Interactive comment on “Determination limits for cosmogenic 10 Be and their importance for geomorphic applications”

Abstract. When using cosmogenic nuclides to determine exposure ages or denudation rates in rapidly evolving landscapes, challenges arise related to the small number of nuclides that have accumulated in surface materials. Improvements in accelerator mass spectrometry have enabled analysis of samples with low 10Be content (


Introduction
In the last two decades, the use of in situ-produced cosmogenic nuclides for the quantification of denudation processes and the determination of exposure ages of landforms has seen a rapid expansion (Balco, 2011;Granger et al., 2013).This development is due to advances in the technique and to the wide range of geological environments in which the method can be applied.Comprehensive summaries of the method can be found in 5 Anderson et al. (1996), Bierman and Steig (1996), Granger et al. (1996), von Blanckenburg (2005), Balco (2011), and Granger et al. (2013).Among the suite of cosmogenic nuclides that can be used to study geomorphic processes (e.g., 10 Be, 26 Al, 36 Cl, 3 He, and 21 Ne), in situ-produced 10 Be is the most widely used, especially for the quantification of denudation rates (von Blanckenburg, 2005).For simplicity and clarity, we will focus our discussion on in situ 10 Be produced in the target mineral quartz only, although similar concepts can be applied to 10 other cosmogenic nuclides.The broad expansion of 10 Be applications includes studies that extend the limits of the technique by analyzing nuclide concentrations in environments where some of the assumptions inherent to the method are not always satisfied.These studies explore, for example, landscapes that are not in erosional steadystate, e.g.due to recent glaciation (Wittmann et al., 2007), settings where different rock types do not contribute quartz equally (Safran et al., 2005;Torres Acosta et al., 2015), and environments prone to mass failures or with 15 non-uniform sediment supply (Niemi et al., 2005;Binnie et al., 2006;Yanites et al., 2009;Kober et al., 2012;McPhillips et al., 2014;Savi et al., 2014;Schildgen et al., 2016).It is particularly challenging to apply these techniques in environments where the cosmogenic nuclide content is low.For example, the occurrence of deepseated landslide or debris-flow events in rapidly eroding landscapes may result in admixing of low cosmogenic nuclide concentration material into fluvial sediments (Niemi et al., 2005;Yanites et al., 2009;Kober et al., 2012;20 Savi et al., 2014).Likewise, recently exposed bedrock surfaces contain low 10 Be due to their short exposure time to cosmic rays (Licciardi et al., 2009;Schaefer et al., 2009;Schimmelpfenning et al., 2014;Savi et al., 2016).In other cases, scarcity of the target mineral within the collected material can limit the total amount of nuclides in the sample.Difficulties encountered with low 10 Be-content samples are related to the technical limits of the Accelerator Mass Spectrometer (AMS), which can precisely measure 10 Be/ 9 Be ratios down to 10 -15 or 10 -16 (Stone, 25 1998), as well as technical limits and cleanliness issues related to the laboratory where the samples are prepared (e.g., Balco, 2011).
Many AMS facilities provide the lower limit at which they can precisely measure nuclide ratios (e.g., Rood et al., 2010).However, this number does not account for laboratory cleanliness and contamination that may be introduced during sample preparation and chemical procedures, which are reflected in laboratory blanks.30 Nevertheless, several recent studies demonstrated that it is possible to date very young landforms (such as moraines of the Little Ice Age) if a series of strict sampling and laboratory standards (including low laboratory blank measurements) are met (Licciardi et al., 2009;Schaefer et al., 2009;Schimmelpfenning et al., 2014).As such, the adoption of a standardized procedure for defining measurement thresholds is timely and valuable in the field of geomorphic studies.Despite the existence of a procedure commonly used in analytical chemistry to define a 35 statistically significant threshold for low-concentration samples (e.g., Long and Winefordner, 1983), there have been so far no clear guidelines on how to apply this procedure to cosmogenic nuclide data.
In this paper, we describe this method and evaluate its application for 10 Be studies.The method is based on laboratory blank measurements, thereby not only accounting for the AMS detection limit, but also for the Earth Surf.Dynam. Discuss., doi:10.5194/esurf-2017-30, 2017 Manuscript under review for journal Earth Surf.Dynam.Discussion started: 30 May 2017 c Author(s) 2017.CC-BY 3.0 License.cleanliness and contamination that may occur in the laboratory.As an example, we use the 10 Be measurements of 57 samples and 61 laboratory blanks to address (1) the determination of statistically significant limits that define the lower threshold for quantifying exposure ages or denudation rates, and (2) the influence of the approach used for the calculation of this threshold on the quantitative use of the samples.
2 State of the Art 5

