## 1. Introduction

[2] Paleoseismic records of earthquake sequences arise only at some points along faults because they require specific geological conditions such as a continuous and datable sedimentary record that is thick enough to capture multiple earthquake disturbances, and that is deposited near an active fault. Because of this, many faults have limited or irregular spatial paleoseismic coverage. Thus paleo-sites are essentially a point process, and like earthquake epicenters, often cannot reveal much information about rupture dimensions or variability. However, they provide essential empirical mean earthquake rates that are crucial to seismic hazard assessment.

[3] If a fault is assumed to behave according to a characteristic earthquake model [e.g., *Schwartz and Coppersmith*, 1984; *Wesnousky*, 1994], where fault segments repeatedly rupture segments in a similar fashion, then one paleoseismic site along a fault segment is representative of that segment's earthquake recurrence distribution. However, if one hypothesizes a broader range of possible ruptures [e.g., *Field and Page*, 2011] that overlap, branch, or that have variable magnitudes [*Weldon et al.*, 2004] with different recurrence distributions, then interpreting paleoseismic information at one point on a segment can become more complicated. Indeed, sites where overlapping ruptures are thought to occur, like Wrightwood and Pallet Creek on the San Andreas fault [*Biasi and Weldon*, 2009], have paleoseismic series that cannot be reproduced by any one recurrence distribution even after 50 · 10^{6} attempts [*Parsons*, 2008a], signaling a more complex process. Additionally, earthquake sequences may change character, branching into long-term cycles of increased or diminished activity owing to fault interactions [e.g., *Marzocchi and Lombardi*, 2008] that obey different recurrence distributions.

[4] Paleoseismic observations reveal a number of earthquakes above an observable surface slip threshold (assumed proportional to magnitude) in a period. This empirical information is extremely valuable to earthquake forecasters who need a rate to make probability calculations. The variability in earthquake recurrence intervals due to inconsistency in the rupture process, which gives rise to aleatory uncertainty in hazard modeling, is perhaps best represented by paleoseismic observations. Two sources of epistemic uncertainty must be accounted for before a paleoseismic rate constrains a probability calculation: (1) dating uncertainty (usually radiocarbon dating), and (2) the effects of undersampling that can cause a time-limited historical or paleoseismic record to preferentially reflect the shortest intervals and miss the longest ones [*Stein and Newman*, 2004]. Dating uncertainty can be addressed by bootstrapping across the possible event time ranges (sampling a uniform PDF determined by the reported uncertainties) [e.g., *Ellsworth et al.*, 1999; *Biasi et al.*, 2002]. Undersampling has been accounted for by Monte Carlo sampling from long-tailed recurrence distributions [e.g., *Console et al.*, 2008; *Parsons*, 2008a]; this has been necessary because the arithmetic mean of observed interevent times is not likely to represent the true average recurrence because the means of distributions thought to represent earthquake occurrence are all skewed to the right of their modes, and it requires many samples to capture that.

[5] In this paper I present a method to estimate the long-term mean and confidence bounds on the earthquake rate at a point when a segmented and/or characteristic earthquake rupture concept is not assumed. This application is to be explored for the Uniform California Earthquake Rupture Forecast version 3 (UCERF3); prior California forecasts have segmented faults by characteristic ruptures [e.g., *Field et al.*, 2009]. Results are given as interevent times (*T*) rather than earthquake rates (1/*T*), because *T* is more intuitive and more often reported in the paleoseismic literature. The primary result of interest for UCERF3 is the range of allowable interevent times rather than the mean.