Two-stage multi-source precipitation fusion method based on deep learning and bayesian model

Abstract

The application discloses a two-stage multi-source precipitation fusion method based on deep learning and a Bayesian model, which comprises collecting precipitation products and preprocessing; time alignment of the precipitation products and ground station observation data, and extraction of quality control marks of each ground station observation data; construction of multiple space-time sample units, and normalization into a tensor structure respectively; construction of a deep learning model based on a combination of a three-dimensional convolutional neural network and a convolutional long short-term memory network, input of the normalized precipitation products into the deep learning model, and obtaining of multi-source precipitation initial prediction results; construction of a weighted Gaussian mixture model by using a Bayesian model average, input of the multi-source precipitation initial prediction results, obtaining of dynamic optimal weights, fusion of the dynamic optimal weights and the multi-source precipitation initial prediction results, and finally obtaining of fused precipitation products. The method has high space-time resolution and statistical robustness, and provides data support for input of a refined hydrological model.

Description

Technical Field

[0001] This invention relates to the fields of hydrology and meteorology, specifically to a two-stage multi-source precipitation fusion method based on deep learning and Bayesian models. Background Technology

[0002] Precipitation, as a core factor in the hydrological cycle and climate change, has a significant impact on flood forecasting, water resource allocation, agricultural production, and ecological environment security. Accurately obtaining spatiotemporal distribution information of precipitation is fundamental to hydrological forecasting and meteorological disaster prevention and mitigation. Due to the significant spatiotemporal heterogeneity and suddenness of precipitation, the three main observation methods currently relied upon all have limitations. Ground meteorological station observations can provide high-precision precipitation records, often used as "true values" for calibration and verification; however, due to the sparse distribution of stations, they cannot fully reflect the spatial differences in regional precipitation. Weather radar can provide high-resolution continuous monitoring over a large area, and its reflectivity factor can be converted into precipitation intensity through empirical formulas, which can be used for quantitative precipitation estimation (QPE). However, the inversion results are biased due to factors such as topographic obstruction, ground clutter, and long-distance attenuation. Satellite remote sensing products such as CMORPH and GPM can achieve long-term global coverage, providing support for large-scale climate and hydrological research. However, their precipitation inversion relies on microwave and infrared observations, and the estimation accuracy is easily affected by cloud microphysical characteristics and orbital coverage limitations, resulting in systematic deviations from ground-based measurements.

[0003] To overcome the limitations of single data sources, multi-source precipitation fusion has become a research hotspot in recent years. Traditional methods often employ interpolation or weighted averaging to linearly superimpose different data sources, but these methods struggle to reflect complex nonlinear relationships and lack sensitivity to extreme precipitation events. Statistical regression methods can correct for systematic biases to some extent, but they also fail to capture the spatiotemporal evolution patterns of multi-source data. Deep learning methods, due to their powerful feature extraction capabilities, can perform nonlinear modeling of multi-source precipitation and have made some progress, but these methods are often considered "black box" models, lacking characterization of prediction uncertainties and prone to overfitting under extreme weather conditions. Probabilistic statistical methods, such as Bayesian model averaging, can weight and fuse multiple prediction results in a probabilistic sense, providing robust statistical inference, but their performance is still limited in complex precipitation contexts due to a lack of in-depth modeling of high-dimensional spatiotemporal features.

[0004] In summary, existing multi-source precipitation fusion methods generally suffer from the inability to simultaneously achieve both spatiotemporal accuracy and uncertainty quantification, especially under extreme precipitation events where prediction performance is unsatisfactory. To address this, this invention proposes a two-stage multi-source precipitation fusion method based on deep learning and Bayesian models. This method utilizes precipitation data from multiple sources as the primary input data, extracts complex spatiotemporal features from multi-source observations through a custom deep learning model, and generates sub-prediction results by combining meteorological parameters (such as temperature, cloud cover, and wind speed). Subsequently, a Bayesian model averaging method is used to dynamically weight the multi-source predictions in a probabilistic sense, ultimately obtaining a fused precipitation estimate that possesses both high spatiotemporal resolution and statistical robustness, thus effectively overcoming the shortcomings of existing methods. Summary of the Invention

[0005] The purpose of this invention is to provide a two-stage multi-source precipitation fusion method based on deep learning and Bayesian models. This method can make full use of the complementary information from multi-source precipitation observations such as stations, radar and satellites to improve the spatiotemporal accuracy and uncertainty quantification capabilities of precipitation.

