Retrieving Chlorophyll-a Concentrations in Baiyangdian Lake from Sentinel-2 Data Using Kolmogorov–Arnold Networks

Abstract

This study pioneers the integration of Sentinel-2 satellite imagery with Kolmogorov–Arnold networks (KAN) for the evaluation of chlorophyll-a (Chl-a) concentrations in inland lakes. Using Baiyangdian Lake in Hebei Province, China, as a case study, a specialized KAN architecture was designed to extract spectral features from Sentinel-2 data, and a robust algorithm was developed for Chl-a estimation. The results demonstrate that the KAN model outperformed traditional feature-engineering-based machine learning (ML) methods and standard multilayer perceptron (MLP) deep learning approaches, achieving an R² of 0.8451, with MAE and RMSE as low as 1.1920 μg/L and 1.6705 μg/L, respectively. Furthermore, attribution analysis was conducted to quantify the importance of individual features, highlighting the pivotal role of bands B3 and B5 in Chl-a retrieval. Furthermore, spatio-temporal distributions of Chl-a concentrations in Baiyangdian Lake from 2020 to 2024 were generated leveraging the KAN model, further elucidating the underlying causes of water quality changes and examining the driving factors. Compared to previous studies, the proposed approach leverages the high spatial resolution of Sentinel-2 imagery and the accuracy and interpretability of the KAN model, offering a novel framework for monitoring water quality parameters in inland lakes. These findings may guide similar research endeavors and provide valuable decision-making support for environmental agencies.

1. Introduction

Inland lakes are vital to both human societies and ecological systems, supplying essential resources such as drinking water, irrigation, and fisheries. They also play a critical role in ecosystem balance, regional climate regulation, groundwater recharge, and flood risk mitigation [1,2]. Chlorophyll-a (Chl-a) concentration serves as a key indicator of phytoplankton biomass and eutrophication, offering a direct measure of primary productivity and water quality. Consequently, accurate monitoring of Chl-a holds considerable significance for assessing ecological health and conducting comprehensive water quality surveillance.

Traditional assessments of Chl-a primarily rely on field sampling, which is costly, time-consuming, potentially hazardous, and limited in its capacity to achieve broad spatial coverage and continuous monitoring. In contrast, satellite remote sensing can provide large-scale, noninvasive water quality data with high spatial and temporal resolutions. With ongoing advancements in sensor performance and accessibility, the utilization of remote sensing data for quantitative Chl-a retrieval has gained increasing prevalence [3]. Nevertheless, the complexity of optical properties in inland waters, coupled with pronounced regional and seasonal variations, frequently results in a highly non-linear relationship between Chl-a concentrations and spectral signals.

Machine learning (ML) algorithms, such as random forest (RF) [4,5], extreme gradient boosting (XGBoost) [6,7], and support vector machine (SVM) [8,9], have been extensively applied in water quality retrieval, particularly for addressing non-linear relationships. The performance of these ML models largely depends on the quality and suitability of input features, rendering feature engineering a critical step. Various band combinations and band-based indices have been proposed to enhance the extraction of Chl-a signal patterns from spectral data [10]. Examples include the normalized difference chlorophyll index (NDCI) [11], maximum chlorophyll index (MCI) [12], enhanced three-band index [13], GrB2 index [14], and near-infrared to red ratio [15,16]. These approaches can achieve satisfactory performance in specific study areas or with certain sensors [17]. However, adapting or reconstructing the feature engineering process is often necessary when the sensor type or research region changes, which can limit the model’s robustness and generalizability.

The emergence of deep learning presents a promising avenue for retrieving Chl-a. A deep neural network (DNN) leverages multiple layers of non-linear activation functions to automatically extract complex features from raw inputs, reducing dependence on prior knowledge and feature engineering. This approach ensures the global applicability of extracted Chl-a features and enhances the model’s capacity to model complex functions [18,19]. With high-resolution remote sensing data, the increased spectral purity and pixel-level information content further facilitate convolutional neural network (CNNs) and similar architectures in effectively extracting local spectral and spatial features [20]. As a result, deep learning models have demonstrated significant potential in Chl-a retrieval, offering automated feature extraction, robust generalization, and high predictive accuracy.

However, current applications of deep learning in Chl-a retrieval are still subjected to several challenges. First, deep neural networks typically require large amounts of training data, making them challenging to obtain in practical settings. Second, while multilayer perceptron (MLP)-based frameworks can represent complex input–output mappings, they provide limited interpretability regarding the contribution of specific features to Chl-a estimates [21,22]. Achieving both high accuracy and automated feature extraction while maintaining clear interpretability of the relationship between spectral characteristics and Chl-a remains an unresolved research question.

Recent advancements in artificial intelligence have introduced baseline deep models based on the Kolmogorov–Arnold network (KAN) representation theory [23,24]. These models have demonstrated strong interpretability and robustness under limited data conditions and are increasingly being applied across various domains [25,26]. We employ a Kolmogorov–Arnold network (KAN) for per-pixel Chl-a retrieval because the task is inherently spectral. Predictions are governed by per-band reflectance values rather than spatial context; hence, the convolutional inductive bias of CNNs offers limited advantage. In contrast, KANs are designed to operate on vector inputs, aligning with the input structure of multispectral or hyperspectral data. More importantly, KANs provide built-in, function-level interpretability: each edge is parameterized by a learnable univariate kernel function, enabling direct extraction of band-wise activation profiles and edge-wise attributions. This architecture affords an explicit understanding of how individual spectral bands and their non-linear compositions influence Chl-a predictions, capabilities that typically require post hoc interpretability tools when using MLPs or CNNs. These properties (task–model alignment, built-in interpretability, and data-efficient learning) make KAN a theoretically grounded and practically robust choice for inland water Chl-a estimation.

