Water Multi-Parameter Sampling Design Method Based on Adaptive Sample Points Fusion in Weighted Space

: The spatial representativeness of the in-situ data is an important prerequisite for ensuring the reliability and accuracy of remote sensing product retrieval and veriﬁcation. Limited by the collection cost and time window, it is essential to simultaneously collect multiple water parameter data in water tests. In the shipboard measurements, sampling design faces problems, such as heterogeneity of water quality multi-parameter spatial distribution and variability of sampling plan under multiple constraints. Aiming at these problems, a water multi-parameter sampling design method is proposed. This method constructs a regional multi-parameter weighted space based on the single-parameter sampling design and performs adaptive weighted fusion according to the spatial variation trend of each water parameter within it to obtain multi-parameter optimal sampling points. The in-situ datasets of three water parameters (chlorophyll a, total suspended matter, and Secchi-disk Depth) were used to test the spatial representativeness of the sampling method. The results showed that the sampling method could give the sampling points an excellent spatial representation in each water parameter. This method can provide a fast and efﬁcient sampling design for in-situ data for water parameters, thereby reducing the uncertainty of inversion and the validation of water remote sensing products.


Introduction
Inland water worldwide has undergone tremendous changes from the pressure of climate change and human activities [1,2]. With the development of remote sensing technology, remote sensing data have been widely used in water monitoring, development, and water resources assessment [3][4][5][6][7]. More and more water remote sensing products are applied based on different sensors or spatial and temporal scales. For example, Tortini has used the multiple satellite data and the surface extent estimated data of the Moderate Resolution Imaging Spectrometer (MODIS) to produce the water remote sensing dataset of 347 lakes and reservoirs [8]; Wang released the global inland water remote sensing Forel-Ule index data set from 2000 to 2018 based on MODIS data [9]. These water remote sensing products have been widely used in water pollution, environmental monitoring, eutrophication, etc. [10][11][12][13][14]. However, many uncertainties exist in the quality of water remote sensing products retrieved by different inversion algorithms. Therefore, scientific evaluation of these water remote sensing products is the basis for applying water remote sensing products. The validation of water remote sensing products independently evaluates their uncertainty by comparing them with reference data (relative true values) [15,16]. The validation of water remote sensing products includes essential technical methods, such as spatial sampling design, scaling conversion, validation strategy, etc. [17]. Among them, as significant work in validation, the scientificity and reliability of spatial sampling design directly influence the quality of the in-situ data [15,18]. 1 1 Figure 1. The geographical location of the two experimental areas in this study, the sampling area, is in the red box: Nanyi Lake (a) and Boston Lake (b). Nanyi Lake is located in the southeastern part of Anhui Province, China. Its center coordinates are 118 • 56 E and 31 • 05 N. It is 20 km long and 17 km wide and has a water area of 189 km 2 . Bosten Lake is located in the northwestern part of Xinjiang Province, China. It is the largest inland freshwater lake in China. Its location is between 86 • 40 ∼ 87 •

Experimental Data
The experimental data included remote sensing image data and in-situ data. The former included Sentinel-2 and GaoFen (GF) high-resolution satellite remote sensing image data, and the latter was a water multi-parameter dataset measured by various instruments. The specific information is shown in Table 1.

Preprocessing of Remote Sensing Dataset and In-Situ Dataset
Remote sensing images were used from multiple satellites as prior knowledge in this paper. The sentinel-2 satellite remote sensing images were obtained through the European Union's Earth observation program (https://scihub.copernicus.eu, accessed on 28 August 2021). The GaoFen satellite remote sensing data were obtained through the China Centre for Resources Satellite Data and Application (https://www.cresda.com, accessed on 10 September 2021). Firstly, radiometric and atmospheric corrections are carried out on remote sensing images. Then, the GF data are resampled to 10 m, and the effect of outliers is removed. The water extraction algorithm was used to separate land and water. Erosion Operation in Mathematical Morphology was used to remove the influence of the shore area. Third, the in-situ dataset is divided into two parts. The in-situ dataset of Nanyi Lake includes three water parameter data of Chl-a, TSM, and SD of 15 sampling points. The collected water parameters were Chl-a, TSM, and SD. The in-situ dataset of Bosten Lake includes three water quality parameter data of 32 sampling points, and 16 sampling points are arranged using the adaptive weight sampling method and the systematic sampling method, respectively. The spatial distribution of sampling points is shown in Figure 1. In field measured data. A water parameter spectrometer directly measured the Chl-a; the TSM was measured by conventional drying, baking, and weighing methods; the SD was measured by the Secchi disk [37]. The preliminary statistics of the measured data are shown in Table 2.

