along the big bend, southern California

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. B4, PAGES , APRIL 10, 1998 Viscoelastic coupling model of the San Andreas Fault along the big bend, southern California J. C. Savage and M. Lisowski 1 U.S. Geological Survey, Menlo Park, California Abstract. The big bend segment of the San Andreas fault is the 300-km-long segment in southern California that strikes about N65øW, roughly 25 ø counterclockwise from the local tangent to the small circle about the Pacific-North America pole of rotation. The broad distribution of deformation of trilateration networks along this segment implies a locking depth of at least 25 km as interpreted by the conventional model of strain accumulation (continuous slip on the fault below the locking depth at the rate of relative plate motion), whereas the observed seismicity and laboratory data on fault strength suggest that the locking depth should be no greater than 10 to 15 km. The discrepancy explained by the viscoelastic coupling model which accounts for the viscoelastic response of the lower crust. Thus the broad distribution of deformation observed across the big bend segment can be largely associated with the San Andreas fault itself, not subsidiary faults distributed throughout the region. The Working Group on California Earthquake Probabilities [1995] in using geodetic data to estimate the seismic risk in southern California has assumed that strain accumulated off the San Andreas fault is released by earthquakes located off the San Andreas fault. Thus they count the San Andreas contribution to total seismic moment accumulation more than once, leading to an overestimate of the seismicity for magnitude 6 and greater earthquakes in their Type C zones. 1. Introduction The big bend segment of the San Andreas fault is the 300- km-long section of this fault extending from 116ø30'W to 119ø20'W (Figure 1). The strike of this segment is rotated about 25 ø counterclockwise from the local tangent to the small circle about the Pacific-North American pole of rotation. The accommodation of the relative motion across the big bend segment of the San Andreas fault is distributed over a 100-km-wide zone roughly centered on the fault trace [Snay et al., 1996]. In the conventional model of strain accumulation (see Appendix) the fault is represented by a vertical cut in an elastic half-space. No slip occurs on this cut above the locking depth, but below that depth uniform slip at the secular rate of relative plate motion obtains. If the broad zone of deformation observed along the big bend is to be explained by the conventional model of strain accumulation, a locking depth of at least 25 km [Eberhart- Phillips et al., 1990] is required, a depth somewhat greater than might be expected given the dependence of strength upon depth [Gilbert et al., 1994]. Furthermore, seismicity is generally confined to the uppermost km along the big bend segment of the San Andreas fault [Hill et al., 1990, Figure 5.9B], and continuous slip (fault creep) is presumed to relieve stress at greater depths, a presumption roughly in accord with depth dependence of strength referred to above. On that basis the locking depth would be expected to be km, and even the 25-km lower bound [Eberhart-Phillips et al., 1990] on the locking depth along the big bend segment of the fault would appear to be too large. 1 Now at Hawaiian Volcano Observatory, Hawaii National Park. This paper is not subjecto U.S. copyfight. Published in 1998 by the American Geophysical Union Paper number 98JB The viscoelastic coupling model [Thatcher, 1983] of strain accumulation offers an explanation of this apparent inconsistency. The Earth model employed in viscoelastic coupling theory is an elastic layer (lithosphere) of thickness H overlying a viscoelastic (Maxwell solid) half-space (asthenosphere). The fault is represented by a vertical cut through the elastic layer, and periodic abrupt slip events with uniform slip on the cut throughout the elastic layer but not into the viscoelastic half-space are imposed. The earthquake cycle then consists of periodic abrupt slip events (earthquakes) in the elastic layer followed by deformation associated with the relaxation of the asthenosphere from the load imposed by the lithospheric slip event. Savage [1990] showed that the deformation at the free surface produced by the viscoelastic coupling model in the interseismic interval is completely equivalent to deformation that would be produced in an elastic half-space by prescribed slip over different depth intervals on the vertical cut which represents the fault: No interseismic slip occurs in the depth interval 0 to H, but a spatially uniform slip rate is prescribed over each of the depth intervals H to 3H, 3H to 5H... (2n-1)H to (2n+ 1)H... The temporal dependence of the slip rate in the depth interval (2n-1)H to (2n+ 1)H is the product of a (n-l) degree polynomial in t and the exponential exp(-i. tt/2 l), where t is the time since the last earthquake and!.t and 1 are the rigidity and viscosity of the Maxwell solid that represents the asthenosphere. The details of the equivalent half-space model are given in the appendix. For reasonable viscosities in the Maxwell-solid representation of the asthenosphere the earthquake perturbation does not penetrate deeply into the equivalent half-space, and the slip rates prescribed for the deeper intervals are essentially uniform in time, being equal to the secular rate of plate motion. Moreover, the slip rate in the shallowest asthenosphere depth interval, H to 3H, decreases exponentially (exp(-i. tt/2 l)) with time after the 7281

