Exposure dating of precariously balanced rocks

Greg Balco∗,a,e, Matthew D. Purvanceb,c,e, Dylan Roodd,e

a Berkeley Geochronology Center, 2455 Ridge Road, Berkeley CA 94709 USAb Seismological Laboratory, University of Nevada - Reno, Reno, NV 89557

c Itasca Consulting Group, Minneapolis, MN 55401d Center for Accelerator Mass Spectrometry, Lawrence Livermore National Laboratory, Livermore CA USA

eAuthors contributed equally. Listed in alphabetical order.

Abstract

Precariously balanced rocks (PBRs) are freestanding boulders that are precarious orfragile in the sense that they could be toppled by relatively low-amplitude earthquakeground motion. They are important in paleoseismology because their continued exis-tence limits the amplitude of ground motion experienced at their location during theirlifetime. In order to make quantitative use of PBRs for seismic hazard studies, one mustdetermine when they attained their present state of fragility, that is, the point in timewhen the contact between the rocks and the pedestals on which they rest was exhumedfrom surrounding soil and the rock became vulnerable to earthquake ground motions.Cosmogenic-nuclide exposure dating can be used for this purpose, but is complicatedbecause nuclide production occurs throughout exhumation of the PBR, so the apparentexposure age of any part of the rock surface exceeds the time that the rock has actuallybeen precariously balanced. Here we describe a method for determining the length oftime that a PBR has been fragile by measuring cosmogenic-nuclide concentrations atseveral locations on the PBR surface, and linking them together with a forward modelthat accounts for nuclide production before, during, and after exhumation of the PBR.Fitting model to data yields the rate and timing of rock exhumation and thus the lengthof time the rock has been fragile. We use this method to show that an example PBR insouthern California has been fragile for 18.1 ± 2.5 ka.

Key words: Precariously balanced rocks, paleoseismology, cosmogenic nuclides,exposure dating, beryllium-10

1. Precariously balanced rocks 1

As pointed out by Shaler (1896) and numerous subsequent researchers, delicate 2

or fragile geologic features are important in paleoseismology because they would not 3

survive significant ground motions, so the fact that they are still present limits the am- 4

plitude of shaking during their lifetime. These features include, among others, balanced 5

∗Corresponding author. Tel. 510.644.9200 Fax 510.644.9201Email address: balcs@bgc.org (Greg Balco )

Preprint submitted to Quaternary Geochronology April 19, 2010

rocks, rock stacks or piles, precipitous cliffs, arches, hoodos, and some speleothems. 6

In this paper we are concerned with precariously balanced rocks (PBRs), which are a 7

special class of these features that consist of a single freestanding boulder resting on a 8

stone pedestal in such a way that it could be toppled by relatively low amplitude earth- 9

quake ground motions (e.g., Brune, 1996). Subsequently we use the term ’fragile’ to 10

describe this condition. Specifically, fragility is the overturning potential as a function 11

of ground motion amplitude, and one advantage of PBRs over other fragile features is 12

that one can quantitatively determine their fragility by numerical calculations as well as 13

field and laboratory experiments (Anooshehpoor et al., 2004; Purvance et al., 2008a). 14

In order to make quantitative use of PBR fragilities in seismic hazard studies, one 15

must determine when the PBRs formed. By ‘formation’ of a PBR we mean the time 16

when the freestanding boulder and its pedestal became free from surrounding regolith, 17

thus making it possible for the boulder to shift and perhaps topple in an earthquake. 18

Brune (1996) identified numerous PBRs in southern California and pointed out that 19

their existence appeared inconsistent with existing seismic hazard estimates, but could 20

only estimate when they formed by noting that the presence of rock varnish suggested 21

at least thousands of years of exposure. Subsequently, Bell et al. (1998) attempted to 22

estimate the formation age of some of these PBRs by rock varnish microstratigraphy 23

and 36Cl exposure dating. Correlation of rock varnish microlaminations to late-glacial 24

climate yielded minimum exposure ages of 10,500 - 14,500 years for at least some parts 25

of the rock surfaces, and apparent 36Cl exposure ages from a single sample associated 26

with each PBR were 13,000 - 73,000 years. Purvance et al. (2008b) assumed, based on 27

these observations, that similar boulders in southern California had been fragile for at 28

least 10,000 years, and combined this assumption with measurements of their fragilities 29

to show quantitatively that the existence of these PBRs was inconsistent with the 2002 30

