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Article

Comparing and Optimizing Four Machine Learning Approaches to Radar-Based Quantitative Precipitation Estimation

by
Miaomiao Liu
1,
Juncheng Zuo
1,
Jianguo Tan
2,3,* and
Dongwei Liu
4
1
College of Marine Science and Ecological Environment, Shanghai Ocean University, Shanghai 201306, China
2
Shanghai Eco-Meteorological and Satellite Remote Sensing Center, Shanghai 200030, China
3
Key Laboratory of Urban Meteorology, China Meteorological Administration, Beijing 100089, China
4
Shanghai Meteorological Information and Technical Support Center, Shanghai 200030, China
*
Author to whom correspondence should be addressed.
Remote Sens. 2024, 16(24), 4713; https://doi.org/10.3390/rs16244713
Submission received: 31 October 2024 / Revised: 29 November 2024 / Accepted: 13 December 2024 / Published: 17 December 2024
(This article belongs to the Special Issue Advances in Oceanic Dynamics by SAR and Numeric Model in Tropical Cyclone)
Browse Figures
Figure 1
<p>Distribution of automatic weather stations (blue dots) and the Qingpu radar (red triangle) in Shanghai.</p> "> Figure 2
<p>Schematic diagram of the 5 × 5 radar range bin data above the automatic weather station.</p> "> Figure 3
<p>Workflow diagram for the relationship between the Z and R model, SVM, GBDT, RFR, and the KAN deep learning model for single-variable and multivariable precipitation estimation.</p> "> Figure 4
<p>Single-variable KAN deep learning neural network architecture.</p> "> Figure 5
<p>Multivariable KAN deep learning neural network architecture.</p> "> Figure 6
<p>Comparison of the estimation effects of two Z-R relationships.</p> "> Figure 7
<p>Scatter density plots of estimated vs. actual precipitation for five single-variable models: (<b>a</b>) Z = 270.81 R<sup>1.09</sup>; (<b>b</b>) SVM; (<b>c</b>) RF; (<b>d</b>) GBDT; and the (<b>e</b>) KAN deep learning method. The black solid line represents the ideal scenario where estimated values are perfectly aligned with observed values (<span class="html-italic">y = x</span>), while the red solid line indicates the actual relationship between estimated and observed values, highlighting the bias between them.</p> "> Figure 8
<p>Map of radar reflectivity and spatial distribution of univariate precipitation estimates using five different models at 06:00 UTC on 24 June 2024: (<b>a</b>) radar reflectivity; (<b>b</b>) Z = 270.81 R<sup>1.09</sup>; (<b>c</b>) Support Vector Machine model; (<b>d</b>) Random Forest model; (<b>e</b>) Gradient Boosting Decision Tree model; and (<b>f</b>) KAN deep learning model.</p> "> Figure 8 Cont.
<p>Map of radar reflectivity and spatial distribution of univariate precipitation estimates using five different models at 06:00 UTC on 24 June 2024: (<b>a</b>) radar reflectivity; (<b>b</b>) Z = 270.81 R<sup>1.09</sup>; (<b>c</b>) Support Vector Machine model; (<b>d</b>) Random Forest model; (<b>e</b>) Gradient Boosting Decision Tree model; and (<b>f</b>) KAN deep learning model.</p> "> Figure 9
<p>Scatter density plots of estimated vs. actual precipitation for four multivariable models: (<b>a</b>) SVM (multivariable); (<b>b</b>) GBDT (multivariable); (<b>c</b>) RF (multivariable); and (<b>d</b>) KAN deep learning method (multivariable). The red solid line represents the ideal scenario where estimated values are perfectly aligned with observed values (<span class="html-italic">y = x</span>), while the black solid line indicates the actual relationship between estimated and observed values, highlighting the bias between them.</p> "> Figure 10
<p>Map of radar reflectivity and spatial distribution of multivariable precipitation estimates using four different models at 06:00 UTC on June 24, 2024: (<b>a</b>) radar reflectivity map; (<b>b</b>) Support Vector Machine model; (<b>c</b>) Random Forest model; (<b>d</b>) Gradient Boosting Decision Tree model; and (<b>e</b>) KAN deep learning model.</p> ">
Versions Notes

Abstract

:
To improve radar-based quantitative precipitation estimation (QPE) methods, this study investigated the relationship between radar reflectivity (Z) and hourly rainfall intensity (R) using data from 289 precipitation events in Shanghai between September 2020 and March 2024. Two Z-R relationship models were compared in terms of their fitting performance: Z = 270.81 R1.09 (empirically fitted relationship) and Z = 300 R1.4 (standard relationship). The results show that the Z = 270.81 R1.09 model outperforms the Z = 300 R1.4 model in terms of fitting accuracy. Specifically, the Z = 270.81 R1.09 model more effectively captures the nonlinear relationship between radar reflectivity and rainfall intensity, with a higher degree of agreement between the fitted curve and the observed data points. This model demonstrated superior performance across all 289 precipitation events. This study evaluated the performance of four machine learning approaches while incorporating five meteorological features: specific differential phase shift (KDP), echo-top height (ET), vertical liquid water content (VIL), differential reflectivity (ZDR), and correlation coefficient (CC). Nine QPE models were constructed using these inputs. The key findings are as follows: (1) For models with a single-variable input, the KAN deep learning model outperformed Random Forest, Gradient Boosting Decision Trees, Support Vector Machines, and the traditional Z-R relationship. (2) When six features were used as inputs, the accuracy of the machine learning models improved significantly, with the KAN deep learning model outperforming other machine learning methods. Compared to using only radar reflectivity, the KAN deep learning model reduced the MRE by 20.78%, MAE by 4.07%, and RMSE by 12.74%, while increasing the coefficient of determination (R2) by 18.74%. (3) The integration of multiple meteorological features and machine learning optimization significantly enhanced QPE accuracy, with the KAN deep learning model performing best under varying meteorological conditions. This approach offers a promising method for improving radar-based QPE, particularly considering seasonal, weather system, and precipitation stage differentiation.

