# A Novel Distance Estimation Method for Near-Field Synthetic Aperture Interferometric Radiometer

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## Abstract

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## 1. Introduction

## 2. Near-Field SAIR Signal Model

- $\mathcal{K}$ is the Boltzmann constant;
- ${D}_{i}$ and ${D}_{j}$ are the antenna directivities of each radiometer receiver channel;
- ${B}_{i}$ and ${B}_{j}$ are the equivalent noise bandwidth of the receiver channel;
- ${G}_{i}$ and ${G}_{j}$ are the gain of the receiver channel;
- ${T}_{B}$ is the brightness temperature of the target scene;
- ${T}_{r}$ is the equivalent noise temperature of the receiver channel;
- ${F}_{{n}_{i}}$ and ${F}_{{n}_{j}}$ are the normalized antenna radiation pattern of each radiometer receiver channel;
- ${r}_{ij}$ is the fringe washing function between the ith and jth receiver that accounts for spatial decorrelation effects;
- $\Delta R$ is the distance difference between the target to the two channels;
- k is the radiometric signal wave number.

## 3. Proposed Distance Estimation Method for Near-Field SAIR

#### 3.1. Basic Concepts of Distance Estimation Method

#### 3.2. Iterative Distance Estimation Method

- Initialize at an initial annealing temperature ${T}_{0}$ and generating a random initial solution ${x}_{0}$ as the initial solution and calculating the corresponding objective function $E\left({x}_{0}\right)$. The initial solution ${x}_{0}$ represents the initial target–instrument distance ${R}_{0}$, and the corresponding objective function $E\left({x}_{0}\right)$ is the reciprocal of MAG.
- Set the cooling rate ${t}_{k}$ for this temperature.
- Apply a random perturbation to the current target–instrument distance ${x}_{t}$ to generate a new solution ${x}_{t+1}$ in the current domain and calculate the corresponding objective function $E\left({x}_{t+1}\right)$.$$\begin{array}{c}\hfill \Delta E=E\left({x}_{t+1}\right)-E\left({x}_{t}\right)\end{array}$$
- According to the Metropolis criterion, the distance ${x}_{(t+1)}$ can be received as the current solution or does not need to be calculated according to the following probabilities.$$\begin{array}{c}\hfill P\left\{Accept{x}_{(t+1)}\right\}=\left\{\begin{array}{cc}exp\left[\frac{-\Delta E}{{t}_{k}}\right],\hfill & \Delta E>0\hfill \\ 1,\hfill & \Delta E\le 0\hfill \end{array}\right.\end{array}$$
- Repeat steps 2, 3, and 4 above until the equilibrium at the current temperature is reached.
- Lower the temperature and repeat the iterative process.
- Determine if the temperature reaches the termination temperature; if yes, then terminate the algorithm; otherwise, return to step 2.

## 4. Experimental Analysis

#### 4.1. One-Dimensional Imaging

#### 4.2. Two-Dimensional Imaging

#### 4.3. Real Measurement Imaging Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The flowchart of the simulated annealing algorithm for the optimization problem of the target–instrument distance estimation.

**Figure 5.**Iterative process of the 1D target imaging distance estimation based on simulated annealing algorithm.

**Figure 6.**The one-dimensional line array imaging results. (

**a**) Inversion results 1; (

**b**) inversion results 2; (

**c**) inversion results 3.

**Figure 8.**Iterative process of the 2D target imaging distance estimation based on simulated annealing algorithm.

**Figure 9.**The two-dimensional array near-field imaging results. (

**a**) Original extended source scene; (

**b**) inversion results 1; (

**c**) inversion results 2; (

**d**) inversion results 3.

**Figure 10.**The pictures of real imaging experiments. (

**a**) Experimental scenario; (

**b**) photograph of real extended source; (

**c**) photograph of SAIR system; (

**d**) hexagonal antenna array structure.

**Figure 11.**Iterative process of target imaging distance estimation based on simulated annealing algorithm.

**Figure 12.**The near-field imaging results of real measurement imaging experiment. (

**a**) Result 1; (

**b**) result 2; (

**c**) result 3.

**Table 1.**MAG of near-field imaging results at different estimated distances of one-dimensional line array.

Result | MAG | FuncValue | Estimated Distance (m) |
---|---|---|---|

1 | 0.0107 | 93.46 | 0.500 |

2 | 0.0178 | 56.13 | 0.818 |

3 | 0.0308 | 32.43 | 1.086 |

**Table 2.**The optimal estimated distance of near-field imaging results from different initial estimated distances of one-dimensional line array.

