Fig 1.
This figure shows the locations of the stations on the 25th of January 2019 that were used in this research. The first figure shows the locations of the 1PD, the second of the 2PD and the third of the 3PD. The base map is public domain and is available via: https://www.naturalearthdata.com/.
Fig 2.
Predictions made using synthetic data.
Maps of the synthetic data generated by using Eq (1) for various values of N. The base map is public domain and is available via: https://www.naturalearthdata.com/.
Fig 3.
Visualisation of a Gaussian Process.
In Fig 3 we show an illustration of the methodology described in Section 3. Fig 3A shows the prior realisation of 8 ensemble members, which have not been conditioned on the data. Fig 3B shows the realisation of ensemble members, which have now been conditioned on the high-fidelity data (green dots), but not yet on the low-fidelity data(gray dots). Fig 3C) illustrates the corresponding power spectra, with the reference (or ‘true’) spectrum in black, and the ensemble spectra in corresponding blue. Note that the ensemble spectra still deviate quite significantly from the reference. Fig 3D shows the realisation of ensemble members, which have now been conditioned on the high-fidelity data (green dots) as well as on the low-fidelity data(gray dots), but without a noise model. Fig 3E illustrates the corresponding power spectra. Note that the ensemble spectra still deviate quite significantly from the reference and show high-frequency noise. Fig 3F shows the realisation of ensemble members, which have now been conditioned on the high-fidelity data (green dots) as well as on the low-fidelity data(gray dots), now with a noise model. Fig 3G illustrates the corresponding power spectra. Note that the ensemble spectra now deviate much less from the reference and that the high-frequency noise has effectively been removed.
Fig 4.
The regression error under different noise levels.
The accuracy of the interpolation depends on the true noise level and the estimated noise level. However, as can be seen from this illustration, the regression error is robust against changes in the estimated noise level.
Fig 5.
Fitting performance based on weather complexity.
This figure shows the RMSE values for different combinations of 1PD, 2PD and 3PD for various levels of spatial variability of the data. For each combination in the picture, the RMSE is computed five times. Each time a different random noise is added to the synthetic data. It shows that for low spatial variability they all perform equally well, but after approximately 1.5 oscillations per degree 1PD performs significantly worse than the others.
Fig 6.
Temperature predictions for N = 1.5.
Fig 6A shows the true synthetic temperature field and the following figures the predictions made by using 1PD, 12PD and 123PD respectively. The locations of first-party stations are indicated with a black square, second-party stations with a white circle and the third-party stations with a white triangle. These maps show that for a low N 1PD, 12PD and 123PD are all able to make a prediction that resembles the true temperature. However, it also shows that the more parties are used, the more accurate the prediction becomes. This can clearly be seen in the north of the country. The base map is public domain and is available via: https://www.naturalearthdata.com/.
Fig 7.
Uncertainty predictions for N = 1.5.
The figures show the uncertainty of the predictions made by using 1PD, 12PD and 123PD from Fig 6 respectively. In the first map there are some large dark green areas where the uncertainty is high. This is because the density of stations is very low there. When we add 2PD, these dark green areas disappear and the overall uncertainty decreases a lot, almost by 1.4°Celsius. Adding 3PD decreases the uncertainty even more, but only by a small amount. The base map is public domain and is available via: https://www.naturalearthdata.com/.
Fig 8.
Temperature predictions for N = 7.
Fig 8A shows the true synthetic temperature field and the following figures the predictions made by using 1PD, 12PD and 123PD respectively. The second map indicates that 1PD is not able to make an accurate prediction of the temperature field at all. The following map shows that using 12PD delivers a much better prediction already, but still has some spotty predictions in the areas with a lower station density. The last map shows that adding 3PD helps predict the areas where 12PD have a low station density. The base map is public domain and is available via: https://www.naturalearthdata.com/.
Fig 9.
Uncertainty predictions for N = 7.
The figures show the uncertainty of the predictions made by using 1PD, 12PD and 123PD from Fig 8 respectively. We see that 1PD has a very high average uncertainty and has a high uncertainty in almost the whole country. Using 12PD reduces that average uncertainty a lot, but there are still large areas with high uncertainty. By adding 3PD we see that these areas with high uncertainty are greatly reduced, resulting in an even lower average uncertainty. The base map is public domain and is available via: https://www.naturalearthdata.com/.
Fig 10.
Temperature predictions for real world data.
Temperature predictions for real temperature data from the 25th of January 2019. These figures show the predictions made by using 1PD, 12PD and 123PD respectively. The base map is public domain and is available via: https://www.naturalearthdata.com/.
Fig 11.
Uncertainty predictions for real world data.
Uncertainty predictions for the predictions shown in Fig 10. The base map is public domain and is available via: https://www.naturalearthdata.com/.