A Generic 1D Forward Modeling and Inversion Algorithm for TEM Sounding with an Arbitrary Horizontal Loop

Abstract

We present a generic 1D forward modeling and inversion algorithm for transient electromagnetic (TEM) data with an arbitrary horizontal transmitting loop and receivers at any depth in a layered earth. Both the Hankel and sine transforms required in the forward algorithm are calculated using the filter method. The adjoint-equation method is used to derive the formulation of data sensitivity at any depth in non-permeable media. The inversion algorithm based on this forward modeling algorithm and sensitivity formulation is developed using the Gauss–Newton iteration method combined with the Tikhonov regularization. We propose a new data-weighting method to minimize the initial model dependence that enhances the convergence stability. On a laptop with a CPU of i7-5700HQ@3.5 GHz, the inversion iteration of a 200 layered input model with a single receiver takes only 0.34 s, while it increases to only 0.53 s for the data from four receivers at a same depth. For the case of four receivers at different depths, the inversion iteration runtime increases to 1.3 s. Modeling the data with an irregular loop and an equal-area square loop indicates that the effect of the loop geometry is significant at early times and vanishes gradually along the diffusion of TEM field. For a stratified earth, inversion of data from more than one receiver is useful in noise reducing to get a more credible layered earth. However, for a resistive layer shielded below a conductive layer, increasing the number of receivers on the ground does not have significant improvement in recovering the resistive layer. Even with a down-hole TEM sounding, the shielded resistive layer cannot be recovered if all receivers are above the shielded resistive layer. However, our modeling demonstrates remarkable improvement in detecting the resistive layer with receivers in or under this layer.

Appendix

A horizontal electrical dipole in a layered earth has both the TE and TM modes, but only the TE mode contributes to the vertical magnetic field. Combining the work of Ward and Hohmann (1987) and the work of Key (2009), in an N + 1 layered earth, the vertical magnetic field in the mth layer of the TE mode due to a finite-length horizontal electric dipole at the depth of z _s in the nth layer has the following expression:

$$H_{z} = \frac{{I{\text{d}}s}}{4\pi }\frac{y}{R}\int_{0}^{\infty } {\left( {A_{{m,z_{s} }} e^{{ - u_{m} (z - z_{m} )}} + B_{{m,z_{s} }} e^{{u_{m} \left( {z - z_{m + 1} } \right)}} + \delta_{m,n} e^{{ - u_{n} |z - z_{s} |}} } \right)\frac{{\lambda^{2} }}{{u_{n} }}J_{1} \left( {\lambda R} \right){\text{d}}\lambda } .$$

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In the layers above the nth layer (m < n):

$$A_{{m,z_{s} }} = B_{{m,z_{s} }} e^{{u_{m} (z_{m} - z_{m + 1} )}} R_{\text{TE}}^{m} .$$

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For the first layer (m = 1), $A_{1} = 0$. In the layers below the nth layer (m > n):

$$B_{{m,z_{s} }} = A_{{m,z_{s} }} e^{{ - u_{m} (z_{m + 1} - z_{m} )}} r_{\text{TE}}^{m} .$$

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For the last layer (m = N + 1), $B_{N + 1} = 0$. $r_{TE}^{m}$ (m ≥ n) has the following expression:

$$r_{TE}^{m} = \frac{{Y_{m} - \tilde{Y}_{m + 1} }}{{Y_{m} + \tilde{Y}_{m + 1} }} ,$$

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with $Y_{m} = u_{m} /i\omega \mu_{m}$ and the following recursive relationship

$$\begin{aligned} \tilde{Y}_{N + 1} = Y_{N + 1} , \hfill \\ \tilde{Y}_{m} = Y_{m} \frac{{\tilde{Y}_{m + 1} + Y_{m} \tanh \left[ {u_{m} \left( {z_{m + 1} - z_{m} } \right)} \right]}}{{Y_{m} + \tilde{Y}_{m + 1} \tanh \left[ {u_{m} \left( {z_{m + 1} - z_{m} } \right)} \right]}} \hfill \\ \end{aligned} ,$$

