Interpolation of China’s Nearshore Sea Surface Temperature Based on Information Diffusion with Small Sample Sizes

Addressing the issue of data sparsity and gaps caused by missing values, this study employs an information diffusion approach to effectively spread information from sparse sample points to monitoring locations. By thoroughly extracting insights from a limited dataset, it achieves more precise interpolation outcomes. To validate the superiority of the information diffusion interpolation technique under conditions of sparse samples, we utilize sea surface temperature (SST) data from the offshore waters of China as a case study. We compare three interpolation methods: Kriging, Gaussian information diffusion, and asymmetric information diffusion. The calculations and comparisons of interpolation results are conducted across varying sample sizes. The findings indicate that in situations with relatively sparse samples, asymmetric information diffusion yields the most favorable results, with Kriging and Gaussian diffusion exhibiting comparable performance. In cases of extremely sparse samples, asymmetric information diffusion yields the lowest interpolation error, followed by Gaussian diffusion, while Kriging performs the least effectively. Thus, when confronted with sample sparsity, the application of the information diffusion interpolation method can yield notably improved results.


Introduction
Sea surface temperature (SST) plays a pivotal role within the global climate system.Monitoring and analyzing variations in SST enable a better comprehension of climate change trends and patterns, subsequently facilitating predictions of future climate shifts [1][2][3].However, due to the limited distribution of observation stations, data scarcity is prevalent in certain regions.In such contexts, interpolation techniques become crucial tools for obtaining comprehensive and accurate distributions of SST.
Interpolation methods aim to predict values at unknown locations based on existing data points using appropriate mathematical models or algorithms.Commonly used interpolation methods for SST include Kriging, inverse distance weighting, spline interpolation, and Lagrange interpolation, which generally fulfill most requirements.Yet, under extremely sparse sample conditions, their accuracy significantly diminishes [4][5][6].For modern neural networks, larger datasets often enhance model performance and generalization capabilities.However, in scenarios with small sample sizes, neural networks struggle to achieve satisfactory performance.
To address the challenge of scarce sample information, Huang Chongfu proposed a method and algorithm model for small-sample data diffusion based on information matrices and the concept of information diffusion.This was applied to estimating seismic intensity [7][8].Currently, information 2 diffusion finds extensive use in risk assessment for small samples.DU combined information diffusion models with machine learning to estimate typhoon storm surge disaster losses [9].Zhang employed an information diffusion model to conduct comprehensive risk assessments of agricultural drought disasters in the main grain-producing areas of Jilin Province, China [10].Wang utilized a generalized information diffusion model to evaluate and analyze post-earthquake road damage situations [11].Altogether, the information diffusion model plays a significant role in small-sample risk assessment [12][13].
In this paper, information diffusion theory is incorporated into interpolation techniques.Known observation points are diffused into fuzzy sets to represent information for unknown points.A suitable fuzzy input-output system is established, enabling accurate interpolation of environmental elements such as SST using a limited amount of sparse sample data.Experimental comparative results validate the superiority of the information diffusion method under scenarios of sample scarcity, providing an effective approach to resolving real-world environmental issues.This research is expected to deepen the application of spatial interpolation methods in cases of insufficient data, offering valuable reference for scientific research and decision-making in fields like environmental monitoring and climate studies.

Principles of Information Diffusion
The theory of information diffusion is a methodology introduced by Professor Huang Chongfu for handling incomplete data and uncertain information [14].According to this theory, when the sample size is exceedingly small, a known sample point can be transformed into the form of a fuzzy set to represent a sample point without observed data. Let be a domain space, where R controls the relationship between X andU .The non-diffusion estimate is defined as follows: Where  is the correlation operator and ( ) Where represents the absolute difference between the estimated correlation and the actual correlation.
The working principle of information diffusion theory is shown in Figure 1.

Information Diffusion Interpolation Concept
In the 'input-output' system, let M X Y =be the parent set, where y represent the input and output sets, respectively.If ( , ) f x y is the probability density formula of the parent set M, the conditional probability density of y given xu = is: In this 'input-output' system, if the input is xu = , the above equation can be used to determine the probability of the output y .This means that providing the system with an 'input' allows the corresponding 'output' to be presented in the form of probability.
According to fuzzy set theory, when a 'input-output' system is provided with a fuzzy set A as input, the corresponding output fuzzy set B can be obtained.The membership function of fuzzy set B corresponds to the probability density function of the 'output'.In other words, the membership function of fuzzy set B is: Therefore, in the aforementioned 'input-output' system, by providing any given input, it is possible to obtain all possible outputs and represent them using fuzzy set B. This represents the fuzzy mapping relationship between inputs and outputs in the 'input-output' system [15][16].
In cases of limited data availability, the probability density function ( , ) f x y obtained through conventional methods is likely to have significant errors.However, employing the information diffusion method allows for the thorough extraction of information from a small amount of available data, leading to an approximate estimation of the overall probability density distribution.

