# An Adaptive Non-Uniform Vertical Stratification Method for Troposphere Water Vapor Tomography

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}. Additionally, severe weather can negatively affect the accuracy of the tomographic results. The results also showed that the accuracy of the tomographic results was reduced with increasing altitude. Moreover, the performance of the tomographic water vapor fields below 3000 m was improved by the proposed approach.

## 1. Introduction

## 2. Principle of GNSS Tomography

_{3}and k’

_{2}are the atmospheric refractive index constants with the value of 3.776 × 10

^{5}K

^{2}/hPa and 16.52 K/hPa, respectively; R

_{v}is the specific gas constant of water vapor with a value of 461.495 J/K/kg; T

_{m}is the weighted mean tropospheric temperature, calculated from the numerical integration using the meteorological measurements [24]; SWD refers to the slant wet delay (SWD), which can be expressed with the following Equation [10]:

_{w}is the wet mapping function, and the global mapping function (GMF) is as proposed by Böhm et al. [25]; e and ϕ are the satellite elevation angle and the azimuth angle, respectively; ${G}_{EW}^{w}$ and ${G}_{NS}^{w}$ refer to the wet delay gradients in the east-west and north-south directions, respectively; R refers to the residual unmodeled delay between the satellite and the receiver; and ZWD represents the zenith wet delay (ZWD), which is the portion of the zenith tropospheric delay (ZTD) except the zenith hydrostatic delay (ZHD), as follows:

_{S}refers to the surface station pressure in hPa; φ

_{S}is the station latitude; and h

_{S}is the station geodetic height in km.

^{3}; and s represents the distance of satellite signal traveled in the troposphere in m. As atmospheric water vapor is continuously distributed in the atmosphere, it is necessary to discretize the research area in most cases. Under the assumption that the water vapor density in each discrete voxel is a constant during a tomographic period, the small influence of geometric delay in signal propagation is neglected. In discrete form, Equation (5) becomes:

_{i}refers to the slant water vapor of the i th satellite signal in the research area; l

_{ij}represents the distance traveled by the i th satellite signal in the j th voxel, which can be calculated based on two adjacent intersections between the signal and grid faces; and ρ

_{j}represents the water vapor density of the j th voxel.

_{m}

_{×n}and b

_{m}

_{×1}represent the coefficient matrix consisting of distance values and the column vector made up of SWV observations, respectively; x

_{n}

_{×1}represents the column vector of unknown water vapor density values; and m and n are the number of SWV observations and number of discrete voxels subdividing the research area, respectively.

_{j}and ρ

_{k}represent the water vapor density of the j th layer and the k th layer, respectively; h

_{j}and h

_{k}are the height of the j th layer and the k th layer, respectively; and H refers to the water vapor scale height, which can be calculated by the following formula:

^{2}that can be obtained from PWV; and ρ

_{S}represents the surface station humidity in g/m

^{3}.

_{n}

_{×n}and V

_{v}

_{×n}are the coefficient matrices of horizontal constraints and vertical constraints, respectively; and v refers to the number of vertical constraints.

^{k}is the k th iterative solution; n is the total number of voxels; λ is the relaxation factor, which gives the weight of the correction term computed for each voxel with respect to the initial value; a

_{i,j}represents the element in i th row and j th column of the coefficient matrix of the tomographic equation; and b

_{i}refers to the i th row element in the column vector of the observed value.

## 3. Adaptive Non-Uniform Exponential Stratification Method

#### 3.1. Modeling of Vertical Distribution Characteristics of Atmospheric Water Vapor

_{h}refers to the water vapor density at height h in g/m

^{3}; ρ

_{0}refers to the water vapor density parameter at the surface; D

_{v}represents the parameter of vertical distribution characteristic of water vapor; and H stands for the water vapor scale height in m.

_{v}is the specific gas constant of water vapor with a value of 461.495 J/K/kg; T refers to the temperature in K; and P

_{w}represents the vapor pressure in hPa, which can be calculated according to the Goff–Gratch formula, as follows [30]:

_{d}refers to the dew-point temperature in K; and T

_{1}is a constant with a value of 273.16 K.

