# Improving Inversion Quality of IP-Affected TEM Data Using Dual Source

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Forward Modeling of IP-Affected TEM Response

^{2}= −1; ω represents the angular frequency; σ

_{0}represents conductivity of the earth at zero-frequency; m represents the chargeability, and 0 ≤ m ≤1; c represents the exponent describing the variation of phase with frequency, and 0 ≤ c ≤ 1; τ represents the IP relaxation time constant.

_{i}, m

_{i}, c

_{i}, τ

_{i}) and the z-coordinate of its top interface Z

_{i}. Assuming a time-harmonic factor of e

^{−iωt}, the governing equations can be written as

**B**represents the magnetic field,

**E**represents the electric field, μ represents the permeability, and

**J**represents the source current density vector. The magnetic vector potential

**A**in the Coulomb gauge ∇·

**A**= 0 can be expressed by

_{0}is the 0-order Bessel functions of the first kind, r is the separation of the receiver from the dipole source,

**r**is the position vector from the dipole source to the receiver. The kernel $\widehat{A}$ can be obtained by recursion of each layer, and then the electromagnetic fields

**E**and

**B**in the frequency domain can be obtained using Equations (4) and (5).

#### 2.2. Singularity of IP-Affected TEM Data Inversion

#### 2.3. TEM Data Acquisition Using Dual Source

#### 2.4. Joint Inversion of Dual-Source TEM Responses

**d**represents the TEM responses, the superscript ‘obs’ and ‘fit’ represent the observation and modeling responses, respectively, the subscript ‘1’ indicates that the dataset is acquired with the first offset, and the subscript ‘2’ indicates that the dataset is acquired with another offset,

**C**is the weight matrix,

_{m}**m**is the model parameter vector,

**D**is the regularization matrix; generally, a smoothness or roughness matrix is adopted to improve the ill condition of the inversion problem. The particle swarm optimization is adopted for global optimization of the objective function (9) to recover the geoelectrical parameters.

## 3. Results

#### 3.1. Synthetic Data Test

**d**represents the TEM responses, the superscript ‘obs’ and ‘fit’ represent the observation and modeling responses respectively, the subscript ‘1’ indicates that the dataset is acquired with the first offset, and the subscript ‘2’ indicates that the dataset is acquired with another offset, N is the number of elements in vector

**d**. We can see that the true solution can be obtained occasionally, while the parameters of each inversion fluctuate greatly with a stable small fitting error. It indicates that all the ten inversion models are algebraic approximate solutions of the inversion problem, actually proves the singularity of inversion using the single central-loop dataset.

#### 3.2. A Field Example: Baiyun Gold Deposit

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Validation of the modeling results. (

**a**) The Lornex model; (

**b**) The Copper cities model; (

**c**) The Kidd Creek model.

**Figure 3.**The TEM respones of the equivalent three-layer models. (

**a**) The Central loop; (

**b**) Offset = 80 m; (

**c**) Offset = 120 m.

**Figure 5.**The inversion parameters of Model 3 using single source. (

**a**) The layer thickness; (

**b**) The resistivity; (

**c**) The chargeability; (

**d**) The frequency exponent; (

**e**) The time constant; (

**f**) The relative fitting error.

**Figure 6.**The inversion parameters of Model 3 using dual source. (

**a**) The layer thickness; (

**b**) The resistivity; (

**c**) The chargeability; (

**d**) The frequency exponent; (

**e**) The time constant; (

**f**) The relative fitting error.

**Figure 8.**Fitting curves and fitting error of different stations. (

**a**) Fitting curves of station 850; (

**b**) Fitting curves of station 1500; (

**c**) Fitting curves of station 2950; (

**d**) Fitting errors of all stations.

**Figure 9.**The inversion cross-sections. (

**a**) The resistivity; (

**b**) The chargeability; (

**c**) The frequency exponent; (

**d**) The time constant.

**Figure 10.**The logging curves. (

**a**) The resistivity logging curves; (

**b**) The TEM logging curves of 25–31 channels.

Model | σ (S/m) | τ (s) | c | 1 − m |
---|---|---|---|---|

Lornex | 7.90 × 10^{−3} | 1.00 × 10^{−4} | 0.160 | 0.540 |

Copper cities | 6.45 × 10^{−3} | 6.90 × 10^{−3} | 0.280 | 0.580 |

Kidd Creek | 6.40 × 10^{−2} | 3.08 × 10^{−2} | 0.306 | 0.089 |

Parameters | h (m) | ρ (Ω·m) | m | c | τ (s) |
---|---|---|---|---|---|

Layer 1 | 100 | 400 | 0.06 | 0.55 | 10^{−2} |

Layer 2 | 50 | 100 | 0.06 | 0.55 | 10^{−2} |

Layer 3 | - | 800 | 0.60 | 0.55 | 10^{−2} |

Parameters | h (m) | ρ (Ω·m) | m | c | τ (s) |
---|---|---|---|---|---|

Layer 1 | 100 | 517 | 0.70 | 0.06 | 3.96 × 10^{−3} |

Layer 2 | 50 | 195 | 0.66 | 0.36 | 3.98 × 10^{−3} |

Layer 3 | - | 251 | 0.26 | 0.26 | 3.26 × 10^{−3} |

Parameters | h (m) | ρ (Ω·m) | m | c | τ (s) |
---|---|---|---|---|---|

Layer 1 | 100 | 800 | 0.05 | 0.55 | 10^{−3} |

Layer 2 | 50 | 100 | 0.30 | 0.55 | 10^{−3} |

Layer 3 | - | 800 | 0.05 | 0.55 | 10^{−3} |

Terms | Parameters | Terms | Parameters |
---|---|---|---|

Instrument | SMARTem24 | Receiver type | Crone probe |

Current | 23 A | Receiver area | 3850 m^{2} |

Time Base | 50 ms | Stack times | 512 |

Ramp time | 122 μs | Receive signal | dB_{z}/dt |

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**MDPI and ACS Style**

Zhi, Q.; Wu, J.; Li, X.; Wang, X.; Deng, X.
Improving Inversion Quality of IP-Affected TEM Data Using Dual Source. *Minerals* **2022**, *12*, 684.
https://doi.org/10.3390/min12060684

**AMA Style**

Zhi Q, Wu J, Li X, Wang X, Deng X.
Improving Inversion Quality of IP-Affected TEM Data Using Dual Source. *Minerals*. 2022; 12(6):684.
https://doi.org/10.3390/min12060684

**Chicago/Turabian Style**

Zhi, Qingquan, Junjie Wu, Xiu Li, Xingchun Wang, and Xiaohong Deng.
2022. "Improving Inversion Quality of IP-Affected TEM Data Using Dual Source" *Minerals* 12, no. 6: 684.
https://doi.org/10.3390/min12060684