For consistency with eqution (3), m cn e ssumed to depend on the slip y m ¼ m 0 þ Dm f ðd r Þ; ð5þ where m 0 is the friction coefficient of the fult fter lrge slip nd Dm represents the vrition of the friction coefficient in the strengthening nd wekening process. The function f(d r ) is given y nlogy with eqution (3) s f ðd r Þ ¼ d r exp 1 d r ; d r < d rc ; d rc d rc f ðd r Þ ¼ exp d ð6þ r d rc ; d r > d rc : L [13] One wy to ccommodte the vriility of norml stress is to enforce constrint condition either (1) tht the norml displcement of the fult surfce is zero if the fult surfces re in contct or (2) tht the norml stress is zero if the fult surfces re seprted. For generl nonliner nlysis, however, this method is computtionlly difficult ecuse the norml stress is step function of norml displcement. To circumvent this difficulty, we dopt penlty method widely used in structurl nlysis [Hughes, 1987] y introducing n imginry spring etween the fult surfces in the norml direction. The stress nd displcement reltion of this spring my e expressed y Ds n ¼ s n ½1 expð d n =d nc ÞŠ; ð7þ where d n is the norml displcement of the fult surfce nd d nc is the chrcteristic norml displcement which cn e viewed s prmeter. Figure 1 shows typicl norml stress vrition s function of norml displcement normlized y the chrcteristic norml displcement d nc. Note tht s long s d nc is selected to e sufficiently smll compred to the criticl sher displcement d rc, reltion (7) fits the ehvior of contct pressure, i.e., the contct pressure ecomes infinitely lrge when the surfces re slightly compressed towrd ech other nd pproches zero when they re slightly seprted (in greement with the step function ehvior descried ove). In the numericl nlysis, the prmeter d nc is selected optimlly so tht further reduction of d nc results in n insignificnt numericl difference. In other words, our method converges to the result tht would e otined using method with no surfce interpenetrtion or opening. The mjor dvntge of this technique is tht the finl coupled frictionl constitutive reltion, which reltes the norml nd sher stress to the displcements of the fult surfce, is differentile everywhere, nd therefore stndrd Newton-Rphson itertion cn e used for solving the resulting nonliner eqution tht governs fult slip motion. It should e emphsized, though, tht this method does not ssume ny specil new physicl interction in the fult-norml direction; it is merely numericl convenience tht llows ccurte nd efficient clcultion of the norml stress. 3. Vritionl Boundry Integrl Formultion [14] Boundry integrl eqution methods hve een widely used to study the instility of fult slip motion. In this frmework, fults re modeled s surfces of displcement discontinuities in n elstic medium, nd the stresses on the fults re relted to the surfce displcements vi generl solutions for disloctions in the hlf-spce [e.g., Chinnery, 1963]. The resulting integrl equtions re solved y discretizing the fult surfce with finite elements. In this pper, we present generl vritionl oundry integrl formultion of this type of prolem. This method cn e used to study fults of ritrry geometry in generl nisotropic 4of18

5 Figure 2. Schemtic illustrtion of trnsforming hlfspce prolem into prolem in n infinite spce. () A crck extending downwrd t n olique ngle from free surfce. This is the configurtion we wish to consider. () The surfce nd the crck which cn e viewed together s prt of three-dimensionl crck emedded in n infinite medium. (c) A surfce step which cn lso e viewed s prt of three-dimensionl crck. elstic medi. In ddition, the vritionl formultion results in oundry integrl equtions with smller degree of singulrity tht cn e treted numericlly with high precision. The detils of this development re descried y Xu nd Ortiz [1993], Xu et l. [1995], nd Xu [2000], so only rief summry is given here with focus on its ppliction to erthquke fult systems. [15] Consider generl fult (displcement discontinuity) in n infinite elstic medium, nd let d(x, t) denote the displcement discontinuity of the fult surfce. The totl energy of the system my e expressed s ½dðx; tþš ¼ W½dðx; tþšþv ½dðx; tþš P½dðx; tþš; ð8þ where W is the elstic strin energy of the fult system, V is the potentil energy ssocited with the constitutive frictionl reltion etween the fult surfces, nd P is the externl work due to externl remote tectonic loding, which cn e otined y integrtion on the fult surfce y using the principle of superposition. Only long-term preseismic