Method
The water multi-parameter sampling design method proposed is based on the existing water single-parameter sampling design method. Therefore, comparing the current spatial sampling design methods is necessary to evaluate their effectiveness and representativeness in water sampling design. This paper studies the spatial sampling design using high-

Spatial Distribution of Water Parameters
Understanding the spatial distribution of water parameters is essential for research sampling design. Because the spatial distribution of the parameters changes with time, in most cases, prior knowledge of various water parameters cannot be obtained in time. Remote sensing images have the characteristics of fast information acquisition and short cycles. Therefore, it is an effective method to get the spatial distribution of the parameters by using remote sensing images near the sampling time [38][39][40][41]. To ensure the broad applicability of the sampling design method, the sensitive band or the combination of sensitive bands for water parameters on remote sensing images was applied to represent the spatial distribution characteristics of the parameters [42]. In this paper, taking the experimental sample area of Nanyi Lake as an example, a sampling design study was carried out on the three parameters of Chl-a, TSM, and SD. Previous studies have shown that the near-infrared/red, red/green, and green bands can be used as remote sensing feature bands for Chl-a, TSM, and SD [43][44][45][46][47]. Therefore, this paper uses Band 8/Band 3, Band 4/Band 3, and Band 3 of Sentinel-2 satellites as the sensitive bands of Chl-a, TSM, and SD and generates the spatial distribution characteristics map of the water parameters.
To describe the heterogeneity of the spatial distribution characteristics of each water parameter, this paper uses K-means to perform spatial clustering on the spatial distribution characteristic map of each water parameter [48,49]. The optimal number of clusters for K-means clustering was determined using the sum of squared errors (SSE). SSE is the sum of the squared errors of each observation and its cluster center, which measures the closeness within a category [50]. As shown in Formula (1): where represents all points in the th category, is the number of categories, and ( , ) represents the Euclidean distance between the points in and the cluster center . Figure 2. Flowchart of the method used in this paper.

Spatial Distribution of Water Parameters
Understanding the spatial distribution of water parameters is essential for research sampling design. Because the spatial distribution of the parameters changes with time, in most cases, prior knowledge of various water parameters cannot be obtained in time. Remote sensing images have the characteristics of fast information acquisition and short cycles. Therefore, it is an effective method to get the spatial distribution of the parameters by using remote sensing images near the sampling time [38][39][40][41]. To ensure the broad applicability of the sampling design method, the sensitive band or the combination of sensitive bands for water parameters on remote sensing images was applied to represent the spatial distribution characteristics of the parameters [42]. In this paper, taking the experimental sample area of Nanyi Lake as an example, a sampling design study was carried out on the three parameters of Chl-a, TSM, and SD. Previous studies have shown that the near-infrared/red, red/green, and green bands can be used as remote sensing feature bands for Chl-a, TSM, and SD [43][44][45][46][47]. Therefore, this paper uses Band 8/Band 3, Band 4/Band 3, and Band 3 of Sentinel-2 satellites as the sensitive bands of Chl-a, TSM, and SD and generates the spatial distribution characteristics map of the water parameters.
To describe the heterogeneity of the spatial distribution characteristics of each water parameter, this paper uses K-means to perform spatial clustering on the spatial distribution characteristic map of each water parameter [48,49]. The optimal number of clusters for K-means clustering was determined using the sum of squared errors (SSE). SSE is the sum of the squared errors of each observation and its cluster center, which measures the closeness within a category [50]. As shown in Formula (1): where C i represents all points in the ith category, k is the number of categories, and d 2 (x, m i ) represents the Euclidean distance between the points in C i and the cluster center m i .

Comparison of Water Single-Parameter Sampling Methods
High-quality single-parameter sampling design results are the premise of multiparameter spatial sampling design. To verify the effectiveness of different single water parameter sampling design methods, this paper selects and evaluates six water parameter spatial sampling methods, including three conventional methods (random sampling, systematic sampling, stratified sampling) and three intelligent optimization methods based on objective function (SSA, GA, PSO). The specific information of each sampling design method is shown in Table 3. Among them, the objective function of spatial sampling design based on an intelligent optimization algorithm is the minimization of mean square error (MSE). MSE is the mean square error between the kriging interpolation surface of the sampling points and the spatial distribution characteristic map of the water parameters [51,52]. As shown in Formula (2): where V i represents the pixel value of the spatial distribution map of the parameters,V i represents the estimated value of sampling area obtained by kriging interpolation surface, and N represents the number of pixels of the sampling area.

