validation of using gumbel probability plotting to estimate gutenberg-richter seismicity parameters...

Validation of using Validation of using GumbelGumbel Probability Plotting to estimate Probability Plotting to estimate Gutenberg-Richter Gutenberg-Richter

seismicityseismicity parameters parametersTo be presented at To be presented at AEESAEES

Annual Conference at Canberra in Annual Conference at Canberra in November 2006November 2006

Dr Dion Weatherly Mike Turnbull

G-R Vs EVTG-R Vs EVT It is common to characterize earthquake It is common to characterize earthquake

seismicity of a region by specifying values for the seismicity of a region by specifying values for the a and b parameters of the Guttenberg-Richter a and b parameters of the Guttenberg-Richter seismicity model: seismicity model: N(m ≥ M)N(m ≥ M) = 10= 10(a – b m)(a – b m)

Estimations of these parameters can be derived Estimations of these parameters can be derived from a number of statistical processes.from a number of statistical processes.

In situations where a comprehensively complete In situations where a comprehensively complete catalogue of earthquake events is not available, catalogue of earthquake events is not available, methods provided by the statistics of extreme methods provided by the statistics of extreme events (the so-called events (the so-called extreme value theoryextreme value theory (EVT)) (EVT)) have been applied, using reduced variate have been applied, using reduced variate probability plottingprobability plotting..

EVT DistributionsEVT Distributions

The generalized EVT cumulative The generalized EVT cumulative distribution function (cdf) reduces to distribution function (cdf) reduces to one of three specific one of three specific Fisher TippettFisher Tippett distributions, depending on the value distributions, depending on the value chosen for its three parameters, ξ, chosen for its three parameters, ξ, θ(> 0), and k(> 0). θ(> 0), and k(> 0).

Fisher & Tippett, 1928Fisher & Tippett, 1928 Fisher Tippett Type 1:Fisher Tippett Type 1:

Pr[X ≤ x]Pr[X ≤ x] = exp{-exp{-1/θ(x – ξ)}}= exp{-exp{-1/θ(x – ξ)}}

Fisher Tippett Type 2:Fisher Tippett Type 2:Pr[X ≤ x]Pr[X ≤ x] = 0= 0 where x < ξwhere x < ξ = exp{-exp{-(1/θ(x – ξ))k}}= exp{-exp{-(1/θ(x – ξ))k}} where x ≥ ξ.where x ≥ ξ.

Fisher Tippett Type 3:Fisher Tippett Type 3:Pr[X ≤ x]Pr[X ≤ x] = {-exp{-(1/θ(ξ - x))k}}= {-exp{-(1/θ(ξ - x))k}} where x ≤ ξwhere x ≤ ξ

= 1= 1 where x > ξwhere x > ξ

The Gumbel DistributionThe Gumbel Distribution The Type 2 distribution is often referred to The Type 2 distribution is often referred to

as the as the FréchetFréchet distribution. distribution.

The Type 3 distribution is often referred to The Type 3 distribution is often referred to as the as the WeibullWeibull distribution. distribution.

The Type 1 distribution is mostly referred The Type 1 distribution is mostly referred to as the to as the GumbelGumbel distribution, but is distribution, but is sometimes referred to as the sometimes referred to as the log-Weibulllog-Weibull distribution. In this presentation it will be distribution. In this presentation it will be referred to as the Gumbel distribution. referred to as the Gumbel distribution.

To EV or not to EV – that is the To EV or not to EV – that is the question!question!

There are two common criticisms made, arguing There are two common criticisms made, arguing that the probability plotting method of analysing that the probability plotting method of analysing extreme events to estimate regional seismicity is extreme events to estimate regional seismicity is of little value to practical seismology. These of little value to practical seismology. These criticisms are that:criticisms are that:

1.1. The extreme value methods only assess the few The extreme value methods only assess the few maximum value events and ignore the many maximum value events and ignore the many other important smaller events.other important smaller events.

2.2. The various methods of determining the plotting The various methods of determining the plotting positions used to calculate the reduced variate positions used to calculate the reduced variate are arbitrary in nature. Therefore the choice of are arbitrary in nature. Therefore the choice of plotting position algorithm can be used to plotting position algorithm can be used to manipulate the results. manipulate the results.

