## Abstract

An objective technique is presented to estimate tropical cyclone intensity using the relevance vector machine (RVM) and deviation angle distribution inhomogeneity (DADI) based on infrared satellite images of the northwest Pacific Ocean basin from China’s *FY-2C* geostationary satellite. Using this technique, structures of a deviation-angle gradient co-occurrence matrix, which include 15 statistical parameters nonlinearly related to tropical cyclone intensity, were derived from infrared satellite imagery. RVM was then used to relate these statistical parameters to tropical cyclone intensity. Twenty-two tropical cyclones occurred in the northwest Pacific during 2005–09 and were selected to verify the performance of the proposed technique. The results show that, in comparison with the traditional linear regression method, the proposed technique can significantly improve the accuracy of tropical cyclone intensity estimation. The average absolute error of intensity estimation using the linear regression method is between 15 and 29 m s^{−1}. Compared to the linear regression method, the average absolute error of the intensity estimation using RVM is between 3 and 10 m s^{−1}.

## 1. Introduction

In recent decades, tropical cyclone track prediction has been greatly improved. However, because of the difficulties in estimating tropical cyclone intensity, advances in intensity prediction are not quite evident.

Currently, tropical cyclone intensity estimation mainly depends on satellite observations. Polar-orbiting weather satellites and geostationary weather satellites were launched in the 1960s and 1970s. Since then, researchers have attempted to use satellite data to estimate tropical cyclone intensity (Fett 1964; Sadler 1964; Fritz et al. 1966; Erickson 1967), and, currently, the Dvorak technique is most widely used for estimating intensity (Dvorak 1972, 1975). The Dvorak technique estimates tropical cyclone intensity using tropical cyclone cloud structures derived from visible and infrared satellite images (Dvorak 1984; Dvorak and Smigielski 1995; Velden et al. 2006). In the late 1980s, the World Meteorological Organization (WMO) recommended the Dvorak technique as the world’s primary intensity forecasting tool. But the technique is both subjective and time intensive. Particularly, its intensity estimation accuracy mainly depends on the experience of the user. Based on the original Dvorak technique, Velden et al. (1998) and Olander et al. (2002), respectively, proposed the objective Dvorak technique (ODT) and the advanced objective Dvorak technique (AODT). Although these two techniques reduce manual intervention, they are not suitable for intensity estimation of weak tropical cyclones. Since then, AODT has been further improved. After considering how the cloud-top height of the troposphere decreases with latitude (Kossin and Velden 2004), the estimation error of intensity of the Dvorak technique was reduced by 10% compared with AODT (Olander et al. 2004). The latest ODT is the advanced Dvorak technique (ADT; Olander and Velden 2007). This scheme expands both ODT and AODT and relaxes their restrictions. The accuracy of ADT estimation depends on tropical cyclone center positioning based on a single-channel infrared satellite image. When the eye of a tropical cyclone or spiral rainband is shielded by cirrus clouds, it is difficult to automatically locate the center position (Olander and Velden 2009). Therefore, Olander and Velden (2009) modified ADT once again by comparing the differences between the geostationary satellite infrared window (IRW) and water vapor channel (WV) brightness temperature values in the strong convective regions of a tropical cyclone. This allows for the estimation of tropical cyclone intensity by using a linear regression technique (IRWV).

In addition to the Dvorak-type techniques, researchers have explored other tropical cyclone intensity estimation techniques based on geostationary satellite data. For example, Kossin et al. (2007) estimated the maximum wind speed radius and critical wind radius using linear regression techniques based on infrared satellite image data. In recent years, the deviation angle variance (DAV) technique, based on infrared brightness temperature data, has been used (Piñeros et al. 2008, 2011). However, this technique performs poorly when there is strong wind shear (Piñeros et al. 2011). Best-track data from the National Hurricane Center (NHC) have been used to improve the original DAV technique (Ritchie et al. 2012, 2014). Fetanat and Homaifar (2013) have used the *k*-nearest neighbor technique to estimate intensity via geostationary satellite azimuth brightness temperature profile data from historical tropical cyclones. As the infrared brightness temperature slopes in the tropical cyclone eyewall, there is significant negative correlation with tropical cyclone intensity. Sanabia et al. (2014) have also estimated tropical cyclone intensity by computing the multipoint cloud-top slopes of the inner core of tropical cyclones with infrared brightness temperature.

