scholarly journals Relation of dislocation Love numbers and conventional Love numbers and corresponding Green's functions for a surface rupture in a spherical earth model

2013 ◽  
Vol 193 (2) ◽  
pp. 717-733 ◽  
Author(s):  
Wenke Sun ◽  
Jie Dong
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Earth Model ◽  
Surface Rupture ◽  
Love Numbers ◽  
Spherical Earth

Determining dislocation love number of vertical displacement using GPS observations: case study of 2011 Tohoku-Oki earthquake (Mw 9.0)

2020 ◽  
Vol 222 (2) ◽  
pp. 965-977
Author(s):  
Junyan Yang ◽  
Wenke Sun
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Vertical Displacement ◽  
Fault Slip ◽  
Earth Model ◽  
Love Number ◽  
The Earth ◽  
Love Numbers ◽  
Vertical Displacements

SUMMARY The concept of determining the dislocation Love numbers using GNSS (Global Navigation Satellite System) data and calculating the corresponding Green's functions is presented in this paper. As a case study, we derive the dislocation Love number h of vertical displacement by combining 1232 onshore GPS data and 7 GPS-Acoustic data with the 2011 Tohoku-Oki earthquake (Mw 9.0). Three fault-slip distributions are used to compare and verify the theory and results. As the GPS stations are only located in Japan Island and along the Japan trench, we use the theoretical vertical displacements of a spherically layered Earth structure to constrain the low-order signal. The L-curve and an a priori preliminary reference skill are applied in the inversion method. Then, the GPS-observed vertical displacement changes are used to invert for the vertical displacement dislocation Love numbers h based on three different fault-slip models. Our results indicate that the estimated dislocation Love numbers $h$ fluctuate significantly from the earth model (i.e. the preliminary reference earth model), especially for the $h_{n1}^{32}$ component, and these changes in $h_{n2}^{12}$ and $h_{n0}^{33} - h_{n0}^{22}$ are relatively small. The vertical displacements derived from the inversion results agree well with the GPS vertical observations. The inverted dislocation Love numbers are considered to profile the regional structure which differs from the mean 1-D heterogeneous structure of the Earth, and the corresponding Green's functions of four independent dislocation sources describe a more reasonable seismic deformation field.

Deformation of a spherical, viscoelastic, and incompressible Earth for a point load with periodic time change

2020 ◽  
Vol 222 (3) ◽  
pp. 1909-1922 ◽  
Author(s):  
He Tang ◽  
Jie Dong ◽  
Lan Zhang ◽  
Wenke Sun
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Point Load ◽  
Point Sources ◽  
Earth Model ◽  
Surface Mass ◽  
Planetary Scale ◽  
Periodic Loading ◽  
Love Numbers ◽  
Analytical Time

SUMMARY Planetary-scale mass redistributions occur on Earth for certain spatiotemporal periods, and these surface mass changes excite the global periodic loading deformations of a viscoelastic Earth. However, the characteristics of periodic viscoelastic deformations have not been well investigated even in a simple earth model. In this study, we derive the semi-analytical Green's functions (fully analytical Love numbers) for long-standing point sources with given periods using a modified asymptotic scheme in a homogeneous Maxwell spherical earth model. Here, the asymptotic scheme is needed in order to obtain accurate semi-analytical time-dependent Green's functions. The amplitudes and phases of the Green's functions may be biased if only the series summations of the Love numbers are used because the influence of viscoelasticity is degree-dependent. We compare the viscoelastic and elastic periodic Green's functions with different material viscosities and loading periods and investigate the amplitude increase percentage and phase delay of the periodic displacement and geoid change. For example, our analysis revealed that the viscosity increases the amplitude by 40–120 per cent and delays the phase approximately −100° to 60° for the displacement and geoid change when bearing a 10-yr loading period, assuming a viscosity of 1018 Pa s and a shear modulus 4 × 1010 Pa.

A point dislocation in a layered, transversely isotropic and self-gravitating Earth — Part II: accurate Green's functions

2019 ◽  
Vol 219 (3) ◽  
pp. 1717-1728 ◽  
Author(s):  
J Zhou ◽  
E Pan ◽  
M Bevis
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Transversely Isotropic ◽  
Earth Model ◽  
Transversely Isotropic Media ◽  
Love Numbers ◽  
Isotropic Media ◽  
Point Dislocation ◽  
High Degree ◽  
Harmonic Degree

SUMMARY We present an accurate approach for calculating the point-dislocation Green's functions (GFs) for a layered, spherical, transversely-isotropic and self-gravitating Earth. The formalism is based on the approach recently used to find analytical solutions for the dislocation Love numbers (DLNs). However, in order to make use of the DLNs, we first analyse their asymptotic behaviour, and then the behaviour of the GFs computed from the DLNs. We note that the summations used for different GF components evolve at different rates towards asymptotic convergence, requiring us to use two new and different truncation values for the harmonic degree (i.e. the index of summation). We exploit this knowledge to design a Kummer transformation that allows us to reduce the computation required to evaluate the GFs at the desired level of accuracy. Numerical examples are presented to clarify these issues and demonstrate the advantages of our approach. Even with the Kummer transformation, DLNs of high degree are still needed when the earth model contains very fine layers, so computational efficiency is important. The effect of anisotropy is assessed by comparing GFs for isotropic and transversely isotropic media. It is shown that this effect, though normally modest, can be significant in certain contexts, even in the far field.

scholarly journals A Python framework for efficient use of pre-computed Green's functions in seismological and other physical forward and inverse source problems

