Model types

Type hierarchy

The module defines the SeisModel type and subtypes of this specify the kind of model (i.e., symmetry, nature of basis function, etc.).

The current type hierarchy is as follows:

• SeisModel: Abstract type of all seismic models
  • SeisModel1D: Abstract type of all one-dimensonal (radially-symmetric) seismic models
    • SteppedLayeredModel: Type defined by layers with contstant properties.
    • LinearLayeredModel: Type defined by layers with constant gradient.
    • PREMPolyModel: Type defined by polynomials.

As you can see, there are currently three types of models implemented, all 1D models, with polynomial, linear or constant basis within each layer.

SteppedLayeredModel

A SteppedLayeredModel is composed of a series of layers with constant properties in each layer. The radii define the top of each layer.

An example of this is Weber et al.’s (2011) moon model:

SteppedLayeredModel <: SeisModel1D

See the constructor for details on creating SteppedLayeredModels.

LinearLayeredModel

A LinearLayeredModel is described by radial nodes, between which the properties vary linearly. Discontinutities are added by setting two subsequent nodes to have the same radii but different properties.

For example, a two-layer model for a planetesimal with radius 1 km and a liquid core at 0.3 km, with linearly varying properties, could be given by:

julia> using SeisModels

julia> m = LinearLayeredModel(r=[0, 0.3, 0.3, 1], vp=[1, 1, 1.8, 1], vs=[0, 0, 1, 0.7])
LinearLayeredModel(1.0, 4, [0.0, 0.3, 0.3, 1.0], [1.0, 1.0, 1.8, 1.0], [0.0, 0.0, 1.0, 0.7], Float64[], false, Float64[], Float64[], Float64[], Float64[], Float64[], false, Float64[], Float64[])

You can see that the nodes are used to define a constant lower layer and linearly-varying upper layer:

LinearLayeredModel <: SeisModel1D

See the constructor for details on creating LinearLayeredModels.

PREMPolyModel

PREMPolyModels are parameterised (like PREM) by polynomials in each layer, considering the normalised radius $x=r/a$, where $a$ is the surface radius. Clearly, SteppedLayeredModels and LinearLayeredModels are specific cases of a polynomial parameterisation.

For example, PREM’s outer core density in g/cm³ is given by $12.5815 - 1.2638x - 3.6426x^2 - 5.5281x^3\,.$

One could imagine a single-layer planet whose velocities are described by

where $V_\mathrm{S} = V_\mathrm{P}/\alpha$, and $\alpha = 1.7$.

julia> radii = [500]    # Layer tops in km
1-element Array{Int64,1}:
 500

julia> vps =   reshape( # Ensure we have a matrix with 1 column and 4 rows
               [3       # x^0 coefficients in km/s
                0       # x^1
               -2       # x^2
               -1],     # x^3
                :, 1)
4×1 Array{Int64,2}:
  3
  0
 -2
 -1

julia> α = 1.7
1.7

julia> vss = vps./α
4×1 Array{Float64,2}:
  1.7647058823529411
  0.0
 -1.1764705882352942
 -0.5882352941176471

julia> m = PREMPolyModel(r=radii, vp=vps, vs=vss)
PREMPolyModel(500.0, 1, [500.0], [3.0; 0.0; -2.0; -1.0], [1.7647058823529411; 0.0; -1.1764705882352942; -0.5882352941176471], Array{Float64}(undef,0,0), false, Array{Float64}(undef,0,0), Array{Float64}(undef,0,0), Array{Float64}(undef,0,0), Array{Float64}(undef,0,0), Array{Float64}(undef,0,0), false, Array{Float64}(undef,0,0), Array{Float64}(undef,0,0), NaN)

Note that the polynomial coefficients appear as the row of a matrix, and each layer occupies a column.

julia> using Plots: plot, plot!, vline!, savefig

julia> p = plot(xlabel="Radius (km)", ylabel="Velocity (km/s) or Density (g/cm^3)", title="PREMPolyModel example", framestyle=:box, grid=false)
Plot{Plots.GRBackend() n=0}

julia> for (f, name) in zip((vp, vs), ("Vp", "Vs"))
           plot!(p, r->f(m, r), 0, surface_radius(m), label=name)
       end

julia> vline!(p, m.r, line=(0.5,:dash), label="Layer tops")
Plot{Plots.GRBackend() n=3}

julia> savefig("PREMPolyModel_example.svg")

PREMPolyModel <: SeisModel1D

val_x = x[i,1] + (r/a)*x[i,2] + (r/a)^2*x[i,3] ... (r/a)^order*x[i,order+1]

See the constructor for details on creating PREMPolyModels.

Attenuation

Attenuation in PREMPolyModels is specified as in PREM (see equation (3) in section 6 on page 309). That means, the shear and bulk quality factors ($Q_\mu$ and $Q_\kappa$ respectively) control the effective velocities, which will vary by wave frequency.

By default, velocities for PREMPolyModels are returned at the reference frequency, which can be found with reffrequency and is given in Hz. To obtain velocities at other frequencies, just pass a target frequency to the relevant function via the freq keyword argument.

In this example, we ask for the Voigt average isotropic shear-wave velocity at radius 4500 km for a frequency of 0.01 Hz (equally, 100 s period):

julia> vs(PREM, 4500, freq=0.01)
6.840542771852348

Conversion

Conversion can be performed between any of the above model types. Note that in some cases (e.g., SteppedLayeredModel to LinearLayeredModel or LinearLayeredModel to PREMPolyModel) no information is lost, whilst in others (e.g., PREMPolyModel to LinearLayeredModel or to SteppedLayeredModel) the converted model can only be an approximation to the original.

To convert between model types, simply pass one model to the constructor of another. For example:

julia> m = PREM;

julia> m2 = LinearLayeredModel(PREM);

julia> typeof(m2)
LinearLayeredModel

julia> ≈(mass(m), mass(m2))
false

See the conversion section for details.