GA sampling MSE
The objective function is minimized by genetic operations, such as selection, crossover, and variation of different initial sampling points [57,58]

SSA sampling
The objective function is minimized by allocating sampling locations randomly [36,59,60]

PSO sampling
The objective function is to minimize collaboration and information sharing among individuals in the group [61]

Water Multi-Parameter Sampling Method
The water multi-parameter sampling design needs to comprehensively consider the spatial distribution of various parameters. Therefore, to balance the local spatial representation of multiple parameter sampling points, we propose an adaptive sampling points fusion method in multi-parameter weighted space in this paper. The basic idea is to use the sampling points of multiple groups of parameters to construct a local weighted space. The multi-parameter sampling points in the weighted space are obtained by weighted fusion of the single-parameter sampling points. The weighted space refers to the nonoverlapping polygonal areas (or lines) constructed adaptively with multiple parameter sampling points as vertices in the local sampling area. An example of the weighted space is shown in Figure 3. When there are two water parameters, the weighted space is a line segment (a, b). When there are three or more water parameters, the weighted space is a polygonal area (c, d). The basic framework of multi-parameter sampling design: where Q(x, y) represents adaptive sampling points, w 1 , w 2 · · · w n represents weight factor, P 1 (x 1 , y 1 ), P 2 (x 2 , y 2 ) · · · P n (x n , y n ) represents each parameter sampling points. Among them, the size of each weight factor determines the position of the adaptive sampling points in the weighted space and the ability to balance the sampling points of each water parameter. In conventional methods, equal weights are usually used to construct weighting factors, but equal weights are more suitable for the case where the spatial variation of each parameter is consistent. Since the spatial variation characteristics of various water parameters in the weighted space are different and in order to determine the optimal adaptive sampling point in the weighted space, this paper uses the spatial variation characteristics of each water parameter in the weighted space to construct the weights. The faster the characteristics change, the stronger the spatial heterogeneity and it should be given a higher weight. In this paper, directional derivatives are used to describe the rate of spatial variation of the water parameters at the sampling point [62]. The solution process is as follows: The weight coefficient w depends on the directional derivatives of P i (x, y) and Q(x, y), and the calculation method is as follows: The → QP i is expressed as: The azimuth ϕ i of → QP i is: The directional derivative f i (ϕ i ) of P i in the direction of → QP i is: The weight w i is expressed as: The multi-parameter sample point locations are solved by the ordinary least square method: where α is the scaling factor. A simple numerical example is shown to explain the solution process of the adaptive weight sampling design. First, as shown in Figure 3d, P 1 , P 2 , P 3 are Chl-a, TSM, and SD sampling points, respectively; Q is the adaptive weight sampling point. Their coordinates and values are assumed for: Secondly, since the water parameters show a very obvious continuous change in the water, the rate of change of Chl-a, TSM and SD values near the sampling points is assumed to be linear with the distance. Therefore, the change functions of the three water parameters are: where r is the Euclidean distance between the other sampling points and the sampling points of the three water parameter. The advantage of the above assumption is that the rate of change of the water parameters in any direction near the sampling point is constant. In other words, its directional derivative is the first derivative of the change function: (12) Finally, the adaptive sampling point is solved by its Euclidean distance relationship with the three water parameter sampling points:

Evaluation Method
Relative precision (RP mean ) is used to evaluate the representativeness of sampling, and it is defined as the ratio of the mean of the sample data to the mean of all the data in the sampling [23]. The formula is as follows: where mean(x i ) represents the sample mean of the ith sampling method, mean(x b ) represents the average of all data in the sampling area. The root means square error (RMSE) was utilized to assess the effectiveness of the sampling methods. RMSE is the root mean square error between the kriging interpolation surface of the sampling points and the spatial distribution characteristics map of the parameters [63]. The formula for RMSE is as follows: where V i represents the pixel value of the spatial distribution characteristics map of the water parameters,V i represents the estimated value of the sampling area obtained by kriging interpolation, and N represents number of pixels of the sampling area. The Spearman correlation coefficient (R) was applied to evaluate the correlation between the kriging interpolation surface of the in-situ data and remote sensing image [64]. The formula is as follows: where n is the number of sample points, and d i is the grade difference between the value on the remote sensing image and the kriging interpolation surface of the in-situ data value. In this paper, 1000 sampling points are uniformly selected in the sampling area. In order to evaluate the representativeness of the measured data over the range. The average relative error (MRE) is used to evaluate the representativeness of the in-situ data in the range. Ideally, the water parameter data collected should be equally spaced across the range. It can make the in-situ data more representative in the experimental area. Therefore, we assume a set of uniformly distributed simulated sampled data as the standard value, and the MRE assess the error of the in-situ data and the standard value [65]. The lower the MRE, the better the distribution of the in-situ data in the range. The formula is shown in Formula (12): where V i represents the in-situ data, S i represents the simulated data uniformly distributed in the range, and n is the number of the in-situ data. Figure 4 reveals the trend of SSE and the spatial distribution characteristics of the three parameters. The SSE decreases with the increase in clusters K. When K > 4, the SSE of change tends to be flat, and the SSE does not improve much as the number of clusters K increases. Therefore, we determined that the number of clusters of each parameter is four. From Figure 4b-d, each class of Chl-a was relatively discrete. The spatial distribution of each category of the TSM and SD was relatively concentrated, and there is heterogeneity in the spatial distribution characteristics of each water parameter. The Spearman correlation coefficient (R) was applied to evaluate the correlation between the kriging interpolation surface of the in-situ data and remote sensing image [64]. The formula is as follows:

Spatial Distribution of Various Water Parameters
where is the number of sample points, and is the grade difference between the value on the remote sensing image and the kriging interpolation surface of the in-situ data value. In this paper, 1000 sampling points are uniformly selected in the sampling area.
In order to evaluate the representativeness of the measured data over the range. The average relative error (MRE) is used to evaluate the representativeness of the in-situ data in the range. Ideally, the water parameter data collected should be equally spaced across the range. It can make the in-situ data more representative in the experimental area. Therefore, we assume a set of uniformly distributed simulated sampled data as the standard value, and the MRE assess the error of the in-situ data and the standard value [65]. The lower the MRE, the better the distribution of the in-situ data in the range. The formula is shown in Formula (12): where represents the in-situ data, represents the simulated data uniformly distributed in the range, and is the number of the in-situ data. Figure 4 reveals the trend of SSE and the spatial distribution characteristics of the three parameters. The SSE decreases with the increase in clusters K. When K > 4, the SSE of change tends to be flat, and the SSE does not improve much as the number of clusters K increases. Therefore, we determined that the number of clusters of each parameter is four. From Figure 4(b-d), each class of Chl-a was relatively discrete. The spatial distribution of each category of the TSM and SD was relatively concentrated, and there is heterogeneity in the spatial distribution characteristics of each water parameter.

