Magnitude Distribution Of Earthquakes Research Articles

Deviation of magnitude frequency distribution of earthquakes from the Gutenberg-Richter law: detection of precursory anomalies prior to large earthquakes

We present a measure that represents the degree of deviation of the magnitude frequency distribution of earthquakes from the Gutenberg-Richter (GR) law. The magnitude frequency distribution of earthquakes has been considered to obey the GR law. Actual distributions, however, often deviate considerably from this law. Let n i , N, M 0 and b be the number of events with magnitude M i ∼ M i + ΔM, total number of earthquakes, lower limit of the magnitude and b value of the GR formula, respectively. Then the measure of deviation is given using Kullback-Leibler's mean information as C= ∑ i p(X i)ln[ p(X i) q(X i ] X i=M i−M 0 where p(X i) = n i N and q(X i) = b ln 10 exp (−b ln 10 X i) . Function q( X i ) denotes the GR distribution and thus this measure gives the degree of deviation of the magnitude distribution from the GR law. When the magnitude distribution of earthquakes fits the GR formula more closely, the ‘ C’ value becomes smaller and is zero if the distribution obeys the GR law exactly. Conversely, if the distribution deviates greatly from the GR law, the ‘ C’ index shows a large value. We have investigated the temporal variation of ‘ C’ values before and after large earthquakes and swarm earthquakes in Japan. Results show considerable variations of ‘ C’ values before and after large earthquakes. Prior to large earthquakes, ‘ C’ values become small, which indicates that magnitude distributions fit better to the GR law. The ‘ C’ indices show drastic change after mainshocks. During swarm earthquakes, the variation of ‘ C’ values is rather large, but ‘ C’ values are smaller than ordinary earthquakes. In some swarms the ‘ C’ indices reach a maximum at the time of the greatest activity. This measure is a useful tool with which to observe the characteristic properties of the seismic activity and may be used for detection of precursory phenomena of large earthquakes.

Changes in frequency–size relationship from small to large earthquakes

THE constant 'b value' observed in frequency–magnitude distributions of earthquakes has been taken as an indication of self-similarity at all magnitudes. Hence, earthquake properties should scale uniformly in the same way from small to large earthquakes. It has often been observed, however, that the seismic moment released in small earthquakes scales differently with rupture length than it does for large events1,2, where the crossover between small and large events is defined at a rupture dimension equal to the down-dip width of the seismogenic zone. A possible explanation for this contradiction is that frequency–size distributions are biased by small earthquakes. Small events dominate most global earthquake catalogues because of the short time-period covered. Another source of bias in size distributions is the saturation of earthquake magnitudes for large events. Here we correct biases in calculations of b values and present evidence for a change in b value in frequency–size distributions. We find that a break in self-similarity, from small to large earthquakes, occurs at a point where the dimension of the event equals the down-dip width of the seismogenic layer.

An application of Bayesian (ABIC) smoothing methods to estimating space and time variations in the magnitude distributions of earthquakes

On the basis of Akaike's Bayesian information criterion (ABIC), a new method of estimating temporal and spacial variations in the magnitude-frequency relation (b value) is developed. The space-time volume studied is divided into a number of segments having equal volume, for which b values are estimated. The smoothness of the estimated b value is guaranteed by the introduction of a prior distribution. The parameters of the prior distribution are chosen in such a way as to minimize the ABIC. This method is employed to study the temporal and spatial variations of b values for microearthquake activity in the Kanto, Tokai and Tottori areas, where seven earthquakes having magnitudes of M6.0 and larger have recently taken place. States of temporal variations of b values are classified into three groups; increase, decrease and nochange. The space-time volume of decrease accounts for approximately 10% of the total volume. For a period before the occurrence of earthquakes, the decrease appears more frequently than might be expected. It is concluded that the decrease in the b value is a promising candidate to act as a precursory phenomena.

