
In this study, the TC dataset is derived from Joint Typhoon Warning Center (JTWC) besttrack dataset, which contains the information of TC location and intensity from 2000 to 2018 over the WNP basin at a 6h interval. The TC surrounding environment is derived from NCEP final analysis data in a 6h interval with a horizontal resolution of 1° × 1°. The NCEP Global Ocean Data Assimilation System (GODAS) pentad dataset is employed to calculate the OHC (Saha et al., 2006). The GODAS has a constant zonal resolution of 1° and meridional grid of 1/3°, and has 40 levels with a 10m resolution in the upper 200 m.

The maximum wind speed (MWS) of TC is chosen as the intensity metric. The selected samples in this study cover all TCs reaching the intensity of tropical storm (MWS exceeds 17 m s^{−1}). As the time interval of JTWC dataset is 6 h, to make full use of the time resolution of the data, the IR of TC is defined as the average of 12 and 24h intensity change rate:
$$ {\rm{IR}}(t) = \frac{{[V(t + {\text{6}})  V(t  {\text{6}})] \times 2 + [V(t + {\text{12}})  V(t  {\text{12}})]}}{2} , $$ (1) where IR is the intensification rate, and V is the maximum wind speed. The IR is evaluated by using the central difference. The number of eligible samples is 6307. Figure 1 displays the probability density function of IR in WNP basin during 2000–2018. It is similar to the global probability density function during 1982–2009 (Bhatia et al., 2019). As shown in Fig. 1, 24h intensity changes of TCs are generally normally distributed and are mainly concentrated in the interval of −20 to 20 knots. There are more intensifying cases than weakening cases, because most of the weakening cases happen when a TC approaches land but such areas are relatively small compared to the open ocean. The TC has enough time to develop over the ocean and decays quickly when it approaches land.
Figure 1. Probability density function (PDF) of the intensification rates [knots (24 h)^{−1}] for TCs in WNP during 2000−2018.
In Kaplan and DeMaria (2003), the RI was defined as the maximum surface wind speed increase of 30 knots over 24h period. In this study, all cases are categorized into five groups in terms of their IR: RI [IR ≥ 30 knots (24 h)^{−1}]; SI [slowly intensifying, 10 ≤ IR < 30 knots (24 h)^{−1}]; N [neutral, −10 ≤ IR < 10 knots (24 h)^{−1}]; SW [slowly weakening, −30 ≤ IR < −10 knots (24 h)^{−1}]; and RW [rapidly weakening, IR < −30 knots (24 h)^{−1}]. Table 1 shows the number and proportion of each group of samples. From the perspective of climate statistics, the probability of TC RI over the WNP is about 13.6%. Figure 2 shows the geographic distribution of all five groups of IRs represented by different colors. Intensifying cases (SI and RI) are located mostly equatorward of 20°N and weakening cases (SW and RW) are mainly located in the northwest of intensifying cases. Neutral cases (N) are more evenly distributed among weakening and intensifying groups. As the occurrence of TC RI has a clear correlation with the TC location, we use the difference between longitude and latitude (Lon − Lat) to describe this geographical distribution feature. The larger the Lon − Lat value, the more the position of the TC is to the southeast. Statistically significant differences at the 95% confidence level are found among different IR groups for the Lon − Lat.
Group IR [knots (24 h)^{−1}] Case number Percentage (%) RI IR ≥ 30 856 13.6 SI 10 ≤ IR < 30 1809 28.7 N −10 ≤ IR < 10 2018 32 SW −30 ≤ IR < −10 1144 18.1 RW IR < −30 480 7.6 All 6307 100 Table 1. Categories of TC intensity change rate [knots (24 h)^{−1}], definitions, sample sizes, and percentage for the WNP basin from 2000 to 2018

Before further analysis, an 11day running mean is applied to the reanalysis data to extract the low frequency environment. The selection of environmental atmospheric and oceanic variables is subjective according to factors that are well known to affect the TC intensity change. In this study, the chosen variables are SST, temperature, relative vorticity, divergence, relative humidity, specific humidity, vertical velocity, geographical distribution (Lon −Lat), OHC, vertical wind shear, and vertical zonal wind shear. Briefly, the SST, temperature, relative vorticity, divergence, relative humidity, specific humidity, and vertical velocity are calculated by the area average of corresponding low frequency fields within a range of 10° × 10° at the TC location. The Lon−Lat is the difference between the longitude and latitude of the TC center. The OHC is defined as the lowfrequency sea potential temperature integrated vertically downward at a depth of 300 m within a range of 10° × 10° at the TC location. The vertical wind shear (the vertical zonal wind shear) is calculated by subtracting 850hPa total (zonal) wind speed from 200hPa total (zonal) wind speed, averaged within a range of 10° × 10° at the TC center.
