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Bureau of Reclamation
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INTRODUCTION |
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 TOP INTRODUCTION EFFECT OF CENSORING...SIMPLIFIED PSHA ANALYSISMONTE CARLO SIMULATIONS OF...3D SITE EFFECTSVARIATION OF PEAK SPECTRAL...DISCUSSIONCONCLUSIONSACKNOWLEDGMENTSREFERENCES |
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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 well-established. The duration of balanced-object persistence is a necessary quantity to make statistical inferences about the rate of exceedance or nonexceedance of ground-motion amplitudes.
The relative paucity of well-constrained age estimates for balanced rocks is a significant limitation on the statistical inferences that can be derived from balanced rocks on ground-motion amplitude rates. However, Bell et al. (1998) provide sufficient age-dating constraints to make the Mojave Desert balanced-rock sites a test bed for statistical testing.
Two classes of information must be estimated from balanced rocks to allow direct comparisons with PSHA results, ground-motion rate information, and ground-motion amplitude information. Fragility relationships between toppling and ground-motion parameters provide a means to estimate ground-motion amplitude information. Housner (1963) indicated that without estimates of the prior population of balanced objects, it's not possible to infer ground-motion 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 ground-motion 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 ground-motion amplitudes. The surviving balanced rocks are left-censored because they experienced ground-motion 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 ground-motion parameters that the surviving balanced rocks were subjected to. Ground-motion 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 0-100% of the pre-earthquake population over some period of time. This is a form of interval censoring, but clearly indicates we need supplementary balanced-rock rate information to constrain ground-motion rates.
The duration that a balanced object exists with a quantifiable fragility behavior provides constraints on ground-motion 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 balanced-rock sites, although it is generally assumed that they have persisted for at least 10,000-20,000 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 ground-motion amplitudes as a function of frequency or annual exceedance probability (AEP). Without balanced-rock persistence-duration information and balanced-rock genesis-rate information, there is not sufficient information to compare against nonexceedance rate estimates from PSHAs for balanced-rock 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 time-duration context is needed to convert exceedances into a useful binomial-censored 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 balanced-rock parameters, particularly balanced-rock 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, balanced-object toppling can be a function of ground-motion 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, balanced-object 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 balanced-rock fragility dynamics but rather focuses on statistical inference approaches that are general enough to incorporate any balanced-object fragility behavior that is appropriate.
We devise statistical tests that are relatively insensitive to or explicitly account for the multimodal, nonlinear nature of PHA-toppling relationships, to test if the persistence of the balanced rocks in Brune (1996, 1999) is consistent with site-specific PSHA and the ergodic assumption. We conduct a site-specific PSHA for the Mojave sites using site-specific shear-wave velocity estimates from Abbott et al. (2001) to construct 30-m-average shear-wave velocities (Vs30) for the ground-motion prediction relation of Boore et al. (1997) that explicitly accounts for Vs30, instead of using the lumped stiff-soil/rock and soft-soil 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 ground-motion 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 site-specific ground motion associated with repeated characteristic earthquakes on the San Andreas fault, a result that directly contradicts the conclusions of Anderson and Brune (1999).
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EFFECT OF CENSORING (SURVIVAL/TOPPLING) ON PHA STATISTICS |
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TOPINTRODUCTIONEFFECT OF CENSORING...SIMPLIFIED PSHA ANALYSISMONTE CARLO SIMULATIONS OF...3D SITE EFFECTSVARIATION OF PEAK SPECTRAL...DISCUSSIONCONCLUSIONSACKNOWLEDGMENTSREFERENCES |
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), of 0.55, consistent with empirical ground-motion
relations (Abrahamson and Silva
1997; Sadigh et al.
1997, and Boore et
al. 1997) for M 
6, but larger than empirical
ln() for larger (M > 7) earthquakes. We use the ground-motion
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
strike-slip 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 balanced-rock fragility curve using a mean toppling PHA of 0.25 and a natural-log-PHA-toppling 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 motion-distribution for surviving rocks. Similarly, the mean toppling PHAs are larger than the mean ground-motion PHAs because the fragility curve censors most of the lower tail of the ground-motion distribution. A greater proportion of rocks survive as Vs30 increases because PHA decreases with increasing Vs30. Thus, the assumption that balanced-rock mean PHA corresponds to actual ground-motion 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.
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SIMPLIFIED PSHA ANALYSIS |
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TOPINTRODUCTIONEFFECT OF CENSORING...SIMPLIFIED PSHA ANALYSISMONTE CARLO SIMULATIONS OF...3D SITE EFFECTSVARIATION OF PEAK SPECTRAL...DISCUSSIONCONCLUSIONSACKNOWLEDGMENTSREFERENCES |
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) = 0.4) relative to stiff soil
sites (ln() = 0.6). However, for the purposes of illustration the
standard ln() 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 slip-rate 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 balanced-rock 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 1857-type earthquakes on the San Andreas fault for
a 10,000-year exposure period, consistent with the inferred age of the
rocks (Bell et al.
1998).
The naïve conclusion is that the agreement between site-specific PSHA and balanced-rock toppling-acceleration 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 balanced-rock constraints and PSHA predictions requires explicit consideration of the residence time and life-cycle characteristics of balanced rocks.
