﻿ 瞬变电磁全期视电阻率的弦截无限逼近算法
 石油地球物理勘探  2019, Vol. 54 Issue (6): 1371-1375  DOI: 10.13810/j.cnki.issn.1000-7210.2019.06.022 0
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### 引用本文

ZENG Qingning, LUO Ying, LIU Shuai, LONG Chao. A secant infinite approximation algorithm for the full-time apparent resistivity in transient electromagnetic method. Oil Geophysical Prospecting, 2019, 54(6): 1371-1375. DOI: 10.13810/j.cnki.issn.1000-7210.2019.06.022.

### 文章历史

① 桂林电子科技大学认知无线电与信息处理教育部重点实验室, 广西桂林 541004;
② 桂林电子科技大学信息与通信学院, 广西桂林 541004

A secant infinite approximation algorithm for the full-time apparent resistivity in transient electromagnetic method
ZENG Qingning①② , LUO Ying , LIU Shuai , LONG Chao①②
① Key Laboratory of Cognitive Radio and Information Processing, Ministry of Education, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China;
② College of Information and Telecommunications, Guilin University of Electronic Technology, Gui-lin, Guangxi 541004, China
Abstract: In the geology exploration with the transient electromagnetic method (TEM), the apparent resistivity calculation is quite important as the appa-rent resistivity may intuitively indicate underground geological bodies.However, conventional approximation formulas usually lead a large error in the apparent resistivity calculation.According to the characteristics of implicit functions of apparent resistivity for half-space central-loop TEM, we propose in this paper a secant infinite approximation algorithm for the full-time apparent resistivity in the transient electromagnetic method, in which the secant method is used to calculate the roots of non-linear functions.The proposed algorithm avoids complicated derivative calculations of implicit functions.Simulation and real data examples show that the proposed algorithm has faster convergence than the bisection algorithm and can calculate with arbitrary accuracy the full-time apparent resistivity for half-space central-loop TEM.
Keywords: transient electromagnetic method(TEM)    apparent resistivity    inversion    secant method
0 引言

1 全期视电阻率

 $V(t)=\frac{\mu_{0} I S_{\mathrm{r}}}{4 a t} F(z)$ (1)

 $F(z)=\frac{1}{z^{2}}\left[3 \operatorname{erf}(z)-z\left(3+2 z^{2}\right) \frac{2}{\sqrt{\pi}} \mathrm{e}^{-\frac{z^{2}}{2}}\right]$ (2)
 $z=\frac{a}{2} \sqrt{\frac{\mu_{0}}{\rho t}}$ (3)

 $g(\rho)=\frac{\mu_{0} I S_{\mathrm{r}}}{\sqrt{\pi} a t} F(z)-V_{0}(t)$ (4)

ρ的求解问题转化为求取非线性函数g(ρ)的解。由式(3)及F(z)的上凸特性，函数g(ρ)在(0, ρ0]内单调递增，在[ρ0, +∞)内单调递减，其中

 $\rho_{0}=\frac{\mu_{0} a^{2}}{4 z_{0}^{2} t}$ (5)

2 弦截求根法

 图 1 弦截法示意图

 $y=\frac{f\left(x_{i-1}\right)-f\left(x_{i-2}\right)}{x_{i-1}-x_{i-2}}\left(x-x_{i-2}\right)+f\left(x_{i-2}\right)$

 $x_{i}=x_{i-2}-f\left(x_{i-2}\right) \frac{x_{i-1}-x_{i-2}}{f\left(x_{i-1}\right)-f\left(x_{i-2}\right)}$ (6)
3 全期视电阻率的弦截逼近算法

(1) 任意给定视电阻率ρ的精度ε＞0，任取ρ(1)、ρ(2)∈[ρ0, ρb]；令i=2，运用式(2)~式(4)计算g[ρ(i)]和g[ρ(i-1)]。

(2) 令i=:i+1，计算g[ρ(i)]。

(3) 计算

 \begin{aligned} \rho(i)=& \rho(i-2)-g[\rho(i-2)] \times \\ & \frac{\rho(i-1)-\rho(i-2)}{g[\rho(i-1)]-g[\rho(i-2)]} \end{aligned}

(4) 若|ρ(i)-ρ(i-1)|＜ε，则迭代停止，ρ(i)即为所求之解；否则，转步骤(2)继续进行迭代计算。

4 理论模型计算效果分析

4.1 均匀半空间

4.2 层状模型

5 应用实例

 图 2 过水仓测线的视电阻率反演剖面
6 结束语

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