slip ehvior is modeled here, so the kinetic energy is ignored. By modeling the displcement s continuous distriution of disloction loops nd using the known expression of interction energy etween two disloction loops [Lothe, 1982], the elstic strin energy for generl nisotropic solids cn e otined s [Xu, 2000] W½dðx; tþš ¼ 1 Z Z Z 1 2p 16p 2 e i ðn rd i Þ S S R 1 z; n rd j zš ðn rd i Þ 1 z; z ðz; zþ z; n rd j 2 z ej djds 1 ds 2 ð9þ where S represents the Erth s surfce nd fult plne; ( ) 1 nd ( ) 2 denote two different points on the surfce; R is the distnce etween these two points; e i, i = 1, 2, 3, re Crtesin sis vectors; n is the unit norml on the surfce S; z represents ny unit vector in the plne perpendiculr to R (vector etween the two points), nd j is the polr ngle of z within the plne. The components of the second-rnk tensor in the nottion (, ) re defined s (, ) jk = i c ijkl 1, where c ijkl re elstic constnts. [16] The evolution of fult slip D(x, t) cn e solved y rendering the totl energy [D(x, t)] sttionry through the vritionl principle. The Euler eqution reltes the stress on the fult to the displcement of the fult surfce, exctly corresponding to the integrl eqution tht is widely dopted in previous studies. We use six-node tringulr elements to discretize the fult surfce. The resulting eqution is nonliner due to the nonliner frictionl constitutive reltion nd is solved y Newton-Rphson itertion. A uniform slip rte oundry condition t the ottom of the fult is enforced y Lgrnge multipliers. As common prctice, we choose the strting point s homogeneous stedy stte nd only the vrition of the stress nd slip displcement re of concern in the nlysis. The instntneous slip rte of ech node is clculted y V ¼ dnþ1 r d n r Dt ; ð10þ where d r n+1 nd d r n re slip displcements t the n + 1 nd n time steps. Dt is the dptive time step tht is djusted ccording to the mximum slip rte on the fult, so tht the mximum slip displcement increment on the fult t ech step does not exceed criticl vlue. The computtion stops when the mximum slip rte ecomes infinitely lrge, nd the time step ecomes infinitely smll, corresponding to slip instility t the loction where mximum slip rte occurs. [17] We note tht the ove formultion is only pplicle for fult system in n infinite elstic medium. To include the Erth s trction-free surfce, it is typicl to modify the elstic strin energy sed on the generl solution of disloction loops in hlf-spce [Bcon nd Groves, 1970; Eshely, 1979]. Alterntively, we propose strightforwrd pproch tht llows for the use of the existing vritionl oundry integrl method for hlf-spce prolems. In this pproch, the Erth s surfce is modeled s prt of very lrge crck emedded in n infinite solid. As the size of this crck is selected to e much lrger thn ny relevnt size of the fult configurtion, the prolem cn then e solved in n equivlent system tht is essentilly com- Figure 3. Schemtic geometry of dip-slip fult illustrting the definitions of vrious quntities to e used in the text nd Figures of18

6 Figure 4. Sequence of sher stress on the fult. () Norml fult. () Thrust fult. The down-dip coordinte is denoted y r; the free surfce is on the left of ech plot (t r/d = 0), nd the se of the fult (r/d = 1, t which loding is pplied) is on the right. Erliest stress snpshots re close to zero over most of the fult; the stress peks move to the left with incresing time. Note tht the stress evolution is different for the two fults, with higher qusi-sttic crck propgtion speed for the thrust fult thn for the norml fult. posed of the fult plne nd the crck emedded in the infinite solid. This pproch nd its dvntge of solving hlf-spce prolems tht involve nonplnr surfces re schemticlly shown in Figure 2. The resulting free surfce is trction-free s desired. 4. Physicl Model [18] In this work, we focus on nlysis of the geometric effects on erthquke nucletion on dip-slip fults. Consider plnr dip-slip fult tht intersects the Erth s surfce, s shown in Figure 3. The dip ngle of the fult is denoted y, nd the down-dip length of the fult is denoted y D. The Crtesin coordintes re selected so tht the z xis is norml to the Erth s surfce nd the y xis is long the intersection of the fult nd the Erth s surfce. The lithosphere is ssumed homogeneous liner elstic plte lying on top of relxed sthenosphere. The effects of tectonic loding re pproximted in the following mnner: the fult is driven y imposing uniform slip rte, equl to the long-term plte velocity V se t the fult ottom. This