A Combined Linear–Nonlinear Short-Term Rainfall Forecast Method Using GNSS-Derived PWV

Abstract

Short-term rainfall forecast using GNSS-derived tropospheric parameters has gradually become a research hotspot in GNSS meteorology. Nevertheless, the occurrence of rainfall can be attributed to the impact of various weather factors. With only using tropospheric parameters retrieved from GNSS (such as ZTD or PWV) for linear forecast, it could be challenging to describe the process of rainfall occurrence accurately. Unlike traditional linear algorithms, machine learning can construct better the relationship between various meteorological parameters and rainfall. Therefore, a combined linear–nonlinear short-term rainfall forecast method is proposed in this paper. In this method, the PWV time series is first linearly fitted using least squares, and rainfall events are determined based on the PWV value, PWV variation, and PWV variation rate. Then, a support vector machine (SVM) is used to establish a nonlinear rainfall forecasting model using the PWV value, air temperature, air pressure, and rainfall. Finally, the previous two rainfall forecast methods are combined to obtain the final rainfall event. To evaluate the accuracy of the proposed method, experiments were conducted utilizing the temperature, pressure, and rainfall data from ERA5. The experimental results show that, compared to existing short-term rainfall forecast models, the proposed method could significantly lower the false alarm rate (FAR) of rainfall forecasts without compromising the true detection rate (TDR), which were 26.33% and 98.66%, respectively. In addition, the proposed method was verified using measured GNSS and meteorological data from Yunmao City, Guangdong, and the TDR and FAR of the verified results were 100% and 20.2%, respectively, which were proven to apply to actual rainfall forecasts.

1. Introduction

Rainfall is an integral part of the water cycle [1], and extreme rainfall events are among the most destructive weather events worldwide. Rainfall events are also closely associated with disasters, such as droughts, floods, landslides, and urban waterlogging [1]. Thus, the accurate, timely, and reliable prediction of rainfall events has considerable economic and social importance [2].

Meteorological departments commonly use numerical weather prediction (NWP) models to provide rainfall predictions for the upcoming 1–3 d [3], which is the improvement in this article. There is a strong correlation between rainfall and atmospheric water vapor content, which can be quantified by precipitable water vapor (PWV)—a measurement of the amount of precipitation resulting from the condensation of the water vapor in the air column per unit cross-sectional area from the surface of the Earth to the top of the atmosphere [4,5,6]. Radiosondes and water vapor radiometers can detect atmospheric water vapor, but they have certain limitations [7]. Radiosondes, for example, can provide water vapor products with a high vertical resolution; however, the distance between stations is approximately 200 to 300 km, and only two to four radiosonde events can be performed per day. Therefore, it is not easy to detect small-scale and short-period rainfall events with radiosonde balloons [8]. Although water vapor radiometers can provide water vapor products with high temporal resolution, because of the high cost of equipment and the tendency to be affected by clouds and rain, they have not been widely adopted [9]. Hence, traditional water vapor detection methods are often employed for data calibration and accuracy verification [10,11].

Signals from the Global Navigation Satellite System (GNSS) passing through the troposphere are affected by the presence of water vapor, resulting in a delay effect, which can be converted into PWV through calculations. For the first time, Bevis et al. (1992) used GNSS measured data to retrieve PWV at the station, thereby extending the application of GNSS to meteorology [12]. The PWV products acquired with GNSS technology feature high precision (1–2 mm), short sampling intervals (1 s), high spatial resolution (within 10 km), and all-weather performance [13]. With the continuous development of GNSS meteorology in recent years, GNSS-derived PWV has become a widely used tool for climate change research [14,15], ENSO monitoring [10,16], typhoon speed and path estimation [17], drought monitoring index improvement [18], and other fields. Additionally, some progress has been made in short-term rainfall forecasts using GNSS-derived water vapor products; at present, short-term rainfall prediction models suitable for tropical [4], subtropical [19,20], and temperate zones [21] have been established. However, these models only considered a limited number of rainfall forecast factors, which resulted in a low true detection rate (TDR) and a relatively high false alarm rate (FAR) for rainfall forecasts. Zhao et al. (2020) proposed an improved rainfall forecast model by incorporating rainfall forecast factors (PWV value, the variation amount, and variation rate of PWV and ZTD (zenith total delay) [22]. The TDR of this model for rainfalls is above 90%, and the FAR is below 30%. However, this model neglected the impacts of various meteorological parameters (e.g., air temperature and air pressure) on rainfall events.

