3.1 Artificial-data approach

We used artificial data to constrain the temporal discrepancy, which we denote τ, between the youngest age sampled on a surface (a_min) and the actual timing of surface abandonment (t_aban):

Our experiments are designed to be representative of natural alluvial-fan surfaces but the results are more widely applicable to any abandoned depositional surface that has been dated.

In the absence of additional information (e.g. the existence of an independent constraint, such as a younger alluvial-fan surface with an intermediate age), the abandonment of a surface could have occurred at any time between the youngest sampled age, a_min, and the present, or within a particular time window after a_min. In the example case (Fig. 1), the data in sample set 1 would require a time window, τ, of 1.1 kyr placed immediately after the youngest age to overlap with the correct timing of surface abandonment, t_aban. However, for sample set 2, a 14.4 kyr window is required. For natural cases, the abandonment timing is unknown; we know the temporal discrepancy between a_min and t_aban in these artificial-data examples because we impose t_aban. An advantage of the artificial-data approach is that knowledge of t_aban allows us to quantify τ in every tested scenario, which in turn enables us to determine probability distributions of τ. Conceptually, the size or τ, i.e. the proximity of a_min to t_aban, depends on the number of surface ages obtained, n. The greater the number of samples, the closer the youngest sampled age is likely to come to the abandonment age (Fig. 2a). The size of τ also depends on the total duration of surface activity, which we denote as T. If n ages are randomly sampled from a longer time span, T, then a_min is likely to fall farther from t_aban (Fig. 2b).

Figure 2Schematic surface with a period of activity, T (orange bar), abandoned at t_aban, and randomly sampled with n ages (circles). (a) If n increases, the youngest sampled age, a_min, is likely to fall closer to t_aban. (b) If T increases, the youngest sampled age, a_min, is likely to fall farther from t_aban, even if the same number of ages are sampled.

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Our artificial-data experiments simulate surfaces with durations of activity, T, between 10 and 50 kyr, sampled with numbers of surface ages, n, between 2 and 10. These values are representative of natural alluvial-fan surfaces and typical dating studies involving a small number of ages. For each combination of T and n, we randomly sampled a set of surface ages 10 000 times, allowing us to constrain the probability that a_min falls within a certain temporal window (τ) of t_aban in each scenario.

Figure 3Example results of the artificial-data experiments for a surface active from 80 to 50 ka (T=30 kyr). A number of ages, n, were randomly sampled from the surface 10 000 times. (a) Frequency distributions of resulting mean sampled age, $\stackrel{\mathrm{‾}}{a}$. (b) Frequency distributions of the youngest sampled age, a_min. (c) Cumulative frequency distributions of τ normalized to a sum of 1. (d) Selected percentiles of τ plotted against n.

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3.2 Implementation

We first implemented our experiments using discrete sampling within a spreadsheet. For each surface, we created a list of selectable surface ages spanning the total period of surface activity, T, and placed at equal intervals of 0.1 kyr. For the example case (Fig. 1), this would mean a list of selectable ages of 80.0, 79.9, 79.8 ka, etc., to a minimum value of 50.0 ka. We chose periods of surface activity, T, equal to 10, 20, 30, 40, and 50 kyr. From each list, we randomly selected n unique values and repeated this exercise 10 000 times for each integer value of n between 2 and 10. For example, if n=6 and T=20 kyr, then we extracted 10 000 different datasets, each comprising six randomly selected ages, from the 20 kyr long list of selectable ages available at 0.1 kyr intervals. This process is analogous to random sampling of six boulders for cosmogenic nuclide exposure dating on an alluvial-fan surface that formed over a 20 kyr period and deposited a selectable boulder once every 100 years. We extracted 10 000 sets of ages for each of the 45 different combinations of T (five unique values) and n (nine unique values). For each dataset, we calculated the mean value of the selected ages, $\stackrel{\mathrm{‾}}{a}$, and the time difference between the youngest age and the abandonment time, τ. We then determined cumulative frequency distributions of τ in each scenario with a given combination of T and n.