Iteration Curve | Initial Estimated Distance (m) | Optimal Estimated Distance (m) | Iteration Curve | Initial Estimated Distance (m) | Optimal Estimated Distance (m) |
---|---|---|---|---|---|

1 | 0 | 1.0472 | 6 | 0.5 | 1.0860 |

2 | 0.1 | 1.0317 | 7 | 0.6 | 1.0500 |

3 | 0.2 | 1.0639 | 8 | 0.7 | 0.9716 |

4 | 0.3 | 1.0173 | 9 | 1 | 1.0000 |

5 | 0.4 | 1.0215 | 10 | 1.5 | 1.0566 |

**Table 3.**MAG and RMSE of near-field imaging results at different estimated distances of two-dimensional array.

Result | MAG | RMSE | FuncValue | Estimated Distance (m) |
---|---|---|---|---|

1 | 0.0215 | 0.6078 | 46.51 | 0.200 |

2 | 0.0734 | 0.3604 | 13.62 | 0.488 |

3 | 0.1794 | 0.1299 | 5.574 | 1.012 |

**Table 4.**The optimal estimated distance of near-field imaging results from different initial estimated distances of two-dimensional line array.

Iteration Curve | Initial Estimated Distance (m) | Optimal Estimated Distance (m) | Iteration Curve | Initial Estimated Distance (m) | Optimal Estimated Distance (m) |
---|---|---|---|---|---|

1 | 0 | 1.0441 | 6 | 0.5 | 1.0120 |

2 | 0.1 | 1.0324 | 7 | 0.6 | 1.0134 |

3 | 0.2 | 0.9675 | 8 | 0.7 | 0.9529 |

4 | 0.3 | 1.0236 | 9 | 1 | 1.0000 |

5 | 0.4 | 1.0540 | 10 | 1.5 | 1.0365 |

**Table 5.**MAG of near-field imaging results for different estimated distances in real measurement imaging experiment.

Result | MAG | FuncValue | Estimated Distance (m) |
---|---|---|---|

1 | 0.0587 | 17.03 | 0.8000 |

2 | 0.1436 | 6.964 | 1.6770 |

3 | 0.2378 | 4.205 | 1.9488 |

**Table 6.**The optimal estimated distance of near-field imaging results for different initial estimated distances with a real imaging system.

Iteration Curve | Initial Estimated Distance (m) | Optimal Estimated Distance (m) | Iteration Curve | Initial Estimated Distance (m) | Optimal Estimated Distance (m) |
---|---|---|---|---|---|

1 | 0 | 2.0429 | 6 | 1 | 2.0891 |

2 | 0.2 | 2.0156 | 7 | 1.2 | 2.0655 |

3 | 0.4 | 1.9557 | 8 | 1.5 | 2.1091 |

4 | 0.6 | 2.1159 | 9 | 2 | 2.0000 |

5 | 0.8 | 1.9488 | 10 | 3 | 1.9852 |

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**MDPI and ACS Style**

Hu, H.; Zhu, D.; Hu, F.
A Novel Distance Estimation Method for Near-Field Synthetic Aperture Interferometric Radiometer. *Remote Sens.* **2023**, *15*, 1795.
https://doi.org/10.3390/rs15071795

**AMA Style**

Hu H, Zhu D, Hu F.
A Novel Distance Estimation Method for Near-Field Synthetic Aperture Interferometric Radiometer. *Remote Sensing*. 2023; 15(7):1795.
https://doi.org/10.3390/rs15071795

**Chicago/Turabian Style**

Hu, Hao, Dong Zhu, and Fei Hu.
2023. "A Novel Distance Estimation Method for Near-Field Synthetic Aperture Interferometric Radiometer" *Remote Sensing* 15, no. 7: 1795.
https://doi.org/10.3390/rs15071795