⚡ electrical engineering

Adaptive constrained first-arrival tomography driven by uphole velocity priors for complex near-surface imaging

This paper proposes an adaptive constrained first-arrival tomography method that integrates uphole velocity priors with a novel horizontally–vertically decoupled Gaussian regularization to overcome sparse ray coverage and ill-posed artifacts, thereby significantly improving near-surface velocity modeling accuracy and static corrections for complex environments like China's Loess Plateau.

Original authors: Huo Yuanyuan, Yang Rui, Qi Hui, Li Xin

Published 2026-09-01

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CC BY 4.0

Original authors: Huo Yuanyuan, Yang Rui, Qi Hui, Li Xin

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To see what lies beneath the ground, geophysicists often listen to the Earth's own voice. They send sound waves into the soil and rock, recording how long it takes for the echoes to return. By measuring these travel times, they can build a map of the underground layers, revealing where oil or gas might be trapped. However, this process relies on a crucial assumption: that the sound waves travel through the ground in a predictable way. When the surface itself is chaotic, that assumption breaks down. In places like China's Loess Plateau, the ground is a rugged, uneven landscape of deep gullies and soft, shifting soil. This complex near-surface layer acts like a distorted lens, bending and scrambling the sound waves before they even reach the deeper targets. If the map of this shallow layer is wrong, the entire picture of the deeper reservoir becomes a blur, making it nearly impossible to find the resources hidden below.

For decades, scientists have tried to correct these distortions using a technique called tomography, which is essentially a way of reconstructing an image from many different angles. But in the Loess Plateau, the conditions are so difficult that the standard tools fail. The terrain is so rough and the access to the land so restricted by nature reserves and old mines that the sound waves cannot travel through the ground from enough different directions. The data becomes too sparse, leaving huge gaps in the information. When researchers try to fill these gaps with traditional math, the result is often a blurry, stretched-out image where the boundaries between rock layers disappear. It is as if trying to solve a puzzle with half the pieces missing; the picture that emerges is distorted, and the errors can be massive, sometimes off by hundreds of meters.

A team of researchers, led by Huo Yuanyuan and Yang Rui, has developed a new approach to fix this problem by bringing in a piece of information that was previously underutilized: the direct measurements taken from small holes drilled into the ground. These "uphole" surveys provide a precise, absolute speed of sound at specific points, acting as fixed anchors in the chaotic data. The researchers realized that while the sound waves traveling across the surface were too sparse to build a complete map on their own, these direct measurements could guide the process. They created a new mathematical method that blends the sparse surface data with these solid ground truths. Instead of forcing the entire underground model to be smooth and uniform, which often hides the real geological features, their method allows the model to stay sharp and detailed right next to the measurement points, while gently smoothing out the areas in between where data is missing.

The key to their success was designing a system that treats the ground differently in different directions. In the Loess Plateau, the speed of sound changes very quickly as you go deeper, but changes much more slowly as you move sideways. Traditional methods treated these two directions the same, which caused the map to look like a series of concentric circles around the measurement points, a visual error known as a "bullseye" artifact. The new method uses a flexible weighting system that respects the natural shape of the rock layers. It allows the precise information from the holes to influence the surrounding area strongly in the vertical direction, preserving the sharp boundaries between layers, while letting the influence fade more gradually sideways. This prevents the map from becoming artificially smooth or distorted, ensuring that the thin, critical layers of rock are not blurred out.

To test if this idea worked, the team first built a computer model that perfectly mimicked the difficult conditions of the Loess Plateau, complete with deep valleys and rapidly changing soil layers. They simulated the sparse data conditions found in real surveys, where the number of sound wave paths is extremely low. In these tests, the traditional method produced a velocity map with a significant error, missing the true speed of the rock by an average of 15.4 meters per second. The new method, however, reduced that error to just 5.3 meters per second, a massive improvement that brought the model much closer to reality. Crucially, it successfully preserved the sharp edges of the rock layers that the old method had smoothed away, proving that it could see the details that were previously invisible.

The researchers then took their method to the real world, applying it to a 3D seismic survey in the Weibei Oilfield. This area is a perfect example of the challenges they aimed to solve, with a rugged surface and a shallow, thin oil reservoir that is only a few meters thick. The survey had been severely limited by environmental restrictions, leaving the data extremely sparse. When they applied their new technique, the results were striking. The corrected map of the near-surface layer showed a clear, accurate picture of the low-speed soil layers, whereas the old method had overestimated the speed of the ground by a wide margin. This accuracy translated directly into better images of the deeper oil reservoir. The seismic reflections, which were previously jumbled and hard to read, became clear and continuous, revealing the true shape of the underground structures.

To ensure their method wasn't just getting lucky, the team tested it against a "blind" well—a real hole drilled into the ground that was not used to create the map. They compared the speed of sound predicted by their model against the actual speed measured in that well. The difference was only 12 meters per second, a remarkably small error for such a complex environment. This confirmed that the method was not just fitting the data to look good, but was actually discovering the true physical properties of the ground. Furthermore, they developed a way to automatically tune the settings of their method without needing to know the answer in advance. By looking at how well the seismic waves lined up after correction, they could find the perfect balance for their model, making the process reliable and repeatable for future surveys.

The implications of this work extend beyond just one oilfield. The method offers a robust solution for any area where the ground is difficult and the data is scarce. Whether it is the shifting sands of a desert or the rocky debris at the foot of a mountain, the ability to anchor a model with a few direct measurements and let the math fill in the gaps intelligently changes what is possible. It turns a situation that was once considered too broken to fix into one that can be mapped with high precision. By respecting the natural anisotropy of the earth—its tendency to behave differently in different directions—and by using the few reliable data points as strong guides, the researchers have provided a new way to see clearly through the noise. This approach does not require expensive new equipment or endless amounts of data; it simply uses the information already available in a smarter, more adaptive way, opening the door to finding resources in some of the most challenging landscapes on Earth.