Cosmogenic nuclide techniques in Earth Surface Sciences
In situ cosmogenic nuclides are produced when secondary cosmic rays (formed by nuclear reactions in the atmosphere) collide with target minerals at or near the Earth's surface (Nishiizumi et al., 1989;Lal, 1991;Brown, 1992;von Blanckenburg, 2005;Dunai, 2010).Being a rare cosmogenic radioactive nuclide with a half-life of 1.39 Myr (Chmeleff et al., 2010;Korschinek et al., 2010), 10 Be is not naturally present in Earth surface materials.10 Because the production rate of 10 Be decreases approximately exponentially with depth, nuclide accumulation mostly occurs within the upper few meters of the surface (Lal, 1991;Dunai, 2010).For this reason, 10 Be is widely used for two main categories of geomorphological research: (1) dating of exposed or buried surfaces and sediment (e.g., Ivy-Ochs and Kober, 2008;Granger, 2006) and ( 2) quantification of recent (typically millennial-scale) denudation rates (e.g. von Blanckenburg, 2005;Granger et al., 2013).In these applications, the 10 Be content is 15 measured in a sample collected from an exposed surface (e.g., bedrock or a large boulder) or shielded deposit, or from a sample of sediment collected from a river-bed or sedimentary deposit (Dunai et al., 2000;Gosse and Phillips, 2001;von Blankenburg, 2005;Nishiizumi et al. 2007;Dunai, 2010).The 10 Be content of a sample is directly related to the time that the sample has been exposed to cosmic rays (minus those lost due to decay) and is inversely related to the erosion rate (Nishiizumi et al., 1989;Lal, 1991).Hence, the 10 Be content of a sample 20 depends on both the nuclide´s concentration and the mass of the target mineral (e.g., quartz) from which 10 Be is extracted.

From AMS ratios to denudation rates and exposure ages
Once Be is extracted from the sample, the 10 Be (and 9 Be) atoms contained in a target mineral are detected with an AMS.This instrument uses a source of accelerated ions and a series of magnets to separate different chemical 25 elements with similar atomic masses (e.g.Dewald et al., 2013), which can be interpreted in terms of a 10 Be/ 9 Be ratio based on a known measurement standard (Rugel et al., 2016).To calculate the number of 10 Be atoms in a sample, the derived 10 Be/ 9 Be ratio is multiplied by the amount of 9 Be in the sample, which is normally the known amount added as 9 Be carrier during sample preparation.Most rock types contain the trace metal 9 Be in the lower ppm-range (Rudnick and Gao, 2004), which makes the total contribution of natural 9 Be negligible relative to the 30 amount added by the carrier.Recent studies, however, have demonstrated that in some rock types the natural 9 Be is high enough that it must be quantified and included when converting 10 Be/ 9 Be ratios to 10 Be content (Portenga et al., 2015;Corbett et al., 2016).
Common chemical preparation procedures include the processing of one or more procedural blanks (referred to as laboratory blanks hereafter), containing the 9 Be carrier only, to help quantify any contamination that may 35 Earth Surf.Dynam. Discuss., doi:10.5194/esurf-2017-30, 2017 Manuscript under review for journal Earth Surf.Dynam.Discussion started: 30 May 2017 c Author(s) 2017.CC-BY 3.0 License.
occur during sample preparation and to account for any 10 Be within the carrier.To obtain the number of 10 Be atoms in the sample related to in-situ production, a blank correction or blank subtraction is commonly performed.

Blank types and blank corrections
Laboratory blanks are typically introduced during the Be separation stage right before total sample dissolution.
Samples are likely to undergo different preparation histories (e.g., different amounts of acid used for dissolution, 5 different time-spans needed for evaporation, etc.) that could result in differing amounts of 10 Be contamination (Balco, 2011), which are reflected in the laboratory blanks (see supplementary material, text S5 and S6).In addition, the amount of 10 Be contamination introduced with the 9 Be carrier from commercial Be solutions is not negligible (Balco, 2011;Granger et al., 2013).Merchel et al. (2008) reported 10 Be/ 9 Be ratios between 10 -14 to 10 - 15 for a range of commercially available Be solutions that may be used as carrier.Deep-mined phenakite or beryl 10 minerals can alternatively be used to produce carrier solutions, in which case 10 Be/ 9 Be ratios in the range of 10 -16   have been obtained (e.g.Stone, 1998;Schaefer et al., 2009;Merchel et al., 2013;Portenga et al., 2015;Corbett et al., 2016).
Machine or instrument blanks (generally related to AMS measurements) indicate the precision at which the AMS can measure the 10 Be/ 9 Be ratio (Balco, 2011).This latter kind of blank is uninfluenced by contamination that 15 occurs during the chemical procedure in the laboratory and can provide information about the sensitivity of the measuring process.In accelerator mass spectrometry, cross-contamination due to long-term memory of the AMS measurements is in the order of 0.1‰ (Rugel et al., 2016), so that the machine background is commonly neglected (Currie, 2008).It follows that most of the contamination stems from the laboratory processing of the samples (Balco, 2011) and thus we focus only on laboratory blanks hereafter.20

