Fundamental Concepts of Probabilistic Seismic Hazard Analysis
1
Author
Frank Scherbaum
Title
Fundamental Concepts of Probabilistic Seismic Hazard Analysis
Description
The book provides a computational, multi-perspective approach to understanding the underlying concepts of Probabilistic Seismic Hazard Analysis (PSHA). Written entirely in Mathematica, it seamlessly integrates explanatory text, interactive exercises, and dynamic walk-through examples.
Category
Books & Supplements
Keywords
Seismic Hazard Analysis, Probability Theory, Earthquake Seismology, Earthquake Engineering
URL
http://www.notebookarchive.org/2025-02-5zwyxr9/
DOI
https://notebookarchive.org/2025-02-5zwyxr9
Date Added
2025-02-13
Date Last Modified
2025-02-13
File Size
60.72 megabytes
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Rights
CC BY-NC-SA 4.0
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Frank Scherbaum
Fundamental Concepts of
Probabilistic Seismic Hazard Analysis
Fundamental Concepts of
Probabilistic Seismic Hazard Analysis
Probabilistic Seismic Hazard Analysis
Version 1.0
Foreword
Foreword
This idea for this book has grown over the course of many years during which I have worked on projects related to seismic hazard analysis. My academic training in hazard related issues was originally from a deterministic perspective. Therefore, some of the concepts used in probabilistic seismic hazard analysis (PSHA) were rather new (and partially counterintuitive) to me when - around the turn of the century- I was invited to join the PEGASOS project (Abrahamson et al., 2002), a SSHAC level 4 probabilistic seismic hazard assessment for the sites of four nuclear power plants in Switzerland. During that time I started to develop some code for seismic hazard analysis and some training material in Mathematica which were both originally intended for my own education. Over the years and stimulated by the different projects I worked in since then, the material continued to grow in the number of topics it covered. Gradually, it migrated into the material for a course in PSHA which I have been teaching inside and outside of academia for several years. When reading the current book as a standalone text, it needs to be stressed up front that it addresses PSHA in ways which are different from what the reader might see in other books and courses on this topic.
First, it focuses on concepts, not on procedures. What I mean by this is that, in contrast to how I originally learned it, I do not begin to explain PSHA as a sequence of steps which start with the characterisation of the seismic sources, followed by the characterization of the ground-motion model, etc. Based on my own learning and teaching experience I have come to the conclusion that there is enormous benefit to focus on the underlying concepts first and only subsequently discuss the procedures (the “recipes”). This may be considered simply a matter of taste, but procedures can be memorized by simple repetition without having to understand the rational for the individual building blocks. I believe that the mechanics of doing seismic hazard analysis should become obvious from the underlying concepts and not the other way around. As a consequence of this perspective, this book is not aimed at a complete in-depth coverage of all aspects of all real-life PSHAs, nor at discussing the implementation of PSHA in existing computer codes. It is my hope though that readers who appreciate the approach taken here, will also understand why a particular way of implementation may or may not make sense.
Second, in order to accommodate different cognitive preferences and different levels of background knowledge, I introduce the key concepts in PSHA, namely the concepts from which hazard curves are derived, from a number of different perspectives. This was driven by trying to acknowledge that “Learners have different strategies, approaches, patterns of abilities, and learning styles that are a function of the interaction between their heredity and their prior experiences”, (National Research Council, 2000). My goal has also been to first develop the core concepts of seismic hazard analysis from the rather detailed analysis of very simple, but easily understandable “toy problems”. Subsequently, the discussion is extended to more realistic situations.
Third, people interested in PSHA come from many different backgrounds with very different curricula. In addition, seismic hazard analysis draws from many different fields such as Probability Theory, Earthquake Seismology, Engineering Seismology, and more. As a consequence of this diversity, the material is aimed at seismic hazard analysis novices and at readers having some background knowledge in parts of the fields relevant for PSHA. After an introduction it starts out by covering those aspects of probability theory, earthquake seismology, and engineering seismology which are prerequisite for PSHA. For readers only trying to refresh their memory on particular aspects of these topics, the chapters can be consulted in essentially any order or even skipped at first reading by readers with prior experience in probability theory, earthquake seismology, and engineering seismology. The core material is contained in chapter 5, which discusses the fundamental principles of seismic hazard analysis guided by the question “How safe do you want to be?”
