A model for inversion of hyperspectral characteristics of phosphate content in mural plaster based on fractional-order differential algorithm

Abstract

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 R² value of 0.861. This study created an efficient FOD-based model for estimating phosphate in mural plaster.

Introduction

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 crystallization^10,11. The deterioration of porous ancient Mural Plaster layer materials is primarily caused by two factors: one is the chemically erosive impact of saline solutions within the material on the building materials; the other is the cumulative effect of diurnal variations in temperature and humidity in the environment, leading to a recurrent crystallization-dissolution process. Among these, the latter is the predominant factor causing the expansion and fracturing of the plaster layer’s pores. Considering that phosphates can form dodecahydrate under specific hydrothermal conditions, this could lead to recurring crystallization-dissolution effects¹². The physical properties of phosphates make them one of the salts that pose a threat to cultural heritage, thus, a deeper understanding of this process is crucial for developing effective conservation measures. Among existing methods for salt content analysis in mural samples, Spectrometer Diagnostics stands out as a non-contact, non-destructive technique^13,14,15. In the analysis of mural salt content, the spectrometer provides critical insights into chemical composition, spatial distribution, concentration estimates, crystallization monitoring, environmental impacts, and non-destructive assessment, revealing these aspects through spectral reflection and absorption^{16,17,18,19,20,21}. Therefore, utilizing spectral data captured by the spectrometer and applying quantitative inversion models, the salt content within the Mural Plaster layer can be accurately diagnosed.

The application of hyperspectral remote sensing technology in predicting salt content, though promising, exhibits a degree of variability in accuracy as documented by sources^22,23,24. To enhance the sensitivity of spectral bands, mathematical differential transformations have been extensively adopted in the realm of spectral data mining. Presently, research on the salt content in Mural Plaster primarily revolves around constructing inversion models using integer-order differentiation and salt spectral indices^25,26,27. Nevertheless, it has been observed that integer-order differentiation (such as first and second-order differentiation) can introduce noise and compromise some useful information, thereby affecting the performance of estimation models²⁸. In contrast, fractional-order differentiation offers a more flexible processing framework capable of capturing finer spectral variations, particularly demonstrating superior performance in handling nonlinear and complex physical processes. Current research on the latest advancements and applications of fractional-order differentiation methods reveals that, with the development of computing technologies and advancements in data processing algorithms, an increasing number of studies are exploring the application of fractional-order differentiation in spectral analysis, especially in the quantitative remote sensing research of soil, vegetation, and other surface characteristics²⁹.

Guo et al.²⁵ amalgamated first and second-order derivatives with salinity indices to formulate a spectral inversion model for sulfates in mural plasters. Similarly, Zhong et al.³⁰ employed the reciprocal of first-order derivatives to identify bands with significantly enhanced spectral characteristics, thereby establishing a hyperspectral predictive model for soil nickel content. Upon comparing simple linear regression, multiple linear stepwise regression, and partial least squares regression (PLSR), Hou et al.³¹ ascertained that the second-order differential PLSR model attained the highest modeling precision and sensitivity within the 640–790 nm wavelength spectrum. Despite the simplicity and ease of implementation of integer-order differential methods, they may neglect crucial information when analyzing complex spectral data³². In contrast, fractional-order differentiation provides a more adaptable framework capable of detecting subtler spectral variations, particularly excelling in the analysis of nonlinear and intricate physical processes³³.Recent investigations into the latest technological advancements and applications of fractional-order differentiation methods reveal that with the evolution of computational technologies and the advancement of data processing algorithms²⁹,an increasing number of studies have commenced exploring the application of fractional-order differentiation in spectral analysis, particularly in the quantitative remote sensing research of soil, vegetation, and other surface characteristics. Although fractional-order differentiation holds theoretical advantages, determining the most suitable order of differentiation, managing uncertainties and complexities during computation, and integrating these methodologies with existing remote sensing data processing pipelines remain challenges that warrant further investigation and exploration.

Compared to integer-order differentials, fractional-order differentials describe more accurate models and are broadly applied in fields such as signal analysis, weather forecasting, image processing, biomedicine, fractal theory, and automatic control^{34,35,36,37,38,39}. However, it is only in recent years that scientists have applied fractional-order differential models to the preprocessing of soil spectral data. Hong et al.⁴⁰conducted an analysis of soil samples collected from the Han River Plain in Wuhan to investigate the impact of fractional-order differentiation on the predictive performance of Partial Least Squares Support Vector Machine (PLS-SVM) and Partial Least Squares Regression (PLSR) models in determining soil organic matter. They discovered that the highest predictive accuracy with PLS-SVM occurred at a fractional order of 1.25, indicating that finely tuned fractional adjustments can more precisely explore the correlations between spectral data and soil organic matter. However, the high computational complexity of PLS-SVM may necessitate additional computational resources, which could be a disadvantage under resource-limited conditions. In contrast, Wang et al.⁴¹ examined salinized soils within the Xinjiang Ebinur Wetland National Nature Reserve using a BP neural network model that integrates fractional-order differentiation with grey relational analysis. They noted that the model achieved its highest predictive accuracy at a fractional order of 1.2, highlighting the effectiveness of grey relational analysis in enhancing data processing capabilities, especially when dealing with complex or incomplete data sets. Nonetheless, the sensitivity of grey relational analysis to data noise and the complexity of parameter adjustment remain significant limitations. Further expanding the application of Fractional Order Derivative (FOD) techniques, Zhang et al.’s⁴² research in Northwest China involved fractional-order derivative preprocessing at intervals of 0.05 order, systematically analyzing the correlation between soil organic matter content and spectral data. Their findings suggest that FOD processing between 1.05 and 1.45 orders demonstrates the strongest correlation with soil organic matter prediction, offering novel perspectives and methodologies for soil science research. However, frequent adjustments of the fractional order may increase data processing complexity and redundancy. In summary, recent studies have found that fractional-order differential methods can more finely delineate the changes in the reflectance spectrum of Mural Plaster samples, uncovering more hidden information and enhancing the accuracy of inversion models.Based on the aforementioned analyses, scholars have extensively utilized the Fractional-Order Differential (FOD) method for the quantitative inversion of salt content in hyperspectral remote sensing, conducting considerable research and exploration. However, the majority of scholars typically resort to conventional integer-order differentials and spectral transformations when selecting salt-sensitive bands in ancient Mural Plaster. The potential of innovatively applying fractional-order differential methods to extract salt content features on the surface of salt-affected murals, more intricately delineating changes in the reflectance of ancient Mural Plaster, and thus mining more hidden information from spectral data to enhance the accuracy of inversion models, awaits validation through this empirical study.