oundry condition ws dopted from one used in the nlysis of slip sptiotemporl complexity on strike-slip fult [Rice, 1993]. Although the present method is pplicle for generl three-dimensionl configurtions, for simplicity, only plin strin is considered in this Figure 5. Sequence of the vrition of norml stress on the fult. () Norml fult. () Thrust fult. The down-dip coordinte is denoted y r; the free surfce is on the left of ech plot (t r/d = 0), nd the se of the fult (r/d = 1, t which loding is pplied) is on the right. Erliest stress snpshots re close to zero over most of the fult; the stress peks move from zero with incresing time. Note tht the norml stress increment hs different sign nd mplitude for the norml nd thrust fults. 6of18

7 Figure 6. Sequence of the distriution of fult slip. () Norml fult. () Thrust fult. The free surfce is on the left of ech plot (t r/d = 0), nd the se of the fult (t which loding is pplied) is on the right. Erliest slip snpshots re close to zero over most of the fult; the slip increses with incresing time. The qusi-sttic crck propgtes frther updip on the thrust fult thn on the norml fult in n equivlent mount of time. preliminry study. Thus the displcement nd sher stress on the slip plne re ssumed to e uniform long the y xis. In the numericl nlysis, periodic condition is pplied in the y xis direction so tht the computtion cn e reduced to finite domin. The detils of this tretment re given y Xu nd Ortiz [1993]. It should e noted tht while we impose the slip rte t the se of the fult, the prtitioning of the resulting stress lod into increments of sher nd norml stress on the fult is clculted result of the model nd depends on the symmetric fult geometry s well s the qusi-sttic dvnce of the crck on the fult. [19] We first discuss in reltive detil the rupture of 45 dipping norml nd thrust fults to introduce the quntities tht we will nlyze nd to explin some of the physicl mechnisms ehind the differences etween thrust nd norml fult nucletion. We will then investigte how nucletion loction, nucletion time, nd mount of preseismic surfce slip depend on the frictionl nd geometricl prmeters of our model. Except s noted in the text nd figures, our model prmeters re selected to e s n = 100 MP, m 0 = 0.6, V se = m/s, nd d nc = 0.02 m. [20] Physicl nd geometricl prmeters re selected to correspond to typicl fults nd erthquke events. Some of them cn e either determined y lortory friction experiments or djusted y n oserved ground deformtion versus time curve. It is noteworthy to mention tht Figure 7. Sequence of the distriution of fult slip rte. () Norml fult. () Thrust fult. The free surfce is on the left of ech plot (t r/d = 0), nd the se of the fult (t which loding is pplied) is on the right. Erliest slip rte snpshots re close to zero over most of the fult; the slip rte increses monotoniclly with incresing time. Note tht the thrust fult reches its instility (the pek of the slip rte snpshots) erlier nd frther updip thn the norml fult. 7of18

8 Figure 8. Loction of unstle slip for the norml nd thrust fult. The free surfce is on the left of ech plot (t r/d = 0), nd the se of the fult (t which loding is pplied) is on the right. The nucletion loction is indicted y the divergence the V mx /V se curve. The nucletion point is frther updip for the thrust fult (mrked on the horizontl xis with n cross) thn for the norml fult (mrked on the horizontl xis with circle). lortory mesurements of the chrcteristic slip-wekening distnce L re typiclly the 0/ mm rnge. This result, however, is otined y inducing smll slip on lortory-prepred fult surfces tht re smooth t wvelengths in the order of millimeters [Dieterich, 1981; Biegel et l., 1989]. Lrger vlues of L pper to e justified to ccommodte scling to lrger wvelengths of fult roughness. Although it is still uncler how this Figure 9. Time for unstle slip to occur for the norml nd thrust fult. The horizontl xis is the totl se slip normlized y the down-dip fult length, which is proxy for the elpsed time in the simultion. The time of nucletion is indicted y the divergence of the V mx /V se curve. This time is erlier for the thrust fult (mrked on the horizontl xis with cross) thn for the norml fult (mrked on the horizontl xis with circle). Figure 10. Mximum slip rte s function of surfce slip for the norml nd thrust fult. This plot indictes tht the surfce slip t the time of rupture nucletion on the thrust fult (mrked with cross on the horizontl xis) is lower thn the surfce slip t the time of rupture nucletion on the norml fult (mrked with circle on the horizontl xis). prmeter should e selected for relistic fult system, we follow conventionl pproch y selecting L in the order of centimeters [Rice, 1993; Kto nd Hirsw, 1997; Scholz, 1998]. 