Seismological Research Letters
 © 2007 by the Seismological Society of America
INTRODUCTION
Brune (1996, 1999) proposed using balanced rocks located at a variety of distances from faults, some with relatively high slip rates (e.g., the San Andreas fault), as proxy paleogroundmotion indicators of peak acceleration. Shi et al. (1996) derived simplified relationships between peak horizontal accelerations (PHA) and toppling probabilities to convert pseudostatic field measurements of rock pedestal stability to limits on PHA consistent with rock stability and persistence. Anderson and Brune (1999) concluded that the persistence of balanced rocks 1030 km from the San Andreas fault was inconsistent with the ergodic assumption implicit in probabilistic seismic hazard analyses (PSHA) derived using the approach of Cornell (1968). The existence of a modest population of balanced rocks on the Mojave side of the San Andreas fault affords an opportunity to test the hypothesis of Anderson and Brune (1999) that the persistence of these balanced rocks is inconsistent with sitespecific PSHA and the ergodic assumption. Through Monte Carlo simulations of balancedrock life cycles we show that the existing empirical groundmotion relation that explicitly accounts for site velocity (Boore et al. 1997) is consistent with PSHA assumptions (Cornell 1968) and the persistence of balanced rocks east of the San Andreas fault in the Mojave Desert (Brune 1996, 1999).
In this paper, discussion is not limited to balanced rocks, because a wide variety of balanced objects can provide useful information on ground motions. For example, various types of monuments have the advantage that the duration that the monument has remained standing is documented in the historical record, and the fragility of the monument can be wellestablished. The duration of balancedobject persistence is a necessary quantity to make statistical inferences about the rate of exceedance or nonexceedance of groundmotion amplitudes.
The relative paucity of wellconstrained age estimates for balanced rocks is a significant limitation on the statistical inferences that can be derived from balanced rocks on groundmotion amplitude rates. However, Bell et al. (1998) provide sufficient agedating constraints to make the Mojave Desert balancedrock sites a test bed for statistical testing.
Two classes of information must be estimated from balanced rocks to allow direct comparisons with PSHA results, groundmotion rate information, and groundmotion amplitude information. Fragility relationships between toppling and groundmotion parameters provide a means to estimate groundmotion amplitude information. Housner (1963) indicated that without estimates of the prior population of balanced objects, it's not possible to infer groundmotion amplitude statistics solely from a surviving population of such objects after the occurrence of an earthquake. However, this doesn't mean that statistical information about groundmotion characteristics cannot be obtained from a surviving population of balanced objects. To do so, we need to employ statistical approaches that explicitly account for censoring. In statistics, censoring occurs when the value of an observation is only partially known.
Balanced rocks are represented by two types of censoring. The population of surviving balanced rocks represents a left censoring of groundmotion amplitudes. The surviving balanced rocks are leftcensored because they experienced groundmotion accelerations, velocities, and durations, etc. less than the thresholds required to cause toppling, but we don't know the amplitudes of the differences between the critical values required for toppling and the actual values of the groundmotion parameters that the surviving balanced rocks were subjected to. Groundmotion rate information is censored because the total population of balanced rocks through time is unknown. While we are completely ignorant of the total population of balanced rocks prior to an earthquake, we know that the number of observed surviving balanced rocks lies within the interval of 0100% of the preearthquake population over some period of time. This is a form of interval censoring, but clearly indicates we need supplementary balancedrock rate information to constrain groundmotion rates.
The duration that a balanced object exists with a quantifiable fragility behavior provides constraints on groundmotion rates. Since balanced rocks are the product of differential erosion, their fragility behavior evolves with time toward states of increased fragility and eventual static collapse. In this paper we first consider a simplified static fragility curve, then we evaluate cases of continuous increases of fragility with age. Specific estimates of durations that balanced rocks have existed with particular fragility characteristics are not available from the Mojave balancedrock sites, although it is generally assumed that they have persisted for at least 10,00020,000 years (Bell et al. 1998; Anooshehpoor, personal communication). The combination of amplitude constraints (fragility curves), time constraints (time period that balanced objects have not toppled), and the rate that balanced rocks are created are required to extract information about the nonexceedance of groundmotion amplitudes as a function of frequency or annual exceedance probability (AEP). Without balancedrock persistenceduration information and balancedrock genesisrate information, there is not sufficient information to compare against nonexceedance rate estimates from PSHAs for balancedrock sites.
