1 2 3 4 Typhoon-Position-Oriented Sensitivity Analysis. 5 Part I: Theory and Verification 6 7 8 9 10 11 12 13 14 15 Journal of the Atmospheric Sciences 16 (Accepted on Feb. 6, 2013). 17 18 19 20 Kosuke Ito 21 National Taiwan University, Taipei, Taiwan and 22 Japan Agency for Marine-Earth Science and Technology, Yokohama, Japan 23 24 Chun-Chieh Wu* 25 National Taiwan University, Taipei, Taiwan
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28 *Corresponding author address: Chun-Chieh Wu, Department of 29 Atmospheric Sciences, National Taiwan University, No. 1, Sec. 4, 30 Roosevelt Rd., Taipei 106, Taiwan. 31 E-mail: [email protected]
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32 Abstract
33 A new sensitivity analysis method is proposed for the ensemble
34 prediction system in which a tropical cyclone (TC)-position is taken as a
35 metric. Sensitivity is defined as a slope of linear regression (or its
36 approximation) between state variable and a scalar representing the TC
37 position based on ensemble simulation. The experiment results illustrate
38 important regions for ensemble TC track forecast. The typhoon-position-
39 oriented sensitivity analysis (TyPOS) is applied to Typhoon Shanshan
40 (2006) for the verification time of up to 48 hours. The sensitivity field of
41 the TC central latitude with respect to the vorticity field obtained from
42 large-scale random initial perturbation is characterized by a horizontally
43 tilted pattern centered at the initial TC position. These sensitivity signals
44 are generally maximized in the middle troposphere and are far more
45 significant than those with respect to the divergence field. The results are
46 consistent with the sensitivity signals obtained from existing methods. The
47 verification experiments indicate that the signals from TyPOS
48 quantitatively reflect an ensemble-mean position change as a response to
49 the initial perturbation. Another experiment with Typhoon Dolphin (2008)
50 demonstrates the long-term analysis of forecast sensitivity up to 96 hours.
51 Several additional tests have also been carried out to investigate the
52 dependency among ensemble members, the impacts of using different
53 horizontal grid spacing, and the effectiveness of ensemble-Kalman-filter-
54 based perturbations.
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55 1. Introduction
56 In terms of disaster prevention and mitigation associated with
57 tropical cyclones (TCs), the accuracy of track forecasts is critical because
58 forecasts of the intensity and the rainfall amount can result in large errors if
59 the track forecast is inadequate (Wu and Kuo 1999; Chan 2010). During
60 past decades, the physical processes governing the movement of TCs have
61 been intensively investigated (Wu and Emanuel 1993; Elsberry 1995; Chan
62 2010). The 48-h track error has been almost halved due to increased
63 observations and many improvements in numerical weather prediction
64 systems (National Hurricane Center, 2012). However, “busted” cases in
65 which track forecast errors exceed 1000 km in a 72-hour forecast still
66 occasionally occur (Yamaguchi et al. 2009, 2012; Japan Meteorological
67 Agency 2010). Therefore, reducing track forecast errors remains one of the
68 major issues for TC researchers and operational centers.
69 The errors in weather forecasts are mainly associated with two
70 factors: imperfections of the numerical model and errors in the initial
71 conditions (Yoden 2007). For a given model, additional observations from
72 critical regions are supposed to reduce uncertainties in the initial conditions
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73 and lead to a better forecast. For the purpose of reducing errors in the track
74 forecast, aircraft-borne observations have been ‘targeted’ in the synoptic
75 environment of TCs. At present, operational surveillance observations of
76 TCs have been conducted by the National Oceanic and Atmospheric
77 Administration (NOAA) in the Atlantic basin since 1997 (Aberson and
78 Franklin 1999; Aberson 2003) and by the Dropwindsonde Observations for
79 Typhoon Surveillance near the Taiwan Region (DOTSTAR) project (Wu et
80 al. 2005) for selected western North Pacific TCs since 2003. In the summer
81 of 2008, the issue of targeted observation was further explored in the
82 Observing System Research and Predictability Experiment (THORPEX)
83 Pacific Asian Regional Campaign (T-PARC; Elsberry and Harr 2008;
84 Weissmann et al. 2011; Chou et al. 2011; Wu et al. 2012a,b; Huang et al.
85 2012).
86 Several techniques for targeted observation guidance have been
87 developed and compared to each other to identify “sensitive" regions
88 (Majumdar et al. 2006; Reynolds et al. 2007; Wu et al. 2009a). These
89 techniques include the total energy singular vector (SV) method used by
90 the Navy Operational Global Atmospheric Prediction System (NOGAPS;
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91 Peng and Reynolds 2006; Chen et al. 2009), the European Centre for
92 Medium-Range Weather Forecasts (ECMWF; Buizza et al. 2007), the
93 Japan Meteorological Agency (JMA; Yamaguchi et al. 2009), and Yonsei
94 University (YSU; Kim and Jung 2009; Kim et al. 2011); the adjoint-derived
95 sensitivity steering vector (ADSSV) guidance (Wu et al. 2007, 2009b;
96 Chen et al. 2011); the ensemble transform Kalman filter (ETKF; Majumdar
97 et al. 2006, 2011); the analysis of ensemble deep-layer mean (DLM) wind
98 variance (Aberson 2003); and the Conditional Nonlinear Optimal
99 Perturbation (CNOP) method (Mu et al. 2009). While TC track forecasts
100 have been improved with additional observations in sensitive areas (e.g.,
101 Chou and Wu 2008; Yamaguchi et al. 2009; Chou et al. 2011), some
102 differences can be found among different methods (Majumdar et al. 2006;
103 Wu et al. 2009a). These differences stem from the different approaches to
104 detect the sensitivity on TC motion as well as the different numerical
105 models and configurations used (Wu et al. 2009b; Kim et al. 2011).
