Scale-dependent truncated Lévy modeling of Gamma-ray log increments for characterizing stratigraphic heterogeneity in IODP well logs

Gamma-ray (GR) well logs record lithologic variability across multiple scales, but their increment statistics are often analyzed under stationary Lévy assumptions. In this study, we introduce a scale-dependent framework based on the gradually truncated Lévy flight model to characterize nonstationary increment behavior in 23 International Ocean Discovery Program (IODP) wells. This approach treats the stability index \(\:\alpha\:\) and truncation scale \(\:{l}_{c}\) as functions of lag scale, combining Lévy-stable core distributions with exponential truncation of large deviations. A robust estimation pipeline is developed that integrates core-distribution fitting, tail-sensitive truncation detection, and profile-likelihood optimization. Mean squared displacement (MSD) analysis reveals a crossover from super-diffusive (β > 1) to sub-diffusive (β < 1) regimes at scales of approximately 0.8–1.2 m, reflecting changes in dominant stratigraphic controls. α and \(\:{l}_{c}\)exhibit scale-dependent evolution, from interface-dominated increments at small scales to bounded variability in larger stratigraphic units. The proposed framework enhances stochastic modeling by extending traditional Gaussian assumptions and provides a versatile tool for quantifying anomalous diffusion and stratigraphic heterogeneity in well-log data. The Python implementation is open-source, ensuring reproducibility and flexibility in broader geological contexts.