## 1. Introduction

The primary purpose of this study is to develop a flexible algorithm based on fuzzy logic concepts to provide early assessment and prediction of afternoon thunderstorms (TS_{A}s) in northern Taiwan. Convective weather phenomena, and especially intense thunderstorms, occupy a very important part of the Taiwan Central Weather Bureau’s (CWB) forecast and warning responsibilities. The prediction of thunderstorms (if, where, and when they will occur) is one of the most difficult tasks for a weather forecaster. It is particularly challenging in Taiwan, which is a mountainous island characterized by the Central Mountain Range (CMR), which runs the length of the island in a north-northeast–south-southwest orientation at an average height of about 2 km with peaks over 4 km (Fig. 1). Mountains not only generate local circulation patterns but also interact with large-scale low-level winds to produce localized convective storms (Akaeda et al. 1995; Li et al. 1997). Numerous investigators (e.g., Fuelberg and Biggar 1994; Huntrieser et al. 1997) have shown several preconvective large-scale indices useful for thunderstorm prediction based on thermodynamic and kinematic features such as stability, wind shear, and relative humidity. Thunderstorm climatologies based on reflectivity and lightning data have also been useful for improving the time and space accuracy of thunderstorm predictions (Shafer and Fuelberg 2006; Saxen et al. 2008). However, the challenge still exists to accurately forecast thunderstorms on small spatiotemporal scales. The challenge is amplified by sparse observational data (Weckwerth 2000) and the limited capability of numerical models to provide accurate forecasts on small time and space scales (Lynn et al. 2001).

A previous paper by Lin et al. (2011), utilizing many of the same data as in this paper, examined the characteristics and evolution of afternoon thunderstorms over Taiwan under weak synoptic-scale forcing (synoptically undisturbed). As will be described later, it was found that sea-breeze flows and anabatic–katabatic flows play an important role in moistening the boundary layer for the initiation of TS_{A}s; similar findings have been reported by Johnson and Bresch (1991) and Chen and Li (1995). The initial convective activity is frequently enhanced when thunderstorm outflows collide with terrain, sea breezes, or other outflows (Szoke et al. 1985; Wilson and Schreiber 1986; Jou 1994). Studies by Johnson and Bresch (1991), Lin and Kuo (1996), and Chen et al. (2001) have shown the maximum storm activity in Taiwan peaks during the afternoon along the slopes of the mountains rather than at higher elevations farther inland. Based on 4 yr of radar and lightning datasets, Lin et al. (2011) showed that the maximum thunderstorm frequencies were during afternoon hours and located in a narrow strip along the lower western slopes of the mountains, parallel to the orientation of the mountains. Lin’s study also found some preconvective features associated with the occurrence of TS_{A}s from both surface and sounding observations that will be utilized in this present study.

Although the aforementioned studies show the capabilities of individual preconvective factors to assess the occurrence of TS_{A}s, it is expected that the accuracy of temporal and spatial predictions would be improved if local thermodynamic and kinematic factors were simultaneously integrated with thunderstorm climatology. In practice, an adequate tool to straightforwardly integrate preconvective factors is essential and important to quantitatively provide the forecast information in real-time operations.

Many methodologies have been proposed that go beyond those already used for nowcasting thunderstorms to allow quantitative precipitation forecasts (QPF). These include, for example, limited-area NWP models, analog forecasting (e.g., Obled et al. 2002; Panziera et al. 2011), Lagrangian extrapolation (e.g., Turner et al. 2004; Germann et al. 2006), and blending of NWP model output with radar (Atencia et al. 2010). Another approach is to use fuzzy logic, which is conceptually a very simple and valuable tool that can be used to avoid the difficulties of establishing complex relationships between weather systems and environmental features. The technique has the potential to be applied in linear and nonlinear control systems (e.g., Klir and Folger 1988; Kosko 1992) and has the ability to perform probabilistic analyses by using temporal and spatial information (e.g., Mueller et al. 2003; Cho et al. 2006). Fuzzy logic approaches have been extensively utilized in atmospheric science, including radar-based cloud-particle typing (Vivekanandan et al. 1999; Liu and Chandrasekar 2000), measures of the boundary layer depth (Bianco and Wilczak 2002), nonprecipitating echo detection and mitigation (Berenguer et al. 2006; Cho et al. 2006; Gourley et al. 2007), bounded weak-echo region detection (Lakshmanan 2000; Pal et al. 2006), and lightning prediction (Kuk et al. 2012). The fuzzy logic technique is also adapted for nowcasting thunderstorms in real-time operations, for example the Auto-Nowcast System (ANC; Mueller et al. 2003), which is a data fusion system composed of several feature detection algorithms that ingest all available operational datasets and provide automated detection of features and precursor signatures relevant to thunderstorm initiation, growth, and decay.