This paper addresses these two This paper addresses these two criticismscriticisms

1.1. Via counter-arguments, Via counter-arguments,

2.2. and by providing a demonstration that and by providing a demonstration that the reduced variate probability plotting the reduced variate probability plotting method in conjunction with Gumbel method in conjunction with Gumbel statistics of extreme events can statistics of extreme events can reproduce accurate estimates of reproduce accurate estimates of a prioria priori seismicity parameters used to generate seismicity parameters used to generate synthetic earthquake calenders. synthetic earthquake calenders.

Our analysis consists of two parts.Our analysis consists of two parts. Firstly we demonstrate that the probability Firstly we demonstrate that the probability

plotting method estimates to within 2% accuracy, plotting method estimates to within 2% accuracy, the Gumbel parameters of a synthetic dataset the Gumbel parameters of a synthetic dataset constructed with constructed with a prioria priori values of these values of these parameters. parameters.

Secondly we apply the Gumbel method to analyse Secondly we apply the Gumbel method to analyse synthetic seismicity calendars generated from a synthetic seismicity calendars generated from a Gutenberg-Richter distribution with prescribed a- Gutenberg-Richter distribution with prescribed a- and b-values. and b-values.

Our results testify that the Gumbel method Our results testify that the Gumbel method accurately estimates the a and b values of the accurately estimates the a and b values of the underlying G-R source distribution, via statistical underlying G-R source distribution, via statistical analysis of only the extreme values of the analysis of only the extreme values of the synthetic catalogues.synthetic catalogues.

Are important data being ignored? Are important data being ignored?

Statistical analysis aims to provide an accurate model for a given set of Statistical analysis aims to provide an accurate model for a given set of observations, using some assumptions about the underlying process observations, using some assumptions about the underlying process giving rise to the observations.giving rise to the observations.

In the case of regional seismicity, one assumes the underlying process In the case of regional seismicity, one assumes the underlying process gives rise to a Gutenberg-Richter frequency-magnitude distribution: gives rise to a Gutenberg-Richter frequency-magnitude distribution: N(m ≥ M)N(m ≥ M) = 10= 10(a – b m)(a – b m)

a two-parameter model determining the average rate of seismicity (a a two-parameter model determining the average rate of seismicity (a value) and the scaling of recurrence intervals with given earthquake value) and the scaling of recurrence intervals with given earthquake magnitude (b value).magnitude (b value).

For a particular region, one aims to estimate the values for these two For a particular region, one aims to estimate the values for these two parameters via curve fitting of the observed historical seismicity.parameters via curve fitting of the observed historical seismicity.

Since the dataset of observations is invariably only a small subset (or Since the dataset of observations is invariably only a small subset (or sampling) of the seismic history and the observations may contain errors sampling) of the seismic history and the observations may contain errors (e.g. imprecise magnitude determination or poor detection level) one (e.g. imprecise magnitude determination or poor detection level) one cannot expect to obtain an arbitrarily accurate estimation of the model cannot expect to obtain an arbitrarily accurate estimation of the model parameters.parameters.

Are important data being ignored?Are important data being ignored?

It is well-known that estimated values for any statistical It is well-known that estimated values for any statistical model parameters may be significantly skewed when model parameters may be significantly skewed when using a dataset which does not contain adequate using a dataset which does not contain adequate samples of the full range of observable values.samples of the full range of observable values.

Seismicity particularly suffers from this limitation as Seismicity particularly suffers from this limitation as historical seismic catalogues are typically complete for historical seismic catalogues are typically complete for large magnitudes (the extreme values of the G-R large magnitudes (the extreme values of the G-R distribution) but incomplete or non-existent for smaller distribution) but incomplete or non-existent for smaller magnitudes.magnitudes.

Historical earthquake catalogues are biased Historical earthquake catalogues are biased towards extreme valuestowards extreme values

Are important data being ignored?Are important data being ignored? The Fisher-Tippett probability distributions are The Fisher-Tippett probability distributions are

specifically formulated to model the extreme data specifically formulated to model the extreme data values that are invariably found in samples extracted values that are invariably found in samples extracted from underlying source distributions.from underlying source distributions.

EV distributions provide a parameterisation for the EV distributions provide a parameterisation for the extreme values that is related to the parameters of the extreme values that is related to the parameters of the source distribution, while taking into account the source distribution, while taking into account the inherent bias towards extreme values in the dataset inherent bias towards extreme values in the dataset under analysis.under analysis.