5. Results 5.1. Comprison of 45 Dipping Norml nd Thrust Fult [21] We strt y modeling 45 dipping thrust fult nd 45 dipping norml fult. These fults hve identicl physicl properties nd initil conditions, with the exception of the sign of the slip rte imposed on the se of the fult. One of the key physicl processes tht determine the nucletion ehvior of dip-slip fults is illustrted in Figure 4. Figure 4 shows snpshots of the sher stress increment long the fult t sequentil points in time. Note tht the incrementl sher stress is initilly uniformly zero everywhere on the fult. However, due to loding from elow, the sher stress uilds up ner the se of oth the norml nd thrust fult. This stress uildup is followed y sher stress drop s the lower prt of the fult egins to slip nd trverse its slipwekening pth. Over time, the sher stress pek migrtes up the fult towrd the Erth s surfce, tking the form of qusi-sttic rupture front. At the sme time, due to the effect of the Erth s surfce nd the fult s symmetric geometry, the norml stress lso experiences n increment [Brune, 1996; Nielsen, 1998; Oglesy et l., 1998; Shi et l., 1998; Brune nd Anooshehpoor, 1999; Oglesy et l., 2000, 2000]. As shown in Figure 5, s the rupture front on the norml fult ners the Erth s surfce, the norml stress increses (i.e., ecomes less negtive nd therefore less compressive) hed of the rupture front nd decreses ehind the rupture front. In other words, the norml fult is unclmped hed of the rupture front, nd clmped in the more rpidly slipping region ehind the rupture front. 8of18

9 Figure 11. Loction of unstle slip for vrious chrcteristic sliding distnce L. () Norml fult. () Thrust fult. See the cption of Figure 8 for explntion of xes. L mkes little difference in the nucletion loction of the norml fult, while the nucletion loction move updip with incresing L on the thrust fult. The opposite pttern is seen in the cse of the thrust fult. This effect leds (through eqution (4)) to corresponding chnge in the frictionl sher stress in Figure 4: As the norml fulting rupture front ners the surfce, its pek sher stress level decreses, while for the thrust fult the pek sher stress increses. [22] Snpshots of slip (Figure 6) nd slip rte (Figure 7) long the norml nd thrust fults show some of the effects of the norml nd sher stress evolution. In prticulr, we see lrger preseismic slip nd higher slip rtes t the Erth s surfce for the norml fult thn for the thrust fult. This difference is clerly due to the decresed compressive stress in the norml fulting cse reltive to the thrust fulting cse. Another ovious feture in Figure 7 is the onset of seismic instility, illustrted y the concentrtion of high slip velocity in smll re on oth the norml nd thrust fults. The loction t which this runwy slip velocity occurs is more esily seen in Figure 8, which displys the pek slip velocity (scled to the se slip velocity) s function of position on the fult. The thrust fult experiences unstle slip frther updip (0.28 of the fult from the free surfce) thn the norml fult (0.39 of the fult from the free surfce). [23] It is importnt to note tht while our qusi-sttic models show stress drop nd slipping region propgting updip, this is not the sme thing s dynmic rupture propgtion: our stress drop propgtes t low speed in stle fshion. Only fter the onset of instility in slip is true dynmic rupture reched, nd n erthquke egins. At this time, our qusi-sttic modeling method reks down, nd we cese to descrie the evolution of the system. At this point fully dynmic modeling method is required to chrcterize Figure 12. Time for unstle slip to occur for vrious chrcteristic sliding distnce L. () Norml fult. () Thrust fult. See the cption of Figure 9 for explntion of the xes. The time until unstle slip increses with incresing L for oth norml nd thrust fults. 9of18