2 7282 SAVAGE AND LISOWSKI: VISCOELASTIC COUPLING MODEL A4-, POUCE GNEISS PAJUELA BALD '"" i PORTAL PALMDALE TEHACHAPI LEO.^, 10 km SOLEDAD BENI, A.N,D REAS LOCKED BURN' RID GAR, WHITAKER WARM SPR! TOM H / UES I TE IH SAN GABRIEL "%... PICO NCER 1994 s M6.7, MTGLEASN r 1987 a 5.9 ELS M6.0 M=6.0 Figure 1. Map of the big bend of the San Andreas fault (heavy sinuous diagonal line) in southern California showing the Palmdale (grey inverted triangles), San Gabriel (open triangles), and Tehachapi (black triangles) trilateration networks. The open circles show the initial points for the fault segments shown in Table 1. Lighter sinuous lines show the Garlock (G.), Elsinore (E.), and San Jacinto (S.J.) faults. Stars show epicenters of the important earthquakes in the interval. Notice the proximity of trilateration station Pico NCER to the epicenter of the Northridge earthquake (1994, M=6.7). preceding earthquake. Thus it is possible late in the seismic cycle for the viscoelastic coupling model to produce surface deformation, which, inverted on the basis of an elastic halfspace model, would appear to involve no slip on the fault in the depth interval 0 to H, very low slip rates on the fault in the H to 3H depth interval, and a slip rate near the secular plate rate below 3H. Interpreted in terms of the conventional model, that deformation would be attributed to a locking depth near 3H, 3 times the thickness of the seismogenic (elastic) layer. The following conventions are employed in this paper. Extensional strain is reckoned positive. Uncertainties quoted in the text are standard deviations, but error bars in figures represent 2 standard deviations on either side of the plotted point. 2. Fault Models Feigl et al. [1993, Table 5] employed a conventional fault model to represent the San Andreas fault system. The fault trace was approximated by rectilinear segments (Table 1), and the fault surface was approximated by downward vertical extensions of those segments. The San Andreas fault itself is represented by seven such segments (SA1 through SA7), the first and last of which are semi-infinite. The Gatlock fault (GAR) is represented by a single 160-km-long segment, and the Elsinore (ELS) and San Jacinto (SJC) faults are represented by semi-infinite segments. The initial points for the San Andreas segments SA4, SA5, SA6, and SA7, the Gatlock segment (GAR), the San Jacinto segment (SJC), and the Elsinore (ELS) are shown in Figure 1; the initial points for Table 1. Fault Segments for Dislocation Modeling Segment Origin Length Azimuth Latitude Longitude km Right-Lateral Slip Rate mm/yr Locking Depth mm/yr SA1 36ø36 ' 121o13 ' oo N42.0øW SA2 36ø36 121o13 ' 89.3* S38.1øE 34 1 SA3 35o58 ' 120o35 ' 158.3' S42.6øE SA4 34o55 ' 119o23 ' 97.2* S73.4øE SA5 34o40 ' 118o22 ' 117.1' S61.7øE SA6 34o10 ' 117o15 ' 65.0* S73.4øE SA7 34o00 ' 116o35 '* oo S47.0øE GAR 34o49 ' 118o56 ' N57.0øE ELS 33o54 ' 117o40 ' oo S55.0øE 5 15 SJC 34o18 ' 117o32 ' oo S48.0øE * These values correct apparentypographical errors in Feigl et al. [1993, Table 5].

3 ß SAVAGE AND LISOWSKI: VISCOELASTIC COUPLING MODEL 7283 :':':"?ii i:11;i!il :ii:i!?i!:!ii:i:iiii?:!ii:iii:iii?i i! i,::,:: ::i,::::: ::::::::::::::::::::::::::::::::::::::::! i!i!: ii :}iii iiiiii il;!!iii!i i!:i!ii:11 :i!!i! i ;i ii!;:/ :-i j i '._....:.. (a) function Figure 3b. of Clearly velocities the model predicted accounts by the for Feigl most model of the in,:,,:¾-:?:i::i::i::i::i::i:i,i,i::i:i:: ii'.-'i. '.. /! observed deformation, but there are systematic local residuals [... : : i deformation. [Feinl et at., We 1993] show which here that require the local Feigl source model also of ,. : : >.:.....:.,,,:,,...,..:.:..:.... accounts for most of the deformation observed in the? i ;.:::::. '"'::?"i"i::'""--"-:: "" '" '" '"' '""" ' " :--. i:ii ' ' i-i. i!!! ii ' i'! i... ' trilateration networks of Figure 1. We investigate a few I ":"::: '""!i!.':'."..i ' "'"'"" '"'"' 0::..::: --"N10, variants ofthemodelbutconcludeth!1 ' i ' '"'... ' '.'-.!-":: % ::.!.!':'::: :: :: ::.Y "...":...!.! I -":: ::,::,::,,:,-."::':',::,,::-:.'":':X :::: '!!!!ii%:: :: 117 ;..x.. ' "... '"::-: ' ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::... ', taken significantly into account strain [Shen accumulation et al., 1996]. other faults Notice must that be the Feigl modelwi thout the Garlock fault segm eat is in effect [--'10 i!i a model of plate motion. The fault system extends to infinity ::i...,j::--':'..:'?...: i!¾.':':,!ii.!::i ':..,:., '% both to the northwest (SA1 segment) and to the southeast o i....--\...!.!...?. ""L -N...:" : ::.:--. " i VNDN i ' N '7i 33o.. v m!, \ '..,..! e,u.: 2^,x.,l '.ax.., '-,. \ø't.,:3 Mexico ' ' ' ' 0 Feigl et al. [1993, Table 4] Figure 2. (a) Contour map of engineering shear strain accumulation rate (nrad/yr) implied by the Feigl model of the San Andreas fault system (Table 1). The heavy shaded line is the California state boundary, and the heavy white lines are the fault segments of the Feigl model. (b) Contour map of engineering shear strain rate (nrad/yr) from release of the accumulated slip deficit on the Feigl model prorated over the interseismic interval. the other San Andreas segments (SA1, SA2, and SA3) lie north of the area covered in Figure 1. The locking depth and the right-lateral slip rate below that locking depth are shown in'table 1 for each segment. We will refer to this model of the San Andreas fault system as the Feigl model. Figure 2a shows the surface trace of the Feigl model as constructed from the fault segments and the surface shear strain accumulation rates predicted from the model as calculated from dislocation theory [Okada, 1985]. Feigl et al. [1993] showed that the Feigl model accounted for most of the deformation observed in a network of 40 Global Positioning System (GPS) stations that covered most of southern California. The locations of those GPS stations are shown in Figure 3a, and the velocities (relative to the Pacific plate) observed at those stations are plotted as a 10 " ) t ' e_"; &,l.t= I m.. I' East I i o 1 &,, Calculated Velocity, m yr Fibre 3. (a) Map of the GPS network established by Feigl et al [1993] in southern California. Sinuous grey lines represent faults. Stations on the perimeter of the network are identified by name. (b) Plot of the velocity (measured relative to the Pacific plate) of GPS stations observed by Feigl et al. [1993, Table 4] versus the velocity predicted by the Feigl model. The open circles represent the north component of velocity relative to the Pacific plate and the solid circles the east component. The error bars represent 2 standard deviations on either side of the plotted point. The solid line has a slope of 1 and a 0 intercept; the dashed line is the best linear fit to the data. I

