Up and down deconvolution in complex geological scenarios

Summary The interferometric approach, often referred to as up-down deconvolution, is well-known and applied to attenuate free-surface-related multiples in seabed surveys. In a horizontally layered medium, the upgoing wavefield can be expressed as the convolution of the downgoing wavefield with the earth’s reflectivity for each plane-wave component in which the wavefields are decomposed. Consequently, deconvolving the downgoing wavefield from the upgoing wavefield, gives us the earth’s reflectivity response for a certain plane-wave component. It has been experimentally observed that deconvolution carried out one plane-wave component at a time gives reasonable results, even in the presence of complex subsurface structures, provided that the seabed is relatively flat. When these geological conditions are not satisfied, the same problem can be formulated in terms of interferometric redatuming using multidimensional deconvolution. The inverse problem that performs the deconvolution is usually ill-conditioned and, hence, must be regularized. We propose to mitigate numerical instability by adding a regularization term on the solution norm and by introducing lateral weighted regularization to exploit the similarity across different receiver and source locations.

The Numerical Solution of Integral Equations on the Half-Line

Numerial Methods for Volterra Integral Equations with Singular Kernels

Seismic interferometry (SI) allows retrieval of virtual-source responses at positions of receivers, where no actual source is shot, by cross-correlating (CC) the seismic responses between receivers. The theory requires a boundary of subsurface sources to retrieve the surface reflection response. With reflection data acquired with both sources and receivers at the Earth’s surface, the retrieved virtual-source reflection responses suffer from non-physical arrivals and amplitude errors that may be significant. Instead of using the common CC method, we propose an approximate method to apply SI by multidimensional deconvolution (MDD). The method is data-driven and does not require a priori information about the subsurface. Numerical results show that, although its effect on the non-physical arrivals is limited, the MDD method clearly improves the retrieved amplitudes by flattening the spectrum and balancing the illumination of the virtual-source responses. Therefore, the virtual-source response retrieved by MDD is a better estimate of the reflection response than the response retrieved by CC. The additional reflection data retrieved by MDD have higher potential of filling in possibly missing source data in the original dataset.

Seismic reflection data can be redatumed to a specified boundary in the subsurface by solving an inverse (or multidimensional deconvolution) problem. The redatumed data can be interpreted as an extended image of the subsurface at the redatuming boundary, depending on the subsurface offset and time. We retrieve target-enclosed extended images by using two redatuming boundaries, which are selected above and below a specified target volume. As input, we require the upgoing and downgoing wavefields at both redatuming boundaries due to impulsive sources at the earth’s surface. These wavefields can be obtained from actual measurements in the subsurface, they can be numerically modeled, or they can be retrieved by solving a multidimensional Marchenko equation. As output, we retrieved virtual reflection and transmission responses as if sources and receivers were located at the two target-enclosing boundaries. These data contain all orders of reflections inside the target volume but exclude all interactions with the part of the medium outside this volume. The retrieved reflection responses can be used to image the target volume from above or from below. We found that the images from above and below are similar (given that the Marchenko equation is used for wavefield retrieval). If a model with sharp boundaries in the target volume is available, the redatumed data can also be used for two-sided imaging, where the retrieved reflection and transmission responses are exploited. Because multiple reflections are used by this strategy, seismic resolution can be improved significantly. Because target-enclosed extended images are independent on the part of the medium outside the target volume, our methodology is also beneficial to reduce the computational burden of localized inversion, which can now be applied inside the target volume only, without suffering from interactions with other parts of the medium.

