## 1. Introduction

Remotely sensed measurements from meteorological satellite instruments play an extremely important role in studying the earth’s climate (Arkin and Ardanuy 1989; Ferraro et al. 1996). Data from passive microwave satellite instruments are now widely used to derive climatological values of various atmospheric water cycle components (Klepp and Bakan 2000). Microwave radiometers such as the Special Sensor Microwave Imager (SSM/I) have been shown to yield reliable information concerning instantaneous precipitation rates. This is because the microwave signal, which is the upwelling signal from the surface, can penetrate through cloud tops and directly detect the presence of precipitation particles within and below the clouds (Petty and Krajewski 1996).

Many approaches to the estimation of the rainfall rate and cloud parameters from passive microwave measurements have been proposed (Romanov 1999; Dietrich et al. 2000). The Satellite Research Laboratory National Environmental Satellite, Data, and Information Service (NESDIS) of the National Oceanic and Atmospheric Administration (NOAA) uses an empirically derived function of the lower frequencies of the SSM/I to predict vertically polarized 85.5-GHz brightness temperatures expected in the absence of precipitation (Grody 1991; Ferraro et al. 1994; Petty and Krajewski 1996). The depression of the observed brightness temperature relative to this expected brightness temperature is the scattering index (*SI*) and is mapped empirically to rain intensity (Ferraro and Marks 1995). Conceptually similar scattering-based precipitation retrieval algorithms have been formulated by other authors (e.g., Kidd and Barrett 1990) in terms of the simple difference between vertically polarized 85.5- and 37.0-GHz frequencies of the SSM/I. Kidd (1998) presented the polarization-corrected temperature (PCT) technique for the delineation and retrieval of rainfall over the land and ocean from passive microwave data.

The passive microwave precipitation retrievals can often be separated into two groups. One is the emission-based method, and the other is scattering based (Spencer et al. 1989). The former is the best known, primarily based on work by Wilheit et al. (1977) at 19.35-GHz frequency of brightness temperature, and can only be employed over ocean areas. Similar applications can be found in the work of Wilheit and Chang (1980), Wu and Weinman (1984), Wilheit et al. (1991), Liu and Curry (1997), and Klepp and Bakan (2000). The scattering method is usually used over land or ocean (Spencer et al. 1989). The concepts of *SI* for the rainfall retrieval originate from Grody (1991), who developed a global *SI* at 85.5 GHz using SSM/I data. Further refinement of the technique was described by Grody (1991), Ferraro and Marks (1995), Ferraro et al. (1996), and Mishra et al. (2009). These studies indicated that the SI index is highly variable with region and season.

This study focuses on the retrievals of the rainfall rate over land in Taiwan during tropical cyclone (typhoon) events. Typhoons often affect the western North Pacific region (including the Philippines, Taiwan, Japan, Korea, China, and others). Enormous flood damage in Taiwan is frequently caused by typhoons between May and October every year. A technology called support vector machine (SVM), which was developed by Vapnik (1995), has been in widespread use. SVM is a concept in statistics and computer science for a set of related supervised learning methods that analyze data and recognize patterns, used for classification and regression analysis (Joachims 2002). SVM has been extended to solve nonlinear regression estimation problems, such as techniques known as support vector regression (SVR), which have been shown to exhibit excellent performance (Vapnik et al. 1997) (the details of the SVR algorithm can be found in section 2).

This study employed the SVR and an extension of SVR-based scattering index over land (SIL) algorithms and compared them with the traditional regression and SIL approaches for hourly rainfall rate retrievals. The microwave data sources originate from the SSM/I and the rainfall rates are measured by the automatic meteorological gauges during typhoons. The SSM/I comprises a seven-channel, four-frequency, linearly polarized, passive microwave radiometric system (Raytheon Systems Company 2000). It measures atmospheric, ocean, and terrestrial microwave brightness temperatures at 19.35, 22.23, 37.0, and 85.5 GHz (in the following context shortened as 19, 22, 37, and 85 GHz, respectively). All frequencies except the water vapor absorption frequency at 22 GHz are dual polarized. The presented methodology is applied for rainfall rate retrievals at the Tanshui River basin in northern Taiwan. These SVR-based methods developed a new relationship between the microwave measurements and the surface rainfall rate in the basin.

## 2. Support vector regression

The SVR formulation follows the principle of structural risk minimization seeking to minimize an upper bound of the generalization error rather than minimize the prediction error on the training set (Chen and Wang 2007).

