Gutenberg-Richter Magnitude-frequency Research Articles

Stress transfer in earthquakes, hazard estimation and ensemble forecasting: Inferences from numerical simulations

Observations indicate that earthquake faults occur in topologically complex, multi-scale networks driven by plate tectonic forces. We present realistic numerical simulations, involving data-mining, pattern recognition, theoretical analyses and ensemble forecasting techniques, to understand how the observable space–time earthquake patterns are related to the fundamentally inaccessible and unobservable dynamics. Numerical simulations can also help us to understand how the different scales involved in earthquake physics interact and influence the resulting dynamics. Our simulations indicate that elastic interactions (stress transfer) combined with the nonlinearity in the frictional failure threshold law lead to the self-organization of the statistical dynamics, producing 1) statistical distributions for magnitudes and frequencies of earthquakes that have characteristics similar to those possessed by the Gutenberg–Richter magnitude–frequency distributions observed in nature; and 2) clear examples of stress transfer among fault activity described by stress shadows, in which an earthquake on one group of faults reduces the Coulomb failure stress on other faults, thereby delaying activity on those faults. In this paper, we describe the current state of modeling and simulation efforts for Virtual California, a model for all the major active strike slip faults in California. Noting that the Working Group on California Earthquake Probabilities (WGCEP) uses statistical distributions to produce earthquake forecast probabilities, we demonstrate that Virtual California provides a powerful tool for testing the applicability and reliability of the WGCEP statistical methods. Furthermore, we show how the simulations can be used to develop statistical earthquake forecasting techniques that are complementary to the methods used by the WGCEP, but improve upon those methods in a number of important ways. In doing so, we distinguish between the “official” forecasts of the WGCEP, and the “research-quality” forecasts that we discuss here. Finally, we provide a brief discussion of future problems and issues related to the development of ensemble earthquake hazard estimation and forecasting techniques.

A Power-Law Function for Earthquake Interarrival Time and Magnitude

The question of time-dependent seismic hazard models is still an open one. While most hazard studies assume stationarity of seismicity, there has been some debate on the relative merits of Poissonian and non-Poissonian recurrence models, and opinions about the viability of the seismic gap hypothesis also vary. Previous attempts to treat seismic hazard as time-dependent have, however, concentrated on large earthquakes, which do not always control the hazard at a site. In this study, earthquake interarrival times are studied for several regions in Japan and Greece. It is found that a lognormal distribution provides a good model and that seismicity can be represented by the equation ln IAT = a + b M ± c where ln IAT is the log interarrival time of earthquakes exceeding magnitude M and a, b , and c are regional constants. This power law is clearly related to the normal Gutenberg-Richter magnitude-frequency law, but actually contains more information. This law provides a basis for time-dependent seismic hazard analysis in which the whole earthquake catalog is used, rather than just the largest events. A question still remains as to whether c (the standard deviation) is significantly dependent on magnitude. Manuscript received 10 January 2000.

Site-specific Seismic Hazard Estimation in the Main Seismogenic Zones of North Algeria

— The procedure developed by Kijko and Sellevoll (1989, 1992) and Kijko and Graham (1998, 1999) is used to estimate seismic hazard parameters in north Algeria. The area-specific seismic hazard parameters that were calculated consist of the b value of the Gutenberg–Richter frequency–magnitude relation, the activity rate λ(M) for events above the magnitude M, and the maximum regional magnitude Mmax. These parameters were calculated for each of the six seismogenic zones of north Algeria. The site-specific seismic hazard was calculated in terms of the maximum possible PGA at hypothetical engineering structures (HES), situated in each of the six seismogenic zones with coordinates corresponding with those of the six most industrial and populated cities in Algeria.

SCALING LAWS OF SEISMIC EVENTS — A MODEL WITH FRACTAL GEOMETRY

Most models of earthquakes attempt to reproduce the observed scaling laws of seismic events: the Gutenberg-Richter frequency magnitude distribution, but not the Omori law for aftershocks and the multifractal distribution of hypocenters location. Many of these models are based on the idea of Self-Organized Criticality (SOC). These are dynamic systems which organize themselves into a transitional state and can reproduce the Gutenberg-Richter distribution, but generally do not reproduce the space-time distribution. Here, we suggest a model based on a fractal geometry: the two sides of a fault are modeled by means of a fractal surface. As a first step, one of them is slipped of a random amount with periodic boundary conditions, then new contact points between the surfaces are found. The area surrounded by these points is assumed to be proportional to the area of the earthquake. The size distribution of events is in good agreement with the observed Gutenberg-Richter law and the local fluctuations of the b value are explained in terms of variations of the fractal dimension of the surface. Also the multifractal distribution of earthquakes in space is well-reproduced with global properties not depending on the fractal dimension of the surface. However, we are not able to obtain something similar to the Omori law simply because we do not control the time evolution of the model.

