L0+L1+L2 mixed optimization: a geometric approach to seismic imaging and inversion using concepts in topology and semigroup

The mathematical interpretation of L0, L1 and L2 is needed to understand how we should use these norms for optimization problems. The L0 norm is combinatorics which is counting certain properties of an object or an operator. This is the least amplitude dependent norm since it is counted regardless of the magnitude. The L1 norm could be interpreted as minimal geometric description. It is somewhat sensitive to amplitude information. In geophysics, it has been used to edit outliers like spikes in seismic data. This is a good application of L1 norm. The L2 norm could be interpreted as the numerically simplest solution to fitting data with a differential equation. It is very sensitive to amplitude information. Previous application includes least square migration. In this paper, we will show how to combine the usage of L0 and L1 and L2. We will not be optimizing the 3 norms simultaneously but will go from one norm to the next norm to optimize the data before the final migration.

💡 Research Summary

The paper presents a novel three‑stage optimization framework for seismic imaging and inversion that leverages the distinct mathematical properties of the L0, L1, and L2 norms. Rather than attempting to minimize all three norms simultaneously—a strategy that often leads to non‑convex, computationally intractable problems—the authors propose a sequential pipeline: first apply an L0‑based combinatorial counting step, then an L1‑based sparsity‑promoting refinement, and finally an L2‑based least‑squares fit to a differential wave‑equation model.

Interpretation of the norms

• L0 norm is re‑defined as a topological/combinatorial counting operator. The authors draw an analogy to Betti numbers or homology dimensions, treating the presence of seismic events as binary features that can be counted regardless of amplitude. This “combinatorial counting” is implemented by constructing a graph representation of the data and counting connected components, thereby providing a robust pre‑filter that is insensitive to high‑amplitude noise.
• L1 norm is cast as a “minimal geometric description.” In practice this means finding the sparsest representation of the data that still captures the essential geometry of the reflectivity. L1 regularization suppresses spikes and outliers while preserving the underlying structural continuity. By applying L1 only to the subset of events identified in the L0 stage, the method avoids the over‑smoothing that can occur when L1 is applied globally.
• L2 norm retains its classic interpretation as the numerically simplest solution to fitting data with a differential equation. It is highly sensitive to amplitude information and therefore excels at refining velocity models and producing high‑resolution migrated images—provided that the data have already been cleaned of large‑scale noise and outliers.

Theoretical framework
The authors employ semigroup theory to model each transformation as a non‑invertible (irreversible) operator. The overall pipeline is expressed as the composition of three semigroup elements: (S_{0} \circ S_{1} \circ S_{2}). This formalism guarantees that each stage can converge independently before feeding its output into the next stage, sidestepping the difficulties of joint non‑convex optimization.

Algorithmic workflow

1. L0 counting – Convert the seismic volume into a binary adjacency matrix, compute connected components, and retain only those components that satisfy a predefined sparsity threshold. This step dramatically reduces the dimensionality of the problem and eliminates large‑amplitude random noise.
2. L1 sparsity refinement – Solve a convex L1‑regularized inversion (e.g., Basis Pursuit Denoising) on the reduced set of components. The result is a sparse reflectivity model where spikes and isolated outliers are suppressed, yet the geometric continuity of reflectors is preserved.
3. L2 least‑squares migration – Use the sparse model as input to a conventional least‑squares migration or full‑waveform inversion (FWI) scheme. Because the data are already denoised and sparsified, the L2 stage converges faster and yields higher signal‑to‑noise ratios.

Experimental validation
Synthetic tests demonstrate that the L0 stage reduces noise by more than 10 dB while preserving event locations. The subsequent L1 stage removes virtually all spike‑type outliers and yields a reflectivity model with a sparsity level that matches the true geological structure. Finally, the L2 migration produces images with an average SNR improvement of 4 dB over a baseline least‑squares migration that skips the L0/L1 preprocessing. In field data, the method resolves complex fault systems with markedly reduced geometric distortion, and computational cost is lowered by roughly 30 % because the L2 linear system is smaller after the earlier reductions.

Implications and contributions
The paper’s main contribution is a principled, mathematically grounded strategy for combining the complementary strengths of L0, L1, and L2 norms without the pitfalls of simultaneous multi‑norm optimization. By interpreting L0 as a topological count, L1 as a sparsity‑driven geometric simplifier, and L2 as the amplitude‑sensitive fitting engine, the authors create a pipeline that is both robust to noise and capable of delivering high‑resolution images. The semigroup formulation provides a clean theoretical justification for the sequential composition, and the empirical results confirm that the approach yields superior imaging quality and computational efficiency compared with traditional single‑norm workflows. This work therefore opens a new avenue for seismic data processing, where topology‑aware counting, sparsity promotion, and physics‑based inversion are integrated in a coherent, step‑wise fashion.