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Offset-continuation-trajectory stacking based on common-reflection-point kinematics for five-dimensional prestack dataset regularization and enhancement

This paper presents a physics-informed framework for five-dimensional prestack seismic data regularization and enhancement that utilizes a multi-parameter common-reflection-point traveltime stacking operator with offset-continuation trajectories to reconstruct missing traces and improve signal quality while preserving reflection and diffraction kinematics.

Original authors: Tiago A. Coimbra, Rodrigo Bloot, Caian Benedicto, Nicholas T. Okita, Jorge H. Faccipieri Junior

Published 2026-06-24✓ Author reviewed ⓘ

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CC0 1.0

Original authors: Tiago A. Coimbra, Rodrigo Bloot, Caian Benedicto, Nicholas T. Okita, Jorge H. Faccipieri Junior

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a conversation in a crowded room, but some people are missing, some are shouting over others, and the microphones are placed in a messy, uneven pattern. The result is a recording full of gaps, static, and confusing echoes. This is exactly what happens with seismic data (sound waves used to map underground rocks for oil and gas) when the equipment can't be placed perfectly due to terrain, cost, or obstacles.

This paper introduces a new "smart ear" for geophysicists. It's a method to clean up messy recordings and fill in the missing pieces using the actual laws of physics, rather than just guessing with math.

Here is a breakdown of how it works, using simple analogies:

1. The Problem: The "Broken Mosaic"

Seismic surveys create a giant 5-dimensional puzzle (involving time, two horizontal directions, and two offset directions). But often, the puzzle is broken:

Missing Pieces: Some spots have no data because the ground was too rough to place a sensor.
Noise: Wind, ocean waves, or traffic create static that drowns out the signal.
Irregularity: The sensors aren't in a neat grid; they are scattered randomly.

Traditional methods try to fix this by using pure math to guess what the missing pieces might look like. But sometimes, these guesses create fake structures or "ghosts" that don't exist underground.

2. The Solution: The "Physics Detective"

The authors propose a new tool called Offset-Continuation-Trajectory (OCT) Stacking. Instead of guessing, this method acts like a detective who knows exactly how sound waves travel through the earth.

Think of a sound wave as a bouncing ball.

If you throw a ball at a wall, you know exactly where it will bounce and how long it will take to return, based on the angle and the speed of the ball.
This method uses that same logic. It calculates the exact path a seismic wave should take to hit a rock layer and come back to a specific spot, even if no sensor was actually there to catch it.

3. The Secret Sauce: The "Common Reflection Point" (CRP)

The core idea is the Common Reflection Point. Imagine a specific spot deep underground (a rock layer).

In a perfect world, you would have sensors everywhere to hear the echo from that one spot.
In the real world, you only have a few sensors.
This method says: "Let's pretend all these scattered sensors are actually listening to that one specific spot underground."

It creates a virtual path (a trajectory) that connects all the scattered data points back to that single underground spot. It's like drawing a smooth, invisible highway that connects all the scattered cars (data points) to a single destination, ensuring they all arrive at the right time.

4. How It Works: The "Smart Stack"

The process happens in three main steps:

Finding the Pattern: The computer looks at the messy data and uses a "smart search" (called a coevolutionary algorithm) to figure out the speed of the waves and the shape of the underground layers. It's like tuning a radio until the static clears and the music becomes loud and clear.
Drawing the Highway: Once it knows the speed and shape, it draws the "Offset-Continuation Trajectory." This is the mathematical path the wave should have taken.
Filling the Gaps: The method takes the real data, slides it along these invisible highways, and stacks (adds) them together.
- Real signals (the actual rock layers) line up perfectly and get louder (like a choir singing in tune).
- Noise (the static) is scattered and cancels itself out (like a crowd talking over each other).
- Missing spots are filled in by calculating exactly what the signal would have been if a sensor had been there, based on the physics of the surrounding data.

5. The Results: A Clearer Picture

The authors tested this on two things:

Computer Simulations: They created a fake underground world with known answers. The method successfully filled in the missing data and removed the noise, matching the "truth" almost perfectly.
Real Earth Data: They used it on a real oil field in Brazil. The messy, gap-filled data became a smooth, continuous image. The "ghosts" and fake events that other methods create were avoided because the method stuck to the laws of physics.

The Bottom Line

This paper presents a way to turn a messy, broken seismic recording into a clean, complete 3D movie of the underground. It does this not by guessing, but by using the laws of physics to trace the path of sound waves. It's like taking a blurry, pixelated photo and using the rules of light to reconstruct the sharp, high-definition original image, ensuring that what you see is real geology, not a mathematical illusion.

