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Society of Exploration Geophysicists International Exposition and 80th Annual Meeting 2010, SEG 2010

作者： Morency, Christina Luo, Yang Tromp, Jeroen Department of Geosciences Princeton University United States Department of Geosciences and Program in Applied and Computational Mathematics Princeton University United States

ISBN: (纸本)9781617389801

The key issues in CO2 sequestration monitoring involve accurate monitoring, from the injection stage to prediction & verification, of CO2 movement over time for environmental considerations. A natural non-intrusive monitoring technique is referred to as "4D seismics", which involves 3D time-lapse seismic surveys. The success of monitoring the CO2 movement is subject to a proper description of the physical properties of the structure. We realize time-lapse migrations comparing acoustic, elastic, and poroelastic simulations of 4D seismic imaging to characterize the storage zone, based solely upon the first arrival traveltime anomaly arising from the injection of CO2. This approach highlights the influence of using different physical theories on interpreting seismic data, and, more importantly, on extracting the CO2 signature from the seismic wave field. Simulations are performed using a spectral-element method, which allows for highly accurate results. Biot? Equations are implemented to account for poroelastic effects. The sensitivity of observables to the model parameters is quantified based upon finite-frequency sensitivity kernels calculated using an adjoint method. © 2010 SEG.

关键词： Carbon dioxide

A spectral-infinite-element solution of Poisson’s equation: An application to self gravity

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arXiv 2017年

作者： Gharti, Hom Nath Tromp, Jeroen Department of Geosciences Princeton University PrincetonNJ08544 United States Program in Applied & Computational Mathematics Princeton University PrincetonNJ United States

We solve Poisson’s equation by combining a spectral-element method with a mapped infinite-element method. We focus on problems in geostatics and geodynamics, where Earth’s gravitational field is determined by Poisson’s equation inside the Earth and Laplace’s equation in the rest of space. Spectral elements are used to capture the internal field, and infinite elements are used to represent the external field. To solve the weak form of Poisson/Laplace equation, we use Gauss-Legendre-Lobatto quadrature in spectral elements inside the domain of interest. Outside the domain, we use Gauss-Radau quadrature in the infinite direction, and Gauss-Legendre-Lobatto quadrature in the other directions. We illustrate the efficiency and accuracy of the method by comparing the gravitational fields of a homogeneous sphere and the Preliminary Reference Earth Model (PREM) with (semi-)analytical solutions. Copyright © 2017, The Authors. All rights reserved.

关键词： Earth (planet)

Sensitivity kernels for inferring Lorentz stresses from normal-mode frequency splittings in the Sun

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arXiv 2020年

作者： Das, Srijan Bharati Chakraborty, Tuneer Hanasoge, Shravan M. Tromp, Jeroen Department of Geosciences Princeton University PrincetonNJ United States Department of Astronomy and Astrophysics Tata Institute of Fundamental Research Mumbai India Department of Geosciences and Program in Applied & Computational Mathematics Princeton University Princeton NJ United States

Departures from standard spherically symmetric solar models, in the form of perturbations such as global and local-scale flows and structural asphericities, result in the splitting of eigenfrequencies in the observed spectrum of solar oscillations. Drawing from prevalent ideas in normal-mode coupling theory in geophysical literature, we devise a procedure that enables the computation of sensitivity kernels for general Lorentz stress fields in the Sun. Mode coupling due to any perturbation requires careful consideration of self- and cross-coupling of multiplets. Invoking the isolated-multiplet approximation allows for limiting the treatment to purely self-coupling, requiring significantly less computational resources. We identify the presence of such isolated multiplets under the effect of Lorentz stresses in the Sun. Currently, solar missions allow precise measurements of self-coupling of multiplets via "a-coefficients" and the cross-spectral correlation signal which enables the estimation of the "structure coefficients". We demonstrate the forward problem for both self-coupling (a-coefficients) and cross-coupling (structure coefficients). In doing so, we plot the self-coupling kernels and estimate a-coefficients arising from a combination of deep-toroidal and surface-dipolar axisymmetric fields. We also compute the structure coefficients for an arbitrary general magnetic field (real and solenoidal) and plot the corresponding "splitting function", a convenient way to visualize the splitting of multiplets under 3D internal perturbations. The results discussed in this paper pave the way to formally pose an inverse problem, and infer solar internal magnetic fields. Copyright © 2020, The Authors. All rights reserved.