USGS National Seismic Hazard Maps. 31

Measurements of PBR fragility could be used more effectively for seismic hazard 32

studies if the formation age of the PBRs could be more precisely determined. Al- 33

though the rock varnish microstratigraphy provides clear minimum ages for the sur- 34

face residence time of some parts of the PBRs, single apparent cosmogenic-nuclide 35

exposure ages yield little information about the time the PBRs became fragile, because 36

of several complications. The most important complication is that, as discussed be- 37

low, these PBRs formed by exhumation from beneath a regolith layer. Because some 38

cosmogenic-nuclide production occurs below the surface, cosmogenic nuclides begin 39

to accumulate well before any part of a PBR is actually exhumed. Also, if a PBR was 40

exhumed slowly, many parts of its surface could have been exposed at the surface for 41

a long time before the rock actually became fragile. An additional complication is that 42

typical PBRs are 1-2 meters in size, a dimension similar to the characteristic attenua- 43

tion length for cosmic rays at the Earth’s surface. As the PBR becomes exposed, it acts 44

to partially obstruct the cosmic ray flux at most locations on the surface of the PBR or 45

its pedestal, and this shielding effect changes over time as the soil surface lowers. In 46

this paper we deal with both of these complications by developing a quantitative model 47

that predicts cosmogenic-nuclide concentrations at any point on the surface of a PBR 48

as a function of its exhumation history. Given several measurements of cosmogenic- 49

nuclide concentrations at varying heights on the surface of the PBR and its pedestal, 50

one can find an exhumation history that best fits the measurements, and thus determine 51

2

the formation age of the PBR. We then use this method to date an example PBR from 52

southern California using measurements of cosmogenic 10Be. 53

2. Forward model for cosmogenic-nuclide production in a PBR 54

2.1. Model setup 55

Precariously balanced rocks in desert regions of southern California, such as our 56

example boulder discussed below (Fig. 1), are characteristic of regions where rounded 57

granite boulders are formed below the ground surface by weathering along joint sur- 58

faces, and are subsequently exhumed by stripping of the surrounding regolith (Twidale, 59

1982; Oberlander, 1972). Thus, these PBRs are corestones formed below the surface 60

and exposed to the cosmic-ray flux by gradual exhumation from beneath a regolith 61

blanket (Fig. 2). As a result we require a model expression that describes changes in 62

the concentration of a cosmogenic nuclide during and after this process. We will write 63

this expression for the nuclide we will use in a subsequent example, 10Be in quartz, but 64

it applies to any cosmogenic nuclide formed primarily by spallation and to a relatively 65

small extent by muon interactions. 66

The model has four unknown parameters that we will later find by fitting model 67

to data: ε0,sp and ε0,µ, erosion rates prior to the time the uppermost point on the PBR 68

became exposed, averaged over time periods pertinent to production by spallation and 69

muon interactions, respectively (cm a−1; see additional discussion below); t0, the time 70

that the uppermost point on the PBR became exposed (years before present); and ε1, 71

the lowering rate of the soil surface during PBR exhumation (cm a−1). Given these 72

parameters we can compute the time ttip (years before present) at which the contact 73

points between the PBR and the pedestal become exposed, so that the rock becomes 74

fragile: ttip = t0 − h/ε1, where h is the height of the PBR (cm) (Fig. 2). 75

Cosmogenic-nuclide concentrations prior to PBR exhumation. We are describing 76

10Be concentrations in a number of samples collected from the surface of the PBR 77

or pedestal at various heights; zi is the vertical distance (cm) that sample i lies below 78

the top of the PBR. At time t0, we assume that 10Be concentrations are in equilibrium 79

with some steady erosion rate. As discussed in more detail below, 10Be concentrations 80

attributable to different production mechanisms may not be in equilibrium with the 81

same erosion rate. This gives: 82

N10,0,i =P10,spe−ziρ/Λsp

λ10 + ε0,spρ/Λsp+

3∑j=1

P10, je−ziρ/Λ j

λ10 + ε0,µρ/Λ j(1)

Here N10,0,i is the 10Be concentration (atoms g−1) in quartz in sample i at time t0; 83

λ10 is the 10Be decay constant (4.99 × 10−7 a−1), Λsp is the effective attenuation length 84

for spallogenic production (177 g cm−2 for consistency with the shielding calculation 85

described below; see discussion in Gosse and Phillips, 2001), and ρ is the rock density 86

(g cm−3). Note that by using a single value of ρ we are assuming that the PBR and the 87

rock fragments and/or regolith that surrounded it in the past have the same density. As 88

no information exists regarding the density of no-longer-extant material, this seems to 89

be the most sensible approach. P10,sp is the surface 10Be production rate due to spal- 90

lation (atoms g−1 a−1); in the subsequent example we estimate P10,sp using the scaling 91