^{3}; a and b are the concrete values of ρ

_{0}and D

_{v}after fitting, respectively; and H represents the water vapor scale height in m.

#### 3.2. Adaptive Non-Uniform Exponential Stratification Method

_{b}refers to the water vapor density at the top boundary of the tomographic region in g/m

^{3}, and it can usually be determined according to the prior information or radiosonde data. Therefore, the height interval of the first stratification is [0, D] under this model, and the numerical interval of water vapor density in the first stratification can be represented as [a, a∙e

^{b}

^{∙(-D/H)}]. After dividing the first stratification, the remaining stratifications are equally divided according to the water vapor density, and the numerical interval of the water vapor density of each remaining stratification can hence be expressed as:

_{b}represent the same meanings as in Equation (16).

_{b}represent the same meanings as in Equation (17); and m refers to the number of iterations that divide the bottom stratifications with D m as the minimum height interval.

_{b}and m represent the same meanings as in Equation (18); and H

_{i}refers to the height interval of the lowest stratification in the remaining (L-m) stratifications in m. If H

_{i}is greater than D m, terminate the iteration and the non-uniform exponential stratification is completed; otherwise, the lowest stratification in the remaining (L-m) stratifications will be divided unceasingly with D m as the height interval and continue to iterate.

## 4. Results and Validations

#### 4.1. Processing Strategy

#### 4.2. Vertical Stratification Strategy

^{3}(between stratification 1 and stratification 2), and the minimum value of the differences was 0.111 g/m

^{3}(between stratification 12 and stratification 13); the difference between the maximum and the minimum was 5.126 g/m

^{3}. In schemes (b) to (f), in which the vertical resolution was divided based on the ANES, the mean water vapor density of each stratification decreased linearly with increasing stratification, and the differences between the maximum and the minimum difference value derived from the six schemes were 1.383 g/m

^{3}, 1.501 g/m

^{3}, 1.583 g/m

^{3}, 2.206 g/m

^{3}, 2.302 g/m

^{3}, respectively. The above computations obtained from radiosonde data indicate that the traditional uniform stratification leads to a large difference in the difference value of the mean water vapor density between two adjacent stratifications, which is represented by a large difference in the difference value between bottom stratifications compared with a small difference between top stratifications, while the ANES can effectively avoid this problem.

^{3}, while in scheme (f), four values are greater than 2 g/m

^{3}, and two of them are greater than 3 g/m

^{3}.

#### 4.3. Tomographic Experiments and Evaluation of the ANES Approach

^{3}, while in the comparison of the MAE, scheme (f) had the advantage because of the lowest MAE value. Overall, in the comparison of the schemes based on the ANES approach, the statistical results of scheme (d) exhibited a higher accuracy than the other schemes. Further, it should be noted that the minimum height intervals of scheme (b), (c), and (d) were 350 m, 375 m, and 400 m, respectively. However, scheme (d) displayed a smaller RMSE than those of scheme (b) and (c). This phenomenon indicates that smaller height intervals do not correspond to higher accuracy. A reason for this may be that a smaller height interval reduces the height of voxels at the bottom boundary, which decreases the number of voxels passed by GNSS signals under the condition that the geometrical distribution of satellite rays is constant.

^{3}, indicating that its accuracy was the lowest in the ANES schemes, which was still lower than the RMSE of scheme (a) with a value of 1.321 g/m

^{3}. Scheme (a) had a value of 0.167 g/m

^{3}in the bias, while the worst bias value of the ANES was −0.104 g/m

^{3}appearing in scheme (b). The MAE value of the uniform stratification was 0.022 g/m

^{3}greater than that of scheme (b), which was the maximum MAE value in the schemes using the ANES. Figure 6 further validates the advantages of the ANES approach by showing the correlation between the tomographic reconstructions of the six schemes and the radiosonde data in the form of scatter plots. Among the six schemes, the scatter plot of scheme (a) had both the largest RMSE and the lowest R

^{2}, revealing that the correlation between the tomographic reconstructions of the traditional stratification approach and the radiosonde data is lower than that of ANES. Moreover, by observing the scatter plots, it can be seen that scheme (a) showed distinct gaps in the water vapor density data retrieved by tomography, which demonstrates the inadequacy of the traditional stratification approach mentioned in Section 4.2.