SPATIO-TEMPORAL SEISMICITY IN AN ELASTIC BLOCK LATTICE MODEL

An elastic block lattice model is proposed to simulate the spatio-temporal seismicity and stress patterns in the Earth's brittle crust. The famous Gutenberg-Richter magnitude-frequency law in seismology is reproduced. The synthetic catalogs generated by this model are analyzed by using a linked stress release model, which incorporates the stress transfer and spatial interactions. The results highlight the triggering mechanism of earthquake occurrence and the evidence that the crust may lie in a near-critical or self-organized critical state due to the long-range spatial interaction of elastic stress. The spatio-temporal complexity of seismicity is closely related to both nonlinear dynamics of faults and heterogeneities in a seismic region.

Assessment of Seismicity Model for Hong Kong Region

The seismicity model, the Gutenberg-Richter magnitude-frequency relationship. has been developed for Hong Kong region. Detailed investigation and statistical tests on the completeness of the earthquake catalogue and the regional variation of the seismicity have been carried out. It has been found that in the catalogue compiled in this study the historical records on land since 1895 are likely to be complete for surface-wave magnitude M = 4 % or greater. and that there is no record before 1920 in offshore area about 100 km from coastal lines. Almost all historical records before 1894 are on land. indicating that the historical records may be complete only on land. The earthquake spatial distribution is not uniform on land. The number of earthquakes with M = 6 or greater per year in the last 100 years is much larger than that in the previous historical time (1500–1894). It has also been found that the regional variation of the seismicity is a significant factor likely to affect the seismic hazard in Hong Ko...

The statistical mechanics of earthquakes

We review recent theoretical developments on the physics of earthquakes. In particular, we focus on the rise of the statistical mechanical view of earthquakes as a kind of ‘phase transition’. This view is appealing in light of the well known scaling relations such as the Gutenberg-Richter magnitude frequency and Omori's law of aftershock decay. Scaling relations such as these, which are in reality power laws, are known to be associated with dynamical systems residing near a critical point in the state space of the system. These second-order critical points are associated with second-order transitions, which are a result of gradual changes of the controlling parameters. At the same time, characteristic earthquakes, which involve the entire fault segment sliding nearly at once, are more reminiscent of a first-order transition, which is characterized by sudden widespread changes in the physical state of the system. In this paper, we review these ideas and show how recent developments are leading to a view of earthquake fault systems based on modern statistical mechanics.

Precursory seismicity changes preceding moderate and large earthquakes in the Garm region, Central Asia

This study reports on variations in seismic activity preceding moderate and large earthquakes in a highly active portion of the India/Eurasia collision zone. The study is based on the catalog of earthquake locations from the Complex Seismic Expedition (CSE) network in the Garm region, Central Asia. The local seismic network was established in the mid-1950s and recorded and located over 85,000 earthquakes from 1955 to 1988. Two main approaches were used to evaluate variations in seismic activity associated with 12 moderate- to large-magnitude ( ML = 5.0 to 6.3) mainshocks that occurred in the study area: (1) a qualitative approach, involving examination of maps, cross sections, space-time plots, and earthquake histograms for the area identified as the rupture area of the mainshock and (2) a quantitative approach based on magnitude-frequency relations estimated for small space-time windows within the seismic network. In addition to temporal variations in seismicity, three long-term gaps were identified: (1) in the western Peter the First Range, near the city of Garm; (2) in the central Peter the First Range; and (3) in the northeastern part of the study area, approximately coinciding with a gap identified by other authors using different approaches. The potential magnitude of future gap-filling events is estimated to be in the range 6.7 to 7.0. Evidence of precursors has been found to varying extent for 9 out of the 12 mainshocks studied. These include quiescence or low activity before the mainshock, foreshock activity before the mainshock, and a sequence of low-activity-high-activity-quiescence before the mainshock. For the quantitative analysis, the level of seismic activity in small space-time windows is quantified by the parameter A 2, defined as the A value at magnitude 2 in the Gutenberg-Richter magnitude-frequency relation log N = A - bM . The A 2 values were calculated for 30 12 × 12-minute spatial cells with a moving time window of five years. From this analysis, only one earthquake, the M = 6.3 Dzhirgital earthquake of 26 October 1984, shows a significant decrease (by a factor of 45%) in activity level A 2. The results of the quantitative analysis also show strong spatial variations: the central portion of the study area, which includes the Peter the First Range fold-thrust belt, shows high, but relatively stable, seismic activity; the Tien Shan basement uplift in the north has a tendency of decreasing activity; and the northeastern part of the study area shows increasing activity. The October 1984 earthquake is located near the transition between the two tendencies.

