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.

Randomly irregular waves are difficult to incorporate into engineering design. The response of the structure is usually quite complicated and often involves other factors than wave action. Thus it is frequently not possible to analytically determine the statistical characterization of the response directly from the statistical properties of the waves. In problems of this type» simulation techniques have often been the only successful method for determining solutions* These techniques have been used in a wide range of problems in physics, operations research, and other fields, wherever random factors were involved in a complicated interaction with other factors» Basically, simulation techniques are procedures whereby artificial data having imposed statistical properties is generated by some computational means. Usually this is done in a digital computer. The artificial data is fed into the problem and the re^onse calculated. By doing this with enough data, the equivalent of many years, or even centuries, of experience with the problem can be produced. Such factors as maximum response, or the number of times some critical value is attained, can be determined by inspection or by monitoring the output with the computer. Simulation techniques have the advantage of working for fairly complicated situations, but the disadvantage of often Assoc. Prof, of Engineering Geoscience, Univ. of Calif., Berkeley, Calif. requiring sizeable amounts of computer time. Simulation procedures have not been used extensively in coastal engineering and ocean wave problems although several of the oil companies have used the techniques. The following study was undertaken to make the procedures more available to the engineer working with ocean wave problems and to investigate possibls ways to increase the efficiency and the realism of the ocean wave and force simulations produced. The conventions used in defining spectral density are not standardised. Differences of v and 2 show up in various papers depending on whether onesided or two-sided spectral densities are used and whether frequencies are expressed in radians or cycles per unit time. All derivations in the following analysis will be based on the two-sided, cycles-per-unit-time spectral density relations. These will be converted to one-sided relations, where appropriate, by multiplying by 2 and taking the integration from zero to infinity instead of from minus infinity to plus infinity. The exact mathematical definitions are given in the table of notation at the back of the paper. The sea su./face elevations will be assumed to be a stationary, ergodic stochastic process produced by the addition of many infinitesimal wavelets each with a random phase. By the usual random theory of ocean waves, this 2 3 leads to a Gaussian process. '

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