Knowledge of the population of toppled objects would provide a second type of censored information (binomial censoring) that represents positive exceedance information. As discussed in O'Connell (2005), the combination of exceedance and nonexceedance information can be sufficient in itself to provide statistically useful constraints on amplitude frequency (flood frequency in the specific examples in O'Connell 2005). Again a timeduration context is needed to convert exceedances into a useful binomialcensored dataset. For instance, if n toppled objects with fragility information could be associated with toppling occurring over a time interval T, this information can be used with appropriate uncertainties in n and T to estimate peak amplitude frequency (O'Connell 2005). Toro and Cornell (2006) have begun exploring application of the approach in O'Connell et al. (2002) to the statistical analysis of balanced rocks. However, in this paper we take a different approach to illustrate some important statistical concepts and use Monte Carlo simulations to investigate the sensitivity of statistical inferences on various balancedrock parameters, particularly balancedrock survival duration and simplified fragility behavior.
It has been clearly established that the toppling of balanced objects often involves nonlinear dynamics (Yim et al. 1980a,b; Zhang and Makris 2000; Makris and Zhang 2001). For example, Yim et al. (1980a) found that small changes in vertical motions could cause objects not to topple at higher horizontal accelerations, although the same objects toppled at lower horizontal accelerations. Further, balancedobject toppling can be a function of groundmotion parameters other than PHA, including duration, spectral accelerations, peak velocities, etc. However, for the purposes of the statistical formulation focus of this paper we only consider toppling as a function of PHA. Even for PHA, balancedobject fragility curves can be complex and have multiple modes (Yim et al. 1980a,b; Zhang and Makris 2000; Makris and Zhang 2001), but even if fragility curves are complex, their shapes can be explicitly accounted for in Bayesian parametric and nonparametric frequency estimation procedures (O'Connell et al. 2002; O'Connell 2005) or Monte Carlo simulations. However, this paper does not dwell on the details of balancedrock fragility dynamics but rather focuses on statistical inference approaches that are general enough to incorporate any balancedobject fragility behavior that is appropriate.
We devise statistical tests that are relatively insensitive to or explicitly account for the multimodal, nonlinear nature of PHAtoppling relationships, to test if the persistence of the balanced rocks in Brune (1996, 1999) is consistent with sitespecific PSHA and the ergodic assumption. We conduct a sitespecific PSHA for the Mojave sites using sitespecific shearwave velocity estimates from Abbott et al. (2001) to construct 30maverage shearwave velocities (Vs30) for the groundmotion prediction relation of Boore et al. (1997) that explicitly accounts for Vs30, instead of using the lumped stiffsoil/rock and softsoil classifications implemented in Abrahamson and Silva (1997) and Sadigh et al. (1997). While we show that mean PHA hazard curves are consistent with the existence of balanced rocks in Brune (1996, 1999), it's not clear if the balanced rocks represent mean, modal, or some other statistic correlated with PHA. Consequently, we delve into the possible statistical relationships between balanced rocks and PHA. We evaluate the impact of censoring (survival versus toppling) on groundmotion statistics. Then, assuming that the current existence of the balanced rocks implies persistence of the balanced rocks over the Quaternary, we show that persistence of balanced rocks in their current locations is consistent with sitespecific ground motion associated with repeated characteristic earthquakes on the San Andreas fault, a result that directly contradicts the conclusions of Anderson and Brune (1999).
EFFECT OF CENSORING (SURVIVAL/TOPPLING) ON PHA STATISTICS
Each class of balanced rock (toppled or surviving) has its own population of groundmotion amplitudes with means and variances that are different from the means and variances associated with the total population of input groundmotion amplitudes (PHA is used to represent groundmotion amplitude for simplicity here). We construct several simple simulations to illustrate the resulting biases in the estimates of the mean PHA that result from using sample means and variances of either the surviving or toppled population of rocks. For the purposes of illustration we use a natural log PHA standard deviation, ln(σ), of 0.55, consistent with empirical groundmotion relations (Abrahamson and Silva 1997; Sadigh et al. 1997, and Boore et al. 1997) for M ∼ 6, but larger than empirical ln(σ) for larger (M > 7) earthquakes. We use the groundmotion relation of Boore et al. (1997) that allows specification of Vs30 to illustrate the effects of site velocity on mean PHA for sites with Vs30 of 760 m/s, 1,250 m/s, and 2,250 m/s. The scenario strikeslip earthquake magnitude is M 7.8 and the site is located 20 km from the closest portion of the surface trace of the fault (figure 1).
