On the Limitation of Receiver Functions Method: Beyond Conventional Assumptions & Advanced Inversion Techniques Hrvoje Tkalčić RSES, ANU Acknowledgment: RSES Seismology & Mathematical Geophysics Group members
Some passive-source receiverbased methods Teleseismic travel times Teleseismic receiver functions (P and S) Body wave earthquake interferometry Ambient noise dispersion Interstation method surface wave dispersion..
Basin structure and geometry from nuclear blasts waveforms and teleseismic travel times Rodgers et al. PAGEOPH 26; Tkalčić et al., BSSA 28
The benefits of RFs + SWD Earth Vs structure can be inverted using: 1. Receiver functions (RF) 2. Surface wave dispersion curves 3. RF + dispersion curves (jointly) or other datasets Different approaches to modeling Forward modeling Linearized inversion Grid-search Non-linear inversion with optimization Multi-step approach
IRFFM (Interactive RF Forward Modeling) Multi-step approach Linearized Inversion Tkalčić et al., JGR 26 Chen et al., JGR 21 Stipčević et al., GJI 211 Tkalčić et al., GJI 211; GJI 212
Lithospheric structure of Saudi Arabia, China, Australia & Croatia from multi-step modeling of RFs and SWs SE China (RFs combined with tomography) Croatia and Adriatic Sea SE Australia (RFs combined with ambient noise)
Advantages and limitations of Advantages RFs A way to invert for Vs structure under a single station Sensitive to gradients (discontinuities) in Vs velocity A needed complement to crustal tomography RF + SW dispersion curves (jointly) or other datasets Limitations of conventional methods Information limited to a volume beneath a single station Insensitive to absolute velocity unless SW are added Simplifications/assumptions often cannot explain real Earth (1. lack of data, 2. anisotropy, 3. dipping layers, 4. non-uniqueness and noise in the data)
1. Exploiting seismic signal and noise in an aseismic environment to constrain crustal structure c. CAPRA station CP8 Initial IRFFM2 model RF only inversion Best models from joint inversion Preferred model from joint inversion Observed RF 5 1 15 Time (s) Depth (km) 1 2 3 4 Velocity (km/s) 4.5 4. 3.5 3. Group velocity data from noise Phase velocity data from interstation method Group velocity dispersion curve from RF modelling Phase velocity dispersion curve from RF modelling Group velocity dispersion curve from joint modelling Phase velocity dispersion curve from joint modelling 3. 3.5 4. 4.5 Velocity (km/s) d. CAPRA station CP3 2.5 1 2 5 1 2 5 Period (s) Initial IRFFM model average model: RF analysis Best models: joint inversion Final average model: joint inversion 5 1 15 time Young et al., GJI 212 depth 1 2 3 4 3. 3.5 4. 4.5 velocity velocity 4.5 4.4 4.3 4.2 4.1 4. 3.9 3.8 3.7 3.6 3.5 3.4 3.3 3.2 3.1 3. 2.9 2.8 2.7 2.6 2.5 Group velocity data from noise Phase velocity data from interstation method Group velocity dispersion curve from RF modelling Phase velocity dispersion curve from RF modelling Group velocity dispersion curve from joint modelling Phase velocity dispersion curve from joint modelling 1 2 5 1 2 5 period
2. Dipping Moho From Stipčević, PhD Thesis, paper in preparation Moho depth determined using H κ (above) and NA (right) method
From Stipčević, PhD Thesis, paper in preparation
Stations with similar results obtained using H-κ and NA methods Stipčević et al., in preparation
Stations for which there is a large difference between the H-κ i NA results Stipčević et al., in preparation
Moho dip determined using NA algorithm Stipčević et al., in preparation
Dipping Moho synthetic experiment Starting model : Two 2km thick layers in the crust Moho at 4 km with 2 dip & 27 strike Synthetics are calculated using Fredrickson and Bostock method assuming an isotropic medium, and synthetic RFs are determined by deconvolution. These synthetic RF data are then linearly stacked and inverted for Earth structure using NA method introduced in Exercise 6.
Dipping Moho synthetic experiment Starting model: Two 2km thick layers in the crust Moho at 4 km with 2 dip & 27 strike Synthetics are calculated using the Fredrickson and Bostock method assuming an isotropic medium and synthetic RFs are determined by deconvolution.