Normal Information Diffusion Function
Information diffusion aims to effectively propagate information from sample points to monitoring locations in space, and an effective diffusion function is pivotal for the system.Based on molecular motion, Huang Chongfu derived a simple normal information diffusion function: In the equation, i x stands for the sample point, i u represents the monitoring point, and h denotes the diffusion coefficient.
In the equation, n represents the number of components corresponding to the sample point, a is min{ } x , and b is max{ } x .If the samples involve multiple input and output components, the corresponding h can be obtained using the same logic.The normal information diffusion function takes each sample point as the center and uniformly spreads its information to the surrounding grid points while decreasing with distance.Taking the twodimensional normal information diffusion function as an example: In the equation, the exponent part follows the standard equation of a circle, indicating that the sample point information diffuses uniformly in all directions.If it is a three-dimensional normal diffusion function, it adheres to the standard formula for a sphere, hence the name 'normal information diffusion formula'.

Asymmetric Information Diffusion Function
The diffusion effectiveness of the normal information diffusion function to some extent relies on the distribution of sample points, especially for data with characteristics of normal distribution.However, it has limitations in handling non-uniform and non-normally distributed samples in natural settings.Therefore, adjustments to the diffusion function can be made based on the distribution of sample points.Using an ellipse as a model, the direction of faster diffusion aligns with the major axis of the ellipse, while the direction of slower diffusion aligns with the minor axis.This extends the original circular diffusion pattern of the normal diffusion function to an 'elliptical' form of diffusion [17][18].The schematic diagram of elliptical information diffusion is shown in Figure 2.

Figure 2. Schematic Representation of Elliptical Information Diffusion
Taking two-dimensional interpolation as an example, the formula for the elliptical information diffusion function is as follows: Where k represents the slope of the ellipse's major axis, defined as the rotation coefficient, and  represents the square of the ratio between the length of the major axis and the length of the minor axis, defined as the scaling coefficient.
The rotation coefficient is determined by the distribution of sample points.Assuming the equation of the line passing through the major axis of the ellipse is In the above equation,

Establishing Fuzzy Relationships
This experiment utilizes latitude and longitude as input samples x and y, and temperature as the output sample z.Considering the sparse samples as the set ,( , , )

S x y z x y z =
, the steps for establishing the fuzzy relationship matrix in the 2D information diffusion interpolation model are as follows: (1) Based on the components of the sparse sample set S, select an appropriate step size and establish the domain space corresponding to x, y, and z: (2) Apply the information diffusion formula to expand the information from sparse sample points to the monitoring point space, obtaining all the information increments at each monitoring point in U V W  .Given that the input is two-dimensional and the output is one-dimensional, the three- dimensional information diffusion formula should be utilized.The formula for three-dimensional normal information diffusion is as follows:

y w w q x y z h h h
The formula for three-dimensional elliptical information diffusion is as follows: In the equation, , and i  represent the scaling coefficients, while i k represents the slope of the major axis of the ellipse.(3) After propagating the information of sample points to monitoring points, it is necessary to sum up the gains of all sample points at the monitoring points.Utilizing Equation (15), the information matrix Q can be obtained.

( , , )
x y z represents the matrix obtained by spreading the information of sample points i s to the monitoring point spaceU V W  using Equation ( 13) or ( 14), and summing it up results in the three-dimensional information matrix.
(4) After generating the information matrix Q, utilize Equation ( 16) to obtain the fuzzy relationship matrix R:

Handling Inputs and Outputs
Upon obtaining the fuzzy relationship matrix R based on the sparse sample set S, mapping given inputs to outputs through the fuzzy relationship matrix R is possible.However, appropriate fuzzification and defuzzification of the system's inputs and outputs are necessary: (1) As the experiment's input consists of a two-dimensional sample point (longitude and latitude), prior to performing fuzzy mapping, the inputs need to undergo fuzzification: (1 )( 1), ( , ) 0, else (2) Given the input fuzzy set A and the fuzzy relationship matrix R, use Equation ( 18) to calculate the output fuzzy set B: (3) After obtaining the output fuzzy set B, utilize Equation ( 19) for defuzzification to achieve the interpolation result:

Interpolation Method Comparison
In order to validate the superiority of the information diffusion method for interpolating small sample data, reanalysis data from the South China Sea Institute of Oceanology, jointly established by the Institute of Oceanology of the Chinese Academy of Sciences (CAS) and the CAS Center for Ocean Mega-Science, is employed.The data covers the geographic range of 0°N to 40°N in latitude and 100°E to 130°E in longitude, spanning the eastern part of Asia and the western part of the Pacific Ocean, encompassing diverse geographical and climatic features.In this experiment, interpolation is carried out for each month.Due to space reasons, this paper presents the experimental results in December when the temperature difference is larger.Experiment 1: Monthly average temperature data for December 2019 is selected.The data coverage is from 0°N to 40°N in latitude and 100°E to 130°E in longitude.The resolution is 0.5° by 0.5°, resulting in a grid of 80 points in the longitudinal direction and 60 points in the latitudinal direction, totaling 4800 grid points.After excluding land points, there are 2647 remaining grid points.Five sampling experiments are conducted, with each experiment selecting 1% of the grid points (26 sample     Based on the information presented in Table 1 and Figure 4, under a 1% sampling condition, it is evident that the Asymmetric Fuzzy Diffusion (AFD) interpolation method has the lowest root mean square error (RMSE) at 1.764 and the highest correlation coefficient (R) at 0.952.Therefore, the AFD interpolation method performs the best, followed by the Isotropic Fuzzy Diffusion (IFD) method, while the traditional Ordinary Kriging (OK) interpolation method exhibits the poorest performance.
Experiment 2:The data, time, and region remain the same as in Experiment 1.However, the sampling points are increased to 5%, which involves randomly selecting around 130 samples from the  The root mean square error (RMSE) and correlation coefficient (R) between the interpolated sea surface temperature field and the actual field are presented in Table 2.  Based on the information from Table 2 and Figure 6, it can be observed that under a 5% sampling condition, the Asymmetric Diffusion Interpolation method exhibits the lowest Root Mean Square Error (RMSE) of 0.886 and the highest correlation coefficient (R) of 0.985.The Isotropic Diffusion method and the traditional Ordinary Kriging (OK) interpolation method show similar performance with a minor difference.Kriging interpolation is a classic and effective method in spatial interpolation [19][20].However, based on the comparison of the results from the two experiments, it is evident that in scenarios with sparse samples, the asymmetric diffusion interpolation method exhibits the best interpolation performance.It demonstrates the smallest errors and the highest correlation coefficients.Particularly, even in cases of extremely sparse samples, the asymmetric diffusion interpolation method consistently maintains smaller errors, outperforming both normal diffusion interpolation and Kriging interpolation methods.In other months, the interpolation method based on information diffusion can still achieve good interpolation results.

Conclusion
This paper addresses the issue of limited distribution of sea surface temperature observation stations, which leads to data gaps and interpolation challenges.A small-sample interpolation method based on information diffusion is proposed.By spreading the information of sparse sample points to the monitoring point space, accurate interpolation of environmental elements such as sea surface temperature is achieved.In the experiments, we compare the effectiveness of three methods: ordinary kriging interpolation, normal diffusion-based interpolation, and asymmetric diffusion-based interpolation.
The experimental results demonstrate the clear superiority of the information diffusion interpolation method in handling interpolation issues with small sample data.Through this study, we explore the application of information diffusion theory in interpolating environmental elements and validate its effectiveness in scenarios with sparse samples.This method holds promise as a viable and effective approach to address real-world environmental challenges, providing valuable insights for scientific research and decision-making in fields like environmental monitoring and climate studies.Future research could further investigate the potential application of this method in interpolating other environmental elements and advance the theoretical and practical exploration of information diffusion interpolation methods.

1 y
kx b =+ , the direction in which sample points diffuse faster is certainly along the line a .Therefore, the likelihood of sample points being near ICFOST-2023 Journal of Physics: Conference Series 2718 (2024) 012021IOP Publishing doi:10.1088/1742-6596/2718/1/0120215 the line is high, meaning that the sum of the squared distances between each sample point and the line is minimized.This allows us to calculate and using Equation (11): . The scaling coefficient  reflects the flatness of the ellipse.In this context, it can be represented by the ratio of the maximum distance from each sample point to the short axis line b to the maximum distance from the sample point to the major axis line a.
of Physics: Conference Series 2718 (2024) 012021 points) as known observation points, and the data at the remaining grid points are considered missing.The interpolation methods used include Ordinary Kriging (OK), Isotropic Fuzzy Diffusion (IFD), and Anisotropic Fuzzy Diffusion (AFD).The sample point distribution from five sampling experiments under a 1% sampling condition is shown in Figure 3.

Figure 3 .
Figure 3. Distribution of Sample Data (1% Sampling)The root mean square error (RMSE) and correlation coefficient (R) between the interpolated sea surface temperature field and the actual field are presented in Table1.Table1.RMSE and R of Different Interpolation Methods (1% Sampling)
.1088/1742-6596/2718/1/012021 8 data points as known observation points.The remaining grid points are considered as missing data points.Different interpolation methods are used, and this process is repeated five times.The sample point distribution from five sampling experiments under a 5% sampling condition is shown in Figure5.

Figure 5 .
Figure 5. Distribution of Sample Data (5% Sampling)The root mean square error (RMSE) and correlation coefficient (R) between the interpolated sea surface temperature field and the actual field are presented in Table2.Table2.RMSE and R of Different Interpolation Methods (5% Sampling)
Schematic Representation of the Working Principle of Information Diffusion Theory The formula for calculating the diffusion coefficient is as follows: 15)

Table 2 .
RMSE and R of Different Interpolation Methods (5% Sampling)