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.

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

2019 ◽  
Author(s):  
Sebastian Heimann ◽  
Hannes Vasyura-Bathke ◽  
Henriette Sudhaus ◽  
Marius Paul Isken ◽  
Marius Kriegerowski ◽  
...  
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Application Programming Interface ◽  
Forward Modelling ◽  
Field Equations ◽  
Wide Range ◽  
Geophysical Processes ◽  
Inverse Source Problems ◽  
Storage Format ◽  
Numerical Calculation Method

Abstract. The finite physical source problem is usually studied with the concept of volume and time integrals over Green's functions (GF), 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, on-the-fly re-calculation of synthetic GFs in each iteration of an optimisation is time-consuming and impractical. Therefore, pre-calculation and efficient storage of synthetic GFs on a dense grid of source to receiver combinations enables efficient look-up and utilisation of GFs in time critical scenarios. We present a Python-based framework and toolkit – Pyrocko-GF – that enables pre-calculation of synthetic GF stores, which are independent of their numerical calculation method and GF transfer function. The framework integrates a suite of established numerical forward-modelling codes in seismology, and can incorporate new user-specified GF calculation methods. Pyrocko-GF defines an extensible GF storage format suitable for a wide range of GF types, handling especially elasticity- and wave propagation problems. The framework assists with visualisations, quality control and 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.

scholarly journals Beyond the Rigid Lid: Baroclinic Modes in a Structured Atmosphere

2017 ◽  
Vol 74 (11) ◽  
pp. 3551-3566 ◽  
Author(s):  
Jacob P. Edman ◽  
David M. Romps
Keyword(s):  
Gravity Wave ◽  
Gravity Waves ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Mode Decomposition ◽  
Wide Range ◽  
Tropospheric Heating ◽  
Baroclinic Modes

Abstract The baroclinic-mode decomposition is a fixture of the tropical-dynamics literature because of its simplicity and apparent usefulness in understanding a wide range of atmospheric phenomena. However, its derivation relies on the assumption that the tropopause is a rigid lid that artificially restricts the vertical propagation of wave energy. This causes tropospheric buoyancy anomalies of a single vertical mode to remain coherent for all time in the absence of dissipation. Here, the authors derive the Green’s functions for these baroclinic modes in a two-dimensional troposphere (or, equivalently, a three-dimensional troposphere with one translational symmetry) that is overlain by a stratosphere. These Green’s functions quantify the propagation and spreading of gravity waves generated by a horizontally localized heating, and they can be used to reconstruct the evolution of any tropospheric heating. For a first-baroclinic two-dimensional right-moving or left-moving gravity wave with a characteristic width of 100 km, its initial horizontal shape becomes unrecognizable after 4 h, at which point its initial amplitude has also been reduced by a factor of 1/π. After this time, the gravity wave assumes a universal shape that widens linearly in time. For gravity waves on a periodic domain the length of Earth’s circumference, it takes only 10 days for the gravity waves to spread their buoyancy throughout the entire domain.

Sensitivity kernels for seismic Fresnel volume tomography

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. U35-U46 ◽  
Author(s):  
Yuzhu Liu ◽  
Liangguo Dong ◽  
Yuwei Wang ◽  
Jinping Zhu ◽  
Zaitian Ma
Keyword(s):  
Green's Functions ◽  
Heterogeneous Media ◽  
Green’S Functions ◽  
Dominant Frequency ◽  
Sensitivity Kernel ◽  
Distribution Ranges ◽  
Band Limited ◽  
Homogeneous Media ◽  
Limited Sensitivity ◽  
Fresnel Volume Tomography

Fresnel volume tomography (FVT) offers higher resolution and better accuracy than conventional seismic raypath tomography. A key problem in FVT is the sensitivity kernel. We propose amplitude and traveltime sensitivity kernels expressed directly with Green’s functions for transmitted waves for 2D/3D homogeneous/heterogeneous media. The Green’s functions are calculated with a finite-difference operator of the full wave equation in the frequency-space domain. In the special case of homogeneous media, we analyze the properties of the sensitivity kernels extensively and gain new insight into these properties. According to the constructive interference of waves, the spatial distribution ranges of the monochromatic sensitivity kernels in FVT differ from each other greatly and are [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] periods of seismic waves, respectively, for 2D amplitude, 3D amplitude, 2D traveltime, and 3D traveltime conditions. We also have a new understanding of the relationship between raypath tomography and FVT. Within the first Fresnel volume of the dominant frequency, the band-limited sensitivity kernels of FVT in homogeneous media or smoothly heterogeneous media are very close to those of the dominant frequency. Thus, it is practical to replace the band-limited sensitivity kernel with a few selected frequencies or even the single dominant frequency to save computation when performing band-limited FVT. The numerical experiment proves that FVT using our sensitivity kernels can achieve more accurate results than traditional raypath tomography.

Dynamic Green’s functions in anisotropic piezoelectric, thermoelastic and poroelastic solids

1994 ◽  
Vol 447 (1929) ◽  
pp. 175-188 ◽  
Keyword(s):  
Time Domain ◽  
Unit Sphere ◽  
Green's Functions ◽  
Green’S Functions ◽  
Fundamental Solutions ◽  
Anisotropic Elasticity ◽  
Field Equations ◽  
One Dimensional ◽  
Time Harmonic ◽  
Point Forces

A procedure is described to generate fundamental solutions or Green’s functions for time harmonic point forces and sources. The linearity of the field equations permits the Green’s function to be represented as an integral over the surface of a unit sphere, where the integrand is the solution of a one-dimensional impulse response problem. The method is demonstrated for the theories of piezoelectricity, thermoelasticity, and poroelasticity. Time domain analogues are discussed and compared with known expressions for anisotropic elasticity.

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.

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.

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