### a. SVR algorithm

**x**is the input vector,

*y*is the output, and

**Φ**,

**x**is mapped into a feature space in which a linear estimate function is defined as (Schölkopf et al. 2000; Zhang et al. 2004)where

*b*is the threshold in SVR.

*f*with a small risk. Vapnik (1995) suggested using the following regularized risk functional to obtain a small risk:where

*C*is a penalty parameter,

*ɛ*is a small positive number,

*l*is the number of support vectors, and

**x**

_{i}represents the so-called support vectors determined from the training data (Yang and Wang 2008).

### b. SVR versus traditional regression

The heuristic behind the SVR algorithm is quite different from that of the commonly used regression modeling for prediction. This latter approach uses a weighted least squares algorithm; that is, the prediction is based on construction of a regression line as the best fit through the data points by minimizing a weighted sum of the squared distances to the fitted regression line. SVR, in contrast, tries to model the input variables by finding the separating boundary, called hyper plane, to reach classification of the input variables: if no separation is possible within a high number of input variables, the SVR algorithm still finds a separation boundary for classification by mathematically transforming the input variables by increasing the dimensionality of the input variable space (Verplancke et al. 2008).

### c. Constructing SVR

*ɛ*in Eq. (2) is specified by a user-defined parameter

*ɛ*and

*ɛ*= 0.05 is set herein. However, the parameter

*C*varies from case to case. There are no general rules in this respect. This study proposed a hierarchical flowchart for determining the

*C*in order to find the suitable value. The evaluation performance is root-mean-square error (RMSE), defined aswhere

*j*,

*j*, and

*N*is the number of hourly records. In general, the smaller the RMSE value, the better the performance of the predicted outcomes is.

Figure 1 illustrates the flowchart of SVR parameter training, which involves a series of repetitive calculation steps. As seen in the figure, this study sets the initial penalty value (*C*_{0}) = 1, and maximum of *C*(*C*_{max}) = 100. In step 3, negative retrieval values might appear, and we set them to 0.0. In step 6, the incremental of *C*(Δ*C*) was set to 1. Finally, the suitable *C* value is selected by the superior record of RMSE(*C*).

## 3. Study area

The Tanshui River basin, located in northern Taiwan, is in the subtropical zone (see Fig. 2) and covers a total area of 2726 km^{2}. The spatial range of the watershed covers 24.43°–25.19°N and 121.20°–121.86°E. Metropolitan Taipei is located in the Tanshui River basin and has a population of about 6 million. In the summer and fall seasons, tropical cyclones with torrential rain occur frequently because of the attributes of the subtropical climate. The torrential rain, combined with the steep mountain slopes, makes the time of floodwater convergence very short—about 3–6 h—and causes flash floods with huge disasters (Hsu et al. 2003).

The Tanshui River basin comprises three major tributaries, including the Tahan River, Hsintien River, and Keelung River. The Tahan River originates from Mt. Pinten and joins Hsintien River at Chiangchutsui, which is the starting point of the Tanshui River. The length of the Tahan River is 135 km with a drainage area of 1163 km^{2} and an average slope of ^{2} and an average slope of ^{2} and has an average slope of

## 4. Application

This study examines the features of passive microwave imagery and develops a usable scheme for rainfall retrievals during typhoons. The procedures are shown in Fig. 3. In step 1, the measurements from the SSM/I satellite comprise the brightness temperatures at 19, 22, 37, and 85 GHz. In step 2, the temperature data records for the specific spatial region will be searched. Then the surface data from automatic rain gauges is collected and the average rainfall rate over the watershed is calculated, respectively, in steps 3 and 4. In step 5, when defining the attributes, all the attributes are of numerical type, while target is the rainfall rate. To compare the accuracy of rainfall retrievals based on traditional regression, SVR, SIL, and an extension of SVR-based methods, these methods are analyzed and compared in steps 8–10.

### a. Data and preprocessing

This study consists of 70 typhoons affecting the watershed over the past 12 years (1997–2008), as can be seen in Table 1. The two kinds of data collected were the automatic rain gauge data over the watershed and the SSM/I satellite microwave data during typhoons.

Typhoon events studied.

#### 1) Data from rain gauges

The collection of surface data is from automatic rain gauges, including hourly typhoon precipitation in the reservoir watershed. Complete data of the hourly typhoon rain data of the reservoir watershed are available from the Water Resources Agency (WRA). The 16 rain gauges (see Fig. 2) distributed over the watershed are listed in Table 2. The mean and standard deviation of the annual rainfalls for these gauges are 2866 and 811 mm, respectively, with a mean elevation and standard deviation of 497 and 530 m. This study found both topography and climatic conditions belong to high variability within this basin.