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where tanh is the hyperbolic tangent function. $R_{TE}^{m}$ (m ≤ n) has the following relationship:

$$R_{TE}^{m} = \frac{{Y_{m} - \hat{Y}_{m - 1} }}{{Y_{m} + \hat{Y}_{m - 1} }} ,$$

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with the following recursive relationship

$$\begin{aligned} \hat{Y}_{1} = Y_{1} , \hfill \\ \hat{Y}_{m} = Y_{m} \frac{{\hat{Y}_{m - 1} + Y_{m} \tanh \left[ {u_{m} \left( {z_{m + 1} - z_{m} } \right)} \right]}}{{Y_{m} + \hat{Y}_{m - 1} \tanh \left[ {u_{m} \left( {z_{m + 1} - z_{m} } \right)} \right]}} \hfill \\ \end{aligned} .$$

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In the layer containing the source (m = n), the following relationships exist:

$$A_{{n,z_{s} }} = R_{TE}^{n} \left[ {B_{{n,z_{s} }} e^{{u_{n} (z_{n} - z_{n + 1} )}} + e^{{u_{n} (z_{n} - z_{s} )}} } \right] ,$$

(34a)

$$B_{{n,z_{s} }} = r_{TE}^{n} \left[ {A_{{n,z_{s} }} e^{{ - u_{n} (z_{n + 1} - z_{n} )}} + e^{{ - u_{n} (z_{n + 1} - z_{s} )}} } \right] .$$

(34b)

Solving the equations above, we get

$$\left[ {\begin{array}{*{20}c} {A_{{n,z_{s} }} } \\ {B_{{n,z_{s} }} } \\ \end{array} } \right] = \frac{1}{{1 - R_{TE}^{n} r_{TE}^{n} e^{{ - 2u_{n} (z_{n + 1} - z_{n} )}} }}\times\left[ {\begin{array}{*{20}c} {R_{TE}^{n} \left( {r_{TE}^{n} e^{{u_{n} (z_{n} - 2z_{n + 1} + z_{s} )}} + e^{{u_{n} (z_{n} - z_{s} )}} } \right)} \\ {r_{TE}^{n} \left( {R_{TE}^{n} e^{{ - u_{n} (z_{n + 1} - 2z_{n} + z_{s} )}} + e^{{ - u_{n} (z_{n + 1} - z_{s} )}} } \right)} \\ \end{array} } \right] .$$

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Then, using the continuation relationship at the layer interfaces, for the layers above the nth layer (m < n), $B_{{m,z_{s} }}$ has the following expression:

$$B_{{m,z_{s} }} = \frac{{Y_{m} \left( {1 + R_{TE}^{m + 1} } \right) + Y_{m + 1} \left( {1 - R_{TE}^{m + 1} } \right)}}{{2Y_{m} }}\times\left[ {B_{{m + 1,z_{s} }} e^{{ - u_{m + 1} (z_{m + 2} - z_{m + 1} )}} + \delta_{(m + 1),n} e^{{ - u_{n} (z_{s} - z_{n} )}} } \right] ,$$

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and for the layers below the nth layer (m > n), A _m has the following expression:

$$A_{{m,z_{s} }} = \frac{{Y_{m} \left( {1 + r_{TE}^{m - 1} } \right) + Y_{m - 1} \left( {1 - r_{TE}^{m - 1} } \right)}}{{2Y_{m} }}\times\left[ {A_{{m - 1,z_{s} }} e^{{ - u_{m - 1} (z_{m} - z_{m - 1} )}} + \delta_{(m - 1),n} e^{{ - u_{n} (z_{n + 1} - z_{s} )}} } \right] .$$

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To make sure the stability of the numerical calculation, the hyperbolic tangent function “tanh” in Eq. (A-5) and (A-7) is calculated through a negative exponential function “exp”:

$$\tanh \left[ {u_{m} \left( {z_{m + 1} - z_{m} } \right)} \right] = \frac{{1 - \exp \left[ { - 2u_{m} \left( {z_{m + 1} - z_{m} } \right)} \right]}}{{1 + \exp \left[ { - 2u_{m} \left( {z_{m + 1} - z_{m} } \right)} \right]}} .$$

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