# Time-Dependent Earthquake Probability Maps

## Summary

The USGS has historically developed time-independent models of earthquake occurrence that are based on the assumption that the probability of the occurrence of an earthquake in a given period of time follows a Poisson distribution. Probabilities calculated in this way require only knowledge of the mean recurrence time. Results of these calculations do not vary with time (i.e., results are independent of the time since the last event) and are a reasonable basis for the earthquake resistant provisions in building codes and long-term mitigation strategies.

In contrast, time-dependent models of earthquake occurrence are based on the assumption that the probability of occurrence of an earthquake in a given time period follows a renewal model, that is a lognormal, Brownian Passage Time (BPT), or other probability distribution in which the probability of the event depends on the time since the last event (Appendix A [of full document]). In addition to the mean frequency (or recurrence time) of earthquakes, these models require additional information about the variability of the frequency of events (the variance or standard deviation), and the time of the last event. The time-dependent models are intuitively appealing because they produce results broadly consistent with the elastic rebound theory of earthquakes. The USGS and CGS are beginning to develop these types of hazard products as new geologic and seismic information regarding the dates of previous events along faults becomes available.

In application, both the time-independent and time-dependent models also depend on assumptions about the magnitude-frequency characteristics of earthquake occurrence, the simplest of which is the “characteristic earthquake model” in which all large earthquakes along a particular fault segment are assumed to have similar magnitudes, average displacements, and rupture lengths. More complicated models include Gutenburg-Richter magnitude-frequency distributions and multi-segment ruptures. In as much as time-dependent models require more input parameters and assumptions as contrasted with time-independent models, there is not yet the same degree of consensus about the methods and results for these calculations.

Both time-independent and time-dependent hazard calculations require moment-balanced models that are consistent with the global plate rate models and slip rates determined on individual faults. Geologists can estimate the average slip rates on faults in California from offset geologic features that have been dated using radiometric dating techniques. At sites along some faults we know the approximate times of past events extending hundreds or thousands of years into the past, but we do not know the magnitudes of or the length of faults involved in these past earthquakes. A fundamental constraint that we apply to candidate earthquake occurrence models, commonly called “moment balancing,” is the requirement that over the long term, the displacements from the earthquakes sum to the observed slip rate all along the fault. Models that permit smaller earthquakes will generally contain more frequent earthquakes in order to add up to the total slip rate.