It was Fisher and Tippet (1928) who proved that no It was Fisher and Tippet (1928) who proved that no matter what source probability distribution data is matter what source probability distribution data is derived from, the distribution of extreme data values derived from, the distribution of extreme data values will necessarily converge to one of the three forms will necessarily converge to one of the three forms shown in the introductory slides.shown in the introductory slides.

Are important data being ignored?Are important data being ignored? The perception that extreme value methods ignore important small value The perception that extreme value methods ignore important small value

data is false.data is false.

EV methods are designed to model the distribution of extreme values EV methods are designed to model the distribution of extreme values accurately, not the distribution of non-extreme values. Including these accurately, not the distribution of non-extreme values. Including these later values in the analysis would be erroneous.later values in the analysis would be erroneous.

Since the dataset of extreme values is complete, one does not suffer from Since the dataset of extreme values is complete, one does not suffer from the finite sampling issues when estimating the parameters of the EV the finite sampling issues when estimating the parameters of the EV distribution.distribution.

It must be emphasised that EV methods make allowance for the bias It must be emphasised that EV methods make allowance for the bias towards extreme values in the original dataset. This is codified in the towards extreme values in the original dataset. This is codified in the relationship between EV model parameters and those of the source relationship between EV model parameters and those of the source distribution.distribution.

Thus it is possible, by analysing a catalogue of extreme values, to Thus it is possible, by analysing a catalogue of extreme values, to accurately estimate the parameters of the source distribution.accurately estimate the parameters of the source distribution.

Given the indisputable bias towards large magnitudes in seismic Given the indisputable bias towards large magnitudes in seismic catalogues, EV methods are well-suited for modelling regional seismicity. catalogues, EV methods are well-suited for modelling regional seismicity.

The probability integral transformation The probability integral transformation (PIT) theorem (PIT) theorem

The theorem of The theorem of probability integral probability integral transformationtransformation states that any cumulative states that any cumulative distribution function, considered as a function distribution function, considered as a function of its random variable X, is itself a of its random variable X, is itself a uniformuniform random variable on the closed interval (0,1) random variable on the closed interval (0,1) (Bury, 1999, p 25).(Bury, 1999, p 25).

F(X; θ) = U where θ represents parameters, F(X; θ) = U where θ represents parameters, either known or not yet determined.either known or not yet determined.

The probability integral transformation The probability integral transformation (PIT) theorem(PIT) theorem

A consequence of this theorem is that all A consequence of this theorem is that all possible values of X are equally likely. So possible values of X are equally likely. So that any sample variate F(xthat any sample variate F(xii; θ) derived ; θ) derived from the parent distribution F(X; θ) can be from the parent distribution F(X; θ) can be expressed in the form:expressed in the form:

F(xF(xii; θ) = u; θ) = uii

where uwhere uii is a value in the closed interval is a value in the closed interval (0, 1), and where all values of u(0, 1), and where all values of uii are are equally likely.equally likely.


A important corollary of the probability A important corollary of the probability integral transformation theorem is integral transformation theorem is that:that:

xxii = F = F-1-1(u(uii; θ) ; θ)


This corollary has two important This corollary has two important applications in practice:applications in practice:

1.1. simulating random observations, simulating random observations, andand

2.2. Probability plotting.Probability plotting.

Simulating random variates Simulating random variates

The corollary of the probability integral The corollary of the probability integral transformation theorem provides a means transformation theorem provides a means of simulating random variates from any of simulating random variates from any known probability distribution.known probability distribution.

By substituting random numbers uBy substituting random numbers uii from the from the closed (0, 1) interval into the inverse of closed (0, 1) interval into the inverse of the distribution’s cumulative distribution the distribution’s cumulative distribution function, independent identically function, independent identically distributed random variates can be distributed random variates can be generated.generated.

