1. Introduction
Radar is an important tool for monitoring the weather and producing short-range forecasts by linking rain microphysics parameters to rain types and intensity (Zhang 2016). In the early era of radar development with single polarization, only the radar reflectivity (Zh) was available to estimate the rain rate (R) utilizing the well-known Z–R relationship derived from power-law forms to relate Zh and R. The coefficients of the Z–R relation were found to depend upon the observed drop size distribution (DSD), which is affected by rain “type” (e.g., convective vs stratiform), season, and location resulting in over 200 proposed Z–R relationships (e.g., Doviak and Zrnić 1993; Rosenfeld and Ulbrich 2003).
The subsequent development of weather radar polarimetry provided polarimetric radar data (PRD) with additional microphysical information that has the potential to lead to more comprehensive knowledge regarding the shape, composition, and phase of hydrometeors (e.g., Zhang et al. 2019). Polarimetric variables such as differential reflectivity (ZDR) and specific differential phase (KDP), when combined with Zh, enhance the reliability of R estimates, reducing errors from the range of 30%–40% to ∼15% (Brandes et al. 2002). PRD has also enabled DSD retrievals by incorporating two or more radar variables. These retrievals have been widely used for the calculation of rain physics parameters (Zhang 2016), parameterization of cloud microphysics (e.g., Haddad et al. 1996; Seifert and Beheng 2006; Zhang et al. 2006; Hsieh et al. 2009), and estimation of latent heating (Nelson et al. 2016; Naren Athreyas et al. 2020). This added knowledge has led to improvements in weather forecasts (Lim and Hong 2010; Y. Zhang et al. 2020)
The deterministic retrievals providing DSD parameters of slope (Λ) and intercept terms (N0) using the definition of ZDR [i.e., ZDR = 10 log10(Zdr), with the reflectivity ratio of Zdr = Zh/Zυ] and Zh are often obtained by assuming two-parameter exponential or constrained-gamma DSD models (Zhang et al. 2001). However, DSD parameters are nonphysical variables with skewed non-Gaussian distributions (Cao et al. 2010; Zhang 2016). Mahale et al. (2019) proposed that the physical parameters of mass-weighted mean diameter (Dm) and liquid water content (W) could be used for DSD retrieval with a two-parameter exponential DSD model. These two parameters (Dm and W) can be derived from model physical parameters (number concentration and mixing ratio) that tend to more closely follow a Gaussian distribution than the DSD parameters Λ and N0. These physics-based inversion retrievals were conducted using the parameterized PRD operators derived as function of the physical parameters (Mahale et al. 2019).
Even the use of PRD, empirical formula-, and physics-based inversion retrievals have inherent limitations. For example, power-law forms (i.e., empirical formulas) yield linear relations only in the logarithmic domain, cannot handle nonlinear terms, and may not optimally account for measurement and/or model errors (Zrnić et al. 2006; Zhang 2016; Zhang et al. 2019). While the retrievals from physics-based inversions may perform well when the underlying physical assumptions are valid, these methods will not work suitably in real precipitation cases when significant errors exist. In addition, since physics-based inversion cannot account for any bias in the data, small errors may significantly deteriorate its performance (Zrnić et al. 2006; Mahale et al. 2019). This study was conducted to overcome the weaknesses of these two existing methods by applying the deep neural network approach.
In our approach, one machine learning (ML) method, the deep neural network (DNN) technique, was applied to obtain R and other physical parameters from PRD. ML has an advantage as it can include nonlinear terms and additional input and output variables; this technique has the potential to improve regression and classification problems of multidimensional and multivariety data (e.g., Alimissis et al. 2018; Lang et al. 2021; J. Zhang et al. 2020; Chang-Hoi et al. 2021; Sarker 2021). Another advantage of this ML approach is that the results can be obtained quickly once the training is completed allowing the approach to be easily employed for operational purposes. Previous studies have utilized different forms of ML for quantitative precipitation estimation. For example, Kusiak et al. (2013) demonstrated that the neural network produced the best performance among various ML models in estimating rain from Zh alone. Chen et al. (2020) used multilayer perceptrons to estimate rain rate over a vast area from satellite-observed multichannel radiances. However, their approach had a crucial constraint that assumed radar-retrieved R as the ground truth. In our study, we utilized a similar DNN algorithm to that of Chen et al. (2020) and included small-scale errors contained in observed radar variables. Specifically, several types of observational errors (i.e., measurement error and model bias) were included in the training and testing data for estimating physical parameters and R. The DNN results from this approach were compared with two existing methods (i.e., physics-based inversion and empirical formula).
The organization of the paper is as follows. Section 2 describes the data and its partitioning process. Section 3 discusses the three retrieval methods: physics-based inversion, empirical formula, and DNN. Section 4 analyzes the performance of the three methods on disdrometer data by including measurement and model errors. Section 5 applies the empirical and DNN methods to radar data for rain estimation and DSD retrievals. The main conclusions and discussion are provided in section 6.
