Marchenko System in Inverse Scattering

• The Marchenko system is a set of coupled integral equations that convert spectral or reflection data into kernels, focusing functions, and potentials across various fields.
• It employs algebraic and separable formulations to reduce the inverse scattering problem to structured linear systems, facilitating efficient numerical reconstruction.
• In seismic applications, the method reconstructs full wavefields and internal multiples, enabling advanced redatuming and virtual-source imaging techniques.

The Marchenko system is a context-dependent but structurally coherent construction in inverse scattering. Across quantum scattering, seismic redatuming, and integrable systems, it denotes coupled integral equations and reconstruction formulas that transform spectral or reflection data into kernels, focusing functions, Green’s functions, or potentials. In the fixed-partial-wave Schrödinger setting, the system is built from the Marchenko equation

$F(x,y)+L(x,y)+\int_x^{+\infty}L(x,t)F(t,y)\,dt=0,$

with the potential recovered from

$V(r)=-2\frac{dL(r,r)}{dr},$

while in geophysics it links single-sided reflection data to focusing functions $f_1^\pm$ and Green’s functions $G^\pm$, and in integrable-systems work it appears as a GLM-type system whose solution reconstructs the potentials in a Lax operator (Khokhlov, 2021, Wapenaar et al., 2020, Unlu, 23 Jul 2025).

1. Fixed-$l$ inverse scattering and the classical half-line formulation

In the radial Schrödinger setting, the Marchenko system addresses the inverse problem for a fixed partial wave $l$. The direct equation is written as

$$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0,$$

and the scattering data are

$$\left\{S(q),(0<q<\infty), \tilde{q}_{j}, M_{j}, j=1,\ldots,n_b\right\},$$

with $S(q)=e^{2i\delta(q)}$, bound-state poles $q=\tilde q_j$, and asymptotic constants $V(r)=-2\frac{dL(r,r)}{dr},$0. The corresponding Marchenko kernel is

$V(r)=-2\frac{dL(r,r)}{dr},$1

where

$V(r)=-2\frac{dL(r,r)}{dr},$2

Once $V(r)=-2\frac{dL(r,r)}{dr},$3 is known, solving the Marchenko equation for $V(r)=-2\frac{dL(r,r)}{dr},$4 and differentiating its diagonal completes the reconstruction (Khokhlov, 2021).

A closely related half-line formulation appears in the Morse-potential study, where the Marchenko system is the “full set of equations and constructions that connect scattering data to the potential” for a one-dimensional Schrödinger operator on $V(r)=-2\frac{dL(r,r)}{dr},$5. There the integral equation is

$V(r)=-2\frac{dL(r,r)}{dr},$6

with

$V(r)=-2\frac{dL(r,r)}{dr},$7

and the reconstruction formula is again

$V(r)=-2\frac{dL(r,r)}{dr},$8

That paper also introduces a differential companion,

$V(r)=-2\frac{dL(r,r)}{dr},$9

to relate perturbations of a model kernel and a model potential (Selg, 2015).

This classical usage fixes the essential meaning of the Marchenko system: it is not only one kernel equation, but the full inversion chain from scattering data to a kernel and from that kernel to the potential.

2. Algebraic and separable formulations for quantum scattering

A major development is the conversion of the Marchenko equation into a purely algebraic problem. For arbitrary orbital angular momentum $f_1^\pm$0, the kernel is rewritten through auxiliary functions

$f_1^\pm$1

and a differential operator $f_1^\pm$2 satisfying

$f_1^\pm$3

This yields

$f_1^\pm$4

so that the fixed-$f_1^\pm$5 kernel is reduced to combinations of the functions $f_1^\pm$6 (Khokhlov, 2021).

The key numerical step is a separable approximation in a triangular wave basis. With $f_1^\pm$7, one writes

$f_1^\pm$8

and hence

$f_1^\pm$9

The unknown transformation kernel is expanded similarly,

$G^\pm$0

which reduces the integral equation to the linear system

$G^\pm$1

In this form, the Marchenko equation becomes a family of matrix problems, one for each grid slice $G^\pm$2 (Khokhlov, 2021).