We construct a balancedrock fragility curve using a mean toppling PHA of 0.25 and a naturallogPHAtoppling standard deviation of 0.55 (figure 2), and sample from it using a uniform random number generator with 10,000 trials to obtain simulations of outcomes for the three Vs30 scenarios in table 1. Specifically, in each trial we generate a random PHA and then generate another uniform random number between 0 and 1 to sample from figure 2 to determine if the rock topples or survives. As expected, the standard deviations of surviving and toppling rocks are smaller than the actual PHA standard deviation because the fragility curve splits (censors) the populations into two distinct groups. The surviving mean PHAs are systematically smaller than the actual ground motion PHAs because the fragility curve censors most of the upper tail of the ground motiondistribution for surviving rocks. Similarly, the mean toppling PHAs are larger than the mean groundmotion PHAs because the fragility curve censors most of the lower tail of the groundmotion distribution. A greater proportion of rocks survive as Vs30 increases because PHA decreases with increasing Vs30. Thus, the assumption that balancedrock mean PHA corresponds to actual groundmotion PHA results in significant biases of 36% for a nominal National Earthquake Hazards Reduction Program (NEHRP) B site condition (Vs30 of 760 m/s), 29% for a Vs30 of 1,250 m/s, and 22% for a Vs30 of 2,250 m/s.
Table 1 shows that for a known population of balanced objects observed at many distinct independent sites (sites located at sufficient distances from each other so that ground motions become uncorrelated at the periods of interest), the mean PHA associated with the surviving balanced objects underestimates the actual mean groundmotion PHA by > 20%, with the bias increasing with increasing mean PHA. Restricting balancedobject analyses to a single site does not provide multiple independent observations from a single earthquake, but multiple highly correlated observations. It's necessary to obtain observations from sites sufficiently separated that groundmotion coherence is negligible (statistical independence) to make inferences about groundmotion variability from a single earthquake. A collection of spatially independent sites that have been exposed to repeated earthquakes provides the spatial independence necessary to evaluate the ergodic assumption from repeated earthquakes. Thus, the Mojave balanced rock sites east of the San Andreas fault (Brune 1996, 1999) provide a critical mass (eight independent sites) of balanced rocks to conduct statistical investigations of probabilistic groundmotion predictions.
SIMPLIFIED PSHA ANALYSIS
We calculate a 1%ina100year PSHA analysis using the approach of Cornell (1968) that accounts for the Mojave site Vs30 estimates from Abbott et al. (2001) to compare to Brune's (1999) PSHA with his rock toppling PHAs estimated for the Mojave sites (figure 1). Abbott et al. (2001) showed that Mojave balanced rock sites (figure 3) all have Vs30 values that place the sites in NEHRP class B (Vs30 > 760 m/s). Since the rock pedestals are located outside the shallow lowvelocity grus regions where the velocities were estimated in Abbott et al. (2001), as indicated in the photographs in Brune (1999), a Vs30 = 1,250 m/s, corresponding to the lower subgrus velocity of the two sitevelocity profiles in Abbott et al. (2001) was used in the PSHA analyses to represent typical site conditions beneath the rock pedestals. RodriguezMarek et al. (2001) found that competent rock sites had lower PHA dispersion (ln(σ) = 0.4) relative to stiff soil sites (ln(σ) = 0.6). However, for the purposes of illustration the standard ln(σ) of 0.55 from Boore et al. (1997) is used to calculate PHA hazard.