[0006] To achieve the above functions, this invention designs a two-stage multi-source precipitation fusion method based on deep learning and Bayesian models, executing the following steps S1-S5 to complete the multi-source fused precipitation estimation for the target area:

[0007] Step S1: Collect precipitation products from different sources for the target area;

[0008] Step S2: Preprocess precipitation products from different sources to obtain precipitation products with consistent format, extent, and spatial resolution;

[0009] Step S3: Time-align the precipitation products and ground station observation data, and extract the quality control indicators for each ground station observation data; construct multiple spatiotemporal sample units based on the precipitation products, with each ground station as the center, and normalize them into tensor structures respectively;

[0010] Step S4: Construct a deep learning model based on the combination of a 3D convolutional neural network and a convolutional long short-term memory network. Input the normalized precipitation products into the 3D convolutional neural network to extract spatiotemporal features. Combine with other meteorological parameters to generate comprehensive input features, which are then input into the convolutional long short-term memory network. During the model training process, the quality control flags obtained in step S3 are used to weight the loss function, giving higher weights to high-quality station data, and finally obtaining the initial prediction results of multi-source precipitation.

[0011] Step S5: Construct a weighted Gaussian mixture model using the Bayesian model average. Use the initial multi-source precipitation prediction results output by the deep learning model in Step S4 as the input of the weighted Gaussian mixture model to obtain the dynamic optimal weights. Fuse the dynamic optimal weights with the initial multi-source precipitation prediction results to obtain the fused precipitation product and complete the multi-source fused precipitation estimation for the target area.

[0012] Beneficial effects: Compared with the prior art, the advantages of the present invention include:

[0013] First, based on the characteristics of different precipitation products, the format of precipitation products is converted using the NumPy and GDAL libraries in Python. A dynamic pruning strategy based on climate characteristics and geographic information is used for pruning, and bilinear interpolation is employed for spatial downscaling to obtain a unified spatial resolution. Then, a collaborative preprocessing method for multi-source precipitation fusion is used to normalize all input data into a unified five-dimensional tensor structure suitable for subsequent model input. The normalized precipitation data is trained using a custom deep learning model combining a custom three-dimensional convolutional neural network and a convolutional long short-term memory network. Spatiotemporal features extracted from the custom deep learning model are fused and post-processed, and combined with other meteorological parameters (such as temperature, cloud cover, and wind speed) for multi-channel input to further improve the accuracy of precipitation prediction, obtaining initial multi-source precipitation prediction results. Finally, the initial multi-source precipitation prediction results output by the custom deep learning model are used as input to a sub-model. A weighted Gaussian mixture model is constructed using Bayesian model averaging to obtain the dynamically optimal weighted multi-source precipitation prediction results, ultimately yielding the fused precipitation. This method leverages the powerful nonlinear mapping capabilities of deep learning to fully explore the complex spatiotemporal characteristics and evolutionary patterns in multi-source precipitation data, generating high-quality initial forecasts. Utilizing the excellent probabilistic and statistical properties of Bayesian model averaging, multiple initial forecasts are dynamically weighted and averaged, combining the advantages of both methods. This not only improves the accuracy of deterministic forecasts but also provides valuable probabilistic forecast information (uncertainty quantification). By constructing a multi-channel data cube, the model can simultaneously "see" the distribution patterns of radar, satellite, and other multi-source data within the spatiotemporal range around the site, effectively utilizing their complementary information. The Bayesian model averaging framework can automatically learn and assign credibility weights to different data sources under different conditions, ensuring that the fusion result remains relatively stable even if a data source experiences systematic bias or temporary failure. Attached Figure Description