This research applies the FOD method to reduce high-dimensional data and construct intelligent interpretation models for different phosphate concentrations in Dunhuang mural plaster samples. Firstly, a spectrometer is employed for non-contact, non-destructive detection of salt content in murals, circumventing the high costs, inefficiency, and need for onsite sampling inherent to traditional methods. Subsequently, the study leverages spectral technology to establish models that accurately reflect the phosphate content in murals. By processing Mural Plaster spectral data with FOD, the study analyzes the impact of varying phosphate concentrations on mural samples, selects sensitive bands through significance testing, and constructs an inversion model using Partial Least Squares Regression. Ultimately, a spectral inversion model for phosphate in Mural Plaster based on FOD is proposed, aiming to more precisely estimate the phosphate content in Mural Plaster, thus providing a scientific method and basis for the preservation of cultural relics.

Experiment area and sample preparation

Ancient murals were selected for study due to their significant historical value and characteristic salt damage phenomena. The research site is located at the Mogao Caves in Dunhuang City (situated approximately 25 km southeast of Dunhuang City, longitude: 94.662° E, latitude: 40.142° N). This region is marked by substantial annual temperature fluctuations , with an average temperature of 11.3 °C, peaking at 44 °C (primarily in July and August), and plummeting to as low as − 27 °C (mainly in November and December). The multi-year average relative humidity stands at 27%, with an annual average precipitation of 39.19 mm, increasing by approximately 2.746 mm every decade. July experiences the highest rainfall, while January has the least. Summer temperatures average around 32.5 °C at their highest and 17.5 °C at their lowest⁴³. The Mogao Caves area has recently witnessed an increasing trend in rainfall, coupled with a rise in the frequency of extreme precipitation events. In the last 29 years, seven instances of extreme rainfall have occurred, notably in 2011, 2019, and 2021, with precipitation reaching 38.6 mm, 31.3 mm, and 29.1 mm, respectively. These rainfall events have caused loosening, erosion, and rock falls at the Mogao Caves, simultaneously serving as the primary source of environmental water in the region⁴⁴. Figure 1 illustrates the typical types of salt damage found in the murals of the study area. Panel (a) depicts efflorescence, characterized by the crystallization of salts on the surface of the mural, colloquially known as “white frost”. Panel (b) presents the phenomenon of friable alkali, a state of looseness in the mural plaster layer caused by the action of soluble salts. Panel (c) reveals fissures, indicative of misalignment and cracking within the mural. Lastly, panel (d) shows crazing, which refers to the fine network of surface cracks on the mural.

Figure 1

Typical salt damage types found in mural paintings of the study area: (a) Salt efflorescence, (b) Alkaline effervescence, (c) Fissures, (d) Crazing.

The methodology for sample preparation is detailed in the study conducted by Bi⁴⁵, while the desalination process adheres to the GB/T50123-2019 standard. Initially, the materials used for sample fabrication, namely levigated clay, sand, and wheat straw, underwent desalination. The purpose of desalination was to prevent inherent salts in the raw materials from interfering with the experiment. The specific procedure involved: weighing a certain amount of dried and crushed soil samples that were sieved through a 0.56 mm mesh, mixing them with deionized water at a 1:5 mass ratio, and stirring clockwise for 0.5 h using a stirrer to ensure complete dissolution of the salts within the soil. After a 24-h settling period, the clear supernatant was removed with a syringe, leaving the soil residue, which was then stirred again with deionized water. This desalination process was repeated five times. The soil sample’s electrical conductivity was measured to convert ion content, and desalination was deemed complete when the total salt content in the soil did not exceed 0.1%. The desalinated soil samples were then dried, crushed, sieved, and stored for later use. The desalinated levigated clay, sand, and wheat straw were mixed in a mass ratio of 64:36:3, to which 20% of the total mass of distilled water was added. This was followed by the mold preparation of sample blocks. The uniformly mixed raw materials were spread into molds, leveled on the surface, and vibrated to expel excess air within the samples. Subsequently, the samples were dried in an oven at a temperature of 90 °C. The specimens were subjected to a drying process at 90 °C for a duration of two hours, aimed at achieving a state of maximal desiccation. This particular temperature was selected to substantially reduce the moisture content within the samples, thereby approximating a condition of complete dryness. The initial moisture content was controlled at 20% during the sample preparation process, as this moisture level in the mixture of levigated clay and sand is most suitable for application on cave cliff surfaces. Thus, it is possible that the moisture content of the actual Mural Plaster layers in the caves might also fluctuate around this value.

Research workflow overview

To elucidate our methodological approach comprehensively, Fig. 2 delineates the holistic workflow for establishing the predictive model based on the FOD method for mural plaster phosphate content using high-spectral feature inversion. This figure serves as a visual framework that systematically outlines each step from data acquisition to the final implementation of the model.

Figure 2

Workflow of the FOD-PLSR method.

Data source and methodology

Data collection

The aim of this experiment is for the non-contact diagnosis of salt content on the surface of Mural Plaster materials to be conducted using hyperspectral technology, and for a pertinent model to be developed accordingly. First, five sets of Mural Plaster samples with varying concentrations are to be designed and prepared (a total of 50 pieces), and their spectral data are to be collected. The initial step involved preparing five different concentrations of dodecahydrate disodium hydrogen phosphate solutions, specifically at 0.608 mol/L, 0.808 mol/L, 1.008 mol/L, 1.208 mol/L, and 1.408 mol/L. These solutions represented a gradient of erosive conditions from the lowest to the highest, sequentially termed as minimal concentration erosion, low concentration erosion, medium concentration erosion, high concentration erosion, and maximal concentration erosion. Temperature control was maintained using a THD-0506/1015 high-precision low-temperature constant temperature water bath. The mural plaster samples were immersion exposed to different concentrations of the dodecahydrate disodium hydrogen phosphate solution at 32.5 °C through capillary action, simulating the real-world capillary absorption effect of Mural Plaster from groundwater seepage. Subsequently, the samples were thoroughly air-dried at room temperature, and their surface spectral reflectance was measured using an ASD-FieldSpec4HI-RES terrestrial spectrometer.

The spectral analysis was conducted in Beijing at 6 p.m. in June 2023, with each session preceded by a white board calibration to eliminate dark current interference, maintaining a consistent height for the probe. Since the testing was performed in a laboratory, atmospheric effects on the results were negligible. To characterize the impact of indoor lighting on the spectral data, measurements were taken on the same sample block in both a well-lit room and a dark room. As indicated in Fig. 3, the spectral data within the wavelength range of 350–700 nm exhibited more pronounced noise in brightly lit environments. Consequently, to mitigate the influence of lighting on data collection, all tests were uniformly conducted in a dark room. Furthermore, to avoid interference from surface roughness and cracks on the sample blocks, measures were taken to distance the testing from potential sources of disruption. Data points were collected from five different locations on each sample’s surface and then averaged for analysis.

