REFERENCE MODELS

Introduction

To a very good first order approximation, the earth is made up of concentric spherical shells, and its average internal structure can be described in terms of variations of properties as a function only of depth. Measurements of travel times and amplitudes of seismic waves generated by large earthquakes and observed at distant stations contain information about the elastic and anelastic properties of the medium through which they travel. These measurements can be used to build models of seismic velocity, density, and attenuation structure with depth through mathematical inversion. Earth properties are observed to change laterally and vertically, but change most strongly with depth (lateral velocity variations are at most 10% laterally compared with 500% vertically) thus Earth structure can be well approximated by a 1-dimensional (spherically symmetric) model of elastic velocities, density, and attenuation as a function of depth (Fig. 1). Different 1D reference seismological earth models have been constructed using different data types, different parametrisation, and computation procedures. Knowing what choices have been made in the construction of a 1D seismological reference earth model is important when one wants to use such a model, depending on the application.

Figure 1 – 1-Dimensional Earth structure and velocities for the Preliminary Reference Earth Model or PREM (Dziewonski and Anderson, 1981). a. Radial Earth structure showing the major divisions. Solid lines mark discontinuities due to phase changes or compositional changes. b. Corresponding depth-dependent velocity structure for compressional and shear-waves (Vp and Vs, respectively). c. Depth-dependent density structure.

The differing sensitivities of the seismic observables used and the scatter in the measurements caused by lateral velocity structure (Fig. 2) mean that the resulting velocity structure is only an approximation of the true Earth and not representative of any real physical structure, mineral assemblage, or location. Caution must be taken when 1D reference models are used to interpret results from other disciplines, such as measurements or calculations of elastic properties of different mineral assemblages.

Figure 2. IASP91 travel-time curves with travel-times measured from test events used to verify the model. Velocity varies most strongly with depth, thus arrival times roughly follow curves with distance for each wave. However, despite the source parameters of the test events used here being well known, the scatter in arrival times demonstrates that a 1-Dimensional model is only an approximate representation of the Earth. Figure 6 from Kennett and Engdahl (1991).

While reference models are approximations of the real Earth, they are needed for, among others, the following purposes:

• Determining earthquake locations, which involves converting times to distances and thus requires an understanding of wave propagation velocities.
• Identifying different kinds of seismic waves (phases) on seismic records is guided by calculations of predicted travel-times
• Predicting ray paths requires knowing the velocity structure
• Modelling the propagation of waves from earthquakes requires an understanding of the velocity structure
• Construction of 3D seismic models requires a starting model for the inversion process. Indeed, most of the non-linearity resides in the 1D model.
• Reference models are used to forward model standard travel-times and waveforms against which to compare observations to help identify variability
• Providing a reference for the interpretation of mineral physics experimental and computational data
• Providing a reference structure to inform dynamic modelling

Many current seismological studies rely upon reference models. While global 3D models of the Earth's mantle have been developed for over 30 years (e.g. Dziewonski et al., 1977, Dziewonski, 1984), 1D models are still commonly used.

Historical background

The earliest representation of Earth structure was a simple layered model (Table 1) derived from travel-time curves displaying how the time taken for waves to arrive following an earthquake varied with distance e.g. Gutenberg or the Jeffreys-Bullen Tables (Jeffreys 1940, Bullen 1942). These tables were compiled from observations from 1930 to 1939 when the global network of seismometers was very limited. Nonetheless, the International Seismological Centre used the Jeffreys-Bullen tables in their earthquake location process until 1991 (Kennett and Engdahl, 1991).

TABLE1 Table 1: Regions in Jeffreys-Bullen earth model. After Stein and Wysession (2009).

Later compilations of body wave travel-time tables determined that the Jeffreys-Bullen tables were 2-4 seconds slow and that, due to strong upper mantle heterogeneity, velocities at short distances needed revision (Herrin et al., 1968).

Constructing a model

Constructing a reference model requires a number of choices that depend on: the type of data, computational ease, and the goal of the model. Models are calculated from different datasets and frequencies and may be represented in different mathematical ways. Therefore, different models do not all represent the Earth structure in the same way.

Models are commonly constructed using either long period or short period teleseismic data (Fig. 3). Short period data are sensitive to only small volumes of the Earth thus can be biased by sampling, but can carry precise information about the Earth and are sensitive to the deep Earth. Meanwhile, long period waves are sensitive to larger volumes or, in the case of normal modes, the whole Earth. The frequencies of seismic data used to construct the model affect how the resulting model should be used. Seismic data can be supplemented by other measurements, such as astronomical data. See Table 2 and Data section for more detail.

Figure 3 – Seismic data types used to construct reference models. a. Body waves travel through the Earth, therefore, they can sample all depths although lateral resolution is limited by the location of sources and receivers. Body waves are relatively short period (~1s for P waves, ~10-30 s for S waves) and sensitive to the structure along narrow ray paths (only the narrow region that the wave travels through). b. Surface waves are trapped at the surface of the Earth thus are most sensitive to shallower structure, although longer period surface waves can resolve deeper structure. Surface waves typically have periods of 20 s to 250 s, their wavelengths are longer and so they sample broader regions. They provide sampling of the upper mantle in oceanic basins, which are unresolved by body waves due to the lack of seismic stations in the oceans. c. Normal modes (also know as Free Oscillations) are whole earth oscillations where the whole Earth deforms at some harmonic order at very long periods. Normal modes result from the superposition of surface waves and are either twisting (toroidal modes) or undulating motions (spheroidal modes). They are sensitive to whole earth structure (including density) with little spatial bias but have limited depth resolution. From Stein and Wysession (2009) (a and b) and IRIS website (c).

TABLE2 Table 2 – Data type summary. More information in Data Type section.

As sampling of the Earth is not geographically even with the vast majority of seismometers being on the mainland, models relying on body waves are often inherently biased towards structure beneath the continents. Meanwhile, normal mode data record whole earth structure without geographic bias. Model construction also involves the decision of where to place the dividing layers and discontinuities, and which to include. Data guides the depth of discontinuities and layers, but models may include different numbers of layers based on what is observed in the data, the purpose of the model, and the choice of the modellers.

Reference models can be defined in different ways, and are often parameterised as mathematical functions e.g. the core velocities of the model IASP91 (Kennett and Engdahl, 1991) are described by a quadratic polynomial. While mathematical parameters likely do not represent the Earth, such parameterisations can allow models to be interpolated at different depths more accurately.

As well as showing lateral and vertical heterogeneities, the Earth is anelastic and anisotropic. Seismic waves are attenuated as they travel through the Earth, which affects observed amplitudes, and also makes wave velocity frequency dependent, which is known as dispersion (Fig. 4). Anisotropy causes waves to travel at different speeds dependent on their direction of travel or their polarization (Fig. 5). These properties are necessary features of reference models to properly represent seismic data. The first 1D reference model to include a realistic representation of attenuation, as well as radial anisotropy down to 200 km depth was PREM (Dziewonski and Anderson, 1981).

Figure 4 – The effect of anelasticity (attenuation) on wavespeed for a ‘standard linear solid’. This anelastic dispersion calculation demonstrates the need to account for anelasticity when comparing velocities across frequencies. a. Internal friction (attenuation) as a function of frequency for a band-limited "constant Q" model. b. Group and Phase velocities vary as a function of frequency due to attenuation c. Corresponding surface wave attenuation factor (energy lost per cycle) as a function of frequency demonstrating that the highest frequencies are the most strongly attenuated. From Liu et al., (1976).