GEMPAK: A FORTRAN-77 program for calculating Gumbel's First, Third, and Mixture upper earthquake magnitude distributions employing maximum likelihood estimation

Abstract Progam GEMPAK (consisting of three subprograms, viz. GEMFID, GEMTID, and GEMMID) estimates the parameters of Gumbel's First ( G 1), Third ( G 3), and Mixture ( G mix) upper earthquake magnitude extremal asymptotic distributions respectively. The G 1 [ A and B ], G 3 [ W , U , and K ], and G mix [ A , B , W , U , K , and P ] parameters are determined utilizing the maximum likelihood technique by numerical maximization of the ln (likelihood) function employing the Newton-Raphson procedure. All three subprograms also calculate the return periods at selected magnitude classes regarded as risky as well as determining the risk probability at specific design intervals or economic lifetimes.

Influence of fractal flaw distributions on rock deformation in the brittle field

Abstract The geometrical distribution of flaws plays a crucial role in the physical behaviour of geological materials under stress. Flaws are present in the earth on all scales, from microcracks to plate-rupturing faults. They may be distributed on one characteristic length scale (e.g. joints, ‘characteristic’ earthquakes), or more commonly exhibit scale-invariance over a specified range of sizes. Scale-invariance implies that the discrete length distribution in a finite range is a power law of negative exponent D , where 1 ≤ D < 3. Fault systems where motion is concentrated on a dominant fault (e.g. San Andreas) have D ≈ 1, but more diffuse fault systems have D near 2. D is one of the fractal dimensions of the fracture system. The length distribution of faults or microcracks may be inferred from the slope b of the log-linear frequency—magnitude distribution of earthquakes, or laboratory-scale acoustic emissions, since it can be shown that D = 3 b/c . The scaling factor c depends on the relative time constants of the seismic event and the recording instrument, and is usually equal to 3/2. b is found experimentally to be negatively correlated with the stress intensity on the dominant flaw, which depends in turn on the applied stress and the flaw length. Thus a fracture mechanics model of rock failure which includes a range of flaw sizes can be tested by seismic monitoring. We describe a fracture mechanics model of rock failure for a variety of styles of deformation, ranging from elastic failure to quasi-static cataclastic flow, and predict the time-dependence of D and the seismic b -value at different times up to and including failure. Critical coalescence of microcracks during dynamic failure (e.g. earthquake foreshocks) occurs when D = 1 ( b = 0.5); random processes (e.g. cataclastic flow, background seismicity) are associated with D = 2 ( b = 1); positive feedback in the concentration of stress on the dominant flaw (e.g. during strain softening and shear localisation) occurs when D < 2 ( b < 1); negative feedback in stress concentration (e.g. during the early stages of dilatancy), and where a highly diffuse fracture system is produced, occurs at low stress intensities and is associated with D > 2 ( b > 1). It has long been a goal of structural geologists to measure stress on rocks, since most geometrical signatures of deformation are strain-related. We show that stress is not usually as significant in rock fracture as stress intensity, and furthermore that the geometric signature of the length distribution of microcracks is well-correlated with the stress intensity.

Properties of earthquakes generated by fault dynamics.

We present a model for fault dynamics consisting of a uniform chain of blocks and springs pulled slowly across a rough surface. The only nonlinear element of our model is a slip-stick friction force between the blocks and the surface. We find that this model gives rise to events of all sizes. Our numerical evaluation of the distribution of earthquake magnitudes results in power-law spectrum similar to what is observed in nature. Like certain other dissipative dynamical systems, the observed large fluctuations in earthquake magnitude persist because the system is in a state of marginal stability.

上限をもつ地震の規模別度数分布について

上限をもつ地震の規模別度数分布について

A Bayesian method for estimating earthquake magnitude distribution and changes in the distribution with time and space in New Zealand

Abstract A Bayesian procedure for examining whether the time-space variation of the magnitudefrequency relation conforms with the exponential distribution or not is proposed. The relation in each time-space segment is obtained as a histogram. The smoothness of the estimated histogram is guaranteed by the introduction of prior distributions of the parameters in time, space, and magnitude. The parameters of the prior distribution are chosen to minimise Akaike's Bayesian Information Criterion (ABIC). This procedure is applied to three earthquake catalogues in New Zealand: those for the Wellington Seismograph Network, the Pukaki Network, and the New Zealand Network respectively. For both the Wellington and Pukaki Networks, departure from the exponential distribution is the main effect and the time variation is secondary. For the New Zealand Network, where the magnitude- frequency relation appears to be an exponential distribution, the variation of b over time and space suggests a relation between periods of active seismicity and periods of high b-value, not locally, but overall in the main seismic region.