Spatial Representative Comparison of Various Sampling Design Methods
Sampling points can reflect the overall distribution characteristics of water parameters in the sampling area. The representativeness of the sampling points can be represented by the difference between the statistical data of the sampling point and the statistical data of the population (RP mean ). The RP mean is used to analyze the representativeness of sampling points in statistical theory in this paper. This paper used six sampling methods to conduct 100 simulated samplings on the spatial distribution characteristic maps of the three water parameters. The number of sampling points was set to 15. The RP mean value of each simulated sample was extracted as shown in Figure 5. It can be concluded that the interquartile ranges (IQR) of SSA in the three water parameters are 0.0051, 0.0076 and 0.0070, and the results are all smaller than other sampling methods. Its range is also smaller than other sampling methods, which indicates the representativeness of the SSA sampling points are less volatile. Then, although the median value of stratified sampling in Chl-a is slightly higher than that of the SSA, in TSM and SD, the median of the SSA is closer to one than other methods. Random and GA methods have too high or too low outliers, and the representativeness of their sampling results is relatively unstable. Therefore, the RP mean results show that the sampling points obtained by the SSA method are well represented in the characteristic distribution maps of the three water parameters.   SSA (a, b, c) and Chl-a TSM and SD sampling points (d, e, f). After this initial phase, the RMSE decreased steadily and gradually stabled after 10,000 iterations. No further reduction in the RMSE was achieved after about 14,000 iterations, indicating that the sampling design reached the optimal solution. The RMSE for the three water parameters was reduced from 0.0537, 0.119, and 0.01 to In order to further evaluate the spatial representativeness of various sampling methods, the RMSE between the kriging interpolation of sampling points and the spatial distribution characteristics maps of the three water parameters was used as the evaluation index. The RMSE value of each simulated sampling is recorded, as shown in Figure 6. The RMSE box plot of a sample with good representability should have the following characteristics: (1) the median and IQR should be small; (2) no outliers [23]. From Figure 6, in the characteristic distribution map of the three water parameters, the median of the RMSE of the SSA sampling method was 0.0135, 0.0216, and 0.0055, respectively. Compared with other sampling methods, the median of the SSA sampling method was the smallest. In addition, the IQR of the SSA method was significantly better than the other sampling methods. It showed that the SSA sampling method has good spatial representation and stability. The random, stratified, and GA all had local outliers, indicating that their sampling results were unstable. The result of the systematic sampling method after a fixed interval was more dependent on the heterogeneity of the sampling area. If the sampling area has firm heterogeneity, the representativeness of its sampling points is relatively weak. The median and the IQR of the PSO sampling method are second only to the SSA and outperform other sampling methods, and the sampling results also have good stability. Therefore, it can be concluded that the SSA method is more suitable for the water single-parameter optimal sampling method.   Figure 7 presents the decrease in RMSE during SSA (a, b, c) and Chl-a TSM and SD sampling points (d, e, f). After this initial phase, the RMSE decreased steadily and gradually stabled after 10,000 iterations. No further reduction in the RMSE was achieved after about 14,000 iterations, indicating that the sampling design reached the optimal solution. The RMSE for the three water parameters was reduced from 0.0537, 0.119, and 0.01 to  Figure 7 presents the decrease in RMSE during SSA (a, b, c) and Chl-a TSM and SD sampling points (d, e, f). After this initial phase, the RMSE decreased steadily and gradually stabled after 10,000 iterations. No further reduction in the RMSE was achieved after about 14,000 iterations, indicating that the sampling design reached the optimal solution. The RMSE for the three water parameters was reduced from 0.0537, 0.119, and 0.01 to 0.0128, 0.0196, and 0.006. It showed that the prediction accuracy of the three water parameters was improved by 76%, 83%, and 50%. At the same time, we found that the sampling points of the three water parameters were evenly distributed in each category of the sampling area. This showed that the SSA improves the representativeness of the sampling points of the three water parameters. 0.0128, 0.0196, and 0.006. It showed that the prediction accuracy of the three water parameters was improved by 76%, 83%, and 50%. At the same time, we found that the sampling points of the three water parameters were evenly distributed in each category of the sampling area. This showed that the SSA improves the representativeness of the sampling points of the three water parameters.  Figure 8 shows the multi-parameter weighted space and sampling points of the adaptive weight sampling design method. Table 4 lists the RMSE of Chl-a sampling points, TSM sampling points, SD sampling points, adaptive weight sampling points, the centroid of the triangle, and the incentre of the triangle in the distribution characteristic map of the three water parameters. The optimization results show that sampling points of each parameter only had the smallest RMSE on its spatial distribution characteristic map. In contrast, the RMSE on the spatial distribution characteristic map of other water parameters was higher, indicating that the single-parameter sampling points were less representative of the other water parameters' spatial distribution characteristic map. The RMSE of adaptive weight sampling points in the spatial distribution characteristic map of the three wa-  Figure 8 shows the multi-parameter weighted space and sampling points of the adaptive weight sampling design method. Table 4 lists the RMSE of Chl-a sampling points, TSM sampling points, SD sampling points, adaptive weight sampling points, the centroid of the triangle, and the incentre of the triangle in the distribution characteristic map of the three water parameters. The optimization results show that sampling points of each parameter only had the smallest RMSE on its spatial distribution characteristic map. In contrast, the RMSE on the spatial distribution characteristic map of other water parameters was higher, indicating that the single-parameter sampling points were less representative of the other water parameters' spatial distribution characteristic map. The RMSE of adaptive weight sampling points in the spatial distribution characteristic map of the three water parameters were 0.0156, 0.0216, and 0.0065. Compared with the optimal sampling points for single-parameter, the spatial representativeness of the adaptive weight sampling points was slightly reduced. Still, its advantage was that it could maintain relatively high spatial representativeness simultaneously among the three water parameters and demonstrated the ability to balance the spatial distribution of multiple water parameters.