As mentioned above, the current tropical cyclone intensity estimation techniques are improvements over the original Dvorak technique, but deficiencies remain. Some researchers use linear regression techniques to estimate tropical cyclone intensity (Kossin et al. 2007; Olander and Velden 2009). However, this may result in significant error when the sample size is not big enough. Recently, some researchers have found that nonlinear models seem promising in estimating tropical cyclone intensity (Piñeros et al. 2011; Ritchie et al. 2012; Jiang 2012; Sanabia et al. 2014). Numerous studies show that intensity changes in a tropical cyclone have nonlinear physical relationships with vertical wind shear (DeMaria 1996; Zehr 2003). These nonlinear models often require some subjective experience to determine the optimal parameters within the models. As both linear regression techniques and current nonlinear models have some disadvantages in intensity estimation, we need to consider more efficient indicating factors that can be used to describe tropical cyclone intensity. Recently, more and more researchers have identified the potential use of cloud-top brightness temperatures (Piñeros et al. 2011; Ritchie et al. 2012; Jiang 2012; Sanabia et al. 2014) in tropical cyclone intensity estimation.

In fact, tropical cyclone intensity estimation can be intrinsically considered as data fitting. There are many methods for data fitting, in which a black-box method is often used. Black-box methods are used to find the relationship between the input variables and output variables gradually through analyzing limited samples and fitting the unknown function. The most common black-box methods involve the use of artificial neural networks, such as a back propagation neural network (BPNN; McCulloch and Pitts 1943) and a radical basis function neural network (RBFNN; Moody and Darken 1989). Because the network node is similar to a human’s brain nerve cells, a neural network can simulate the brain’s incentive function to perform complex data analyses. The basic requirement of data fitting is that it has high accuracy in prediction, which depends on large amounts of sample data. More sample data leads to better predictions. However, the amount of sample data cannot increase indefinitely under normal circumstances. To get better predictive results with limited sample data, a new machine learning method called support vector machine (SVM) was proposed by Vapnik et al. (1997) and widely used for classification, fitting, and pattern recognition. Recently, a new machine learning algorithm, relevance vector machine (RVM), was put forward. Compared with SVM, RVM can reduce the calculation burden of the kernel function and relaxes the conditions required for selecting the kernel function. RVM can also avoid subjective operation in the parameter adjustment process.

In this paper, deviation angle distribution inhomogeneity (DADI) is used as the indicating factor for tropical cyclone intensity (maximum surface wind speed). DADI is computed using the geostationary infrared satellite brightness temperatures of the inner core of a tropical cyclone. The RVM, which has excellent nonlinear modeling ability even for small samples, was used to relate DADI to the intensity of different types of tropical cyclones. DADI and tropical cyclone intensity are, respectively, used as the input and output of RVM. Certain numbers of samples with information about DADI and tropical cyclone intensity are used to train RVM in order to build a tropical cyclone intensity estimation model.

## 2. Methodology and data

### a. Methodology

#### 1) Tropical cyclone structure description based on deviation angle

Figure 1a shows an axisymmetric graph whose gradient direction and tangent direction are mutually perpendicular. This can be used to investigate the quasi-axisymmetric structure of the inner core of a typical tropical cyclone. The deviation angle (see Fig. 1b) is the angle between the radial line and the gradient direction of a point .

The life cycle of a tropical cyclone can be divided into three stages: early, mature, and dissipation. The cloud structure of a cyclone is usually shaped from disorganized to axially symmetric with increasing intensity. For example, the infrared satellite images of different development stages of Typhoon Talim (2005) are shown in Figs. 2a–c. They reveal that Talim’s cloud structure during its mature stage approaches an axisymmetric circle, but in other stages the cloud structure is disorganized. We calculated the tropical cyclone deviation angle matrix with the above method based on infrared satellite imagery, and then used a deviation angle histogram to describe the structure of the tropical cyclone.

