On Optimizing Hyperspectral Inversion of Soil Copper Content by Kernel Principal Component Analysis

Abstract

Heavy metal pollution not only causes detrimental effects on the environment but also poses threats to human health; thus, it is crucial to monitor the heavy metal content in the soil. Hyperspectral technology, characterized by high spectral resolution, rapid response, and non-destructive detection, is widely employed in soil composition monitoring. This study aims to investigate the effects of dimensionality reduction methods on the performance of hyperspectral inversion. To this end, 56 soil samples were collected in Daye, with the corresponding hyperspectral data acquired by the advanced ASD Fieldspec4 instrument. We employed the linear dimensionality reduction method, i.e., the principal component analysis (PCA), and non-linear method in terms of kernel PCA (KPCA) with polynomial, radial basis function (RBF), and sigmoid kernels to reduce the dimensionalities of original spectral reflectance and that processed by first-derivative transformation (FDT). Building upon this foundation, we applied the Adaptive Boosting (AdaBoost) algorithm for inverting the soil copper (Cu) content. The performance of each inversion model was evaluated by evaluation indices in terms of the coefficient of determination (R²), root-mean-square error (RMSE), and residual prediction deviation (RPD). The results revealed that the KPCA with polynomial kernel function applied to the FDT-based spectra could yield the optimal inversion accuracy, with corresponding R², RMSE, and RPD being 0.86, 21.47 mg·kg⁻¹, and 2.72, respectively. This study demonstrates that applying the FDT with KPCA processing can significantly improve the accuracy of the hyperspectral inversion for soil Cu content, providing a potential approach for monitoring heavy metal pollution using hyperspectral technology.

Full text

Citation: Guo, F.; Xu, Z.; Ma, H.; Liu,
X.; Gao, L. On Optimizing
Hyperspectral Inversion of Soil
Copper Content by Kernel Principal
Component Analysis. Remote Sens.
2024,16, 2914. https://doi.org/
10.3390/rs16162914
Academic Editors: Eyal Ben-Dor and
Dominique Arrouays
Received: 13 May 2024
Revised: 7 July 2024
Accepted: 7 August 2024
Published: 9 August 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
remote sensing
Article
On Optimizing Hyperspectral Inversion of Soil Copper Content
by Kernel Principal Component Analysis
Fei Guo 1,2,3, Zhen Xu 4, * , Honghong Ma 1,2,3, Xiujin Liu 1,2,3 and Lei Gao 5
1Institute of Geophysical & Geochemical Exploration, Chinese Academy of Geological Sciences,
Langfang 065000, China; guofei@mail.cgs.gov.cn (F.G.); mahonghong@mail.cgs.gov.cn (H.M.);
liuxiujin@mail.cgs.gov.cn (X.L.)
2Key Laboratory of Geochemical Cycling of Carbon and Mercury in the Earth’s Critical Zone, Chinese
Academy of Geological Sciences, Langfang 065000, China
3Geochemical Research Center of Soil Quality, China Geological Survey, Langfang 065000, China
4Department of Electronic and Information Engineering, Shantou University, Shantou 515063, China
5School of Economics, Shandong University of Technology, Zibo 255000, China;
21121403034@stumail.sdut.edu.cn
*Correspondence: xuzhen@stu.edu.cn
Abstract: Heavy metal pollution not only causes detrimental effects on the environment but also poses
threats to human health; thus, it is crucial to monitor the heavy metal content in the soil. Hyperspectral
technology, characterized by high spectral resolution, rapid response, and non-destructive detection,
is widely employed in soil composition monitoring. This study aims to investigate the effects of
dimensionality reduction methods on the performance of hyperspectral inversion. To this end,
56 soil samples were collected in Daye, with the corresponding hyperspectral data acquired by the
advanced ASD Fieldspec4 instrument. We employed the linear dimensionality reduction method,
i.e., the principal component analysis (PCA), and non-linear method in terms of kernel PCA (KPCA)
with polynomial, radial basis function (RBF), and sigmoid kernels to reduce the dimensionalities of
original spectral reflectance and that processed by first-derivative transformation (FDT). Building
upon this foundation, we applied the Adaptive Boosting (AdaBoost) algorithm for inverting the
soil copper (Cu) content. The performance of each inversion model was evaluated by evaluation
indices in terms of the coefficient of determination (R
2
), root-mean-square error (RMSE), and residual
prediction deviation (RPD). The results revealed that the KPCA with polynomial kernel function
applied to the FDT-based spectra could yield the optimal inversion accuracy, with corresponding
R
2
, RMSE, and RPD being 0.86, 21.47 mg
·
kg
−1
, and 2.72, respectively. This study demonstrates that
applying the FDT with KPCA processing can significantly improve the accuracy of the hyperspectral
inversion for soil Cu content, providing a potential approach for monitoring heavy metal pollution
using hyperspectral technology.
Keywords: hyperspectral reflectance; soil copper (Cu) content; first derivative transformation (FDT);
principal component analysis (PCA); kernel PCA (KPCA); Adaptive Boosting (AdaBoost)
1. Introduction
Mining activities are widely recognized as major contributors to the accumulation
of heavy metals in soil [
1
]. The complex interaction between heavy metals and soil mi-
croorganisms, combined with the propensity for heavy metal accumulation [
2
,
3
], leads to
varying degrees of soil pollution and ultimately results in the deterioration of soil quality.
This degradation exerts detrimental effects on the ecosystem, which, in effect, poses a threat
to human health. While copper (Cu) stands as an essential element for optimal growth and
development for both plants and animals, the excessive amounts of Cu content, however,
can impede their growth [
4
,
5
]. Therefore, it is crucial to assess the extent and distribution
of Cu pollution in the soil.
Remote Sens. 2024,16, 2914. https://doi.org/10.3390/rs16162914 https://www.mdpi.com/journal/remotesensing

Remote Sens. 2024,16, 2914 2 of 21
The conventional approach for examining soil heavy metal contamination involves
field sampling and subsequent laboratory chemical analyses. The results are then used
to conduct geostatistical interpolation, which showcases the spatial distribution of heavy
metals [
6
,
7
]. Although such a method provides a more accurate representation of the spatial
distribution of soil heavy metals, it is labor-intensive, time-consuming, and financially
expensive [
8
–
10
]. In contrast, visible and near-infrared reflectance (VNIR) hyperspec-
tral spectroscopy has gained widespread prominence due to its speed, affordability, and
non-destructive nature in acquiring high spectral resolution and continuous spectral infor-
mation [
11
,
12
]. Consequently, the spectral information offers a comprehensive depiction
of the soil component’s status. Hyperspectral technology presents a promising potential
solution for the detection of heavy metal contents in the soil [13–15].
Recent advances in hyperspectral technology for soil components detection have
yielded promising results, including the soil carbon content, organic matter levels, and
the presence of heavy metals. For example, Kemper and Sommer successfully predicted
concentrations of various metals using stepwise multiple linear regression (MLR) and
artificial neural network (ANN) methods, demonstrating the viability of spectra for this
purpose in mining-contaminated soils [
9
]. Jarmer et al. employed partial least-squares
regression (PLSR) on reflectance spectra to analyze nitrogen and organic carbon contents,
proposing this method as a rapid screening tool for spatial assessment [
10
]. George et al.
successfully achieved SOC (soil organic carbon) content prediction by combining SOC-
sensitive spectral indices and reflectance transformations with an ANN mode [
11
]. Kooistra
et al. found that spectral pre-processing methods could enhance model performance and
robustness for detecting soil Cd and Zn contamination [
13
]. Viscarra et al. developed
PLSR-based calibration models for predicting soil properties from spectra, showcasing
the potential of diffuse reflectance spectroscopy for efficient soil analysis [
15
]. Song et al.
found that VIS/NIR spectra outperformed MIR spectra in predicting toxic metal levels
for agricultural soils when it comes to using univariate and partial least-squares (PLS)
models [16].
Despite the aforementioned advancements, challenges persist in addressing the com-
plex spectral responses of heavy metals in soil [
17
]. In the context of inverting soil Cu
content, the spectral response of Cu across the visible and infrared spectra is characterized
by several absorption features, each providing valuable information for remote sensing and
spectroscopic analysis [
18
–
20
]: In the visible range, Cu can cause broad absorption features
in the blue–green region (around 450–520 nm). This often results in a reddish or brownish
color in Cu-rich soils, which can indicate its relative concentration in the soil. This alone
may not offer clear differentiation from other soil minerals and organic matter. Moving into
the near-infrared region, Cu displays distinct absorption features around 830–870 nm and
940–980 nm due to electronic transitions. However, such features can be relatively weak
and may be masked by other soil components. Furthermore, in the shortwave infrared
spectra, Cu can influence the shape and position of absorption features related to clay
minerals and organic matter. Specific Cu-OH vibrations may occur around 1400 nm and
2200–2300 nm, but these effects are quite subtle.
It is important to note that the spectral response of Cu in soil is often complicated by
its interactions with other soil components, such as organic matter, iron oxides, and clay
minerals [
21
]. As a consequence, the spectral signatures of Cu might be masked or altered.
Additionally, the intensity and position of Cu-related absorption features can vary with Cu
concentration in the soil, with higher concentrations generally leading to stronger spectral
features, though this relationship is not always linear [
12
]. Additionally, soil moisture,
texture, and other environmental factors can also influence the spectral response of Cu in
the soil, adding further complexity to its spectral features [
22
]. Due to these complexities, it
is challenging, if not impossible, to conduct direct spectral detection of Cu, especially at
lower contents.
Given these challenges, the selection of an appropriate inversion model utilizing VNIR
hyperspectral data plays a critical role in enhancing the accuracy of inverting the Cu content.