This study evaluates the KAN model in Baiyangdian Lake (Hebei Province, China), a representative inland lake test site. As one of the largest freshwater lakes in northern China, Baiyangdian exhibits diverse aquatic habitats, varying degrees of anthropogenic influence, and complex water-optical properties. These attributes designate it as a strong proxy for wider inland water systems and enable the generalization of our findings to other environments with similar ecological characteristics.

The objectives of this study included the following:

1.

To develop a robust KAN model for retrieving Chl-a concentrations in inland lakes and validate its performance.

2.

To identify the most influential spectral variables in retrieving Chl-a concentrations within Baiyangdian using attribution scores and to further forge a theoretical foundation for model interpretability.

3.

To investigate the driving factors underlying water quality dynamics during 2020–2024 by generating remote sensing-based Chl-a maps spanning this time frame and analyzing their spatio-temporal variations. Figure 1 illustrates the overall technical framework and workflow of this research.

2. Materials and Methods

2.1. Study Area

Baiyangdian Lake, located in Hebei Province, China, spans from 115°45 E to 116°07 E and from 38°44 N to 38°59 N. As the largest freshwater lake in northern China, it covers approximately 366 km² and has an average annual water storage capacity of around 1.32 billion cubic meters. In recent years, comprehensive environmental management efforts have substantially enhanced the lake’s water environment carrying capacity. However, challenges such as eutrophication and diffuse pollution from rural areas persist, necessitating sustained monitoring and management interventions.

Shaoche Dian and the waters of Quantou Village form the hydrological core of Baiyangdian Lake, accounting for approximately 18% of its open-water surface. Recent real-time buoy observations indicate that these sub-basins capture the full north–south gradients in salinity and nutrient concentrations and encompass key habitat types, including reed fringes, open pelagic zones, and semi-enclosed bays, thereby reflecting the lake’s overall ecological heterogeneity. They also experience the heaviest anthropogenic loading: Shaoche Dian receives $\sim 1.4$ million visitors yr⁻¹, accompanied by intensive motorized boat traffic, while Quantou Village is adjacent to $\sim 7500$ ha of peri-urban agricultural land and supports a resident population exceeding 9600 people. Collectively, these areas account for $>35\%$ of diffuse nutrient inputs to the lake (Baiyangdian Environmental Protection Bureau, 2024). Therefore, monitoring at these sites provides a sensitive and representative basis for assessing lake-wide ecological dynamics and the effectiveness of ongoing restoration efforts.

2.2. In Situ Data

Between 2023 and 2024, five field surveys were conducted at Baiyangdian Lake under clear, cloudless conditions, with a calm, ripple-free water surface to ensure standardized spectral measurements. The locations of the sampling sites are shown in Figure 2, and detailed information on their coordinates, survey dates, and sample numbers is provided in

Table 1. Sampling dates, point counts, and value ranges (μg/L).

Sample Date	Image Date	Number of Points	Min. Value	Max. Value
1 June 2023	31 May 2023	20	5.02	20.4
30 August 2023	30 August 2023	34	3.39	17.6
31 May 2024	31 May 2024	21	5.15	21.90
26 June 2024	25 June 2024	14	3.91	19.9
23 September 2024	23 September 2024	15	4.501	18.3

. Each site’s latitude and longitude were recorded using a GPS device. Furthermore, above-water radiometric methods were employed to acquire water body spectral data using an ASD Hand-Held2 (ASD Inc., Boulder, CO, USA) field spectroradiometer during the local time window of 10:00–15:00. Simultaneously, at each spectral measurement station, water samples were collected at a depth of 20–30 cm below the surface for subsequent Chl-a analysis in the laboratory with an L5S UV-Vis spectrophotometer (Hebei Shenglang Environmental Testing Co., Ltd., Langfang, China).

From an initial set of 126 Chl-a samples, spatial homogeneity was assessed using a 3 × 3 pixel window. Outliers in water quality parameters were then removed using the interquartile range (IQR) method, resulting in 104 Chl-a data points for model development. Descriptive analysis of the Chl-a measurements showed a mean concentration of 9.21 μg/L, with values ranging from as low as 3.39 μg/L to as high as 21.90 μg/L. According to the trophic state classification standards for lakes established by the Organization for Economic Co-operation and Development [27], these results indicate that the studied region remains affected by some degree of eutrophication.

2.3. Sentinel-2 Data

Sentinel-2 Level-2A (L2A) surface-reflectance imagery was obtained from the ESA Copernicus Open Access Hub. For each monthly in situ campaign, we selected the nearest cloud-free scene acquired under comparable meteorological conditions (low wind and no precipitation), preferably on the same day, or, if unavailable, within a narrow $\pm 5$-day window. This ensured temporal and spectral comparability at the monthly scale. All spectral bands were resampled to a common 10 m spatial resolution in SNAP and exported to ENVI format. Subsequent preprocessing (band stacking, mosaicking, cropping, and water masking) was performed in ENVI v5.3 (Harris Geospatial Solutions, Broomfield, CO, USA). The water extent of Baiyangdian was delineated using the normalized difference water index (NDWI) [28]; seasonal analyses used the spatial intersection of seasonal water masks to maintain a consistent spatial support. Although the Sen2Cor processor provides bottom-of-atmosphere (BOA) reflectance, we further applied scene-wise reflectance normalization to improve cross-date spectral consistency over optically complex inland waters. This additional step mitigates residual spectral heterogeneity that can remain after atmospheric correction, including minor aerosol-model mismatches, adjacency effects from bright shorelines, thin-cloud or sun-glint residues, and variations in sun–sensor geometry. By reducing artificial between-scene variability, this procedure improves the robustness of the subsequent inversion. For optical modeling, BOA reflectance $R\left(\lambda \right)$ was converted to remote-sensing reflectance $R_{rs}\left(\lambda \right)$ following Shenglei et al. [29]:

$$R_{rs}\left(\lambda \right)={\displaystyle \frac{R\left(\lambda \right)-min\left(R_{\mathrm{swir}}\right)}{\pi }},$$