However, not all environmental fields are critical to the TC RI. In order to figure out the key environmental variables associated with the TC RI, our strategy is carried out in two steps. First, for variables that we list above, the significance t test is employed to determine whether the difference between the RI group and other groups is significant. It is worth mentioning that TCs in the RW group are generally close to the land and terrain areas (Fig. 2). The IR in this group is mainly affected by surface land and terrain friction. Therefore, the environmental field difference between RI and RW is not considered when selecting RI predictors. Figure 3 shows the temperature profile difference between the RI group and other groups as an example. The red dot at each vertical level indicates that the temperature difference at that level is statistically significant at the 95% confidence level. It can be seen that the temperature difference between the RI and SW groups is the most obvious (Fig. 3c). In addition, according to the distribution of red dots and the difference value, significant difference appears in the bottom and upper layers of the atmosphere. This result demonstrates that the more unstable atmospheric condition is favorable for the growth of the TC intensity, which makes a larger probability of RI. Notably, the differences of lowfrequency environmental vorticity and divergence fields between the RI group and other groups are statistically insignificant at all vertical levels (figure omitted). This is consistent with the result of Kaplan and DeMaria (2003). They pointed out that the difference of areaaveraged 850hPa vorticity between the RI and nonRI groups is not significant. By using the method of significance test, following variables are primarily selected as key environmental variables affecting the RI: SST (T_{S}), 200hPa temperature (T_{0}), 500hPa relative humidity (RH500), 700hPa specific humidity (SH700), 400hPa vertical velocity (ω400), vertical zonal wind shear (VUS), vertical wind shear (VWS), OHC, and Lon − Lat.
Figure 3. Differences (∆T) between the lowfrequency temperature composites of the RI and (a) SI, (b) N, (c) SW from 2000 to 2018 within a 10° × 10° area centered at the TC center. Red dot indicates that the difference is statistically significant at 95% confidence level.
Next, the box difference index (BDI) method is applied on the variables passing the significance test. BDI is a nondimensional number used to describe the difference between two sets of data (Peng et al., 2012). The BDI between two groups is defined as:
$$ {\rm BDI} = \frac{{{M_{\rm A}}  {M_{\rm B}}}}{{{\sigma _{\rm A}} + {\sigma _{\rm B}}}}, $$ (2) where M and σ denote the mean and standard deviation of the variable, and the subscripts “A” and “B” represent different groups of the same variable (e.g., the SST in RI and SW groups). The larger the difference between the mean and the smaller the sum of the standard deviation between two groups, the larger the BDI, and the variable is more instinctively different between two groups. Although different variables have different units, the relative importance to the TC RI can be revealed by the ranking of BDI. Figure 4 shows the BDI of key environmental variables between the RI and other groups. As both T_{S} and T_{0} show large differences (Figs. 4a, b), a difference between SST and temperature at 200 hPa (T_{S} − T_{0}) actually has the largest BDI. This parameter reflects the atmospheric instability and is the major contributor to the TC maximum potential intensity (Emanuel, 1986; Holland, 1997). According to the ranking of BDI, six variables are chosen as TC RI predictors: T_{S} − T_{0}, OHC, VUS, Lon − Lat, ω400, and RH500.

In Section 2, six variables (T_{S} − T_{0}, OHC, VUS, Lon −Lat, ω400, and RH500) are chosen as TC RI predictors by using BDI method. Here, the least squares method is employed to fit the relationship between the environmental field and the IR. A statistical model for the estimated intensification rate of TC is constructed with options considering or not considering initial TC intensity, and they are named as IR_{en} and IR_{e}, respectively. In the form of exponential multiplication, IR_{e} is defined as:
$$\begin{aligned}[b] {\rm {IR_e}} & = {({T_{\rm S}}  {T_0})^a} \times {\rm OHC}^b \times {\rm RH500}^c \times {\text{ω}} {400^d} \\ & \times {\rm VUS}^e \times {({\rm Lon  Lat})^f} ,\end{aligned} $$ (3) where coefficients a–f can be determined based on the least squares method. For convenience, natural logarithm will be applied at both sides of Eq. (3). Before applying logarithm operations, it is necessary to adjust the value of predictors and IR into nondimensional numbers and ensure that they are all positive. If each term at the righthand side of Eq. (3) (e.g., OHC^{b}) is less than 1, the final product will be less than 1. This implies that all predictors are not conducive to the TC intensification at this time so that TC intensity will decrease. On the contrary, if each term at the righthand side of Eq. (3) is larger than 1, TC intensity tends to increase.