Blank corrections
To perform a blank correction, the number of 10 Be atoms contained in the blank is subtracted from the number of 10 Be atoms contained in the sample.When the number of 10 Be atoms in a sample is significantly greater than in the blanks, the value and variability of multiple blank measurements has little impact on the blank-corrected result.
However, in the case of low 10 Be content in a sample, the blank correction constitutes a large subtraction and 25 variations among individual blanks become important.Some 10 Be in the blank may originate from processes that would affect an entire batch of samples that are processed together, such as the 10 Be contained in the carrier or in the stock chemicals used (see supplementary material, text S5).In these cases, a single blank per batch probably provides a good measure of the 10 Be contamination.Other sources, such as cross-contamination from poor laboratory practices or insufficient cleaning 30 of reusable labware result in variable contamination among samples of the same batch and using a single blank per batch may thus be inadequate.Bierman et al. (2002) provide details of replicate blank measurements from 53 batches of samples processed at the cosmogenic-nuclide target preparation laboratories of the University of Vermont, where two blanks were processed per batch.The good agreement between these blank pairs suggests, at least in that laboratory, that inter-batch contamination is not an issue.However, this is a point that needs to be 35 addressed for each laboratory and perhaps at the individual-user level.

General statistical background
Following the International Union of Pure and Applied Chemistry (IUPAC) definition, there is a minimum sample concentration that can be determined to be statistically different from an analytical blank in every analytical procedure (Long and Winefordner, 1983).The term "statistically different" implies the application of a statistical 5 approach that tries to answer the question "what is the lowest sample concentration that can be reliably distinguished from a blank?" (Currie, 1968;Long and Winefordner, 1983;McKillup and Darby Dyar, 2010;Schrivastava and Gupta, 2011;Bernal, 2014).This question can also be formulated as "what is the upper value of the blank distribution (i.e. the distribution of all available 10 Be blank measurements) that ensures a reliable distinction between blank and sample amounts?" 10 For cosmogenic studies, the previous question can be translated into the following null hypothesis (Fig. 1): "The number of 10 Be atoms in a given sample is not distinguishable from that within the blank(s)", which must be tested at a fixed confidence interval.Here, we give the example of a 1-tail test, because we are interested in defining an upper limit for the blank distribution.For variables that are normally distributed, the most common values used for confidence intervals, calculated according to equation (1) below, are ± kσ, with k = 1, 2, or 3 (McKillup and 15 Darby Dyar, 2010), and σ being the standard deviation of the distribution.Whenever the lower limit of the sample's 10 Be value, including the uncertainty interval, lies outside of a chosen confidence interval, the null hypothesis can be rejected, and we can infer that the sample is statistically distinguishable from the blank (Fig. 1).It follows that the choice of the confidence interval to use for the definition of a statistically distinguishable value defines the determination limit (McKillup and Darby Dyar, 2010) and, consequently, if a sample can be distinguished from 20 the blanks (Fig. 1).

Limits of detection (LOD) and quantification (LOQ)
The IUPAC and the ACS (American Chemical Society) recommend, by generalizing the formulas first introduced by Currie in 1968, calculating the limits of determination with the following equation: (1) 25 where and are the mean and standard deviation of a normal distribution of all blank measurements (Analytical Methods Committee, 1987), which ideally contains at least 20 individual blank measurements (Bernal, 2014).When k=3, Eq. ( 1) defines the LOD (Limit of Detection), known as "the lowest concentration, or amount, of an analyte that can be detected with reasonable confidence for a given analytical procedure" (Analytical Methods Committee, 1987;Mocak et al., 1997), according to: 30 . ( This definition of the LOD fixes the confidence interval for the blank distribution, assumed to be Gaussian, at 3σ (99.9% for 1-tail test).Alternatively, when k=10 in Eq. ( 1), the LOQ (Limit of Quantification) is defined as the lower limit above which every analytical sample can be used in a quantitative way with a confidence interval larger than 99.9999% (Analytical Methods Committee, 1987): 35 . (3) Earth Surf.Dynam. Discuss., doi:10.5194/esurf-2017-30, 2017 Manuscript under review for journal Earth Surf.Dynam.Discussion started: 30 May 2017 c Author(s) 2017.CC-BY 3.0 License.
From these definitions of the LOD and LOQ, it is clear that the variability of the blank measurements (expressed through the variance, σ 2 ) has a large influence on the final value of the limits.In particular, the LOD and LOQ will be relatively close to the mean value of the blank distribution when it has a low variance, whereas the LOD and LOQ will be substantially larger than the mean when the blank distribution has pronounced variance.