Simulating random Gumbel variatesSimulating random Gumbel variates For example (Bury, 1999, p 268), the cdf of the Gumbel For example (Bury, 1999, p 268), the cdf of the Gumbel

distribution may be expressed in the formdistribution may be expressed in the form

F(x; µ, σ) = exp{-exp{-1/ σ (x - µ)}} = uF(x; µ, σ) = exp{-exp{-1/ σ (x - µ)}} = u

By inversionBy inversionx = µ - σln(-ln(u))x = µ - σln(-ln(u))

Therefore, simulated random variates xTherefore, simulated random variates xii from the from the Gumbel distribution can be generated using the Gumbel distribution can be generated using the following formula, where ufollowing formula, where uii is a random number on the is a random number on the closed interval (0, 1).closed interval (0, 1).

xxii = µ - σln(-ln(u = µ - σln(-ln(uii)) ))

Gumbel Probability plottingGumbel Probability plottingManipulation of the equationManipulation of the equation

xxii = µ - σln(-ln(u = µ - σln(-ln(uii))))

produces the linear relationproduces the linear relation

-ln(-ln(p-ln(-ln(pii)))) = 1/σ (x= 1/σ (xii - µ) - µ)

where the u notation has been replaced by a p, where the u notation has been replaced by a p, for reasons that will become clear as we for reasons that will become clear as we progress.progress.

Gumbel Probability plottingGumbel Probability plotting This relationThis relation

-ln(-ln(p-ln(-ln(pii)))) = 1/σ (x= 1/σ (xii - µ) - µ) provides a potential means of testing whether a set of n provides a potential means of testing whether a set of n experimental observations {xexperimental observations {xii}}nn is a sample from a Gumbel is a sample from a Gumbel distribution.distribution.

If the If the reduced variatesreduced variates {-ln(-ln(p {-ln(-ln(pii))}))}nn are plotted against the are plotted against the experimental observations {xexperimental observations {xii}}nn, and a straight line graph results, , and a straight line graph results, then the postulated Gumbel parent distribution is confirmed, and then the postulated Gumbel parent distribution is confirmed, and ordinary linear regression can be used to estimate the parameters ordinary linear regression can be used to estimate the parameters σ and µ from the slope and intercept.σ and µ from the slope and intercept.

There is one difficulty in accomplishing this task. There is one difficulty in accomplishing this task. In any real In any real experimental situation the observations {xexperimental situation the observations {xii}}nn are known, are known, but the n reduced variates {-ln(-ln(pbut the n reduced variates {-ln(-ln(pii))}))}nn cannot be cannot be calculated exactly because the calculated exactly because the plotting positionsplotting positions {p {pii}}nn are are unknownunknown..

Gumbel Probability plottingGumbel Probability plotting The only things that can be assumed regarding The only things that can be assumed regarding

the pthe pii values is that they are in the closed values is that they are in the closed interval (0, 1), and that each value has equal interval (0, 1), and that each value has equal likelihood of presence.likelihood of presence.

This information suggests a widely used, but This information suggests a widely used, but controversial, method for producing artificial controversial, method for producing artificial plotting positions that can be substituted for plotting positions that can be substituted for the actual ones.the actual ones.

The method used to determine the substitute The method used to determine the substitute plotting positions can be described as follows.plotting positions can be described as follows.

Gumbel Probability plottingGumbel Probability plottingThe n observations are first ordered and ranked The n observations are first ordered and ranked

according to their relative values. Depending on according to their relative values. Depending on the requirements of the particular situation this the requirements of the particular situation this ranking may be in ascending or descending ranking may be in ascending or descending order. The examples described here will use order. The examples described here will use ascending order. The ordered observations are ascending order. The ordered observations are notated asnotated as

{x{xii}*}*nn ≡ {x ≡ {x11 ≤ x ≤ x22 ≤ x ≤ x33 ≤ … ≤ x ≤ … ≤ xn-2n-2 ≤ x ≤ xn-1n-1 ≤ x ≤ xnn}}

where xwhere x11 is the smallest valued variate, x is the smallest valued variate, xnn is the is the largest valued variate, and the subscript values largest valued variate, and the subscript values are the variate ranks.are the variate ranks.

Gumbel Probability plottingGumbel Probability plotting The next step is the controversial part of The next step is the controversial part of

the method.the method.

The rank value of the mThe rank value of the mthth ordered variate ordered variate is used to determine an artificial plotting is used to determine an artificial plotting position quantile pposition quantile pmm for that variate. for that variate.

There is no single definitive formula There is no single definitive formula or equation for doing this. However, or equation for doing this. However, there are guidelines for doing so.there are guidelines for doing so.

Gumbel Probability plottingGumbel Probability plottingGumbel (1958) expressed the following five conditions as requirements that substitute Gumbel (1958) expressed the following five conditions as requirements that substitute

plotting positions should necessarily fulfil.plotting positions should necessarily fulfil.