Following Cao et al. (2003) and Petersen et al. (1996), the Mojave segment of the San Andreas fault was assigned sliprate scenarios of 23, 30, and 37 mm/yr and a characteristic magnitude of M 7.4. Earthquakes within the Mojave Desert region shown in figure 4 were used to establish background earthquake recurrence for a maximum background earthquake magnitude of 6.5 (figure 5). The existence of the balanced rocks is consistent with repeated M 7.4 earthquakes on the Mojave segment of the San Andreas fault and M 5.5+ background earthquakes in the region, particularly when the censoring bias is used to adjust the toppling accelerations (figure 6). As noted by Anooshehpoor et al. (2004), dynamic toppling accelerations are typically 30% higher than quasistatic toppling accelerations. Consequently, the combined effects of censoring and toppling dynamics further shift the balancedrock toppling PHA for comparison to the PSHA results as indicated in figure 6. This implies that these balanced rocks persisted over about 48 characteristic M 7.4 1857type earthquakes on the San Andreas fault for a ∼ 10,000year exposure period, consistent with the inferred age of the rocks (Bell et al. 1998).
The naïve conclusion is that the agreement between sitespecific PSHA and balancedrock topplingacceleration estimates supports using the ergodic assumption in PSHA. Figure 6 shows there is no inconsistency, as indicated by Brune (1999) and Anderson and Brune (1999), between balanced rocks and PSHA methodology, once actual site conditions are used in a PSHA. However, a realistic comparison of balancedrock constraints and PSHA predictions requires explicit consideration of the residence time and lifecycle characteristics of balanced rocks.
MONTE CARLO SIMULATIONS OF BALANCEDROCK LIFE CYCLES
We assume that the balanced rocks in the Mojave Desert have persisted over many cycles of large San Andreas fault earthquakes, based on the estimated ages of the balanced rocks (Bell et al. 1998) and the relatively short return period of 210 years for M 7.4 Mojavesegment San Andreas earthquakes (Petersen et al. 1996; Cao et al. 2003). The Monte Carlo simulation approach models the life cycles of a collection of balanced rocks, starting with an initial population of 20 balanced rocks, by subjecting the balancedrock population to 4,000 cycles of San Andreas Mojavesegment M 7.4 earthquakes, and finds the combinations of ground motion and balancedrock lifecycle parameters that allow a nonzero number of rocks to persist through all 4,000 earthquake cycles. To do this we must specify the lifetime of balanced rocks (rate they are destroyed due to static instability associated with erosive processes), their timedependent fragility behavior, and the rate at which new balanced rocks are created.
The Mojave Desert balanced rocks are a product of fracturecontrolled differential erosion of granitic boulders (Bell et al. 1998). Even in the absence of earthquake loadings, it's necessary to continually create new balanced rocks through differential erosion to replace rocks that have toppled as they reach their static stability limits. For instance, to maintain a population of about 20 balanced rocks that have a fragility of 20,000 years, it's necessary to create a new balanced rock about every 1,000 years as older rocks are retired through erosion to their static stability limit. So to use the 44 balanced rocks from the Mojave Desert in Brune (1999) in a probabilistic analysis it's necessary to postulate the average steadystate rate that new balanced rocks are created. The existence of the balanced rocks combined with earthquake loadings places a lower bound on the rate that new balanced rocks are created and is investigated numerically in the rock lifecycle simulations.
We simulate the persistence or extinguishing of a population of balanced rocks over 800,0001,200,000year durations of 4,000 Mojavesegment M 7.4 San Andreas earthquakes at a distance of 20 km from the San Andreas fault by varying groundmotion and balancedrock parameters. The balancedrock and earthquake parameters considered are the erosion rate that creates new balanced rocks in terms of an effective birth interval in years and the time interval in years over which rocks degrade statically toward collapse, as listed in table 2, and the three timedependent fragilitycurve scenarios in figure 7.
We calculate the most appropriate sitespecific estimate of mean PHA for a M 7.4 strikeslip earthquake at 20 km using Boore et al. (1997) with a site Vs30=1250 m/s to establish the minimum mean PHA used in the simulations (table 3). The earthquake parameters include the mean San Andreas Mojavesegment M 7.4 return period, mean PHA, and mean PHA ln(σ), as listed in table 3. The simulations vary all the balancedrock and earthquake parameters over the ranges in tables 2 and 3. Balanced rocks are statically created and destroyed based on the parameters in table 2. For each earthquake, PHAs are randomly generated consistent with the groundmotion parameters in table 3, and the balanced rocks either topple or survive depending on whether the PHAs exceed the fragility PHA limits in figure 7 as a function of time within the balanced rocks' life cycles. If during any of the 4,000 earthquake cycles no balanced rocks survive, the simulation is terminated and flagged as a nonpersistent balancedrock outcome. Only simulation parameter combinations that maintained a nonzero number of surviving balanced rocks through all 4,000 earthquake cycles were flagged as persistent.
