Short-term Rainfall Prediction Method for Beidou Navigation System Based on LSTM Network

Abstract

The present invention discloses a short-term rainfall prediction method for a Beidou navigation system based on an LSTM network, comprising the following steps: First, divide the atmospheric precipitable water data into seasonal categories according to the rainy season and the non-rainy season; construct and train an LSTM network prediction model; obtain the real-time atmospheric precipitable water data of an observation point and input it into the pre-trained LSTM network prediction model to output the predicted rainfall probability value one hour later for this station; convert the predicted rainfall probability values of several individual stations into a graph structure according to spatial information, convert the graph structure into an adjacency matrix, and re-input the adjacency matrix into the pre-trained LSTM network prediction model to obtain a networked prediction result presented in the form of a graph; use a multilinear fitting algorithm to extract the predicted value at any position from the networked prediction result; adopt a distance weighting strategy to weight the predicted value at any position to obtain the final prediction result and output it.

Description

Technical Field

[0001] The present invention relates to the technical field of remote sensing inversion of navigation satellites, and specifically refers to a short-term rainfall prediction method for the Beidou navigation system based on the LSTM network. Background Art

[0002] High-precision short-term rainfall prediction is not only related to the selection of prediction quantities, but also closely related to the prediction algorithm. Currently, the short-term rainfall prediction algorithms based on Beidou technology mainly include two types: threshold-based and machine learning-based prediction algorithms. The idea of the threshold-based prediction algorithm is to use mathematical statistical methods to set prediction thresholds according to parameters such as prediction success rate, so as to predict rainfall. However, the existing threshold-based algorithms only consider the linear relationship between the prediction quantity and rainfall, while ignoring their non-linear relationship, resulting in a low prediction success rate; the machine learning-based prediction algorithms mainly perform simulation training through a large amount of historical data. Although the prediction success rate can be improved to a certain extent, it relies on a huge amount of data. If the historical data is less and the training amount is insufficient, the prediction accuracy will be severely reduced.

[0003] In addition, the existing short-term rainfall prediction algorithms mainly adopt a node-based prediction model. The node-based prediction model only considers the time correlation of the prediction quantity of a single station, not only has a small prediction coverage area, but also has a relatively high prediction error rate and omission rate. Due to the fluidity of the atmosphere, especially the short-term rainfall caused by severe weather such as typhoons, the spatial fluidity is stronger. Therefore, in order to effectively increase the prediction coverage area and spatial resolution, and reduce the prediction error rate and omission rate, not only the time correlation of the monitoring station itself needs to be considered, but also the spatial correlation of neighboring monitoring stations needs to be taken into account. In addition, the existing methods use continuous time series when training the rainfall prediction model, ignoring the seasonal rainfall characteristics of rainfall. Experiments show that there are significant differences in rainfall characteristics between the rainy season and the non-rainy season. For example, in the rainy season, a slight change in the atmospheric precipitable water can cause rainfall, while in the non-rainy season, a larger change amount is required, resulting in a large deviation in the prediction accuracy of the existing method's rainfall prediction model. Summary of the Invention

[0004] Aiming at the deficiencies of the existing technology, the present invention proposes a short-term rainfall prediction method for the Beidou navigation system based on the LSTM network, which can effectively overcome the problem that the current node-based prediction algorithm is difficult to effectively expand the prediction coverage area, resulting in a relatively high prediction error rate and omission rate.

[0005] To solve the above technical problems, the technical solution of the present invention is as follows:

[0006] A short-term rainfall prediction method for the Beidou navigation system based on the LSTM network, comprising the following steps:

[0007] Step 1: Download the relevant rainfall files over other stations around the observation point on the IGS (International GNSS Service) official website. Calculate the precipitable water vapor data over the monitoring station through the BeiDou precise point positioning technology. Conduct a correlation analysis between the precipitable water vapor data over each station and the rainfall at the observation station, and also perform a correlation analysis between the precipitable water vapor data and rainfall of the prediction station itself. The results are presented using the Spearman correlation coefficient.

