Review A complete and easily accessible means of ... · Quaternary Geochronology 3 (2008) 174–195 Review A complete and easily accessible means of calculating surface exposure ages

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Quaternary Geochronology 3 (2008) 174–195

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Review

A complete and easily accessible means of calculating surface exposureages or erosion rates from 10Be and 26Al measurements

Greg Balcoa,�,1, John O. Stonea, Nathaniel A. Liftonb, Tibor J. Dunaic

aQuaternary Research Center and Department of Earth and Space Sciences, University of Washington, Mail Stop 351310, Seattle, WA 98195-1310, USAbGeosciences Department, University of Arizona, Tucson, AZ 85721, USA

cUniversity of Edinburgh, School of GeoSciences, Institute of Geography, Drummond Street, Edinburgh EH8 9XP, UK

Received 2 June 2007; received in revised form 14 November 2007; accepted 10 December 2007

Available online 15 December 2007

Abstract

We codify previously published means of calculating exposure ages and erosion rates from 10Be and 26Al concentrations in rock

surfaces, and present a single complete and straightforward method that reflects currently accepted practices and is consistent with

existing production rate calibration measurements. It is intended to enable geoscientists, who wish to use cosmogenic-nuclide exposure

age or erosion rate measurements in their work to: (a) calculate exposure ages and erosion rates; (b) compare previously published

exposure ages or erosion rate measurements on a common basis; (c) evaluate the sensitivity of their results to differences between

published production rate scaling schemes. The method is available online at http://hess.ess.washington.edu.

r 2008 Published by Elsevier Ltd.

Keywords: Cosmogenic-nuclide geochronology; Beryllium-10; Aluminum-26; Exposure dating; Erosion rate measurements

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

2. Description of the exposure-age calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

2.1. System architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

2.2. Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

2.2.1. Direct observations vs. calculated values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

2.2.2. Shielding factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

2.2.3. Nuclide concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

2.2.4. AMS measurement standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

2.2.5. Elevation and atmospheric pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

2.2.6. Importance of reporting direct measurements and observations as well as derived ages and erosion rates . . . 180

2.3. Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

2.4. Physical constants and input parameters used throughout the calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

2.4.1. Effective attenuation length for spallation in rock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

2.4.2. Decay constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

2.5. Production-rate scaling schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

2.6. Production rate calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

2.6.1. The calibration data set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

2.6.2. Fit of the scaling schemes to the calibration data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

e front matter r 2008 Published by Elsevier Ltd.

ageo.2007.12.001

ing author. Tel.: 1 206 221 2579.

ess: [email protected] (G. Balco).

ess: Berkeley Geochronology Center, 2455 Ridge Road, Berkeley, CA 94709, USA.

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2.7. Exposure ages and erosion rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

2.8. Error propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

2.8.1. Formal uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

2.8.2. Formal uncertainties vs. difference in results from different scaling schemes. . . . . . . . . . . . . . . . . . . . . . . . 192

3. Significant compromises and cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

3.1. Height–pressure relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

3.2. Geometric shielding of sample sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

3.3. Very thick samples or subsurface samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

3.4. Cross-sections for nuclide production by fast muon interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

3.5. Application to watershed-scale erosion rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

4. Future improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Appendix A. Supplementary data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

1. Introduction

In this paper we describe a complete method forcalculating surface exposure ages and erosion rates frommeasurements of the cosmic-ray-produced radionuclides10Be and 26Al in surface rock samples. It is available onlinevia any web browser at the following URL: http://hess.ess.washington.edu/.

This method codifies previously published proceduresfor carrying out the various parts of the calculation. Theimportance of this contribution is not that we presentsignificant improvements over previous calculation schemes,but that we have combined them with all publishedproduction rate calibration measurements in an internallyconsistent fashion, and made the resulting method easilyaccessible via an online system. This system is intendedto enable geoscientists who seek to use cosmogenic-nuclideexposure ages or erosion rate measurements in their workto: (a) calculate exposure ages and erosion rates; (b) comparepreviously published exposure ages or erosion rate measure-ments on a common basis; (c) evaluate the sensitivity oftheir results to differences between published production ratescaling schemes. This contribution is part of the CRONUS-Earth project, an initiative funded by the U.S. NationalScience Foundation whose goal is to improve Earth scienceapplications of cosmogenic-nuclide geochemistry. It alsoreflects collaboration with the CRONUS-EU project, aninitiative with similar goals funded by the European Union.

This project is motivated in the first place by the fact thatthe number of applications of cosmogenic-nuclide mea-surements, as well as the number of papers published onthe subject, is growing rapidly. These studies are no longerbeing carried out exclusively by specialists in cosmogenic-nuclide geochemistry, but by Earth scientists who wish toapply cosmogenic-nuclide methods in a wide variety ofstudies. These methods are still in active development, so avariety of data-reduction procedures, reference nuclideproduction rates, and production rate scaling schemes existin the literature. Many of these schemes are at least in partinconsistent with each other, and yield different results forthe same measurements of nuclide concentrations. Theeffect of this has been that published exposure age and

erosion rate data sets lack a common basis for comparison.For example, even without regard to the absolute accuracyof any of the exposure age calculation methods relative tothe true calendar year time scale, the variety of inconsistentmethods makes it difficult to directly compare the results ofany two exposure-dating studies. This, in turn, is a seriousobstacle for paleoclimate research or any other broaderresearch task which relies on synthesizing the results ofmany studies. We seek to address this situation byproviding an easily accessible means of comparing newand previously published exposure ages or erosion rates ina consistent fashion.The primary goal of this system is to provide an

internally consistent result that reflects commonly acceptedpractices. At present, it is impossible to evaluate whether ornot it will always yield the ‘right answer,’ that is, forexample, the correct calendar age for exposure-datingsamples of all locations and ages. There are still manyuncertainties in the present understanding of nuclideproduction rates and scaling factors, and there are certainto be future improvements in understanding the physicsbehind production rate calculations as well as in the qualityand coverage of production rate calibration measurements.This means that the exposure age that we infer from aparticular measurement of nuclide concentration, at aparticular location, will change as production rate calcula-tion methods improve. In nearly all cases, the exposure agecalculated with this system from a published 10Be or26Al measurement will differ from the age reported in theoriginal paper. Future versions of this system that reflectimproved understanding of nuclide production rates will,in turn, yield different results from the present version. Ourgoal is to ensure that, at any time, it will be possible torecalculate old and new measurements with a commonmethod.The second goal of this project is to standardize the

entire array of published production rate scaling schemesto a common set of calibration measurements. Papersreporting cosmogenic-nuclide measurements commonlyparameterize nuclide production rates by a referenceproduction rate at sea level and high latitude. This hidesthe important fact that a stated reference production does

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ARTICLE IN PRESSG. Balco et al. / Quaternary Geochronology 3 (2008) 174–195176

not in fact reflect a single direct measurement at sea leveland high latitude, but a normalization of many geographi-cally scattered measurements. Thus, a reference productionrate implies both a set of calibration measurements and ascaling scheme used to normalize them. When one seeks tomeasure the erosion rate or exposure age at a new site, thescaling scheme used to scale the reference production rateto the new site must be the same as the scaling scheme thatwas used to normalize the calibration measurements. Ifthese two parts of the calculation are not consistent, theexposure age or erosion rate will be incorrect. Further-more, in order to compare exposure ages for the samesite that result from two different scaling schemes, onemust make sure that the two scaling schemes havebeen normalized to the same calibration data set. Both ofthese standardizations require a systematic compilationof calibration data, which has not been widely available todate. Here we have compiled such a data set, and used it toensure not only that the results of using a single scalingscheme are internally consistent, but also that the resultsfrom the entire suite of scaling methods are based on asingle calibration data set.

Finally, the system we describe here is intended to serveas a foundation for future improvements. Overall, theobjective of the CRONUS-Earth and CRONUS-EUprojects is to improve the accuracy of exposure age anderosion rate measurements in two ways: first, by betterunderstanding the physical processes involved in cosmo-genic-nuclide production; second, by improving the pro-duction rate calibration data set to better constrainproduction rates and scaling methods. The online exposureage and erosion rate calculators are intended to provide anoutlet for the results of this work.

2. Description of the exposure-age calculator

2.1. System architecture

The exposure age and erosion rate calculator is basedon MATLAB software. MATLAB itself provides a high-level programming language designed for mathematicalcomputation. It is useful for this purpose because: (i) itminimizes the need for low-level coding of numericalmethods; (ii) it is commonly used by geoscientists; (iii)MATLAB code is easier to understand than lower-levelprogramming languages.

We have chosen to use a central server, rather thandistributing a standalone application that runs on a user’spersonal computer (e.g., CALIB: Reimer et al., 2004),because: (i) the web-based input and output scheme isplatform-independent; (ii) the existence of only a singlecopy of the code minimizes maintenance effort and ensuresthat out-of-date versions of the software will not remain incirculation; (iii) the fact that all users are using the samecopy of the code at a particular time makes it easy to traceexactly what method was used to calculate a particular setof results.

