Regional/Single Station Zenith Tropospheric Delay Combination Prediction Model Based on Radial Basis Function Neural Network and Improved Long Short-Term Memory
Abstract
:1. Introduction
1.1. Motivations
1.2. Contributions
1.3. Organization
2. Materials and Methods
2.1. Study Region and Datasets
2.2. Methods
2.2.1. K-Means Clustering-Assisted RBF Neural Network Region ZTD Modeling
- (1)
- Randomly select k objects, which indicate the initial centers of the k clusters to be divided. The number of k can be preferred by the k-fold cross-check or bootstrap method. In this paper, the value of k is selected as 1.
- (2)
- Calculate the distance between each point and the center point and find the center with the shortest distance as the new center point of each cluster.
- (3)
- Calculate the average value (centroid) of all objects in each cluster as the new center point of each cluster.
- (4)
- Calculate the distance between all objects and the new k centers again and redistribute all objects to each cluster according to the nearest distance principle.
- (5)
- Repeat the above steps until all cluster centers remain unchanged (the distance between the newly generated cluster and the previous cluster is less than a set threshold). This is the end of clustering.
2.2.2. Real-Time Parameters Updating LSTM Single-Station ZTD Modeling
- (1)
- The actual time series is extended to , where n is the time series length, k is the sample dimension, n – k + 1 is the number of samples, and is the training data label. X is normalized:
- (2)
- Initialize network parameters and set super parameters:
- (3)
- Calculate what information needs to be forgotten from the cell state at time t – 1.
- (4)
- Calculate which input information can be left in the cell state at time t.
- (5)
- Calculate the cell state at time t.
- (6)
- Calculate the network output at time t.
- (7)
- Calculate the errors between the predicted values and the true values of all samples.
- (8)
- Update parameters in real time according to online observation data. The new samples, and , perform the forward operation of the LSTM shown in Steps 3–6 to obtain the predicted value . When the data are collected, they can be used as the true value label of the predicted value to calculate the overall error:Then, the BPTT algorithm is used to update the model parameters to :
2.2.3. Regional/Single Station ZTD Combination Model
2.2.4. Accuracy Evaluation Criteria
3. Results
3.1. Regional Modeling Results
3.2. Single Station Modeling Results
3.3. Regional/Single Station Combination Modeling Results
4. Discussion
4.1. Regional Modeling
4.2. Single Station Modeling
4.3. Regional/Single Station Combination Modeling
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
GNSS | Global navigation satellite system |
GPS | Global positioning system |
ZTD | Zenith tropospheric delay |
ZHD | Zenith hydrostatic delay |
ZWD | Zenith wet delay |
BP | Back propagation |
LSTM | Long short-term memory |
LSTM E/D | Long short-term memory encoder decoder |
RBF | Radial basis function |
K-RBF | RBF neural network assisted by the K-means cluster algorithm |
R-LSTM | LSTM of real-time parameter updating |
KR-RBF-LSTM | K-RBF and R-LSTM |
RMSE | Root-mean-square error |
STD | Standard deviation |
MAE | Mean absolute error |
MAPE | Mean absolute percentage error |
R2 | Coefficient of determination |
TTC | Training time consumption |
PWV | Precipitable water vapor |
NWP | Numerical weather prediction |
ML | Machine learning |
ANFIS | Adaptive network-based fuzzy inference system |
ANN | Artificial neural network |
LSSVM | Least-squares support vector machine |
PCA | Principal component analysis |
ICA | Independent component analysis |
GGOS | Global geodetic observing system |
CNN | Convolutional neural network |
KNN | K-nearest neighbor |
