熵-PSO双驱优化的改进巴特沃斯重力向下延拓方法

doi:10.11947/j.AGCS.2025.20250137

测绘学报 ›› 2025, Vol. 54 ›› Issue (9): 1620-1632.doi: 10.11947/j.AGCS.2025.20250137

熵-PSO双驱优化的改进巴特沃斯重力向下延拓方法

王翰¹^,²(), 肖云²^,³(), 关怀魁¹^,², 孙玮萱¹^,²

^1.长安大学地质工程与测绘学院，陕西　西安　710054
^2.空间基准全国重点实验室，陕西　西安　710054
^3.西安测绘研究所，陕西　西安　710054

收稿日期:2025-03-31 修回日期:2025-08-01 出版日期:2025-10-10 发布日期:2025-10-10
通讯作者: 肖云 E-mail:wh_021112@163.com;2262164268@qq.com
作者简介:王翰（2002—），男，硕士生，研究方向为重力数据分析与处理。E-mail：wh_021112@163.com
基金资助:
国家重点研发计划(2021YFB3900604);国家自然科学基金(42404008);长安大学研究生科研创新实践项目(300103725040)

An improved Butterworth gravity downward continuation method driven by entropy-PSO dual optimization

Han WANG¹^,²(), Yun XIAO²^,³(), Huaikui GUAN¹^,², Weixuan SUN¹^,²

^1.School of Geological Engineering and Geomatics, Chang'an University, Xi'an 710054, China
^2.National Key Laboratory of Space Reference, Xi'an 710054, China
^3.Xi'an Institute of Surveying and Mapping, Xi'an 710054, China

Received:2025-03-31 Revised:2025-08-01 Online:2025-10-10 Published:2025-10-10
Contact: Yun XIAO E-mail:wh_021112@163.com;2262164268@qq.com
About author:WANG Han (2002—), male, postgraduate, majors in gravitational field data analysis and processing. E-mail: wh_021112@163.com
Supported by:
The National Key Research and Development Program of China(2021YFB3900604);The National Natural Science Foundation of China(42404008);Scientific Innovation Practice Project of Postgraduates of Chang'an University(300103725040)

摘要/Abstract

摘要：

针对重力向下延拓中高频噪声放大与解的不稳定性难题，本文提出一种融合迭代巴特沃斯滤波与粒子群优化（particle swarm optimization，PSO）的智能延拓方法。该方法引入改进巴特沃斯函数重构向下延拓算子，结合迭代补偿机制实现频谱残差修正；设计基于信息熵理论的粒子群双参数协同优化方法，实现滤波器截止频率与阶次的自适应寻优；提出适应度反馈驱动的惯性权重动态调整策略，以平衡全局探索能力与局部精确解的搜索效率。试验结果表明，本文方法相对于混合域迭代Tikhonov正则化方法与改进导数迭代法，在不同噪声水平下延拓精度均保持较好稳定性，为海空重力数据处理提供了一种有效解决方案。

关键词: 向下延拓, 波数域, 巴特沃斯滤波器, 粒子群优化算法, 迭代法

Abstract:

In response to the challenges of high-frequency noise amplification and instability in downward continuation solutions influenced by gravity, this paper proposes an intelligent continuation method that integrates iterative Butterworth filtering with particle swarm optimization (PSO). The method introduces an improved Butterworth function to reconstruct the downward continuation operator and employs an iterative compensation mechanism to correct spectrum residuals. A particle swarm dual-parameter collaborative optimization method based on information entropy theory is designed to achieve adaptive optimization of the filter's cutoff frequency and order. Furthermore, a fitness feedback-driven dynamic adjustment strategy for inertia weight is proposed to balance global exploration capability and the efficiency of searching for local precise solutions. Experimental results demonstrate that compared to the hybrid domain iterative Tikhonov regularization method and improved derivative iterative method, the proposed method maintains good stability in continuation accuracy under varying noise levels, providing an effective solution for processing marine and aerial gravity data.