Remote Sens. 2024,16, 2914 3 of 21
At present, a wide range of hyperspectral inversion models has been used to estimate soil
characteristics. Both linear models such as the PLSR [
15
,
23
,
24
] and non-linear models in-
cluding support vector machine (SVM) [
25
], random forest (RF) [
26
], and ANN [
27
,
28
] play
crucial roles in predicting the soil properties. Among those models, the Adaptive Boosting
(AdaBoost) model stands as a highly successful boosting approach that has outperformed
in various applications, leading to its widespread use across diverse fields [29]. Neverthe-
less, the potential of the AdaBoost algorithm in inverting ] soil properties remains largely
unexplored. Therefore, there lies a profound significance and practicality in investigating
the performance of this algorithm when it comes to estimating the content of soil elements,
particularly the Cu content.
The implementation of a well-suited inversion model utilizing VNIR hyperspectral
data plays a critical role in enhancing the accuracy of the inversion process [
23
]. However,
the inversion accuracy of soil element content is constrained by various factors, including
but not limited to spectral measurement, spectral preprocessing, and dimensionality re-
duction [
12
,
30
]. Regarding spectral preprocessing, it is generally employed to eliminate
or minimize signal noise as well as enhance desired features. However, it is imperative
to recognize that distinct preprocessing methods yield varying results, which, in turn,
exert distinct impacts on the accuracy of the inversion model [
31
]. Notably, the spectral
preprocessing method, such as the second derivative (SD) [
32
], Savitzky–Golay smoothing
(SG) [
7
], and orthogonal signal correction (OSC) [
33
], has demonstrated significant roles in
improving accuracies of hyperspectral inversion models. Furthermore, the first-derivative
transformation (FDT) has gained considerable attention in the field of spectral pretreatment
due to its advantages in eliminating baseline interference, reducing background distortions,
resolving overlapping individual peaks, and enhancing spectral resolution and sensitivity.
Therefore, the FDT was applied as the spectral preprocessing method in this study.
Spectral preprocessing can enhance the accuracy of model inversion to some extent, but
it cannot address the problem of the “curse of dimensionality” in hyperspectral data [
12
,
34
].
The collected spectral data contain hundreds or thousands of variables, which presents
significant challenges in modeling and analysis. Many of these spectral features may be
redundant, noisy, or unrelated to soil properties of interest. Including all these variables in
an inversion model can lead to overfitting even if they are preprocessed. To tackle such
an issue, the key approach lies in dimensionality reduction, which allows for focusing on
the most relevant spectral information while discarding redundancies. Several studies
have indicated that reducing the spectral variables through the careful selection of input
variables and effective feature parameter extraction can remarkably enhance the inversion
performance of hyperspectral models [
35
]. Genetic algorithm (GA) [
6
,
36
] and principal
component analysis (PCA) [
37
] have been widely employed in various research studies and
play a critical role in reducing data dimensions and improving model accuracy. From this
point, it is a crucial step to apply dimensionality reduction to mitigate overfitting, improve
model interpretability, and enhance performance in the inversion modeling for predicting
soil properties by hyperspectral data [35,36].
The selection of the dimensionality reduction method depends on the dataset prop-
erties and modeling objectives. Among various methods, principal component analysis
(PCA) has demonstrated effectiveness in reducing data dimensionality and enhancing
model accuracy [
38
,
39
]. The PCA is a linear dimensionality reduction method; on the
other hand, the soil spectra often exhibit inherent nonlinearity due to complex interactions
between the soil components and electromagnetic radiation. As a consequence, the PCA
may not fully capture these nonlinear relationships, leading to the limited representation of
the spectral data. To address such an issue, the kernel PCA (KPCA), a nonlinear extension
of the PCA, can be applied to capture nonlinear relationships [
40
,
41
]. The kernel function
is the core of KPCA, which enables the nonlinear mapping process of original data into a
feature space, significantly influencing effectiveness in capturing the nonlinear structure of
the spectral data.

Remote Sens. 2024,16, 2914 4 of 21
At present, three kernel functions, namely the polynomial, radial basis function (RBF),
and sigmoid kernels, are commonly employed in the KPCA algorithm. It is worth noting
that there may not be a single universally best kernel function for all cases. Especially when
it comes to inverting soil compositions, the choice of the most suitable kernel function
depends on the specific characteristics of the spectral data and the underlying relationships
between the spectral features and the content of soil composition. As a result, the perfor-
mance of the KPCA-based inversion model would be affected by the selection of kernel
function to a certain extent. Based on this consideration, it is essential to systematically
evaluate and compare the influence of different kernel functions on the performances of
KPCA-based inversion models.
In this study, we aimed to develop a PCA/KPCA-AdaBoost-based inversion model
for predicting Cu content in soil samples using VNIR hyperspectral data. By leveraging the
spectral reflectance and the corresponding soil Cu content, we sought to build an inversion
model that effectively captures the complex relationships between spectral features and
Cu content. Also, we herein assess the impact of spectral preprocessing in terms of FDT
on the accuracy of the proposed inversion model. This assessment can provide insights
into the effectiveness of FDT in enhancing the spectral features and improving inversion
performance. Furthermore, we explore the influence of different kernel functions employed
in KPCA, including polynomial, RBF, and sigmoid kernels, on the accuracy of the proposed
inversion model. By comparing the performance of the inversion model using different
kernel functions, we aim to identify the most suitable one for inverting Cu content for the
specific study area and, further, to optimize the model’s ability to capture the underlying re-
lationships between spectral data and Cu content. Through the approach mentioned above,
this study can certainly provide insights for environmental monitoring and management.
2. Materials and Methods
2.1. Study Area and Sampling Points
Daye City, a county-level municipality in Hubei Province, China, is located in the
southeastern part of the province on the southern banks of the midstream section of the
Yangtze River [
42
]. It lies between 114
◦
31
′
to 115
◦
20
′
East longitude and 29
◦
40
′
to 30
◦
15
′
North latitude (Figure 1). The city is situated on the northern fringes of the hilly terrain of
the Mufu Mountains, with a topography that slopes southwards, lowers towards the north,
and remains relatively flat in the east and west. The main topographic features include hills,
mountains, and plains [
25
]. With an elevation ranging from 120 to 200 m [
12
], Daye City has
a typical subtropical humid monsoon climate characterized by distinct seasonal changes,
abundant sunlight, rainfall, and warmth in each season, and a long frost-free period. The
region is known as the birthplace of Chinese bronze culture and possesses a wealth of
mineral resources and numerous large- and medium-scale mines. Historical evidence
reveals that as early as 3000 years ago, the pioneers of China began Cu mining and smelting
in Daye, giving rise to an ancient civilization rich in the art of bronze metallurgy. However,
it is important to recognize that these mining and smelting activities have contributed to
the contamination of the surrounding soil. Therefore, the agricultural land located in the
study area may have the risk of exceeding the Cu content to some extent.
In line with the study area’s characteristics and research objectives, 56 surface soil
samples were collected from agricultural land (0–20 cm) surrounding the mining region,
with a uniform sampling density of one sample per 500 to 800 m. The detailed sampling
strategy for this study can be also found in Figure 1, which was designed to capture the
unique characteristics of the study area: The uniform density ensured comprehensive cov-
erage of the study area, allowing for a systematic assessment of Cu distribution throughout
the agricultural lands. Such an approach is effective in capturing spatial variations in
contamination levels, which can be influenced by factors such as proximity to mining sites,
topographical changes, and soil type differences. Given the long history of Cu mining and
the presence of numerous mines in the study area, the regions surrounding Cu mine sites
were considered high-risk zones for elevated Cu concentrations. By focusing on these re-

Remote Sens. 2024,16, 2914 5 of 21
gions, the study aimed to assess the impact of both historical and ongoing mining activities
on soil Cu levels. Furthermore, uniform sampling is dense enough to capture meaningful
variations in Cu levels across the landscape, which also allows for more accurate interpola-
tion between sampling points, thereby providing a more comprehensive understanding of
Cu distribution across the entire study area. Lastly, this sampling process ensures that both
heavily contaminated and relatively unaffected areas are equally represented in the dataset,
providing a more accurate overall demonstration of Cu contamination in the agricultural
lands of the study area.
Remote Sens. 2024, 16, x FOR PEER REVIEW 5 of 23
sites, topographical changes, and soil type differences. Given the long history of Cu min-
ing and the presence of numerous mines in the study area, the regions surrounding Cu
mine sites were considered high-risk zones for elevated Cu concentrations. By focusing
on these regions, the study aimed to assess the impact of both historical and ongoing min-
ing activities on soil Cu levels. Furthermore, uniform sampling is dense enough to capture
meaningful variations in Cu levels across the landscape, which also allows for more accu-
rate interpolation between sampling points, thereby providing a more comprehensive un-
derstanding of Cu distribution across the entire study area. Lastly, this sampling process
ensures that both heavily contaminated and relatively unaffected areas are equally repre-
sented in the dataset, providing a more accurate overall demonstration of Cu contamina-
tion in the agricultural lands of the study area.
It is noteworthy that the collection, handling, and processing of the soil samples fol-
lowed the Specification of the Land Quality Geochemical Assessment Standard (DZ/T
0295-2016). The main soil types in the study area are paddy soil and red soil. Each soil
sample consisted of three sub-samples, each weighing more than 1000 g. All samples were
carefully dried, avoiding exposure to sunlight and moisture. Subsequently, the soil sam-
ples were then ground and passed through a 10-mesh nylon screen with a diameter of 2
mm to remove plant residues, rocks, and large debris [12,43]. After the aforementioned
processing, those soil samples were then divided into two parts: one for indoor spectral
testing and the other for heavy metal measurement in the chemical laboratory.
Figure 1. Overview of the study area and sampling points.
2.2. Data Determination
A portion of processed soil was sent to the Institute of Geophysical and Geochemical
Exploration (IGGE) to assess the soil Cu content. The analysis was performed using
plasma mass spectrometry, following the method described in WSBB/001-2019, which al-
lows for the determination of 31 trace elements. The detection limit for Cu in the soil was
set at 1.0 μg/g. Moreover, to ensure the accuracy and reliability of measurement, the la-
boratory implemented quality control measures by incorporating primary soil reference
Figure 1. Overview of the study area and sampling points.
It is noteworthy that the collection, handling, and processing of the soil samples
followed the Specification of the Land Quality Geochemical Assessment Standard (DZ/T
0295-2016). The main soil types in the study area are paddy soil and red soil. Each soil
sample consisted of three sub-samples, each weighing more than 1000 g. All samples were
carefully dried, avoiding exposure to sunlight and moisture. Subsequently, the soil samples
were then ground and passed through a 10-mesh nylon screen with a diameter of 2 mm to
remove plant residues, rocks, and large debris [
12
,
43
]. After the aforementioned processing,
those soil samples were then divided into two parts: one for indoor spectral testing and the
other for heavy metal measurement in the chemical laboratory.
2.2. Data Determination
A portion of processed soil was sent to the Institute of Geophysical and Geochemical
Exploration (IGGE) to assess the soil Cu content. The analysis was performed using plasma
mass spectrometry, following the method described in WSBB/001-2019, which allows for
the determination of 31 trace elements. The detection limit for Cu in the soil was set at
1.0
µg/g
. Moreover, to ensure the accuracy and reliability of measurement, the laboratory
implemented quality control measures by incorporating primary soil reference materials
(SRMs) during the analytical process. It is worth noting that the study adhered to the
quality requirements established by [44], thus validating the obtained experimental data.
Another portion of processed soil was utilized to acquire soil spectral reflectance using
ASD FieldSpec4 spectroradiometers (Analytical Spectral Device, Inc., Boulder, CO, USA).

