# Empirical Orthogonal Function Analysis and Modeling of Global Tropospheric Delay Spherical Harmonic Coefficients

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## Abstract

**:**

## 1. Introduction

## 2. Data

#### 2.1. ERA-5 Data

_{n}is the gravity constant, with g

_{n}= 9.80665 m/s

^{2}.

^{−7}, and φ is the latitude of the grid. Finally, we get the ellipsoid height through geoid correction as follows:

_{N}is the geoidal undulation that derived from the Earth Gravitational Model 2008 (EGM2008) [29].

#### 2.2. Global Tropospheric Delay SH Coefficients Set

_{0}of each grid point at mean sea level are obtained by exponential function, the formula is as follows:

_{0}at each time is expanded in spherical harmonics up to degree 15, and the SH coefficient of altitude correction coefficient at each time is obtained. The SH function is defined as:

_{0}and β), P

_{nm}is the normalized associated Legendre function of degree n and order m, φ and λ are the latitude and longitude, respectively. A

_{nm}and B

_{nm}are spherical harmonic coefficients. The spherical harmonic coefficients are solved by the least square method. The SH coefficients at a single time are stored or released in sets.

^{−4}cm, RMSE: 1.97 cm), which can better represent the original data, indicating that the spherical harmonic coefficient set can be used as the basic experimental data.

## 3. Construction of the EOF-Based Model

#### 3.1. Empirical Orthogonal Function and Analysis

#### 3.1.1. EOF Method

_{i}(k) is the ith basis function of SH_set (k, h), which reflects the relevant information between the spherical harmonic coefficients. A

_{i}(k) is the correlation coefficient of Ui(k), representing the change in the SH_set (k, h) over time (such as annual, quarterly, and daily changes). m is the number of basic functions or correlation coefficient functions.

_{i}(h) are computed using Equations (8) and (9). The cumulative contribution percentage of the ith EOF component relative to the total variance and the first m EOF components [30] can be calculated according to the following.

_{i}is the variance in the ith EOF component.

#### 3.1.2. Timing Characteristics of A_{i}(h)

_{i}(h). As such, coefficient A

_{i}(h) reflects the average variation in the tropospheric SH coefficients. The chart shows that coefficient A

_{i}(h) exhibits obvious annual and semiannual cycles, and coefficients A

_{3}(h) and A

_{4}(h) also exhibit obvious quarterly variations. Through high-precision modeling of coefficient A

_{i}(h), the SH coefficient is accurately inverted, and the high-precision tropospheric delay can then be obtained quickly and efficiently.

_{i}(h), we performed the FFT analysis of the first four orders coefficient A

_{i}(h) from 2015 to 2019 and focused on certain positions to obtain more details, as shown in Figure 2. The chart reveals that A

_{i}(h) is dominated by both annual, semiannual, diurnal, and semidiurnal variations. In addition, A

_{1}(h) exhibits certain 1/3-year variation, A

_{2}(h) experiences certain 1/4-year variation, and A

_{3}(h) and A

_{4}(h) reveal certain 1/3-year and1/4-year variations.

#### 3.2. EGtrop Model Construction

_{i}(h) exhibits obvious periodic characteristics, and the periods of the different parameters are distinct. Therefore, we use trigonometric functions with different periods to model EGtrop. The specific fitting equation is as follows:

_{i}, b

_{i}, c

_{i}, d

_{i}, e

_{i}and f

_{i}are all unknown parameters, which can be solved by the least-squares method. Please refer to Appendix A for the detailed establishment process.

_{i}(h) can be achieved with a few coefficients, and then, the reconstruction of the SH coefficients also can be realized by using Equation (7). Only by giving doy and Hod, the user can obtain 256 tropospheric SH coefficients and the global tropospheric delay based on the mean sea level can be obtained by using the SH function. The global altitude correction coefficient is obtained by the same method. Finally, the ZTD at the specified position is obtained by Equation (5).

_{i}(h).

## 4. Results

^{mod}and ZTD

^{dat}are the model value and the reference value, respectively. M is the number of observations.