[0008] Step 2: Obtain the historical precipitable water vapor data from the ERA5 (ECMWF, European Centre for Medium-Range Weather Forecasts) official website. Process it using the Seasonal Autoregressive Integrated Moving Average model (SARIMA) to extract the seasonal components and trends in the data. Use the processed data and seasonal component information as inputs to train the LSTM network prediction model. Considering that the most direct predictor affecting short-term rainfall comes from the temporal variation characteristics over the monitoring station, predictions need to be made separately for each station, and the prediction results are presented in the form of single nodes.

[0009] Step 3: Use the Gaussian diffusion model method to find the optimal transformation method. Combine the prediction results of a single monitoring station with spatial information to form a graph, and fuse the information of multiple spatially related nodes, such as the longitude and latitude of the stations, the distances between the monitoring stations, etc. Transform the graph structure into the adjacency matrix of the LSTM network, and use the constructed prediction image as the input of the LSTM network prediction model to re-predict. The prediction results are still presented in the form of a graph, thus achieving a networked prediction result.

[0010] Step 4: Use the multi-linear fitting algorithm to extract the prediction results at any position from the networked prediction results, thereby achieving accurate prediction at any position within the region and effectively expanding the prediction coverage. In addition, adopt a distance-weighted strategy to weight the fitting points, which can effectively improve the prediction accuracy of the prediction points. Therefore, through the above process, the prediction coverage can be effectively increased, the prediction error rate and omission rate can be reduced, and the short-term rainfall prediction requirements can be met.

[0011] After being processed by the above rainfall prediction algorithm, it can meet the high-precision short-term rainfall prediction at any position within the network, solve the problems that the existing node-based prediction algorithms are difficult to effectively expand the prediction coverage, and have relatively high prediction error rates and omission rates, and effectively expand the application scope and scenarios of short-term rainfall prediction.

[0012] The present invention has the following characteristics and beneficial effects:

[0013] Adopting the above technical solution, first consider the spatial correlation between stations, collect the data of precipitable water vapor (PWV) in the atmosphere above the prediction station and its surrounding stations, and analyze the correlation between the PWV data and rainfall of the prediction station itself and its surrounding stations; in addition, considering the seasonal characteristics of rainfall, analyze the data set in the rainy season and non-rainy season; use the prediction algorithm based on the LSTM network to perform real-time prediction on the entire area, realizing the prediction structure from nodes to the network; combine the multi-factor weighting strategy to perform single-point fitting on any node in the network to form a node-network-node type prediction model. This model can effectively overcome the problem that the current node-type prediction algorithm is difficult to effectively expand the prediction coverage, resulting in relatively high prediction error rate and omission rate. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0015] Figure 1 It is the flowchart of the short-term rainfall prediction method for the Beidou navigation system based on the LSTM network in this embodiment.

[0016] Figure 2 It is the flowchart of the LSTM network rainfall prediction model considering spatial correlation in this embodiment.

[0017] Figure 3 It is the flowchart of constructing the LSTM network rainfall prediction model considering seasonal correlation through ERA5 in this embodiment.

[0018] Figure 4 It is the flowchart of constructing the node-network-node type prediction model based on the LSTM network in this embodiment. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0019] It should be noted that, without conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other.

[0020] In the description of the present invention, it should be understood that the orientation or positional relationships indicated by the terms "center", "longitudinal", "lateral", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. are based on the orientation or positional relationships shown in the drawings. These are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation. Therefore, it should not be construed as a limitation to the present invention. In addition, the terms "first", "second", etc. are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly specifying the quantity of the indicated technical features. Thus, features defined with "first", "second", etc. may explicitly or implicitly include one or more of such features. In the description of the present invention, unless otherwise specified, the meaning of "a plurality" is two or more.

[0021] In the description of the present invention, it should be noted that unless otherwise clearly specified and limited, the terms "installed", "connected", "coupled" should be understood in a broad sense. For example, it may be a fixed connection, a detachable connection, or an integral connection; it may be a mechanical connection or an electrical connection; it may be directly connected or indirectly connected through an intermediate medium, and it may be the communication inside two elements. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood through specific circumstances.

[0022] The present invention provides a short-term rainfall prediction method for a Beidou navigation system based on an LSTM network, as Figure 1 shown, which includes the following steps:

[0023] Step 1: Obtain the historical atmospheric precipitable water data of the observation point at one-hour intervals and process it through a seasonal autoregressive integrated moving average model, so as to seasonally divide the atmospheric precipitable water data into the rainy season and the non-rainy season.

