Simple computer code for estimating cosmic-ray shielding by oddly shaped objects Greg Balco a,* a Berkeley Geochronology Center, 2455 Ridge Road, Berkeley CA 94709 USA Abstract This paper describes computer code that estimates the effect on cosmogenic-nuclide production rates of arbitrarily shaped obstructions that are partially or completely opaque to cosmic rays. This is potentially useful for cosmogenic-nuclide exposure dating of geometrically complex landforms. The code has been validated against analytical for- mulae applicable to objects with regular geometries. It has not yet been validated against empirical measurements of cosmogenic-nuclide concentrations in samples with the same exposure history but different shielding geometries. Keywords: Cosmogenic nuclide geochemistry, Exposure-age dating, Monte Carlo integration, Precariously balanced rocks 1. Importance of geometric shielding of the cosmic-ray flux to exposure dating 1 Cosmogenic-nuclide exposure dating is a geochemical method used to determine 2 the age of geological events that create or modify the Earth’s surface, such as glacier 3 advances and retreats, landslides, earthquake surface ruptures, or instances of erosion 4 or sediment deposition. To determine the exposure age of a rock surface, one must 5 i) measure the concentration of a trace cosmic-ray-produced nuclide (e.g., 10 Be, 26 Al, 6 3 He, etc.); and ii) use an independently calibrated nuclide production rate to interpret 7 the concentration as an exposure age (see review in Dunai, 2010). 8 Typically (Dunai, 2010), estimating the production rate at a sample site employs 9 simplifying assumptions that i) the sample is located on an infinite flat surface, and ii) 10 any topographic obstructions to the cosmic-ray flux can be represented as an apparent 11 horizon below which cosmic rays are fully obstructed, and above which they are fully 12 admitted. These assumptions are adequate for a very wide range of useful exposure- 13 dating applications, but they fail when the sample site is located on or near objects 14 whose dimensions are similar to the cosmic ray particle attenuation length in rock 15 (order 1 meter). 16 The main reason this assumption fails is referred to in this paper as ‘geometric 17 shielding.’ Geometric shielding describes the effect that when a sample is surrounded 18 * Corresponding author. Tel. 510.644.9200 Fax 510.644.9201 Email address: [email protected](Greg Balco ) Preprint submitted to Quaternary Geochronology October 3, 2013
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Simple computer code for estimating cosmic-ray shieldingby oddly shaped objects
Greg Balcoa,∗
a Berkeley Geochronology Center, 2455 Ridge Road, Berkeley CA 94709 USA
Abstract
This paper describes computer code that estimates the effect on cosmogenic-nuclideproduction rates of arbitrarily shaped obstructions that are partially or completely opaqueto cosmic rays. This is potentially useful for cosmogenic-nuclide exposure dating ofgeometrically complex landforms. The code has been validated against analytical for-mulae applicable to objects with regular geometries. It has not yet been validatedagainst empirical measurements of cosmogenic-nuclide concentrations in samples withthe same exposure history but different shielding geometries.
Heimsath, A., Chappell, J., Dietrich, W., Nishiizumi, K., Finkel, R., 2000. Soil produc- 407
tion on a retreating escarpment in southeastern Australia. Geology 28 (9), 787–790. 408
Kubik, P., Reuther, A., 2007. Attenuation of cosmogenic 10Be production in the first 409
20 cm below a rock surface. Nuclear Instruments and Methods in Physics Research 410
B 259, 616–624. 411
Lal, D., 1991. Cosmic ray labeling of erosion surfaces: in situ nuclide production rates 412
and erosion models. Earth Planet. Sci. Lett. 104, 424–439. 413
Lal, D., Chen, J., 2005. Cosmic ray labeling of erosion surfaces II: Special cases of 414
exposure histories of boulders, soil, and beach terraces. Earth and Planetary Science 415
Letters 236, 797–813. 416
Mackey, B., Lamb, M., 2013. Deciphering boulder mobility and erosion from cosmo- 417
genic nuclide exposure dating. Journal of Geophysical Research - Earth Surface 118, 418
184–197. 419
Masarik, J., Beer, J., 1999. Simulation of particle fluxes and cosmogenic nuclide pro- 420
duction in Earth’s atmosphere. Journal of Geophysical Research 104, 12099–12111. 421
Masarik, J., Wieler, R., 2003. Production rates of cosmogenic nuclides in boulders. 422
Earth and Planetary Science Letters 216, 201–208. 423
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Plug, L., Gosse, J., McIntosh, J., Bigley, R., 2007. Attenuation of cosmic ray flux in 424
temperate forest. Journal of Geophysical Research 112, F02022. 425
Rood, D., Anooshehpoor, R., Balco, G., Biasi, G., Brune, J., Brune, R., Grant Ludwig, 426
L., Kendrick, K., Purvance, M., Saleeby, I., 2012. Testing seismic hazard models 427
with be-10 exposure ages for precariously balanced rocks. American Geophysical 428
Union 2012 Fall Meeting, San Francisco, CA. Abstract 1493571. 429
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Figures 430
Figure 1: Photograph and shape model of a precariously balanced rock in southern California, USA (Balcoet al., 2011). The shape model was photogrammetrically generated: tape markers (visible in photo) wereattached to the rock, the rock was photographed from multiple directions, and the software package Photo-modeler was used to generate a representation of the rock as a set of triangular facets.