3

scheme of Stone (2000) as implemented in Balco et al. (2008), and the global calibra- 92

tion data set of Balco et al. (2008). The parameters P10, j (atoms g−1 a−1) and Λ j (g 93

cm−2) that describe the depth-dependence of production due to muons are site-specific. 94

We computed them as follows: first, we calculated the 10Be production rate at the site 95

due to muons as a function of depth, P10,µ(z) (atoms g−1 a−1) using a MATLAB imple- 96

mentation, described in Balco et al. (2008), of the method of Heisinger et al. (2002b,a). 97

To obtain the P10, j and Λ j we then fit these results with a simplified expression: 98

P10,µ(z) =3∑

j=1

P10, je−zρ/Λ j (2)

The purpose of Equation (1) is to parameterize the 10Be concentration-depth re- 99

lationship prior to PBR exhumation using the smallest possible number of unknown 100

parameters. To do this, we take advantage of the fact that, even if the erosion rate is 101

unsteady and the actual nuclide concentration at a given depth below the surface varies 102

with time, the form of the depth-concentration relationship for 10Be produced by a par- 103

ticular mechanism does not change. Thus, it can be parameterized by a single apparent 104

erosion rate. For example, the nuclide concentration due to spallation as a function 105

of depth under the assumption of steady erosion is given by the first term in Equation 106

(1) above; the only unknown parameter is the steady erosion rate ε0,sp. If the erosion 107

rate changed, the assumption of steady erosion would fail. This would not change the 108

functional form of the depth-concentration relationship, and the depth-concentration 109

relationship could still be described by some value of ε0,sp, but this value would no 110

longer be equal to the true erosion rate – it would lie between the new and old ero- 111

sion rates. After some time, the depth-concentration profile would reach equilibrium 112

with the new erosion rate and the value of ε0,sp needed to describe the profile would 113

again be equal to the actual erosion rate. This process of equilibration to a change in 114

erosion rate has a time scale defined by an effective half-life − ln (0.5)/(λ10 + ε/Λ), 115

where ε is the (new) erosion rate and Λ is the effective attenuation length for the pro- 116

duction mechanism in question. Because this depends on the value of Λ, the nuclide 117

concentrations attributable to a particular production mechanism will change at dif- 118

ferent rates. Thus, if the erosion rate is unsteady, at any given time different erosion 119

rate values are needed to describe the concentration-depth relationship for the different 120

production pathways. This has two important consequences. First, because the very 121

existence of PBRs, as discussed above, implies an unsteady erosion rate, we cannot 122

describe the 10Be concentrations prior to PBR exposure with only a single apparent 123

erosion rate for all production pathways. Thus, we use separate erosion rates to pa- 124

rameterize nuclide concentrations attributable to production by spallation and muon 125

interactions (note that using only a single erosion rate to parameterize the nuclide con- 126

centration due to muons is still somewhat oversimplified; we discuss this in more detail 127

below). Second, the best-fitting values of ε0,sp and ε0,mu are unlikely to reflect the actual 128

erosion rate at time t0, or at any other time for that matter. Subsequently we treat them 129

as nuisance parameters. 130

Cosmogenic-nuclide production during sample exhumation. We next consider nu- 131

clide production in a sample from the PBR surface between the time the top of the PBR 132

is exhumed and the time the sample is exhumed. This reflects the rate ε1 at which the 133

4

soil surface drops during PBR exhumation, and the duration of this period of time is 134

zi/ε1. The nuclide concentration developed in sample i during this time period N10,1,i 135

(atoms g−1) is: 136

N10,1,i =

∫ zi/ε1

0P10,spS i(tε1)e−λ10tdt +

3∑j=1

P10, j

λ10 +ε1ρΛ j

[1 − e

−

(λ10+

ε1ρΛ j

)(ziε1

)](3)

Here the function S i(z) is a sample-specific function describing the fast neutron 137

shielding factor at the site of sample i as a function of depth below the soil surface 138

(dimensionless; the shielding factor is the ratio of the production rate at the shielded 139

location to that at a point on an infinite, unobstructed flat surface). As discussed in the 140

next section, once the top of the PBR breaks the surface, the PBR itself partially shields 141

samples on the side of the PBR or on the pedestal. In this case the shielding geometry 142

becomes quite complicated and must be computed numerically. Because of the much 143

greater penetration depth of muons than of fast neutrons, we have ignored this effect in 144

computing production by muons. As we discuss in detail in the next section, we found 145

that we could approximate the numerical estimates of S i(z) by: 146

S i(z) = S 0,ie−zρ/Li (4)

where S 0,i (dimensionless) and Li (g cm−2) are sample-specific constants. With this 147

approximation, 148

N10,1,i =P10,spS 0,i

λ10 +ε1ρLi

[1 − e

−

(λ10+

ε1ρLi

)(ziε1

)]+

3∑j=1

P10, j

λ10 +ε1ρΛ j

[1 − e

−

(λ10+

ε1ρΛ j

)(ziε1

)](5)

