3502 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 6, JUNE 2016 Generalized Empirical Formulas of Threshold Distance to Characterize Cyclone–Cyclone Interactions Yuei-An Liou, Senior Member, IEEE, Ji-Chyun Liu, Meng-Xi Wu, Yueh-Jyun Lee, Chi-Han Cheng, Ching-Ping Kuei, and Rong-Moo Hong

Abstract—This paper presents a framework of using remote their distribution and variation, are valuable information for sensing imagery and image processing techniques to analyze the disaster prevention [1], [2]. The satellite data that provide interaction among or tropical cyclones (TCs) with and images were used to analyze the cloud structure and dynam- without the involvement of mid-tropical depressions (TDs) or tropical storms (TSs). The cyclone–cyclone interaction (named the ics of typhoons [3], [4], [7]–[11]. So far as the dynamics is Fujiwhara effect) could happen when two TCs are located at a concerned, if two tropical cyclones (TCs) come close to each distance within 1400 km. However, when there exists a mid-TD other, cyclone–cyclone interaction or the Fujiwhara effect [5] between the two TCs even with a distance of more than 1400 km, would occur. There are several types of cyclone–cyclone in- but usually within a distance of 1500–2700 km, the mid-TD may teractions, such as complete merge (CM), partial merge (PM), induce impacts on the two cyclones, defined as a TD-inlaid cyclone– cyclone interaction. The existence of the mid-TD significantly complete straining out (CSO), partial straining out (PSO), and complicates weather forecasting due to difficulty in modeling its elastic interaction (EI) [6]. Fujiwhara effects are related to influence on the two cyclones. In this paper, we proposed inno- the intensity of the two TCs, which could be determined by vative empirical formulas of threshold distance to characterize the size, height difference, and self-rotation of TCs. TCs are cyclone–cyclone interactions either with or without the involve- regarded as stronger if they have larger sizes, bigger height ment of the TD(s). We generalize empirical formulas for interac- tion distances between TCs and TD/TS related to current-intensity difference, and faster rotation speed, and vice versa. If two number, size factor, height difference, and rotation factor. To val- stronger TCs come close, they start to interact with each other idate the appropriateness of the generalized empirical formulas, at the distance around 1400 km. While two TCs are weaker, we choose and analyze seven sets of two- events. It is they need to come closer to around 1100 km before influencing shown that the generalized formulas successfully predicted the each other. In contrast, the interaction would happen at the impact of mid inlaid TC/TS quantitatively. The mid-TD’s effects on the two cyclones represent a new aspect and a significantly distances between 1200 and 1300 km if one TC is stronger and improved understanding for the prediction of more complicated the other is weaker. Nevertheless, TC–TC interaction becomes cyclone–cyclone interactions. Note that the empirical formulas are more complicated when mid-tropical depressions (TDs) appear derived from a very small set of events and that they must be and start to interact with the two TCs. verified with a sufficient data set for future practical use. The interactions between mid-TDs and TCs are considered Index Terms—Cyclone–cyclone interaction, empirical formula, as a new type of cyclone–cyclone interaction. Studying the satellite cloud image. variety of cyclone–cyclone interactions is useful for improving I. INTRODUCTION weather forecasting. In previous research, in the North Pacific area, the mid-TS Bopha located between the two typhoons HE impacts of climate variability and global warming Saomai and Wukong (2000) presented an unusual movement in on the occurrence of tropical storms (TSs), including T the west. With the use of the piecewise potential vorticity (PV) inversion, Wu et al. [12] quantitatively evaluated TS Bopha Manuscript received November 13, 2015; revised December 26, 2015; that moved southward due to the circulation associated with accepted January 13, 2016. Date of publication February 8, 2016; date of cur- rent version April 27, 2016. This work was supported by the Ministry of Science . The flow field associated TS Bopha was and Technology under Grants MOST 104-2111-M-008-004 and 104-2221- mainly steered by typhoon Soamai, and typhoon Wukong also E-008-067. affected the motion of TS Bopha [12]. Recently, when mid-TDs Y.-A. Liou is with the Center for Space and Remote Sensing Research, National Central University, Taoyuan 32097, (e-mail: yueian@csrsr. appeared between two cyclones, typhoons Tembin and Bolaven ncu.edu.tw). (2012), the TCs were still located far from each other. Due to J.-C. Liu and M.-X. Wu are with the Department of Electrical Engineering, the fact that the TDs are smaller and lower than the TCs, the Chien Hsin University, Taoyuan 320, Taiwan (e-mail: [email protected]). Y.-J. Lee is with the Department of Information Management, Chien Hsin upward convection between TCs and TDs happened in addi- University, Taoyuan 32097, Taiwan (e-mail: [email protected]). tion to attraction. The individual interactions among mid-TDs C.-H. Cheng is with the School of Hydrometeorology, Nanjing University of and TCs, depression–cyclone interactions, cause indirective Information Science & Technology, Nanjing 210044, . C.-P. Kuei and R.-M. Hong are with the Department of Electronics Engi- cyclone–cyclone interaction even as the distance between the neering, Chien Hsin University of Science and Technology, Taoyuan 32097, two typhoons is more than 1500 km [13]. Hence, observations Taiwan. and quantitative analysis on a more variety of cyclone–cyclone Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. interactions are needed so that the TD-inlaid cyclone–cyclone Digital Object Identifier 10.1109/TGRS.2016.2519538 interactions may be better interpreted.