To test whether 10 000 iterations are sufficient to produce reliable statistics and whether the discretization of ages has an important effect, we repeated all our artificial-data experiments using a non-discrete approach in a MATLAB script. We defined T as a time range, from within which any point in time could be randomly sampled, i.e. tens of thousands of “selectable” surface ages were available rather than hundreds of discrete values. Performing 100 000 iterations with the MATLAB script produced results that are indistinguishable from the discrete spreadsheet-based approach with 10 000 iterations. All data analyses are provided by D'Arcy et al. (2019) in an online data repository. Finally, we explore the assumptions and limitations of our analyses in Sect. 5.3.

3.3 Experimental assumptions

In designing our artificial-data experiments, we make several assumptions. First, surface ages are randomly selected from the total period of surface activity. Therefore, when constructing our experiments, we assume that when ages are obtained from real geomorphic surfaces, they are randomly point-sampling the full time span of surface formation, and that this time span represents a uniform probability distribution of selectable ages. A uniform probability of ages may not be realistic in certain natural cases, for example, if boulders on an alluvial-fan surface are spatially clustered by age and all samples are taken from one part of the surface. Second, the entire period of surface activity is assumed to be available for sampling, i.e. no subset of the surface history is missing as a result of processes like burial or erosion. Third, all selectable ages within the period of surface activity have an equal likelihood of being sampled; this implies that the surface formed with a constant deposition rate and there are no pulses of activity that increase the probability of sampling a particular age. Finally, we do not explicitly factor in processes like nuclide inheritance, erosion, or incomplete exposure, which can affect exposure ages derived from cosmogenic nuclides. We consider the implications of all these assumptions for natural datasets in Sect. 5.3.

4.1 Random sampling of surface ages

To illustrate the results of our experiments, we first present one artificial-data scenario in Fig. 3 in which the surface is formed between 80 and 50 ka (i.e. T=30 kyr) and is randomly sampled with $n=\mathrm{2},\mathrm{4}$, 6, or 8 ages (with 10 000 repeat experiments for each value of n). Figure 3a shows how a frequency distribution of the mean value of all sampled ages, $\stackrel{\mathrm{‾}}{a}$, changes with n. The distribution is centred on the true average surface age of 65 ka and narrows as a greater number of ages are sampled. If only two ages are sampled, then $\stackrel{\mathrm{‾}}{a}$ can occupy almost any age within the full period of surface activity. As n increases, $\stackrel{\mathrm{‾}}{a}$ tends to fall closer to 65 ka. The distribution of $\stackrel{\mathrm{‾}}{a}$ approaches a normal distribution as n increases. This observation is compatible with the central limit theorem and the law of large numbers: $\stackrel{\mathrm{‾}}{a}$ converges on the true average surface age as the number of samples increases, despite the dataset randomly sampling a linear series.

Figure 4The probable abandonment window, τ, as a function of the number of boulder ages, n. Data are shown for different probabilities, P (panels), and durations of surface activity, T (colours). Parameter k is a decay constant that depends on P (see text for details).

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A frequency distribution can also be plotted for the youngest age, a_min, randomly selected from the surface (Fig. 3b). If only two ages are obtained, then the youngest can fall almost anywhere between 50 and 80 ka, although the distribution is asymmetric and younger values of a_min occur slightly more frequently than older values. As n increases, the distribution of a_min shifts towards 50 ka such that when n=8, a_min falls within 5–10 kyr of t_aban (i.e. τ is equal to 5–10 kyr) in the majority of sampling experiments. Cumulative frequency distributions of τ reveal how close the youngest sampled age comes to the known timing of surface abandonment (Fig. 3c). For example, if only two ages are obtained, then in 60 % of experiments, τ≤12 kyr, i.e. the youngest age falls somewhere within 12 kyr of abandonment. If six ages are obtained, then in 90 % of experiments, τ≤10 kyr. Any percentile of τ can be measured from Fig. 3c, allowing τ to be plotted against n (Fig. 3d). As a greater number of ages are obtained, the value of τ associated with a given percentile decreases, i.e. the youngest sampled age comes closer to the timing of surface abandonment. However, the decrease in τ is non-linear and diminishes with increasing n. For example, as n increases from two to four ages, the 95th percentile of τ falls from ∼23 to ∼16 kyr, but collecting another two ages (n=6) only reduces τ to ∼12 kyr. The 95th percentile of τ falls below 10 kyr when n exceeds seven ages. In other words, if seven ages are randomly sampled from a surface, abandonment will have occurred within 10 kyr after the youngest age in 95 % of cases.