Cosmogenic-nuclide production after sample exhumation. We next consider nu- 149

clide production after the sample location has been completely exhumed. The duration 150

of this period is t0 − zi/ε1. The nuclide concentration developed in sample i during this 151

time N10,2,i (atoms g−1) is: 152

N10,2,i =P10,spS 0,i

λ10 +εs,iρ

Λsp

[1 − e

−

(λ10+

εs,iρΛsp

)(t0−

ziε1

)]+

3∑j=1

P10, j

λ10

[1 − e

−(λ10)(t0−

ziε1

)](6)

Here εs,i is the erosion rate of the PBR surface at sample i after it is exposed. We 153

do not solve for this, so we must estimate it from other evidence, and we allow for 154

the possibility that it may differ among the samples. Note that we ignore the effect 155

of surface erosion on the nuclide concentration due to muons. In addition, Λsp is 156

strictly the effective attenuation length for spallogenic production under an infinite flat 157

surface, so this expression is somewhat oversimplified for samples that lie on the sides 158

of the PBR. These simplifications are unimportant as long as the surface erosion rate is 159

relatively slow, on the order of a few cm during the time the PBR has been exposed. 160

Total cosmogenic-nuclide concentration at the present time. The 10Be concentra- 161

tion in sample i at the present time N10,i (atoms g−1) is the sum of these contributions, 162

adjusted for radioactive decay: 163

5

N10,i = N10,0,ie−t0λ10 + N10,1,ie−(t0−zi/ε1)λ10 + N10,2,i (7)

Figure 3 shows example height-nuclide concentration profiles predicted by Equa- 164

tion (7) for two contrasting exhumation scenarios. These emphasize two important 165

points. First, the apparent exposure age of any part of the PBR (the exposure age cal- 166

culated on the basis of a single period of exposure at the present production rate) can be 167

wildly in excess of the age that the PBR actually became fragile. Second, for any partic- 168

ular sample location (in fact, for two sample locations simultaneously), very different 169

exhumation histories can yield identical nuclide concentrations. Thus, to determine 170

the actual exhumation history, one must collect a number of samples over enough of a 171

height range to adequately characterize the height-concentration profile. 172

2.2. Attenuation of the cosmic-ray flux by the PBR 173

This section describes how we obtained the parameters S 0,i and Li that describe 174

the shielding of each sample by the PBR during various stages of exhumation. First, 175

we developed a three-dimensional shape model for the PBR and pedestal using a pho- 176

togrammetric method. We marked a number of points on the PBR and photographed it 177

from multiple directions (Figure 1). We then used the software package PhotoModeler 178

(http://www.photomodeler.com) to identify common points in multiple photographs and 179

photogrammetrically determine the relative positions of the camera positions and these 180

common points. This process results in a set of three-dimensional (relative) coordi- 181

nates for points lying on the PBR and pedestal surfaces. As PBRs mostly have convex 182

shapes, we used the surface triangulation capability of PhotoModeler to fit a convex 183

hull to points lying on the PBR and each pedestal block. This results in a represen- 184

tation of the PBR and each pedestal block as a closed, convex surface consisting of 185

numerous triangular facets (Figure 4). An estimate of the accuracy of this method is 186

provided by Anooshehpoor et al. (2007), who used it to estimate the volume of test 187

rocks that could subsequently be weighed; they found that estimated and measured 188

rock volumes differed by a few percent or less. Although this accuracy estimate is 189

not precisely applicable to the shielding calculation here, it shows that uncertainty in 190

the PBR shape model is most likely minor in comparison to other uncertainties in the 191

shielding calculation. 192

We located the samples in the same coordinate system as the shape model using the 193

same photogrammetric software. For each sample location, we calculated the shielding 194

factor S (dimensionless) using a Monte Carlo integration, as follows. 195

First, we represented the cosmic-ray flux at the sample location by a series of rays, 196

with randomly generated azimuth φ and horizon angle θ, terminating at the sample site. 197

Values of φ are uniformly distributed between 0 and 2π, and values of θ are distributed 198

between 0 and π/2, with probability of given θ proportional to sin2.3 θ (see discussion 199

of the angular distribution of the cosmic-ray flux in Gosse and Phillips, 2001). For 200

each (φ, θ) pair, we determined whether a ray with that orientation intersected the PBR 201

or pedestal before arriving at the sample location. In order to take account of sample 202

thickness in this procedure, we defined the sample locations to be the center of the 203

sample that was actually collected, somewhat interior to the surface of the PBR. Thus, 204

all rays pass through at least some part of the PBR or pedestal. We calculated the 205