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A variety of empirical methods were utilized to crudely characterize the cloud vertical structure [18] based on cloud effective temperatures and optical depths with 3.7- [14] or 3.8-μm [18] detectors. In particular, the ice or ice- covered surfaces on cloud were observed by using the short- wave infrared (SIR) channels [14]–[19]. In this paper, the SIR channel with 3.7 μm was also used to detect the cloud images of the typhoon to observe the cloud surfaces in depth. The 3-D profiles of typhoons were constructed using the image recon- struction technique from Multifunctional Transport Satellite (MTSAT) satellite cloud image data. The SIR channel with 3.7 μm was applied to detect the satellite cloud images of the typhoon to observe the cloud surfaces in depth at first. The satellite cloud image was sliced in a vertical plane taking the line profile. The line profile was represented with cloud top temperature by toggling the profile mode of an integrated display and processing (iDAP) system [20]. After slicing the cloud image in sequence, a series of vertical surfaces were produced for rendering the volume. The mesh-amplitude model was used to produce the structure of the sliced plane from a satellite image and 3-D cloud images. In this paper, generalized empirical formulas of threshold distance that were assumed to be functions of the interaction distances between TC and TD/TS related to the current inten- sity (CI) number, size factor, height difference, and rotation factor were proposed to predict the cyclone–cyclone interac- tions. Seven two-typhoon cases were chosen, including typhoon Ma-on and TS Tokage (2011), typhoons Fitow and Danas (2013), typhoons Nari and Wipha (2013), typhoons Francisco and Lekima (2013), typhoons Tembin and Bolaven (2012), typhoons Usagi and Pabuk (2013), and typhoon Saomai and TS Bopha (2000). Four parameters, including distance, size, height-difference, and rotation among two typhoons and mid- TDs, wereused to expound the variety of cyclone–cyclone inter- action. Finally, cloud images of typhoons, tracks of typhoons, Fig. 1. Six-time cloud images of typhoon Ma-on and TS Tokage (2011) at 3-D profiles of typhoons, and time series of interactions were (a) T1 =02:32, July 15; (b) T2 =09:32, July; (c) T3 =16:32, July 15; used for the present analysis. More variety of cyclone–cyclone (d) T4 =23:32, July 15; (e) T5 =06:32, July 16; and (f) T6 =20:32, July 16. interactions that are very important for the weather prediction models and forecasts were observed and quantified. To demon- strate the advantages of the generalized empirical formulas of TS Bopha (2000). The detailed descriptions and analysis of the cyclone–cyclone interactions over the empirical formulas of seven two-typhoon images are described as follows. typical cyclone–cyclone interactions, the empirical formulas for Case 1. Typhoon Ma-on and TS Tokage (2011): The typhoon typical cyclone–cyclone interactions are first introduced in the Ma-on occurred in the northwest of and following section. on July 11, 2011. From MTSAT cloud images, the six-time cloud images of typhoon Maon and TS Tokage taken at 02:32, July 15; 09:32, July 15; 16:32, July 15; 23:32, July 15; 06:32, II. ANALYSIS OF SEVEN TWO-TYPHOON IMAGES July 16; and 20:32, July 16 were shown in Fig. 1. During To investigate the behaviors of the typical cyclone–cyclone the period of T1 − T3, both typhoon Ma-on and TS Tokage interactions and the TD-inlaid cyclone–cyclone interactions, existed, and at T4, the ice-covered surfaces with a length of six two-typhoon cases from years 2011 to 2013 and one two- 220–270 km were observed as shown in Fig. 1. From the typhoon case from year 2000 composed the seven events which typhoon track maps presented in Fig. 2, typhoon Ma-on went were divided into two main groups: Group 1 with typical west and TS Tokage moved northeast within T1 − T3,and cyclone–cyclone interactions, including typhoon Ma-on and typhoon Ma-on was changed to the north after merging with TS Tokage (2011), typhoons Fitow and Danas (2013), ty- TS Tokage within T5 − T6. The distances between typhoon phoons Nari and Wipha (2013), and typhoons Francisco and Ma-on and TS Tokage were within 600 to 1250 km, and the Lekima (2013); and Group 2 with TD-inlaid cyclone–cyclone height difference between the typhoon and TS was 4 km. The interactions, including typhoons Tembin and Bolaven (2012), TS rotated counterclockwise. This case belongs to the CM of typhoons Usagi and Pabuk (2013), and typhoon Saomai and cyclone–cyclone interaction basically. 3504 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 6, JUNE 2016

Fig. 2. Tracks of typhoon Ma-on and TS Tokage (2011). Fig. 4. Tracks of typhoons Fitow and Danas (2013).