Figure 3

Difference in spectral data test between high light and dark room environments.

Moreover, in our experiments, we subjected the raw materials to a desalination process with the aim of eliminating any potential interference from the salts present in the materials. This was done to ensure that the changes induced by phosphates could be accurately reflected through conductivity measurements. The primary intention behind this was to employ conductivity as an indicator for evaluating the phosphate content in soil samples. Against this backdrop, we posited a hypothesis that variations in the phosphate content within the samples would be closely correlated with changes in conductivity. This perspective is corroborated by the literature. Specifically, Xiaoguang Zhang et al.⁴⁶ and Xiangyu Ge et al.⁴⁷have indicated in their research that the conductivity of soil solutions can effectively reflect the total salt content in the soil. Although these studies do not directly associate phosphate content with conductivity, they furnish compelling evidence that conductivity can indeed mirror the salt content in soil, thereby substantiating our hypothesis.

Statistical analysis of electrical conductivity data in mural plaster

The coefficient of variation is employed to articulate the degree of relative fluctuation or dispersion in conductivity measurement values.The electrical conductivity data of mural plaster collected under the five different conditions were statistically described using data density distribution, mean, standard deviation (SD), minimum value (Min), first quartile (Q1), third quartile (Q3), and coefficient of variation (CV). The first and third quartiles (Q1 and Q3) are indicated with red dashed lines, while the mean is denoted with a blue dashed line. OC1 to OC5 represent the five conditions ranging from minimal to maximal concentration erosion. The coefficient of variation (CV), also known as the dispersion coefficient, is a standardized measure of the dispersion of a probability distribution, defined as the ratio of the data’s standard deviation to its mean. The formula for calculating the coefficient of variation (CV) is as follows:

$$CV = \frac{\sigma }{u} \times 100\%$$

(1)

where CV denotes the Coefficient of Variation (expressed as a percentage), symbolized by σ, which represents the standard deviation of the sample (measured in ms m⁻¹).The symbol u corresponds to the mean value of the sample (measured in ms m⁻¹).Data is considered to have low concentration variability when CV ≤ 15%, moderate variability when 15% < CV ≤ 35%, and high variability when CV > 35%²⁸.

Spectral data preprocessing

Inevitably, the spectral data collected exhibit noise spikes, necessitating the application of smoothing preprocessing to ensure a more consistent progression of variations. Preprocessing of spectral data is crucial for diminishing high-frequency noise, outliers, as well as interference from internal, external spectral elements, and environmental factors. This enhances the accuracy and reliability of the spectral signals, which are pivotal for quantitative analysis, research, and the precision of the models developed^16,30. Initially, in processing the Mural Plaster spectral reflectance data, we removed the data from the 350–399 nm and 2401–2500 nm bands, which typically suffer from significant noise interference due to equipment limitations and detector sensitivity deficiencies²⁹. Moreover, our dataset exhibited spectral overlaps in the 710–805 nm and 1450–1800 nm bands. Despite attempts to mitigate these overlaps using Savitzky–Golay filtering and moving average techniques, these efforts were not wholly successful. Given the potential adverse impact of such spectral overlaps on the accuracy of our analysis, we ultimately decided to eliminate these specific data segments. Following this, the Savitzky–Golay filter was applied to smooth the reflectance spectrum data of 50 simulated plaster layer samples, utilizing 21 window points and a second-order polynomial⁴⁸.

$$y_{i}^{\prime } = \frac{1}{\Delta }\sum\limits_{j = - m}^{m} {c_{j} } \cdot y_{i + j}$$

(2)

where $y_{i}^{\prime }$ represents the smoothed reflectance spectrum at position i,with $y_{i + j}$ indicating the points in the original data series,and $c_{j}$ denoting the coefficients in the convolution kernel.These coefficients are ascertained through polynomial fitting.The variable m is half the size of the window, $\Delta$ typically 1, unless the intervals between data points are non-uniform.

Research methodology

Fractional order differentiation

Fractional Order Differentiation (FOD) represents an extension of traditional integer-order differentiation and manifests in various forms within the realms of mathematics and engineering⁴⁹, such as Riemann–Liouville (R–L), Lévy, Weyl, Caputo, and Grünwald–Letnikov (G–L)⁵⁰.The Grünwald–Letnikov approach, due to its proficiency in handling discrete data, finds extensive application in the fields of engineering and science. A principal advantage of the Grünwald–Letnikov method lies in its foundational basis on the difference definition of differentiation, which facilitates its direct and convenient application in numerical computations, especially pertinent when dealing with discrete data such as spectral signals²⁹.Consequently, pursuant to the Grünwald–Letnikov algorithm, the fractional order (v) of f(x) within the wavelength domain [a, t] can be delineated as depicted in Eq. (3). $\Gamma$ denotes the Gamma function $\left( {\Gamma \left( v \right) = \left( {v - 1} \right)!} \right)$.

$$d^{v} f(x) = \mathop {\lim }\limits_{h \to 0} \frac{1}{{ h^{v} }}\sum\limits_{n = 0}^{[(t - a)/h]} {( - 1)^{n} } \frac{\Gamma (v + 1)}{{n!\Gamma (v - n + 1)}}f\left( {x - nh} \right)$$

(3)

where v is the fractional order, h is the step size, t and a is the upper and lower bounds of fractional order differentiation.

In this experiment, assuming the function f(x) as a one-dimensional spectral signal with a band range of [a, t], where x ∈ [a, t] divided by the differential step length h. Given that the ASD FieldSpec® 4 Hi-Res Spectrometer’s retention interval is 1 nm, the differential step length can be set to h = 1. Consequently, the expression for the v-th order fractional differentiation of the function f(x) can be derived from Eq. (3) as follows:

$$\begin{aligned} \frac{{d^{v} f(x)}}{{dx^{v} }} & \approx f(x) + ( - v)f(x - 1) + \frac{( - v)( - v + 1)}{2}f(x - 2) \\ & \quad + \frac{( - v)( - v + 1)( - v + 2)}{6}f(x - 3) + \cdots + \frac{\Gamma ( - v + 1)}{{n!\Gamma ( - v + n + 1)}}f(x - n) \\ \end{aligned}$$

(4)

where ν is the the fractional order, x is the wavelength.

In our study, the domain of v spans from 0 to 2, with increments of 0.1. When v equals 0, the spectrum corresponds to the original, undifferentiated spectrum, while the calculations of the first and second derivative spectra are executed at v values of 1 and 2, respectively.