Compared with traditional linear models, the artificial intelligence (AI) theory (such as support vector machine (SVM), artificial neural network (ANN), and adaptive neuro-fuzzy inference system (ANFIS)) could deliver more accurate rainfall prediction results [8,23,24,25]. The AI theory has therefore been widely applied to research rainfall predictions [26,27,28,29,30,31,32]. In a study by Manandhar et al. (2019), an SVM model was trained based on GNSS PWV data and various meteorological parameters to classify rainfall and non-rainfall events [29]. Liu et al. (2019) built a rainfall prediction model between several meteorological parameters and rainfall data based on the backpropagation neural network (BPNN), which achieved a TDR higher than 96% and a FAR of approximately 40% [30]. Benevides et al. (2019) constructed a nonlinear autoregressive exogenous neural network model (NARX) to detect heavy rainfall events using GNSS observation and various meteorological data [33]. Le et al. (2020) developed a rainfall prediction model for the next 48 h using the nonlinear autoregressive neural network (NANN) [27]. The correlation coefficient between the measured and model-predicted rainfall time series could reach 0.90. Abdullah et al. (2021) successfully applied the support vector regression model to forecast 18 months rainfall in Bogor city, Indonesia [31]. Numerous studies have demonstrated the possibility of employing machine learning to build nonlinear relationships between various meteorological parameters and rainfall events with relatively high accuracy. Nevertheless, the choice of optimal key parameters and the determination of parameter combinations are crucial for the accuracy of the rainfall prediction model. Thus, although nonlinear rainfall prediction models based on machine learning are more accurate than linear models, the rainfall FAR is still relatively high for this type of models.

To address the above problems, a combined linear–nonlinear short-term rainfall forecast method is proposed in this paper. In this method, the low rainfall FAR and the nonlinear model’s high rainfall TDR of the traditional linear model are combined to effectively lower the rainfall FAR without compromising TDR. Concerning linear rainfall forecast, the least-squares principle was utilized to fit the PWV time series linearly, and the influences of PWV value, PWV variation, and PWV variation rate on rainfall event forecast were simultaneously considered. Concerning nonlinear rainfall forecast, machine learning was utilized for rainfall forecast, while considering the influences of seasonal factors. The cross-grid search algorithm was also employed to determine the optimal combination of machine learning parameters. Experimental tests were conducted using the ERA5 data—the fifth-generation atmospheric reanalysis of the global climate from the European Centre for Medium-Range Weather Forecasting (ECMWF)—and three measured GNSS datasets for Yunmao City, Guangdong, to demonstrate that the proposed short-term rainfall forecast model combined linear–nonlinear methods of this work can provide superior TDR and FAR for rainfall prediction than the existing models.

2. Theory and Data

2.1. Data Description

In this study, three GNSS collocated meteorological stations in Yunmao City, Guangdong, China, were selected as examples for the experiment. Located in southern China, Guangdong with borders 20.15° N–25.52° N and 109.75° E–117.33° E, where the annual average temperature and precipitation are 22.3 °C and 1777 mm over the period of 1956–2006 [34], respectively. This region is characterized by tropical and subtropical monsoon climates. It is estimated that 80% of all precipitation falls from April to September of a given year and that the dry season lasts from October to March the following year [19,34]. Because the period of the measured data was limited to 10 November 2020–4 February 2021 (86 days), this paper also incorporated air temperature, air pressure, rainfall, and other measured data from ERA5 provided by the ECMWF for the experiment. The spatial and temporal resolution of the ERA5 dataset were determined as 0.25° × 0.25° and 1 h, respectively. This dataset can be downloaded from https://cds.climate.copernicus.eu/ accessed on 15 September 2021. The locations of GNSS stations and the ERA5 grid points generally do not coincide; thus, the ERA5 grid data must be interpolated to the locations of GNSS stations by bilinear interpolation. In addition, the determination of the GNSS collocated meteorological stations followed the principle of the vertical and horizontal distances being within 100 m and 3 km, respectively, between GNSS and meteorological stations [5]. The geographical location and data information for the three GNSS collocated meteorological stations are detailed in Figure 1 and

Table 1. Geographical information of the three ground-based GNSS collocated meteorological stations and statistics during the experimental data period.