#### 2) RVM and DAGCOM

Current tropical cyclone intensity estimation models are mainly built on traditional linear regression methods and asymptotic theory, which leads to significant errors when the numbers of samples are small. Nonlinear modeling techniques show promise because the relationship between tropical cyclone intensity and its influencing factors is usually nonlinear. RVM is divided into a regression model and a classification model (Tipping 2001). This paper uses a regression model and the Matlab toolbox (http://www.Miketipping.com/sparsebayes.htm) to estimate tropical cyclone intensity (Tipping 2001).

In this paper, the gray gradient co-occurrence matrix (Hong 1984) was generalized to the deviation angle gradient co-occurrence matrix (DAGCOM). It is called DAGCOM because we replace the gray gradient with the deviation angle gradient in the original gray gradient co-occurrence matrix. A total of 15 statistical parameters (small gradient advantage, big gradient advantage, deviation angle distribution inhomogeneity, gradient distribution inhomogeneity, energy, mean deviation angle value, mean gradient value, deviation angle standard deviation, gradient standard deviation, correlation, deviation angle entropy, gradient entropy, mixed entropy, differential distance, and opposite differential distance) in Table 1 related to tropical cyclone intensity were calculated by DAGCOM. An element in a DAGCOM is defined as the number of pixels whose deviation angle value and deviation angle gradient value are, respectively, *x* and *y* in a normalized deviation angle image and a normalized deviation angle gradient image . The deviation angle image can be obtained by employing the method outlined in section 2a(1). Based on a deviation angle image , the deviation angle gradient image can be obtained by

The origin of the DAGCOM is shown in the top-left corner of the matrix. Gradient values increase toward the right and the deviation angle values increase downward. Normalized DAGCOM is shown by .

#### 3) Tropical cyclone intensity estimation with RVM and DAGCOM

Based on the calculation of parameters for the tropical cyclone structure, RVM is used to build models between these parameters and tropical cyclone intensity (maximum surface wind speed). A parameter of DAGCOM most closely correlated to the tropical cyclone intensity is called the intensity indicating factor.

Tropical cyclones are divided into two categories in this study: eyed tropical cyclones (ETCs) and noneyed tropical cyclones (NTCs). ETCs are those with a higher intensity grade and that have an obvious eye in the mature stage. NTCs are those without an obvious eye and include tropical storms. By calculating the DAGCOM within a radius of 65 pixels (~325 km), experiment results indicate that the average absolute error of the tropical cyclone intensity estimation is almost the same as that within a radius of 40 pixels (~200 km). Therefore, the DAGCOM within a radius of 40 pixels (~200 km) was calculated in order to reduce computation time.

##### (i) Intensity estimation for an NTC

For NTCs including tropical storms (TSs), we referred to the technique used by Piñeros et al. (2008), which considers each pixel point to be the reference point used to calculate the deviation angle between the reference point and each point in the infrared satellite image, respectively. First, for a NTC infrared satellite image whose size is *N* × *N* pixels, we get *N* × *N* deviation angle matrixes. Then, 15 parameters (Table 2) of each deviation angle matrix were calculated, to get *N* × *N* × 15 deviation angle gradient co-occurrence matrix parameter matrixes. Finally, the median, minimum, and mean of the parameter matrices were calculated. Models between the above three values and maximum surface wind speed were built by RVM to estimate NTC intensity.

##### (ii) Intensity estimation for an ETC

We adopted two schemes to estimate the intensity of ETCs. First, because the eye area is obvious, we used the eye area center point as the reference point. The deviation angle between the reference point and each point in the infrared satellite image were calculated in turn. We get *N* × *N* deviation-angle matrixes from *N* × *N* eyed tropical cyclone satellite images, and then computed 15 parameters (Table 1) of the deviation angle matrix. Finally, a nonlinear model between the 15 parameters and the maximum surface wind speed was built based on RVM to estimate the ETC intensity, which is the same intensity estimation scheme we applied to NTCs.