2. Data and evaluation metrics
a. Disdrometer data
The rain microphysics quantities from a two-dimensional video disdrometer (2DVD) for the 12-yr period 2006–17 were analyzed for our initial model evaluations. The 2DVD can directly measure DSDs, from which R, Dm, and W can be calculated/measured, in 1-min intervals. The radar variables (i.e., ZH, ZDR, KDP) were calculated by numerical integration of T-matrix scattering amplitudes at the S band over the rain DSD (Waterman 1968; Vivekanandan et al. 1991; Zhang et al. 2001; Ryzhkov et al. 2011), and R, Dm, and W by numerical integration across the 2DVD observed DSD. Here, ZH symbolizes Zh in the logarithmic domain as ZH = 10 log10(Zh). The 2DVD observing system for this study was located at the University of Oklahoma’s Kessler Atmospheric and Ecological Field Station (KAEFS; 34.98°N and 97.53°W), about 44.75 km southwest (210°) of the KTLX radar. During this 12-yr period, a total of 616 320 DSDs were collected over 428 rain events.
b. Radar data
Polarimetric radar measurements from the S-band KTLX Weather Surveillance Radar-1988 Doppler (WSR-88D, also known as NEXRAD), with a beamwidth of 0.95° and a range resolution of 250 m, were used to compare the performance of each method for the dates corresponding to 2DVD observations from 2013 onward to 2017 to ensure availability of polarimetric variables. The NEXRAD level-II data were obtained from the National Centers for Environmental Information (www.ncei.noaa.gov) for only the lowest (i.e., 0.5°) elevation angle, approximately 508 m above the 2DVD location. These data included several mesoscale convective systems containing both stratiform and convective rain. Note that radar data are often prone to sources of contamination—such as ground clutter and anomalous propagation (e.g., Habib et al. 2009; Hubbert et al. 2009; Lukach et al. 2017; Wu et al. 2021)—and calibration biases, particularly for ZDR (Ryzhkov et al. 2005a; Zrnić et al. 2006; Lee et al. 2021; Sanchez-Rivas and Rico-Ramirez 2022). Additionally, the presence of hail within a sampling volume can lead to incorrect retrievals of rain DSD parameters since the methods discussed herein are tailored only for rain.
c. Data collection
To minimize the error from the 1-min 2DVD data, unrealistic (e.g., rain rate over 1000 mm h−1) and nonmeaningful (e.g., rain rate below 0.1 mm h−1) periods of rain events were excluded. More specifically, only those rain events with R between 0.1 and 100 mm h−1, a total drop count larger than 50, and ZDR between 0.2 and 4.5 dB that continued for at least 20 min were considered. Note that if interruptions (i.e., ZDR less than 0.2 dB, larger than 4.5 dB, or R below 0.1 mm h−1) between the rain events were less than 10 min apart, they were considered as the same rain event. The consideration for interruptions was made to account for the limited spatial coverage of the 2DVD and prevent division of a storm into different datasets. The ZDR range was limited because the exponential DSD model does not accurately represent very small and large drops from natural rain (Ulbrich 1983; Kozu and Nakamura 1991; Tokay and Short 1996). While constrained-gamma models often better represent small drops (Zhang et al. 2001), exponential DSD were assumed because most numerical weather prediction models use single or double moment microphysics schemes (Lin et al. 1983; Milbrandt and Yau 2005; Morrison et al. 2005; Liu et al. 2022).
The rain events were divided into groups based on the observation date for training (70%), validation (15%), and testing (15%), respectively. A total of 617–850 rain events, consisting of 42 462–81 832 data points, were selected based on the aforementioned criteria. The amount of included data differs due to exclusion of points when additional errors were added or introduced in subsequent analyses. The training and validation sets were used in the training process, and all results were calculated using testing data.
Additional adjustments via filtering and averaging were applied to the raw radar data to account for the increased error sources and different spatiotemporal resolution between the observations. The radar data are updated in approximately 5-min intervals in contrast to the 1-min DSD data from the 2DVD, and the radar resolution volumes are about 107 larger than the sampling volume of the 2DVD (Cao et al. 2008). First, a five-point median filter along the radial was used to remove outliers and then, five-gate moving average to smooth the data. Second, a correlation coefficient filter (ρhv > 0.97) was applied to minimize the influence of hail and nonmeteorological objects (e.g., ground clutter and beam blockage) that tend to be associated with greatly reduced ρhv (e.g., Straka et al. 2000). Subsequently, the Zh and Zdr (in 5-min intervals) were averaged over five gates (two gates before and two after the estimated corresponding location with missing values excluded). The 2DVD-observed DSDs (which are provided in 1-min intervals) were modified using the corresponding time and the next 4 min of 2DVD observations. Within these 5 min of DSD data, both the maximum and minimum values of each size bin were discarded and the mean of the remaining three was used to retrieve the Dm, W, and R using numerical integration. These procedures were conducted to lessen the impact of the 1) different spatiotemporal resolutions between the radar and 2DVD, resulting in reduced fluctuation error, and 2) the time lag resulting from the observation height differences between the 2DVD and radar.
d. Three types of error
A schematic diagram on how fitting, measurement, and model errors were included and applied to form training, validation, and testing datasets based on 2DVD observations. The black lines indicate the procedures to form the datasets, and the red lines the error categories for retrievals. The “TMM num. integration” denotes the T-matrix method with numerical integration.