10 Figure 13. Mximum slip rte s function of surfce slip for vrious chrcteristic sliding distnce L. () Norml fult. () Thrust fult. See the cption of Figure 10 for explntion of the xes. The surfce slip t the time of instility is lrger for the norml fult thn for the thrust fult, nd oth times increse with incresing L. the unstle propgtion of rupture nd slip, nd the resultnt ground motion. [24] Interestingly, even though the slip instility occurs frther from the fult se in the thrust fult, it tkes less time for the thrust fult to rech this instility. Figure 9 displys the pek slip velocity s function of the se displcement scled to the fult size, which is relted through the constnt se slip velocity to the mount of time since the strt of the model. The thrust fult reches unstle slip t reltive se displcement of , corresponding to 390 yers t our sl slip rte. The norml fult reches instility 50 yers lter. Thus the propgtion of the nucletion zone updip is fster for the thrust fult thn the norml fult. A finl quntity of interest is the mount of preseismic surfce slip t the time of instility nucletion. Figure 10 shows the mximum fult velocity s function of the mount of surfce slip, scled to the frictionl strengthening distnce. As ws shown in the plots of slip (Figure 6) nd slip rte (Figure 7), the thrust fult hs significntly less surfce slip (0.16) t the onset of its instility thn the norml fult (0.54) t the onset of its instility Effect of Vritions in Criticl Slip-Wekening Distnce L [25] The first frictionl prmeter we vry is L, the slip distnce over which frictionl stress drops from its pek to its stedy sliding vlue. As L increses, so does the frcture energy expended, so we would expect the nucletion process to proceed more slowly for lrger vlues of L. The loctions of unstle slip for norml nd thrust fult for vrious vlues of L re shown in Figure 11. For the norml fult, L hs very smll effect on the loction of slip instility, except t very lrge vlues (for which there is no instility nucletion). In contrst, the thrust fult shows stronger, ut still rther smll dependence of nucletion position on slip-wekening distnce: lrger Figure 14. Loction of unstle slip for vrious criticl displcement d rc. () Norml fult. () Thrust fult. See the cption of Figure 8 for explntion of the xes. The loction of slip instility moves down dip with incresing d rc in similr mnner for oth norml nd thrust fults. 10 of 18

11 Figure 15. Time for unstle slip to occur for vrious criticl displcement d rc. () Norml fult. () Thrust fult. See the cption of Figure 9 for explntion of the xes. The time to nucletion of slip instility increses with incresing d rc in similr mnner for oth norml nd thrust fults, with lrger times for norml fults thn thrust fults. L leds to nucletion frther updip from the fult se, with L = 300 cm leding to no nucletion. Figure 12 shows tht the time of instility nucletion is strongly ffected y L for oth norml nd thrust fults, with oth fults hving longer time to instility for lrger L. Figure 13 shows tht the preseismic surfce slip ccumultion y the time of instility increses with incresing L for oth norml nd thrust fults Effect of Vritions in Criticl Slip-Strengthening Distnce [26] In ddition to modifying the slip-wekening distnce L, we my lso increse the frcture energy y modifying the criticl slip-strengthening distnce d rc. Figure 14 compres nucletion loctions for fults with vrying d rc nd shows n effect tht is opposite tht of L: s d rc increses, the loction of unstle slip moves frther down the fult, wy from the free surfce. However, s in the cse of L, we would expect tht incresing d rc would mke the nucletion process more difficult nd require more time. The results for nucletion time er out these predictions, s shown in Figure 15, which shows tht nucletion time increses with d rc for oth norml nd thrust fults. Note tht the effect of d rc is lrger for the thrust fult thn for the norml fult, so tht for very lrge vlues of d rc, the thrust fult nucletes lter thn the norml fult. Finlly, Figure 16 shows tht the mount of preseismic surfce slip increses quite strongly with incresing d rc Effect of Vritions in Frictionl Coefficient (Strength) Drop [27] The finl frictionl property tht we vry is the drop in frictionl coefficient from its mximum vlue to its stle Figure 16. Mximum slip rte s function of surfce slip for vrious criticl displcement d rc. () Norml fult. () Thrust fult. See the cption of Figure 10 for explntion of the xes. The mount of surfce slip t the time of slip instility increses with incresing d rc for oth norml nd thrust fults, with lrger surfce slip on the norml fult thn on the thrust fult. 11 of 18

