Marchenko Equation Research Articles

Tutorial: unified 1D inversion of the acoustic reflection response.

ABSTRACTAcoustic inversion in one‐dimension gives impedance as a function of travel time. Inverting the reflection response is a linear problem. Recursive methods, from top to bottom or vice versa, are known and use a fundamental wave field that is computed from the reflection response. An integral over the solution to the Marchenko equation, on the other hand, retrieves the impedance at any vertical travel time instant. It is a non‐recursive method, but requires the zero‐frequency value of the reflection response. These methods use the same fundamental wave field in different ways. Combining the two methods leads to a non‐recursive scheme that works with finite‐frequency bandwidth. This can be used for target‐oriented inversion. When a reflection response is available along a line over a horizontally layered medium, the thickness and wave velocity of any layer can be obtained together with the velocity of an adjacent layer and the density ratio of the two layers. Statistical analysis over 1000 noise realizations shows that the forward recursive method and the Marchenko‐type method perform well on computed noisy data.

Marchenko imaging based on self-adaptive traveltime updating

Marchenko imaging obtains the subsurface reflectors using one-way Green’s functions, which are retrieved by solving the Marchenko equation. This method generates an image that is free of spurious artifacts due to internal multiples. The Marchenko imaging method is a target-oriented technique; thus, it can image a user specified area. In the traditional Marchenko method, an accurate velocity model is critical for estimating direct waves from imaging points to the surface. An error in the velocity model results in the inaccurate estimation of direct waves. In turn, this leads to errors in computation of one-way Green’s functions, which then affects the final Marchenko images. To solve this problem, in this paper, we propose a self-adaptive traveltime updating technique based on the principle of equal traveltime to improve the Marchenko imaging method. The proposed method calculates the time shift of direct waves caused by the error in the velocity model, and corrects the wrong direct wave according to the time shift and reconstructs the correct Green’s functions. The proposed method improves the results of imaging using an inaccurate velocity model. By comparing the results from traditional Marchenko and the new method using synthetic data experiments, we demonstrated that the adaptive traveltime updating Marchenko imaging method could restore the image of geological structures to their true positions.

On the double-pole and two-soliton solutions of the Harry Dym equation

The double-pole solution of the initial value problem for the Harry Dym equation (HDE) was obtained by using the inverse scattering transform (IST) method, i.e., solving the associated Gelfand–Levitan–Marchenko (GLM) equation. Further, we show that the double-pole solution can be viewed as some proper limits of the two-soliton solution obtained by Riemann–Hilbert problem (RHP) and IST method, respectively.

Marchenko multiple elimination of a laboratory example

SUMMARY The Marchenko multiple elimination (MME) scheme is derived from the coupled Marchenko equations. It is proposed for filtering primary reflections with two-way traveltime from the measured acoustic data. The measured acoustic reflection data are used as its own filter and no model information or adaptive subtraction is required to apply the method. The data obtained after MME are better suited for velocity model construction and artefact-free migration than the measured data. We apply the MME scheme to a measured laboratory data set to evaluate the success of the method. The results suggest that the MME scheme can be the appropriate choice when high-quality pre-processing is performed successfully.

Data-driven control over short-period internal multiples in media with a horizontally layered overburden

SUMMARY Short-period internal multiples, resulting from closely spaced interfaces, may interfere with their generating (bandlimited) primaries, and hence they pose a long-standing challenge in their prediction and removal. A recently proposed method based on the Marchenko equation enables removal of the entire overburden-related scattering by means of calculating an inverse transmission response. However, the method relies on time windowing and can thus be inexact in the presence of short-period internal scattering. In this work, we present a detailed analysis of the impact of band-limitation on the Marchenko method. We show the influence of an incorrect first guess, and that adding multidimensional energy conservation and a minimum phase principle may be used to correctly account for both long- and short-period internal multiple scattering. The proposed method can currently only be solved for media with a laterally invariant overburden, since a multidimensional minimum phase condition is not well understood for truly 2-D and 3-D media. We demonstrate the virtue of the proposed scheme with a complex acoustic numerical model that is based on sonic log measurements in the Middle East. The results suggest not only that the conventional scheme can be robust in this setting, but that the ‘augmented’ Marchenko method is superior, as the latter produces a structural image identical to one where the finely layered overburden is missing. This is the first demonstration of a data-driven method to account for short-period internal multiples beyond 1-D.