4 7284 SAVAGE AND LISOWSKI: VISCOELASTIC COUPLING MODEL (SA7, Elsinore, and San Jacinto segments). The fact that a 34 mm/yr slip rate has been imposed at depth on this fault system requires that at large distances from the fault the block southwest of the fault system (the Pacific plate) must move at 34 mm/yr relative to the part of the block to the northeast (North American plate) similarly remote from the fault system. Thus the Feigl model should account for a substantial fraction (34/48) of the total relative plate motion (about 48 mm/yr across southern California). In this context the big bend in the San Andreas fault is an impediment to the relative plate motion: Right-lateral slip alone on the big bend will not completely accommodate the relative plate motion. Deformation of three trilateration networks (Palmdale, San Gabriel, and Tehachapi; see Figure 1) spanning the central section of the big bend segment of the San Andreas fault has been measured by repeated surveys over the interval. In each survey the distances (shown by lines in Figure 1) between the same pairs of intervisible geodetic monuments were measured. The deformation is defined by the observed changes in those distances. Savage and Lisowski [1995] have plotted the measured distances in the Palmdale, San Gabriel, and Tehachapi networks as a function of time and shown that the data are well represented by linear fits to those plots. That is,.the deformation is adequately described by the rates of change in distance (dl/dt) of the individual lines. Here we use these values of dl/dt, with standard deviations determined from the residuals from the linear fit, as the fundamental measure of deformation. We used dislocation modeling to calculate the values of dl/dt predicted by the Feigl model for the lines in the Palmdale, San Gabriel, and Tehachapi trilateration networks. In Figure 4 we have plotted the values of dl/dt actually observed against those predicted values. In Figure 4a all of the data are shown, whereas Figures 4b, 4c, and 4d show the data for individual networks. Notice that the Palmdale data (Figure 4c), which involve relatively short lines, are shown at a different scale. Agreement between model and observation would require that the plotted points lie along the lines (observed = calculated) of slope 1 shown in the figures. In general most of the data do lie along those straight lines. A least squares fit to the data in Figure 4a indicates a slope of and intercept of That slope is significantly different from the expected slope of 1. The five data points that lie significantly off the line (observed = calculated) of slope 1 are labeled in Figure 4 to identify the observations involved; four of those lines are in the San Gabriel network. The sum of the squares of the 71 normalized residuals (Z{(dLi/dt)obs-(dLi//dt)ca c }2/( i2 where ( i is the Figure 4. Plot of the observed values of dl/dt in the three trilateration networks versus the value of dl/dt calculated from the Feigl model of the San Andreas fault (Table 1). The error bars represent 2 standard errors on either side of the plotted points. (a) All data, (b) data from the Tehachapi network only, (c) data from the Palmdale network only, and (d) data from the San Gabriel network only.

5 SAVAGE AND LISOWSKI: VISCOELASTIC COUPLING MODEL 7285 E E i, i, i, i, i, i, i, i, i, i, i 15-km Loc Figure 5. Plot of the observed values of dl/dt in the three trilateration networks versus the value of dl/dt calculated from the Feigl model of the San Andreas fault (Table 1) with all 25 km locking depths changed to 15 km. The error bars represent 2 standard errors on either side of the plotted points. standard error in observing dli/dt)is 320 (dimensionless), whereas the sum of the squares of the 71 normalized line length change rates (Z{(dLi/dt)ob s } 2/( i2) is Thus the Feigl model reduces the total variance by 80%. In Figure 5 we show a similar comparison of observed values of dl/dt with those calculated from a dislocation model identical to that in Table 1 except locking depths of 25 km have been replaced by locking depths of 15 km. A linear least squares fit to the data in Figure 5 indicates a slope of 0.613_ and an intercept of , whereas the expected values are 1 and 0. It is clear that this change in locking depth has caused the trend of the data points to deviate significantly from the expected slope 1 trend. The sum of the squares of the 71 normalized residuals is 1030, 3 times larger than that for the Feigl model described above. The implication is that a locking depth of at least 25 km is required along the big bend segment of the San Andreas fault segment if the Feigl model is to explain the data. In comparing the Feigl model to the data, we have not adjusted parameters to improve the fit. Here we attempt to improve the fit by adjusting the model parameters to fit the data. Because the viscoelastic coupling model of strain accumulation suggests that, if the data are inverted treating the Earth as a uniform half-space, different slip rates would obtain at different depth intervals on the fault (see Appendix), we have attempted to invert the observed values of dl/dt to determine the equivalent distribution of slip with depth in a uniform half-space Earth model. To do this we use the fault model of Table 1 except that the locking depths for the SA4, SA5, and GAR segments have been set at 50 km rather than the values 25, 25, and 10 km, respectively. The aseismic slip rates on those three segments at shallower depths were then determined by a least squares fit to the observed values of dl/dt in the Palmdale, San Gabriel, and Tehachapi trilateration networks. To avoid discontinuities in slip within the area where deformation was measured, we required the slip distribution to be the same on the SA4 and SA5 segments. Even so some discontinuity will be involved due to the slightly different (~10 ø) strikes for the two segments. There are four slip rates to be determined the