The software consists of two main components: a set ofweb pages that act as the user interface to the software, anda set of MATLAB functions (‘m-files’) that check inputdata, carry out calculations, and return results. In thispaper, we describe the general features of the calculationmethod, that is, the key equations, parameter values, andreference data, and discuss important assumptions andlimitations of the methods. A detailed description of all theMATLAB code, including full mathematical descriptionsof the calculations as well as a discussion of the accuracy ofthe numerical methods, is accessible through the web siteand is included as supplementary material with this article.The underlying MATLAB code is freely accessible

through the web site (subject to the terms of the GNUGeneral Public License, version 2, as published by the FreeSoftware Foundation). We intend that the existing func-tions can serve as building blocks for users who wish tocarry out more elaborate calculations than are possibleusing the relatively restricted set of input forms.

2.2. Inputs

2.2.1. Direct observations vs. calculated values

Table 1 shows the measurements and observationsneeded to calculate an exposure age or an erosion ratefrom 10Be or 26Al concentrations. In general, the calculatoris designed to take only direct observations and measure-ments as input to the exposure age and erosion ratecalculations. Our goal has been to make sure that allderived or calculated quantities are produced internallywithin the calculation, so that exposure ages or erosionrates computed with this system cannot be renderedinconsistent by differences in how the input data weregenerated. However, there are two exceptions to this rule:the ‘shielding factor’ and the 26Al and 10Be concentrations.Here we discuss these two input parameters, as well asthe different options for entering site elevations and/oratmospheric pressures, in more detail.

2.2.2. Shielding factor

The calculation of corrections for topographic orgeometric obstructions that reduce the cosmic-ray flux tothe sample site takes place outside the main exposure ageor erosion rate calculation. That is, the calculators take asinput a derived ‘shielding factor’ rather than directobservations of the horizon geometry, which presents thepossibility that shielding factors computed using differentmethods could introduce an inconsistency in the results.We have done this for two reasons. First, there is nostandard format for recording exposure geometry, and thecomplexity of the horizon description required for anaccurate computation differs greatly between differentsamples. Second, we are aware of only very few publicationsthat include detailed horizon descriptions (most report onlycalculated shielding factors), so if this information wererequired as input it would be difficult to recalculate exposureages or erosion rates from most published measurements.

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Table 1

Measurements and observations needed to calculate an exposure age or erosion rate

Field Units Comments

Sample name Text

Latitude Decimal degrees South latitudes are negative

Longitude Decimal degrees West longitudes are negative; longitudes should be between �180� and 180�

Elevation (atmospheric

pressure)

m (hPa) Sample elevation can be specified as either meters above sea level or as mean atmospheric

pressure at the site. If elevation is given, one must also select an atmosphere approximation

to use for calculating the atmospheric pressure. Two are available: the standard atmosphere

equation with geographically variable surface pressure and 1000mbar temperature (see text

for details) and one designed for Antarctica (see Stone, 2000, for discussion)

Sample thickness cm

Sample density g cm�3

Shielding correction Nondimensional,

between 0 and 1

Ratio of the production rate at the obstructed site to the production rate at a site at the same

location and elevation, but with a flat surface and a clear horizon. We provide a separate

calculator to obtain the shielding correction from the measured horizon geometry

Erosion rate cmyr�1 The erosion rate of the sample surface inferred from independent evidence, to be taken into

account when computing the exposure age. Only required for exposure age calculations

Nuclide concentrations atoms g�110Be and 26Al concentrations in quartz in the sample. Should be normalized to the10Be standard of Nishiizumi (2002) and the 26Al standard of Nishiizumi (2004) (see text for

discussion). Should account for laboratory process and carrier blanks

Uncertainties in nuclide

concentrationsatoms g�1 1-standard error analytical uncertainties in the measured nuclide concentrations. Should

account for all sources of analytical error, including AMS measurement uncertainty, Al or Be

concentration measurement uncertainty, and blank uncertainty

G. Balco et al. / Quaternary Geochronology 3 (2008) 174–195 177

Corrections for horizon obstructions are generallycomputed using the approach introduced by Nishiizumiet al. (1989) and nearly universally adopted thereafter,which assumes that the angular distribution of the cosmicradiation responsible for 26Al and 10Be production de-creases in proportion to cosð2:3ÞðyÞ, where y is the zenithangle. Gosse and Phillips (2001) discuss the choice of theexponent in more detail. One then determines the fractionof the total cosmic-ray flux that is obstructed by integratingthis function over the portion of the upper hemisphere thatis below the horizon imposed on the sample site by thesurrounding topography. The ‘horizon’ seen by the sampleis usually taken to be that imposed by obstructions that aremore than several effective attenuation lengths for spallo-genic production—that is, at least several meters—thick. Inpractice this means that this method of calculating theshielding factor is used to account for both topographicfeatures at the scale of tens of meters or greater, and forsituations in which the sample is located in the center of agently dipping surface that is at least meters to tens ofmeters wide. We provide a separate online calculator,external to the main exposure age or erosion ratecalculation, for calculating shielding corrections accordingto this set of assumptions.

This approach conceals a number of major simplifica-tions, which are insignificant for most sample sites, butbecome important in three situations: (i) when the samplesite is surrounded by small obstructions with spacing andthickness comparable to the attenuation length for

spallogenic production; (ii) when sample sites are heavilyshielded, that is, a significant fraction of the upperhemisphere is obstructed; (iii) when sample sites are steeplydipping ð4�30�Þ or strongly convex. In these situations,our present approach could cause future systematic errorsif an improved method of calculating geometric shieldingwere introduced, but existing publications did not containenough information to fully recalculate shielding factorsusing the improved method. We suggest that authorsworking in unusual geometric situations report not onlycalculated shielding factors, but detailed descriptions of thegeometry of their sample sites as well.

2.2.3. Nuclide concentrations

The calculator takes as input a nuclide concentrationðatoms g�1Þ, rather than a raw isotope ratio measurement.Users must convert the Be or Al isotope ratios measured byAMS into nuclide concentrations before input. In otherwords, we separate the calculation of the nuclide concen-tration itself from the computation of an exposure age orerosion rate from that nuclide concentration. Calculatingthe nuclide concentrations requires not only the raw Be orAl isotope ratio of the sample, but also information aboutthe quantity and isotope ratio of the carrier solutions used,a variety of Al and Be concentration measurements madeon samples and carrier solutions, the concentrations ofstable and radio-isotopes in reagent and process blanks,and the isotope ratios of primary and secondary standardsused in the AMS measurements. Deriving the nuclide

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-20

-15

-10

-5

0

5

10

15

20

0 1000 2000 3000 4000

Station elevation (m)

100

(1 -

P10

,mod

el /

P 10,

sta

tion)

Standard atmosphereNCEP surface pressure and 1000 mb temperature

Fig. 1. Effect of atmospheric pressure–elevation approximations on

production rate estimates. The data on this figure show 10Be production

rates estimated for the locations of meteorological stations in the NCAR

World Monthly Surface Station Climatology data set (see text for details).

Stations south of 601S are not included. The y-axis shows the percentage

difference between the 10Be production rates P10;model and P10;station, where

P10;model is the production rate given the mean atmospheric pressure

estimated from the location and elevation of the station and a global

pressure–elevation relationship, and P10;station is the production rate given

the mean atmospheric pressure actually recorded at the station. The St

scaling scheme is used here; other scaling schemes yield equivalent results.

For the open circles, P10;model was calculated using the ICAO standard

atmosphere, which systematically underestimates production rates at high

elevations. For the closed circles, P10;model was estimated using our default

atmospheric pressure–elevation relationship (that draws sea level pressure

and 1000mbar temperature from the NCEP reanalysis, as described in the

text). Our default elevation-pressure relationship reduces scatter and

corrects the systematic bias. We removed obvious errors from the

climatology data (for example, where one or more months were missing

from the annual average, or where the station elevation was grossly in

error), but did not make a comprehensive effort to screen the data set

further. Thus, some of the outliers in this figure probably reflect errors in

the reported station pressures.

G. Balco et al. / Quaternary Geochronology 3 (2008) 174–195178

concentrations in the sample quartz from this informationdepends too heavily on internal laboratory standards andprocedures for us to provide a standard interface. We haveprovided an outline of how to calculate nuclide concentra-tions from isotope ratio measurements in the onlinedocumentation. Also, we anticipate that future efforts tostandardize the reporting format for AMS results maymake it possible to move part or all of this calculationonline.

2.2.4. AMS measurement standards

An additional aspect of nuclide concentration measure-ments that is important to the exposure age and erosioncalculations is the fact that Be and Al isotope ratiomeasurements are made by comparison to a referencestandard with a defined isotope ratio. In order to maintainconsistency between calculated exposure ages or erosionrates and the calibration data set, nuclide concentrationssubmitted to the calculator must be normalized to areference standard that is compatible with the referencestandard used in the calibration measurements. Themeasurements in our calibration data set are referencedto standards that are compatible with the Be and Alisotope ratio standards described in Nishiizumi (2002,2004) (henceforth, the 2002 Nishiizumi Be standard and the2004 Nishiizumi Al standard). If measurements of nuclideconcentration submitted to the online calculators are alsonot compatible with these standards, the resulting exposureages or erosion rates will be incorrect. These standards, orcompatible ones, are widely used at many AMS facilities.However, results from some AMS laboratories may needto be renormalized before they are submitted to the onlinecalculators.