106 One potential way for further elaborating the sensitivity analysis
107 method is to employ TC position as a metric, whereas existing methods use
108 only TC steering flow (ADSSV) or total energy norms (SV, ETKF and
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109 CNOP) at the verification time. The metrics employed in the existing
110 studies are presumably relevant to TC motion, and several studies have
111 shown that these methods can help capture the effect of synoptic features
112 and binary interactions (e.g., Peng et al. 2007; Wu et al. 2007).
113 Nevertheless, it is worth investigating a more direct evaluation of the
114 sensitivity field on TC tracks by using TC position as a metric.
115 In this study, we propose a typhoon-position-oriented sensitivity
116 analysis (TyPOS), which directly takes a TC position as a metric, for the
117 ensemble prediction system. This technique is based on the ensemble-based
118 sensitivity analysis method (Martin and Xue 2006; Ancell and Hakim 2007;
119 Torn and Hakim 2008, 2009; Liu et al. 2008; Gombos and Hansen 2008;
120 Sugiura 2010; Aonashi and Eito 2011). It has been applied to the sensitivity
121 analysis of the minimum sea level pressure (SLP) and the root mean square
122 error of SLP of TCs (Torn and Hakim, 2009) as well as the 1000-hPa
123 potential vorticity (Gombos et al. 2012). The ensemble-based sensitivity
124 analysis has also been used to correct the displacement error suitable for
125 real world observations using the data assimilation technique (Aonashi and
126 Eito, 2011). These approaches motivate us to calculate the sensitivity field
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127 by employing TC position directly as a metric. Using this metric allows us
128 to bypass the requirement of defining the metric associated with TC motion,
129 and to perform sensitivity analysis without constructing tangent linear and
130 adjoint models.
131 In this study, we apply TyPOS to Typhoons Shanshan (2006) and
132 Dolphin (2008). The synoptic sensitivity field affecting the TC motion is
133 shown and verification tests are conducted to evaluate the response of the
134 TC displacement to initial perturbations. This work is the first attempt to
135 quantify the direction and distance of the TC vortex displacement directly
136 by using the sensitivity field. A companion paper (hereafter referred to as
137 Part II) will describe comparisons with other methods and provide physical
138 interpretations of the sensitivity field in detail.
139
140 2. Theoretical background
141 a. Review of ensemble-based sensitivity
142 TyPOS is based on the ensemble-based sensitivity analysis recently
143 developed by several researchers (Martin and Xue 2006; Ancell and Hakim
144 2007; Torn and Hakim 2008, 2009; Liu et al. 2008; Gombos and Hansen
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145 2008; Sugiura 2010; Aonashi and Eito 2011). Here, we briefly outline this
146 sensitivity analysis technique in a generalized manner.
147 Let x and j be an n degree-of-freedom input vector and output
148 scalar, respectively. The input vector represents the perturbations in the
149 physical variables at the initial time and the output scalar is a forecast
150 metric. Considering m realizations of the input vector,
151 xi(x1,i,x2,i,...,xn,i), where i(i1,2,...,m) represents the index of each
152 ensemble member, with corresponding realizations of the output scalar,
153 ji , we define matrix X and vector j as follows: 1 T 154 X x1,x2,...,xm, (1) m1 1 T 155 j j1,j2,...,jm. (2) m1
156 Matrix is composed of xi and the vector is composed of ji
157 where indicates deviations from ensemble mean values, that is,
158 xi xi x and ji ji j . The angle bracket denotes the
159 ensemble mean and superscript T denotes the transpose. Note that the
160 deviations are not necessarily infinitesimal, while Ancell and Hakim (2007)
161 introduced the assumptionJ/xJ/x.
162 The generalized ensemble-based sensitivity is defined as follows
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163 (Gombos et al., 2008):
j 164 λx X j , (3)
165 where the superscript cross ( ) indicates the Moore-Penrose generalized
166 inverse matrix1, which can be computed from the equation below.
167 X VDUT , (4)
168 where X UDVT is the singular vector decomposition of X . The
169 generalized inverse matrix D is formed by replacing every nonzero
170 diagonal entry of D by its reciprocal and transpose the resulting matrix.
j 171 In this paper, the vector λ x is referred to as “sensitivity” and the k-
λ j (k1,2,...,n) 172 th component of is denoted as xk . The reason why
173 is called the “sensitivity” is that it represents the generalized solution of
174 inverse problem associated with a first-order relationship between j and
175 X ,
176 j Xλˆ, (5)
177 which means the vector λˆ multiplied by initial perturbations in physical
1 A Moore-Penrose generalized inverse matrix of X is defined as a matrix satisfying all of the
following four criteria: XX X X, X XX X, (XX )* XX and (X X)*X X, where the asterisk
represents the adjoint matrix. The matrix exists and is unique. See more details in Menke (1989).
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178 variables equals to perturbations in the forecast metric at the verification
179 time.