[0024] Specifically, as Figure 2 shown, in this embodiment, products such as observation files, navigation files, and clock errors provided on the IGS official website are used to obtain relevant data above the observation site and other surrounding sites (within a range of 50 km radius centered on the observation site). The data is solved through Beidou precise point positioning technology to obtain the atmospheric precipitable water data of each site. The correlation analysis is performed between the atmospheric precipitable water data of the prediction station itself and the rainfall, and the correlation analysis is also performed between the atmospheric precipitable water data of each site and the rainfall of the prediction station. The results are presented by the Spearman correlation coefficient to represent the spatial correlation of rainfall.

[0025] Download the observation files, navigation files, clock error files, ephemeris files, and antenna phase center files of HKLM, HKSS, HKSC, HKST, and HKOH stations in the Hong Kong Special Administrative Region of China from the IGS official website from January 1 to December 31, 2022, and calculate the atmospheric precipitable water data over each station. Take the HKOH station as the observation station and conduct correlation analysis on the atmospheric precipitable water data with HKLM, HKSS, HKSC, and HKST stations respectively; conduct correlation analysis on the atmospheric precipitable water data of the HKOH station itself and the rainfall, and the results are presented by the Spearman correlation coefficient.

[0026] Among them, the correlation coefficient of HKOH&HKOH is 0.52827, the correlation coefficient of HKSC&HKOH is 0.52478, the correlation coefficient of HKSS&HKOH is 0.62210, the correlation coefficient of HKST&HKOH is 0.59285, and the correlation coefficient of HKLM&HKOH is 0.62666. It can be seen that the relationship between the atmospheric precipitable water data and rainfall of adjacent stations is moderately correlated.

[0027] Furthermore, the method of dividing the atmospheric precipitable water data by the seasonal autoregressive integrated moving average model is as follows: determine the seasonal cycle in the data and determine the seasonal autoregressive order , the seasonal moving average order and the seasonal differencing times . By performing seasonal differencing on the atmospheric precipitable water data, make the data stationary on the seasonal cycle, combine the autoregressive order and moving average terms of the non-seasonal part, and establish a complete seasonal autoregressive integrated moving average model. Use this model to perform seasonal decomposition on the data, extract the seasonal components and analyze their fluctuation characteristics. According to the distribution characteristics of the seasonal components, set reasonable thresholds to divide the rainy season and non-rainy season.

[0028] The mathematical expression of the seasonal autoregressive integrated moving average model is as follows:

[0029] ;

[0030] Among them, the parameter is the non-seasonal autoregressive order, is the non-seasonal moving average order, d is the one-step differencing times, is the seasonal autoregressive order, is the seasonal moving average order, is the seasonal differencing times, is the seasonal / cycle length, is the lag operator, is the seasonal lag operator, is the observed value at time t, is the white noise term, and is a delay polynomial operator.

[0031] Step 2: Construct an LSTM network prediction model, and use the atmospheric precipitable water data in the rainy season and the atmospheric precipitable water data in the non-rainy season obtained in Step 1 as inputs to train the LSTM network prediction model.

[0032] Specifically, as Figure 3 shown, the LSTM network prediction model includes an input gate, a forget gate, a memory cell, and an output gate. To improve the model performance, the model is initialized with weights in this embodiment. The data after seasonal analysis is used as training samples to train the model and obtain the output results.

[0033] Furthermore, the forget gate decides which information should be forgotten from the cell state, and uses the sigmoid activation function to determine the retention degree of each cell state. Its mathematical expression is:

[0034] ;

[0035] where is the output of the forget gate at the th time step, is the sigmoid function, is the weight matrix of the forget gate, is the hidden state of the previous time step, is the input of the current time step, , is the bias term of the forget gate;

[0036] The input gate includes a sigmoid layer and a tanh layer. The sigmoid layer decides which values will be updated, and the tanh layer is used to create a new candidate value vector. The input gate is used to decide which new information will be stored in the cell state. Its mathematical expressions are as follows:

[0037] ;

[0038] ;

[0039] where, is the output of the input gate at the th time step, is the candidate cell state at the th time step, and are the weight matrices of the input gate and the candidate cell state respectively, and Specifically, they are the bias terms of the input gate and the candidate cell state;

[0040] The memory cell (Cell State) is the memory part of the LSTM network, carrying information about the time series. Its mathematical expression is as follows:

[0041] ;

[0042] Among them, is the cell state at the th time step, is the cell state at the previous time step, and are the outputs of the forget gate and the input gate respectively.