Method and study design 5
To be able to statistically distinguish a sample with low 10 Be content from a blank, a proper confidence interval needs to be chosen and related to the distribution of the laboratory blanks.In 1987, the Analytical Method Committee suggested to follow the IUPAC recommendations (i.e., equations ( 2) and ( 3)).This recommendation, however, assumes a normal distribution of values, which is rarely the case when dealing with low concentrations of an analyte (Currie, 1972;Bernal, 2014).As such, following this assumption could result in slightly different 10 confidence intervals for the calculated determination limits, as we will illustrate later in Sect. 4. Importantly, if the number of blanks used for the calculation of the determination limits is low (e.g., less than 20), or if the distribution of the measured blank values is not normal, the use of µ and σ for the estimation of the LOD and LOQ defined by the IUPAC's equations may be not appropriate (Long and Winefordner, 1983;Analytical Methods Committee, 1987;Currie, 1968;1972;Bernal, 2014).15 In the case of a low number of blank measurements, which may not accurately constrain laboratory contamination, we overcome this limitation by including additional blank measurements collected over a long time period from the same laboratory.However, this approach assumes that the whole laboratory process is unchanging and that long-term variations in the blanks are representative of the variation within a single batch.In case of nonnormal distributions, to estimate the determination limits at fixed confidence intervals we use an alternative 20 approach to Eq. ( 2) and (3).This approach entails (a) identifying which distribution best describes the measured blank data, and (b) calculating the percentile of that distribution relative to the chosen confidence interval.For example, if the best-fit to the blank data is represented by a Negative Binomial distribution, fixing the confidence interval at 99.9% (equivalent to the LOD) would require that only samples with a number of 10 Be atoms above the 99.9 th percentile of the blank distribution can be considered to be significantly different from the blanks.As such, 25 once having determined the distribution that best describes the blank ensemble, the determination limits are given by the percentiles equivalent to the desired confidence level.

Dataset
To test these two approaches (IUPAC recommendations versus distribution percentiles) and evaluate their implications for geomorphic studies, we use one set of 61 blank values (Table 1) and one set of 57 sample values 30 (35 of which are published in Savi et al., 2016) (Table 2).Blanks and samples were prepared at the Helmholtz Laboratory of the Geochemistry of Earth Surface (HELGES) at GFZ Potsdam (von Blanckenburg et al., 2016) between July 2013 and May 2016.The set of 61 blanks includes eight blanks simultaneously processed with the batches of samples over a few 1-month periods between 2014 and 2016.These eight blanks were processed by the same operator, whereas the long-term blanks were prepared by different users.After having converted 10 Be/ 9 Be 35 ratios from the AMS into the number of 10 Be atoms, we: (1) calculate the LOD and LOQ values for both normal

Determination limits (LOD and LOQ) for cosmogenic 10 Be
Although a minimum of 20 values is generally needed for a statistically significant description of a data 5 ensemble (Long and Winefordner, 1983;Analytical Methods Committee, 1987;Bernal, 2014), for comparison purposes, we determined the LOD and LOQ values for two sets of blanks: (1) the eight blanks processed together with the samples by one operator, and ( 2) the 61 long-term blank values processed in the laboratory by multiple users.In this way, we are able to explore a real situation and observe the effects of the different approaches on the calculation of the LOD and LOQ values.We are aware that the use of the LOQ as a lower threshold may be overly 10 restrictive for cosmogenic studies, because such a conservative confidence interval (i.e., 99.9999%) is commonly not needed in this field.However, for the completeness of this study, we discuss all the different results and their implications in Sect. 4 and 5.
For both blank datasets, we calculated the LOD and LOQ using Eq. ( 2) and (3) assuming a Gaussian distribution, as well as by fitting a negative binomial curve to the blank distribution using the Matlab TM  15   Distribution Fitting toolbox (within the Statistic and Machine Learning toolboxes).The latter requires calculating percentiles, which can be performed using the same toolbox.With this approach, we can compare the effects of having different blank distributions (Fig. 2).To assess whether or not a sample falls below the calculated limits, we compare the lower number of 10 Be atoms contained in the sample (i.e., the number of uncorrected 10 Be atoms in a sample minus its uncertainty, generally expressed by 1σ) (Table 2) to the various LOD and LOQ values.20

