USING PROBABILITY DENSITIES TO EVALUATE SLIP RATES AND UNCERTAINTY

Methods for evaluating geologic slip rates and their uncertainty using probability densities are described, including: (1) appropriate parameter choices for representing physical quantities, (2) propagation of measurement uncertainty, (3) handling of model uncertainty, and (4) combining alternative probability densities. Probability densities provide a completely general way of describing physical systems and parameters; their use doesn’t necessarily imply a high level of precision, nor a detailed understanding of the physical processes involved.

The choice of coordinates affects how probability densities are computed. Physical quantities taking on only positive values are called Jeffreys parameters, and include fault offset, feature age, or slip rates; those taking on both positive and negative values are called Cartesian parameters, and include paleoearthquake dates, or earthquake magnitudes. Jeffreys parameters are better represented by loguniform or lognormal probability distributions, while Cartesian parameters are better represented by uniform or Gaussian distributions. It generally is a modeling error to use a Gaussian distribution to represent a Jeffreys parameter if the variance is large.

Estimating slip rates from geologic measurements requires a physical model. Measurement uncertainty will be propagated into the probability density for the computed slip rate, which can be determined analytically. However, it usually is simpler to evaluate the slip rate density with Monte Carlo sampling using random draws of the measured input densities, computing the modeled slip rate for each draw, and then making a histogram of the results. It also is important to consider how well any model describes physical reality. If models are incomplete or otherwise wrong, then model predictions likely won’t match measured results even with error-free data.

Alternative probability densities for slip rates or paleoearthquake dates obtained at multiple trench sites can be combined. Disjoint densities can be combined using an averaging operation (disjunction), while independent densities can be combined using a multiplicative operation (conjunction). For Jeffreys parameters such as slip rate, it is the volumetric probability densities that are multiplied together for the conjunction operation.