The direct spectral element method for the calculation of synthetic seismograms in self-gravitating, spherically symmetric planets

Abstract

This paper describes the implementation of the direct solution method (DSM) using radial spectral elements for the calculation of synthetic seismograms in self-gravitating, spherically symmetric, non-rotating, anelastic, and transversely isotropic Earth models. In contrast to previous implementations of the DSM that used a potential formulation within fluid regions, we use a displacement formulation throughout. It is this feature that allows us to extend the DSM to account fully for self-gravitation along with arbitrary fluid stratification. Our code, $\texttt{DSpecM1D}$, is benchmarked against the normal mode summation code $\texttt{MINEOS}$ as well as the direct radial integration code $\texttt{YSpec}$. Agreement between the codes is excellent for both elastic and anelastic models.

Figures (6)

• Figure 1: Comparison of vertical displacement spectra at location $(80^\circ \text{\,N}, 0^{\circ} \text{\,E})$ using the Bolivia 1994 source, for a 100 h time-series, up to maximum degree 100. The model is transversely isotropic, non-dispersive and non-attenuating PREM, without an ocean layer. DSpecM1D (red dashed line) is compared with the results from the direct radial integration method (YSpec, blue line). Between the two codes, the average relative misfit is 0.007%, the maximum relative misfit is 0.32%. In the inset axes we analyse more closely two portions of the spectra. The black dashed line in the inset axes indicates the frequency for the modes as computed by MINEOS, the corresponding modes are indicated above the peaks.
• Figure 2: Comparison of displacement traces at $(80.0^{\circ} \text{\,N}, 0.0^{\circ} \text{\,E})$ for the Bolivia 1994 source, in a non-attenuating, non-dispersive, transversely isotropic PREM without an ocean layer. Calculated for all modes between 1.5 mHz and 21mHz, maximum degree 300. A half-cosine taper was applied with corner frequencies $(1.9,2.0)$ and $(20.0,20.1)$ mHz. The average relative differences between DSpecM1D and YSpec (MINEOS) are written on each plot in black (green), for each component respectively. The maximum relative difference (across all three components) was 0.4% for YSpec and 7.3% for MINEOS.
• Figure 3: Comparison of acceleration traces at TLY station, located at $(51.68^{\circ} \text{\,N}, 103.64^{\circ} \text{\,E})$, for the China source, in an attenuating, dispersive, transversely isotropic PREM without an ocean layer. Calculated for all modes up to 51mHz, maximum degree 750. A half-cosine taper was applied with corner frequencies $(0.1,0.2)$ and $(50.0,50.5)$ mHz. The average relative differences between DSpecM1D and YSpec (MINEOS) are written on each plot in black (green), for each component respectively. The maximum relative difference (across all three components) was 1.5% for YSpec and 3.3% for MINEOS.
• Figure 4: Record section for vertical and northward motion, up to 500 mHz, maximum degree 7500. The Earthquake source is the moment tensor for the Bolivia 1994 Earthquake, placed at $(0^{\circ} \text{\,N}, 0^{\circ} \text{\,E})$. A half-cosine taper applied with corner frequencies $(0.1,5.0)$ and $(450.0,500.0)$ mHz. Major seismic phase arrival times are given by the red lines, as calculated by TauP crotwell1999taup.
• Figure 5: Average relative error computed in the time domain for a 500 minute time-series, compared to the exact solution as a function of element size. The time series is computed at station TLY for the Bolivia 1994 Earthquake. The relative error is calculated using the formula in the text. Maximum frequency is 50mHz, up to degree 750. The half-cosine taper was applied with corner frequencies $(0.1, 0.15)$, $(45.0, 50.0)$ mHz.