Blank correction methods
Samples were corrected for blanks after having assessed which samples can be reliably distinguished from the blanks.Based on the data ensemble and study design mentioned in Sect.3.1, three different procedures for the blank correction can be applied: (1) a single-batch blank correction, (2) an average-blank correction, and (3) a long-term laboratory blank correction.25 For the single-batch blank correction, the number of 10 Be atoms calculated from a single blank processed along with the sample batch is subtracted from each sample within the batch.In this case, the AMS uncertainties associated with each sample and the individual blank measurement are then used for error propagation (Peters, 2001).In the average-blank correction, all the blanks processed in multiple batches by one operator over a limited time frame (i.e., the eight blanks used for the calculation of the determination limits) are used to obtain a 30 representative value of 10 Be atoms for the blanks (e.g., the value that best describes the blank distributionsee below).This representative value is then subtracted from all samples.In contrast, the long-term laboratory blank correction subtracts a 10 Be value obtained from all the blanks measured at the laboratory over a longer time span from each sample (this value is obtained from the 61 blank values from which the determination limits are calculated).As such, the average blank correction accounts for operator-specific abilities, any temporal 35 For the average-blank correction and the long-term laboratory blank correction methods, the representative value of 10 Be atoms used for the blank subtraction can either be the mean of the blank distribution, or its median, 5 depending on which parameter best characterizes the blank ensemble.The uncertainty associated with the chosen representative value that is then used for error propagation procedures is given by the standard error of the mean (εµ) or the standard error of the median (εm) of the blank frequency distribution, calculated as follows (Evans, 1942;Peters, 2001): (4) 10 (5) where σ is the standard deviation of the blank ensemble, n is the number of blank measurements in the blank distribution, C is the width of the bins used for the blank distribution, and f is the number of blanks falling in the C-bin that includes the median (Figure S2 and Table S2).

Determination limits
The distribution of the eight blanks is characterized by a mean (µ) of 0.76 x 10 4 10 Be atoms, a standard deviation (σ) of 0.39 x 10 4 10 Be atoms, and a median of 0.70 x 10 4 10 Be atoms.The standard error of the mean is 0.14 x 10 4 10 Be atoms and the standard error of the median is 0.09 x 10 4 10 Be atoms (for C=2000 and f=3 in Eq. ( 5)).The 61 20 long-term blanks are characterized by a mean of 1.72 x 10 4 10 Be atoms, a standard deviation of 1.96 x 10 4 10 Be atoms, and a median of 0.93 x 10 4 10 Be atoms.The standard error of the mean is 0.25 x 10 4 10 Be atoms and the standard error of the median is 0.16 x 10 4 10 Be atoms (for C=2000 and f=5 in Eq. ( 5)).
For both sets of blank values (subscript "8" for the eight blanks and subscript "61" for the 61 long-term blanks), the LOD and LOQ values calculated following the IUPAC recommendations (subscript "N", assuming the blanks 25 are normally distributed) are always smaller than the ones calculated using the distribution percentiles (subscript "NB", assuming the blanks are best described by a negative binomial distribution) (Table 3, Fig. 3).In particular, for the long-term blank ensemble, which is non-normal and strongly positively skewed (Fig. 2), the LOD61,N calculated with Eq. ( 2) corresponds to a confidence interval of 98.9% of the actual distribution rather than 99.9%.
The same is true for the eight-blank ensemble, where the LOD8,N estimated using Eq. ( 2) corresponds to a 30 confidence interval of 99.3% (rather than 99.9%) because the values are non-normally distributed (Fig. 2).

Reliability of the samples
The number of samples falling below the different determination limits (Table 3, Fig. 3) varies depending on which blank ensemble is chosen and which approach is used to calculate the determination limit (i.e., IUPAC recommendations versus distribution percentiles).For the eight blanks processed by one operator, the approach used to calculate the determination limits does not significantly influence their values nor the number of samples 5 falling below these thresholds (Fig. 3).In contrast, for the 61 long-term laboratory blanks, the approach used for the calculation of the limits has a strong impact on the values of the limits and consequently also on the number of samples that can be quantitatively used to calculate exposure ages or denudation rates (Table 3, Fig. 3).

