3D Seismic Tomographic Inversion as A Data Assimilation Problem
With the recent advancement in numerical modeling of seismic wave propagation in 3D heterogeneous media, we can now solve the seismic tomography problem by the variational data assimilation approach widely used in allied fields such as meteorology and oceanography. The observed seismograms recorded at various stations for various events can be assimilated into the full wave equation which describes the physical processes of wave propagation through 3D heterogeneous media. For a cost function that quantifies the difference between the observed data and synthetics generated for an initial earth model, in terms of either waveform, traveltime or amplitude, we show that its variation with respect to the earth structural model parameters, i.e. the Fréchet derivatives, can be constructed by the interactions of the forward wavefield s(x,t) and the so-called adjoint wave field s†(x,t) The adjoint wavefield s†(x,t) is generated by using time-reversed signal at receivers as fictitious, simultaneous sources. The popular 'banana-doughnut' kernels which depicts the sensitivity of structural parameters to individual elements, as well as the Fréchet derivatives can be computed by the adjoint methods for both 1D and 3D earth models. Utilizing the Fréchet derivatives in optimization methods, we can improve our knowledge of the earth seismic structure by iteratively minimizing the cost function. We show an example of the application of this technique to tomographic inversions in southern California, where new updated 3D models have proved to capture both the tectonics and surface geology of the region.
Adjoint Tomography of the Southern California Crust
Adjoint tomography utilizes 3D simulations of seismic wave propagation in conjunction with a tomographic technique based on adjoint methods. We begin with an initial 3D model of shear and compressional wavespeeds for southern California provided by the Southern California Earthquake Center (SCEC; model CVM-H), extending to a depth of 60~km. We use the spectral-element method to simulate 140 good-quality local earthquakes, each recorded by as many as 160 stations. We compute misfits between observed and synthetic seismograms by using a new automated time-window selection algorithm that picks any time window within which the data and 3D synthetics are reasonably similar (e.g., P, S, Love, and Rayleigh waves). For each record with a measurement, we compute an adjoint source that is used to create an adjoint wavefield. The interaction between the adjoint wavefield and the regular wavefield forms the gradient of the misfit function for one event. These gradients are combined using a source subspace projection method to compute a model update. We present a seismic wavespeed model for the southern California crust and uppermost mantle. Over the course of 16 iterations, we have applied net changes in excess of ± 30% from the initial 3D model. With each iteration, the changes in wavespeeds have improved the data fit, and we are able to include additional seismograms whose fits to the data for previous model iterations were too poor for selection. The tomographic results compare well with surface geology, the most striking features being the low wavespeeds of the southern San Joaquin basin, the high wavespeeds beneath the western Transverse Ranges, the low wavespeeds in the Coast Ranges, the low wavespeeds in the eastern Mojave region, and the sharp contrast at the eastern front of the Sierra Nevada due to volcanism in the Coso Junction area. Several dramatic improvements of three-component seismic waveforms highlight the power of the iterative approach.
Full Waveform Inversion for Upper-Mantle Structure in the Australasian Region Based on the Spectral-Element and Adjoint Methods
We present results from a full seismic waveform tomography for upper-mantle structure in the Australasian region. The tomographic images are based on high-quality data collected over the past 15 years at several temporary arrays and permanent stations. Fundamental- and higher-mode surface waves as well as S body waves and multiple reflections are used. The centrepieces of our methodology are envelope and phase misfits computed from time-frequency transforms of the seismograms. They allow us (1) to extract a large amount of robust information that is quasi- linearly related to Earth structure, (2) to separate phase and amplitude information and (3) to measure time- and frequency-dependent misfits for all types of elastic waves including interfering phases that are common at shorter epicentral distances. These characteristics make the envelope and phase misfits well-suited for high- resolution full waveform tomography. We derive Fréchet kernels for the envelope and phase misfits using the adjoint method in conjunction with a recently developed and highly efficient spectral-element code that operates in a spherical section. The tomographic problem itself is solved by iteratively minimising the phase and - to a lesser extent - the envelope misfit. To ensure the convergence of the misfit minimisation we use a smoothed ray-tomographic image of the Australasian region. In the course of the iteration we decrease the dominant period from 100 s to 50 s, and we reach acceptable results after 10 to 15 iterations. In the upper 100 km of our current tomographic model we observe S wave speed variations of ± 15 % that reduce to ± 3.5 % at 500 km depth where the resolution starts to become poor. The principal tectonic features of the Australasian region, are clearly visible, but also small-scale variations within them can be distinguished. Resolution tests indicate that lateral S wave speed variations as small as 2° × 2° can be detected.
