Information about past climate, tectonics, and landscape evolution is often obtained by dating geomorphic surfaces comprising deposited or aggraded material, e.g. fluvial fill terraces, alluvial fans, volcanic flows, or glacial till. Although surface ages can provide valuable information about these landforms, they can only constrain the period of active deposition of surface material, which may span a significant period of time in the case of alluvial landforms. In contrast, surface abandonment often occurs abruptly and coincides with important events like drainage reorganization, climate change, or landscape uplift. However, abandonment cannot be directly dated because it represents a cessation in the deposition of dateable material. In this study, we present a new approach to inferring when a surface was likely abandoned using exposure ages derived from in situ-produced cosmogenic nuclides. We use artificial data to measure the discrepancy between the youngest age randomly obtained from a surface and the true timing of surface abandonment. Our analyses simulate surface dating scenarios with variable durations of surface formation and variable numbers of exposure ages from sampled boulders. From our artificial data, we derive a set of probabilistic equations and a MATLAB tool that can be applied to a set of real sampled surface ages to estimate the probable period of time within which abandonment is likely to have occurred. Our new approach to constraining surface abandonment has applications for geomorphological studies that relate surface ages to tectonic deformation, past climate, or the rates of surface processes.
Geomorphological studies that link the formation of landforms to past changes in climate or tectonic deformation depend on the accurate dating of surfaces comprising aggraded or deposited material. Surfaces commonly targeted for dating include alluvial fans, fluvial fill terraces, glacial till, pediments, and volcanic flows, among others. For example, the ages of fluvial fill terraces and alluvial-fan surfaces have been used to (i) decipher how erosion and sedimentation have responded to past hydroclimate changes (Owen et al., 2014; Schildgen et al., 2016; Tofelde et al., 2017); (ii) derive time-integrated slip rates for active faults (e.g. Frankel et al., 2007, 2011; Gosse, 2011; Hughes et al., 2018); and (iii) quantify the rates of surface processes such as weathering, landform erosion, or channel avulsion and incision (Schildgen et al., 2012; Regmi et al., 2014; Bufe et al., 2017; D'Arcy et al., 2018).
A common assumption is that a geomorphic surface can be represented by a single formation age. Surfaces are usually point-sampled and dated in multiple locations, e.g. by cosmogenic nuclide exposure dating of surface boulders. Typically, sampling is limited to a small number of (often fewer than 10) large, stable surface boulders, which exhibit no evidence of weathering, rotation, or disturbance. From the set of exposure ages obtained, an average surface age can be calculated with an uncertainty that reflects both analytical uncertainty and the spread of sampled ages. However, many geomorphic surfaces are active for an extended period of time, during which material is continually deposited until the surface is abandoned (e.g. Savi et al., 2016; Denn et al., 2017; Foster et al., 2017). Alluvial-fan surfaces provide one example. Rather than being formed instantaneously, fan surfaces are typically active for thousands or tens of thousands of years before being abandoned when the channel avulses or incises (e.g. Dühnforth et al., 2007). This prolonged period of activity results in a meaningful spread in ages collected from a single surface (e.g. Owen et al., 2011). For any geomorphic surface with a non-negligible period of formation, a set of surface ages will capture a portion of the full time span over which that surface was active. An average of those ages will sit somewhere within the true time span of surface deposition, whereas the maximum age might approximate the onset of surface activity, and the minimum age might approximate the timing of surface abandonment.
In some cases, the timing of surface abandonment may be a more useful constraint than an average surface age. In contrast to surface deposition, abandonment occurs at a particular moment in time (e.g. coinciding with a switch to incision) and so can, in principle, be defined with greater precision. For surfaces with an extended period of formation, the timing of abandonment is more likely to coincide with events of interest such as reorganization of a drainage network (Bufe et al., 2017); changes in climate, sediment supply, or base level (Steffen et al., 2009; Tofelde et al., 2017; Mouslopoulou et al., 2017; Brooke et al., 2018); or tectonic deformation such as faulting, uplift, or subsidence (e.g. Frankel et al., 2007, 2011; Ganev et al., 2010). Abandonment ages would also benefit any study that uses surface exposure dating to measure the rates of post-depositional processes, such as in situ weathering (e.g. White et al., 1996, 2005; D'Arcy et al., 2015, 2018), the topographic decay of landforms (e.g. Hanks et al., 1984; Andrews and Bucknam, 1987; Spelz et al., 2008), or channel avulsion and incision (e.g. Schildgen et al., 2012; Finnegan et al., 2014; Malatesta et al., 2017). Yet the abandonment of a surface represents a cessation in the deposition of dateable material, and therefore cannot be directly dated. Instead, the timing of abandonment must be inferred. Some studies make assumptions about when geomorphic surfaces were abandoned based on independent information such as palaeoclimate records (e.g. Macklin et al., 2002; Cesta and Ward, 2016); others assume that the youngest sampled surface ages fall close to the timing of surface abandonment (e.g. Sarıkaya et al., 2015; Foster et al., 2017; Ratnayaka et al., 2018; Clow et al., 2019). These assumptions risk interpretations that are circular (in the former case) or potentially inaccurate (in the latter case), highlighting the need for a robust method to quantitatively infer the timing of surface abandonment from a set of sampled surface ages.