#### 4.1. Global Tropospheric Delay SH Coefficients Set Validation

#### 4.2. Verification of SH Coefficient for EGtrop Model

#### 4.3. Verification of the Tropospheric Delay for EGtrop Model

## 5. Conclusions

- 1.
- This study adopts a spherical harmonic function to fit the tropospheric delay calculated with global ERA-5 meteorological data at each time, and an SH coefficients dataset is obtained in the calculation, which is convenient for EOF decomposition and formula fitting. It is verified that the SH_set yield a good accuracy (ERA-5 ZTD, Bias: −1.0 × 10
^{−4}cm; RMSE: 1.97 cm; IGS ZTD, Bias: 0.08 cm; RMSE: 2.6 cm), indicating the feasibility and reliability of this strategy, which provides a reference for near-real-time model products. - 2.
- Based on the analysis of the SH_set data from 2015 to 2019, it is found that the spherical harmonic coefficients exhibit a certain periodic variation. Based on this phenomenon, this study implements the EOF method and trigonometric functions to establish an SH coefficients model for the SH_set data called EGtrop and combines it with the spherical harmonic function to complete the establishment of the global tropospheric delay model. The results indicate that the accuracy of the new model is higher than that of GPT2w and UNB3m on the different reference data. In addition, through verification of the model accuracy, it is found that the EGtrop model is applicable not only at the global scale but also at the regional scale, and this model yields the advantage of local enhancement.
- 3.
- Compared to GGOS tropospheric delay grid data, the SH_set proposed in this study experiences a slight loss of accuracy, but it greatly reduces the number of parameters and is more convenient for users.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{0}and the corresponding altitude correction coefficient β.

_{0}and β), the global tropospheric spherical harmonic coefficient set (SH_set) is established through the spherical harmonic function (Equation (6)).

_{i}(h)). The EGtrop model is established by time fitting (Equation (12)) the EOF coefficients (A

_{i}(h)).

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**Figure 1.**Time series of the first four orders correlation coefficient from 2015 to 2019. The resolution of the time series is 1-h, and the width of each gray grid is one year in the x-axis direction.

**Figure 2.**Spectral analysis of the four orders correlation coefficients from 2015 to 2019. The block in the figure represents a short-time power spectrum. The blue font represents the period corresponding to the high-power spectrum.

**Figure 3.**Time series diagram of the first four orders EOF parameters and fitting parameters. Red dots indicate the fitting data, and cyan dots indicate the metadata.

**Figure 4.**Tropospheric delay sequence diagram solved by ERA-5 and SH_set. Red spots indicate the tropospheric delay of the ERA-5 solution, and green spots indicate the tropospheric delay of the SH_set solution.

**Figure 5.**Error distribution map of the SH_set data compared to the global IGS stations in 2018. The left side of the picture is the Bias distribution diagram, and the right side is the RMSE distribution diagram.

**Figure 6.**Scatter plots of observational data versus modeled values of SH coefficients for the period 2015−2019. The blue-green box shows the first spherical harmonic coefficient. The correlation coefficient (R), RMSE (RMS) and Bias (Mean) are also shown in the panels.

**Figure 7.**Time series of SH coefficients between EGtrop and SH_set for the period 2015−2019. Cyan spots represent SH coefficients provide by SH_set, and red spots represent SH coefficients derived by EGtrop.

**Figure 8.**Error distribution map of each model compared to the global ERA-5 ZTD product over 2020. The left side of the picture is the Bias distribution diagram, and the right side is the RMSE distribution diagram. From top to bottom are the error distributions of the EGtrop, GPT2w and UNB3m.

**Figure 9.**Monthly Bias and RMSE for different models tested by radiosonde in 2020. Grey cyan, green and orange represents EGtrop, GPT2w and UNB3m, respectively. At the top of the figure is the distribution of the monthly average Bias of different models. The lower part of the figure shows the distribution of the monthly average RMSE of different models.