Published intercomparison studies suggest some situa-tions where renormalization may be needed, and wesummarize these briefly here. However, this is not anexhaustive list, and users who need further informationshould consult the AMS laboratory where their measure-ments were made. With regard to 26Al measurements,Nishiizumi (2004) and Wallner et al. (2000) compared Alisotope ratio standards in use at various AMS laboratories.These studies suggest that 26Al measurements made againstsome standards in use at ETH-Zurich and University Kolnmay need to be adjusted for use with the online calculators.With regard to 10Be measurements, several AMS labora-tories use, or have used in the past, a National Instituteof Standards and Technology (NIST) Be isotope ratiostandard whose certified value is not consistent withthe 2002 Nishiizumi Be standard. Thus, 10Be measure-ments referenced to the NIST standard may need to berenormalized before they are submitted to the calculators.Nishiizumi et al. (2007) discuss this issue in detail. Inaddition, there are likely to be significant changes in10Be reference standards in common use in the near future.Nishiizumi et al. (2007) remeasured the isotope ratio of thematerial used for the 2002 Nishiizumi Be standards andfound that it differed from the value stated in Nishiizumi

(2002). This is likely to lead to a change in the reporting of10Be measurements at many AMS laboratories.

2.2.5. Elevation and atmospheric pressure

The calculators provide several different means ofspecifying the sample elevation. The sample elevation isimportant because it determines the atmospheric depthat the site, which is the environmental factor that hasthe largest effect on nuclide production rates. As theatmospheric depth, not the elevation, is the factorcontrolling production rates, the measurement that isactually needed to compute the production rates is themean atmospheric pressure at the site during the period of

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180 240 300 0 60 120 180

0

-60

-30

30

60

-60

-30

0

30

60

P10,model within 5% of P10,station

P10,model / P10,station > 1.05

P10,model / P10,station < 0.95

Fig. 2. Geographic representation of the accuracy of the atmospheric pressure–elevation relationship used in the online calculator. The points on the map

are the locations of meteorological stations in the NCAR World Monthly Surface Station Climatology data set (see text for details). The same data are

shown in Fig. 1. The small dots show stations where the 10Be production rate estimated from the location of the station and our default atmospheric

pressure–elevation relationship (that draws sea level pressure and 1000mbar temperature from the NCEP reanalysis, as described in the text) is within 5%

of the 10Be production rate estimated from the mean atmospheric pressure actually recorded at the station. White circles show stations where the

production rate calculated using our pressure–elevation relationship is more than 5% lower than the production rate calculated from the actual station

pressure; gray circles show stations where it is more than 5% higher. Stations south of 601S are not included. As discussed in the caption for Fig. 1, we did

not make a comprehensive effort to screen the data set for errors. Thus, this figure is intended only as a general guideline to identify regions where the

default pressure–elevation relationship in the calculators may result in inaccurate exposure ages.


exposure. Unfortunately, this is difficult to determineaccurately because: (i) sample sites are rarely located adjacentto long-term weather observing stations, and (ii) most samplesites are much older than instrumental records of atmo-spheric pressure, so the modern observations are unlikely toreflect the true mean pressure throughout their full exposureduration. The site elevation, on the other hand, is easy tomeasure accurately. Thus, most studies report the elevationsof sample sites, and use a standard pressure–elevationrelationship, typically the ICAO standard atmosphere, todetermine the site pressure. The standard atmosphere isdesigned to approximate the typical atmospheric densitystructure at mid-latitudes; however, many features of globalair circulation cause persistent geographic variations, as wellas temporal changes associated with long-term climate andsea-level change, in the pressure–height relationship. Thismeans that applying the standard atmosphere universallywill result in important errors in exposure ages and erosionrates in many parts of the world. Stone (2000), Dunai (2000),Farber et al. (2005), and Staiger et al. (2007) discuss this inmore detail.

This calculator allows users to enter either atmosphericpressure or elevation for their sample sites. If elevation isentered, it provides a choice of atmosphere approximationsto convert elevation to atmospheric pressure. For sites inAntarctica, users should choose a height–pressure relation-ship from Radok et al. (1996); Stone (2000) discusses therelationship between this and the standard atmosphere indetail. For sites outside Antarctica, we provide a defaultatmosphere approximation that uses the basic formula ofthe standard atmosphere, but incorporates geographically

variable mean sea level pressure and 1000mbar temperaturefields as a means of capturing regional variations in theheight–pressure relationship. The mean sea level pressureand 1000mbar temperature fields are from the NCEP–NCAR reanalysis (www.cdc.noaa.gov/ncep_reanalysis/).Henceforth, we refer to this scheme as ‘our defaultheight-pressure relationship.’ We evaluated this schemeby comparison to actual mean annual pressures observedat 2873 meteorological stations in the National Center forAtmospheric Research, World Monthly Surface StationClimatology data set (NCAR Data Set ds570.0; http://dss.ucar.edu/datasets/ds570.0/). The 10Be production ratecalculated using our default height–pressure relationship iswithin 5% of the 10Be production rate calculated from theobserved pressure at 96% of the stations (Figs. 1 and 2). Inaddition, our default height–pressure relationship correctsa systematic underprediction of the production rate at highelevations that results from applying the ICAO standardatmosphere globally (Fig. 1), and slightly improves the fitof some scaling schemes to the production rate calibrationdata set relative to the ICAO standard atmosphere.Stations where our default height–pressure relationshipdoes a poor job of predicting the measured atmosphericpressure are mainly located in high-relief continental areas,for example in the Andes and Central Asia (Fig. 2). Thereare many such areas, however, so users who requireaccurate exposure-dating results are best served by makinga serious effort to determine the mean atmosphericpressure at their sites from nearby station data, ratherthan from global approximations (e.g., Farber et al., 2005).It is also important to note that we have made no effort to

www.cdc.noaa.gov/ncepreanalysis/

http://dss.ucar.edu/datasets/ds570.0/

http://dss.ucar.edu/datasets/ds570.0/


account for temporal variations in atmospheric pressure.Staiger et al. (2007) discuss this issue in detail.

2.2.6. Importance of reporting direct measurements and

observations as well as derived ages and erosion rates

Finally, the fact that methods of calculating exposureages and erosion rates are certain to change in futurehighlights the critical importance of reporting raw ob-servations as well as derived ages or erosion rates. Wecannot emphasize enough that the results derived from aparticular set of measurements will be superseded by futureimprovements, and if a study does not report enoughinformation to update the results using improved methods,it will be effectively useless to future researchers. Table 1provides a checklist of the observations and measurementsthat are needed to calculate an exposure age or erosionrate, and the input web pages provide links to samplespreadsheets that contain all the necessary information. Inpractice, we suggest that publications that report exposureages and erosion rates provide equivalent spreadsheets as atable or appendix.

Table 2

Results of an exposure age calculation


Exposure age Years Reports five expo

Internal uncertainty Years The internal uncer

it is the same for

External uncertainty Years Reports five value

Thickness scaling factor Nondimensional Ratio of the prod

Shielding factor Nondimensional This value is subm

completeness

Surface production rate

due to muonsatoms g�1 yr�1 The production ra


due to spallation for the

St scaling scheme

atoms g�1 yr�1 The production ra

it. Production rate

plot on the single

Table 3

Results of an erosion rate calculation


Erosion rate g cm�2 yr�1 and mMyr�1 Reports five erosio

Internal uncertainty mMyr�1 The internal uncert

is the same for all s

External uncertainty mMyr�1 Reports five values

Shielding factor Nondimensional This value is submi


due to muonsatoms g�1 yr�1 The production rat


due to spallation for the

St scaling scheme

atoms g�1 yr�1 The production rate

2.3. Outputs

The exposure age and erosion rate calculations returntwo sets of results:

(1)

sure

tain

all s

s co

uctio

itte

te fr

te ch

var

-sam

n rat

ainty

calin

corr

tted

e fro

cha

Version information. The result page identifies theversion of each component of the software that wasactive at the time. Users should keep track of theseversion numbers as a record of exactly what calculationmethod was used.

(2)
Results of the calculation. Tables 2 and 3 describe these.We discuss the difference between internal and externaluncertainties in detail in Section 2.8.
2.4. Physical constants and input parameters used

throughout the calculations

The physical constants and parameters that are used inthe calculation fall into three categories: reference nuclideproduction rates derived from calibration measurements,which are described in the next section; constants specific

ages corresponding to the five scaling schemes

ty depends only on measurement error in the nuclide concentration, so

caling schemes

rresponding to the five scaling schemes

n rate in the thick sample to the surface production rate

d by the user (see Table 1)—we re-report it in the results for

om muons is common to all the scaling schemes

anges with time in the other four scaling schemes, so we do not report

iations for the time-dependent scaling schemes are shown in a separate

ple results page

es corresponding to the five scaling schemes

depends only on measurement error in the nuclide concentration, so it

g schemes

esponding to the five scaling schemes

by the user—we re-report it in the results for completeness

m muons is common to all the scaling schemes

nges with time in the other four scaling schemes, so we do not report it


to particular parts of the calculation, which are describedin the documentation for individual MATLAB functions;and parameters that are widely used throughout thecalculations, which we describe here.

