Scarp degraded by linear diffusion: Inverse solution for age

Under the assumption that landforms unaffected by drainage channels are degraded according to the linear diffusion equation, a procedure is developed to invert a scarp profile to find its “diffusion age.” Diffusion age, having dimension [length]², is the product of diffusivity times chronological age. The second moment of scarp slope grows linearly with age. This fact, together with an assumption about initial scarp shape, allows the inverse determination of diffusion age. Age found assuming a vertical initial scarp is termed “apparent age”; any nonvertical initial scarp profile has a nonzero initial apparent age. True diffusion age differs from apparent age by a fraction of scarp offset squared. The inverse procedure applied to synthetic data yields the following rules of thumb. Evidence of initial scarp shape has been lost when apparent age reaches twice its initial value. If a scarp is formed by two events, the inversion gives their offset-weighted-mean age with an error that is a fraction of offset squared. A scarp that appears to have been formed by one event may have been formed by two with an interval between them as large as apparent age. After scarps of two fault traces have diffused to appear as one, the error in inferred age may be as large as half the apparent age. Variation of apparent age along strike would indicate multiple fault traces. The simplicity of scarp profile measurement and this inversion makes profile analysis attractive. If linearity of the flow law, time for a free face to be reduced to the angle of repose, and variation of diffusivity with climate and material could be established, profile analysis would become a reliable dating technique.