Significance test

The p-value is employed to ascertain the statistical significance of the correlation between two variables⁵¹. Herein, the Pearson correlation coefficient r serves as a metric to gauge the linear relationship between absorption intensity (reflectance at specific wavelengths) and the phosphate concentration in mural plaster. By calculating the Pearson correlation coefficient r from the sample data and converting it into a t-statistic, we can quantitatively evaluate the relationship between hyperspectral features and phosphate concentrations. This procedure involves determining the degrees of freedom for the t-distribution, typically the number of samples minus two, to account for the parameters estimated when deriving r. Subsequently, the p-value is computed using the cumulative distribution function (CDF) of the t-distribution to gauge the probability of observing the given correlation under the null hypothesis of no correlation. Should the p-value fall below a predetermined threshold of significance (p < 0.01), it would indicate that the correlation between the hyperspectral features and phosphate concentration is statistically significant, thereby affirming that spectral characteristics can serve as a reliable indicator for inferring phosphate concentrations.

$$t = \frac{{r\sqrt {n - 2} }}{{\sqrt {1 - r^{2} } }}$$

(5)

$$p - {\text{value}} = 2 \times \left( {1 - {\text{CDF}} \left( {t,{\text{df}} = n - 2} \right)} \right)$$

(6)

where n denotes the number of data points, r represents the Pearson correlation coefficient, t is the transformed t-statistic, a metric predicated on the assumption that the data is derived from a normal distribution, This statistic adheres to a t-distribution with n − 2 degrees of freedom under the null hypothesis of no correlation. The function CDF (t, df = n − 2) is the cumulative distribution function (CDF), which yields the probability that the t-statistic under a t-distribution is less than or equal to the observed t value. The p-value is a probabilistic measure utilized to ascertain statistical significance. The term df = n − 2 corresponds to the number of data points available for estimation, while CDF denotes the cumulative distribution function for the t-distribution.

Partial least squares regression

In this study, the objective is to establish an inversion model to explore the relationship between conductivity of Mural Plaster and spectrally significant bands using machine learning method.A Partial Least Squares regression model was employed to construct the model as delineated in Eqs. (7) and (8)⁵².

Initially, a compendium of fifty spectral data sets was garnered from mural plasters subject to a spectrum of erosive concentrations. These data underwent fractional-order differentiation processing within the range of 0.1–2.0 orders. Subsequently, the correlation coefficients between the original spectral data at 0.0 order and the post-fractional-order differentiated data with the electrical conductivity were calculated. Wavelengths that achieved a significance level of 0.01 were identified. Utilizing these discriminant wavelengths as predictors and the mural plaster’s conductivity as the response variable, a Partial Least Squares Regression (PLSR) model was formulated. To enhance the precision of the model, the data underwent standardization and mean-centering pre-processing steps. The model’s architecture was underpinned by the Kernel PLS algorithm, setting the ceiling for principal components at seven. Optimal numbers of components were ascertained through stochastic cross-validation, and specific model alert parameters were established (thresholds for calibration and validation residual variance ratios were set at 0.5 and 0.75 respectively, with a cap on residual variance increase fixed at 6%) to assure the model’s steadfastness and dependability.

$$X = T \times P^{T} + E$$

(7)

$$Y = U \times Q^{T} + F$$

(8)

where X represents the matrix of spectrally significant bands, i.e., the matrix of explanatory variables; Y denotes the electrical conductivity of Mural Plaster, the matrix of response variables. T and U are the score matrices for explanatory and response variables, respectively, while P and Q represent the loading matrices for these variables. E and F are the residual matrices, signifying the aspects of the model that remain unexplained.

Evaluation

This paper employs accuracy assessment metrics to evaluate the efficacy of the predictive modeling^28,30,53. The computation formulas for R², RMSE (Root Mean Square Error), and MAE (Mean Absolute Error) are delineated as follows in Eqs. (9)–(11).

$${\text{R}}^{2} = \left( {\frac{{\sum\limits_{i = 1}^{{\text{n}}} {\left( {{\text{y}}_{{\text{i}}} - \overline{{\text{y}}} } \right)} \left( {{\text{z}}_{{\text{i}}} - \overline{{\text{z}}} } \right)}}{{\sqrt {\sum\limits_{{{\text{i}} = 1}}^{{\text{n}}} {\left( {{\text{y}}_{{\text{i}}} - \overline{{\text{y}}} } \right)^{2} } \cdot \sum\limits_{i = 1}^{{\text{n}}} {\left( {{\text{z}}_{{\text{i}}} - \overline{{\text{z}}} } \right)^{2} } } }}} \right)^{2}$$

(9)

$$RMSE = \sqrt {\frac{{\sum\limits_{{{\text{i}} = 1}}^{n} {\mathop {\left( {\mathop y\nolimits_{i} - \mathop z\nolimits_{i} } \right)}\nolimits^{2} } }}{n}}$$

(10)

$$MAE = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left| {\mathop y\nolimits_{i} - \mathop {\mathop y\nolimits_{i} }\limits^{ \wedge } } \right|}$$

(11)

where n represents the number of mural plaster samples; $y_{i}$ Represents the EC measurement value of the i-th mural plaster sample ;$\overline{y }$ Represents the average of the measured values of all mural plaster samples; $\widehat{{{\text{y}}_{{\text{i}}} }}$ represents the predicted value for the i-th sample of Mural Plaster. $z_{i}$ Represents the predicted EC value of the i-th mural plaster sample; $\overline{z}$ Represents the average EC predicted value of all mural floor samples;The performance metrics of the model encompass the coefficient of determination for the calibration dataset, denoted by $R_{c}^{2}$,the coefficient of determination for the validation dataset, represented by $RMSE_{c}$, the coefficient of determination of the validation data set is expressed as $R_{v}^{2}$, the root mean square error for the validation dataset, articulated as $RMSE_{v}$,and the mean absolute error, delineated as MAE. Among them, $R^{2}$ is used to evaluate the model fitting degree.The closer the value is to 1, the higher the model accuracy.RMSE and MAE are used to evaluate the stability of the model.The closer the value is to 0, the better the RMSE and MAE are.

Results

Statistical description of mural plaster electrical conductivity

The conductivity values of the Mural Plaster samples are a direct reflection of the variations in salt content within the same material, exhibiting a certain degree of variability. As illustrated in Fig. 4, the range of the samples spans from 2.27 to 4.86 ms m⁻¹.The average and standard deviation of the sample’s Electrical Conductivity (EC) values are 3.50 ± 0.75 ms m⁻¹. The data’s first quartile (Q1) and third quartile (Q3) are respectively 2.96 ms m⁻¹ and 4.14 ms m⁻¹, indicating that the EC values for 50% of the samples are encompassed between these two figures. Additionally, the coefficient of variation (CV) is 22%, signifying that the relative variability of the EC values is at a moderate level.

Figure 4

Statistical description of the electrical conductivity (EC) values in mural plaster.

