Statistical Methods and Modeling of Seismogenesis

Nikolaos Limnios, Eleftheria Papadimitriou, George Tsaklidis

• Publisher: Wiley-ISTE
• Year: 2021
• ISBN:
  9781119825043
• Edition: 1
• Topic: Social Sciences
• Subtopic: Social Science Research & Methodology
• Index: Social Sciences

The study of earthquakes is a multidisciplinary field, an amalgam of geodynamics, mathematics, engineering and more. The overriding commonality between them all is the presence of natural randomness.

Stochastic studies (probability, stochastic processes and statistics) can be of different types, for example, the black box approach (one state), the white box approach (multi-state), the simulation of different aspects, and so on. This book has the advantage of bringing together a group of international authors, known for their earthquake-specific approaches, to cover a wide array of these myriad aspects. A variety of topics are presented, including statistical nonparametric and parametric methods, a multi-state system approach, earthquake simulators, post-seismic activity models, time series Markov models with regression, scaling properties and multifractal approaches, selfcorrecting models, the linked stress release model, Markovian arrival models, Poisson-based detection techniques, change point detection techniques on seismicity models, and, finally, semi-Markov models for earthquake forecasting.

1
Kernel Density Estimation in Seismology

Stanisław LASOCKI

Institute of Geophysics, Polish Academy of Sciences, Warsaw, Poland

1.1. Introduction

The basis of statistical seismology is the probabilistic distribution of earthquake parameters. Unfortunately, the models of probabilistic distributions of earthquake parameters are mostly unknown, and those which are thought to be known are often disproved by more thorough testing.

On the other hand, increasing quality and decreasing prices of seismic monitoring systems, increasing density of seismic networks and the development of event picking tools cause the current seismic catalogs to be large and rising. This situation opens up the area for model-free approaches to probabilistic estimation and statistical inferences, which to be accurate, require considerable sample sizes.

The kernel density estimation is a model-free estimation of probability functions of continuous random variables. The estimation is carried out solely from sample data.

There is no attempt here to comprehensively present the kernel density estimation method. A detailed description and the discussion of the method can be found in the textbooks by Silverman (1986), Wand and Jones (1995), Scott (2015) and in a multitude of high-level research papers. The method is also implemented in Matlab, R and Python, among others, and in many statistical packages. In this chapter, the author presents how this method has been applied to selected problems of seismology. This presentation begins with a short and simplified theoretical introduction. The kernel density estimation has been fast developing from both theoretical and practical sides; hence, the techniques presented here do not aspire to be optimal. There is plenty of space for future modifications and developments.

In the case of a univariate random variable X, and a constant kernel, the kernel estimator of the actual probability density function (PDF) of X, f_X(x), takes the form:

[1.1]

where {x_i}, i = 1,.., n is the sample data, K(•) is the kernel function, which is a PDF symmetric about zero, and h is the bandwidth, whose value decides how much smoothing has been applied to the sample data.

In the presented seismological applications of the kernel density estimation, we use the normal kernel function:

[1.2]

For this kernel function, the kernel estimates of PDF and the cumulative distribution function (CDF,

) are, respectively:

[1.3]

[1.4]

where Ф(·) is the CDF of standard normal distribution.

As it will follow, the kernel method is used, among others, to estimate the magnitude distribution, whose distribution is exponential-like or light-tailed. Because of that, the tail values are sparse in a sample. Such sparsity can result in spurious irregularities in the estimate on tails, if a constant bandwidth is used. We can alleviate this problem by using an adaptive kernel with variable bandwidth. Because the estimates of magnitude distribution functions serve in the probabilistic seismic hazard analysis (PSHA), the quality of the estimate...

Table of contents

• Cover
• Table of Contents
• Title Page
• Copyright
• Preface
• 1 Kernel Density Estimation in Seismology
• 2 Earthquake Simulators Development and Application
• 3 Statistical Laws of Post-seismic Activity
• 4 Explaining Foreshock and the Båth Law Using a Generic Earthquake Clustering Model
• 5 The Genesis of Aftershocks in Spring Slider Models
• 6 Markov Regression Models for Time Series of Earthquake Counts
• 7 Scaling Properties, Multifractality and Range of Correlations in Earthquake Time Series: Are Earthquakes Random?
• 8 Self-correcting Models in Seismology: Possible Coupling Among Seismic Areas
• 9 Markovian Arrival Processes for Earthquake Clustering Analysis
• 10 Change Point Detection Techniques on Seismicity Models
• 11 Semi-Markov Processes for Earthquake Forecast
• List of Authors
• Index
• End User License Agreement