The fuzzy approach developed in this study is similar to that of Berenguer et al. (2006). However, the membership functions for the fuzzy logic are objectively derived from the statistics of observed TS_{A} characteristics, as stated in Lin et al. (2011). The fuzzy methodology is described in section 2, as well as an explanation of the predictand and identification of predictors from past studies. In section 3 the frequency distribution and conditional probability functions for 28 predictors (except persistence) are developed. In section 4, the construction of the membership functions and associated weights for each predictor is described. Results and discussions for the performance of the proposed fuzzy logic algorithm on independent data and a simplified fuzzy logic approach in section 5 are followed by conclusions of results in section 6.

## 2. Methodology

Data, including surface stations and sounding observations during the warm season (May–October) from 2005 to 2008 analyzed in Lin et al. (2011), are used to highlight the preconvective characteristics of TS_{A} and construct the algorithms to quantitatively assess the probability of occurrence of TS_{A}. The fuzzy logic algorithm developed in this paper to forecast TS_{A} utilizes sounding and surface station observations. The occurrence of TS_{A}s is based on radar observations. The sounding and surface station data are analyzed to identify predictors prior to thunderstorm initiation (preconvective predictors). The membership functions for the fuzzy logic are derived from the individually observed pre-TS_{A} parameters and the proper weights for these membership functions are also determined based on maximizing the critical success index (CSI) skill score (Donaldson et al. 1975). The reason for using CSI skill scores was based primarily on the prevalent use of this skill score in the broad literature for assessing the accuracy of thunderstorm forecasts (e.g., Huntrieser et al. 1997; Mitchell et al. 1998; Mueller et al. 2003; Mazur et al. 2009). There are two observed datasets collected in the analysis, one (*calibration* dataset, May–October 2005–08) is used to develop the fuzzy logic algorithm, and the other (*validation* dataset, May–October 2009–10) is used for the performance evaluation of the fuzzy logic algorithm on the prediction of TS_{A}s.

### a. Predictand

In this study, in order to investigate the preconvective predictors that were primarily related to the local geography (orographic effect and local circulation, etc.), cases influenced by synoptic-scale perturbations (fronts, tropical cyclones, etc.) located within 1.5° latitude or longitude of Taiwan were eliminated based on examination of daily weather maps and weather outlooks issued by the CWB. Accordingly, from the 4 yr (2005–08) of warm season (May–October) data, 277 days out of a total of 736 were selected as synoptically undisturbed days.

The predictand for this study is the occurrence of radar reflectivity ≥40 dB*Z* over an area ≥10 km^{2} that persisted ≥30 min between 1200 and 2100 local standard time (LST) in northern Taiwan on a synoptically undisturbed day. Northern Taiwan in this study is defined by three regions, Taipei City, New Taipei City, and Keelung City, which are shown in Fig. 1. Radar data were collected from four operational Doppler radars whose observing coverage ranges cover all of Taiwan Island. The radars were installed by the CWB and the data have been available since 2001. The reflectivity observations from the individual radars were then combined to generate 3D reflectivity mosaic grids (Zhang et al. 2005). The mosaic grid has a spatial resolution of 0.0125° on the latitude–longitude coordinate system and a 10-min update cycle. Before the mosaic was developed, reflectivity observations were quality controlled (Chang et al. 2009) to remove nonprecipitation echoes. The complex terrain around the radars results in beam blockage in some directions and ground clutter in other directions and creates difficulties in using radar to characterize thunderstorms over the CMR and eastern Taiwan. Chang et al. (2009), following the procedures of others (Krajewski and Vignal 2001; Overeem et al. 2009), conducted a comprehensive study using the Taiwan radars and rain gauges to identify radar beams that are blocked or partially blocked by terrain. Chang’s study used radar reflectivity data from the four CWB radars and rainfall data from at least 370 rain gauges for the 3-yr period from 2005 to 2007. These data were used to identify the blocked beams and produce hybrid scans for each radar. Hybrid scans are a set of radar bins, from the lowest elevation angle, that do not have significant blockage or clutter (O’Bannon 1997; Maddox et al. 2002). In the current study, reflectivity observations are quality controlled by the hybrid scans constructed by Chang et al. (2009). These data were then used to identify and characterize TS_{A}s.

Of these 277 synoptically undisturbed days, 148 had TS_{A}s (TS_{A} days) in northern Taiwan and 127 did not (non-TS_{A} days), and there were no radar data for 2 days. Lin et al. (2011) documented that the hourly average rainfall for the 277 undisturbed days significantly increased during the afternoon (1200–2100 LST) and reached a maximum between 1500 and 1700 LST. Comparison of spatial distributions between reflectivity frequencies of ≥40 dB*Z* and cloud-to-ground (CG) lightning frequencies during 1200–2100 LST also showed close correspondence. These afternoon maxima suggest that the rainfall is associated with thunderstorms (Johnson and Bresch 1991; Lin and Kuo 1996; Chen et al. 2001). As mentioned above, the period of forecasting if TS_{A}s occurred in this study is during 1200–2100 LST.