180 If the number of ensemble members exceeds the degrees-of-freedom,
181 i.e., m n , a purely over-determined problem and the inverse of (XTX)-1
ˆ j 182 exists, the vector λ λ x is the slope of regression line obtained from
183 minimizing deviations as shown in Fig. 1 (Menke, 1989). In other words,
λ j j x 184 xk represents the ratio of changes in to the change in k based on
j 185 ensemble simulations. The vector λ x is the same as the adjoint sensitivity
186 if the time evolution of initial perturbation is completely equivalent to that
187 described by the tangent linear model and is a continuous function of
188 x (Ancell and Hakim, 2007).
189 If the degrees-of-freedom of exceeds the number of ensemble
190 members, i.e., m n , a purely under-determined problem, the vector
ˆ j 191 λ λ x represents the solution which satisfies Eq. (5) and yields the
ˆ L 192 minimum of λ in the 2 norm (Menke 1989). This type of solution is
193 termed as a minimum length solution and is frequently used in tackling
194 inverse problems. Although the vector can still be interpreted as a
195 slope with respect to satisfying j Xλˆ , the additional condition
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ˆ 196 (minimization of λ ) is introduced to fully determine all the n-elements of
197 the sensitivity field from m realizations. Thus, we do not expect that this
198 sensitivity would appropriately reflect the impact of changes in the input
199 vector, x , on changes in the metric, j , when the number of ensemble
200 members, m , is far smaller than the degrees-of-freedom of , n .
201 If the inverse matrix of XTX is known, the sensitivity can be
202 obtained without executing the singular value decomposition in equation
203 (4). By multiplying both sides of equation (5) by XT, we obtain
T Tˆ 204 XjXXλ. (6)
205 Since an element in the a -th row and the b-th column of XTX is 1 m 206 xa,ixb,i , can be replaced by the ensemble covariance matrix. m1i1
207 In particular, when the ensemble covariance matrix is diagonal, the
208 equations (3) and (4) are reduced to the simplified equation below:
1 m xk,iji 209 λj m1i1 , (7) xk 2 k
2 2 210 where k represents ensemble variance .
211
2 Note that Eq. (7) is used as a definition of ensemble-based sensitivity for both diagonal and off-diagonal matrix in Ancell and Hakim (2007).
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212 b. TyPOS
213 TyPOS is an application of the ensemble-based sensitivity analysis
214 designed for the ensemble forecast system. The input vector x represents
215 perturbations in the physical variables at the initial time and the output
j 216 scalar is a scalar representing the vortex position such as longitude and
217 latitude of the TC center at the verification time. TyPOS does not assume
218 that the perturbations are infinitesimal. It fits our objective of employing
219 TC position as a metric because it is not a differentiable function when
220 reproduced in a dynamical model.
221 One problem is that the ensemble-based sensitivity is not expected to
222 be a good indicator for sensitivity when the number of ensemble members
223 is far smaller than the degrees-of-freedom of . If we consider the
224 sophisticated three-dimensional numerical model with degrees-of-freedom
225 n', it is not practical to perform simulations as much as times.
226 To resolve this problem, we consider a reduced-dimension space in
227 which the n-elements ( n n') of used for sensitivity analysis are
228 sampled with larger grid spacing compared with that of the original
229 numerical model in this work. The horizontal grid spacing can be greater
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230 than a few hundred kilometers if the aim is dealing with the synoptic
231 features. In this work, the grid spacing used to define the sensitivity field
232 differs from the grid spacing used to define the forecast metric. The grid
233 points on the coarse mesh used for sensitivity calculation are referred to as
234 the reduced grid points, while the grid points on the original model mesh
235 used to perform the time integration are referred to as the model grid points.
236 Here, initial perturbations in the reduced grid points are interpolated into
237 the model grid points by cubic spline. The use of a basis function
238 corresponding to the fast developing modes is another option aside from
239 employing the large-scale perturbations, which will be discussed in section
240 6b.
241 TyPOS has the potential to quantify the displacement distance and
242 direction of ensemble-mean TC position. Suppose that the sensitivity
243 approximates the ratio of changes in j to the change in xk , ensemble-
244 mean change in the output metric as a response to the initial perturbation is
245 evaluated as follows ( denotes the perturbation relative to the control
246 run):
1~j 247 jx'x', (8) S
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248 where denotes the summation over all the model grid points, and x'
249 represents the changes in the initial state defined at the model grid points.
250 In order to complete the manipulation between the initial perturbations
251 defined on model grid points and the sensitivity defined on reduced grid
252 points, we introduce the interpolated sensitivity field defined on the model
~j 253 grid points, x' , and the scaling factor S. The scaling factor is defined as
~ 254 Sλ λ (in L1 norm), which is based on the assumption that the influence
255 of the initial changes in a reduced grid point is equivalent to the integrated
256 influence of uniform initial condition changes in corresponding model grid
257 points, i.e., it has a large value when the reduced grid points are coarsely
258 sampled from the model grid points. Eq. (8) is useful for evaluating the
259 magnitude of displacement as a response to changes in the initial state
260 variables and for determining the priority of the targeted observations,
261 although the expected displacement distance should be regarded as an
262 approximation. This is due to the following reasons: the truncation error
263 inherent in the projection, dependency on perturbation magnitudes, and the
264 assumption that impacts are calculated by simple linear superposition of
265 impacts from single grid points.
