Natural Time, Nowcasting and the Physics of Earthquakes: Estimation of Seismic Risk to Global Megacities

John B Rundle1,2,3,4, Molly Luginbuhl1, Alexis Giguere1, Donald L. Turcotte3

1 Department of Physics University of California, Davis, CA 2 Santa Fe Institute Santa Fe, NM 3 Department of Earth and Planetary Science University of California, Davis, CA 4 Open Hazards Group Davis, CA

Abstract

Natural Time (“NT”) refers to the concept of using small earthquake counts, for example of M>3 events, to mark the intervals between large earthquakes, for example M>6 events. The term was first used by (Varotsos et al., 2005) and later by (Holliday et al., 2006) in their studies of earthquakes. As we discuss in this paper, it is particularly useful in describing complex stochastic nonlinear systems characterized by fat-tail statistics rather than Gaussian normal statistics. In this paper we discuss ideas and applications arising from the use of NT to understand earthquake dynamics. The usual end-user applications of fault-based studies are often applied to risk of a particular geographic location, so it seems best to start the analysis with that geographic region. Rather than focus on an individual earthquake faults, we have found it more productive to focus on a defined local geographic region surrounding a particular location. This local region is considered to be embedded in a larger regional setting from which we accumulate the relevant statistics. From this different philosophical point of view, we first discuss methods to use NT, counts of small earthquakes, to evaluate the current state of a regional collection of faults. We then use these concepts to first discuss the nucleation physics of large earthquakes. We introduce the idea of nowcasting, a term originating from economics and finance. The goal of nowcasting is to determine the current state of the fault system, or put another way, the current state of progress through the earthquake cycle. This is in contrast to forecasting, which is the calculation of probabilities of future large earthquakes. Finally, we apply the nowcasting idea to the practical development of methods to estimate the current state of risk for dozens of the world’s seismically exposed megacities, defined as cities having populations of over 1 million persons. We compute a ranking of these cities based on their current nowcast value, and discuss the advantages and limitations of this approach. We note explicitly that the nowcast method is not a model, in that there are no free parameters to be fit to data. Rather, the method is simply a presentation of statistical data, which the user can interpret.

2 Introduction

Natural time is a term first used by (Varotsos et al., 2005, 2011)and subsequently by (Holliday et al., 2006). It builds on the idea that driven threshold systems such as earthquake fault systems often display a power-law distribution of event sizes or magnitudes. While these bursts of activity are observed at all scales, the largest events are usually of most interest. For earthquakes, these largest events are the magnitude 6+ events that cause the most damage and injuries. Interspersed between these largest events are many smaller events of varying sizes and magnitudes.

Taken together, these small and large events are distributed in a scale-invariant power-law statistical distribution of magnitude. The Gutenberg-Richter magnitude- frequency law (Gutenberg and Richter, 1942; Scholz, 1990) is a simple model of this distribution which is found to be applicable over large spatial domains and over long time intervals. The GR model has two parameters, a and b, which must be fit to the observed data: 𝐿𝑜𝑔$% 𝑁= 𝑎−𝑏𝑀

(1) Here N is the number or frequency of earthquakes having magnitudes larger than M.
Typically, b ~ 1.

Over smaller spatial domains and shorter time intervals, the actual statistics of the observed number or frequency of earthquakes can depart considerably from the simple model (1). A good example is shown in Figure 1a,b . On the left in Figure 1a we see
Figure 1. a) Map of earthquakes having magnitude 𝑀≥6.5 near San Diego since 1970. Circle centered on San Diego has radius 𝑅= 400 km. b) GR number-magnitude statistics. The upper blue square symbols are all earthquakes 𝑀≥3 for the region as a whole since 1970. The lower green circles are all earthquakes 3 ≤𝑀< 6.5 since the last 𝑀≥6.5 earthquake, which was the M7.2 El Major-Cucapah earthquake on 4/4/2010.

3

a map of a large region surrounding the city of city of San Diego, USA, between latitudes 24o N and 43o N Latitude, and between 128o W and 110o W Longitude. In the center of the map is a circle of radius 400 km surrounding the city of San Diego. We then construct the Gutenberg-Richter (GR) number-magnitude statistics in Figure 1b. The statistics represented by th

…(Full text truncated)…