Scientific Journal of Earth Science December 2012, Volume 2, Issue 4, PP.109-115

A numerical Calculating Method for Apparent Resistivity of TEM Sounding Xiaoping Wu 1, 2#, Yunpeng Wu 3 1. China Non-ferrous Metals Resource Geological Survey, Beijing 100012, P.R. China 2. Beijing Donia Resources Co. Ltd., Beijing 100012, P.R. China 3. College of Computer Science & Technology of Chengdu University of Information Technology, Chengdu 610225, P.R. China #Email: wuxiaoping198@126.com

Abstract For the advantage of Stable signal and large detective depth of the Time Domain Electromagnetic Sounding, it is widely used currently for Mineral and Geological Structure Exploration. But in general, it is supposed some extremely conditions for processing and explaining the observed results. And it is very difficultly to fit that conditions we supposed. So we cannot get the exact processing and explaining of the observed results. We suggested a method by using Numerical Differential Calculating to obtain the apparent resistivity of TEM sounding in this paper. It can get the results close to the real Geo-Electrical section. We can simplify the explanations of the observed results. And we can get the Model Parameters for the inverse calculating easily. Keywords: TEM Sounding; Numerical Calculating; Numerical Differential; Apparent Resistivity; K Model

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1 电偶极源的时间域电磁响应及视电阻率的数值计算 如图 1 所示为一个放置在均匀半无限空间地面的电偶极子示意图[1][3][4][7][8]，当使用阶跃波在电偶极子中 激励时，位于地面Ｐ点的垂直磁场的时间域电磁响应表达式为：

Bz (t ) t

u

3PE  sin   2 r

2 r

4

2

  (u ) 

2t ，   2

0

u (1 

u2 3

)e

u

2

2

  

（1）

2 t  107

（2）

 (t )   NS

Bz (t ) t



[5][7][8]

3 NSPE  sin   2 r

4

2

u2

u (1  )e   (u )   3 

u

2

2

  

（3）

du u d   ，  d  dt 2t

du du d u   dt d dt 2t 我们可以将（3）式两边同时对时间 t 求导即得：

NSPE  sin  5 u 2 2 d  (t )  ue dt 2 2 r 4t - 110 http://www.j-es.org/

（4）

（5）

V1 

d 2 (t ) d  (t ) ， V2  dt 2 dt

V2 u 2  7  V1 2t

（6）

u 2  2t

V2 7 V1

（7）

u2 

r 2 0

（8）

2t



r 2 0 V 2t (2t 2  7) V1

（9）

（9）式中可以看到，均匀半空间电阻率值与观测点接收线圈感应电动势随时间的一阶、二阶导数值有 关，而我们实际观测了感应电动势随时间的变化曲线，因此，我们可以采用数值微分的方法来求取接收线圈 中感应电动势随时间的一阶和二阶微分，然后，利用（9）式即可以直接得到均匀半空间的电阻率。也就是 可以直接定义这一电阻率为非均匀半空间情形下的视电阻率。

2 传统方式的时间域晚期视电阻率与数值计算结果比较 传统的时间域电磁方法的视电阻率定义的基础是根据均匀半无限空间的电磁表达式，即（1）式，利用 泰勒级数将指数部分展开得出。晚期视电阻率的表达式为[8][5][12][9][22][25]：

  r P   ( Bz )  0  0 E  4 t  5tBz (t )  l s

2

3

（10）

-8

10

-10

10

-12

10

-14

10

-16

10

-18

10

-20

10

-22

10

-3

10

-2

10

-1

10

0

10 时间

1

10

2

10

3

10

10

2

10

1

10

0

10 -3 10

-2

10

-1

10

0

10 时间

1

10

2

10

3

10

3

10

2

10

1

10 -3 10

-2

10

-1

10

0

10 时间

1

10

2

10

3

10

3 实际应用分析 图 5 所示为一个 K 模型正演计算的电磁响应曲线图。模型参数为一个三层 K 模型：

1  100M

h1  60m

 2  200M

h2  40m

3  100M 电磁相应曲线

0

10

-1

10

-2

10

-3

10

-4

10

-6

10

-5

10 时间

-4

10

10

2

10

1

10 -6 10

-5

10 时间

-4

10

4 结语 时间域电磁测深方法因其较大的勘探深度而在当今向深部勘探挺进的时期被广泛的应用。但由于其观测 结果的复杂性，给解释工作带来很大的困难。我们通过采用数值微分的方法来计算时间域电磁测深的视电阻 - 113 http://www.j-es.org/

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【作者简介】 吴小平（1964- ） ，男，湖南澧县人，博士，主要从事时间域电磁方法研究。Email: wuxiaoping198@126.com

- 115 http://www.j-es.org/

A numerical Calculating Method for Apparent Resistivity of TEM Sounding

For the advantage of Stable signal and large detective depth of the Time Domain Electromagnetic Sounding, it is widely used currently for Mi...