(1)

where $R_{rs}\left(\lambda \right)$ is the corrected remote-sensing reflectance, $R\left(\lambda \right)$ is the L2A BOA reflectance, and $min\left(R_{\mathrm{swir}}\right)$ denotes the minimum among the considered SWIR bands. The constant $\pi$ was set to 3.14. As shown in Figure 3, the corrected spectra agree more closely with in situ measurements and exhibit typical features of turbid inland waters, ensuring the suitability of the processed imagery for subsequent Chl-a retrieval.

2.4. Methodology

2.4.1. Principles of the KAN Algorithm

The Kolmogorov–Arnold representation theorem, which KAN is founded upon, asserts that any continuous multivariate function defined on a bounded domain can be expressed as a finite composition of univariate continuous functions combined with additive operations. More formally, for a smooth function $f:{[0,1]}^n\to \mathbb{R}$,

$$\begin{array}{c}\hfill f\left(x\right)=f\left(x_1,\cdots ,x_n\right)=\sum _{q=1}^{2n+1}{\mathrm{\Phi }}_q\left(\sum _{p=1}^n{\varphi }_{q,p}\left(x_p\right)\right),\end{array}$$

(2)

where ${\mathrm{\Phi }}_{q,p}:[0,1]\to \mathbb{R}$ and ${\mathrm{\Phi }}_q:\mathbb{R}\to \mathbb{R}$. This theorem indicates that any function can be represented using only univariate functions and summation. Equation (2) suggests that by identifying suitable univariate functions ${\mathrm{\Phi }}_{q,p}$ and ${\mathrm{\Phi }}_q$, the learning task can be effectively inverted and predicted. To implement Equation (2) in a neural network, KAN must be designed such that its parameters explicitly correspond to the univariate functional forms. Since all learned functions are univariate, each one-dimensional function can be parameterized as a B-spline curve with learnable local B-spline basis coefficients:

$$\begin{array}{cc}\hfill \mathrm{\Phi }\left(x\right)& =\sum _{i=0}^n{c}_i{B}_i\left(x\right).\hfill \end{array}$$

(3)

Having defined the KAN prototype with a computational graph specified by Equation (2), the next step is to generalize it to arbitrary depths and widths. For an input dimension $n_{\mathrm{in}}$ and an output dimension $n_{\mathrm{out}}$, a KAN layer can be represented as a matrix of 1D functions:

$$\begin{array}{c}\hfill \mathrm{\Phi }=\left\{{\mathrm{\Phi }}_{q,p}\right\},p=1,2,\dots ,n_{\mathrm{in}},q=1,2,\dots ,n_{\mathrm{out}},\end{array}$$

(4)

where each ${\mathrm{\Phi }}_{q,p}$ has trainable parameters. By stacking KAN layers, the hierarchical composition implied by Equation (2) is achieved. For a concrete example and intuitive understanding, refer to the left panel of Figure 4. In that example, a KAN network approximates the classic water index, the normalized difference water index (NDWI) [28] initialized with two input nodes, five hidden nodes, and one output node.

The shape of a KAN is represented by an integer array:

$$\begin{array}{c}\hfill [n_0,n_1,\dots ,n_L],\end{array}$$

(5)

where $n_i$ is the number of nodes in the $i\mathrm{th}$ layer of the computational graph. The $i\mathrm{th}$ neuron in layer l is denoted by $(l,i)$ and its activation value is represented by $x_{(l,i)}$. Between layer l and $l+1$, there are $n_l{n}_{l+1}$ activation functions. Each function connecting $(l,i)$ to $(l+1,j)$ is ${\mathrm{\Phi }}_{(l,j,i)}$. Prior to activation, the input is $x_{(l,i)}$, and upon activation, it becomes ${\mathrm{\Phi }}_{(l,j,i)}\left(x_{(l,i)}\right)$. The activation value of neuron $(l+1,j)$ is the sum of all incoming activated values:

$$\begin{array}{c}\hfill x_{(l+1,j)}=\sum _{i=1}^{n_l}{\mathrm{\Phi }}_{(l,j,i)}\left(x_{(l,i)}\right),j=1,\dots ,n_{l+1}.\end{array}$$

(6)

In the matrix form, this can be expressed as follows:

$$\begin{array}{c}\hfill x_{l+1}=\underset{{\mathrm{\Phi }}_l}{\underbrace{\left(\begin{array}{cccc}{\mathrm{\Phi }}_{(l,1,1)}(·)& {\mathrm{\Phi }}_{(l,1,2)}(·)& \dots & {\mathrm{\Phi }}_{(l,1,n_l)}(·)\\ {\mathrm{\Phi }}_{(l,2,1)}(·)& {\mathrm{\Phi }}_{(l,2,2)}(·)& \dots & {\mathrm{\Phi }}_{(l,2,n_l)}(·)\\ ⋮& ⋮& \ddots & ⋮\\ {\mathrm{\Phi }}_{(l,n_{l+1},1)}(·)& {\mathrm{\Phi }}_{(l,n_{l+1},2)}(·)& \dots & {\mathrm{\Phi }}_{(l,n_{l+1},n_l)}(·)\end{array}\right)}}{x}_l,\end{array}$$