least squares fit, right-lateral slip in the km and km depth intervals both on the SA4-5 (combined SA4 and SA5) segment and on the GAR segment. The best fit slip rates are shown in Table 2 (Model A), and the fit of the observed values of dl/dt to the calculated values are shown in Figure 6. A linear least squares fit to the data in Figure 6a indicates a slope of and an intercept , in good agreement with the expected values of 1 and 0. The sum of the squares of the 71 normalized residuals is 266, a modest improvement over the value of 320 found for the Feigl model. Model A is at least marginally consistent with no significant slip down to 50 km depth on the SA4-5 segment and 10 mm/yr left-lateral slip on the GAR segment at all depths below 10 km. There are three prominent outliers in Figure 6: Pico NCER to Warm Spr and Whitaker (Figures 6a and d) in the San Gabriel network and Tenhi-Verde (Figure 6c) in the Palmdale network. Pico NCER (Figure 1) lies directly above the northwest corner of the coseismic rupture of the Northridge earthquake (1994 M6.7 in Figure 1) [Hudnut et al., 1996, Figure 1 ]. All of the measurements used in deducing dl/dt for the lines out of Pico NCER were made before the 1994 earthquake, and the anomalous values of dl/dt for those two lines are presumably due to preearthquake strain accumulation on the blind thrust associated with the Northridge event. The Tenhi-Verde anomaly in the Palmdale network (Figure 6c) is unexplained. We have constructed another least squares solution for the big bend segment of the San Andreas fault (Figure 7). In that solution we have included the slip rates below 50 km depth on the SA4-5, and GAR segments as unknowns, for a total of six unknowns. Moreover, in this solution (Model B in Table 2) the anomalous data (lines Pico NCER-Warm Spr, Pico NCER- Whitaker, and Tenhi-Verde) were omitted. A least squares fit to the data in Figure 6a indicates a slope of and intercept of , in close agreement with the expected slope of 1 and intercept of 0. The sum of the squares of the 68 normalized residuals is 170 (compare 266 for Model A and 320 for the Feigl model, both of which fits included the three anomalous lines Pico NCER to Whitaker and Warm Springs and Tenhi-Verde), whereas the sum of the squares of the 68 normalized observations is 1598 (dimensionless). Thus the fit reduces the total variance by almost 90%. The slip rates for Models A and B do not differ significantly (Table 2). Table 2. Right-Lateral Slip Rates with Standard Deviations on the San Andreas (Segments S.A.4 and S.A.5) and Garlock faults Segment SlipRate km km 50-oo km mm/yr mm/yr mm/yr Model A (deep slip rates specified) SA GAR Model B (deep slip rates unconstrained) SA GAR _ Quoted uncertainties are standard deviations.

6 7286 SAVAGE AND LISOWSKI: VISCOELASTIC COUPLING MODEL i ß i, i, i! i, i ß i ß i, J.8 i! i i i ß i ß i i i i i i i i,. Model 6' T '1 T Tt Pico NCER-Warm Spr r' 4 8!hei.. tli ; E.R.,Whl ir.! f. Diorite-Saw Ecc ). a... (,b J ' I, I, I, I, I. 8 I. I. I.,.,.,.,.,.,. 2 Palmdale 6 San 4 2 o -2 -> -1-2 '4t V I t Plcø NcER'war* spr -3-6 / /"'cø "CER'Whltaker (,d! Figure 6. Same as Figure 4 except the calculated values of dl/dt are from Model A. The dislocation modeling is unable to resolve the slip distribution in intervals as narrow as 2H in depth. The models do indicate that slip in the depth interval H to 3H is small (e.g., 8+3 mm/yr) compared to the plate velocity V= 34 mm/yr and that slip (2_+17 mm/yr) in the depth interval 3H to 5H is probably less than V. Thus the overall conclusion is that slip is significantly less than V=34 mm/yr down to at least 30 km and possibly down to 50 km. Because the dislocation models A and B are not significant improvements over the original Feigl model, we use the Feigl model in what follows. grad/yr. Thus the Feigl model indicates that the San Andreas fault system contributes to shear strain accumulation over the whole of southern California. Most of the shear strain accumulation shown in Figure 2a will be released by rupture of the San Andreas, San Jacinto, Elsinore, and Garlock faults. Figure 2b shows the expected shear strain release due to coseismic slip on those faults and postseismic relaxation prorated over the interseismic interval. The slip deficit accumulated above the locking depth in the Feigl model is released in the coseismic (depth 0 to H) and postseismic intervals. The shear strain associated with that release is calculated from a dislocation model with 3. Shear Strain Accumulation slip confined to the depth interval from the surface to the locking depth in Table 1 for each fault segment. That shear The rate of accumulation of the total shear strain (¾=œ1-œ2 strain is then divided by the duration of the interseismic where el and e2 are the principal strain rates) predicted by the interval to give the coseismic and postseismic strain release Feigl model is shown in Figure 2a. Notice that the prorated over the interseismic interval. An equivalent engineering (twice the tensor) shear strain rate is shown in the figure. The maximum predicted shear strain rate along the San Andreas and San Jacinto faults is greater than 0.32 grad/yr, and the rate of accumulation is greater than 0.08 grad/yr within a corridor extending 50 km on either side of calculation is simply to find the shear strain accumulation rate due to slip above the locking depth in the dislocation model at the slip rates in Table 1 for the various segments. The prorated coseismic-postseismic, total shear strain rate found in this way is shown in Figure 2b. As expected, Figures those faults. Moreover, appreciable (-0.02 grad/yr) 2a and 2b are very similar, particularly close to the faults. accumulation of shear strain reaches the Nevada border and However, not all of the accumulated strain can be released by extends out into the Pacific Ocean. The accumulation rate the coseismic and postseismic slip because the big bend within the Los Angeles basin (LA in Figure 2) is about 0.08 segment of the San Andreas fault is not properly oriented to

