AUGUST 2013 I T O A N D W U 2525 Typhoon-Position-Oriented Sensitivity Analysis. Part I: Theory and Verification KOSUKE ITO Department of Atmospheric Sciences, National Taiwan University, Taipei, Taiwan, and Japan Agency for Marine-Earth Science and Technology, Yokohama, Japan CHUN-CHIEH WU Department of Atmospheric Sciences, National Taiwan University, Taipei, Taiwan (Manuscript received 1 November 2012, in final form 2 February 2013) ABSTRACT A new sensitivity analysis method is proposed for the ensemble prediction system in which a tropical cy- clone (TC) position is taken as a metric. Sensitivity is defined as a slope of linear regression (or its approx- imation) between state variable and a scalar representing the TC position based on ensemble simulation. The experiment results illustrate important regions for ensemble TC track forecast. The typhoon-position- oriented sensitivity analysis (TyPOS) is applied to Typhoon Shanshan (2006) for the verification time of up to 48 h. The sensitivity field of the TC central latitude with respect to the vorticity field obtained from large-scale random initial perturbation is characterized by a horizontally tilted pattern centered at the initial TC position. These sensitivity signals are generally maximized in the middle troposphere and are far more significant than those with respect to the divergence field. The results are consistent with the sensitivity signals obtained from existing methods. The verification experiments indicate that the signals from TyPOS quantitatively reflect an ensemble-mean position change as a response to the initial perturbation. Another experiment with Typhoon Dolphin (2008) demonstrates the long-term analysis of forecast sensitivity up to 96 h. Several additional tests have also been carried out to investigate the dependency among ensemble members, the impacts of using different horizontal grid spacing, and the effectiveness of ensemble-Kalman-filter-based perturbations. 1. Introduction Meteorological Agency 2010). Therefore, reducing track forecast errors remains one of the major issues for TC In terms of disaster prevention and mitigation asso- researchers and operational centers. ciated with tropical cyclones (TCs), the accuracy of track The errors in weather forecasts are mainly associated forecasts is critical because forecasts of the intensity and with two factors: imperfections of the numerical model the rainfall amount can result in large errors if the track and errors in the initial conditions (Yoden 2007). For forecast is inadequate (Wu and Kuo 1999; Chan 2010). a given model, additional observations from critical re- During past decades, the physical processes governing gions are supposed to reduce uncertainties in the initial the movement of TCs have been intensively investigated conditions and lead to a better forecast. For the purpose (Wu and Emanuel 1993; Elsberry 1995; Chan 2010). The of reducing errors in the track forecast, aircraft-borne 48-h track error has been almost halved owing to in- observations have been ‘‘targeted’’ in the synoptic en- creased observations and many improvements in nu- vironment of TCs. At present, operational surveillance merical weather prediction systems (National Hurricane observations of TCs have been conducted by the National Center 2012). However, ‘‘busted’’ cases in which track Oceanic and Atmospheric Administration (NOAA) in forecast errors exceed 1000 km in a 72-h forecast still the Atlantic basin since 1997 (Aberson and Franklin occasionally occur (Yamaguchi et al. 2009, 2012; Japan 1999; Aberson 2003) and by the Dropwindsonde Ob- servations for Typhoon Surveillance near the Taiwan Region (DOTSTAR) project (Wu et al. 2005) for se- Corresponding author address: Chun-Chieh Wu, Department of Atmospheric Sciences, National Taiwan University, 1, Sec. 4, lected western North Pacific TCs since 2003. In the Roosevelt Rd., Taipei 10617, Taiwan. summer of 2008, the issue of targeted observation was E-mail: [email protected] further explored in the Observing System Research and DOI: 10.1175/JAS-D-12-0301.1 Ó 2013 American Meteorological Society Unauthenticated | Downloaded 09/26/21 09:37 PM UTC 2526 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 70 Predictability Experiment (THORPEX) Pacific Asian 2012). The ensemble-based sensitivity analysis has also Regional Campaign (T-PARC; Elsberry and Harr 2008; been used to correct the displacement error suitable for Weissmann et al. 2011; Chou et al. 2011; Wu et al. 2012a,b; real-world observations using the data assimilation tech- Huang et al. 2012). nique (Aonashi and Eito 2011). These approaches moti- Several techniques for targeted observation guidance vate us to calculate the sensitivity field by employing TC have been developed and compared to each other to position directly as a metric. Using this metric allows us to identify ‘‘sensitive’’ regions (Majumdar et al. 2006; bypass the requirement of defining the metric associated Reynolds