The plotting position should be such that all observations can be plotted.The plotting position should be such that all observations can be plotted.

The plotting position should lie between the observed frequencies (m – 1)/n and m/n The plotting position should lie between the observed frequencies (m – 1)/n and m/n and should be universally applicable, i.e., it should be distribution-free. This and should be universally applicable, i.e., it should be distribution-free. This excludes the probabilities of the mean, median, and modal mexcludes the probabilities of the mean, median, and modal mthth value which differ for value which differ for different distributions.different distributions.

The return period of a value equal to or larger than the largest observation, and the The return period of a value equal to or larger than the largest observation, and the return period of a value smaller than the smallest observation, should approach n, return period of a value smaller than the smallest observation, should approach n, the number of observations. This condition need not be fulfilled by the choice of the the number of observations. This condition need not be fulfilled by the choice of the mean and median mmean and median mthth value. value.

The observations should be equally spaced on the frequency scale, i.e., the The observations should be equally spaced on the frequency scale, i.e., the difference between the plotting positions of the (m + 1)th and the mdifference between the plotting positions of the (m + 1)th and the mthth observation observation should be a function of n only, and independent of m. This condition … need not be should be a function of n only, and independent of m. This condition … need not be fulfilled for the probabilities at the mean, median, or modal mfulfilled for the probabilities at the mean, median, or modal mthth values. values.

The plotting position should have an intuitive meaning, and ought to be analytically The plotting position should have an intuitive meaning, and ought to be analytically simple. The probabilities at the mean, modal, or median msimple. The probabilities at the mean, modal, or median mthth value have an intuitive value have an intuitive meaning. However, the numerical work involved is prohibitive meaning. However, the numerical work involved is prohibitive [at the time of [at the time of writing. Current computing capabilities now make these calculations routine]writing. Current computing capabilities now make these calculations routine]..

Gumbel Probability plottingGumbel Probability plotting The simplest approach is to assume that The simplest approach is to assume that

the value of the plotting position quantile the value of the plotting position quantile is equal to its fractional position in the is equal to its fractional position in the ranked list, m/n. This would assign the ranked list, m/n. This would assign the quantile 1/n to the smallest plotting quantile 1/n to the smallest plotting position and n/n = 1 to the largest.position and n/n = 1 to the largest.

This is unsatisfactory because it leaves This is unsatisfactory because it leaves no room at the upper end for values no room at the upper end for values greater than the largest variate observed greater than the largest variate observed thus far.thus far.

Gumbel Probability plottingGumbel Probability plotting

Most plotting position formulae are Most plotting position formulae are ratios of the formratios of the form

(m ± a)/(n ± b)(m ± a)/(n ± b)

where the addends and subtrahends where the addends and subtrahends are chosen to improve estimates in are chosen to improve estimates in the extreme tails of the postulated the extreme tails of the postulated distribution.distribution.

Gumbel Probability plottingGumbel Probability plotting Gumbel recommended the following quantile Gumbel recommended the following quantile

formulation, which calculates the mean frequency of formulation, which calculates the mean frequency of the mthe mthth variate. variate.

ppmm = m / (n + 1) = m / (n + 1)

This formulation ensures that any plotting position is as This formulation ensures that any plotting position is as near to the subsequent one as it is to the previous.near to the subsequent one as it is to the previous.

It also produces a symmetrical sample cdf in the sense It also produces a symmetrical sample cdf in the sense that the same plotting positions will result from the that the same plotting positions will result from the data regardless of whether they are assembled in data regardless of whether they are assembled in ascending or descending order.ascending or descending order.

Gumbel Probability plottingGumbel Probability plotting A more sophisticated formulation is A more sophisticated formulation is

ppmm = (m – 0.3) / (n + 0.4) = (m – 0.3) / (n + 0.4)

This formulation approximates the median of the This formulation approximates the median of the distribution free estimate of the sample variate distribution free estimate of the sample variate to about 0.1% and, even for small values of n, to about 0.1% and, even for small values of n, produces parameter estimations comparable to produces parameter estimations comparable to the results obtained by the results obtained by maximum likelihood maximum likelihood estimationsestimations (Bury, 1999, p 43). (Bury, 1999, p 43).