6

total length of each ray l (cm) that lay within the PBR or pedestal, and computed an 206

attenuation factor s (dimensionless) for that ray according to s = exp (−lρ/Λp) where 207

Λp is 208 g cm−2 (see Gosse and Phillips (2001) for discussion. This value for Λp 208

corresponds to a value for Λsp of 177 g cm−2). The shielding factor S for that sample 209

location is then the average value of s over all iterations. We found that approximately 210

1000 iterations were required to converge on a reproducible value for S . To build up 211

the function S i(z) required to evaluate Equation (3), we repeated this calculation with 212

the regolith surface defined to be at various heights above the sample location (Figure 213

5). In this situation, rays terminating at the sample location experience attenuation by 214

both regolith and rock. As discussed above, we assumed that regolith and rock have 215

the same density. Figure 6 shows that S i(z) conforms to Equation (4) for all samples 216

on the GV1 PBR. This is also true for all other PBRs we have investigated up to this 217

point. 218

2.3. Model fitting 219

We fit the model to the measured nuclide concentrations by choosing values of the 220

free parameters ε0,sp, ε0.mu, ε1, and t0 that minimized the chi-squared misfit M between 221

observed and predicted nuclide concentrations: 222

M =n∑

i=1

[N10,i − N10,m,i

σN10,m,i

]2(8)

where N10,m,i is the measured 10Be concentration in sample i (atoms g−1), σN10,m,i 223

is its one-standard-error uncertainty (atoms g−1), and n is the number of samples. As 224

all of the sample locations are exposed at the present time, the parameters are subject to 225

the constraint that the total amount of exhumation t0ε1 must be greater than the distance 226

that the lowest sample lies below the PBR top (note also that if this constraint is not met, 227

Equations (5) and (6) will yield unphysical results for some samples). In addition, for 228

better computational efficiency we limited the exhumation rate ε1 to be less than 10 cm 229

a−1. Any exhumation rate higher than this can equivalently be considered instantaneous 230

exhumation relative to PBR exposure ages of thousands to tens of thousands of years. 231

We cannot show analytically that this objective function and constraints always 232

yield a unique minimum. The objective function is not convex. However, we explored 233

this issue for the example data described below by randomly varying the starting guess 234

over a large range of parameter space; we found that the optimization scheme always 235

converged on a single minimum. 236

2.4. Potential weaknesses in this model 237

First, the assumption in this model that is most likely to fail in some natural cases 238

is that the entire PBR is exhumed steadily at a fixed exhumation rate ε1. We show later 239

that this model does a good job of fitting measured nuclide concentrations from the 240

GV1 PBR, but it is likely that many PBRs emerge in multiple pulses of rapid exhuma- 241

tion separated by periods of relative stability. In this case one might not fit measured 242

7

nuclide concentrations with a steady-exhumation model. If one were to allow for un- 243

steady exhumation, many more parameters would be required to describe nuclide accu- 244

mulation during exhumation, and the problem would resemble the problem of identify- 245

ing multiple earthquake slip events from a cosmogenic-nuclide profile on a fault scarp 246

that has been addressed by, among others, Mitchell et al. (2001) and Benedetti et al. 247

(2002). 248

Second, the model assumes that the orientation of the PBR was the same before 249

exhumation that it is at present. Thus, our approach would fail if the PBR had tipped 250

or rolled into its present orientation as it was exhumed. This issue can most likely be 251

dealt with in the process of selecting PBRs by carefully looking at their geometry and 252

rock fabric orientation before attempting to date them. 253

A third potential weakness is that in some cases fitting the model to data relies fairly 254

heavily on the accuracy of nuclide production rates by muons. When the erosion rate 255

prior to PBR exhumation is relatively slow, production by muons prior to the start of 256

