2D+1D Inversion in Seismology & Scattering

• 2D+1D inversion is a strategy that splits a 3D inverse problem into a joint 2D lateral analysis and independent 1D depth inversions.
• It is applied in surface-wave seismology and inverse scattering to reduce high-dimensional problems into multiple low-dimensional ones, boosting computational efficiency.
• Implementations include variational Bayesian methods, rj-McMC sampling, and Fourier-based 2.5-D formulations, each balancing tractability with potential trade-offs in lateral continuity and uncertainty quantification.

In the cited literature, 2D+1D inversion denotes inverse formulations in which two spatial dimensions are treated jointly while a third dimension is handled separately, either as a subsequent set of 1D inversions, as a Fourier synthesis variable, or as the axial variable of a Radon parameterization. In surface-wave seismology, the split is explicit: a period-by-period 2D tomography yields maps of phase or group velocity $c_T(x,y)$, and these maps are then converted into independent 1D depth inversions for $v_s(z)$. In inverse scattering, the synonymous 2.5-D formulation models the object in 2D but retains 3D transmitter–receiver physics through a 1D Fourier integral. In seismic Radon theory, functions on $\mathbb{R}^2\times\mathbb{R}$ are reconstructed from integrals over parabolic or hyperbolic hypersurfaces (Zhao et al., 21 Jul 2025, Hidayetoglu et al., 2022, Chihara, 2019).

1. Conceptual forms of 2D+1D inversion

The main mathematical architectures that appear under the 2D+1D label are structurally distinct but share the same dimensional separation. In Zhao et al., the inversion is split into a 2D lateral tomography at each period and a family of independent 1D local depth inversions. In the fast-integration formulation of 2.5-D inverse scattering, the object is assumed invariant in one spatial direction, so the field is decomposed into 2D Helmholtz problems indexed by a 1D Fourier variable. In Chihara’s seismic-type Radon setting, the unknown is a function on $(x_1,x_2,y)\in\mathbb{R}^2\times\mathbb{R}$, and the data are integrals over hypersurfaces parameterized by 2D slope-like variables and a 1D intercept-like variable (Zhao et al., 21 Jul 2025, Hidayetoglu et al., 2022, Chihara, 2019).

Formulation	2D component	1D component
Surface-wave tomography	$c_T(x,y)$ maps at each period	local inversion for $v_s(z)$
2.5-D inverse scattering	$(x,z)$-plane Helmholtz solves	Fourier integration in $k_y$
Seismic-type Radon inversion	$(x_1,x_2)$ offsets	vertical variable $y$

This shared separation is computationally consequential. The 2D stage usually captures lateral structure, while the 1D stage either restores depth dependence or synthesizes the missing dimension. A plausible implication is that 2D+1D inversion is best understood not as a single algorithm but as a recurring dimensional-reduction strategy whose exact meaning depends on the forward operator.

2. Two-step 2D+1D inversion in surface-wave seismology

In the two-step surface-wave framework summarized by Zhao et al., the inversion is divided into two stages. First, for each period $v_s(z)$0, one solves a 2D tomography problem for a map $v_s(z)$1 of surface-wave velocities, with forward operator $v_s(z)$2 and travel-time data $v_s(z)$3. Second, at each horizontal location $v_s(z)$4, the period-dependent means and uncertainties are assembled into a local dispersion curve $v_s(z)$5, which is inverted for a 1D shear-velocity profile $v_s(z)$6 (Zhao et al., 21 Jul 2025).

In the variational 2D+1D inversion, the first step is a Bayesian linear inverse problem with Gaussian observational errors and a smoothness-promoting prior,

$v_s(z)$7

Variational inference minimizes the Kullback–Leibler divergence by maximizing the ELBO, or equivalently minimizing the deterministic objective

$v_s(z)$8

The second step is a Metropolis–Hastings McMC inversion with likelihood

$v_s(z)$9

The alternative rj-McMC 2D+1D inversion parameterizes the first-step phase-velocity map by Voronoi cells and samples over both cell values and model dimension $\mathbb{R}^2\times\mathbb{R}$0. The second step then uses an independent 1D rj-McMC inversion of the local dispersion curve. In both schemes, the decisive simplification is the factorization

$\mathbb{R}^2\times\mathbb{R}$1

that is, zero lateral covariance.