(7)

where ${\mathrm{\Phi }}_l$ refers to the function matrix corresponding to the $l^{\mathrm{th}}$ KAN layer. A general KAN network consists of L layers. Given an input vector $x_0\in {\mathbb{R}}^{n_0}$, the KAN output can be obtained by the following:

$$\begin{array}{c}\hfill \mathrm{KAN}\left(x\right)=({\mathrm{\Phi }}_{L-1}\circ {\mathrm{\Phi }}_{L-2}\circ \cdots \circ {\mathrm{\Phi }}_1\circ {\mathrm{\Phi }}_0)\left(x\right).\end{array}$$

(8)

The Kolmogorov–Arnold representation theorem enables the construction of efficient mapping from multivariate to univariate functions. The NDWI example shows that a basic KAN network can factor results back to the original green and infrared nodes and generate stable output weights through training and iteration. Evidently, this network architecture fully exploits the learnability of underlying functions, facilitating KAN to excel in both function representation and prediction tasks. These formulations forge a solid mathematical foundation for the Chl-a retrieval studied here.

2.4.2. Interpretability of KAN

Consider a KAN network as detailed in Equation (1). Let $E_{(l,i,j)}$ denote the standard deviation of the activations on the edge $(l,i,j)$, and $N_{(l,i)}$ represent the standard deviation of the activations at node $(l,i)$. Then, a node attribution score $A_{(l,j)}$ and an edge attribution score $B_{(l,i,j)}$ are defined. These scores are computed iteratively from the output layer back to the input layer, allowing for determining the contribution of each input feature and edge to the model’s output. All output layer dimensions $A_{(L,i)}$ are initialized to 1:

$$\begin{array}{c}\hfill A_{(L,i)}=1,i=0,1,\dots ,n_L-1.\end{array}$$

(9)

For edge attribution scores, the formula can be expressed as follows:

$$\begin{array}{c}\hfill B_{(l-1,i,j)}=A_{(l,j)}{\displaystyle \frac{E_{(l,j)}}{N_{(l+1,j)}}}.\end{array}$$

(10)

For node attribution, the formula can be expressed as follows:

$$\begin{array}{c}\hfill A_{(l-1,i)}=\sum _{j=0}^{n_l}{B}_{(l-1,i,j)},l=L,L-1,\dots ,1.\end{array}$$

(11)

2.4.3. KAN Network Design for Chl-a Retrieval

Herein, a KAN-based model was employed to predict Chl-a concentrations. The overall workflow comprises three primary steps, including data preprocessing, model construction, and model performance evaluation. During the data preprocessing phase, 104 valid spectral records were collected from the observation dataset, which were subsequently divided into training, validation, and test sets following a 7:2:1 ratio. The training and validation sets were adopted for model building and parameter tuning, while the test set was reserved for the final performance assessment. To enhance training efficiency and stability, the observed Chl-a concentrations were normalized using the min–max normalization technique, serving as the target output of the model.

For model construction, reflectance values from eight Sentinel-2 bands (B2–B8A) were used as input features. Extensive five-fold cross-validation indicated that a shallow network with two hidden layers achieved the best bias–variance trade-off for this relatively small tabular dataset. The first hidden layer comprised 16 neurons (i.e., $2\times$ the input dimensionality), providing sufficient capacity to capture higher-order feature interactions, while the second layer compressed the representation to 8 neurons to reduce overfitting and facilitate subsequent pruning. Model parameters were optimized using the L-BFGS algorithm with an initial learning rate of 10⁻²; training proceeded until convergence.

Following training, the model was pruned to enhance interpretability. Attribution scores were computed for each neuron and connection, and a threshold of 10⁻² was applied. This value was selected because (a) it corresponded to the “elbow” of the attribution-score distribution, below which further reductions in the threshold yielded only marginal additional sparsity, and (b) ablation studies confirmed that stricter thresholds (e.g., ≤10⁻³) did not improve sparsity but reduced validation $R^2$ by more than $1\%$. Accordingly, all nodes and edges with attribution scores below ${10}^{-2}$ were deemed negligible and removed.

Following pruning, the simplified KAN model was applied to the test set for prediction performance assessment. Subsequently, a statistical analysis of the remaining nodes and edges’ attribution scores was conducted, which facilitated a quantitative identification of the most critical spectral features for Chl-a concentration prediction. Overall, this analysis provides both theoretical and practical references for subsequent feature interpretation and water quality monitoring.