Station	Longitude/°	Latitude/°	Height/m	ERA5	GNSS
YM01	111.5087	22.5915	147.76	1 September 2017–31 August 2020	10 November 2020–4 February 2021
YM02	111.2309	22.3889	586.55	1 September 2017–31 August 2020	10 November 2020–4 February 2021
YM03	110.7017	22.2286	94.00	1 September 2017–31 August 2020	10 November 2020–4 February 2021

.

2.2. GNSS-Derived PWV

GNSS observation was first processed using GAMIT/GLOBK software (v10.5) [35], and the sampling rate and the elevation cut-off angle were 30 s and 10°, respectively. In addition, a global mapping function was introduced [36] and the FES 2004 model was also considered to obtain accurate ZTD data (http://holt.oso.chalmers.se/loading/ accessed on 22 August 2022). ZTD comprises two components, namely the zenith hydrostatic delay (ZHD) and the zenith wet delay (ZWD). The ZHD can be calculated using the Saastamoinen model [37]:

$$ZHD=\frac{0.002277\times P_{sr}}{1-0.00266\times cos(2\omega )-0.00028\times H_{eg}}$$

(1)

where $P_{sr}$ refers to the surface air pressure (hPa), and $\omega$ and $H_{eg}$ are the latitude (^o) and elevation of the station (m), respectively. ZHD can thus be separated from ZTD, and then ZWD can be obtained, which can be further converted into PWV:

$$PWV=\frac{{10}^6}{{\ell }_w\times R_t\times (c_2+c_3/T_m)}\times ZWD$$

(2)

where ${\ell }_w$ is the density of liquid water (g/m³); $R_t$, $c_2$, and $c_3$ are constant terms; and $T_m$ is the weighted average temperature of the atmosphere. In this paper, the neural network (NN) model was adopted to calculate $T_m$ [38]. The NN model was validated using data collected for more than 40 years from 201 RS stations worldwide. It was determined that the NN model provides better calculation accuracy for $T_m$ than the GPT2W [39], GTm [40], GTm-I, and PTm-I models [41].

2.3. Introduction of the SVM Model Theory

The SVM model is primarily composed of two parts:

(1) Regression function optimization—the function of SVM is to construct a regression function hyperplane that fits the changing trend of the target vector with the least possible error:

$$\phi (n)=w^T\times \theta (n)+t$$

(3)

where $n_i$ is the sample set input variable ($i\in 1,\cdots ,l$), $m_j$ is the sample set output variable ($j\in 1,\cdots ,l$), $w$ and $t$ are parameters, and $\theta (n)$ is a nonlinear function. Vapnik and Lerner (1963) defined the loss function $E_{\epsilon }$ based on the insensitive loss function $\epsilon$ [42].

$$E_{\epsilon }(m_j)=\{\begin{array}{ll}0& if\left|m_j-(w^T\times \theta (n_i)+t)\right|<\epsilon \\ \left|m_j-(w^T\times \theta (n_i)+t)\right|-\epsilon ,& if\left|m_j-(w^T\times \theta (n_i)+t)\right|\ge \epsilon \end{array}$$

(4)

Support vector regression (SVR) is then transformed into the following optimization problem:

$$min[\frac{1}{2}{\Vert w\Vert }^2+C{\displaystyle \sum _{i=1}^l({\xi }_i,{\xi }_i^{\ast })}],\hspace{1em}\hspace{1em}s.t.\{\begin{array}{l}{m}_j-(w^T\times \theta (n_i)+t)\le \epsilon +{\xi }_i^{}\\ -m_j+(w^T\times \theta (n_i)+t)\le \epsilon +{\xi }_i^{\ast }\\ {\xi }_i,{\xi }_i^{\ast }\ge 0(i,j=1,2,\cdots ,l)\end{array}$$

(5)

where ${\xi }_i,{\xi }_i^{\ast }$ are relaxation factors, which refer to the upper and lower bounds of the training error subject to the error threshold $\epsilon$. $C$ is the penalty factor, which refers to the degree of error correction during error classification.

