arXiv subjects

Morten Jakobsen

Publications and source records attributed to Morten Jakobsen.

11 recordsLinked to original sources

arXiv2026

Frequency Domain Biot--Allard Equations for Isotropic and Anisotropic Poroelastic Media: Two-field formulations and iterative splitting

We present a frequency-domain formulation of Biot's dynamic poroelastic equations with frequency-dependent dissipation (Biot--Allard) for anisotropic, heterogeneous media with memory effects. Two equivalent two-field representations--a displacement-pressure and a velocity-pressure-rate formulation--enable stabilized iterative splitting. While coupling operators generally lack an adjoint or skew-adjoint relationship at finite frequencies, the velocity--pressure-rate representation restores a skew-adjoint structure in the quasi-static limit. We prove continuity of the coupling operators and coercivity of the diagonal blocks, essential for convergence of the L-stabilized splitting scheme. The frequency-domain setting eliminates convolutional memory terms, incorporates attenuation and dispersion via complex-valued parameters, and reduces the time-dependent problem to a family of elliptic boundary-value problems suited for parallel computation and multi-frequency inversion. A conforming Galerkin finite element discretization preserves block structure, and numerical experiments confirm robustness and capture frequency-dependent attenuation. To illustrate discretization independence, we include a large-scale wave simulation using a pseudo-spectral method. This work provides a rigorous and efficient framework for modeling wave phenomena in complex porous media.

math.NA↗

arXiv2026

Verification of reciprocity in anisotropic poroelastic wave simulation using symmetric Strang splitting

Poroelastic wave simulations are important for many applications relating fluid flow and wave characteristics in porous rock formations. Reciprocity is a key physical property of wave propagation in porous media that is important for such applications, even when viscous dissipation is present. However, numerical poroelastic simulations often fail to reproduce reciprocal responses because the discretization does not preserve the balance between reversible wave dynamics and irreversible fluid-solid drag. To address this, we formulate the Biot equations in terms of a continuous evolution operator split into a reversible (skew-adjoint) wave part and an irreversible (self-adjoint, non-positive) Darcy part, including the leading-order Johnson-Koplik-Dashen correction. This structure clarifies why reciprocity holds in the continuous equations and how it is easily broken in discrete form. Guided by this interpretation, we construct a symmetric second-order Strang-splitting scheme with half-step source injection. The method conserves energy in the reversible subsystem, treats Darcy dissipation unconditionally stably, and retains Courant limits similar to elastic solvers. Using a staggered pseudo-spectral discretization, we model multimode propagation in 2D VTI media and obtain cross-component reciprocity with a relative L2 misfit approaching machine precision, demonstrating that the discrete scheme inherits the symmetry properties of the continuous evolution operator.

math.NA↗

arXiv2026

Cayley-transform analysis and numerical validation of the convergent Born series for the Helmholtz equation

We develop an operator-theoretic framework for the Convergent Born Series (CBS) method applied to the Lippmann--Schwinger equation for high-frequency Helmholtz problems. In contrast to the Fourier-based analysis of Osnabrugge et al., our approach expresses the preconditioned Lippmann--Schwinger iteration entirely in terms of the resolvent of a self-adjoint background operator. This leads to a unitary Cayley-transform representation of the CBS iteration operator, from which we derive basis-independent bounds on its numerical range and a general convergence criterion valid on arbitrary bounded domains and for complex-valued wave numbers. Because the analysis does not rely on an explicit Green's function in the Fourier domain, the Cayley-transform framework extends naturally to a broader class of frequency-domain wave and diffusion equations whose fundamental solutions are not available in closed form. We further incorporate smoothly tapered complex-wavenumber absorbing layers that preserve the self-adjoint structure of the reference operator and enhance the contractivity of the iteration without modifying the differential operator. In addition to this theoretical generalization, we present a detailed numerical validation in which CBS solutions are benchmarked against PML-based finite-difference wavefield simulations. These experiments demonstrate that the operator-theoretic CBS formulation delivers accurate and stable results across a broad range of contrasts and frequencies, thereby significantly extending the applicability and theoretical foundation of the CBS method beyond previously analyzed settings.

math.NA↗

arXiv2026

Elastic waveform inversion for double-couple microseismic source estimation in vertically fractured transversely isotropic media