关键词： Chemical reactions

Cosserat elasticity as the weak-field limit of Einstein-Cartan relativity

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Physical Review D 2024年第10期109卷 104052-104052页

作者： Matthew Maitra Jeroen Tromp Institut für Geophysik ETH Zürich Sonneggstrasse 5 Zürich 8092 Switzerland Department of Geosciences Program in Applied and Computational Mathematics Princeton University Princeton 08540 NJ United States

The weak-field limit of Einstein-Cartan (EC) relativity is studied. The equations of EC theory are rewritten such that they formally resemble those of Einstein general relativity (EGR); this allows ideas from post-Newtonian theory to be imported without essential change. The equations of motion are then written both at first post-Newtonian (1PN) order and at 1.5PN order. EC theory’s 1PN equations of motion are found to be those of a micropolar/Cosserat elastic medium, along with a decoupled evolution equation for nonclassical, spin-related fields. It seems that a necessary condition for these results to hold is that one chooses the nonclassical fields to scale with the speed of light in a certain empirically reasonable way. Finally, the 1.5PN equations give greater insight into the coupling between energy-momentum and spin within slowly moving, weakly gravitating matter. Specifically, the weakly relativistic modifications to Cosserat theory involve a gravitational torque and an augmentation of the gravitational force due to a dynamic mass moment density with an accompanying dynamic mass moment density flux, and new forms of linear momentum density captured by a dynamic mass density flux and a dynamic momentum density.

关键词： Alternative gravity theories Solar system & its planets

Preconditioned BFGS-based uncertainty quantification in elastic full waveform inversion

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arXiv 2020年

作者： Liu, Qiancheng Beller, Stephen Lei, Wenjie Peter, Daniel Tromp, Jeroen Department of Geosciences Princeton University PrincetonNJ08544 United States Thuwal Saudi Arabia Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Full Waveform Inversion (FWI) has become an essential technique for mapping geophysical subsurface structures. However, proper uncertainty quantification is often lacking in current applications. In theory, uncertainty quantification is related to the inverse Hessian (or the posterior covariance matrix). Even for common geophysical inverse problems its calculation is beyond the computational and storage capacities of the largest high-performance computing systems. In this study, we amend the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm to perform uncertainty quantification for large-scale applications. For seismic inverse problems, the limited-memory BFGS (L-BFGS) method prevails as the most efficient quasi-Newton method. We aim to augment it further to obtain an approximate inverse Hessian for uncertainty quantification in FWI. To facilitate retrieval of the inverse Hessian, we combine BFGS (essentially a full-history L-BFGS) with randomized singular value decomposition to determine a low-rank approximation of the inverse Hessian. Setting the rank number equal to the number of iterations makes this solution efficient and memory-affordable even for large-scale problems. Furthermore, based on the Gauss-Newton method, we formulate different initial, diagonal Hessian matrices as preconditioners for the inverse scheme and compare their performances in elastic FWI applications. We highlight our approach with the elastic Marmousi benchmark model, demonstrating the applicability of preconditioned BFGS for large-scale FWI and uncertainty quantification. Copyright © 2020, The Authors. All rights reserved.

