Downward Continuation Iterative Regularization Solution Based on Quasi⁃Optimal Regularization Factor Set

LIU Meng; WANG Zhengtao

doi:10.13203/j.whugis20230127

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Downward Continuation Iterative Regularization Solution Based on Quasi⁃Optimal Regularization Factor Set

Geodesy and Survey Engineering | 更新时间：2026-11-05

- Downward Continuation Iterative Regularization Solution Based on Quasi⁃Optimal Regularization Factor Set
- Geomatics and Information Science of Wuhan University Vol. 50, Issue 12, Pages: 2450-2459(2025)
- 作者机构：
  
  1.武汉大学测绘学院，湖北武汉，430079
  2.武汉大学地球空间环境与大地测量教育部重点实验室，湖北武汉，430079
- 作者简介：
- 基金信息：
- DOI：10.13203/j.whugis20230127
  CLC： P223
- Received：31 October 2024，
  
  Published：05 December 2025
- 稿件说明：
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刘猛,王正涛.基于拟最优正则化因子组的航空重力向下延拓迭代求解方法[J].武汉大学学报（信息科学版）,2025,50(12):2450-2459.

LIU Meng,WANG Zhengtao.Downward Continuation Iterative Regularization Solution Based on Quasi⁃Optimal Regularization Factor Set[J].Geomatics and Information Science of Wuhan University,2025,50(12):2450-2459.
刘猛,王正涛.基于拟最优正则化因子组的航空重力向下延拓迭代求解方法[J].武汉大学学报（信息科学版）,2025,50(12):2450-2459. DOI：10.13203/j.whugis20230127

LIU Meng,WANG Zhengtao.Downward Continuation Iterative Regularization Solution Based on Quasi⁃Optimal Regularization Factor Set[J].Geomatics and Information Science of Wuhan University,2025,50(12):2450-2459. DOI：10.13203/j.whugis20230127

摘要

迭代Tikhonov正则化和迭代Landweber正则化法是有效解决向下延拓不适定问题的两种迭代正则化算法。针对迭代正则化法中迭代次数和正则化参数选取问题，首先提出拟最优正则化因子组概念，系统分析二者在拟最优正则化因子组序列中的分布关系；然后提出一种迭代正则化法中最优正则化因子组的选取策略，并推导出L曲线法确定与迭代次数对应最优正则化参数的公式；最后通过重力场模型的模拟实验与传统向下延拓方法进行比对，验证了所提正则化因子组选取依据的可靠性。

Abstract

Objectives

This paper addresses a critical limitation hindering the practical application of widely used iterative regularization methods

such as Tikhonov regularization and Landweber regularization. The lack of a clear strategy for optimally pairing iteration counts with regularization parameters.

Methods

The concept of a quasi-optimal regularization factor set is constructed. By analyzing the distribution of iteration counts and regularization parameters within these sets

a strategy for selecting the best regularization factor group is proposed. Furthermore

a formula is presented to determine the optimal regularization parameter corresponding to a given iteration count using the L-curve method.

Results

Analysis of the extension error and the variation within quasi-optimal factor sets reveals a significant correlation between optimal regularization parameters and iteration counts. When the iteration count exceeds 10

the extension solutions corresponding to different quasi-optimal factor sets become nearly identical. For sufficiently large iteration counts

any quasi-optimal factor set yields similar effects. Compared with the traditional Tikhonov regularization

the proposed iterative method based on the new selection strategy produces smoother extension solutions with smaller errors. However

in regions with sharp data variations

some high-frequency signals may be filtered out as noise

leading to no significant improvement in extension performance there.

Conclusions

The proposed iterative regularization algorithm incorporates the novel selection strategy

and generates smoother and more accurate extension solutions than the traditional Tikhonov method.The results demonstrate its reliability and practical utility.

关键词

Keywords

references

CARPENTER C . Machine-Learning Method Determines Salt Structures from Gravity Data ［J］. Journal of Petroleum Technology ， 2021 ， 73 （ 2 ）： 70 - 71 .

YE Z H ， ZHANG Z H ， WÜNNEMANN B ， et al . High-Precision Downward Continuation of the Potential Field Based on the D-Unet Network ［J］. Frontiers in Earth Science ， 2022 ， 10 ： 897055 .

