Marchenko Equation Research Articles

High-Order Block Toeplitz Inner-Bordering method for solving the Gelfand–Levitan–Marchenko equation

We propose a high precision algorithm for solving the Gelfand–Levitan–Marchenko equation. The algorithm is based on the block version of the Toeplitz Inner-Bordering algorithm of Levinson’s type. To approximate integrals, we use the high-precision one-sided and two-sided Gregory quadrature formulas. Also we use the Woodbury formula to construct a computational algorithm. This makes it possible to use the almost Toeplitz structure of the matrices for the fast calculations. To the best of our knowledge, this is the first algorithm to solve this problem with an order of accuracy higher than the second.

Fast nonlinear Fourier transform algorithm for reconstruction of optical data from nonlinear spectra of the Manakov system.

We propose a high-precision algorithm for solving the three-component Gelfand-Levitan-Marchenko equations (GLME) associated with the Manakov system, which describes the behavior of light waves through the optical fibers. The algorithm generalizes the high-order generalized Toeplitz inner-bordering method for solving the two-component GLME associated with the nonlinear Schrödinger equation. Numerical experiments have shown that the proposed algorithm makes it possible to increase the accuracy of solving the GLME associated with Manakov system up to the sixth order.

Right and left inverse scattering problems formulations for the Zakharov–Shabat system

Abstract We consider right and left formulations of the inverse scattering problem for the Zakharov–Shabat system and the corresponding integral Gelfand–Levitan–Marchenko equations. Both formulations are helpful for numerical solving of the inverse scattering problem, which we perform using the previously developed Toeplitz Inner Bordering (TIB) algorithm. First, we establish general relations between the right and left scattering coefficients. Then we propose an auxiliary kernel of the left Gelfand–Levitan–Marchenko equations, which allows one to solve the right scattering problem numerically. We generalize the TIB algorithm, initially proposed in the left formulation, to the right scattering problem case with the obtained formulas. The test runs of the TIB algorithm illustrate our results reconstructing the various non-symmetrical potentials from their right scattering data.

Inverse scattering transform algorithm for the Manakov system

A numerical algorithm is described for solving the inverse spectral scattering problem associated with the Manakov model of the vector nonlinear Schrödinger equation. This model of wave processes simultaneously considers dispersion, nonlinearity and polarization effects. It is in demand in nonlinear physical optics and is especially perspective for describing optical radiation propagation through the fiber communication lines. In the presented algorithm, the solution to the inverse scattering problem based on the inversion of a set of nested matrices of the discretized system of Gelfand–Levitan–Marchenko integral equations, using a block version of the Levinson-type Toeplitz bordering algorithm. Numerical tests carried out by comparing calculations with known exact analytical solutions confirm the stability and second order of accuracy of the proposed algorithm. We also give an example of the algorithm application to simulate the collision of a differently polarized pair of Manakov optical vector solitons.

Full wavefield inversion with multiples: Nonlinear Bayesian inverse multiple scattering theory beyond the Born approximation

Imaging earth’s structures is fundamental to the earth’s interior, yet how to reconstruct earth’s heterogeneities using full-waveform inversion of full wavefield data that includes primary reflections and multiples remains enigmatic. I develop a nonlinear Bayesian approach for full-waveform inversion method with multiple scattering. Instead of using single scattering Born approximation to formulate the sensitivity kernels, I develop multiple scattering sensitivity kernels using multiple scattering-based Green’s functions. It is based on the Lippmann-Schwinger integral and Marchenko methods, for which the Green’s functions are retrieved from reflection data by solving a Marchenko equation. To estimate the uncertainty of velocities, I apply a Bayesian framework to the inverse problem. Our results indicate that the method does not depend on the earth’s scattering potential and the full-waveform inversion method with multiple scattering is a good alternative approach to image the earth’s structures.

The algebraic structure of the non-commutative nonlinear Schrödinger and modified Korteweg–de Vries hierarchy

We prove that each member of the non-commutative nonlinear Schrödinger and modified Korteweg–de Vries hierarchy is a Fredholm Grassmannian flow, and for the given linear dispersion relation and corresponding equivalencing group of Fredholm transformations, is unique in the class of odd-polynomial partial differential fields. Thus each member is linearisable and integrable in the sense that time-evolving solutions can be generated by solving a linear Fredholm Marchenko equation, with the scattering data solving the corresponding linear dispersion equation. At each order, each member matches the corresponding non-commutative Lax hierarchy field which thus represent odd-polynomial partial differential fields. We also show that the cubic form for the non-commutative sine–Gordon equation corresponds to the first negative order case in the hierarchy, and establish the rest of the negative order non-commutative hierarchy. To achieve this, we construct an abstract combinatorial algebra, the Pöppe skew-algebra, that underlies the hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by compositions, endowed with the Pöppe product—the product rule for Hankel operators pioneered by Ch. Pöppe for classical integrable systems. Establishing the hierarchy members at non-negative orders, involves proving the existence of a ‘Pöppe polynomial’ expansion for basic compositions in terms of ‘linear signature expansions’ representing the derivatives of the underlying non-commutative field. The problem boils down to solving a linear algebraic equation for the polynomial expansion coefficients, at each order.