The 16 rain gauges and their annual rainfalls and elevations. Data from WRA website: http://gweb.wra.gov.tw/wrhygis/.

The most frequently adopted algorithms in estimating average rainfall within a certain watershed are the arithmetical averaging method, Thiessen polygons method, height balance polygons method (HBPM), and isohyetal method (Chow et al. 1988; Tseng and Chou 2002). To calculate the average hourly precipitation of all 16 rain gauges representing the rainfall amount over the watershed, the HBPM was employed here because of the topographical properties. The main difference between the HBPM and the others is that weighted functions are calculated according to the elevations of rain gauges; namely, topographic factors are involved in the estimation. The steps of the height balance polygons method include 1) digitize rain stations and watershed polygon information, 2) triangulate irregular networks, 3) estimate central points of elevation between two rain stations, 4) connect all elevation of central points of triangulated irregular networks, 5) create height balance polygons, 6) analyze intersection polygons between height balance polygons and watershed polygons, and 7) compute elevation weights of HBPM. More details can be seen in Chow et al. (1988) and Tseng and Chou (2002).

#### 2) Data from SSM/I instruments

The data records from SSM/I satellite instruments covering the extent of the watershed for these years during typhoons were searched. The spatial sampling interval is 25 km for the four frequencies. The microwave data is in coordinated universal time (UTC) must be switched to local time. The time transfer formula to the studied site can be expressed as “local time = UTC + 8 h.”

According to the collected data, the attributes include polarized brightness temperatures at 19, 22, 37, and 85 GHz while the target is rain rate (mm h^{−1}). The 620 hourly tuples (i.e., records) were available, including 324 records for nonrainfall and 296 rainy records. Here, the overall tuples were classified into two subsets: training datasets (1997–2004, comprising 46 typhoons) and validation datasets (2005–08, 24 typhoons). It should be noted that the satellite measurement is derived from an instantaneous observation, while the gauge data is an hourly accumulation. Therefore, the two collocated estimates/measurements are inherently different. Because of the limitations of temporal resolutions of the raw data from the rain gauges, this study assumes that the instantaneous satellite observations are the same within the specific hour.

### b. Methods

This study compares the retrievals by the conventional regression approaches in methods 1 and 2 and the SVR approach in method 3. Meanwhile, to have confidence in the proposed SIL–SVR method (i.e., method 5), it is compared with that using the global rainfall and scattering index relationship (i.e., method 4). Table 3 summarizes the five methods.

Lists of methods and coefficients.

#### 1) Method 1: Single-channel regression

_{X}is the polarized brightness temperature at channel

*X*, and

*a*

_{1}–

*a*

_{4}are the regression parameters. The above regressions estimate the best-fitting regression for predicting the output field, based on the input fields. Table 3 lists their coefficients in regressions.

#### 2) Method 2: Multichannel linear regressions (MLR)

_{19V}, TB

_{22V}} and {TB

_{19V}, TB

_{22V}, TB

_{85V}} are selected to be the same as in Eqs. (14) and (17) (see method 4) of SIL approach, respectively. In addition, methods 2.3 and 2.4 chose all vertically and horizontally polarized channels, respectively. These methods can be expressed aswhere

*b*

_{1}–

*b*

_{3},

*c*

_{1}–

*c*

_{4},

*d*

_{1}–

*d*

_{5}, and

*e*

_{1}–

*e*

_{4}are the regression coefficients in methods 2.1–2.4, respectively. These regressions estimate the best-fitting regression for predicting the output field, based on the input fields. Also, Table 3 lists their coefficients in regressions.

#### 3) Method 3: SVR

As described in section 2, during the training, the weights of (*b*) gradually converge to values in which the input vectors produce output values as close as possible to the desired target output. Note that the penalty *C* in Eq. (2) affects the trade-off between complexity and proportion of nonseparable samples. If *C* is too small, it will increase the number of training errors and have underfitting. If *C* is too large, it will lead to a high penalty for nonseparable points and overfit (Cherkassky and Mulier 2007). Figure 4 shows the sensitivity analysis of *C* from 1 to 100 in methods 3.1–3.4. As seen in the figure, the superior RMSEs obtained in methods 3.1–3.4 were (3.283, 2.972, 2.876, 3.032) at *C* = (7, 3, 5, 7), respectively. Therefore, this study adopts these *C* values for each model analysis.