2.4.1. Effective attenuation length for spallation in rock

We take the effective attenuation length for productionby high-energy spallation in rock (henceforth, Lsp) to be160 g cm�2 always. Gosse and Phillips (2001) reviewmeasurements of Lsp in detail.

2.4.2. Decay constants

The absolute isotope ratios assigned to the 2002 NishiizumiBe standard and the 2004 Nishiizumi Al standard, to which wehave normalized our calibration measurements, also implyparticular values for the 26Al and 10Be decay constants. Thus,our choice of values for the decay constants is determined byour choice of measurement standards. These values are 4:62�10�7 and 9:83� 10�7 yr�1 for 10Be and 26Al, respectively(Nishiizumi, 2002, 2004). Note that the redetermination byNishiizumi et al. (2007) of the isotope ratio of the 2002Nishiizumi Be standard also implies a larger value for the10Be decay constant. Thus, the value of the 10Be decayconstant adopted here may be incorrect. However, it is stillrequired in the context of the online calculator to ensureconsistency between 10Be production rates and 10Be measure-ments that are both referenced to the 2002 Nishiizumistandards. As noted by Nishiizumi et al. (2007), adopting therevised values for the isotope ratio of the Be reference standardand the 10Be decay constant would result in a negligible changeto relatively young exposure ages (order 104 yr), becauserestandardization of nuclide concentrations at calibration sitesand unknown sites will offset each other, but would resultin a significant change to older (order 106 yr) exposure ages.To summarize, this issue is of limited importance for manyexposure-dating applications, but is potentially significant forlong exposure times and slow erosion rates. Future modifica-tions to the online calculators will most likely be required totake account of this and other measurements of the 10Be decayconstant.

2.5. Production-rate scaling schemes

Calculating cosmogenic-nuclide production rates at asample site requires two things: first, a scaling scheme thatdescribes the variation of the production rate with time,location, and elevation; and second, a reference productionrate at a particular time and place, usually taken to be thepresent time, sea level, and high latitude. This referenceproduction rate is not measured directly, but is determinedby: (i) measuring nuclide concentrations in surfaces ofknown exposure age, which yields a set of local, time-averaged production rates; (ii) using the scaling scheme toscale these local, time-averaged production rates to thepresent time at sea level and high latitude; (iii) averaging theresulting set of reference production rates to yield a bestestimate of the true value. Given a particular set of

calibration measurements, each scaling scheme yields oneand only one best estimate of the reference production rate,and each scaling scheme will yield a different such estimate.In other words, we know the local, time-integrated,production rates at the calibration sites accurately, and thenseek to tie them together using a set of scaling equations thathas one free parameter, the reference production rate. Foreach scaling method, there is a different value of thatparameter that best reproduces the measured local produc-tion rates. The important point is that a reference sea-levelhigh-latitude production rate is not an absolute or indepen-dent constant, but implies a particular set of calibrationmeasurements and a particular scaling scheme.In this section, we describe the scaling schemes for

different nuclide production pathways that we use in thecalculators. In the following section, we describe the set ofcalibration measurements used to obtain the referenceproduction rates for each scaling scheme. We make noattempt here to describe the individual scaling schemesin more than general terms; readers are referred to thesupplementary material, the online documentation, and thesource papers for complete details.Production of 26Al and 10Be takes place by three

mechanisms: high-energy spallation, negative muon cap-ture, and fast muon interactions, each of which varydifferently with time and location. We use only a singlescaling scheme for nuclide production by muons. Wecalculate production rates due to fast muon interactionsaccording to Heisinger et al. (2002b), and production ratesdue to negative muon capture according to Heisinger et al.(2002a). As suggested in these papers, we scale muonproduction rates for elevation using energy-dependentatmospheric attenuation lengths from Boezio et al.(2000). We do not consider magnetic field effects on themuon flux; thus, production due to muons does not varywith latitude or with time. This simplification does notintroduce significant inaccuracies because the productionrate due to muons varies only by ca. 15% over the relevantrange of cutoff rigidity values. This is much smaller thanthe variation in production due to muons with altitude(an approximate doubling every 2000m). As production bymuons is never more than a few percent of total surfaceproduction of 26Al or 10Be, this simplification does nothave a significant effect on the exposure age or erosion ratecalculations.On the other hand, we use five different scaling schemes

for nuclide production by high-energy spallation. These arelisted in Table 4. This results in a suite of five differentexposure age or erosion rate results for each sample.Presenting multiple scaling schemes is intended to accom-plish two things: first, to give an idea of how strongly aconclusion drawn from exposure age or erosion rate resultsdepends on the assumptions of the particular scalingscheme that was used; second, to help in future develop-ment of scaling methods by identifying locations and ageswhere different scaling schemes diverge significantly, andtherefore where development efforts ought to be targeted.

Table 4

List of scaling schemes for spallogenic production. The column marked ‘ID’ gives the two-letter code that identifies each scaling scheme in tables, figures,

the supplementary material, the online documentation, and much of the MATLAB code

ID References Description

St Lal (1991), Stone (2000) Based on the latitude–altitude scaling factors of Lal (1991), as recast as functions of latitude and

atmospheric pressure by Stone (2000). The scaling factor is a function of geographic latitude and

atmospheric pressure. Does not take account of magnetic field variations—the nuclide production

rate is constant over time

De Desilets et al. (2006) The scaling factor is a function of cutoff rigidity and atmospheric pressure. Production rates vary with

time according to magnetic field changes

Du Dunai (2001) The scaling factor is a function of cutoff rigidity and atmospheric pressure. Production rates vary with

time according to magnetic field changes

Li Lifton et al. (2005) The scaling factor is a function of cutoff rigidity, atmospheric pressure, and a solar modulation

parameter. Production rates vary with time according to changes in solar output as well as changes in

the Earth’s magnetic field

Lm Lal (1991), Stone (2000),

Nishiizumi et al. (1989)

An adaptation of the Lal (1991) scaling scheme that accommodates paleomagnetic corrections.

Production rates vary with time according to magnetic field changes. Based on the paleomagnetic

correction described in Nishiizumi et al. (1989)


The simplest scaling scheme that we use, that of Stone(2000) following Lal (1991) (henceforth, the ‘St’ scheme)describes variation in spallogenic production rates withlatitude and atmospheric pressure, and assumes that theproduction rate is constant through time. This scalingscheme was developed by Lal (1991), who described thevariation of nuclide production rates with altitude andlatitude. Stone (2000) then recast these scaling factors asfunctions of atmospheric pressure rather than altitudewithout changing the basic scaling relationships (note thatStone, 2000 also corrected a typographical error in Lal,(1991, Table 1)). This was the earliest production-ratescaling scheme available, and is by far the most commonlyused in the existing literature. It fits the existing calibrationdata set as well or better than any other (see discussionbelow). However, the surface cosmic-ray flux depends onthe strength of the Earth’s magnetic field, and the strengthof the Earth’s magnetic field is known to have varied in thepast. Hence, production rates must have varied over time,and the St scaling scheme cannot account for this variation.This deficiency, as well as the fact that measurements of themodern near-surface cosmic-ray flux are now moreextensive than the early data sets used by Lal, led severalresearchers to develop alternative scaling methods: theseare the scaling schemes of Dunai (2001) (henceforth, ‘Du’),Desilets et al. (2006) (‘De’), and Lifton et al. (2005)(‘Li’). These differ from the St scheme in two importantways: first, they predict a significantly different elevationdependence for spallogenic production; second, theyaccount for changes in production rates due to magneticfield changes (De, Du, and Li) and solar variability (Li).Finally, in an effort to separate the difference in altitudescaling from the difference between the time-dependent andnon-time-dependent scaling schemes, we have also imple-mented a fifth scaling scheme that retains the altitude

scaling of Lal (1991), but adds a simple paleomagneticcorrection based on Nishiizumi et al. (1989) (the ‘Lm’scaling scheme).We determine the reference production rates for all the

scaling schemes using a single calibration data set (discussedbelow), and we use a single set of magnetic field reconstruc-tions as input to the time-dependent scaling schemes. Themagnetic field reconstruction is similar to that used by Liftonet al. (2005), with the addition of spherical harmonic fieldmodels for the last 7000 years from Korte and Constable(2005a). The scaling schemes that take account of magneticfield effects define production rate variations not as afunction of primary magnetic field properties, but as afunction of geomagnetic cutoff rigidity (the minimum energythat an incoming primary particle must have to generate aparticle cascade in the atmosphere at a particular location),and the different scaling schemes calculate the cutoff rigidityfrom the magnetic field reconstruction in different ways. Notethat the Lm scaling scheme cannot accommodate long-itudinal variation in cutoff rigidity, because the Lal (1991)scaling factors on which it is based are defined as a functionof latitude rather than cutoff rigidity. Thus, the sphericalharmonic magnetic field model we use for the Holocene mustbe reduced to an geocentric dipole field in applying thisscaling scheme. Table 5 summarizes the magnetic fieldreconstructions as well as the methods of calculating pastvalues of cutoff rigidity.The difference between exposure ages and erosion rates

calculated using the different scaling schemes varies withlocation, elevation, and the duration of surface exposure.In general terms, the major differences are as follows:

(1)
The magnetic field reconstructions that we use in thecalculators portray the recent magnetic field strength asunusually high. This predicts that, at low latitudes

ARTICLE IN PRESS

Table 5

Magnetic field models used for the paleomagnetically corrected scaling schemes; method of calculating the cutoff rigidity from the magnetic field model.

