The multilevel fast multipole method for forward modelling the multiply scattered seismic surface waves

Çakır, Özcan

doi:10.1111/j.1365-246x.2006.02928.x

Cited by 10 publications

(10 citation statements)

References 30 publications

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“…Bapat et al ., ]. Its concept has only been sparsely used by the geophysical community with applications reported for the 3‐D direct current resistivity problem [ Blome et al ., ], the seismic wave propagation problem [ Çakir , ; Çakır , ], and the 3‐D electromagnetic induction problem [ Ren et al ., , ]. As for gravity modeling problems, only geometrically simple Earth models with 3‐D structured grids were considered [ May and Knepley , ].…”

Section: Introductionmentioning

confidence: 99%

Fast 3‐D large‐scale gravity and magnetic modeling using unstructured grids and an adaptive multilevel fast multipole method

et al. 2017

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A novel fast and accurate algorithm is developed for large‐scale 3‐D gravity and magnetic modeling problems. An unstructured grid discretization is used to approximate sources with arbitrary mass and magnetization distributions. A novel adaptive multilevel fast multipole (AMFM) method is developed to reduce the modeling time. An observation octree is constructed on a set of arbitrarily distributed observation sites, while a source octree is constructed on a source tetrahedral grid. A novel characteristic is the independence between the observation octree and the source octree, which simplifies the implementation of different survey configurations such as airborne and ground surveys. Two synthetic models, a cubic model and a half‐space model with mountain‐valley topography, are tested. As compared to analytical solutions of gravity and magnetic signals, excellent agreements of the solutions verify the accuracy of our AMFM algorithm. Finally, our AMFM method is used to calculate the terrain effect on an airborne gravity data set for a realistic topography model represented by a triangular surface retrieved from a digital elevation model. Using 16 threads, more than 5800 billion interactions between 1,002,001 observation points and 5,839,830 tetrahedral elements are computed in 453.6 s. A traditional first‐order Gaussian quadrature approach requires 3.77 days. Hence, our new AMFM algorithm not only can quickly compute the gravity and magnetic signals for complicated problems but also can substantially accelerate the solution of 3‐D inversion problems.

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Section: Introductionmentioning

confidence: 99%

Fast 3‐D large‐scale gravity and magnetic modeling using unstructured grids and an adaptive multilevel fast multipole method

et al. 2017

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“…Despite its great effectiveness, the FMM (or MLFMM) has found relatively few applications in seismology (i.e. Pollitz 2002; Fujiwara 2000, 1998; Çakır 2006).…”

Section: Introductionmentioning

confidence: 99%

Forward modelling the multiply scattered 2.5-D teleseismicPwaves accelerated by the multilevel fast multipole method

Çakır

2009

Geophysical Journal International

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S U M M A R YThe present study introduces an algorithm to compute the forward modelling of multiple scattering. The propagating medium is defined, in general, by 3-D heterogeneities embedded in a 1-D layered structure, assumed as the reference structure. The 3-D formulation is consequently modified to suit 2.5-D scattering modelling. The basic formulation relies on the Green's function method and employs (1) the reflectivity method (RM) to acquire the 1-D multilayered, half-space Green's function, (2) the multilevel fast multipole method (MLFMM) to gain speed and (3) the generalized minimal residual method (GMRES) to iteratively solve the linear system in a memory efficient manner. The heterogeneities are modelled as point scatterers, creating a secondary wavefield that superimposes on the waves in the reference structure. The algorithm is efficient since the CPU time increases logarithmically with the size of the model. The test cases included various anomalous lithosphere structures, ranging from a simple point scatterer to a suture zone.

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“…This case corresponds to x ∈ Ω F , for which one has y 3 > x 3 since only the case y 3 = 0 needs to be considered for enforcing (5). Taking the partial Fourier transform of (13) and exploiting (16a) and (18a) therein, one arrives at…”

Section: Derivation In the Fourier Spacementioning

confidence: 99%

“…Elastodynamic FMM formulations in the frequency domain have been investigated in e.g. [5] and [16] for the computation of seismic waves. The methodology of [16] has later been improved in [6] for homogeneous semi-infinite elastic propagation domains, through incorporation of recent advances of FMM implementations for Maxwell equations [10], allowing to run BEM models of size up to N = O(10 6 ) on a single-processor PC.…”

Section: Introductionmentioning

confidence: 99%

A new Fast Multipole formulation for the elastodynamic half-space Greenʼs tensor

Chaillat

Bonnet

2014

Journal of Computational Physics

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Abstract. In this article, a version of the frequency-domain elastodynamic Fast Multipole-Boundary Element Method (FM-BEM) for semi-infinite media, based on the half-space Green's tensor (and hence avoiding any discretization of the planar traction-free surface), is presented. The half-space Green's tensor is often used (in non-multipole form until now) for computing elastic wave propagation in the context of soil-structure interaction, with applications to seismology or civil engineering. However, unlike the fullspace Green's tensor, the elastodynamic half-space Green's tensor cannot be expressed using derivatives of the Helmholtz fundamental solution. As a result, multipole expansions of that tensor cannot be obtained directly from known expansions, and are instead derived here by means of a partial Fourier transform with respect to the spatial coordinates parallel to the free surface. The obtained formulation critically requires an efficient quadrature for the Fourier integral, whose integrand is both singular and oscillatory. Under these conditions, classical Gaussian quadratures would perform poorly, fail or require a large number of points. Instead, a version custom-tailored for the present needs of a methodology proposed by Rokhlin and coauthors, which generates generalized Gaussian quadrature rules for specific types of integrals, has been implemented. The accuracy and efficiency of the proposed formulation is demonstrated through numerical experiments on single-layer elastodynamic potentials involving up to about N = 6×10 5 degrees of freedom. In particular, a complexity significantly lower than that of the non-multipole version is shown to be achieved. IntroductionThe main advantage of the boundary element method (BEM) is that only the domain boundaries (and possibly interfaces) are discretized, leading to a reduction of the number of degrees of freedom (DOFs) relative to domain-discretization methods. Moreover, elastodyamic field equations and radiation conditions are exactly satisfied by the formulation [26], which avoids cumulative effects of grid dispersion and makes the BEM well suited for modeling wave propagation in unbounded domains. However, the standard BEM [3] leads to fully-populated matrices, which results in high computational complexity (O(N 2 ) per iteration using an iterative solver such as GMRES, where N denotes the number of boundary degrees of freedom (DOFs) of the BE model) and severe problem size limitations induced by the O(N 2 ) size of the influence matrix.The advent of accelerated BE methodologies has dramatically improved the capabilities of BEMs for many areas of application, in a large part owing to the rapid development of the Fast Multipole Method (FMM) over the last 15-20 years [29]. Such approaches have resulted in considerable solution speedup, memory reduction and model size increase. The FMM inherently relies on iterative solvers (usually GMRES), so as to avoid actual computation and storage of the fully-populated influence matrix, and is known to require O(N log N )...

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The multilevel fast multipole method for forward modelling the multiply scattered seismic surface waves

Cited by 10 publications

References 30 publications

Fast 3‐D large‐scale gravity and magnetic modeling using unstructured grids and an adaptive multilevel fast multipole method

Fast 3‐D large‐scale gravity and magnetic modeling using unstructured grids and an adaptive multilevel fast multipole method

Forward modelling the multiply scattered 2.5-D teleseismicPwaves accelerated by the multilevel fast multipole method

A new Fast Multipole formulation for the elastodynamic half-space Greenʼs tensor

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