GP | Gaussian processes |
ERA5 | Fifth-generation European Center for Medium-range Weather Forecast reanalysis |
PPP | Precision point positioning |
RT-PPP | Real-time precision point positioning |
RTK | Real-time kinematic positioning |
CORS | Continuously-operating reference station |
PPP-RTK | Integer ambiguity resolution-enabled precise point positioning |
IGS | International GNSS Service |
DOY | Day of the year |
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Station Number | RMSE/mm | Increasing Rate/% | MAE/mm | Increasing Rate/% | R2 | Increasing Rate/% | TTC/s | Increasing Rate/% |
---|---|---|---|---|---|---|---|---|
K-RBF/R-LSTM/KR-RBF-LSTM | Imp1/Imp2 | K-RBF/R-LSTM/KR-RBF-LSTM | Imp1/Imp2 | K-RBF/R-LSTM/KR-RBF-LSTM | Imp1/Imp2 | K-RBF/R-LSTM/KR-RBF-LSTM | Imp1/Imp2 | |
1 | 10.85/6.74/5.25 | 51.63/22.09 | 9.11/5.92/4.38 | 51.94/26.10 | 0.46/0.98/0.90 | 49.05/−8.18 | 5.31/0.18/5.49 | −3.30/−96.70 |
2 | 6.16/6.18/3.95 | 35.86/36.10 | 5.01/5.31/3.36 | 33.07/36.75 | 0.76/0.97/0.96 | 20.97/−1.80 | 5.26/0.18/5.44 | −3.31/−96.69 |
3 | 8.07/5.74/4.29 | 46.83/25.28 | 6.46/4.98/3.49 | 45.95/29.85 | 0.61/0.98/0.92 | 33.72/−6.23 | 4.87/0.19/5.06 | −3.70/−96.30 |
4 | 12.01/6.83/5.26 | 56.22/22.93 | 10.04/5.88/4.40 | 56.16/25.09 | 0.42/0.98/0.92 | 54.66/−6.00 | 4.46/0.18/4.63 | −3.84/−96.16 |
5 | 7.01/5.96/3.74 | 46.72/37.35 | 5.77/5.09/3.13 | 45.80/38.59 | 0.75/0.98/0.96 | 22.02/−2.24 | 5.17/0.21/5.38 | −3.88/−96.12 |
6 | 7.73/6.28/4.62 | 40.28/26.52 | 6.39/5.50/3.79 | 40.59/31.05 | 0.60/0.97/0.89 | 32.81/−8.05 | 4.56/0.18/4.74 | −3.77/−96.23 |
7 | 7.50/5.73/3.85 | 48.65/32.82 | 6.25/4.92/3.23 | 48.28/34.33 | 0.68/0.98/0.95 | 27.86/−3.27 | 4.27/0.18/4.45 | −4.05/−95.95 |
8 | 8.48/5.85/4.23 | 50.04/27.63 | 6.98/5.05/3.52 | 49.53/30.26 | 0.61/0.98/0.93 | 34.27/−5.03 | 4.99/0.18/5.18 | −3.54/−96.46 |
9 | 11.15/6.11/4.62 | 58.56/24.37 | 9.27/5.22/3.86 | 58.36/26.05 | 0.50/0.98/0.93 | 46.79/−4.64 | 5.20/0.18/5.38 | −3.30/−96.70 |
10 | 9.55/6.23/5.15 | 46.03/17.22 | 7.90/5.56/4.19 | 46.98/24.69 | 0.48/0.97/0.86 | 44.23/−11.73 | 4.69/0.18/4.87 | −3.68/−96.32 |
11 | 6.08/5.89/3.77 | 38.02/35.98 | 4.89/5.13/3.16 | 35.51/38.48 | 0.75/0.98/0.94 | 20.18/−3.37 | 4.20/0.19/4.38 | −4.28/−95.72 |
12 | 8.31/5.70/4.23 | 49.07/25.73 | 6.92/4.93/3.55 | 48.64/27.86 | 0.60/0.98/0.90 | 33.53/−7.74 | 4.58/0.19/4.76 | −3.92/−96.08 |
13 | 5.66/6.15/3.81 | 32.57/37.98 | 4.55/5.32/3.22 | 29.19/39.51 | 0.78/0.97/0.95 | 18.46/−2.40 | 4.58/0.19/4.77 | −3.90/−96.10 |
mean | 8.35/6.11/4.37 | 47.70/28.48 | 6.89/5.29/3.64 | 47.20/31.29 | 0.61/0.98/0.92 | 33.51/−5.43 | 4.78/0.18/4.96 | −3.71/−96.29 |
min | 5.66/5.70/3.74 | 33.94/34.47 | 4.55/4.92/3.13 | 31.20/36.43 | 0.42/0.97/0.86 | 51.54/−11.73 | 4.20/0.18/4.38 | −4.28/−95.94 |
max | 12.01/6.83/5.26 | 56.22/22.93 | 10.04/5.92/4.40 | 56.16/25.68 | 0.78/0.98/0.96 | 19.10/−2.24 | 5.31/0.21/5.49 | −3.30/−96.20 |
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Yang, X.; Li, Y.; Yu, X.; Tan, H.; Yuan, J.; Zhu, M. Regional/Single Station Zenith Tropospheric Delay Combination Prediction Model Based on Radial Basis Function Neural Network and Improved Long Short-Term Memory. Atmosphere 2023, 14, 303. https://doi.org/10.3390/atmos14020303
Yang X, Li Y, Yu X, Tan H, Yuan J, Zhu M. Regional/Single Station Zenith Tropospheric Delay Combination Prediction Model Based on Radial Basis Function Neural Network and Improved Long Short-Term Memory. Atmosphere. 2023; 14(2):303. https://doi.org/10.3390/atmos14020303
Chicago/Turabian StyleYang, Xu, Yanmin Li, Xuexiang Yu, Hao Tan, Jiajia Yuan, and Mingfei Zhu. 2023. "Regional/Single Station Zenith Tropospheric Delay Combination Prediction Model Based on Radial Basis Function Neural Network and Improved Long Short-Term Memory" Atmosphere 14, no. 2: 303. https://doi.org/10.3390/atmos14020303