from October 4 to 7: at 23:32, October 4; 11:32, October 5; 23:32, October 5; 11:32, October 6; 23:32, October 6; and 11:32, October 7. The tracks of typhoons Fitow and Danas are plotted in Fig. 4. Both typhoons Fitow and Danas moved toward the northwest within T1 − T6. The distance between typhoons Fitow and Danas started at 1600 km, which is more than the 1400 km of typical cyclone–cyclone interaction. The results showed that the typical interaction could not occur between typhoons Fitow and Danas. However, typhoon Danas moved quickly, and became weaker after touching the mainland. Although the distance between the two typhoons Fitow and Danas was decreased to 1200 km, as well as no mid- TD presence, the cyclone–cyclone interaction did not happen. Case 3. Typhoons Nari and Wipha (2013): Fig. 5(a)–(f) shows the six-time cloud images of typhoons Nari and Wipha (2013) taken from October 11 to 13: at 11:32, October 11; 23:32, October 11; 11:32, October 12; 23:32, October 12; 11:32, October 13; and 23:32, October 13. The tracks of typhoons Nari and Wipha (2013) are plotted in Fig. 6. At T1, typhoons Nari and Wipha separated with a distance of 1680 km, which is more than the 1400 km of typical cyclone–cyclone interaction. The situations showed that the typical interaction could not occur between typhoons Nari and Wipha. Typhoons Nari and Wipha have broken away from each other within T1 − T6. moved toward the west, and moved toward the north; the distance between typhoons Nari and Wipha was increased up to 2700 km, as well as no mid-TD presence. Therefore, the cyclone–cyclone interaction did not happen. Case 4. Typhoons Francisco and Lekima (2013): The fourth- case typhoons are Francisco and Lekima (2013). Fig. 7(a)–(f) shows the six-time cloud images of typhoons Francisco and Lekima taken from October 22 to 25: at 23:32, October 22; 11:32, October 23; 23:32, October 23; 11:32, October 24; Fig. 3. Six-time cloud images of typhoons Fitow and Danas (2013) at (a) T1 = 23:32, October 24; and 11:32, October 25. The tracks of ty- 23: 32, October 4; (b) T2 =11:32, October 5; (c) T3 =23:32, October 5; (d) T4 =11:32, October 6; (e) T5 =23:32, October 6; and (f) T6 =11:32, phoons Francisco and Lekima (2013) are plotted in Fig. 8. Both October 7. typhoons Francisco and Lekima moved toward the northwest and north simultaneously within T1 − T6. The distance between Case 2. Typhoons Fitow and Danas (2013): Next, typhoons typhoons Francisco and Lekima was about 2250 km at first, Fitow and Danas (2013) are discussed. Fig. 3(a)–(f) shows which is more than the 1400 km of typical cyclone–cyclone the six-time cloud images of typhoons Fitow and Danas taken interaction. The results showed that the typical interaction could LIOU et al.: EMPIRICAL FORMULAS OF THRESHOLD DISTANCE TO CHARACTERIZE TC–TC INTERACTIONS 3505

Fig. 5. Six-time cloud images of typhoons Nari and Wipha (2013) at (a) T1 = Fig. 7. Six-time cloud images of typhoons Francisco and Lekima (2013) at 11: 32, October 11; (b) T2 =23:32, October 11; (c) T3 =11:32, October 12; (a) T1 =23:32, October 22; (b) T2 =11:32, October 23; (c) T3 =23:32, (d) T4 =23:32, October 12; (e) T5 =11:32, October 13; and (f) T6 =23:32, October 23; (d) T4 =11:32, October 24; (e) T5 =23:32, October 24; and October 13. (f) T6 =11:32, October 25.

Fig. 6. Tracks of typhoons Nari and Wipha (2013). Fig. 8. Tracks of typhoons Francisco and Lekima (2013). not occur between typhoons Francisco and Lekima. However, mid-TD appearance still. Therefore, the cyclone–cyclone inter- moved northeast quickly, and the distance action did not happen. However, after T6, typhoons Fransico between the two typhoons was decreased to 1100 km with no and Lekima began to interact slightly. 3506 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 6, JUNE 2016

Fig. 10. Tracks of typhoons Tembin and Bolaven (2012) and two mid-TDs in the period of August 21–22.

