Alternative Iterative Scheme

• Alternative Iterative Scheme is a variant process that modifies standard iterative methods by embedding adaptive corrections and PSF estimation to compensate for irregular sampling and ill-conditioning.
• It incorporates algorithmic steps like multidimensional deconvolution and iterative PSF updates to correct blurred data and recover high-fidelity wavefields.
• The scheme demonstrates significant improvement in artifact suppression and amplitude recovery, achieving convergence in a similar number of iterations as classical methods while addressing nonideal conditions.

An alternative iterative scheme is a variant or modification of a standard iterative process designed to achieve improved robustness, convergence, or applicability in specific regimes or data configurations. Such schemes are widely developed in numerical analysis, scientific computing, optimization, and inverse problems, particularly when the standard iteration fails to accommodate non-ideal conditions such as incomplete data, irregular geometry, ill-posedness, or poor conditioning. The following exposition focuses on the class of alternative iterative schemes with particular attention to the PSF-corrected iterative Marchenko scheme for handling imperfectly sampled seismic data, as a paradigmatic example (IJsseldijk et al., 2020), with complementary insights from related advancements in alternative iterative algorithms.

1. General Concept and Motivation

Classical iterative schemes (e.g., Richardson, Picard, stationary splitting, Krylov subspace methods) are typically designed under idealized assumptions: sufficient regularity, full and regular data sampling, or well-conditioned operators. However, practical scenarios frequently violate these assumptions, resulting in severe degradation of convergence and accuracy. An alternative iterative scheme introduces modifications at the algorithmic or algebraic level to compensate for such non-idealities.

Motivating scenarios include:

• Irregular or imperfectly sampled data: Acquisition mechanisms may yield data with missing samples or nonuniform spacing, leading to systematic "blurring" or distortion in inversion algorithms (notably in seismic imaging).
• Ill-conditioning or non-uniqueness: The underlying linear or nonlinear problems may have near-zero singular values or large null spaces, as in the presence of strong multiple scattering or redundancy.
• Nonlinearities or nonstationarities: Standard fixed-point or perturbative schemes might stagnate or diverge unless adapted.

The key principle in alternative iterative schemes is the explicit incorporation of data- or model-dependent correction mechanisms—often in the form of deblurring, preconditioning, or projection steps—within each iteration to recover the target solution as if the forward problem were ideally posed.

2. The PSF-Corrected Iterative Marchenko Scheme

A canonical example is provided by the alternative iterative Marchenko scheme for addressing irregularly sampled seismic data (IJsseldijk et al., 2020). The classical Marchenko method reconstructs focusing and Green's functions in the subsurface from surface reflection data, invoking two integral equations discretized as summations over a regular grid of sources. Under nonideal (irregular) sampling, these summations no longer yield accurate integral approximations, and the retrieved wavefields suffer "blurring" and artifacts.

The PSF-corrected iterative scheme resolves this by embedding the point-spread function (PSF) theory into the iterative loop:

• PSF Definition: The PSF quantifies the deviation from perfect sampling, encoding the blurring kernel arising when integration domains are sampled irregularly.
• Modified Update Equations: Each Marchenko iteration is reformulated so that the blurred (i.e., convolved) fields replace the standard ones on the left-hand side:

$$\hat{G}^-(x_A, x_R, \omega) + \hat{f}_1^-(x_R, x_A, \omega) = \sum_i R(x_R, x_S^{(i)}, \omega)\, f_1^+(x_S^{(i)}, x_A, \omega)\, S(\omega)$$

$$\hat{G}^+(x_A, x_R, \omega) - [\hat{f}_1^+(x_R, x_A, \omega)]^* = -\sum_i R(x_R, x_S^{(i)}, \omega)\, [f_1^-(x_S^{(i)}, x_A, \omega)]^*\, S(\omega)$$

Here, $\hat{G}^-$, $\hat{G}^+$, $\hat{f}_1^-$, and $\hat{f}_1^+$ denote quantities convolved with the appropriate PSFs, $\Gamma^+$ and $\Gamma^-$. The essential correction is that the retrieved fields are first blurred by the PSFs before matching to the data summations; the unblurring (deblurring) is performed after each iteration via multidimensional deconvolution (MDD).

• Iteration Cycle: Estimation of the PSFs themselves is embedded within the fixed-point loop, so that the blurring kernels are iteratively updated along with the wavefields.
• Deblurring Step: Once the blurred quantities are obtained, the "true" focusing and Green's functions are recovered by inverting the convolution, typically via regularized (Tikhonov or water-level) MDD.

