Reverse time migration with an exact two way illumination compensation

Reverse time migration (RTM) has been widely used for imaging complex subsurface structures in oil and gas exploration. However, because only the adjoint of the forward Born modeling operator is applied to the seismic data in RTM, the output migration profile is biased in terms of the amplitude. To help partially balance the amplitude performance, the RTM image can be preconditioned with the inverse of the diagonal of the Hessian operator. Yet, existing preconditioning methods do not correctly consider the receiver-side effects, assuming that the receiver coverage is infinite or the velocity model is constant. We therefore provide a comparative study aiming to give a clearer understanding on the importance of incorporating the receiver-side effects by developing a frequency-domain scattering-integral reverse time migration (SI-RTM). In the proposed SI-RTM, the diagonal of the Hessian operator is explicitly computed in its exact formulation, and the source-side wavefield and receiver-side Green’s functions are obtained by solving the two-way wave equation. The computational cost is relatively affordable when compared with the more expensive least-squares RTM. In the comparative counterpart, the diagonal of the Hessian operator is approximated by the source-side illumination. We perform two synthetic numerical examples using an overthrust model and a complex reservoir model; the final migration images were significantly improved when the receiver-side effects were accurately considered. A third application of SI-RTM on one field data set acquired from the East China Sea further demonstrates the importance of incorporating the receiver-side effects in normalizing the RTM image. Findings of this study are expected to provide a theoretical basis for improving the ability of RTM imaging of subsurface structures, thereby critically advancing the application of geophysical techniques for imaging complex environments.

The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(\Omega,f)$

Let $\Omega\subset \mathbb{C}^{n}$ be a bounded $m$-hyperconvex domain, where $m$ is an integer such that $1\leq m\leq n$. Let $\mu$ be a positive Borel measure on $\Omega$. We show that if the complex Hessian equation $H_m (u) = \mu$ admits a (weak) subsolution in $\Omega$, then it admits a (weak) solution with a prescribed least maximal $m$-subharmonic majorant in $\Omega$.

Poincaré Inequality for a Mesh-Dependent 2-Norm on Piecewise Linear Surfaces with Boundary

We establish several useful estimates for a non-conforming 2-norm posed on piecewise linear surface triangulations with boundary, with the main result being a Poincaré inequality. We also obtain equivalence of the non-conforming 2-norm posed on the true surface with the norm posed on a piecewise linear approximation. Moreover, we allow for free boundary conditions. The true surface is assumed to be C 2 , 1 C^{2,1} when free conditions are present; otherwise, C 2 C^{2} is sufficient. The framework uses tools from differential geometry and the closest point map (see [G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, Partial Differential Equations and Calculus of Variations, Lecture Notes in Math. 1357, Springer, Berlin (1988), 142–155]) for approximating the full surface Hessian operator. We also present a novel way of applying the closest point map when dealing with surfaces with boundary. Connections with surface finite element methods for fourth-order problems are also noted.

Different Types of Curvature and Their Vanishing Conditions

In the present paper, I studied different types of Curvature like Riemannian Curvature, Concircular Curvature, Weyl Curvature, and Projective Curvature in Quarter Symmetric non-Metric Connection in P-Sasakian manifold. A comparative study of a manifold with a Riemannian connection is done with a P-Sasakian Manifold. Conditions for vanishing for different types of curvature are also a part of the study. Some necessary properties of the Hessian operator are discussed with respect to all curvatures as well.

Migration deconvolution using domain decomposition

Least-squares migration (LSM) is an effective technique for mitigating blurring effects and migration artifacts generated by the limited data frequency bandwidth, incomplete coverage of geometry, source signature, and unbalanced amplitudes caused by complex wavefield propagation in the subsurface. Migration deconvolution (MD) is an image-domain approach for least-squares migration, which approximates the Hessian operator using a set of precomputed point spread functions (PSFs). We introduce a new workflow by integrating the MD and the domain decomposition (DD) methods. The DD techniques aim to solve large and complex linear systems by splitting problems into smaller parts, facilitating parallel computing, and providing a higher convergence in iterative algorithms. The following proposal suggests that instead of solving the problem in a unique domain, as conventionally performed, we split the problem into subdomains that overlap and solve each of them independently. We accelerate the convergence rate of the conjugate gradient solver by applying the DD methods to retrieve a better reflectivity, which is mainly visible in regions with low amplitudes. Moreover, using the pseudo-Hessian operator, the convergence of the algorithm is accelerated, suggesting that the inverse problem becomes better conditioned. Experiments using the synthetic Pluto model demonstrate that the proposed algorithm dramatically reduces the required number of iterations while providing a considerable enhancement in the image resolution and better continuity of poorly illuminated events.

Optimal Łojasiewicz–Simon inequalities and Morse–Bott Yang–Mills energy functions

For any compact Lie group 𝐺 and closed, smooth Riemannian manifold ( X , g ) (X,g) of dimension d ≥ 2 d\geq 2 , we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal 𝐺-bundle over 𝑋 supporting a connection with L p L^{p} -small curvature, when p > d / 2 p>d/2 , to the case of a connection with L d / 2 L^{d/2} -small curvature. We prove an optimal Łojasiewicz–Simon gradient inequality for abstract Morse–Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis (2019), principally by removing the hypothesis that the Hessian operator be Fredholm with index zero. We apply this result to prove the optimal Łojasiewicz–Simon gradient inequality for the self-dual Yang–Mills energy function near regular anti-self-dual connections over closed Riemannian four-manifolds and for the full Yang–Mills energy function over closed Riemannian manifolds of dimension d ≥ 2 d\geq 2 , when known to be Morse–Bott at a given Yang–Mills connection. We also prove the optimal Łojasiewicz–Simon gradient inequality by direct analysis near a given flat connection that is a regular point of the curvature map.

A remark on the existence of positive radial solutions to a Hessian system

<abstract><p>We give new conditions for the study of existence of positive radial solutions for a system involving the Hessian operator. The solutions to be obtained are given by successive-approximation. Our interest is to improve the works that deal with such systems at the present and to give future directions of research related to this work for researchers.</p></abstract>

A remark on the existence of positive radial solutions to a Hessian system

<abstract><p>We give new conditions for the study of existence of positive radial solutions for a system involving the Hessian operator. The solutions to be obtained are given by successive-approximation. Our interest is to improve the works that deal with such systems at the present and to give future directions of research related to this work for researchers.</p></abstract>