For a given time period, the magnetic field model itself is the same for all of the scaling schemes. However, the method of calculating the cutoff rigidity and

hence the scaling factor from the field model differs between scaling schemes

ID Time period

(103 yr BP)

Field model Derivative of field model Method of calculating cutoff

rigidity RC

De 0–7 Spherical harmonic field model

of Korte and Constable (2005a)

RC from trajectory tracing (Lifton et al.,

2008)

Directly assigned

7–12 GADa with dipole moment from

Yang et al. (2000)

M=M0, the ratio of past to present dipole

moment, by normalizing to 1950

Definitive Geomagnetic Reference Field

(DGRF)

Eq. (19) of Desilets and Zreda

(2003), using geographic latitude

as input

12–800 GAD with dipole moment from

SINT800 record (Guyodo and

Valet, 1999)

’’ ’’

4800 GAD with mean intensity of

SINT800 record

’’ ’’

Du 0–7 Spherical harmonic field model


Magnetic inclination and horizontal field

strength

Dunai (2001, Eq. (2))


Yang et al. (2000)

M=M0, by normalizing to 1950 DGRF Dunai (2001, Eq. (1))


SINT800 record

’’ ’’


SINT800 record

’’ ’’

Li 0–7 Spherical harmonic field model


RC from trajectory tracing (Lifton et al.,

2008)

Directly assigned


Yang et al. (2000)

M=M0, by normalizing to 1950 DGRF Eq. (6) of Lifton et al. (2005),

using geographic latitude as

input


SINT800 record

’’ ’’


SINT800 record

’’ ’’

Lm 0–7 Spherical harmonic field model


Pole position and dipole moment derived

from approximating the complete field by

a geocentric dipole (Korte and Constable,

2005b)

Eq. (1) of Dunai (2001), using

geomagnetic latitude as input


Yang et al. (2000)

M=M0, by normalizing to 1950 DGRF Eq. (1) of Dunai (2001), using

geographic latitude as input


SINT800 record

’’ ’’


SINT800 record

’’ ’’

aGAD: geocentric axial dipole.


where paleomagnetic field changes are important,surfaces that have been exposed for a longer timewould have been subjected to a higher averageproduction rate. As most of the calibration sites haveexposure ages of 10–20 ka, a non-time-dependentscaling scheme will predict a lower production ratethan a paleomagnetically corrected scheme for samplesolder than the calibration sites, and a higher produc-tion rate for samples younger than the calibration sites.This means that for low-latitude sites, the St scalingscheme will yield older exposure ages than the otherscaling schemes for samples older than the calibrationsites, and younger ages than the other scaling schemes

for samples younger than the calibration sites. Fig. 3shows this effect.

(2)
Nuclide production rates depend more strongly onelevation in the De, Du, and Li scaling schemes than inthe St and Lm scaling schemes. In effect the scalingschemes are fixed to each other by the calibrationmeasurements at moderate elevations, so the De, Du,and Li schemes predict higher production rates at highelevations, and lower production rates at low eleva-tions, than the St and Lm schemes. Thus, De, Du, andLi will yield older exposure ages than St and Lm at lowelevations, and younger ages at high elevations. Fig. 4shows this effect.

ARTICLE IN PRESS

Fig. 3. Difference in exposure ages generated by the St and De scaling schemes. Here we have calculated exposure ages according to the St and De scaling

schemes for a range of latitudes, elevations, and ages. The plot shows three-dimensional contours of the ratio tSt=tDe, where tSt is the exposure age

according to the St scheme and tDe is the exposure age according to the De scheme. The intermediate surface is the tSt=tDe ¼ 1 contour, that is, the set of

locations and ages where the two schemes yield the same exposure age. The darker surface is the tSt=tDe ¼ 1:1 contour, that is, the set of locations and ages

where the St scaling scheme yields exposure ages 10% higher than the De scaling scheme. The lighter surface is the tSt=tDe ¼ 0:9 contour, that is, the set of

locations and ages where the St scaling scheme yields exposure ages 10% lower than the De scaling scheme. The white circles show the locations and ages

of samples in the calibration data set. This figure is drawn for 1001W longitude; other meridians would share the same overall characteristics but differ in

detail. The important features of this comparison are that tSt=tDe increases with increasing age because of the paleomagnetic effect, and with increasing

elevation because of the different elevation dependences of the production rate.

Fig. 4. Difference in exposure ages generated by the Lm and De scaling schemes. The axes, colors, symbols, and assumptions are the same as in Fig. 3: the

intermediate surface shows tLm=tDe ¼ 1, the dark surface shows tLm=tDe ¼ 1:1, and the light surface shows tLm=tDe ¼ 0:9. Both the Lm and De schemes

account for paleomagnetic variations, so the difference between exposure ages calculated with these two schemes does not depend strongly on age.

However, Lm and De have different elevation dependences, so tLm=tDe increases with elevation.



Other differences between the results produced by thedifferent scaling schemes are smaller, more complex inspace and time, and largely stem from: (a) the differentmethods of calculating cutoff rigidities from the magneticfield reconstructions, and (b) small differences in theplacement of the latitudinal ‘knees,’ that is, the criticalvalues of cutoff rigidity at which the cosmic-ray flux beginsto change rapidly.

Note that several of the source papers (Dunai, 2000;Lifton et al., 2005) for the individual scaling schemesinclude plots and contour maps that compare differentscaling methods. These comparisons all fix the scalingschemes being compared so that they yield the sameproduction rate at the reference location of sea level andhigh latitude. Here we have fixed all of the scaling schemesnot to a single value at sea level and high latitude, but toa common set of geographically widespread calibrationmeasurements. Thus, using the online calculators tocompare different scaling schemes will not duplicatepreviously published comparisons. The geographic andtemporal variation in predicted production rates will besimilar, but the absolute differences in production rates orexposure ages will not.

2.6. Production rate calibration

2.6.1. The calibration data set26Al and 10Be production rates due to muons are fully

specified from muon flux observations and interactioncross-sections by Heisinger et al. (2002a,b). Productionrates due to spallation, on the other hand, must becalibrated by measuring nuclide concentrations in sitesof known age. We used a set of calibration measure-ments that is similar to that used by Stone (2000). Thisincludes only published measurements, from Nishiizumiet al. (1996, 1989), Gosse and Klein (1996), Gosse et al.(1995), Stone et al. (1998a), Larsen (1996), Kubiket al. (1998), Kubik and Ivy-Ochs (2004), and Farberet al. (2005). Gosse provided additional data on samplelocations and geometries that are not reported in Gosseet al. (1995) or Gosse and Klein (1996). Of these studies,Nishiizumi et al. (1989), Gosse et al. (1995), Larsen (1996),and Kubik et al. (1998) included 26Al as well as 10Be mea-surements, but we did not use the 26Al measurements fromKubik et al. (1998) due to the possibility of a standardiza-tion inconsistency with the other measurements (Nishiizu-mi, 2004). We have corrected a standardization error inStone et al. (1998a) and renormalized their results to the2002 Nishiizumi Be standards. To the best of our knowl-edge, all the measurements in the calibration dataset are consistent with the 2002 Nishiizumi Be standardsand the 2004 Nishiizumi Al standards. In this work, wehave done our best to apply recent improvements inthe radiocarbon time scale (Reimer et al., 2004) to thelimiting radiocarbon ages for some of the calibration sites.The calibration data set appears in the supplementarymaterial.

