The Dunhuang murals, a significant part of Chinese heritage, have suffered deterioration primarily due to environmental and chemical factors, notably salt damage. This study proposes a sophisticated method that synergizes Fractional Order Differentiation (FOD) and Partial Least Squares Regression (PLSR) to accurately invert the phosphate content in the mural plaster of the Dunhuang paintings. The focal points of the research include: (1) To address the issue of information loss and reduced modeling precision caused by integer-order differentiation algorithms, the FOD method is employed for preprocessing spectral data. This approach ensures the fine spectral differences in the phosphate content of the mural plaster are precisely captured, (2) Utilizing PLSR, the study models the spectral bands identified at a significance level of 0.01 with measured conductivity values, thereby enabling the precise prediction of the phosphate content in the murals. The research outcomes reveal: (1) The FOD method can elucidate the nonlinear characteristics and variation patterns of the mural samples in the spectral data. As the order increases from zero to two, the number of spectral bands meeting the 0.01 significance test initially decreases and then increases. The highest absolute value of the positive correlation coefficient is observed at the 1.9 order, corresponding to the 2077 nm band, (2) For predicting the phosphate content in the murals, the model at the 1.9 order is most suitable for inversion. This model, after cross-validation, achieves a maximum R 2 value of 0.861. This study created an efficient FOD-based model for estimating phosphate in mural plaster.Ancient mural sites are historical annotations, with the murals of Dunhuang in China embodying the pinnacle of contemporary human civilization and its developmental achievements, holding immense historical significance [1][2][3][4][5] . Regrettably, due to the passage of time, these murals have endured environmental adversities, leading to fading, efflorescence, erosion by windblown sand, and mold infestation, among other detriments. The detachment of the mural's pigment layer from the Mural Plaster, and the plaster from the cliff support layer, has severely impacted both the aesthetic and historical value of these murals 6,7 .In recent years, the frequent occurrence of extreme climatic conditions has precipitated various problems in some representative cave murals, such as fissures, cracking, flaking, pulverization, and scratches. Among these challenges, the irreversible and enduring damage caused by climate change to the porosity and integrity of salt-affected murals is particularly concerning 8,9 . Hence, the implementation of preventative diagnostic techniques for salt damage in cultural artifacts equates to administering an essential mural health assessment, pivotal for their enduring preservation and heritage. Extensive research has shown that changes in the thermo-hydric environmental conditions are a significant cause of mural deterioration due to salt crystal...