Hyperspectral characteristics of simulated mural plaster under different salt concentration erosion

The spectral reflectance characteristics of salt-laden mural samples are determined by a confluence of factors including the composition of the materials, salt concentration, and moisture content. The extraction of characteristic bands supports the inversion of salt content^54,55. As depicted in Fig. 5, the colored spectral data represent the average reflectance spectrum of the mural plaster samples, while the grey shading indicates the standard deviation. The spectral data maintain a consistent shape and trend across samples with varying salt concentrations, indicating spectral similarity among the samples. However, the levels of reflectance exhibit discernible differences, reflecting the impact of salt concentration on reflectance spectrum properties.

Figure 5

Spectral data of mural samples under different conditions.

Observations in Fig. 5 reveal that within the 400–710 nm range (visible light spectrum), the reflectance spectrum exhibits a rapid increase with wavelength, where the spectral data demonstrates a steep slope. Between 805 and 1400 nm, reflectance continues to rise with increasing wavelength, albeit at a diminished rate. Despite fluctuations within the 2200–2400 nm range (a larger portion of the shortwave infrared spectrum), the overall trend is a decline. When investigating the impact of varying salt concentrations on the reflectance spectrum of mural samples, it is noted that reflectance typically decreases before increasing with higher salt concentrations, aligning with the findings of Guo et al²⁵. However, under conditions of maximum salt concentration, a significant change occurs with a notable decrease in reflectance spectrum, potentially due to two factors. Firstly, at maximum concentration, the deterioration of mural samples is most severe, increasing porosity and surface roughness, thereby enhancing the sample’s water absorption⁵⁶, changes that may not be as evident at medium or low concentrations. Secondly, in natural environments, phosphates readily absorb surrounding moisture, increasing the sample’s humidity. Generally, an increase in soil humidity leads to a reduction in soil reflectance, especially noticeable at lower humidity levels⁵⁷. Hence, the observed decrease in reflectance under high concentration conditions might be attributed to increased sample humidity. Furthermore, asymmetric absorption valleys near 1420 nm and 1940 nm, with the latter being more pronounced in depth and width, are observed. The former is due to specific vibrational absorption characteristics of water in the sample at 1420 nm²⁹, where the presence of phosphates might indirectly influence this absorption valley by affecting the soil’s moisture adsorption and retention properties. The latter might be attributed to the typical absorption characteristics of phosphates in the shortwave infrared region (SWIR), related to the vibrational modes of the phosphate oxygen bonds⁵⁸.

Fractional order differentiation results of spectral data

Given the rich high-dimensional information in spectral data and the difficulty in capturing sensitive bands and features⁵⁹, the fractional order differentiation (FOD) calculation method detailed in the Research Methodology chapter under the subsection Fractional Order Differentiation was utilized on the samples. This method allows for the control of differential step length, thereby enhancing the accuracy of salt content detection. Following the methodology proposed by Jie Zhang et al.²⁸, the interval and step length were set at [0–2] and 0.1, respectively. Figure 6 illustrates the average spectrum of the Mural Plaster samples.

Figure 6

Average fractional-order derivative spectra of mural plaster samples. The range of orders spans from 0 to 2, with a step size of 0.1. Colored spectral data represent the average spectra of mural plaster samples, while the gray shadow indicates their standard deviation.

The results indicate that following the application of Fractional Order Derivative (FOD) processing, three distinct hygroscopic valleys were identified near 1400 nm, 1900 nm, and 2200 nm, which are recognized as water absorption characteristics²⁹. Although the mural plaster samples subjected to phosphate degradation experiments were naturally air-dried, the experimental process increased the porosity and surface roughness of the mural samples, thereby enhancing their water absorption capacity⁵⁶. Additionally, phosphates tend to absorb moisture from their surroundings under natural conditions, thus increasing the humidity of the samples. Consequently, these wavelengths associated with moisture absorption may contain information pertinent to phosphates.

As the order of differentiation increases, a significant reduction in the morphology of the spectral data is observed. At 0.5 order, the reflectance across the entire spectral range fluctuates around zero, beginning to exhibit negative values; by 1.0 order, the reflectance across the full spectrum has fallen below 0.005. Beyond 0.6 order differential spectra, numerous observable fluctuations in the spectral data emerge, amplifying the differences between spectra. However, further increases in order do not allow this differentiation to expand indefinitely; the trend slows down after 1.1 order, and post-1.4 order, the spectral data become blurred and overlapped, indicating that at higher orders of differentiation, the distinctiveness of the spectral data is obscured, and clear differences between them are lost. In the development of the phosphate content inversion model for mural plaster, employing fractional-order differentiation effectively broadens the differences between spectral data and enhances spectral feature information, thereby improving the model’s accuracy and robustness. This can be explained by the mathematical theory of the G–L method. Given the presence of peaks and valleys of certain widths in the reflectance, when the sampling step is smaller than these features, the differences are amplified during the calculation, thus enhancing spectral information. Inevitably, differential operations can introduce noise significantly different from adjacent bands or further amplify short-interval reflectance peaks and valleys, thereby introducing high-frequency noise²⁸.

After subjecting the mural plaster samples to phosphate erosion experiments at varying concentrations, changes in salt concentration, moisture content, porosity, and surface roughness were observed to varying degrees. In Fig. 6, aside from the three absorption valleys characterized by water absorption properties, several significant reflective peaks emerge around 400 nm, 870 nm, 1000 nm, 2050 nm, and 2400 nm as the order increases, becoming more pronounced in high-order differential spectra. This characteristic may relate to the phosphate or organic matter composition within the mural plaster samples⁶⁰. Simultaneously, the width of the absorption peaks and valleys narrows, and the features of specific substances become less pronounced relative to lower-order FOD. Between 805 and 1300 nm, reflectance is generally low, which may suggest interactive effects between phosphates and certain minerals or organic materials within the mural plaster samples⁶¹. This also demonstrates that differential spectroscopy effectively integrates information from the original spectrum where the red edge position is not prominent. Comparing the differences between first and second-order derivatives reveals that the use of integer-order differentiation alone often leads to significant information loss²⁸. Compared to the original spectrum, the FOD-processed spectrum more finely delineates spectral changes, enhancing the resolution of spectral data and sharpening the spectral absorption valleys.

The results demonstrate that, firstly, as the order transitions from 0 to 2, the differences in reflectance spectrum increase, with the differential values gradually approaching zero. This phenomenon can be elucidated through the mathematical theories of the G–L method. Due to the presence of peaks and valleys of certain widths in the reflectance, when the sampling step is smaller than these widths, such differences are amplified during the computation, thereby enhancing the spectral information. Inevitably, differential operations introduce substantial noise, especially when adjacent frequency bands vary greatly, or short intervals of reflective peaks and valleys are again magnified, leading to high-frequency noise intrusion²⁸.