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Figure 2: Intersection of a representative cosmic ray path with the shape model shown in Figure 1. The shapemodel is shown as a wireframe mesh. A near-vertical cosmic ray path to a sample location on the pedestalbelow the precariously balanced rock enters the top of the rock, exits the bottom of the rock, and enters thepedestal to reach the sample location. Facets that intersect the ray path are highlighted by gray fill. Theportion of the cosmic ray path that traverses rock is highlighted by a thickened line.
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Figure 3: Example of convergence of the Monte Carlo integration estimate of the shielding factor (for thesample location shown in Figure 2) as the sample size increases. Typically the shielding factor converges ona stable value after ∼ 500 iterations.
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0 0.5 1 1.50
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R = 1.5 m
Distance from center of sphere (m)
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Along horizontal radius
Figure 4: Validation of Monte Carlo integration code against analytical formulae of Lal and Chen (2005) forgeometric shielding factors inside a hemispherical boulder with radius 1.5 meters. The solid and dashed linesare analytical results for samples along vertically-oriented and horizontally-oriented, respectively, radii, andin part reproduce Figure 4 of Lal and Chen (2005). The circles show shielding factors at the same locationscalculated by the Monte Carlo integration with N = 1000. MATLAB code to generate this figure is includedin the supplemental information.
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0 50 100 150 200 2500
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Depth below surface (g cm-2)
160 g cm-2
170 g cm-2
Figure 5: Validation of Monte Carlo integration against calculations by Dunne et al. (1999) of productionrates on and beneath a flat surface obstructed by a thick obstacle spanning an azimuth of 180°and horizonangle of 50°. The obstructed horizon both reduces the production rate at the surface relative to that foran unshielded surface (by a factor of 0.79 in this example) and also causes the cosmic-ray flux at the siteto be more vertically collimated, which increases the apparent attenuation length for subsurface production.The dark line shows shielding factors (the production rate relative to the production rate on an unshielded flatsurface at the same location) predicted by Dunne et al. (1999) for this horizon geometry, and the circles showequivalent shielding factors predicted by the Monte Carlo integration with N = 2000. The dashed line showsan exponential depth dependence with the shorter attenuation length expected for an unshielded surface(here 160 g cm−2) to highlight the fact that the Monte Carlo results correctly reproduce the attenuationlength prediction of Dunne et al. MATLAB code to generate this figure is included in the supplementalinformation.
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0 50 100
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exhumation history
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Shielding factor
Figure 6: The right-hand panel shows a shape model for a precariously balanced rock near Lake Los Angeles,CA, described by Rood et al. (2012). Black circles show sample locations. The center panel shows geometricshielding factors calculated by Monte Carlo integration for these samples. The left panel shows cosmogenic10Be concentrations measured by Rood et al. (2012) in these samples (black circles) as well as 10Be concen-trations predicted by a forward model based on these shielding factors and a best-fitting exhumation history(using the method of Balco et al. (2011)) (open circles).
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0 20 40 60 80 100 120 140 160 180
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Regolith thickness above sample (cm)
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160 g cm-2
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Regolith thickness
Figure 7: Example calculation showing the variation in shielding factor for the sample shown in Figure 2with increasing thickness of regolith above the sample location. This reproduces the calculation of Balcoet al. (2011) (also note correction in Balco et al., 2012). The solid line shows an exponential function fit tothe Monte Carlo results; this fitting procedure is used to derive the sample-specific constants in Table 1 ofBalco et al. (2011) . The dashed line shows the variation in spallogenic production rate with depth belowthe surface expected for an infinite flat surface (exponentially decreasing with an attenuation length of 160 gcm−2). The top of the precariously balanced rock is 169 cm above the sample, so when the sample is coveredby 169 cm of regolith, the PBR no longer protrudes above the surface and the sample experiences the sameshielding that it would ordinarily experience below a flat surface. MATLAB code to generate this figure isincluded in the supplemental information.