### b. Predictors (fuzzy variables)

Predictors to be used in this study are partially based on the study by Lin et al. (2011), which used the same 4-yr dataset and definition of TS_{A} days. Surface station and radiosonde data from northern Taiwan were examined to explore possible differences between the 148 TS_{A} days and 127 non-TS_{A} days. Figure 1 shows the locations of the surface stations and sounding sites. Kinematic and thermodynamic differences between the TS_{A} and non-TS_{A} days extracted from the Lin et al. (2011) analysis are summarized in Table 1. The TS_{A} days have earlier onset times of sea breeze from the northwest coast and predominately northwesterly winds in the Taipei Basin in contrast to easterly wind on the non-TS_{A} days. It is also shown that the dewpoints at the northwest coast and inland are higher than along the northeast coast. Accordingly, the sea breeze moving in from the northwest facing coast down the Danshui River valley is particularly responsible for this favorable situation. The temperature distribution is higher during the mornings on the TS_{A} days at these three surface stations and one sounding site, as well as the dewpoint (or vapor pressure) observations. These differences indicate that moister and warmer air could be transported inland via sea breeze.

Differences between the 148 TS_{A} and 127 non-TS_{A} days in northern Taiwan for surface stations and the composite sounding from Panchiao.

Figure 2 contrasts morning surface wind and thermodynamic differences between the TS_{A} and non-TS_{A} days based on analysis from Lin et al. (2011). Two additional surface stations (Keelung and Sijhih, with black circle symbols in Fig. 1) installed by the Taiwan Environmental Protection Administration (EPA) have been added to improve on the earlier Lin et al. (2011) analysis. For mitigating the terrain effects, vapor pressures and wind vectors are derived by interpolating the spatial patterns from the surface stations and automatic meteorological stations whose elevations are below 300 m through the kriging method. Kriging has been used effectively to interpolate meteorological data (e.g., Earls and Dixon 2007; Mirás-Avalos et al. 2007). Figure 2a shows the maximum frequency of TS_{A} activity tended to occur parallel to the orientation of the mountains. The sea breeze from the northwest coast down the Danshui River valley converges in the Taipei Basin with the sea breeze from the northeast coast moving down the Keelung River valley. This sea-breeze pattern on the TS_{A} days apparently differs from that on the non-TS_{A} days (Fig. 2b). Therefore, significant anabatic flows (Esteban and Chen 2008) could be found on the windward side of the Snow Mountain Range (SMR; Fig. 1) that converge near the south of the Taipei Basin, which is consistent with the location of maximum frequency of occurrence for TS_{A} activity. Simultaneously, the different sea-breeze patterns caused the corresponding distributions of the vapor-pressure differences; it also indicates that increased moisture is being primarily transported inland via the sea breeze from the Danshui River valley, as was also stated by Chen et al. (2007).

Wind and thermodynamic parameters from the soundings launched daily from Panchiao at approximately 0800 LST (0000 UTC) were also compared between the TS_{A} and non-TS_{A} days (see Table 1). In Table 1, the composite sounding profiles were obtained for the 148 TS_{A} days and the 127 non-TS_{A} days (Lin et al. 2011). Before the composite soundings were combined, an interpolation at fixed levels was carried out in steps of 25 hPa except for standard levels (1000, 925, 850, 700, 500, 400, 300, 200, and 100 hPa). Significantly, there is no convective available potential energy (CAPE; Moncrieff and Miller 1976) in the composite sounding and more convective inhibition (CIN; Colby 1984) for the non-TS_{A} days. From the vertical profiles of averagely anomalous temperature and dewpoint temperature shown in Fig. 2, it is evident that the TS_{A} days are warmer and moister. The higher temperatures and moisture in the low layers would provide more instability and the greater moisture at midlevels would reduce the entrainment of dry air into growing cumulus (Chen et al. 2001; Zehnder et al. 2006). Similar results were also documented in Fuelberg and Biggar (1994).

Further results from Lin et al. (2011) of wind velocities in the lowest 6 km of the Panchiao sounding showed that the most common wind direction was southwest for the TS_{A} days, and average wind speeds at altitudes between 0 and 6 km on the TS_{A} days were weaker than on the non-TS_{A} days (Table 1). This generally agrees with the research stated by Carleton et al. (2008) who reported a greater influence of local land surface conditions on deep convection for the weaker flow days.

Based on information from Table 1 and previous studies (Table 2), a set of 29 predictors were selected to develop the fuzzy logic algorithm to forecast TS_{A} days. Since it is difficult to determine the importance of surface temperature to the occurrence of TS_{A}, relative humidity, which includes both the influence of temperature and dewpoint [*T* and *T _{d}* are in degrees Celsius and RH is in percent; e.g., Lawrence 2005; WMO 2008], is utilized. In addition, dewpoint and vapor pressure represent the water vapor content in the boundary layer. Water vapor plays an especially important role in mountainous areas via changes in precipitation and cloud amount (Richardson et al. 2003), which can be taken as the precipitable water at the surface level (WMO 1986). Under the conditions of constant pressure, dewpoint and vapor pressure are mutual functions by

*e*is the actual vapor pressure in hPa; e.g., WMO 2008; Markowski and Richardson 2010) derived from the Clausius–Clapeyron equation (e.g., Marvin 1909; Washburn 1924; Whipple 1927). Moreover, the dewpoint depression from sounding data represents not only the moist synoptic conditions but also implies the height of the lifting condensation level (LCL) from

_{s}*Z*is in meters; Lawrence 2005).