#### 4) Method 4: SIL

_{19V}and TB

_{22V}) and the actual TB

_{85V}(Grody 1991). That is, the 19- and 22-GHz frequencies are relatively unaffected by scattering, and therefore are used for estimation of 85-GHz brightness temperature during nonscattering conditions (Ferraro et al. 1996; Mishra et al. 2009). The

*SI*index consists of two steps: in the first step region-specific SI is developed using a combination of the 19 and 22 GHz over land; that is,where

*F*is the 85-GHz-frequency brightness temperature under the theoretic conditions without scattering. The values of

*g*

_{1},

*g*

_{2},

*g*

_{3}, and

*g*

_{4}are regressed by dataset over the land region under nonrainy conditions. Then,

*h*

_{1}and

*h*

_{2}are regression coefficients. For nonraining situations the value of SI should ideally be less than or equal to zero. From this step onward, the SI index can be used to separate the scattering and nonscattering signals for a given dataset.

*d*of the scattering index should be determined. To test the different thresholds, the RMSE is employed to assess the retrieval outcomes. Figure 6 shows the RMSE of a 2 K threshold or less (i.e., 0 and 1 K) is roughly constant at 5.47 mm. Therefore,

*d*= 2 K was selected. The derivation SI formula is

#### 5) Method 5: SIL–SVR

*F*formulae, this study employed the same inputs {TB

_{19V}, TB

_{22V}} as in method 4. Meanwhile, to identify whether the SI value is greater than the threshold

*d*, the same

*d*= 2 K as in method 4 was used. In step 3 of Fig. 7, the parameter settings with respect to

*C*and

*ɛ*are the same as in method 3.1 to construct the SIL–SVR. Figure 8 demonstrates the results of training and validation of SIL–SVR for

*F*values (i.e., the expected nonscattering 85 GHz). The derivation formula is derived as

### c. Results and comparisons

Figure 9 illustrates the results obtained from validation for rainfall retrievals in all methods. First, from the figure of the validation parts, the RMSEs of methods 1.1–1.7 (see Fig. 9a) range from 3.277 to 3.549. The best one occurs at method 1.4. In method 2 (see Fig. 9b), which performed MLR, the RMSE values were (3.348, 3.149, 3.133, 3.166) for methods 2.1–2.4, respectively, while in method 3 (i.e., SVR), RMSEs of (3.286, 2.969, 2.874, 3.030) were for methods 3.1–3.4, respectively. In contrast, method 3.3, which comprises 19, 22, 37, and 85 GHz, has good outcomes among methods 2 and 3, depending on the small errors in RMSE. This demonstrates that the solutions using the SVR have better performance than regression. Further, in methods 4 and 5, which employ the SIL-based approach to achieve rainfall retrievals, a better RMSE of 2.811 is obtained by method 5 (SIL–SVR), compared to 3.001 by method 4 (SIL).

### d. Methods evaluation by skill scores of rain/nonrain

According to the above analysis, the methods 1.4, 2.3, 3.3, 4, and 5 demonstrated better retrievals of rainfall rates in their corresponding methods. Therefore, these five methods were selected for advanced evaluations. In this section, two categorical statistics measures, including bias ratio and equitable threat score (ETS), are used, computed from the elements of this rain/no-rain contingency table. As seen in Table 4, the term categorical refers to the yes/no nature of the forecast verification. Then for each retrieval, each verification time is scored as falling under one of the four categories of correct nonrain forecasts, false alarms, misses, or hits (denoted as *Z*, *F*, *M*, and *H*, respectively) (McBride and Ebert 2000). In the ideal situation of perfect forecast, both *M* and *F* are equal to 0. By considering retrievals of the rainfall rate greater than a certain threshold, the problem can be broken down into a series of 2 × 2 contingency tables, each for a different threshold value (Hall et al. 1999).

Rain contingency table.

#### 1) Definitions of skill scores

If bias ratio > 1.0, the model overpredicts rain occurrence; otherwise, the model underpredicts rain occurrence.

ETS ranges from −⅓ to 1, with a value of 1 indicating perfect correspondence between predicted and observed rain occurrence.

#### 2) Evaluation

Figure 10 shows the bias ratio as a function of the threshold value, which ranges from 0.1 to 20 mm h^{−1}, chosen to separate the rain/no-rain events. As mentioned above, if the bias ratio > 1.0, the model overpredicts the rain occurrence; otherwise, the model underpredicts the rain occurrence. In the figure, the straight line “bias ratio = 1” divided the figure into two regions. For both methods 1.4 and 2.3, the threshold is less than about 3.8 mm h^{−1} (bias ratio > 1.0), and it has a relatively higher bias (about 1.85 and 1.77 at the threshold of 0–1 mm h^{−1}) than other methods. This means that the two methods strongly overestimate the light rains. For method 3.3, it has approximately horizontal lines with a bias ratio between 1 and 3 mm h^{−1}. For methods 4 and 5, the threshold is equal to about 2 and 3.5 mm h^{−1}, respectively, when bias ratio = 1.0. In the region of bias ratio < 1.0, method 3.3 demonstrates close to the bias ratio of 1.0. On the contrary, methods 1.4, 2.3, and 4 are relatively far from the horizontal line. This means that these three methods strongly underestimate the peak values.

