The Magnitude Threshold and Missing and Pseudo Links in Markov Chains

IF 1.9 4区 地球科学 Q2 GEOCHEMISTRY & GEOPHYSICS
F. A. Nava, Q. J. Gutiérrez
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引用次数: 0

Abstract

A crisp step function is not an adequate threshold for studies of Markovian occurrence of large earthquakes, because it can lead to missing or pseudo links in an observed sequence that should be a Markov chain. A more realistic threshold is a fuzzy one where there is a transition magnitude band, located between those magnitudes that are too small for the earthquakes to be part of a Markovian process and those who are certainly large enough for the earthquakes to be part of it, where earthquakes may or may not be part of the process. This fuzzy threshold is described by a membership function that gives the probability of an earthquake with a given magnitude belonging to the process. We propose a membership function with probabilities in the transition band proportional to the seismic moment. To estimate empirical transition probabilities when considering a fuzzy magnitude threshold, we propose a counting strategy for the observed transitions and justify it through Monte Carlo simulations. The counting strategy is illustrated by application to the model from a previous seismic study of the Japan area by testing, through Monte Carlo simulations, how well the counting strategy results resemble optimum estimations of the transition probabilities. The simulations are also used to study the behavior of three Markovianity measures, and it is found that the peak values of these measures are not useful in identifying the true transition band, but that this band may be better identified by using the whole set of values taken by each measure for different transition band models. As an illustration, the measures were applied to real data from the previous study, a short set corresponding to a single realization, and found that the behavior of the measures does not agree with those expected from a crisp threshold, but agree, within the limitations of the data, with either a fuzzy threshold going from zero probability for magnitudes \(M\le 6.9\) to probability one for \(M\ge 7.2\) or from zero probability for magnitudes \(M\le 7.0\) to probability one for \(M\ge 7.2\).

Abstract Image

马尔可夫链中的幅度阈值以及缺失和伪链接
对于大地震马尔可夫发生过程的研究来说,清晰的阶跃函数并不是一个合适的阈值,因为它可能导致本应是马尔可夫链的观测序列中出现缺失或伪链接。一个更现实的阈值是一个模糊阈值,在这个阈值中存在一个过渡震级带,它位于那些震级太小,地震不可能是马尔可夫过程的一部分,以及那些震级足够大,地震肯定是马尔可夫过程的一部分的震级之间,在这个过渡震级带中,地震可能是过程的一部分,也可能不是过程的一部分。这个模糊阈值由一个成员函数来描述,该函数给出了给定震级的地震属于该过程的概率。我们提出了一种成员函数,其过渡带中的概率与地震力矩成正比。在考虑模糊震级阈值时,为了估算经验过渡概率,我们提出了一种观测到的过渡的计数策略,并通过蒙特卡罗模拟证明了这一策略的正确性。通过蒙特卡罗模拟测试计数策略结果与过渡概率最佳估计值的相似程度,并将计数策略应用于之前日本地区地震研究的模型,以说明计数策略。模拟还用于研究三种马尔可夫性度量的行为,结果发现,这些度量的峰值对于识别真正的过渡带并无用处,而通过使用不同过渡带模型的每种度量所取的整套值,可以更好地识别过渡带。为了说明这一点,我们将这些测量值应用于之前研究中的真实数据,这是一个与单次实现相对应的简短集合,结果发现这些测量值的行为与清晰阈值的预期行为并不一致,但在数据的限制范围内,它们与模糊阈值一致,模糊阈值从幅度为零的概率(M\le 6.9)的概率为零,到(M\ge 7.2)的概率为一,或者(M\le 7.0)的概率为零,到(M\ge 7.2)的概率为一。
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来源期刊
pure and applied geophysics
pure and applied geophysics 地学-地球化学与地球物理
CiteScore
4.20
自引率
5.00%
发文量
240
审稿时长
9.8 months
期刊介绍: pure and applied geophysics (pageoph), a continuation of the journal "Geofisica pura e applicata", publishes original scientific contributions in the fields of solid Earth, atmospheric and oceanic sciences. Regular and special issues feature thought-provoking reports on active areas of current research and state-of-the-art surveys. Long running journal, founded in 1939 as Geofisica pura e applicata Publishes peer-reviewed original scientific contributions and state-of-the-art surveys in solid earth and atmospheric sciences Features thought-provoking reports on active areas of current research and is a major source for publications on tsunami research Coverage extends to research topics in oceanic sciences See Instructions for Authors on the right hand side.
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