We consider a one-dimensional stochastic model of sediment deposition in which the complete
time history of sedimentation is the sum of a linear trend and a fractional Brownian motion
wH(t) with self-similarity parameter H ∈ (0, 1). The thickness of the sedimentary layer as a
function of time, d(t), looks like the Cantor staircase. The Hausdorff dimension of the points
of growth of d(t) is found. We obtain the statistical distribution of gaps in the sedimentary
record, periods of time during which the sediments have been eroded. These gaps define
sedimentary unconformities. In the case H = 1/2 we obtain the statistical distribution of
layer thicknesses between unconformities and investigate the multifractality of d(t). We show
that the multifractal structures of d(t) and the local time function of Brownian motion are
identical; hence d(t) is not a standard multifractal object. It follows that natural statistics
based on local estimates of the sedimentation rate produce contradictory estimates of the
range of local dimension for d(t). The physical object d(t) is interesting in that it involves the
above anomalies, and also in its mechanism of fractality generation, which is different from
the traditional multiplicative process.