We used the following chain of reasoning in averagingthe calibration measurements to determine the referenceproduction rate and its uncertainty for each scalingscheme. First, we assume that all the individual measure-ments at a particular calibration site are scattered onlybecause of measurement error. Thus, we scale all themeasurements from a single site to the reference location,and take the error-weighted mean of the resulting estimatesof the reference production rate to arrive at a summaryestimate of the reference production rate measured at thatsite. However, we do not know if scatter among thereference production rates estimated at different sites isthe result of measurement uncertainty, dating errors indetermining the actual exposure ages of the calibrationsites, or inaccuracies in the scaling scheme. Lackingadditional knowledge, we simply take the average andstandard deviation of the set of summary values from allthe sites as the reference production rate to be used in thecalculators. This results in a relatively large uncertainty inthe reference production rate that conflates randommeasurement errors, errors in dating or geologic inter-pretation at the calibration sites, errors in estimating meanatmospheric pressure at the calibration sites, and inaccura-cies in the scaling methods or the magnetic field recon-structions. The practical importance of this is that thecalculators will overestimate the external uncertainty inexposure ages or erosion rates from sites that are closeto the calibration measurements in location and age. Inreality, the closer an unknown site is to a calibration site,the smaller the production rate uncertainty ought to be, butwe have not attempted to capture this effect.This is a different averaging procedure than is used in

Stone (2000). In that work, each sample was equallyweighted; here we weight each site equally. This change inthe averaging procedure, the adjustments to some cali-brated radiocarbon dates, and additional samples fromLarsen (1996) and Farber et al. (2005) account for the smalldifference between the reference production rates for the Stscaling scheme used here (e.g., 4:96� 0:43 atoms g�1 yr�1

for spallogenic 10Be production) and in Stone (2000)ð4:99� 0:3 atoms g�1 yr�1Þ.The 26Al calibration data set is much smaller than the

10Be calibration data set, consisting of only three sites fromNorth America between 371N and 431N latitude. Thereference 26Al production rates inferred from the 26Al dataset agree with the 10Be production rates inferred from thefull 10Be data set in that the 26Al/10Be production ratio isin all cases indistinguishable from the established value of6.1 (Table 6). However, presumably because the 26Al dataset is geographically restricted, the uncertainty in thereference 26Al production rates inferred from the 26Al cali-bration data set is much smaller than the uncertainty in thereference 10Be production rates, and would probablyunderestimate scaling uncertainties if applied globally.Thus, for the exposure age and erosion rate calculators,we actually use reference production rates and uncertain-ties for 26Al that we obtain by multiplying the reference

ARTICLE IN PRESS

Table 6

Reference production rates for spallation Pref ;sp ðatoms g�1 yr�1Þ inferred from the calibration data set for the five scaling schemes, and reduced chi-square

statistics describing the fit of each scaling scheme and corresponding reference production rate to the individual calibration measurements. The averaging

procedure for computing the reference production rates weights all calibration sites, rather than all samples, equally in order to reduce over-weighting of

sites where more samples were measured. The reduced chi-squared values, on the other hand, reflect the goodness of fit to the entire data set, not the site

averages alone

ID 10Be 26Ala 26Alb 3He

Pref ;sp w2R Pref ;sp w2R Pref ;sp w2Rc Pref ;sp w2R

St 4:96� 0:43 (9%) 2.8 30:6� 1:6 (5%) 2.9 30:2� 2:6 (9%) 2.8 107� 16 (15%) 9.1

De 4:88� 0:56 (11%) 13 29:8� 1:3 (4%) 2.9 29:8� 3:4 (11%) 2.9 114� 14 (12%) 6.7

Du 4:90� 0:56 (11%) 10 29:8� 1:3 (4%) 2.9 29:9� 3:4 (11%) 2.9 113� 15 (13%) 7.2

Li 5:39� 0:52 (10%) 10 32:1� 1:8 (6%) 3.1 32:9� 3:2 (10%) 3.1 122� 14 (11%) 6.1

Lm 4:84� 0:41 (8%) 3.6 29:7� 1:3 (4%) 2.8 29:6� 2:5 (8%) 2.8 106� 14 (13%) 8.4

aCalculated from 26Al calibration data set.bCalculated from 10Be calibration data set with 26Al/10Be ¼ 6.1. This value is used in the 26Al calculators.cWith respect to the 26Al data set.


10Be production rates inferred from the 10Be calibrationdata set by 6.1. This yields reference 26Al production ratesthat are indistinguishable from those inferred directly fromthe 26Al measurements, but larger uncertainties that reflectthe geographically more comprehensive 10Be data set.

One potential weakness in the calibration data set is thatmost of the source papers reported only site elevations andsimply used the ICAO standard atmosphere to obtainmean atmospheric pressures. We have improved thissomewhat by using our default pressure–altitude relation-ship described above in Section 2.2. Farber et al. (2005)are an exception: they calculated the mean air pressure attheir sites from nearby meteorological station records. Abetter effort to determine mean air pressures for all thecalibration sites from nearby station data or regionalclimate models, as well as to account for past changes in airpressure at ice-marginal sites, might reduce scatter in thedata set. Staiger et al. (2007) discuss this issue in moredetail.

2.6.2. Fit of the scaling schemes to the calibration data

We can to some extent evaluate the quality of a scalingscheme by looking at how well it fits the calibration dataset (Table 6, Figs. 5 and 6). The two scaling schemes basedon the Lal (1991) scaling factors (St and Lm) fit the10Be calibration data marginally better than the threescaling schemes that are based on more recent neutronmonitor measurements (De, Du, and Li). However, thenominal difference in fit arises mainly because these latterschemes do not do as good a job of matching a singlecalibration study, that of Farber et al. (2005) in Peru, to the

Fig. 5. Fit of the five scaling schemes to the 10Be calibration data set. The le

scaling the local, time-integrated production rate obtained from each individua

against sample elevation. The right-hand panel shows the error-weighted me

production rate at each calibration site, plotted against the average elevation o

mean and standard deviation of the summary values for all sites. These ar10Be measurements.

mid- and high-latitude sites. This is potentially important,because this site is at high elevation and low latitude wherescaling schemes differ the most, and comparing this sitewith high-latitude sites is a critical test of the scalingmethods. However, this site is also located in an area wherethe pressure–height relationship is very different from thestandard atmosphere, so how one determines the meanatmospheric pressure makes a significant difference in theresults of the fitting exercise. Thus, we are hesitant to takethe Peru results as favoring the St and Lm scaling schemeswithout additional measurements from low latitudes.Overall, this highlights the importance of locating addi-tional calibration sites at low latitudes.In general, none of the scaling schemes yield a

statistically acceptable fit to the calibration data set. Inpart this must reflect inaccuracies in the scaling schemesthemselves, but with the present data set it is impossible toseparate scaling scheme errors from possible systematicerrors in the calibration measurements themselves, whichmight arise from errors in the independent dating or fromsystematic offsets between measurements made at differentlaboratories.The fact that the Peru calibration data are an outlier

with respect to the De, Du, and Li scaling schemessignificantly increases the nominal uncertainty in thereference production rates corresponding to these schemes(Table 6). This suggested that we might be overestimatingthe uncertainty in the reference production rate by relyingtoo heavily on the Peru data. In an effort to bring moredata to bear on this question, we looked at the available3He calibration data set as well. Although we are not

ft-hand panels show reference production rates for spallation derived by

l calibration sample to sea level, high latitude, and the present time, plotted

an and standard error computed from all the estimates of the reference

f all the samples at that site. The horizontal line and gray band show the

e the values we use to calculate exposure ages and erosion rates from

ARTICLE IN PRESS

0 1 2 3 4 0 1 2 3 4

Elevation (km)

7

6

5

4

3

7

6

5

4

7

6

5

4

7

6

5

4

7

6

5

4

Ref

eren

ce 10

Be

prod

uctio

n ra

te (

atom

s g-1

yr-1

)

All samples Site averages

Sierra Nevada (Ni89)

Kofels (Ku98,Ku04)

Titcomb Basin (Go95)

An Teallach (St98)

New Jersey (La96)

Peru (Fa05)

Lake Bonneville (Go96)

Water targets (Ni96)

St

De

Du

Li

Lm



presenting a 3He exposure age calculator in the presentwork, these measurements provide additional informationabout scaling uncertainties from an entirely independentdata set. The 3He calibration data consist of measure-ments from Kurz et al. (1990), Cerling and Craig (1994),Licciardi et al. (2006, 1999), Dunai and Wijbrans (2000),and Ackert et al. (2003). We used only the samples fromKurz et al. (1990) with good surface preservation, and alsodiscarded samples from very young lavas that had largeuncertainties. We followed the example of these and otherauthors in assuming that there is no 3He production bymuons. We did not include the data from Blard et al.(2006), because they argued persuasively that their methodof accounting for a grain-size dependence in 3He concen-trations in crushed minerals rendered their results incon-sistent with the previous measurements that were madewithout regard for grain size. The scaling schemes do not fitthe 3He data set as well as the 10Be data set, and yieldslightly larger uncertainties (11–15%) in the referenceproduction rate (Table 6 and Fig. 7). A detailed discussionof the reasons for this scatter is well beyond the scope of thispaper. Regardless, the scatter in reference production ratesinferred from the 3He data set suggests that we have notoverestimated the uncertainty in the 26Al and 10Be pro-duction rates by relying too heavily on the Peru calibrationsite.

2.7. Exposure ages and erosion rates

The exposure age calculator solves the equation:

N ¼ SthickSGPref ;sp;Xx

Z T

0

SXxðtÞ expð�ltÞ exp��t

Lsp

� �dt

þ Pm

Z T

0

expð�ltÞ exp��t� z=2

Lm

� �dt (1)

for the exposure age T. Here N is the measured nuclideconcentration in the sample ðatoms g�1Þ, Sthick is the thick-ness correction (nondimensional), SG is a geometricshielding correction (nondimensional), Pref ;sp;Xx is thereference production rate due to spallation for scalingscheme Xx ðatoms g�1 yr�1Þ, SXxðtÞ is the scaling factor(nondimensional) for scaling scheme Xx, which may ormay not vary over time depending on the scaling scheme, lis the decay constant for the nuclide in question ðyr�1Þ,� is an independently determined surface erosion rateðg cm�2 yr�1Þ, Lsp is the effective attenuation length forspallogenic production ðg cm�2Þ, Pm is the surface produc-tion rate in the sample due to muons ðatoms g�1 yr�1Þ, z isthe sample thickness ðg cm�2Þ, and Lm is an effectiveattenuation length for production by muons ðg cm�2Þ.