We equate the percentiles of τ in Fig. 3c with the probability, P, of abandonment occurring within a time window defined by τ. Thus, if P=0.9, the window of time τ (placed immediately after a_min) is large enough that in 90 % of our experiments, the true timing of surface abandonment would fall within it. This is equal to the 90th percentile of  τ, which would be 7.5 kyr for the scenario T=30 kyr and n=8, for example (Fig. 3d). Note that in this scenario, τ does not imply that the surface was abandoned exactly 7.5 kyr after a_min, but rather that there is a 90 % likelihood that abandonment occurred anywhere within a 7.5 kyr window after a_min. The probable window of abandonment, τ, increases with P because a larger window of time is required to capture the true timing of abandonment in a greater proportion of cases.

At the same time, τ is inversely and non-linearly related to the sample size of ages obtained, n (Fig. 4). The dependencies between τ and n, T, and P are illustrated in Fig. 4 for all tested scenarios that are representative of natural alluvial-fan surfaces (n=2 to 10; T=10 to 50 kyr), with probabilities between 0.50 and 0.95. For example, if six ages are obtained from a surface that formed over a 30 kyr duration, then τ=12 kyr for P=0.95 (Fig. 4a). In other words, in 95 % of cases the youngest of six ages obtained from such a surface will fall within 12 kyr of the true timing of abandonment. Only in 5 % of cases does the discrepancy between a_min and abandonment exceed 12 kyr. If P decreases to 0.5 (Fig. 4f), then τ decreases to 3 kyr for this particular scenario (n=6 and T=30 kyr).

Figure 5(a) Variation in the decay constant, k, as a function of the probability, P. Error bars show the standard error in k when Eq. (2) is fitted to the data in Fig. 4. The regression corresponds to Eq. (3). (b) Variation in τ₀ as a function of T for different values of P (indicated by colours). Linear regressions are fitted corresponding to Eq. (4). Inset: variation in m as a function of P. An exponential regression is fitted corresponding to Eq. (5).

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The results of our artificial-data experiments (Fig. 4) can be described by one equation that allows τ to be calculated for any scenario:

Here, the parameter k is a decay constant with a negative value that increases exponentially with P (Fig. 5a):

Constants a, b, and c can be derived empirically using our artificial data. Note that we calibrate all our equations with time in kiloyears (kyr).

$a=-\mathrm{0.425}\pm \mathrm{0.029}$, $b=\mathrm{0.011}\pm \mathrm{0.011}$, $c=\mathrm{2.830}\pm \mathrm{0.885}$

The parameter τ₀ increases linearly with T, but with a slope that increases exponentially with P (Fig. 5b), and can therefore be described by a pair of relationships:

Parameters m₀, g, and h are constants with values again determined empirically from our artificial-data experiments:

Given that τ₀ signifies the value of τ as n tends towards infinity, it represents the limit of precision that can be obtained on the abandonment period, τ, when inferring the timing of surface abandonment with this probabilistic method. For the scenarios shown in Fig. 5b, which represent reasonable values of T for natural alluvial fans and desirable values of P, τ₀ varies from a few centuries to ∼10 kyr.