1. Introduction

Quantitative precipitation estimation (QPE) plays a critical role in meteorology and water resource management, particularly in applications such as flood forecasting and water resource allocation [1,2,3,4,5]. Compared to ground-based automatic stations, radar-based QPE offers a distinct advantage by estimating precipitation from reflectivity, thereby providing broader observational coverage and a higher spatial resolution. This makes radar-based methods particularly valuable for large-scale precipitation monitoring [6,7,8,9].
One of the first methods employed in radar QPE was the Z-R relationship, which establishes an empirical link between radar reflectivity (Z) and rainfall rate (R) [10]. However, the validity of the Z-R relationship is highly dependent on meteorological conditions, and its applicability varies significantly across different environments, limiting its universal use [11,12,13,14]. Studies have shown that the parameters defining the Z-R relationship can exhibit substantial variation across spatial and temporal scales, reflecting the complexity of precipitation microphysics [15]. These variations are especially pronounced under large-scale meteorological conditions, where differences in particle size, shape, and phase directly influence the accuracy of the Z-R relationship [16]. Additionally, this relationship shows notable discrepancies when applied to warm rainfall and stratiform cloud precipitation, suggesting that a one-size-fits-all approach may lead to estimation bias [17]. Furthermore, radar data are susceptible to interference from noise, false echoes, and topographical effects, while rain gauge data can be influenced by instrument malfunctions or environmental obstructions [18,19,20].
In response to these challenges, there has been a growing trend in the development of multivariable models that incorporate additional radar parameters, such as KDP (differential phase shift) and ZDR (differential reflectivity) [21,22,23]. These parameters enable a more precise representation of precipitation’s microphysical properties, such as particle shape, size, and phase, thus addressing the limitations of traditional univariate models and improving the reliability and accuracy of QPE estimates.
As the dimensionality and complexity of radar data increase, traditional Z-R relationships often struggle to meet the demands of processing multidimensional datasets, which has led to a surge in interest in machine learning techniques for QPE research. Machine learning approaches, including Support Vector Machines (SVMs), Random Forests (RFs), and Gradient Boosting Decision Trees (GBDTs), have shown promising results across various applications. For instance, integrating wavelet transforms with SVMs has significantly enhanced the performance of precipitation estimations [24]. Similarly, the optimization of RF models in the mountainous regions of southern Ecuador has notably improved estimation accuracy [25], and terrain-weighted RF models, which integrate vertical radar reflectivity profiles with topographical information, have further increased the reliability of precipitation estimates [26].
In recent years, deep learning techniques have become increasingly important in the field of QPE. The use of Attention–UNet networks, combined with wavelet transforms and feature selection strategies, has significantly improved the identification of extreme precipitation events [27]. Additionally, multiparameter deep learning models, such as QPE-Net, which are based on dual-polarization radar data, have optimized estimation accuracy for varying precipitation intensities through the use of customized loss functions. The application of Gated Recurrent Unit (GRU) neural networks, integrated with echo-top height data, has enhanced the ability to predict both the spatial and temporal patterns of intense rainfall, particularly outperforming traditional Z-R relationships in capturing spatial distribution [28]. Furthermore, deep convolutional neural networks (CNNs), utilizing multiscale convolutional operations, have been applied to polarized radar data during landfall typhoons, achieving high-precision precipitation estimates by learning the complex nonlinear relationships between multidimensional radar observations and ground-based precipitation measurements [29]. However, despite these advancements, the challenge of selecting the optimal model and dynamically adjusting it to accommodate the variability in meteorological conditions remains a significant hurdle in current research.
To address the identified challenges, this study implements stringent quality control of radar and rainfall data to ensure accuracy, incorporating five key radar features: KDP (differential phase), ZDR (differential reflectivity), VIL (vertically integrated liquid), ET (echo top), and CC (correlation coefficient). Based on these features, we developed nine quantitative precipitation estimation (QPE) models, including five univariate models and four multivariate machine learning models, to determine the optimal solution. The machine learning models employed include Support Vector Machine (SVM), Random Forest (RF), Gradient Boosting Decision Tree (GBDT), and the KAN (Knowledge-Aware Network) deep learning model. Initially, we selected the best-performing model from the nine candidates and then conducted a systematic analysis of its performance under varying meteorological conditions to further optimize its accuracy. Through this comprehensive optimization and evaluation process, the purpose of this research is to significantly improve the precision of estimating the linkage between radar reflectivity and the intensity of hourly rainfall, thus providing a stronger technical foundation for quantitative precipitation estimation.
This paper is structured as follows: Section 2.1 and Section 2.2 provide an introduction to methods of data collection and preprocessing, including detailed descriptions of data quality control and training set construction. Section 2.3 outlines the design of the nine models, comprising five univariate models (traditional Z-R relationship, RF, GBDT, SVM, and the KAN deep learning model) and four multivariable machine learning models. Section 2.4 introduces the evaluation index. Section 3 discusses the precipitation estimation results based on the Z-R relationship and machine learning models and offers a thorough examination of the performance of each model.