The important details of this method as follows:

(1)
Production by spallation is taken to have an exponen-tial depth dependence with a single attenuationlength.
(2)
As discussed above in Section 2.5, production bymuons is taken to be constant in time.
(3)
Production by muons is taken to be unaffected bytopographic shielding. This is acceptable because whereproduction by muons is important, that is, at depthsgreat enough so that spallogenic production is insig-nificant, the muon flux is highly collimated and lessaffected by geometric shielding than the surface muonflux. This assumption would fail for near-vertical facesor otherwise heavily shielded sites.
(4)
The depth-dependence of production by muons issimplified to be exponential with a single attenuationlength. This is justified because sites that can beaccurately exposure dated are by definition those whereerosion is slow. If erosion is slow, production by muonsaccounts for a small fraction of the measured nuclideconcentration and this simplification is irrelevant.
(5)
The depth dependence of production by muons is takento be linear within the sample, that is, we take theproduction rate due to muons at the center of thesample to be the production rate in the entire sample.By similar reasoning as (4), this is acceptable forsamples thinner than ca. 20–30 cm.
Eq. (1) cannot be solved analytically, so must be solvednumerically. Furthermore, as SXxðtÞ is defined piecewise,the corresponding integral must be computed numerically.The details of the solution method, as well as the step sizeand accuracy of the integration methods, appear in thedocumentation for the MATLAB code (available throughthe web site) and in the supplementary material. We alsouse Eq. (1) for determining the reference productionrates from the calibration data set but solve for Pref ;sp;Xx

instead. This can be done analytically except that the firstintegral in Eq. (1) must still be calculated numerically. Weuse the same integration method as in the exposure agecalculation.The erosion rate calculator solves the equation:

N ¼ SthickSGPref ;sp;Xx

Z 10

SXxðtÞ expð�ltÞ exp��t

Lsp

� �dt

þ

Z 10

Pmð�tþ z=2Þ expð�ltÞdt (2)

for the erosion rate �. There are two key differencesbetween this and Eq. (1): first, the erosion rate calculationretains the full depth dependence of production by muonsrather than treating it as an exponential with a singleattenuation length; second, the integration method ismodified somewhat to be numerical with respect to changesin the production rate due to spallation with time only, andto use an exact formula with respect to depth andradioactive decay. Once again, Eq. (2) cannot be solvedanalytically and a numerical solution method must be used.Most erosion rates calculated from 10Be and 26Al con-

centrations in the existing literature were calculated using

ARTICLE IN PRESS

0 1 2 3 4 0 1 2 3 4

40

30

20

Ref

eren

ce 26

Al p

rodu

ctio

n ra

te (

atom

s g-1

yr-1

)

St

De

Du

Li

Lm

All samples Site averages

Sierra Nevada (Ni89)Titcomb Basin (Go95)New Jersey (La96)

40

30

20

40

30

20

40

30

20

40

30

20

Elevation (km)

Fig. 6. Fit of the five scaling schemes to the 26Al calibration data set. The details of this figure are the same as for Fig. 5. As in Fig. 5, the horizontal line

and gray band show the mean and standard deviation of the summary values for all sites. However, these are not the values actually used to calculate

exposure ages and erosion rates from 26Al measurements. See the text for details.


ARTICLE IN PRESS

1

100

150

200St

Oregon (Lic99)Iceland (Lic06)Western US (Ce94)Hawaii (Ku90)Patagonia (Ack03)Canary Islands (Du00)

De Du

Li

100

150

200

50

100

150

200Lm

1

Elevation (km)

Ref

eren

ce 3 H

e pr

oduc

tion

rate

(at

oms

g-1 y

r-1)

Site averages

0 2 3 2 30

Fig. 7. Fit of the five scaling schemes to the 3He calibration data set. Only an error-weighted mean and standard error for each site, and not the full set of

measurements from each site, are shown. Note that each of the source papers includes data from multiple sites.


the equation of Lal (1991):

N ¼P0

lþ�

Lsp

(3)

where P0 is the surface production rate. This assumes thatthe depth dependence of nuclide production is that ofspallation only, and disregards the fact that production bymuons decreases less rapidly with depth. As pointed outby Stone et al. (1998b) and Granger et al. (2001), giventhat the other assumptions of the method are satisfied,erosion rates calculated using Eq. (3) underestimate thetrue erosion rate by at least a few percent in all cases, andby several tens of percent for low-elevation sites. Thus, theerosion rates calculated using the present method (Eq. (2))will be systematically higher than many erosion rate

measurements in the existing literature. Fig. 8 gives anidea of the significance of this offset.

2.8. Error propagation

2.8.1. Formal uncertainties

The challenge in providing a realistic uncertainty forcalculated exposure ages and erosion rates is that thereare few data available to establish the accuracy of manyparts of the calculation. In principle, we ought to includethree separate uncertainties in calculating the uncertaintyon an exposure age or erosion rate: (i) uncertainties innuclide concentration measurements, (ii) uncertaintiesin the scaling schemes, and (iii) uncertainties in theinput parameters to the scaling schemes, in particular thereference production rate, the atmospheric pressure atthe site, and the magnetic field reconstruction. As the

ARTICLE IN PRESS

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

10-5 10-4 0.001 0.01 0.1 1

εsp (g cm-2 yr-1)

ε μ / ε

sp

Sea level

1000 m

4000 m

2000 m

εsp (m Myr-1)

0.1 1 10 100 1000 10000

Fig. 8. Difference between calculated erosion rates from a high-latitude

sample that take account of subsurface nuclide production by muons (�m;Eq. (2)) and those that do not (�sp; Eq. (3)). The two different x-axis scales

are related by a density of 2:65 g cm�3.


analytical standards to which our calibration measure-ments are normalized are associated with specific values ofthe 10Be and 26Al decay constants, we do not take accountof uncertainty in the decay constants. Also, the decayconstant uncertainty makes a negligible contribution to thetotal uncertainty in most exposure dating and erosion rateapplications.

Of these three important sources of uncertainty, themagnitude of the measurement uncertainty is directlyavailable from the AMS measurement, but in practice itis difficult to separate uncertainty in the scaling schemeitself from uncertainty in the input parameters to thescaling scheme. First, the source papers describing thescaling schemes vary widely in their treatment of scalingfactor uncertainty. Lifton et al. (2005) include formaluncertainties for all their parameters, but Lal (1991), Dunai(2001), and Desilets et al. (2006) only provide generaluncertainty estimates. Second, the scaling uncertaintiesdescribed in these source papers reflect the fit of the scalingschemes to measurements of the modern cosmic-ray fluxusing neutron monitors or film emulsions. Cosmic raysactually responsible for production reactions in rock mayhave a different energy spectrum from those recorded byneutron monitors, so it is not clear whether scaling factorsthat accurately reproduce neutron monitor measurementsare equally accurate for scaling production rates in rock.The calibration data set provides a test of this to someextent, but at present it is not extensive or consistentenough to quantitatively separate errors in the scalingscheme from errors in the calibration measurementsthemselves, the independent dating of the calibration sites,or the atmospheric pressure estimates.

We have chosen to deal with this by taking the scalingschemes and the magnetic field reconstructions used as

input to the scaling schemes as precise, and arguing thatthe uncertainty introduced by this simplification will bemanifested in the scatter among reference production ratesinferred from different calibration sites. We then usethe magnitude of this scatter to assign an uncertainty tothe reference production rate that includes the effectsof uncertainties in the scaling scheme and the inputparameters to the scaling scheme. In other words, we areassuming that the scatter among reference production ratesderived from the individual calibration sites is a goodrepresentation of the combined uncertainty in the calibra-tion data themselves, the scaling schemes, and the magneticfield reconstructions. This approach is straightforward andrelatively conservative, and has the advantage that thecalibration measurements from which the uncertainties arederived are similar in nature to the exposure age anderosion rate measurements for which the online calculatoris intended. Also, disregarding the formal uncertaintiesprovided in the published magnetic field reconstructionsthat we use is relatively unimportant to the total un-certainty estimate. This is because, as we are interestedin the average production rate during a long period ofexposure, point-to-point uncertainties in past magneticfield strength are minimized. We note that the uncertaintyin the reference production rates derived in this way is infact similar in magnitude ð�10%Þ to the scaling uncertain-ties suggested in the source papers for the scaling schemes.Overall, we believe that the available data do not supporta more complicated approach to error propagation atpresent. The main disadvantage of this approach, asdiscussed above, is that in principle we ought to knowproduction rates more accurately at locations that are closeto the calibration sites in time and space, but we cannotdo this with the present method.We report two separate uncertainties for each exposure

age or erosion rate calculation. First, the ‘internal uncer-tainty’ takes only measurement uncertainty in the nuclideconcentration into account. This is useful in situations whereone wishes to compare exposure ages or erosion rates derivedfrom 26Al and 10Be measurements on samples from a singlestudy area. For example, when asking whether exposure agesof adjacent boulders on a single moraine agree or disagree,one should use the internal uncertainty. Second, the ‘externaluncertainty’ adds the uncertainty in the reference nuclideproduction rate for spallation (derived from the scatter in thescaled calibration measurements as described above), andthe uncertainty in the nuclide production rate by muons(derived from the cross-section measurements in Heisingeret al. 2002a,b). One should use the external uncertainty whencomparing exposure ages from widely separated locations,or for comparing exposure ages to ages generated byother dating techniques. Finally, with regard to comparingexposure ages derived from different cosmogenic nuclides,26Al and 10Be exposure ages calculated here are based on thesame production model and calibration data set, so 26Al and10Be exposure ages from closely spaced sites may becompared using the internal uncertainty. 3He, 21Ne, 36Cl,


or 14C exposure ages, on the other hand, are based ondifferent production models and calibration data, so theexternal uncertainty should be used when comparing themto 26Al or 10Be ages calculated here.