Accuracy Evaluation of In-Situ Dataset of Water Parameters
To assess the spatial representativeness of the adaptive weight sampling design method, we used the in-situ dataset to verify the sampling method. In the water experiment of Nanyi Lake, this sampling method was applied to design 15 sampling points to assess the effectiveness of the sampling method. In the water experiment of Bosten Lake, 16 sampling points were designed using systematic sampling and adaptive weight sampling design method to prove the spatial representativeness of the sampling method. Figure 9 displays the 15 sampling points and the MRE of the in-situ and simulated data in Nanyi Lake. The simulated data are a set of optimal values obtained by assuming that the data of the sampling points are uniformly distributed in the range. The range of the simulated values is the maximum and minimum values of the in-situ data. The simulated values of the other sampling points were evenly distributed in the value. The MRE of the in-situ and simulated data for the three water parameters was 6.76%, 5.72%, and 2.67%. The errors were within an acceptable range (10%), indicating that the in-situ data were uniformly distributed and had a good representationin the range.

Accuracy Evaluation of In-Situ Dataset of Water Parameters
To assess the spatial representativeness of the adaptive weight sampling design method, we used the in-situ dataset to verify the sampling method. In the water experiment of Nanyi Lake, this sampling method was applied to design 15 sampling points to assess the effectiveness of the sampling method. In the water experiment of Bosten Lake, 16 sampling points were designed using systematic sampling and adaptive weight sampling design method to prove the spatial representativeness of the sampling method. Figure 9 displays the 15 sampling points and the MRE of the in-situ and simulated data in Nanyi Lake. The simulated data are a set of optimal values obtained by assuming that the data of the sampling points are uniformly distributed in the range. The range of the simulated values is the maximum and minimum values of the in-situ data. The simulated values of the other sampling points were evenly distributed in the value. The MRE of the in-situ and simulated data for the three water parameters was 6.76%, 5.72%, and 2.67%. The errors were within an acceptable range (10%), indicating that the in-situ data were uniformly distributed and had a good representationin the range. data in Nanyi Lake. The simulated data are a set of optimal values obtained by assuming that the data of the sampling points are uniformly distributed in the range. The range of the simulated values is the maximum and minimum values of the in-situ data. The simulated values of the other sampling points were evenly distributed in the value. The MRE of the in-situ and simulated data for the three water parameters was 6.76%, 5.72%, and 2.67%. The errors were within an acceptable range (10%), indicating that the in-situ data were uniformly distributed and had a good representationin the range.    Figure 10 displays the 15 sampling points and the MRE of the in-situ data and simulated data in Bosten Lake. On the one hand, The MRE of the adaptive weight sampling method was 6.24%, 9.63%, and 4.68%, while the MRE of the systematic sampling method was 17.15%, 29.54%, and 6.54%. The MRE of Chl-a, TSM, and SD improved by 10.91%, 19.91%, and 1.86%. On the other hand, taking SD as an example, the range of system sampling points was [340-330], while the range of the multi-parameter sampling points was [240,366], and the breadth of the range of the latter was improved by 27%. The results showed that the adaptive weight sampling points were more uniformly distributed and had more capacity in the range than the systematic sampling points. method was 6.24%, 9.63%, and 4.68%, while the MRE of the systematic sampling method was 17.15%, 29.54%, and 6.54%. The MRE of Chl-a, TSM, and SD improved by 10.91%, 19.91%, and 1.86%. On the other hand, taking SD as an example, the range of system sampling points was [340-330], while the range of the multi-parameter sampling points was [240,366], and the breadth of the range of the latter was improved by 27%. The results showed that the adaptive weight sampling points were more uniformly distributed and had more capacity in the range than the systematic sampling points. To further evaluate the spatial representativeness of the adaptive weight sampling method, we extracted the sensitive bands of the three water parameters in the synchronously observed remote sensing images during on-site sampling and calculated the spatial distribution characteristic maps of the three water parameters. The interpolation surface of the sampling area is obtained by performing kriging interpolation on the in-situ data of the sampling points. Finally, in this paper, 1,000 sampling points are simultaneously extracted from the spatial distribution characteristics map of water parameters and the kriging interpolation surface for correlation analysis; the results are shown in Figure  11. The correlation coefficients of the systematic sampling method were 0.45, 0.54, and 0.61, while the correlation coefficients of the adaptive weight sampling method were 0.60, Figure 10. There are multi-parameter sampling points in Bosten lake (a-c) and the MRE of the in-situ and simulated data (d-f). Black dots are simulated data, red dots are adaptive weight sampling points, and blue dots are system sampling points.
To further evaluate the spatial representativeness of the adaptive weight sampling method, we extracted the sensitive bands of the three water parameters in the synchronously observed remote sensing images during on-site sampling and calculated the spatial distribution characteristic maps of the three water parameters. The interpolation surface of the sampling area is obtained by performing kriging interpolation on the in-situ data of the sampling points. Finally, in this paper, 1,000 sampling points are simultaneously extracted from the spatial distribution characteristics map of water parameters and the kriging interpolation surface for correlation analysis; the results are shown in Figure 11. The correlation coefficients of the systematic sampling method were 0.45, 0.54, and 0.61, while the correlation coefficients of the adaptive weight sampling method were 0.60, 0.65, and 0.60. The correlation coefficients of Chl-a and TSM are relatively improved by 0.13 and 0.11, while the correlation coefficients of SD are equal. The results show that the correlation of adaptive weight sampling points in the three water parameters is better than the systematic sampling method. Meanwhile, we find that the systematic sampling points have better spatial representation in SD, but the spatial representation of Chl-a and TSM is insufficient. The adaptive weight sampling design method has a similar correlation among the three water parameters, indicating that it can balance the spatial distribution of three water parameters. Thus, it is reasonable to conclude that the adaptive weight sampling design method is significantly better than the systematic sampling method.