References for predictors used in the fuzzy logic approach.

As mentioned above, the 29 predictors used in constructing the fuzzy logic approach are 1) vapor pressure (VPRE), relative humidity (HUMD), wind direction (WDIR), and wind speed (WDSD) at surface stations Danshui, Taipei, and Keelung; 2) CAPE, dewpoint depression (*T* − *T _{d}*), and wind directions and speeds at standard levels 1000, 925, 850, 700, and 500 hPa from the Panchiao sounding; and 3) persistence (PRST), which assumes the same weather as the previous day.

In practice, the environmental conditions for the occurrence of thunderstorms could not be fully considered in the proposed fuzzy logic approach. Chen et al. (2009) have stated that the continuous occurrence of afternoon convection under weak synoptic forcing was not determined by any single factor but depended on the combination of multiple environment conditions. In the period of the 148 TS_{A} days, only 24 (16.2%) days were single TS_{A} days (i.e., there was no preceding or following TS_{A} day). For the non-TS_{A} days, the percentage of continuous undisturbed days was as high as 110 (86.6%) days. A similar finding (82.5%) was also reported by Chen et al. (2009). As mentioned previously, it is assumed that persistence could be an auxiliary factor to mitigate the uncertainties in the preconvective environment from sparse observations and the limited number of predictors.

### c. Fuzzy logic

The overall procedure of developing the fuzzy logic approach for forecasting Taiwan TS_{A}s is illustrated in Fig. 3. The more detailed explanations on each step will be stated in the following sections. Below, we outline the steps found in Fig. 3.

- Predictor distribution functions (step 2)—For each of the 28 predictors (step 1, except persistence), determine the preconvective time history of the frequency of predictor values (called predictor distribution functions, factor curves, or frequency distribution curves) separately for the TS
_{A}and non-TS_{A}days (section 3a and Fig. 4). - Conditional probability curves (step 3)—For the TS
_{A}days, convert the above predictor distribution functions to conditional probability curves with values between 0 and 1 (see section 3b and Fig. 5). Simultaneously, subtract the probability of TS_{A}days from 1 and that is the probability for the non-TS_{A}days. - Membership functions (step 4)—Convert the conditional probability curves for each predictor to membership functions (section 4a and Fig. 6), primarily a simple piecewise linear activity.
- Predictor weight—Determine weights for each of the predictor membership functions based on the weights that give the maximum skill score (CSI) for predicting TS
_{A}s. The weights are determined separately for the surface stations and sounding [step 5, see section 4b(1) and Tables 4 and 5]. Correspondingly, a set of three predictors, SOUNDING, STATION, and PERSISTENCE in steps 6 and 7 [see section 4b(2) and Fig. 7] was established to represent the large-scale environment (sounding predictors), local influences (surface stations predictors) and persistence. In the same way, the weights for these three predictors (step 8) were based on maximizing the skill score (CSI) for forecasting TS_{A}s. - Fuzzy forecast—From step 8, use both the individual predictor weights and three category weights to obtain a likelihood value for TS
_{A}. For this purpose, a likelihood of > 0.5 is considered a*yes*forecast for TS_{A}.

## 3. Predictor functions

Before determination of the fuzzy logic membership functions, the frequency and probability of TS_{A} for each predictor is determined. The purpose of the predictor functions is to convert numerical values of observed predictors during the morning to the probability of thunderstorms that afternoon or evening. For example, what is the probability of an afternoon or evening thunderstorm if the morning sounding has a CAPE of 500 m^{2} s^{2}?

### a. Predictor distribution functions

*X*(vapor pressure, wind direction, CAPE, etc.), conditional to the day type

_{k}*s*(TS

_{A}and non-TS

_{A}days) can be expressed as follow (Berenguer et al. 2006):where

*f*(

_{k,s}*x*) is the frequency; subscript

*k*indicates the different parameters;

*X*=

_{k}*x*and the day has been classified as type

*s*; and

*n*(day type

*= s*) is the total number of days classified as day type

*s*.

Figure 4 shows the sample frequency distribution curves for predictors on the TS_{A} and non-TS_{A} days from 0800 to 1200 LST. At coastal station Danshui, the vapor pressure (Fig. 4a) gradually increases with time on the TS_{A} days, but no significant temporal variation is found on the non-TS_{A} days. The wind direction at Danshui (Fig. 4b) shifts from south-southeasterly to northwesterly for both the TS_{A} and non-TS_{A} days; however, the shift is more distinct on the TS_{A} days, suggesting the passage of the sea-breeze front. The coastal station of Keelung (Fig. 4c) shows a wind shift from southwest to northeast during the morning, which also suggests the passage of a sea-breeze front that is more pronounced on the TS_{A} days. This suggests, as discussed in section 2b, the sea breeze passing Danshui and moving down the river valley toward the Taipei Basin combined with the sea breeze moving down the Keelung River valley often creates convergence (Fig. 2a) that likely plays an important role in the occurrence of TS_{A}s.

