We present a new method that allows paleomagnitude and paleorupture length to be estimated quantitatively given a measurement of earthquake rupture displacement at a point along a fault. Rupture displacement typically varies along a rupture profile such that a point paleoseismic displacement measurement constrains the average displacement only to within a factor of three or so. We used previously published results summarizing rupture variability and then applied a graphical method of identifying the relative likelihoods among a suite of magnitudes, one of which must have caused the measured displacement. Results were developed for displacement observations from 1 to 6 m using a magnitude range of 6.0 ≤ M ≤ 8.0. Probabilities of rupture lengths for a given displacement were developed at the same time. Although smaller earthquakes can cause ground rupture, we show that they would not strongly influence likelihoods for 1 m and larger observed displacements. Displacements less than 1 m are also of potential interest but will require extension of the method to include the declining probability that smaller magnitudes produce ground rupture. We also consider application of length distributions inferred from a displacement measurement to correlation of rupture evidence between sites. Dating evidence alone, even when excellent, does not provide a physical basis to relate rupture at one site to rupture at another. Ruptures, however, have an expected length, and thus do provide a physical basis for correlation. We present probability of correlation curves for given rupture lengths, which may be combined with probabilities of rupture length to obtain a probability of correlation given a point displacement. Applications for quantitative probabilities of magnitude and length given a paleoseismic displacement measurement include probabilistic seismic hazard analyses, where probabilities of magnitude and length must be assigned to branches in the analysis.

Abstract The geologic evolution of southern California and the spreading in the mouth of the Gulf of California are known well enough to constrain a model for the formation of the Gulf of California. The San Andreas fault has been increasingly active during the past ~17 m.y., culminating at its present slip rate of ~35 mm/yr by 4 to 5 Ma. Another ~15 mm/yr of right-lateral shear is inferred to exist on recently activated faults near the southern California coast. These include the predominantly strike-slip, northwest-trending, transpeninsular Agua Blanca and Elsinore faults and the east-trending zone of convergence within the western Transverse Ranges. The total relative plate rate across the southern California region is modeled at ~53 mm/yr, which is comparable to global kinematic plate model rates. At the Tamayo spreading center in the mouth of the Gulf, magnetic anomaly patterns record a constant spreading rate of ~48 mm/yr for the most recent ~4 m.y., though evidence exists for a proto-gulf prior to 10 Ma. Our present understanding of the deformation in the vicinity of southern California is difficult to reconcile with a plate rate near 48 mm/yr, suggesting 5 to 7 mm/yr of additional deformation near the mouth of the Gulf, probably largely the result of continental extension on the southeastern side of the mouth of the Gulf. We suggest the following model for the formation of the Gulf of California. From 28 to 17 Ma, most of the relative plate motion occurred near the continental rise, although active strike-slip faulting within the southern California Borderland is inferred by the rotation of the crust that is now the western Transverse Ranges. From 17 to 4.5 Ma, the proto-gulf developed through oblique rifting in direct relation to the increasingly active San Andreas system. About 150 km of San Andreas-related shear occurred during this time. Shear across the Borderland and rotation of the western Transverse Ranges continued. From 4.5 to 1 Ma, both the Gulf and the San Andreas fault were fully active. Transpeninsular faults are thought to have supplied Borderland shear related to the continuing rotation of the western Transverse Ranges, initially on faults located south of those currently active. During the past million years, the transpeninsular shear has been confined to the northernmost portion of the peninsula and, as a result, rotation of the western Transverse Ranges greatly diminished in rate or ceased and high rates of contraction initiated there. This history is more protracted than most workers have proposed, and represents a migration of Pacific-North America plate-related activity steadily from the continental rise into the North American continent.