PBR exhumation and production by all processes after all samples have been exhumed 257

both tend to flatten the nuclide concentration-height profile and thus have somewhat 258

offsetting effects. Resolving these two similar effects requires an accurate represen- 259

tation of the relative importance of production by spallation and muons. Production 260

rates due to muons are less well established than production rates due to spallation at 261

present, which may lead to unrecognized systematic errors. A related issue is that we 262

have represented production due to muons by a sum of several exponentials with dif- 263

ferent attenuation lengths, but parameterized the nuclide concentration due to muons 264

prior to PBR exhumation by a single steady erosion rate. This may be oversimplified 265

for some situations. 266

Finally, resolving nuclide inventories produced by spallation and muon interactions 267

also depends on the accuracy of the shielding calculations. The calculations described 268

above accurately implement the assumption that all cosmic-ray particles come from 269

outside the immediate region and their interactions with rock can be characterized by 270

a single attenuation coefficient, but this assumption explicitly disregards the energy 271

distribution of cosmic rays as well as any secondary particle production. At present it 272

is not clear how important these effects are. 273

3. An example from Grass Valley, southern California 274

3.1. The Grass Valley PBR 275

The GV1 PBR shown in Figure 1 is located in the central San Bernardino Moun- 276

tains (SBM) of southern California, specifically within the Big Bear block of the SBM 277

between the Santa Ana and North Frontal thrust faults of the San Andreas fault sys- 278

tem. The SBM consist of crystalline basement rocks including Precambrian gneiss 279

and schist intruded by two generations of Mesozoic granitic plutons (Barth, 1990). 280

The PBR itself is granite. The geomorphology of the SBM is dominated by a high- 281

elevation, low-relief plateau, covered by deeply weathered granite saprolite, that is 282

in places deeply incised by river canyons. The low-relief plateau is believed to have 283

been formed by Miocene weathering prior to uplift of the mountain block (Oberlan- 284

der, 1972). Stratigraphic evidence shows that uplift postdated the Miocene and may 285

8

have been as recent as 2-3 Ma (Blyth et al., 2000; Spotila et al., 1998). This uplift 286

event presumably initiated river canyon incision which has since then been propagat- 287

ing headward into the low-relief interior of the range. The GV1 site lies within this 288

low-relief, high-elevation zone, within a small (ca. 1 km2) watershed whose lower 289

reaches are littered with PBRs and loose granite boulders, but whose upper reaches re- 290

main covered by regolith and saprolite and display few boulders. An abrupt steepening 291

of the channel slope separates these two zones. This geomorphic arrangement suggests 292

that time-transgressive stripping of regolith and saprolite, presumably at the leading 293

edge of headward-propagating incision into the central parts of the range, is currently 294

proceeding upward from the bottom of the watershed. 295

The GV1 PBR is located adjacent to a dry wash that forms the central drainage of 296

the watershed. We measured the 10Be concentration in a sample of fluvial sediment 297

from this wash (Table 1). Given a mean watershed elevation of 1530 m, the local 298

bedrock density of 2.67 g cm−3, and the ’St’ scaling scheme and erosion rate calculation 299

method of Balco et al. (2008), this concentration implies a watershed-averaged steady 300

bedrock erosion rate of 82.8 ± 6.4 m Ma−1. This generally agrees with the observations 301

of Binnie et al. (2007) that basin-scale erosion rates for watersheds within the high- 302

elevation, low-relief plateau are 50-130 m Ma−1. However, as discussed above, the 303

distribution of PBRs, other granite boulders, and regolith within this basin suggests 304

that erosion in the basin is both spatially and temporally variable, so the steady-state 305

assumption required to compute an apparent basin-scale erosion rate estimate is not 306

valid. 307

Much of the surface of GV1 is covered with rock varnish. We did not investigate 308

rock varnish microstratigraphy at this site. However, studies of varnish microstratigra- 309

phy at other sites in Southern California with similar climate implied that no surface 310

erosion had taken place for 10,000-15,000 years (Bell et al., 1998). The summit of the 311

PBR is not varnished and has a corrugated surface with ∼2-3 cm of relief, suggesting 312

some surface erosion. Thus, we assumed εs = 0 for samples from the varnished sides 313

of the PBR and pedestal, and εs = 2 ×10−4 cm a−1 for the uppermost sample (Table 1). 314

3.2. 10Be measurements 315

We collected three samples from the GV1 PBR, at the top surface and at two loca- 316

tions on the sides, and one sample from the pedestal (Figure 1). We crushed and sieved 317

rock samples to extract the 0.25-0.5 mm size fraction and separated and purified quartz 318

by selective etching in HF. We then added a carrier solution derived from deep-mined 319

beryl with a 10Be/9Be ratio of 2 ± 2 × 10−16, dissolved the quartz, extracted Be by SiF4 320

evaporation and column chromatography, and measured the Be isotope ratio by accel- 321

erator mass spectrometry (AMS). We carried out both chemical preparation and AMS 322

analysis at the Center for Accelerator Mass Spectrometry, Lawrence Livermore Na- 323

tional Laboratory. Rood et al. (2010) describe the AMS measurement procedure. Total 324

carrier and process blanks were 33,000 ± 15,000 atoms, 0.2-1.5 % of the total number 325

of atoms measured in each sample. Table 1 and Figure 7 show the 10Be concentrations. 326

