Stabilization of Vertical Gradient Calculation Against Random Noise in Potential Field Data Using Tikhonov Regularization

Accurate calculation of vertical gradients in potential field data plays a vital role in geophysical surveys and subsurface explorations. These gradients, which are derived from gravity or magnetic field measurements, serve to enhance the resolution of anomaly boundaries and help differentiate between geological structures located at various depths. By sharpening the edges of anomalies, vertical gradients allow geophysicists to better understand the geometry and distribution of subsurface features such as faults, intrusive bodies, or mineral deposits. However, calculating these gradients is highly sensitive to the presence of random noise, which is an inevitable component of real-world data. This sensitivity becomes increasingly problematic when computing higher-order vertical gradients, where noise amplification can dominate the signal, leading to unstable, distorted signals and hence unreliable interpretations.
This study addresses the critical need to stabilize vertical gradient calculations in the presence of noise. It begins by analyzing the standard method of calculating vertical gradients in the frequency domain using Fourier transforms. While this approach is efficient and mathematically straightforward, it inherently increases the influence of high-frequency components, many of which represent noise rather than useful geological information. As a result, the final gradient estimates obtained through this method may be contaminated, especially in areas with complex geological settings or low signal-to-noise ratios. To overcome these limitations, the paper investigates two advanced stabilization techniques: Tikhonov regularization and upward continuation filtering.
Tikhonov regularization is a well-established method for addressing ill-posed inverse problems. In this context, it introduces a stabilizing constraint that penalizes overly sharp or erratic variations in the gradient estimates. This is achieved through the minimization of a cost function that includes both a data fidelity term and a regularization term that suppresses rapid changes. The strength of this regularization is controlled by a parameter that must be carefully selected. The paper evaluates two methods for determining this parameter: the norm C method and Morozov’s discrepancy principle. These approaches help ensure a balance between suppressing noise and preserving meaningful geological detail in the gradient results.
Upward continuation filtering is another effective technique explored in this study. It works by projecting the data to a higher elevation, which naturally smooths out high-frequency (i.e., shallow or noisy) components. This filtering effect reduces the influence of noise and results in more stable and geologically interpretable gradient estimates, particularly for deeper structures.
The effectiveness of these methods is demonstrated using both synthetic and real geophysical datasets. A synthetic model involving a Bouguer anomaly at a 10-meter depth shows that Tikhonov regularization and upward continuation outperform conventional Fourier-based gradient calculations in terms of accuracy and noise resistance. Furthermore, real-world gravity data from Slovakia are used to validate the practical utility of these approaches. The improved clarity in detecting geological features such as faults and shear zones, underscores the value of stabilized gradient methods.
In conclusion, the study highlights the importance of using stabilization techniques for vertical gradient computation. Both Tikhonov regularization and upward continuation significantly improve the reliability and interpretability of gradient estimates, making them essential tools for modern geophysical data analyses and subsurface mapping.