Key words: downward extension, wave number domain, Butterworth filter, particle swarm optimization algorithm, iterative approach

中图分类号:

P223

王翰, 肖云, 关怀魁, 孙玮萱. 熵-PSO双驱优化的改进巴特沃斯重力向下延拓方法[J]. 测绘学报, 2025, 54(9): 1620-1632.

Han WANG, Yun XIAO, Huaikui GUAN, Weixuan SUN. An improved Butterworth gravity downward continuation method driven by entropy-PSO dual optimization[J]. Acta Geodaetica et Cartographica Sinica, 2025, 54(9): 1620-1632.

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种群规模	运行时间/s	性能指标	种群规模	运行时间/s	性能指标
10	5.45	2636	80	13.76	527
30	7.33	1318	100	20.89	615
50	11.43	791	150	50.67	1054

学习因子	运行时间/s	性能指标	学习因子	运行时间	性能指标
0.5	12.45	2636	2	13.87	123
1	13.77	703	3	12.74	1582
1.5	13.96	878	5	12.32	791

Z/km	Max/mGal	Min/mGal	Mean/mGal	SD/mGal
0	1.121	0.002	0.058	0.135
1	0.503	0.009	0.006	0.105

噪声水平/（%）	最小熵值	截断频率	阶次
2	15.55	0.0027	19
5	15.76	0.0023	20
10	15.83	0.0021	32

延拓方法	迭代次数	噪声水平/（%）	残差最大值/mGal	残差最小值/mGal	Mean/mGal	SD/mGal	RE/（%）	RMSE/mGal	Time/s
方法1	100	2	0.006	－0.007	3.61×10^－6	0.002	7.834	0.011	7.56
		5	0.018	－0.017	1.24×10^－5	0.004	7.845	0.011	8.73
		10	0.032	－0.037	1.87×10^－5	0.007	7.889	0.011	8.89
方法2	100	2	0.050	－0.054	3.22×10^－5	0.008	7.643	0.011	5.53
		5	0.088	－0.085	5.42×10^－5	0.012	7.865	0.021	5.59
		10	0.120	－0.117	6.52×10^－5	0.013	8.356	0.027	6.97
方法3	100	2	0.010	－0.028	9.71×10^－6	0.002	5.452	0.008	15.76
		5	0.013	－0.029	1.50×10^－5	0.019	5.970	0.008	17.98
		10	0.019	－0.043	3.02×10^－5	0.020	6.251	0.010	16.53

熵-PSO双驱优化的改进巴特沃斯重力向下延拓方法

An improved Butterworth gravity downward continuation method driven by entropy-PSO dual optimization

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图/表 28

参考文献 26

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延拓方法	迭代次数	噪声水平/（%）	残差最大值/mGal	残差最小值/mGal	Mean/mGal	SD/mGal	RE/（%）	RMSE/mGal	Time/s
方法1	500	2	27.638	－46.268	12.886	4.674	16.442	4.621	10.74
		5	27.320	－46.359	12.883	4.798	16.454	4.625	10.53
		10	29.021	－45.775	12.888	4.842	16.459	4.654	11.32
方法2	500	2	24.141	－43.405	13.378	4.659	15.585	5.212	8.97
		5	24.974	－43.784	13.648	4.752	15.754	5.301	9.57
		10	22.336	－45.281	14.213	4.942	15.946	5.793	8.99
方法3	500	2	28.483	－41.795	13.597	3.725	13.534	3.722	19.91
		5	28.502	－42.422	13.610	3.890	13.874	3.806	20.89
		10	28.851	－44.790	13.815	3.8841	14.356	3.882	19.47

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Z/km	Min/mGal	Max/mGal	Mean/mGal	SD/mGal
0	131.630	－69.473	13.556	24.059
3	86.053	－48.757	13.556	17.758

噪声水平/（%）	最小熵值	截断频率	阶次
2	17.57	0.000 6	22
5	17.60	0.000 4	24
10	17.70	0.000 4	44