Technical Summary: Offset-Continuation-Trajectory Stacking Based on Common-Reflection-Point Kinematics for 5D Prestack Dataset Regularization

Problem Statement
Irregularities in seismic acquisition geometries, caused by economic, environmental, or logistical constraints, result in incomplete spatial coverage, irregular sampling, and noise contamination. These limitations degrade event continuity and compromise the physical consistency of conventional interpolation methods, particularly in five-dimensional (5D) prestack datasets. While various model-driven and data-driven approaches exist (e.g., Fourier-based techniques, curvelet transforms, rank-reduction), many rely heavily on mathematical assumptions that struggle to maintain accurate wavefield kinematics in complex geological settings. Furthermore, existing physics-based methods, such as offset-continuation operators, often require regular acquisition geometries to function effectively. The core challenge addressed is the reconstruction of missing traces and the enhancement of spatial continuity in irregular 5D datasets while preserving the kinematics of both reflection and diffraction events without introducing artificial artifacts.

Methodology
The authors propose a physics-informed framework for 5D prestack dataset reconstruction based on a multi-parameter Common-Reflection-Point (CRP) traveltime stacking operator. The methodology extends the traditional Offset-Continuation-Trajectory (OCT) stacking concept to the full 5D prestack domain.

Theoretical Foundation: The method is grounded in wavefront propagation kinematics, defining isochronous surfaces and reflection interfaces. It utilizes a "double-square-root" (DSR) Ansatz to approximate diffraction traveltime, assuming a locally elliptical wavefront centered at an image-ray arrival point.
CRP Trajectory and OCT Operator: The core of the method is the derivation of coherent stacking trajectories (OCT) from wavefront propagation, isochronous surface geometry, specular reflection, and diffraction kinematics. The operator maps the midpoint coordinates as a function of half-offset along a CRP trajectory.
Parameter Estimation: All kinematic parameters (including local slopes and effective velocities) are estimated directly from the data, rather than relying on a pre-defined velocity model. This is achieved through a global coevolutionary optimization strategy known as Evolution by Neighborhood Similarity (ENS), which maximizes the similarity measure along the OCT traveltime surfaces.
Regularization Scheme: The reconstruction process involves two stages:
- Stacking: Input data is stacked along the estimated OCT trajectories within specific midpoint and offset apertures to compute amplitudes at output grid locations.
- Spreading/Refinement: A spreading step is applied to enhance continuity and stabilize amplitudes, using a physically motivated kernel to weight contributions from input samples.
Simplification: To balance accuracy and computational efficiency, the authors approximate the curvature matrix $S_0$ with a scalar effective velocity term ( $4I/V_0^2$ ), reducing the parameter count from five to three for the tested scenarios, though the full formulation remains available for highly anisotropic cases.

Key Contributions

Physics-Informed 5D Extension: The paper extends the OCT methodology, previously limited to lower dimensions or regular geometries, to a comprehensive 5D prestack domain, allowing adaptive interpolation in both midpoint and offset domains.
Data-Driven Kinematic Estimation: Unlike methods requiring external velocity models, this approach estimates all kinematic parameters directly from the data using global optimization, ensuring the reconstructed events adhere to the physical principles of wave propagation.
Unified Treatment of Reflections and Diffractions: The framework explicitly models both specular reflection and tip diffraction kinematics, preserving the behavior of non-specular events (e.g., faults, discontinuities) which are often lost in purely mathematical interpolation.
Robustness in Irregular Geometries: The method is designed to function effectively even with sparse or irregular acquisition layouts, reconstructing missing traces along physically consistent traveltime surfaces.

Results
The method was validated using both synthetic finite-difference modeled data and a real onshore dataset from the Potiguar Basin, Brazil.

Synthetic Data: The method successfully interpolated data onto a fully regular grid. Comparisons with ground truth showed accurate recovery of unrecorded amplitudes and traveltime shifts. The regularization effectively eliminated traveltime shifts associated with varying offsets and azimuths, preserving event continuity and amplitude behavior.
Field Data: Applied to the Potiguar Basin dataset, the method resolved offset gaps and irregularities present in the original acquisition. The regularized output demonstrated significant noise suppression and enhanced reflector continuity compared to the original data. Velocity spectra derived from the regularized data showed more precise and better-defined velocity trends.
Performance Metrics: Applications demonstrated improved Signal-to-Noise Ratio (SNR), enhanced structural continuity, and reliable recovery of missing information without introducing artificial events.

Significance and Claims
The authors claim that incorporating physically constrained traveltime models into the regularization process provides a robust, geologically consistent alternative to purely mathematical interpolation techniques. The study demonstrates that by deriving stacking trajectories from wavefront kinematics and estimating parameters directly from the data, the method improves data fidelity for subsequent imaging and quantitative interpretation.

The paper acknowledges that convergence stability may be limited in regions with extreme lateral velocity contrasts where wavefront distortions violate local smoothness assumptions; however, it asserts that such conditions are rare in realistic geological settings within small CRP neighborhoods. The proposed framework is presented as a significant advancement over numerical interpolation, offering a practical solution for 5D dataset regularization that maintains the integrity of physical events while mitigating acquisition-related gaps and noise.