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15
16 \section{MT inversion: 2D TE Mode}\label{sec:forward 2DMT TEMode}
17 In this section we present the forward model for the magnetotelluric (MT)\index{magnetotelluric}\index{MT}inversion case~\cite{ChaveJones2012a}
18 where we particularly looking into the two-dimensional case of the electrical field polarization\index{electrical field polarization}.
19 For this case the horizontal electrical field $E_x$\index{electrical field} and the magnetic field $H_y$\index{magnetic field} component
20 for a given frequency $\omega$ are measured and the horizontal impedance $Z_{xy}$\index{impedance} is recorded as the ratio of
21 electrical and magentical field:
22 \begin{equation}\label{ref:2DMTTE:EQU:1}
23 E_x = Z_{xy} \cdot H_y
24 \end{equation}
25 It is common practice to record the impedance using the apparent resistivity $\rho_a$\index{resistivity!apparent} and the
26 phase $\phi$ as
27 \begin{equation}\label{ref:2DMTTE:EQU:2}
28 Z_{xy} = (\rho_a \cdot \omega \cdot \mu) ^{\frac{1}{2}} \cdot e^{\mathbf{i} \phi}
29 \end{equation}
30 where $\mu$ is the permeability (e.g. $mu = 4 \pi \cdot 10^{-7} \frac{Vs}{Am}$). Notice that the impedance is independent of the scale of the $E_x$ and $E_y$.
31 The electrical field $E_x$ as function of depth $z$ and horizontal coordinate $y$ is given as the solution of the
32 PDE
33 \begin{equation}\label{ref:2DMTTE:EQU:3}
34 - E_{x,kk} + \mathbf{i} \omega \mu \sigma \cdot E_x = 0
35 \end{equation}
36 where $\mathbf{i}$ is the complex unit, and $k=0$ and $k=1$ correspond to the $y$ and $z$ direction.
37 $\sigma$ is the unknown conductivity to be calculated through the inversion. The domain
38 of the PDE is comprised of the subsurface region in which the conductivity is to be calculated
39 and a sufficient high airlayer in which the conductivity is assumed to be zero.
40
41
42 It is assumed that $E_x$ takes the value $1$ at the top of the domain $\Gamma_0$ representing an incoming
43 electro-magnetic wave front. On the other faces of the domain homogeneous Neuman conditions
44 are set to model the assumption that the electrical field is constant away from the domain.
45 \begin{equation}\label{ref:2DMTTE:EQU:4}
46 H_y = - \frac{1}{\mathbf{i} \omega \mu} E_{x,z}
47 \end{equation}
48
49 The impedance $Z_{xy}^{(s)}$ is measured at certain points ${\mathbf x}^{(s)}$. The defect
50 of the measurements for the prediction $E_x$ is given as
51 \begin{equation}\label{ref:2DMTTE:EQU:5}
52 J^{MT}(\sigma) = \frac{1}{2}\sum_{s} \int_{\Omega}
53 w^{(s)} \cdot \| E_x - Z_{xy}^{(s)} \cdot H_y \|^2 d{\mathbf x}
54 \end{equation}
55 where $w^{(s)}$ is a spatially dependent weighting faction. Here we use the Gaussian profile
56 \begin{equation}\label{ref:2DMTTE:EQU:6}
57 w^{(s)}({\mathbf x}) = \frac{w^{(s)}_0}{2\pi (\eta^{(s)})^2} \cdot e^{-\frac{\|{\mathbf x} - {\mathbf x}^{(s)} \|^2}{2(\eta^{(s)})^2}}
58 \end{equation}
59 where $\eta^{(s)}$ is measuring the spatial validity