2.4.4. Accuracy Verification

Standardized quantitative metrics are necessary to accurately assess the model’s retrieval performance and ensure comparability among various models. Herein, three universally recognized indices, including root mean square error (RMSE), mean absolute error (MAE), and the coefficient of determination ($R^2$), were employed. Among them, $R^2$ and the slope approach unity as the model’s theoretical generalization capacity improves, while smaller RMSE and MAE values indicate higher model accuracy. The calculation formulas for these three metrics are as follows:

$$\begin{array}{c}\hfill \mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2},\end{array}$$

(12)

$$\begin{array}{c}\hfill \mathrm{MAE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|y_i-{\widehat{y}}_i\right|,\end{array}$$

(13)

$$\begin{array}{cc}\hfill R^2& =1-{\displaystyle \frac{{\sum }_{i=1}^n{(y_i-{\widehat{y}}_i)}^2}{{\sum }_{i=1}^n{(y_i-{\overline{y}}_i)}^2}},\hfill \end{array}$$

(14)

where n represents the total number of Chl-a samples used for testing; $y_i$ denotes the measured Chl-a concentration; ${\widehat{y}}_i$ is the predicted Chl-a concentration; ${\overline{y}}_i$ refers to the mean Chl-a concentration. As $R^2$ and the slope approach unity and as RMSE and MAE decrease, the model’s predictive performance and accuracy improve, providing a robust basis for evaluating and comparing different retrieval models.

3. Results

3.1. Comparison with Other Models

In this study, five representative models for Chl-a concentration retrieval were constructed and compared, involving three machine learning models (SVM, XGBoost, and RF) and two deep learning models (DNN and CNN). The hyperparameters of SVM, XGBoost, and RF were optimized utilizing a randomized search strategy to determine their optimal configurations. For the deep learning models, the DNN was designed with three hidden layers (comprising 128, 256, and 32 neurons, respectively) and an initial learning rate of ${10}^{-3}$, and the Adam optimizer was employed. The CNN model uses a $5\times 5$ pixel patch centered on each target pixel to form a $5\times 5\times 8$ spatio-spectral input constructed from the eight raw Sentinel-2 reflectance bands (B2–B8A). All bands are co-registered to a common 10 m grid, cloud/shadow and land pixels are masked, reflection or replicate padding is applied at image borders to preserve the full $5\times 5$ context, and each channel is standardized using training-set statistics. The network then applies a $3\times 3$ convolutional kernel in two hidden convolutional layers with 8 and 16 filters, respectively (ReLU activations; no pooling to retain local context), followed by a fully connected layer with 32 neurons to produce a scalar Chl-a prediction. The CNN is optimized with Adam (initial learning rate ${10}^{-3}$), and all models are trained and evaluated on the same dataset with early stopping on the validation split to mitigate overfitting.

Upon the determination of their optimal parameters, the proposed KAN model was compared with the five baseline models (CNN, DNN, SVM, RF, and XGBoost) for performance evaluation. Figure 5 shows the comparison between measured and predicted values, and

Table 2. Comparative performance of different algorithms.

Algorithms	${\mathit{R}}^2$	MAE (μg/L)	RMSE (μg/L)
KAN	0.8451	1.1920	1.6705
CNN	0.7242	2.2288	1.6601
DNN	0.7888	1.6807	1.9505
SVM	0.7503	1.8552	2.1207
RF	0.7698	1.7724	2.0364
XGBoost	0.7464	1.4067	2.1372

presents the quantitative evaluation indices. As shown in

Table 3. Engineered features and their references.

Feature	Reference
$(R_{rs}\left(665\right)-R_{rs}\left(560\right))/(R_{rs}\left(665\right)+R_{rs}\left(560\right))$	[12]
$(R_{rs}\left(705\right)-R_{rs}\left(665\right))/(R_{rs}\left(705\right)+R_{rs}\left(665\right))$	[11]
$R_{rs}\left(740\right)/R_{rs}\left(705\right)$	[13]
$R_{rs}\left(705\right)/R_{rs}\left(665\right)$	[4]
$R_{rs}\left(560\right)/R_{rs}\left(665\right)$	[14]
$R_{rs}\left(842\right)/R_{rs}\left(560\right)$	[30]
$(R_{rs}{\left(665\right)}^{-1}-R_{rs}{\left(705\right)}^{-1})/(R_{rs}{\left(740\right)}^{-1}-R_{rs}{\left(705\right)}^{-1})$	[13]

, the KAN model achieved superior performance in terms of $R^2$, MAE, and RMSE, yielding $R^2=0.8451$, MAE = 1.1920 μg/L, and RMSE = 1.6705 μg/L, which were markedly better than the results of other models. These findings demonstrate that the KAN model maximizes the fit between measured and predicted Chl-a concentrations while effectively reducing prediction bias and uncertainty. This strong performance underscores the potential of KAN in high-precision Chl-a retrieval tasks in remote sensing applications.

3.2. Comparison with Machine Learning Algorithms Using Domain-Specific Engineered Features

Unlike many lake water-quality retrieval studies that boost the signal-to-noise ratio (SNR) by handcrafting spectral indices, we deliberately restricted the KAN inputs to the eight raw Sentinel-2 reflectance bands (B2–B8A). First, hand-engineered indices impose strong priors and can obscure higher-order, subtle non-linear interactions; in contrast, the KAN’s adaptive functional bases learn such relationships directly from the data. Second, indices tuned to a particular optical regime often lack portability across lakes with differing biogeophysical conditions. Training on raw bands improves cross-system transferability, allowing a single trained KAN to be applied across multiple water types without redesigning features. Third, attribution scores computed on raw bands map directly to physically observed quantities, facilitating sensor selection and management decisions, whereas mixing composite indices would dilute causal interpretability and inflate model complexity. Nevertheless, to quantify the incremental value of handcrafted features and to provide a comparable benchmark, we also applied the same feature-engineering scheme to three representative machine-learning baselines and evaluated them on the same dataset.