A field data example of Marchenko multiple elimination

Internal multiple reflections have been widely considered as coherent noise in measured seismic data, and many approaches have been developed for their attenuation. The Marchenko multiple elimination (MME) scheme eliminates internal multiple reflections without model information or adaptive subtraction. This scheme was originally derived from coupled Marchenko equations, but it was modified to make it model independent. It filters primary reflections with their two-way traveltimes and physical amplitudes from measured seismic data. The MME scheme is applied to a deepwater field data set from the Norwegian North Sea to evaluate its success in removing internal multiple reflections. The result indicates that most internal multiple reflections are successfully removed and primary reflections masked by overlapping internal multiple reflections are recovered.

Complex short-pulse solutions by gauge transformation

In this article we apply the gauge transformation method for solving the Heisenberg ferromagnet equation by using the NLS Jost solutions to derive solutions of the complex short-pulse equation from the Jost functions of the complex sine-Gordon equation. The matrix triplet method is used to compute the most general reflectionless complex short-pulse solutions by solving the Marchenko integral equation for the complex sine-Gordon equation.

Theory of Multidimensional Delsarte–Lions Transmutation Operators. II

The differential-geometric and topological structures related to the Delsarte transmutation operators and the Gelfand–Levitan–Marchenko equations that describe these operators are studied by using suitable differential de Rham–Hodge–Skrypnik complexes. The correspondence between the spectral theory and special Berezansky-type congruence properties of the Delsarte transmutation operators is established. Some applications to multidimensional differential operators are presented, including the three-dimensional Laplace operator, the two-dimensional classical Dirac operator, and its multidimensional affine extension associated with self-dual Yang–Mills equations. The soliton solutions are discussed for a certain class of dynamical systems.

Data-driven internal multiple elimination and its consequences for imaging: A comparison of strategies

We have compared three data-driven internal multiple reflection elimination schemes derived from the Marchenko equations and inverse scattering series (ISS). The two schemes derived from Marchenko equations are similar but use different truncation operators. The first scheme creates a new data set without internal multiple reflections. The second scheme does the same and compensates for transmission losses in the primary reflections. The scheme derived from ISS is equal to the result after the first iteration of the first Marchenko-based scheme. It can attenuate internal multiple reflections with residuals. We evaluate the success of these schemes with 2D numerical examples. It is shown that Marchenko-based data-driven schemes are relatively more robust for internal multiple reflection elimination at a higher computational cost.

The soliton solutions for the Wadati–Konno–Ichikawa equation

By taking the Wadati–Konno–Ichikawa (WKI) equation as an example, this paper refines the Olmedilla’s technique of solving the Gelfand–Levitan–Marchenko (GLM) equation with one higher-order pole. In order to solve the GLM equation, the Laurent’s series and residue theorem are used, and the formulae of the Nth order soliton solution and Nth order bound-state soliton solution are expressed in a unified determinant form. Moreover, these formulae are simpler and more explicit to calculate solutions of the nonlinear integrable equations, compared with the results of Olmedilla.

Solitons: Conservation laws and dressing methods

We review some of the fundamental notions associated with the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary differential equations that are associated to discrete or continuous integrable systems. The Riccati equation associated to a given continuous integrable system is also solved and hence suitable conserved quantities are derived. The notion of the Darboux–Bäcklund transformation is introduced and employed in order to obtain soliton solutions for specific examples of integrable equations. The Zakharov–Shabat dressing scheme and the Gelfand–Levitan–Marchenko equation are also introduced. Via this method, generic solutions are produced and integrable hierarchies are explicitly derived. Various discrete and continuous integrable models are employed as examples such as the Toda chain, the discrete nonlinear Schrödinger model, the Korteweg–de Vries and nonlinear Schrödinger equations as well as the sine-Gordon and Liouville models.