**Table 1.**Summary of the variance through decomposition of SH coefficients under the first six-order EOF mode.

EOF Mode | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Variances (%) | 99.9503 | 0.0184 | 0.0040 | 0.0031 | 0.0029 | 0.0014 |

Cumulative var. (%) | 99.9503 | 99.9687 | 99.9727 | 99.9758 | 99.9787 | 99.9801 |

Year | Bias [cm] | RMSE [cm] | ||||
---|---|---|---|---|---|---|

Min | Max | Mean | Min | Max | Mean | |

2015 | −5.78 | 4.03 | −1.1 × 10^{−4} | 0.38 | 6.64 | 1.98 |

2016 | −5.97 | 3.78 | −1.0 × 10^{−4} | 0.40 | 6.91 | 2.01 |

2017 | −5.99 | 3.79 | −1.1 × 10^{−4} | 0.39 | 6.79 | 1.93 |

2018 | −6.28 | 3.72 | −1.1 × 10^{−4} | 0.41 | 7.01 | 1.97 |

2019 | −6.00 | 3.73 | −1.1 × 10^{−4} | 0.42 | 6.74 | 1.96 |

Mean | −1.0 × 10^{−4} | 1.97 |

Bias [cm] | RMSE [cm] | |||||
---|---|---|---|---|---|---|

Min | Max | Mean | Min | Max | Mean | |

EGtrop | −10.84 | 6.04 | −0.25 | 1.06 | 11.69 | 3.79 |

GPT2w | −9.20 | 16.11 | −1.02 | 1.19 | 15.79 | 4.32 |

UNB3m | −13.28 | 17.32 | 3.11 | 1.06 | 17.72 | 6.60 |

**Table 4.**Error statistics of the tropospheric delay models compared to the ZTD derived from IGS and Radiosonde.

Data | Area | Error [cm] | Asia | Europe | Oceania | Africa | North America | South America | Antarctica | |
---|---|---|---|---|---|---|---|---|---|---|

Model | ||||||||||

IGS ZTD | EGtrop | Bias | 1.73 | −0.27 | 2.93 | 0.36 | 1.33 | 0.90 | 1.05 | |

RMSE | 4.97 | 3.14 | 3.66 | 3.09 | 3.98 | 4.09 | 3.36 | |||

GPT2w | Bias | −0.11 | −0.33 | 0.17 | 0.41 | −0.30 | 1.01 | 0.02 | ||

RMSE | 4.55 | 3.37 | 3.53 | 3.05 | 4.04 | 4.19 | 2.45 | |||

UNB3m | Bias | −1.11 | −2.44 | 4.44 | 2.30 | 2.19 | 0.51 | 8.70 | ||

RMSE | 6.29 | 4.32 | 6.27 | 4.82 | 5.28 | 5.44 | 9.48 | |||

Radiosonde ZTD | EEtrop | Bias | 0.89 | 0.06 | 3.24 | −0.27 | 1.33 | −0.52 | 0.96 | |

RMSE | 3.69 | 3.54 | 3.89 | 3.74 | 4.02 | 4.36 | 2.66 | |||

GPT2w | Bias | −0.95 | 0.72 | 2.38 | −0.23 | −0.41 | −0.32 | −0.13 | ||

RMSE | 3.49 | 3.52 | 3.20 | 3.59 | 3.88 | 4.43 | 2.64 | |||

UNB3m | Bias | −0.10 | −0.42 | 8.70 | 5.22 | 0.63 | −0.60 | 9.00 | ||

RMSE | 3.65 | 4.03 | 9.21 | 7.44 | 4.22 | 6.35 | 9.57 |

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**MDPI and ACS Style**

Ma, Y.; Liu, H.; Xu, G.; Lu, Z.
Empirical Orthogonal Function Analysis and Modeling of Global Tropospheric Delay Spherical Harmonic Coefficients. *Remote Sens.* **2021**, *13*, 4385.
https://doi.org/10.3390/rs13214385

**AMA Style**

Ma Y, Liu H, Xu G, Lu Z.
Empirical Orthogonal Function Analysis and Modeling of Global Tropospheric Delay Spherical Harmonic Coefficients. *Remote Sensing*. 2021; 13(21):4385.
https://doi.org/10.3390/rs13214385

**Chicago/Turabian Style**

Ma, Yongchao, Hang Liu, Guochang Xu, and Zhiping Lu.
2021. "Empirical Orthogonal Function Analysis and Modeling of Global Tropospheric Delay Spherical Harmonic Coefficients" *Remote Sensing* 13, no. 21: 4385.
https://doi.org/10.3390/rs13214385