Inverse problems

Uncertainty and errors in mapping data to model

The mapping from data to model encapsulates the core of inverse problems. Within the framework of first-order perturbation theory, these partial derivatives of misfit functions with respect to model parameters are calculated upon a large number of solutions to the forward problem, which poses a formidable computational task. Using AXISEM, we developed an efficient technique to deliver time- and frequency-dependent waveform kernels which represent the raw mapping between seismogram and structure. We compute these kernels upon raw Green tensors in the frequency domain, allowing for flexible posteriori adjustments such as filtering, misfit selection, model parameterization, time windowing. In particular, this allows for using diffracted waves and generally any fraction of a seismogram into large-scale multi-frequency tomography at any desired seismic frequency. This is computationally feasible only since we explore the dimensional reduction within the axisymmetric spectral-element method, and thus the crux of these 3D kernels simply lies in rotating and convolving the generic once-and-for-all 2D wavefields for a given source-receiver pair. Such time-dependent kernels describe the instantaneous 3-D region of perturbations with respect to a reference model that a particular instant within a seismogram may “feel”. I have applied the method to constraining the dependence of time-dependent 3-D seismic sensitivity on different misfit functions, source radiation patterns, frequency ranges, earthquake depths, epicentral distances, azimuths, and respectively addressed the sensitivity of source- and receiver-side region as well as numerical errors and inaccuracies in selecting time windows. The effect of these choices on the data-to-model mapping is significant, in many cases larger than the effect of different background models.
In a attempt to quantify the effect of large heterogeneities on waveforms and tomographic imaging, we compute synthetic seismograms for models in the core-mantle boundary region under- and overlain by a spherical symmetric Earth. The statistics of misfit anomalies are computed using the reference symmetric spherical Earth model. We analyze the impact of three-dimensional propagation on these statistics and the validity range of the Born approximation when trying to recover the structure and the amplitude of the various anomalies.

Optimizing tomographic data selection & processing

(T. Nissen-Meyer, A. Fournier)

To enhance robustness and resolution of imaging capabilities for a given region of interest, we strive to quantify and automate some of the data selection and processing tasks for large-scale tomographic inversions by analyzing the nature and parameter dependencies of seismic sensitivity kernels. This is achieved by formulating an optimization problem for a given 3D region, and computing spatio-temporal seismic sensitivity kernels for seismograms upon an earthquake as a function of source frequency and depth, radiation pattern, receiver component, epicentral distance, azimuth, time windows and filtering of the seismogram, misfit parameters (e.g. traveltime versus waveforms), and model parameterization (wavespeeds versus elastic moduli). The optimization problem is explicitly multi-modal, in that we strive to find all possible extrema in the sensitivity within this multi-dimensional parameter space, and as such can only be tackled via brute-force or Bayesian approaches. However, our method’s capability to consider all these free parameters independently after the computation of the respective forward solution is used to construct a primary region-of-interest assessment of optimal data selection: Given a location of geophysical interest and sufficient data coverage, we readily supply a set of e.g. phases, frequency ranges, epicentral distances, and receiver components that maximizes sensitivity and resolution and may then be used in a tomographic inversion. In the light of immense data availability (e.g. USArray), this may serve as a valuable tool to efficiently select and process data which offer best possible resolution, or predict optimal illumination strategies for future array or ocean deployments.