Discussion
The spatial sampling design is an essential prerequisite for ensuring the validity and rationality of in-situ data for the inversion and validation of water remote sensing products. More and more researchers pay attention to the spatial representation of sampling points research. The field measurement of water parameters mainly includes fixed station and shipboard measurements. Some scholars have researched water multi-parameter sampling design, but they are primarily used to select long-term fixed measurement points [36]. Different from measurement at fixed stations, shipboard measurement faces more challenges. On the one hand, the sampling design will make necessary changes to factors, such as weather, ships, and waterways. This requires sampling designs to be fast, efficient, and stable. On the other hand, the number of sampling points is limited due to the limitations of economic cost and the time window of satellite-ground synchronization observation. It is usually necessary to collect multiple water quality parameters at one sampling point. However, there is a specific heterogeneity in the spatial distribution of different water quality parameters. Designing sampling points to improve their spatial representation in different water parameters under a limited number of sampling points to meet the needs of the inversion and verification of remote sensing satellite products is also a key problem to be solved in this paper. Therefore, this paper focuses on the sampling design problem in the shipboard water parameter measurement tests. We propose a water multi-parameter sampling design method based on adaptive sample points fusion in multi-parameter weighted space. The results show that it effectively solved the problem of multi-parameter spatial representation under the condition of heterogeneous spatial distribution characteristics. According to our calculations, taking the Nanyi Lake sampling area as an example, the algorithm's time complexity is one-fourth to one-sixth of the UCK sampling method. The fast and efficient sampling design capability enables the sampling plan to be redesigned quickly in variable environmental conditions. This makes the method more suitable for sudden needs, such as algal blooms, post-disaster assessment, and other applications. This dramatically improves the ability of the shipboard measure-