Finally, this book was completely written in Mathematica. It contains a combination of textbook material, interactive exercises and interactive walk-through tutorials which make heavy use of Mathematica´s dynamic interactivity. This type of interactivity is different from the pre-generated interactivity often found in electronic books, where it comes in the form of pre-generated animations or movies. While these elements are a step forward from the traditional paper-textbook mindset, which come as a one way conversation in which the narrative content is fixed by the author and static, it still maintains the role of the reader as pure recipient, because the interactivity is pre-generated and all the assumptions which go into an interactive figure for example are still fixed by the author. When I learn a new topic, I usually have a lot of “What if?”-questions, which often lead me to programming to explore these questions by small scale simulations. In particular with topics such as PSHA, I find such exercises very helpful to develop some intuition for the behaviour of a system in simple situations. The interactive material here is at least partially aimed at addressing “What if?”-questions, which is possible because the interactivity in Mathematica notebooks can be computed and does not have to be pre-generated. This allows the reader to take on a completely new role in learning because assumptions going into an interactive element (“knowledge apps”) do not need to be fixed, which allows the reader to interactively explore "What if?"-questions. In such an environment, the author takes on the role of a curator of information while the readers become an active part in the learning process who may even design their own exercises.
Since my first attempts to write Mathematica code for seismic hazard analysis, which goes back to the Wolfram Summer School 2008, Mathematica has come a long way. Wolfram Cloud did not exist then and to make use of Mathematica’s interactive features required either a copy of Mathematica or the CDF player. As a consequence, to use the first fragments of this material required quite a bit of enthusiasm and CDF-file-juggling on the student’s side. Now, however, with the possibility to run Mathematica notebooks in the Wolfram Cloud from anywhere or read them directly in the CDF player, independent of the existence of a Mathematica licence on the reader’s side, I could finally make this the book on PSHA which I would have liked to have had when I started to become involved in this topic. I hope you’ll find it useful.
Berlin, February 2025
Frank Scherbaum
fscherbaum@mac.com
Acknowledgments
Acknowledgments
References
References
Table of Contents
Table of Contents
Fundamentals of Probability Theory for PSHA
Fundamentals of Probability Theory for PSHA
Fundamentals of Earthquake Seismology for PSHA
Fundamentals of Earthquake Seismology for PSHA
Fundamentals of Engineering Seismology for PSHA
Fundamentals of Engineering Seismology for PSHA
Fundamental principles of seismic hazard analysis
Fundamental principles of seismic hazard analysis
5.1 Simulation models for ground-motion generation using urn games
5.1 Simulation models for ground-motion generation using urn games
5.2 Building hazard curves from the contribution of individual seismic sources
5.2 Building hazard curves from the contribution of individual seismic sources
5.3 The hazard integral and its visual interpretation
5.3 The hazard integral and its visual interpretation
5.4 Monte Carlo approach to seismic hazard analysis
5.4 Monte Carlo approach to seismic hazard analysis
5.5 The world is not a disc
5.5 The world is not a disc
A final word
A final word
Introduction
Don´t be afraid to do simple things
Robin Adams
The main goal of this book is to develop a basic conceptual understanding for the key aspects of modern Probabilistic Seismic Hazard Analysis (PSHA). The epigraph, which I picked up from Robin Adams, one of the pioneers of observational seismology in the 20th century, has stuck to my memory ever since I attended one of his training courses for young seismologists more than 30 years ago. It can be considered as one of the mottos for the approach taken here, namely to first develop concepts for rather simple situations in which most aspects are easy to understand. Subsequently, these concepts can be extended to more realistic situations. It has been my experience that this transition from the “toy world” to the “real world” can at least partially done rather intuitively.
Probabilistic seismic hazard analysis (PSHA) deals with probabilistic models for seismically generated ground motion and the assessment of the corresponding uncertainties. It draws from a number of disciplines such as probability theory, earthquake seismology and earthquake engineering. The fact that the analysis of any type of natural hazard is concerned with situations which are characterized by large amounts of uncertainties and randomness makes uncertainty assessment a key element of hazard analysis. In the context of seismic hazard it is for example the spatial distribution of future earthquakes which is uncertain, as are their occurrence rates, their source properties, and their site specific shaking characteristics. As a consequence, modern probabilistic seismic hazard analysis (PSHA) is usually accompanied by a systematic uncertainty analysis, sometimes involving different experts or expert groups. Different experts might provide different estimates for the building blocks of a hazard model, which in themselves carry uncertainties. In other words, uncertainties are everywhere and they appear in different flavors. Uncertainties are called aleatory (from alea the Latin word for dice) if they appear as an intrinsic and inseparable aspect of the process under study. They are referred to as epistemic (from the Greek word ϵπιστϵμωσ related to knowledge) if they are caused simply from the lack of knowledge. Epistemic uncertainties can be reduced, at least in principle, by gathering more knowledge while aleatory can not. This makes important differences for the way these can be treated in practice.