[0014] Figure 1 This is a flowchart of a two-stage multi-source precipitation fusion method based on deep learning and Bayesian models provided in an embodiment of the present invention;

[0015] Figure 2 This is a coarse-resolution precipitation distribution map of various precipitation products provided according to embodiments of the present invention;

[0016] Figure 3 This is a fine-resolution precipitation distribution map of various precipitation products provided according to embodiments of the present invention;

[0017] Figure 4 This is a fused precipitation distribution map provided according to an embodiment of the present invention. Detailed Implementation

[0018] The present invention will be further described below with reference to the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and should not be used to limit the scope of protection of the present invention.

[0019] The two-stage multi-source precipitation fusion method based on deep learning and Bayesian models provided in this embodiment of the invention refers to... Figure 1 Perform the following steps S1-S5 to complete the multi-source fusion precipitation estimation for the target area:

[0020] Step S1: Collect precipitation products from different sources for the target area;

[0021] The specific steps of step S1 are as follows:

[0022] Step S1.1: Locate the official website for each satellite or reanalysis dataset;

[0023] Step S1.2: User registration and login;

[0024] Step S1.3: Locate the precipitation dataset corresponding to the target area;

[0025] Step S1.4: Download precipitation data to complete the collection of precipitation products.

[0026] Step S2: Preprocess precipitation products from different sources to obtain precipitation products with consistent format, extent, and spatial resolution;

[0027] The specific steps of step S2 are as follows:

[0028] Step S2.1: Using the numpy and gdal libraries in Python, convert the different formats of precipitation products to TIF format.

[0029] Step S2.2: For the spatial dimension of each precipitation product after format conversion, adaptive cropping is performed according to the spatial resolution and quality of the different precipitation products after conversion. For the temporal dimension of each precipitation product after format conversion, missing and outlier data are removed. The coarse resolution precipitation distribution map of each precipitation product is referenced. Figure 2 , Figure 2(a) is a coarse-resolution precipitation distribution map of the precipitation product CMORPH; (b) is a coarse-resolution precipitation distribution map of the precipitation product GPH; and (c) is a coarse-resolution precipitation distribution map of the precipitation product Radar.

[0030] Step S2.3: Based on the spatial resolution of each precipitation product after format conversion, use bilinear interpolation to perform spatial downscaling, ensuring consistent spatial resolution across all precipitation products. This results in precipitation products with consistent format, extent, and spatial resolution. The fine-resolution precipitation distribution maps for each precipitation product are referenced... Figure 3 , Figure 3 (a) is a fine-resolution precipitation distribution map of the precipitation product CMORPH; (b) is a fine-resolution precipitation distribution map of the precipitation product GPH; and (c) is a fine-resolution precipitation distribution map of the precipitation product Radar.

[0031] Step S3: Time-align the precipitation products and ground station observation data, and extract the quality control indicators for each ground station observation data; construct multiple spatiotemporal sample units based on the precipitation products, with each ground station as the center, and normalize them into tensor structures respectively;

[0032] The specific steps of step S3 are as follows:

[0033] Step S3.1: Time-align all precipitation products and ground station observation data to UTC time;

[0034] Step S3.2: For the ground station observation data, routine missing values ​​and obvious physical anomalies are removed. A spatiotemporal nearest neighbor-based interpolation method is used to fill in some random missing values. At the same time, the quality control flag of each ground station observation data is recorded and retained. The quality control flag is used to identify the reliability level of the data and will be used to screen high-quality data in subsequent steps S3.3, S4 and S5.

