One of the fundamental problems of interpreting the sedimentary record is reconstructing the original heights of palaeotopographical features such as bedforms or river channels. This requires an understanding of the relationship between topography and set thickness, but at present an exact theory exists only for periodic topography of uniform height. The applicability of this simple theory is severely limited by the random variability characteristic of many sedimentary systems. In this paper, we develop an exact theory for the probability-density function (PDF) of sets generated from topography of random height. We focus on the limiting case of zero net deposition in order to provide a lower bound for the set thickness, and derive an analytical set-thickness PDF that is determined by one parameter of the PDF for topographical height. This parameter, β, measures the breadth of the tail of the topographical PDF. The mean set thickness is 0.8225 β for bedforms and 1.645 β for river channels. If the topographical height is gamma distributed, the preservation ratio, defined as (mean thickness of preserved sets)/(mean topographical height), is 0.8225 r2 for bedforms and 1.645 r2 for river channels, where r is the coefficient of variation (standard deviation/mean) of the generating topography.
In a comparison with data from laboratory current ripples, our analytical predictions compare well with observations of both mean set thickness and thickness distribution. The preservation ratio for the ripples is about 0.5, whilst measured dune heights give a preservation ratio of about 0.12. Depth data from two modern braided streams yield preservation ratios ranging from 0.4 to 0.75. As more data on the distributions of topographical height in modern environments become available, calculations such as these should help provide reliable error bounds for quantitative topographical reconstruction.