Remote Sens. 2024,16, 2914 6 of 21
The spectroradiometers covered a wavelength range from 350 to 2500 nm, with a sampling
interval of 1.4 nm from 350 to 1100 nm and a 2 nm interval from 1000 to 2500 nm [
45
]. This
level of detail enables the capture of fine spectral features that may be indicative of specific
soil characteristics.
To ensure accurate and consistent measurements, great care was taken in preparing
the samples and controlling the measurement environment. The soil samples were first
screened to remove large particles and ensure homogeneity. They were then placed in clear
glass containers with dimensions of approximately 9 cm in diameter and 2 cm in depth.
This shallow depth helps to minimize shadowing effects and ensures even illumination
across the sample surface. All measurements were conducted in a dark room to eliminate
interference from ambient light, providing a controlled environment crucial for precise
spectral analysis.
The spectroradiometers were positioned approximately 7 cm above the sample surface
and centered over the soil samples, which were evenly distributed in dishes. This consistent
positioning is vital for maintaining measurement geometry and ensuring comparable
results across all samples. Before taking measurements, a thorough calibration process was
followed to ensure the highest possible accuracy. This process began with a 30 min warm-
up period to allow the instrument to reach a stable operating temperature. Following this, a
sequence of calibration steps was performed, including dark current acquisition to account
for internal electronic noise, optimization of instrument settings, and white reference
correction to calibrate against a standardized white BaSO
4
panel. The measurements were
conducted in a dark room with a stable 50 W halogen lamp as the light source, mounted
at a 15
◦
angle and positioned 50 cm away, without any obstructions [
12
,
46
]. This setup
ensures consistent, even illumination across the sample surface while minimizing specular
reflection that could interfere with the diffuse reflectance measurements of interest. For each
soil sample, ten individual spectral curves were measured. These multiple measurements
were then averaged to reduce random noise and improve the overall signal-to-noise ratio
of the data.
After the raw spectral measurements were collected, the data underwent further
processing to prepare them for analysis. The averaged spectral curves were subjected to a
resampling procedure, resulting in a final output of 2151 spectral bands for each sample,
with a consistent interval of 1 nm across the entire measured range. This resampling to
a uniform spectral resolution facilitates subsequent data analysis and allows for direct
comparison between different soil samples.
Figure 2presents the measurement results of Cu content and corresponding spectral
reflectance for 56 soil samples. From Figure 2a, it is clearly observed that the distribution
of Cu content in the soil is highly uneven. Some soil samples show significantly higher
Cu content, while others exhibit notably lower levels. As a result, there is a high degree
of heterogeneity in the collected soil Cu content. This marked variability may stem from
multiple factors, including parent material composition, land-use patterns, environmental
pollution levels, and local geological conditions. Figure 2b illustrates that the spectral re-
flectance and its variation demonstrate complex patterns versus the wavelength. Although
the general shape of the spectral curves is similar across all samples, there are evident
differences in reflection intensity at various wavelengths. These variations are not solely
influenced by Cu content but are likely closely related to other physicochemical properties
of the soil, such as organic matter content, particle size distribution, moisture content, and
the presence of other minerals. It is the combined effect of these multiple factors that result
in such diverse spectral reflectance characteristics.
Given the high heterogeneity of the soil Cu content and the multifaceted factors
influencing spectral reflectance, it is difficult, if not impossible, to estimate the soil Cu
content from spectra directly. This complexity emphasizes the need to develop a reliable
inversion model, coupled with suitable processing methods, for predicting Cu content in
the soil. The following sections elucidate the proposed approach in detail.

Remote Sens. 2024,16, 2914 7 of 21
Remote Sens. 2024, 16, x FOR PEER REVIEW 7 of 23
(a) (b)
Figure 2. The measurement results of 56 soil samples: (a) Cu contents and (b) spectral reflectance.
Each color represents a soil sample, and the color scheme is consistent between (a) and (b).
Given the high heterogeneity of the soil Cu content and the multifaceted factors in-
fluencing spectral reflectance, it is difficult, if not impossible, to estimate the soil Cu con-
tent from spectra directly. This complexity emphasizes the need to develop a reliable in-
version model, coupled with suitable processing methods, for predicting Cu content in
the soil. The following sections elucidate the proposed approach in detail.
2.3. Methodology
2.3.1. Workflow
The flow chart of this study is depicted in Figure 3. Initially, we proceeded with the
spectral reflectance with the FDT processing. Subsequently, we employed both linear and
nonlinear dimensionality reduction methods, i.e., the PCA and KPCA, on both the spectral
reflectance and that processed by the FDT, wherein the KPCA utilized polynomial, RBF,
and sigmoid kernel functions to reduce the dimensionality of the two groups of spectral
data (namely the original spectra and FDT-processed spectra). The resulting principal
components (PCs) were then used as input variables for inverting the soil Cu content us-
ing the PCA/KPCA-AdaBoost-based inversion model. Finally, we investigated the influ-
ence of different dimensionality reduction methods on the estimation accuracy for the Cu
content to determine the optimal PCA/KPCA-AdaBoost-based inversion model.
Figure 3. The flowchart for the establishment of PCA/KPCA-AdaBoost-based inversion model.
Figure 2. The measurement results of 56 soil samples: (a) Cu contents and (b) spectral reflectance.
Each color represents a soil sample, and the color scheme is consistent between (a) and (b).
2.3. Methodology
2.3.1. Workflow
The flow chart of this study is depicted in Figure 3. Initially, we proceeded with the
spectral reflectance with the FDT processing. Subsequently, we employed both linear and
nonlinear dimensionality reduction methods, i.e., the PCA and KPCA, on both the spectral
reflectance and that processed by the FDT, wherein the KPCA utilized polynomial, RBF,
and sigmoid kernel functions to reduce the dimensionality of the two groups of spectral
data (namely the original spectra and FDT-processed spectra). The resulting principal
components (PCs) were then used as input variables for inverting the soil Cu content using
the PCA/KPCA-AdaBoost-based inversion model. Finally, we investigated the influence of
different dimensionality reduction methods on the estimation accuracy for the Cu content
to determine the optimal PCA/KPCA-AdaBoost-based inversion model.
Remote Sens. 2024, 16, x FOR PEER REVIEW 7 of 23
(a) (b)
Figure 2. The measurement results of 56 soil samples: (a) Cu contents and (b) spectral reflectance.
Each color represents a soil sample, and the color scheme is consistent between (a) and (b).
Given the high heterogeneity of the soil Cu content and the multifaceted factors in-
fluencing spectral reflectance, it is difficult, if not impossible, to estimate the soil Cu con-
tent from spectra directly. This complexity emphasizes the need to develop a reliable in-
version model, coupled with suitable processing methods, for predicting Cu content in
the soil. The following sections elucidate the proposed approach in detail.
2.3. Methodology
2.3.1. Workflow
The flow chart of this study is depicted in Figure 3. Initially, we proceeded with the
spectral reflectance with the FDT processing. Subsequently, we employed both linear and
nonlinear dimensionality reduction methods, i.e., the PCA and KPCA, on both the spectral
reflectance and that processed by the FDT, wherein the KPCA utilized polynomial, RBF,
and sigmoid kernel functions to reduce the dimensionality of the two groups of spectral
data (namely the original spectra and FDT-processed spectra). The resulting principal
components (PCs) were then used as input variables for inverting the soil Cu content us-
ing the PCA/KPCA-AdaBoost-based inversion model. Finally, we investigated the influ-
ence of different dimensionality reduction methods on the estimation accuracy for the Cu
content to determine the optimal PCA/KPCA-AdaBoost-based inversion model.
Figure 3. The flowchart for the establishment of PCA/KPCA-AdaBoost-based inversion model.
Figure 3. The flowchart for the establishment of PCA/KPCA-AdaBoost-based inversion model.
2.3.2. Spectral Pretreatments
In this study, the FDT was employed to preprocess the original spectral data. The FDT
could enhance the spectral features by highlighting the regions where the reflectance is
changing rapidly with respect to wavelength. The resulting FDT spectra had peaks and
valleys that corresponded to the inflection points in the original spectral reflectance. This
could help to identify spectral features related to specific soil components or properties.
These features were often more pronounced and easier to interpret than the original spectra.