The same program was first carried out explicitly for the $G^\pm$3 case. There the kernel depends only on $G^\pm$4, and the expansion coefficients are obtained as linear functionals of the scattering data on a finite momentum interval. In particular, the paper states that “for the zero orbital angular momentum, a linear expression of the kernel expansion coefficients is obtained in terms of the Fourier series coefficients of a function depending on the momentum $G^\pm$5 and determined by the scattering data on the finite range of $G^\pm$6” (Khokhlov, 2021). The later fixed-$G^\pm$7 generalization preserves this linear dependence and emphasizes the practical resolution constraint: “The kernel expansion coefficients are determined by the scattering data in the finite range $G^\pm$8” (Khokhlov, 2021).

In this algebraic usage, the Marchenko system is therefore a structured discretization whose unknowns are kernel coefficients rather than wavefunctions. The method makes the trade-off between spatial step $G^\pm$9 and required momentum range $l$0 explicit.

3. Single-sided seismic Marchenko systems

In exploration seismology, the Marchenko system is a single-sided wavefield-reconstruction framework. Its inputs are the reflection response $l$1 at an acquisition surface and a smooth macro model that supplies a direct focusing term. The central objects are the focusing functions $l$2 and the Green’s functions $l$3. In compact notation, the iterative focusing relations are

$l$4

and once the focusing functions are known, the Green’s functions follow from

$l$5

This system underlies redatuming by multidimensional deconvolution, double focusing, virtual seismology, double dereverberation, and transmission-compensated Marchenko multiple elimination (Wapenaar et al., 2020).

The same literature stresses that the Marchenko system is not merely a retrieval of primaries. It uses all orders of internal multiples, reconstructs both downgoing and upgoing fields at virtual sources and receivers, and can redatum the data while removing overburden effects without a detailed model of the overburden reflectivity (Wapenaar et al., 2020). In that sense, the seismic Marchenko system is an operational adaptation of Marchenko inverse scattering to single-sided acquisition.

A further extension replaces point focusing by time focusing on a depth level, leading to virtual plane-wave responses. In that formulation, the focusing condition at the focal depth is

$l$6

and the resulting coupled system retrieves areal Green’s functions $l$7 rather than point-source responses. The paper emphasizes that this permits “multiple-free imaging using only a one-dimensional sampling of the targeted model at a fraction of the computational cost of standard Marchenko schemes” (Meles et al., 2017).

This seismic usage widens the meaning of the Marchenko system from a kernel-reconstruction equation to a family of coupled single-sided integral representations for focusing, redatuming, and virtual-source synthesis.

4. Evanescent, refracted, and PDE-based generalizations

A frequent simplification in seismic Marchenko practice is the restriction to propagating waves. That restriction is not fundamental. In horizontally layered media, the Marchenko method has been extended to evanescent waves by retaining the terms in the propagation invariants that survive in the evanescent regime. For the case propagating at the surface and evanescent at the focal depth, the paper derives the new relation

$l$8

and from a second focusing construction obtains

$l$9

This reduces the problem to a single effective Marchenko equation for $l$0 (Wapenaar, 2020).

The same study is explicit about a practical asymmetry. It states that, in theory, both upward and downward decaying components can be retrieved, but “the retrieval of the upward decaying component appears to be very sensitive to model errors, but the downward decaying component, including multiple reflections, can be retrieved in a reasonably stable and accurate way” (Wapenaar, 2020). A common misconception is therefore that the evanescent extension yields equally robust access to all non-propagating components; the paper argues instead for a more limited practical regime.

A complementary reinterpretation treats Marchenko focusing functions as solutions of a PDE-constrained wavefield-reconstruction problem. There the focusing fields $l$1 satisfy the acoustic wave equation

$l$2

together with vanishing conditions as $l$3, the focal-plane condition

$l$4

and a one-way condition at the focal plane. The least-squares solution

$l$5

yields focusing functions containing “evanescent, refracted, and trapped waves (tunneling)” (Hajjaj et al., 2022).

This PDE-based viewpoint suggests that the Marchenko system can be understood not only as an integral-equation formalism driven by reflection data, but also as a constrained wavefield construction whose exact multi-dimensional solutions contain strong evanescent content.