Citation: Journal of Hydrometeorology 24, 11; 10.1175/JHM-D-22-0166.1
Second, the statistical Gaussian distributed external errors were introduced to the radar variables that were calculated in the previous step in the measurement error category (Fig. 1). The selected standard deviation of the errors for ZH was 1 dBZ and 0.2 dB for ZDR as assumed in Mahale et al. (2019). No additional errors were included in the estimators.
Third, for the model error category, errors arising from 2DVD observations and DSD modeling were included (Fig. 1). The 2DVD observations inherently contain measurement errors and model errors arise from differences between the modeled exponential DSD and the measured rain DSD. These errors were accounted for by using the 2DVD observed DSD to directly calculate the radar variables, physical parameters, and R for the analysis. Figure 1 depicts the aforementioned processes for each error type in a diagram.
e. Evaluation metrics
3. Retrieval methods for physical parameters and rain rate
a. Physics-based inversion
The physics-based inversion method utilizes the parameterized PRD operators [Eqs. (1) and (2)] taken from Mahale et al. (2019). The entire procedure is formulated based on the definitions and model relations between the physical parameters and radar variables. The process is composed of two steps: the calculation of physical parameters from parameterized PRD operators and the retrieval of microphysics variables from the calculated physical parameters. The former process is the direct inversion of Eqs. (1) and (2), using the root of the nonlinear function. For a given Zdr, the best representation of Dm was calculated from the relation in Eq. (2). Then, the calculated Dm and given Zh were used to calculate W from Eq. (1), and these Dm and W values were used to calculate the DSD parameters from Eqs. (3) to (5). Subsequently, the DSD was used to calculate R, using numerical integration. Since this approach does not utilize additional data, include error terms, or adjust the relations for different types and forms of data, the equations derived from the physical models do not always represent real situations with observational errors. Thus, as will be subsequently discussed, this technique has limitations in analyzing real data.
b. Empirical formula
Empirical formulas are commonly used for operational purposes due to their intuitiveness and simplicity (May et al. 1999; Brandes et al. 2004; Ryzhkov et al. 2022). Both power-law and polynomial forms were adopted for rain-rate estimation and calculation of physical parameters (e.g., Ryzhkov et al. 2005b; Cao et al. 2010). In this study, both forms were formulated and evaluated. The former was found to perform better than the latter when adequate weighting was used. While the polynomial forms (Zhang 2016) are often applied using simulation data, power-law forms are most commonly used when errors are introduced. Thus, only power-law forms are used hereafter. The empirical formulas were derived using the least squares fit in the known covariance method with appropriate weighting. The weighting is necessary to accommodate for less frequent large values of physical parameters and R. Here, estimators with different powers were tested as weights. The analyses showed that R to the power of 1–1.5 produced the best results (Tables 4 and 5 in Ho 2022). For simplicity, estimators with R to the power of 1.5 were used for all estimators in this study. The empirical formulas for the physical parameters and R for the errors associated with fitting, adding measurements, and with model are shown in Table 1; a description of these errors is provided in section 4. The table indicates that Dm mainly relies on the positive exponent of Zdr for estimation. The exponent of Zh is small for all error types of Dm estimation, ranging from 0.011 to 0.028. This finding is because Dm is a property of the drop sizes, which is strongly informed by Zdr owing to the robust relationship between raindrop size and shape (e.g., Kumjian 2013). Although Dm is often derived as a function of Zdr only, the results indicate that including Zh provided better performance (not shown) compared to the use of Zdr alone. The power-law forms for W and R have a positive exponent for Zh and a negative exponent for Zdr. The equation for those two estimators are similar in that W is the third and R is the ∼3.67th DSD moment.
Empirical formulas of Dm, W, and R for fitting, adding measurement, and with model errors. All formulas were derived using 2DVD data at the Kessler Farm Field Laboratory, Oklahoma, for the period 2006–17.
Empirical formulas are relatively easy to calculate given observations and a general form (e.g., power law), and they provide interpretable relations between the variables. In addition, power-law formulas can be derived for different sets of data, providing more flexibility compared to the previous physics-based inversion method. Minimizing errors by least squares and using proper weighting also take some observational errors and prior probability density function (PDF) into account, providing advantageous properties. However, for optimal results, the error in the logarithm domain needs to be Gaussian distributed. In addition, statistical and/or ML methods typically can be more useful in dealing with observational errors and prior PDF. It is noted that the linear relationship between the radar variables and rain rate in the logarithmic domain does not show accurate results in the linear domain (Zhang et al. 2019).
c. Deep neural network
Various forms and types of ML and deep learning (DL) algorithms have been developed since the advent of artificial neural networks in McCulloch and Pitts (1943). ML, encompassing all algorithms that form an output from a nonexplicit training process, is widely used for various purposes. Recent advancements in computational resources have led to the increased prevalence of DL, which are more complex and computationally expensive due to the hierarchical structures composed of numerous layers. Reichstein et al. (2019) claimed that DL methods are more suitable for solving problems for Earth systems compared to the traditional forms of ML. While even more advanced DL methods incorporating statistical and/or more rigorous spatiotemporal information are available, the complexity of the model does not guarantee improved results (Schultz et al. 2021). In this study, DNN, which is a rather simple form of DL, was used for both DSD retrieval and QPE purposes.