Accurate characterization of microseismic events during fluid injection in sedimentary formations is essential to mitigate environmental risks. The source mechanism for microseismic events related to a slip on a fault plane is given by a double-couple. Waveform inversion has emerged as a promising technique for estimating the moment tensor and the position vector of double-couple sources. In most applications of waveform inversion for the moment tensor of double-couple sources, the formation is typically assumed to be isotropic or, less frequently, transversely isotropic. Modification of the moment-tensor representation to account for anisotropy created by aligned vertical fractures in transversely isotropic formations has not been included while inverting microseismic waveform data. In this study on synthetic microseismic data, we present a waveform inversion algorithm that includes this modification, considering the formation in the focal region to be vertically fractured transversely isotropic (VFTI) and possessing orthorhombic symmetry. Since VFTI media lack rotational symmetry, no assumptions have been made about the orientation of the fault plane where the slip occurred. The moment tensor of double-couple sources is formulated in terms of the elastic parameters of the VFTI medium and geometrical parameters which are slip magnitude, slip angle, fault dip, and azimuth angle of the fault-normal. Source inversion is treated as a local optimization problem, and we invert for the source location and the geometrical parameters. These geometrical parameters are more directly constrained by seismic data than the moment tensor components and offer geologically meaningful insights. This approach enhances microseismic monitoring in fractured formations and can be extended to more complex anisotropic media, such as monoclinic systems.

physics.geo-ph↗

arXiv2024

A History-dependent Dynamic Biot Model

In this work, we consider a fully dynamic Biot model that includes memory effects due to evolving permeability. Time integrals are used to account for the change in structure. We propose an iterative splitting scheme for this model, extending the fixed-stress split for the quasi-static Biot. We use finite elements in space and a backward Euler discretization in time. The performance of the method is demonstrated through a numerical experiment

math.NA↗

arXiv2024

Efficient scattering approach to seismic full-waveform inversion in anisotropic elastic media with variable density

This paper introduces a novel matrix-free approach for full waveform inversion in anisotropic elastic media, incorporating density variation through the utilization of the distorted Born iterative method. This study aims to overcome the computational and storage challenges associated with the conventional matrix-based distorted Born iterative inversion method while accurately capturing the subsurface's anisotropic properties and density variations. An elastic integral equation is utilized to account for the anisotropic nature of elastic wave propagation, enabling more precise modeling of subsurface complexities. This integral equation is efficiently solved by a fast Fourier transform accelerated Krylov subspace method. Leveraging the integral equation with the distorted Born approximation, a linear relationship between the scattered wavefield and the model parameter perturbation is formulated for an integrated inversion scheme. To address the inherent ill-posedness of each linear inversion step, we formulate the normal equation with a regularization term. This is achieved by minimizing an objective function using the generalized Tikhonov method. Therefore, we can find an adequate solution for the inverse scattering problem by solving the normal equation. Following the physical interpretation of Green's function, the Fr{é}chet and adjoint operators within the normal equation can be employed in a matrix-free manner, allowing for significant improvement of the computational efficiency and memory demand without compromising accuracy. The proposed matrix-free full waveform inversion framework is thoroughly validated through extensive numerical experiments on synthetic datasets, showcasing its ability to reconstruct complex anisotropic structures and accurately recover stiffness parameters and density.

physics.geo-ph↗

arXiv2023

3D induction log modelling with integral equation method and domain decomposition preconditioning

The deployment of electromagnetic (EM) induction tools while drilling is one of the standard routines for assisting the geosteering decision-making process. The conductivity distribution obtained through the inversion of the EM induction log can provide important information about the geological structure around the borehole. To image the 3D geological structure in the subsurface, 3D inversion of the EM induction log is required. Because the inversion process is mainly dependent on forward modelling, the use of fast and accurate forward modelling is essential. In this paper, we present an improved version of the integral equation (IE) based modelling technique for general anisotropic media with domain decomposition preconditioning. The discretised IE after domain decomposition equals a fixed-point equation that is solved iteratively with either the block Gauss-Seidel or Jacobi preconditioning. Within each iteration, the inverse of the block matrix is computed using a Krylov subspace method instead of a direct solver. An additional reduction in computational time is obtained by using an adaptive relative residual stopping criterion in the iterative solver. Numerical experiments show a maximum reduction in computational time of 35 per cent compared to solving the full-domain IE with a conventional GMRES solver. Additionally, the reduction of memory requirement for covering a large area of the induction tool sensitivity enables acceleration with limited GPU memory. Hence, we conclude that the domain decomposition method is improving the efficiency of the IE method by reducing the computation time and memory requirement.

math.NA↗

arXiv2023

Integral equation method for microseismic wavefield modelling in anisotropic elastic media