et al. 2007; Wu et al. 2009a). These techniques with TC motion and to perform sensitivity analysis with- include the total energy singular vector (SV) method out constructing tangent linear and adjoint models. used by the Navy Operational Global Atmospheric Pre- In this study, we apply TyPOS to Typhoons Shanshan diction System (NOGAPS; Peng and Reynolds 2006; (2006) and Dolphin (2008). The synoptic sensitivity field Chen et al. 2009), the European Centre for Medium- affecting the TC motion is shown and verification tests Range Weather Forecasts (ECMWF; Buizza et al. 2007), are conducted to evaluate the response of the TC dis- the Japan Meteorological Agency (JMA; Yamaguchi placement to initial perturbations. This work is the first et al. 2009), and Yonsei University (YSU; Kim and Jung attempt to quantify the direction and distance of the TC 2009; Kim et al. 2011); the adjoint-derived sensitivity vortex displacement directly by using the sensitivity field. steering vector (ADSSV) guidance (Wu et al. 2007, A companion paper (hereafter referred to as Part II) will 2009b; Chen et al. 2011); the ensemble transform Kalman describe comparisons with other methods and provide filter (ETKF; Majumdar et al. 2006, 2011); the analysis physical interpretations of the sensitivity field in detail. of ensemble deep-layer-mean (DLM) wind variance (Aberson 2003); and the conditional nonlinear optimal 2. Theoretical background perturbation (CNOP) method (Mu et al. 2009). While TC track forecasts have been improved with additional a. Review of ensemble-based sensitivity observations in sensitive areas (e.g., Chou and Wu 2008; TyPOS is based on the ensemble-based sensitivity Yamaguchi et al. 2009; Chou et al. 2011), some differ- analysis recently developed by several researchers ences can be found among different methods (Majumdar (Martin and Xue 2006; Ancell and Hakim 2007; Torn et al. 2006; Wu et al. 2009a). These differences stem from and Hakim 2008, 2009; Liu et al. 2008; Gombos and the different approaches to detect the sensitivity on TC Hansen 2008; Sugiura 2010; Aonashi and Eito 2011). motion as well as the different numerical models and Here, we briefly outline this sensitivity analysis technique configurations used (Wu et al. 2009b; Kim et al. 2011). in a generalized manner. One potential way for further elaborating the sensi- Let x and j be an n degree-of-freedom input vector tivity analysis method is to employ TC position as a and output scalar, respectively. The input vector repre- metric, whereas existing methods use only TC steering sents the perturbations in the physical variables at the flow (ADSSV) or total energy norms (SV, ETKF, and initial time and the output scalar is a forecast metric. CNOP) at the verification time. The metrics employed Considering m realizations of the input vector, x 5 in the existing studies are presumably relevant to TC i (x , x , ..., x ), where i(i 5 1, 2, ..., m) represents the motion, and several studies have shown that these 1,i 2,i n,i index of each ensemble member, with corresponding m methods can help capture the effect of synoptic features realizations of j , we define matrix X and vector j as and binary interactions (e.g., Peng et al. 2007; Wu et al. i follows: 2007). Nevertheless, it is worth investigating a more di- rect evaluation of the sensitivity field on TC tracks by 1 X 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi(dx , dx , ... , dx )T, and (1) using TC position as a metric. m 2 1 1 2 m In this study, we propose a typhoon-position-oriented sensitivity analysis (TyPOS), which directly takes a TC 1 j 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi(dj , dj , ... , dj )T . (2) position as a metric, for the ensemble prediction system. m 2 1 1 2 m This technique is based on the ensemble-based sensi- tivity analysis method (Martin and Xue 2006; Ancell and Matrix X is composed of dxi and the vector j is composed Hakim 2007; Torn and Hakim 2008, 2009; Liu et al. 2008; of dji, where d indicates deviations from ensemble-mean Gombos and Hansen 2008; Sugiura 2010; Aonashi and values; that is, dxi 5 xi 2 hxi and dji 5 ji 2 h ji. The angle Eito 2011). It has been applied to the sensitivity analysis brackets denote the ensemble mean and superscript T of the minimum sea level pressure (SLP) and the root- denotes the transpose. Note that the deviations are not mean-square error of SLP of TCs (Torn and Hakim 2009) necessarily infinitesimal, while Ancell and Hakim (2007) as well as the 1000-hPa potential vorticity (Gombos et al. introduced the assumption ›J/›x ’ dJ/dx. Unauthenticated | Downloaded 09/26/21 09:37 PM UTC AUGUST 2013 I T O A N D W U 2527 The generalized ensemble-based sensitivity is defined as follows (Gombos and Hansen 2008): lj [ X1 x j, (3) where the plus sign superscript indicates the Moore– Penrose generalized inverse matrix,1 which can be computed from the equation below: 1 1 X 5 VD UT , (4) 1 where X 5 UD VT is the singular value decomposition 1 of X.