and Bolaven were slow. From Fig. 9(a), it is evident that the TD1 appeared between typhoons Tembin and Bolaven, while the development of TD2 is evident in Fig. 9(b). Therefore, two mid-TDs formed in the period of T3 − T4 are shown in Fig. 9(c) and (d). The interactions between and TD1 and between and TD2 could be found by inspect- ing the convection between them. Considering the existence of mid-TDs, typhoon Tembin and TD1 located about 910–980 km apart with a height difference of 2 km at T2 − T6, and typhoon Bolaven and TD2 come close to about 780–920 km apart with a height difference of 3.5 km at T2 − T4. Therefore, two sets of depression–cyclone interactions occurred, and they could be assumed as a combined depression–cyclone interaction. During the period of T5 − T6,TD1 and TD2 merged together and formed a new TD. This newly formed TD provides convection to typhoons Tembin and Bolaven as shown in Fig. 9(e) and (f). The tracks of typhoons Tembin and Bolaven (2012) are plotted in Fig. 10. Typhoon Tembin moved toward the west, while the Fig. 9. Six-time cloud images of typhoons Tembin and Bolaven (2012) at typhoon Bolaven moved toward the northwest within T1 − T6. (a) T1 =12:32, August 21; (b) T2 =17:32, August 21; (c) T3 =21:32, August 21; (d) T4 =01:32, August 22; (e) T5 =05:32, August 22; and Both TD1 and TD2 occurred in T2. The direction of the (f) T6 =09:32, August 22. movement of TD1 changed counterclockwise to the south after interacting with typhoon Tembin within T2 − T6. On the other Case 5. Typhoons Tembin and Bolaven (2012): In our pre- hand, the direction of the TD2 movement was changed coun- vious research [13], we studied typhoons Tembin and Bolaven terclockwise and directed to typhoon Bolaven within T2 − T4. (2012) to examine whether the Fujiwhara effect would happen Thus, the two sets of TC–TD interactions can be demonstrated. when two typhoons spread apart over 1500 km. It was demon- Although the distances between typhoons Tembin and strated that the two typhoons Tembin and Bolaven interacted Bolaven were within 1500 to 1600 km, more than the 1400 km with each other through upward convection and attraction of typical cyclone–cyclone interaction, the results showed that between two typhoons and the two TDs, respectively. The two mid-TDs that occurred between the two typhoons inter- merge of the two TDs caused the typhoons to start rotating acted with each other through upward convection and attraction and generate strengthened power. Hence, from this case, it is between two typhoons and TDs, respectively. The merge of the suggested that the TDs have impacts on the typhoons, and this two mid-TDs caused the typhoons to start rotating and generate impact may be considered as a new type of Fujiwhara effect. strengthened power. The findings of the mid-TD effects on two To investigate and understand how TDs interact with TCs, we typhoons represent a new aspect of the interaction between mid- choose more cases with the appearance of mid-TDs and longer TD and typhoon. Due to the fact that higher/larger typhoons distances between the two typhoons. Tembin and Bolaven and the lower/smaller TD had come close Fig. 9(a)–(f) shows the six-time cloud images of typhoons to each other, the two sets of TC–TD interactions existed in Tembin and Bolaven (2012) taken from August 21 to 22: at the processing. Typhoon Tembin has increased in intensity and 12:32, August 21; 17:32, August 21; 21:32, August 21; 01:32, made a U-turn, and Bolaven was intensified and changed its August 22; 05:32, August 22; and 09:32, August 22. During the direction by the effect of combined interactions. It could belong period of T1 − T6, the movements of both typhoons Tembin to the TD-inlaid cyclone–cyclone interaction. LIOU et al.: EMPIRICAL FORMULAS OF THRESHOLD DISTANCE TO CHARACTERIZE TC–TC INTERACTIONS 3507

Fig. 12. Tracks of typhoons Usagi and Pabuk (2013).

toward the northwest in T1 − T6, while the typhoon Pabuk moved toward the west in T2 − T4. Both TD and TD (Pabuk) existed in T2 − T3. The TD rotated counterclockwise. The direction of the TD (Pabuk) movement was changed clockwise to the north after interacting with TD in T4 − T6. Thus, the mid- TD affected either one of the two typhoons representing a new aspect of the leave among two typhoons. It could also belong to the TD-inlaid cyclone–cyclone interaction. Case 7. Typhoon Saomai, TS Bopha, and Typhoon Wukong (2000): TS Bopha moved westward from September 7 to 9, 2000, then had a dramatic southward turn on September 9, and finally continued its unusual southward course parallel to the east coast of Taiwan until September 11. TS Bopha showed an unusual movement, mainly steered by the circulation associated with typhoon Saomai to TS Bopha’s east. With the use of the piecewise PV inversion, Wu et al. [12] quantitatively evalu- ated how TS Bopha moved southward due to the circulation associated with typhoon Saomai, the flow field associated TS Fig. 11. Six-time cloud images of typhoons Usagi and Pabuk (2013) at Bopha steered by typhoon Soamai, and the motion of TS Bopha (a) T =00:32, September 20; (b) T =10:32, September 20; (c) T = 1 2 3 affected by the TC Wukong. 20: 32, September 20; (d) T4 =07:32, September 21; (e) T5 =17:32, September 21; and (f) T6 =03:32, September 22. It has been reported in the previous research works (see Figs. 2, 5, 6, and 7) [12] that, when the cloud images were Case 6. Typhoons Usagi and Pabuk (2013): Fig. 11(a)–(f) kept well, they can provide accurate information of the tracks shows the six-time cloud images of typhoons Usage and Pabuk of typhoon Saomai, TS Bopha, and typhoon Wukong. The (2013) taken from September 20 to 22: at 00:32, September 20; tracks of typhoon Saomai, TS Bopha, and typhoon Wukong in 10:32, September 20; 20:32, September 20; 07:32, September September 7 to 12 are plotted in Fig. 13. The distances in six- 21; 17:32, September 21; and 03:32, September 22. During time intervals among typhoon Saomai, TS Bopha, and typhoon the period of T1 − T6, the movements of both typhoons Usage Wukong are presented in Fig. 14. Typhoon Saomai moved and Pabuk were slow. In Fig. 11(a), it is evident that the TD toward the northwest in D1 − D6, while typhoon Wukong (without a specific name) appeared beside typhoon Usage and moved toward the west in D1 − D4. TS Bopha started in D1 located apart from typhoon Usage with a distance of 2100 km. and changed direction in D2 − D3. Then, TS Bopha interacted In Fig. 11(b), the development of another TD (Pabuk) was ob- with typhoon Wukung moving with a zigzag path. Meanwhile, served, and the TD (Pabuk) located apart from typhoon Usage TS Bopha interacted with typhoon Saomai and turned to the with a distance of 1700 km. However, both TD and TD (Pabuk) south. Since these two interactions, the TS Bopha related to existed closely in a distance of 800 km with a height difference typhoon Saomai rotated counterclockwise. Thus, the mid-TS of 2 km. Therefore, the two TDs interacted and attracted each Bopha affected either one of the two typhoons. This could also other in T3 as shown in Fig. 11(c). At T4, TD and TD (Pabuk) belong to the TD-inlaid cyclone–cyclone interaction. merged together and formed typhoon Pabuk in Fig. 11(d). Then, In Fig. 14, the distances between typhoons Saomai and two typhoons Usage and Pabuk moved individually as shown Wukong were within 2700–3100 km. Without TS Bopha, it is in Fig. 11(e) and (f). The tracks of typhoons Usage and Pabuk said that cyclone–cyclone interaction would not occur if the (2013) and TDs are plotted in Fig. 12. Typhoon Usage moved distance between the two cyclones was more than 1400 km. 3508 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 6, JUNE 2016