Nonlinearly bandlimited signals

In this paper, we study the inverse scattering problem for a class of signals that have a compactly supported reflection coefficient. The problem boils down to the solution of the Gelfand–Levitan–Marchenko (GLM) integral equations with a kernel that is bandlimited. By adopting a sampling theory approach to the associated Hankel operators in the Bernstein spaces, a constructive proof of existence of a solution of the GLM equations is obtained under various restrictions on the nonlinear impulse response (NIR). The formalism developed in this article also lends itself well to numerical computations yielding algorithms that are shown to have algebraic rates of convergence. In particular, the use Whittaker–Kotelnikov–Shannon sampling series yields an algorithm that converges as whereas the use of Helms and Thomas (HT) version of the sampling expansion yields an algorithm that converges as for any m > 0 provided the regularity conditions are fulfilled. The complexity of the algorithms depend on the linear solver used. The use of conjugate-gradient (CG) method yields an algorithm of complexity per sample of the signal where N is the number of sampling basis functions used and is the number of CG iterations involved. The HT version of the sampling expansions facilitates the development of algorithms of complexity (per sample of the signal) by exploiting the special structure as well as the (approximate) sparsity of the matrices involved. The algorithms are numerically validated using Schwartz class functions as NIRs that are either bandlimited or effectively bandlimited. The results suggest that the HT variant of our algorithm is spectrally convergent for an input of the aforementioned class.

Handling short-period scattering using augmented Marchenko autofocusing

SUMMARY Marchenko autofocusing constructs arbitrary acoustic wavefields inside an unknown heterogeneous medium from a single-sided, bandlimited in practice, reflection response by solving the Marchenko equation. Presence of short-period scattering leads to erroneous solutions and therefore incorrect medium characterization. We show that augmenting this equation with energy conservation and minimum phase conditions enables to correct these erroneous solutions, and thus it removes the long-/short-period timescale separation and solves this long standing problem.

Virtual plane-wave imaging via Marchenko redatuming

Marchenko redatuming is a novel scheme used to retrieve up- and down-going Green's functions in an unknown medium. Marchenko equations are based on reciprocity theorems and are derived on the assumption of the existence of so called focusing functions, i.e. functions which exhibit time-space focusing properties once injected in the subsurface. In contrast to interferometry but similarly to standard migration methods, Marchenko redatuming only requires an estimate of the direct wave from the virtual source (or to the virtual receiver), illumination from only one side of the medium, and no physical sources (or receivers) inside the medium. In this contribution we consider a different time-focusing condition within the frame of Marchenko redatuming and show how this can lead to the retrieval of virtual plane-wave responses, thus allowing multiple-free imaging using only a 1 dimensional sampling of the targeted model. The potential of the new method is demonstrated on a 2D synthetic model.

The continuous classical Heisenberg ferromagnet equation with in-plane asymptotic conditions. I. Direct and inverse scattering theory

We develop the direct and inverse scattering theory of the linear eigenvalue problem associated with the classical Heisenberg continuous equation with in-plane asymptotic conditions. In particular, analyticity of the scattering eigenfunctions and scattering data, and their asymptotic behaviours are derived. The inverse problem is formulated in terms of Marchenko equations, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is provided.

Partial differential systems with non-local nonlinearities: generation and solutions.

We develop a method for generating solutions to large classes of evolutionary partial differential systems with non-local nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogous to the Marchenko integral equation in integrable systems. We show explicitly how this can be achieved through several examples, including reaction-diffusion systems with non-local quadratic nonlinearities and the nonlinear Schrödinger equation with a non-local cubic nonlinearity. In each case, we demonstrate our approach with numerical simulations. We discuss the effectiveness of our approach and how it might be extended.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