2. Methodology

2.1. Data Collection

All data in this paper were provided by Shanghai Meteorological Information and Technical Support Center. The research area is located in Shanghai, a coastal city in Eastern China at the mouth of the Yangtze River, positioned between longitudes of 121°29′ and 122°12′E and latitudes of 31°14′ and 31°53′N, making it one of the largest cities in China. The region is characterized by a subtropical monsoon climate, marked by four distinct seasons, with warm, humid conditions in summer and cool, dry conditions in winter. Annual precipitation is primarily concentrated during the Meiyu season (June to July) and autumn, resulting in frequent and intense rainfall events. For precipitation data measured by automatic weather stations, Figure 1 illustrates the spread of automatic weather stations within the study area. Precipitation data were recorded using a tipping bucket rain gauge with a temporal resolution of 1 min, after which the data were accumulated to calculate the hourly rainfall amount (R). The tipping bucket rain gauge operates by collecting rainfall until a predetermined volume is reached, triggering the tipping of the bucket and recording an event. After each tip, the counter logs a rainfall event, and the number of tips per minute is used to achieve a 1-minute temporal resolution. This study focused on analyzing 289 precipitation events that occurred between September 2020 and March 2024.
Radar data were obtained from the Qingpu dual-polarization radar system (CINRAD/SAD) in Shanghai, which operates with an effective detection range of 230 km. The radar system was configured to use the VCP21 observation mode, ensuring high temporal and spatial resolution for precipitation monitoring. The time for each radar volume scan is approximately 6 min, so 10 radar data points can be obtained per hour. Six radar features were selected for this study: reflectivity (Z), specific differential phase shift (KDP), vertically integrated liquid water content (VIL), echo-top height (ET), correlation coefficient (CC), and differential reflectivity (ZDR). Radar reflectivity was derived by calculating the 5 × 5 grid average of the maximum value from the three lowest elevation scans (Figure 2). Radar reflectivity was derived by calculating the 5 × 5 grid average of the maximum value from the three lowest elevation scans (Figure 2). This smoothing technique, specifically the 5 × 5 grid averaging, helps mitigate the influence of the range effect, ensuring more accurate precipitation estimation. The vertically integrated liquid water content (VIL) was computed using the relationship between reflectivity (Z) and liquid water content (LWC), where VIL is given by the following formula:
VIL = 3.44 ∗ 10−6 ∗ Z4/7
where Z is the radar reflectivity in dBZ, and the constant 3.44 × 10−6 is an empirically derived coefficient.
These features provide distinct information on precipitation: Z is primarily used to estimate the size and concentration of precipitation particles, as it is proportional to the sixth power of particle diameter and the number concentration. However, its accuracy can be influenced by factors such as particle shape, phase (e.g., liquid or solid), and distribution, making it a useful but indirect indicator of precipitation characteristics. ZDR provides information about the shape and orientation of raindrops. It represents the ratio of reflectivity between the horizontal and vertical polarizations, which is influenced by the degree of flattening of raindrops caused by air resistance. ZDR is particularly useful for distinguishing between spherical particles, such as hail or snow, and non-spherical raindrops, as well as for identifying the type and phase of hydrometeors. KDP is primarily associated with rainfall rate, as it reflects the phase shift per unit distance caused by non-spherical raindrops, and it can indirectly provide insights into particle characteristics such as shape and orientation. CC evaluates the homogeneity and phase correlation of precipitation particles, serving as an indicator of particle consistency and the presence of mixed-phase hydrometeors. VIL represents the total mass of liquid water content in a column of atmosphere from the ground to the top of the radar-detected precipitation echo, providing an estimate of the potential precipitation intensity. ET (echo top) indicates the maximum altitude at which precipitation-sized particles are detected by the radar, serving as an indicator of storm intensity and vertical development of precipitation systems. Table 1 summarizes the descriptions and units of these six radar parameters related to precipitation.

2.2. Data Preprocessing

Before using machine learning methods for QPE, the quality control of the radar and rainfall gauge data was performed.
For improved radar data accuracy, we excluded data from radar blind spots, where blind spots are defined as regions with no data coverage due to factors such as terrain obstruction, antenna scanning limitations, and signal attenuation. Additionally, we selected valid samples with reflectivities greater than 10 dBZ and filtered out weak echoes and non-precipitation targets. A fuzzy logic algorithm was employed to remove non-meteorological echoes, constructing a T-type membership function based on the statistical characteristics of echo parameters [30]. The algorithm assigns different weights according to the overlap rate of feature distributions, ultimately calculating an aggregation value between 0 and 1. By analyzing the distribution of aggregation values for meteorological and non-meteorological echoes, non-meteorological echoes were successfully removed. The formula is as follows:
A j = ∑ i = 1 n W i ∗ μ i ∑ i = 1 n W i
where Aj represents the aggregation value for the element at the index j scatterer, Wi is the weight for the parameter at index i, and μ i is the membership degree for the parameter at index i.
For precipitation data, the following inclusion criteria were used to ensure accuracy: (1) Minute-level precipitation data were screened using quality control codes to ensure accuracy and reliability. These codes identify and filter anomalies, such as missing data, physical inconsistencies, or instrument errors, by assessing data completeness, consistency, and validity. (2) There must be no missing data for the entire hour, enabling the accumulation of minute-level precipitation into hourly precipitation. (3) The K-Nearest Neighbor (KNN) algorithm was applied for spatial consistency checks, selecting nearby stations within a 15 km radius to perform extreme value checks. The maximum (Rmax) and minimum (Rmin) precipitation values were calculated for each station’s nearby stations. For precipitation events lasting T ≥ 3 h, if Rmax > R > Rmin was satisfied for more than two-thirds of the data, the station’s data were retained. For events lasting T < 3 h, the station’s data were retained only if Rmax > R > Rmin was consistently met throughout the event.
Radar data and rainfall measurements were matched based on the principles of temporal consistency and spatial correspondence. Data from September 2020 to March 2024 were used to divide the data into training and testing groups using a 4-to-1 ratio. Additionally, 25% of the training set was randomly selected as a validation set to ensure the model evaluation was reliable and accurate. To maintain the same scaling between training and testing phases and avoid inconsistencies in data distribution, features were standardized. Data standardization was performed by adjusting the data to achieve a mean of zero and a standard deviation of one, using the following formula:
X norm   = X − μ σ
where X represents the original data, μ represents the average of the feature in the training set, and σ denotes the feature’s standard deviation within the training set.