7 SAVAGE AND LISOWSKI: VISCOELASTIC COUPLING MODEL 7287 All Model B 6 i, i, i, i, i, i, i. i, Te E E Diorite-Sawmill. / (a) -6,/. -4 Diorite-Sawmill (b) -8 i ß i ß i, i ß i ß i, i ß i ß i I i I i i i i 8 I, i i i, i, i, i, i, i, Palmdale 6 E E c ß ' ß! Figure 7. Same as Figure 4 except the calculated values of dl/dt are from Model B. (d) 6 8 fully accommodate the relative motion between the North American and Pacific plates. The rate of accumulation of this residual shear strain is shown in Figure 8a. That accumulation rate is calculated from slip on the Feigl fault model at all depths (0 to infinity) at the rates given in Table 1 for each segment. The orientation of the principal compression rate for that shear strain accumulation is shown in Figure 8b. Although the strain pattern is complicated at the ends of each of the fault segments in the Feigl model, the pattern is principally a northeast-southwest contraction in the far field. That is, the the residual strain accumulation corresponds largely to the fault normal compression along the big bend, compression which cannot be accommodated by right- lateral slip on the San Andreas. The Working Group on California Earthquake Probabilities [1995, equation 8] approximated the rate of seismic moment accumulation by dmoldt = 2 g H A dadt = g! H ( dy / dt )da, (1) where g is the crustal rigidity, H is the seismogenic depth, œ is the tensor shear strain averaged over the area, ¾ is the total (engineering) shear strain at a point, A is surface area, and t is time. (Savage and Simpson [1997] have argued that (1) underestimates the actual moment accumulation rate appreciably. Moreover, the uncertainty in H [Ward, 1994, p. 1297] implies a large uncertainty in the moment rate.) In this way they distribute seismic moment accumulation across the area rather than concentrating it directly on the faults upon which it will actually be released. The advantage of this representation is that it allows the seismic moment accumulation to be inferred from the observed strain accumulation. We have calculated by numerical integration (20 km by 20 km grid) the total seismic moment accumulation over the 220 km by 220 km area over which the strain field is mapped in Figures 2a, 2b, and 8a. For g=3x10 lø N/m 2 and H=I 1 km, the moment rates are found to be 5.67x1018 N m/yr for Figure 3a, 5.57x10 8 N m/yr for Figure 3b, and 2.06x10 8 N m/yr for Figure 8a. Notice that these moment rates are not additive since the total shear strain rate itself is not additive (Figure 2a - Figure 2b : Figure 8a): Although the shear components ¾ and ¾2 are themselves additive, ¾ =(¾12+ e22) 1/2 is a nonlinear combination and is not. Apparently, 36% (2.06/5.67) of the shear strain accumulated along the San Andreas fault system must be released by rupture of faults other than the San Andreas, Elsinore, San Jacinto, and Garlock segments in the Feigl model. 4. Discussion The observed deformation along the big bend section of the San Andreas fault is consistent with a locking depth of 25 km or more along that segment of the fault. (The best

8 7288 SAVAGE AND LiSOWSKI: VISCOELASTIC COUPLING MODEL 400 t.:.....-::..-:::, ::.-...., *":' *...' i.... " " ".' ;ø',- : : ' :Z::ii::::::::::::i:.:: : ':'.' :":i:,:.. ::::::i::i :: i i :: :: i... :- :: ii::i::i::i:/' -, i'?,?... :,a' ;... % :: ::"': ' '" 'u iii::'!',!::': ' '!... ', " " -:,...:.....:.:.: : ii::iiii-' "': :. ' ', ß " :.:.....:.'::'".':' -: :..-. '., : If the interoccurrence time is about 5 times the asthenosphere ::i::!iii::::i.-:.-"!::::::iii!!i::i::i i :... '... relaxation time (Xo=2.5), the apparent slip rate in the second ":: : :"": '"' ':: ' ; the depth interval 0 to H is locked during the interseismic interval. Figure 9 shows the apparent slip rates expected for the first four asthenosphere depth intervals (H to 3H, 3H to 5H, 5H to 7H, and 7H to 9H) as a function of time for five different values of x o (see the appendix for the method of calculation). If the earthquake interoccurrence time is less than twice the asthenosphere relaxation time (xo<l), the apparent slip rates below 3H remain at essentially the secular slip rate V, and the slip rate in the uppermost asthenosphere (depth H to 3H) decays exponentially (Figure 9a). Thus late in the cycle the depth interval H to 3H will appear to be slipping at a rate somewhat below the secular rate V, and the apparent locking depth in the interpretation of the conventional model would be somewhat greater than H. depth interval (3H to 5H) in the asthenosphere shows a significant variation over the interseismic interval, but the deeper (>5H) depth intervals appear to continue to slip at about the secular slip rate V (Figure 9b). Late in the cycle the depth interval H to 3H will appear to be slipping at a rate of only about 0.2 V, and the apparent locking depth.in the interpretation of the conventional model would be about 3H. If the interoccurrence time is about 10 times the asthenosphere relaxation time (Xo=5) the earthquake perturbation extends down to 13H, and the apparent slip rates 2OO I ß r =2 / (a) t ) ½ Figure 8. (a) Rate of accumulation (nrad/yr) of engineering shear strain that will not be released by slip on the San Andreas, Elsinore, Gatlock, and San Jacinto faults. (b) Orientation of the principal contraction axes for the strain field above. conventional model that we found is the same as the Feigl 3.5 I \ model in Table 1 except that all 25 km locking. depths are 3.0, H to $H 2.5 changed to 30 km.) A plausible explanation of that apparent deep locking depth is provided by the viscoelastic coupling 2.0 i'",... $H to SH model of strain accumulation: The viscoelastic relaxation of : ' '- $H to?h the asthenosphere following an earthquake can be represented ?H to 9H by an equivalent elastic half-space model in which slip at 1.0 I depth on the fault is spatially uniform over individual depth o.o o intervals but varies with time [Savage, 1990]. The important parameters in the viscoelastic coupling model are H, the Figure 9. Slip rates for the elastic half-space model thickness of the elastic layer, T interval between equivalent to the viscoelasticoupling model plotted as a earthquakes, and Xo=gTI2 l, which is half of the earthquake function of the ratio t/r/' (where t is time after the preceding interoccurrence time measured in units of the asthenosphere earthquake and T is the earthquake interoccurrence time) for relaxation time (B/I.t). The depth intervals over which the the depth intervals H to 3H, 3H to 5H, 5H to 7H, and 7H to 9H slip rate is spatially uniform in the equivalent elastic halfspace model are H to 3H, 3H to 5H... (2n-1)H to (2n+ 1)H, at various values of T measured in units of the asthenosphere relaxation time Ti/I.t o.o,.o o.o.o];' /./'X 'x X",,.o' ',,... (.) o.o o.,.o