Here, we introduce a new probabilistic approach to constraining when a depositional surface was abandoned based on what is known about its activity. We use artificial data to randomly point-sample the ages of virtual surfaces in scenarios that are representative of studies dating natural geomorphic landforms such as alluvial fans. We quantify how close the youngest obtained age is likely to fall within the true timing of abandonment, depending on the overall period of surface activity and the number of samples collected. From these artificial data, we derive a set of probabilistic equations and a MATLAB tool that can be applied to real geomorphic surfaces to estimate when they were abandoned. Finally, we demonstrate the application of these equations and the MATLAB tool to natural surfaces with a case study of dated alluvial-fan surfaces in Baja California, Mexico.
Here, we present a hypothetical example of a dated alluvial-fan surface to illustrate why the timing of abandonment may, in some cases, be more useful than an average of sampled surface ages. Consider an alluvial-fan surface that was active for a 30 kyr time span, starting at 80 ka and ending at 50 ka, when the surface was abandoned due to fan incision (Fig. 1). In this example, deposition occurred on the fan surface throughout a period of climatic stability and abandoned when the climate changed, and we make the assumption that there is an equal likelihood of obtaining any age within the entire period of deposition. A distribution of surface ages can be obtained by point-sampling the fan surface; an approach analogous to studies that measure the exposure ages of boulders atop landforms. We present two arbitrarily selected possible outcomes in Fig. 1, where six surface ages are obtained. In scenario 1, the ages are distributed relatively evenly through time, producing a mean age of 65.8 ka, which closely approximates the true average surface age of 65 ka, with a standard deviation of 10.5 kyr. In scenario 2, the ages obtained are unevenly distributed through time, producing a slightly older mean surface age (71.4 ka) and a smaller standard deviation (5.2 kyr). These scenarios are plotted against time in Fig. 1b as data points and kernel density plots, and they resemble equivalent natural datasets (e.g. Owen et al., 2014).
Sample set 2 is more tightly clustered than sample set 1, despite being less representative of the average surface age, illustrating that greater clustering of ages is not necessarily an indicator of accuracy. Furthermore, neither average age corresponds to any meaningful event. The fan surface was equally active for the entire period between 80 and 50 ka, the average ages sit within a period of climatic and depositional stability, and the peaks in the kernel density plots are artefacts created by randomly sampling a linear series.
In contrast, the abandonment of the fan surface does occur at a precise
moment in time when deposition ends at 50 ka. In this example, abandonment
coincides with an abrupt change in climate that triggered an incision event
(cf., Simpson and Castelltort, 2012), so it is arguably a more informative
target for dating than an average age that imprecisely approximates the
mid-point in the duration of surface deposition. However, the abandonment of
the surface represents a cessation in the deposition of dateable material,
so its timing instead must be inferred from what is known about the surface
activity. Given that (i) the sampled ages constrain the time span over which
the surface was formed and (ii) abandonment occurred sometime after the
youngest age, it could be assumed that the youngest sampled age best
approximates abandonment. In scenario 1, the youngest age falls within
This question cannot currently be answered for a natural dataset, yet the ability to reliably estimate when a surface was abandoned has important implications for many geomorphological applications (see Sect. 1). In this study, we use artificial data to simulate natural surfaces that undergo active deposition over variable periods of time and dated with a limited number of surface ages. These artificial data are analogous to studies that date natural geomorphic surfaces, for example, with cosmogenic nuclide exposure ages obtained from sampled boulders on alluvial fans or fluvial fill terraces. However, unlike field-based datasets, artificial data enable us to constrain the time difference between the youngest age obtained from a geomorphic surface and the true timing of surface abandonment, which is generally unknowable for natural surfaces. There are several additional advantages to taking an artificial-data approach. First, we can repeat the random sampling of surface ages (e.g. as depicted in Fig. 1) many times to probabilistically determine where the youngest sampled age tends to fall with respect to abandonment. Second, we can prescribe the surface parameters, meaning the exact timing of abandonment and the full period of surface activity are known. Third, we can select surface properties that are representative of natural geomorphic surfaces and numbers of samples commonly obtained in geomorphic studies. Fourth, we can perform a thorough quantification of the uncertainties in our analyses. For the above reasons, the artificial-data approach allows us to derive a set of equations and develop a MATLAB tool that can then be applied to natural datasets (a set of surface ages) to determine the probability of surface abandonment occurring within a specified window of time.