(2) Optimization problem solving—the optimization problem can be solved by introducing Lagrange multipliers, and the regression function is as follows:

$$\phi (n)={\displaystyle \sum _{i=1}^l({\delta }_i-{\delta }_i^{\ast })}\theta {(n_i)}^T\times \theta (n_i)+t$$

(6)

The input variables corresponding to the non-zero Lagrange multipliers (${\delta }_i-{\delta }_i^{\ast }$) are the support vectors, so Equation (6) can be transformed into:

$$\phi (n)={\displaystyle \sum _{k=1}^r({\delta }_k-{\delta }_k^{\ast })}K(n_k,n)+t$$

(7)

where $n_k$ is the support vector, $K(n_k,n)$ is the kernel function, and three types of commonly used kernel functions include polynomial, Sigmoid, and radial basis function (RBF) kernels. In this paper, the most commonly used Gaussian Radial Basis function (GRBF) was the kernel function [43].

3. Rainfall Forecast Model Combined Linear–Nonlinear Rainfall Method

Considering that single linear or nonlinear models were established in previous rainfall forecast studies, this study innovatively proposes a rainfall forecast model combined linear–nonlinear method based on GNSS PWV data. This method involved first using GNSS-derived PWV to construct a linear model, then using an SVM to construct a nonlinear rainfall forecast model with PWV, temperature, and other meteorological parameters. Finally, the results of both forecast methods were combined to determine the final rainfall event. As illustrated in Figure 2, the experimental design diagram for the method presented in this paper includes three parts: a linear method-based short-term rainfall forecast using PWV data, a nonlinear method-based rainfall forecast using meteorological parameters, and a rainfall forecast model based on the combined linear–nonlinear method.

3.1. Linear Method-Based Short-Term Rainfall Forecast using GNSS-Derived PWV

• Determination of rain forecast factors

Several related studies [4,19,20,21] have shown that the variation amount and rate of variation PWV are important factors that affect rainfall forecasts. Consequently, the variation amount and variation rate of PWV were selected as the rainfall forecast factors in this study, along with the PWV value, as a new rainfall predictor. Given the humid subtropical climate with four distinct seasons along the southeastern coast of China [44], this study also considered the effect of seasonal factors on rainfall forecasts when establishing the model.

• Calculation of the PWV variation amount and variation rate

The PWV variation amount and variation rate are keys to constructing a linear short-term rainfall forecast model. As a first step, the processing windows for the PWV time series were determined, and then the PWV variation time series in each window was fitted using the least squares principle to substitute for the highly nonlinear original sequence. Next, the maximum and minimum PWV values within each window were determined. The PWV variation amount and variation rate within each window were calculated based on these values and the corresponding time. Because most rainfall events occur after the continuous increase in PWV, the amount and rate of variation in PWV are greater than 0. Finally, the above process was repeated to calculate the PWV variation amount and variation rate of the long PWV time series in each sliding window.

• Determination of the optimal thresholds for forecast factors

Using the highest TDR and lowest FAR as criteria, the optimal thresholds for PWV value, PWV variation amount, and PWV variation rate were determined using historical rainfall data to compare the three parameters in different seasons. In most existing studies, threshold values for PWV variation amount and variation rate were established from experience [19,20]. However, the thresholds determined using this approach are often not optimal for PWV variation amount and variation rate. As the percentile method can automatically determine the value of the variables corresponding to different percentiles in an unknown dataset [22,45], the percentile method was introduced in this paper to determine the thresholds for the PWV value, PWV variation amount, and PWV variation rate [46]. Additionally, a large number of experiments were performed to determine the optimal percentile of each predictor.

• Establishment of a linear short-term rainfall forecast model with PWV values

A linear short-term rainfall forecast model was established, considering the PWV value, PWV variation amount, PWV variation rate, and seasonal factors. The early forecast of rainfall events consisted of those within the next 2–6 h.

3.2. Nonlinear Rainfall Forecast Based on SVM

• Data preprocessing

In this paper, multiple variables were selected to construct the nonlinear rainfall forecast model by season. For constructing the SVM-based short-term rainfall prediction model, input parameters included the PWV for the previous hour, air temperature (T), air pressure (P), and rainfall data, and output parameters are rainfall data for the next hour. Because PWV, T, P, and rainfall data have different dimensions and large numerical differences, if the sample data is directly input into the SVM model for training, the weigh will become extremely large through the accumulator, making convergence difficult. Therefore, it is necessary to normalize the input parameters before training the model. For more specific information regarding the process of normalization, please refer to Xu et al. (2012) [47].