Technical Summary: Adaptive Constrained First-Arrival Tomography Driven by Uphole Velocity Priors

Problem Statement
High-precision seismic imaging of shallow, thin interbedded reservoirs, particularly within the complex near-surface environments of China's Loess Plateau, faces severe challenges. These environments are characterized by rugged topography, laterally variable low-velocity zones (LVZs), and restricted acquisition layouts (due to nature reserves and abandoned mines) that result in ultra-low effective data folds (often < 10). Under these conditions, seismic rays in the shallow 0–200 m interval propagate nearly vertically, creating a critical lack of crossing raypaths. This scarcity leads to extensive null spaces in the Jacobian matrix and severe singularities in the Hessian matrix during conventional first-arrival traveltime tomography. Consequently, standard inversion workflows suffer from non-uniqueness, producing anomalous LVZ stretching artifacts, blurred velocity boundaries, and significant long-wavelength static errors. Furthermore, traditional isotropic regularization strategies fail to preserve sharp vertical interfaces while accommodating lateral variations, often resulting in "bullseye" artifacts or unphysical cross-layer smoothing.

Methodology
The authors propose an adaptive constrained first-arrival tomography method that integrates absolute uphole velocity priors into the inversion process. The core technical framework involves:

Reformulated Objective Function: The standard Tikhonov objective function is modified to include a third term representing adaptive constraints based on uphole survey data. The measured absolute uphole velocities are converted to slowness and embedded as prior information ( $m_{prior}$ ).
Horizontally–Vertically Decoupled Gaussian Kernel: To address the anisotropy of sedimentary near-surface media (where vertical gradients exceed lateral variations), the authors design a novel spatial weighting matrix ( $W$ ). Unlike traditional isotropic Gaussian kernels, this matrix uses separable horizontal ( $\sigma_h$ ) and vertical ( $\sigma_v$ ) decay radii. The vertical radius is fixed to match grid spacing to prevent unphysical smoothing across sharp boundaries, while the horizontal radius is optimized to control lateral constraint propagation.
Energy Normalization: A trace-ratio energy normalization principle is applied to balance the gradient energy between the data misfit term and the prior constraint term. This dynamically scales the constraint strength ( $\gamma$ ) to ensure numerical stability and robust convergence.
Data-Driven Parameter Optimization: To eliminate subjective trial-and-error in selecting the Gaussian influence radius ( $\sigma_h$ ), the authors propose a quantitative criterion: maximizing the stacked Root-Mean-Square (RMS) amplitude within a target horizon time window (e.g., the Chang 3 reservoir). This creates a closed-loop optimization framework that does not require a known ground truth model.
Synergistic Regularization: The method establishes a spatial synergy where the prior constraints act as "anchors" near uphole locations to preserve sharp gradients, while the global smoothing operator acts as a "transmission mechanism" to propagate these stable solutions into ray-blind far-field zones, preventing global drift.

Key Contributions

Anisotropic Kernel Design: The development of a horizontally and vertically decoupled Gaussian kernel specifically tailored to the sedimentary anisotropy of loess tablelands, effectively suppressing bullseye artifacts and over-smoothing.
Mathematical Conditioning: The integration of uphole slowness data into the objective function to inject positive values into the diagonal of the ill-conditioned Hessian matrix, mathematically resolving null-space singularities under ultra-low-fold conditions.
Objective Parameter Calibration: The establishment of a data-driven, closed-loop optimization framework based on stacked RMS amplitude maximization and blind-well verification, providing a reproducible method for determining optimal regularization parameters in field production.

Results

Synthetic Experiments: In synthetic models mimicking loess tableland conditions with effective folds < 10, the proposed method reduced the global velocity Root-Mean-Square Error (RMSE) by 65.6% (from 15.4 m/s to 5.3 m/s) compared to unconstrained tomography. It successfully preserved sharp vertical velocity boundaries and eliminated "bullseye" artifacts.
Single-Well Robustness: Even under extreme single-well constraints, the method reduced prediction errors by 47.4% compared to unconstrained methods, demonstrating robustness in data-poor environments.
Field Application (Weibei Oilfield): Applied to a 3D dataset from the Weibei Oilfield, the method calibrated an optimal Gaussian influence radius of 500 m. Blind-well verification (Well No. 3) confirmed a velocity prediction error of merely 12 m/s.
Imaging Quality: The constrained inversion produced a near-surface velocity model that accurately characterized the LVZ bottom boundary (300–500 m/s range) and significantly improved the continuity of reflectors and the clarity of reflection wave packages in stacked seismic sections compared to conventional methods.

Significance and Claims
The paper claims that this method provides a robust technical framework for high-fidelity static corrections and near-surface imaging in complex, ultra-low-fold environments. By fundamentally addressing the non-uniqueness of traveltime inversion through adaptive, anisotropic constraints, the approach enables the construction of reliable velocity models where traditional methods fail. The authors emphasize that the method is computationally efficient (adding less than 2% overhead per iteration) and generalizable to other complex near-surface scenarios (e.g., desert dunes, piedmont moraines) where sparse absolute measurements are available to anchor ray-path blind zones. The study concludes that the proposed workflow transforms parameter tuning from an empirical process into an objective, data-driven system, unlocking the imaging potential of historical datasets with restricted acquisition geometries.

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