We used artificial data to constrain the temporal discrepancy, which we
denote
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,
Schematic surface with a period of activity,
Our artificial-data experiments simulate surfaces with durations of
activity,
Example results of the artificial-data experiments for a surface
active from 80 to 50 ka (
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,
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
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.
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.
The probable abandonment window,
A frequency distribution can also be plotted for the youngest age,
We equate the percentiles of
At the same time,
The results of our artificial-data experiments (Fig. 4) can be described by
one equation that allows
The parameter
Equation (2) can be solved for
The artificial data indicate that, for example, six randomly distributed ages
will span
A regression can be fitted to the distributions in Fig. 6, taking the form
Given that Eqs. (2) through (6) are probabilistic (i.e.
Box plots showing the fraction of the total period of fan
activity,
To solve for
Schematic demonstrating how to infer the timing of surface
abandonment from a set of sampled ages.
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
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
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,
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.
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.
Two 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).
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
For both distributions of fan surface ages, we used Eqs. (2) through (6) to
calculate probable abandonment windows,
Our estimates of when the Laguna Salada fans were abandoned have important
climatic implications. Spelz et al. (2008) speculated that the aggradation
and incision of the fan surfaces was partly controlled by past climate
changes, and there is growing evidence that alluvial systems can be highly
sensitive hydroclimate recorders (D'Arcy et al., 2017a, b; Terrizzano et al.,
2017; Tofelde et al., 2017; Ratnayaka et al., 2018; Wickert and Schildgen,
2019). We explore this idea by comparing the surface age data with two
palaeoclimate proxy records (Fig. 8d): the GRIP ice core
The obtained Q7 ages clearly coincide with the broadly interglacial
conditions of MIS 7, so we interpret that the surface was deposited
throughout this stage. Our statistical analyses indicate that the Q7 surface
was abandoned – in this case due to fan incision – during the subsequent MIS
6 and coinciding with a climatic transition to more glacial conditions.
Indeed, 71 % of the area beneath the Q7
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
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
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.
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,
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
Finally, our approach derives the true period of surface activity,
The uncertainty in
Our study uses artificial data to simulate depositional geomorphic surfaces that form over a non-negligible time span, and are subsequently dated with exposure ages on a set of randomly sampled boulders. We investigate scenarios that are representative of natural alluvial fans, which are commonly targeted for surface dating; however, our results may be more broadly applicable to other depositional landforms that form over protracted periods of time. Our findings suggest that, for a variety of different purposes, inferring the timing of surface abandonment may provide more informative and more precise interpretations than taking an average of measured surface ages. We use our artificial data to derive a set of probabilistic equations that can be applied to a distribution of real sampled surface ages to estimate a period of time within which abandonment is likely to have occurred with a given probability. These equations account for site-specific variables including the number of ages and the duration of activity for a particular surface and our artificial-data approach can be used to generate a probability distribution of likely abandonment ages. We provide a MATLAB script that generates a probability distribution of abandonment ages for a given surface and furthermore allows for the uncertainty associated with measured ages to be incorporated. The ability to constrain the timing of surface abandonment has useful applications for geomorphological studies that relate surface ages to tectonic deformation (e.g. deriving fault slip rates), climate (e.g. reconstructing past hydroclimate changes), or the rates of surface processes (e.g. weathering and landform evolution), a subset of which we demonstrate using a case study of alluvial-fan surfaces in the Laguna Salada Basin, Mexico. The statistical framework we introduce in this paper offers a new method of probabilistically estimating when a surface was abandoned, which can complement and enhance interpretations of any distribution of sampled ages obtained from surfaces that experienced a non-negligible period of deposition.
The MATLAB script used to analyse the data is provided in the Supplement.
All data in this article are presented in the
main paper and are freely available online via the Figshare repository (D'Arcy et al., 2019);
The supplement related to this article is available online at:
MKD, JMT, and TFS conceived of the idea for this article. MKD performed the analyses and wrote the main text. TFS created the probabilistic sampling MATLAB script and contributed to interpretations. JMT contributed to the statistical analyses and PDN contributed to climatic interpretations. All authors commented on and contributed to the final article.
They authors declare that they have no conflict of interest.
We thank Luca Malatesta, an anonymous reviewer, and the associate editor Simon Mudd for constructive comments that improved the quality of the article.
Mitch K. D'Arcy was supported by an Alexander von Humboldt postdoctoral fellowship, a “Research Focus Earth Sciences” grant awarded by the University of Potsdam, and by the Emmy-Noether-Programm of the Deutsche Forschungsgemeinschaft, grant number SCHI 1241/1-1, awarded to Taylor F. Schildgen. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.
This paper was edited by Simon Mudd and reviewed by Luca C. Malatesta and one anonymous referee.