Dipping Moho Synthetic Experiment 2 km 36 km Earth model This is a result of the NA inversion when laterally homogeneous horizontal layers are assumed. It is also assumed that the Moho is horizontal (not dipping). (Exercise 1 in synopsis)
Dipping Moho Synthetic Experiment 2 km 4km Earth model Moho: D~17 ; S~262 Now inverting for the Moho dip and orientation
3. A multi-step approach including polarization anisotropy 1. 3. 2. 4. Tkalčić et al., JGR 26
4. Non-uniqueness etc. Different approaches to inverse problems Forward modeling Linearized inversion Grid-search Non-linear inversion with optimization Multi-step approach Non-linear inversion with the Bayesian framework Transdimensional Bayes framework hierarchical Bayes theorem: p(m d obs ) p(d obs m) p(m)
The importance of knowing the data noise in trans-dimensional formulation! est = 15 1 6 x 14 5 5 1 5 1 1! est = 3 Number of samples 4 2 5 1 15 2 25 Number of cells x 1 4 5 5 1 5 1 1! est = 8 Number of Samples 1 5 1 15 2 25 Number of cells 1 x 14 5 5 1 5 1 Number of samples 5 5 1 15 2 25 Number of cells
Hierarchical Models Relationship between data noise and model complexity Treating data noise σ as an unknown in the problem Data noise is uncorrelated Likelihood function unknown p(d m) 1 ( 2πσ 2 ) & exp ( d g(m) N '( 2σ 2 2 ) + * + unknown unknown Data noise = measurement uncertainty + modeling uncertainty
Covariance Matrix of Noise in Data Data noise is correlated Misfit Φ(m) = [d g(m)] T C D 1 [d g(m)] Likelihood p(m d) = 1 (2π) N C D % exp Φ(m) ( & ' 2 ) *
Noise Parameterization How do we parameterize C D? $ 1 r r 2... r N 1 ' & ) & r 1 r r N 2 ) C D = σ 2 & r 2 r 1 r N 3 ) & ) & ) %& r N 1 r N 2 r N 3... 1 () 2 parameters : Magnitude of noise σ Correlation of noise r
Synthetic experiment Receiver Function.4.3.2.1!.1 Noisy Receiver Func8on!.2!5 5 1 15 2 25 3 Time (s) Magnitude and correlation of noise are treated as unknowns Depth (km) Depth(km) 1 2 3 4 5 True model 6 2 3 4 5 Solution is a large ensemble of models distributed according to the target distribution Vs (km/s) Vs(km/s) Bodin et al., JGR 212
Synthetic experiment Receiver Function.4.3.2.1!.1!.2!5 5 1 15 2 25 3 True model Mean Noisy Receiver Func8on Time (s) Number of models Magnitude and correlation of noise are unknown 1 1 1 Depth(km) 2 3 4 2 3 4 2 3 4 Different ways to look at the solution 5 5 5 6 2 3 4 5 Vs(km/s) 6 2 3 4 5 Vs(km/s) 6 p(transition) Bodin et al., JGR 212
Synthetic experiment Receiver Function.4.3.2.1!.1!.2!5 5 1 15 2 25 3 True model Mean Noisy Receiver Func8on Time (s) Solu8on for maximum number of models Magnitude and correlation of noise are unknown 1 1 Probability distribution at 31 km Depth(km) 2 3 4 Depth(km) 2 3 4 Ensemble Solution True Model Average Solution Maximun Solution 5 5 6 2 3 4 5 Vs(km/s) 6 2 3 4 5 Vs(km/s) 2 2.5 3 3.5 4 4.5 5 Vs (Km/s)
Synthetic experiment Depth(km) Receiver Function 1 2 3 4 5.4.3.2.1!.1!.2!5 5 1 15 2 25 3 True model 1 Noisy Receiver Func8on Time (s) 5 1 15 2 25 3 35 4 45 Number of layers Number of models Prior Posterior True Model Algorithm is able to recover the complexity of the model and the level of data noise 6 2 3 4 5 Vs(km/s).1.2.3!.7.75.8.85.9.95 r
Synthetic experiment Receiver Function.4.3.2.1!.1!.2!5 5 1 15 2 25 3 True model Noisy Receiver Func8on Time (s) Magnitude and correlation of noise are unknown Trade-offs between parameters 1 1 28 Depth(km) 2 3 4 5 2 3 4 5 Moho Depth Moho depth (km) 3 32 34 6 2 3 4 5 Vs(km/s) 6 2 3 4 5 Vs(km/s) 2 2.5 3 3.5 4 4.5 5 Vs in the last layer of the crust
Application to field data Bodin et al., JGR 212
Application to Joint Inversion Dispersion curve for Rayleigh waves Group Velocity Receiver Function 4 3.5 3 2.5 2.4.2 1 2 3 4 5 6 7 8 9 1 Period (s) Receiver Function!.2!5 5 1 15 2 25 3 Time(s) d = d RF + d dis " C = C RF % D D $ ' C dis # D & Algorithm naturally weights the information brought by each data type.
Application to Joint Inversion (synthetic test) Receiver Function Dispersion Joint Inversion 1 1 1 2 2 2 Depth(km) 3 Depth(km) 3 4 Depth(km) 3 4 5 4 5 6 5 6 2 3 4 5 Vs(km/s) p(discontinuity) 7 2 3 4 5 Vs(km/s) p(discontinuity) 6 2 3 4 5 Vs(km/s) Bodin et al., JGR 212
Probability Application to Joint Inversion WOMBAT data from SE Australia: RFs + ambient noise dispersion 1 Only RF Only SWD Joint (RF+SWD) Average models 5 5 5 5 1 1 1 1 Depth(km) 15 Depth(km) 15 Depth(km) 15 Depth(km) 15 2 2 2 2 25 2.5 3 3.5 4 Vs(km/s) 25 2.5 3 3.5 4 Vs(km/s) 25 2.5 3 3.5 4 Vs(km/s) RF SWD Joint 25 2.5 3 3.5 4 Vs(km/s) Bodin et al., JGR 212
Future research Modelling multiple geophysical datasets will be approached through the transdimensional hieararchical Bayesian framework, where the number of free parameters and the data noise will be treated as unknowns in the inversion. Various simplifications that hinder the progress in crustal and lithospheric imaging using passive-source data and permanent/temporary seismic receivers will be gradually incorporated in the Bayesian inversion strategy this includes, but is not limited to: anisotropy, dipping layers and 3D structure, noisy data, etc. This is a general strategy that can be applied to other types of inverse problems in Earth Science and for imaging various parts of the Earth s crust.