• Optimal combination of SVM key parameters (Penalty and Kernel)

With SVM, the nonlinear problem in the low-dimensional space is converted into a high-dimensional feature space. A linear regression function is constructed through the kernel function such that the original data is linearly separable in the high-dimensional space. The commonly used kernel functions include polynomial, sigmoid, and RBF. The advantages of high-speed learning, lack of the local minima problems, and the simple and fixed three-layer architecture comparing with multilayer perception [48,49] are sufficient reasons for choosing it as the kernel function. The most common Gaussian RBF was employed in this paper [43]:

$$K(x_I,x_J)=\exp (-\gamma {\Vert x_I-x_J\Vert }^2)$$

(8)

where $\gamma$ is the kernel function and $\gamma$> 0.

The SVM used the penalty factor to overcome the tradeoff between algorithm complexity and misclassified samples to achieve good generalization performance. A smaller penalty factor $C$ will reduce the penalty for outliers. Therefore, this study utilized cross-validation and a grid search algorithm to optimize the parameter combination [50]. In this model, the optimal combination of the two parameters was determined using exponential growth. The range of parameters was set as follows: $C=2^{-10},2^{-9.5},2^{-9},\cdots ,2^9,2^{9.5},2^{10},\gamma =2^{-10},2^{-9.5},2^{-9},\cdots ,2^9,2^{9.5},2^{10}$. A total of 41 × 41 pairs of parameters were combined from this data range, and 1681 cross-validations were performed on this basis. Each sample subset was trained and predicted according to cross-validation theory. Using the ranking of the average classification accuracy of the cross-validation method, the parameter combination with the highest classification accuracy was selected as the optimal model parameters. With the combination of cross-validation and grid search, the optimal combination of the penalty factor and kernel parameter can be established.

3.3. A Linear–Nonlinear Rainfall Forecast Method and Evaluation

To fully leverage the low FAR of the linear rainfall forecast model and the high TDR of the nonlinear rainfall prediction model, this study combined the predictions of the two methods to determine the final rainfall event. It should be noted that a correct rainfall prediction event can be predicted by either of the two rainfall forecast models, whereas an incorrect rainfall prediction event is one that either model can predict, but no rainfall actually occurs. Accordingly, the evaluation indicators for the combined linear–nonlinear rainfall forecast model in this paper may be outlined, namely, TDR and FAR. These two indicators were used to evaluate the advantages and disadvantages of the proposed model. The calculation formulas for TDR and FAR are, respectively as follows:

$$\begin{array}{l}TDR=\frac{N_{tr}}{N_{total}}\\ FAR=\frac{N_{ff}}{M_{total}}\end{array}$$

(9)

where $N_{tr}$ is the number of correct rainfall predictions, $N_{total}$ is the actual number of rainfalls, $N_{ff}$ is the number of incorrect rainfall predictions, and $M_{total}$ is the number of rainfall predictions.

4. Verification and Evaluation

4.1. ERA5-Provided Weather Data Verification

To verify the interpolation accuracy of PWV and rainfall data from ERA5 to GNSS stations, hourly PWV collected during the same period (10 November 2020–4 February 2021) was selected as the measured data, and the ERA5 grid data was interpolated to each GNSS station using bilinear interpolation. Figure 3 shows the probability density maps of PWV from ERA5 and those measured from the three GNSS stations (YM01, YM02, and YM03) in Yunmao City. The figure shows that the PWV provided by ERA5 exhibit good consistency with the estimated data, and there is no indication of systematic deviation. At the three GNSS stations, the root-mean-square error (RMSE) of PWV was all less than 3 mm, and the correlation coefficient was above 0.87. The above results indicate that the data interpolation from ERA5 to the GNSS stations had a relatively high level of accuracy. Additionally, the average coincidence rate between rainfalls interpolated from the ERA5 to the stations and actual rainfalls was above 88%, indicating that the rainfall data was also accurate. Therefore, the interpolated data from ERA5 to the GNSS stations could be used to construct rainfall models.