The first section of this chapter collects results from [240] which gives a further contribution to the analysis of the hp-version of the boundary element method (BEM) by presenting a more general result for Dirichlet and Neumann problems than [21] allowing the use of a general geometric mesh refinement on the polygonal boundary Γ. Here as in [240] we prove the exponential convergence of the hp- version of the boundary element method by exploiting only features of the solutions of the boundary integral equations. The key result in this approach is an asymptotic expansion of the solution of the integral equations in singularity functions reflecting the singular behaviour of the solutions near corners of Γ. With such expansions we show that the solutions of the integral equations belong to countably normed spaces. Therefore these solutions can be approximated exponentially fast in the energy norm via the hp- Galerkin solutions of those integral equations. This result is not restricted to integral equations which stem from boundary value problems for the Laplacian but applies to Helmholtz problems as well. Further applications are 2D crack problems in linear elasticity. For numerical experiments with hp-version (BEM) see [165, 340].

Surface-related multiples remain an obstacle to accurate subsurface imaging across many acquisition settings. Multi-dimensional deconvolution (MDD) offers an effective way to remove these surface-related multiples and the source signature, but its performance can deteriorate when the source wavelet or overburden structure is complex. To improve the stability and accuracy of MDD, we reformulate it within a plug-and-play regularization framework. In this approach, an implicit prior, implemented through a denoiser acting as a proximal operator, guides the inversion process. This enables the use of advanced denoisers to suppress incoherent noise while preserving geologically meaningful events, leading to cleaner wavefield retrieval and a more reliable reflection response. Test on a synthetic dataset constructed from drill-bit noise shows that the proposed approach yields cleaner reflection responses and fewer imaging artifacts than standard MDD. Application to the Volve field data further demonstrates the robustness and practical value of the method, indicating its potential for stable and high-fidelity seismic imaging.

By integrating the electric field integral equation and the transmission line equation, an iterative solution for the electromagnetic field coupling to buried wires is obtained. At first we establish the integral equation which the difference between solutions of the integral equation and the telegraph equation satisfies. Then the solution of the telegraph equation is used to approximate the solution of this integral equation. Every following step of iteration is an improvement on the transmission line solution, and with several iterations, a well approximation to the solution of electric field integral equation can be obtained.

We consider a special case of the fourth Painlevé equation given by d 2 ƞ / dξ 2 = 3 ƞ 5 + 2ξ ƞ 3 + (1/4ξ 2 - v - 1/2 ) ƞ , (1) with v a parameter, and seek solutions ƞ (ξ; v ) satisfying the boundary condition ƞ (∞)=0. (2) Equation (1) arises as a symmetry reduction of the derivative nonlinear Schrödinger (DNLS) equation, which is a completely integrable soliton equation solvable by inverse scattering techniques. Solutions of equation (1), satisfying (2), are expressed in terms of the solutions of linear integral equations obtained from the inverse scattering formalism for the DNLS equation. We obtain exact ‘bound state’ solutions of equation (1) for v = n , a positive integer, using the integral equation representation, which decay exponentially as ξ→ ± ∞ and are the first example of such solutions for the Painlevé equations. Additionally, using Bäcklund transformations for the fourth Painlevé equation, we derive a nonlinear recurrence relation (commonly referred to as a Bäcklund transformation in the context of soliton equations) for equation (1) relating ƞ (ξ; v ) and ƞ (ξ; v + 1).

Numerical solution of two-dimensional stochastic Fredholm integral equations on hypercube domains via meshfree approach