The computational and inferential consequences are explicit. Zhao et al. report that this dimensionality reduction replaces one problem of dimension $\mathbb{R}^2\times\mathbb{R}$2 by many problems of dimension $\mathbb{R}^2\times\mathbb{R}$3, which decreases memory and per-sample cost and improves stochastic mixing in the local inversions. At the same time, the final 3D model “typically shows discontinuities at grid-cell boundaries, and overestimated posterior variances.” In the British Isles example, the direct-3D PSVI required $\mathbb{R}^2\times\mathbb{R}$4 iterations $\mathbb{R}^2\times\mathbb{R}$5 samples $\mathbb{R}^2\times\mathbb{R}$6 forward solves, whereas the 2D+1D MH-McMC workflow required $\mathbb{R}^2\times\mathbb{R}$7 million solves, and the 2D+1D rj-McMC approach required “$\mathbb{R}^2\times\mathbb{R}$8 hundreds of millions of solves.” The direct-3D method also preserved better lateral continuity and achieved lower normalized misfit than the 2D+1D methods (Zhao et al., 21 Jul 2025).

3. 2.5-D inverse scattering as a 2D+1D formulation

In the 2.5-D inverse-scattering formulation, the object is assumed invariant in the $\mathbb{R}^2\times\mathbb{R}$9-direction, but the transmitters and receivers remain fully 3D. The starting point is the 3D Helmholtz equation

$(x_1,x_2,y)\in\mathbb{R}^2\times\mathbb{R}$0

with scattered field

$(x_1,x_2,y)\in\mathbb{R}^2\times\mathbb{R}$1

Fourier transformation along the invariant direction gives a family of 2D Helmholtz problems,

$(x_1,x_2,y)\in\mathbb{R}^2\times\mathbb{R}$2

with modified wavenumber $(x_1,x_2,y)\in\mathbb{R}^2\times\mathbb{R}$3, and the 2.5-D Green’s function is recovered by inverse Fourier synthesis (Hidayetoglu et al., 2022).

The numerical bottleneck is the defining 1D integral. It is difficult for two reasons given explicitly in the paper: the infinite integration range $(x_1,x_2,y)\in\mathbb{R}^2\times\mathbb{R}$4, and the presence of logarithmic singularities and algebraic branch points. Since $(x_1,x_2,y)\in\mathbb{R}^2\times\mathbb{R}$5 has branch points at $(x_1,x_2,y)\in\mathbb{R}^2\times\mathbb{R}$6, and $(x_1,x_2,y)\in\mathbb{R}^2\times\mathbb{R}$7 near $(x_1,x_2,y)\in\mathbb{R}^2\times\mathbb{R}$8, a naive uniform discretization requires “tens of thousands of points” around the branch points.

The paper therefore develops a sequence of contour and variable transformations: an angular-spectrum substitution $(x_1,x_2,y)\in\mathbb{R}^2\times\mathbb{R}$9, a stationary-phase approximation around the saddle $c_T(x,y)$0, an exact steepest-descent contour defined by a constant-phase condition, and finally a mapping to $c_T(x,y)$1 for Gauss–Legendre quadrature. On the resulting contour, the $c_T(x,y)$2-point Gauss–Legendre rule converges exponentially, with error bounded by

$c_T(x,y)$3

The reported practical effect is substantial: “Typical $c_T(x,y)$4 already achieves $c_T(x,y)$5 relative error,” and the reduction from $c_T(x,y)$6 to $c_T(x,y)$7 quadrature points yields a $c_T(x,y)$8 speed-up in Green’s-function synthesis (Hidayetoglu et al., 2022).

This formulation is called “2.5-D” in the paper itself, but it is explicitly described there as the “2D+1D” inversion approach. The label is therefore not merely geometric; it denotes a solver architecture in which the missing dimension is synthesized analytically or semi-analytically rather than discretized directly.

4. Seismic-type 2D+1D Radon inversions

Chihara studies higher-dimensional seismic-type Radon transforms on the full space $c_T(x,y)$9. The model 2D+1D parabolic transform is

$v_s(z)$0

while the corresponding hyperbolic transform is the $v_s(z)$1 specialization of $v_s(z)$2. These operators arise in seismology as generalizations of parabolic and hyperbolic Radon transforms (Chihara, 2019).

The inversion is not stated abstractly; it is reduced step by step to the classical Radon transform on $v_s(z)$3. In the parabolic case one introduces an auxiliary function

$v_s(z)$4

after which $v_s(z)$5 becomes exactly the standard hyperplane Radon transform of $v_s(z)$6. Classical inversion, in the sense of Palamodov or Helgason, is then applied, and undoing the change of variables yields the explicit formula

$v_s(z)$7

The hyperbolic case follows the same template, but with a different power-variable substitution. The validity of these formulas is tied to explicit hypotheses: $v_s(z)$8 must be even in each horizontal coordinate about $v_s(z)$9; in the hyperbolic case it must also be even in $(x,z)$0 and satisfy $(x,z)$1; and the paper often assumes $(x,z)$2. Under these conditions, uniqueness follows from the injectivity of the classical Radon transform, and stability is inherited through Sobolev-type estimates of the form

$(x,z)$3

The paper also identifies the reduction to the usual one-parameter stack formulas of seismic practice when $(x,z)$4 (Chihara, 2019).

