This paper introduces optimally-blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations. To quantify the approximation errors when we modify the inner products, we generalize the Pythagorean eigenvalue theorem of Strang and Fix. The proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and nonuniform meshes as well as for different polynomial orders and continuity of the basis. The optimally-blended schemes improve the convergence rate of the method by two orders with respect to the fully-integrated Galerkin method. The proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems.

We develop and analyze quadrature blending schemes that minimize the dispersion error of isogeometric analysis up to polynomial order seven with maximum continuity in the span. The schemes yield two extra orders of convergence (superconvergence) on the eigenvalue errors, while the eigenfunction errors are of optimal convergence order. Both dispersion and spectrum analysis are unified in the form of a Taylor expansion for eigenvalue errors. The resulting schemes increase the accuracy and robustness of isogeometric analysis for wave propagation as well as the differential eigenvalue problems. We also derive an a posteriori error estimator for the eigenvalue error based on the superconvergence result. We verify with numerical examples the analysis of the performance of the proposed schemes.

Geophysical inversion attempts to estimate the distribution of physical properties in the Earth's interior from observations collected at or above the surface. Inverse problems are commonly posed as least-squares optimization problems in highdimensional parameter spaces. Existing approaches are largely based on deterministic gradient-based methods, which are limited by nonlinearity and nonuniqueness of the inverse problem. Probabilistic inversion methods, despite their great potential in uncertainty quantification, still remain a formidable computational task. In this paper, I explore the potential of deep learning methods for electromagnetic inversion. This approach does not require calculation of the gradient and provides results instantaneously. Deep neural networks based on fully convolutional architecture are trained on large synthetic datasets obtained by full 3-D simulations. The performance of the method is demonstrated on models of strong practical relevance representing an onshore controlled source electromagnetic CO 2 monitoring scenario. The pre-trained networks can reliably estimate the position and lateral dimensions of the anomalies, as well as their resistivity properties. Several fully convolutional network architectures are compared in terms of their accuracy, generalization, and cost of training. Examples with different survey geometry and noise levels confirm the feasibility of the deep learning inversion, opening the possibility to estimate the subsurface resistivity distribution in real time.

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Vertical seismic profile (VSP) is one of the technologies for monitoring hydrocarbon production and CO2 geosequestration. However, quantitative interpretation of time‐lapse VSP is challenging due to its irregular distribution of source‐receiver offsets. One way to overcome this challenge is to use full waveform inversion (FWI), which does not require regular offsets. We present a workflow of elastic FWI applied to offset vertical seismic profile data for the purpose of identification and estimation of time‐lapse changes introduced by injection of 15,000 t of CO2‐rich gas mixture at 1.5 km depth. Application of this workflow to both synthetic and field data shows that elastic FWI is able to detect and quantify the time‐lapse anomaly in P wave velocity with the magnitude of 100–150 m/s.

We develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation. The rules are optimal in the sense that they only require two quadrature points per element to minimize the dispersion error [8], and they are equivalent to the optimized blending rules we recently described. Our approach further simplifies the numerical integration: instead of blending two three-point standard quadrature rules, we construct directly a single two-point quadrature rule that reduces the dispersion error to the same order for uniform meshes with periodic boundary conditions. Also, we present a 2.5-point rule for both uniform and non-uniform meshes with arbitrary boundary conditions. Consequently, we reduce the computational cost by using the proposed quadrature rules. Various numerical examples demonstrate the performance of these quadrature rules.

Distributed acoustic sensing (DAS) is a rapidly developing technology particularly useful for the acquisition of vertical seismic profile (VSP) surveys. DAS data are increasingly used for seismic imaging, but not for estimating rock properties. We propose a workflow for estimating elastic properties of the subsurface using full waveform inversion (FWI) of DAS VSP data. Whereas conventional borehole geophones usually measure three components of particle velocity, DAS measures a single quantity, which is an approximation of the strain or strain rate along the fiber. Standard FWI algorithms are developed for particle velocity data, and hence their application to DAS data requires conversion of these data to particle velocity along the fiber. This conversion can be accomplished by a specially designed filter. Field measurements show that the conversion result is close to vertical particle velocity as measured by geophones. Elastic time-domain FWI of a synthetic multi-offset VSP dataset for a vertical well shows that the inversion of the vertical component alone is sufficient to recover elastic properties of the subsurface. Application of the proposed workflow to a multioffset DAS dataset acquired at the CO2CRC Otway Project site in Victoria, Australia reveals salient subhorizontal layering consistent with known geology of the site. The inverted VP model at the well location matches the upscaled VP log with a correlation coefficient of 0.85.