Revised estimates for the probabilities of earthquakes following observation of unreliable precursors

abstract This paper treats, in a statistical manner, the class of geophysical observations that sometimes are precursors to a large earthquake, but at other times occur apparently unrelated to any earthquakes in time or space. Each such observation is handled by associating it with the largest ensuing earthquake in a definite time and spatial window; then the probability that the observation is useful is estimated from the difference between this extreme value distribution and the long-term average extreme value distribution for the same region. This probability that the precursor is useful can then be incorporated into calculations of the revised probability of an earthquake following observation of one (a trivial case) or several unreliable precursors to derive the revised probability of a significant earthquake. The model takes full account of the magnitude distribution of earthquakes and of discrete levels of precursor observations. The United States earthquake prediction program is still many years from being able to obtain the full benefits of the model. However, the model suggests useful statistics which the program should gather on unreliable precursors.

A seismicity model for New Zealand

A model for the distribution of shallow earthquake magnitudes in and around New Zealand is presented. The model comprises 9 onshore and 8 offshore regions, selected for their general tectonic homogeneity. For each region, values of the Gutenberg and Richter seismicity parameters a and b are estimated using seismological, geological and tectonic data. Smooth transitions between regions are achieved using concepts from fuzzy set theory. The model is well-suited to seismic risk computations.

Maximum entropy and the magnitude distribution

abstract The principle of maximum entropy is used to determine the distribution of earthquake magnitudes. Given no upper bound to magnitude, the maximum entropy principle generates the Gutenberg and Richter recurrence relation. When an upper bound m 1 is assumed, the principle leads to the truncated exponential distribution proposed by Cornell and Vanmarcke (1969), N ( M ) = T e β M − e − β m 1 1 − e − β m 1 in which N denotes the number of earthquakes with magnitude equal to or greater than M , and T is the total number of earthquakes with nonnegative magnitudes. This relationship agrees with observed data both at small magnitudes, where it coincides with the Gutenberg and Richter relation, and at large magnitudes, where it fits the data remarkably well. Further, given only the mean rate of occurrence, the principle leads to a Poisson temporal distribution of earthquakes.

Some properties of earthquake frequency distributions

Various statistical parameters are calculated for frequency—magnitude distributions of earthquakes. Theoretical expressions are given for the mean, median, quartiles, variance and skewness and applied to four different earthquake sequences.

7.有珠山の地震活動-地震の上限とm(b)の値について : 日本火山学会1978年春季大会

7.有珠山の地震活動-地震の上限とm(b)の値について : 日本火山学会1978年春季大会

On the magnitude distribution and prediction of earthquakes : Maaz, R, Centr. Earth Phys. Inst. Jena, DDR Purcaru, G, Min. Res. Inst. Bucharest, R 25R. Rev. Roum. Geol. Et Geogr. Serie de Geoph. V17, N2, 1973, P155–163

On the magnitude distribution and prediction of earthquakes : Maaz, R, Centr. Earth Phys. Inst. Jena, DDR Purcaru, G, Min. Res. Inst. Bucharest, R 25R. Rev. Roum. Geol. Et Geogr. Serie de Geoph. V17, N2, 1973, P155–163

A statistical argument for the existence of a discontinuity in some subduction zones

The slope b of the magnitude distribution of earthquakes is proportional to the inverse of the mean magnitude in a region. Systematic changes of b with depth in some subduction zones are interpreted as evidence that the downgoing slabs are not continuous down to depths of 600 km. The Tonga-Kermadec and Japan subduction zones exhibit no significant variation of b with depth and are therefore proposed as prototypes of continuous subduction zones.

地震の規模別頻度の地域性について (II)

地震の規模別頻度の地域性について (II)