[0043] The output gate determines the value of the hidden state, which is the information of the cell state and the output related to the task at the current time step. The mathematical expression of the output gate is:

[0044] ;

[0045] ;

[0046] Among them, is the output of the output gate at the th time step, is the hidden state at the th time step, is the weight matrix of the output gate, is the bias term of the output gate.

[0047] Step 3: Obtain the real-time precipitable water vapor data at the observation point and input it into the pre-trained LSTM network prediction model to output the rainfall probability prediction value one hour later at this station.

[0048] Specifically, as Figure 4 shown, the precipitable water vapor data over the atmosphere at each station is processed through the seasonal autoregressive integrated moving average model, and the processed data is used as the input. The LSTM network prediction model trained in Step 2 is used to make predictions for each station separately. The precipitable water vapor data over the atmosphere at each station can be obtained with the assistance of the Beidou precise point positioning technology. The tropospheric wet delay is solved using the relationship between the total tropospheric delay and the tropospheric wet delay and the tropospheric hydrostatic delay. The specific calculation formula is as follows:

[0049] ;

[0050] Among them, and are the total tropospheric delay and the tropospheric hydrostatic delay respectively;

[0051] Through the above process, the accurate tropospheric wet delay can be obtained. . Calculate the precipitable water vapor in the atmosphere according to the corresponding relationship between the precipitable water vapor in the atmosphere and the tropospheric wet delay. The specific formula is as follows:

[0052] ;

[0053] In the formula, represents the tropospheric wet delay; Π is a dimensionless conversion factor, and the calculation formula is as follows:

[0054]

[0055] In the formula, ρ w represents the density of liquid water vapor; R w is the specific gas constant of water vapor; T m represents the weighted average temperature of the atmosphere. In this embodiment, in this embodiment, R w = 461.51 J / K / kg; and K3 are the atmospheric refraction constants respectively, and the specific values are: 16.48 K / hPa and (3.776 ± 0.014) × 105 K2 / hPa; T m represents the weighted average temperature of the atmosphere, which can be calculated by using an empirical model.

[0056] Step 4: Convert the rainfall probability prediction values of several individual stations into a graph structure according to the spatial information, and convert the graph structure into an adjacency matrix, and re-enter the adjacency matrix into the pre-trained LSTM network prediction model to obtain a networked prediction result presented in the form of a graph.

[0057] Specifically, combine the prediction results of individual stations with spatial information to form a graph, and fuse the information of multiple spatially related nodes. When converting the monitoring station information into a graph structure, collect the latitude and longitude positions, wind direction, wind speed, etc. of each monitoring station, use the geographical distance between stations as the weight of the edge, and the prediction results of each station as the weight of the vertex to construct a graph structure, and convert the graph structure into the adjacency matrix of the LSTM network, where the matrix elements can be expressed as:

[0058] ;

[0059] In the formula, represents the correlation coefficient between the i-th and j-th monitoring stations, represents the distance between two monitoring stations, and are respectively and diffusion parameters in the direction.

[0060] Use the constructed predicted image as the input of the LSTM network prediction model to perform prediction again. The prediction result is still presented in the form of a graph, so as to achieve a networked prediction result.

[0061] Step 5: Use the multi-linear fitting algorithm to extract the predicted values at any position from the networked prediction result.

[0062] Specifically, the multi-linear fitting algorithm, the specific formula is as follows:

[0063] ;

[0064] In the formula, y represents the predicted value at any position, including the geographical location of the monitoring station, wind direction, and wind speed, are model parameters, represents the model error term. The parameter estimation of the model usually uses the least squares method, which estimates the model parameters by minimizing the sum of squared residuals:

[0065]

[0066] Among them, is the predicted value of the monitoring station; is the predicted value obtained after networking.

[0067] Step 6: Adopt a distance weighting strategy to weight the predicted values at any position, and obtain and output the final prediction result.