12 Figure 17. Loction of unstle slip for different fult strength drops. () Norml fult. () Thrust fult. See the cption of Figure 8 for explntion of the xes. The point of instility moves down dip with incresing strength drop on the thrust fult ut is reltively insensitive to strength drop on the norml fult. sliding vlue (Dm). The effect of on nucletion loction is rther complicted. As shown in Figure 17, the nucletion loction is reltively insensitive to Dm for the norml fult. However, for the thrust fult, incresing Dm leds to the nucletion loction moving down dip. Furthermore, the effect ppers to sturte out for high Dm. Note tht for extremely low Dm, no instility tkes plce. The effect of Dm on time of nucletion, however, is much clerer, s is shown in Figure 18. The time until instility increses significntly for oth norml nd thrust fults. Figure 19 shows tht the mount of preseismic surfce slip decreses with incresing Dm Effect of Fult Length [28] In relistic fulting model, fults of different down-dip lengths would penetrte different regions of the Erth s lithosphere, nd thus hve different frictionl properties. However, it is worthwhile to investigte whether the fult length cn hve n effect on the nucletion process even in the sence of these depthdependent frictionl effects. Figure 20 shows tht fult length hs very little effect on the reltive loction of instility nucletion. However, s shown in Figure 21, the time until unstle slip increses mrkedly s the fult length increses. This result is not surprising, given tht longer fult mens tht the nucletion zone hs to trvel longer distnce efore it reches the reltively constnt reltive instility loction seen in Figure 20. The mount of preseismic surfce slip decreses with incresing fult length, s shown in Figure 22. This effect is most likely due to the incresing distnce from the stedily slipping fult se Effect of Fult Dip [29] The finl phenomenon tht we investigte is the dependence of the nucletion process on fult dip. Becuse Figure 18. Time for unstle slip to occur for different fult strength drops. () Norml fult. () Thrust fult. See the cption of Figure 9 for explntion of the xes. The time to slip instility increses on oth norml nd thrust fults with incresing strength drop. 12 of 18

13 Figure 19. Mximum velocity s function of surfce slip for different fult strength drops. () Norml fult. () Thrust fult. See the cption of Figure 10 for explntion of the xes. The mount of preseismic surfce slip decreses with incresing strength drop on oth the norml nd thrust fults. the effect of fult symmetry on the time dependence of norml stress is strongly dip-dependent [Oglesy et l., 1998, 2000, 2000], we might expect strong effect of fult dip on the nucletion process. However, s shown in Figure 23, fult dip hs reltively smll effect on the nucletion loction. Interestingly, though, the effect of fult dip is opposite for norml nd thrust fults. For norml fults, the instility loction moves updip s the dip ngle steepens. For thrust fults, however, the instility loction moves slightly down dip s the dip ngle steepens. As shown in Figure 24, the time of instility is lso only slightly ffected y dip ngle, ut in this cse, the onset time decreses with steepening dip ngle for oth fults. The most complicted reltionship is etween the mount of preseismic surfce slip nd dip ngle (Figure 25). The thrust fult s surfce slip decreses monotoniclly with steepening dip ngle. However, the norml fult s surfce slip increses etween dip ngles of 30 nd 45 nd then decreses etween 45 nd 60. As will e discussed in the next section, these different ehviors cn e interpreted in the context of the competing effects of sher nd norml stress ner the surfce of dip-slip fult. 6. Discussion [30] Our results cn e roken into three mjor ctegories: the effects of fult slip direction (norml versus thrust), frictionl fctors (slip-wekening distnce, slip-hrdening distnce, nd strength drop), nd geometricl fctors (fult length nd fult dip). These generl effects re summrized in Tles 1 nd 2. Some of the results re reltively esily interpreted, nd others re more complicted. Mny mjor fetures of our models re rodly consistent with previous work on verticl strike-slip fults. Stedy slip t the fult se results in stress concentrtion t the se, which then Figure 20. Loction of unstle slip for vrious fult lengths. () Norml fult. () Thrust fult. See the cption of Figure 8 for explntion of the xes. The loction (reltive to the down-dip length of the fult) of instility nucletion is reltively insensitive to fult length for norml nd thrust fults. 13 of 18