DNN, often referred to as multilayer perceptrons in the literature, can be described as a collection of links consisting of artificial neurons or perceptrons (e.g., Murtagh 1991; Taravat et al. 2015; Chen et al. 2020). The biggest advantage that DNN provides is the ability to include nonlinear terms and flexibilities that are often lacking in previous deterministic or statistical approaches (Alzubaidi et al. 2021). DNN, like other ML methods, requires a set of hyperparameters for optimal results (Yang and Shami 2020). The structure and design of the DNN model consist of choosing the model architecture and activation function, selecting unified input and output size and shape, and choosing an additional set of parameters, such as learning rate and batch size.
DNN schematic diagram and table of hyperparameters.
Citation: Journal of Hydrometeorology 24, 11; 10.1175/JHM-D-22-0166.1
DNN has advantageous properties in addition to those described previously, such as the feasibility of including additional input/output variables and spatiotemporal information. In addition, the automatic adjustments of weight and bias terms help find optimal solutions when various observational errors exist in the input/output variables. However, DNN requires precautions when choosing the training data. For example, if the model is trained using stratiform rain only, it will have difficulties retrieving higher values of estimators for convective rain because convective rain contains both low and high rain rates while light rain dominates in stratiform rain. Also, the use of even the most updated ML does not always provide better results over conventional statistical methods in all situations and areas of research due to issues such as data preprocessing, detrending, computational complexity and overfitting (Makridakis et al. 2018). In fact, statistical methods showed better results in cases when abundant prior information is known about the topic and the input variables were limited and established (Rajula et al. 2020). Thus, both the quantity and quality of the training data matters heavily for the performance of DNN.
4. Evaluation using 2DVD data
The three methods (physics-based inversion, empirical formula, and DNN) were compared and evaluated using the ground-truth 2DVD data. The three variables (Dm, W, and R) were estimated for two temporal domains (instantaneous and rain-event average) and three error cases (fitting, adding measurement error, and with model error). In the fitting error category, no additional errors were added to the parameterized operators, and only fitting errors from each method exist. The addition of measurement error randomly assigns Gaussian error with standard deviation of 1 dBZ for ZH and 0.2 dB for ZDR. The model error category utilizes the variables that were numerically integrated from the 2DVD-observed DSD and T-matrix-calculated scattering amplitudes for S-band radar; errors come from the differences between 2DVD observations and physics-based models. The comparison of retrieved and ground-truth (i.e., values derived from the 2DVD-observed DSD) Dm, W, and R are displayed in 2D histograms to understand the distribution (Fig. 3). The residuals are illustrated as boxplots (Fig. 4) for the three variables, three methods, and three error sources. The relative bias and RMSE were also calculated.
A two-dimensional histogram for the (a)–(i) fitting, (j)–(r) measurement error, and (s)–(aa) model error categories. The figure is divided into three columns based on the retrieval method used: (left) physics-based, (center) empirical, and (right) DNN. Within each section (i.e., 3 × 3 panel group), the top row shows the comparison for Dm, the middle for W, and the bottom for R. The color bars for all parameters are on a logarithmic scale, and the x and y axes for W and R are also on a logarithmic scale.
Citation: Journal of Hydrometeorology 24, 11; 10.1175/JHM-D-22-0166.1
Boxplot of Dm (mm), W (g m−3) and R (mm h−1) for the physics-based inversion (P), empirical formula (E), and DNN (D) methods in the instantaneous domain. There are three error sources, (a)–(c) fitting, (d)–(f) adding measurement, and (g)–(i) with model errors to instantaneous 2DVD data for the period 2006–17. The numbers of instantaneous 2DVD data vary depending on the error sources, 11 577–13 037 min. The lower and upper boundaries of each box denote the interquartile range (25% and 75%, respectively), and the horizontal line inside the box marks the median. The lower and upper error lines mark 1.5 × interquartile range, and dots mark outlier values beyond the 1.5 interquartile range. The y-axis tick marks are adjusted accordingly.
