Water depth estimation method of KAN network based on self-adaption and physical constraint

Abstract

The invention relates to a water depth estimation method of a KAN network based on self-adaption and physical constraint, and belongs to the technical field of remote sensing data processing. Grades are divided according to different water body turbidity, different KAN expert networks are adopted for different grades to learn the complex nonlinear relation between B3 and B4 wave band reflectivity and water depth, and KAN initial model parameters corresponding to different grades of water bodies are obtained through independent self-adaptive training; taking the B3 and B4 wave band reflectivity after normalization processing as input, selecting a corresponding KAN network model according to the water turbidity grade, and outputting predicted water depth; and training the KAN network by using the physical constraint loss function L, and finally extracting an interpretable mathematical inversion formula of the wave band reflectivity and the water depth after the training is completed. Through deep fusion of B3 / B4 waveband spectral characteristics of the KAN, a remote sensing technical solution with high precision, low cost and strong adaptation is provided for dynamic monitoring of the water depth of the shallow water area.

Description

Technical Field

[0001] This invention relates to a water depth estimation method based on adaptive and physically constrained KAN networks, belonging to the field of remote sensing data processing technology. Background Technology

[0002] Water depth information is the core foundational data for marine surveying, integrated coastal zone management, underwater topography analysis, waterway planning and maintenance, and nearshore ecological protection. The accuracy and efficiency of its acquisition directly determine the scientific nature of decision-making and the safety of implementation in scenarios such as marine resource development, disaster prevention and mitigation (e.g., storm surge inundation prediction) and underwater engineering construction (e.g., cross-sea bridge pile foundation construction).

[0003] In the development of water depth measurement technology, traditional measurement methods mainly rely on sonar depth sounding (such as single-beam and multi-beam sonar) and shipboard surveying. These methods can achieve high-precision water depth acquisition at the centimeter to decimeter level through direct contact detection. However, they have limitations that are difficult to overcome in practical applications: First, the contradiction between cost and efficiency is prominent. The costs of equipment purchase, ship leasing, and manpower maintenance are high, and a single measurement can only cover a local area. For vast shallow seas, river deltas, or remote islands and reefs, it is difficult to achieve rapid and large-scale mapping. Second, they have poor environmental adaptability. In nearshore shallow waters (water depth < 5m), they are easily interfered with by bottom obstacles (such as reefs and silt). In severe sea conditions (such as strong winds and huge waves) or complex terrain (such as coral reef areas), they cannot even carry out operations, resulting in "blank areas" and "fragmentation" of water depth data in large water areas.

[0004] With the maturity of aerospace remote sensing technology, water depth inversion methods based on optical satellite imagery have become an important alternative technology for water depth estimation in shallow water areas (water depth ≤ 20m) due to their advantages such as wide coverage (a single Landsat image covers approximately 34,000 km²), fixed observation period (Landsat 8 / 9 revisit period of 16 days), and low data acquisition cost (open source sharing). Among them, the visible light band of the Landsat series satellites is significantly sensitive to the optical properties of water bodies, especially the B3 band (green band, wavelength 0.53-0.59 μm) and the B4 band (red band, wavelength 0.64-0.67 μm), which are key core bands for shallow water inversion. The green band, due to weak absorption and scattering by water, has strong water penetration capabilities and can effectively capture the reflection signals of the seabed sediment; its reflectivity changes are directly correlated with water depth. The red band, due to the significant attenuation effect of water, exhibits a strong nonlinear response relationship between reflectivity and water depth, and can serve as an important constraint condition for water depth inversion. Although optical remote sensing inversion technology has achieved some applications, existing methods still face multi-dimensional technical bottlenecks, making it difficult to meet the high-precision requirements in complex water environments.

[0005] 1. The problem of "dimensional redundancy and insufficient robustness" in multi-band modeling: Current mainstream methods generally adopt Landsat B1-B4 multi-band joint modeling. Although it attempts to improve the generalization ability of the model by increasing the number of bands, it leads to data dimensional redundancy. This not only increases the complexity and computational cost of the early data preprocessing (such as atmospheric correction and band registration), but also makes the model overly dependent on the integrity of bands: when there are cloud cover, cloud shadows, or band data missing due to sensor failure, the model cannot operate normally, the inversion accuracy drops sharply, and the robustness is seriously insufficient.