4.2 Total period of surface formation

Equation (2) can be solved for τ (using the parameterization of Eqs. 3 through 5) with knowledge of only the number of ages sampled, n, and the total period of surface formation, T, as well as a chosen probability, P. We are able to parameterize τ using artificial data because we know the value of T in our experiments. However, when sampling natural depositional surfaces, T is unknown and instead only the span of sampled ages, a_max−a_min, can be measured. This span might approximate T, but some fraction of time will not be captured. Fortunately, our artificial-data experiments also allow us to determine what fraction of T is captured by a_max−a_min in scenarios of varying n (Fig. 6).

The artificial data indicate that, for example, six randomly distributed ages will span ∼70 % of the total time span of surface activity, T, in the average case. In the 1 % of worst (most clustered age) cases, a_max−a_min will only represent ∼30 % of T, and in the 1 % of best (least clustered age) cases it will represent more than 95 % of T. In half of all experiments for n=6 (from P25 to P75), a_max−a_min falls within 60 %–85 % of T. There is a diminishing improvement with an increasing number of sampled ages, such that by n=10 the average span of ages has only increased to ∼80 % of T and 50 % of all experiments fall between 75 % and 90 % of T. An order of magnitude more ages (hundreds) would be needed for the mean a_max−a_min to come within 95 % of the full period of surface activity.

A regression can be fitted to the distributions in Fig. 6, taking the form

Parameters q, r, and s are empirical coefficients derived graphically from our artificial data. For the mean case (the solid black line in Fig. 6), they take the values

Equation (6) is also fitted to ±1 standard deviation (σ) above and below the mean values in Fig. 6 (dashed black lines). For 1σ above the mean, parameters q, r, and s take the values

For 1σ below the mean, parameters q, r, and s take the values

Equation (6) can therefore be used to estimate the size of T in the average case ±1σ bounds, given the measured span of ages collected from a surface. Equations (2) through (6) are thus calibrated using our artificial data, and can be used to probabilistically calculate the window of time during which any dated surface was likely abandoned.

4.3 Application to measured surface ages

Given that Eqs. (2) through (6) are probabilistic (i.e. P is a variable), our artificial-data approach can be used to infer a probability distribution of abandonment ages from a set of measured surface ages. We illustrate the steps involved in applying Eqs. (2) through (6) and the MATLAB script to real data in Fig. 7.

Figure 6Box plots showing the fraction of the total period of fan activity, T, captured by the span of sampled boulder ages in the artificial-data experiments, a_max−a_min plotted against the number of sampled ages, n. Each box represents 10 000 experiments. As a greater number of ages are sampled, the span of the set of ages is more likely to capture a greater fraction of T, although with diminishing returns for increasing n. Black lines show exponential regressions corresponding to Eq. (6) and fitted to the mean values (solid) and ±1 standard deviation (dashed).

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To solve for τ at a discrete probability value (P), T is first calculated with Eq. (6) and τ is then calculated using Eq. (2), with parameters defined in Eqs. (3) through (5) (Fig. 7a). These discrete values of τ can be converted into a probability distribution by calculating the density of P within fixed increments of τ. For example, in Fig. 7a, 50 % of the probability of abandonment falls within a relatively small window of time (the light blue bar for P=0.5), whereas a longer window of time is required to contain an additional 45 % of the probability of abandonment (the light pink bar for P=0.95). Thus, the density of P is greater within the window τ for P=0.5, and this density diminishes as τ and P increase. The MATLAB script (provided in the Supplement) enables determination of the continuous probability distribution of τ. After generating artificial data based on n ages and a duration of deposition T (from Eq. 6), the script calculates the density of P within fixed increments of τ. If the sampled surface ages are known with exact precision, then the resulting distribution of τ values provides a probability distribution of times that directly postdate the youngest age and yield a probability distribution of surface-abandonment ages (Fig. 7a).

