# Joint Inversion of Atmospheric Refractivity Profile Based on Ground-Based GPS Phase Delay and Propagation Loss

^{*}

^{†}

## Abstract

**:**

_{1}, c

_{2}) and the vertical altitude (h

_{1}, h

_{2}). We apply the NSGA-II (Non-dominated Sorting Genetic Algorithm II), a multiobjective optimization algorithm, to achieve the goals of joint optimization inversion in the inverting process, and compare this method with traditional individual inversion methods. The anti-noise ability of joint inversion is investigated under the noiseless condition and adding noise condition, respectively. The numerical experiments demonstrate that joint inversion is superior to individual inversion. The adding noise test further suggests that this method can estimate synthesized parameters more efficiently and accurately in different conditions. Finally, a set of measured data is tested in the new way and the consequence of inversion shows the joint optimization inversion algorithm has feasibility, effectiveness and superiority in the retrieval of the refractivity profile.

## 1. Introduction

## 2. Forward Model and Parameterization Schemes

#### 2.1. Forward Model-Excess Phase Delay

**O**is the geocenter,

**R**and

**T**represent the receiver and transmitter, r

_{1}and r

_{2}are the distances from the geocenter to receiver and transmitter, respectively. Using Snell's law and substituting dr = dl cosα, the phase delay can be expressed as:

_{1}− ϕ

_{2}+ α, where ϕ

_{1}and ϕ

_{2}is the zenith of the ray, α is the bending angle. Thus, we can obtain the excess phase delay from the given refractivity index profile n(r):

#### 2.2. Forward Model-Propagation Loss

_{0}. Using the Fourier ‘split-step’ solution to calculate all the results of x > x

_{0}, the following observing formula is obtained:

#### 2.3. Four-Parameter Model

_{1}is the gradient of the refractivity profile from the receiver to the altitude h

_{1}which is a variable parameter, and c

_{2}is the gradient of the refractivity profile from the altitude h

_{1}to the altitude h

_{1}+ h

_{2}which is always larger than 160 N-unit/Km. The gradient of the refractivity profile from altitude h

_{1}+ h

_{2}to an altitude of 6 Km can be obtained from the CIRA + Q model (The CIRA86a with Q_UoG model) [21].

_{3}is the gradient of the modified refractivity profile from the receiver to the altitude h

_{3}and c

_{4}is the gradient of the modified refractivity profile from the altitude h

_{3}to the altitude h

_{3}+ h

_{4}. The atmosphere above h

_{3}+ h

_{4}is always regarded as the standard situation.

_{1}, c

_{2}, h

_{1}, h

_{2}), Equation (2) will transform M into N and we will change the inputting parameters of the propagation loss model accordingly in the process of joint inversion.

## 3. Inversion Algorithm

_{1},c

_{2},h

_{1},h

_{2})

^{T}, and the cost functions are defined as:

## 4. Results

#### 4.1. Ideal Condition

_{1}.

Parameters | Inversion Slope c_{1} | Height h_{1} | Inversion Slope c_{2} | Height h_{2} | |
---|---|---|---|---|---|

Units | N-units/m | m | N-units/m | m | |

Lower Bound | −0.1 | 50 | −0.4 | 250 | |

Upper Bound | 0 | 150 | 0 | 350 | |

True Value | −0.02 | 100 | −0.2 | 300 | |

NSGA-II | 0% | −0.0227 (13.5%) | 99.8246 (−1.75%) | −0.199 (−0.5%) | 298.7839 (−0.4%) |

3% | −0.0155 (−22.5%) | 95.9541 (−4.04%) | −0.1990 (−0.5%) | 306.0764 (2.03%) | |

5% | −0.0121 (−39.5%) | 94.7610 (−5.239%) | −0.2008 (0.4%) | 306.0295 (2.01%) | |

7% | −0.0254 (27%) | 98.9927 (−1.007%) | −0.2007 (0.35%) | 300.4304 (0.143%) | |

10% | −0.0146 (−27%) | 89.1687 (−10.83%) | −0.1959 (−2.05%) | 302.2124 (0.74%) | |

GA-Delay | 0% | −0.0576 (188%) | 109.5935 (9.59%) | −0.1969 (−1.55%) | 281.8778 (−6.04%) |

3% | −0.0519 (159.50%) | 91.2888 (−8.71%) | −0.1723 (−13.85%) | 332.9521 (10.98%) | |

5% | −0.0361 (80.50%) | 115.5426 (15.54%) | −0.1865 (−6.75%) | 305.7128 (1.90%) | |