邓凯亮，黄谟涛，吴太旗，等 . 利用高阶径向导数带限模型进行重力向下延拓计算［J］. 武汉大学学报（信息科学版）， 2024 ， 49 （ 3 ）： 442 - 452 .

DENG Kailiang ， HUANG Motao ， WU Taiqi ， et al . Downward Continuation of Gravity Using the Band-Limited Models for High-Order Radial Derivatives of Gravity Anomaly ［J］. Geomatics and Information Science of Wuhan University ， 2024 ， 49 （ 3 ）： 442 - 452 .

郭金运，张鸿飞，李真，等 . 多源船载重力异常数据的联合再处理：以墨西哥湾为例［J］. 武汉大学学报（信息科学版）， 2025 ， 50 （ 7 ）： 1277 - 1290 .

GUO Jinyun ， ZHANG Hongfei ， LI Zhen ， et al . Joint Reprocessing of Multi-Sources Shipborne Gravity Anomalies： A Case Study of Gulf of Mexico ［J］. Geomatics and Information Science of Wuhan University ， 2025 ， 50 （ 7 ）： 1277 - 1290 .

CLARKE G K C . Optimum Second-Derivative and Downward-Continuation Filters ［J］. Geophysics ， 1969 ， 34 （ 3 ）： 424 - 437 .

FEDI M ， FLORIO G . A Stable Downward Continuation by Using the ISVD Method ［J］. Geophysical Journal International ， 2002 ， 151 （ 1 ）： 146 - 156 .

KERN M ， SCHWARZ K P . A Comparison of Direct and Indirect Numerical Methods for the Downward Continuation of Airborne Gravity Data ［C］// Vistas for Geodesy in the New Millennium ， Berlin， Heidelberg，Germany ， 2002 .

FORSBERG R . A New Covariance Model for Inertial Gravimetry and Gradiometry ［J］. Journal of Geophysical Research： Solid Earth ， 1987 ， 92 （ B2 ）： 1305 - 1310 .

陈欣，翟国君，暴景阳，等 . 航空重力向下延拓的最小二乘配置Tikhonov正则化法［J］. 武汉大学学报（信息科学版）， 2018 ， 43 （ 4 ）： 578 - 585 .

CHEN Xin ， ZHAI Guojun ， BAO Jingyang ， et al . Least Squares Collocation-Tikhonov Regularization Method for the Downward Continuation of Airborne Gravity Data ［J］. Geomatics and Information Science of Wuhan University ， 2018 ， 43 （ 4 ）： 578 - 585 .

DAMPNEY C N G . The Equivalent Source Technique ［J］. Geophysics ， 1969 ， 34 （ 1 ）： 39 - 53 .

KERN M . An Analysis of the Combination and Downward Continuation of Satellite， Airborne and Terrestrial Gravity Data ［D］. Calgary ： University of Calgary ， 2003 .

A Comparison of Different Downward Continuation Methods for Airborne Gravity Data ［J］. Chinese Journal of Geophysics ， 2004 ， 47 （ 6 ）： 1143 - 1149 .

TIKHONOV A I . Solution of Incorrectly Formulated Problems and the Regularization Method ［J］. Soviet Math Dokl . 1963 ， 5 （ 1 ）： 115 - 128 .

LI Y G ， DEVRIESE S G R ， KRAHENBUHL R A ， et al . Enhancement of Magnetic Data by Stable Downward Continuation for UXO Application ［J］. IEEE Transactions on Geoscience and Remote Sensing ， 2013 ， 51 （ 6 ）： 3605 - 3614 .

ABEDI M ， GHOLAMI A ， NOROUZI G H . A Stable Downward Continuation of Airborne Magnetic Data： A Case Study for Mineral Prospectivity Mapping in Central Iran ［J］. Computers & Geosciences ， 2013 ， 52 ： 269 - 280 .

XU P L . Determination of Surface Gravity Anomalies Using Gradiometric Observables ［J］. Geophysical Journal International ， 1992 ， 110 （ 2 ）： 321 - 332 .

徐新强，赵俊 . 航空重力向下延拓的多参数正则化方法［J］. 武汉大学学报（信息科学版）， 2020 ， 45 （ 7 ）： 956 - 963 .