TABLE I RESULTS OF CYCLONE–CYCLONE INTERACTIONS

Fig. 13. Tracks of typhoon Saomai, TS Bopha, and typhoon Wukong (2000).

Fig. 14. Distances in six-time intervals among typhoon Saomai, TS Bopha, and typhoon Wukong (2000).

Considering the existence of TS Bopha, typhoon Wukong and TS Bopha located about 2050–1500 km apart at D1 − D4, and typhoon Saomai and TS Bopha came closer to about Fig. 15. Typical cyclone–cyclone interactions. 1500–1200 km apart at D1 − D6. Obviously, two individ- ual TC–TS interactions happened. Note that the interaction between typhoon Wukong and TS Bopha was weaker and the interaction between typhoon Saomai and TS Bopha was stronger. It is the reason why typhoon Saomai caused TS Bopha to make a circulation. This analysis is consistent with the valuable results of the previous works which stated that “the influence of Bopha on Saomai and Wukong on Bopha are both considerably smaller than the influence of Saomai on Bopha” Fig. 16. Variety of cyclone–cyclone interactions. and “the presence of Saomai plays the major role in leading to the southward movement of Bopha” [12]. From the results III. GENERALIZED CYCLONE–CYCLONE INTERACTIONS of a variety of cyclone-to-cyclone interactions in Table I, three two-typhoon images, including typhoons Tembin and Bolaven There are five types of cyclone–cyclone interactions, i.e., (2012), typhoons Usagi and Pabuk (2013), and typhoon CM, PM, CSO, PSO, and EI as depicted in Fig. 15(a)–(e). Saomai, TS Bopha, and typhoon Wukong (2000), belong to The variety of cyclone–cyclone interactions are graphically the TD-inlaid cyclone–cyclone interactions. Table I also sum- illustrated in Fig. 16(a)–(c), representing interactions between marizes the behaviors of typical cyclone–cyclone interactions TC and TD, among mid-TD and TCs, and among two mid-TDs for Group 1, including typhoon Ma-on and TS Tokage (2011), and TCs, respectively. typhoons Fitow and Danas (2013), typhoons Nari and Wipha The cases mentioned in the previous section are primarily (2013), and typhoons Francisco and Lekima (2013) indicated helpful to advance our qualitative understanding of the va- as cases (1)–(4), respectively. riety of cyclone–cyclone interactions. In contrast, empirical LIOU et al.: EMPIRICAL FORMULAS OF THRESHOLD DISTANCE TO CHARACTERIZE TC–TC INTERACTIONS 3509

