# A New Approach to Defining Uncertainties for MODIS Land Surface Temperature
## 1 Introduction
In sparse or mixed vegetation pixels, the satellite-observed LST is the average temperature of the sunlit and shadow components of the canopy, understorey, and the soil surface--which is influenced by both the satellite viewing geometry and the position of the sun at the time of observation. For global 1-km observations such variability within a pixel is difficult to assess, and as such, any uncertainty analysis is challenging for every component of the retrieval. Uncertainty information is critical for many of the applications utilising these datasets, a prime example being data assimilation. With LST being integral to the surface energy budget, satellite-derived LST data have been used as inputs into data assimilation routines [21; 22; 23; 24] to improve the estimate of model state and prognostic variables. The strengths of both satellite observations and model predictions are assessed given the respective weightings of the model and observation uncertainties enabling data assimilation to apply appropriate corrections to the model. The broad distinction between random and systematic uncertainties is critical since an underlying assumption is that the differences between the model and observations are unbiased [22]. Uncertainty budgets have been estimated for LST products in several previous studies [14; 15; 25; 26; 27] with some being applied in an operational capacity or in the generation of first climate data records for LST [28]. In general, these budgets have been developed based on how different conditions impact the retrieval and estimating these uncertainties on a pixel level within the retrieval process. While these are welcome steps forward in understanding the LST data, there has been no previous attempt to quantify the uncertainties into effects whose errors differ in terms of their correlation properties. This is important, since unless errors can be appropriately categorised by their correlation properties then propagation of uncertainties from native resolution to higher-level gridded products is not possible with any degree of accuracy. The approach we take in this study is a significant advance on what has generally been done for LST datasets to date, since we can demonstrate a method for propagating the LST pixel-level uncertainties to gridded products utilising the knowledge on the correlation properties of the errors. LST is a relatively new Essential Climate Variable (ECV) having been included for the first time in the 2016 Global Climate Observing System (GCOS) Implementation Plan [29]. Improving uncertainty estimation for the generation of LST CDRs is a key action for this ECV. Failing to account for the properties of error correlation can lead to uncertainty underestimation thus affecting the interpretation of the CDR [18]. Our study addresses this challenge ensuring an approach that can be used with more confidence in the generation of CDRs for LST. This paper introduces a comprehensive and consistent approach to determining an uncertainty budget for LST products consistent with advances recently made for sea surface temperature (SST) retrievals [30; 31]. In generating this uncertainty model a new LST product for MODIS has been produced. This differs both in the LST retrieved for MODIS compared with current operational products [5; 32] and in the estimation of the LST uncertainty. The algorithm itself and the framework of uncertainties are introduced in Section 2. There are numerous studies that have presented validation of MODIS LST [33; 34; 32], but none have evaluated the uncertainty budgets. In Section 3 we present validation of both LST and the uncertainty budget. Propagation of these uncertainties to averaged LST products are considered in Section 4. Finally, the overall product and uncertainty approach are discussed in Section 5, with limitations and further work postulated.