4.2. Validation of the Linear Method-Based Rainfall Forecast Model using GNSS-Derived PWV

Several existing studies [4,19,20,21] have demonstrated that GNSS PWV can be used for short-term rainfall forecasts, but none of them have demonstrated the feasibility of this method in Yunmao City. Therefore, this paper first verified the feasibility of the linear rainfall forecast model in Yunmao City.

In this study, verification was performed using the ERA5 data for 1 September 2017–31 August 2020 at three stations in Yunmao City. In the rainfall prediction experiment based on GNSS PWV variation amount and variation rate, the percentile method was introduced to determine the optimal thresholds for PWV variation amount and variation rate. Numerous experiments have revealed that the inter-annual variations in the optimal percentiles of PWV variation amount and variation rate at each station are consistent. Figure 4 shows the optimal percentile chart comparing PWV variation amount and variation rate in different seasons. This figure demonstrates that, in most cases, the optimal percentile for PWV variation amount and variation rate is 10. Thus, this paper used the percentile value mentioned above for rainfall experiments.

Figure 5 shows the statistical results of rainfall predictions at each season made by the linear rainfall forecast model for each of the three stations from September 2017 to August 2020. It can be seen from the figure that the TDR and FAR of the rainfall forecast model constructed by the linear method were similar in the same season in different years, where the TDR was above 85% and the FAR was below 30%, except in winter. Additionally, the difference between TDR and FAR at the three stations was relatively small, indicating that the rainfall forecast model constructed based on the linear method has good applicability and stability.

Table 2. Statistical results of TDRs and FARs of the linear method-based rainfall forecast model in different seasons at the three stations (%/number of rainfalls).

Station	TDR					FAR
	Spring	Summer	Autumn	Winter	Average	Spring	Summer	Autumn	Winter	Average
YM01	87.5/69	88.2/104	85.0/37	87.6/16	87.4	25.8	25.7	29.9	46.4	27.9
YM02	85.2/76	88.3/115	84.9/46	98.0/16	87.3	24.9	26.4	28.7	45.3	27.5
YM03	88.2/79	88.4/123	85.3/44	89.6/15	87.9	28.4	24.6	25.2	49.2	27.2
Aver.	86.9	88.3	85.0	91.8	87.5	26.3	25.5	27.8	46.9	27.5

presents the statistical results of TDR and FAR of the rainfall forecast model constructed using the linear method at three stations in various seasons. According to the statistical results, the method applies to all seasons. The TDRs of the three stations were all above 87%, and the differences were relatively small. The FAR was relatively high in the winter, but it was consistently below 30% in the other three seasons. Additionally, the TDR in summer is obviously higher than it in spring and autumn, which reached 88.3%. In contrary, the FAR in summer is lower than it in the other seasons, which was 25.5%, and the FAR in winter is the highest with 46.9%. The main reason for this was that there were relatively few actual measurements of rainfalls in winter, and the number of erroneous rainfall predictions was close to the number of actual rainfall events. Therefore, the winter FAR as calculated according to Equation (9) was relatively high. Additionally, this method can maintain a low rainfall FAR to a large extent, but its average rainfall TDR was only 87.5%, which is relatively low.

4.3. Construction of the Nonlinear Method-Based Rainfall Forecast Model

Rainfall is the result of the combined influence of a variety of meteorological factors. Using changes in tropospheric parameters alone cannot provide a clear indication of changes in water vapor. Unlike constructing a linear rainfall forecast model, machine learning experiments can account for the effects of multiple meteorological variables (PWV, T, P, and rainfall) on short-term rainfalls. Therefore, this paper introduced the SVM method to construct a nonlinear rainfall forecast model. The model used PWV, T, P, and rainfall of the previous hour as input parameters and trained the model by season to estimate the rainfall in the next hour. Figure 6 shows the TDRs and FARs of internal and external coincidence accuracies of the nonlinear rainfall forecast model at the three stations. The figure shows that both the internal and external coincidence accuracies of the model are relatively high, where the TDR is above 98% and 94%, respectively. Additionally,

Table 3. The data period used in the internal and external experiments in different seasons.