XU Xinqiang ， ZHAO Jun . A Multi-parameter Regularization Method in Downward Continuation for Airborne Gravity Data ［J］. Geomatics and Information Science of Wuhan University ， 2020 ， 45 （ 7 ）： 956 - 963 .

SCHROTER T ， TAUTENHAHN U . On the Optimality Regularization Methods for Solving Linear Ill-Posed Problems ［J］. Journal of Analysis and Its Applications ， 1994 ， 13 （ 4 ）： 297 - 710 .

曾小牛，李夕海，韩绍卿，等 . 位场向下延拓三种迭代方法之比较［J］. 地球物理学进展， 2011 ， 26 （ 3 ）： 908 - 915 .

ZENG Xiaoniu ， LI Xihai ， HAN Shaoqing ， et al . A Comparison of Three Iteration Methods for Downward Continuation of Potential Fields ［J］. Progress in Geophysics ， 2011 ， 26 （ 3 ）： 908 - 915 .

LIU X G ， LI Y C ， XIAO Y ， et al . Downward Continuation of Airborne Geomagnetic Data Based on Two Iterative Regularization Methods in the Frequency Domain ［J］. Geodesy and Geodynamics ， 2015 ， 6 （ 1 ）： 34 - 40 .

BLAKELY R J . Potential Theory in Gravity and Magnetic Applications ［M］. New York ： Cambridge University Press ， 1995 .

MARTINEC Z . Stability Investigations of a Discrete Downward Continuation Problem for Geoid Determination in the Canadian Rocky Mountains ［J］. Journal of Geodesy ， 1996 ， 70 （ 11 ）： 805 - 828 .

SCHWARZ K P ， SIDERIS M G ， FORSBERG R . The Use of FFT Techniques in Physical Geodesy ［J］. Geophysical Journal International ， 1990 ， 100 （ 3 ）： 485 - 514 .

GOLUB G H ， HEATH M ， WAHBA G . Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter ［J］. Technometrics ， 1979 ， 21 （ 2 ）： 215 - 223 .

HOERL A E ， KENNARD R W . Ridge Regression： Biased Estimation for Nonorthogonal Problems ［J］. Technometrics ， 2000 ， 42 （ 1 ）： 80 - 86 .

黄谟涛，邓凯亮，欧阳永忠，等 . 海空重力测量及应用技术研究若干进展［J］. 武汉大学学报（信息科学版）， 2022 ， 47 （ 10 ）： 1635 - 1650 .

HUANG Motao ， DENG Kailiang ， OUYANG Yongzhong ， et al . Development and Study in Marine and Airborne Gravimetry and Its Application ［J］. Geomatics and Information Science of Wuhan University ， 2022 ， 47 （ 10 ）： 1635 - 1650 .

MA T ， WU M P ， HU X P ， et al . A Regularization Method for Downward Continuation of Potential Fields ［J］. Applied Mechanics and Materials ， 2011 ， 135/136 ： 132 - 139 .

钟炀，刘辉 . 基于BP神经网络的磁异常向下延拓方法［J］. 舰船科学技术， 2023 ， 45 （ 18 ）： 133 - 138 .

ZHONG Yang ， LIU Hui . Downward Extension Method of Magnetic Anomaly Based on BP Neural Network ［J］. Ship Science and Technology ， 2023 ， 45 （ 18 ）： 133 - 138 .

徐世浙 . 迭代法与FFT法位场向下延拓效果的比较［J］. 地球物理学报， 2007 ， 50 （ 1 ）： 285 - 289 .

XU Shizhe . A Comparison of Effects Between the Iteration Method and FFT for Downward Continuation of Potential Fields ［J］. Chinese Journal of Geophysics ， 2007 ， 50 （ 1 ）： 285 - 289 .

HANSEN P C ， O’LEARY D P . The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems ［J］. SIAM Journal on Scientific Computing ， 1993 ， 14 （ 6 ）： 1487 - 1503 .

LANDWEBER L . An Iteration Formula for Fredholm Integral Equations of the First Kind ［J］. American Journal of Mathematics ， 1951 ， 73 （ 3 ）： 615 - 624 .

SCHERZER O . A Modified Landweber Iteration for Solving Parameter Estimation Problems ［J］. Applied Mathematics and Optimization ， 1998 ， 38 （ 1 ）： 45 - 68 .

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