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266 As stated in Torn and Hakim (2008), the reliability of sensitivity
267 signals in TyPOS can be measured using a t-test. The t-test is relevant to the
268 rejection of the null hypothesis that the slope of the output metric, j , with
269 respect to the single variable, xk , is zero (Wilks 2005). A higher
270 confidence level is obtained with the increasing number of ensemble
271 members or with statistically smaller deviations from the regression line if
272 the values of the slope are equal. It is critical to distinguish meaningful
273 signals from statistical noise, in particular, when the number of ensemble
274 simulations is far smaller than the degrees-of-freedom of initial
275 perturbations as demonstrated in section 6a.
276
277 c. Summary of TyPOS
278 Before applying the proposed method to a real-case calculation, the
279 features and procedures of TyPOS are summarized here. In this
280 methodology, TC position is taken as a forecast metric, and thus it is a
281 direct method for a given ensemble prediction system. In addition to the
282 clear objectivity, this method has several advantages. The method provides
283 the possibility to evaluate the ensemble-mean vortex displacement
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284 direction and distance without using the tangent linear and adjoint models.
285 To alleviate the burden posed by the enormous computational resources
286 needed for obtaining the significant signals, we consider the reduced-
287 dimension space and, if needed, to distinguish meaningful signals from
288 noise using t-tests.
289 The procedure of TyPOS is as follows: (i) Generating m realizations
290 with n degrees-of-freedom in the reduced grid points and projecting them
291 onto the model variable space by cubic spline interpolation, (ii) performing
292 the ensemble simulations m times after adding interpolated perturbation
293 to the reference initial state, (iii) calculating the sensitivity by using the m
294 realizations of the resultant metric that represents the TC position [Eqs. (3)
295 and (4), or Eq. (6)], (iv) (optional) evaluating the displacement of the
296 vortex position by multiplying the changes in the initial state variable with
297 the sensitivity field after the interpolation into the model grid points [Eq.
298 (8)], and (v) (optional) conducting t-tests to distinguish reliable signals
299 from noise.
300 TyPOS can directly reflect the sensitivity field of the metric relevant
301 to TC position for a given model and initial perturbations if the number of
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302 ensemble members is sufficiently large. However, TyPOS is still subject to
303 a variety of uncertainties in terms of specifying initial perturbations. In this
304 study, limitations arise from the fact that dynamically balanced large-scale
305 perturbations are employed as initial perturbations. A fine scale feature is
306 briefly discussed in section 6, although detailed examination is beyond the
307 scope of this work.
308 Note that TyPOS is based on the linear regression obtained from
309 ensemble simulation and the perturbation is not necessarily infinitesimal.
310 Therefore, the validity of TyPOS is not limited to the time scale of tangent
311 linear assumption as in any singular vector- or adjoint-based method.
312 Nevertheless, care should be taken in interpreting the sensitivity, and the
313 influence of nonlinear growth of perturbations requires further study.
314
315 3. Experimental Design
316 a. Model Configuration and time schedule
317 In order to assess the feasibility of TyPOS, we apply this method to
318 Typhoon Shanshan, for which sensitivity analyses based on different
319 targeted methods have already been conducted (Wu et al. 2009a,b;
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320 Reynolds et al. 2009; Chen et al. 2011) and Typhoon Dolphin, for which
321 the spread of ensemble forecast TC track was reported to be large (from
322 JMA 2009). The verification time is set to 12, 24 and 48 hours for
323 Shanshan in order to compare with the existing studies, and to 24, 48 and
324 96 hours for Dolphin in order to investigate its applicability to long-term
325 sensitivity analyses with large uncertainties in track forecasts.
326 In this study, the Advanced Research version of Weather Research
327 and Forecast (ARW-WRF) model version 2.2.1 is employed to conduct the
328 numerical simulation. The model configuration is the same as the doubly-
329 nested domain calculations of Wu et al. (2010). The coarse domain
330 (Domain 1) has 150x120 grid points (longitude by latitude) with the
o o 331 horizontal grid spacing of 54 km centered at 120.0 E and 27.5 N, while the
332 fine-meshed domain (Domain 2) has 120 x 150 grid points with the
o o 333 horizontal grid spacing of 18 km centered at 124.9 E and 27.1 N. The
334 model contains 35 vertical levels in the terrain following sigma coordinate.
335 The domains and TC position from the JMA best track are shown in Fig. 2.
336 The prognostic state variables are perturbation of potential
337 temperature, geopotential and dry air mass in a column from the basic state
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338 as well as horizontal and vertical wind fields. Six mixing ratios (water
339 vapor, cloud water, cloud ice, rain, snow and graupel) are also
340 prognostically calculated based on the WRF single moment six-class
341 graupel microphysics scheme (Hong et al. 2004; Hong and Lim 2006).
342 Other parameterization schemes include the rapid radiative transfer model
343 scheme (Mlawer et al. 1997) for longwave radiation, the simple shortwave
344 scheme Dudhia (1989) for shortwave radiation, and the YSU planetary
345 boundary layer scheme (Hong et al. 2006). The Grell-Devenyi ensemble
346 scheme (Grell and Devenyi 2002) is used for the parameterization of
347 cumulus convection in both Domains 1 and 2.
