The scale-dependent and non-linear relationship between topographic metrics and rock strength

Abstract:

It is commonly accepted that the mechanical strength of rock masses influences erosional processes. While strength can be quantified in the field or laboratory for small rock pieces, the integration of larger scale discontinuities (fractures, bedding planes) remains challenging. Because discontinuities lower the bulk strength of rock masses, the difference between small scale measurements and landscape scale properties is thought to be significant. However, we currently lack observational and quantitative data in order to quantify the scale-dependent variability in rock strength. Instead, topographic metrics are often used as proxies for the forcing of landscape change and variation in rock properties at larger spatial scales. While we often assume that there is a positive relationship between rock strength and topographic metrics, we lack knowledge of how topography is quantitatively related to changes in strength at the appropriate scale.

Here we evaluate the contribution of rock strength to topographic form in the Topatopa Mountains, southern California, USA, which exhibit a strong gradient in the maximum burial depth of Cretaceous through Plio-Pleistocene clastic sedimentary rocks exposed at the surface and for which long-term erosion rates and climate conditions are spatially uniform. Field measures of rock strength including subsurface S-wave velocities, Schmidt hammer hardness values, and Geological Strength Index observations, are used to demonstrate a three- to ten-fold increase in strength with increasing stratigraphic age. We find that for rocks of a given stratigraphic age, outcrop-scale shear strength is an order of magnitude lower than shear strength of intact rock pieces. Topographic metrics including channel steepness, local relief, hillslope relief, and slope demonstrate a non-linear relationship with shear strength at both spatial scales. This relationship suggests that channel and hillslope gradients, as well as relief, initially increase rapidly with increasing strength, but become progressively less sensitive to rock strength variations with the exposure of stronger rocks. This may explain the observed limiting values of topographic form or “threshold” conditions, and cautions interpretation of steady-state landscapes based on topography alone.