5. Related dimensional-reduction inversions

A closely related strategy appears in electromagnetic induction inversion for trench-like structures invariant in one spatial dimension. Dierckx et al. begin from a 3D Green’s-kernel formulation for the apparent conductivity,

$(x,z)$5

and then assume invariance in $(x,z)$6. Integrating out the scaled variable $(x,z)$7 yields a 2D response function $(x,z)$8 and cumulative response $(x,z)$9, which are then used in a multilayer 2D forward model. The corresponding inversion employs a non-linear Tikhonov objective with both $k_y$0-smoothing and $k_y$1-sparsity,

$k_y$2

For synthetic trench data with SNR $k_y$3 dB and $k_y$4 dB, the reported 2D inversion gives $k_y$5 and $k_y$6 of only a few percent for $k_y$7, whereas 1D errors remain $k_y$8, and position errors are typically $k_y$9 m versus $(x_1,x_2)$0 m for the 1D approach (Dierckx et al., 2018).

Other papers illustrate adjacent, though not identically labeled, dimensional inversions. In momentum-density spectroscopy, the Direct Inversion Method reconstructs a 2D plane projection $(x_1,x_2)$1 from $(x_1,x_2)$2 measured 1D Compton profiles by solving a single Tikhonov-regularized linear system,

$(x_1,x_2)$3

with derivative penalties on the discrete 2D grid. The authors describe it as a stable, one-shot inversion “without iterative loops,” and note that a $(x_1,x_2)$4 reconstruction from 10 profiles runs in a few minutes on a modern PC (Ketels et al., 2021).

In axisymmetric orientation analysis, Kloza and Elliott show that the projection of a 3D orientation distribution to 2D can be cast as an Abel transform. After introducing $(x_1,x_2)$5 and $(x_1,x_2)$6, they write

$(x_1,x_2)$7

and obtain the exact inverse

$(x_1,x_2)$8

They also derive a matrix relation between 2D Chebyshev moments $(x_1,x_2)$9 and 3D Legendre moments $y$0, including explicit closed forms for both $y$1 and $y$2 (Kloza et al., 2023).

These examples suggest a broader family of dimensional-reduction inversions in which one dimension is marginalized, projected, or reintroduced analytically while the core unknown is reconstructed on a 2D domain.

6. Assumptions, limitations, and methodological trade-offs

The principal limitation of two-step 2D+1D surface-wave inversion is not merely reduced resolution but the explicit neglect of lateral covariance. Zhao et al. attribute “discontinuities at grid-cell boundaries” and larger posterior standard deviations to this assumption, while also noting that the reduction to many low-dimensional problems is what makes McMC tractable in the first place. Their comparison with direct-3D inversion therefore frames 2D+1D not as a universally inferior method, but as a trade-off between tractability and lateral coherence (Zhao et al., 21 Jul 2025).

In the 2.5-D inverse-scattering setting, the simplification depends on a strict invariance assumption in one spatial direction. Even after this reduction, the Fourier integral is singular enough that direct quadrature is impractical; the method relies on contour deformation and analyticity. The paper also notes special care near grazing configurations $y$3 and in very near-field regimes $y$4 (Hidayetoglu et al., 2022).

For seismic Radon inversions, exact inversion formulas come with exact symmetry and vanishing assumptions. Chihara’s results require evenness in each horizontal coordinate, and in the hyperbolic case also evenness in $y$5 together with $y$6. These are not incidental regularity conditions; they are part of the mechanism by which the generalized transform is reduced to the classical Radon transform (Chihara, 2019).

The related dimensional inversions show analogous constraints. The electromagnetic-induction method assumes the low-induction-number approximation and invariance in one spatial dimension (Dierckx et al., 2018). The Direct Inversion Method for Compton inversion does not enforce positivity of $y$7, and its extension to $y$8 is described as “straightforward in principle” but dependent on a careful choice of regularizer to preserve Fermi-surface discontinuities (Ketels et al., 2021). The Abel inversion of axisymmetric orientation distributions requires genuine axisymmetry; the paper states that any non-uniform $y$9-dependence destroys the Abel form (Kloza et al., 2023).

Taken together, the literature indicates that 2D+1D inversion is most effective when the dimensional split is physically justified by symmetry, invariance, or acquisition geometry. When those assumptions hold, the split can deliver explicit inversion formulas, non-iterative solvers, or major computational savings. When they do not hold, the same split can suppress lateral correlations, distort uncertainty quantification, or force restrictive priors.