348 To reproduce a realistic TC vortex, we performed a one-week data
349 assimilation with the EnKF technique prior to the sensitivity analysis. This
350 data assimilation procedure starts from the field of National Centers for
351 Environmental Prediction (NCEP) final analysis. The WRF-based EnKF
352 data assimilation system used in this study is similar to the system in Meng
353 and Zhang (2007, 2008a,b) and has been developed to adjust TC central
354 position and pressure in order to fit the best track (Wu et al., 2010). In this
355 study, the number of ensemble members in the EnKF system is 100. We
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356 regard 0000 UTC 15 September 2006 (1200 UTC 11 December 2008) as
357 the initial time of the sensitivity analysis for Shanshan (Dolphin) and
358 regard ensemble-mean state of the EnKF analysis field at this moment as
359 the initial reference state. For simplicity, we assume that this initial time is
360 the same as the observation time (i.e., no lead-time forecast experiment).
361 For the presentation of results, the number of hours beginning from the
362 initial time of the sensitivity analysis is used. The time schedule of the
363 experiment is summarized in Fig. 3. In fact, we calculate the sensitivity
364 both of latitude and longitude at the model grid point where the minimum
365 pressure is achieved at the verification time. However, for concise
366 presentation, we will present the only sensitivity of the latitude (longitude)
367 for Shanshan (Dolphin) unless otherwise noted.
368
369 b. Perturbations for sensitivity analysis
370 In order to construct the sensitivity field, u and v at the initial
371 time are perturbed. As a first step, the magnitude is simply determined so
-1 - 372 that the standard deviation becomes an uniform value of 1.8 m s (1.4 m s
1 373 ) for Shanshan (Dolphin), which is the same as the area-averaged spread of
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374 the 100 EnKF members in the middle troposphere at the initial time
375 (figures not shown).
376 In this study, initial perturbations are generated independently in
377 each reduced grid point and thus t-tests are applied independently in each
378 reduced grid point except in section 6b. The benefit of using the
379 uncorrelated perturbations is that we can calculate the sensitivity based on
380 the simplified equation (7). Independent Gaussian random numbers in each
381 reduced grid point are generated by using Mersenne Twister method and
382 Box-Mullar Transform (Box and Muller 1958; Matsumoto and Nishimura
383 1998) and are added to the initial reference state of these variables. It is
384 used to construct the sensitivity field with respect to u and v. Although the
385 relative vorticity and divergence are not prognostic state variables
386 in this model, these are essential variables for clear physical interpretation.
387 Thus, the sensitivity field with respect to u and v is converted to one with
388 respect to relative vorticity and divergence as in Kleist and Morgan
389 (2005) and Wu et al. (2007).
390 The horizontal reduced-grid spacing is set to 540 km regardless of
391 the model grid spacing. This coarse grid spacing used in this study is
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392 presumably sufficient in resolving synoptic features of a few thousand
393 kilometers, while not reflecting the fine-scale structures of the sensitivity
394 field. The vertical grid points are sampled on = 0.99885, 0.8445, 0.5060,
395 0.2879, 0.101. The potential temperature and geopotential fields are also
396 changed so that the perturbation fields are close to geostrophic and
397 hydrostatic balances as a response to initial perturbations of vorticity. For
398 instance, the standard deviation of perturbations in potential temperature
2 -2 399 (geopotential) is 0.12 K (30.1 m s ) at 20N and = 0.8445, and 0.06 K
2 -2 400 (14.2 m s ) at 20N and = 0.5060 for Shanshan. In order to obtain
401 temperature and geopotential fields, we follow the same procedure as in
402 Komaromi et al. (2011). We first calculate the perturbation in stream
403 function by solving Poisson’s equation using a relaxation method and then
404 calculate the temperature and geopotential perturbation field consistent
405 with the stream function.
406 The number of ensemble simulations ( m ) is 800. However, 170
407 simulations for Dolphin are not used for sensitivity analysis because the
408 maximum wind speed of the cyclone is reduced below the tropical storm
-1 409 intensity of 17.2 m s during their lifetime. This threshold is introduced to
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410 strictly avoid false detection of TCs. The degrees-of-freedom of the initial
411 perturbations (n) is 1800 for Shanshan (15x12x5 reduced grid points in the
412 zonal, meridional and vertical direction for the two components, u and
413 v ) and 3600 for Dolphin (24x15x5x2). Therefore, the problems are
414 formulated as underdetermined ones.
415
416 4. Sensitivity analysis for Shanshan
417 a. Synopsis and simulated track
418 Figure 4 shows the center position of Shanshan based on the JMA
419 best track. It formed to the west of Guam and moved northwestward and
420 started to recurve on 13 September, turning northward from its original
421 westward path. The TC later began to accelerate northeastward on 16
422 September. During the period of the current numerical experiment, there
423 are several synoptic features around the TC region. Figures 5a,d show that
424 the location of Shanshan was between the midlatitude trough over
425 northern-central China and the anti-cyclonic flow at the initial time (0000
426 UTC 15 September). On 16 September, Shanshan moved swiftly as it
427 started to interact with a midlatitude trough over eastern China (Figs. 5b,e).
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428 On 17 September, Shanshan has become embedded in the midlatitude
429 trough in the 500-hPa fields (Figs. 5c,f).
430 The simulated tracks of ensemble members from 0000 UTC 15
431 September to 0000 UTC 17 September are overlaid with TC symbols in Fig.
432 4. This shows that the TC central position is reproduced with a left-of-track
433 bias. Ensemble spreads of the forecasted TC central longitude and latitude
434 at 0000 UTC 17 September are 0.422 and 0.384 degree (a distance of 39
435 km and 43 km), respectively.