Some aspects of the process under consideration might also be more uncertain than others, independent on the type of uncertainty. This naturally leads to the need for the quantification of different degrees of uncertainties. A common way in which this is done is in terms of probabilities. As a consequence, before we even start to think about details of a seismic hazard model in terms of seismicity models, earthquake source properties, site effects, or how to calculate hazard curves, we have to make sure that we have a basic knowledge in probability theory. This will be covered in chapter 2.
Fundamental aspects of earthquake seismology and earthquake engineering are very briefly touched upon in chapters 3 and 4, respectively, but only to a degree which I consider necessary from a seismic hazard analysis perspective. This is then covered in detail in chapter 5, which presents the derivation of hazard curves, one of the key outputs of any PSHA, from multiple perspectives, offering both theoretical and practical insights. From a probability-theoretical standpoint, ground motion can be treated as a random variable (RV) with hazard curves serving as one of many ways to represent its distribution function. Alternatively, from a more pragmatic perspective, hazard curves will be viewed as the outcome of a fictitious long-term observational process. By assuming nature to be consistently observable over extended periods, one can track how frequently certain ground motion levels are exceeded, essentially treating hazard analysis as a systematic bookkeeping exercise. Chapter 5 introduces these and other viewpoints, aiming to provide at least one approach that resonates with every reader. The discussion is kept concise, allowing the material to serve as the foundation for a “crash course” that can be covered in just a few days.
2
Introduction to probability theory for PSHA
Certainty is what disappears when you begin to understand a problem.
Nicolas Deichmann
A common way to describe uncertainties, independent of their nature, is in terms of probabilities. Challenging in the context of modern probabilistic hazard analysis is the fact that different interpretations regarding the meaning of probability are used in combination. Although this might seem counterintuitive at first glance, this combination of concepts is one pillar of the foundations of PSHA. Under conditions which are discussed below (Kolmogorov’s axioms) this is completely acceptable. The current section is not a comprehensive discussion of all possible probability concepts but focused on those which are essential to obtain a conceptual understanding of modern probabilistic hazard analysis.
2.1. Probability concepts
2
.1
. Probability concepts2.2. Joint occurrence of random events
2
.2
. Joint occurrence of random events2.3. Random variables
2
.3
. Random variables2.4. Discrete probability distributions
2
.4
. Discrete probability distributions2.5. Continuous probability distributions
2
.5
. Continuous probability distributions2.6. Transformation of univariate random variables
2
.6
. Transformation of univariate random variables2.7. Jointly distributed random variables
2
.7
. Jointly distributed random variablesReferences
References
3
Earthquake Seismology Primer for PSHA
When the earth shakes, the foundations of human arrogance are tested.
Seneca
Probabilistic seismic hazard analysis draws from a number of different fields of science, one of the most obvious being earthquake seismology. Earthquakes are the cause of the seismic hazard but it does not hurt to remind us again and again that what constitutes the actual threats are effects of earthquakes such as strong ground shaking or tsunamis, not the earthquakes themselves. Here we will concentrate on the shaking hazard. In this chapter I will touch upon those aspects of earthquake physics, wave propagation and site effects, which are needed to understand the essential building blocks of probabilistic seismic hazard models at a very basic level. More advanced aspects are covered in the second part of this book.
3.1. The earthquake source
3
.1
. The earthquake source3.2. Seismic waves
3
.2
. Seismic waves3.3. Measures of earthquake size and shaking intensity
3
.3
. Measures of earthquake size and shaking intensity3.4. Ground Motion
3
.4
. Ground MotionReferences
References
Related Demonstrations
Related Demonstrations
4
Fundamental Aspects of Engineering Seismology for PSHA
Even the strongest walls will shake, but those built with wisdom will stand.
