A method of spherical harmonic analysis in the geosciences via hierarchical Bayesian inference

The problem of decomposing irregular data on the sphere into a set of spherical harmonics is common in many fields of geosciences where it is necessary to build a quantitative understanding of a globally varying field. For example, in global seismology, a compressional or shear wave speed that emerges from tomographic images is used to interpret current state and composition of the mantle, and in geomagnetism, secular variation of magnetic field intensity measured at the surface is studied to better understand the changes in the Earth's core. Optimization methods are widely used for spherical harmonic analysis of irregular data, but they typically do not treat the dependence of the uncertainty estimates on the imposed regularization. This can cause significant difficulties in interpretation, especially when the best-fit model requires more variables as a result of underestimating data noise. Here, with the above limitations in mind, the problem of spherical harmonic expansion of irregular data is treated within the hierarchical Bayesian framework. The hierarchical approach significantly simplifies the problem by removing the need for regularization terms and user-supplied noise estimates. The use of the corrected Akaike Information Criterion for picking the optimal maximum degree of spherical harmonic expansion and the resulting spherical harmonic analyses are first illustrated on a noisy synthetic data set. Subsequently, the method is applied to two global data sets sensitive to the Earth's inner core and lowermost mantle, consisting of PKPab-df and PcP-P differential traveltime residuals relative to a spherically symmetric Earth model. The posterior probability distributions for each spherical harmonic coefficient are calculated via Markov Chain Monte Carlo sampling; the uncertainty obtained for the coefficients thus reflects the noise present in the real data and the imperfections in the spherical harmonic expansion

Transdimensional inversion of receiver functions and surface wave dispersion

International audienceWe present a novel method for joint inversion of receiver functions and surface wave dispersion data, using a transdimensional Bayesian formulation. This class of algorithm treats the number of model parameters (e.g. number of layers) as an unknown in the problem. The dimension of the model space is variable and a Markov chain Monte Carlo (McMC) scheme is used to provide a parsimonious solution that fully quantifies the degree of knowledge one has about seismic structure (i.e constraints on the model, resolution, and trade-offs). The level of data noise (i.e. the covariance matrix of data errors) effectively controls the information recoverable from the data and here it naturally determines the complexity of the model (i.e. the number of model parameters). However, it is often difficult to quantify the data noise appropriately, particularly in the case of seismic waveform inversion where data errors are correlated. Here we address the issue of noise estimation using an extended Hierarchical Bayesian formulation, which allows both the variance and covariance of data noise to be treated as unknowns in the inversion. In this way it is possible to let the data infer the appropriate level of data fit. In the context of joint inversions, assessment of uncertainty for different data types becomes crucial in the evaluation of the misfit function. We show that the Hierarchical Bayes procedure is a powerful tool in this situation, because it is able to evaluate the level of information brought by different data types in the misfit, thus removing the arbitrary choice of weighting factors. After illustrating the method with synthetic tests, a real data application is shown where teleseismic receiver functions and ambient noise surface wave dispersion measurements from the WOMBAT array (South-East Australia) are jointly inverted to provide a probabilistic 1D model of shear-wave velocity beneath a given station

Long-wavelength topography and multiscale velocity heterogeneity at the core-mantle boundary

The structure of the lowermost mantle and the core-mantle boundary (CMB) has profound implications for Earth's evolution and current-day dynamics. Whilst tomographic studies of VS show good agreement in the lowermost mantle, consensus as to VP and especially CMB radius has not yet been reached. We perform a hierarchical Bayesian inversion for VP in the lowermost 300 km of the mantle and the radius of the CMB using differential travel time data. Concurrent with finding VP perturbations of 0.56% RMS amplitude that spatially agree with previous studies in areas of low posterior variance, we find 4.5 km RMS amplitude CMB radius perturbations with a broadly north-south hemispherical character, with spherical harmonic power evenly distributed between degrees 1–3. These results suggest that CMB radial processes are set by a longer scale process than the VP perturbations

Testing the limits of virtual deep seismic sounding via new crustal thickness estimates of the Australian continent

© 2019 The Author(s) . We apply virtual deep seismic sounding (VDSS) to data collected from both permanent and temporary seismic stations in Australia with the goal of examining (i) the resilience of the method to the presence of complex lithospheric structure and (ii) the effectiveness of different approaches for estimating bulk crustal properties (namely thickness and Vp). Data from the permanent station WRAB in the Northern Territory is ideal for benchmarking VDSS (large number and favourable distribution of recorded earthquakes), with the results from several approaches agreeing on a thickness of 40-42 km. Application of VDSS to data from the temporary BILBY array, a linear distribution of broadband stations that traverses central Australia, shows that strong Moho reflections can be retrieved with as few as two earthquakes even at the transition between crustal blocks of different character and in the presence of thick sedimentary basins. Crustal thickness varies between 36 and 54 km and compares well with the reflectivity character of nearby deep seismic reflection lines. Furthermore, we find that off-line estimates of crustal thickness, calculated by binning the source regions according to back-Azimuth, produce values of crustal thickness that are consistent with the regional geology. Overall, we find that VDSS is a powerful technique for estimating crustal thickness and velocity due to its insensitivity to complex short-wavelength structure and requirement of a small number earthquakes to produce a stable result. However, not all schemes tested for extracting bulk crustal properties appear to be robust and stringent data quality checking is still required during implementation

Testing the limits of virtual deep seismic sounding via new crustal thickness estimates of the Australian continent

© 2019 The Author(s) . We apply virtual deep seismic sounding (VDSS) to data collected from both permanent and temporary seismic stations in Australia with the goal of examining (i) the resilience of the method to the presence of complex lithospheric structure and (ii) the effectiveness of different approaches for estimating bulk crustal properties (namely thickness and Vp). Data from the permanent station WRAB in the Northern Territory is ideal for benchmarking VDSS (large number and favourable distribution of recorded earthquakes), with the results from several approaches agreeing on a thickness of 40-42 km. Application of VDSS to data from the temporary BILBY array, a linear distribution of broadband stations that traverses central Australia, shows that strong Moho reflections can be retrieved with as few as two earthquakes even at the transition between crustal blocks of different character and in the presence of thick sedimentary basins. Crustal thickness varies between 36 and 54 km and compares well with the reflectivity character of nearby deep seismic reflection lines. Furthermore, we find that off-line estimates of crustal thickness, calculated by binning the source regions according to back-Azimuth, produce values of crustal thickness that are consistent with the regional geology. Overall, we find that VDSS is a powerful technique for estimating crustal thickness and velocity due to its insensitivity to complex short-wavelength structure and requirement of a small number earthquakes to produce a stable result. However, not all schemes tested for extracting bulk crustal properties appear to be robust and stringent data quality checking is still required during implementation

Candy wrapper for the Earth's inner core

Recent global expansion of seismic data motivated a number of seismological studies of the Earth’s inner core that proposed the existence of increasingly complex structure and anisotropy. In the meantime, new hypotheses of dynamic mechanisms have been put forward to interpret seismological results. Here, the nature of hemispherical dichotomy and anisotropy is re-investigated by bridging the observations of PKP(bc-df) differential travel-times with the iron bcc/hcp elastic properties computed from first-principles methods.The Candy Wrapper velocity model introduced here accounts for a dynamic picture of the inner core (i.e., the eastward drift of material), where different iron crystal shapes can be stabilized at the two hemispheres. We show that seismological data are best explained by a rather complicated, mosaic-like, structure of the inner core, where well-separated patches of different iron crystals compose the anisotropic western hemispherical region, and a conglomerate of almost indistinguishable iron phases builds-up the weakly anisotropic eastern side