Based on the elastic rebound theory, major earthquakes on active faults exhibit temporal memory characteristics. To describe this temporal dependency, researchers have proposed renewal models, with the Brownian passage time （BPT） model being widely adopted due to its clear physical significance. However, the BPT model assumes a fixed critical stress state for earthquake occurrence, which contradicts the observed randomness in characteristic earthquake magnitudes. This study presents an improved Brownian passage time （IBPT） model that addresses the limitations of the traditional BPT model by incorporating both loading process randomness and critical state uncertainty.

The IBPT model is founded on elastic rebound theory with enhanced assumptions. Unlike the BPT model, which only considers random disturbances during the steady loading process, the IBPT model simultaneously accounts for: ① random perturbations in the stress accumulation process, and ② uncertainty in the critical physical state required for earthquake occurrence. The model assumes that seismic moment accumulation includes both steady loading and random loading disturbances, while the critical physical state for characteristic earthquakes is not fixed but exhibits certain uncertainty. This dual consideration reflects the complexity of fault loading processes, involving both tangential and normal stress components with random perturbations.

The IBPT model introduces a significant innovation by assuming that seismic moment release during characteristic earthquakes is normally distributed, contrasting with earlier models that applied normal distribution assumptions to characteristic earthquake magnitudes. This assumption has stronger physical basis since seismic moment directly relates to energy release and fault displacement. The model incorporates three variability coefficients: overall recurrence interval variability α, critical state randomness contribution α₁, and loading process randomness contribution α₂, related through error propagation theory as α²＝α₁²＋α₂².

To determine model parameters, this study compiled coseismic displacement data from 54 recurring earthquake events across 17 active faults in mainland China. Through statistical analysis of normalized dimensionless seismic moment data calculated from these displacement measurements, the critical state variability coefficient α₁ was determined to be 0.23. Combined with the established total variability coefficient α＝0.34 from previous studies, the loading variability coefficient α₂ was calculated as approximately 0.25.

The probability density function of the IBPT model is derived through joint consideration of both randomness sources. When only critical state randomness is considered, the recurrence interval follows a normal distribution with the same coefficient of variation as seismic moment. When combined with loading process randomness, the total probability is calculated using the law of total probability, yielding a more comprehensive distribution function.

The IBPT model was applied to the Luhuo segment of the Xianshuihe fault as a case study. This 140-km-long segment has experienced historical major earthquakes in 1816 （M_S7½） and 1973 （M_S7.6）, with an average recurrence interval of 157 years. Using the following parameters: μ＝157 years, α＝0.34, α₁＝0.23, and ∆T＝10 years, probability calculations were performed and compared with traditional BPT model results.

Comparative analysis reveals that IBPT and BPT models yield generally consistent results, but with notable differences in specific ranges. For normalized elapsed time T_e＜0.5\overlineT , IBPT shows slightly higher earthquake probabilities than BPT. Within the range 0.5\overlineT ＜T_e＜1\overlineT , IBPT yields slightly lower probabilities than BPT. However, for T_e＞1\overlineT , IBPT probabilities become substantially higher than those predicted by BPT, suggesting that the IBPT model provides more conservative earthquake probability estimates when the elapsed time since the previous event is considerably long. Given the current elapsed time of 52 years for the Luhuo segment, the 10-year earthquake probability is 0.008 using IBPT versus 0.004 using BPT, both indicating low probability for the near future.

The IBPT model offers several advantages over the traditional BPT model: First, it provides clearer physical meaning by considering both tangential and normal stress loading effects on fault systems; Second, it better accounts for the unpredictability of characteristic earthquake magnitudes through critical state uncertainty; Third, it maintains compatibility with BPT results while providing more comprehensive physical representation; Fourth, the model parameters can be determined through statistical analysis of paleoseismic displacement data, making it practically applicable.

The model’s consideration of critical state uncertainty aligns with observed variability in characteristic earthquake magnitudes and coseismic displacements. This represents a more realistic approach compared to the fixed critical state assumption in traditional models. The incorporation of seismic moment rather than magnitude as the primary physical parameter provides stronger theoretical foundation based on fault mechanics.

The IBPT model not only provides clearer physical meaning, but also yields recurrence probability calculations that exhibit only small discrepancies from BPT model results. Notably, the IBPT model does not underestimate seismic hazard; rather, when the elapsed time exceeds one mean recurrence interval, it yields probability estimates significantly higher than those predicted by the BPT model. This characteristic makes the IBPT model particularly well-suited for long-term seismic hazard assessment applications.

This study contributes to earthquake hazard assessment methodology by providing a more physically comprehensive model for characteristic earthquake recurrence. Future research should focus on parameter determination methods and validation across diverse fault systems to further establish the model’s effectiveness in seismic hazard evaluation.