##### (iii) Intensity estimation for mixed types of tropical cyclones

For mixed types (NTC and ETC) of tropical cyclones, we used the intensity estimation scheme of NTCs to estimate intensity.

### b. Implementation of the proposed tropical cyclone intensity estimation technique

The proposed technique is implemented as follows.

Step 1—Based on infrared satellite imagery that contains a tropical cyclone, the corresponding deviation angle matrix is calculated as in section 2a(1).

For NTCs, each pixel point is used as the reference point to calculate the corresponding deviation angle matrix.

For ETCs, each pixel point and eye area center point are, respectively, used as reference points to calculate the corresponding deviation angle matrices.

Step 2—DAGCOM is obtained as in section 2a(2), and 15 statistic parameters are calculated by DAGCOM and Table 1.

Step 3—Noneyed typhoon data (2367 infrared satellite images) are used to choose the optimal intensity, indicating the factor from the DAGCOM. DADI is chosen as the optimal intensity indicating factor by the above method.

Step 4—RVM is used to relate the tropical cyclone intensity (maximum surface wind speed) to DADI for eyed typhoons (ETPs), noneyed typhoons (NTPs), TSs, and mixed tropical cyclones (MTCs), respectively.

Step 5—The linear regression method and average absolute error are used to evaluate the performance of the proposed technique.

### c. Data

The data presented are derived from infrared (10.3 *μ*m) satellite images with a spatial resolution of 5 km per pixel, captured at 30-min intervals from the China *FY-2C* geostationary satellite over the northwest Pacific basin. Tropical cyclone best-track data from the *Yearbook of Tropical Cyclone* [China Meteorological Administration (CMA) 2007, 2008, 2009, 2010, 2011], published by the China Meteorological Press, were used for verification of the estimated tropical cyclone intensities. These data were archived at 6-h intervals. To match center locations and wind speed estimates to the 30-min temporal resolution of the satellite data, the best-track data were linearly interpolated to match the satellite temporal resolution. The northwest Pacific study uses infrared satellite images that include existing tropical cyclones from the 2005–09 typhoon seasons and comprises a total of 4275 unique hourly images (Table 2). According to the standard (GB/T 19201–2006) for Chinese tropical cyclones (http://baike.baidu.com/link?url=pY-lOv_diz-7yvKloFN7xO7xLsdyvEG6PxyTAjudVi8wNKcqFuxRhS8n1uXA9JsBg3v5E0QCQyf-iJDz2aHD_K), a tropical cyclone is divided into six categories: supertyphoon (51 m s^{−1}+), severe typhoon (41.5~50.9 m s^{−1}), typhoon (32.7~41.4 m s^{−1}), severe tropical storm (24.5~32.6 m s^{−1}), tropical storm (17.2~24.4 m s^{−1}), and tropical depression (10.8~17.1 m s^{−1}). Here, the maximum surface wind speed near the tropical cyclone center is used as the intensity estimation for a tropical cyclone. The resulting dataset included eight supertyphoons (SUTs; 51 m s^{−1}+), two severe typhoons (STPs; 41.5~50.9 m s^{−1}), five typhoons (TPs; 32.7~41.4 m s^{−1}), and seven TSs (17.2~32.6 m s^{−1}). Furthermore, ETPs (32.7 m s^{−1}+) consisted of 699 images, NTPs (32.7 m s^{−1}+) 2367 images, and TSs (17.2~32.6 m s^{−1}) 1209 images. The above dataset is listed in Table 2. The size of the original infrared satellite image was 2288 × 2288 pixels, so we cropped a 130 × 130 pixels image from the original infrared satellite image that included tropical cyclone cloud structures as analysis objects.

