Conclusions

Apparent ages of low temperature thermochronometers generally decrease with depth below the surface. When basement terranes are uplifted, this variation produces an age-elevation dependence. Therefore, it may be possible to recover the provenance elevation of the erosional products of exhumed fault-blocks. This paper introduced the Cumulative Age Distribution as a tool for comparing observed detrital age distributions with hypsometric predictions. The CAD is the cumulative density function of the measured ages. Arguably the most important advantage of the CAD over alternative approaches is that it visualizes the detrital sample without the need for data smoothing. (Unequal) measurement uncertainties are incorporated in a hypsometrically predicted CAD by numerically simulating the sampling process. The statistical variability caused by the combined effects of limited sample size and analytical uncertainty can be estimated and visualized with a bootstrapped confidence interval of the predicted CAD.

If (1) apatite concentration is uniform across the entire drainage basin, (2) measurement uncertainties are small, and (3) erosion is uniform across the entire drainage basin, then a hypsometrically weighted CAD of detrital AFT data has the same shape as the PAZ curve of the basement. Under the aforementioned assumptions, it would be possible to use the CAD as a tool for paleo-relief reconstruction, by studying sequential CADs through time, where the modern CAD is used to calibrate the older ones (i.e., convert cumulative percentages to meters), thus taking the approach suggested by Stock and Montgomery [1996] one step further. Unfortunately, very few if any field areas fulfill all three requirements. If assumption (3) is not valid, the method is still useful, because testing assumption (3) yields useful quantitative geomophological information. The CADs of the White Mountains reveal that sediments in the currently active Marble Creek are derived from a single point source, but composite samples of the entire alluvial fan are derived from the whole catchment, with the largest contributions from the base of the range and lower contributions from higher up the drainage.

Table 1: Definition of acronyms used in the literature and in this paper

term	definition
PDF	Probability Density Function
CDF	Cumulative Density Function
SPDF, CSPDF	Synoptic Probability Density Function and Cumulative Synoptic Probability Density Function, respectively
SPDF$_{z}$	Hypsometry, i.e. the probability (or area) of any given elevation within a catchment
SPDF$_{t}$	This curve is obtained from n age measurements by: (1) assigning a Gaussian distribution to each age measurement with standard deviation equal to the measurement uncertainty, (2) ``stacking'' all these Gaussian distributions, and (3) normalizing the area under the resulting curve
CSPDF$_{z}$, CSPDF$_{t}$	Cumulative versions of SPDF$_{z}$ and SPDF$_{t}$, respectively
SPDF$_{z^*}$, SPDF$_{t^*}$, CSPDF$_{z^*}$ and CSPDF$_{t^*}$	In the nomenclature of Ruhl and Hodges [2005], z$^*$ and t$^*$ are the normalized versions of z and t, respectively; i.e. z$^*$ = z/(max(z)-min(z)) and t$^*$ = t/(max(t)-min(t)). Thanks to the normalization, elevations and ages can be directly compared with each other. In this paper, the asterisks are dropped and it is clear from the context whether the normalized or the raw variables are used
CSPDF$_{t^*m}$	A ``model CSPDF$_t$-curve''. If n measurements were used to calculate SPDF$_t$, then CSPDF${_t}$ is calculated by randomly selecting n points from SPDF$_z$. First, a nominal normally distributed ``analytical uncertainty'' like that of the actual grain ages is associated with each of the n elevations, the Gaussian distributions are summed, and the resulting curve is normalized to form a CSPDF$_{t^*m}$ curve
(Observed) CAD	``Cumulative Age Distribution'', a discrete step-function with the sorted detrital age measurements on the x-axis and their respective ranks on the y-axis
Predicted CAD	Is calculated by randomly selecting a large number (e.g. k = 1000) of points from SPDF$_{z}$, and assigning a synthetic ``measurement error'' to each of them by adding a random number from the error distribution of one of the n measurement uncertainties of the ``real'' measurements which were used for the construction of SPDF$_{t}$ and CSPDF$_{t}$. The k ``synthetic measurements'' are then sorted and plotted as a CAD, i.e. a discrete step-function with the sorted measurements on the x-axis and their respective ranks on the y-axis. Thus, the predicted CAD is practically identical to the CSPDF$_{t^*m}$