^{3}, respectively, while the minimum RMSEs were 0.943, 0.680, 0.687, 0.558, 0.662 and 0.561 g/m

^{3}, respectively. Evidently, the corresponding RMSE of the proposed method in this paper was smaller than the traditional method when the radiosonde data were taken as a reference. In addition, by observing the RMSE time series, it can be seen that higher RMSE values were generally present at UTC 12:00 of DOY 213 and UTC 12:00 of DOY 237. This was related to the strong winds and torrential rains caused by typhoons in Hong Kong from DOY 213 to DOY 214 and DOY 237 to DOY 238.

_{T}represents the retrieval of water vapor density by water vapor tomography in g/m

^{3}; and D

_{RS}represents the water vapor density calculated from radiosonde data in g/m

^{3}.

^{3}, while that of scheme (a) was consistently greater than 2 g/m

^{3}. In the region from 3000 to 6000 m, the RMSE and the relative error of the tomographic reconstructions were significantly decreased by the new approach. Therefore, the above comparison further demonstrates the marked superiority of the proposed method in the lower boundary layer.

## 5. Discussion

## 6. Conclusions

^{3}. The statistical results were analyzed, and the results show that severe weather, such as typhoons, will affect the accuracy of the tomographic results to some extent. Moreover, by comparing the vertical water vapor profile reconstructed by tomography with the radiosonde data, it was found that although all the schemes could accurately reflect the vertical distribution of atmospheric water vapor, it is difficult to use the water vapor density field reconstructed by tomography to compare with the radiosonde data with a high vertical resolution, especially under severe weather conditions. Statistical results for each stratification were studied, and the results showed that the tendencies between the RMSE and the corresponding RE were opposite with increasing altitude, which indicated that it became gradually difficult to realize the retrieval of water vapor density using tomography, as the moisture content decreased exponentially with the increasing altitude.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Fitting schematic diagram of the average radiosonde water vapor profile in the study area.

**Figure 2.**Geographical distribution of Global Navigation Satellite System (GNSS) receivers and radiosonde station in the tomographic area.

**Figure 3.**Vertical resolution according to six stratification schemes: (

**a**) uniform stratification with height interval of 812 m; (

**b**) ANES with minimum height interval of 350 m; (

**c**) ANES with minimum height interval of 375 m; (

**d**) ANES with minimum height interval of 400 m; (

**e**) ANES with minimum height interval of 425 m; (

**f**) ANES with minimum height interval of 450 m.

**Figure 4.**Mean water vapor density of each stratification and difference value between two adjacent stratifications derived from six stratification schemes, where (

**a**) derived from uniform stratification; (

**b**–

**f**) derived from ANES with minimum height intervals of 350, 375, 400, 425 and 450 m, respectively.

**Figure 5.**Box plots of the difference value of mean water vapor density between two adjacent stratifications derived from the six stratification schemes, where (

**a**) derived from uniform stratification; (

**b**–

**f**) derived from ANES with minimum height intervals of 350, 375, 400, 425 and 450 m, respectively.

**Figure 6.**Scatter plots of water vapor density between radiosonde and tomography derived from the six stratification schemes, where (

**a**) derived from uniform stratification; (

**b**–

**f**) derived from ANES with minimum height intervals of 350, 375, 400, 425 and 450 m, respectively.

**Figure 7.**Time series of the RMSE of the tomographic reconstructions derived from the six schemes using radiosonde data as a reference for 31 days (DOY 213 to DOY 243).