Discretization of inverse scattering on a half line

AbstractWe solve inverse scattering problem for Schrödinger operators with compactly supported potentials on the half line. We discretize S‐matrix: We take the value of the S‐matrix on some infinite sequence of positive real numbers. Using this sequence obtained from S‐matrix, we recover uniquely the potential by a new explicit formula, without the Gelfand–Levitan–Marchenko equation.

Focusing functions correction in Marchenko imaging with deep learning and transfer learning

The Marchenko method is a subsurface approach to calculate Green's functions from the subsurface to the surface. It requires only reflection response from the surface and direct arrivals from the selected subsurface focal points to the surface. Marchenko imaging can use these obtained Green's functions to create images for all the focal points (image points). Focusing functions are defined to relate Green's functions with the surface reflection response. Estimating focusing functions is the first step in the calculation of Marchenko imaging. However, the estimation of incorrect direct arrivals with the initial model can lead to errors in the focusing functions, thus affecting the effect of the image. In addition, Marchenko imaging is not an efficient imaging method, especially when the imaging area is particularly large, since the number of image points corresponds to the times of solving Marchenko equations. Therefore, we introduce a deep learning and transfer learning approach for the focusing functions correction. We construct a neural network and train the model in a small imaging region by deep learning. Then update this model for extended area by transfer learning which enables the network to work on new data dissimilar to the training data. Hence, our work can quickly obtain accurate focusing functions in a larger region instead of training an extra model for correcting new focusing functions. Numerical examples shows that our network not only helps to reduce errors in the focusing functions, but also can be applied to cope with other limitations of the Marchenko method like travel time error and imperfectly sampled reflection response.

Energy-independent complex single-P-wavesNNpotential from the Marchenko equation

We extend our previous results of solving the inverse problem of quantum scattering theory (Marchenko theory, fixed-$l$ inversion). In particular, we apply an isosceles triangular-pulse function set for the Marchenko equation input kernel expansion in a separable form. The separable form allows a reduction of the Marchenko equation to a system of linear equations for the output kernel expansion coefficients. We show that in the general case of a single partial wave, a linear expression of the input kernel is obtained in terms of the Fourier series coefficients of $q^{1-m}(1-S(q))$ functions in the finite range of the momentum $0\leq q\leq\pi/h$ [$S(q)$ is the scattering matrix, $l$ is the angular orbital momentum, $m=0,1,\dots,2l$]. Thus, we show that the partial $S$--matrix on the finite interval determines a potential function with $h$-step accuracy. The calculated partial potentials describe a partial $S$--matrix with the required accuracy. The partial $S$--matrix is unitary below the threshold of inelasticity and non--unitary (absorptive) above the threshold. We developed a procedure and applied it to partial-wave analysis (PWA) data of $NN$ elastic scattering up to 3 GeV. We show that energy-independent complex partial potentials describe these data for single $P$-waves.

Data-driven suppression of short-period multiples from laterally varying thin-layered overburden structures

Marchenko multiple elimination methods remove all orders of overburden-generated internal multiples in a data-driven way. In the presence of thin beds, however, these methods have been shown to underperform. This is because the underlying inverse problem requires the information about short-period internal multiple (SPIM) imprint on the inverse transmission to be correctly constrained. This has been addressed in 1.5D media with energy conservation and minimum-phase reconstruction. Extending the applications to two dimensions and, hence, making the step toward field data were believed to be hampered by the need for a multidimensional minimum-phase reconstruction which is (1) not unique and (2) no algorithm has been found to perform this in practice on band-limited data. Here, we address both of these problems with an approach that includes solving the Marchenko equation with a trivial constraint, evaluating the energy conservation condition of its solutions to find the spatially dependent error syndrome, using the 1.5D minimum-phase reconstruction for each shot gather to find the spatially dependent constraint, and finally using that inside another run of the Marchenko equation solver to find a much-improved result. We find that the method works because in 2D media the expression of SPIMs in the inverse transmission coda is approximately 1.5D. We then investigate a class of models and synthetic data sets to verify where the 1.5D approximation starts breaking down. Our analysis indicates that this approach could perform very well in settings with moderate lateral variations, which also is where the (short-period) internal multiples are most difficult to differentiate from primary reflections.