2.3. Model Selection and Optimization

In this study, nine models were used to enhance the precision of precipitation estimation, including the model describing the relationship between radar reflectivity (Z) and rainfall rate (R), and three types of machine learning algorithms: SVM (Support Vector Machine), GBDT (Gradient Boosting Decision Tree), and RFR (Random Forest Regression). In addition, we developed a Knowledge-Aware Network (KAN) model that incorporates prior knowledge to further optimize the estimation reliability of precipitation estimation. The process is illustrated in Figure 3.

2.3.1. Z-R Relationship

The correlation between radar reflectivity (Z) and precipitation intensity (R) poses a substantial challenge in radar-based quantitative precipitation estimation. This research aimed to determine the local correlation between these two variables by applying the least-squares approach to analyze radar reflectivity and ground-based rainfall measurements collected in Shanghai from September 2020 to March 2024.

2.3.2. Support Vector Machine (SVM)

In this study, Support Vector Regression (SVR) with the Radial Basis Function (RBF) kernel was employed to model the relationship between radar reflectivity (Z) and precipitation (R), mapping the data into higher-dimensional feature spaces to capture complex nonlinear patterns that may not be apparent in the original lower-dimensional space [30]. Prior to training, both radar reflectivity and precipitation data were standardized to ensure consistency across features, thereby improving model performance. The primary hyperparameters, penalty parameter (C), and epsilon (ϵ), were optimized using a grid search approach, with the best combination found to be C = 10 and ϵ = 1 through 3-fold cross-validation. In addition to the univariate SVR model, a multivariate model was implemented, incorporating five additional radar features—KDP (specific differential phase), ZDR (differential reflectivity), VIL (vertically integrated liquid), ET (echo top), and CC (correlation coefficient)—which enhanced the model’s accuracy and generalization ability.

2.3.3. Gradient Boosting Decision Tree (GBDT)

Gradient Boosting Decision Tree (GBDT) is an ensemble learning technique that sequentially constructs multiple weak learners, typically decision trees, to enhance the model’s overall performance [31]. Each tree is designed to correct the bias made by its predecessor, resulting in an improved prediction through a weighted combination of outputs. GBDT employs a gradient descent algorithm to progressively minimize estimation bias, making it particularly suitable for nonlinear regression problems. To optimize radar precipitation estimation, we utilized a grid search approach to fine-tune the model’s hyperparameters, including the number of decision trees, the maximum depth of the trees, and the learning rates. A three-fold cross-validation was performed to identify the optimal set of hyperparameters, followed by an evaluation of model performance on the validation and test datasets. For the univariate model, the best hyperparameter configuration includes a learning rate of 0.01, a maximum depth of 3, and 300 decision trees. In contrast, the optimal parameter settings for the multivariate models consist of a learning rate of 0.1, a maximum depth of 5, and 100 decision trees, with the operation of the multivariate model being similar to that of the univariate model.

2.3.4. Random Forest (RF)

Random Forest Regression (RF) mitigates overfitting and enhances robustness in radar-based quantitative precipitation estimation (QPE) by generating multiple decision trees in parallel, effectively modeling complex, high-dimensional, and noisy radar datasets [32]. To optimize model performance, grid search is used to tune key hyperparameters, including the number of trees, maximum tree depth, and minimum samples for splitting nodes. The optimal hyperparameters are selected through 3-fold cross-validation and evaluated on both the validation and test sets. For the univariate model, the optimal combination is as follows: maximum depth = 5, minimum samples for splitting = 2, and 200 trees. For the multivariate model, the optimal parameters are as follows: unlimited maximum depth, minimum samples for splitting = 5, and 200 trees. The operation of the multivariate model is identical to that of the univariate model.

2.3.5. KAN (Knowledge-Aware Network) Deep Learning Model

With the aim of improving precipitation estimation accuracy, this study introduced the Knowledge-Aware Network (KAN) model proposed by Ziming Liu in 2024 [33]. By incorporating domain knowledge into the feature extraction process and combining prior knowledge with deep learning, the KAN significantly improves the effectiveness of estimating precipitation. The model relies on the Kolmogorov–Arnold representation theorem and incorporates learnable activation functions and parameterized spline functions to enhance the expressiveness of the network. Regularization is used to manage model complexity and mitigate overfitting. The KAN model is structured with a series of layers—an initial input layer, followed by two hidden layers, and concluding with an output layer—with each layer interconnected through nonlinear transformations. The activation function used by SiLU is defined as follows:
SiLU x = x · σ x = x 1 + e − x
where σ(x) is the sigmoid function. SiLU combines linear and nonlinear properties. Due to the uneven and severely right-skewed distribution of hourly precipitation R, we applied a logarithmic transformation log(1 + R) to reduce bias and normalize the data distribution. In the model training process, we used the Adam optimization algorithm set at a learning rate of 0.001, the optimization function employed MSE, and the batch size was 512. We used Bayesian optimization to minimize the RMSE of precipitation estimation, and finally we saved the weight of the best performing model and used reverse transformation to restore the estimated value to the original scale.
First, we examined how radar reflectivity (Z) affects the accuracy of precipitation estimation. To capture the long-term trends and distribution characteristics and to handle extreme values and nonlinear information, we generated four input features through feature engineering: trend component (Z_trend), bin number (Z_binned), logarithmic transformation (Z_log), and square root transformation (Z_sqrt). The input layer of the model based on single reflectivity consists of four neurons, with two hidden layers containing 256 and 128 neurons, and the model ends in an output layer with a single neuron. This model converged after 144 training epochs. Figure 4 illustrates the processing flow of the single-variable Knowledge-Aware Network (KAN) deep learning method.
Subsequently, we incorporated multiple key meteorological variables into the model, including Z, VIL, ZDR, KDP, ET, and CC, to assess their impact on the accuracy of precipitation estimation (Figure 5). The network architecture consists of an input layer with 24 neurons, followed by four layers in total. The two hidden layers are made up of 256 and 128 neurons, and the output layer has one neuron. This model achieved rapid convergence within 25 training epochs. Figure 5 illustrates the processing flow of the multivariable Knowledge-Aware Network (KAN) deep learning method.
Due to the diversity and complexity of precipitation processes, using a unified KAN deep learning model for QPE may lead to accuracy limitations. To enhance the accuracy of precipitation estimation, it is necessary to classify different meteorological conditions and review the performance outcomes of the KAN deep learning model under these conditions. The classification methods for meteorological conditions are as follows:
  • The classification of precipitation processes by season organizes the year into four distinct periods: spring, occurring from March to May; summer from June to August; autumn from September to November; and winter, spanning from December to February of the subsequent year. The seasonal changes in KAN deep learning during these periods is discussed.
  • Weather System Classification: Precipitation events are categorized by the weather system, including typhoon-induced precipitation (affected by typhoons), Meiyu precipitation (during the Meiyu period), frontal precipitation (occurring in spring and autumn with a duration of more than 3 h), and convective precipitation in summer (short-duration precipitation excluding typhoons and Meiyu). The differences in KAN deep learning under these weather systems are analyzed.
  • Precipitation Intensity Classification: Based on the median hourly rainfall intensity, precipitation is divided into three categories—light rain is characterized by a precipitation rate that ranges from 0.1 to 1.5 mm per hour, moderate rain ranges from 1.6 to 6.9 mm/h, and heavy rain is classified as 7 to 14.9 mm/h. KAN deep learning for different precipitation intensities is investigated.
  • Duration-Based Classification: Precipitation processes are classified by duration into short (1–6 h), medium (7–12 h), and long-duration events (greater than 12 h). The variation in KAN deep learning with precipitation duration is examined.
By categorizing precipitation processes based on these criteria, this study aimed to assess the capability of the KAN model under various meteorological conditions, thereby improving the accuracy of QPE.