Blank corrections and error propagation
Not surprisingly, for samples with low 10 Be content (i.e., < 10 6 10 Be atoms), the corrected 10 Be values vary 10 depending on which blank correction method is applied.Considering only the samples that can be reliably distinguished from the blanks (i.e., above LOD values), the single-batch blank correction commonly yields the highest sample 10 Be content.These values, however, are not always associated with the lowest uncertainty, which is in some cases given by the average blank correction method (Table S2).The maximum difference in 10 Be content between these two correction methods is associated with the sample with the lowest 10 Be content, and 15 amounts to 13.6%.Within the average blank correction method, the choice of the mean (0.76 x 10 4 10 Be atoms) or median (0.70 x 10 4 10 Be atoms) as the representative value for the blank subtraction has little influence on the final 10 Be content of the samples (maximum difference < 2.5%).Also, the difference in the propagated uncertainties among samples is negligible (Table S2).
In general, samples corrected with the single-batch blank correction and the average blank correction methods 20 yield higher values with lower uncertainties than the ones calculated with the long-term blank correction method.
When this latter correction method is applied, the choice of the mean (1.72 x 10 4 10 Be atoms) or median (0.93 x 10 4 10 Be atoms) as the representative value for the blank subtraction yields slightly different results.The maximum difference in 10 Be content obtained when using the mean or median is associated with the sample with the lowest 10 Be content, and amounts to 6.5%.Also in this case, the difference between the propagated uncertainties among 25 samples is negligible (Table S2).

Discussion
Our results highlight that whether or not a sample measurement lies above the chosen determination limit depends on: (1) the distribution of blanks, (2) the type of applied determination limit (LOD or LOQ), and (3) the approach used for its calculation (i.e., IUPAC recommendations versus distribution percentiles).All samples for 30 which the 10 Be content is lower than the chosen determination limit cannot be considered as reliably distinguishable from the laboratory blanks.Therefore, these values can only be used to limit the age (i.e., younger than the youngest detectable age) or denudation rate (i.e., faster than the fastest detectable rate) recorded by a sample.In this sense, the LOD or LOQ value can be considered as the youngest detectable age or fastest detectable rate measurable from a data ensemble.35 Earth Surf.Dynam. Discuss., doi:10.5194/esurf-2017-30, 2017 Manuscript under review for journal Earth Surf.Dynam.Discussion started: 30 May 2017 c Author(s) 2017.CC-BY 3.0 License.

Carrier quality, blank distribution, and determination limits
The results of our analysis highlight that the choice of the blank ensemble from which to calculate the determination limits is among the most important factors in defining the final results and the number of samples that can be statistically distinguished from the blanks.
For the eight-blank ensemble, blank values are relatively low and show low variance, which leads to low 5 corresponding LOD8 and LOQ8 values.It follows that most of the samples lie above the determination limits (although 20% of samples cannot be distinguished from blanks when the LOQ8 is chosen as the threshold).These determination limits, being based on only eight blanks, are representative of the individual operator's ability and the contamination that occurred in the laboratory during sample processing.Compared to the eight-blank ensemble, the long-term laboratory blanks have larger variance and show values spanning two orders of 10 magnitude.As such, the relatively high LOD61 and LOQ61 values imply that at least 16 samples, and up to 29 samples cannot be reliably distinguished from the blanks (Fig. 3, Table 3).
In general, the use of the long-term laboratory blanks (being based on many blank measurements) guarantees more reliable values for the statistics of the blank distribution and for the calculation of the determination limits; as such, they may be preferred.Nevertheless, when the long-term blank ensemble shows a large variance, the 15 assumption of unchanging laboratory conditions is unlikely to be valid, and the blank measurements are unlikely to be representative of the variation occurring within a single batch.Under these circumstances, and when there is an acceptable number of blank measurements available (at least 20; Bernal, 2014), a set of blanks obtained from a single operator over a shorter time interval may be favoured for the calculation of the threshold.As such, and especially when samples with low 10 Be content are expected, processing more than one blank per batch (Fig. 4) 20 guarantees better constrained (i.e., statistically based on a larger number of measurements) determination limits.
It is important to remark that choosing a subset of blanks for the calculation of the determination limits does not reflect the long-term history of the laboratory.This approach can only be used to limit the ages or erosion rates of the samples processed in the same time-span of when the subset of blanks was processed.Also, in the case of lowconcentration samples, the effort and/or expense of acquiring low 10 Be/ 9 Be carrier, which typically defines the 25 lower attainable limit for the laboratory blank measurements, is most likely worthwhile.When estimating the determination limits, it is also important to consider whether to follow the IUPAC recommendations (i.e., assuming a Gaussian distribution), or to use the distribution percentiles for a more accurate estimation of the determination limits (Fig. 4).

Increasing the precision of AMS measurements 30
In general, samples with larger uncertainties have a higher chance of falling below the determination limit.
Accordingly, apart from improvements in laboratory cleanliness, another way to decrease the uncertainty on the 10 Be content of a sample would be to increase the precision of the AMS measurement.This could be done by increasing the amount of sample processed during the chemical procedure, by reducing the loss of quartz during the chemical treatment of the samples (see supplementary material, text S4), or by increasing the AMS current 35 during measurement (Schaefer et al., 2009;Rugel et al., 2016;Fig. 4).Whether to increase the amount of sample to process, or to use specific techniques to minimize quartz loss during Be-purification may depend on the laboratory standards, costs, processing-times, and the feasibility of acquiring large amounts of sample.