5. Sampling, acquisition geometry, and background-model dependence

The discretized seismic Marchenko system assumes regularly sampled and co-located sources and receivers. When those assumptions fail, the discrete sums no longer approximate the underlying surface integrals. A point-spread-function reformulation addresses this by introducing kernels such as

$l$6

and

$l$7

so that the irregularly sampled Marchenko outputs are interpreted as blurred versions of the ideal wavefields and are then deblurred by multidimensional deconvolution (IJsseldijk et al., 2020). The practical conclusion is stated directly: “By removing the requirement for perfect geometries, the Marchenko method can be more widely applied to field data” (IJsseldijk et al., 2020).

The role of the background velocity model is likewise more specific than is sometimes assumed. In the full-field focusing study, the background model enters through the direct arrival and the window $l$8, not through the full multiple-scattering dynamics, which are carried by the measured reflection response $l$9. The reported numerical tests show that the retrieved Green’s functions “give correlation coefficients with the exact Green’s function larger than 90% on average except near the edges of the receiver aperture,” and also that “the Marchenko focusing algorithm produces refracted waves only if the initial velocity model used for the iterative scheme is sufficiently detailed to model the refracted waves” (Kiraz et al., 2023).

This practical literature establishes two limits of the Marchenko system. First, sampling irregularity acts as a geometric blur rather than a conceptual breakdown of the theory. Second, reflected and multiply reflected components are largely data-driven, whereas refracted components depend directly on what the initial background model can generate.

6. Integrable systems, reflectionless operators, and generalized GLM systems

In integrable-systems work, the Marchenko system becomes a GLM-type inverse problem associated with a Lax operator. For the first-order energy-dependent system

$$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0,$$0

the generalized Marchenko system is a $$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0,$$1 matrix integral system with scalar kernels $$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0,$$2, and the bound-state information is encoded by a pair of matrix triplets $$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0,$$3 and $$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0,$$4. The discrete spectrum then enters through

$$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0,$$5

which is the first-order, energy-dependent analogue of the classical sum of exponentials in scalar Marchenko theory (Aktosun et al., 2022).

The same architecture is used for the DNLS II or Chen–Lee–Liu system. There the paper explicitly refers to a “Marchenko system of linear integral equations,” again with a full kernel built from reflection coefficients and two matrix triplets. Solving that system reconstructs the potentials $$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0,$$6, while the simple phase evolution of the scattering data under the Lax flow yields time-dependent DNLS II solutions (Unlu, 23 Jul 2025). In reflectionless cases, the matrix-triplet formalism produces closed-form formulas for soliton solutions.

A third-order analogue appears for the Sawada–Kotera equation. There the Marchenko system is the GLM equation

$$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0,$$7

with kernel

$$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0,$$8

Its solution yields the $$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0,$$9-soliton potential

$$\left\{S(q),(0<q<\infty), \tilde{q}_{j}, M_{j}, j=1,\ldots,n_b\right\},$$0

and the paper states that this Marchenko equation is the “GLM (Gel'fand–Levitan–Marchenko) integral equation” Kaup had sought for the third-order problem (Aktosun et al., 13 Jun 2026).

A more spectral-operator-oriented use appears in the Marchenko representation of reflectionless Jacobi and Schrödinger operators. There the reflectionless class is parametrized by a Herglotz function $$\left\{S(q),(0<q<\infty), \tilde{q}_{j}, M_{j}, j=1,\ldots,n_b\right\},$$1 and a measure $$\left\{S(q),(0<q<\infty), \tilde{q}_{j}, M_{j}, j=1,\ldots,n_b\right\},$$2, and in the Schrödinger case the associated moment functions satisfy

$$\left\{S(q),(0<q<\infty), \tilde{q}_{j}, M_{j}, j=1,\ldots,n_b\right\},$$3

This infinite system provides a Marchenko-type description of reflectionless potentials and is used to prove real analyticity in a strip (Hur et al., 2014).

Taken together, these formulations show that the Marchenko system is not tied to a single equation or a single physical setting. Its invariant content is the conversion of spectral data into a linear integral or moment system whose solution reconstructs the underlying medium, operator, or field.