The correlation between fractional order derivative (FOD) results and salt content

In this section, we employ Fractional Order Derivative (FOD) technology to analyze the correlation between phosphate concentration in Mural Plaster and spectral data. The fundamental objective of this method is to demonstrate the unique advantages of FOD over traditional integer-order derivatives in processing such data²⁸. By meticulously annotating the maximum absolute values of the correlation coefficients and their corresponding wavelengths under different orders, we not only clearly identify the efficiency of FOD in data analysis but also lay a significant foundation for subsequent data modeling endeavors. Overall, the primary contribution of this study lies in empirically validating the potential of FOD to enhance the precision of spectral data analysis, offering a novel perspective and methodology for research in related fields.

The optimal correlations between the fractional order differential spectra and the phosphate content in Mural Plaster, along with their corresponding spectral bands, are comprehensively displayed in Table 1. Notably, in the range of 0.1–0.7 order differential, the best correlation coefficients are positive, with the corresponding wavelength being near 408 nm. In the range of 0.8–1.6 order differential, the optimal correlation coefficients are negative, with respective wavelengths at 874 nm and 1054 nm. For the 1.8–2.0 order differential, the best correlation coefficients are positive again, with the highest positive correlation coefficient at the 1.9 order, corresponding to a wavelength of 2077 nm.

Table 1 Optimal correlation between the fractional order differential spectra and phosphate content in mural plaster, along with corresponding spectral bands.

The correlation coefficient can intuitively reflect the linear correlation between reflectance spectrum and the phosphate content of Mural Plaster; a higher correlation signifies a more sensitive spectral response, which in turn suggests more accurate predictive modeling outcomes. The correlation coefficients of the various orders of differential spectra and the severity of the salt damage are as depicted in Fig. 7. Figure 7 elucidates the significant differences in the correlation between the differential spectra of varying orders and the phosphate content of the Mural Plaster. Within these differential spectra of different orders, one can observe distinct patterns of correlation with the phosphate content of the Mural Plaster. Initially, it is noticeable that the maximum absolute values of the correlation coefficients fluctuate minimally and follow a similar trend from 0.1 to 0.7 order. In the visible light region (approximately 400–700 nm) and the near-infrared region (805–1450 nm and 1800–2400 nm), the correlation coefficients transition from positive to negative as the order increases. Furthermore, the adjacent bands of the correlation coefficients exhibit considerable volatility, becoming increasingly chaotic, which indicates that higher-order differentials capture more complex non-linear relationships. Lastly, the Mural Plaster phosphate has several significant wavelength ranges: 400–490 nm, 840–890 nm, 1150–1170 nm, 1380–1390 nm, 2060–2090 nm, 2190–2210 nm, and 2240–2290 nm. As the order incrementally rises from zero to first order, the overall trend of the number of bands that satisfy the 0.01 significance level test under spectral transformation is to initially decrease and then increase with the order. According to Table 1, the maximum absolute value of the positive correlation coefficient occurs at the 1.9 order, with the corresponding band at 2077 nm. In our study, the selection of characteristic bands was predicated upon the magnitude of the absolute values of the Pearson correlation coefficients. Specifically, we initially computed the correlation coefficients between the phosphate concentrations in the mural plaster and the absorption intensities across various bands. Subsequently, we ranked these coefficients in descending order based on their absolute values. Its first six characteristic bands (2077, 849, 874, 2063, 2064, and 402 nm) exhibit high consistency with the advantageous wavelength ranges of the Mural Plaster phosphate. In contrast, the characteristic bands extracted at integer orders (1.0 and 2.0 orders), although partially coinciding at 1.0 order (874, 847, 848, 854, 862, and 875 nm), differ significantly at 2.0 order (2077, 586, 819, 849, 1373, and 2063 nm).This suggests that fractional-order differentiation is more effective in capturing spectral characteristics related to the phosphate content of Mural Plaster.

Figure 7

Correlation coefficients between differential spectra at different orders and the phosphate content in Mural Plaster.

Table 2 elaborates in detail on the sensitive bands for phosphate in mural plaster, identified based on the Fractional Order Derivative (FOD) spectral selection (p < 0.01). This further examination of the spectral data, post fractional order differentiation, reveals the impact on the correlation with the salt content in Mural Plaster.

Table 2 Delineates the sensitive spectral bands for phosphate in mural plaster, identified through fractional order derivative (FOD) spectral selection, with a significance level of p < 0.01.

In conclusion, when establishing a spectral inversion model for mural plaster phosphate, 1.9 order differential spectral analysis may be the optimal choice. However, to ensure the accuracy and reliability of the model, it is still necessary to conduct a comprehensive comparison, considering the results of different orders of differential spectral analysis, as well as their capability in revealing the relationship between reflectance spectrum and plaster phosphate content.

Model construction of mural plaster phosphate monitoring from feature bands extracted by the FOD algorithm

To validate the efficacy of fractional order differential processing in spectral monitoring of phosphate in Mural Plaster, this study, referencing²⁸, conducted a spectral analysis of both the original reflectance spectra and the 0.1–2.0 order differential spectra (with an interval of 0.1 order). This analysis was performed to construct a phosphate monitoring model for Mural Plaster. The study utilized a feature selection method based on FOD band analysis and a significance test based on correlation coefficients, selecting specific bands as predictive variables for the model.

In exploring the performance of the Partial Least Squares Regression model (FOD-PLSR) for mural phosphates under varying concentration erosion conditions, we utilized significance analysis (p < 0.01) to select the number of spectral bands for model construction. This section computes the Mean Absolute Error (MAE), R-squared (R²), and Root Mean Square Error (RMSE) for both the calibration and validation datasets (as shown in Table 3). From Table 3, we can discern a comparison of the model’s accuracy under integer order (first and second order differential spectra) and fractional order differentiation.The units for RMSE and MAE are ms m⁻¹.

Table 3 Monitoring models and validation results for phosphate content in mural plaster.

As Table 3 illustrates, the model manifests distinct performance characteristics at different orders of differentiation. Notably, from the perspective of integer-order differentials, the second-order differential spectral model exhibits a superior coefficient of determination (R² value) compared to the first-order differential spectrum, achieving the peak value of 0.751 within the validation dataset. This outcome accentuates the relative superiority of the second-order differential spectrum in predicting phosphate content over its first-order counterpart.

In the analysis employing fractional-order differentiation, the performance of the model exhibited variations with increasing orders of differentiation. It is noteworthy that at the 1.9 order of differentiation, the model attained an optimal R² value of 0.815. This value not only surpasses the performance of the first and second integer-order differential models but also exceeds that of the original spectral data. This finding corroborates the efficacy of fractional-order differentiation in refining data processing, particularly at the 1.9 order, where the model demonstrated exemplary performance within the validation dataset.