Remote Sens. 2024,16, 2914 8 of 21
Moreover, the FDT was particularly useful for reducing the baseline drifts, background
noise, and illumination variations in hyperspectral data, as it is less sensitive to these factors
compared to the original reflectance spectra. As a result, the FDT facilitated a more precise
identification of characteristic wavelength bands and resulted in a significant improvement
in the predictive efficacy of the model [45].
Figure 4displays the FDT-processed spectral curves of collected soil samples as a
function of wavelength, while the corresponding original spectral reflectance is presented
in Figure 2. Notably, three prominent absorption peaks, as observed in the vicinity of 1400,
1900, and 2200 nm of FDT processed spectra, are attributable to the absorption properties
of soil clay minerals [
47
,
48
]. Additionally, it is worth mentioning that wavelengths ranging
from 350 to 399 nm and 2450 to 2500 nm were excluded due to their comparatively lower
signal-to-noise ratio (SNR) [
24
]. Consequently, after the removal of these fringe bands,
2050 bands were retained for each sample, thus ensuring the integrity of the analysis.
Remote Sens. 2024, 16, x FOR PEER REVIEW 8 of 23
2.3.2. Spectral Pretreatments
In this study, the FDT was employed to preprocess the original spectral data. The
FDT could enhance the spectral features by highlighting the regions where the reflectance
is changing rapidly with respect to wavelength. The resulting FDT spectra had peaks and
valleys that corresponded to the inflection points in the original spectral reflectance. This
could help to identify spectral features related to specific soil components or properties.
These features were often more pronounced and easier to interpret than the original spec-
tra. Moreover, the FDT was particularly useful for reducing the baseline drifts, back-
ground noise, and illumination variations in hyperspectral data, as it is less sensitive to
these factors compared to the original reflectance spectra. As a result, the FDT facilitated
a more precise identification of characteristic wavelength bands and resulted in a signifi-
cant improvement in the predictive efficacy of the model [45].
Figure 4 displays the FDT-processed spectral curves of collected soil samples as a
function of wavelength, while the corresponding original spectral reflectance is presented
in Figure 2. Notably, three prominent absorption peaks, as obs erved in the vicin ity of 1400,
1900, and 2200 nm of FDT processed spectra, are attributable to the absorption properties
of soil clay minerals [47,48]. Additionally, it is worth mentioning that wavelengths rang-
ing from 350 to 399 nm and 2450 to 2500 nm were excluded due to their comparatively
lower signal-to-noise ratio (SNR) [24]. Consequently, after the removal of these fringe
bands, 2050 bands were retained for each sample, thus ensuring the integrity of the anal-
ysis.
Figure 4. The spectral reflectance of 56 soil samples after being processed by the FDT. Each color
represents a soil sample, and the color scheme is consistent with Figure 2.
2.3.3. Spectral Dimensionality Reduction
The PCA is widely adopted for analyzing and streamlining the high-dimensional da-
taset. The core idea of PCA is to reduce the dimensionality of a dataset comprised of an
abundance of interconnected variables while maintaining as much of the original infor-
mation and variance as possible. In high-dimensional data scenarios, the PCA identifies
multiple sets of orthogonal vectors in data space via matrix transformations. This process
transforms spectral data, which includes variables with multicollinearity, into a fresh set
Figure 4. The spectral reflectance of 56 soil samples after being processed by the FDT. Each color
represents a soil sample, and the color scheme is consistent with Figure 2.
2.3.3. Spectral Dimensionality Reduction
The PCA is widely adopted for analyzing and streamlining the high-dimensional
dataset. The core idea of PCA is to reduce the dimensionality of a dataset comprised of
an abundance of interconnected variables while maintaining as much of the original infor-
mation and variance as possible. In high-dimensional data scenarios, the PCA identifies
multiple sets of orthogonal vectors in data space via matrix transformations. This process
transforms spectral data, which includes variables with multicollinearity, into a fresh set
of uncorrelated variables that form a linear combination of the original independent vari-
ables. However, simply increasing the number of PCs does not necessarily yield enhanced
results. Typically, the first few PCs can encapsulate a significant portion of the variance
in the original dataset. Hence, the selection of preserved PC numbers is generally based
on the amount of cumulative variance attributable to the specific portion within the total
variance. Nevertheless, the amplification of the preserved PCs can increase the amount
of information, but it does not inherently improve the inversion accuracy. Therefore, this
study determined the optimal number of preserved PCs based on their impact on the per-
formance of estimating the Cu content. Further, to evaluate the influence of FDT processing

Remote Sens. 2024,16, 2914 9 of 21
on the inversion performance, the PCA was conducted on the original and FDT-processed
spectral data.
The KPCA represents a nonlinear approach to data processing that extends the tradi-
tional PCA algorithm. Its fundamental concept revolves around projecting the initial data
from the input space to a high-dimensional feature space through a nonlinear mapping,
typically achieved using kernel functions. The most commonly employed kernel functions
include the polynomial (Poly), radial basis function (RBF), and sigmoid kernels. Once
the data are mapped to the feature space, KPCA applies the PCA algorithm to compute
the PCs by solving an eigenvalue problem using the covariance matrix of the mapped
data. This allows KPCA to capture nonlinear relationships in the original data and ex-
tract meaningful features. By selecting a subset of the PCs, the KPCA can be used for
dimensionality reduction, projecting the data onto a lower-dimensional subspace while
preserving the most important nonlinear structures. The kernel functions play a pivotal
role in the KPCA algorithm. They furnish a method to implicitly map the input data to a
high-dimensional feature space without explicitly computing the coordinates in that space.
The kernel functions employed in this study are enumerated as follows.
The polynomial kernel is a kernel function that measures the similarity between two
vectors by computing their inner product raised to a specified degree
d
. It serves as a
representation of the similarity between these vectors. Essentially, the polynomial kernel
takes into account not only the similarity between vectors within the same dimension but
also across different dimensions. When employed in machine learning algorithms, this
property enables the consideration of feature interaction. The polynomial kernel is defined
as follows:
k(x,y)= (γxTy+c0)d(1)
where
x
and
y
are the input vectors, and
d
is the kernel degree; if
c0=
0, the kernel
is homogeneous.
The RBF kernel function facilitates the computation of the radial basis function (RBF)
kernel across a pair of vectors, which is defined as follows:
k(x,y)=exp(−γ∥x−y∥2)(2)
If γ=σ−2, the kernel is known as the Gaussian kernel of variance σ2.
The sigmoid kernel function facilitates the computation of the sigmoid kernel value,
also referred to as the hyperbolic tangent or multilayer perceptron. In neural networks, this
kernel is of vital significance, as it commonly serves as an activation function for neurons.
We express the sigmoid kernel function as follows:
k(x,y)=tanh(γxTy+c0)(3)
where γis known as slope, and c0is known as intercept.
Although the parameter optimization could potentially enhance the performance of
the KPCA-based inversion model, this study did not conduct specific optimization for the
kernel parameters. Instead, the kernel parameters listed in Table 1were utilized to establish
the KPCA-based inversion models to ensure the generalizability of each inversion model
while maintaining comparability across different models.
Table 1. Kernel parameters for the applied kernel functions.
Kernel Functions Polynomial Kernel RBF Kernel Sigmoid Kernel
Kernel parameters γ=1
nfeatures
;
c0=
1;
d=
3
γ=1
nfeatures γ=1
nfeatures ;c0=1
nfeatures represents the number of features for the employed dataset.

Remote Sens. 2024,16, 2914 10 of 21
2.3.4. Model Construction
Boosting is a powerful ensemble learning algorithm for both classification and re-
gression problems, which iteratively updates the weights of the base classifiers based on
changes in sample weights, resulting in a high-performance model [
29
]. Among its variants,
AdaBoost stands out as one of the most successful algorithms that has resolved numerous
practical issues [
49
]. In this study, AdaBoost was utilized to invert Cu content in the soil.
Some of the potential benefits of using the AdaBoost algorithm in this context include
improved accuracy by combining multiple weak learners, robustness to noise, the ability to
capture non-linear relationships, and a degree of interpretability through the individual
weak learners used in the ensemble [
50
]. Further, the AdaBoost method is less prone to
overfitting and has fewer parameters, reducing the need for extensive parameter tuning in
applications. Since the AdaBoost model does not limit the types of weak learners, different
learning algorithms can be used to construct weak classifiers. Moreover, compared to
the bagging algorithm and random forest algorithm, AdaBoost fully considers the weight
of each classifier, leading to high accuracy. A step-by-step procedure of the AdaBoost
algorithm is summarized in Algorithm 1.
Algorithm 1. The procedure outline for the AdaBoost algorithm
1. Initialize sample weights
Assign equal weights to all training samples, typically initialized as 1/N, where N is the
total number of samples.
2. For each iteration t = 1, 2,..., T
(1)
Train a weak classifier: Fit a weak classifier (e.g., decision stump or decision tree) to
the training data, taking into account the sample weights, to minimize the weighted
classification error.
(2) Calculate the weighted error of the weak classifier: Compute the weighted error εt
by summing the weights of the misclassified samples, with
εt=Σ(wi∗I(yi=ht(xi))), where wiis the weight of sample i,I() is the indicator
function, yiis the true label, and ht(xi)is the predicted label by the weak classifier.
(3) Compute the coefficient αtfor the weak classifier: αt=0.5 ∗ln((1−εt)/εt), where
ln()
is the natural logarithm, and
αt
represents the importance or weight of the weak
classifier in the final ensemble.
(4) Update the sample weights: Increase the weights of the misclassified samples and
decrease the weights of the correctly classified samples.
wi=wi·exp{αt·I([yi=ht(xi)])}for misclassified samples.
wi=wi·exp(−αt·I(yi=ht(xi))) for correctly classified samples.
(5)
Normalize the sample weights: Normalize the updated weights so that they sum up
to 1.
wi=wi/Σ(wi)for all samples.
3.
Combine the weak classifiers: The final AdaBoost classifier
H(x)
is a weighted combination
of all weak classifiers.
H(x)=sign(Σ(αt∗ht(x))),
where sign() is the sign function that returns +1 and −1 for positive and negative values,
respectively.
4. Make predictions
(1) For a new input sample x, compute the predictions of all the weak classifiers.
(2) Combine the predictions using the weighted sum: Σ(αt∗ht(x)).
(3) Apply the sign function to obtain the predicted class label: sign(Σ(αt∗ht(x))).