9 SAVAGE AND LISOWSKI: VISCOELASTIC COUPLING MODEL 7289 in the uppermost two asthenosphere depth intervals become relatively small by the end of the interseismic interval (Figure 9c). The apparent locking depth in the interpretation of the conventional model at the end of the interseismic cycle would then be about 5H. For larger values of Xo the perturbation extends deeper, and the slip rates in the uppermost four depth intervals in the asthenosphere are relatively small at the end of the interseismic interval (Figures 9d and 9e). In terms of the equivalent half-space model then the locking depth late in the seismic cycle appears to be 9H or greater. The conventional (i.e., single locking depth in a halfspace) dislocation model fits to the data for the big bend segment of the San Andreas fault require a locking depth in excess of 25 km. We interpret this as implying viscoelastic relaxation in the lower crust. That is, the deformation should be interpreted in terms of the viscoelastic coupling model. The recurrence time T for the big bend segment is 150 to 200 years [Working Group on California Earthquake Probabilities, 1995], and the segment last ruptured in Thus for the trilateration data discussed here 0.6 < t/t < 0.8. An appropriate value for the thickness H of the seismogenic layer along the big bend is 10 km [Hill et al., 1990, Figure 5.9B]. The apparent locking depth of 25 km as interpreted by the conventional model then corresponds to little slip in the H to 3H depth interval. This condition is roughly satisfied for 2.5<Xo<5 (Figures 9b and 9c), corresponding to asthenosphere relaxation times (q/p)of 15 to 40 years, values consistent with other estimates of asthenosphere relaxation time (e.g., 25 years found by Thatcher [1983] and years by Li and Rice [1987]). Pollitz and Sacks [1992] have argued for a much shorter (2.5 years) asthenosphere relaxation time, but a relaxation time that short would imply an apparent locking depth in excess of 100 km, too deep to be consistent with the trilateration data discussed here. The Feigl model (Table 1 and Figure 2) of the San Andreas fault system provides a fairly good representation of the observed deformation along the big bend segment of the San Andreas fault (Figures 3 and 4). As explained above, the 25- km locking depth implied by that model should be understood as an apparent locking depth and interpreted in terms of the half-space equivalent of the viscoelasti coupling model. The important point is that deep apparent locking depths should be 'expected late in the interseismic interval, and dislocation models should not be rejected-simply because the apparent locking depth is greater than the seismogenic zone thickness (10 to 15 km). The velocity field predicted by the model of Feigl et al. [1993] is shown in Figure 10 referred to two different points. Figure 10a shows the velocity field referred to a point near Mercury, Nevada, a location that is presumably part of the stable North American plate. The velocity vectors along the big bend segment of the San Andreas fault cut across the strike of that fault similar to the velocity field observed by Shay et al. [1996]. Figure 10b % ' O ß,',',' SO0 9. OO -SO SO0 Figure 10. Deformation implied by the Feigl model of the San Andreas fault system (Table 1). The heavy shaded line is the California state boundary, and the lighter shaded lines are the fault segments of the Feigl model of the San Andreas fault system (Table 1). The axes show distance in kilometers from Palmdale, California. (a) Velocity field measured relative to a point near Mercury, Nevada (open circle at northeast comer of velocity field). The vector in the southwest corner represents a velocity of 31.3 mm/yr. (b) Velocity field measured relative to a point near Palmdale, California (center of velocity field plot). The vector in the southwest corner represents a velocity of 14.1 mm/yr. (c) Orientation of the axis of principal contraction. loo -loo -.oo -soo -loo o loo soo

10 7290 SAVAGE AND LISOWSKI: VISCOELASTIC COUPLING MODEL shows the velocity field referred to a point near Palmdale, California, which lies along the big bend segment. In that plot one can see better how the velocity field follows the local strike of the San Andreas fault as was shown in the velocity field observed by Lisowski et al. [1991]. Figure 10c shows the orientation of the principal contraction axis as predicted by the Feigl model. Eberhart-Phillips et al. [1990] observed that along the big bend segment of the San Andreas fault the contraction axis makes an angle of close to 45 ø with the local fault strike, but farther from the fault the contraction axis tends to be oriented at 45 ø to the local tangent to the small circle drawn about the North America-Pacific pole of rotation. This is explained by the geometry of the fault model: close to the fault the local contribution (slip at depth on the local segment) dominates, but farther away segments beyond the big bend, which are'more nearly parallel to the tangent to the small circle, also contribute. The sharp corners in the fault configuration at either end of the big bend segment seem to influence the orientation of the contraction axis at surprisingly large distances from those corners (Figure 10c). The fact that the line lengths measured out of Pico NCER (Figure 1) show a greater north-south contraction rate (more negative dl/dt) than would be predicted by either the model of Feigl et al. [1993] (Figure 3) or our Model A (Figure 6) suggests that an anomalous motion in the epicentral