Figures 4d–g show the predictor distributions for the interior station at Taipei. The values of vapor pressure (Fig. 4d) on the TS_{A} days generally peaked and were higher than on the non-TS_{A} days. The relative humidity gradually decreased (Fig. 4e) with increasing temperature (not shown); however, the humidity values were higher on the TS_{A} days than on the non-TS_{A} days during the morning and early afternoon. Notably during the later morning the wind direction (Fig. 4f) on the TS_{A} days was northwesterly and it was east-northeasterly on the non-TS_{A} days. Further, most of the wind speeds on the TS_{A} days were smaller than 3 m s^{−1} (Fig. 4g), implying that weak wind speed might lead to longer exposure of air to the concentrated heating and encourage the development of thunderstorms (Tucker and Crook 2005).

Figures 4h–j show the predictor distributions derived from the Panchiao soundings launched at 0800 LST. The peak value for CAPE on the TS_{A} days was 1000 m^{2} s^{2}, which is higher than the 250 m^{2} s^{2} peak on the non-TS_{A} days (Fig. 4h). The frequencies for values greater than 1000 m^{2} s^{2} were higher on the TS_{A} days. The CAPE values for the non-TS_{A} days in Table 1 and Fig. 4h seem to be contradictory. However, in Table 1, the CAPE was computed from the composite sounding (see section 2b), while the CAPEs in Fig. 4h were calculated from individually daily *T* and *T _{d}* profiles. Regardless, they both are suggestive of the importance of higher CAPE for the TS

_{A}days. While CIN was not considered in this study, the composite sounding (Table 1) suggests it should be considered in the future. The TS

_{A}days have a higher frequency of southwest winds (Fig. 4i) with smaller dewpoint depression above the surface layer (Fig. 4j); this suggests less dry air entrainment during the developing stage of the cumulus clouds (Chen et al. 2001), which is conducive to the initiation of TS

_{A}s.

### b. Conditional probability curves

*X*, the conditional probability curve of a day is affected by a certain day type

_{k}*s*when

*X*

_{k}*= x*; the expression is (Berenguer et al. 2006)where

*p*(

_{k,s}*x*) is the probability and

*n*(

*X*

_{k}*= x*) is the total number of day type

*s*when

*X*

_{k}*= x*. The conditional probability curves of preconvective features and the variations can be calculated with different values of

*x.*In this study,

*p*(

_{k,s}*x*) indicates the conditional probability curve for the TS

_{A}days. On the contrary, 1–

*p*(

_{k,s}*x*) exhibits the conditional probability curve for the non-TS

_{A}days.

Sample conditional probability curves for the surface stations and sounding are shown in Fig. 5. For example, the wind direction at station Danshui on the TS_{A} days (Fig. 5b) has a high probability of shifting from southeast to northwest, indicating the passage of the sea breeze. Similarly, the conditionally probability curves for other surface stations indicate weather features for the TS_{A} days similar to those already discussed in section 3a.

The conditional probability curve for CAPE (Fig. 5h) shows that the probability of a TS_{A} day increases with increasing CAPE and reaches 1.0 when CAPE is greater than or equal to 1750 m^{2} s^{2}. The TS_{A} days also tended to have southwest winds (Fig. 5i) with low dewpoint depressions (Fig. 5j) above the surface layer, which would also be favorable for thunderstorm initiation.

Based on the conditional probability curves, favorable preconvective conditions for each predictor for each hour for the initiation and development of TS_{A} are summarized in Table 3. Favorable is defined as the probability above 0.5. These predictor values are then candidates for discriminating between the TS_{A} and non-TS_{A} days, but there are still some uncertainties to be applied alone in assessing the occurrence of TS_{A}*.* Accordingly, we have chosen to combine them in an algorithm based on fuzzy logic concepts.

Favorable (probability > 0.5) hourly preconvective conditions and predictors for TS_{A} occurrence using conditional probability curves.

## 4. The fuzzy logic approach

Based on fuzzy logic concepts (Berenguer et al. 2006), we now utilize the above conditional probability curves to develop an algorithm to discriminate the preconvective characteristics between the TS_{A} and non-TS_{A} days. The fuzzy logic approach is based on two basic processes: *fuzzification* and *composition* (Cornman et al. 1998). The essence of the fuzzification step is to determine membership functions for each predictor. The membership function is a simple piecewise linear functions based on the conditional probability curves developed in section 3b. The composition step determines the weights to be assigned to each membership function (e.g., Heske and Heske 1996; Liu and Chandrasekar 2000; Berenguer et al. 2006).

### a. Fuzzification (membership functions)

Conditional probability curves *p _{k,s}*(

*x*) quantify the degree of confidence that a day, when

*X*

_{k}*= x*, will be of a certain day type

*s*(TS

_{A}and non-TS

_{A}day). The shape of membership functions should be similar to the shape of conditional probability curves. Because there is some degree of subjectivity involved in producing membership functions, they are usually defined as simple piecewise linear curves. The most common membership functions are triangular, trapezoidal, piecewise linear, and have Gaussian distribution (e.g., Lakshmanan 2000; Berenguer et al. 2006).