Remote Sens. 2024,16, 2914 11 of 21
The AdaBoost algorithm iteratively trains weak classifiers, assigns them weights based
on their performance, and updates the sample weights to focus on the misclassified samples.
The final classifier is a weighted combination of all the weak classifiers, where weights are
determined by their individual accuracies. By following the aforementioned procedure,
AdaBoost creates a strong classifier that can effectively classify new samples based on the
combined predictions of the weak classifiers.
2.3.5. Accuracy Validation
To evaluate the estimation performance and predictive accuracy of the inversion
models for Cu content in the soil, three evaluation metrics in terms of the coefficient of
determination (R
2
), root-mean-square error (RMSE), and residual prediction deviation
(RPD) were utilized. These metrics are commonly adopted in similar studies, where R
2
measures the proportion of variance in the dependent variable that is predictable from the
independent variables; RMSE provides the standard deviation of the prediction errors or
residuals, offering a measure of how far the predicted values are from the observed values;
and RPD is the ratio of the standard deviation of the observed values to the RMSE [
8
,
51
].
The representations for these evaluation metrics are presented below:
R2=1−∑n
i=1(yi−ˆ
yi)2
∑n
i=1(yi−yi)2(4)
RMSE =s1
n
n
∑
i=1
(ˆ
yi−yi)2(5)
RPD =SD
RMSE (6)
where
yi
and
ˆ
yi
represent measured and predicted content separately of samples in the
validation set,
yi
represents the mean of samples,
n
represents the number of samples, and
SD
represents the standard deviation of samples. It should be noted that both the coefficient
of determination and RPD are dimensionless metrics, whereas the RMSE is expressed in
the same unit as the measured Cu content, i.e., mg·kg−1in this case.
Generally, a robust model is characterized by high R² and RPD but by a low RMSE. R²
and RPD are frequently used to evaluate the accuracy of inversion performance, while the
RMSE is dependent on the range of measured values [
52
,
53
]. The interpretation of these
metrics is as follows:
(1)
R² is a measure of the proportion of the variance in the dependent variable that is
predictable from the independent variable (s). An R² value close to 1 indicates a high
goodness of fit, while a value close to 0 suggests a poor fit;
(2)
RMSE represents the standard deviation of the prediction residuals and provides a
measure of the average magnitude of the errors. A lower RMSE indicates a better
model fit;
(3)
An RPD value greater than 2.0 indicates an excellent inversion performance. An RPD
value between 1.4 and 2.0 suggests the ability to distinguish between high and low
values. An RPD value less than 1.4 represents an unsuccessful inversion performance.
3. Results
3.1. Statistic Analysis of Cu Content in Soil
The soil samples were divided into two groups: 38 samples for calibration and 18 sam-
ples for validation. Table 2provides a statistical summary of the soil Cu content character-
istics. As seen, the overall dataset revealed an average Cu content of 67.89 mg/kg. This
average exceeds that of the validation subset but is less than the calibration subset’s mean. It
is noteworthy that the natural background level of soil Cu content, as reported in the China
Soil Elements Background Values by the China National Environmental Monitoring Centre,
is 92 mg/kg. Significantly, 21.4% of the samples surpassed the national pollution threshold

Remote Sens. 2024,16, 2914 12 of 21
levels. Furthermore, the highest observed Cu content exceeded the natural background
level, and a coefficient of variation above 0.72 indicates a considerable spatial variability in
the Cu content distribution across the study area.
Table 2. Statistical descriptions of soil Cu content (mg·kg−1).
Soil Cu (mg·kg−1) Number Min Max Median Mean SD 1CV 2
Calibration set 38 21.68 320.86 73.03 90.43 67.89 0.75
Validation set 18 29.81 284.73 73.38 86.10 58.36 0.68
Whole dataset 56 21.68 320.86 73.03 89.04 64.48 0.72
1SD: standard deviation; 2CV: coefficient of variation.
3.2. Inversion Accuracy without Dimensionality Reduction
Before applying the dimensionality reduction in terms of PCA and KPCA processing,
we should first focus on the hyperspectral inversion performance using the spectral data
without applying any dimensionality reductions. To this end, the AdaBoost-based inversion
models were established in accordance with the original and FDT-processed spectra. The
corresponding results are presented in Figure 5and serve as the baseline for comparison
with the following dimensionality reduction methods (PCA and KPCA).
Remote Sens. 2024, 16, x FOR PEER REVIEW 13 of 23
Figure 5. The validation results of AdaBoost-based inversion models with (a) original spectra; (b)
FDT-processed spectra. The black diagonal line represents the “1:1 line”, indicating perfect agree-
ment between predicted and observed values.
The results in Figure 5 reveal notable differences between the performance when us-
ing the original spectra versus the FDT-processed spectra for predicting Cu content in the
soil. The inversion model based on the original spectra displayed a limited predictive per-
formance, as evidenced by a minor R² value of 0.14. This subpar performance may be at-
tributed to the presence of substantial noise and interferents in the original spectra, which
hindered the ability of the inversion model to capture the underlying relationship be-
tween the spectral features and Cu contents. In contrast, the inversion model constructed
using the FDT-processed spectra showed an improved R² of 0.24, suggesting that the FDT
preprocessing was effective in removing unwanted noise and enhancing the informative
spectral features for Cu content prediction but to a limited extent.
Nonetheless, it is worth noting that neither the original spectra nor FDT-processed
spectra achieved a desirable coefficient of determination. Furthermore, the RPD values in
both cases were consistently below 1.4, signifying an overall inadequate performance in
inverting soil Cu content. These unsatisfactory results underscore the significance of em-
ploying dimensionality reduction, such as PCA and KPCA, in this particular spectral anal-
ysis pertaining to Cu content. The diminished R² values hint at the possibility of redun-
dant or irrelevant features within the raw spectral data, impeding the model’s predictive
capability. By implementing PCA and KPCA to extract the most informative principal
components or nonlinear features, it is anticipated that subsequent inversion models may
attain enhanced performance. This dimensionality reduction step can aid in noise filtra-
tion, multicollinearity elimination, and the identification of key spectral signatures that
exhibit the highest correlation with Cu content. Further details are expounded upon be-
low.
3.3. Inversion Accuracy with PCA Processing
Next, the AdaBoost-based model for inverting the Cu content was developed with
the PCA-processed spectra. The independent variables of the inversion model were the
preserved PCs obtained after applying PCA to the original and FDT-processed spectra,
while the dependent variables were the soil Cu content. On this foundation, a comparative
analysis of the inversion performance under different numbers of PCs was conducted to
determine the optimal number of preserved PCs. The methodology for selecting the opti-
mal number of principal components in PCA/KPCA-based inversion models was as fol-
lows:
(1) Cumulative explained variance: The individual and cumulative explained variance
of the PCs were first calculated. This process was continued until the cumulative
explained variances reached 99.99%, which resulted in a large number of potential
PCs;
Figure 5. The validation results of AdaBoost-based inversion models with (a) original spectra; (b) FDT-
processed spectra. The black diagonal line represents the “1:1 line”, indicating perfect agreement
between predicted and observed values.
The results in Figure 5reveal notable differences between the performance when
using the original spectra versus the FDT-processed spectra for predicting Cu content in
the soil. The inversion model based on the original spectra displayed a limited predictive
performance, as evidenced by a minor R² value of 0.14. This subpar performance may
be attributed to the presence of substantial noise and interferents in the original spectra,
which hindered the ability of the inversion model to capture the underlying relationship
between the spectral features and Cu contents. In contrast, the inversion model constructed
using the FDT-processed spectra showed an improved R² of 0.24, suggesting that the FDT
preprocessing was effective in removing unwanted noise and enhancing the informative
spectral features for Cu content prediction but to a limited extent.
Nonetheless, it is worth noting that neither the original spectra nor FDT-processed
spectra achieved a desirable coefficient of determination. Furthermore, the RPD values
in both cases were consistently below 1.4, signifying an overall inadequate performance
in inverting soil Cu content. These unsatisfactory results underscore the significance of
employing dimensionality reduction, such as PCA and KPCA, in this particular spectral
analysis pertaining to Cu content. The diminished R² values hint at the possibility of redun-
dant or irrelevant features within the raw spectral data, impeding the model’s predictive
capability. By implementing PCA and KPCA to extract the most informative principal
components or nonlinear features, it is anticipated that subsequent inversion models may
attain enhanced performance. This dimensionality reduction step can aid in noise filtration,