The simulation results depend strongly on the shape of the timedependent fragility curve and birth interval (rate) that new balanced rocks are created. A linear fragility curve (table 4) allows the largest range of groundmotion and balancedrock parameters to be consistent with the persistence of balanced rocks in the Mojave Desert. Linear fragility allows the largest mean PHA, largest ln(σ), longest rock lifetimes (20,000 years), and slowest rock birth rates (two new balanced rocks created over 20,000 years) consistent with the existence of the balanced rocks in the Mojave Desert (Brune 1996, 1999). As fragility develops more rapidly with age (table 5), the largest PHA (0.4 g) becomes inconsistent with the existence of the balanced rocks in the Mojave Desert and shortest lifetimes, highest birth rates, and longest return periods are required to allow existence of balanced rocks for large mean PHA and large ln(σ) scenarios (table 5). The scenario where fragility develops most rapidly (table 6) excludes all mean PHAs larger than 0.3 g and only allows the longest lifetimes and largest ln(σ) for mean PHAs of 0.2 g or less. The one attribute that is independent of fragilitycurve scenario is the persistence of balanced rocks in the Mojave Desert from Brune (1996, 1999) consistent with the sitespecific estimates of mean PHA and ln(σ) from Boore et al. (1997). The doubleexponential fragilitycurve scenario does require the highest birth rate (table 6) to be consistent with a lifetime of 20,000 years and the groundmotion parameters of Boore et al. (1997). However, the birth rate in table 6 would result in a peak population of eight independent rock sites in the absence of earthquake loadings, and there are eight separate sites in Brune (1999), so it's clear the balancedrock birth rates used in table 6 are reasonable and certainly not excessive.
If the rock site ln(σ) = 0.4 results of RodriguezMarek et al. (2001) from the 1994 M 6.7 Northridge and 1989 M 7.0 Loma Prieta earthquake prove appropriate in general, tables 4, 5, 6 indicate that even larger mean PHAs than predicted by Boore et al. (1997) for a Vs30 = 1,250 m/s are allowed by the persistence of the balanced rocks in the Mojave Desert in Brune (1999). A ln(σ) of 0.4 consistent with RodriguezMarek et al. (2001) allows mean PHAs 50% larger than Boore et al. (1997) to be consistent with the persistence of balanced rocks in the Mojave Desert. This becomes a progressively more important issue as annual exceedance probability (AEP) become smaller in PSHA investigations.
3D SITE EFFECTS
In addition to balanced rocks, important engineered structures, such as concrete dams, tend to be situated on the stiffest materials in a region. Using 3D finitedifference groundmotion simulations with correlatedrandom velocity fluctuations in the top 2 km of the crust, O'Connell (1999) found systematic reduction of mean PHA and PHA dispersion with increasing local site velocities. Here a deterministic 3Dvelocity structure more relevant to the conditions at the Mojave Desert balancedrock sites is used to compare groundmotion responses near a lateral discontinuity between rock and grus. A 3D cylindrically symmetric model in plan view (figure 8) has a 1kmradius core of rock with Vs = 1,250 m/s at the surface surrounded by a progressively increasing depth of grus with Vs = 300 m/s at the surface (figures 9 and 10), with a Vs vertical velocity gradient of 0.5 m/s/m in both the rock and grus portions of the model located above 2.5 km depth. A 10m node spacing is used in the shallow (< 0.2) km portion of the model that was increased to 30m node spacing at depths > 0.2 km using the 3D finitedifference approach of Liu and Archuleta (2002). The model is 8 km wide and 3.5 km deep. The 3D variablespacing viscoelastic finitedifference method of Liu and Archuleta (forthcoming) is used with the attenuation parameters indicated in table 7 to simulate ground motions to 5.5 Hz associated with a uniformamplitude, plane SHwave incident 10 degrees from vertical incidence in the homogenous depth region of the model (> 2.5 km).
Acceleration seismograms were extracted from the SHwavepolarized horizontal component from 90 sites located at 4degree intervals in each of 43 receiver rings at a range of radii from the center of the model (figure 8). A 5mradius increment was used for the receiver rings within 50 m of the rockgrus contact. The thinnest grus, 10 m, was next to the contact. Grus thickness increased in 10m increments at increasing distances from the contact (figure 10). Acceleration response spectra were calculated for a wide range of periods using 5% damping. There is a substantial contrast in short period (0.2 s) and longer period (1 s) responses near the rockgrus contact (figure 10).