Figure 7Schematic demonstrating how to infer the timing of surface abandonment from a set of sampled ages. (a) Probable abandonment windows, τ, are calculated using Eq. (2) for discrete values of P (coloured bars). A continuous probability distribution of τ is equal to the density of P within each discrete increment of τ (dashed line); this can be generated using the MATLAB tool provided. (b) In reality, a_min is not perfectly known and has an associated age uncertainty that must be accounted for. (i) The ±3σ uncertainty in a_min provides a distribution of probable values of a_min. (ii) The distribution of a_min values is discretized. In the MATLAB tool, we have set this discretization to be one-tenth of the 1σ uncertainty in the youngest age, a_min, to provide a highly resolved result (note that the illustration here shows much wider discretization bins for ease of visualization), but this discretization value can be modified. The discrete window of τ used to calculate the density of P is set to the same width. (iii) Probability distributions of τ are calculated for each potential value of a_min (as per panel a), and weighted according to the probability distribution of a_min values. (iv) The weighted, temporally shifted, τ distributions are then summed to produce a final probability distribution of surface abandonment timing that incorporates uncertainty into the youngest age.

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However, real surface ages have associated uncertainties that must also be incorporated into the estimated abandonment ages (Fig. 7b). The MATLAB tool is designed to incorporate this uncertainty, and is explained in the following steps. First, we use ±3σ uncertainty in a_min to characterize the probability distribution of potential a_min values. In the example schematic (Fig. 7b) we assumed a normal distribution, as is typical for exposure ages of individual boulders, but alternative distributions could be used. This distribution of a_min values is then discretized, and separate probability distributions of τ are calculated for each potential value of a_min, i.e. repeating Fig. 7a. The resulting, temporally shifted probability distributions of τ are weighted according to the probability distribution of a_min and summed, producing an overall probability distribution of likely abandonment ages that accounts for uncertainty in the youngest age (Fig. 7c). If the uncertainty in a_min is small compared to τ calculated using Eq. (2), then incorporating age uncertainty will have little impact on the resulting probability distribution of abandonment ages. If the uncertainty on a_min is large, it will have a greater influence on the final probability distribution of abandonment ages.

5.1 Implications for surface dating

Our artificial data provide new information about what measured ages represent when collected from aggraded surfaces that formed over non-negligible time spans. Crucially, our findings indicate that averages of sampled surface ages are likely to be imprecise representations of the mid-point of surface formation, which may not coincide with a particular external forcing event (Fig. 1). In contrast, surface abandonment typically occurs at a discrete moment in time and is more likely to coincide with external forcing events such as changes in climate or tectonics. By using artificial data, we have derived a set of probabilistic equations for inferring when a surface was likely to have been abandoned, based on a distribution of randomly sampled surface ages. These equations can complement and enhance interpretations based on any dataset comprising surface ages. The spreadsheet and the MATLAB tool allow for quantification of the full probability distribution of τ and the MATLAB tool additionally allows for the incorporation of uncertainty in the youngest age, a_min.

While a distribution of ages is required for dating surfaces that have formed over extended periods of time, our analyses reveal that an increasing number of ages yield diminishing returns for constraining the timing of abandonment (Figs. 3d and 4) and the total duration of surface activity (Fig. 6). An appropriate number of surface ages will depend on the desired precision, but our results indicate that there is little to be gained by obtaining more than six to seven ages per surface (Figs. 3, 4, and 6), assuming no outliers, for the purposes of most geomorphological studies. To obtain substantially more information about a surface, an order of magnitude more ages would be required. As explained in Sect. 4.1, τ₀ represents the maximum precision with which the abandonment age can, in principle, be inferred. For many natural surfaces, τ₀ can range from a few centuries to ∼10 kyr (Fig. 5b), depending on the period of surface activity and the desired probability. Our methodology thus provides a new way to quantify how precisely a distribution of surface ages can be interpreted. This limit to precision should be considered in addition to the age uncertainty associated with cosmogenic nuclide exposure dating; both are important considerations when inverting landforms and sedimentary deposits for palaeo-environmental information (Foreman and Straub, 2017).

When sampling in the field, it may be advantageous to target different parts of an aggraded surface to capture as much of its period of activity as possible. This strategy applies to surfaces upon which the locus of deposition has systematically migrated during deposition. For example, if channel migration on an alluvial-fan surface resulted in a portion of its overall history being recorded in particular parts of the surface (e.g. Savi et al., 2016; Schürch et al., 2016; D'Arcy et al., 2017a, b), then greater spatial coverage would capture a greater range of ages. However, if each deposition event followed a random trajectory on the surface, resulting in all potentially selectable ages being spatially mixed, then it would be unnecessary to distribute sampling locations across the surface.