Remote Sens. 2024,16, 2914 13 of 21
multicollinearity elimination, and the identification of key spectral signatures that exhibit
the highest correlation with Cu content. Further details are expounded upon below.
3.3. Inversion Accuracy with PCA Processing
Next, the AdaBoost-based model for inverting the Cu content was developed with
the PCA-processed spectra. The independent variables of the inversion model were the
preserved PCs obtained after applying PCA to the original and FDT-processed spectra,
while the dependent variables were the soil Cu content. On this foundation, a comparative
analysis of the inversion performance under different numbers of PCs was conducted
to determine the optimal number of preserved PCs. The methodology for selecting the
optimal number of principal components in PCA/KPCA-based inversion models was
as follows:
(1)
Cumulative explained variance: The individual and cumulative explained variance
of the PCs were first calculated. This process was continued until the cumulative ex-
plained variances reached 99.99%, which resulted in a large number of potential PCs;
(2)
Iterative model building and evaluation: Starting with the first PC, we incrementally
built inversion models using an increasing number of principal components (1 to
n
,
where
n
is the number of PCs needed to reach 99.99% cumulative explained variance).
For each iteration, we used the current set of PCs as input variables for the Cu
inversion model and evaluated the performance using metrics in terms of
R2
, PRD,
and RMSE;
(3)
Optimal selection: By comparing the inversion accuracy across all established inver-
sion models, we determined the number of PCs that resulted in the highest accuracy
(lowest RMSE, highest R², or highest RPD) as the optimal choice.
The aforementioned methodology balances the need to retain sufficient information,
and it allows for data-driven decision making rather than arbitrary cutoffs. Furthermore,
this approach considers both the explained variance and mode performance while avoiding
overfitting. The results suggest that using all preserved PCs corresponding to 99.99%
cumulative explained variances as input variables did not always lead to the best inversion
results. The underlying reason for this is that as the number of preserved PCs increased,
noise was introduced into the inversion model to a certain extent.
The inversion results from using the optimal preserved PCs are presented in Figure 6,
and the validation outcomes of the AdaBoost-based inversion models are also illustrated in
Figure 6.
Remote Sens. 2024, 16, x FOR PEER REVIEW 14 of 23
(2) Iterative model building and evaluation: Starting with the first PC, we incrementally
built inversion models using an increasing number of principal components (1 to ,
where  is the number of PCs needed to reach 99.99% cumulative explained
variance). For each iteration, we used the current set of PCs as input variables for the
Cu inversion model and evaluated the performance using metrics in terms of R,
PRD, and RMSE;
(3) Optimal selection: By comparing the inversion accuracy across all established
inversion models, we determined the number of PCs that resulted in the highest
accuracy (lowest RMSE, highest R², or highest RPD) as the optimal choice.
The aforementioned methodology balances the need to retain sufficient information,
and it allows for data-driven decision making rather than arbitrary cutoffs. Furthermore,
this approach considers both the explained variance and mode performance while avoid-
ing overfitting. The results suggest that using all preserved PCs corresponding to 99.99%
cumulative explained variances as input variables did not always lead to the best inver-
sion results. The underlying reason for this is that as the number of preserved PCs in-
creased, noise was introduced into the inversion model to a certain extent.
The inversion results from using the optimal preserved PCs are presented in Figure
6, and the validation outcomes of the AdaBoost-based inversion models are also illus-
trated in Figure 6.
Figure 6. The validation results of PCA-AdaBoost-based inversion models with (a) original spectra;
(b) FDT-processed spectra, where  represents the number of optimal preserved PCs. The black
diagonal line represents the “1:1 line”, indicating perfect agreement between predicted and ob-
served values.
The results in Figure 6 reveal the PCA-AdaBoost-based inversion model achieved the
optimal predictive accuracy, regardless of using original or FDT-processed spectra, when
13 preserved PCs were employed. Additionally, the results indicate that FDT processing
can enhance the performance of the PCA-AdaBoost-based inversion model to a certain
extent. Specifically, the model with FDT processing yielded an R2 of 0.60, an RMSE of 35.53
mg·kg−1, and an RPD of 1.63, respectively. In contrast, the AdaBoost-based inversion
model with the original spectra exhibited inferior performance, with the corresponding
R2, RMSE, and RPD values being 0.53, 38.89 mg·kg−1, and 1.53, respectively. The overall
results indicate that the inversion model employing PCA processing exhibited inferior
performance for estimating the Cu content, suggesting that the linear dimensionality re-
duction method is unable to effectively capture the nonlinear relationship between the
spectral data and Cu content, thereby leading to suboptimal predictive performance.
3.4. Inversion Accuracy of KPCA Dimensionality Reduction Methods
Figure 6. The validation results of PCA-AdaBoost-based inversion models with (a) original spec-
tra; (b) FDT-processed spectra, where
npc
represents the number of optimal preserved PCs. The
black diagonal line represents the “1:1 line”, indicating perfect agreement between predicted and
observed values.
The results in Figure 6reveal the PCA-AdaBoost-based inversion model achieved the
optimal predictive accuracy, regardless of using original or FDT-processed spectra, when
13 preserved PCs were employed. Additionally, the results indicate that FDT processing

Remote Sens. 2024,16, 2914 14 of 21
can enhance the performance of the PCA-AdaBoost-based inversion model to a certain
extent. Specifically, the model with FDT processing yielded an R
2
of 0.60, an RMSE of
35.53 mg
·
kg
−1
, and an RPD of 1.63, respectively. In contrast, the AdaBoost-based inversion
model with the original spectra exhibited inferior performance, with the corresponding
R
2
, RMSE, and RPD values being 0.53, 38.89 mg
·
kg
−1
, and 1.53, respectively. The overall
results indicate that the inversion model employing PCA processing exhibited inferior
performance for estimating the Cu content, suggesting that the linear dimensionality
reduction method is unable to effectively capture the nonlinear relationship between the
spectral data and Cu content, thereby leading to suboptimal predictive performance.
3.4. Inversion Accuracy of KPCA Dimensionality Reduction Methods
KPCA is a powerful methodology for non-linear dimensionality reduction. In this
study, we implemented three distinct kernel functions to effectively reduce the dimensional-
ity of both the original and FDT-processed spectral data. Subsequently, we estimated the Cu
content using the obtained data and proceeded to compare the corresponding accuracies.
The detailed analysis results are provided in the following.
3.4.1. Polynomial Kernel
The implementation of a polynomial kernel was employed in KPCA to reduce the di-
mensionalities of both the original and FDT-processed spectral data. Subsequently, distinct
preserved PCs were utilized as input variables of the AdaBoost-based inversion model to
invert Cu content. Then, a comparison was made across the achieved inversion accura-
cies. The evaluation results, obtained under the optimal number of principal components,
are presented in Figure 7. Furthermore, the estimation results of the inversion model
considering different spectral types are also illustrated in Figure 7.
Remote Sens. 2024, 16, x FOR PEER REVIEW 15 of 23
KPCA is a powerful methodology for non-linear dimensionality reduction. In this
study, we implemented three distinct kernel functions to effectively reduce the dimen-
sionality of both the original and FDT-processed spectral data. Subsequently, we esti-
mated the Cu content using the obtained data and proceeded to compare the correspond-
ing accuracies. The detailed analysis results are provided in the following.
3.4.1. Polynomial Kernel
The implementation of a polynomial kernel was employed in KPCA to reduce the
dimensionalities of both the original and FDT-processed spectral data. Subsequently, dis-
tinct preserved PCs were utilized as input variables of the AdaBoost-based inversion
model to invert Cu content. Then, a comparison was made across the achieved inversion
accuracies. The evaluation results, obtained under the optimal number of principal com-
ponents, are presented in Figure 7. Furthermore, the estimation results of the inversion
model considering different spectral types are also illustrated in Figure 7.
Figure 7. The validation results of Poly-KPCA-AdaBoost based inversion model with (a) original
spectra; (b) FDT-processed spectra, where  represents the number of optimal preserved PCs.
The black diagonal line represents the “1:1 line”, indicating perfect agreement between predicted
and observed values.
From Figure 7, it is evident that the utilization of 14 preserved PCs resulted in optimal
prediction accuracy when using the original spectral dataset. On the other hand, in the
case of the FDT-processed spectra, the inversion accuracy reached its peak with the utili-
zation of 15 preserved PCs. It is noteworthy that the FDT processing could significantly
enhance the predictive capability. The respective values of R2, RMSE, and RPD for the
KPCA-AdaBoost-based inversion model improved from 0.69, 31.41 mg·kg−1, and 1.86 for
the original spectral case to 0.86, 21.47 mg·kg−1, and 2.74 for the FDT-processed spectral
case.
3.4.2. RBF Kernel
When the spectral dimensionality was reduced by KPCA utilizing the RBF kernel
function, the subsequent accuracy of its prediction under the optimal preserved PCs was
achieved as is presented in Figure 8. To further visualize the relationship between the
estimated soil Cu content and the measured Cu content, we refer to Figure 8, which show-
cases a scatterplot.
Figure 7. The validation results of Poly-KPCA-AdaBoost based inversion model with (a) original
spectra; (b) FDT-processed spectra, where
npc
represents the number of optimal preserved PCs. The
black diagonal line represents the “1:1 line”, indicating perfect agreement between predicted and
observed values.
From Figure 7, it is evident that the utilization of 14 preserved PCs resulted in optimal
prediction accuracy when using the original spectral dataset. On the other hand, in the case
of the FDT-processed spectra, the inversion accuracy reached its peak with the utilization
of 15 preserved PCs. It is noteworthy that the FDT processing could significantly enhance
the predictive capability. The respective values of R
2
, RMSE, and RPD for the KPCA-
AdaBoost-based inversion model improved from 0.69, 31.41 mg
·
kg
−1
, and 1.86 for the
original spectral case to 0.86, 21.47 mg·kg−1, and 2.74 for the FDT-processed spectral case.
3.4.2. RBF Kernel
When the spectral dimensionality was reduced by KPCA utilizing the RBF kernel
function, the subsequent accuracy of its prediction under the optimal preserved PCs was
achieved as is presented in Figure 8. To further visualize the relationship between the