[0035] Step S3.3: Based on the quality control indicators obtained in step S3.2, select qualified ground stations. Taking each selected ground station as the center, dynamically extract a three-dimensional data cube with a size of T×H×W from each precipitation product. At the same time, use the C different precipitation product data as C channels to form a spatiotemporal sample unit.

[0036] Step S3.4: Divide all generated spatiotemporal sample units according to time order and normalize them into a five-dimensional tensor structure in the form of (number of samples × time step × longitude × latitude × data source).

[0037] Step S4: Construct a deep learning model based on the combination of a 3D convolutional neural network and a convolutional long short-term memory network. Input the normalized precipitation products into the 3D convolutional neural network to extract spatiotemporal features. Combine with other meteorological parameters to generate comprehensive input features, which are then input into the convolutional long short-term memory network. During the model training process, the quality control flags obtained in step S3 are used to weight the loss function, giving higher weights to high-quality station data, and finally obtaining the initial prediction results of multi-source precipitation.

[0038] The specific steps of step S4 are as follows:

[0039] Step S4.1: Calculate the minimum and maximum values ​​of each spatiotemporal sample unit obtained in step S3 for each channel. Then, use min–max normalization to linearly map the original precipitation product data to the [0,1] interval. The normalization formula is:

[0040] ;

[0041] Where P represents the raw precipitation product data. These are the minimum and maximum values ​​of the corresponding channels in the training set, respectively; P' is the result after normalization.

[0042] Step S4.2: Systematically select and optimize hyperparameters such as kernel size and number, max pooling layer stride, learning rate, batch size, and training epochs for the 3D convolutional neural network;

[0043] Step S4.3: Extract local structural features of precipitation products in the spatiotemporal dimension using a three-dimensional convolutional neural network, given input sample X∈R. T×H×W×C Where T represents the time step, H×W is the spatial dimension, and C is the number of data source channels; the convolution process of a 3D convolutional neural network is as follows:

[0044] ;

[0045] In the formula, This represents the value of the input tensor at the time dimension (t+p), spatial location ((i+q),(j+r)), and channel c; This represents the value of the output feature tensor at time dimension t, spatial location (i,j), and channel k. This represents the weight parameters of the convolutional kernel in the time direction p, the spatial direction (q, r), the input channel c, and the output channel k. This represents the bias term corresponding to output channel k; represents the non-linear activation function; C represents the number of input channels; P, Q, and R represent the dimensions of the convolutional kernel in the temporal, spatial height, and spatial width directions, respectively.

[0046] Step S4.4: After the 3D convolutional neural network, a tree-based feature importance assessment method is used to screen key meteorological features. The spatiotemporal features extracted by the 3D convolutional neural network are concatenated with other meteorological parameters (such as surface temperature, cloud cover, wind speed, etc.) to form a comprehensive feature set. Then, the gradient boosting decision tree (XGBoost) model is used to fit this comprehensive feature set, and the gain importance score of each feature is calculated. Finally, the features with the top 80% of the cumulative importance scores are selected and defined as key meteorological features. These selected key meteorological features are fused with precipitation data. Key meteorological features are screened through feature selection and importance assessment, and these features are fused with precipitation data to generate comprehensive input features. The fused comprehensive input features are then input into the convolutional long short-term memory network.

[0047] ;

[0048] ;

[0049] ;

[0050] ;

[0051] ;

[0052] In the formula, Let be the input feature tensor at time t; Let be the hidden state tensor at time t; The state of the memory cell at time t; , , These are the input gate, the forget gate, and the output gate, respectively. Use the Sigmoid activation function; The hyperbolic tangent function is represented by *; convolution operation is indicated by *. This represents element-wise multiplication; , These are the convolution weights for the input and hidden states of the candidate cell states used to update the memory cell states, respectively. , , These are the convolution weights for the input, hidden state, and memory unit corresponding to the input gate, respectively; , , These are the convolution weights for the input, hidden state, and memory unit corresponding to the forget gate, respectively; , , These are the convolution weights for the input, hidden state, and memory unit corresponding to the output gate, respectively; , , , These are the bias terms for the input gate, forget gate, memory unit, and output gate, respectively.

