# A New Approach to Defining Uncertainties for MODIS Land Surface Temperature
## 1 Introduction  
Land surface temperature (LST) is a basic determinant of the terrestrial thermal behaviour, as it controls the effective radiating temperature of the Earth's surface. It is influenced by land/atmosphere boundary conditions and exerts control over the partitioning of energy into the component turbulent heat fluxes. There is now an archive of LST data records from satellites [1; 2; 3; 4; 5; 6] produced from both thermal infrared [7; 8; 9] and passive microwave instruments [10] on multiple low-earth orbit and geostationary observing platforms [11; 12; 13]. The LST for each pixel is the mean radiative skin temperature of an area of land resulting from the mean balance of solar heating and land-atmosphere cooling fluxes. Any inference of LST as measured from satellite is of limited value without the accompaniment of an uncertainty estimate. Indeed, all LST observations will have an associated uncertainty estimate, since effects such as atmospheric attenuation and variability of surface emissivities are not known to sufficient accuracy; and the appropriate examination of these uncertainties is a necessary step in developing an LST retrieval scheme. Furthermore, pixel-level uncertainty fields are becoming a mandatory requirement in provision of operational LST products, such as those from the Sea and Land Surface Temperature Radiometer(SLSTR) instrument on-board Sentinel-3 [1], the MOderate Resolution Imaging Spectroradiometer (MODIS) instrument on-board Terra and Aqua [14; 15], and the Spinning Enhanced Visible InfraRed Imager (SEVIRI) on-board Meteosat [16]. The Joint Committee for Guides in Metrology quote an uncertainty as a parameter associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measurand that is the value of the particular quantity to be measured [17]. Although often used interchangeably, here we use the term \"uncertainty\" rather than \"error\". The term \"error\" is defined as the measured value minus the true value of the measurand [17]. In most real-world situations the error is not known since the true value cannot be measured, whereas the uncertainty does not rely on a unique true value. An exception is when the measured value can be compared with a reference value of negligible uncertainty [18]. In simple terms, \"error\" is the \"wrongness\" of the measured value, and \"uncertainty\" is the \"doubt\" on the value of the measurement given an estimate of the error distribution [18]. It has been conventional to categorise uncertainties into two broad categories: random and systematic [19; 20]. Random uncertainties, or in other words uncertainties due to uncorrelated error effects, can be defined as \"the result of a measurement minus the mean that would result from an infinite number of measurements of the same measurand carried out under repeatability conditions\" [17] providing an estimate of the precision of the observations. Truly random uncertainties are uncorrelated, whereas pseudo-random uncertainties--which may appear to be randomly scattered over large spatial and temporal domains--may be correlated on either synoptic to seasonal scales or on scales greater than this. Systematic uncertainties, on the other hand, are defined as \"the mean that would result from an infinite number of measurements of the same measurand carried out under repeatability conditions minus a true value of the measurand\" [17]. All inputs to an LST algorithm contribute to the overall uncertainty of the retrieved value for each pixel. The uncertainty analysis should thus take into account the expected performance of the algorithm under varying surface and atmospheric conditions. Satellite-derived observations of LST for each pixel represent the aggregated surface temperatures of all the land cover features within the instantaneous field of view: for bare soil surfaces this corresponds to the soil surface temperature; whereas for dense vegetation this can be interpreted as the canopy surface temperature.