关键词： Uncertainty analysis

Minuscule corrections to near-surface solar internal rotation using mode-coupling

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arXiv 2023年

作者： Das, Srijan Bharati Kashyap, Samarth G. Oktay, Deniz Hanasoge, Shravan M. Tromp, Jeroen Department of Geosciences Princeton University PrincetonNJ United States Department of Astronomy and Astrophysics Tata Institute of Fundamental Research Mumbai India Department of Computer Science Princeton University PrincetonNJ United States Department of Geosciences and Program in Applied & Computational Mathematics Princeton University PrincetonNJ United States

The observed solar oscillation spectrum is influenced by internal perturbations such as flows and structural asphericities. These features induce splitting of characteristic frequencies and distort the resonant-mode eigenfunctions. Global axisymmertric flow — differential rotation — is a very prominent perturbation. Tightly constrained rotation profiles as a function of latitude and radius are products of established helioseismic pipelines that use observed Dopplergrams to generate frequency-splitting measurements at high precision. However, the inference of rotation using frequency-splittings do not consider the effect of mode-coupling. This approximation worsens for high-angular-degree modes, as they become increasingly proximal in frequency. Since modes with high angular degrees probe the near-surface layers of the Sun, inversions considering coupled modes could potentially lead to more accurate estimates of rotation very close to the surface. In order to investigate if this is indeed the case, we perform inversions for solar differential rotation, considering coupling of modes for angular degrees 160 ≤ ` ≤ 300 in the surface gravity f-branch and first-overtone p modes. In keeping with the character of mode coupling, we carry out a non-linear inversion using an eigenvalue solver. Differences in inverted profiles for frequency splitting measurements from MDI and HMI are compared and discussed. We find that corrections to the near-surface differential rotation profile, when accounting for mode-coupling effects, are smaller than 0.003 nHz and hence are insignificant. These minuscule corrections are found to be correlated with the solar cycle. We also present corrections to even-order splitting coefficients, which could consequently impact inversions for structure and magnetic fields. © 2023, CC BY.

关键词： Rotation

Cosserat elasticity as the weak-field limit of Einstein–Cartan relativity

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arXiv 2024年

作者： Maitra, Matthew Tromp, Jeroen Institut für Geophysik ETH Zürich Sonneggstrasse 5 Zürich8092 Switzerland Department of Geosciences and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08540 United States

The weak-field limit of Einstein–Cartan (EC) relativity is studied. The equations of EC theory are rewritten such that they formally resemble those of Einstein General Relativity (EGR);this allows ideas from post-Newtonian theory to be imported without essential change. The equations of motion are then written both at first post-Newtonian (1PN) order and at 1.5PN order. EC theory’s 1PN equations of motion are found to be those of a micropolar/Cosserat elastic medium, along with a decoupled evolution equation for non-classical, spin-related fields. It seems that a necessary condition for these results to hold is that one chooses the non-classical fields to scale with the speed of light in a certain empirically reasonable way. Finally, the 1.5PN equations give greater insight into the coupling between energy-momentum and spin within slowly moving, weakly gravitating matter. Specifically, the weakly relativistic modifications to Cosserat theory involve a gravitational torque and an augmentation of the gravitational force due to a ‘dynamic mass moment density’ with an accompanying ‘dynamic mass moment density flux’, and new forms of linear momentum density captured by a ‘dynamic mass density flux’ and a ‘dynamic momentum density’. Copyright © 2024, The Authors. All rights reserved.

关键词： Equations of motion

Inferring solar differential rotation through normal-mode coupling using bayesian statistics

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arXiv 2021年

作者： Kashyap, Samarth Ganesh Bharati Das, Srijan Hanasoge, Shravan M. Woodard, Martin F. Tromp, Jeroen Department of Astronomy and Astrophysics Tata Institute of Fundamental Research Mumbai India Department of Geosciences Princeton University PrincetonNJ United States NorthWest Research Associates Boulder Office 3380 Mitchell Lane BoulderCO United States Department of Geosciences and Program in Applied & Computational Mathematics Princeton University PrincetonNJ United States

Normal-mode helioseismic data analysis uses observed solar oscillation spectra to infer perturbations in the solar interior due to global and local-scale flows and structural asphericity. Differential rotation, the dominant global-scale axisymmetric perturbation, has been tightly constrained primarily using measurements of frequency splittings via "a-coefficients". However, the frequency-splitting formalism invokes the approximation that multiplets are isolated. This assumption is inaccurate for modes at high angular degrees. Analysing eigenfunction corrections, which respect cross coupling of modes across multiplets, is a more accurate approach. However, applying standard inversion techniques using these cross-spectral measurements yields a-coefficients with a significantly wider spread than the well-constrained results from frequency splittings. In this study, we apply Bayesian statistics to infer a-coefficients due to differential rotation from cross spectra for both f-modes and p-modes. We demonstrate that this technique works reasonably well for modes with angular degrees ` = 50 − 291. The inferred a3−coefficients are found to be within 1 nHz of the frequency splitting values for ` > 200. We also show that the technique fails at ` © 2021, CC BY.