Citation: Journal of Hydrometeorology 24, 11; 10.1175/JHM-D-22-0166.1
a. Comparison of instantaneous estimate performance
Before quantifying the statistical errors, we examined the distributions of one-to-one comparisons of the results. Figure 3 depicts 27 2D histograms (i.e., the three parameters, the three error types, and the three methods) for fitting (Figs. 3a–i), adding measurement errors (Figs. 3j–r), and with model errors (Figs. 3s–aa). In the category of fitting error (Figs. 3a–i), the empirical methods show a broader distribution, with a slight bias (Figs. 3b,e,h). The other two methods showed small errors. When adding measurement error (Figs. 3j–r), all three methods and parameters display a much broader distribution. Overall, since random errors were added, the majority of the points were still aligned along the one-to-one line, and no clear differences in higher rain rates were noticed among the three methods. The DNN method showed relatively small errors (Figs. 3l,o,r). In the category of model error (Figs. 3s–aa), the physics-based inversion method clearly shows broader distributions and biases compared to the other two methods (Figs. 3s,v,y). For W and R, the retrievals from the physics-based inversion are skewed toward the actual values and deviate from the one-to-one line. In contrast, DNN generally shows comparable estimates, with the least dispersion and narrowest distributions, for all 27 subplots, with notable benefits when model errors were introduced (Figs. 3u,x,aa). The Dm retrievals show almost a straight line, and there is little bias in the W and R retrievals. It is noted that wider spread and/or underestimation are more evident in the higher rain rates (R > 30 mm h−1) range, probably due to underrepresentation in the training dataset. While wider spread is observed across all three methods, DNN especially has difficulties retrieving larger values, with no retrieved points above 3 g m−3 for W and 80 mm h−1 for R. Further modifications in the training data and/or introduction of constraints should be made to improve retrievals of heavy rain and extreme events when utilizing ML algorithms.
The estimation errors of Dm, W, and R were calculated by comparing the 2DVD observed truth from the estimates using the physics-based inversion, empirical formula (empirical hereafter), and DNN methods. The error distributions categorized into the three error categories (i.e., fitting, adding measurement, and with model errors) in the instantaneous domain are shown in Fig. 4. The number of 2DVD observations in the testing data vary (11 577–12 934 min) depending on the error sources. In the fitting error category (Figs. 4a–c), the interquartile ranges (i.e., between the 25th and 75th percentiles) of estimation errors of the three variables are nearly zero for the physics-based inversion and DNN methods. Errors using the empirical method are meaningfully larger for all three variables. For Dm, the interquartile range of errors using the empirical method is much larger than those using the other two methods, and the outliers are generally positive (Fig. 4a). For W and R (Figs. 4b,c), the interquartile range from the empirical method is centered near zero; however, most of the outliers are significantly negative. This larger error in the empirical method is due to the polynomial form of the parameterized operators in the fitting error category, while power-law forms were used for the empirical formulas. Nevertheless, all methods showed small errors for all three estimators in the fitting error category.
With the addition of measurement error, estimation errors increased for all methods and variables; the maximum error values increased up to 5 times (Figs. 4d–f). The interquartile ranges appear larger in Dm (Fig. 4d) and remain relatively narrow and close to zero in W and R for all methods (Figs. 4e,f). The outliers are distributed in both positive and negative directions with a small bias. In the category with model error, the range of outliers increased in Dm (Fig. 4g) and decreased in W and R (Figs. 4h,i) compared to values in the measurement error category. The errors for the three estimators increased, revealing clear differences in performance between the three methods. The physics-based inversion deteriorated heavily, with the interquartile range of Dm distributed from −0.4 to −0.6 mm, showing a very large negative bias, with 0.2–0.3 g m−3 interquartile range for W, and 2–3 mm h−1 for R with positive bias. The empirical method errors had positively distributed outliers for Dm and negatively distributed for W and R. While the range of outliers increased for DNN in both positive and negative regions, the interquartile range and the median stayed near zero. In summary, considering all three methods, three variables, and three error sources, DNN performed comparable or better than the physics-based inversion and empirical methods with less than 2% bias even when measurement and model errors were introduced.
The relative bias and RMSE of the parameter estimates for the three variables, the three methods, and the three error sources are shown in Table 2. In the fitting error category (top third of Table 2), using the physics-based inversion and DNN methods, bias and RMSE were found to be less than or equal to 1% in all three variables. By contrast, there were large errors in the empirical method: the bias increased to 4.3% and the RMSE increased to 17.1% in W. Adding measurement errors, the magnitude of the bias and RMSE increased appreciably for all variables and methods (middle third of Table 2) with the lone exception of the bias for the physics-based inversion method, which remained at −0.1%. Overall, the errors of the three methods were comparable for Dm estimation. For W and R, however, the magnitudes of the biases were around 1% using DNN and 6.5%–10.3% using the other two methods. The RMSE values were very large; the physical method showed about 77%, and the empirical and DNN methods showed close to 60%. When model errors were added (bottom third of Table 2), the errors increased appreciably in the physics-based inversion method compared to when measurement errors were added but decreased in the empirical and DNN methods with a substantial reduction in bias. Note that if the constrained-gamma model is used for the physics-based inversion, which often better represents natural rain (Zhang et al. 2001), the relative RMSE reduces about 40% when model errors exist. Compared with the empirical method, the DNN method showed smaller errors in general. Overall, the results from Table 2 indicate that the physical method produces large errors in the three variables, with DNN displaying the smallest bias and RMSE especially when adding model errors.