436
437 b. Ensemble-based sensitivity
438 Figure 6 shows the sensitivity field of the TC central latitude with
439 respect to the vorticity field at three vertical levels ( = 0.8445, 0.5060,
440 0.2879) for the verification time of 12, 24 and 48 hours. The values are
441 horizontally interpolated into the model grid points with a cubic spline. The
442 circles superposed in these figures indicate the t-test results. The sensitivity
443 related to the vorticity field is less significant at = 0.99885 and =
444 0.101 than at = 0.8445, 0.5060, 0.2879 (figures not shown). The signals
445 are the strongest at the verification time of 48 h, which reflects the fact that
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446 the signals are relevant to the ratio of the change in the TC central latitude
447 at the verification time as a response to the unit initial change. The
448 evolution of the initial perturbation brings about large ensemble-mean
449 changes in the TC central latitude, particularly associated with the swift
450 passage as the TC interacts with midlatitude troughs. Another notable
451 feature is that the major signals are sufficiently far away from the lateral
452 boundary, implying that the regional model is able to reasonably reproduce
453 the TC track, at least within this time scale.
454 Here, we briefly outline the structure of the sensitivity field. Further
455 investigations on the physical interpretation and comparison of other
456 methods will be presented in Part II. Figure 6 shows that the largest
457 sensitivity near the TC center appears at = 0.5060, with values
458 generally larger than those at = 0.2879. These are consistent with
459 results from previous sensitivity studies for Shanshan obtained with the
460 ADSSV and SV methods (Wu et al. 2009b; Reynolds et al. 2009), implying
461 that TC motion is more sensitive to changes in the middle troposphere.
462 For the sensitivity field at the verification time of 12 h, the north-
463 south dipole pattern appears to be centered at the initial TC position. A
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464 vertically coherent structure is present from the lower troposphere to the
465 middle troposphere with the largest sensitivity occurring in the latter. This
466 dipole pattern is to some extent relevant to the latitudinal displacement of
467 the vortex. This implies that the appropriate positioning of the vortex is an
468 important element for improving short-term scale track predictions (Ito et
469 al. 2013). It is notable that the sensitivity signal of meridional velocity in
470 ADSSV (Wu et al. 2007)3 at the verification time of 12 h exhibits a west-
471 east dipole pattern in contrast to the signals in TyPOS. Physical
472 interpretation of this difference will also be investigated in Part II.
473 The sensitivity field exhibits a horizontally tilted pattern centered at
474 the initial TC position with respect to the vorticity field in the middle
475 troposphere. This pattern is typically found in ADSSV-based sensitivity
476 analyses (Wu et al. 2007, 2009b; Chen et al. 2011). It is intriguing that the
477 negative sensitivity signals are well developed from east to southwest
478 relative to the initial TC position with extended verification time, which is
479 consistent with the stronger sensitivity signals for Shanshan obtained from
480 ADSSV (Wu et al. 2009b; Chen et al. 2011). The confidence level is above
3 To avoid misunderstanding, note that the directions of the vectors in TyPOS and ADSSV have different physical implications.
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481 99.9%. This feature is also detected in the SV-based sensitivity analysis at
482 the recurvature stage for Shanshan (Reynolds et al. 2009).
483 Besides the horizontally tilted pattern in the middle troposphere, the
484 sensitivity field is elongated northwestward in the upper layer at the
485 verification time of 48 h, presumably due to the strong southeastward wind.
486 This feature is also detected and investigated as a sign of interactions
487 between the TC and midlatitude trough from sensitivity signals of ADSSV
488 (Wu et al. 2009b) and SV (Reynolds et al. 2009). Under close inspection,
489 our signal around the midlatitude trough is not distinct in the middle
490 troposphere in contrast to the ADSSV- and moist SV-based sensitivity
491 signals.
492 The confidence levels of the sensitivity signals with respect to the
493 divergence field are mostly below 99% in the lower, middle and upper
494 troposphere (figures not shown). These sensitivity signals are not organized
495 in synoptic scale, suggesting that the vorticity field is more relevant to the
496 TC motion at least in comparison to the divergence field (consistent with
497 results in Wu et al. 2007). As mentioned above, the sensitivity signals
498 obtained from the existing SV- and ADSSV-based methods are fairly
27
499 consistent with the signals in TyPOS, although some differences are also
500 identified.
501
502 c. Verification experiment
503 To ensure that the sensitivity signals obtained from TyPOS are
504 relevant to the motion of Shanshan, additional ensemble experiments are
505 conducted. One hundred members are randomly sampled from eight
506 hundred members used for the sensitivity calculation. This set of 100
507 ensemble members is referred to as CTRL. We perform ensemble
508 experiments in which three patterns of positive vorticity are added to each
509 initial state of CTRL in the prescribed region. To conduct an ensemble run,
510 the positive vorticity perturbations (without the divergence) are
511 transformed into horizontal velocity perturbations (u,v), which are the
512 model prognostic variables, via calculations of the stream function by
513 solving Poisson’s equation. The amplitude of the initial perturbation of u
514 and v is adjusted to the size of standard deviation consistent with the
515 sensitivity analysis.