5.2 Case study: alluvial fans in the Laguna Salada Basin, Mexico

Here, we use a case study of alluvial-fan surfaces in the Laguna Salada Basin, Mexico, to demonstrate how our findings can be applied to real surfaces to gain new information about when they were abandoned.

Figure 8Two alluvial-fan surfaces in the Laguna Salada Basin, northern Baja California, Mexico. Left column shows the Q4 surface and right column shows the Q7 surface, after Spelz et al. (2008). (a) Locations of surface boulders sampled for ¹⁰Be cosmogenic nuclide exposure dating. (b) Boulder exposure ages recalculated after Spelz et al. (2008) (white circles) and mean surface ages ±1σ (yellow stars). (c) Probability distributions of τ calculated using the MATLAB tool and incorporating uncertainty in a_min (black). For illustrative purposes, probability distributions of τ are shown if uncertainty in a_min is not incorporated (red dashed). (d) Selected palaeoclimate proxies: the GRIP ice core δ¹⁸O record from Greenland (blue; Johnsen et al., 1997) and the LR04 global benthic δ¹⁸O stack (black; Lisiecki and Raymo, 2005). Marine isotope stages (MIS) are indicated with boxes.

The Laguna Salada Basin is a half-graben in northern Baja California, Mexico. This basin contains well preserved alluvial fans eroded from the neighbouring Sierra El Mayor and Sierra de Los Cucapah, with at least eight generations of distinct fan surfaces formed by a sequence of aggradation and incision cycles. The ages of two of these fan surfaces – mapped as Q4 and Q7 – were estimated by Spelz et al. (2008) using ¹⁰Be exposure ages of stable surface boulders with no evidence of erosion or disturbance (Fig. 8). We used the CREp calculator (Martin et al., 2017) to update the exposure age estimates of Spelz et al. (2008) using the time-corrected Lal/Stone scaling scheme (Lal, 1991; Stone, 2000), the ERA40 atmosphere model (Uppala et al., 2005), the atmospheric ¹⁰Be-based virtual dipole moment (VDM) geomagnetic database of Muscheler et al. (2005) and Valet et al. (2005), and the current global reference sea level high latitude (SLHL) ¹⁰Be production rate of 4.13±0.20 g⁻¹ yr⁻¹ in the ICE-D database (Martin et al., 2017). We assume a sample density of 2.7 g cm⁻³ and no boulder erosion. The oldest age measured on the Q4 surface was excluded as an outlier by Spelz et al. (2008), and we maintain this interpretation. The remaining exposure ages span from 14.4±1.1 to 32.1±2.9 ka for Q4 (n=5) and 188.6±22.7 to 246.9±13.7 ka for Q7 (n=6) (Fig. 8b, yellow bars). On both fan surfaces, the dispersion of ages greatly exceeds the age uncertainty, suggesting that each surface was deposited over an extended period of time.