Remote Sens. 2024,16, 2914 15 of 21
estimated soil Cu content and the measured Cu content, we refer to Figure 8, which
showcases a scatterplot.
Remote Sens. 2024, 16, x FOR PEER REVIEW 16 of 23
Figure 8. The validation results of RBF-KPCA-AdaBoost-based inversion model with (a) original
spectra; (b) FDT-processed spectra, where  represents the number of optimal preserved PCs.
The black diagonal line represents the “1:1 line”, indicating perfect agreement between predicted
and observed values.
Both results in Figure 8 demonstrate that the performance of the inversion model
with the RBF kernel is not at an ideal level, though the FDT processing continues to im-
prove inversion results to some extent. Specifically, within the AdaBoost-based inversion
model, the utilization of the original spectra, accompanied by 13 principal components,
resulted in an optimal R2 of 0.46, an RMSE of 41.61 mg·kg−1, and an RPD of 1.40. By con-
trast, when employing the FDT-processed spectra, it was found that the most beneficial
PCs to retain were the first 18 ones. This refinement led to an enhancement in the accuracy
of the inversion model, resulting in evaluation indices of 0.52, 39.4 mg·kg−1, and 1.48.
3.4.3. Sigmoid Kernel
Next, the sigmoid kernel was employed to reduce the dimensions of both the original
and FDT-processed spectra. Subsequently, we scrutinized the performance of the KPCA-
AdaBoost-based inversion model in estimating the Cu content. The results show that the
number of optimal preserved PCs varied according to the spectral type. The optimal pre-
served PC and corresponding evaluation are presented in Figure 9, while Figure 9 visually
illustrates the scatterplot of estimated Cu content versus the measured Cu content.
Figure 9. The validation results of the Sigmoid-KPCA-AdaBoost-based inversion model with (a)
original spectra; (b) FDT-processed spectra, where  represents the number of optimal preserved
PCs. The black diagonal line represents the “1:1 line”, indicating perfect agreement between pre-
dicted and observed values.
The results in Figure 9 suggest that for the original spectra, the application of 10 pre-
served PCs yielded the optimal inversion performance. The corresponding R2, RMSE, and
Figure 8. The validation results of RBF-KPCA-AdaBoost-based inversion model with (a) original
spectra; (b) FDT-processed spectra, where
npc
represents the number of optimal preserved PCs. The
black diagonal line represents the “1:1 line”, indicating perfect agreement between predicted and
observed values.
Both results in Figure 8demonstrate that the performance of the inversion model with
the RBF kernel is not at an ideal level, though the FDT processing continues to improve
inversion results to some extent. Specifically, within the AdaBoost-based inversion model,
the utilization of the original spectra, accompanied by 13 principal components, resulted
in an optimal R
2
of 0.46, an RMSE of 41.61 mg
·
kg
−1
, and an RPD of 1.40. By contrast,
when employing the FDT-processed spectra, it was found that the most beneficial PCs to
retain were the first 18 ones. This refinement led to an enhancement in the accuracy of the
inversion model, resulting in evaluation indices of 0.52, 39.4 mg·kg−1, and 1.48.
3.4.3. Sigmoid Kernel
Next, the sigmoid kernel was employed to reduce the dimensions of both the original
and FDT-processed spectra. Subsequently, we scrutinized the performance of the KPCA-
AdaBoost-based inversion model in estimating the Cu content. The results show that
the number of optimal preserved PCs varied according to the spectral type. The optimal
preserved PC and corresponding evaluation are presented in Figure 9, while Figure 9
visually illustrates the scatterplot of estimated Cu content versus the measured Cu content.
Remote Sens. 2024, 16, x FOR PEER REVIEW 16 of 23
Figure 8. The validation results of RBF-KPCA-AdaBoost-based inversion model with (a) original
spectra; (b) FDT-processed spectra, where  represents the number of optimal preserved PCs.
The black diagonal line represents the “1:1 line”, indicating perfect agreement between predicted
and observed values.
Both results in Figure 8 demonstrate that the performance of the inversion model
with the RBF kernel is not at an ideal level, though the FDT processing continues to im-
prove inversion results to some extent. Specifically, within the AdaBoost-based inversion
model, the utilization of the original spectra, accompanied by 13 principal components,
resulted in an optimal R2 of 0.46, an RMSE of 41.61 mg·kg−1, and an RPD of 1.40. By con-
trast, when employing the FDT-processed spectra, it was found that the most beneficial
PCs to retain were the first 18 ones. This refinement led to an enhancement in the accuracy
of the inversion model, resulting in evaluation indices of 0.52, 39.4 mg·kg−1, and 1.48.
3.4.3. Sigmoid Kernel
Next, the sigmoid kernel was employed to reduce the dimensions of both the original
and FDT-processed spectra. Subsequently, we scrutinized the performance of the KPCA-
AdaBoost-based inversion model in estimating the Cu content. The results show that the
number of optimal preserved PCs varied according to the spectral type. The optimal pre-
served PC and corresponding evaluation are presented in Figure 9, while Figure 9 visually
illustrates the scatterplot of estimated Cu content versus the measured Cu content.
Figure 9. The validation results of the Sigmoid-KPCA-AdaBoost-based inversion model with (a)
original spectra; (b) FDT-processed spectra, where  represents the number of optimal preserved
PCs. The black diagonal line represents the “1:1 line”, indicating perfect agreement between pre-
dicted and observed values.
The results in Figure 9 suggest that for the original spectra, the application of 10 pre-
served PCs yielded the optimal inversion performance. The corresponding R2, RMSE, and
Figure 9. The validation results of the Sigmoid-KPCA-AdaBoost-based inversion model with (a) orig-
inal spectra; (b) FDT-processed spectra, where
npc
represents the number of optimal preserved PCs.
The black diagonal line represents the “1:1 line”, indicating perfect agreement between predicted and
observed values.
The results in Figure 9suggest that for the original spectra, the application of 10 pre-
served PCs yielded the optimal inversion performance. The corresponding R
2
, RMSE, and

Remote Sens. 2024,16, 2914 16 of 21
RPD values were 0.66, 33.14 mg
·
kg
−1
and 1.76, respectively. Conversely, the inversion
model utilizing the FDT-processed spectra surpassed that using original spectra in terms of
estimation accuracy, demonstrating a noteworthy improvement. This enhancement was
achieved by effectively incorporating 13 preserved PCs, resulting in evaluation indices of
0.72 (R2), 30.26 mg·kg−1(RMSE), and 1.93 (PRD), respectively.
3.5. Spatial Distribution of Soil Cu Contents
Geostatistics, grounded in the theory of regionalized variables, is pivotal for revealing
spatial structures. In the field of soil science, a primary application of geostatistics involves
estimating and mapping soil attributes in unsampled regions. In this study, the inverse-
distance weighting (IDW) method, a prototypical algorithm in geostatistics, is employed
to delineate the spatial arrangement and variability of Cu content. This preference is
attributable to its computational efficiency and straightforward implementation [54,55].
Figure 10 depicts the spatial distribution of Cu content within the study area, mapped
and simulated using the IDW method. It contrasts the experimental chemical analysis
values of Cu with the predicted values from both original and FDT processed spectra, em-
ploying various dimensionality reduction techniques within the AdaBoost-based inversion
model. While the geochemical maps of all predicted values mirror the general trend of the
interpolated measured values, notable deviations are observed in Figure 10(b1,b3,c3) within
the high-value zone (223.22 to 284.73 mg/kg). Similarly, slight differences are evident in the
low-value zone (29.81 to 45.02 mg/kg), especially in Figure 10(b1,b3,b4,c1). The compara-
tive analysis in Figure 10(c2) highlights the closest resemblance to the geochemical maps of
the measured values. Furthermore, the spatial distribution analysis of soil Cu content sug-
gests that areas with notably high Cu levels are predominantly located in the northeastern
section of the study area. This enrichment trend exceeds the benchmarks set by the China
Soil Elements Background Values. On the other hand, the southern and southwestern
regions of the study area are characterized by comparatively low Cu concentrations.
Remote Sens. 2024, 16, x FOR PEER REVIEW 17 of 23
RPD values were 0.66, 33.14 mg·kg−1 and 1.76, respectively. Conversely, the inversion
model utilizing the FDT-processed spectra surpassed that using original spectra in terms
of estimation accuracy, demonstrating a noteworthy improvement. This enhancement
was achieved by effectively incorporating 13 preserved PCs, resulting in evaluation indi-
ces of 0.72 (R2), 30.26 mg·kg−1 (RMSE), and 1.93 (PRD), respectively.
3.5. Spatial Distribution of Soil Cu Contents
Geostatistics, grounded in the theory of regionalized variables, is pivotal for reveal-
ing spatial structures. In the field of soil science, a primary application of geostatistics
involves estimating and mapping soil attributes in unsampled regions. In this study, the
inverse-distance weighting (IDW) method, a prototypical algorithm in geostatistics, is em-
ployed to delineate the spatial arrangement and variability of Cu content. This preference
is attributable to its computational efficiency and straightforward implementation [54,55].
Figure 10 depicts the spatial distribution of Cu content within the study area, mapped
and simulated using the IDW method. It contrasts the experimental chemical analysis val-
ues of Cu with the predicted values from both original and FDT processed spectra, em-
ploying various dimensionality reduction techniques within the AdaBoost-based inver-
sion model. While the geochemical maps of all predicted values mirror the general trend
of the interpolated measured values, notable deviations are observed in Figure 10b1,b3,c3
within the high-value zone (223.22 to 284.73 mg/kg). Similarly, slight differences are evi-
dent in the low-value zone (29.81 to 45.02 mg/kg), especially in Figure 10b1,b3,b4,c1. The
comparative analysis in Figure 10c2 highlights the closest resemblance to the geochemical
maps of the measured values. Furthermore, the spatial distribution analysis of soil Cu
content suggests that areas with notably high Cu levels are predominantly located in the
northeastern section of the study area. This enrichment trend exceeds the benchmarks set
by the China Soil Elements Background Values. On the other hand, the southern and
southwestern regions of the study area are characterized by comparatively low Cu con-
centrations.
Figure 10. The study area of soil Cu content spatial distribution map: (a) the measured values; (b1)
the predicted values by the Ori-PCA-AdaBoost-based inversion model; (b2) the predicted values by
the Ori-Poly-KPCA-AdaBoost-based inversion model; (b3) the predicted values by the Ori-RBF-
KPCA-AdaBoost-based inversion model; (b4) the predicted values by the Ori-Sigmoid-KPCA-Ada-
Boost-based inversion model; (c1) the predicted values by the FDT-PCA-AdaBoost-based inversion
model; (c2) the predicted values by the FDT-Poly-KPCA-AdaBoost-based inversion model (c3); the
Figure 10. The study area of soil Cu content spatial distribution map: (a) the measured values; (b1) the
predicted values by the Ori-PCA-AdaBoost-based inversion model; (b2) the predicted values by the
Ori-Poly-KPCA-AdaBoost-based inversion model; (b3) the predicted values by the Ori-RBF-KPCA-
AdaBoost-based inversion model; (b4) the predicted values by the Ori-Sigmoid-KPCA-AdaBoost-
based inversion model; (c1) the predicted values by the FDT-PCA-AdaBoost-based inversion model;
(c2) the predicted values by the FDT-Poly-KPCA-AdaBoost-based inversion model (c3); the predicted
values by the FDT-RBF-KPCA-AdaBoost-based inversion model; (c4) the predicted values by the
FDT-Sigmoid-KPCA-AdaBoost-based inversion model.