Block Toeplitz Inner-Bordering method for the Gelfand–Levitan–Marchenko equations associated with the Zakharov–Shabat system

Abstract We propose a generalized method for solving the Gelfand–Levitan–Marchenko equation (GLME) based on the block version of the Toeplitz Inner-Bordering (TIB). The method works for the signals containing both the continuous and the discrete spectra. The method allows us to calculate the potential at an arbitrary point and does not require small spectral data. Using this property, we can perform calculations to the right and to the left of the selected starting point. For the discrete spectrum, the procedure of cutting off exponentially growing matrix elements is suggested to avoid the numerical instability and perform calculations for soliton solutions spaced apart in the time domain.

Direct, wave-equation-based modeling of Marchenko-type focusing functions.

Focusing functions are wavefields that focus at a single point when injected into a source-free volume from its boundary. Focusing functions that are related to the Marchenko equation can be injected from an open part of the boundary only, while vanishing on the remaining boundary. Building on this property, a method for direct, wave-equation-based modeling of Marchenko-type focusing functions in arbitrarily complex media is presented. The method naturally extends conventional frequency-domain modeling. While the numerical examples are for one dimension, a similar concept should, in principle, be applicable to the general three-dimensional case.

Imaging of thrust structures under Colombian foothills landscape through Marchenko approach

The Marchenko imaging approach successfully deals with multiple scattering during seismic migration, obtaining clear images in structurally complex geological zones below at a specific depth. Through the Marchenko equation, the method retrieves Green's function of a virtual source in profundity using reflections recorded on the surface and a smoothed velocity model. According to the theorem of reciprocity, Green's function represents the medium's response to a source on the surface recorded by a virtual receiver in-depth and decomposes it into a down-going wavefield and an up-going wavefield. Both wave fields' deconvolution provides the reflection response of virtual sources and virtual receivers repositioned in the subsurface, mimicking redatumming. The result is an image free of multiple internal reflections from the area below virtual both sources and receivers. After successfully testing with synthetic seismograms generated in a plane-horizontal layers model, we applied the Marchenko algorithm to seismic data of the Upper Magdalena Valley project using an available velocity model. Compared to an image supplied by a Kirchhoff pre-stack depth migration algorithm, the new image correctly positions the seismic events avoiding misleading interpretations in locating complex structures.

Sparse Inversion for the Iterative Marchenko Scheme of Irregularly Sampled Data

The Marchenko method is a data-driven way that makes it possible to calculate Green’s functions from virtual points in the subsurface using the reflection data at the surface and requiring only a macro velocity model. This method requires collocated sources and receivers. However, in practice, irregular sampling of sources or receivers will cause gaps and distortions in the obtained focusing functions and Green’s functions. To solve this problem, this paper proposes to integrate a sparse inversion into the iterative Marchenko scheme. Specifically, we add sparsity constraints to the Marchenko equations and apply the sparse inversion during the iterative process. To reduce the strict requirements on acquisition geometries, our work deals with the situation in which the sources are subsampled where the integrations are carried out over the receivers, while the existing point spread function method solves the situation where the receivers are subsampled. We make a step to handle both situations at the same time by integrating this method with our work because of the same iterative framework. Our new method is applied to a two-dimensional numerical example with irregularly sampled data. The result shows that it can effectively fill gaps in the obtained focusing functions and Green’s functions in the Marchenko method.

Grassmannian Flows and Applications to Nonlinear Partial Differential Equations

We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto a coordinate chart of the underlying Grassmann manifold. Our method relies on extending this idea to the infinite dimensional case. The key is an integral equation analogous to the Marchenko equation in integrable systems, that represents the coodinate chart map. We show explicitly how to generate such solutions to scalar partial differential equations of arbitrary order with nonlocal quadratic nonlinearities using our approach. We provide numerical simulations that demonstrate the generation of solutions to Fisher--Kolmogorov--Petrovskii--Piskunov equations with nonlocal nonlinearities. We also indicate how the method might extend to more general classes of nonlinear partial differential systems.

Integrability of Local and Nonlocal Non-commutative Fourth Order Quintic Nonlinear Schrodinger Equations

We prove integrability of a generalised non-commutative fourth order quintic nonlinear Schrodinger equation. The proof is relatively succinct and rooted in the linearisation method pioneered by Ch. Poppe. It is based on solving the corresponding linearised partial differential system to generate an evolutionary Hankel operator for the `scattering data'. The time-evolutionary solution to the non-commutative nonlinear partial differential system is then generated by solving a linear Fredholm equation which corresponds to the Marchenko equation. The integrability of reverse space-time and reverse time nonlocal versions, in the sense of Ablowitz and Musslimani, of the fourth order quintic nonlinear Schrodinger equation are proved contiguously by the approach adopted. Further, we implement a numerical integration scheme based on the analytical approach above which involves solving the linearised partial differential system followed by numerically solving the linear Fredholm equation to generate the solution at any given time.