Using the Gumbel Distribution to model Using the Gumbel Distribution to model extreme earthquakes extreme earthquakes

Cinna Lomnitz (1974) showed that if an homogeneous Cinna Lomnitz (1974) showed that if an homogeneous earthquake process with cumulative magnitude earthquake process with cumulative magnitude distributiondistribution

F(m; β) = 1 - eF(m; β) = 1 - e-βm-βm; m ≥ 0; m ≥ 0is assumed, whereis assumed, where

β is the inverse of the average magnitude of β is the inverse of the average magnitude of earthquakesearthquakes

in the region under consideration; and ifin the region under consideration; and ifα is the average number of earthquakes per year α is the average number of earthquakes per year

above magnitude 0.0above magnitude 0.0then y, the maximum annual earthquake magnitude, will then y, the maximum annual earthquake magnitude, will

be distributed according to the following Gumbel cdf.be distributed according to the following Gumbel cdf.

G(y; α, β) = exp(-α exp(-β y)); y ≥ 0G(y; α, β) = exp(-α exp(-β y)); y ≥ 0

Using the Gumbel Distribution to model Using the Gumbel Distribution to model extreme earthquakesextreme earthquakes

Using the PIT theorem, simulated Using the PIT theorem, simulated maximum yearly earthquakes can be maximum yearly earthquakes can be generated using the inversion generated using the inversion formulaformula

yyii = -(1/β) ln((1/α) ln(1/u = -(1/β) ln((1/α) ln(1/uii)) ))


The conversion factors to transform from The conversion factors to transform from the Bury to the Lomnitz versions of the the Bury to the Lomnitz versions of the Gumbel formulation are as follows.Gumbel formulation are as follows.

α = exp(µ/σ)α = exp(µ/σ)

β = 1/σβ = 1/σ Conversely:Conversely:

σ = 1/βσ = 1/β

µ = (1/β) ln(α) µ = (1/β) ln(α)


Manipulation of the Lomnitz formulation Manipulation of the Lomnitz formulation produces the following linear relationproduces the following linear relation

-ln(-ln(p-ln(-ln(pii)) )) = βy= βyii - ln(α) - ln(α)

where p represents the plotting position, where p represents the plotting position, and the left hand expression is the and the left hand expression is the reduced variate that can be used to plot reduced variate that can be used to plot earthquake data that are postulated as earthquake data that are postulated as being drawn from a Gumbel distribution.being drawn from a Gumbel distribution.

Demonstration of Gumbel Probability Demonstration of Gumbel Probability Plotting Plotting

Gumbel Probability Plot using i/(n+1) for the Gumbel Probability Plot using i/(n+1) for the plotting positionsplotting positions

Gumbel Plot using plotting position G(y) = j/(N+1) y = 1.3132x - 3.7419

R2 = 0.9987

-4

-2

0

2

4

6

8

0 2 4 6 8 10

Extreme Magnitude

Red

uce

d V

aria

te -

ln(-

ln(G

(y))

Demonstration of Gumbel Probability Demonstration of Gumbel Probability Plotting Plotting

Gumbel Probability Plot using (i-0.3)/(n+0.4)Gumbel Probability Plot using (i-0.3)/(n+0.4) for the for the plotting positionsplotting positions

Gumbel Plot using plotting position G(y) = (j - 0.3)/(N+0.4) y = 1.3199x - 3.7627

R2 = 0.9986

-4

-2

0

2

4

6

8

0 2 4 6 8 10

Extreme Magnitude

Red

uce

d V

aria

te -

ln(-

ln(G

(y))

Demonstration of Gumbel Probability Demonstration of Gumbel Probability PlottingPlotting

Parameter estimation

α β

pi=i/(n+1) pi=(i-0.3)

/(n+0.4)

pi=i/(n+1) pi=(i-0.3)

/(n+0.4)

Average 45.75 46.76 1.35 1.36

StdDev 3.49 3.60 0.03 0.03

StdError 1.10 1.14 0.0095 0.0095

Rel Error 2.4% 2.4% 0.7% 0.7%

ExactValue 48.00 48.00 1.37 1.37

Parameter estimations using Gumbel Probability Plotting

Demonstration of Gumbel Probability Demonstration of Gumbel Probability PlottingPlotting

It is evident that both plotting methods can It is evident that both plotting methods can estimate estimate αα and and β β within standard relative errors within standard relative errors of 2.4% and 0.7% respectively, of 2.4% and 0.7% respectively, if sufficient if sufficient trials are madetrials are made..

