**5. Discussion**

#### *5.1. Accuracy of the Albedo Background*

In the data assimilation algorithm, the background field is a preliminary estimate of target parameters, which is obtained from historical and empirical data, and reflects the general variation trend of the estimator [54]. Our EnKF algorithm results were obtained based on the optimal estimation of the current time. The background field trend introduces an observation increment in the optimal weight, so its accuracy is very important. We conducted spatial registration before constructing the background; after processing, the projection mode and resolution of MODIS data were consistent with Landsat. MODIS data actually provides background fields and dynamic models (preliminary estimates), so errors do exist, but they are allowed under the assimilation algorithm. Surface albedo presents a certain regularity in the growing season. We averaged the historical MODIS data, instead of the contemporary MODIS data, as the albedo background field because MODIS data were missing across our study area [55]. Consider the AU-DaP site as an example: The background field albedo has the same change trend as the original MODIS albedo, but with no missing data and an overall smoother structure. To this effect, the background field is more suitable for data assimilation.

The background field encompasses periodic albedo characteristics, but the albedo is overestimated or underestimated due to the accuracy of MODIS products (especially in heterogeneous pixels) [56]. Under the assimilation scheme, however, the background field only provides an initial prediction and the weight of background data is determined according to its accuracy. When the background accuracy is low, the weight is decreased and the weight of the TM observation is increased. Therefore, the estimation results gradually approach the field observations as the quantity of TM observations increases; using the mean albedo of the time series as a background field is feasible and effective.

#### *5.2. Errors Induced by TM Albedo Estimation*

In this study, TM albedos derived by direct estimation were utilized as albedo observation data. The TM albedos were used to adjust model-based predictions to bring them as close as possible to actual measured values. The quality of the TM data directly affects the accuracy of the final estimations. Previous researchers have thoroughly validated TM albedos derived by direct estimation [18]. In this study, we also compared the TM albedo with the ground measured albedo for several FluxNet sites. Figure 9 shows that the surface albedo estimated from TM images is highly accurate (maximum error < 0.03) and can be used as the input observation value for the data assimilation algorithm.

The accuracy of inversion is affected by the accuracy, quantity, and time distribution of TM data, where quantity and time distribution complement each other. In this study, we used L1T data with less than 10% cloud contamination for our estimations. There were approximately 5–11 TM data points available at the selected sites, which was sufficient for albedo estimation by data assimilation. If the quantity of available TM data in one year falls below 5 points, the albedo during the time series may not be updated for long periods of time, and the accuracy of the estimations degrades. Several high-quality observations are needed to ensure accurate estimations. Other high-resolution satellite images such as HJ1A/B or Sentinel-2 can be supplemented to resolve this problem for those areas where high-resolution observations are limited.

**Figure 9.** Directly estimated TM albedos and field albedos including all available TM images used in this study.

#### *5.3. Error Setting in the Data Assimilation Algorithm*

In the data assimilation system, the observation-model error is critical for land surface albedo estimation. We calculated observation error from field observations and TM data and calculated model error from field observations and the albedo background. Field observation data are key in terms of error determination. For each site, we used the field observation data to generate errors resulting in high estimation accuracy. When extending the model to regional use where no field observation data was available, we attempted to generate a "common" error by averaging the error of the 18 ground stations and inputting it to the assimilation scheme. The test results on the AU-DaP site are shown in Figure 10. Errors caused by TM data are relatively small and uniform, and using the mean errors instead of the original observation error did not affect the estimation results significantly. In effect, the common error from different sites represents the averaged field condition and can be used to account for large areas.

**Figure 10.** Estimated results using mean error of 18 stations as data assimilation input at AU-DaP. Replacing original observation error with mean error does not significantly affect the estimation results.

#### *5.4. Capability of Capturing Abrupt Variations in Land Surface Albedo*

Abrupt changes in surface albedo are often due to drastic changes in surface characteristics over a short period, which are difficult to capture using remote sensing data [57]. The background field as obtained from MODIS historical period data can reflect abrupt changes to a certain extent, but the effect is not obvious. If TM data are available during the period of abrupt change, the model prediction is updated and the estimated results are closer to the observed values; the abrupt change in surface albedo can be captured effectively.

In our time-series albedo estimation for farmland stations, surface albedo jump occurred on the 230th day at the DE-Kli site, the 150th day at the IT-Bci site, and the 230th and 250th days at the FR-Gri site. There are TM data available before and after these days, and the trajectory of the estimated results was corrected in time after adding the observed data. On the 170th day at the DE-Kli site, the 140th day at the IT-Bci site, and the 245th day at the FR-Gri site, there were no TM data available to update the model prediction and only the background field trajectory could be used for forward predictions. The abrupt variation in surface albedo is difficult to show and the estimated results contain deviations. The jump is more severe during snowy periods (especially snowfall and melting), where the surface albedo deviation may be higher than 0.5 and the EnKF algorithm cannot be applied effectively without support from TM data. To operate the surface albedo data assimilation algorithm, it is necessary to secure more high-quality satellite data as input observations. With support from more observations, the data assimilation algorithm is better able to capture abrupt variations and produce accurate estimations.