TABLE II TABLE III CI NUMBER THRESHOLD DISTANCES FOR TYPICAL CYCLONE–CYCLONE INTERACTIONS

B. Generalized Cyclone–Cyclone Interactions Cyclone–cyclone interactions are generally applied for an- alyzing two nearly equal-sized cyclones with similar heights. Conventionally, the cyclone–cyclone interactions include cap- ture (merge or strain) and escape (elastic) related to the distance between the two cyclones. However, when the cyclone and TS, or cyclone and TD, occur and come closer, the interactions between the two weather systems are different from the cyclone– cyclone interactions. Since the sizes of the two weather systems formulas that could describe the interactions of concern are are not equal and TS and TD are smaller and lower than crucial as they are able to improve quantitatively the numerical cyclones, the upward convection between the cyclone and TS or modeling of weather forecasting, particularly as the formulas TD could happen. The upward convections enhance the cyclone could characterize not only the typical but also the generalized and sustain its rotation. The size ratio, height difference, and cyclone–cyclone interactions. In general, the typical cyclone– rotation could be involved in addition to the distances. As the cyclone interactions are associated with the intensity of in- situations in Fig. 16(b) and (c), the mid-TD or TDs locate dividual cyclones, which is represented by the CI number, between two TCs, and two sets of TC–TD interactions happen. corresponding to the maximum wind speed at the center and By neglecting the mutual coupling between two sets of TC–TD, the intensity on the sea surface as given in Table II. In this two individual threshold distances of (2) could be additive. section, empirical formulas for characterizing the generalized Under such circumstance, we write the empirical formulas cyclone-to-cyclone interactions are described. for threshold distances d1 and d2 proposed to examine if the two sets of TC–TD interactions exist between TC1 (CI1) and A. Typical Cyclone–Cyclone Interactions mid-TD (CId) and between TC2 (CI2) and mid-TD (CId), respectively Typical cyclone–cyclone interactions involve two cyclones,   TC1 (CI1) and TC2 (CI2). The cyclone–cyclone interaction CI CI d =1, 000 + 100 1 + d F (2) usually happens when two TCs are located at a distance about 1 4 4 1 1100–1400 km. We write the empirical formula for the thresh-   CI2 CId old distance d0 to decide if the two cyclones, TC1 and TC2 d2 =1, 000 + 100 + F2 (3) (km), interact with each other as 4 4   CI CI where d =1, 000 + 100 1 + 2 (1) 0 4 4 1 1 F1(q1, Δh1,τ1)= × × τ1 (4) q1 Δh1 For example, as there exist two stronger TCs and CI1 = CI2 = 4,wehaved0 = 1400 km. As there exist two medium TCs and 1 1 F2(q2, Δh2,τ2)= × × τ2 (5) CI1 = CI2 =2, d0 = 1200 km is obtained. When two weaker q2 Δh2 TCs exist and CI1 = CI2 =1, d0 = 1050 km is found. In this paper, four two-typhoon cases were chosen, including where typhoon Ma-on and TS Tokage (2011), typhoons Fitow and q1,2 size factor 1, 2 and q1,2 =(size(TD)/size(TC1,2)); Danas (2013), typhoons Nari and Wipha (2013), and typhoons Δh1,2 related height difference between TC1,2 and TD, Francisco and Lekima (2013). Their corresponding threshold and Δh1,2 =((hTC − hTD)/(hTC )); distances for interaction may be then calculated by using (1). 1,2 1,2 τ1,2 related rotation factor; They were found to be 1150, 1303, 1188, and 1238 km, τ1,2 =+1 counterclockwise-related rotation between TC1 respectively, which were shorter than the observed distances and TD; of 1250, 1600, 1680, and 2250 km between TC and TC/TS, τ1,2 = −1 clockwise-related rotation between TC1 and TD. respectively, for the four investigated cases. This indicates that there exists no typical cyclone–cyclone interaction for the four Given the empirical formulas for the threshold distances d1 cases (see Table III). and d2, the empirical formula for the threshold distance D to 3510 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 6, JUNE 2016