Background/Objectives: Integral transforms and iterative techniques play a central role in the theory and solution of integral equations. In this work, generalized special functions, namely Prefunctions and Extended Prefunctions are used to obtain exact and approximate solutions of Volterra integral equations of the second kind. These functions are parameterized by 𝛼 and extend the analytic structure of exponential, hyperbolic and trigonometric functions. These functions are effective tools in solving differential and integral equations. Methods: In this study successive approximations method is used to derive closed-form solutions for two classes of Volterra integral equations with different kernels. For the kernel 𝐾(𝑥, 𝑡) = 𝑥 − 𝑡 with 𝑓 (𝑥, 𝛼) = 𝑝𝑠𝑖𝑛(𝑥, 𝛼) the iterative sequence yields a generalized solution expressed as a series of shifted Prefunctions, reducing to 𝑦(𝑥, 0) = 1 2 (𝑝𝑠𝑖𝑛ℎ(𝑥) + 𝑠𝑖𝑛(𝑥)) for 𝛼 =0 confirming the consistency of the method with known classical results. For the convolution kernel 𝐾(𝑥, 𝑡) = 𝑒−(𝑥−𝑠) with 𝑓(𝑥, 𝛼) = 𝑝𝑠𝑖𝑛ℎ(𝑥, 𝛼) the Picard iteration process leads to the compact representation 𝑦(𝑥, 𝛼) = 𝑝𝑠𝑖𝑛ℎ(𝑥, 𝛼) + 𝑠𝑖𝑛(𝑥, 𝛼 + 1). Findings: Numerical illustrations and graphical results confirm uniform convergence of the successive approximations to the exact solution for various values of 𝛼. For a large value of 𝛼 convergence rate is higher. Novelty: Prefunction -based methods provide generalized frameworks for solving Volterra integral equations and open further directions for applications in mathematical physics and engineering models. 2020 Mathematics Subject Classification: 45Dxx, 34A12. Keywords: Integral Equations; Prefunctions; Successive Approximation Method; Convergence; Picard Iterates

C15 - ON THE INTERACTION BETWEEN CUBIC ANHARMONICITY AND A SECOND-ORDER ELECTRIC MOMENT IN THE OPTICAL ABSORPTION OF CRYSTALS

Multi-Dimensional Deconvolution is a data-driven method that is at the center of key seismic processing applications - from suppressing multiples to inversion-based imaging. When posed in an interferometric context, it can grant access to overburden-free seismic virtual surveys at a given datum in the subsurface. As such, it constitutes an essential processing operation that achieves multiple imaging objectives simultaneously in redatuming or target-oriented imaging: e.g., suppressing multiples, removing complex overburden effects, and retrieving amplitude consistent image gathers for impedance inversion. Despite its potential, the deconvolution process relies on the solution of an ill-conditioned linear inverse problem sensitive to noise artifacts due to incomplete acquisition, limited sources, and band-limited data. Typically, this inversion is performed in the Fourier domain where the estimation of optimal regularization parameters hinders accurate waveform reconstruction. We reformulate the problem in the time domain - long believed to be computationally intractable - and introduce several physical constraints that naturally drive the inversion towards a reduced set of reliable, stable solutions. This allows to successfully reconstruct the overburden-free reflection response beneath a complex salt body from noise-contaminated data.

An iterative solution of a particular non-linear Fredholm integral equation is considered. The method is based on the reduction of the equation for the residual at any stage of the iteration to a linear integral equation for the required perturbation, using Newton' s method. This equation is then solved by using a low order expansion in terms of Chebyshev polynomials. The numerical integrations required in the basic iterative process are carried out by the powerful Clenshaw-Curtis quadrature prescription. The non-linear integral equation considered arises from the unstable two-point boundary value problem for a tubular chemical reactor. The capacity of the present method in dealing with such numerical difficulties is illustrated in representative calculations. Comparisons are made with conventional solutions of the parent differential equation and also with attempts at direct iterative solution of the integral equation by standard Neumann-Liouville series and finally with the direct algebraic approach.

Multidimensional singular integral equations (SIEs) play a key role in many areas of applied science such as aerodynamics, fluid mechanics, etc. Solving an equation with a singular kernel can be a challenging problem. Therefore, a plethora of methods have been proposed in the theory so far. However, many of them are discussed in the simplest cases of one–dimensional equations defined on the finite intervals. In this study, a very efficient method based on trigonometric interpolating polynomials is proposed to derive an approximate solution of a SIE with a multiplicative Cauchy kernel defined on the Euclidean plane. Moreover, an estimation of the error of the approximated solution is presented and proved. This assessment and an illustrating example show the effectiveness of our proposal.