3.3. Results, discussion, and conclusions. 327

We used the ‘fmincon’ function of the MATLAB Optimization Toolbox to fit model 328

to measurements. Table 2 shows the values of all the fixed parameters needed to eval- 329

9

uate the model equations. The best-fitting values for the free parameters are ε0,sp = 2.0 330

×10−3 cm a−1, ε0,µ = 3.3 ×10−3 cm a−1, ε1 = 10 cm a−1, and t0 = 18.2 ka. These results 331

imply ttip = 18.1 ka. As noted above, the optimal value of ε1 is the maximum value we 332

allowed in the optimization, that is, the best-fitting parameters imply essentially instan- 333

taneous exhumation within years to decades. This exhumation rate is much faster than 334

the apparent basin-scale erosion rate in the watershed above the site. However, as noted 335

above, the PBR is located adjacent to a channel that is locally incising into saprolite and 336

regolith. As channel erosion in arid landscapes such as this one is commonly episodic 337

and associated with large flood events, it is certainly possible that the channel could 338

incise 2-3 meters vertically and/or laterally within a series of floods over several years. 339

In addition, it is easily possible to construct scenarios where low background erosion 340

rates in most of the watershed coupled with relatively recent, faster exhumation of part 341

of the watershed produce 10Be concentrations in stream sediment consistent with the 342

value we observed. Thus, a relatively slow apparent basin-scale erosion rate does not 343

present any conflict with the observation of rapid local exhumation 18 ka at the PBR 344

site. 345

Figure 7 shows the fit of model to data. It is clear that the model closely duplicates 346

the main features of the data, but the best-fitting parameters yield a value for the misfit 347

statistic M of 39. If we believed that the only potential source of difference between 348

modeled and predicted nuclide concentrations was measurement uncertainty, this value 349

for M would imply a low probability that the model adequately explained the measure- 350

ments. However, in computing M we have not included several known uncertainties in 351

the model, which means we cannot interpret the value of M as a statistical significance 352

level. For example, including uncertainties in the shielding calculation at the level of a 353

few percent would greatly reduce the minimum value of M without affecting the best- 354

fitting value of ttip. Errors in the shielding calculation of this magnitude, although hard 355

to quantify, are likely, so we conclude that the model most likely adequately fits the 356

data even though we do not have a basis for calculating the degree of confidence in this 357

conclusion. 358

We estimated the uncertainty in ttip by a Monte Carlo simulation in which we varied 359

only the measured nuclide concentrations by independently sampling the N10,m,i from 360

normal distributions with mean and standard deviation given by the values in Table 1. 361

Figure 8 shows the results. This yields an internal uncertainty (i.e., including mea- 362

surement uncertainties only) of 1.9 ka. Including the production rate uncertainty from 363

Balco et al. (2008) yields a total uncertainty of ± 2.5 ka. We conclude that the GV1 364

PBR has been fragile for 18.1 ± 2.5 ka. 365

4. Acknowledgements. 366

This study could not have been carried out without the collaboration of many re- 367

searchers who are part of a broader study of southern California PBRs, in particular 368

Rasool Anooshehpoor, Jim Brune, Lisa Grant Ludwig, and Katherine Kendrick. In ad- 369

dition, MP thanks Lesley Perg for ideas and suggestions early in the project. Funding 370

for this work was provided by Southern California Earthquake Center Award no. 10118 371

(to Lisa Grant Ludwig), a Lawrence Livermore National Laboratory Lawrence Scholar 372

Program Fellowship (to DR), NSF Postdoctoral Fellowship Award no. EAR-0948350 373

10

(to DR), and by the Ann and Gordon Getty Foundation. Portions of this work were per- 374

formed under the auspices of the US Department of Energy by Lawrence Livermore 375

National Laboratory under Contract DE-AC52-07NA27344. 376

11

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13

Tables 441

Table 1: 10Be concentrations and sample-specific parameters for GV1 samples. Sample GV-SED consists ofchannel sediment from the dry wash adjacent to the PBR: this sample is not used to constrain the age of thePBR and we used it only to estimate the basin-scale erosion rate in the watershed upstream of the site. Seetext for details.

Distance below Sample [10Be]1 Li AssumedSample name PBR top zi (cm) thickness (cm) (atoms g−1) S 0,i (g cm−2) εs,i (cm a−1)

GV2-3 0 4.5 688 ± 16 0.967 177 2 × 10−4

GV2-2 69 4 410.3 ± 6.7 0.900 178 0GV2-4 117 3.5 207.6 ± 4.4 0.511 175 0GV2-1 169 5 163.3 ± 3.8 0.350 137 0

GV-SED 115.6 ± 3.5

1 Normalized to the isotope ratio standards of Nishiizumi et al. (2007).

Table 2: Site-specific parameters for GV1 PBR.