There are 15 statistical parameters in the DAGCOM. In the study, we used noneyed typhoon data (2367 infrared satellite images) to choose the optimal intensity-indicating factor from the DAGCOM.

## 3. Results

### a. The relationship between DADI and tropical cyclone intensity

The results from the model based on DAGCOM and RVM show that DADI (*T*_{3}; see Table 1) has the strongest relevance to tropical cyclone intensity (Fig. 3). Therefore, the DADI is chosen as the indicating factor to estimate the tropical cyclone intensity. Figure 4 shows the original infrared satellite image, the deviation angle histogram, and a pseudocolor map of the DADI for Typhoon Talim in its early, mature, and dissipation stages. In the deviation angle histogram, the *X* axis shows the deviation angle and the *Y* axis indicates the probability density. In the pseudocolor map of the DADI, the *X* and *Y* axes represent the width and height of the infrared satellite image, respectively. The size of both the original infrared satellite image and the pseudocolor map of the DADI is 130 × 130 pixels (within a radius of 325 km). The bar on the right of the pseudocolor map shows the logarithm value of the DADI. Taking each point in the infrared satellite image as a reference point, the deviation angle between each point in the infrared satellite image and the reference point were calculated. Then, a deviation angle matrix was obtained, based on which the DADI was then obtained. The above operations were repeated, and a DADI matrix was developed. To distinguish this matrix, we use different colors to represent the values of the DADI in Fig. 4. The pseudocolor map of the DADI with the same color scale is shown in Fig. 5.

As shown in Fig. 4, we found that a deviation angle histogram can be used to describe the structural characteristics of a tropical cyclone during its different development stages. When a tropical cyclone is in its early stage, the cloud structure is disorganized, and the deviation angle histogram is spread out (Fig. 4a). When a tropical cyclone is in its mature stage (Fig. 4b), the corresponding histogram shows a large peak at the 0° angle and very little spread. Most of the deviation angles approach or are equal to zero. The probability density of the deviation angle histogram approaches a normal distribution. The structure of a tropical cyclone during the dissipation stage (Fig. 4c) is messy, and the deviation angle probability density distribution approaches uniformity. The spread of the histogram in Fig. 4c once again increases. The pseudocolor map (Figs. 4a–c) of the DADI can be used to intuitively describe the structure of a tropical cyclone across its whole life cycle. From Fig. 5, we find that the DADI rises to a peak and then gradually descends. What is more, the peak arrives when a tropical cyclone is in its mature stage.

### b. Intensity estimation for tropical cyclones with different intensities

Four experiments were conducted to estimate the intensity of tropical cyclones: 1) ETPs (32.7 m s^{−1}+), 2) NTPs (32.7 m s^{−1}+), 3) TSs (17.2~32.6 m s^{−1}), and 4) MTCs (17.2 m s^{−1}+). Two-thirds of the sample was used to train the RVM and one-third to verify the built intensity estimation model. The average absolute error of the above tropical cyclone intensity estimation results are shown in Table 3. In Table 3, boldfaced values show the minimum of the average absolute errors. The average absolute error, using the mean, median, and minimum of the DADI separately as intensity-indicating factors, is similar. Therefore, in the future, only one of the mean, median, or minimum of the DADI is sufficient to estimate the intensity of a tropical cyclone. Next, we chose the best result for ETP, NTP, TS, and MTC from Table 3. Absolute error histograms are shown in Figs. 6a–j. The *X* and *Y* axes of each histogram, respectively, represent the absolute error (error interval is 1 m s^{−1}) and frequency.

Table 3 shows that the average absolute error by RVM is smaller than that of linear regression. Figure 6 shows that the error histogram of RVM is concentrated, but the error histogram of linear regression is dispersive. What is more, the error histogram of RVM is closer to zero than is the linear regression. According to Table 3 and Fig. 6, we find that the proposed method, compared to linear regression, shows an obvious improvement in tropical cyclone intensity estimation. The average absolute errors for both ETPs (taking each point of the infrared satellite image as a reference point) and NTPs are about 8 m s^{−1} or so, and the average absolute error for TSs is about 3 m s^{−1}. However, the average absolute errors for both ETPs (taking the center of the tropical cyclone as a reference point) and MTCs are about 10 m s^{−1}. For ETP, NTP, TS, and MTC cases, the average absolute error using the linear regression method is inferior to that of RVM.