The Non-Commutative Korteweg—De Vries Hierarchy and Combinatorial Poppe Algebra

We give a constructive proof, to all orders, that each member of the non-commutative potential Korteweg-de Vries hierarchy is a Fredholm Grassmannian flow and is therefore linearisable. Indeed we prove this for any linear combination of fields from this hierarchy. That each member of the hierarchy is linearisable, and integrable in this sense, means that the time evolving solution can be generated from the solution to the corresponding linear dispersion equation in the hierarchy, combined with solving an associated linear Fredholm equation representing the Marchenko equation. Further, we show that within the class of polynomial partial differential fields, at every order, each member of the non-commutative potential Korteweg--de Vries hierarchy is unique. Indeed, we prove to all orders, that each such member matches the non-commutative Lax hierarchy field, which is therefore a polynomial partial differential field. We achieve this by constructing the abstract combinatorial algebra that underlies the non-commutative potential Korteweg-de Vries hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by the set of all compositions endowed with the Poppe product. This product is the abstract representation of the product rule for Hankel operators pioneered by Ch. Poppe for integrable equations such as the Sine-Gordon and Korteweg-de Vries equations. Integrability of the hierarchy members translates, in the combinatorial algebra, to proving the existence of a `Poppe polynomial' expansion for basic compositions in terms of `linear signature expansions'. Proving the existence of such Poppe polynomial expansions boils down to solving a linear algebraic problem for the expansion coefficients, which we solve constructively to all orders.

Wavefield focusing using a generalised, potentially asymmetric homogeneous Green’s function

Marchenko-type integrals typically relate so-called focusing functions and Green’s functions via the reflection response measured on the open surface of a volume of interest. Originating from one dimensional inverse scattering theory, the extension to two and three dimensions set in motion various new developments regarding imaging in complex materials. This extension, however, is based on wavefield decomposition inside the volume and a truncated medium state, i.e. a version of the medium that is reflection-free underneath the focusing location, suggesting that evanescent, refracted and diving waves cannot be included in the representation. We elaborate on a new derivation for Marchenko-like integrals that (i) extends the concept of wavefield focusing by using a generalised homogeneous Green’s function, (ii) is based on partial differential equations and thereby allows for additional insights and a new physical intuition for Marchenko equations, (iii) unifies wavefield focusing for open and closed boundary systems, (iv) does not require wavefield decomposition or a truncated medium state, thus including the full wavefield Green’s function, (v) enables using forward modelling to obtain, e.g., Marchenko-type, time-compact focusing functions. We place a particular focus on the latter point, illustrating and investigating how to solve the underlying partial differential equations for various types of focusing functions. This paves the way for a deeper understanding of focusing functions as well as advanced full wavefield Marchenko schemes. While the derivations are generally presented for the 3D case, we show numerical examples in 1D.

A distributional Gelfand–Levitan–Marchenko equation for the Helmholtz scattering problem on the line

We study an inverse scattering problem for the Helmholtz equation on the whole line. The goal of this paper is to obtain a Gelfand–Levitan–Marchenko (GLM)-type equation for the Jost solution that corresponds to the 1D Helmholtz differential operator. We assume for simplicity that the refraction index is of compact support. Using the asymptotic behavior of the Jost solutions with respect to the wave-number, we derive a generalized Povzner–Levitan representation in the space of tempered distributions. Then, we apply the Fourier transform on the scattering relation that describes the solutions of the Helmholtz scattering problem and we derive a generalized GLM equation. Finally, we discuss the possible application of this new generalized GLM equation to the inverse medium problem.

Marchenko Green’s Function Retrieval in Layered Elastic Media from Two-Sided Reflection and Transmission Data

By solving a Marchenko equation, Green’s functions at an arbitrary (inner) depth level inside an unknown elastic layered medium can be retrieved from single-sided reflection data, which are collected at the top of the medium. To date, it has only been possible to obtain an exact solution if the medium obeyed stringent monotonicity conditions and if all forward-scattered (non-converted and converted) transmissions between the acquisition level and the inner depth level were known a priori. We introduce an alternative Marchenko equation by revising the window operators that are applied in its derivation. We also introduce an auxiliary equation for transmission data, which are collected at the bottom of the medium, and a coupled equation, which is based on both reflection and transmission data. We show that the joint system of the Marchenko equation, the auxiliary equation and the coupled equation can be succesfully inverted when broadband reflection and transmission data are available. This results in a novel methodology for elastodynamic Green’s function retrieval from two-sided data. Apart from these data, our approach requires P- and S-wave transmission times between the inner depth level and the top of the medium, as well as two angle-dependent amplitude scaling factors, which can be estimated from the data by enforcing energy conservation.