TABLE IV CALCULATION RESULTS OF TD-INLAID CYCLONE–CYCLONE INTERACTIONS

quantitatively define the TC–TC interaction can be written as measured distance between typhoon Usage and TD was longer   than the threshold distance (1250 km) for interaction. There is CI1 CId no interaction between typhoon Usage and TD. In contrast, the D = d + d =2, 000 + 100 + F 1 2 4 4 1 interaction between typhoon Pabuk and TD happened because   the observed distance between typhoon Pabuk and TD was CI2 CId + 100 + F2. (6) shorter than the threshold distance (1400 km) for interaction. 4 4 However, the observed distance (2300 km) between typhoons a. Case 1: In Table IV, there are four two-typhoon cases Usage and Pabuk was shorter than the threshold distance (2650 for analysis. The first case (1) is about typhoon Ma-on and TS km) for interaction. Thus, the TC-to-TC interaction existed Tokage. It has been once discussed in Table III, which showed between typhoons Usage and Pabuk. no interaction between the two weather systems. However, d. Case 4: For the fourth case (7) in Table IV, two sets of when this case was reexamined by using (2) and (4), the two TC-to-TS interactions happened between typhoons Saomai and weather systems exhibited interaction between each other since Wukong and the TS Bopha. The interaction between typhoon the measured distance (1250 km) between them was shorter Saomai and TS existed since the measured distance (1200 km) than the threshold distance (1600 km) for interaction. This in- between the two weather systems was shorter than the threshold dicates that the empirical formulas of threshold distances could distance (1500 km) for interaction. It is observed that the inter- provide accuracy determination to characterize the behavior of action between typhoon Wukong and TS Bopha was weaker the interactions. because the measured distance (1500 km) between the two b. Case 2: The second case (5) in Table IV is about two weather systems was close to the threshold distance (1450 km) sets of TC–TD interactions for typhoons Tembin and Bolaven. for interaction. The measured distance (2700 km) between ty- phoons Saomai and Wukong was close to the threshold distance Typhoon Tembin and TD1 located 910 km apart with a size ratio factor of 0.44 and a related height factor of 0.82, and (2950 km) of interaction. Thus, there was a slight TC-to-TC interaction between typhoons Saomai and Wukong. typhoon Bolaven and TD2 came close to 780 km apart with a size ratio factor of 0.44 and a related height factor of 0.72. Both of the measured distances between typhoon Tembin and IV. DISCUSSION AND CONCLUSION TD1 and between typhoon Bolaven and TD2 were shorter than the threshold distances (1450 and 1520 km) for interaction, Generalized cyclone–cyclone interactions that involved typ- respectively. In addition, the combined depression–cyclone in- ical cyclone–cyclone interactions and TD-inlaid cyclone– teractions were presented. TD1 and TD2 merged together and cyclone interactions were introduced in this paper.The empirical formed a new TD. This newly formed TD provided convections formulas of threshold distances for interactions between TC to typhoons Tembin and Bolaven and caused the TC-to-TC and TC and between TC and TD/TS related to the CI number, interaction. size factor, height difference, and rotation factor were pro- c. Case 3: For the third case (6) in Table IV, two typhoons posed to quantitatively explain the generalized cyclone–cyclone Usage and Pabuk and a TD were examined. The TD located interactions. Seven practical two-typhoon cases were chosen apart from typhoon Usage with a distance of 1600 km. The TD to demonstrate the consistency of the applications of empiri- and typhoon Pabuk existed closely in a distance of 800 km. The cal formulas for threshold distances. The proposed empirical LIOU et al.: EMPIRICAL FORMULAS OF THRESHOLD DISTANCE TO CHARACTERIZE TC–TC INTERACTIONS 3511 formulas can be applied to real-time analysis and prediction to understand the TC–TD/TS interactions for improving the system and to quantitatively evaluate the impact of mid inlaid forecast of severe weather system. When the distances between TC/TS on TCs. The results obtained from this analysis have a two cyclones are more than the baseline of 1400 km, the typical great potential to improve the weather forecasting. interaction could not occur basically. However, as the mid-TD The results of the four two-typhoon cases, including ty- appeared and the mid-TD has impacts on the two cyclones, an phoon Ma-on and TS Tokage (2011), typhoons Fitow and alternative cyclone–cyclone interaction could happen and can Danas (2013), typhoons Nari and Wipha (2013), and typhoons be referred as a TD/TS-inlaid cyclone–cyclone interaction. The Francisco and Lekima (2013), were obtained. The measured generalized cyclone–cyclone interaction could happen when distances (1250, 1600, 1680, and 2250 km) between TC and the mid-TD located within two cyclones and the two cyclones TC/TS are longer than the corresponding threshold distances are apart within the distance of 1600–2700 km. Such extension (1150, 1303, 1188, and 1238 km) for interaction, indicating of the distances between cyclone and cyclone allows one to pre- that the four two-typhoon cases could not have typical cyclone- dict the weather in advance. The generalized cyclone–cyclone to-cyclone interactions. Second, the results of the four two- interaction can be represented with the empirical formulas and typhoon events with the TD-inlaid cases, including typhoon quantified, and is very important for the weather prediction Ma-on and TS Tokage (2011), typhoons Tembin and Bolaven models and forecasts. 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[16] H. L. Huang et al., “Inference of ice cloud properties from high spec- Ji-Chyun Liu was born in Taiwan in 1951. He ob- tral resolution infrared observations,” IEEE Trans. Geosci. Remote Sens., tained the B.S, M.S., and Ph.D. degrees from Chung- vol. 42, no. 4, pp. 842–853, Apr. 2004. Cheng Institute of Technology, Taoyuan, Taiwan, in [17] G. Hong et al., “The sensitivity of ice cloud optical and microphysical 1974, 1981, and 1993, respectively. passive satellite retrievals to cloud geometrical thickness,” IEEE Trans. From 1981 to 2003, he was an Instructor, Associate Geosci. Remote Sens., vol. 45, no. 5, pp. 1315–1323, May 2007. Professor, and Professor in the Chung Cheng Insti- [18] P. Minnis et al., “CERES edition 2 cloud property retrievals using TRMM tute of Technology. He has been a Professor of the VIRS and Terra and Aqua MODIS data: Part I: Algorithms,” IEEE Trans. Chien Hsin University of Science and Technology, Geosci. 