The bias and RMSE of the error estimation for the three variables (Dm, W, and R) and three methods—physics-based inversion (P), empirical (E), and DNN (D)—in the instantaneous domain according to the three error sources, fitting error, adding measurement error, and with model error. The unit is percent (%). The values in bold indicate the smallest bias and RMSE.
b. Rain-event average
The previous analysis was repeated by summing the continuous rain period for at least 20 min from the instantaneous 2DVD data (Fig. 5). There are 127 rain-event-average cases in the 2DVD data. Since rainy periods vary from 22 to 950 min in duration, there are differences in the error distributions between the instantaneous and rain-event-average domains. For the fitting error category (Figs. 5a–c), the physics-based inversion and DNN methods show nearly zero error for the three variables; the empirical method shows minor errors. By adding measurement error (Figs. 5d–f), the errors slightly increase for the three methods and variables, with 0.04 mm in Dm, −0.013 g m−3 in W, and −0.24 mm h−1 in R from empirical formulas, which display the largest boxes out of the three methods. The interquartile ranges of the error are located around zero, and the median value is also near zero. Since Gaussian errors were added to the input variables, the averaging process reduced errors significantly when rain-event-average values were compared. Overall, the performance of all three methods shows limited extent and number of outliers, and spread of residuals. In the category where model errors are involved, the difference between the three methods is noticeable (Figs. 5g–i). In the physics-based inversion method, it is evident that the median value of the residual is 0.45 mm in Dm, −0.11 g m−3 in W, and −1.14 mm h−1 in R, all three of which are far larger in magnitude than those from the other two methods. In the DNN method, the interquartile ranges and the median of W and R are relatively small and centered near zero, indicating a very accurate estimation (Figs. 5h,i).
As in Fig. 4, but for the rain-event-average domain. The number of the rain-event-average cases is 127.
Citation: Journal of Hydrometeorology 24, 11; 10.1175/JHM-D-22-0166.1
The relative bias and RMSE estimated for the three variables, methods, and error sources using 127 cases in the rain-event-average domain are shown in Table 3. For the fitting error category, the physics-based inversion and DNN methods produce almost perfect results for all three variables, but the empirical method produces an RMSE of 5.9% in W and 3.9% in R estimation (top third of Table 3). For the adding measurement error category, only the empirical method displayed a notable Dm bias and RMSE of −2.2% and 5.2%, respectively (middle third of Table 3). The DNN method is superior for estimating W and R, with a bias less than 2%, compared to 6%–8% from the physics-based inversion and empirical methods, and about 4% reduction in RMSE. When model errors are involved, there is a large difference in bias and RMSE between methods (bottom third of Table 3). In the physics-based inversion method, the absolute value of the bias and RMSE error was 25% or more in all three variables. In the empirical method, errors decrease significantly compared to the physics-based inversion method, but the RMSE error is 19.8% in W and 14.3% in R estimation. The error is additionally reduced by about 4% in RMSE, with less than 2% bias in the DNN method.
5. Verification using radar observations
As seen in the previous section, the performance of the physics-based inversion retrieval degraded greatly when model errors were included. Consequently, the physics-based inversion method is not used in the following analysis. Here, previously developed empirical and DNN methods based on 2DVD data were applied to real radar data. Then, both the empirical and DNN methods were trained and evaluated directly using radar observation data.
a. Model training with 2DVD data
The aforementioned trained models using 2DVD data were applied to radar data. Due to differences in observation data (S-band radar and 2DVD), such as the spatiotemporal resolution of the two systems, there is an inherent bias between the training (2DVD) and testing (radar) data, which varies depending on the elevation and distance from the radar (Tokay et al. 2020). The bias of ZH and ZDR were calculated by comparing the mean difference between the radar and 2DVD observations over the 2DVD site. The radar variables were calibrated by conducting bias correction of −3.0 dBZ for ZH and −0.1 dB for ZDR with the radar observations being smaller than the 2DVD. Table 4 displays the estimation results for the three variables with and without bias correction. For W and R, large biases were noticed in the estimation results without bias correction. The RMSE using DNN method before and after bias correction showed similar or increased values for all three estimators with 28.1%, 109.8%, and 118.1% without bias correction, and 29.5%, 144.9%, and 159.5% with bias correction for Dm, W, and R, respectively. The biases decreased significantly, however, from −44.0% to −6.2% for W, and from −42.7% to −1.0% for R. For Dm, the empirical with bias correction performed the best, while no meaningful differences are noticed with and without bias correction using DNN. The slight increase of bias after bias correction for Dm retrieval and the increase of RMSE are most likely due to statistical causes, such as nonlinear relations and nonnormal distribution between the radar variables and microphysics parameters (i.e., physical parameters and rain rate). In addition, due to the large mean difference (i.e., difference between radar variables from radar observations and the 2DVD-observed DSD), large values from radar observations may have increased even more due to the bias correction, resulting in higher RMSE than prior to bias correction.
