Corrigendum to “A stochastic model of geomorphic risk due to episodic river aggradation and degradation” [Engineering Geology 309 (2022) 106845] (Engineering Geology (2022) 309, (S0013795222003301), (10.1016/j.enggeo.2022.106845))
Journal
Engineering Geology
Journal Volume
333
Journal Volume
333
Start Page
107504
ISSN
00137952
Date Issued
2024-05
Author(s)
Abstract
In our paper titled “Stochastic Model of Geomorphic Risk due to Episodic River Aggradation and Degradation (2022),” we introduced and applied a stochastic process that we called the “Gamma-subordinated Ornstein-Uhlenbeck (GSOU)” process, derived formulas for its time-evolving properties, and conducted Monte Carlo simulations using a discretized approximation of the process. In relation to these elements, we recently identified three issues in need of correction, which we address in the present Corrigendum. First, an important and relevant prior reference had escaped our attention. In a book chapter, Poczynek et al. (2020) examined an equivalent stochastic process, and similarly derived its time-evolving properties. We wish here to correct this omission. Secondly, we identified an error in our derivation of the time-evolving variance, for which we provide a correction below. Third, we found that the simple Euler scheme discretization used for our earlier Monte-Carlo simulations does not produce the correct solution properties for the initial transient. To address this problem, we therefore provide a corrected method, which extends to the GSOU process the exact solution approach of Gillespie (1992, 1996). In light of these corrections, we have corrected Figs. 3d and e in the paper to reflect accurate information. The results of Figs. 10 and 11 are also slightly affected by these corrections. However, the changes are so small that they are not discernible at the scale of the figures. We therefore do not provide corrected figures, but provide instead two Tables listing numerical values for the original and corrected results. The other parts of the paper and its conclusions are not affected by the corrections. Correction with omitted reference We correct our paper with the omitted reference after Eq. (7): The formula for the time-evolving mean is identical to the one given by Poczynek et al. (2020) for an equivalent stochastic process (an Ornstein-Uhlenbeck process delayed by a Gamma subordinator). Time-evolving variance of the Gamma-subordinated OU process Eq. (8) in the paper should be corrected into [Formula presented] where the first term was originally omitted but contributes to the transient evolution of the variance. The derivation of the equation in Appendix B (Eqs. (B.11), (B.12)) should be corrected into: According to the properties of Itō integral and Itō isometry (Øksendal, 2000) the time-evolving variance can also be calculated as [Formula presented] Integrating with the pdf of Eq. (B.7), we obtain for the time-evolving variance the result [Formula presented]which converges asymptotically towards a finite long-term variance [Formula presented] Correction to the Monte-Carlo simulation scheme Although the Euler (or Euler-Maruyama) method that we used to perform Monte-Carlo simulations of the GSOU process is widely applied and converges for many stochastic processes, in the case of the GSOU process we found that it does not produce the correct solution properties, even at very fine discretization. This problem, however, can be addressed by adapting to the GSOU process the exact simulation approach of Gillespie (1992, 1996). Accordingly, Eq. (4) in the paper should be corrected to the formula [Formula presented] based on analogizing the GSOU process for given [Formula presented] to the OU process and the formula for the OU process proposed by Gillespie (1992, 1996). Revised Figures The panels d and e in Fig. 3 are corrected into:[Formula presented] The results of Figs. 10 and 11 are also slightly affected by these corrections. However, the changes are not visible at the scale of the printed figures. Therefore, we do not provide corrected figures but summarize instead the corrected numerical values in Tables C1 and C2.
Publisher
Elsevier B.V.
Type
corrigendum