Membership functions perform the conversion of measurement data into scaled, unitless numbers that indicate the correspondence or ‘‘membership level’’ of the data to the desired predictand of TS_{A}. Values of membership functions are converted into a likelihood *L _{k,s}*(

*x*) between 0 (lowest grade of membership) and 1 (highest grade of membership). In the fuzzy logic approach, the membership function is a crucial component and is usually defined or determined by knowledge and experience (Lakshmanan 2000; Shao 2000; Mueller et al. 2003; Berenguer et al. 2006). Figure 6 presents these membership functions subjectively determined by linearizing in a piecewise manner the conditional probability curves in Fig. 5. The membership functions for wind directions were kept as their conditional probability curves due to the difficulties in ascertaining the nonlinear relationships among varied wind directions.

_{k}### b. Composition (weights)

*Y*) for that day (TS

_{s}_{A}and non-TS

_{A}days). This is expressed by the equation below:where

*L*(

_{k,s}*x*) is the likelihood of membership function for each predictor

_{k}*x*and its

_{k}*W*is the membership function weight for type

_{k,s}*s*(TS

_{A}and non-TS

_{A}day). For each day, each of the 28 membership functions and the persistence rule were weighted and summed to obtain the final likelihood value

*Y*(steps 5–8 in Fig. 3). In other words, the forecast of TS

_{s}_{A}is obtained by converting the predictors to dimensionless likelihood values using the membership functions shown in Fig. 6, weighting the importance of each likelihood value to the forecast, and summing. The likelihood value of TS

_{A}varies from 0 to 1. For this study we used a likelihood value >0.5 as a forecast for a TS

_{A}day.

*h*,

*m*, and

*f*are defined as hits, misses, and false alarms, respectively. The hits and misses represent correct and incorrect predictions, while days incorrectly considered as TS

_{A}days are false alarms.

#### 1) Surface station and sounding weights

The practice here was to iterate the weights between 0 and 1 in steps of 0.1 for each predictor. The number of iterations would be impossibly large if all 29 predictors were considered at once (11^{29}). Thus, the weights are determined separately for the surface stations (11^{12}) and soundings (11^{16}) (step 5 in Fig. 3a). Table 4 shows the weights only for the surface station membership functions from 0800 to 1200 LST. The predictor with the highest consistent weight from hour to hour was the Taipei wind speed closely followed by the vapor pressure at Danshui. The weights for these two predictors dipped substantially at 1000 LST when the weights for wind direction and speed at Keelung spiked; as discussed previously, this may be an indication of the sea breeze passing Keelung and heading down the Keelung River valley. At the same time (1000 LST) the weight for the wind direction at Taipei also spiked, which is possibly an indication of the importance of the sea breeze reaching Taipei from the Danshui River valley. After 1000 LST, when the sea breezes have passed Danshui and Keelung, the weights for the vapor pressure at Taipei and Danshui increased, possibly indicating the importance of increased water vapor behind the sea-breeze front. Again as discussed in section 2b, moisture was mainly brought inland from the Danshui River valley. In addition, the possibility of TS_{A} is further increased by low wind speeds in the Taipei Basin, which allows for longer periods for heating of the air and increased instability (Tucker and Crook 2005). Low wind speeds also provide a better environment for development of a strong anabatic circulation along the mountain slopes that surround the Taipei Basin.

Hourly weights of surface station predictors for the occurrence of TS_{A}s and associated CSI values obtained from the fuzzy logic approach. Predictors with higher weights are set in boldface.

Table 5 shows the weights for the sounding membership functions at 0800 LST (0000 UTC), and surprisingly *T*–*T _{d}* at 1000 hPa received the most weight for the 16 sounding weights instead of CAPE, as expected. However, in Table 1 and Fig. 2, the importance of dewpoint depression in the 850–650-hPa layer from the composite sounding profiles for the TS

_{A}days was not captured by the weights from the individual TS

_{A}-day sounding (Table 5). This result implies that the large difference in the dewpoint depression in the 850–650-hPa layer between the TS

_{A}and non-TS

_{A}days possibly resulted from the variations of extreme dry environments on a number of the non-TS

_{A}days. The weights are substantially adequate to reflect the importance of the variations of dewpoint depression on the prediction of TS

_{A}s.

As in Table 4, but for sounding predictors.

#### 2) Weights for combined station, sounding, and persistence

In a combined step, a set of three predictors was established for determining the final likelihood of a TS_{A} day. The 29 predictors were reduced to a set of three (steps 6 and 7 in Fig. 3b), representing the large-scale environment, local-scale influences, and persistence. In this step, the 12 surface station predictors were combined into one predictor representing local influences (STATION), the 16 sounding predictors were combined into a second predictor representing the large-scale environment (SOUNDING), and persistence (PERSISTENCE) was the third predictor. Similar to the earlier procedure for determining weights for the surface station and sounding predictors, the weights for these three predictors (*W*_{STATION}, *W*_{SOUNDING}, *W*_{PERSISTENCE}) were iterated through all possible weight combinations (11^{3}) to determine which weights resulted in the highest skill score for forecasting TS_{A}s (step 8 in Fig. 3b). Figure 7 presents the weights for these three predictors (SOUNDING, STATION, and PERSISTENCE) for each forecast hour and the resulting skill scores are shown at the top of the figure. Again, a forecast for a TS_{A} day is a likelihood value > 0.5.