14 Figure 21. Time for unstle slip to occur for vrious fult lengths. () Norml fult. () Thrust fult. See the cption of Figure 9 for explntion of the xes. The time for unstle slip strongly increses with incresing fult length for oth norml nd thrust fults. propgtes updip until slip instility is reched [Tse nd Rice, 1986; Rice, 1993; Shizki, 2002]. Incresing the slip-wekening distnce nd slip-hrdening distnce oth increse the time to instility, due to oth the resultnt increse in frcture energy nd the fct tht s these quntities increse, the inherent instility of the system is decresed [Scholz, 2002]. Likewise, n increse in the strength drop increses the time to filure y incresing the energy hump to e overcome for instility to occur. Finlly, the lrger surfce slip in the norml fulting cse (reltive to the thrust fulting cse) cn e explined y the reduced norml clmping stress ner the free surfce in this cse [Brune, 1996; Nielsen, 1998; Oglesy et l., 1998]. [31] Other effects of the slip direction nd frictionl prmeters on the depth of instility nucletion re somewht more difficult to interpret. It is not ovious why the norml fult hs its instility nucletion point frther down dip thn the thrust fult, even though the time until the instility is greter for the norml fult. These results men tht the propgtion of the nucletion region is fster for the thrust fult thn for the norml fult. One possile explntion for this ehvior cn e found in the work of Oglesy et l. [1998, 2000, 2000], who show tht hed of the slipping region, the compressionl norml stress should decrese on norml fult, nd increse on thrust fult. Conversely, inside the slipping region, the compressionl norml stress should increse on norml fult, nd decrese on thrust fult. Therefore it would pper tht the thrust fult should present stronger rrier to rupture hed of the slipping region, ut would hve higher slip inside the slipping region. These two effects would pper to hve opposite consequences for the nucletion process, ut it is possile tht the incresed slip for the thrust fult is the lrger of the two effects. Thus these effects could led to n overll iding of the Figure 22. Mximum slip rte s function of surfce slip for vrious fult lengths. () Norml fult. () Thrust fult. See the cption of Figure 10 for explntion of the xes. The mount of preseismic surfce slip decreses with incresing fult length for oth norml nd thrust fults. 14 of 18

15 Figure 23. Loction of unstle slip for vrious fult dip ngles. () Norml fult. () Thrust fult. See the cption of Figure 8 for explntion of the xes. The instility loction moves updip with incresing fult dip on the norml fult, ut moves slightly down dip with incresing fult dip on the thrust fult. rupture process nd fster rupture propgtion for the thrust fult. [32] The effects of some frictionl prmeters on nucletion re lso puzzling. Incresing L nd d rc should oth increse the frcture energy, nd thus increse the criticl ptch size for instility [Dy, 1982]. Thus the nucletion region would hve to propgte frther updip for this lrger criticl ptch size to e reched. This effect is roughly wht is seen in the dependence on L. However, n increse in d rc produces the opposite effect: nucletion moves frther down dip, even though the nucletion time still increses. The decrese in preseismic surfce slip with incresing Dm cn e understood y noting tht s Dm increses, the fult experiences incresed resistnce to slip s it trverses the strengthening prt of the constitutive lw. Thus slowly slipping regions of the fult (such s the free surfce) tend to slip less for the sme mount of driving force. [33] The effects of fult geometry on the nucletion process re lso menle to interprettion in the context of erlier dynmic work on dip-slip fults. If there is preferred reltive loction for nucletion on oth the norml nd thrust fults (i.e., certin frction of the down-dip extent of the fult), it is not surprising tht the effect of fult length on the reltive loction of the onset of instility should e smll, s it is in our models. However, incresing the fult length mens tht it tkes longer for the slipping region to propgte to tht loction, leding to the incresed time to instility ssocited with incresing fult length. Similrly, incresed fult length leds to greter distnce etween the free surfce nd the stedily slipping fult se, nd resultnt smller driving force ner the surfce. This effect explins the smller mount of preseismic surfce slip for longer fults. [34] The effects of fult dip serve to illustrte the trnsition etween symmetry nd symmetry in the fult geometry, nd the resulting effects on the nucletion process. As dip ngle increses, the nucletion point moves updip on the norml fult, nd down dip on the thrust fult. This result is resonle ecuse for shllow dips, the Figure 24. Time for unstle slip to occur for vrious fult dip ngles. () Norml fult. () Thrust fult. See the cption of Figure 9 for explntion of the xes. The time for instility nucletion slightly decreses with incresing dip for oth norml nd thrust fults. 15 of 18