For both distributions of fan surface ages, we used Eqs. (2) through (6) to calculate probable abandonment windows, τ, for different values of P. For example, on the Q4 fan surface with an a_min of 14.4±1.1 ka, τ= 3.3 kyr when P=0.5, suggesting a 50 % probability that the surface was abandoned within 3.3 kyr after a_min, i.e. between 14.4 and 11.1 ka. The size of τ increases with P, as explained in Sect. 4.1, such that τ=12.0 kyr for the Q4 surface when P=0.95, i.e. the abandonment window ranges from 14.4 to 2.4 ka. The full probability distribution of τ is highly asymmetric (Fig. 8c, red dashed lines). Of course, the uncertainty in a_min must also be accounted for. To do so, we used the MATLAB script with the required inputs – T and n; the desired number of iterations, a_min; and the 1σ uncertainty in a_min – to derive a continuous probability distribution of τ for each fan surface that incorporates age uncertainty (Fig. 8c, solid black lines). The probability distributions of τ incorporating age uncertainty, derived for both the Q4 and Q7 surfaces, illustrate how the likelihood of surface abandonment is distributed over time for two representative natural datasets. On the Q4 surface, the measured age uncertainty is small compared to τ, so the resulting τ distribution has an asymmetric shape that is primarily determined by the form of Eq. (2) and our artificial-data calibration (Figs. 3 and 4). The majority of the Q4 τ distribution occupies a short time span that is smaller than the spread of sampled surface ages; this result supports our reasoning that the timing of surface abandonment can, in some cases, be constrained more precisely than a representative age of surface formation (see Sect. 2). The age uncertainty in a_min is significantly larger on the older Q7 surface and therefore dominates the probability distribution of τ, giving it a wider and more symmetrical shape despite the greater number of measured ages, n. This result underscores the importance of accounting for age uncertainty when using our equations to infer the likely timing of surface abandonment.

5.2.2 Tectonic and weathering implications

The results in Fig. 8 also have tectonic implications. The Laguna Salada fans are dissected by fault scarps related to the Laguna Salada fault and the Cañada David detachment; the largest Q7 scarp has an offset of 9.9 m (Spelz et al., 2008). Typically, studies divide the fault offset by the mean surface age (which for Q7 is 215.9 ka) to estimate a time-averaged slip rate, which would be 0.046 mm yr⁻¹ in this example. However, as a scarp can only accumulate displacement once the surface has been abandoned, i.e. when it is no longer being resurfaced, the estimated age of abandonment may be a more appropriate timescale for determining a displacement rate. Accumulating a 9.9 m offset since 177 ka (the most likely abandonment age, Fig. 8c) would produce a time-averaged slip rate of 0.056 mm yr⁻¹ an increase of 22 %. Following this logic, the probability distribution of τ could be translated into a probability distribution of time-averaged slip rates. For the Q4 fan surface, calculating a slip rate with a most likely abandonment age (e.g. 12.7 ka) instead of the mean surface age (23.3 ka) would result in an even larger increase in the calculated displacement rate of 83 %. Underestimating fault slip rates by this magnitude could have important implications for tectonic and fault hazard analyses.

Figure 9(a) Measured surface ages for the Laguna Salada alluvial-fan surfaces, following Fig. 8b. (b) Probability distributions of τ calculated using Eq. (2) and incorporating age uncertainty, where T is derived from the measured spread of surface ages using the mean case in Fig. 6 (black curves) and ±1σ uncertainties in T (red and blue curves).

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Spelz et al. (2008) also measured the diffusional decay of fault scarp geometry over time, and used the calculated mean fan surface ages to derive time-integrated scarp mass diffusivities between ∼0.01 and 0.10 m² kyr⁻¹. Intriguingly, the authors interpreted these diffusivities to be anomalously low. More realistic values can be obtained by, again, using the estimated surface abandonment ages to calculate scarp mass diffusivity, rather than average surface ages. This approach would result in faster diffusion rates, as Spelz et al. (2008) expected, while recognizing that a fault scarp can only form and erode once a fan surface has been abandoned.

The alluvial fans of the Laguna Salada Basin provide a representative example of natural, aggraded geomorphic surfaces that are formed over a non-negligible period of activity and are dated with a small set of exposure ages that randomly sample the duration of surface activity. This case study demonstrates that our artificial-data approach can provide valuable constraints on the timing of surface abandonment based on a set of exposure ages, which can improve interpretations involving palaeoclimate, tectonics, and landform evolution.

