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Modeling of Magnetotelluric Static Shift Based on Non-Uniform Meshes Finite Difference Method
DOI: 10.12677/AG.2018.88148, PDF, HTML, XML, 下载: 634  浏览: 6,720  科研立项经费支持

Abstract: Because of the existence of shallow inhomogeneous bodies, static shift occurs in magnetotelluric sounding curves, which makes the interpretation of magnetotelluric data very difficult, or even impossible. It is very important to identify static shift in the interpretation of magnetotelluric data. In this paper, in order to compute the two-dimensional magnetotelluric responses of static shift model, non-uniform grids finite difference method was adopted for numerical results. From the electric field and magnetic field based on the differential equation, the linear equation for magnetotelluric numerical modeling was derived by non-uniform meshes finite difference method. By the numerical simulation for the geo-electric model with shallow inhomogeneous body, the character of static shift can be obtained and the magnetotelluric responses computed by non-uniform girds finite difference method are effective.

1. 引言

2. 非均匀网格差分正演算法

2.1. 边值问题

$\frac{\partial }{\partial y}\left(\frac{1}{\text{i}\omega \mu }\frac{\partial {E}_{x}}{\partial y}\right)+\frac{\partial }{\partial z}\left(\frac{1}{\text{i}\omega \mu }\frac{\partial {E}_{x}}{\partial z}\right)+\sigma {E}_{x}=0$ (1)

$\frac{\partial }{\partial y}\left(\frac{1}{\sigma }\frac{\partial {H}_{x}}{\partial y}\right)+\frac{\partial }{\partial z}\left(\frac{1}{\sigma }\frac{\partial {H}_{x}}{\partial z}\right)+\text{i}\omega \mu {H}_{x}=0$ (2)

$\nabla \cdot \left(\tau \nabla u\right)+\lambda u=0$ (3)

$u={E}_{x}$$\tau =\frac{1}{\text{i}\omega \mu }$$\lambda =\sigma$ (4)

$u={H}_{x}$$\tau =\frac{1}{\sigma }$$\lambda =\text{i}\omega \mu$ (5)

Figure 1. Boundary conditions of 2D geo-electric model

1) TE极化模式的外边界条件

a) 上边界 $z={z}_{\mathrm{min}}$ 离地面足够远，使异常场在边界上为零，即以该处的 $u$ 为1单位

${u|}_{{z}_{\mathrm{min}}}=1$ (6)

b) 下边界 $z={z}_{\mathrm{max}}$ 为第三类边界条件，即

${\left(\frac{\partial u}{\partial z}+\sqrt{-\text{i}\omega \mu \sigma }u\right)|}_{z={z}_{\mathrm{max}}}=0$ (7)

c) 取左右边界 $y={y}_{\mathrm{min}}$$y={y}_{\mathrm{max}}$ 离局部不均匀体足够远，电磁场在 ${y}_{\mathrm{min}}$${y}_{\mathrm{max}}$ 上左右对称，其上的边界条件是

${\frac{\partial u}{\partial y}|}_{y={y}_{\mathrm{min}}}={\frac{\partial u}{\partial y}|}_{y={y}_{\mathrm{max}}}=0$ (8)

2) TM极化模式外边界条件

a) 上边界 $z={z}_{\mathrm{min}}$ 直接取在地面上，并以该处的 $u$ 为1单位，则有

${u|}_{z={z}_{\mathrm{min}}}=1$

2) 下边界 $z={z}_{\mathrm{max}}$ 的边界条件，同TE极化模式，见式(7)。

3) 左右边界 $y={y}_{\mathrm{min}}$$y={y}_{\mathrm{max}}$ 的边界条件，同TE极化模式，见式(8)。

2.2. 差分线性方程组导出

Figure 2. Discretization of 2D geo-electric model

${\left[\frac{\partial }{\partial y}\left(\tau \frac{\partial u}{\partial y}\right)\right]}_{i,j}\approx \frac{1}{\Delta y}\left[{\left(\tau \frac{\partial u}{\partial y}\right)}_{i+\frac{1}{2},j}-{\left(\tau \frac{\partial u}{\partial y}\right)}_{i-\frac{1}{2},j}\right]\approx \frac{{\tau }_{i,j}+{\tau }_{i+1,j}}{2}\frac{{u}_{i+1,j}-{u}_{i,j}}{\Delta {y}^{2}}-\frac{{\tau }_{i,j}+{\tau }_{i-1,j}}{2}\frac{{u}_{i,j}-{u}_{i-1,j}}{\Delta {y}^{2}}$ (9)

${\left[\frac{\partial }{\partial z}\left(\tau \frac{\partial u}{\partial z}\right)\right]}_{i,j}\approx \frac{1}{\Delta z}\left[{\left(\tau \frac{\partial u}{\partial y}\right)}_{i,j+\frac{1}{2}}-{\left(\tau \frac{\partial u}{\partial y}\right)}_{i,j-\frac{1}{2}}\right]\approx \frac{{\tau }_{i,j}+{\tau }_{i,j+1}}{2}\frac{{u}_{i,j+1}-{u}_{i,j}}{\Delta {z}^{2}}-\frac{{\tau }_{i,j}+{\tau }_{i,j-1}}{2}\frac{{u}_{i,j}-{u}_{i,j-1}}{\Delta {z}^{2}}$ (10)