Lao Tzu (Tao Te Ching)
This chapter discusses the fundamental aspects of engineering seismology, as far as they are relevant for understanding the basis of PSHA. The focus in engineering seismology is considerably different from the topics commonly referred to as classical earthquake seismology, which were covered in chapter 3. In contrast to classical earthquake seismologists, engineering seismologists focus on strong ground motion of engineering relevance and care less about weak motion. For this reason, engineering seismology is sometimes also referred to as strong-motion seismology. As a consequence, ground motion in engineering seismology is commonly recorded with accelerometers, which in contrast to classical velocity- or displacement transducers, are not subject to saturation once the shaking, e. g. close to a strong earthquake source, gets really strong. In weak-motion seismology, a seismogram can usually be seen as output of a linear system, which e. g. for strong motion recorded on soft sediments is no longer guaranteed. Engineering seismology also becomes non-linear when it comes to spectral concepts. In contrast to classical (linear-system based) seismology, where Fourier spectra are the basis of nearly all spectral concepts (e. g. the theoretical models for source spectra), in the context of seismic hazard analysis, ground motion is usually thought of in terms of the response spectral values, which describe the maximum response amplitudes of SDOF oscillators as a function of their oscillator frequencies under seismic loading. The reason for this is simply that in contrast to Fourier spectra, response spectra are better predictors of the response of buildings than Fourier spectra. Seismic response spectra, which describe the maximum response amplitudes of SDOF oscillators as a function of their oscillator frequencies, are actually the most common engineering tool to describe the action of buildings under seismic loading. Response spectra are non-linear systems and their theoretical relationship to Fourier spectra contains some rather counterintuitive aspects. Here, where the purpose is simply to understand the principles behind PSHA, it suffices to introduce the different types of response spectra used in PSHA and discuss what the calculation of these requires in terms of ground-motion observation and seismogram processing.
4.1 The response spectrum concept
4
.1
The response spectrum concept4.2. Strong motion observation and processing
4
.2
. Strong motion observation and processingReferences
References
5
Fundamental principles of seismic hazard analysis
Engineers design for ground motion, not for earthquakes
Seismic hazard analysis deals with the hazard generated by strong ground shaking. Earthquakes or other sources of seismic waves are the reason for the hazard, but it is the effect, the corresponding ground motion, that actually constitutes the hazard. This is an important distinction which cannot be overemphasized. Seismic hazard is a site dependent quantity, the properties of which are controlled by physical processes at the source, by properties of the medium through which the waves propagate and by local site conditions. A brief introduction into the seismological aspects of seismic hazard analysis can be found in chapter 3.
Although seismic hazard analysis can be explained procedurally as a sequence of steps which start with the characterisation of the seismic source(s), followed by the characterisation of the ground motion, I believe (very much in line with the epigraph) that it facilitates the understanding to approach seismic hazard analysis conceptually by thinking about ground motion first. The way ground motion is parametrized in the context of seismic hazard analysis differs considerably from the way it is treated in classical seismology. In engineering seismology and seismic hazard analysis, the most popular ground motion intensity parameters are related to response spectral amplitudes. This might be an unfamiliar concept to some non-engineers who might be more familiar with Fourier spectra. Those readers are referred to chapter 4 for a brief introduction into the concept of elastic response spectrum and the processing steps necessary to derive them. Here, seismic hazard will be addressed from a ground motion perspective and furthermore from within a probabilistic framework. Therefore, the type of hazard analysis which is discussed below can be referred to as probabilistic seismic hazard analysis, often simply called PSHA.
Loosely speaking, PSHA is aimed at the determination of statistical properties of ground shaking which is expected to occur at a site of interest. The adjective "probabilistic" in PSHA refers to the fact that, mathematically speaking, the chosen ground motion intensity parameter is treated as a random variable (RV). The concept of a random variable is the key mathematical concept in PSHA. An introduction into this and other essential aspects of probability theory, needed to understand the fundamental concepts of PSHA, is given in chapter 2. In contrast to a normal variable, a random variable does not have a specific value but is described by a distribution function which defines a range of values together with a "likelihood" to obtain a particular value. In the probabilistic framework, the goal of seismic hazard analysis can be stated as the attempt to derive a model for the distribution function of a random variable. From this function, all desired representations, for example the often used hazard curve, can be derived by simple mathematical operations.
In this chapter you will be made familiar with the concept of hazard curves from a number of different perspectives. In other words, you will be made familiar with hazard curves in very different representations. The reason for taking this approach is to acknowledge that humans have different cognitive preferences and different ways of learning. Some of us learn best through theoretical derivations, some through incremental examples, some through synoptic ones, and so forth. I hope that the diversity of perspectives taken here has something to offer for everybody.
5.1. Simulation models for ground-motion generation using urn games
5
.1
. Simulation models for ground-motion generation using urn games5.2. Building hazard curves from the contribution of individual seismic sources
5
.2
. Building hazard curves from the contribution of individual seismic sources5.3. The hazard integral and its visual interpretation
5
.3
. The hazard integral and its visual interpretation5.4. Monte Carlo approach to seismic hazard analysis
5
.4
. Monte Carlo approach to seismic hazard analysis5.5 The world is not a disc
5
.5
The world is not a disc5.6 A final word
5.6 A final word
References
References
Cite this as: Frank Scherbaum, "Fundamental Concepts of Probabilistic Seismic Hazard Analysis" from the Notebook Archive (2025), https://notebookarchive.org/2025-02-5zwyxr9
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