This procedure eliminates the requirement for perfect source sampling in the Marchenko redatuming workflow, demonstrating marked improvements in artifact suppression and amplitude recovery even under realistic (degraded) field data (IJsseldijk et al., 2020).

3. Mathematical Structure and Algorithmic Steps

The alternative iterative Marchenko scheme can be distilled into the following procedural steps per iteration $k$:

1. Initialization: Compute the initial downgoing focusing function $f_1^+,(0)$ as the direct arrival Green's function in a background model; set the initial upgoing focusing function and Green's functions to zero.
2. Imperfect Data Update: Apply the physically measured (irregular) reflection data to update the blurred upgoing focusing and downgoing Green's functions.
3. PSF Estimation: Estimate the downgoing and upgoing PSFs, $\Gamma^{+,(k+1)}$ and $\Gamma^{-,(k+1)}$, typically by constructing temporal inverse operators relative to the current focusing functions and integrating over the irregular geometry.
4. PSF Application: Convolve the computed PSFs with the current focusing and Green's functions to yield the updated blurred fields.
5. Deblurring (MDD step): Invert the PSF convolution, employing regularized multidimensional deconvolution to retrieve the current best approximations of the true up/down Green’s functions and focusing functions.
6. Focusing/Green’s Separation: Extract physical wavefields by appropriate time-gating/separation.
7. Convergence Check: The iteration is terminated either after a fixed number of steps or when the change in focusing functions falls below a prescribed (typically small) threshold.

The algorithm inherits the stability properties of the classical Marchenko scheme provided the MDD inversions are appropriately regularized for numerical robustness.

4. Quantitative Performance and Comparative Analysis

Numerical demonstration on 2D layered acoustic models with irregular 50% source decimation confirms the effectiveness of the alternative iterative scheme (IJsseldijk et al., 2020):

• Artifact Suppression: The PSF-corrected scheme achieves over 75% reduction in sampling artifacts (as visually quantified) compared to standard iteration on irregular data.
• Amplitude Recovery: Retrieved amplitudes and wavefield events match within ±10% of the reference (fully-sampled) results, contrasting with severe amplitude distortion in the standard approach.
• Convergence: The modified scheme converges in approximately 12 iterations, which is similar to the standard scheme on perfectly sampled data.
• Computational Cost: Cost per iteration increases due to multiple (two) MDD inversions and explicit PSF computations, but remains tractable.

The method enables Marchenko redatuming applications directly on field data characterized by acquisition gaps or irregularity, a significant extension beyond the classical method's requirements.

5. Broader Context: Other Alternative Iterative Schemes

The principle of embedding problem-specific correction terms or adaptive procedures within iterative schemes is widely adopted in many domains:

• Adaptive Iterative Linearized Galerkin methods use a posteriori estimators to balance linearization and discretization errors, embedding alternative stopping and refinement policies into the iterative framework (Heid et al., 2019).
• Auxiliary Schemes for Large-Kernel Operators (e.g., Maxwell or grad–div systems) append secondary terms to the primary operator, restoring Laplace-like spectral properties and enabling robust application of classical solvers (Lu, 2021).
• Residual-Minimization Corrections exploit low-dimensional "correction spaces"—often based on problem asymptotics or Krylov subspaces—after an initial fixed-point or stationary update, yielding acceleration without full abandonment of the underlying split or source iteration (Bardin et al., 2024).
• Randomized Block Iterative Updates introduce stochasticity or blockwise projections to address high-dimensional or poorly conditioned systems (Xiang et al., 2017).

The unifying principle is the explicit design of the iterative template to encode physical or algebraic knowledge about the nonideal features of the data, operator, or sampling, with convergence rigorously established via spectral, residual, or variational analysis.

6. Significance and Extensions

Alternative iterative schemes, by incorporating adaptive correction, compensation for geometric or operator nonidealities, or explicit deblurring, extend the practical and theoretical reach of iterative algorithms. The PSF-corrected iterative Marchenko scheme exemplifies how such methodologies can render otherwise impracticable inverse-problem solvers robust to sampling or acquisition defects, with rigorous justification and significant empirical gains.

Potential extensions include:

• Joint estimation schemes coupling data-driven and model-driven corrections,
• Hybrid iterative methods combining deblurring, adaptive regularization, and data augmentation,
• Application to nonlinear or high-dimensional inverse problems,
• Integration with machine-learned PSFs or adaptive step control,
• Parallelization and acceleration via multi-resolution or block strategies.

The systematic development of alternative iterative schemes continues to underpin advances across inverse problems, scientific computing, and data assimilation, where ideal conditions are seldom attained and domain knowledge must be encoded within the algorithmic core (IJsseldijk et al., 2020).