Research ArticleGEOLOGY

Time scale bias in erosion rates of glaciated landscapes

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Science Advances  05 Oct 2016:
Vol. 2, no. 10, e1600204
DOI: 10.1126/sciadv.1600204
  • Fig. 1 Schematic of a time series of landscape-scale erosion.

    Schematic of a time series of landscape-scale erosion highlighting the erosional pulses (blue bars) and hiatuses. In this framework, the functional dependence of estimated erosion rates on averaging time scale is determined by the probability distributions of the magnitudes of erosional pulses and hiatuses (fig. S1 and Supplementary Note 1).

  • Fig. 2 Worldwide compilation of erosion rate estimates over a range of averaging time scales.

    (A) Estimated erosion rates in glaciated landscapes show marked increase with a decrease in averaging time scale, which correlates with age and is characterized by an inverse power-law trend (Materials and Methods). My, million years; NW, Northwest. (B) Estimated erosion rates in landscapes where glacial processes are not dominant. The colored boxes around the square markers indicate the SE in averaging time scale and erosion rate estimates (table S2). The colored dashed lines indicate the best power-law fit to the data of estimated erosion rates and the averaging time scale (Materials and Methods). The dashed black lines denote a slope of −1, which could be expected because of plotting a rate against time (45).

  • Fig. 3 Scaling exponents that characterize the inverse power-law trend of estimated erosion rates on averaging time scale.

    Plot showing the estimated scaling exponents (including SE) of the best-fitting power laws, which describe the time scale bias in erosion rates of glaciated landscapes. Results indicate that the scaling exponents across a wide range of glaciated landscapes lie within a relatively narrow range of −0.71 to −0.11.

  • Fig. 4 Deterministic versus probabilistic interpretation of inverse power-law trend of estimated erosion rates.

    (A) Representative time series of the magnitudes of erosion with age for the deterministic model, where the magnitudes of erosion decline as a power-law function with age. (B) Representative time series of the magnitudes of erosion with age for the probabilistic model. The magnitudes of erosional pulses in this simulation were held constant with age and were separated by erosional hiatuses drawn from a truncated Pareto distribution (tail index, 0.5; upper bound, 200 ky). (C) Estimated erosion rates for both the time series shown in (A) (deterministic; gray markers) and (B) (probabilistic; red markers) as a function of increasing averaging time scale. In these simulations, the estimated erosion rates average from some time in the past to the present, such that the averaging time scale is equal to the age. Both models result in an inverse power-law relationship of estimated erosion rates with averaging time scale (or age), but for different reasons. For the deterministic model, the power-law trend occurs because the magnitudes of erosional pulses are imposed to decrease as a power-law function of age, and averaging time scale correlates with age. For the probabilistic model, the power-law trend occurs because of the intrinsic dependence of estimated erosion rates on averaging time scale for heavy-tailed erosional hiatuses. We generated 1000 independent realizations of the probabilistic model and estimated the slope and the intercept of the best-fitting power law between estimated erosion rates and averaging time scales less than the upper bound. The black and gray lines show the best-fitting power law function with a mean power-law slope (−0.50) along with the SE (0.004) evaluated by averaging over these 1000 different realizations. The estimated erosion rates saturate for time scales greater than the upper bound on erosional hiatuses (shaded gray area).

  • Fig. 5 Effect of heavy-tailed erosional hiatuses on masking changes in magnitudes of erosional pulses with age.

    (A) Representative time series of magnitudes of erosional pulses showing a twofold increase at 50 ky. (B) Estimated erosion rates as a function of the averaging time scale for the cases when the upper bound on erosional hiatuses was 200 ky and the erosional pulses had a 2-fold (blue markers) and 10-fold (red markers) increase at 50 ka. The estimated erosion rates average from some time in the past to the present, such that the averaging time scale is equal to the age. The estimated erosion rates for the cases with an increase in the magnitudes of erosional pulses are indistinguishable from the control case for averaging time scales that are shorter than the longest possible hiatus (red markers plot on top of the gray and blue markers). The black and gray lines show the best-fitting power-law function with mean power-law slope (−0.50) along with the SE (0.004) evaluated by averaging over these 1000 different realizations. (C) Plot showing the results of time scale dependence of estimated erosion rates where time series of erosion with equal magnitudes of erosional pulses (10 mm) and the same truncated Pareto distribution of erosional hiatuses (tail index, 0.5; upper bound, 200 ky) were spatially averaged (Materials and Methods). The estimated erosion rates average from some time in the past to the present, such that the averaging time scale is equal to the age. Results show that the time scale bias in estimated erosion rates is reduced with increasing number of series that are spatially averaged, that is, large values of n. The best-fitting power-law slope, along with SE resulting from averaging across multiple realizations, is indicated in the legend, where the power-law fits to estimated erosion rates were made for averaging time scales between 10 and 200 ky (upper bound on erosional hiatuses).

  • Fig. 6 Multiscale climate variability during the last 5 million years.

    The δ18O carbonate record (blue curve; inset) from marine basins (65) shows global cooling associated with the onset of late Plio-Pleistocene glaciations. Increasing δ18O values reflect cooler temperatures and greater continental ice volume and vice versa. The power spectral density (red curve; Materials and Methods) of this climate proxy indicates that apart from the major climatic periodicities at 100 and 40 ky (paced by variation in solar insolation), several nested characteristic time scales exist down to the smallest scale as shown by the power-law decay of the spectral density (66). We hypothesize that this climate variability leads to heavy-tailed erosional hiatuses and thus a time scale bias in estimated erosion rates of glaciated landscapes.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/10/e1600204/DC1

    Supplementary Materials and Methods

    fig. S1. Effect of heavy-tailed erosional hiatuses on estimated erosion rates.

    fig. S2. Illustration of the effect of heavy-tailed erosional hiatuses on overshadowing a systematic increase in magnitudes of erosional pulses with age.

    fig. S3. Cumulative erosion versus averaging time scale for glaciated and fluvial landscapes.

    fig. S4. Numerical simulations highlighting the effect of varying magnitudes of erosional pulses.

    fig. S5. Location maps of data for fluvial landscapes.

    fig. S6. Location maps of data for glaciated landscapes.

    table S1. Landscape setting and fitted scaling exponents.

    table S2. Tabulated data used in this study.

    References (87130)

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Materials and Methods
    • fig. S1. Effect of heavy-tailed erosional hiatuses on estimated erosion rates.
    • fig. S2. Illustration of the effect of heavy-tailed erosional hiatuses on overshadowing a systematic increase in magnitudes of erosional pulses with age.
    • fig. S3. Cumulative erosion versus averaging time scale for glaciated and fluvial landscapes.
    • fig. S4. Numerical simulations highlighting the effect of varying magnitudes of erosional pulses.
    • fig. S5. Location maps of data for fluvial landscapes.
    • fig. S6. Location maps of data for glaciated landscapes.
    • table S1. Landscape setting and fitted scaling exponents.
    • table S2. Tabulated data used in this study.
    • References (87130)

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