Notably the weights in Fig. 7 for the large-scale environment (sounding predictors) decreased over time, while the weights for the local-scale influences (station predictors) increase. Correspondingly, the CSI value slowly increases from 0.709 at 0800 LST to 0.760 at 1200 LST. This indicates the importance of the synoptic scale in establishing the general conditions necessary for thunderstorms (Adang and Gall 1989; Fuller and Stensrud 2000) and the importance of station predictors in identifying local influences that trigger the TS_{A}s. Doswell (1987) proposed that convective systems depended primarily on large-scale processes for developing a suitable thermodynamic structure, while mesoscale processes act mainly to initiate convection.

## 5. Results and discussions

### a. Evaluation

Table 6 shows a comparison of the fuzzy logic algorithm performance with the skill of CWB forecasts issued for the 4-yr period of interest. Most of the CSIs are greater than 0.7 with PODs > 0.9 and FARs < 0.25 by fuzzy algorithm forecasts. The CWB forecasters issue a daily forecast for TS_{A}s at 1030 LST, so these forecasts were compared with fuzzy algorithm forecasts at 1000 LST. The CSI value for the CWB forecasts was 0.602 compared to 0.719 for the fuzzy algorithm. There was a tendency for the forecasters to overforecast the occurrence of TS_{A}s.

Hourly skill scores of the fuzzy logic algorithm and CWB forecasters for the calibration dataset (May–October 2005–08).

An independent dataset from the warm seasons (May–October) from 2009 to 2010 was used to evaluate the performance of the fuzzy logic algorithm that was developed with the 2005–08 dataset. There are 118 days selected as synoptically undisturbed days. In northern Taiwan, 45 TS_{A} days are identified. Table 7 shows the hourly skill scores for the independent dataset from 0800 to 1200 LST. Most PODs are greater than 0.85, with FARs greater than 0.35 resulting in the CSI values in the range of 0.54–0.60, which are, not surprising, lower than those upon which the fuzzy algorithm was developed. Again the CSI for the CWB forecasters was lower (0.473) with a tendency to overforecast the number of the TS_{A} days relative to the fuzzy algorithm.

As in Table 6, but for the validation dataset (May–October 2009–10).

### b. Determination of an optimal threshold

The procedure to this point was to use a *Y _{s}* value (likelihood of a TS

_{A}day) > 0.5 as a yes forecast for a TS

_{A}day. We now consider if there is a more optimum threshold value for

*Y*that would increase the skill scores for forecasting TS

_{s}_{A}s. Figure 8 shows the results of this experiment where skill scores for forecasting TS

_{A}days are displayed as a function of likelihood threshold values between 0.5 and 0.7 and the forecast hour. The skill scores consistently increase to about 0.8 as the threshold value increases to about 0.54 at 1200 LST and to 0.57 at 0900 LST. The hourly optimal decision likelihood values based on the fuzzy logic algorithm’s performance for forecasting TS

_{A}s (Fig. 8) are listed in Table 8.

The hourly optimal decision likelihood values based on the fuzzy logic algorithm performance for forecasting TS_{A}s.

The 2009–10 dataset was used to test the more optimum threshold values of 0.57 at 0900 LST and 0.54 at 1200 LST, as shown in Fig. 8, that were obtained from the 2005–08 dataset. Consequently, skill scores increased from 0.542 (Table 7) to 0.612 at 0900 LST and from 0.594 (Table 7) to 0.660 at 1200 LST, respectively. The results indicate the desirability of using these higher threshold values in future operational use.

### c. Reduction of membership functions

It is recognized that there may be a high correlation between some of the predictors and some of the others would have low scientific basis for being a predictor of TS_{A}. Thus, a reduced set of predictors may result in a fuzzy algorithm with skill scores similar to the 29-predictor version. We illustrate this point by selecting 12 predictors (11 membership functions and the persistence rule) for the 1200 LST forecast time. The 11 membership functions selected were felt to best represent important large-scale environmental predictors from the sounding (stability and its impact on dry entrainment into cumulus clouds) and surface mesoscale predictors such as moist sea-breeze air from the Danshui and Keelung River valleys converging into the Taipei Basin. The results of this experiment are shown in Table 9.

The skill scores from all predictors (CSI-1, CSI-2, CSI-3, and CSI-4) and simplified predictors (CSI-2′, CSI-3′, and CSI-4′) at 1200 LST.