Further experiments were conducted to enhance the performance of three traditional machine learning algorithms (SVM, RF, and XGBoost) on Chl-a retrieval by integrating carefully selected feature engineering approaches. These engineered features were derived from the original Sentinel-2 bands, leveraging commonly used spectral bands and remote sensing indices from previous studies on similar environments and water bodies (see Table 3). The engineered features included various band ratios, normalized difference indices, and empirically chosen sensitive band combinations. Such features have been extensively validated in the literature as beneficial for retrieving water-related parameters.

Figure 6 illustrates the comparisons between measured and predicted values using these feature-engineered machine learning algorithms. Upon training and validation on the expanded feature sets (see

Table 4. Comparative performance of feature-engineered machine learning algorithms.

Algorithms	${\mathit{R}}^2$	MAE (μg/L)	RMSE (μg/L)
SVM	0.7888	1.7023	1.9827
RF	0.8025	1.5849	1.8862
XGBoost	0.8220	1.0017	1.7898

), XGBoost demonstrated the best performance among these three models, achieving $R^2=0.8220$, RMSE = 1.7898 μg/L), and MAE = 1.0017 μg/L. Meanwhile, XGBoost slightly outperformed RF ($R^2=0.8025$, RMSE = 1.8862 μg/L, and MAE = 1.5849 μg/L) and SVM ($R^2=0.7888$, RMSE = 1.9827 μg/L, and MAE = 1.7023 μg/L).

Despite these improvements, feature-engineered traditional machine learning methods still lag behind the proposed KAN model. Notably, KAN achieved superior accuracy in estimating Chl-a concentrations without the need for additional, labor-intensive feature engineering. This result further highlights the model’s robustness and adaptability, suggesting that KAN can effectively capture and represent essential spectral characteristics without relying on domain expertise or extensive prior knowledge.

3.3. Model Interpretability Analysis

Feature attribution analysis facilitated deeper comprehension of the relative importance of each spectral band in Chl-a concentration retrieval. Figure 7 illustrates that pruning reduces the KAN model’s hierarchical structure to two hidden layers and three nodes. This streamlined architecture simplifies the calculation of attribution scores for each input band and its associated edges. These scores quantify the contribution of each feature to the Chl-a predictions, emphasizing the critical influence of key bands on achieving high retrieval accuracy.

Table 5. Attribution scores for each Sentinel-2 band.

Band	B2	B3	B4	B5	B6	B7	B8	B8A
Attribution Scores	0.485	0.497	0.294	1.007	0.428	0.017	0.101	0.199

provides detailed attribution scores. As shown in the table, B5 and B3 exert the strongest influence on Chl-a retrieval, presenting respective scores of 1.007 and 0.497. The prominence of band B5 coincides with a notable fluorescence peak at approximately 710 nm, commonly associated with Chl-a. Similarly, the substantial contribution of band B3 aligns with increased reflectance characteristics observed in phytoplankton-rich waters. In contrast, bands B2, B6, and B4 exhibit moderate importance (attribution scores of 0.485, 0.428, and 0.294, respectively), while bands B8, B8A, and B7 contribute the least (0.101, 0.199, and 0.0178, respectively). This suggests that the latter bands yield comparatively weaker signals or increased noise in capturing Chl-a fluorescence features, which is consistent with their limited spectral sensitivity.

Analysis of the hierarchical tree structure’s edges offers additional insights into inter-band relationships (

Table 6. Attribution scores for selected edges and nodes in the KAN.

$\mathbf{\Phi }\left(\mathit{x}\right)$	${\mathbf{\Phi }}_{0,1,1}$	${\mathbf{\Phi }}_{0,1,2}$	${\mathbf{\Phi }}_{0,2,2}$	${\mathbf{\Phi }}_{0,1,3}$	${\mathbf{\Phi }}_{0,2,3}$	${\mathbf{\Phi }}_{0,1,4}$	${\mathbf{\Phi }}_{0,2,4}$	${\mathbf{\Phi }}_{0,1,5}$
Score	0.485	0.124	0.373	0.041	0.253	0.111	0.897	0.029
	${\mathrm{\Phi }}_{0,2,5}$	${\mathrm{\Phi }}_{0,1,6}$	${\mathrm{\Phi }}_{0,1,7}$	${\mathrm{\Phi }}_{0,1,8}$	${\mathrm{\Phi }}_{0,2,8}$	${\mathrm{\Phi }}_{1,1,1}$	${\mathrm{\Phi }}_{1,1,2}$	${\mathrm{\Phi }}_{2,2,1}$
	0.399	0.017	0.101	0.119	0.090	0.276	0.897	0.999

). For example, edges ${\mathrm{\Phi }}_{0,2,4}$, ${\mathrm{\Phi }}_{0,1,1}$, and ${\mathrm{\Phi }}_{0,2,2}$ present attribution scores of 0.8973, 0.485, and 0.373, respectively, underscoring the essential roles of B5, B2, and B3. These high-contributing edges indicate non-linear interactions among these key spectral bands, thereby facilitating the model to more accurately capture the dynamic changes in Chl-a. In contrast, edges ${\mathrm{\Phi }}_{0,1,4}$ and ${\mathrm{\Phi }}_{0,2,3}$ yield relatively low attribution scores of 0.040 and 0.253, respectively, reflecting their limited impact on the final predictions. At the hidden layer level, the second-layer node ${\mathrm{\Phi }}_{1,1,2}$ achieves a notably high attribution score of 0.897, highlighting its critical role in synthesizing complex multi-band interactions and underlying Chl-a variation mechanisms. Additionally, the single node at the final output layer attains an attribution score of 0.999, which further emphasizes its pivotal role in aggregating information and guiding final predictions.