[0053] Step S5: Construct a weighted Gaussian mixture model using the Bayesian model average. Use the initial multi-source precipitation prediction results output by the deep learning model in Step S4 as the input of the weighted Gaussian mixture model to obtain the dynamic optimal weights. Fuse the dynamic optimal weights with the initial multi-source precipitation prediction results to obtain the fused precipitation product and complete the multi-source fused precipitation estimation for the target area.

[0054] The specific steps of step S5 are as follows:

[0055] Step S5.1: Construct a weighted Gaussian mixture model. The initial multi-source precipitation prediction results output by the deep learning model are transformed into a normal distribution using the Box-Cox transformation, and then used as input to the weighted Gaussian mixture model. The weighted Gaussian mixture model is as follows:

[0056] ;

[0057] ;

[0058] in, is the posterior probability density function of the merged precipitation, y is the predicted merged precipitation variable, and G is the time series precipitation observation data of length T after being filtered by the quality control criteria in step S3. It is the predicted value of the kth precipitation product based on precipitation observation data; It is the posterior probability of the predicted k-th precipitation product given the precipitation observation data; Let y be the posterior distribution of y given the predicted precipitation product and the observed precipitation data;

[0059] Step S5.2: Construct a log-likelihood function to estimate the average weights and variance of the Bayesian model that maximizes the likelihood, assuming conditional probabilities. Following a normal distribution, the log-likelihood function is expressed as follows:

[0060] ;

[0061] in, These are the unknown Bayesian model mean parameters, i.e., the weights and variances of each precipitation product; It is the average weight of the Bayesian model for the k-th precipitation product; It is the predicted value of the k-th precipitation product at time step t; It is the predicted fused precipitation variable at time step t. and These are the mean and standard deviation of a normal distribution;

[0062] Step S5.3: Use the expectation-maximization algorithm to find the numerical solution of the maximum value of the log-likelihood function, thereby obtaining the average variance and optimal weights of the Bayesian model, as shown in the following formula:

[0063] ;

[0064] ;

[0065] ;

[0066] in, It is a latent variable, specifically depending on the performance of the k-th precipitation product in the i-th iteration of the expectation-maximization algorithm at time t. and These are the weights and standard deviations of the k-th precipitation product in the i-th iteration. It is the predicted value of the k-th precipitation product at time step t. g k Represents the Gaussian function. It is the predicted fused precipitation variable at time step t;

[0067] Step S5.4: Use inverse distance interpolation to adjust the weights. Spatial interpolation and normalization are performed to obtain the grid dynamic weights;

[0068] Step S5.5: The optimal weights are fused with the initial multi-source precipitation prediction results obtained in Step S4 to obtain the final fused precipitation product, referring to... Figure 4 .

[0069] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. A two-stage multi-source precipitation fusion method based on deep learning and Bayesian models, characterized in that, Perform the following steps S1-S5 to complete the multi-source fusion precipitation estimation for the target area: Step S1: Collect precipitation products from different sources for the target area; Step S2: Preprocess precipitation products from different sources to obtain precipitation products with consistent format, extent, and spatial resolution; Step S3: Time-align the precipitation products and ground station observation data, and extract the quality control indicators for each ground station observation data; Centered on each ground station, multiple spatiotemporal sample units are constructed based on precipitation products and normalized into tensor structures respectively; Step S4: Construct a deep learning model based on the combination of a 3D convolutional neural network and a convolutional long short-term memory network. Input the normalized precipitation products into the 3D convolutional neural network to extract spatiotemporal features. Combine with other meteorological parameters to generate comprehensive input features, which are then input into the convolutional long short-term memory network. During the model training process, the quality control flags obtained in step S3 are used to weight the loss function, giving higher weights to high-quality station data, and finally obtaining the initial prediction results of multi-source precipitation. Step S5: Construct a weighted Gaussian mixture model using the Bayesian model average. Use the initial multi-source precipitation prediction results output by the deep learning model in Step S4 as the input of the weighted Gaussian mixture model to obtain the dynamic optimal weights. Fuse the dynamic optimal weights with the initial multi-source precipitation prediction results to obtain the fused precipitation product and complete the multi-source fused precipitation estimation for the target area.