[0006] 2. The problem of "weak nonlinear fitting ability" of traditional statistical models: Traditional statistical models based on empirical formulas such as single-band linear regression and multi-band ratios (such as B3 / B4) can only describe the simple linear or piecewise linear relationship between "spectral reflectance and water depth". However, in actual water bodies, factors such as turbidity (suspended matter content), bottom sediment type (sand, mud, coral) and water composition (chlorophyll, colored soluble organic matter) will cause a complex nonlinear relationship between reflectance and water depth. Traditional models cannot accurately capture this intrinsic relationship. In heterogeneous water bodies (such as estuarine freshwater areas and nearshore aquaculture areas), the inversion error often exceeds 20%, which is difficult to meet the engineering-level accuracy requirements.

[0007] 3. The "Data Dependence and High Cost" Problem of Deep Learning Models: In recent years, research has introduced deep learning architectures such as CNN (Convolutional Neural Network) and U-Net to solve the problem of nonlinear fitting. Although these models have significantly improved the fitting ability, they have the characteristic of "data hunger" - they need to rely on thousands or even tens of thousands of measured water depth data for labeled training. Moreover, the model has a large number of parameters (usually more than a million) and a long training cycle (requiring high-performance GPU support for several days to several weeks), resulting in high computational costs. For small water areas (such as small lakes and near-shore aquaculture ponds) or remote areas with scarce measured data (such as some islands and reefs in the South China Sea), the models are difficult to apply due to the inability to meet the data and hardware requirements. Summary of the Invention

[0008] The purpose of this invention is to overcome the above-mentioned shortcomings and provide a water depth estimation method based on adaptive and physical constraints KAN (Kolmogorov-Arnold Networks). By using a dynamic function selection mechanism guided by environmental classification and a physical constraint loss function, this invention solves the problems of insufficient accuracy and poor adaptability of existing technologies in complex water scenarios.

[0009] The technical solution adopted in this invention is as follows: The water depth estimation method based on adaptive and physically constrained KAN networks includes the following steps: S1. Acquire multi-band remote sensing image data containing B3 and B4 bands, as well as measured water depth data of the target water body; S2. Preprocess the acquired data; S3. Classify the water turbidity into different levels and use different KAN expert networks to learn the complex nonlinear relationship between reflectivity and water depth in the B3 and B4 bands for different levels. Independently and adaptively train the KAN expert networks corresponding to each level to obtain the initial model parameters of the KAN network corresponding to different levels of water bodies. S4. Using the normalized reflectance of the B3 and B4 bands as input, select the corresponding KAN network model according to the turbidity level of the water body, and output the predicted water depth. S5. Using the physical constraint loss function L, the KAN network is trained. After training, the interpretability of the KAN network is utilized to analyze and merge the mathematical expressions of all activation functions in the network, and finally extract an interpretable mathematical inversion formula for the reflectivity of the B3 and B4 bands and the water depth D.

[0010] In the above method, the preprocessing in step S2 includes atmospheric correction, water masking, extraction of water environmental parameters, radiometric calibration, and normalization.

[0011] The water masking and data normalization steps described in step S2 are as follows: After radiometric calibration, the bands are masked. A water mask is used to set the pixel values ​​of non-water areas to 0, retaining only the pixel values ​​of water areas. The retained areas are then normalized and numerically transformed. The specific calculation formula is as follows: , In the formula, This refers to remote sensing image data that has undergone radiometric calibration. f a The offset coefficient for image normalization; f b The scaling factor for image normalization; It is the band reflectance data after normalization processing.

[0012] The extraction of water environment parameters mentioned in step S2 is as follows: Based on the measured data, the target water body is divided into three levels according to turbidity: low turbidity: suspended solids content < 50 mg / L; medium turbidity: 50 mg / L ≤ suspended solids content < 150 mg / L; high turbidity: suspended solids content ≥ 150 mg / L.

[0013] In step S3, different levels of turbidity in water bodies are classified and different KAN expert networks are used. For water bodies with high turbidity, weight coefficients are introduced. The reflectance of the B3 and B4 bands is multiplied by their respective weight coefficients as input for training, and the weight coefficients are optimized through training.