2.4. Evaluation Metrics

To thoroughly examine and measure the performance outcomes of the nine models in radar-based quantitative precipitation estimation, this study employed five commonly used evaluation metrics: Mean Relative Error (MRE), Coefficient of Determination (R2), Root-Mean-Square Error (RMSE), and Mean Absolute Error (MAE). These metrics offer an in-depth assessment of the models’ bias, error distribution, and fitting performance, ensuring an objective assessment of model accuracy. The specific formulas for these metrics are as follows:
M R E = 1 n ∑ i = 1 n P i − O i O i ∗ 100 %
R 2 = 1 − ∑ i = 1 n O i − P i 2 ∑ i = 1 n O i − O ¯ 2
R M S E = 1 n ∑ i = 1 n P i − O i 2
M A E = 1 n ∑ i = 1 n P i − O i
where P i represents the estimated value for the i observation, Oi denotes the observed value for the i observation, and n indicates the complete count of samples.

3. Results and Discussion

3.1. Comparison of the Estimation Effects of Two Z-R Relationships

This study investigated the relationship between radar reflectivity (Z) and hourly rainfall intensity (R) using data from 289 precipitation events in Shanghai between September 2020 and March 2024. Two Z-R relationship models were compared in terms of their fitting performance: Z = 270.81 R1.09 (empirically fitted relationship) and Z = 300 R1.04 (standard relationship). The results are shown in Figure 6, indicating that the former model outperformed the latter in terms of fitting accuracy. Specifically, the former more effectively captured the nonlinear relationship between radar reflectivity and rainfall intensity, with a higher degree of agreement between the fitted curve and observed data points. This model demonstrated superior performance across all 289 precipitation events.

3.2. Evaluation of Single-Variable Precipitation Estimation Reliability

In this research, we assessed the effectiveness of five models for utilizing radar observations to determine precipitation estimates. These models include the traditional Z-R relationship (Z = 270.81 R1.09) and four machine learning algorithms: RF (single variable), SVM (single variable), GBDT (single variable), and KAN deep learning (single variable). To comprehensively compare the performance of each model, we employed four evaluation metrics: MRE, R2, MAE, and RMSE (Table 2). Additionally, scatter density plots (Figure 7) were created to visually illustrate the performance of each model.
Overall, while the traditional Z-R relationship has a solid foundation in precipitation estimation, machine learning methods demonstrated significant advantages in handling complex precipitation processes. Among them, the KAN deep learning model stood out, significantly improving the estimation reliability and reliability of precipitation estimation.
This study investigated a significant rainfall event that occurred at 06:00 on 24 June 2024, to evaluate and compare the spatial precipitation outputs generated by five meteorological models in the Shanghai region. Figure 8a illustrates the radar reflectivity recorded during the event, while the subsequent panels showcase the spatial precipitation distributions simulated by the respective models. Utilizing the Radial Basis Function interpolation method, discrete precipitation datasets were transformed into continuous grids and subsequently benchmarked against observed radar reflectivity data (Figure 8). The analysis reveals that machine learning models demonstrate distinct advantages in capturing precipitation distribution patterns, particularly in areas characterized by intense rainfall. When compared to the conventional Z = 270.81 R1.09 formula, machine learning models exhibited significantly improved accuracy in replicating the spatial variability in precipitation, resulting in a higher reliability of estimations. Among the evaluated models, the KAN deep learning approach consistently outperformed its counterparts. The KAN model demonstrated superior alignment with radar observations, particularly in defining the boundaries and core intensities of heavy rainfall zones. Moreover, it delivered smoother spatial transitions, effectively addressing the localized estimation inconsistencies prevalent in other models. Additionally, the KAN model excelled in representing precipitation’s spatial heterogeneity, providing higher accuracy in regions of intense rainfall while maintaining realistic predictions in transitional and moderate rainfall zones. These results highlight the KAN model’s exceptional predictive capacity and robustness, establishing it as an effective tool for modeling complex precipitation scenarios with greater spatial precision.