We actually calculate the uncertainties by assuming thatthe uncertainties in the input parameters are normal andindependent, and that the result is linear with respect to allof the uncertain parameters, and adding in quadrature inthe usual fashion (e.g., Bevington and Robinson, 1992).The main disadvantage of this method is that it does notcapture the fact that the actual uncertainties in our resultsare not symmetrical around the central value. The fact thatwe cannot incorporate non-ideal probability distributionsfor the input parameters is a secondary disadvantage,although it is mitigated by the fact that there is littleevidence to suggest whether or not the uncertainty in theseinput parameters is in fact asymmetric or otherwiseunusual. Keeping this issue in perspective relative to actualgeological applications of cosmogenic-nuclide measure-ments, we are not aware of any studies where the differencebetween asymmetrical and symmetrical uncertaintieswould affect the conclusions of the study.

2.8.2. Formal uncertainties vs. difference in results from

different scaling schemes

Finally, we discuss the relationship between the formaluncertainty we report for an exposure age calculated usingone of the five scaling schemes, and the spread among theresults from all the scaling schemes. The purpose ofreporting results according to five different scaling schemesis so that users can identify situations where exposure agesare or are not sensitive to the major assumptions that varybetween scaling methods. If all the scaling schemes yieldthe same exposure age—which is mainly true for sites ofsimilar location and age to the calibration sites—the usercan have more confidence in the accuracy of the result.Note that the fact that we have estimated the uncertaintiesin all the reference production rates from the samecalibration data set requires that the results of the differentscaling schemes will nearly always overlap within theirexternal uncertainties.

In practice, there is no compelling reason for a user toprefer the results from any particular scaling scheme, andthere is no strong argument to recommend any one of thescaling schemes for reporting published results. Further-more, there is no reason to believe that the results from thedifferent scaling schemes are randomly distributed, soaveraging them will not yield a more accurate age estimate.As we have already discussed, the most important point toremember when reporting results from these calculators isthat users should always report all the input data as well asthe derived exposure ages or erosion rates. If all the inputdata appear in a paper, it is easy for authors, reviewers,or readers to recalculate the ages or erosion rates usingany present or future scaling scheme, and thus evaluatehow sensitive the authors’ conclusions are to productionrate scaling assumptions.

3. Significant compromises and cautions

Several aspects of calculating exposure ages or erosionrates that we have discussed above involve simplificationsor parameterizations for parts of the calculation that:(i) are not well understood physically; (ii) are wellunderstood, but not well calibrated by existing data;(iii) must be simplified to make the calculation methodcomputationally manageable. In some cases these compro-mises maintain the accuracy of the results for mostapplications, but reduce accuracy for certain unusualgeometric situations or exposure histories. In other cases,we do not know the effect of these compromises on theaccuracy of the results. Here we review the importantsimplifications in our method that may lead to inaccurateresults in some situations.

3.1. Height–pressure relationships

The strong dependence of production rate on atmo-spheric pressure makes this one of the most importantpotential inaccuracies for exposure dating. Users who areworking in areas where the pressure–height relationshipdiffers significantly from the standard atmosphere shouldconsider using pressure measurements from local meteor-ological stations, or regional climate model output, ratherthan a global approximation. Stone (2000), Licciardi et al.(2006), Staiger et al. (2007), and others discuss this in moredetail.

3.2. Geometric shielding of sample sites

The geometric situation at and near a sample site affectsnuclide production at the site in two ways. First, shieldingdue to large-scale topography reduces the cosmic-rayflux that arrives at the sample site. Second, differencesbetween the small-scale geometry of the sample site and theinfinite flat surface usually assumed in production ratecalculations are predicted to reduce the production rate inthe sample due to secondary particle leakage (e.g., Dunneet al., 1999; Masarik and Wieler, 2003; Lal and Chen,2005). This means that both the surface production rateitself and the production rate—depth profile may differfrom the ideal at heavily shielded, steeply dipping, orseverely concave or convex sample locations. In keepingwith common practice, as discussed above, we greatlysimplify this part of the calculation by using only a singleshielding factor that takes account of large-scale topo-graphic obstructions only and relies on a standardizedangular distribution of cosmic radiation at the surface.Our method may have systematic inaccuracies for samplescollected on steeply dipping surfaces (greater than approxi-mately 301) or in heavily shielded locations (e.g., at thefoot of cliffs or in slot canyons). Users who seek tocollect samples in these pathological situations shouldconsider this issue further. Dunne et al. (1999), Masarik


and Wieler (2003), and Lal and Chen (2005) discuss it inmore detail.

3.3. Very thick samples or subsurface samples

Parts of the exposure age and erosion rate calculationsinvolve linear or exponential approximations for the depthdependence of nuclide production by muon interactions.These approximations improve the speed of the calculationwithout sacrificing accuracy for the vast majority ofapplications in which a relatively thin ðo�20 cmÞ sampleis collected at the surface, but they may be inaccurate ifused to calculate ages or erosion rates from samples thateither are very thick or were collected well below thesurface. The user interface is not designed to supportsubsurface samples, so users could only encounter thislimitation if, for example, they attempted to simulate theeffects of sample depth below the surface by applying alarge topographic shielding factor (this approach wouldalso cause errors because of the fact that the topographicshielding factor is only applied to spallogenic production).However, users who seek to adapt the MATLAB sourcecode for depth-profile applications should investigate thisissue closely.

3.4. Cross-sections for nuclide production by fast muon

interactions

Cross-sections for fast muon reactions depend stronglyon muon energy. The values of these cross-sections that weuse (from Heisinger et al., 2002b) were measured at muonenergies 470GV. As the mean energy of muons at groundlevel is near 10GV, this requires a large extrapolation inmuon energy. It appears that production rates predicted bythese cross-section measurements overestimate 26Al and10Be concentrations measured in deep rock cores (Stoneet al., 1998b, unpublished measurements by Stone), but thereason for this mismatch is unclear. Thus, our method maysystematically overestimate erosion rates when they areextremely high (greater than ca. 0:5 cmyr�1).

3.5. Application to watershed-scale erosion rates

Many erosion rate studies seek to infer watershed-scaleerosion rates from cosmogenic-nuclide concentrations inriver sediment (e.g., von Blanckenburg, 2006; Bierman andNichols, 2004). The method described and implementedhere is designed for calculating surface erosion rates ata particular site and not for calculating basin-scaleerosion rates. A strictly correct calculation of the basin-scale erosion rate requires a complete representation ofthe basin topography, which cannot be submitted to thepresent calculators. If supplied with the mean latitude andelevation of the watershed, however, the method describedhere will yield approximately correct results. For water-sheds that do not span a large elevation range andotherwise satisfy the assumptions of the method, the error

arising from using this approximation instead of a strictlycorrect areally averaged production rate will be only a fewpercent. In reality, this uncertainty is likely to be smallrelative to the uncertainty contributed by the many otherassumptions that are required to calculate a basin-scaleerosion rate. However, users who seek very accurate basin-scale erosion rates, or are working in high-relief basins,should consider this in more detail.

4. Future improvements

It is certain that the calculation methods that we haveused here will be superseded by improved methods infuture. The most important improvements, which are thesubject of active research at present, are likely to be:(i) improvements in scaling production rates for elevation,(ii) improved models of past magnetic field variations;(iii) more accurate and more geographically widespreadcalibration measurements. Future versions of this calculator,that take these improvements into account, will yielddifferent results for the same measurements of nuclideconcentrations. Once again, this highlights the importanceof reporting all the observations and measurements thatare needed to recalculate published results using improvedmethods.

Acknowledgments

The contributions of G. Balco, J. Stone, and N. Lifton tothis work are part of the CRONUS-Earth project, andwere supported by NSF Grant EAR-0345574. T. Dunai’scontribution to this work was part of the CRONUS-EUproject and was supported by EU Grant MC-RTN-511927.Dan Farber, Bob Finkel, Brent Goehring, Kuni Nishiizumi,Lewis Owen, Bill Phillips, and Joerg Schaefer providedvaluable help in developing and testing the overallapproach, the MATLAB code, and the user interface.The manuscript was reviewed by Fred Phillips and ananonymous reviewer. John Gosse was the editor.

Appendix A. Supplementary data

The supplementary data associated with this article canbe found in the online version at doi:10.1016/j.quageo.2007.12.001.

Editorial handling by: J.C. Gasse

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