The bias, RMSE, and bias factor of the error estimation from radar data using 2DVD trained empirical (E) and DNN (D) methods. The three variables (Dm, W, and R) without and with bias correction (w/o bias correction and w/bias correction, respectively) are shown. The unit is percent (%). The values in bold indicate the smallest bias and RMSE.
Note that RMSE is often affected by a small number of large errors and may not represent overall deterioration or improvement. It is noticed that a large increase of errors was observed for both empirical and DNN methods when tested on radar data, but the overall bias/bias factor may be reduced by adjusting the mean difference (i.e., bias correction) in the two different datasets. The bias factor for the empirical method showed similar results with previous publications (Brandes et al. 2002). A large number of outliers were observed, due to both observational and additional physical reasons. Additional errors in the instantaneous domain were expected, due to an increase in the observational errors and sources of contamination that can occur in low-level radar data. Note that the mean difference between the 2DVD and radar observation differs heavily based on the distance between the two and may require accurate bias correction for each location of various distances (and terrains) for accurate estimations (Tokay et al. 2020).
b. Directly training with real radar data
The temporal domains were separated into instantaneous and rain-event average (Table 5). Table 5 displays the related statistics after excluding outlier cases induced from the difference in spatiotemporal volume between the two observational tools, ground clutter, and hail cases. For bias and RMSE in the instantaneous domain, the DNN method showed improved performance compared to the empirical method (top half of Table 5). The DNN method had near perfect performance in terms of bias factor of 1.0 for W and R and the bias was −2.2% for W and 3.3% for R. The RMSE of Dm was 14.4% in the DNN method, which was slightly better than 15.1% in the empirical method. For W and R, RMSE values were about 65% in the DNN method and 80%–85% in the empirical method. The high relative bias and RMSE implies that the empirical method requires correct calibration.
The bias, RMSE, and bias factor of the error estimation for the three variables (Dm, W, and R) from the radar data were trained and tested using the empirical (E) and DNN (D) methods. The values are shown in two temporal domains, instantaneous and rain-event average. Units of bias and RMSE are percent (%). The values in bold indicate the smallest bias and RMSE.
In the rain-event-average domain, most extreme outliers disappeared as the continuous rain rates were combined into the same rain event. RMSE values decreased significantly in this domain; these reduced by half in all three variables and both methods (bottom half of Table 5). By contrast, the bias and bias factor were similar or rather increased. The bias from the DNN method was ∼5%, indicating very good performance.
The results in Table 5 are displayed after removing the outlying points. From the estimation results, outliers with the largest deviations from ground observations were sorted. In a total of 606 points in testing data, about 66 points showed large errors, contributing to more than half of the total error. The outliers that consist of about 10.9% of the testing data were causing more than 67% of the rain estimation errors for all training methods. It is noted that sources of contamination include but are not limited to ground clutter and anomalous propagation, hail contamination, heterogeneous rainfall, heavy rain, and infrequent large drops. The outliers resulting from the previously mentioned physical reasons do not show significant improvements regardless of the methods used. Additional analysis and incorporation of these outliers seems to be needed.
There were 22 rain-event-average cases for the testing period of January–July 2017. Figure 6 displays testing results of Dm, W, and R using the empirical and DNN methods in the rain-event-average domain. Overall, the deviations from observations were relatively small in Dm (Fig. 6a) and large in W and R (Figs. 6b,c, respectively). When comparing the two methods, except for a few cases, the DNN method showed better performance than the empirical method. It is noted that a large amount of positive deviation appears in all three variables produced by the empirical method in numerous cases (e.g., 6th, 17th cases). The DNN method yielded a value close to the observation.
Testing results of 22 rain-event-average cases trained with radar data using empirical and DNN methods, (a) Dm, (b) W, and (c) R. The solid line denotes 2DVD observation, blue dots indicate DNN estimation, and red × marks indicate empirical estimation.
Citation: Journal of Hydrometeorology 24, 11; 10.1175/JHM-D-22-0166.1
6. Concluding remarks
Numerous studies have utilized various ML algorithms solely for rain-rate estimation using radar data (e.g., Kusiak et al. 2013; Chen et al. 2020; Shin et al. 2021), and the effect of different types of observational errors (i.e., measurement and model) on the estimations were not fully discussed. The present study evaluated the radar retrievals of three rain microphysics parameters (Dm, W, and R) from a DNN model through comparison against two conventional methods (i.e., physics-based inversion and empirical formula). During the period 2006–17, the ground-truth 2DVD located at KAEFS and radar data obtained from the nearby KTLX radar were analyzed. The 2DVD data were temporally divided into 5-min intervals to match the temporal resolution of the radar data. The empirical and DNN methods were trained and tested on different data; partitioning of training, validation, and testing periods was 70%, 15%, and 15% of the 2DVD data and 80%, 10%, and 10% of the radar data, respectively.