ETS measures the number of retrieval fields that match the observed threshold amount. Figure 11 illustrates the ETS as a measure of the relative skill for a distribution of rainfall amounts. For methods 1.4 and 2.3, the ETS score is equal to about zero at the threshold of 0.1 mm h^{−1}, and increases ETS values until maximal ETS value of about 0.6 at the threshold = 5 mm h^{−1}. For method 3.3, the ETS value begins at 0.2 at threshold = 0.1 mm h^{−1}, reaches the maximum of ETS (i.e., 0.6) at threshold = 2 mm h^{−1}, and then decreases the ETS scores smoothly until threshold = 20 mm h^{−1}. Moreover, methods 4 and 5 appear two peaks at thresholds of (0.1, 3) mm h^{−1} and (0.1, 5) mm h^{−1}, respectively.

*k*is the threshold index between the rainfall ranges; bias ratio

_{k}and ETS

_{k}are the scores of bias ratio and ETS at the assigned threshold

*k*, respectively; and

*O*is the number of observations at assigned threshold

_{k}*k*. Table 5 lists the average bias ratio and ETS scores of these rainfall levels.

Range of rain strengths and calculation of the average bias ratio and ETS in different rain levels.

## 5. Discussion and conclusions

This paper focuses on addressing the rainfall retrieval problem for quantitative precipitation forecast over land during tropical cyclones. This study applies a machine learning technique—support vector regression (SVR)—for rainfall rate retrievals. The feasibility of SVR and SIL–SVR by comparing them with traditional regression method and SIL approaches is examined. The SSM/I on board the satellites and WRA measurements of Taiwan were used to estimate quantitative precipitation over the Tanshui River basin in Taiwan. This study collected data from 70 typhoons affecting the studied watershed over the years 1997–2008. The measurements of SSM/I sensor comprised the terrestrial brightness temperatures at 19, 22, 37, and 85 GHz.

The retrievals were compared in terms of three measures (RMSE, bias ratio, and ETS). The RMSE results were obtained from five methods. Methods 1.1–1.7 performed single-channel regression, while methods 2.1–2.4 and methods 3.1–3.4 were run by MLR and SVR, respectively. Moreover, methods 4 and 5 were the SIL and SIL–SVR, respectively. In terms of RMSE, the retrievals of methods 1.4, 2.3, 3.3, 4, and 5 demonstrated better retrievals of rainfall rates in their corresponding methods. Further, this study evaluated different rainfall strengths, denoted as “light rain,” “heavy rain,” “torrential rain,” and “pouring rain.” This study found the following.

- For bias ratio, in light rain, method 4 having average bias-ratio scores close to 1.0 gives better estimation than other methods, meaning that SIL approach achieves better retrievals in this level. In heavy rain, torrential rain, and pouring rain situations, method 3.3 yields superior bias-ratio scores to those of other methods, meaning that SVR algorithm achieves better retrievals in rainfall > 5.4 mm h
^{−1}. Moreover, method 3.3 gets the best one among all methods in overall average bias ratio. - For ETS, methods 4 and 5 having average bias-ratio scores give better estimation in light rain and heavy rain, respectively. This implies that SIL and SIL–SVR approaches achieve better retrievals in both levels. In torrential rain and pouring rain situations, method 3.3 yields superior ETS scores to those of other methods, meaning that SVR algorithm achieves better retrievals in rainfall > 8.3 mm h
^{−1}. Also, this study found method 5 gets the best one among all methods in whole average ETS.

Consequently, the results showed the approaches using the SVR and SIL–SVR have superior products to traditional regressions and SIL. This is because the SVR techniques are good at identifying and learning correlated patterns between the input datasets and corresponding target values.

## Acknowledgments

The support under Grant NSC100-2111-M-464-001 by the National Science Council, Taiwan, is greatly appreciated. The authors would like to acknowledge data provided by the National Oceanic and Atmospheric Administration (NOAA). The authors are also indebted to the three reviewers for their valuable comments and suggestions.

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