Prior to applying natural logarithm to Eq. (3), a procedure is applied to make sure that the values of all nondimensional terms at both sides of Eq. (3) are positive. First, we calculate the first quartile and the third quartile for each predictor in the RI group. Next, we normalize the predictors by dividing each predictor with the first quartile. A special treatment needs to be carried out for the vertical velocity and vertical zonal wind shear as they are negative. By adding a maximum absolute value in the two terms before the normalization, we ensure that the two terms are both positive. Given the range of IR from −100 to 100 (Fig. 1), we simply normalize this lefthand side term of Eq. (3) by adding 100 to its original value and then dividing 100 so that the nondimensional value is ranging between 0 and 2. Below are specific treatments for each term in Eq. (3):
$$\begin{aligned} & {\rm IR}^* = 0.01 \times {\rm IR} + {\text{1}}, \quad {\rm IR_e^* } = 0.01 \times {\rm IR_e} + {\text{1}} \\ & {({T_{\rm S}}  {T_0})^*} = ({T_{\rm S}}  {T_0})/{\text{80}} \\ & {\rm OHC}^* = {\rm OHC}/7{\text{673}} \\ & {\rm RH}{500^*} = {\rm RH500}/50 \\ & {\rm {\text{ω}}} {400^*} = ({\text{ω}} 400 + 0.3)/0.27 \\ & {\rm VUS}^* = ({\rm VUS} + 20)/22 \\ & {({\rm Lon  Lat})^*} = ({\rm Lon  Lat})/1{\text{12}} \end{aligned} . $$ (4) After the normalization procedure, natural logarithm is applied at both sides of Eq. (3). Thus, we have
$$\begin{aligned}[b] \ln {\rm {IR_e^*}} = & a \ln {({T_{\rm S}}  {T_0})^*} + b \ln {\rm OHC}^* + c \ln {\rm RH}{500^*} \\ + & \,\,d \ln {\text{ω}} {400^*} + e \ln {\rm VUS}^* + f \ln {({\rm Lon  Lat})^*} . \end{aligned} $$ (5) The least squares method is further applied to make the residual term between the fitted IR_{e} and observed IR as small as possible. Mathematically, it is expressed as:
$$ I = \sum\limits_{n = 1}^N {{{( {\ln {\rm IR}^*}  \ln {\rm IR_e^*}} )}^2} \to \min , $$ (6) where N (= 6307) is the sample size of the training data. After calculating partial derivatives with respect to a–f, the solution of Eq. (6) can be transformed into following sixelement linear equations:
$$ {\left( \begin{array}{*{20}{l}} &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({\Delta}T)}^*} \times \ln {{({\Delta}T)}^*}} } &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({\Delta}T)}^*} \times \ln C^*}} &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({\Delta}T)}^*} \times \ln q}^* } & {\displaystyle\sum\limits_{n = 1}^N {\ln {{({\Delta}T)}^*} \times {\text{ω}}}^* } &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({\Delta}T)}^*} \times \ln {U^*}}} &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({\Delta}T)}^*} \times \ln {{({\rm {\Delta} \varphi })}^*}} } \\ &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({\Delta}T)}^*} \times \ln C^*}} &{\displaystyle\sum\limits_{n = 1}^N {\ln {C^*} \times \ln {C^*}} } &{\displaystyle\sum\limits_{n = 1}^N {\ln q^* \times \ln {\rm {C}^*}} } & {\displaystyle\sum\limits_{n = 1}^N {{\text{ω}}^* \times \ln C^*}} &\displaystyle\sum\limits_{n = 1}^N {\ln U^*} \times \ln {C^*} &{\displaystyle\sum\limits_{n = 1}^N {\ln C^*} \times \ln {{({\rm {\Delta} \varphi})}^*}} \\ &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({\Delta}T)}^*} \times \ln q^*} } &{\displaystyle\sum\limits_{n = 1}^N {\ln q^* \times \ln C^*}} &{\displaystyle\sum\limits_{n = 1}^N {\ln q^* \times \ln q^*} } & {\displaystyle\sum\limits_{n = 1}^N {{\text{ω}}^* \times \ln q^*} } &{\displaystyle\sum\limits_{n = 1}^N {\ln U^*} \times \ln q^*} &{\displaystyle\sum\limits_{n = 1}^N {q^* \times \ln {{({\rm {\Delta}\varphi})}^*}} } \\ &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({\Delta}T)}^*} \times {\text{ω}}^*} } &{\displaystyle\sum\limits_{n = 1}^N {{\text{ω}}^* \times \ln C^*}} &{\displaystyle\sum\limits_{n = 1}^N {{\text{ω}}^* \times \ln q^*} } & {\displaystyle\sum\limits_{n = 1}^N {{\text{ω}}^* \times {\text{ω}}^*} } &{\displaystyle\sum\limits_{n = 1}^N {\ln {U^*} \times {\text{ω}}^*} } &{\displaystyle\sum\limits_{n = 1}^N {{\text{ω}}^* \times \ln {{({\rm {\Delta} \varphi})}^*}} } \\ &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({\Delta}T)}^*} \times \ln {U^*}} } &{\displaystyle\sum\limits_{n = 1}^N {\ln {U^*} \times \ln C^*}} &{\displaystyle\sum\limits_{n = 1}^N {\ln {U^*} \times \ln q^*} } &{\displaystyle\sum\limits_{n = 1}^N {\ln {U^*} \times {\text{ω}}^*} } &{\displaystyle\sum\limits_{n = 1}^N {\ln {U^*} \times \ln {U^*}} } &{\displaystyle\sum\limits_{n = 1}^N {{U^*} \times \ln {{({ {\Delta} \varphi})}^*}} } \\ &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({\Delta}T)}^*} \times \ln {{({{\Delta}\varphi})}^*}} } &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({{\Delta} \varphi})}^*} \times \ln C^*}} &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({{\Delta}\varphi})}^*} \times \ln q^*} } &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({{\Delta}\varphi})}^*} \times {\text{ω}}^*} } &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({{\Delta} \varphi})}^*} \times \ln {U^*}} } &{\displaystyle\sum\limits_{n = 1}^N {\ln {{({{\Delta} \varphi})}^*} \times \ln {{({{\Delta} \varphi})}^*}} } \end{array} \right) \left( \begin{gathered} \begin{array}{*{20}{l}} a \\ b \\ c\\ d\\ e \end{array} \\ { \;\;{f}} \\ \end{gathered} \right) = \left( \begin{gathered} \begin{array}{*{20}{l}} {\displaystyle\sum\limits_{n = 1}^N {\ln {{({\Delta}T)}^*} \times \ln {\rm IR}^*}} \\ {\displaystyle\sum\limits_{n = 1}^N {\ln C^*} \times \ln {\rm IR}^*} \\ {\displaystyle\sum\limits_{n = 1}^N {\ln {q^*} \times \ln {\rm IR}^*}} \\ {\displaystyle\sum\limits_{n = 1}^N {\ln {\text{ω}}^* \times \ln {\rm IR}^*}} \\ {\displaystyle\sum\limits_{n = 1}^N {\ln U^*} \times \ln {\rm IR}^*} \end{array} \\ \displaystyle\sum\limits_{n = 1}^N {\ln {{({{\Delta} \varphi})}^*} \times \ln {\rm IR}^*} \\ \end{gathered} \right) ,} $$ (7) where ∆T = T_{S} − T_{0}, C = OHC, q = RH500, ω = ω400, U = VUS, and ∆φ = Lon − Lat. By applying all 6307 sample data into Eq. (7), one may obtain the solution for coefficients a–f. Therefore, the probability model of TC RI can be finally written as:
$$ {\rm IR_e} = \Bigg[ {{\left( {\frac{{{T_{\rm S}  {T_0}}}}{{{\text{80}}}}} \right)}^3} \times {{\left( {\frac{{\rm OHC}}{{7{\text{673}}}}} \right)}^{0.1}} \times {{\left( {\frac{{\rm RH500}}{{50}}} \right)}^{0.0{\text{8}}}} \times {{\left( {\frac{{{\text{ω}} 400 + {\text{0}}{\text{.3}}}}{{{\text{0}}{\text{.27}}}}} \right)}^{  0.{\text{02}}}} \times {{\left( {\frac{{\rm VUS + 20}}{{22}}} \right)}^{  0.1{\text{3}}}} \times {{\left( {\frac{{\rm Lon  Lat}}{112}} \right)}^{0.{\text{35}}}}  1 \Bigg] \times 100 . $$ (8) Figure 6 shows the performance of the probability model of TC RI by comparing the relationship between IR_{e} and observed IR based on all cases from 2000 to 2018. It can be seen that the RI probability is getting larger with the increase of IR_{e}. The correlation coefficient between IR_{e} and IR is 0.523 and is significant at the 99% confidence level. In order to further assess the skill of the probability model, the probability of RI in different IR_{e} intervals is calculated (Table 2). When the environmental condition is not conducive to the occurrence of RI (IR_{e} < 0), the probability of RI is very small, about 4.8%. The probability of RI is over 20% when the surrounding environment is good.