First, the training and testing data were taken from the 2DVD data. The estimation results of Dm, W, and R were calculated for three error types (fitting, measurement, and model) in the instantaneous temporal domain. The performance of the three methods was similar for fitting and adding measurement error categories with relatively low bias. However, the estimations deteriorated when model errors were involved. In particular, the bias of the physics-based inversion method was large for Dm, W, and R because this method does not take the error statistics into account. Rain events that occurred continuously for longer than 20 min were grouped into rain-event averages. While the overall results from the rain-event-average cases were similar to those from the analysis of instantaneous data, RMSE fell significantly in the measurement error category; the averaging process compensated for most of the Gaussian errors contained in the input data. These results are consistent with Zhang et al. (2021), where it was discussed that model errors often tend to be much larger than measurement errors, and can affect the retrieval results more significantly.
While not discussed in the figures above, a few additional experiments were conducted using 2DVD data to test the performance when additional radar variables, for example KDP, and/or temporal information were included for retrievals. The inclusion of well-estimated KDP and 20-min temporal information (i.e., from t − 19 to t) prior to the retrievals, showed reduced errors and potential for improvement. However, KDP is a range derivative of differential phase over 9 or 25 range gates, and the error structure is different from that of ZH or ZDR when applied to real radar observations. Also, the different temporal resolution and small size of the radar-2DVD dataset made it difficult to apply temporal information on retrievals using radar observations. Further advanced DL algorithms that have the capability to include spatiotemporal information more efficiently can be utilized to include the entire ray of ΦDP, and/or temporal information more rigorously.
Second, the training and testing data were obtained from two different datasets: 2DVD-trained empirical and DNN methods were applied to the radar observations for testing. Since two different datasets were used for training and testing, bias correction between the two datasets was necessary. It was found that radar observations/retrievals have negative biases compared to 2DVD data due to additional sources of errors and contamination. Two types of bias correction (i.e., with and without) in the training model were examined. When biases in the two datasets were minimized, errors of R estimation decreased by about 40%. Similar results were shown for rain-event-average analysis.
Third, both training and testing were conducted on radar observation data for the empirical and DNN methods. Disdrometer data are observed periodically with much finer spatiotemporal resolution compared to radar. In addition, the bias between the 2DVD and radar varies significantly based on the distance between the two observation locations (Tokay et al. 2020). Thus, direct training from radar data may be helpful. About 50 000 and 3000 min of radar data were used for training and testing, respectively. For instantaneous retrievals and estimation, the bias for the DNN method was 4.9%, −2.2%, and 3.3% for Dm, W, and R, respectively. The bias for the empirical method was higher with 9.0%, 48.8%, 50.1%, for Dm, W, and R, respectively. Similar results were also shown for rain-event-average computations. In sum, the DNN method may provide about 20% improvements of RMSE for R estimation and DSD retrieval, compared to empirical methods when radar data are utilized for both training and testing.
This study reveals the superiority of DNN in radar retrievals of rain rate and physical parameters when observational errors were introduced. Since the DNN algorithm includes nonlinear terms, which are not included in the empirical method, comprehensive representations between the estimators and radar variables are anticipated. However, there are several weaknesses in the present analysis. While the impact of statistical and random errors was reduced using DNN, many of the outliers occur from physical reasons, such as ground clutter, hail contamination, inhomogeneous rain, and particularly heavy rain (which are comparatively much less common and thus comprise many fewer data in the dataset). Additional research is required to develop automated techniques to remove these outliers from the observations.
Future efforts should be focused on testing different DL approaches for rainfall estimation and DSD retrieval to conduct stable estimates for these not as well represented outliers. Recent studies have shown the potential for prediction and understanding of extreme cases using DL algorithms (e.g., Qi and Majda 2019; Davenport and Diffenbaugh 2021), and consideration of physical foundations (e.g., Beucler et al. 2021). DNN is one of the simplest forms of deep learning available at present, with more complex models such as long short-term memory network and convolutional neural networks. It is noted that simple DL methods often show better performance compared to complicated ones (Schultz et al. 2021), thus verification of usage of neural networks for DSD retrieval and QPE was required. In this study, neural networks showed the potential to provide accurate and less biased retrievals and estimates. Thus, integrating vertical profiles and/or spatiotemporal information regarding the entire storm motion, and accounting for the influence of advection due to observation height difference between the radar and ground observations may enhance the retrievals by incorporating additional meteorological conditions and various mixtures of hydrometeors. DNN also has a critical weakness in its inability to exactly describe why and how the errors are decreased compared to the physics-based inversion retrieval or empirical formulas. More efforts to understand the results scientifically are required in the future.
Acknowledgments.
This work was supported by the National Oceanic and Atmospheric Administration (NOAA) under Grant NA16OAR4320115 and the National Science Foundation with Grant AGS-2136161.
Data availability statement.
The 2DVD data are shown on http://weather.ou.edu/∼guzhang/page/Disdrometer_data.html, and the raw data can be obtained by making a request to Dr. Guifu Zhang at guzhang1@ou.edu. The NEXRAD level-II data were obtained from the National Center for Data Collection (www.ncei.noaa.gov).
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