Single- and Double-Sided Marchenko Imaging Conditions in Acoustic Media

In acoustic reflector imaging, we deploy sources and receivers outside a volume to collect a multisource, multioffset reflection response in order to retrieve the internal reflectivity of that volume. It has been shown that Green's functions inside the volume can be retrieved by single-sided wavefield focusing of the acquired reflection data, using so-called focusing functions, which can be computed by solving a multidimensional Marchenko equation. Besides the reflection data, this methodology requires a background model of the propagation velocity. We present several imaging conditions to retrieve the internal reflectivity of an acoustic medium with correct amplitudes and without artifacts, using the Green's functions and focusing functions that are derived from the Marchenko equation. We distinguish three types of imaging: 1) imaging by deconvolution, 2) imaging by double focusing, and 3) imaging by cross correlation. In all cases, reflectors can be approached either from above or from below. Imaging by deconvolution or double focusing requires single-sided illumination (meaning that sources and receivers are deployed at a single boundary above the volume only), whereas imaging by cross correlation requires double-sided illumination (meaning that sources and receivers are placed at two boundaries enclosing the volume). In order to achieve double-sided illumination, the required reflection response at the lower boundary can either be physically recorded or it can be retrieved from the reflection response at the upper boundary. When imaging by deconvolution or double focusing, the internal reflectivity is retrieved solely from primary reflections. When imaging by cross correlation, multiple reflections are focused at the image points, such that they contribute physically to the retrieved reflectivity values. This special feature can be beneficial for imaging weakly illuminated sections of strongly heterogeneous media.

The description of reflection coefficients of the scattering problems for finding solutions of the Korteweg—de Vries equations

The results of inverse scattering problem associated with the initial-boundary value problem (IBVP) for the Korteweg—de Vries (KdV) equation with dominant surface tension are formulated. The necessary and sufficient conditions for given functions to be the left- and right-reflection coefficients of the scattering problem are established. The time-dependence t, t > 0 of each element of the scattering matrix s(k,t) is found in respective sector of the k-spectral plane by expansion formulas which are constructed from the known initial and boundary conditions of the IBVP. Knowing the right-reflection coefficient calculated from the elements of s(k,t), we solve the Gelfand–Levitan—Marchenko (GLM) equation in the inverse problem. Then the solution of the IBVP is expressible through the solution of the GLM equation. The asymptotic behavior at infinity of time of the solution of the IBVP is shown

A tour of Marchenko redatuming: Focusing the subsurface wavefield

Marchenko redatuming can retrieve the impulse response to a subsurface virtual source from the single-sided surface reflection data with limited knowledge of the medium. We illustrate the concepts and practical aspects of Marchenko redatuming on a simple 1D acoustic lossless medium in which the coupled Marchenko equations are exact. Defined in a truncated version of the actual medium, the Marchenko focusing functions focus the wavefields at the virtual source location and are responsible for the subsequent retrieval of the downgoing and upgoing components of the medium's impulse response. In real seismic exploration, where we have no access to the truncated medium, we solve the coupled Marchenko equations by iterative substitution, relying on the causality relations between the focusing functions and the desired Green's functions along with an initial estimate of the downgoing focusing function. We show that the amplitude accuracy of the initial focusing function influences that of the retrieved Green's functions. During each iteration, propagating an updated focusing function into the actual medium can be approximated by explicit convolution with the broadband reflection seismic data after appropriate processing, which acts as a proxy for the true medium's reflection response.

Rayleigh-Marchenko redatuming for target-oriented, true-amplitude imaging

Marchenko redatuming is a revolutionary technique to estimate Green’s functions from virtual sources in the subsurface using only data measured at the earth’s surface, without having to place either sources or receivers in the subsurface. This goal is achieved by crafting special wavefields (so-called focusing functions) that can focus energy at a chosen point in the subsurface. Despite its great potential, strict requirements on the reflection response such as knowledge and accurate deconvolution of the source wavelet (including absolute scaling factor) and co-location of sources and receivers have so far challenged the application of Marchenko redatuming to real-world scenarios. I combine the coupled Marchenko equations with a one-way version of the Rayleigh integral representation to obtain a new redatuming scheme that handles internal as well as free-surface multiples using dual-sensor, band-limited seismic data (with an unknown source signature) from any acquisition system that presents arbitrarily located sources above a line of regularly sampled receivers—for example, ocean-bottom, source-over-spread streamer, and horizontal borehole seismic data. The redatuming scheme is validated using synthetic and field data, and the retrieved subsurface wavefields are used for improved structural imaging and taken as input for the computation of true-amplitude angle gathers, which can lead to more accurate amplitude-versus-angle interpretation and velocity analysis.