[0068] Specifically, adopt a distance weighting strategy to weight the fitting points to obtain a higher-precision prediction result. Given a set of data points , then the weight can be expressed by the following formula:

[0069]

[0070] Among them, represents the spatial position of the monitoring station, represents the predicted value of this station, is the distance between the i-th data point and the predicted position, is the standard deviation. For numerical prediction, the predicted value of the predicted position can be calculated by weighted average:

[0071]

[0072] Among them, n is the total number of monitoring stations, is the weight of the i-th data point, is the predicted value of the i-th monitoring station.

[0073] Finally, save and output the prediction results.

[0074] The embodiments of the present invention have been described in detail above in conjunction with the accompanying drawings, but the present invention is not limited to the described embodiments. For those skilled in the art, without departing from the principles and spirit of the present invention, various changes, modifications, substitutions, and variations can be made to these embodiments including components, and still fall within the protection scope of the present invention.

Claims

1. A short-term rainfall prediction method for Beidou navigation system based on LSTM network, characterized in that, It includes the following steps: Step 1: Obtain the historical precipitable water vapor data of the monitoring station at one-hour intervals and process it through a seasonal autoregressive integrated moving average model, so as to seasonally divide the precipitable water vapor data into rainy seasons and non-rainy seasons; Step 2: Construct an LSTM network prediction model, and use the precipitable water vapor data in the rainy season and the precipitable water vapor data in the non-rainy season obtained in Step 1 as inputs respectively to train the LSTM network prediction model; Step 3: Obtain the real-time precipitable water vapor data of the monitoring station and input it into the pre-trained LSTM network prediction model to output the predicted rainfall probability value one hour later for this monitoring station; Step 4: Convert the predicted rainfall probability values of several individual monitoring stations into a graph structure according to spatial information, convert the graph structure into an adjacency matrix, and re-input the adjacency matrix into the pre-trained LSTM network prediction model to obtain a networked prediction result presented in the form of a graph; Step 5: Use a multilinear fitting algorithm to extract the predicted value at any position from the networked prediction result; Step 6: Use a distance weighting strategy to weight the predicted value at any position to obtain the final prediction result and output it.

2. The short-term rainfall prediction method for the Beidou navigation system based on the LSTM network according to claim 1, wherein, In Step 1, the mathematical expression of the seasonal autoregressive integrated moving average model is as follows: φ p (B)Φ P (B s )(1 - B s ) d (1 - B s ) D x t = θ q (B)Θ Q (B s )ε t ; Among them, the parameter p is the non-seasonal autoregressive order, q is the non-seasonal moving average order, d is the number of one-step differencing times, P is the seasonal autoregressive order, Q is the seasonal moving average order, D is the number of seasonal differencing times, S is the seasonal / cycle length, B is the lag operator, B S is the seasonal lag operator, x t is the observation value at time t, ε t is the white noise term, φ p (B) and θ q (B) are the lag polynomial operators.

3. A short-term rainfall prediction method for a Beidou navigation system based on an LSTM network according to claim 2, characterized in that, The method for the seasonal autoregressive integrated moving average model to divide the precipitable water vapor data is: determine the seasonal period in the data, and determine the seasonal autoregressive order P, the seasonal moving average order Q, and the seasonal differencing times D. By performing seasonal differencing on the precipitable water vapor data, make the data become stationary on the seasonal cycle, combine the autoregressive order and the moving average term of the non-seasonal part, establish a complete seasonal autoregressive integrated moving average model, use this model to perform seasonal decomposition on the data, extract the seasonal component and analyze its fluctuation characteristics, According to the distribution characteristics of the seasonal component, set a reasonable threshold to divide the rainy season and the non-rainy season.