关键词： Eigenvalues and eigenfunctions

Seismic Probes of Solar Interior Magnetic Structure

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Physical Review Letters 2012年第10期109卷 101101-101101页

作者： Shravan Hanasoge Aaron Birch Laurent Gizon Jeroen Tromp Department of Geosciences Princeton University New Jersey 08544 USA Max-Planck Institut für Sonnensystemforschung 37191 Kalenburg-Lindau Germany Georg-August-Universität Institut für Astrophysik D-37077 Göttingen Germany Program for Applied and Computational Mathematics Princeton University New Jersey 08544 USA

Sun spots are prominent manifestations of solar magnetoconvection, and imaging their subsurface structure is an outstanding problem of wide physical importance. Travel times of seismic waves that propagate through these structures are typically used as inputs to inversions. Despite the presence of strongly anisotropic magnetic waveguides, these measurements have always been interpreted in terms of changes to isotropic wave speeds and flow-advection-related Doppler shifts. Here, we employ partial-differential-equation-constrained optimization to determine the appropriate parametrization of the structural properties of the magnetic interior. Seven different wave speeds fully characterize helioseismic wave propagation: the isotropic sound speed, a Doppler-shifting flow-advection velocity, and an anisotropic magnetic velocity. The structure of magnetic media is sensed by magnetoacoustic slow and fast modes and Alfvén waves, each of which propagates at a different wave speed. We show that even in the case of weak magnetic fields, significant errors may be incurred if these anisotropies are not accounted for in inversions. Translation invariance is demonstrably lost. These developments render plausible the accurate seismic imaging of magnetoconvection in the Sun.

关键词： SEISMIC waves -- Speed MAGNETIC structure PARTIAL differential equations DOPPLER effect THEORY of wave motion SUN INTERNAL structure

Internal and external potential-field estimation from regional vector data at varying satellite altitude

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arXiv 2017年

作者： Plattner, Alain Simons, Frederik J. Department of Earth and Environmental Sciences California State University Fresno FresnoCA93740 United States Department of Geosciences Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

When modeling global satellite data to recover a planetary magnetic or gravitational potential field and evaluate it elsewhere, the method of choice remains their analysis in terms of spherical harmonics. When only regional data are available, or when data quality varies strongly with geographic location, the inversion problem becomes severely ill-posed. In those cases, adopting explicitly local methods is to be preferred over adapting global ones (e.g., by regularization). Here, we develop the theory behind a procedure to invert for planetary potential fields from vector observations collected within a spatially bounded region at varying satellite altitude. Our method relies on the construction of spatiospectrally localized bases of functions that mitigate the noise amplification caused by downward continuation (from the satellite altitude to the planetary surface) while balancing the conflicting demands for spatial concentration and spectral limitation. The 'altitude-cognizant' gradient vector Slepian functions (AC-GVSF) were first employed in a preceding paper. They enjoy a noise tolerance under downward continuation that is much improved relative to the 'classical' gradient vector Slepian functions (CL-GVSF), which do not factor satellite altitude into their construction. Furthermore, venturing beyond the realm of their first application, in the present article we extend the theory to being able to handle both internal and external potential-field estimation. Solving simultaneously for internal and external fields in the same setting of regional data availability reduces internal-field artifacts introduced by downward-continuing unmodeled external fields, as we show with numerical examples. We explain our solution strategies on the basis of analytic expressions for the behavior of the estimation bias and variance of models for which signal and noise are uncorrelated, (essentially) spaceand bandlimited, and spectrally (almost) white. The AC-GVSF are optimal linea

关键词： Harmonic analysis