In real situations it may only be possible to In real situations it may only be possible to extract a single useful data set from the extract a single useful data set from the earthquake history. This will limit the precision earthquake history. This will limit the precision of parameter estimation in practice.of parameter estimation in practice.

For single estimations, For single estimations, there is a 95% there is a 95% confidence thatconfidence that αα and and ββ can be estimated can be estimated within two standard deviations of the averages within two standard deviations of the averages quoted in the previous table. That is, quoted in the previous table. That is, within within 15% and 5% respectively15% and 5% respectively..

Demonstration of Gutenberg-Richter Parameter Demonstration of Gutenberg-Richter Parameter estimation using the Gumbel distribution estimation using the Gumbel distribution

The relationships between the The relationships between the Lomnitz-Gumbel parameters α and β Lomnitz-Gumbel parameters α and β and the G-R parameter a and b are :and the G-R parameter a and b are :

ee-β-β = 10 = 10-b-b =>=> b = β logb = β log1010ee

α = 10α = 10aa =>=> a = loga = log1010α α

Simulated G-R CataloguesSimulated G-R Catalogues

Using the transform of the G-R distribution Using the transform of the G-R distribution with with

a = 1.69 and a = 1.69 and b = 0.59, b = 0.59,

eleven synthetic 131 year catalogues of eleven synthetic 131 year catalogues of earthquake events were generated.earthquake events were generated.

For each synthetic catalogue the annual For each synthetic catalogue the annual extreme magnitude events were isolated extreme magnitude events were isolated into data subsets and analysed using into data subsets and analysed using Gumbel plotting.Gumbel plotting.

Gumbel analysis of the G-R synthetic Gumbel analysis of the G-R synthetic cataloguescatalogues

The Gumbel analysis was done using the The Gumbel analysis was done using the ppmm = (m – 0.3) / (n + 0.4) plotting position = (m – 0.3) / (n + 0.4) plotting position formulation.formulation.

Two analyses were performed for each Two analyses were performed for each data subset:data subset:• Using the full data subset.Using the full data subset.• Using a partial data subset with the extreme Using a partial data subset with the extreme

of the extreme values censored.of the extreme values censored.

Gumbel analysis full data set Gumbel analysis full data set Gumbel plot of annual extreme events catalogue 01 y = 1.4207x - 4.1209

R2 = 0.9805

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7

Maximum annual magnitude (M)

Red

uce

d v

aria

te(-

LN

(-L

N((

i-0.

3)/(

N+

0.4)

)

Avg Rel Err

a 1.67 1.8%

b 0.59 1.7%

G-R Parameter estimation

Gumbel analysis truncated data set Gumbel analysis truncated data set

Gumbel plot of truncated annual extreme events catalogue 01 y = 1.2987x - 3.7658

R2 = 0.9939

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7

Maximum annual magnitude (M)

Red

uce

d v

aria

te(-

LN

(-L

N((

i-0.

3)/(

N+

0.4)

)

Avg Rel Err

a 1.69 2.4%

b 0.59 3.4%

G-R Parameter estimation

SummarySummary It has been demonstrated that analysis of multiple It has been demonstrated that analysis of multiple

synthetic earthquake catalogues, derived from a synthetic earthquake catalogues, derived from a Gumbel seismicity model, using Gumbel distribution Gumbel seismicity model, using Gumbel distribution plotting of annual extreme earthquake magnitudes, is plotting of annual extreme earthquake magnitudes, is capable of estimating the capable of estimating the a prioria priori α and β parameters α and β parameters values within a relative error of 2%.values within a relative error of 2%.

There is a 95% confidence that individual estimations of There is a 95% confidence that individual estimations of α and β will be within 15% and 5% respectively of the α and β will be within 15% and 5% respectively of the true value.true value.

Acceptable parameter estimates are obtained using Acceptable parameter estimates are obtained using either full annual extreme data sets, or truncated data either full annual extreme data sets, or truncated data sets with the extreme of the extreme values omitted sets with the extreme of the extreme values omitted from the data plot, but the full data set provides from the data plot, but the full data set provides smaller relative errors.smaller relative errors.

Questions?Questions?

validation of using gumbel probability plotting to estimate gutenberg-richter seismicity parameters...

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