Seismic events at Vulcano (Italy) during 1988–1992

A classification of seismic events is proposed that originate on Vulcano (Aeolian Islands, southern Tyrrhenian Sea) and adjacent regions into four broad classes: (1) shocks that exhibit single-peaked spectra; (2) typical earthquakes with distinct P and S phases; (3) irregular-shaped events with ill-defined to undetectable S phases; and (4) long-period events with dominant frequency peak less than 2 Hz. A hypothesis is put forward that the shocks occurring within the volcanic cone of “La Fossa” are mainly related to intense fumarolic activity of high temperature (more than 650°C in October 1992). Estimation of seismic energy release within two distinct sectors (one including “La Fossa” cone, the other external to it) during 1988–1992 has been developed in order to evaluate the level of seismic activity of the system. Over the time here analysed, the rate of seismic-energy-density release at “La Fossa” was 4.71 × 10 14 erg km −2 yr −1, that is significantly larger than that of the surrounding areas. Furthermore, the energy partition estimate indicates that, within “La Fossa”, seismic energy release associated with rock-failure processes is, on the whole, about 3.6 times larger than that of “fluid-driven” shocks. The latter appear to be concentrated at rather shallow depths (400–1000 m), where geochemical analyses and geothermal surveys indicate the sealing action of hydrothermal minerals. Estimated b values of the Gutenberg-Richter magnitude-frequency relation range between 0.81 (±0.27) and 1.45 (±0.29) within “La Fossa”, whereas a value around 1.6 has been calculated for the “external” sector. The comparison between data presented in this paper and those from recent geophyical and geochemical investigations shows that appreciable increments in the rate of seismicity are accompanied with significant modifications in the chemistry of fumarolic gases and in the ground deformation pattern recorded at the base of the active cone.

Power-law random behaviour of seismic reflectivity in boreholes and its relationship to crustal deformation models

There is good evidence for the fractal nature of some fault systems, both from field observations and the frequency-magnitude distribution of earthquakes. Recently, two models, with fundamentally different dynamics, have been employed to explain seismicity: low-order, non-linear slider block models have successfully explained activity on part of the San Andreas fault system and in the Nankai trough, while high-dimensional self-organised-critical cellular automaton models convincingly reproduce the Gutenberg-Richter frequency magnitude relationship. Output variables from these two models fall, respectively, into two distinct data classes, chaotic (deterministic) and random. Using a predictability algorithm which can distinguish between random and chaotic datasets, we examine the distribution class of in-situ fractures (as inferred from borehole seismic reflectivity logs) in three chalk layers off the south coast of Ireland. We find that fractures are distributed in a non-deterministic manner, consistent with self-organised-critical dynamics. Therefore, we may infer that deterministic seismicity should not occur on structures over the scale range covered by the logs, at least in the end-member case where the frequency-magnitude distribution of earthquakes is controlled by the fractal distribution of faults.

Populations of faults and fault displacements and their effects on estimates of fault-related regional extension

Measured populations of fault displacements, derived from regional seismic, oilfield seismic, coalmine plans and outcrop data show a power law distribution with exponents (− S) of −0.45 to −0.95 for singleline samples across an array of faults. The more negative values indicate relatively larger numbers of smaller faults. An expression for a population of active faults, derived from the Gutenberg-Richter magnitude-frequency relation for earthquakes, is log N = a − b D log D, where D = maximum displacement of a fault, N = number of faults of maximum displacement D or greater, and b D ≊ 1.0 and has the same value as b for the corresponding earthquake population. Populations of ‘dead’ faults existing at the end of a tectonic episode have been numerically derived, using a fault growth model, and satisfy the relation log N = a − E log D where E has a value between 1.6 and 2.0. Numerically derived populations of fault displacements in a dead fault population have slopes of − S where S = E − 1. The contribution of an individual fault to the regional strain varies with the lifetime seismic moment of the fault and is proportional to D 2. Estimation of fault-related extension by summing heaves on faults of a limited size range is valid only if the measured size range of faults accommodates most of the extension. Correction can be made if the S value of the fault displacement population is known.

