*3.3. Inversion Models*

Figure 4 shows the time series of two BELMANIP pixels for different inversion models and their noise value using AVHRR-pre-data. In brackets is shown the relative noise of the time series. For visual purposes, the data is shifted in the y-axis by 0.3 × n in the red and 0.6 × n in the NIR and NDVI for the nth model. The first one is a savanna pixel located in Brazil ( −14.72, −41.75). The red and NIR band show a significant improvement in the noise after the directional effects' correction of ~60% and ~80% respectively, even when using the average method, which is based on the broadest approximation. The result can be appreciated visually, where the seasonal variation of the data can be distinguished after the correction. In the case of the NDVI, however, there is little or no improvement in the noise. When using MODIS parameters, for example, the noise increases. This is due to the intrinsic directional effects' correction of the NDVI computation [18]. The second pixel is a bare ground pixel located in the Algerian Saharan Desert (28.28, −5.03). In this case, there is also a significant decrease in the red and NIR noise when using the different inversion models, but not in the NDVI. The average method, however, significantly increases the noise, evidencing that the approximation it uses might not be viable for non-vegetated sites. These results also show that the assumption that for low NDVI values, the parameters are the same for both bands is reasonable.

To analyze the performance of every individual model in detail, we plotted the distribution of the noise corrections for every pixel considered in this study. Figure 5 shows these distributions in the form of a boxplot, using the different inversion models and for MODIS, AVHRR-pre and AVHRR data. The top and bottom blue edges of the box represent the 25th and 75th percentiles, respectively, while the middle red line shows the median. Points outside the black bars are considered outliers. The green diamond represents the average of the distribution. These values are shown in Table 3.


**Table 3.** Average noise (×<sup>10</sup>4) of the BELMANIP sites' time series obtained before (raw) and after directional effects using the models described for the red and NIR bands and the NDVI, and using MODIS, AVHRR-pre and AVHRR data from top to bottom. The percentage next to every noise value indicates the improvement with respect to the raw data.

**Figure 4.** Time series of two BELMANIP pixels (savanna and barren) for different inversion models and their noise value using AVHRR-pre-data. In brackets is shown the relative noise of the time series. For visual purposes, the data is shifted in the y-axis by 0.3n in the red band, and 0.6n in the NIR and NDVI for the nth model.

Firstly, we can see that the average raw noise of the red and NIR time series is similar for MODIS data (0.020, 0.043) and AVHRR-pre data (0.021, 0.045), but higher than the AVHRR data (0.017, 0.029). This is caused by the large directional errors in MODIS data and the high atmospheric errors in AVHRR-pre-data. Evidence of this can be seen on the NDVI noise. In MODIS, its value is very low (0.016), meaning that the intrinsic directional correction of this index has corrected most of the directional effects. In AVHRR-pre, because the directional effects are not as high, the NDVI correction errors are mostly because of atmospheric uncertainty propagation (0.038).

Secondly, looking at MODIS data (Figure 5, column 1) gives an indication of the quality of the approximations considered by the different models. As is expected, the MODIS inversion provides the best correction (75.7%, 82.3% and 30.2% for the red, NIR and NDVI respectively). Using the Average model, not only is the red and NIR noise improvement lower by ~20%, as compared to the MODIS inversion, but it also has a higher spread of the noise distribution. This is expected considering the broad generalizations of the model. The B1(B2) model shows the second-best performance, only 2% worse than the MODIS inversion, indicating that this a valid approximation that could reduce the computational time while achieving high-quality directional correction. For the NDVI, however, this method shows a significantly smaller improvement (10.0%) than the MODIS model (30.2%). Finally, the Stable method shows to be a valid alternative for the red band, but not for the NIR band. For the NDVI value, it can correct ~5% better than the B1(B2) method.

Analyzing the effects of these models on AVHRR-pre-data (Figure 5, column 2) can now show whether the error provided by their approximations is smaller or higher than the propagated atmospheric error. The MODIS model on AVHRR-pre provides a good correction of the directional effects with ~45.6%, 59.7% in the red and NIR bands, 5% and 3% better than the AVHRR-pre model. This shows that MODIS parameters are preferable to AVHRR-pre derived parameters. The opposite is true for the NDVI. The Average and the B1(B2) model show the worst performances among all of them, with a significant difference with the MODIS model in the red (~8% and 9%) and the NIR (~6% and 3%). This is expected for the Average method but is surprising for the B1(B2) model considering its good performance with MODIS data. It seems like the inversion with two parameters still isn't good enough, despite having reliable assumptions. In the case of the NDVI, the three of them provide a negative improvement (~4%) given that the noise on the raw data is already low. Finally, the Stable method provides the best improvement in the red and NIR bands, with differences of ~9% and 8% respectively, compared to the AVHRR-pre model. This shows that performing the inversion with one parameter provides higher stability to the parameters and therefore a smaller distribution in the corrected noise values, as can be appreciated by the width of the boxplots in Figure 5. In the NDVI, it provides a positive improvement of 2.87%, but it's still lower than using the AVHRR-pre-data.

**Figure 5.** Noise distributions for all the bands considered (rows), using the different inversion models and for MODIS, AVHRR-pre and AVHRR data, (columns 1, 2, and 3, respectively). The green diamonds represent the average of the distribution. The top and bottom blue edges of the box represent the 25th and 75th percentiles, respectively, while the middle red line shows the median. Points outside the black bars are considered outliers.

These results are analogous to the AVHRR data (Figure 5, column 3), but with a significantly smaller difference between the methods. The Stable method, for example, only provides a ~3% and 2% improvement difference with the AVHRR model in the red and NIR bands respectively. The average

method of AVHRR data improves significantly less than with AVHRR-pre-data. This is because atmospheric errors are not as high in this time series and, therefore, the broad assumptions made by the model provide more uncertainty than the propagated atmospheric perturbations.