This detailed examination of the KAN model’s internal mechanisms elucidates the direct impact of individual input bands on Chl-a concentration estimates and reveals stable, meaningful interaction structures across multiple feature extraction layers. The presence of nodes and edges with lower contribution scores provides a rationale for simplifying the model and refining the feature set in subsequent studies. Collectively, this attribution analysis method establishes a robust foundation for enhancing model interpretability and predictive robustness, which also paves novel avenues for probing the relationships between water spectral responses and changes in Chl-a concentrations.

3.4. Spatio-Temporal Variation in Chl-a Concentration

Herein, remote sensing maps of Baiyangdian Lake from 2020 to 2024 were generated using the KAN model, covering the spring (March–May), summer (June–August), and autumn (September–November) seasons. Winter data were excluded due to the cold climate and ice cover in northern China, which hinder effective retrieval. To minimize cloud contamination, we adopted an image-acquisition strategy targeting one scene per 10-day interval that satisfied CLOUDY_PIXEL_PERCENTAGE ≤ 10%. If no suitable scene was available on the target date, we searched within a $\pm 5$-day window and selected the scene with the lowest cloud fraction. When no scene within this window met the threshold, monthly data gaps were filled using inverse-distance-in-time weighted interpolation based on the remaining available scenes. For data processing, the normalized difference water index (NDWI) was employed to delineate the water boundaries and compute the intersection of water areas for each season. The seasonal average products were obtained by accumulating the data and dividing by the number of seasonal images. These seasonal products were then aggregated over the year and divided by three to derive the annual average products. The entire workflow was implemented using Python v3.10.16 GDAL library.

Figure 8 reveals that the overall Chl-a concentration in Baiyangdian Lake remains relatively low, exhibiting pronounced seasonal variation: lowest in spring, intermediate in autumn, and highest in summer. This pattern largely reflects favorable temperature conditions during the summer and autumn months, which foster rapid phytoplankton growth. Similar seasonal dynamics have also been documented in other Chinese lakes [31].

Further analysis shows a declining trend in the lake’s annual mean Chl-a concentration between 2020 and 2022 (annual averages around 7 μg L⁻¹), followed by a marked increase in 2023–2024 (approximately 10 μg L⁻¹). Although the timing aligns with periods of reduced human activity during COVID-19 restrictions and subsequent resumption [32], our dataset does not include independent indicators of anthropogenic pressure. Consequently, the interpretations presented here are correlative rather than demonstrably causal. Therefore, we frame these associations as hypotheses and emphasize the importance of continued monitoring and the systematic collection of pressure indicators. Such data are essential for disentangling the relative contributions of climatic and anthropogenic drivers to inter-annual variability and for supporting more targeted and effective management strategies.

4. Discussion

4.1. Impact of Data and Preprocessing on Chl-a Retrieval

In terms of data acquisition, the number of cloud-free scenes available within a given year (N) fluctuates markedly across years and seasons due to cloudiness and satellite-revisit constraints. When coverage is insufficient, we rely on $\pm 5$-day substitution or within-month time-weighted interpolation. While these methods help fill temporal gaps, they can introduce sampling aliasing and interpolation errors, potentially biasing the annual mean and thereby affecting the interpretation of inter-annual change. To quantify the impact of temporal sampling density, we performed $B=1000$ Monte Carlo resamples for each year at $N\in \{27,24,21,18,15\}$ (month-stratified, without-replacement subsampling, with $\pm 5$-day substitution and within-month time-weighted interpolation applied).

The results (

Table 7. Annual mean Chl-a (μg/L) under different temporal sampling densities.

Year	$N=27$	$N=24$	$N=21$	$N=18$	$N=15$
2024	10.2	10.45	9.85	13.4	12.2
2023	9.8	9.1	9.35	6.9	8.47
2022	7.1	7.8	7.40	8.52	6.5
2021	7.2	6.95	7.38	7.95	6.52
2020	8.6	8.2	8.92	10.2	9.85

) show that the accuracy of the annual mean Chl-a depends strongly on N. When coverage is high (e.g., $N=27$), annual averages are stable and preserve inter-annual rank ordering; as coverage decreases, reliance on substitution/interpolation increases and uncertainty inflates. Our resampling indicates that the standard deviation of the annual mean is $\approx 1.2\mathrm{\mu }\mathrm{g}{\mathrm{L}}^{-1}$ for $N\ge 24$, rising to $\approx 3.5\mathrm{\mu }\mathrm{g}{\mathrm{L}}^{-1}$ for $N<15$. This suggests that, under sparse coverage, part of the apparent “anomalies” may be sampling artifacts rather than genuine biogeochemical change. Accordingly, when reporting annual means, we include the corresponding N and uncertainty estimate (SD/CI), use $N=27$ as the reference baseline, and flag years with $N\le 18$ as unsuitable for trend assessment.

From a data-processing standpoint, we favored the ESA L2A/Sen2Cor workflow, supplemented by scene-wise reflectance normalization and conversion from bottom-of-atmosphere reflectance $R\left(\lambda \right)$ to remote-sensing reflectance $R_{rs}\left(\lambda \right)$. This configuration provides a standardized, globally supported BOA baseline and, after normalization, yields stable cross-date spectra over optically complex inland waters. Alternative atmospheric-correction processors (e.g., ACOLITE, C2RCC, and iCOR) are primarily optimized for coastal/marine conditions or require site-specific parameterization, whereas our objective was to establish an operational and reproducible pipeline applicable across seasons and years. A formal multi-processor intercomparison will be pursued in future work to further assess performance differences.