4. A short-term rainfall prediction method for Beidou navigation system based on LSTM network according to claim 1, characterized in that, The LSTM network prediction model includes an input gate, a forget gate, a memory cell, and an output gate, The forget gate uses a sigmoid activation function to determine the retention degree of each cell state, and its mathematical expression is: f t = σ(W f · [h t-1 , x t + b f ); where, f t is the output of the forget gate at the t-th time step, σ is the sigmoid function, W f is the weight matrix of the forget gate, h t-1 is the hidden state of the previous time step, x t is the input of the current time step, x t ∈R, b f is the bias term of the forget gate; The input gate includes a sigmoid layer and a tanh layer. The input gate is used to determine which new information will be stored in the cell state, and its mathematical expression is as follows: i t = σ(W i ·[h t-1 , x t + b i ); where, i t is the output of the input gate at the t-th time step, is the candidate cell state at the t-th time step, W i and W C are the weight matrices of the input gate and the candidate cell state respectively, b i and b C are the bias terms of the input gate and the candidate cell state respectively; The memory cell is the memory part of the LSTM network, and its mathematical expression is as follows: Among them, C t is the cell state at the t-th time step, and C t-1 is the cell state at the previous time step. f t and i t are the outputs of the forget gate and the input gate respectively; The output gate determines the value of the hidden state. The hidden state is the information of the cell state and the output related to the task at the current time step. The mathematical expression of the output gate is: o t = σ(W o · [h t-1 , x t + b o ); h t = o t *tanh(C t ) where, o t is the output of the output gate at the t-th time step, h t is the hidden state at the t-th time step, W o is the weight matrix of the output gate, b o is the bias term of the output gate.

5. A short-term rainfall prediction method for a Beidou navigation system based on an LSTM network according to claim 1, characterized in that, In Step 3, the method for obtaining the real-time precipitable water vapor data of the monitoring station is: Step 3-1: Obtain the observation data above the monitoring station through Beidou. The observation data includes the total tropospheric delay and the tropospheric hydrostatic delay; Step 3-2: Calculate the tropospheric wet delay ZWD according to the total tropospheric delay and the tropospheric hydrostatic delay. The expression is as follows: ZWD = ZTD - ZHD; where ZTD and ZHD are the total tropospheric delay and the tropospheric hydrostatic delay respectively; Step 3-3: Calculate the precipitable water vapor (PWV) according to the corresponding relationship between the precipitable water vapor and the tropospheric wet delay. The specific formula is as follows: PWV = Π × ZWD; where Π is a dimensionless conversion factor.

6. A short-term rainfall prediction method for Beidou navigation system based on LSTM network according to claim 5, characterized in that The calculation method of the dimensionless conversion factor is as follows: where ρ w represents the density of liquid water vapor; R w is the specific gas constant of water vapor; T m represents the weighted average temperature of the atmosphere.

7. A short-term rainfall prediction method for Beidou navigation system based on LSTM network according to claim 5, characterized in that, The spatial information includes the longitude, latitude, wind direction, and wind speed of the monitoring station.

8. A short-term rainfall prediction method for a Beidou navigation system based on an LSTM network according to claim 7, characterized in that In step 4, the method of converting to a graph structure is: construct a graph structure with the geographical distance between monitoring stations as the weight of the edge and the prediction results of each monitoring station as the weight of the vertex.

9. A short-term rainfall prediction method for Beidou navigation system based on LSTM network according to claim 8, characterized in that, The method of converting the graph structure into an adjacency matrix is: where W ij represents the correlation coefficient between the i-th and j-th monitoring stations, d ij represents the distance between two monitoring stations, σ Y and σ Z are the diffusion parameters in the y and x directions respectively.

10. A short-term rainfall prediction method for a Beidou navigation system based on an LSTM network according to claim 1, characterized in that The multi-linear fitting algorithm, the specific formula is as follows: y = β0 + β1x1 + β2x2 +... + β k x k + ε; where y represents the predicted value at any position, x1, x2,..., x k including the geographical location of the monitoring station, wind direction, wind speed, β1, β2,..., β k are model parameters, and ε represents the model error term.

11. A short-term rainfall prediction method for Beidou navigation system based on LSTM network according to claim 10, characterized in that, The specific method of step 6 is: Given a set of data points (x i , y i ), the weights W i can be expressed by the following formula: where x i represents the spatial location of the monitoring station, y i represents the predicted value of the monitoring station, d i is the distance between the i-th data point and the predicted location, σ is the standard deviation, and for numerical prediction, the predicted value of the predicted location x i can be calculated by weighted average: ​ where n is the total number of monitoring stations, and W i is the weight of the i-th data point, and y i is the predicted value of the i-th monitoring station.

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