5.3 Limitations to the probabilistic approach

Our artificial-data approach and the resulting parameterization of Eqs. (2) through (6) assume that a distribution of surface ages are obtained by randomly sampling the full duration of surface activity. In some cases, this assumption might be realistic. For example, the Q7 surface on the Laguna Salada fans (Fig. 8) was sampled in different places and produced ages spanning all of MIS 7, suggesting the full duration of surface activity might be well represented. If so, our approach could be symmetrically applied to the oldest sampled age to estimate the onset of deposition. In contrast, the Q4 surface was sampled entirely at the fan apex, where enhanced vertical aggradation makes it likely that the earliest deposits from this depositional episode have been buried. In practice, this sampling approach would improve estimates of when abandonment occurred. By clustering the surface ages towards the end of the depositional period, T would effectively shorten, given that our approach derives T empirically from the ages that are actually obtained from a surface, and τ would be constrained more precisely as a result. The total duration of deposition would be biased toward younger ages by the burial of older deposits, but this bias is unimportant when focusing on the timing of abandonment, which is a strength of our approach. However, vertical burial would mean that T (solved with Eq. 6) would no longer represent the total duration of deposition, and it would therefore be inappropriate to use our approach to estimate the onset of deposition.

Like burial, subsequent erosion of part of a surface might hide a fragment of the period of deposition from sampling. The implications of erosion depend on how spatially homogenous the surface is, i.e. whether erosion has randomly eliminated selectable ages from throughout the duration of activity or instead eradicated complete fragments of the time span of activity. Again, erosion would only impede our method of inferring the abandonment age if the youngest part of the duration of activity were destroyed. Given that burial and erosion are site specific, they cannot be universally incorporated into our equations and must be considered on an individual-case basis.

Our approach assumes that all sampled surface ages are true ages. In reality, incorrect ages are sometimes encountered when dating surfaces. For example, cosmogenic nuclide exposure ages may be biased towards older ages as a result of nuclide inheritance, as is interpreted to be the case with the oldest exposure age on the Laguna Salada Q4 fan surface (Fig. 8a). Including old outliers in our analyses would lead to an over-estimation of the size of both T and τ, and therefore unnecessarily imprecise estimates of the abandonment window, but would not change the position of a_min. A more serious error would arise from incorrect young ages, e.g. resulting from erosion or shielding of boulders targeted for cosmogenic nuclide exposure dating. The inclusion of spurious young ages could expand the apparent period of surface activity T past the true timing of abandonment, leading to estimates of τ that are both too large and, more importantly, too young. Therefore, Eqs. (2)–(6) and the MATLAB tool should be applied to clean datasets that do not contain spurious ages, and particularly not spuriously young ages when attempting to calculate abandonment times.

Finally, our approach derives the true period of surface activity, T, from the measured age range a_max−a_min, based on the results of our artificial-data experiments (see Sect. 4.2 and Fig. 6). This step is necessary because the true duration of T is ultimately unknowable for natural surfaces, so we parameterize Eq. (6) using the mean ratio of $\left(a_{\max }-a_{\min }\right)/T$ among our artificial-data experiments. Of course, any given set of real surface ages might happen to capture a greater or smaller fraction of T than the mean case. For this reason, we also provide parameterizations of Eq. (6) for ±1 standard deviation (σ) above and below the mean ratio of $\left(a_{\max }-a_{\min }\right)/T$, thus allowing ±1σ uncertainty in T to be tested. In practice, the uncertainty associated with T has little effect on the probability distributions of τ produced by Eq. (2), and so is likely to be insignificant for most geomorphological applications. To illustrate the sensitivity of τ to the uncertainty in T, we recalculate the probability distributions of τ for the Q4 and Q7 Laguna Salada alluvial-fan surfaces with the MATLAB tool (Fig. 9) using the ±1σ bounds on T (Fig. 6).

The uncertainty in T has a negligible effect on the probability distributions of τ for both the young and precisely dated Q4 surface where the τ distribution is most sensitive to the form of Eqs. (2) through (5), and the older, less precisely dated Q7 surface, where the τ distribution is most sensitive to the measured age uncertainty. This sensitivity analysis demonstrates how the conversion of a_max−a_min to T has little bearing on the estimated timings of surface abandonment. Nonetheless, our artificial-data calibration allows the ±1σ uncertainty in T to be calculated, if desired.