Discussion
The spatial sampling design is an essential prerequisite for ensuring the validity and rationality of in-situ data for the inversion and validation of water remote sensing products. More and more researchers pay attention to the spatial representation of sampling points research. The field measurement of water parameters mainly includes fixed station and shipboard measurements. Some scholars have researched water multi-parameter sampling design, but they are primarily used to select long-term fixed measurement points [36]. Different from measurement at fixed stations, shipboard measurement faces more challenges. On the one hand, the sampling design will make necessary changes to factors, such as weather, ships, and waterways. This requires sampling designs to be fast, efficient, and stable. On the other hand, the number of sampling points is limited due to the limitations of economic cost and the time window of satellite-ground synchronization observation. It is usually necessary to collect multiple water quality parameters at one sampling point. However, there is a specific heterogeneity in the spatial distribution of different water quality parameters. Designing sampling points to improve their spatial representation in different water parameters under a limited number of sampling points to meet the needs of the inversion and verification of remote sensing satellite products is also a key problem to be solved in this paper. Therefore, this paper focuses on the sampling design problem in the shipboard water parameter measurement tests. We propose a water multi-parameter sampling design method based on adaptive sample points fusion in multi-parameter weighted space. The results show that it effectively solved the problem of multi-parameter spatial representation under the condition of heterogeneous spatial distribution characteristics. According to our calculations, taking the Nanyi Lake sampling area as an example, the algorithm's time complexity is one-fourth to one-sixth of the UCK sampling method. The fast and efficient sampling design capability enables the sampling plan to be redesigned quickly in variable environmental conditions. This makes the method more suitable for sudden needs, such as algal blooms, post-disaster assessment, and other applications. This dramatically improves the ability of the shipboard measurements scheme to cope with changes in various conditions.
Although the multi-parameter sampling method has achieved good analysis results, the practical application capabilities need further evaluation. The multi-parameter sampling method needs some constraints to better apply to the actual sampling design. The most crucial issue is the validity of prior knowledge. This paper uses high-resolution remote sensing satellites as prior knowledge to reflect the spatial representation of different water parameters. Due to weather and satellite transit time, we may not be able to obtain the nearest high-resolution satellite remote sensing imagery before field sampling. Therefore, satellite images more days away from the experiment time and even contemporaneous satellite images from other years in history are used for prior knowledge acquisition. For example, we use remote sensing images four days before on-site measurement as the previous knowledge of water sampling in Bosten Lake. Due to the influence of various environmental conditions, the spatial distribution of water parameters will change with time, and this change cannot be effectively simulated and predicted. Therefore, it is necessary to analyze the time scale effect of the spatial distribution of water parameters in combination with the influencing factors of water environment changes to ensure the reliability of the spatial representation of remote sensing images. Moreover, the remote sensing image data may come from different satellites. For example, the remote sensing image data of GF and sentinel-2 satellites are used in this paper. Different satellites have different sensitive bands for water parameters. Different remote sensing satellite data may not be able to use the same band or band combination when describing the spatial representation of water parameters. The wrong band selection may lead to insufficient spatial representation, which will significantly affect the reliability of the sampling design method. The literature analysis of water quality parameter modeling is an excellent way to obtain the sensitive bands of different satellites to water quality parameters [47,66]. Therefore, it is necessary to study the sensitive bands of different satellites to different water parameters to determine the optimal combination of sensitive bands.
Spatial-temporal constraints in sampling design are also factors to consider. This paper pays more attention to evaluating the spatial representation of sampling, ignoring the time constraints. In fact, in the sampling design, the effect of time cannot be overlooked; it directly affects whether the experiment can be completed. For example, to ensure the reliability of the in-situ data, we need to be collected within ±1.5 h of the satellite transit. Usually, we determine the sampling range of the lake surface based on the experience of previous experiments. In fact, under the common constraints of the sampling ship's sailing speed and the dock's position, the area of a single sampling is within a limited range and cannot be estimated empirically. Especially for large inland waters (such as Lake Bosten), it is not easy to perform on-site sampling of the entire water surface within the specified time (±1.5). In this case, it is necessary to consider reducing the sampling points or the sampling area to ensure the quality of in-situ data. Therefore, we should also consider the time and space cost (for example, geographic accessibility, satellite transit time, water area, ship speed, dock location, etc.) of on-site sampling in the sampling design to consider the sampling design methods comprehensively.
To sum up, although the sampling design method proposed in this paper can quantitatively and objectively carry out the experimental design of water multi-parameters, it still faces some problems that need in-depth research. In future research, it is necessary to explore and design more effective and representative multi-parameter sampling design methods based on an in-depth analysis of the reliability of auxiliary data under more constraints to meet the needs of water remote sensing product inversion and verification.

Conclusions
In order to solve the problem of multi-parameter spatial representation in water sampling, this paper proposed a water multi-parameter sampling design method based on adaptive sample points fusion in multi-parameter weighted space, which has obvious advantages compared with the traditional sampling method. Using high-resolution satellite remote sensing images before the experiment as prior knowledge, the spatial distribution characteristics of water parameters can be reflected in time, which provides an effective reference for sampling design methods. On this basis, a multi-parameter sampling design method is constructed through techniques, such as regional weighted space construction and adaptive weight fusion. This method can consider various water parameters' spatial distribution and spatial variation characteristics. The research results show that the sampling results have good spatial representation across multiple water parameters, which can provide effective data support for satellite-ground synchronization observations. Therefore, the water multi-parameter sampling design method can provide an efficient and reliable sampling design method for the inversion and the verification of water remote sensing products. The continuous improvement of sampling methods will be widely used in water remote sensing products, monitoring, and evaluation.

Data Availability Statement:
The satellite data used in this study are in the public domain, available from ESA (https://scihub.copernicus.eu/dhus/#/home, accessed on 13 December 2021) and CRESDA (http://www.cresda.com/CN/, accessed on 13 December 2021). Other data that support the findings of this study are available from the author upon reasonable request.