4.2. Summary of Algorithmic Performance and Interpretability

The proposed KAN model achieved the highest Chl-a retrieval performance $\left(R^2=0.8451,\mathrm{RMSE}=1.6705\mathsf{\mu }\mathrm{g}{\mathrm{L}}^{-1},\mathrm{MAE}=1.1920\mathsf{\mu }\mathrm{g}{\mathrm{L}}^{-1}\right)$, significantly outperforming all other tested models. Handcrafted features improved classical ML baselines (BP, RF, and XGBoost) with $\Delta R^2=+0.0315,+0.0481,+0.0756$, respectively. XGBoost showed the largest reductions in error (MAE $-0.4046$ and RMSE $-0.3465$), highlighting the importance of feature engineering in classical frameworks. Among deep models, a fully connected DNN trained on raw spectra outperformed the original ML baselines but did not exceed feature-engineered ML, likely due to the small dataset. The CNN was slightly inferior overall, plausibly reflecting the limited utility of spatial convolutions for medium-sized lakes, where mixed-pixel effects and environmental interference can diminish the benefits of spatial-context aggregation.

Regarding interpretability, the KAN model introduces an innovative approach by applying learnable activation functions to the weights (edges) and retains a fully connected structure, unlike traditional MLPs that use fixed activation functions at the nodes. Specifically, the computational process of an MLP can be expressed as follows:

$$\mathrm{MLP}\left(x\right)=(W_{L-1}\circ \sigma \circ W_{L-2}\circ \sigma \circ \cdots \circ W_1\circ \sigma \circ W_0)x.$$

Obviously, MLPs treat linear transformations and non-linearities separately, implementing them through W and $\sigma$, respectively. In contrast, KANs handle these components collectively using $\mathrm{\Phi }$. Consequently, traditional linear weight matrices are not utilized; instead, each weight parameter is replaced by a one-dimensional learnable function parameterized by spline functions. In KANs, the nodes aggregate incoming signals without introducing additional non-linearities. Analyzing attribution scores allows for intuitive visualization of each sub-node’s contributions to the parent node. Hence, the critical feature variables B3 and B5 are further identified for Chl-a concentration estimation.

4.3. Limitations and Recommendations

Although the KAN model demonstrates excellent performance on the held-out test set, the dataset size (n = 104) warrants a more rigorous validation protocol to ensure robustness and reproducibility. While KAN attains high predictive accuracy with a degree of interpretability, its use of learnable edge functions yields complex functional compositions that limit full model transparency. Attribution analysis provides valuable insights but remains local and model-dependent, rather than constituting demonstrably causal explanations. Inference is further constrained by data availability and processing choices: the in situ matchup set is relatively small and restricted to a single lake, temporal alignment relies on a narrow $\pm 5$-day window, which introduces uncertainty, and conclusions are conditioned on a specific processing pipeline ($\mathrm{L}2\mathrm{A}\to$ normalization → NDWI mask $\to R\left(\lambda \right)\to R_{rs}\left(\lambda \right)$), under which residual thin-cloud, adjacency, or sun-glint effects may persist. Annual means are sensitive to the density of cloud-free scenes N; under sparse coverage, Monte Carlo analysis indicates markedly wider uncertainty bands, which can affect the interpretation of inter-annual trends.

To validate and stress-test KAN across diverse inland-water systems, future work should incorporate multi-lake, multi-year datasets spanning different optical water types and trophic states, with same-day satellite–in situ matchups whenever feasible. We will employ stratified k-fold cross-validation (e.g., $k=5$–10) and, where appropriate, spatio-temporal blocking or leave-one-lake/leave-one-year designs to obtain more stable performance estimates, mitigate overfitting, and more rigorously assess transferability. Uncertainty should be quantified by systematically varying scene density and preprocessing parameters (e.g., alignment windows and normalization strategies), and communicating predictive uncertainty via confidence intervals and calibration curves alongside standard performance metrics. Finally, cross-sensor evaluations (e.g., Landsat-8/9, PRISMA, and UAV hyperspectral) and open code/data releases will further enhance the robustness, reproducibility, and operational relevance of the proposed approach.

5. Conclusions

This study demonstrates that a KAN can accurately retrieve Chl-a concentrations from Sentinel-2 imagery over optically complex inland lakes, achieving performance comparable to, and in some cases exceeding, that of conventional machine-learning and deep-learning models, while retaining the added benefits of meaningful interpretability. Attribution analysis identifies bands B3 and B5 as the primary spectral predictors. These findings have direct operational value: water-resource agencies can automate Chl-a mapping at a weekly cadence and $10\mathrm{m}$ spatial resolution without heavy reliance on in situ sampling, monitoring programs can prioritize quality control for bands B3/B5 when scheduling acquisitions or assessing scene usability, and environmental managers can embed KAN-derived concentration thresholds into early warning systems to trigger rapid mitigation during bloom-risk periods. For Baiyangdian Lake, the resulting maps from 2020–2024 reveal a pronounced seasonal cycle and a gradual inter-annual improvement consistent with recent restoration efforts, providing a quantitative basis for refining lake nutrient-reduction targets. Looking forward, future work should extend the KAN framework to support multi-parameter retrieval (e.g., TSS, CDOM, and SDD), test its transferability across diverse lake types globally, and integrate the model with real-time data streams, from UAV-based hyperspectral imaging to hydrodynamic forecasts, to build a fully integrated, adaptive decision-support system for inland water management.