2. The two-stage multi-source precipitation fusion method based on deep learning and Bayesian model according to claim 1, characterized in that, The specific steps of step S1 are as follows: Step S1.1: Locate the official website for each satellite or reanalysis dataset; Step S1.2: User registration and login; Step S1.3: Locate the precipitation dataset corresponding to the target area; Step S1.4: Download precipitation data to complete the collection of precipitation products.

3. The two-stage multi-source precipitation fusion method based on deep learning and Bayesian model according to claim 1, characterized in that, The specific steps of step S2 are as follows: Step S2.1: Using the numpy and gdal libraries in Python, convert the different formats of precipitation products to TIF format. Step S2.2: For the spatial dimension of each precipitation product after format conversion, adaptive cropping is performed according to the spatial resolution and quality of the different precipitation products after conversion. For the temporal dimension of each precipitation product after format conversion, missing and outlier data are removed. Step S2.3: For the spatial resolution of each precipitation product after format conversion, use bilinear interpolation to perform spatial downscaling to make the spatial resolution of all precipitation products consistent, and finally obtain precipitation products with consistent format, extent, and spatial resolution.

4. The two-stage multi-source precipitation fusion method based on deep learning and Bayesian model according to claim 1, characterized in that, The specific steps of step S3 are as follows: Step S3.1: Time-align all precipitation products and ground station observation data to UTC time; Step S3.2: For the ground station observation data, perform routine removal of missing values ​​and obvious physical anomalies, and use the spatiotemporal nearest neighbor-based interpolation method to fill in some random missing values. At the same time, record and retain the quality control mark of each ground station observation data. The quality control mark is used to identify the reliability level of the data and will be used to filter the data in subsequent steps S3.3, S4 and S5. Step S3.3: Based on the quality control indicators obtained in step S3.2, select qualified ground stations. Taking each selected ground station as the center, dynamically extract a three-dimensional data cube with a size of T×H×W from each precipitation product. At the same time, use the C different precipitation product data as C channels to form a spatiotemporal sample unit. Step S3.4: Divide all generated spatiotemporal sample units according to time order and normalize them into a five-dimensional tensor structure in the form of (number of samples × time step × longitude × latitude × data source).