Latitude 34.2788° NLongitude 117.2471° WElevation (m) 1510Rock density ρ (g cm−3) 2.67P10,sp (atoms g−1 a−1) 12.81

P10,1 (atoms g−1 a−1) 0.0057P10,2 (atoms g−1 a−1) 0.15P10,3 (atoms g−1 a−1) 0.14L1 (g cm−2) 31L2 (g cm−2) 670L2 (g cm−2) 2300

1 Normalized to the isotope ratio standards of Nishiizumi et al. (2007).

14

Figures 442

GV2-1

GV2-2

GV2-3

GV2-4

h =

15

0 c

m

Figure 1: Precariously balanced rock“GV1” at Grass Valley, CA. Labels show sample locations.

15

t = t0 t0 > t > ttip t = ttip t = 0

Figure 2: Relationship of model time to PBR exhumation history. The PBR forms below the surface and isexposed by stripping of the surrounding regolith. At time t0 the top of the PBR is exposed. At ttip the bottomof the PBR is exposed and the rock is free to topple.

0 20 40 60 80

0

40

80

120

160

Dis

tan

ce

be

low

PB

R to

p (

cm

)

Apparent exposure age (ka)

ε0,sp = ε0,μ = 15 m/Ma

ε1 = 800 m/Ma

t0 = 20 ka

ttip = 18.1 ka

ε0,sp = ε0,μ = 300 m/Ma

ε1 = 30 m/Ma

t0 = 60 ka

ttip = 8.3 ka

Figure 3: Example height-nuclide concentration profiles for various exhumation histories. The model profilesare drawn for the location and elevation of the GV1 PBR, for a hypothetical 150-cm-high PBR that has equalgeometric shielding at all heights. The x-axis is apparent exposure age, which is the nuclide concentrationdivided by the surface production rate. Rapid erosion prior to PBR exhumation, followed by slow exhumationof the PBR, produces a convex profile (dashed line). Slow erosion before exhumation, followed by rapidexhumation, produces a concave profile (solid line). The important point is that although these two scenariosproduce similar nuclide concentrations (in fact, they are identical at two levels), they imply very differentages for the time the PBR became free to topple. Thus, one must measure nuclide concentrations in manysamples from a range of elevations to uniquely determine the tipping age of the PBR.

16

1m

Figure 4: The shape model of the GV1 PBR and pedestal blocks used in the shielding calculations. The viewis the same as in Figure 1.

Figure 5: Shielding effect of PBR and regolith cover on the cosmic ray flux at a representative sample loca-tion. The dark lines denote model cosmic-ray trajectories terminating at the sample location; the lightenedparts of the lines highlight the portion of these trajectories that pass through rock or regolith. Before the sam-ple location is exhumed (left panels), both the PBR and the remaining regolith act to attenuate the cosmic-rayflux; no trajectories reach the sample location directly. After the sample location is exhumed (right panel),some of the trajectories reach the sample location directly and others must pass through the PBR.

17

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

Sh

ield

ing

fa

cto

r S

Mass thickness of regolith above sample site (g cm-2)

GV2-2 (PBR, 69 cm below top)

GV2-4 (PBR, 117 cm below top)

GV2-1 (pedestal, 169 cm below PBR top)

Figure 6: Exponential fits to numerical calculations of the shielding factor for samples below the PBR top as afunction of soil depth above the sample location. The symbols are the results of the Monte Carlo integration;the lines are given by Equation (4) with the parameters from Table 1.

0 200 400 600 800

0

40

80

120

160

200

De

pth

be

low

PB

R to

p (

cm

)

[10Be] (103 atoms g-1)

N10,i (total)

N10,0,i (before exhumation)

N10,1,i (during exhumation)

N10,2,i (after exhumation)

Measured

Figure 7: Measured nuclide concentrations in GV1 samples compared with those predicted by the forwardmodel and best-fitting parameters. The small circles show the components of the total predicted nuclideconcentration from Equations (1), (5), and (6), and the large triangles show the total predicted nuclide con-centration from Equation (7). The dark circles with error bars (1σ) show measured nuclide concentrations;error bars that are not visible are equal to or smaller than the size of the symbols.

18

0 5 10 15 20 25 300

5

10

15

20

25

Age PBR is free to topple ttip (ka)

Fre

qu

en

cy

Figure 8: Results of a 200-point Monte Carlo uncertainty analyis for the age the GV1 PBR became free totopple (ttip). The grey line shows the best-fitting value (18,100 years).

19