area of the Northridge earthquake (1994 M=6.7 in Figure 1) was detected before the earthquake occurred. This is not to imply that a precursor was detected but only that interseismic deformation connected with strain accumulation on the Northridge blind thrust may have been measured. That is, a dense geodetic network in that area might have been able to recognize the existence of the blind thrust. The viscoelastic coupling model implies that strain accumulation associated with a fault will be observed at the surface at relatively large distances from that fault. That strain accumulation will be released partly at the time of fault rupture and partly in the subsequent postseismic relaxation. In addition some strain will be released by local faulting. The Working Group on California Earthquake Probabilities [1995] in using geodetic data to estimate the earthquake risk in southern California has assumed that all of the surface strain accumulation will be released seismically near the location at which it is observed. We show here how this the type A zone. That is, they include both the total moment rate ghl V and the contribution (1) from outside of the type A zone despite the fact that the latter term originates from the very same fault. Thus for the long straight fault they would find a total moment accumulation rate of dmo/dt = ghlv + 2 g/-/l I /(37 / 3t)dx. (2) To the extent that the strain field is totally due to the straight fault, 3T/3t=(VD/ ) / (D2+x 2) (see Appendix) and dmo/dt = gttlv { 1 + (2/ ) arctan(d/h)}, (3) where D is the apparent locking depth (as interpreted by the conventional model) at the present stage of strain accumulation. But the true value of dmo/dt is ghl V. Thus even if D=H, (3) yields 1.5 times the actual fault contribution. If D/H=2.27, a value appropriate to the big bend sector (D=25 km, H=I 1 km), (3) yields 1.74 times the actual contribution. The conclusion of the Working Group on California Earthquake Probabilities [ 1995, p. 399] that their predicted rate of earthquakes of magnitude >6 in southern California is about twice the average experienced over the past 150 years is partly explained by this overestimate of moment rate accumulation in the off-fault (type C) zones (see the plot of Distributed C in Working Group on California Earthquake Probabilities [ 1995, Figure 14]). The seismicity in southern California predicted by the Working Group on California Earthquake Probabilities [1995, p. 379] exceeds the historic rate, and this has led to some speculation of a possible earthquake deficit in southern California [Schiffries and Henyey, 1994]. However, Ward [1994, pp ] has shown that the geodetically measured strain accumulation in southern California is consistent with the historic seismicity. That is, the currently observed rate of accumulation of shear strain is equal to the average coseismic rate of strain release over the past 150 years. Thus, if there is an earthquake deficit in southern California, there is also a strain accumulation deficit. Moreover, the earthquake occurrence rate over the past 60 years seems to be about the same as the rate over the past 200 years: Hutton and Jones [1993] estimated from the magnitude frequency distribution of M<6.5 earthquakes over the past 60 years that the probability of a M>7 earthquake in southern California in the next 30 years was 65+14%. Savage [1994] estimated that the same probability was 67+23% based simply on the number of M>7 earthquakes in southern California in the past 200 years. (The Working Group on California Earthquake Probabilities [1995, p. 420] estimates this probability to be 86+17%.) Finally, the Working Group itself has shown that the observed moment rates for all of assumption may have caused them to overestimate the earthquake risk. Consider a region where a long, straight, strike-slip fault with apparent locking depth D and deep slip rate V is the only source of seismicity. This would be a type A fault in the terminology of the Working Group. Slip at depth on that fault generates a surface velocity field v, which is parallel to the fault trace. The engineering shear strain rate associated with that velocity field is 3v/3x where x is the southern California deduced independently from geodetic data over a 10-year interval, seismic data over a 100-year interval, and geologic data over a 1000-year interval are remarkably distance from the fault. The Working Group on California consistent [Working Group on California Earthquake Earthquake Probabilities [1995] would then calculate the moment rate ghl V for the type A zone (corridor extending a distance H on either side of the fault). (We have not included Probabilities, 1995, p. 399]. Those estimates are relatively free of the assumptions made in constructing the Working Group's seismicity model. The suggested earthquake deficit is here the additional contribution from noncharacteristic a deficit with respect to the Working Group's seismicity earthquakes within the type A zone as that contribution model, not with respec to the data. appears to have been included in the characteristic-earthquake moment rate ghlv used by the Working Goup on California Earthquake Probabilities [1995, p. 399].) In addition, the Working Group would use (1) to calculate the moment rate accumulation for the type C zones which lie on either side of Appendix: Models of Interseismic Deformation Interseismic deformation along a major transform fault is generally attributed to continuous, aseismic slip at depth on