516 Here, we investigate the validity of sensitivity field in the middle
28
517 troposphere with the verification time of 48 h. Two experiments are
518 associated with the strongest sensitivity signals around the initial TC
519 positions and the other experiment is associated with the midlatitude trough
520 in the middle troposphere where differences arise between TyPOS and
521 existing methods. We refer to these three experiments as follows: (i)
522 WTILT is the experiment in which the positive vorticity perturbation is
523 added to the regions consistent with the horizontally tilted pattern west to
524 the initial TC position from = 0.66175 to = 0.4082, (ii) ETILT is the
525 same as WTILT but with the horizontally tilted pattern east to the initial TC
526 position, (iii) TROUGH is the same as WTILT and ETILT but with the
527 perturbation near the midlatitude trough. Figures 7a-c indicate the initial
528 perturbations of additional horizontal wind in ETILT, WTILT and
529 TROUGH, respectively. Based on the definition of sensitivity and Fig. 6f,
530 we can expect an ensemble-mean northward (southward) displacement
531 relative to CTRL in the WTILT (ETILT) experiment and a smaller
532 displacement in the TROUGH experiment.
533 According to equation (8), the expected ensemble-mean changes
534 (Δ
29
535 horizontal wind perturbation at the initial time. As mentioned in section 2b,
536 we multiply the magnitude of perturbations by the sensitivity both of
537 latitude and longitude which are interpolated to the model grid points using
538 a cubic spline.
1~LON~LON1~LON~LON LONuvuv 539 Su vSu v (9) D1D1 D2D2
1~LAT~LAT1~LAT~LAT LATuvuv 540 Su vSu v (10) D1D1 D2D2
541 where the subscripts D1 and D2 represent domain 1 and domain 2, and
S S 542 D1 and D2 are set to constant values of 617.7 and 6273.5, respectively.
543
544 d. Results of the verification experiment
545 Figure 8 shows a typical example of time evolution of perturbation in
546 a member of the ETILT experiment (ETILT_m001) relative to a
547 corresponding member of the control experiment (CTRL_m001).
548 Perturbations are developed in a broader region within the first 12 hours
549 (Fig. 8b). They continue to grow around the TC with maximum values of
-1 550 about 20 m s (Figs. 8c,d), appearing to exhibit a cyclonic circulation
551 southwest to the TC position and an anti-cyclonic circulation northeast at
30
552 the verification time. This is consistent with the southwestward
553 displacement of the TC in ETILT_m001 relative to that in CTRL_m001.
554 Figure 9a shows the ensemble-mean TC positions in different
555 experiments. The members in the WTILT run tend to move northward
556 relative to the CTRL run, while the members in the ETILT run tend to
557 move southward. This finding is consistent with the results of the
558 sensitivity analysis. Ensemble-mean displacement in the TROUGH
559 experiment remains small as expected. It does not necessarily mean that the
560 existing methods overestimate the sensitivity associated with the
561 midlatitude trough in the middle troposphere because the sensitivity is also
562 dependent on the numerical model and the characteristics of initial
563 perturbations. Nevertheless, the results demonstrate that TyPOS accurately
564 reflects the sensitivity of TC position as a response to large-scale initial
565 perturbation for a given ensemble forecast system.
566 Figure 9b shows the frequency distributions of TC central latitude of
567 the 100 members among different ensemble experiments. It is obvious that
568 the ensemble-mean TC position in WTILT (ETILT) is located to the north
569 (south) of the TC position in CTRL, while the distribution overlaps among
31
570 different experiments. The current method is shown to be relevant to the
571 changes in probability density function of TC positions.
572 Figure 9c illustrates the ensemble-mean TC position change relative
573 to CTRL with nonlinear model calculation. The TC displacement estimated
574 from TyPOS quantitatively agrees well with the ensemble-mean
575 displacements in the perturbed experiments (Fig. 9d). We calculate the
576 amplification factor defined as the ratio of total kinetic energy (TKE) of
577 ensemble-mean perturbation integrated over a 1000 km by 1000 km square
578 centered at the TC position at the verification time to that over the entire
579 domain at the initial time. The amplification factors are 1.47 for WTILT,
580 3.73 for ETILT and 0.13 for TROUGH at t = 48 h. It is consistent with the
581 largest development of perturbations near Shanshan in the ETILT
582 experiment.
583 Overall, these features indicate the usefulness of TyPOS. The method
584 can be used to evaluate the expected direction and distance of the vortex
585 displacement as a response to the changes in the initial state and to identify
586 regions that have greater impacts on an ensemble of realizations of the TC
587 forecast position under the probabilistic framework.
32
588
589 5. Sensitivity analysis for Dolphin
590 a. Synopsis and the simulated track
591 Figure 10 shows the center position of Dolphin in the JMA best track.
592 Dolphin turned into a tropical storm west of the Mariana Islands at 0600
593 UTC on 11 December 2008. After turning sharply to the north around 0000
594 UTC on 15 December, it began to move northeastward on 16 December.
595 The simulated tracks of ensemble members from 1200 UTC 11 December
596 to 1200 UTC 15 December are overlaid with TC symbols in Fig. 10.
597 Simulated tracks generally exhibit early recurvature to the north, while TC
598 position varies greatly at the forecast time of 96 h. Ensemble spreads of the
599 forecasted TC central longitude and latitude at 1200 UTC on 15 December
600 are 1.63 and 1.33 degree (a distance of 172 km and 148 km), respectively.
601 Figure 11 shows that Dolphin was embedded in the easterly wind and
602 was located west of the anti-cyclonic flow at 1200 UTC on 11 December
603 (initial time of the sensitivity analysis). The anti-cyclonic flow and the
604 midlatitude trough intensified around 14 December and the simulated TCs
605 swiftly moved northeastward toward the direction of flow and the trough.