Figure 6. Scatter diagrams for IR_{e} calculated by the environmental field versus the IR derived from the JTWC. Different colors represent different TC IR groups. The solid black line is the regression line with a slope of 0.98. The correlation coefficient between IR_{e} and IR is 0.523. The correlation is significant at 99% confidence level.
IR_{e} Case number RI number RI probability (%） < 0 3219 156 4.8 [0, 10) 1636 325 19.9 [10, 20) 1076 296 27.5 ≥ 20 376 79 21 Table 2. The probability of TC RI in different IR_{e} intervals from 2000 to 2018
The forecasting skill of the probability model IR_{e} is evaluated by calculating the Brier score of the forecast result (BS; Wilks, 2005):
$$ {\rm BS} = \frac{1}{N}\sum\limits_{t = 1}^N {{{({f_t}  {o_t})}^2}} , $$ (9) where N represents the total number of the samples, f is the probability of RI calculated by the model IR_{e}, and o is the observed situation (0 represents no RI is observed and 1 represents the occurrence of RI). Therefore, the closer the BS value is to 0, the more skillful the IR_{e} prediction model is. Then, the BS value of IR_{e} is compared with the climatological probability of RI to obtain the Brier skill score (BSS; Wilks, 2005):
$$ {\rm BSS}{\text{ = }}\left( {{\text{1 − }}\frac{{\rm BSM}}{{\rm BSC}}} \right) \times {\text{100}}, $$ (10) where BSM is the Brier score of the probability model IR_{e} and BSC is the Brier score of the climatological forecasts. When calculating the BSC, the probability of RI (f) in Eq. (9) is assumed to be the value of climatological probability of RI. Therefore, the positive value of BSS means that the IR_{e }model is skillful relative to the climatology, vice versa. Figure 7 shows that the BSS value of IR_{e }is 7.3%. This indicates that the RI probability model IR_{e} is skillful.
Figure 7. The skill of two TC RI probability models (IR_{e} and IR_{en}) relative to climatology for dependent samples from 2000 to 2018 (gray bar) and independent samples from 2019 to 2020 (blue bar). The difference between IR_{en} and IR_{e} model is that the TC intensity is taken into account or not.