3.3. Evaluation of Multivariable Precipitation Estimation Accuracy

Building on the single-variable models, we further explored the application of multivariable models to improve the estimation reliability of precipitation estimates. In this research, we utilized four machine learning algorithms: RF (multivariable), SVM (multivariable), GBDT (multivariable), and the KAN model (multivariable). To comprehensively compare the performance of these models, we used four evaluation metrics: MRE, R2, MAE, and RMSE (Table 3). Additionally, scatter density plots (Figure 9) were designed to facilitate a graphical comparison that illustrates the effectiveness of each individual model. The outcomes indicate that the multivariable precipitation estimation reliability of RF, SVM, GBDT, and the KAN model surpasses that of the single-variable models. Among these, Support Vector Machine performed the poorest, while RF and GBDT showed moderate improvements. The KAN model exhibited the highest estimation reliability. Overall, the KAN model exhibited outstanding performance in precipitation estimation, particularly under complex precipitation conditions, outperforming traditional multivariable models such as RF, GBDT, and SVM.
In this study, a significant rainfall event that occurred at 06:00 on 24 June 2024, was selected to analyze the spatial distribution of precipitation in the Shanghai region. In the univariate estimation analysis (Figure 8), the Radial Basis Function interpolation method was applied to convert discrete precipitation data into continuous grid data, which were then compared against actual radar reflectivity. The results demonstrated that machine learning models significantly outperformed the traditional Z = 270.81 R1.09 formula in capturing the spatial variability in precipitation distribution. Among these, the KAN deep learning model exhibited exceptional performance, achieving the highest estimation reliability and spatial continuity, particularly in regions of intense rainfall. In the multivariate estimation analysis (Figure 10), a comparative evaluation of multiple machine learning models, including Support Vector Machine (SVM), Gradient Boosting Decision Tree (GBDT), Random Forest (RF), and the KAN deep learning model, was conducted. The findings revealed that the KAN model once again demonstrated remarkable superiority. Specifically, in regions of heavy rainfall, the KAN model accurately reproduced the boundaries and core intensities of precipitation distribution, showing excellent agreement with actual radar reflectivity. Additionally, its simulation results displayed smoother spatial transitions, effectively mitigating the localized estimation errors often observed in other models. Furthermore, the KAN model maintained high accuracy and continuity in moderate-to-low precipitation regions. Compared to univariate analysis, the multivariate models enhanced the ability to capture the complex characteristics of precipitation, with the KAN model exhibiting outstanding performance under multivariate conditions, thereby confirming its robustness and applicability in handling complex precipitation scenarios. In summary, the analysis of univariate and multivariate estimation demonstrates that the KAN deep learning model consistently exhibits exceptional capabilities in predicting precipitation distribution across various contexts. It not only enhances the simulation accuracy in regions of heavy rainfall but also significantly optimizes overall spatial continuity, providing valuable insights for complex precipitation forecasting.

3.4. Classification of KAN Deep Learning Performance Under Different Meteorological Conditions

Although the KAN deep learning model has demonstrated significant advantages in precipitation estimation, its performance under different meteorological conditions requires further optimization. In this research, a multivariate KAN deep learning model was employed to estimate precipitation amounts across various seasons, weather systems, precipitation intensities, durations, and phases of precipitation events. The results indicate that factors such as seasonality, weather systems, precipitation intensity, and duration have a considerable impact on the model’s estimation reliability (Table 4). These variations mainly stem from the complexity of precipitation processes, which are influenced by changes in ambient temperature levels, atmospheric moisture content, and atmospheric structure.
Temperature directly affects the condensation rate of water vapor within clouds and the phase of precipitation particles (e.g., rain, snow, or hail), thereby modifying the connection between radar reflectivity and precipitation intensity. Variations in humidity influence the size distribution and formation efficiency of precipitation particles, leading to changes in the sensitivity of radar reflectivity to precipitation rate. Additionally, atmospheric stability, stratification (e.g., temperature inversions), and wind field variations contribute to differences in precipitation type and intensity. Seasonal changes significantly impact the microphysical properties of precipitation, particularly in terms of particle size, shape, and distribution. For example, winter precipitation particles tend to be smaller and more uniformly distributed, while summer convective systems often feature a higher particle density and more heterogeneous distribution, resulting in marked differences in estimation reliability between seasons. Furthermore, the characteristics of precipitation vary significantly between weather systems. Typhoons, frontal systems, the Meiyu season, and convective systems each exhibit distinct precipitation patterns. During typhoons and the Meiyu period, precipitation particles tend to be more concentrated and precipitation intensity is higher, whereas in convective systems, the uneven distribution of particle sizes leads to greater variability in estimation reliability. Precipitation intensity is also a key factor affecting model performance. Generally, heavy rain is associated with higher radar reflectivity, while light rain corresponds to lower reflectivity. The duration of precipitation also plays a crucial role in estimation reliability, especially for events lasting longer than 12 h, where the relationship between reflectivity and precipitation becomes more complex. Typically, heavy precipitation is associated with higher radar reflectivity, while light precipitation corresponds to lower reflectivity. However, the duration of precipitation plays a crucial role in the reliability of estimates, particularly for events lasting more than 12 h, where the relationship between radar reflectivity and precipitation becomes more complex. In relatively stable weather systems, such as frontal systems, precipitation tends to be more uniform and consistent, which improves the accuracy of estimation. In contrast, squall line systems, which are fast-moving and uneven frontal systems, are characterized by strong reflectivity echoes at their leading edge and stratiform cloud precipitation behind, posing significant challenges to quantitative precipitation estimation (QPE). In convective weather, the increased heterogeneity of precipitation particles further reduces the reliability of estimates.
These results underscore the impact of intricate meteorological factors on the estimation reliability of the model’s estimations and lay the groundwork for future improvements and refinements in model optimization. Future research should concentrate on improving the model architecture, particularly to enhance the estimation of extreme precipitation under complex meteorological conditions, such as convective weather. In addition, the detailed analysis of the performance of the model under different conditions will help identify its limitations and improve the model’s capacity for broader generalization across different datasets and scenarios, and provide more reliable support for meteorological forecasting, thereby optimizing the model’s adaptability to meet diverse application needs.