${\left[\frac{\partial }{\partial y}\left(\tau \frac{\partial u}{\partial y}\right)\right]}_{i,j}+{\left[\frac{\partial }{\partial z}\left(\tau \frac{\partial u}{\partial z}\right)\right]}_{i,j}+\lambda {u}_{i,j}=0$

$\left(\frac{{\tau }_{i,j}+{\tau }_{i+1,j}}{2}\frac{{u}_{i+1,j}-{u}_{i,j}}{\Delta {y}^{2}}-\frac{{\tau }_{i,j}+{\tau }_{i-1,j}}{2}\frac{{u}_{i,j}-{u}_{i-1,j}}{\Delta {y}^{2}}\right)+\left(\frac{{\tau }_{i,j}+{\tau }_{i,j+1}}{2}\frac{{u}_{i,j+1}-{u}_{i,j}}{\Delta {z}^{2}}-\frac{{\tau }_{i,j}+{\tau }_{i,j-1}}{2}\frac{{u}_{i,j}-{u}_{i,j-1}}{\Delta {z}^{2}}\right)+\lambda {u}_{i,j}=0$ (11)

$\begin{array}{l}\frac{{\tau }_{i,j}+{\tau }_{i+1,j}}{2\Delta {y}^{2}}{u}_{i+1,j}+\frac{{\tau }_{i,j}+{\tau }_{i-1,j}}{2\Delta {y}^{2}}{u}_{i-1,j}+\frac{{\tau }_{i,j}+{\tau }_{i,j+1}}{2\Delta {z}^{2}}{u}_{i,j+1}+\frac{{\tau }_{i,j}+{\tau }_{i,j-1}}{2\Delta {z}^{2}}{u}_{i,j-1}\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(-\frac{{\tau }_{i-1,j}+2{\tau }_{i,j}+{\tau }_{i+1,j}}{2\Delta {y}^{2}}-\frac{{\tau }_{i,j-1}+2{\tau }_{i,j}+{\tau }_{i,j+1}}{2\Delta {z}^{2}}+\lambda \right){u}_{i,j}=0\end{array}$ (12)

$u=\left(\begin{array}{c}{u}_{1}\\ {u}_{2}\\ ⋮\\ \text{}{u}_{\left(i-1\right)×Nz+j}\\ {u}_{\left(i-1\right)×Nz+j+1}\\ ⋮\\ \text{}{u}_{Ny×{N}_{z}-1}\\ \text{}{u}_{Ny×{N}_{z}}\end{array}\right)=\left(\begin{array}{c}{u}_{1,1}\\ {u}_{1,2}\\ ⋮\\ {u}_{i,j}\\ {u}_{i,j+1}\\ ⋮\\ {u}_{Ny,{N}_{z}-1}\\ {u}_{Ny,{N}_{z}}\end{array}\right)$ (13)

$Ku=p$ (14)

2.3. 电磁响应计算

$\begin{array}{l}{Z}_{TE}={E}_{x}/\frac{1}{i\omega \mu }\frac{\partial {E}_{x}}{\partial z}\\ {\rho }_{a}^{TE}=\frac{1}{\omega \mu }{|{Z}_{TE}|}^{2}\\ {\phi }^{TE}=\mathrm{arctan}\frac{\mathrm{Im}\left[{Z}_{TE}\right]}{\mathrm{Re}\left[{Z}_{TE}\right]}\end{array}\right\}$ (15)

Figure 3. The coefficient matrix of finite difference method for 2D magnetotelluric responses

$\begin{array}{l}{Z}_{TM}=-\frac{1}{\sigma }\frac{\partial {H}_{x}}{\partial z}/{H}_{x}\\ {\rho }_{a}^{TM}=\frac{1}{\omega \mu }{|{Z}_{TM}|}^{2}\\ {\phi }^{TM}=\mathrm{arctan}\frac{\mathrm{Im}\left[{Z}_{TM}\right]}{\mathrm{Re}\left[{Z}_{TM}\right]}\end{array}\right\}$ (16)

3. 静位移模型试算分析

Figure 4. 2D geo-electric model with static-shift

(a) ${\rho }_{0}=50\text{}\Omega \cdot \text{m}$ (b) ${\rho }_{0}=5\text{}\Omega \cdot \text{m}$(c) ${\rho }_{0}=500\text{}\Omega \cdot \text{m}$

Figure 5. Pseudo-section map of magnetotelluric responses of the uniform body in TM mode

Figure 6. The resistivity-frequency curves of the uniform body in TM mode

4. 结语

1) 针对各向同性介质中大地电磁场所满足的边值问题，详细推导了二维地电模型大地电磁场响应的有限差分正演算法；在保证计算精度的条件下，本文的非均匀网格剖分方式能减少正演计算时间，提高计算效率。

2) 通过对浅部不均匀体模型的模拟计算，得到了大地电磁的静位移响应规律，同时也验证了非均匀网格有限差分正演算法的准确性。

3) 高阻不均匀体使受影响测点的视电阻率整体向上移动而趋势不变，并且不均匀体电阻率越高，视电阻率向上平移的幅度越大。

4) 低阻不均匀体使受影响测点的视电阻率整体向下移动，并且由于电磁感应现象，单点视电阻率曲线在高频段趋势发生变化。

NOTES

*通讯作者。

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