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Mulder, W.A. (author)To estimate the depth errors in a subsurface model obtained from the inversion of seismic data, the stationary-phase approximation in a two-dimensional constant-velocity model with a dipped reflector is applied to migration with a time-shift extension. This produces two asymptotic solutions: one is a straight line, and the other is a curve....journal article 2024
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Mulder, W.A. (author), Shamasundar, R. (author)Dispersion error analysis can help to assess the performance of finite-element discretizations of the wave equation. Although less general than the convergence estimates offered by standard finite-element error analysis, it can provide more detailed insight as well as practical guidelines in terms of the number of elements per wavelength needed...journal article 2024
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Mulder, W.A. (author)Finite elements with polynomial basis functions on the simplex with a symmetric distribution of nodes should have a unique polynomial representation. Unisolvence not only requires that the number of nodes equals the number of independent polynomials spanning a polynomial space of a given degree, but also that the Vandermonde matrix...journal article 2023
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Mulder, W.A. (author)Finite elements with mass lumping allow for explicit time stepping when modelling wave propagation and can be more efficient than finite differences in complex geological settings. In two dimensions on quadrilaterals, spectral elements are the obvious choice. Triangles offer more flexibility for meshing, but the construction of polynomial...journal article 2023
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Mulder, W.A. (author)Temporal dispersion correction of second-order finite-difference time stepping for numerical wave propagation modelling exploits the fact that the discrete operator is exact but for the wrong frequencies. Mapping recorded traces to the correct frequencies removes the numerical error. Most of the implementations employ forward and inverse...journal article 2023
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Mulder, W.A. (author)When solving the wave equation with finite elements, mass lumping allows for explicit time stepping, avoiding the cost of a lower-upper decomposition of the large sparse mass matrix. Mass lumping on the reference element amounts to numerical quadrature. The weights should be positive for stable time stepping and preserve numerical accuracy....journal article 2022
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Werthmüller, D. (author), Mulder, W.A. (author), Slob, E.C. (author)3-D controlled-source electromagnetic data are often computed directly in the domain of interest, either in the frequency domain or in the time domain. Computing it in one domain and transforming it via a Fourier transform to the other domain is a viable alternative. It requires the evaluation of many responses in the computational domain if...journal article 2021
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Mulder, W.A. (author)ABSTRACTWhen modeling wave propagation, truncation of the computational domain to a manageable size requires nonreflecting boundaries. To construct such a boundary condition on one side of a rectangular domain for a finite-difference discretization of the acoustic wave equation in the frequency domain, the domain is extended on that one side to...journal article 2021
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Mulder, W.A. (author)Simulations of wave propagation in the Earth usually require truncation of a larger domain to the region of interest to keep computational cost acceptable. This introduces artificial boundaries that should not generate reflected waves. Most existing boundary conditions are not able to completely suppress all the reflected energy, but suffice...journal article 2020
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Mulder, W.A. (author)The representation of a force or moment point source in a spectral finite-element code for modelling elastic wave propagation becomes fundamentally different in degenerate cases where the source is located on the boundary of an element. This difference is related to the fact that the finite-element basis functions are continuous across...journal article 2020
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Werthmüller, D. (author), Mulder, W.A. (author), Slob, E.C. (author)Controlled-source electromagnetic (CSEM) surveys are a common geophysical investigationtool in the search for, amongst other, groundwater, hydrocarbons, and minerals. The nu-merical modelling of CSEM data requires the solution of the Maxwell equations. These canbe simplified in the particular case of CSEM, as the frequencies used in surveys are...journal article 2019
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Geevers, S. (author), Mulder, W.A. (author), van der Vegt, J.J.W. (author)We present new and efficient quadrature rules for computing the stiffness matrices of mass-lumped tetrahedral elements for wave propagation modeling. These quadrature rules allow for a more efficient implementation of the mass-lumped finite element method and can handle materials that are heterogeneous within the element without loss of the...journal article 2019
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Geevers, S. (author), Mulder, W.A. (author), van der Vegt, J. J.W. (author)We analyse the dispersion properties of two types of explicit finite element methods for modelling acoustic and elastic wave propagation on tetrahedral meshes, namely mass-lumped finite element methods and symmetric interior penalty discontinuous Galerkin methods, both combined with a suitable Lax–Wendroff time integration scheme. The...journal article 2018
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Mulder, W.A. (author)One way to deal with the storage problem for the forward source wavefield in reverse time migration and full-waveform inversion is the reconstruction of that wavefield during reverse time stepping along with the receiver wavefield. Apart from the final states of the source wavefield, this requires a strip of boundary values for the whole time...journal article 2018
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Geevers, S. (author), Mulder, W.A. (author), van der Vegt, J. J.W. (author)We present a new accuracy condition for the construction of continuous mass-lumped elements. This condition is less restrictive than the one currently used and enabled us to construct new mass-lumped tetrahedral elements of degrees 2 to 4. The new degree-2 and degree-3 tetrahedral elements require 15 and 32 nodes per element, respectively, while...journal article 2018
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Shamasundar, R. (author), Mulder, W.A. (author)Finite elements can, in some cases, outperform finite-difference methods for modelling wave propagation in complex geological models with topography. In the weak form of the finiteelement method, the delta function is a natural way to represent a point source. If, instead of the usual second-order form, the first-order form of the wave equation...journal article 2018
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Ponomarenko, A. V. (author), Kashtan, B. M. (author), Troyan, V. N. (author), Mulder, W.A. (author)Surface waves are often used to estimate a near-surface shear-velocity profile. The inverse problem is solved for the locally one-dimensional problem of a set of homogeneous horizontal elastic layers. The result is a set of shear velocities, one for each layer. To obtain a P-wave velocity profile, the P-guided waves should be included in the...journal article 2017
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Mulder, W.A. (author)The presence of topography poses a challenge for seismic modeling with finite-difference codes. The representation of topography by means of an air layer or vacuum often leads to a substantial loss of numerical accuracy. A suitable modification of the finite-difference weights near the free surface can decrease that error. An existing...journal article 2017
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Kadu, Ajinkya (author), van Leeuwen, Tristan (author), Mulder, W.A. (author)Seismic full-waveform inversion tries to estimate subsurface medium parameters from seismic data. Areas with subsurface salt bodies are of particular interest because they often have hydrocarbon reservoirs on their sides or underneath. Accurate reconstruction of their geometry is a challenge for current techniques. This paper presents a...journal article 2017
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Bharadwaj, Pawan (author), Drijkoningen, G.G. (author), Mulder, W.A. (author), Thorbecke, J.W. (author), Neducza, Boriszlav (author), Jenneskens, R (author)In the near surface with unconsolidated soils, shear-wave properties can often be characterised better and with a higher resolution than compressional-wave properties. To enable imaging ahead of a tunnel-boring machine, we developed a seismic prediction system with a few shear-wave vibrators and horizontal receivers. The boring process is...journal article 2017