The maximum 0.2s PSA response occurs 300 m from the grusrock contact (figure 10), although the ratio of the depth to the rock divided by the wavelength equals 0.83, not onequarter of a wavelength at this position. The second largest 0.2s PSA amplification occurs close to the grus/rock contact (figure 10), although the ratio of the depth to the rock divided by the wavelength equals 0.33 and not onequarter at this position. These deviations from onequarter wavelength amplification reflect the amplification complexity introduced by slightly nonvertical incidence, a firstorder lateral velocity discontinuity at the free surface, and an irregular increasing thickness of grus with increasing distance from the contact that results from the interference of direct S waves with edge waves produced at the rock/grus contact and along the dipping interface between rock and grus. In contrast, the maximum amplification at 1 s occurs at the position in the grus corresponding to a quarter wavelength (∼ 700 m from the rockgrus contact in figure 10). This quantitatively illustrates that even a quarterwavelength approximation is inadequate at short periods to explain amplifications near significant lateral velocity contrasts, much less simpler amplification approximations that employ Vs30. The PSA ln(σ) results in figure 10 nearly mimic the empirical groundmotion findings of RodriguezMarek et al. (2001) that thinsoil shortperiod ln(σ) for soil is about 0.150.2 larger than ln(σ) for rock.
To facilitate comparison with the site amplification investigations of Stirling et al. (2002), the ratio of 0.25 s to 1 s PSA responses are plotted as a function of position relative to the rock grus boundary (figure 11). Within 7.520 m of the rock, where the grus is thinnest, the 3Dsynthetic planeSHwave PSA amplifications at 0.25 s are 25%50% (figure 11), consistent with the same shortperiod amplifications observed by Stirling et al. (2002) adjacent to the Mojave Desert balanced rocks in 715m thicknesses of grus (as indicated by the velocity profiles of Abbott et al. 2001). Thus, the balancedrock sites that are nunataks of rock sticking out of a sea of grus are not likely to be amplified at short periods (< 0.25 s) as suggested by Stirling et al. (2002) precisely because they are not located in grus.
VARIATION OF PEAK SPECTRAL RESPONSE PERIOD WITH MAGNITUDE
Somerville (2003) noted that as earthquake magnitudes increase, rise times increase for constant mean stress drops, resulting in a shifting of peak spectral accelerations to longer periods with increasing magnitude. O'Connell and Ake (2007) use isochrone analyses to show that nearfault ground motions from strikeslip earthquakes will have peak spectral acceleration responses at longer periods than dipslip earthquakes. The available nearfault strikeslip strongmotion data (table 8 and figure 12) are certainly more consistent with maximum spectral accelerations in the 0.350.7 s period range than the 0.20.25 s predictions from Abrahamson and Silva (1997), Boore et al. (1997), and Sadigh et al. (1997). Anooshehpoor et al. (2004) demonstrated that rocktoppling probabilities decrease as ground motions are less enriched in shortperiod energy. Certainly, if the available nearfault strikeslip strongmotion data are relatively depleted of shortperiod energy for strikeslip earthquakes compared to many existing groundmotion relations, it's reasonable to assume that ground motions in the 1535 km distance range would be depleted of shortperiod energy also relative to the spectralshape predictions of Abrahamson and Silva (1997), Boore et al. (1997), and Sadigh et al. (1997).
DISCUSSION
The Monte Carlo balancedrock lifecycle simulations are quite sensitive to the assumed shape of the toppling fragility curve, particularly the rate of degradation of stability over the lifetime of erosion to ultimate static instability. These simulations show that it's possible to make some statistical inferences with prior knowledge of the lifecycle characteristics of a population of balanced objects by statistically simulating balancedobject life cycles over long durations relative to interevent recurrence times. However, some prior population information is still required to make inferences about PHA recurrence rates, although not exclusively in the form indicated by Housner (1963). As demonstrated here, quantitative rates for the static creation and destruction of balanced objects provide sufficient prior information to make quantitative estimates of PHA recurrence. This is information in addition to the durations that balanced objects exist with quantifiable fragility characteristics. Geomorphic investigations beyond the scope of Bell et al. (1998) may quantify the static components of balancedrock creation and destruction.