[0014] The physical constraint loss function L mentioned in step S5 not only includes the mean square error (MSE) between the predicted and measured values, but also introduces a physical consistency constraint term. Its expression is: , in, This is used to penalize predictions that violate the principles of optical depth (first calculate the Pearson correlation coefficient between reflectivity and predicted water depth, then substitute it into the ReLU function; the output is...). (For example, predicting extremely shallow water depths in areas where reflectivity is high). This is a weighting factor.

[0015] In step S5, sparse regularization is used to prune the network during training to simplify its structure.

[0016] The beneficial effects of this invention are: This invention achieves deep integration of the KAN (Kolmogorov-Arnold Network) with the spectral characteristics of the Landsat B3 / B4 bands. On one hand, by utilizing the adaptive feature extraction mechanism of the KAN network, the complex nonlinear mapping relationship between "spectral attenuation law and water depth" can be directly extracted from the reflectance data of the B3 and B4 bands without the need for manual feature design. On the other hand, by combining the physical mechanisms of the high penetration of the B3 band and the attenuation sensitivity of the B4 band, along with the water environment classification module and dynamic function selection mechanism, the stability and accuracy of the model are improved in complex scenarios such as turbid water and heterogeneous bottom sediments. At the same time, the loss function of the KAN network is optimized to further improve the accuracy of the model output results. Ultimately, this provides a "high-precision, low-cost, and highly adaptable" remote sensing technology solution for dynamic water depth monitoring in shallow water areas. Attached Figure Description

[0017] Figure 1 This is a schematic flowchart of the method of the present invention; Figure 2 This is a schematic diagram of the KAN network structure; Figure 3 This is a remote sensing image from an embodiment of the present invention; Figure 4 The results of water depth extraction are shown in the embodiments of the present invention. Detailed Implementation

[0018] The present invention will be further described below with reference to specific embodiments.

[0019] Example 1: A water depth estimation method based on adaptive and physically constrained KAN networks, including the following steps (see...). Figure 1 )as follows: S1. Acquire multi-band remote sensing imagery data containing B3 and B4 bands, as well as measured water depth data of the target water body: The data mainly includes multi-band Landsat 8 / 9 remote sensing image data with B3 and B4 bands, as well as measured water depth data of the target water body.

[0020] S2. Preprocess the acquired data: (1) Image preprocessing, Atmospheric correction: The 6S model is used to eliminate atmospheric effects and obtain surface reflectance data; Water masking: Normalized water index (NDWI) is used to distinguish between water and non-water areas to generate water masking data; Environmental parameter extraction (classification of water body data based on measured data): Based on measured data, the target water body is divided into three levels according to turbidity (low turbidity: suspended solids content < 50 mg / L; medium turbidity: 50 mg / L ≤ suspended solids content < 150 mg / L; high turbidity: suspended solids content ≥ 150 mg / L), which serves as the basis for adaptive adjustment of the model.

[0021] (2) Radiation calibration, The digitized values ​​of the B3 and B4 bands of the Landsat imagery are converted into reflectance to establish a reflectance-water depth relationship. The Landsat image radiometric calibration calculation formula is as follows: , in: B represents the original image data for the band; r scale The scaling factor is used for radiometric calibration to convert the original digital values ​​into radiance, with a scaling factor of 0.011813 for the B3 band and 0.0099612 for the B4 band. r add The offset is used for radiometric calibration and is the reference for adjusting radiance. The band offset for band B3 is -59.06370, and the offset for band B4 is -49.80584. This refers to remote sensing image data after radiometric calibration.

[0022] (3) Water masking and data normalization After radiometric calibration, the bands are masked. A water mask is used to set the pixel values ​​of non-water areas to 0, retaining only the pixel values ​​of water areas. The retained areas are then normalized and numerically transformed. The specific calculation formula is as follows: , In the formula: This refers to remote sensing image data that has undergone radiometric calibration. f a The offset coefficient for image normalization; f b The scaling factor for image normalization; It is the band reflectance data after normalization processing.

