﻿ 基于向下延拓Milne法的重力归一化总梯度法
 石油地球物理勘探  2019, Vol. 54 Issue (6): 1390-1396  DOI: 10.13810/j.cnki.issn.1000-7210.2019.06.025 0
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### 引用本文

SHI Jiaqiang, XIAO Feng, ZHONG Yang. Normalized total gradient of gravity anomaly based on the downward-continuation Milne method. Oil Geophysical Prospecting, 2019, 54(6): 1390-1396. DOI: 10.13810/j.cnki.issn.1000-7210.2019.06.025.

### 文章历史

Normalized total gradient of gravity anomaly based on the downward-continuation Milne method
SHI Jiaqiang , XIAO Feng , ZHONG Yang
College of Geo-Exploration Sciences and Technology, Jilin University, Changchun, Jilin 130026, China
Abstract: The normalized total gradient method of gra-vity anomaly includes multi-calculations of downward continuation and vertical derivative, both of them amplify the high wavenumber components.To sovle the problem, we propose a new idea for the issue in this paper.Two stable algorithms, Milne method of downward continuation and integrated vertical second derivative method, are used for the normalized total gradient of gravity anomaly.The improved method is applied to locate the center of the infinite horizontal cylinder model, and the result is accurate.In the real gravity anomaly profile data processing, compared with the norma-lized total gradient of gravity anomaly based on Taylor series expansion, this proposed method obtains the more accurate location of the mine center, which proves the validity and practicality of the proposed method.
Keywords: normalized total gradient of gravity anomaly    downward-continuation Milne method    integrated vertical second derivative
0 引言

1 方法原理 1.1 重力归一化总梯度法的基本原理

 ${G^{\rm{H}}}(x, z) = \frac{{\sqrt {V_{zx}^2(x, z) + V_{zz}^2(x, z)} }}{{\frac{1}{M}\sum\limits_{i = 1}^M {\sqrt {V_{zx}^2\left( {{x_i}, z} \right) + V_{zz}^2\left( {{x_i}, z} \right)} } }}$ (1)

1.2 向下延拓Milne法的基本原理

 $\begin{array}{c}{V_{z}\left(x, z_{h}\right)=V_{z}\left(x, z_{-3 h}\right)+\frac{4 h}{3}\left[2 V_{z z}\left(x, z_{-2 h}\right)-\right.} \\ {\left.V_{z z}\left(x, z_{-h}\right)+2 V_{z z}\left(x, z_{0}\right)\right]}\end{array}$ (2)

 $\widetilde{V}_{z}\left(u, z_{-n h}\right)=\widetilde{V}_{z}\left(u, z_{0}\right) \mathrm{e}^{-2 \pi n h \sqrt{u^{2}}}$ (3)

1.3 ISVD法的基本原理

 $V_{x x}+V_{x}=0$ (4)

 $\widetilde{V}=\frac{1}{2 \pi \sqrt{u^{2}}} \widetilde{V}_{z}$ (5)

 $V_{x(0)}=\frac{1}{2 \Delta x}\left(V_{(1)}-V_{(-1)}\right)$ (6)

1.4 基于向下延拓Milne法的重力归一化总梯度法的实现步骤

① 以向下延拓深度h为例，利用式(3)在波数域对观测剖面上重力异常的波谱$\widetilde{V}_{z}\left(u, z_{0}\right)$)进行不同高度(h、2h、3h)的向上延拓，得到剖面以上不同高度重力异常的波谱$\widetilde{V}_{z}\left(x, z_{-h}\right), \widetilde{V}_{z}\left(x, z_{-2 h}\right), \widetilde{V}_{z}(x, \left. {{z_{ - 3h}}} \right)$

② 根据式(5)，在波数域对$\widetilde{V}_{z}\left(u, z_{0}\right), \widetilde{V}_{z}(u$, ${z_{- h}}), {\tilde V_z}(u, {z_{ - 2h}})$进行积分，获得重力位的波谱$\widetilde{V}\left(u, z_{0}\right), \widetilde{V}\left(u, z_{-h}\right), \widetilde{V}\left(u, z_{-2 h}\right)$。两次应用式(6)，在空间域分别计算其重力位沿x方向的二阶导数。再将结果代入式(4)，依次得到重力位的垂向二阶导数Vzz(x, z0)、Vzz(x, z-h)、Vzz(x, z-2h)。

③ 将第①步和第②步计算得到的结果代入式(2)，求出下延深度为h的重力异常Vz(x, zh)。

④ 再利用第②步方法，计算Vz(x, zh)的垂向一阶导数Vzz(x, zh)，并应用式(6)直接在空间域计算Vz(x, zh)沿x方向的一阶导数Vzx(x, zh)。

⑤ 将Vzz(x, zh)和Vzx(x, zh)代入式(1)计算GH(x, zh)。

⑥ 修改下延深度，重复以上步骤，获得不同深度的重力归一化总梯度值。

⑦ 最后绘制GH(x, z)等值线剖面，极大值点即是异常体的中心位置。

2 理论模型实验

 图 1 无限长水平圆柱体理论模型不同方法重力归一化总梯度计算结果 (a)理论重力异常曲线；(b)本文方法；(c)泰勒级数展开

 图 2 对含5%噪声和一阶趋势背景场的理论模型数据利用不同方法得到的重力归一化总梯度剖面 (a)重力异常曲线；(b)本文方法；(c)泰勒级数展开

 图 3 对局部重力异常采用不同方法得到的重力归一化总梯度剖面 (a)局部重力异常曲线；(b)本文方法；(c)泰勒级数展开

3 实测重力剖面数据应用

 图 4 实测重力数据计算结果对比 (a)实测布格重力异常和分离的区域(背景)场；(b)局部重力异常(实线)及去噪后的局部重力异常(虚线)；(c)本文方法得到的重力归一化总梯度剖面；(d)基于泰勒级数展开的重力归一化总梯度剖面图中红色马蹄形实线为矿洞位置，红色圆点为重力归一化总梯度剖面中极大值点(即矿洞的中心点)
4 结论与建议