Which limit do I use?
Given the high confidence intervals that are often necessary in particular applications of chemistry, such as when dealing with contaminants (McKillup and Darby Dyar, 2010), the IUPAC and ACS have defined the LOQ as the limit for the quantitative use of the sample data.However, for cosmogenic studies, there is little need for such high confidence intervals and consequently conservative interpretations.As such, the choice of the LOD as 5 the lower threshold is reasonable, and still allows users to distinguish samples from blanks with a reliability of 99.9%.When such high confidence is not required, the users may alternatively use a 2σ confidence interval (i.e., equal to 97.7% for 1-tail test) as suggested by Stuiver and Polach (1977) for radiocarbon measurements.
Accordingly, the LOD97.7 is calculated with k = 2 in Eq. ( 1) if the blank distribution is Gaussian, or using the 97.70 th percentile in the case of a non-normal blank distribution.10 Once calculated, the LOD represents the lowest number of 10 Be atoms that can be distinguished from the blanks and, thus, can be used to limit the ages or the denudation rates shown by the dataset.Our results indicate that, after having established which samples are above the defined threshold, the type of blank correction and the representative value used for the blank subtraction (i.e., mean versus median value), may be important for samples with low 10 Be content, whereas their importance becomes negligible for samples with more than 10 6 10 Be atoms 15 (Fig. 5).However, because the LOD and the results of the blank correction are strongly dependent on the chosen blank ensemble, the final decision on which approach to use is best evaluated case by case (Fig. 4).

Implications for geomorphic applications
With our blank and sample datasets, we demonstrated that depending on which approach is used for the calculation of the determination limits, the number of samples that can reliably be distinguished from blanks may 20 vary strongly.In particular, considering only the LOD values for both subsets of blanks, we showed that the percentage of samples that cannot be reliably distinguished from the blanks varies between 5% and 37% (Fig. 3).
This range shows that for geomorphic applications, the precision of the measurements and the cleanliness of the laboratory procedure can have a strong impact on the final interpretation of the data.For example, by using low 10 Be/ 9 Be-ratio carrier and increasing the AMS current, Schaefer et al. (2009) were able to obtain very low and 25 precise blank values (between 6,000 and 26,000 atoms with 1σ < 10%) and very precise sample measurements (between 70,000 and 1,000,000 atoms with 1σ < 10%).When applying the IUPAC recommendations to this dataset, the LOD is around 58,500 10 Be atoms, implying that all the samples are statistically distinguishable from the blanks and can be used with a confidence of 99.9%.With these highly precise results, the authors measured exposure ages as young as 150 ± 15 years, dating moraines of the Little Ice Age. 30 In a case of similarly young boulders and rapidly denuding tributary catchments on an alluvial fan, Savi et al. (2016) showed that depending on which blank correction method is applied, the propagated error on the final exposure ages or denudation rates can vary up to 20%.Also, considering the LODN as the lower threshold, these authors had 3 samples (about one fifth of a first set of processed samples) that could not be distinguished from the blanks, and could only be used to limit the exposure ages or denudation rates.However, by increasing the amount 35 of sample processed during the chemical procedure for the following set of samples, and thus increasing the precision of the AMS measurements, all of the remaining samples could be statistically distinguished from the blanks with a confidence of 99.9%.These results were interpreted as quantifiable exposure ages as low as 50 ± 8 Earth Surf.Dynam. Discuss., doi:10.5194/esurf-2017-30, 2017 Manuscript under review for journal Earth Surf.Dynam.Discussion started: 30 May 2017 c Author(s) 2017.CC-BY 3.0 License.
years, and denudation rates as fast as 13.8 ± 2.6 mm/yr.This study highlights how it may be possible to measure young exposure ages and fast denudation rates at a very high confidence level.