16 Figure 25. Mximum slip rte s function of surfce slip for vrious fult dip ngles. () Norml fult. () Thrust fult. See the cption of Figure 9 for explntion of the xes. The mount of preseismic surfce slip t the time of slip instility decreses with incresing dip ngle on the norml fult ut is complex function of the dip ngle on the thrust fult. norml fult nucletes t deeper loction (40% of the down-dip extent of the fult) thn the thrust fult (25% of the down-dip extent). As the dip ngle increses, the difference etween thrust nd norml fults should decrese, cusing the norml nd thrust nucletion loctions to converge. An ovious compliction to this result is tht norml fults typiclly hve dip ngles greter thn 45, nd thrust fults hve dip ngles <45. Thus perhps more resonle rel-world comprison is etween the 60 dipping norml fult nd the 30 dipping thrust fult. For the resons discussed ove, the difference in nucletion loction etween the 60 norml fult (35% of the downdip extent) nd the 30 dipping thrust fult (25% of the down-dip extent) is slightly less thn the cse in which two identicl dip ngles re compred. Another potentil compliction is tht in the rel world, dip ngles re often not constnt, with the shllow prt of the fult hving different dip thn the deeper prt. In-depth models of such complicted fult geometries re eyond the scope of the current pper; however, s suggested y S. B. Nielsen (personl communiction, 2004), the effect my e smll for fults in which the nucletion is elow the depth of the chnge in dip, since nucletion is likely dominted y the dip of the fult etween the loding region nd the nucletion point. [35] A somewht counterintuitive result is the complicted dependence etween the mount of preseismic surfce slip nd the fult dip shown in Figure 25. For the thrust fult, the reltionship is reltively simple: s dip ngle increses, the mount of surfce slip decreses. However, the norml fult s surfce slip increses etween 30 nd 45 nd then decreses for steeper dip ngles. This ehvior cn possily e explined y the seprte nd competing effects of the free surfce on the ner-surfce sher nd norml stresses. Oglesy et l. [1998, 2000, 2000] present n nlyticl solution for the chnges in ner-surfce stress on dip-slip fult due to slip on deeper region of the fult. If t nd s n re the sher stress increments t point due to slip somewhere else on the fult, nd is the dip ngle, then the dditionl increments in sher nd norml stress due to the presence of the symmetric ngle with respect to the Erth s surfce re given y Dt ¼ t cos 2 2; Ds n ¼ 4t sin 3 cos : ð11þ [36] The competing effects of these two terms (though the norml nd sher stress dependence on friction) led to complicted pttern of fult wekening ner the surfce due to slip t depth. The reltive fult wekening for prticulr choice of frictionl coefficient is shown y Oglesy et l. [1998, Figure 2; 2000, Figure 2]. The wekening of the thrust fult is consistent over the rnge etween pproximtely 30 nd 75 : the ner-surfce region is decresingly wekened s dip ngle increses. This decrese in the wekening is mnifested in our current study s decrese in the ner-surfce preseismic slip with incresing dip ngle. However, the norml fult s wekening is not monotonic function of dip ngle: it increses etween 0 nd 50 nd then decreses for steeper dip ngles. This pttern exctly mtches the pttern of preseismic surfce slip on the norml fult. 7. Conclusions [37] Through simple two-dimensionl qusi-sttic models of seismic loding nd instility nucletion on dip-slip fults, we find tht the nucletion process differs from tht of more symmetric verticl strike-slip fults in mny wys. Nucletion loction, time to instility, nd the mount of preseismic surfce slip re ll ffected y the direction of slip (norml fulting versus thrust fulting), frictionl prmeters, nd the fult geometry. The results my hve importnt implictions for fult dynmics nd ground mo- Tle 1. Effect of Fult Slip Direction on Nucletion Process Norml Fult Thrust Fult Nucletion loction down dip updip Nucletion time lte erly Surfce slip high low 16 of 18