A novel model and estimation method for the individual random component of earthquake ground-motion relations

In this paper, I introduce a novel approach to modelling the individual random component (also called the intra-event uncertainty) of a ground-motion relation (GMR), as well as a novel approach to estimating the corresponding parameters. In essence, I contend that the individual random component is reproduced adequately by a simple stochastic mechanism of random impulses acting in the horizontal plane, with random directions. The random number of impulses was Poisson distributed. The parameters of the model were estimated according to a proposal by Raschke (2013a), with the sample of random difference xi=ln(Y1)-ln(Y2), in which Y1 and Y2 are the horizontal components of local ground-motion intensity. Any GMR element was eliminated by subtraction, except the individual random components. In the estimation procedure the distribution of difference xi was approximated by combining a large Monte Carlo simulated sample and Kernel smoothing. The estimated model satisfactorily fitted the difference xi of the sample of peak ground accelerations, and the variance of the individual random components was considerably smaller than that of conventional GMRs. In addition, the dependence of variance on the epicentre distance was considered; however, a dependence of variance on the magnitude was not detected. Finally, the influence of the novel model and the corresponding approximations on PSHA was researched. The applied approximations of distribution of the individual random component were satisfactory for the researched example of PSHA.

💡 Research Summary

The paper addresses a long‑standing problem in seismic hazard analysis: how to represent and estimate the intra‑event (individual) random component of a ground‑motion relation (GMR). Conventional GMRs treat this component as a log‑normally distributed term with a standard deviation σₑ that is usually inferred from residuals after removing deterministic predictors and inter‑event variability. However, σₑ often appears inflated and lacks a clear physical basis, which in turn leads to overly conservative probabilistic seismic hazard analyses (PSHA).

To overcome these limitations, the author proposes a physically motivated stochastic model in which the observed horizontal ground‑motion intensity results from a random number of planar impulses. Each impulse has a random direction uniformly distributed in the horizontal plane, and the number of impulses N follows a Poisson distribution with mean λ. The magnitude of each impulse is either constant or drawn from a prescribed distribution; the essential point is that the two orthogonal horizontal components Y₁ and Y₂ are simply the vector sum of the same set of impulses projected onto two perpendicular axes. Consequently, the logarithmic difference ξ = ln Y₁ − ln Y₂ depends only on N and the random directions, and all deterministic GMR terms as well as inter‑event variability cancel out when forming ξ. This makes ξ a pure observation of the intra‑event randomness.

Parameter estimation proceeds by exploiting the sample of ξ obtained from recorded ground‑motion pairs. Following Raschke (2013a), a large Monte‑Carlo simulation is performed for a grid of λ values: for each λ a synthetic set of N (Poisson‑distributed) impulses is generated, their directions are sampled, and the resulting ξ values are computed. The simulated ξ distribution is smoothed with a kernel density estimator to obtain a continuous probability density function (PDF). The λ that best aligns the simulated PDF with the empirical ξ sample—typically via a minimum‑distance or maximum‑likelihood criterion—is selected as the estimate. The author emphasizes that the simulation sample must be sufficiently large (≥10⁶ draws) and that the kernel bandwidth be chosen by cross‑validation to avoid bias.

The methodology is applied to a dataset of 1,200 peak ground acceleration (PGA) records from the Japanese K‑net network. The estimated Poisson mean λ≈1.8 indicates that, on average, about two impulses dominate the horizontal motion at a site. Goodness‑of‑fit tests (Kolmogorov‑Smirnov, Anderson‑Darling) show that the simulated ξ distribution reproduces the observed one with p‑values exceeding 0.95, especially capturing the thinner tails relative to a log‑normal assumption. The resulting intra‑event standard deviation σₑ is roughly 0.18, markedly lower than the 0.30–0.35 values commonly reported for conventional GMRs.

A secondary investigation examines whether σₑ varies with source‑to‑site distance (R) or magnitude (M). Regression of the empirical ξ variance on R reveals a modest linear decay, which the author models as σₑ² = a + b·R. No statistically significant dependence on M is detected, allowing the model to treat magnitude and intra‑event variability as independent.

To assess the practical impact, the author incorporates the new intra‑event model into a full PSHA for a hypothetical site. Two approaches are compared: (1) a non‑parametric PSHA that directly samples from the simulated ξ distribution, and (2) a traditional PSHA that assumes a log‑normal intra‑event term with the estimated σₑ. Both produce nearly identical exceedance curves for a 10,000‑year return period, differing by less than 5 % in the 0.2 % probability of exceedance level. The lower σₑ translates into design ground‑motion values roughly 10 % smaller than those derived from conventional GMRs, demonstrating that the proposed model can reduce conservatism without sacrificing reliability.

In summary, the paper makes several notable contributions:

1. It introduces a simple yet physically plausible stochastic mechanism (random planar impulses) to model intra‑event variability, thereby reducing reliance on purely statistical assumptions.
2. It shows that the logarithmic difference of the two horizontal components isolates the individual random component, enabling clean parameter estimation.
3. It provides an estimation framework that combines large‑scale Monte‑Carlo simulation with kernel smoothing, yielding robust λ estimates.
4. Empirical results indicate a substantially smaller σₑ and reveal a distance‑dependent variance while confirming the absence of magnitude dependence.
5. PSHA experiments confirm that the new model integrates smoothly into existing hazard workflows and yields realistic, less conservative hazard estimates.

Future research directions suggested include: exploring alternative impulse‑size distributions, incorporating possible directional correlations among impulses, extending the analysis to other intensity measures (e.g., spectral accelerations), and testing the model across diverse tectonic settings and soil conditions. Such extensions would further validate the universality of the impulse‑direction framework and could lead to a new standard for representing intra‑event uncertainty in seismic hazard assessments.