Season	Data Period
	Internal Experiment	External Experiment
Spring	1 March 2018–31 May 2018, 1 March 2019–31 May 2019	1 March 2019–31 May 2019, 1 March 2020–31 May 2020
Summer	1 June 2018–31 August 2018, 1 June 2019–31 August 2019	1 June 2019–31 August 2019, 1 June 2020–31 August 2020
Autumn	1 September 2017–30 November 2017, 1 September 2018–30 November 2018	1 September 2018–30 November 2018, 1 September 2019–30 November 2019
Winter	1 December 2017–28 February 2018, 1 December 2018–28 February 2019	1 December 2018–28 February 2019, 1 December 2019–28 February 2020

listed the data period used in the internal and external experiments. Taking spring 2018 as an example to show the difference between the internal and external experiments, the lead one hour T, P, PWV and rainfall in spring 2018 are input into the trained model, which is trained based on the data in spring 2018, and which is defined as the internal experiments. The lead one hour T, P, PWV and rainfall in spring 2019 are input into the trained model, which is trained based on the data in spring 2018, and which is defined as the external experiments. On the other hand, the purpose of internal experiments is to test the performance of the trained model for the internal simulation. Additionally, the trained model is pre-familiar to the data of spring 2018. The purpose of external experiments is to test the performance of the trained model for the external forecast. Additionally, the trained model is unknown and uncorrelated to the data of spring 2019. Therefore, the data and purpose of internal and external experiments are the most important difference in internal and external experiments. Additionally, the FAR of the internal coincidence accuracy of this model is relatively low, whereas the FAR of the external coincidence accuracy is relatively high.

In

Table 4. Statistical results of TDRs and FARs of internal and external coincidence accuracies of the nonlinear method-based rainfall forecast model at the three stations.

Station	Internal Coincidence Accuracy		External Coincidence Accuracy
	TDR	FAR	TDR	FAR
YM01	98.6	12.6	94.4	36.1
YM02	98.6	12.1	95.6	34.9
YM03	98.7	10.9	95.3	28.6
Mean	98.7	11.9	95.1	33.2

, the statistical results are presented for the three stations using the nonlinear rainfall forecast model. As can be seen from the table, the average TDR of both internal and external coincidence accuracy of the model was greater than 95%, whereas the FAR of internal coincidence accuracy was relatively low at only 11.9%. Nevertheless, the FAR of the external coincidence accuracy was relatively high at 33.2%. These results indicate that the rainfall forecast model based on the nonlinear model can greatly improve the TDR for rainfall events, but its FAR is higher than that of the traditional linear forecast method.

4.4. Verification of the Combined Linear–Nonlinear Rainfall Forecast Model

In the above experiments, the linear rainfall forecast model had a low FAR, and the nonlinear rainfall forecast model exhibited a high TDR. To further enhance the TDR of the short-term rainfall forecast model and reduce its FAR, a combined linear–nonlinear rainfall forecast method is proposed in this paper. Figure 7 shows the comparison of using the linear (Lin.), nonlinear (SVM.), and combined linear–nonlinear method (Com.) for short-term rainfall forecast. In addition, the data period of the linear method is same in Section 4.2; and the data periods of the nonlinear and combined linear-nonlinear method are the same, which are listed in Table 3. According to the figure, all three rainfall forecast methods could provide adequate forecast effects, but the linear method exhibited a relatively low TDR, whereas the nonlinear method had a high FAR. However, the combined method proposed in this paper had the highest TDR and lowest FAR at all three stations. A statistical analysis of the TDR and FAR of the three types of methods for rain forecasts is presented in

Table 5. Statistical results of TDRs and FARs of the three rainfall forecast models (%).

Method	Least Squares	SVM	Method of this Paper
TDR	87.3	95.1	98.7
FAR	27.5	33.2	26.3

. Among the three models, the TDR of the proposed method is greater than 98%, and the FAR is lower than 27%, demonstrating the best performance.

4.5. Validation of the Rainfall Forecast Model Based on Measured GNSS Data and Meteorological Parameters

To further verify the accuracy of the combined linear–nonlinear short-term rainfall prediction model proposed in this paper, the measured GNSS and meteorological data from the three stations in Yunmao City from 10 November 2020 to 4 February 2021, were used to external experiments. Additionally, the data over the period of 10 November 2019–4 February 2020 in Table 3 was used to the internal experiment. Because of the absence of rainfall at the YM03 station during this period,

Table 6. Statistical results of TDRs and FARs of the combined linear–nonlinear rainfall forecast model at the YM01 and YM02 stations.

