Some analytical methods for converting thermochronometric age to erosion rate

An analytical method is presented for converting thermochronometric ages to surface erosion or, equivalently, exhumation rate. The method incorporates the two most important thermal processes during cooling by erosion: the dependence of closure temperature on cooling rate and the advection of heat by rock motion toward the Earth's surface. Two thermal models are considered: (1) a steady state model, valid for low erosion rates; and (2) an eroding half-space model, which has no steady state, but captures the transient increase of geothermal gradient with erosion. In each case, it is assumed that data consist of one or more thermochronometric ages, present-day surface geothermal gradient, and topographic information including the elevation at which the age was obtained. Analytical solutions are provided to derive the erosion rate from these data either as an explicit expression for the steady case or as a root-finding problem for the transient case. A graphical method for plotting age against erosion rate and geothermal gradient is presented as a method for solving the root finding problem and for tracking analytical errors in observations of age and surface geothermal gradient. The graphical method is also appropriate for comparing data from different elevations or from different thermochronometric systems. Examples are provided using synthetic data or published data from the literature.