However, it should be pointed out that when the TC intensity is very strong or reaches its maximum potential intensity, even if the surrounding environment is conducive to the intensification, it is still hard for the TC to have RI. On the contrary, in the early stage of the TC development, the influence of environment on the TC IR can be more significant (Gray, 1998). In this study, some RI TC environmental conditions are similar to the SI TC environmental characteristics, such as RH500 and VUS (Figs. 5c, g). The IR of TC is not only related to the surrounding environment, but also to the TC structure (Schubert and Hack, 1982; Montgomery et al., 2006; Stevenson et al., 2018). Since the horizontal resolution of reanalysis data is not high enough to meet the need of TC internal structure study, we divide all samples into four groups according to the MWS of TC and reestablish the TC RI probability model based on Eq. (7):
$$ {\begin{aligned}[b] & {\rm IR_{en}} = \\ & \left\{ {\begin{array}{*{20}{c}} {\left[ {{{\left( {\dfrac{{{T_{\rm S}}  {T_0}}}{{{\text{80}}}}} \right)}^{{\text{1}}{\text{.3}}}} \times {{\left( {\dfrac{{{\rm OHC}}}{{7{\text{673}}}}} \right)}^{0.{\text{08}}}} \times {{\left( {\dfrac{{{\rm RH500}}}{{50}}} \right)}^{0.0{\text{6}}}} \times {{\left( {\dfrac{{\text{ω} 400 + {\text{0}}{\text{.3}}}}{{{\text{0}}{\text{.27}}}}} \right)}^{  0.{\text{06}}}} \times {{\left( {\dfrac{{{\rm VUS} + 20}}{{22}}} \right)}^{  0.{\text{08}}}} \times {{\left( {\dfrac{{{\rm Lon  Lat}}}{{1{\text{12}}}}} \right)}^{0.{\text{2}}}}  1} \right] \times 100} & {34 \leqslant {V_{\max }} < 60} \\ {\left[ {{{\left( {\dfrac{{{T_{\rm S}}  {T_0}}}{{{\text{80}}}}} \right)}^{3.5}} \times {{\left( {\dfrac{{{\rm OHC}}}{{7{\text{673}}}}} \right)}^{0.08}} \times {{\left( {\dfrac{{{\rm RH500}}}{{50}}} \right)}^{0.02}} \times {{\left( {\dfrac{{\text{ω} 400 + {\text{0}}{\text{.3}}}}{{{\text{0}}{\text{.27}}}}} \right)}^{  0.1{\text{2}}}} \times {{\left( {\dfrac{{{\rm VUS} + 20}}{{22}}} \right)}^{  0.18}} \times {{\left( {\dfrac{{{\rm Lon  Lat}}}{{1{\text{12}}}}} \right)}^{0.4}}  1} \right] \times 100}& {60 \leqslant {V_{\max }} < 90} \\ {\left[ {{{\left( {\dfrac{{{T_{\rm S}}  {T_0}}}{{{\text{80}}}}} \right)}^{5.9}} \times {{\left( {\dfrac{{{\rm OHC}}}{{7{\text{673}}}}} \right)}^{0.43}} \times {{\left( {\dfrac{{{\rm RH500}}}{{50}}} \right)}^{0.04}} \times {{\left( {\dfrac{{\text{ω} 400 + {\text{0}}{\text{.3}}}}{{{\text{0}}{\text{.27}}}}} \right)}^{  0.{\text{09}}}} \times {{\left( {\dfrac{{{\rm VUS} + 20}}{{22}}} \right)}^{  0.16}} \times {{\left( {\dfrac{{{\rm Lon  Lat}}}{{1{\text{12}}}}} \right)}^{0.2{\text{5}}}}  1} \right] \times 100}&{ 90 \leqslant {V_{\max }} < 120} \\ {\left[ {{{\left( {\dfrac{{{T_{\rm S}}  {T_0}}}{{{\text{80}}}}} \right)}^{5.5}} \times {{\left( {\dfrac{{{\rm OHC}}}{{7{\text{673}}}}} \right)}^{0.48}} \times {{\left( {\dfrac{{{\rm RH500}}}{{50}}} \right)}^{0.0{\text{8}}}} \times {{\left( {\dfrac{{\text{ω} 400 + {\text{0}}{\text{.3}}}}{{{\text{0}}{\text{.27}}}}} \right)}^{0.{\text{08}}}} \times {{\left( {\dfrac{{{\rm VUS} + 20}}{{22}}} \right)}^{  0.11}} \times {{\left( {\dfrac{{{\rm Lon  Lat}}}{{1{\text{12}}}}} \right)}^{0.{\text{54}}}}  1} \right] \times 100}& {{V_{\max }} \geqslant 120} \end{array}} \right. \end{aligned}.} $$ (11) The probability of RI in each IR_{en} interval is calculated as well (Table 3). When the environmental condition is poor (IR_{en} < 0), the probability of RI calculated by IR_{en} is similar to that of IR_{e}. However, when the environmental condition is conducive to the occurrence of RI (IR_{en} ≥ 20), the forecasting ability of the intensitydependent model IR_{en} is improved significantly. Compared with the IR_{e} model, when IR_{en} is greater than 20, IR_{en} can capture more RI samples from the total and the probability of RI reaches 40.9%, which is much higher than the climatological probability (13.6%). As a result, the BSS value of IR_{en} is 9.6%, which is more skillful than IR_{e} (Fig. 7).