Feedback Between Mountain Belt Growth and Plate Convergence Revealed by Forward and Inverse Tectonic Models
While it is generally assumed that global plate motions are driven by the pattern of convection in the Earth's mantle, the details of that linkage remain obscure. Bouyancy forces associated with subduction of cool, dense lithosphere at convergent zones are thought to provide significant driving force, but the relative magnitudes of other driving and resisting forces are less clear. The ability to consider past as well as present plate motions provides significant additional constraints, because changes in plate motion must be necessarily driven by changes in one or more driving or resisting forces, which may be inferred from independent data. Here we first exploit the capabilities of forward global tectonic models focused on the Andean region to infer plate motion changes as far back as Miocene time. By accurately predicting observed convergence rates between Nazca and South America plates over the past 10 Myrs, we demonstrate that the topographic load of the Andes increases resistive forces between downgoing and overriding plates and thus consumes a significant amount of the driving force available for plate tectonics. This result suggests a strong feedback between mountain belt growth and plate convergence. We then test this notion by performing a numerical inversion of the same model. We use the Automatic Differentiation approach to generate a derivative code that relates convergence of the Nazca/South America plates to gross topography of the Andes mountain belt. We test the derivative code in a simple search algorithm to infer an optimal paleotopography of the Andes at 3.2 Myrs from the well-known history of Nazca/South America plate convergence. Our modeling result is in good agreement with independent published estimates of Andean paleotopography.
An Introduction to Geomagnetic data assimilation: Progress and Issues
Data assimilation was originally developed as a means to initialize weather forecast models in the early days of numerical weather prediction (NWP). In the ensuing 50 years, the accuracy of weather forecasting has improved dramatically. This is due in large part to the continued confrontation of models with observations, which allows for every model improvement and parameter adjustment to be tested by determining their impact on forecast error statistics. Geomagnetic data assimilation is in a similar state of development to the early weather forecasting systems. In this talk, we outline the basics of data assimilation and how it is currently being used to make improvements to geodynamo models. The topics include the estimation of model errors, error correlations, and the use of observing system simulation experiments.
Geomagnetic data assimilation: A quasi-geostrophic approach
The study of the applicability of data assimilation techniques to try and estimate the state and evolution of the earth's core is an emerging research area in geomagnetism. It is mainly motivated by a catalog of observations whose quantity and quality has been continuously increasing over the past few centuries, culminating today with satellite data. This new wealth of data should be analyzed in a dynamically consistent fashion, in line with what has been achieved during the last fifteen years in the area of physical oceanography. An objective of geomagnetic data assimilation is to identify the physical phenomena occurring inside the core which are responsible for the secular variation of the main geomagnetic field. This is not a trivial task, given the sparsity of the data at hand, and its remote origin. Still, it is an effort worth pursuing since it should lead to a more accurate forecast of the secular variation, and enable the reanalysis of historical observations. In this talk, we will cast the core problem in the data assimilation framework, and review the specifics which make it difficult. We will present results obtained with a quasi-geostrophic model of the geomagnetic secular variation on interannual timescales, with an emphasis on the structure and strength of the magnetic field in the core interior.
Strong Magnetic Flux Spots and the Core Flow Near the Core-Mantle Boundary
Observations over the past 400 years show strong magnetic flux spots near the equatorial regions at the core- mantle boundary (CMB) that have not been clearly identified in numerical dynamo simulation. We employ a sequential data assimilation algorithm to integrate surface geomagnetic observations over the past 400 years with our MoSST core dynamics model. The assimilated results show significant changes in the toroidal field near the top of the outer core: there appear strong radial current densities associated with the magnetic flux spots at the CMB. Consequently, the Lorentz force associated with these features is very strong, with a magnitude nearly 50% of the Coriolis force near the CMB. This suggests that the tangential geostrophic approximation may not be applicable in this region. New approaches may therefore need for future core flow inversion from observed secular variation.