Station	YM01	YM02	Mean
TDR	100	100	100
FAR	23.8	16.7	20.2

only lists the statistical results from the proposed rainfall forecast model for the YM01 and YM02 stations. Based on the statistical results, the TDR and FAR rainfall forecast of this model are 100% and 20.2%, respectively. Similar to the results calculated based on the ERA5 data, the statistical results further confirmed the advantage of the method proposed in this paper. Additionally,

Table 7. Statistics on the accuracy of existing rainfall prediction models.

	Index	Temporal Resolution	Considering the Seasons	Predictor	TDR	FAR	Algorithm
Studies
Benevides et al. (2015)		Hourly	No	PWV variation amount and variation rate	75%	60–70%	Least Squares
Yao et al. (2017)		Hourly	No	PWV variation amount, variation rate, and PWV value	80%	66%	Least Squares
Zhao et al. (2018 a)		5 min	No	PWV variation amount and variation rate	>80%	60–70%	Least Squares
Manandhar et al. (2018 a)		5 min	No	PWV variation rate and PWV second derivative	87%	38%	Least Squares
Manandhar et al. (2019)		5 min	No	PWV, solar radiation, DOY, HOD	70%	20%	SVM
Benevides et al. (2019)		15 min	No	PWV, cloud top temperature, air pressure, altitude, relative humidity, surface air pressure, temperature	64%	22%	ANN
Liu et al. (2019)		5 min	No	Air pressure, temperature, DOY, HOD, MOH, PWV, relative humidity	>96%	40%	BP-NN
Zhao et al. (2018 b)		Hourly	No	ZTD variation amount and variation rate	85%	66%	Least Squares
Zhao et al. (2020)		Hourly	Yes	PWV/ZTD variation amount and variation rate, PWV value	96%	29%	Least Squares
Li et al. (2020)		Hourly	Yes	PWV value, PWV increment and its rate, PWV decrement and its rate	95.5%	32.9%	Least Squares
Li et al. (2021)		Hourly	Yes	PWV, ZTD, HOD, P, T, RH, DOY	94.5%	20.8%	BP-NN
Li et al. (2022)		Hourly	Yes	PWV/ZTD value and other 12 factors	99.1%	22.4%	Least Squares
This study		Hourly	Yes	PWV variation amount, variation rate, PWV value, P and T	98.7%	26.3%	Least Squares + SVM

Note: DOY represents the day of year; HOD represents hour of the day; MOH represents the minute of the hour.

shows the comparison statistics of the existing rainfall prediction models in recent years. By comparing the method proposed in this paper with the related studies, it is found that the proposed method could achieve a forecast effect with the relatively low FAR while ensuring the highest TDR.

5. Conclusions

By considering both the advantages and disadvantages of existing rainfall forecast methods, this paper proposed a combined linear–nonlinear short-term rainfall forecast method. This method can fully leverage the advantages of linear and nonlinear rainfall forecast models, thereby further improving the TDR and reducing the FAR of rainfall events. In this method, the linear rainfall prediction model uses the PWV value, the PWV variation amount, and the PWV variation rate as the rainfall predictors, and the nonlinear model uses the PWV value, temperature, pressure, and rainfall as the forecast factors to estimate the rainfall for the next hour. Then, the results of the two forecast models were combined to obtain the final rainfall forecast result. The ERA5 data were used to verify the method proposed in this paper. According to the experimental findings, the average TDR and FAR for the proposed method are 98.7% and 26.3%, respectively, which are better forecast results than using a linear or nonlinear rainfall forecast method alone. In addition, the forecast accuracy of the proposed method was the highest in summer and the lowest in winter. Additionally, the forecast accuracy in spring was slightly higher than autumn. In comparison with existing methods of rainfall forecast, the proposed method achieved the highest TDR and the lowest FAR. Additionally, the proposed method was further verified using the GNSS observation data and meteorological data measured at three stations in Yunmao City. A good rainfall forecast effect was obtained, which further confirmed the feasibility of the proposed method.