IR_{en} Case number RI number RI probability (%) < 0 3206 159 5 [0, 10) 1578 247 15.7 [10, 20) 1090 273 25 ≥ 20 433 177 40.9 Table 3. The probability of TC RI in different IR_{en} intervals from 2000 to 2018
The results above show that the intensitydependent model IR_{en} is skillful relative to forecasts based on climatology for TC samples from 2000 to 2018 (Fig. 7). Further independent verification is carried out to show the ability of IR_{en} model in predicting RI occurrence. Two methods are applied to verify the performance of IR_{en}. First, a verification of the IR_{en} forecast is conducted based on the JTWC data and reanalysis data from 2019 to 2020. Second, the performance of the IR_{en} model on TC Chanthu (2021) is verified. After substituting the TC surrounding environmental field into the IR_{en} model, Table 4 shows the result of independent verification by using samples from 2019 to 2020. Although the 2yr sample size is small, the distribution of RI probability in different IR_{en} interval is similar with that in Table 3. The probability of RI is higher than 33% and 38% when IR_{en} is greater than 20 and 10, respectively. Based on samples from independent cases, the BSS value of IR_{e} and IR_{en} has been calculated again (Fig. 7). Both versions of the probability model are skillful relative to the climatology and IR_{en} has higher BSS. The result suggests that this intensitydependent model has the potential to provide useful information to forecasters. Figure 8 shows a typical recent TC case with RI occurred. The MWS of TC Chanthu increases from 30 to 58 m s^{−1} within 12 h (Fig. 8b). Based on the TC surrounding environment derived from the reanalysis data, we calculate IR_{en} and compare it with the time evolution of TC IR (Fig. 8c). Two curves of the observed IR and calculated IR_{en} have the same trend. During the duration of TC Chanthu, the RI process occurs three times and the model IR_{en} captures twice (take IR_{en} ≥ 20 as the signal for RI). Moreover, in the subsequent development process, there is no false alarm. This means that IR_{en} can capture the characteristic of the surrounding environment well and is a good indicator of IR.
IR_{en} Case number RI number RI probability (%) < 0 347 29 8.3 [0, 10) 229 39 17 [10, 20) 54 21 38.8 ≥ 20 15 5 33.3 Table 4. The probability of TC RI in different IR_{en} intervals based on the independent samples from 2019 to 2020
Figure 8. (a) The best track of TC Chanthu (2021) from 0000 UTC 7 to 1800 UTC 17 September at 24h intervals, (b) time series of the minimum sea level pressure (black line; hPa) and maximum wind speed (red line; m s^{−1}) of TC Chanthu, and (c) time series of the intensity change rate of TC Chanthu (black line) and the value of the intensitydependent IR_{en} model (red line).
Figure 9 evaluates the forecasting skill of the model IR_{en} comprehensively by calculating the probability of detection (POD), probability of false detection (POFD), and Peirce skill score (PSS). The POD is the fraction of the observed number of RI cases that were forecast correctly. The POFD is the ratio of the number of cases that an RI was forecast to occur but did not, divided by the total number of cases that RI did not occur. The PSS was applied in previous studies to show the performance of the linear weighting RI index, which is defined as the difference between the POD and POFD (Kaplan et al., 2010; Shu et al., 2012). In brief, the POD, POFD, and PSS of a perfect model should be 1, 0, and 1 respectively. In this study, the value of IR_{en} greater than 20 is considered as a threshold for TC RI for cases from 2000 to 2018 and TC Chanthu, and 10 for cases from 2019 to 2020 as the 2yr sample size is small. The POD of three groups is greater than 20%, and up to 66.7% for TC Chanthu. The POFD of IR_{en} is performed quite well, which is less than 8%. Shu et al. (2012) developed a linear weighting RI index (RII) for the WNP TC, and the averaged PSS of the RII is about 0.1. The PSS of IR_{en} is greater than 0.15 for these three groups of cases and up to 0.64 for TC Chanthu, indicating that IR_{en} has skill for the WNP TC.
Group  IR [knots (24 h)^{−1}]  Case number  Percentage (%) 
RI  IR ≥ 30  856  13.6 
SI  10 ≤ IR < 30  1809  28.7 
N  −10 ≤ IR < 10  2018  32 
SW  −30 ≤ IR < −10  1144  18.1 
RW  IR < −30  480  7.6 
All  6307  100 