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His research has included microwave circuit de- Yuei-An Liou (S’91–M’96–SM’01) received the sign, wideband filter, wideband amplifier, and an- B.S. degree in electrical engineering from Na- tenna design and measurement. tional Sun Yat-Sen University (NSYSU), Kaohsiung, Taiwan, in 1987 and the M.S.E. degree in electrical engineering (EE), the M.S. degree in atmospheric and space sciences, and the Ph.D. degree in EE and atmospheric, oceanic, and space sciences from the University of Michigan, Ann Arbor, MI, USA, in Yueh-Jyun Lee was born in Taiwan in 1960. He 1992, 1994, and 1996, respectively. received the B.S., M.S., and Ph.D. degrees from From 1989 to 1990, he was a Research Assistant Chung-Cheng Institute of Technology, Taoyuan, with the Robotics Laboratory, National Taiwan Uni- Taiwan, in 1982, 1988, and 1997, respectively. versity. From 1991 to 1996, he was a Graduate Student Research Assistant with He is currently an Assistant Professor of the the Radiation Laboratory, University of Michigan, where he developed land–air Chien Hsin University of Science and Technology, interaction and microwave emission models for prairie grassland. He joined the Taoyuan. His research interests include computer faculty of the Center for Space and Remote Sensing Research (CSRSR) in 1996 network, neural network, digital logic circuit design, and the Institute of Space Sciences in 1997 at the National Central University and embedded system. (NCU), Taoyuan, Taiwan, where he is now a Distinguished Professor. He served as the Division Director of the Science Research Division of the National Space Organization (NSPO) of Taiwan in 2005 and continued to serve as Advisor to NSPO in 2006. From August 2006 to July 2007, he was a Chair Professor Chi-Han Cheng was born in Taiwan in 1980. He and the Dean of the College of Electrical Engineering and Computer Science, received the B.S. degree in civil engineering from Chien Hsin University, Taoyuan. From 2007 to 2010, he served as the Director Chung Yuan Christian University, Taoyuan, Taiwan, of CSRSR, NCU. He is currently the President of the Taiwan Group on Earth in 2002, the M.S. degree in hydrological sciences Observations (2010–) and the President of the Taiwan GIS Center (2014–). His from National Central University, Taoyuan, in 2004, current research activities include GPS meteorology and ionosphere, remote and Ph.D. degree in civil, environmental, and con- sensing of the atmosphere, land surface, and polar ice, and land surface process struction engineering from the University of Central modeling. He is a Principal Investigator on many research projects sponsored Florida, USA, in 2012. by the Ministry of Science and Technology and the Council of Hakka Affairs He is currently an Associate Professor of the of Taiwan. He has over 100 referral papers and more than 200 international Nanjing University of Information Science and conference papers. Technology, Nanjing, China. His research interests Dr. Liou is a referee for IEEE TRANSACTIONS ON GEOSCIENCE AND include hydroclimatic extremes: drought monitoring and analysis, land use REMOTE SENSING (TGRS), International Journal of Remote Sensing, Earth, impacts on energy and water balances, and hydrology remote sensing. Planets, and Space, Remote Sensing of Environment, Journal of Geophysical Research, Annales Geophysicae, etc. He is a member of the Editorial Advisory Board for GPS Solutions since 2001 and of the International Journal of Naviga- Ching-Ping Kuei was born in I-Lan, Taiwan, in tion and Observation since 2011. He serves as an Editor of the Journal of Aero- 1953. He received the B.S. degree in physics from the Chung-Cheng Institute of Technology, Taoyuan, nautics, Astronautics and Aviation since 2009 and an Associate Editor of the Taiwan, in 1975 and the M.S. and Ph.D. degrees IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS in electrical engineering from the University of AND REMOTE SENSING in 2008–2014 and of the Remote Sensing Technology Michigan, Ann Arbor, MI, USA, in 1984 and 1988, and Application from 2011. Also, he is the Lead Guest Editor for special respectively. GPS Solutions Terrestrial, issues ( in 2005 and 2010, IEEE TGRS in 2008, He is currently an Associate Professor of the Atmospheric and Oceanic Sciences Advances in Meteorology in 2014, in 2014 Chien Hsin University of Science and Technology, and 2015, and Atmospheric Research and Remote Sensing in 2015). He was Taoyuan. His research interests include photovoltaic, a recipient of Annual Research Awards from NSC in 1998–2000, First Class microwave measurement, absorbers, and filters. Research Awards from NSC in 2004–2006, and NCU Outstanding Research Awards in 2004 and 2006–2008. He was awarded the “Contribution Award to FORMOSAT3 National Space Mission” by NSPO in 2006. He is a member Rong-Moo Hong was born in Taiwan in 1957. He of the American Geophysical Union, American Meteorological Society, and received the B.S. degree in physics from the Chung- International Association of Hydrological Sciences and a senior member of Cheng Institute of Technology, Taoyuan, Taiwan, the Institute of Electrical and Electronics Engineers, Inc. He was awarded in 1980, the M.S. degree in physics from National Honorary Life Member of the Korean Society of Remote Sensing in 2007. He Taiwan University in 1985, and the Ph.D. degree was elected Foreign Member of the Russian Academy of Engineering Sciences in photoelectron engineering from Ecole Central de in 2008. He was awarded Outstanding Alumni Awards by the University of Nantes, France, in 1996. Michigan Alumni Association in Taiwan and NSYSU in 2008. He was elected He is currently an Associate Professor of the corresponding member/member of the International Academy of Astronautics Chien Hsin University of Science and Technology, in 2009/2014. He was awarded lifetime honor Distinguished Professor of the Taoyuan. His research interests include photovoltaic, NCU in 2010. He was elected fellow of IET in 2015. LED and LD devices, and laser applications.