Remote Sens. 2024,16, 2914 17 of 21
4. Discussion
The precision of hyperspectral inversion for the soil Cu content is influenced not only
by the spectral preprocessing but also by dimensionality reduction method [
56
–
62
]. A
suitable combination of spectral preprocessing and dimensionality reduction can improve
the accuracy and performance of the inversion model. Moreover, prior research has
indicated that fine-tuning the number of preserved PCs can contribute to improving the
performance of the inversion model when applying PCA and its variants [56–58].
In this study, both the original and FDT-processed spectra were subjected to PCA/KPCA
processing to serve as input variables. On this foundation, the AdaBoost-based inversion
model then leveraged these inputs at the optimal PC count to estimate soil Cu content. The
rationale behind this approach was to analyze the impact of linear dimensionality reduction
methods and the non-linear dimensionality reduction method in terms of KPCA with
various kernel functions on the estimation performance of Cu content. The performances
of these inversion models were evaluated using R
2
, RMSE, and RPD metrics, as shown
in Table 3. Additionally, Table 3underscores the optimal number of preserved PCs for
different spectra and dimensionality reduction methods that led to enhanced inversion
accuracy. The scatter plots summarizing the performance of all inversion models are
presented in Figure 11.
Table 3. The prediction accuracies of the AdaBoost-based inversion models under the original and
FDT-processed spectra.
Methods Spectral Type The Number of
Optimal Preserved PCs
Prediction Accuracy
R2RMSE (mg·kg−1) RPD
Non-dimensionality
reduction
Original spectra - 0.14 55.41 1.06
FDT-processed spectra
- 0.24 58.38 0.97
PCA Original spectra 13 0.53 38.89 1.50
FDT-processed spectra
13 0.60 35.85 1.63
Poly-KPCA Original spectra 14 0.69 31.41 1.86
FDT-processed spectra
15 0.86 21.47 2.72
RBF-KPCA Original spectra 13 0.46 41.61 1.40
FDT-processed spectra
18 0.52 39.40 1.48
Sigmoid-KPCA Original spectra 10 0.66 33.14 1.76
FDT-processed spectra
13 0.72 30.26 1.93
Remote Sens. 2024, 16, x FOR PEER REVIEW 18 of 23
predicted values by the FDT-RBF-KPCA-AdaBoost-based inversion model; (c4) the predicted values
by the FDT-Sigmoid-KPCA-AdaBoost-based inversion model.
4. Discussion
The precision of hyperspectral inversion for the soil Cu content is influenced not only
by the spectral preprocessing but also by dimensionality reduction method [56–62]. A
suitable combination of spectral preprocessing and dimensionality reduction can improve
the accuracy and performance of the inversion model. Moreover, prior research has indi-
cated that fine-tuning the number of preserved PCs can contribute to improving the per-
formance of the inversion model when applying PCA and its variants [56–58].
In this study, both the original and FDT-processed spectra were subjected to
PCA/KPCA processing to serve as input variables. On this foundation, the AdaBoost-
based inversion model then leveraged these inputs at the optimal PC count to estimate
soil Cu content. The rationale behind this approach was to analyze the impact of linear
dimensionality reduction methods and the non-linear dimensionality reduction method
in terms of KPCA with various kernel functions on the estimation performance of Cu con-
tent. The performances of these inversion models were evaluated using R2, RMSE, and
RPD metrics, as shown in Table 3. Additionally, Table 3 underscores the optimal number
of preserved PCs for different spectra and dimensionality reduction methods that led to
enhanced inversion accuracy. The scatter plots summarizing the performance of all inver-
sion models are presented in Figure 11.
Table 3. The prediction accuracies of the AdaBoost-based inversion models under the original and
FDT-processed spectra.
Methods Spectral Type
The Number of Op-
timal Preserved PCs
Prediction Accuracy
R2 RMSE (mg·kg−1) RPD
Non-dimensionality
reduction
Original spectra - 0.14 55.41 1.06
FDT-processed spectra - 0.24 58.38 0.97
PCA Original spectra 13 0.53 38.89 1.50
FDT-processed spectra 13 0.60 35.85 1.63
Poly-KPCA Original spectra 14 0.69 31.41 1.86
FDT-processed spectra 15 0.86 21.47 2.72
RBF-KPCA Original spectra 13 0.46 41.61 1.40
FDT-processed spectra 18 0.52 39.40 1.48
Sigmoid-KPCA Original spectra 10 0.66 33.14 1.76
FDT-processed spectra 13 0.72 30.26 1.93
(a) (b)
Figure 11. The validation results of different dimensionality reduction in AdaBoost-based inversion
models with (a) original spectra; (b) FDT-processed spectra. The black diagonal line represents the
“1:1 line”, indicating perfect agreement between predicted and observed values.

Remote Sens. 2024,16, 2914 18 of 21
Different dimensionality reduction methods (PCA and KPCA with various kernels)
corresponded to different numbers of optimal preserved PCs. For example, under the
original spectra, the optimal number of PCs was 13 for the PCA, while it was 10 for the
RBF-KPCA. Moreover, the FDT processing could affect the number of optimal PCs to a
certain extent. Taking PCA as an example, the optimal number of PCs was 13 under the
original spectra, but it increased to 15 after FDT processing, while other KPCA methods
also showed similar changes. Thus, it could be concluded the choice of dimensionality
reduction method and spectral preprocessing both influence the optimal number of PCs.
The prediction performance varies significantly among different dimensionality re-
duction methods: When employing the PCA processing for dimensionality reduction,
the performance of the inversion model did not meet expectations regardless of whether
original or FDT-processed spectra were used. This result suggested that while inversion
models employing PCA for dimensionality reduction have some predictive power, there
is room for further optimization. Interestingly, the inversion model’s accuracy was en-
hanced when FDT-processed spectra were applied, suggesting the FDT preprocessing
step may be enhancing certain spectral features that are beneficial for the AdaBoost-based
inversion model.
The PCA processing was not fully able to meet the precision requirements of a satisfied
inversion model, largely due to the non-linear relationship between the measured spectral
data and Cu content. The PCA is a linear method of reducing dimensionality that identifies
the preserved PCs of the data [
59
,
60
]. Hence, it could not yield optimal results with non-
linear relation data. In contrast, KPCA allows to effectively manage non-linear relationships
by the kernel functions [
61
]. Thus, the efficacy of KPCA is significantly dependent on the
selection of the kernel function, as different kernel functions are appropriate for different
types of data, thereby leading to varied inversion outcomes [62].
In this study, three kernel functions, namely polynomial, RBF, and sigmoid kernels,
were employed, and the corresponding inversion performances under original and FDT-
processed spectra were evaluated. A comparative analysis aimed to identify the most
effective dimensionality reduction method for the study area’s data. Under the original
spectra, RBF-KPCA achieved the highest R
2
of 0.69, outperforming other methods. Through
this comparative study, we also discovered that the FDT-processed spectra could signifi-
cantly improve the accuracy of the inversion model in contrast to the original spectral data.
After FDT processing, the inversion model with Poly-KPCA processing reached an R
2
of
0.86, yielding the best prediction results, as detailed in Table 3. As a result, the dimension-
ality reduction achieved by KPCA-Poly, in conjunction with the FDT-processed spectra,
demonstrated exceptional estimation performance in the AdaBoost-based inversion model,
with an R
2
of 0.86 and an RPD of 2.72. Such high evaluation values indicate that the model’s
validation accuracy is exceptionally predictive. This indicates that selecting appropriate
data processing methods is crucial for improving model performance for specific problems.
The Poly-KPCA-AdaBoost-based inversion model showed excellent capability in
estimating soil Cu content using FDT-processed spectra in the study area. Nevertheless,
further exploration is warranted to enhance the proposed model. A key limitation of the
inversion model stems from the vast array of intricate variables that impact the accuracy
of inversion. Factors such as soil composition, color, type, and the levels of individual
soil elements have diverse impacts on the spectral data, thus affecting the precision of the
inversion model. Moreover, the model’s inversion accuracy depends on several factors,
including the spectral measurement, element measurement, spectral preprocessing, etc.
Therefore, our future research will investigate the transferability of the proposed inversion
model to fully assess its capabilities and broaden its practical use.
5. Conclusions
This study investigated the effects of the PCA and KPCA with polynomial, RBF, and
sigmoid kernels on the inversion performance of Cu content using the spectral data. To
this end, both original and FDT-processed spectral data were utilized for dimensionality

Remote Sens. 2024,16, 2914 19 of 21
reduction. Using these two groups of spectral data as inputs, the AdaBoost-based inversion
models were established, and the hyperspectral inversion was conducted by the proposed
model. Comparing inversion results under various dimensionality reduction methods, it
was found that the polynomial kernel function can enhance the feature extraction, which, in
effect, yields the optimal inversion performance for the soil Cu content. Moreover, the FDT
processing could substantially improve the accuracy of the inversion model. As a result, the
performance of the inversion model based on the transformed spectra surpassed that based
on original spectra, indicating the effectiveness of spectral transformation in mitigating
noise, varying backgrounds, and baseline interference. Ultimately, when employing the
spectral data processed by FTD and KPCA with the polynomial kernel, the AdaBoost-based
inversion model achieved the optimal accuracy at 15 preserved PCs, with R
2
and RPD
values being 0.86 and 2.72, respectively.
The results highlight the substantial potential of soil spectral analysis for estimating
the soil Cu content and monitoring the spatial distribution of heavy metal contamina-
tion. In contrast to the conventional land-quality survey, the soil spectral analysis offers
advantages in terms of time and manpower. Therefore, future research should focus on
further exploring the inversion of heavy metal contents based on soil spectral analysis,
particularly investigating the feasibility of using field spectral measurements to estimate
element contents and assessing the model’s transferability across different environments.
Author Contributions: Conceptualization, F.G. and Z.X.; methodology, F.G.; software, L.G.; valida-
tion, H.M., X.L. and Z.X.; investigation, H.M.; writing—original draft preparation, F.G.; writing—
review and editing, F.G. and Z.X.; visualization, Z.X.; supervision, Z.X.; funding acquisition, F.G. and
Z.X. All authors have read and agreed to the published version of the manuscript.
Funding: This work was supported in part by the Director Foundation of the Institute of Geophysical
and Geochemical Exploration, Chinese Academy of Geological Sciences under Grant AS2019J02;
in part by the National Natural Science Foundation of China under Grant 42101398; in part by the
Geological Survey Project of the China Geological Survey under Grant DD20221770; and in part by
Shantou University Scientific Research Foundation for Talents under Grant NTF20023.
Data Availability Statement: Data for this article can be obtained by contacting the author. The
data are not publicly available due to the data management policies at Chinese Academy of Ge-
ological Sciences. The data contain sensitive information that cannot be shared publicly without
proper authorization.
Conflicts of Interest: The authors declare no conflicts of interest.
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