Frequency-Magnitude Distributions

GRclassic(b, M0)

The classic Gutenberg-Richter law can be expressed as an exponential distribution with rate β and location M0:

\[f(x; \beta, M_{0}) = \beta e^{-\beta (x-M_{0})}, \quad x ≥ M_{0}\]

M0 is the minimum magnitude. β and the b-value b are linked by the following relation: $\beta = blog(10)$.

GRclassic(1.0, 0.5)  # distribution with a b-value of 1 and minimum magnitude of 0.5
bvalue(d)     #  get the b-value
rate(d)       #  get β
location(d)   #  get M0
params(d)     #  get the parameters (b, M0)

fit_mle(D::Type{<:GRclassic}, mags, M0)

Use the maximum likelihood method to determine the GRclassic distribution fitting the input data. The function calculates the b-value with the method of Aki-Utsu.

Arguments

• D: distribution type (see GRclassic(b, M0)).
• mags: array storing magnitude values greater than or equal to the magnitude of completeness M0.
• M0: magnitude of completeness. See Magnitude of completeness for methods that can be used to estimate it.

The standard error affecting the b-value can be estimated with fit_stderr(D::Type{<:GRclassic}, mags)

References

• Aki, 1965. Maximum Likelihood Estimate of b in the Formula logN=a-bM and its Confidence Limits. Belletin of the Earthquake Research Institute, 43, 237-239.
• Utsu 1966. A Statistical Significance Test of the Difference in b-value between Two Earthquake Groups. Journal of Physics of the Earth, 14, 37-40.

fit_binned(D::Type{<:GRclassic}, mags, M0, ΔM)

Determine the GRclassic distribution fitting the input data. This function corresponds to the method of Tinti and Mulargia, which was designed to properly estimate the b-value from binned magnitude values.

Arguments

• D: distribution type (see GRclassic(b, M0)).
• mags: array storing binned magnitude values greater than or equal to the magnitude of completeness M0
• M0: magnitude of completeness. See Magnitude of completeness for methods that can be used to estimate it.
• ΔM: bin width used to group the magnitude values.`

The standard error affecting the b-value can be estimated with fit_stderrbinned(D::Type{<:GRclassic}, mags, ΔM)

References

• Tinti and Mulargia, 1987. Confidence intervals of b values for grouped magnitudes. Bulletin of the Seismological Society of America, 77 (6), 2125–2134.

fit_stderr(D::Type{<:GRclassic}, mags)

Apply the method of Shi and Bolt to calculate the standard error affecting a b-value determined with the method of Aki-Utsu fit_mle(D::Type{<:GRclassic}, mags, M0).

Arguments

• D: distribution type (see GRclassic(b, M0)).
• mags: array storing magnitude values greater than or equal to the magnitude of completeness.

References

• Shi and Bolt, 1982. The standard error of the magnitude-frequency b value. Bulletin of the Seismological Society of America, 72, 1677-1687.

fit_stdbinned(D::Type{<:GRclassic}, mags, ΔM)

Calculate the standard error affecting a b-value estimated with the method of Tinti and Mulargia (fit_binned(D::Type{<:GRclassic}, mags, M0, ΔM)).

Arguments

• D: distribution type (see GRclassic(b, M0)).
• mags: array storing magnitude values greater than or equal to the magnitude of completeness.
• ΔM: bin width used to group the magnitude values.`

References

• Tinti and Mulargia, 1987. Confidence intervals of b values for grouped magnitudes. Bulletin of the Seismological Society of America, 77 (6), 2125–2134.

rand(rng::AbstractRNG, d::UnivariateDistribution)

Generate a scalar sample from d. The general fallback is quantile(d, rand()).

params(d::UnivariateDistribution)

Return a tuple of parameters. Let d be a distribution of type D, then D(params(d)...) will construct exactly the same distribution as $d$.

scale(d::UnivariateDistribution)

Get the scale parameter.

location(d::UnivariateDistribution)

Get the location parameter.

rate(d::UnivariateDistribution)

Get the rate parameter.

bvalue(::UnivariateDistribution)

Get the b-value

pdf(d::UnivariateDistribution, x::Real)

Evaluate the probability density (mass) at x.

See also: logpdf.

logpdf(d::UnivariateDistribution, x::Real)

Evaluate the logarithm of probability density (mass) at x.

See also: pdf.

cdf(d::UnivariateDistribution, x::Real)

Evaluate the cumulative probability at x.

See also ccdf, logcdf, and logccdf.

logcdf(d::UnivariateDistribution, x::Real)

The logarithm of the cumulative function value(s) evaluated at x, i.e. log(cdf(x)).

ccdf(d::UnivariateDistribution, x::Real)

The complementary cumulative function evaluated at x, i.e. 1 - cdf(d, x).

logccdf(d::UnivariateDistribution, x::Real)

The logarithm of the complementary cumulative function values evaluated at x, i.e. log(ccdf(x)).

avalue(::UnivariateDistribution, N::Real, ΔM::Real)

Get the a-value of a frequency-magnitude distribution composed of N events grouped into magnitude bins of width ΔM.