4. Conclusions

This study analyzed 289 precipitation events in the Shanghai area from September 2020 to March 2024. A more adaptive localized Z-R relationship, Z = 270.81 R1.09, was proposed, optimizing quantitative precipitation estimation (QPE) by incorporating various machine learning models. The primary findings of this study can be summarized as follows:
(1)
Limitations of the classical Z-R Relationship: Z = 300 R1.4 significantly underestimated precipitation. In comparison, the locally fitted Z-R relationship Z = 270.81 R1.09 more accurately represented precipitation characteristics in the Shanghai region.
(2)
Superiority of Machine Learning Models: In single-variable input scenarios (using only radar reflectivity (Z)), the KAN deep learning model showed a significantly higher estimation reliability than GBDT, RF, SVM, and the Z-R relationship. All machine learning models outperformed the classical Z-R relationship, showing that deep learning models are more proficient at identifying the intricate nonlinear relationships between radar reflectivity and rainfall intensity.
(3)
Improved Performance with Multivariate Fusion Models: With multivariate inputs (integrating radar reflectivity, specific differential phase, correlation coefficient, echo-top height, differential reflectivity, and vertically integrated liquid water content), the performance of machine learning models improved significantly, with the KAN deep learning model showing the highest estimation reliability. The multivariate model effectively captured the multidimensional physical features influencing precipitation and their complex interrelations, greatly enhancing QPE reliability.
(4)
Advantages of the KAN Deep Learning Model: Whether using single-variable or multivariate inputs, the KAN deep learning model consistently demonstrated the highest precipitation estimation reliability. Its performance was particularly impressive when distinguishing between different seasons, weather systems, precipitation intensities, and phases. The KAN model combines the strengths of physics-driven and data-driven approaches, incorporating meteorological knowledge into the deep learning framework to improve QPE reliability.
In summary, this study reveals the shortcomings of the conventional connection between radar reflectivity (Z) and rainfall measurement (R). The locally fitted Z-R equation and the use of machine learning models incorporating multivariate features significantly enhance QPE reliability. The application of the KAN deep learning model, in particular, offers new insights and methods for improving QPE, resulting in marked improvements in estimation reliability under various meteorological conditions. These findings provide a scientific foundation for future advancements in radar-based QPE techniques and offer valuable technical support for more precise precipitation forecasting.

Author Contributions

M.L. took the lead in drafting the manuscript, contributing significantly to the conceptualization, structure, and initial writing of the paper. J.T. played a crucial role in revising and refining the manuscript, ensuring its accuracy, clarity, and alignment with the journal’s standards. J.Z. provided essential financial support for the publication fees and contributed to the development of the research framework. D.L. offered invaluable guidance in designing and creating the illustrations, ensuring they effectively conveyed the key findings. All authors actively collaborated throughout the research and writing process. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (Grant Nos. 42176012 and 42130402), and Key innovation team of China Meteorological Administration (No. CMA2022ZD09).

