Understanding the element of time in probabilistic submarine slope stability analyses

Probabilistic implementations of limit equilibrium slope stability analysis—submarine or otherwise—typically yield results that are independent of time. If an input variable is truly random in time, for example seismic acceleration as a consequence of earthquakes that follow Poisson distributions, then it is possible to calculate a conditional annual probability of failure. Annual probabilities may also be calculated from empirical data under some conditions. There are two possibilities if none of the variables occur randomly in time: 1) All variables are invariant in time. In this case, the result cannot be transformed into an annual probability of failure. Large time-independent probabilities of failure may be calculated for some slopes that are obviously stable, leading to the apparent contradiction that slopes with high probabilities of failure have not yet failed. This can be explained using a Bayesian interpretation in which a prior estimate of instability is corrected based on the knowledge of current conditions. 2) One or more of the variables (e.g. slope angle) vary steadily over time as a consequence of geologic processes (e.g. tectonic movement). In such a case, the conditional approach employed for Poisson-like processes is inappropriate. Instead, the appropriate procedure is a hazard function approach to calculate an incremental annual probability based upon the changing conditions. One consequence is that the value and rate of change of the probability will vary over time (i.e., the basic shape of the probability distribution changes over time). Thus, it is important to understand the initial state of the slope as well as the geologic processes operating upon it in order to properly estimate the incremental annual probability of failure as a function of time.