Solid Earth ◽  
2019 ◽  
Vol 10 (6) ◽  
pp. 1921-1935 ◽  
Author(s):  
Sebastian Heimann ◽  
Hannes Vasyura-Bathke ◽  
Henriette Sudhaus ◽  
Marius Paul Isken ◽  
Marius Kriegerowski ◽  
...  
Keyword(s):  
Green's Functions ◽  
Heterogeneous Media ◽  
Green’S Functions ◽  
Application Programming Interface ◽  
Earth Model ◽  
Forward Modelling ◽  
Field Equations ◽  
Wide Range ◽  
Geophysical Processes ◽  
Numerical Calculation Method

Abstract. The finite physical source problem is usually studied with the concept of volume and time integrals over Green's functions (GFs), representing delta-impulse solutions to the governing partial differential field equations. In seismology, the use of realistic Earth models requires the calculation of numerical or synthetic GFs, as analytical solutions are rarely available. The computation of such synthetic GFs is computationally and operationally demanding. As a consequence, the on-the-fly recalculation of synthetic GFs in each iteration of an optimisation is time-consuming and impractical. Therefore, the pre-calculation and efficient storage of synthetic GFs on a dense grid of source to receiver combinations enables the efficient lookup and utilisation of GFs in time-critical scenarios. We present a Python-based framework and toolkit – Pyrocko-GF – that enables the pre-calculation of synthetic GF stores, which are independent of their numerical calculation method and GF transfer function. The framework aids in the creation of such GF stores by interfacing a suite of established numerical forward modelling codes in seismology (computational back ends). So far, interfaces to back ends for layered Earth model cases have been provided; however, the architecture of Pyrocko-GF is designed to cover back ends for other geometries (e.g. full 3-D heterogeneous media) and other physical quantities (e.g. gravity, pressure, tilt). Therefore, Pyrocko-GF defines an extensible GF storage format suitable for a wide range of GF types, especially handling elasticity and wave propagation problems. The framework assists with visualisations, quality control, and the exchange of GF stores, which is supported through an online platform that provides many pre-calculated GF stores for local, regional, and global studies. The Pyrocko-GF toolkit comes with a well-documented application programming interface (API) for the Python programming language to efficiently facilitate forward modelling of geophysical processes, e.g. synthetic waveforms or static displacements for a wide range of source models.

Co-seismic internal deformations in a spherical layered earth model

2020 ◽  
Vol 221 (3) ◽  
pp. 1515-1531
Author(s):  
Tai Liu ◽  
Guangyu Fu ◽  
Yawen She ◽  
Cuiping Zhao
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Near Field ◽  
Earth Model ◽  
Coulomb Stress ◽  
Far Field ◽  
Coulomb Stress Changes ◽  
Layered Earth ◽  
Stress Changes ◽  
Internal Deformation

SUMMARY Using a numerical integral method, we deduced a set of formulae for the co-seismic internal deformation in a spherically symmetric earth model, simultaneously taking self-gravitation, compressibility and realistically stratified structure of the Earth into account. Using these formulae, we can calculate the internal deformation at an arbitrary depth caused by an arbitrary seismic source. To demonstrate the correctness of our formulae, we compared our numerical solutions of radial functions with analytical solutions reported by Dong & Sun based on a homogeneous earth model; we found that two sets of results agree well with each other. Our co-seismic internal Green's functions in the near field agree well with the results calculated by the formulae of Okada, which also verifies our Green's functions. Finally, we calculated the Coulomb stress changes on the Japanese Islands and Northeast China induced by the Tohoku-Oki Mw 9.0 earthquake using the methods described above. We found that the effect of layered structure plays a leading role on the near field, while curvature occupies a dominant position on the deep region of the far field. Through a comparison of the Coulomb stress changes at a depth of 10 km on a layered earth model calculated by our method along with the corresponding results of Okada, we found that the discrepancy between them in near field was ∼31.5 per cent, and that of far field was >100 per cent of the signals.

scholarly journals High-frequency global wavefields for local 3D structures by wavefield injection and extrapolation

2020 ◽  
Author(s):  
Marta Pienkowska ◽  
Vadim Monteiller ◽  
Tarje Nissen-Meyer
Keyword(s):  
Wave Propagation ◽  
Large Scale ◽  
Green's Functions ◽  
Green’S Functions ◽  
Body Wave ◽  
Computational Cost ◽  
Spectral Element ◽  
Earth Model ◽  
Local Domain ◽  
Global Database

Summary Earth structure is multiscale, and seismology remains the primary means of deciphering signatures from small structures over large distances. To enable this at the highest resolution, we present a flexible injection and extrapolation type hybrid framework that couples wavefields from a precomputed global database of accurate Green's functions for 1-D models with a local three dimensional (3-D) method of choice (e.g. a spectral element or a finite difference solver). The interface allows to embed a full 3-D domain in a spherically symmetric Earth model, tackling large-scale wave propagation with focus on localized heterogeneous complex structures. Thanks to reasonable computational costs (10k CPU hours) and storage requirements (a few TB for 1 Hz waveforms) of databases of global Green’s functions, the method provides coupling of 3-D wavefields that can reach the highest observable body-wave frequencies in the 1-4 Hz range. The framework is highly flexible and adaptable; alterations in source properties (radiation patterns, source-time function), in the source-receiver geometry, and in local domain dimensions and location can be introduced without re-running the global simulation. The once-and-for-all database approach reduces the overall computational cost by a factor of 5,000-100,000 relative to a full 3-D run, provided that the local domain is of the order of tens of wavelengths in size. In this paper we present the details of the method and its implementation, show benchmarks with a 3-D spectral-element solver, discuss its setup-dependent performance, and explore possible wave-propagation applications.

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