**Figure 8.**Comparison of the RMSE and relative error derived from six schemes in each stratification using radiosonde data as a reference (DOY 213 to DOY 243): (

**a**) comparison of the RMSE in each stratification; (

**b**) comparison of the relative error in each stratification.

**Figure 9.**Comparison of water vapor density profiles derived from six schemes on DOY 222 at UTC 00:00. (

**a**) Derived from uniform stratification; (

**b**–

**f**) derived from ANES with minimum height intervals of 350, 375, 400, 425 and 450 m, respectively.

**Figure 10.**Comparison of water vapor density profiles derived from six schemes on DOY 237 at UTC 12:00. (

**a**) Derived from uniform stratification; (

**b**–

**f**) derived from ANES with minimum height intervals of 350, 375, 400, 425 and 450 m, respectively.

Name of the Strategy | Setting of the Strategy |
---|---|

Cut-off elevation angle | 15° |

Auxiliary IGS stations | BJFS, CHAN, USUD |

Sampling interval for the GPS data | 30 s |

Tropospheric delay correction model | Saastamoinen |

Mapping function | VMF1 |

Ocean tidal model | FES2004 |

Solid tide model | IERS2003 |

Interval of gradient parameters | 2 h |

**Table 2.**Statistical results of the six schemes, with results derived from differences in water vapor density between radiosonde data and tomography.

Statistics | Scheme (a) | Scheme (b) | Scheme (c) | Scheme (d) | Scheme (e) | Scheme (f) |
---|---|---|---|---|---|---|

RMSE (unit: g/m^{3}) | 1.321 | 1.104 | 1.075 | 1.066 | 1.070 | 1.074 |

Bias (unit: g/m^{3}) | 0.167 | −0.07 | −0.104 | −0.09 | −0.08 | −0.07 |

MAE (unit: g/m^{3}) | 0.822 | 0.800 | 0.777 | 0.764 | 0.756 | 0.755 |

**Table 3.**Comparison of tomographic strategies used by developed approaches and their improvements to the accuracy of tomographic reconstructions.

Approach | Area | Duration | Number of GPS Stations | Horizontal Resolution | Vertical Stratification | RMSE (Unit: g/m^{3}) | Percentage | |
---|---|---|---|---|---|---|---|---|

COMMON | IMPROVED | |||||||

ANES (minimum height interval: 400 m) | Hong Kong | 31 days | 19 | 0.09° × 0.09° | ANES | 1.32 | 1.07 | 18.94% |

Tikh-LSQR | Tehran | 10 days | 11 | 0.25° × 0.25° | Uniform (500 m; 1000 m) | 0.82 | 0.40 | 51.22% |

LB-Tikh | Tehran | 10 days | 11 | 0.25° × 0.25° | Uniform (500 m; 1000 m) | 0.82 | 0.49 | 40.24% |

Function-based | North America | 30 days | 17 | 0.20° × 0.20° | Uniform (500 m; 1000 m) | 0.89 | 0.61 | 31.46% |

Integration of MODIS measurements | Xuzhou (China) | 31 days | 5 | 0.13° × 0.14° | Non-uniform | 2.74 | 2.53 | 7.66% |

HFM | Hong Kong | 31 days | 9 | 0.09° × 0.08° | Non-uniform | 1.63 | 1.13 | 30.67% |

Voxel-optimized | Hong Kong | 20 days | 12 | 0.09° × 0.09° | Non-uniform | 1.38 | 1.23 | 10.87% |

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**MDPI and ACS Style**

Wang, H.; Ding, N.; Zhang, W.
An Adaptive Non-Uniform Vertical Stratification Method for Troposphere Water Vapor Tomography. *Remote Sens.* **2021**, *13*, 3818.
https://doi.org/10.3390/rs13193818

**AMA Style**

Wang H, Ding N, Zhang W.
An Adaptive Non-Uniform Vertical Stratification Method for Troposphere Water Vapor Tomography. *Remote Sensing*. 2021; 13(19):3818.
https://doi.org/10.3390/rs13193818

**Chicago/Turabian Style**

Wang, Hao, Nan Ding, and Wenyuan Zhang.
2021. "An Adaptive Non-Uniform Vertical Stratification Method for Troposphere Water Vapor Tomography" *Remote Sensing* 13, no. 19: 3818.
https://doi.org/10.3390/rs13193818