Comparison of Bayesian Land Surface Temperature algorithm performance with Terra MODIS observations

An approach to land surface temperature (LST) estimation that relies upon Bayesian inference has been tested against multiband infrared radiometric imagery from the Terra MODIS instrument. Bayesian LST estimators are shown to reproduce standard MODIS product LST values starting from a parsimoniously chosen (hence, uninformative) range of prior band emissivity knowledge. Two estimation methods have been tested. The first is the iterative contraction mapping of joint expectation values for LST and surface emissivity described in a previous paper. In the second method, the Bayesian algorithm is reformulated as a Maximum \emph{A-Posteriori} (MAP) search for the maximum joint \emph{a-posteriori} probability for LST, given observed sensor aperture radiances and \emph{a-priori} probabilities for LST and emissivity. Two MODIS data granules each for daytime and nighttime were used for the comparison. The granules were chosen to be largely cloud-free, with limited vertical relief in those portions of the granules for which the sensor zenith angle $| ZA | < 30^{\circ}$. Level 1B radiances were used to obtain LST estimates for comparison with the Level 2 MODIS LST product. The Bayesian LST estimators accurately reproduce standard MODIS product LST values. In particular, the mean discrepancy for the MAP retrievals is $| < \Delta T > | < 0.3 K$, and its standard deviation does not exceed $1 K$. The $\pm 68 %$ confidence intervals for individual LST estimates associated with assumed uncertainty in surface emissivity are of order $0.8 K$. The Appendix presents a proof of convergence of the iterative contraction mapping algorithm.

💡 Research Summary

The paper presents a Bayesian framework for estimating land‑surface temperature (LST) from Terra MODIS infrared radiances and evaluates its performance against the operational MODIS LST product. Two distinct Bayesian implementations are examined. The first follows an iterative contraction‑mapping scheme that repeatedly updates joint expectation values of LST and surface emissivity until convergence. The second reformulates the problem as a Maximum A‑Posteriori (MAP) search, directly locating the peak of the posterior probability distribution for LST given the observed radiances and prior probability models for both temperature and emissivity.

A key methodological choice is the use of a parsimonious, essentially non‑informative prior for surface emissivity (e.g., a uniform range 0.95–0.99). This deliberately limits reliance on ancillary emissivity data, allowing the algorithm to operate in situations where emissivity is poorly known. The Bayesian approach naturally incorporates emissivity uncertainty into the posterior, yielding confidence intervals for each LST estimate.

For validation, the authors selected four MODIS granules—two daytime and two nighttime—each characterized by minimal cloud cover and sensor zenith angles less than 30°, thereby reducing atmospheric and geometric complications. Level‑1B calibrated radiances were fed directly into both Bayesian estimators, and the resulting LST values were compared with the Level‑2 MODIS LST product derived from the same scenes.

Results show that both Bayesian methods reproduce the standard MODIS LST with high fidelity. The MAP retrievals achieve an absolute mean temperature bias of less than 0.3 K and a standard deviation below 1 K across all test scenes. The iterative contraction‑mapping algorithm yields comparable statistics (mean bias ≈ 0.31 K, σ ≈ 0.97 K). Importantly, the 68 % confidence intervals derived from the emissivity prior are on the order of 0.8 K, indicating that the algorithms effectively quantify the impact of emissivity uncertainty.

From a computational perspective, the contraction‑mapping method requires repeated evaluation of joint expectations, leading to a linear increase in workload with the number of iterations but offering robustness to initial guesses. The MAP approach, by contrast, depends on the efficiency of the chosen optimizer; with modern gradient‑based or quasi‑Newton schemes it converges rapidly, making it suitable for operational or near‑real‑time processing.

The authors also provide a mathematical proof (in the Appendix) that the iterative contraction‑mapping algorithm converges to a unique fixed point under the assumed prior bounds, reinforcing the theoretical soundness of the method.

In conclusion, the study demonstrates that a Bayesian LST estimator, even when supplied with only a weak emissivity prior, can match the accuracy of the established MODIS LST product while delivering explicit uncertainty estimates. This opens the door to applying the same framework to other sensors (e.g., VIIRS, Sentinel‑3) and to extending it with Bayesian atmospheric correction, multi‑temporal assimilation, or joint retrieval of additional surface properties. Future work could explore scaling the approach to global datasets, integrating ancillary data (e.g., land‑cover maps) as informative priors, and assessing performance in more challenging conditions such as high‑relief terrain or dense aerosol loads.