60 and $w^{(s)}_0$ is confidence of the measurement $s$ (e.g. $w^{(s)}_0$ is set to the inverse of the measurement error. )
61
62 \subsection{Usage}
63
64 \begin{classdesc}{MT2DModelTEMode}{domain, omega, x, Z_XY, eta,
65 \optional{, w0=1}
66 \optional{, mu=4E-7 * PI }
67 \optional{, coordinates=\None}
68 \optional{, fixAtBottom=False}
69 \optional{, tol=1e-8}
70 }
71 \end{classdesc}
72
73 \subsection{Gradient Calculation}
74
75 For the implemtation we set
76 \begin{equation}\label{ref:2DMTTE:EQU:100}
77 E_x = u_0 + \mathbf{i} \cdot u_1
78 \end{equation}
79 which translates teh forward model~\ref{ref:2DMTTE:EQU:3}
80 \begin{align}\label{ref:2DMTTE:EQU:103}
81 - u_{0,kk} - \omega \mu \sigma u_1 = 0 \\
82 - u_{1,kk} + \omega \mu \sigma u_0 = 0
83 \end{align}
84 In weak form this takes the form
85 \begin{equation}\label{ref:2DMTTE:EQU:104}
86 \int_{\Omega}
87 \left(
88 v_{0,k}u_{0,k}
89 + v_{1,k}u_{1,k}
90 + \omega \mu \sigma \cdot ( v_1 u_0 - v_0 u_1) \right) dx =0
91 \end{equation}
92 for all test function $v_i$ with $v_i$ zero on $\Gamma_0$.
93 Using the complex data
94 \begin{equation}\label{ref:2DMTTE:EQU:105}
95 d^{(s)} = - \frac{1}{\mathbf{i} \omega \mu} \cdot Z_{xy}^{(s)} = d^{(s)}_0 + \mathbf{i} \cdot d_1^{(s)}
96 \end{equation}
97 the cost function~\ref{ref:2DMTTE:EQU:6} takes the form
98 \begin{equation}\label{ref:2DMTTE:EQU:106}
99 J^{MT}(\sigma)=
100 \frac{1}{2}\sum_{s} \int_{\Omega} w^{(s)} \cdot \left( (u_0- u_{0,1} \cdot d_0^{(s)} + u_{1,1} \cdot d_1^{(s)} )^2
101 + ( u_1- u_{0,1} \cdot d_1^{(s)} -u_{1,1} \cdot d_0^{(s)} )^2 \right) dx
102 \end{equation}
103
104 We need to calculate the gardient
105 $\frac{\partial J^{MT}}{\partial \sigma}$ of the cost function $J^{{MT}}$ with
106 respect to the susceptibility $\sigma$. We follow the concept as outlined in section~\ref{chapter:ref:inversion cost function:gradient}.
107
108 If $\Gamma_{\sigma}$ denotes the region of the domain where the susceptibility is
109 known and for a given direction $\hat{\sigma}$ with $\hat{\sigma}=0$ on $\Gamma_{\sigma}$ one has
110 \begin{align}\label{ref:2DMTTE:EQU:201aa}
111 \int_{\Omega} \frac{\partial J^{MT}}{\partial \sigma} \cdot \hat{\sigma} \; dx & = &
112 \sum_{s} \int_{\Omega} w^{(s)} \cdot \left( u_0 - u_{0,1} \cdot d_0^{(s)} + u_{1,1} \cdot d_1^{(s)} \right ) \left( \hat{u}_0 - \hat{u}_{0,1} \cdot d_0^{(s)} + \hat{u}_{1,1}\cdot d_1^{(s)} \right ) \\
113 & + & w^{(s)} \cdot \left(u_1- u_{0,1} \cdot d_1^{(s)} -u_{1,1} \cdot d_0^{(s)} \right) \left(\hat{u}_1- \hat{u}_{0,1} \cdot d_1^{(s)} -\hat{u}_{1,1} \cdot d_0^{(s)} \right) \; dx
114 \end{align}
115 with
116 \begin{equation}\label{ref:2DMTTE:EQU:201A}
117 \int_{\Omega}
118 \left(
119 q_{0,k}\hat{u}_{0,k}
120 + q_{1,k}\hat{u}_{1,k}
121 + \omega \mu \sigma \cdot ( q_1 \hat{u}_0 - q_0 \hat{u}_1)
122 + \omega \mu \hat{\sigma} \cdot ( q_1 u_0 - q_0 u_1) \right) dx =0
123 \end{equation}
124 for all $q_i$ with $q_i=0$ on $\Gamma_{0}$. This equation is obtained from equation~(\ref{ref:2DMTTE:EQU:104}).