[0023] S3. Based on different water turbidity levels, different KAN expert networks are used to learn the complex nonlinear relationship between reflectivity and water depth in bands B3 and B4 for each level. The initial model parameters of the KAN network for each level are obtained through independent adaptive training: KAN networks are a type of neural network architecture based on the Kolmogorov-Arnold representation theorem (e.g., Figure 2 Its core idea is to replace the fixed nonlinear activation function (such as ReLU) in traditional neural networks with a learnable, dynamically adjustable univariate function as the activation unit. By explicitly modeling the higher-order interactions of input variables, it significantly improves the model's parameter efficiency and mathematical interpretability while maintaining strong fitting capabilities. That is, any multivariate continuous function can be represented as a combination of a finite number of univariate continuous functions, and its mathematical expression is: , in: It is a function that maps the p-th input to an intermediate variable, where n is the number of input variables and q is the index of the intermediate function; It is another set of univariate functions that combine intermediate summation results.

[0024] Based on this theorem, an L-layer KAN network can be represented as a composite function: , The lth layer The calculation results are as follows, assuming that this layer has n in Input n out One output: , here It is the residual activation function, which is the sum of the basic function b(x) and the spline function spline(x). The specific formula is as follows: , , , In the formula, the weight parameter w is a default trainable parameter.

[0025] Adaptive training: Based on different turbidity levels of the water body, a multi-KAN network architecture is adopted. For low turbidity KAN expert network: Input The network structure is chosen as [2, 8, 1] (containing two input nodes, eight hidden layer neurons, and one output node) to learn complex nonlinear relationships; a medium-turbidity KAN expert network: input... Network structure selection [2, 5, 1], initialization biased logarithmic function; high turbidity KAN expert network: input The network structure is [1, 3, 1], where α and β are initially set to 0.7 and 0.3 respectively as weight coefficients, and the softmax function ensures that α + β = 1. During training, they participate in gradient backpropagation and are automatically adjusted according to the loss function. During training, samples are assigned to corresponding expert networks based on their environment labels, while during inference, the appropriate network is selected based on the real-time environment detection results.

[0026] S4. Using the normalized reflectance of the B3 and B4 bands as input, select the corresponding KAN network model according to the turbidity level of the water body, and output the predicted water depth: After normalization and The band reflectance is used as input, and the corresponding KAN network is selected according to the turbidity level of the water body. The training set and the test set are divided in a 7:3 ratio. The training set contains water body environmental labels for adaptive parameter adjustment.

[0027] S5. Using the physical constraint loss function L, the KAN network is trained. After training, leveraging the interpretability of the KAN network, the mathematical expressions of all activation functions in the network are analyzed and merged, ultimately extracting an interpretable mathematical inversion formula for the reflectivity of the B3 and B4 bands and the water depth D: During iterative model training, a physical guidance mechanism is introduced to establish a physical constraint loss function. The designed loss function L includes not only the mean square error (MSE) between the predicted and measured values, but also a physical consistency constraint term. Its expression is: , in, This is used to penalize predictions that violate the principles of optical depth (first calculate the Pearson correlation coefficient between reflectivity and predicted water depth, then substitute it into the ReLU function; the output is...). (For example, predicting extremely shallow water depths in areas where reflectivity is high). The coefficients are used for weighting. During training, sparse regularization is used to prune the network and simplify its structure. After training, the interpretability of the KAN network is utilized to analyze and merge the mathematical expressions of all activation functions in the network, ultimately extracting an interpretable mathematical inversion formula for the reflectivity of B3 and B4 and the water depth D.

[0028] A region in Shandong Province was selected as the study area. Landsat remote sensing images and corresponding water body data from March, September and October 2024 were used as input data to train the KAN network. An environmental adaptive function selection mechanism was adopted to train specialized KAN network models for different water turbidity conditions.