5. The two-stage multi-source precipitation fusion method based on deep learning and Bayesian model according to claim 4, characterized in that, The specific steps of step S4 are as follows: Step S4.1: Calculate the minimum and maximum values ​​of each spatiotemporal sample unit obtained in step S3 for each channel. Then, use min–max normalization to linearly map the original precipitation product data to the [0,1] interval. The normalization formula is: ； Where P represents the raw precipitation product data. These are the minimum and maximum values ​​of the corresponding channels in the training set, respectively; P' is the result after normalization. Step S4.2: Systematically select and optimize the kernel size and number, max pooling layer stride, learning rate, batch size, and training epochs of the 3D convolutional neural network; Step S4.3: Extract local structural features of precipitation products in the spatiotemporal dimension using a three-dimensional convolutional neural network, given input sample X∈R. T×H×W×C Where T represents the time step, H×W is the spatial dimension, and C is the number of data source channels; the convolution process of a 3D convolutional neural network is as follows: ； In the formula, This represents the value of the input tensor at the time dimension (t+p), spatial location ((i+q),(j+r)), and channel c; This represents the value of the output feature tensor at time dimension t, spatial location (i,j), and channel k. This represents the weight parameters of the convolutional kernel in the time direction p, the spatial direction (q, r), the input channel c, and the output channel k. This represents the bias term corresponding to output channel k; represents the non-linear activation function; C represents the number of input channels; P, Q, and R represent the dimensions of the convolutional kernel in the temporal, spatial height, and spatial width directions, respectively. Step S4.4: After the 3D convolutional neural network, a tree-based feature importance evaluation method is used to screen key meteorological features. The spatiotemporal features extracted by the 3D convolutional neural network are concatenated with other meteorological parameters, including surface temperature, cloud cover, and wind speed, to form a comprehensive feature set. Then, a gradient boosting decision tree model is used to fit this comprehensive feature set, and the gain importance score of each feature is calculated. Finally, the features with the top 80% cumulative importance scores are selected and defined as key meteorological features. These selected key meteorological features are fused with precipitation data to generate comprehensive input features. The fused comprehensive input features are then input into a convolutional long short-term memory network. ； ； ； ； ； In the formula, Let be the input feature tensor at time t; Let be the hidden state tensor at time t; The state of the memory cell at time t; , , These are the input gate, the forget gate, and the output gate, respectively. Use the Sigmoid activation function; The hyperbolic tangent function is represented by *; convolution operation is indicated by *. This represents element-wise multiplication; , These are the convolution weights for the input and hidden states of the candidate cell states used to update the memory cell states, respectively. , , These are the convolution weights for the input, hidden state, and memory unit corresponding to the input gate, respectively; , , These are the convolution weights for the input, hidden state, and memory unit corresponding to the forget gate, respectively; , , These are the convolution weights for the input, hidden state, and memory unit corresponding to the output gate, respectively; , , , These are the bias terms for the input gate, forget gate, memory unit, and output gate, respectively.

6. The two-stage multi-source precipitation fusion method based on deep learning and Bayesian model according to claim 1, characterized in that, The specific steps of step S5 are as follows: Step S5.1: Construct a weighted Gaussian mixture model. The initial multi-source precipitation prediction results output by the deep learning model are transformed into a normal distribution using the Box-Cox transformation, and then used as input to the weighted Gaussian mixture model. The weighted Gaussian mixture model is as follows: ； ； in, is the posterior probability density function of the merged precipitation, y is the predicted merged precipitation variable, and G is the time series precipitation observation data of length T after being filtered by the quality control criteria in step S3. It is the predicted value of the kth precipitation product based on precipitation observation data; It is the posterior probability of the predicted k-th precipitation product given the precipitation observation data; Let y be the posterior distribution of y given the predicted precipitation product and the observed precipitation data; Step S5.2: Construct a log-likelihood function to estimate the average weights and variance of the Bayesian model that maximizes the likelihood, assuming conditional probabilities. Following a normal distribution, the log-likelihood function is expressed as follows: ； in, These are the unknown Bayesian model mean parameters, i.e., the weights and variances of each precipitation product; It is the average weight of the Bayesian model for the k-th precipitation product; It is the predicted value of the k-th precipitation product at time step t; It is the predicted fused precipitation variable at time step t. and These are the mean and standard deviation of a normal distribution; Step S5.3: Use the expectation-maximization algorithm to find the numerical solution of the maximum value of the log-likelihood function, thereby obtaining the average variance and optimal weights of the Bayesian model, as shown in the following formula: ； ； ； in, It is a latent variable, specifically depending on the performance of the k-th precipitation product in the i-th iteration of the expectation-maximization algorithm at time t. and These are the weights and standard deviations of the k-th precipitation product in the i-th iteration. It is the predicted value of the k-th precipitation product at time step t. g k Represents the Gaussian function. It is the predicted fused precipitation variable at time step t; Step S5.4: Use inverse distance interpolation to adjust the weights. Spatial interpolation and normalization are performed to obtain the grid dynamic weights; Step S5.5: The optimal weights are fused with the initial multi-source precipitation prediction results obtained in step S4 to obtain the final fused precipitation product.

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