On the Detectability and Use of Normal Modes for Determining Interior Structure of Mars

Abstract

The InSight mission to Mars is well underway and will be the first mission to acquire seismic data from a planet other than Earth. In order to maximise the science return of the InSight data, a multifaceted approach will be needed that seeks to investigate the seismic data from a series of different frequency windows, including body waves, surface waves, and normal modes. Here, we present a methodology based on globally-averaged models that employs the long-period information encoded in the seismic data by looking for fundamental-mode spheroidal oscillations. From a preliminary analysis of the expected signal-to-noise ratio, we find that normal modes should be detectable during nighttime in the frequency range 5–15 mHz. For improved picking of (fundamental) normal modes, we show first that those are equally spaced between 5–15 mHz and then show how this spectral spacing, obtained through autocorrelation of the Fourier-transformed time series can be further employed to select normal mode peaks more consistently. Based on this set of normal-mode spectral frequencies, we proceed to show how this data set can be inverted for globally-averaged models of interior structure (to a depth of $\sim 250~\mbox{km}$), while simultaneously using the resultant synthetically-approximated normal mode peaks to verify the initial peak selection. This procedure can be applied iteratively to produce a “cleaned-up” set of spectral peaks that are ultimately inverted for a “final” interior-structure model. To investigate the effect of three-dimensional (3D) structure on normal mode spectra, we constructed a 3D model of Mars that includes variations in surface and Moho topography and lateral variations in mantle structure and employed this model to compute full 3D waveforms. The resultant time series are converted to spectra and the inter-station variation hereof is compared to the variation in spectra computed using different 1D models. The comparison shows that 3D effects are less significant than the variation incurred by the difference in radial models, which suggests that our 1D approach represents an adequate approximation of the global average structure of Mars.

Appendices

Appendix A: “Benchmark Inversion”

In the method proposed here, the computation of synthetic spectra relies on an approximation and only serves as a tool for fitting eigenfrequencies. This principally arises because of lack of knowledge of the seismic source. To quantitatively test the proposed method, we shall assume that source parameters (location and mechanism) are perfectly known and proceed to compute the “true” amplitude- and phase-spectrum. This “real spectrum” can be inverted for interior structure and the results compared to those obtained from our approximate method. For the inversion of amplitude and phase, we rely on the Bayesian approach used earlier, but 1) adjust the forward routine by including Yspec for the computation of time series for each new sampled model and 2) re-formulate the likelihood function as follows

$$ {\mathcal{L}}(\mathbf{m})\propto\exp \biggl(-\sum_{i} \biggl(\frac {|{A_{\mathrm{obs}}}(\omega_{i})-A_{\mathrm{syn}}(\omega _{i})|^{2}}{2(\sigma^{A}_{i})^{2}}+\frac{|\phi^{\vee }_{i}|^{2}}{2(\sigma^{\phi}_{i})^{2}} \biggr) \biggr) $$

(5)

where $A_{\mathrm{obs}}$ and $A_{\mathrm{syn}}$ denote observed and synthetic amplitude, respectively, and $\phi^{\vee}$ the angle between observed and synthetic phase, evaluated at the frequency $\omega_{i}$ of the observed peaks. Since all peaks are considered, this automatically includes overtones and noise alongside fundamental-mode peaks. For the purposes of this test, uncertainty in amplitude ($\sigma^{A}$) is set to 30% and uncertainty in phase ($\sigma^{\phi}$) to $\pm\,0.1\,\pi$. Inverting amplitude and phase for the same parameters (Moho depth and lithosphere thickness and temperature) as done previously (Sect. 5) results in the sampled posterior distributions for model and data parameters displayed in Fig. 14A–E. To ensure adequate coverage of the model space, multiple inversions starting with different initial models were run in parallel. Relative to previous results (Fig. 10A–C), sampled model variances are smaller and simply reflects inversion of a larger dataset, i.e., more information. Nevertheless, the results are generally in good agreement with those obtained from the approximate method and attests to proper performance of the latter. In addition, our approximate method is computationally much cheaper ($\sim 0.85~\mbox{s}$ for one forward routine on a single CPU) than the “full” approach ($\sim 200~\mbox{s}$ for one forward routine on a single CPU).

Fig. 14

Inversion results using full amplitude and phase information. (A)–(C) sampled prior and posterior distributions for the main crustal and lithospheric parameters of interest (temperature at the bottom of the lithosphere, Moho thickness, lithospheric thickness). (D)–(F) Corresponding radial profiles of density, P- and S-wave speed. (G)–(H) fitted amplitude (D) and phase (E) spectra. Misfit is evaluated at the frequencies indicated by the red dots

Fig. 15

Slice through a three-dimensional seismic model showing lateral variations in S-wave speed in the crust at a depth of 54 km

Appendix B: Additional 3D–1D Comparison

To additionally test the importance of 3D effects, we analyzed a second model with stronger lateral variations (S-wave speed of ±5%) in the crust following Khan et al. (2018) on top of a 1D mantle. In this model lateral variations in the mantle are neglected because these are unlikely to be seen given the large crustal perturbations combined with decreased sensitivity at greater depth. In the crust, the ratio of S- to P-wave speed is 0.55 and the ratio of density to P-wave speed is 1.2. Repeating the analysis outlined in Sect. 6.3, 3D and 1D spectra are compared in Fig. 16A–E. While the quality of the spectra is generally worse in comparison to the previous 3D model, it is nonetheless possible to pick $\Delta f$ and select fundamental-mode peaks across 20 different stations. As in the previous analysis, $\Delta f$ varies between 0.169 and 0.172 and is thus of the same order as observed previously. This suggests that for the expected long-wavelength 3D variability, peak frequency variability is less important than the variability due to the range of possible 1D models.

Fig. 16

(A) frequency variation among 3D spectra determined at a number of stations across the planet for the model in Fig. 15; (B) zoom-in of (A). (C) frequency variation among 1D spectra at a single station but different radial models (see Fig. 3); (D) zoom-in of (C). Vertical lines and dots in (B) and (D) indicate selected spectral peaks. Spectra are rescaled for comparison. (E) comparison of spectral variation based on globally-averaged models (Fig. 3) and a three-dimensional model (Fig. 15) as a function of frequency. The 3D frequency variation is determined by investigating how the frequency changes across a number of stations distributed over the planet based on the model in Fig. 15; the 1D frequency variation is obtained by considering the frequency change at a single station but different radial models (Fig. 3)

Appendix C: Description of Supplementary Material

Additional material can be found in the Online Resource of this article. It contains figures showing 1) additionally investigated radial models, 2) corresponding dispersion curves and spacing of fundamental modes ($\Delta f$), and 3) comparison of estimated and theoretical $\Delta f$.