[0029] (1) Remote sensing image data, Atmospheric correction and water body masking were performed on the Landsat imagery to obtain the extracted water surface mask vector data (e.g., Figure 3 Based on measured suspended solids concentration data, the water body was divided into three turbidity levels: Low turbidity zone: suspended solids content < 50 mg / L; Medium turbidity range: 50 mg / L ≤ suspended solids content < 150 mg / L; High turbidity area: suspended solids content ≥150mg / L; (2) Establish a model, Landsat remote sensing images from March, September, and October 2024 were radiometrically calibrated, and three KAN expert networks were constructed and trained according to different turbidity levels: a. Low Turbidity KAN Expert Network Network structure [2,8,1] (input) , (8 hidden neurons) Training data: Low turbidity region , Reflectivity and measured water depth; b. Medium Turbidity KAN Expert Network Network structure [2,5,1] (input) , (5 hidden neurons) Training data: Medium turbidity region , Reflectivity and measured water depth; c. High Turbidity KAN Expert Network Network structure [1,3,1] (input) ), Training data: Weighted reflectance and measured water depth in high-turbidity regions. During training, α and β are automatically optimized through gradient backpropagation.

[0030] (3) Training Fitting Formula Each KAN expert network is trained independently, using a physical constraint loss function: , During training, sparse regularization is applied to prune the network, ultimately extracting a targeted and interpretable mathematical formula for each network: low turbidity. Medium turbidity High turbidity .

[0031] (4) Model testing The regression formula obtained after training was substituted into the Landsat remote sensing image from May 2024 to calculate the water depth data for the area in May. The results are as follows: Figure 4 As shown.

Claims

1. A water depth estimation method based on adaptive and physically constrained KAN networks, characterized by: The steps include the following: S1. Acquire multi-band remote sensing image data containing B3 and B4 bands, as well as measured water depth data of the target water body; S2. Preprocess the acquired data; S3. Classify the water turbidity into different levels and use different KAN expert networks to learn the complex nonlinear relationship between reflectivity and water depth in the B3 and B4 bands for different levels. Independently and adaptively train the KAN expert networks corresponding to each level to obtain the initial model parameters of the KAN network corresponding to different levels of water bodies. S4. Using the normalized B3 and B4 band reflectances as input, select the corresponding KAN network model according to the water turbidity level, and output the predicted water depth. S5. Using the physical constraint loss function L, the KAN network is trained. After training, the interpretability of the KAN network is utilized to analyze and merge the mathematical expressions of all activation functions in the network, and finally extract an interpretable mathematical inversion formula for the reflectivity of the B3 and B4 bands and the water depth D.

2. The water depth estimation method based on adaptive and physically constrained KAN networks according to claim 1, characterized in that, The preprocessing described in step S2 includes atmospheric correction, water masking, extraction of water environmental parameters, radiometric calibration, and normalization.

3. The water depth estimation method based on adaptive and physically constrained KAN networks according to claim 2, characterized in that, The water masking and data normalization steps described in step S2 are as follows: After radiometric calibration, the bands are masked. A water mask is used to set the pixel values ​​of non-water areas to 0, retaining only the pixel values ​​of water areas. The retained areas are then normalized and numerically transformed. The specific calculation formula is as follows: ， In the formula, This refers to remote sensing image data that has undergone radiometric calibration. f a The offset coefficient for image normalization; f b The scaling factor for image normalization; It is the band reflectance data after normalization.

4. The water depth estimation method based on adaptive and physically constrained KAN networks according to claim 2, characterized in that, The extraction of water environment parameters mentioned in step S2 is as follows: Based on the measured data, the target water body is divided into three levels according to turbidity: low turbidity: suspended solids content < 50 mg / L; medium turbidity: 50 mg / L ≤ suspended solids content < 150 mg / L; high turbidity: suspended solids content ≥ 150 mg / L.

5. The water depth estimation method based on adaptive and physically constrained KAN networks according to claim 1, characterized in that, In step S3, different levels of turbidity in water bodies are classified and different KAN expert networks are used. For water bodies with high turbidity, weight coefficients are introduced. The reflectance of the B3 and B4 bands is multiplied by their respective weight coefficients as input for training, and the weight coefficients are optimized through training.

6. The water depth estimation method based on adaptive and physically constrained KAN networks according to claim 1, characterized in that, The physical constraint loss function L mentioned in step S5 not only includes the mean square error (MSE) between the predicted and measured values, but also introduces a physical consistency constraint term. Its expression is: ， in, To penalize predictions that violate the principles of optical depth, the Pearson correlation coefficient between reflectivity and predicted water depth is first calculated, then substituted into the ReLU function, and the output is... λ is the tradeoff coefficient, and y is the predicted value. These are measured values.

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