Further, the 1.9 order differential spectrum demonstrates exemplary precision and stability, as evidenced by the Root Mean Square Error (RMSE) of 0.208 and the Mean Absolute Error (MAE) of 0.167 within the calibration dataset. Within the validation dataset, these metrics are 0.337 and 0.279, respectively. These results underscore the superior performance of the 1.9 order differential spectrum model in terms of predictive accuracy and consistency.

Figure 8 showcases the optimal mural plaster phosphate monitoring models established through the selection of sensitive bands based on Fractional Order Derivative (FOD) spectra, with models at (a) 1.0 order, (b) 2.0 order, (c) 1.5 order, and (d) 1.9 order. The color bars within the figure represent the density of the scatter points, achieved through the calculation of Gaussian Kernel Density Estimation (KDE), where the depth of color signifies varying density levels: darker regions denote a higher concentration of data points, indicating greater density, whereas lighter areas suggest fewer data points and lower density. A comparative analysis of model performance reveals that the 1.9 order differential spectral model demonstrates supreme efficacy within the calibration dataset, boasting an R² value of 0.922 and an RMSE of 0.208, and it also excels in the validation dataset with an R² value of 0.815 and an RMSE of 0.337, underscoring its high accuracy and reliability in monitoring phosphate levels in mural plaster. Compared with the first-order and second-order differential models, the second-order model exhibits a slightly lower fit in the calibration set (R² = 0.913) but shows a marked decline in performance in the validation set. In contrast, the first-order model, while achieving an R² of 0.879 in the calibration set, undergoes the most significant performance degradation in the validation set, where R² drops to 0.730. This suggests that when selecting the most appropriate differential order for monitoring phosphate in Mural Plaster, models with excessively high orders may overfit the calibration dataset, thereby compromising their generalizability. In comparison, the 1.5-order differential model, despite slightly inferior performance in the calibration set compared to the 1.9-order, offers a more balanced performance across calibration and validation datasets (R² of 0.917 and 0.792, RMSE of 0.211 and 0.333), making it a promising choice. Considering all factors, the 1.9-order differential model is deemed optimal for monitoring phosphates in Mural Plaster due to its higher accuracy and stability in the validation set.

Figure 8

Optimal mural plaster phosphate monitoring models established based on FOD spectral band selection (drawing upon data from the validation set.): (a) 1.0 order, (b) 2.0 order, (c) 1.5 order, (d) 1.9 order.

Discussion

Performance analysis of fractional order differential method in monitoring phosphate concentration in mural plaster

Integer-order differentiation is a commonly employed method for preprocessing spectral data, with numerous studies having utilized first and second-order differentials to identify sensitive bands for mural salt content and to construct their inversion models, demonstrating its potential applicability²⁵. However, it has also been observed that integer-order differentiation (such as first and second-order differentials) can introduce noise and compromise valuable information, thereby adversely affecting the performance of estimation models²⁸.Furthermore, under the influence of salt damage, the surface and internal structure of the mural plaster undergo significant alterations due to the effects of salinity. The variations in porosity and surface roughness are directly correlated with the extent of salt damage, which markedly impacts the collection and analysis of spectral data⁶². In consideration of the impact of salt damage on mural plaster, particularly its effect on porosity and surface roughness, this study simulates environments of varying salt concentrations to obtain samples with distinct physical properties. Subsequently, spectral data are collected and analyzed to investigate the efficacy of fractional-order differentiation in monitoring phosphate concentrations within mural plaster samples.

Building upon this premise, the present study endeavors to examine the effectiveness of the fractional-order differentiation approach in monitoring phosphate concentrations within mural plaster samples, aiming for a more systematic analysis. By employing Fractional Order Differentiation (FOD) techniques alongside Partial Least Squares Regression models, the study predicts the phosphate content in mural plaster. Spectral preprocessing techniques are paramount for the precise interpretation of spectral data, enhancing beneficial spectral attributes while mitigating interference caused by environmental variability, instrument noise, and the inherent heterogeneity of samples, thereby laying a solid foundation for in-depth spectral analysis and interpretation^63,64.In recent years, fractional-order differentiation (FOD) has demonstrated its unique efficacy in spectral analysis, distinguishing itself from traditional spectral preprocessing techniques such as reciprocal transformation, derivative transformation, and logarithmic reciprocal, among others^28,53,59. The research findings reveal that as the order increases to 0.5, the reflectance corresponding to the entire spectral range exhibits fluctuations around zero, with the onset of negative values; by an order of 1.0, the reflectance across the full spectrum has already diminished to below 0.005. Following the 0.6 order differential spectrum, pronounced undulations in the spectral data become readily observable, accentuating the disparities between different spectral lines. Nonetheless, an escalation in order does not facilitate further expansion of these discrepancies; the trend begins to slow progressively after 1.1 order, and by 1.4 order, the contours of the spectral data become indistinct and overlaid, indicating that at higher orders of differentiation, the delineation of spectral data contours is gradually obfuscated, erasing the clear demarcations between them. These observations align with trends identified in existing literature, though variances may occur in specific details^29,65.In the investigation of the trends between FOD fractional-order results and the correlation with salt content, it was discovered that at the 1.1 order of differentiation, the correlation coefficient between the spectral data and the phosphate content in mural plaster reached its zenith at − 0.572, marking an enhancement of 1.24% and 7.12% over the first and second-order differential spectra, respectively. This outcome underscores the potential of FOD technology in augmenting the correlation between spectral data and target variables, particularly when an apt order of differentiation is selected. As observed in Fig. 7, several advantageous wavelength ranges for mural plaster phosphate include: 400–490 nm, 790–820 nm, 840–890 nm, 1150–1170 nm, 2060–2090 nm, 2190–2210 nm, and 2240–2290 nm.As the order progressively increases from zero to one, there is an initial decrease followed by an increase in the number of spectral bands that meet the 0.01 level of statistical significance under spectral transformation. This phenomenon finds resonance in similar observations made in other studies, such as the one conducted by Zhang Jie et al²⁸.According to Table 1, the highest absolute value of the positive correlation coefficient is observed at the 1.9 order, with the corresponding band at 2077 nm. The top six characteristic bands (2077, 849, 874, 2063, 2064, and 402 nm) exhibit a high degree of consistency with the dominant wavelength range associated with phosphate in mural plaster. In contrast, the feature bands extracted at integer orders (1.0 and 2.0 orders), while partially aligning at the 1.0 order (874, 847, 848, 854, 862, and 875 nm), display significant discrepancies from the dominant wavelengths at the 2.0 order (2077, 586, 819, 849, 1373, and 2063 nm). This suggests that fractional-order differentiation is more effective in capturing spectral features correlated with phosphate content in mural plaster.