With the linear regression method, the average error for NTPs (32.7 m s^{−1}+) is 16 m s^{−1} or so. Similarly, the average absolute error for MTCs (17.2 m s^{−1}+) is 19 m s^{−1}; for TSs (17.2~32.6 m s^{−1}), it is about 20 m s^{−1}; and it is biggest (30 m s^{−1}) for ETPs (32.7 m s^{−1}+). We noticed that the number of ETP samples is smallest (see Table 2). This suggests that the linear regression method results in a bigger intensity estimation error.

For the RVM model, the average absolute error for TSs (17.2~32.6 m s^{−1}) is the least (3 m s^{−1} or so). The average absolute errors for ETPs (taking each point in the infrared satellite image as a reference point; 32.7 m s^{−1}+) and NTPs (32.7 m s^{−1}+) are similar (8 m s^{−1} or so). The average absolute errors for ETPs (taking the center point of the tropical cyclone as a reference point; 32.7 m s^{−1}+) and MTCs (17.2 m s^{−1}+) are also similar (10 m s^{−1}). Compared with the linear regression method, RVM is more efficient for small samples and greatly improves the tropical cyclone intensity estimation.

Three cases [SUT Sepat (No. 0709), STP Hagupit (No. 0814), and TS Dujuan (No. 0912)] were used to verify the performance of the proposed technique. The intensity estimated by RVM was compared with results from CMA, the Japan Meteorological Agency (JMA), the Joint Typhoon Warning Center (JTWC), and the linear regression (LR) method. From Figs. 7a–c, the intensity estimation performance of RVM is shown to be between JMA and JTWC. It is clear that the intensity estimated by LR changes greatly for every case and performs poorly.

## 4. Conclusions

In this study, based on the DADI, which is calculated through the use of infrared satellite imagery from a tropical cyclone, RVM is used to build intensity estimation models for different types of tropical cyclones (ETPs, NTPs, TSs, and MTCs). Experimental results show that, compared with the traditional linear regression method, RVM can greatly improve the accuracy of tropical cyclone intensity estimation. This implies that the nonlinear relationship between tropical cyclone intensity and its indicating factors should be represented in the intensity estimation model. In this sense, nonlinear modeling techniques can yield higher accuracy than those of linear methods in tropical cyclone intensity estimation. A universal intensity estimation model may result in substantial errors in intensity estimation under some circumstances. Experimental results also show that the accuracy of the intensity estimation of RVM is between that of JMA and JTWC.

Although the proposed technique demonstrates a remarkable ability to estimate tropical cyclone intensity, especially in its mature stage, weak tropical cyclones [e.g., tropical depressions (10.8~17.1 m s^{−1})] are not included in this study. Experimental research on estimating the intensity of weak tropical cyclones will be presented in a future publication. In addition, the proposed technique only uses infrared geostationary satellite observations. Water vapor geostationary satellite observations might prove helpful in improving the tropical cyclone intensity estimation’s precision based on the proposed technique. Finally, the proposed technique will be expanded to the Atlantic basin to verify its performance in the future.

## Acknowledgments

This study was supported by the Natural Science Foundation of China (Grants 41575046, 41475059, 40805048, and 11026226), the Project of Commonweal Technique and Application Research of Zhejiang Province of China (Grants 2016C33010 and 2012C23027), the Natural Science Foundation of Zhejiang Province of China (Grant LY13D050001), and the Natural Science Foundation of Shanghai of China (Grant 15ZR1449900). All of the original satellite images in this paper have been provided by National Satellite Meteorological Center, China Meteorological Administration. The best-track data for tropical cyclones are provided by the Shanghai Typhoon Institute of China Meteorological Administration.

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