Analysis of the magnitude-frequency distribution for the 1983–1984 earthquake activity of Campi Flegrei, Italy

The magnitude-frequency distribution of earthquakes is one of the statistical relationships most frequently used to study the seismicity of a region. In this paper the Gutenberg-Richter frequency magnitude relationship is applied to the 1983–1984 earthquake sequence of Campi Flegrei, Italy, in order to estimate the space-time variation of the b-value, and its possible correlation with other geophysical parameters in the sequence. The investigated region has been divided into three areas on the basis of differences in seismic activity: (a) to the west of Pozzuoli, where seismicity occurred mostly as swarms; (b) to the east of Pozzuoli, where events mostly included single, high-energy earthquakes; and (c) in the western sector of the Gulf of Pozzuoli. Moreover, b-value estimates have been carried out according to different hypocentral depths of all seismic events in the sequence. The obtained results have shown strong depth variations of the b-value which might reflect important differences in failure conditions at different depths. Furthermore, a certain complexity in the shape of the obtained distribution functions n( M) has been pointed out. Temporal variations of b-value, computed both for the whole region and the different selected areas, seem to be meaningful because they do not depend on the number of shocks ΔN, the data overlap and the magnitude step ΔM. The rates of change of b-value Δb/ Δt, compared with the changing rates of seismic energy release ΔE/ Δt do not show a consistent correlation, hence the Δb/ Δt curve cannot be interpreted as a precursor to the other in any systematic way.

Scale invariance for a heterogeneous fault model

A very simple model of fault fracturing is presented. The fault is envisaged as a series of faultlets whose area is randomly distributed and whose strength have a Gaussian distribution. Faultlet slip occurs when the loading stress reaches its strength. Large earthquakes are generated by coalescence of smaller cracks. Theb value of the Gutenberg-Richter frequency magnitude relation is then calculated. Ab equal to 1 (scale invariance) is obtained only for events with magnitudeM greater than 3. At lower magnitude the number of events decreases with decreasing magnitude. A similar result can be explained by the experimental observation of an anomalous scaling law of real earthquakes. Here the suggestion is advanced that a similar observation could be interpreted by a multifractal approach.

A modified form of the Gutenberg-Richter magnitude-frequency relation: Maximum likelihood estimation of its parameters

A modified form of the Gutenberg-Richter magnitude-frequency relation: Maximum likelihood estimation of its parameters

A comment on “A modified form of the Gutenberg-Richter magnitude-frequency relation” by J. Lomnitz-Adler and C. Lomnitz

A comment on “A modified form of the Gutenberg-Richter magnitude-frequency relation” by J. Lomnitz-Adler and C. Lomnitz

Reply to A. Kijko's “a comment on ‘a modified form of the Gutenberg-Richter magnitude-frequency relation’”

Reply to A. Kijko's “a comment on ‘a modified form of the Gutenberg-Richter magnitude-frequency relation’”

A comment on “A modified form of the Gutenberg-Richter magnitude-frequency relation” by J. Lomnitz-Adler and C. Lomnitz

A comment on “A modified form of the Gutenberg-Richter magnitude-frequency relation” by J. Lomnitz-Adler and C. Lomnitz

Reply to M. Vered and A. Ben-Menahem's “A comment on ‘A modified form of the Gutenberg-Richter magnitude-frequency relation’”

Reply to M. Vered and A. Ben-Menahem's “A comment on ‘A modified form of the Gutenberg-Richter magnitude-frequency relation’”

A modified form of the Gutenberg-Richter magnitude-frequency relation

abstract A stochastic model of strain accumulation and release at plate boundaries is proposed. The model leads to the generalized Gutenberg-Richter relation In G ( x ) = constant − B exp ( α x ) where G is the cumulative exceedence of a magnitude x . This relation tends asymptotically to the original Gutenberg-Richter relation in the low-magnitude range; at high magnitudes it provides estimates of the probability of occurrence which are significantly more realistic than the Gutenberg-Richter relation. Excellent agreement is obtained with the data of the Chinese earthquake catalog covering a time period of 2,753 years.

本震と最大余震とのマグニチュード差<i>D</i><sub>1</sub>の分布則

The negative binomial distribution is found to show a good fit to the distribution of total number of aftershocks whose magnitudes differ by no more than d from the magnitude of the main shock M0. On the basis of this observation, the probability is derived that the difference Di in magnitude between the main shock and its i-th largest aftershock becomes larger than d.Pr{Di≥d}=∑i-1k=0(α+k-1k)(αbln10)α{a(10db-1)}k/{αbln10+a(10bd-1)}α+kwhere a and α denote the mean of a and the power parameter of the negative binomial distribution, respectively, and a and b indicate thee parameters of the Gutenberg-Richter's magnitude-frequency relation. It is assumed that the Gutenberrg-Richter's relation n(M)dM=a10-b(M-M0)dM holds for aftershocks and that b is constant for every aftershock sequence. we studied the aftershock sequences in the area of Japan and Greece, which have been compiled by Utsu and Papazachos etc., respectively. The distribution of the difference D1 in magnitude between the main shock and its largest aftershock deduced from “negative binomial model” is agreeable with the observation. The value of b deduced from D1 distribution is consistent with the one of aftershock sequences given by Utsu and Papazachos. The observed distributions of D2 and D3 show slight discrepancy from those expected from the model.