11 SAVAGE AND LISOWSKI: VISCOELASTIC COUPLING MODEL 7291 the fault. That is, between earthquakes the uppermost segment of the fault is locked, but slip is thought to occur continuously at greater depths on the fault. In the simplest model of the process [Savage and Burford, 1973] the fault is represented by a long, planar, vertical cut in an elastic halfspace. Continuous slip occurs on the cut below the locking depth D at the secular rate of relative plate motion V; no interseismic slip occurs on the fault above the depth D. The velocity at the surface for this model is v = (V/n) arctan(x/d), and the engineering shear strain rate is (A1) 3T/at=-dv/dx = (V/(nD)) / (1 + (x/d)2), (A2) where x is the horizontal distance from the fault trace. The half width of the strain field is 2D, and half of the plate relative motion is accommodated within that half width. The viscoelastic coupling model involves periodic earthquakes that rupture through the entire lithosphere on a vertical cut in the Earth model that consists of an elastic layer (lithosphere) overlying a viscoelastic half-space (asthenosphere). Savage [1990] showed that the surface deformation for this model is identical to that in an elastic half-space model in which slip is distributed with depth on the vertical cut that represents the fault. The distributed slip velocity b n over the depth interval (2n-1)H to (2n+l)H (where H is the elastic layer (lithosphere) thickness) is uniform in space but does vary with time t after the most recent earthquake. The surface deformation is then v= (bn/ :)[arctan{x/((2n-1)h) }-arctan{x/((2n+l)h)}], (A3) n=l where x is again the horizontal distance from the fault trace, bn+ 1 = bl( :on/n!) (n!/(k!(n-k)!)ak+l(t/t) n'k, (A4) k=l T is the interval between the periodic lithosphere slip events, and a: o is defined below. In most cases of interest b n approaches the secular relative plate velocity V for n>n where N is less than 10, and it is sufficient to replace the sum in (A3) for all terms n>n by (V/ ) arctan[x/((2n+l)h)]. Explicit expressions for the slip velocities (b n) for the first 10 depth intervals (from depth H to depth 21 H) are given below b 1 = V '1: O (1 +g) exp(- : o tit ), b2 = bl :0(t/T + a2) ' b 3 = b 1 ('1:o2/2!) [(t/t)2 + 2 a 2 t/t + a3], b4 = bl 0:03/3!) [(tit )3 + 3 a 2 (t/t)2 + 3 a 3 tit + a4], b5 = bl 0:o4/4!) [(tit a 2 (tit)3+6a3(t/t)2 + 4a4t/T + as], b 6 = b 1 (a:05/5!) [(t/t) 5 + 5a 2 (t/t)4+ 4a 3 (t/t)3+ 6a 4 (t/t) 2 + 4a 5 tit + a6], b7= b 10:o6/6!) [(t/t)6 + 6a 2 (t/t)5 + 15a3 (t/t)4 + 20a 4 (t/t) a 5 (t/t)2 + 6 a 6 t/t + a7], b 8- b 1 ( :o7/7!) [(t/t) 7 +7a 2 (t/r)6+21a3(t/r)5+35a4 (t/t) a 5 (t/t)3 + 21a6 (t/t)2 + 7 a 7 t/t + a8], b9= b! (Xo8/8!) [(t/t)8+ 8a 2 (t/t)7 +28a3(t/T )6+ 56an(t/T ) a 5 (t/t) a 6 (t/t) a 7 (t/t)2 + 8 a 8 t/t + a9], blo = b 1 (Xo9/9!) [(t/t)9+ 9a2(t/T )8+ 36a3(t/T )7+84a4(t/T ) a 5 (t/t) a 6 (t/t) a 7 (t/t) a 8 (t/t) a 9 t/t + alo], where!.t and q are the rigidity and viscosity of the Maxwell solid used to represent the asthenosphere, V is the secular relative plate motion across the fault, and al=l, :o = gt/2q, g = l/(exp(%) - 1), (A5) (A6) a 2 = g, a 3 = g (1+ 2 g), a 4 = g (1 + 6g + 692), a 5 = g (1+ 14g + 36g2+24g3), a 6 - g (l+30g + 150g2+240g3+120g4), a 7 = g (1+ 62g + 540g2+1560g3+1800g ), a 8=g( g g2+8400g3 + 16, ,120 g g6 ), a 9=g(1 +254g g 2 +40, , ,520 g ,120 g6 + 40,320 g?), alo = g ( g + 18,150 g2 +186,480 g ,120 g4 + 1,905,120 g5 + 2,328,480 g6 + 1,451,520 g? + 362,880 gs). References Eberhart-Phillips, D., M. Lisowski, and M.D. Zoback, Crustal strain near the big bend of the San Andreas fault: Analysis of the Los Padres-Tehachapi trilateration networks, California, J. Geophys. Res., 95, , Feigl, K.L., et al., Space geodetic measurement of crustal deformation in central and southern California, , J. Geophys. Res., 98, 21,677-21,712, Gilbert, L.E., C.H. Scholz, and J. Beavan, Strain localization along the San Andreas fault: Consequences for the loading mechanisms, J. Geophys. Res., 99, 23,975-23,984, Hill, D.P., J.P. Eaton, and L.M. Jones, Seismicity , in The San Andreas Fault System, California, edited by R.E. Wallace, U.S. Geol. Surv. Prof Pap. 1515, ,1990. Hudnut, K.W. et al., Co-seismic displacements of the 1994 Northridge, California, earthquake, Bull. Seismol. Soc. Am., 86, S19-S36, Hutton, L.K., and L.M. Jones, Local magnitudes and apparent variations in seismicity rates in southern California, Bull. Seismol. Soc. Am., 83, , Li, V.C., and J.R. Rice, Crustal deformation in great California earthquake cycles, J. Geophys. Res., 92, 11, , Lisowski, M., J.C. Savage, and W.H. Prescott, The velocity field along the San Andreas fault in central and southern California, J. Geophys. Res., 96, , Okada, Y., Surface deformation due to shear and tensile faults in a half-space, Bull. Seisrnol. Soc. Am., 75, , Pollitz, F.F., and I.S. Sacks, Modeling of postseismic relaxation following the 1857 earthquake, southern California, Bull. Seismol. Soc. Am., 82, , Savage, J.C., Equivalent strike-slip earthquake cycles in half-space and lithosphere-asthenosphere Earth models, J. Geophys. Res., 95, , Savage, J.C., Probability of one or more M>7 earthquakes in southern California in 30 years, Geophys. Res. Lett., 21, , Savage, J.C., and R.O. Burford, Geodetic determination of relative plate motion in central California, J. Geophys. Res., 78, , Savage, J.C., and M. Lisowski, Interseismic deformation along the San Andreas fault in southern California, J. Geophys. Res., 100, 12,703-12,717, Savage, J.C., and R.W. Simpson, Surface strain accumulation and the seismic moment tensor, Bull. Seismol. Soc. Am., 87, , Schiffties, C.M., and T.L. Henyey, A possible earthquake deficit in southern California, Geotimes, 39 (6), 4, Shen, Z.-K., D.D. Jackson, and B. X. Ge, Crustal deformation across and beyond the Los Angeles basin from geodetic measurements, J. Geophys. Res., 101, 27,957-27,980, Snay, R.A., M.W. Cline, C.R. Philipp, D.D. Jackson, Y. Feng, Z.-K. Shen, and M. Lisowski, Crustal velocity field near the big bend of

12 7292 SAVAGE AND LISOWSKI: VISCOELASTIC COUPLING MODEL California's San Andreas fault, J. Geophys. Res., 101, , Thatcher, W., Nonlinear strain buildup and the earthquake cycle on the San Andreas fault, J. Geophys. Res., 88, , Ward, S.N., A multidisciplinary approach to seismic hazard in southern California, Bull. Seisrnol. Soc. Am., 84, , Working Group on California Earthquake Probabilities, Seismic hazards in southern California: Probable earthquakes, , Bull. Seisrnol. Soc. Am., 85, , M. Lisowski, Hawaiian Volcano Observatory, P.O. Box 51, Hawaii National Park, HI J. C. Savage, U.S. Geological Survey, MS977, 345 Middlefield Rd., Menlo Park, CA (Received May 15, 1997; revised December 12, 1997; accepted January 8, 1998.)