Data Availability Statement

The data supporting the reported results are not publicly available due to privacy.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Distribution of automatic weather stations (blue dots) and the Qingpu radar (red triangle) in Shanghai.
Figure 1. Distribution of automatic weather stations (blue dots) and the Qingpu radar (red triangle) in Shanghai.
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Figure 2. Schematic diagram of the 5 × 5 radar range bin data above the automatic weather station.
Figure 2. Schematic diagram of the 5 × 5 radar range bin data above the automatic weather station.
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Figure 3. Workflow diagram for the relationship between the Z and R model, SVM, GBDT, RFR, and the KAN deep learning model for single-variable and multivariable precipitation estimation.
Figure 3. Workflow diagram for the relationship between the Z and R model, SVM, GBDT, RFR, and the KAN deep learning model for single-variable and multivariable precipitation estimation.
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Figure 4. Single-variable KAN deep learning neural network architecture.
Figure 4. Single-variable KAN deep learning neural network architecture.
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Figure 5. Multivariable KAN deep learning neural network architecture.
Figure 5. Multivariable KAN deep learning neural network architecture.
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Figure 6. Comparison of the estimation effects of two Z-R relationships.
Figure 6. Comparison of the estimation effects of two Z-R relationships.
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Figure 7. Scatter density plots of estimated vs. actual precipitation for five single-variable models: (a) Z = 270.81 R1.09; (b) SVM; (c) RF; (d) GBDT; and the (e) KAN deep learning method. The black solid line represents the ideal scenario where estimated values are perfectly aligned with observed values (y = x), while the red solid line indicates the actual relationship between estimated and observed values, highlighting the bias between them.
Figure 7. Scatter density plots of estimated vs. actual precipitation for five single-variable models: (a) Z = 270.81 R1.09; (b) SVM; (c) RF; (d) GBDT; and the (e) KAN deep learning method. The black solid line represents the ideal scenario where estimated values are perfectly aligned with observed values (y = x), while the red solid line indicates the actual relationship between estimated and observed values, highlighting the bias between them.
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Figure 8. Map of radar reflectivity and spatial distribution of univariate precipitation estimates using five different models at 06:00 UTC on 24 June 2024: (a) radar reflectivity; (b) Z = 270.81 R1.09; (c) Support Vector Machine model; (d) Random Forest model; (e) Gradient Boosting Decision Tree model; and (f) KAN deep learning model.
Figure 8. Map of radar reflectivity and spatial distribution of univariate precipitation estimates using five different models at 06:00 UTC on 24 June 2024: (a) radar reflectivity; (b) Z = 270.81 R1.09; (c) Support Vector Machine model; (d) Random Forest model; (e) Gradient Boosting Decision Tree model; and (f) KAN deep learning model.
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Figure 9. Scatter density plots of estimated vs. actual precipitation for four multivariable models: (a) SVM (multivariable); (b) GBDT (multivariable); (c) RF (multivariable); and (d) KAN deep learning method (multivariable). The red solid line represents the ideal scenario where estimated values are perfectly aligned with observed values (y = x), while the black solid line indicates the actual relationship between estimated and observed values, highlighting the bias between them.
Figure 9. Scatter density plots of estimated vs. actual precipitation for four multivariable models: (a) SVM (multivariable); (b) GBDT (multivariable); (c) RF (multivariable); and (d) KAN deep learning method (multivariable). The red solid line represents the ideal scenario where estimated values are perfectly aligned with observed values (y = x), while the black solid line indicates the actual relationship between estimated and observed values, highlighting the bias between them.
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Figure 10. Map of radar reflectivity and spatial distribution of multivariable precipitation estimates using four different models at 06:00 UTC on June 24, 2024: (a) radar reflectivity map; (b) Support Vector Machine model; (c) Random Forest model; (d) Gradient Boosting Decision Tree model; and (e) KAN deep learning model.
Figure 10. Map of radar reflectivity and spatial distribution of multivariable precipitation estimates using four different models at 06:00 UTC on June 24, 2024: (a) radar reflectivity map; (b) Support Vector Machine model; (c) Random Forest model; (d) Gradient Boosting Decision Tree model; and (e) KAN deep learning model.
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Table 1. Descriptions of the six radar parameters related to precipitation.
Table 1. Descriptions of the six radar parameters related to precipitation.
Feature NameAbbreviationUintsDescription
Radar reflectivityZdBZSize and density of particles
Differential reflectivityZDRdB
Specific differential phase shiftKDP°/kmRainfall rate and particle type
Correlation coefficientCCDimensionlessParticle uniformity and shape
Vertical liquid water contentVILkg/m2Total liquid water content
Echo-top heightETkmMaximum particle height
Table 2. Comparison of precipitation estimation reliability among five single-variable models.
Table 2. Comparison of precipitation estimation reliability among five single-variable models.
MetricZ-R RelationshipRFGBDTSVMKAN
Deep Learning
TestValidationTestValidationTestValidationTestValidation TestValidation
RMSE
(mm)		4.2234.1573.5083.4043.9663.4393.6373.5093.5143.369
MRE3.8643.1521.3751.3261.7711.6581.5611.5251.2081.018
R20.4250.5250.5920.5790.5500.6500.5710.5790.6190.649
MAE
(mm)		2.1502.1441.8131.6141.9741.8741.8091.6161.6941.473
Table 3. Comparison of the precipitation estimation reliability for four multivariable models: RF (multivariable), SVM (multivariable), GBDT (multivariable), and the KAN model (multivariable).
Table 3. Comparison of the precipitation estimation reliability for four multivariable models: RF (multivariable), SVM (multivariable), GBDT (multivariable), and the KAN model (multivariable).
MetricRFGBDTSVMKAN
Deep Learning
TestValidationTestValidationTestValidationTestTest
RMSE
(mm)		3.7993.5663.4493.4223.6753.4723.1173.017
MRE1.7861.5511.3091.2631.4381.4680.9570.957
R20.5950.6180.6290.6970.5790.6010.7350.755
MAE
(mm)		2.1601.7451.7651.6271.7471.7131.6251.625
Table 4. Comparative evaluation metrics of KAN deep learning model under various weather conditions.
Table 4. Comparative evaluation metrics of KAN deep learning model under various weather conditions.
Main CategorySubcategoryMAE (mm)MRERMSE (mm)R2
SeasonsSpring1.2091.0942.0250.631
Summer1.7871.2783.7590.516
Autumn1.5351.3653.1890.619
Winter0.6870.8941.0640.763
Weather SystemsTyphoon2.2632.6865.1610.372
Frontal System1.4230.5993.0930.642
Meiyu Season3.0771.1094.2260.798
Summer Convective 2.6451.9804.8060.560
Rainfall IntensityHeavy Rain11.0740.91012.2050.241
Moderate Rain1.5751.0982.7470.571
Light Rain1.4721.1762.1440.778
Duration1–6 h2.2491.4153.5550.408
7–12 h1.9100.8242.9450.530
Over 12 h 1.2290.4802.5450.751
Overall1.6250.9573.1170.735
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Liu, M.; Zuo, J.; Tan, J.; Liu, D. Comparing and Optimizing Four Machine Learning Approaches to Radar-Based Quantitative Precipitation Estimation. Remote Sens. 2024, 16, 4713. https://doi.org/10.3390/rs16244713

AMA Style

Liu M, Zuo J, Tan J, Liu D. Comparing and Optimizing Four Machine Learning Approaches to Radar-Based Quantitative Precipitation Estimation. Remote Sensing. 2024; 16(24):4713. https://doi.org/10.3390/rs16244713

Chicago/Turabian Style

Liu, Miaomiao, Juncheng Zuo, Jianguo Tan, and Dongwei Liu. 2024. "Comparing and Optimizing Four Machine Learning Approaches to Radar-Based Quantitative Precipitation Estimation" Remote Sensing 16, no. 24: 4713. https://doi.org/10.3390/rs16244713

APA Style

Liu, M., Zuo, J., Tan, J., & Liu, D. (2024). Comparing and Optimizing Four Machine Learning Approaches to Radar-Based Quantitative Precipitation Estimation. Remote Sensing, 16(24), 4713. https://doi.org/10.3390/rs16244713

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