Boore and Joyner (1997) advocate moving beyond Vs30 to using a quarterwavelength approximation to quantify site effects. The 3Dsynthetic planeSHwave simulation results indicate this approach is likely to work well at sites not located close to large lateral velocity contrasts. However, the 3D siteresponse results (figures 10 and 11) and RodriguezMarek et al. (2001) provide a firstorder quantitative indication of the perils of extrapolating thinsoil responses to rock sites (e.g., not accounting for the actual site conditions beneath the balanced rocks or your site of interest). Both mean PHA and ln(σ) are larger at soil sites adjacent to rockpedestal sites (figures 10 and 11). The balanced rocks in the Mojave Desert in Brune (1999) are consistent with the results of Boore et al. (1997) and RodriguezMarek et al. (2001) that indicate that ground shaking is reduced as a function of increasing site shearwave velocity. However, wellconstrained rock lifetimes, birth rates, and timedependent fragility characteristics are required to further quantify groundmotion constraints based on balancedrock observations.
The difficulties of incorporating balancedrock information into PSHA analyses in many ways parallels the evolution of incorporating paleohydrologic data in flood frequency analyses, with some significant differences. The statistical methodology to incorporate paleohydrologic information in flood frequency analyses (Russell 1982; Stedinger and Cohn 1986) was developed nearly concurrently with the beginning of intensive efforts to collect and analyze paleohydrologic field data, as summarized in Baker et al. (2002). In contrast, balancedrock data have not been rigorously incorporated into PSHA analyses, but have instead been presented in simple comparisons to PSHA hazard curves (Brune 1999; Brune et al. 2006). Quantifying agedating uncertainties is a fundamental requirement in using paleohydrologic data in flood frequency analyses and balanced rocks in PSHA calculations. Relatively simple binary erosion/stability interpretations are most often used with paleohydrologic bound data (Levish 2002), although more rigorous quantitative approaches have been used (Ostenaa and O'Connell 2005). In contrast, considerable effort has been devoted to quantifying balancedobject fragility (Yim et al. 1980a,b; Shi et al. 1996; Zhang and Makris 2000; Anooshehpoor et al. 2000; Makris and Zhang 2001; Anooshehpoor and Brune 2002; Anooshehpoor et al. 2004; Purvance 2004). However, there has not been a rigorous statistical implementation of balancedobject fragility complexities that exist in dynamic toppling behavior, although complex behavior like multimodal toppling states could easily be incorporated into the discrete fragility probability distributions used in flood frequency analyses by O'Connell et al. (2002) and O'Connell (2005). For instance, instead of using the simple mean toppling PHA in the binary toppling/survival fragility curves (figure 7) in the Monte Carlo simulations, we could implement probability distributions within discrete PHA bins that incorporate appropriate toppling probabilities in each bin. Both balancedrock and paleohydrologic approaches started first with simple 1D interpretations of hydrologic models for stepbackwater modeling (O'Connor and Webb 1988; Webb and Jarrett 2002) or seismic wave propagation and site response (Stirling et al. 2002). However, as demonstrated in Denlinger et al. (2002) and O'Connell et al. (2002), rigorous 2D hydraulic modeling is necessary to obtain unbiased peakdischarge/paleohydrologic bound probabilities, and as figures 10 and 11 illustrate, it's necessary to account for realistic site conditions immediately beneath balanced rocks. Paleohydrologicbound information has often been erroneously dismissed as irrelevant because of perceived large discharge estimation uncertainties or as unusable in flood frequency analyses for quantifying the frequencies of low AEP extreme floods (Hosking and Wallis 1997; National Research Council 1999). This led to the Bayesian flood frequency implementations of O'Connell et al. (2002) and O'Connell (2005) to explicitly include complex fragility and agedating uncertainties to obtain rigorous quantitative statistical estimates of flood frequency. Similar statistical advances are needed to extract quantitative PSHA information from balanced rocks. Promising approaches include the Monte Carlo approaches used here or Bayesian approaches (Toro and Cornell 2006).
