LSQR法在位场反演中的分析与评价

doi:10.11720/wtyht.2019.1261

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摘要

LSQR法具有计算效率高、对计算机内存要求低的优点,适合于大规模问题的求解。为探讨其应用于位场反演的稳定性和可靠性,笔者以加入不同噪声的两个合成模型数据为实验对象,比较分析了Tikhonov正则化与LSQR法求解结果,显示直接利用LSQR法求解位场反问题能够得到满意的正则化解,其解模型相对Tikhonov正则化,最大相对误差仅为0.36%,说明直接利用LSQR法求解位场反问题是可行的。将其应用于四川盆地雅安地区重力三维反演,极大地降低计算成本,获取了区内沉积盆及主要断裂分布情况,为页岩气靶区优选提供了有力支撑。

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	梁生贤
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关键词 ：迭代正则化, LSQR法, Tikhonov正则化, 位场反演

Abstract：

The LSQR method has the advantages of high computing efficiency and low memory requirement, and is thus suitable for solving large-scale ill-posed problems. In order to discuss its application in the stability and reliability of potential field inversion, the authors, on the basis of two synthetic models data with different noises as the object of the experiment, studied Tikhonov regularization and LSQR method, and the results show that the LSQR method can satisfy the requirement for solving potential field inversion with the regularization solution. In comparison with Tikhonov regularization, the maximum relative error of the LSQR method is only 0.36%, indicating that the use of LSQR method to solve potential field inverse problems is feasible. It was applied to 3D inversion of gravity in Ya’an area of Sichuan basin, which greatly reduced computation cost and obtained the distribution of sedimentary basins and main faults in the area, thus providing strong support for shale gas target area optimization.

Key words： iterative regularization LSQR method Tikhonov regularization potential field inversion

收稿日期: 2018-07-02 出版日期: 2019-04-10

P631

基金资助:国家重点研发计划项目(2016YFC060308,2018YFC0604103)

作者简介: 梁生贤(1984-),男,工程师,主要从事重、磁、电勘探与研究工作。 Email: liangshengxian626@163.com

引用本文:

梁生贤, 王桥, 焦彦杰, 廖国忠, 郭境. LSQR法在位场反演中的分析与评价[J]. 物探与化探, 2019, 43(2): 359-366.
Sheng-Xian LIANG, Qiao Wang, Yan-Jie JIAO, Guo-Zhong LIAO, Jing GUO. Analysis and evaluation of the potential field inversion using LSQR method. Geophysical and Geochemical Exploration, 2019, 43(2): 359-366.

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https://www.wutanyuhuatan.com/CN/10.11720/wtyht.2019.1261 或 https://www.wutanyuhuatan.com/CN/Y2019/V43/I2/359

初始化:
β₁μ₁=d, α₁ν₁=A^Tμ₁, ω₁=ν₁
m₀=0,

ρ 1 *

=α₁,

φ 1 *

=β₁

for i=1,2,3,…
β_i₊₁μ_i₊₁=Aν_i-α_iμ_i,
α_i₊₁ν_i₊₁=A^Tu_i₊₁-β_i₊₁ν_i,
ρ_i=

ρ i * 2 + β i + 1 2

, c_i=

ρ i * ρ i

, s_i=

β i + 1 ρ i

,
θ_i₊₁=s_iα_i₊₁, φ_i=c_i

φ i *

ρ i + 1 *

=-c_iα_i₊₁,

φ i + 1 *

=s_i

φ i *

,
m_i₊₁=m_i+

φ i ρ i

ω_i, ω_i₊₁=ν_i₊₁-

θ i + 1 ρ i

ω_i
是否达到迭代终止条件。

Table 1 LSQR算法

Fig.1 合成模型Tikhonov正则化和LSQR法L曲线
a—单一模型Tikhonov正则化;b—单一模型LSQR法;c—组合模型Tikhonov正则化;d—组合模型LSQR法

Table 2 合成模型Tikhonov和LSQR正则化求解结果

Fig.2 组合模型数据等值线(等值线间距:0.4 mGal)
a—无误差数据;b—加入10%高斯噪声数据;c—反演模型计算数据

Fig.3 组合模型及其LSQR法反演结果
a—组合模型XY平面切片(深度100 m);b——组合模型XZ剖面切片(Y=1000 m);c—XY平面切片图的反演结果;d—XZ剖面切片图的反演结果

Fig.4 四川雅安地区三维重力反演结果

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