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Fourier Source Spectrum

The source spectrum can be computed through interaction with the SourceParameters type, or the higher level FourierParameters type that holds a SourceParameters instance as a property.

The source spectrum $E(f; \bm{\theta}E)$ is most commonly written in terms of:

\[ E(f; \bm{\theta}_E) = \mathcal{C} M_0 E_s(f; \bm{\theta}_E)\]

where $\mathcal{C}$ is a constant term, to be defined shortly, $M_0$ is the seismic moment, and $E_s(f; \bm{\theta}_E)$ is the source spectral shape.

The most commonly adopted source spectral shape is the $\omega^2$ model that has the form:

\[ E_s(f) = \frac{1}{1 + \left(\frac{f}{f_c}\right)^2}\]

Functionality

StochasticGroundMotionSimulation.fourier_constant — Function
fourier_constant(src::SourceParameters)

Define the constant source term for the Fourier Amplitude Spectrum. The value provided corresponds to Fourier displacement units of cm-s. Constant set to permit distances to be passed in km, densities in t/m^3, and velocities in km/s. The reference distance is set to 1.0 km (and interpreted to be a rupture distance).

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StochasticGroundMotionSimulation.fourier_source_shape — Function
fourier_source_shape(f::Float64, fa::T, fb::T, ε::T, src::SourceParameters) where T<:Real

Fourier amplitude spectral shape for displacement defined by corner frequencies.

See also: fourier_source_shape

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fourier_source_shape(f::Float64, fa::T, fb::T, ε::T, fas::FourierParameters) where T<:Real

Fourier amplitude spectral shape for displacement defined by corner frequencies.

See also: fourier_source_shape

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fourier_source_shape(f::T, m::S, src::SourceParameters) where {S<:Real,T<:Float64}

Source shape of the Fourier Amplitude Spectrum of displacement, without the constant term or seismic moment. This simply includes the source spectral shape.

The nature of the source spectral shape depends upon src.model:

  • :Brune gives the single corner omega-squared spectrum (this is the default)
  • :Atkinson_Silva_2000 gives the double corner spectrum of Atkinson & Silva (2000)
  • :Beresnev_2019 gives a single-corner spectrum with arbitrary fall off rate related to src.n from Beresnev (2019)
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fourier_source_shape(f, m, fas::FourierParameters)

Source shape of the Fourier amplitude spectrum of displacement, without the constant term or seismic moment. Defined using a FourierParameters instance for the source model.

See also: fourier_source_shape

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fourier_source_shape(f::Vector{S}, fa::T, fb::T, ε::T, src::SourceParameters) where {S<:Real,T<:Real}

Fourier amplitude spectral shape for displacement defined by corner frequencies.

See also: fourier_source_shape

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fourier_source_shape(f::Vector{T}, m::S, src::SourceParameters) where {S<:Real,T<:Float64}

Source shape of the Fourier Amplitude Spectrum of displacement, without the constant term or seismic moment. This simply includes the source spectral shape.

The nature of the source spectral shape depends upon src.model:

  • :Brune gives the single corner omega-squared spectrum (this is the default)
  • :Atkinson_Silva_2000 gives the double corner spectrum of Atkinson & Silva (2000)
  • :Beresnev_2019 gives a single-corner spectrum with arbitrary fall off rate related to src.n from Beresnev (2019)
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StochasticGroundMotionSimulation.fourier_source — Function
fourier_source(f::T, m::S, src::SourceParameters) where {S<:Real,T<:Float64}

Source Fourier Amplitude Spectrum of displacement, without the constant term. This simply includes the seismic moment and the source spectral shape.

See also: fourier_source_shape

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fourier_source(f, m, fas::FourierParameters)

Source Fourier Amplitude Spectrum of displacement, without the constant term. This simply includes the seismic moment and the source spectral shape. Defined using a FourierParameters instance for the source model.

See also: fourier_source_shape

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StochasticGroundMotionSimulation.corner_frequency — Function
corner_frequency(m::U, src::SourceParameters{S,T}) where {S<:Float64, T<:Real, U<:Real}

Computes a 3-tuple of corner frequency components, depending upon source spectrum type. By default the single-corner Brune spectrum is considered, but if src.model equals :Atkinson_Silva_2000 then the components of the double-corner spectrum are returned. If some other symbol is passed then the Brune model is returned.

Examples

    m = 6.0
    Δσ = 100.0
    κ0 = 0.035
    fas = FASParams(Δσ, κ0)
    # compute single corner frequency
	src.model = :Brune
    fc, tmp1, tmp2 = corner_frequency(m, src)
    # compute double corner frequencies
	src.model = :Atkinson_Silva_2000
    fa, fb, ε = corner_frequency(m, src)
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