CONCLUSIONS
In the absence of information about the prior static population of balanced objects in the manner envisioned by Housner (1963), we've shown it's sufficient to quantify four types of information to estimate groundmotion recurrence from balanced objects:

The total duration that a balanced object persists (the clock starts when erosion is sufficient that the object can be toppled);

balancedobject fragility in response to ground shaking as a function of time since the onset time that erosion or some other process created a detached pedestal capable of toppling;

pseudostatic rate that balanced objects are created and destroyed independent of earthquake loadings; and

a sufficient sample size of balanced rocks (more than just one or two).
Further, to test PSHA performance, it's necessary to obtain a spatially diverse sampling at sufficient site separations to render ground motions uncorrelated. Particularly near faults, peak ground motion amplitudes can become strongly correlated at site distance separations that are smaller than correlation distances associated with coherent patches of seismic radiation from faults (asperities). For example, mean asperity diameters for M > 6.5 earthquakes are > 6 km (Somerville et al. 1999). Increased peak ground motion amplitudes are likely to be strongly correlated among sites separated by less than 12 km that are located near asperities of M > 6.5 earthquakes. Thus, large populations of rocks distributed over large areas, as in Brune (1999) or Brune et al. (2006), have more statistical information (less correlation and potential for bias) than samples from only a few distinct sites.
Through Monte Carlo simulations of balancedrock life cycles we show that the existing empirical groundmotion relation that explicitly accounts for site velocity (Boore et al. 1997) is consistent with PSHA assumptions (Cornell 1968) and the persistence of balanced rocks east of the San Andreas fault in the Mojave Desert (Brune 1996, 1999). Instead of demonstrating that there is a fundamental inconsistency between PSHA assumptions and the persistence of balanced rocks (Anderson and Brune 1999), the Monte Carlo simulations show quite the opposite. That is, once one accounts for what is physically known about site response in empirical data (Boore et al. 1994, 1997), through theoretical investigations (Boore and Joyner 1997; O'Connell 1999), and 3D wave propagation (figures 10 and 11), there is consistency between the persistence of balanced rocks (Brune 1996, 1999) and PSHA assumptions (Cornell 1968). Empirical strongmotion data (RodriguezMarek et al. 2001) indicate that sites that are truly rock sites don't shake as hard as thin soil sites, and that rock sites have smaller peak amplitude dispersion than stiff soil sites; stiff soil sites form the bulk of the data often used in “rock” ground motion relations (e.g., Abrahamson and Silva 1997; Sadigh et al. 1997; Campbell 1997; Spudich et al. 1999). There is complete consistency between observation (Stirling et al. 2002) and simulation (figures 10 and 11) that site acceleration responses on thin soft grus adjacent to rock pedestals (Abbott et al. 2001) are amplified 25%50% at 4 Hz (Stirling et al. 2002) relative to the responses on the exclusively rockpedestal sites. Nearly all normal faults cataloged in the western United States have hangingwall basins consisting of several hundred meters to several kilometers of lowvelocity sedimentary fill (Zoback 1983) that are likely to strongly amplify hangingwall ground motions relative to footwall sites that typically are located on competent rock over a broad frequency band (O'Connell et al. 2003, 2007). Thus, it is clear that it probably won't be possible to quantify potential differences between hangingwall and footwall ground motions during normal faulting, which could be substantial (Oglesby et al. 1998, 2000; Brune and Anooshehpoor 1999; Bouchon et al. 2000; Brune 2003; Shi et al. 2003; Schürch and Becker 2005; O'Connell et al. 2007), using balanced objects if 3D crustal velocity structure and shallow siteresponse effects are ignored.
Finally, it would be a grave error to use rockresponse results in most built environments because many urban areas are constructed on stiff soils; most ground motion relations (e.g., Abrahamson and Silva 1997; Sadigh et al. 1997; Campbell 1997; Spudich et al. 1999) appear to have appropriate means and log deviations for stiffsoil sites (RodriguezMarek et al. 2001). Possible revisions to estimating groundmotion relations that incorporate balancedrock information will need to rigorously account for the site velocity effects to avoid introducing biases in estimated stiffsoil sites that could result in systematic underestimation of groundmotion amplitudes.
Acknowledgments
We thank Jon Ake and David Boore for helpful reviews. This research was supported by the Bureau of Reclamation Dam Safety Research Program as part of the SITER project.
Footnotes
William Lettis and Associates, Inc.
Bureau of Reclamation