﻿ 改进的布谷鸟MT反演算法
 石油地球物理勘探  2020, Vol. 55 Issue (1): 217-225  DOI: 10.13810/j.cnki.issn.1000-7210.2020.01.025 0
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### 引用本文

WANG Pengfei, WANG Shuming. MT data inversion based on improved cuckoo search algorithm. Oil Geophysical Prospecting, 2020, 55(1): 217-225. DOI: 10.13810/j.cnki.issn.1000-7210.2020.01.025.

### 文章历史

MT data inversion based on improved cuckoo search algorithm
WANG Pengfei , WANG Shuming
Institute of Geophysics&Geomatics, China University of Geosciences(Wuhan), Wuhan, Hubei 430074, China
Abstract: Due to the high non-linearity of the magnetotelluric(MT) data inversion, conventional global optimization algorithms converge slowly and easily to local optimum.To solve this problem, an improved cuckoo search(ICS) algorithm combined with simplex method was proposed in this paper to realize MT data inversion.For cuckoo search(CS) algorithm is advantageous in exploration but dis-advantageous in development, the global optimal solution in particle swarm optimization was introduced to improve the local search performance.Simplex method was also used to improve the bird's nest, in order to enhance optimization precision further.The inversion results of theoretical model and real data demonstrated that ICS has higher stability, faster convergence rate and higher accuracy than CS.
Keywords: magnetotelluric    improved cuckoo search algorithm    particle swarm optimization    simplex method
0 引言

MT数据的反演研究经历了三个发展阶段。最早使用近似反演类方法，如Bostick反演[6]、大地电磁拟地震反演[7]、曲线对比法[8]等；随后发展了各种线性反演方法，如马奎特法[9]、广义逆矩阵法[10]、奥康姆法[11-12]、正则化法[13]等。由于MT反演是高度非线性的，线性反演方法容易陷入局部极小值，且初始模型的选择对反演结果影响很大[14]，目前的反演研究转向非线性方法和仿生算法，如模拟退火法[15]、遗传算法[16]、人工神经网络[17]、粒子群算法[18-19]、人工鱼群算法[20]等。

1 改进布谷鸟算法原理 1.1 基本布谷鸟算法

 图 1 莱维飞行轨迹示意图

 $\boldsymbol{x}_{i}^{(t+1)}=\boldsymbol{x}_{i}^{(t)}+\alpha \boldsymbol{L}(\beta)$ (1)

 $\boldsymbol{L}(\beta)=\frac{\phi \boldsymbol{u}}{|\boldsymbol{v}|^{1 / \beta}}$ (2)

 $\phi=\left[\frac{\Gamma(1+\beta) \sin \frac{\pi \beta}{2}}{2^{\frac{\beta-1}{2}} \beta {\mathit{\Gamma}}\left(\frac{1+\beta}{2}\right)}\right]^{1 / \beta}$ (3)

 $\boldsymbol{x}_{i}^{(t+1)}=\boldsymbol{x}_{i}^{(t)}+\gamma H(P-\varepsilon)\left[\boldsymbol{x}_{j}^{(t)}-\boldsymbol{x}_{k}^{(t)}\right]$ (4)

1.2 改进的布谷鸟(ICS)算法

 $\boldsymbol{x}_{i}^{(t+1)}=\boldsymbol{x}_{i}^{(t)}+\alpha \frac{\phi \boldsymbol{u}}{|\boldsymbol{v}|^{1 / \beta}}\left[\boldsymbol{x}_{i}^{(t)}-\boldsymbol{x}_{\text {best }}^{(i)}\right]$ (5)

 图 2 单纯形法结构示意图

(1) 通过比较各个鸟巢的目标函数值，得到最优鸟巢xg和次优鸟巢xb的目标函数值f(xg)和f(xb)。

(2) 找到最差鸟巢位置的目标函数值f(xs)，对xs执行反射操作，得到反射点xr

(3) 若f(xr)＜f(xg)，说明位置得到改善，继续执行扩张操作，得到扩张点xe；若f(xe)＜f(xg), 则用xe代替xs，否则用xr代替xs

(4) 若f(xr)＞f(xs)，说明位置变得更差，则执行压缩操作，得到压缩点xt; 若f(xt)＜f(xs)则用xt代替xs

(5) 若f(xg)＜f(xr)＜f(xs)，执行收缩操作, 得到收缩点xw；若f(xw)＜f(xs)，则用xw代替xs，否则用xr代替xs

1.3 算法测试

 $\begin{array}{c} F=-20 \exp (-0.2 \sqrt{\frac{1}{d} \sum\limits_{i=1}^{d} x_{i}^{2}})- \\ \exp \left[\frac{1}{d} \sum\limits_{i=1}^{d} \cos \left(2 \pi x_{i}\right)\right]+20+\mathrm{e} \\ x_{i} \in[-32, 32] \end{array}$ (6)

2 理论模型试算

MT反演是求解目标函数最小值的优化问题，由于视电阻率可以直观地反映介质的电阻率，因此对视电阻率数据进行反演的情况居多，而相位数据的反演则相对较少。本文同时对视电阻率和相位进行联合反演，目标函数公式为

 $f=\frac{1}{2} \sum\limits_{i=1}^{N}\left(\frac{\rho_{\mathrm{cal}, i}-\rho_{\mathrm{obs}, i}}{\rho_{\mathrm{obs}, i}}\right)^{2}+\frac{1}{2} \sum\limits_{i=1}^{N}\left(\frac{\theta_{\mathrm{cal}, i}-\theta_{\mathrm{obs}, i}}{\theta_{\mathrm{obs}, i}}\right)^{2}$ (7)

2.1 无噪声理论模型

 图 3 模型A不含噪声理论数据与反演结果对比 (a)视电阻率曲线；(b)相位曲线；(c)模型和反演结果及搜索范围

 图 4 模型B不含噪声理论数据与反演结果对比 (a)视电阻率曲线；(b)相位曲线；(c)模型和反演结果及搜索范围

 图 5 模型C不含噪声理论数据与反演结果对比 (a)视电阻率曲线；(b)相位曲线；(c)模型和反演结果及搜索范围
2.2 含噪声理论数据

 图 6 模型A含10%白噪声理论数据与反演结果对比 (a)视电阻率曲线；(b)相位曲线；(c)模型和反演结果及搜索范围

 图 7 模型B含10%白噪声理论数据与反演结果对比 (a)视电阻率曲线；(b)相位曲线；(c)模型和反演结果及搜索范围

 图 8 模型C含10%白噪声理论数据与反演结果对比 (a)视电阻率曲线；(b)相位曲线；(c)模型和反演结果及搜索范围
2.3 反演结果分析

2.4 ICS与CS算法的对比分析

 图 9 模型C反演目标函数收敛曲线 (a)含噪和不含噪数据ICS法收敛曲线；(b)不含噪数据的ICS法与CS法收敛过程对比

3 实测数据反演

 图 10 内蒙古MT数据两种方法反演结果对比 (a)实测数据与反演模型视电阻率曲线；(b)电阻率反演结果

4 结论

(1) ICS算法是一种性能优异的群智能优化算法，可以有效地对MT数据进行反演，且反演结果稳定、精度较高;

(2) 与传统的布谷鸟算法相比，ICS算法收敛能力更强、寻优精度更高，但反演用时稍多。

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