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MONTE CARLO SIMULATIONS OF BALANCED-ROCK LIFE CYCLES |
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TOPINTRODUCTIONEFFECT OF CENSORING...SIMPLIFIED PSHA ANALYSISMONTE CARLO SIMULATIONS OF...3D SITE EFFECTSVARIATION OF PEAK SPECTRAL...DISCUSSIONCONCLUSIONSACKNOWLEDGMENTSREFERENCES |
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The Mojave Desert balanced rocks are a product of fracture-controlled 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 steady-state 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 life-cycle simulations.
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), as listed in
table 3. The simulations vary
all the balanced-rock 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 ground-motion 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 balanced-rock outcome. Only
simulation parameter combinations that maintained a nonzero number of
surviving balanced rocks through all 4,000 earthquake cycles were flagged as
persistent.
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), 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() 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() for mean PHAs of 0.2 g or less. The
one attribute that is independent of fragility-curve scenario is the
persistence of balanced rocks in the Mojave Desert from Brune
(1996,
1999) consistent with the
site-specific estimates of mean PHA and ln() from Boore et al.
(1997). The double-exponential
fragility-curve scenario does require the highest birth rate
(table 6) to be consistent with
a lifetime of 20,000 years and the ground-motion 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
balanced-rock birth rates used in table
6 are reasonable and certainly not excessive.
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) = 0.4 results of Rodriguez-Marek 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() of
0.4 consistent with Rodriguez-Marek 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 |
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TOPINTRODUCTIONEFFECT OF CENSORING...SIMPLIFIED PSHA ANALYSISMONTE CARLO SIMULATIONS OF...3D SITE EFFECTSVARIATION OF PEAK SPECTRAL...DISCUSSIONCONCLUSIONSACKNOWLEDGMENTSREFERENCES |
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700 m from the rock-grus contact in
figure 10). This
quantitatively illustrates that even a quarter-wavelength 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() results in
figure 10 nearly mimic the
empirical ground-motion findings of Rodriguez-Marek et al.
(2001) that thin-soil
short-period ln() for soil is about 0.15-0.2 larger than ln()
for rock.
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VARIATION OF PEAK SPECTRAL RESPONSE PERIOD WITH MAGNITUDE |
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TOPINTRODUCTIONEFFECT OF CENSORING...SIMPLIFIED PSHA ANALYSISMONTE CARLO SIMULATIONS OF...3D SITE EFFECTSVARIATION OF PEAK SPECTRAL...DISCUSSIONCONCLUSIONSACKNOWLEDGMENTSREFERENCES |
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DISCUSSION |
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TOPINTRODUCTIONEFFECT OF CENSORING...SIMPLIFIED PSHA ANALYSISMONTE CARLO SIMULATIONS OF...3D SITE EFFECTSVARIATION OF PEAK SPECTRAL...DISCUSSIONCONCLUSIONSACKNOWLEDGMENTSREFERENCES |
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Boore and Joyner (1997)
advocate moving beyond Vs30 to using a quarter-wavelength approximation to
quantify site effects. The 3D-synthetic plane-SH-wave simulation results
indicate this approach is likely to work well at sites not located close to
large lateral velocity contrasts. However, the 3D site-response results
(figures 10 and
11) and Rodriguez-Marek et
al. (2001) provide a
first-order quantitative indication of the perils of extrapolating thin-soil
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() are larger at soil sites adjacent to rock-pedestal 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 Rodriguez-Marek
et al. (2001) that
indicate that ground shaking is reduced as a function of increasing site
shear-wave velocity. However, well-constrained rock lifetimes, birth rates,
and time-dependent fragility characteristics are required to further quantify
ground-motion constraints based on balanced-rock observations.
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CONCLUSIONS |
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TOPINTRODUCTIONEFFECT OF CENSORING...SIMPLIFIED PSHA ANALYSISMONTE CARLO SIMULATIONS OF...3D SITE EFFECTSVARIATION OF PEAK SPECTRAL...DISCUSSIONCONCLUSIONSACKNOWLEDGMENTSREFERENCES |
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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 1-2 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 balanced-rock life cycles we show that the existing empirical ground-motion 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 strong-motion data (Rodriguez-Marek 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 rock-pedestal sites. Nearly all normal faults cataloged in the western United States have hangingwall basins consisting of several hundred meters to several kilometers of low-velocity 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 site-response effects are ignored.
Finally, it would be a grave error to use rock-response 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 stiff-soil sites (Rodriguez-Marek et al. 2001). Possible revisions to estimating ground-motion relations that incorporate balanced-rock information will need to rigorously account for the site velocity effects to avoid introducing biases in estimated stiff-soil sites that could result in systematic underestimation of ground-motion amplitudes.
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ACKNOWLEDGMENTS |
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TOPINTRODUCTIONEFFECT OF CENSORING...SIMPLIFIED PSHA ANALYSISMONTE CARLO SIMULATIONS OF...3D SITE EFFECTSVARIATION OF PEAK SPECTRAL...DISCUSSIONCONCLUSIONSACKNOWLEDGMENTSREFERENCES |
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TOPINTRODUCTIONEFFECT OF CENSORING...SIMPLIFIED PSHA ANALYSISMONTE CARLO SIMULATIONS OF...3D SITE EFFECTSVARIATION OF PEAK SPECTRAL...DISCUSSIONCONCLUSIONSACKNOWLEDGMENTSREFERENCES |
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