Conclusions
In this paper we have discussed the challenges related to the use of cosmogenic nuclide techniques in the case of low 10 Be content, which are typically found in samples collected from rapidly eroding landscapes, young 5 surfaces, or when a very limited amount of the target mineral is available for analysis.By adapting a method commonly used in analytical chemistry, we describe an approach to define a lower threshold above which samples with low 10 Be content can be used in a quantitative way, accounting for laboratory cleanliness and contamination that may occur during the chemical procedure.This approach can be applied by the end-user of AMS measurements based on a number of different options for characterizing laboratory blanks.10 In summary, in an ideal situation, the use of at least 20 blank values would guarantee a statistically reliable value for the limit of detection, LOD, which can be considered as the lowest threshold for the quantitative use of cosmogenic nuclide data.When samples with low 10 Be content are expected, the user can process multiple blanks within a single sample batch.As an alternative, one can use a long-term value derived from several laboratory blanks processed over a limited time frame during which laboratory conditions are assumed to be nearly constant.15 When an acceptable number of blank values is available (i.e., minimum 20), finding the distribution that best describes the blank ensemble and using the 99.9 th percentile of that distribution for the calculation of the LOD guarantees a more precise estimate of this threshold at the fixed confidence level.
Our analysis demonstrates the importance of producing low, precise, and reproducible blank measurements, as they reduce the value of the various determination limits, therefore increasing the number of samples that are 20 distinguishable from laboratory blanks.Particularly for samples with low 10 Be content, it is important to report detailed information about the laboratory protocols, blank measurements (including both the measured ratios and the amount of added carrier, in order to calculate the number of 10 Be atoms in the blanks), the value and the procedure used to calculate the chosen determination limit, and the applied blank correction method.
Earth Surf.Dynam.Discuss., doi:10.5194/esurf-2017-30,2017   Manuscript under review for journal Earth Surf.Dynam.Discussion started: 30 May 2017 c Author(s) 2017.CC-BY 3.0 License.and negative binomial blank distributions associated with both the eight-blank ensemble and the 61-blank ensemble; (2) assess the reliability of each single sample by comparing its 10 Be content to the LOD and LOQ values; and (3) perform different blank corrections to obtain the final 10 Be content of our samples.
component of the laboratory background, and the variability that may occur among different sample batches.The long-term blank correction method has the advantage of being based on many measurements (and hence is Earth Surf.Dynam.Discuss., doi:10.5194/esurf-2017-30,2017 Manuscript under review for journal Earth Surf.Dynam.Discussion started: 30 May 2017 c Author(s) 2017.CC-BY 3.0 License.statistically well constrained).However, because it includes variability associated with different operators and potential variations among batches, it will likely overestimate the variability associated with a single operator processing a single batch (or a limited number of batches) of samples.
Detection (LOD) and Limit of Quantification (LOQ) for the eight blank-ensemble (processed along with the samples) and the 61 GFZ long-term blanks.The subscript "N" after LOD and LOQ refers to the normal distribution whereas the "NB" refers to the negative-binomial distribution.

Figure 1 .
Figure 1.Examples of samples with high (a) and low (b) 10 Be content.The black curves describe the distribution of several individual blanks, while the grey curves represent one single sample.As an example, we choose a confidence interval of 95% for setting the determination limit.Samples whose lower uncertainty value (i.e., µS -σS) is below the determination limit are consistent with the null hypothesis (H0), and as such they cannot be distinguished from the 5

Figure 2 .
Figure 2. Probability density function (PDF) and normalized frequency of the eight single blanks (in pink, histogram) approximated with a normal distribution (in pink, solid line) and with a negative binomial distribution (in pink, dashed line).PDF and normalized frequency of the 61 long-term blanks from the GFZ (in violet, histogram) represented by both a normal distribution (violet, solid line) and a negative binomial distribution (violet, dashed line).Probability5

Figure 3 . 5 (
Figure 3. Probability Density Function (PDF) of the 57 samples with calculated determination limits.Note that the samples are best represented by a negative binomial distribution (blue curve; displaying only values < 3.5 x 10 5 atoms for better visibility of the limits).Numbers at the side of the arrows represent the percentage of samples falling below the assigned limit.LOD and LOQ are calculated using Eq.(2) and (3) when using the IUPAC's recommendation 5

Figure 4 .
Figure 4. Schematic view of the steps to perform in case of samples with low 10 Be content.Decreasing the blank values and increasing the precision of the AMS measurements allow to have smaller thresholds and more precise data with small uncertainties.

Figure 5 .
Figure 5.This figure illustrates the difference in values and related uncertainties of the corrected samples associated with the different blank correction methods.a) Samples with low 10 Be atoms (i.e., < 10 5 ) are more sensitive to the applied correction method (note that all samples corrected with the long-term blank correction are not distinguishable from the blanks for values < 10 5 10 Be atoms).b) Samples with high 10 Be atoms (i.e., > 10 6 ) do not show significant differences (note 5

Table 2 . List of the 57 sample values used to test our proposed approach. Note that the column marked as "Minimum 10 Be content" indicates the value to use for the comparison with the LOD and LOQ values.
Earth Surf.Dynam.Discuss., doi:10.5194/esurf-2017-30,2017Manuscriptunder review for journal Earth Surf.Dynam.Discussion started: 30 May 2017 c Author(s) 2017.CC-BY 3.0 License.