7% | −0.0985 (392.50%) | 140.5176 (40.52%) | −0.1665 (−16.75%) | 328.5602 (9.52%) | |

10% | −0.0618 (209.00%) | 124.9692 (24.97%) | −0.1693 (−15.35%) | 273.2624 (−8.91%) | |

GA-Loss | 0% | −0.0208 (4%) | 96.0873 (−3.91%) | −0.1948 (−2.6%) | 309.6933 (3.23%) |

3% | −0.0181 (−9.50%) | 102.2168 (2.22%) | −0.2050 (2.50%) | 296.1529 (−1.28%) | |

5% | −0.0198 (−1.00%) | 93.3431 (−6.66%) | −0.1897 (−5.15%) | 312.0424 (4.01%) | |

7% | −0.01934 (−3.30%) | 97.9463 (−2.054%) | −0.1905 (−4.75%) | 307.5467 (2.52%) | |

10% | −0.0293 (46.50%) | 113.9005 (13.90%) | −0.2106 (5.30%) | 262.6215 (−12.46%) |

**Figure 3.**The retrievals with different Gaussian noise: (

**a**) and (

**b**) are the refractivity profile and corresponding refractivity deviation by NSGA-II; (

**c**) and (

**d**) are the refractivity profile and corresponding refractivity deviation by GA-Delay; (

**e**) and (

**f**) are the refractivity profile and corresponding refractivity deviation by GA-Loss.

#### 4.2. Adding Gaussian Noise

#### 4.3. Real Data Testing

Parameters | Inversion Slope c_{1} | Height h_{1} | Inversion Slope c_{2} | Height h_{2} |
---|---|---|---|---|

Units | N-units/m | m | N-units/m | m |

NSGA-II | −0.0281 | 105.0418 | −0.2013 | 293.8846 |

GA-Delay | −0.0212 | 102.3759 | −0.2313 | 304.9078 |

GA-Loss | −0.0374 | 110.5404 | −0.2187 | 298.8363 |

**Figure 4.**Inversion results of NSGA-II and GA. (

**a**) The refractivity profile. (

**b**) The corresponding refractivity deviation.

#### 4.4. Feasibility

Parameters | Inversion Slope c_{1} | Height h_{1} | Inversion Slope c_{2} | Height h_{2} | |
---|---|---|---|---|---|

Units | N-units/m | m | N-units/m | m | |

True Value | −0.05 | 1100 | −0.3 | 300 | |

400 m | NSGA-II | −0.0501 (0.20%) | 1058.2 (−3.80%) | −0.35 (16.67%) | 312.6988 (4.23%) |

GA-Delay | −0.05 (0.0%) | 1053.1 (−4.26%) | −0.3029 (0.97%) | 309.7630 (3.25%) | |

GA-Loss | −0.0503 (0.60%) | 1148.4 (4.40%) | −0.2723 (−9.23%) | 304.0592 (1.35%) | |

600 m | NSGA-II | −0.05 (0.0%) | 1028.82 (−6.40%) | -0.41 (36.70%) | 289.8588 (−3.38%) |

GA-Delay | −0.044 (-12.00%) | 1012.1 (−7.99%) | -0.4659 (−55.30%) | 293.7786 (−2.07%) | |

GA-Loss | −0.05 (0.0%) | 1029.8588 (−6.38%) | -0.2508 (−16.40%) | 311.7714 (3.92%) |

**Figure 5.**Refractivity absolute deviation under different antenna height: (

**a**) 400 m antenna height. (

**b**) 600 m antenna height.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

GA: | The traditional general genetic algorithm |

GA-Delay: | Retrieval from phase delay by GA |

GA-Loss: | Retrieval from propagation loss by GA |

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**MDPI and ACS Style**

Liao, Q.; Sheng, Z.; Shi, H.
Joint Inversion of Atmospheric Refractivity Profile Based on Ground-Based GPS Phase Delay and Propagation Loss. *Atmosphere* **2016**, *7*, 12.
https://doi.org/10.3390/atmos7010012

**AMA Style**

Liao Q, Sheng Z, Shi H.
Joint Inversion of Atmospheric Refractivity Profile Based on Ground-Based GPS Phase Delay and Propagation Loss. *Atmosphere*. 2016; 7(1):12.
https://doi.org/10.3390/atmos7010012

**Chicago/Turabian Style**

Liao, Qixiang, Zheng Sheng, and Hanqing Shi.
2016. "Joint Inversion of Atmospheric Refractivity Profile Based on Ground-Based GPS Phase Delay and Propagation Loss" *Atmosphere* 7, no. 1: 12.
https://doi.org/10.3390/atmos7010012