In summary, when establishing a spectral inversion model for phosphate in mural plaster, the 1.9 order differential spectral analysis may be the optimal choice. However, to ensure the accuracy and reliability of the model, it is imperative to conduct a comprehensive comparison, taking into account the results of differential spectral analyses at various orders and their capabilities in elucidating the relationship between reflectance spectrum and plaster phosphate content.

Predictive performance and application of the hyperspectral feature inversion model for phosphate content in mural plaster

In this study, we delved further into the potential of fractional-order differentiation techniques to enhance the accuracy of inversion models for phosphate content in mural plaster, by employing Partial Least Squares Regression (PLSR) models to establish quantitative inversion models for phosphate content in mural plaster. PLSR is adept not only at handling nonlinear relationships but also at more effectively identifying and utilizing correlations between variables, thereby augmenting the explanatory power and predictive accuracy of the models^59,66,67. This enables PLSR to more precisely identify and quantify the characteristics of salt damage in mural applications, thereby enhancing the accuracy and efficiency of monitoring efforts. The results indicate that the phosphate content prediction model constructed after 1.9 order FOD processing of spectral data exhibits the best performance, with evaluation metrics of R² = 0.815, RMSE = 0.337, and MAE = 0.279. The model established following first-order differential processing yielded metrics of R² = 0.730, RMSE = 0.371, and MAE = 0.316, while the model resulting from second-order differential processing showed metrics of R² = 0.751, RMSE = 0.363, and MAE = 0.301. These outcomes suggest that the spectral data processing approach provided by FOD demonstrates a clear advantage in this application scenario compared to traditional integer-order differential methods. Previous studies have also affirmed the effectiveness of Fractional Order Differentiation (FOD) in predicting soil salinity and related parameters^47,53,68.

Moreover, it is noteworthy that although the 1.9 order differentiation did not yield the highest number of bands sensitive to phosphate in mural plaster, the model constructed from these bands exhibited the best performance. This could be attributed to the selected bands being more critical and representative. Another possible explanation is that Fractional Order Differentiation (FOD) can capture nonlinear relationships between variables, unveiling the nonlinear relationship between phosphate content and spectral characteristics⁶⁹. Similar research outcomes have been validated in the studies conducted by Zhang Jie et al²⁸.

In summary, the model processed with 1.9 order differentiation demonstrated the highest predictive accuracy in monitoring phosphate content in mural plaster, as illustrated in Fig. 8, providing crucial scientific underpinnings for the conservation and restoration of murals. This model enables precise diagnosis of phosphate damage in murals, facilitating the timely detection and prevention of potential risks such as salt damage. Through accurate monitoring and early intervention, this model contributes significantly to extending the lifespan of precious artistic heritage.

Limitations and future prospects of the study

This research, in selecting sensitive features for phosphate content in Mural Plaster, focused solely on the choice of sensitive spectral bands, without exploring the combination of these bands, such as the construction of salinity indices at different orders. Moreover, the study limited itself to a finite range of fractional orders with a step length of 0.1, and did not delve into a detailed analysis of how variations in fractional order differential spectra might affect remote sensing monitoring of phosphate damage in mural plaster. Furthermore, the current study is primarily concentrated on mural salt damage monitoring under specific samples and conditions. This may limit the model’s generalizability across different types of murals or varying environmental conditions. Future research should consider a broader spectrum of samples and a diversity of environmental conditions to enhance the model’s applicability. Future studies should focus on improving the model’s generalizability, optimizing algorithms and the automation of techniques, and exploring the application of these technologies in other related fields. Such efforts could further advance the application of hyperspectral technology in cultural heritage conservation and other relevant areas. Additionally, this paper utilized only 50 samples of mural plaster. Increasing the number of samples in future experiments could result in more accurate and representative outcomes.

Addressing these limitations, future research directions should include: comprehensive analysis of sensitive bands, not only focusing on individual sensitive bands but also exploring their combinations and interactive effects, and how they collectively impact remote sensing monitoring of phosphate damage; an in-depth exploration of fractional order differential applications, analyzing changes in different fractional order differential spectra in more detail, investigating their impact on monitoring accuracy, and determining more optimized order selections; expanding the sample set and environmental diversity, encompassing a more varied range of samples and different environmental conditions, to enhance the model's generalizability and applicability.

Conclusion

This study has processed spectral data using the Fractional Order Differentiation (FOD) method, delving into the sensitive orders and characteristic bands between reflectance spectrum and phosphate concentration in Mural Plaster under various concentration erosion conditions. Furthermore, we developed an FOD-PLSR model based on fractional order differential spectra, aimed at precisely estimating the phosphate content in Mural Plaster (The code and data used in the study can be found in supplementary information 1 and 2).

• (1) Trends in the Correlation Coefficients between FOD Fractional Orders and Salt Content: Sensitive bands for phosphate content variations are identified at 408, 874, 1054, and 2077 nm (Table 1).

• (2) Correlation between Spectral Reflectance and Phosphate Concentration: The FOD method reveals the nonlinear characteristics and patterns of change in the spectral data of mural samples. The spectral sensitivity, controlled by the weighted order, shows a trend of initially increasing and then decreasing in the number of spectral bands satisfying the 0.01 significance test as the order increases from zero to one. The highest absolute value of positive correlation occurs at 1.9 order, corresponding to the 2077 nm band (Fig. 7, Table 1), with its top six characteristic bands (2077, 849, 874, 2063, 2064, and 402 nm) showing high consistency with known sensitive bands. In contrast, the characteristic bands extracted at integer orders (1.0 and 2.0) partially align at 1.0 order (874, 847, 848, 854, 862, and 875 nm) but differ significantly at 2.0 order (2077, 586, 819, 849, 1373, and 2063 nm).

• (3) Performance Evaluation of the Dunhuang Mural Plaster Phosphate Content Prediction Model: When predicting the behavior of murals under unknown concentration erosion, the 1.9 order model offers the highest inversion accuracy, achieving a maximum R² value of 0.815 after cross-validation. Empirical tests demonstrate that the correlation of fractional order differentiation exceeds that of integer order, with precision improvements of 1.24% and 7.12% over the integer orders (1.0 and 2.0) respectively. This study has successfully developed a hyperspectral feature inversion model for predicting Mural Plaster phosphate content based on the FOD method, exhibiting high efficiency and accuracy in the assessment of salt damage in Dunhuang murals (Fig. 8, Table 3).

In summary,this study has intricately processed spectral data using the fractional order differential method, revealing not only the complex interplay between phosphate concentration and spectral reflectance in mural plaster but also successfully developing an innovative FOD-PLSR model for accurately estimating phosphate content. Our findings demonstrate that the model established using the 1.9 order differential is most effective in predicting phosphate concentration in mural plaster. Furthermore, this research provides not only a scientific basis for the conservation and restoration of mural plaster but also offers new perspectives and methodologies for processing and analyzing spectral data in the future protection of cultural heritage.