  Bayesian inversions followed by estimations of rare event probabilities are
often needed to analyse groundwater hazards. Instead of focusing on the
posterior distribution of model parameters, the main interest lies then in the
distribution of a specific quantity of interest contingent upon these
parameters. To address the associated methodological challenges, we introduce a
two-stage Sequential Monte Carlo approach. In the first stage, it generates
particles that approximate the posterior distribution; in the second stage, it
employs subset sampling techniques to assess the probability of the rare event
of interest. By considering two hydrogeological problems of increasing
complexity, we showcase the efficiency and accuracy of the resulting
PostRisk-SMC method for rare event probability estimation related to
groundwater hazards. We compare the performance of the PostRisk-SMC method with
a traditional Monte Carlo approach that relies on Markov chain Monte Carlo
samples. We showcase that our estimates align with those of the traditional
method, but the coefficients of variation are notably lower for the same
computational budget when targeting more rare events. Furthermore, we highlight
that the PostRisk-SMC method allows estimating rare event probabilities
approaching one in a billion using less than one hundred thousand forward
simulations. Even if the presented examples are related to groundwater hazards,
the methodology is well-suited for addressing a wide range of topics in the
geosciences and beyond.
