The relationship between mountain height and rank in height for a mountainous region is examined. A stochastic differential equation model is derived for the evolution of mountain elevations. The derivation is based o...The relationship between mountain height and rank in height for a mountainous region is examined. A stochastic differential equation model is derived for the evolution of mountain elevations. The derivation is based on simple assumptions about tectonic and erosion processes in mountain elevation dynamics. At any given time, the model yields a CIR-type probability density for mountain heights. As data are often available for mountains of greatest elevation in a region, the tail of the CIR density is studied and compared with mountain height data for the highest mountains in the region. The tail density is proportional to the product of a power of height and an exponential function of height, i.e., h<sup>b-1</sup>exp(-ah) where h is mountain height and a and b are constants. The inverse distribution function of the tail probability density leads to a formula that relates rank in height to the corresponding mountain height. The formula provides, for example, a decreasing sequence of theoretical mountain heights for the region. The derived formula is tested against mountain height data sets for several mountainous regions in the British Isles, Continental Europe, Northern Africa, and North America. The derived formula provides an excellent fit to the mountain height data ranked by height.展开更多
文摘The relationship between mountain height and rank in height for a mountainous region is examined. A stochastic differential equation model is derived for the evolution of mountain elevations. The derivation is based on simple assumptions about tectonic and erosion processes in mountain elevation dynamics. At any given time, the model yields a CIR-type probability density for mountain heights. As data are often available for mountains of greatest elevation in a region, the tail of the CIR density is studied and compared with mountain height data for the highest mountains in the region. The tail density is proportional to the product of a power of height and an exponential function of height, i.e., h<sup>b-1</sup>exp(-ah) where h is mountain height and a and b are constants. The inverse distribution function of the tail probability density leads to a formula that relates rank in height to the corresponding mountain height. The formula provides, for example, a decreasing sequence of theoretical mountain heights for the region. The derived formula is tested against mountain height data sets for several mountainous regions in the British Isles, Continental Europe, Northern Africa, and North America. The derived formula provides an excellent fit to the mountain height data ranked by height.