Table 9 shows, under the CSI-1 column, the skill scores for forecasting TS_{A}s by each predictor alone at 1200 LST. A yes forecast for TS_{A} was possible whenever the conditional probability for that predictor was greater than 0.5, as specified in Fig. 5 and Table 3. The CSI-2 column lists the skill scores for (a) all the surface station predictors combined (CSI = 0.726), (b) all the sounding predictors combined (CSI = 0.702), and (c) persistence (CSI = 0.542). Excluding persistence, a CSI of 0.748 was obtained using all 28 predictors (column CSI-3). Using all 29 predictors combined (column CSI-4) resulted in a CSI of 0.760. The CSI-2′ column shows the skill scores for (a) the six selected surface station predictors (CSI = 0.700), (b) the five selected sounding predictors (CSI = 0.651), and (c) persistence (CSI = 0.542). The resulting CSI values using all selected predictors without (column CSI-3′) and with persistence (column CSI-4′) are 0.744 and 0.757, respectively, essentially the same as when all 29 predictors were used. Thus, clearly indicating a reduced number of predictors is justified.

## 6. Conclusions

The objective of this study was to develop a fuzzy logic algorithm for forecasting TS_{A}s in northern Taiwan. A 4-yr (2005–08) dataset of the synoptically undisturbed days was used to develop the algorithm. Relationships between observations from three surface stations and a sounding established the following scenario favorable for TS_{A}s in northern Taiwan. The large-scale environment as portrayed by a nearby sounding should be unstable with a relatively moist layer between 850 and 650 hPa. A local trigger for initiating thunderstorms under a favorable large-scale environment appears to be the convergence of moist sea-breeze air into the Taipei Basin from two separate valleys that open toward the sea.

Membership functions and weights were developed for 28 predictors derived from three CWB surface stations in northern Taiwan (Danshui, Keelung, and Taipei) and the Panchaio sounding. In addition, the persistence of TS_{A}s from the previous day was used as the 29th predictor. The weights assigned to each predictor involved iterating the individual weights for each predictor through many permutations to identify the weights that produced the highest skill scores for forecasting TS_{A}s using predictors at hourly intervals between 0800 and 1200 LST.

The decreasing weight given to the early morning sounding as the day progressed, in contrast to the increasing weight given to the surface stations, suggests the sounding information delineates if conditions are *generally* favorable for TS_{A}, and the meso-*γ*-scale features, such as baroclinic boundaries produced by local circulations determine specific locations and times for thunderstorm initiation.

The skill scores for forecasting TS_{A}s using all 29 predictors of the 4-yr development dataset were from 0.709 to 0.760 between 0800 and 1200 LST. Skill scores as high as 0.8 were obtained for the fuzzy algorithm by modifying the definition of a yes forecast for TS_{A} from a modified higher-likelihood value. An independent dataset from 2009 and 2010 was used to evaluate the performance of the fuzzy algorithm developed on the dependent dataset. The comparable range of skill scores between 0800 and 1200 LST for the forecast times on the independent dataset was 0.542 to 0.594, considerably lower than for the dependent dataset.

Skill scores for the fuzzy logic algorithm at 1000 LST were compared with the CWB forecasts for TS_{A}s that were made for a similar area issued at 1030 LST. For the 4-yr (2005–08) algorithm development period the fuzzy algorithm had a skill score of 0.719 compared to 0.602 for the CWB forecasters, and a fuzzy algorithm skill score of 0.557 compared to CWB’s 0.473 for the 2-yr (2009–10) independent period. There was a tendency for the forecasters to overforecast the number of TS_{A}s. Thus, the fuzzy logic algorithm approach seems worth continued development and has future potential to serve as guidance material for forecasters.

Because the CSI skill score does not include negative events and provides only one aspect of performance, the use of this score may favor some events more than others, thus making it an inequitable skill score (Gandin and Murphy 1992). For further investigations, the equitable scalar measures will be considered for improving the performance of the algorithm, such as the Heidke skill score (HSS) and the Hanssen–Kuipers skill score (KSS) (Wilks 2006), where both correct forecasts and incorrect forecasts are weighted equally regardless of the frequency of the events.

An experiment was conducted to drastically reduce the number of predictors by selecting a subset of predictors that seemed to represent favorable large-scale conditions and favorable local-scale triggers for initiating thunderstorms. A set of 12 such predictors produced skill scores essentially equal to the full 29-predictor set, demonstrating that simplification of the number of predictors did not diminish the results.

It is reasonable to expect that other predictors may improve the results. Future studies will include additional stability parameters, particularly convective inhibition. Other possible predictors could be shear, time of sea-breeze passage at key surface stations, satellite cumulus cloud IR temperature changes (Roberts and Rutledge 2003), and radar early cumulus cloud detection (Mueller et al. 2003).

We believe that further improvements in forecasting convective storm initiation and evolution can come from better understanding of mesoscale processes via a small field experiment in northern Taiwan. The experiment should focus on the role of sea breezes, wind convergence, and updrafts in the Taipei Basin and updrafts along the slopes of the SMR on storm initiation and evolution.

## Acknowledgments

The authors thank the Central Weather Bureau for providing the datasets and computer resources. Thanks also go to the three anonymous formal reviewers whose comments were particularly helpful for improving the paper. This research is supported by the National Science Council of Taiwan, Republic of China, under Grants 98-2625-M-052-005 and 99-2625-M-052-004-MY3.

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