In this paper, we present a frequency-domain volume integral method to model the microseismic wavefield in heterogeneous anisotropic-elastic media. The elastic wave equation is written as an integral equation of the Lippmann-Schwinger type, and the seismic source is represented as a general moment tensor. The displacement field due to a moment tensor source can be computed using the spatial derivative of the elastodynamic Green's function. The existing matrix-based implementation of the integral equation is computationally inefficient to model the wavefield in a three-dimensional earth. An integral equation for the particle displacement is, hence, formulated in a matrix-free manner through the application of the Fourier transform. The biconjugate gradient stabilized method is used to iteratively obtain the solution of this equation. We apply the numerical scheme to three different models in order of increasing geological complexity and obtain the elastic displacement fields corresponding to the different types of moment tensor sources. The volume integral method has an advantage over the time domain methods in regard to adding multiple sources since it can work with discrete frequencies, one by one, and limit the computational cost. The generated synthetic data can be useful in inversion for the microseismic source and model parameters.

physics.geo-ph↗

arXiv2021

Iterative solution of the Lippmann-Schwinger equation in strongly scattering acoustic media by randomized construction of preconditioners

In this work the Lippmann-Schwinger equation is used to model seismic waves in strongly scattering acoustic media. We consider the Helmholtz equation, which is the scalar wave equation in the frequency domain with constant density and variable velocity, and transform it to an integral equation of the Lippmann-Schwinger type. To directly solve the discretized problem with matrix inversion is time-consuming, therefore we use iterative methods. The Born series is a well-known scattering series which gives the solution with relatively small cost, but it has limited use as it only converges for small scattering potentials. There exist other scattering series with preconditioners that have been shown to converge for any contrast, but the methods might require many iterations for models with high contrast. Here we develop new preconditioners based on randomized matrix approximations and hierarchical matrices which can make the scattering series converge for any contrast with a low number of iterations. We describe two different preconditioners; one is best for lower frequencies and the other for higher frequencies. We use the fast Fourier transform both in the construction of the preconditioners and in the iterative solution, and this makes the methods efficient. The performance of the methods are illustrated by numerical experiments on two 2D models.

physics.comp-ph↗

arXiv2016

Efficient big data assimilation through sparse representation: A 3D benchmark case study in seismic history matching

In a previous work \citep{luo2016sparse2d_spej}, the authors proposed an ensemble-based 4D seismic history matching (SHM) framework, which has some relatively new ingredients, in terms of the type of seismic data in choice, the way to handle big seismic data and related data noise estimation, and the use of a recently developed iterative ensemble history matching algorithm. In seismic history matching, it is customary to use inverted seismic attributes, such as acoustic impedance, as the observed data. In doing so, extra uncertainties may arise during the inversion processes. The proposed SHM framework avoids such intermediate inversion processes by adopting amplitude versus angle (AVA) data. In addition, SHM typically involves assimilating a large amount of observed seismic attributes into reservoir models. To handle the big-data problem in SHM, the proposed framework adopts the following wavelet-based sparse representation procedure: First, a discrete wavelet transform is applied to observed seismic attributes. Then, uncertainty analysis is conducted in the wavelet domain to estimate noise in the resulting wavelet coefficients, and to calculate a corresponding threshold value. Wavelet coefficients above the threshold value, called leading wavelet coefficients hereafter, are used as the data for history matching. The retained leading wavelet coefficients preserve the most salient features of the observed seismic attributes, whereas rendering a substantially smaller data size. Finally, an iterative ensemble smoother is adopted to update reservoir models, in such a way that the leading wavelet coefficients of simulated seismic attributes better match those of observed seismic attributes. (The rest of the abstract was omitted for the length restriction.)

physics.data-an↗

arXiv2016

An Ensemble 4D Seismic History Matching Framework with Sparse Representation Based on Wavelet Multiresolution Analysis

In this work we propose an ensemble 4D seismic history matching framework for reservoir characterization. Compared to similar existing frameworks in reservoir engineering community, the proposed one consists of some relatively new ingredients, in terms of the type of seismic data in choice, wavelet multiresolution analysis for the chosen seismic data and related data noise estimation, and the use of recently developed iterative ensemble history matching algorithms. Typical seismic data used for history matching, such as acoustic impedance, are inverted quantities, whereas extra uncertainties may arise during the inversion processes. In the proposed framework we avoid such intermediate inversion processes. In addition, we also adopt wavelet-based sparse representation to reduce data size. Concretely, we use intercept and gradient attributes derived from amplitude versus angle (AVA) data, apply multilevel discrete wavelet transforms (DWT) to attribute data, and estimate noise level of resulting wavelet coefficients. We then select the wavelet coefficients above a certain threshold value, and history-match these leading wavelet coefficients using an iterative ensemble smoother. (The rest of the abstract is omitted for exceeding the limit of length)

physics.data-an↗