125 With
126 \begin{align}\label{ref:2DMTTE:EQU:202b}
127 Y_0 = & \sum_{s} w^{(s)} \cdot \left( u_0 - u_{0,1} \cdot d_0^{(s)} + u_{1,1} \cdot d_1^{(s)} \right) \\
128 Y_1 = & \sum_{s} w^{(s)} \cdot \left( u_1 - u_{0,1} \cdot d_1^{(s)} - u_{1,1} \cdot d_0^{(s)} \right) \\
129 X_{01} = & \sum_{s} w^{(s)} \cdot \left( u_{0,1} \cdot ( ( d_0^{(s)})^2 + (d_1^{(s)})^2) - u_0 \cdot d_0^{(s)} - u_1 \cdot d_1^{(s)} \right) \\
130 X_{11} = & \sum_{s} w^{(s)} \cdot \left( u_{1,1} \cdot ( ( d_0^{(s)})^2 + (d_1^{(s)})^2) + u_0 \cdot d_1^{(s)} - u_1 \cdot d_0^{(s)} \right)
131 \end{align}
132 and $X_{00}=X_{10}=0$ we can write Equation~(\ref{ref:2DMTTE:EQU:201aa}) as
133 \begin{equation}\label{ref:2DMTTE:EQU:202c}
134 \int_{\Omega} \frac{\partial J^{MT}}{\partial \sigma} \cdot \hat{\sigma} \; dx =
135 \int_{\Omega} Y_i \hat{u}_i + X_{ij} \hat{u}_{i,j} \; dx
136 \end{equation}
137 We then set adjoint function $u_i^*$ with $u_i^*=0$ on $\Gamma_{0}$ as the solution of the variational problem
138 \begin{equation}\label{ref:2DMTTE:EQU:202d}
139 \int_{\Omega}
140 \left(
141 u^*_{0,k} v_{0,k}
142 + u^*_{1,k} v_{1,k}
143 + \omega \mu \sigma \cdot ( u^*_1 v_0 - u^*_0 v_1) \right) dx = \int_{\Omega} Y_i v_i + X_{ij} v_{i,j} \; dx
144 \end{equation}
145 for all $v_i$ with $v_i=0$ on $\Gamma_{0}$. Setting $q=u^*$ in Equation~(\ref{ref:2DMTTE:EQU:201A}) and
146 $v_i=\hat{u}_i$ in Equation~(\ref{ref:2DMTTE:EQU:202d}) Equation~(\ref{ref:2DMTTE:EQU:201A}) gives
147 \begin{equation}\label{ref:2DMTTE:EQU:290}
148 \int_{\Omega} \omega \mu \hat{\sigma} \cdot ( u^*_0 u_1 - u^*_1 u_0) \; dx
149 = \int_{\Omega} Y_i \hat{u}_i + X_{ij} \hat{u}_{i,j} \; dx
150 \end{equation}
151 This gives
152 \begin{equation}\label{ref:2DMTTE:EQU:300}
153 \int_{\Omega} \frac{\partial J^{MT}}{\partial \sigma} \cdot \hat{\sigma} \; dx =
154 \int_{\Omega} \omega \mu \hat{\sigma} \cdot ( u^*_0 u_1 - u^*_1 u_0) \; dx
155 \end{equation}
156 or
157 \begin{equation}\label{ref:2DMTTE:EQU:301}
158 \frac{\partial J^{MT}}{\partial \sigma} = \omega \mu \cdot ( u^*_0 u_1 - u^*_1 u_0)
159 \end{equation}
160

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