33
606
607 b. Ensemble-based sensitivity
608 Figure 12 shows the sensitivity field of the TC central longitude with
609 respect to the vorticity field at three vertical levels ( = 0.8445, 0.5060,
610 0.2879) for the verification time of 24, 48 and 96 h. The values of
611 sensitivity field at the verification time of 24 and 48 h are large as
612 compared to the sensitivity of the forecasted latitude of Shanshan. This
613 suggests that ensemble-mean TC position change is more sensitive to the
614 initial perturbation. The sensitivity with respect to the vorticity field is less
615 significant at = 0.99885 and = 0.101 and the same tendency is
616 observed in the sensitivity with respect to the divergence field as described
617 in section 4 (figures not shown).
618 The largest sensitivity appears at = 0.5060. The values at =
619 0.8445 and = 0.2879 are generally smaller than those at = 0.5060
620 regardless of the verification time. While the confidence level is generally
621 low compared to the case of Shanshan, some sensitivity signals have a
622 confidence level over 90% at the verification time of 96 h.
623 For the sensitivity field at the verification time of 24 h, the east-west
34
624 dipole pattern appears to be centered at the initial TC position. The positive
625 sensitivity area extends from the east of the initial position to southern
626 Taiwan in the middle troposphere at the verification time of 48 and 96 h.
o o 627 The significant signals are found around 120-130 E and 15-25 N, located at
628 the southern part of the westerly jet, which illustrates the influences of the
629 jet on the TC track.
630
631 c. Verification experiment
632 Four verification experiments are conducted to ensure that the
633 sensitivity signals in the middle troposphere are relevant to the TC position
634 at the verification time of 96 h. Referred to as P1, P2, N1 and N2, the
635 experiments correspond to the significant sensitivity signals around the TC.
636 The procedure of the four experiments is the same as that in section 4c, i.e.,
637 the positive vorticity perturbations (without divergence) from =
638 0.66175 to = 0.4082 are added to the initial field of the 100-member
639 ensemble run. The initial perturbations of the horizontal wind are shown in
640 Figures 13a-d. Based on the definition of sensitivity and Fig. 12f, we can
641 expect to observe an ensemble-mean eastward (westward) displacement
35
642 relative to CTRL in P1 and P2 (N1 and N2) experiments.
643 The expected ensemble-mean changes (LON and LAT) as a
644 response to the constant perturbation in the initial state are evaluated
645 according to the equation below:
1~LON~LON LONuv 646 Su v (11) DD
1~LAT~LAT LATuv 647 Su v (12) DD
S 648 where the subscript D represents the domain, and D is set to a constant
649 value of 630.
650
651 d. Results of the verification experiment
652 Figures 14a,b show ensemble-mean tracks of each experiment. It is
653 clear that ensemble-mean displacement at the verification time is consistent
654 with TyPOS. TC tends to move eastward in P1 and P2 experiments relative
655 to the CTRL, while the ones in the N1 and N2 run tend to move westward.
656 These tendencies are connected to earlier (later) recurvature in P1 and P2
657 (N1 and N2) experiments. Figures 14c,d show the frequency distributions
658 of longitude among different ensemble experiments. Although the
36
659 frequency distributions are heavily overlapped, the ensemble-mean value is
660 different among the different experiments.
661 Figure 14e illustrates the ensemble-mean TC position displacement
662 calculated in a nonlinear model. It is qualitatively consistent with the
663 evaluation by using TyPOS (Fig. 14f), with a significant signal only
664 slightly above 80%, indicating a difference from the case of Shanshan
665 which yields a rather quantitative result. Nevertheless, these results show
666 potential of TyPOS for long-term forecast sensitivity analyses. The
667 amplification factors with the verification time of t=96 h are 3.78 for N1,
668 1.01 for N2, 5.06 for P1 and 0.79 for P2. These figures correspond to the
669 large displacement in N1 and P1.
670
671 6. Discussion
672 a. Dependency on ensemble size and resolution
673 It is important to investigate the dependency of the sensitivity
674 analysis on the number of ensemble members. The sensitivity analysis is
675 conducted based on 400, 200 and 100 ensemble members for the same case
676 (Typhoon Shanshan) described in sections 3 and 4. Figure 15 shows the
37
677 sensitivity field at = 0.5060 with respect to the vorticity field,
678 superposed with the results of significance t-test. The north-south dipole
679 pattern at the verification time of 12 h and the horizontally tilted patterns
680 centered at the initial TC position are found in the sensitivity field with
681 fewer number of ensemble members.
682 However, some spurious features appear away from the TC center
683 and the confidence level of the signals decreases with a smaller number of
684 ensemble members. One encouraging result is that noise caused by the
685 decreasing number of ensemble members is not detected as significant in
o o 686 most of the cases, although some signals (e.g., 92 E and 32 N in Fig. 15l)
687 indicate a 90% confidence level.
688 So far, the reduced grid spacing is set to 540km, which is too coarse
689 to investigate fine scale structures. Another sensitivity analysis is
690 performed by halving the horizontal reduced-grid spacing to 270 km for
691 Typhoon Shanshan (2006). With the exception of the reduced-grid spacing,
692 the settings are the same as in the original one described in sections 3 and 4.
693 Figure 16 shows the sensitivity fields with respect to the vorticity fields.
694 The sensitivity fields are consistent with the findings shown in section 4 in
38
695 that the horizontally tilted pattern is found near the initial TC position.
696 However, the sensitivity field is contaminated by statistical noise and the
697 signals are seldom considered significant.
698 This suggests that TyPOS starting from large-scale perturbations can
699 serve as a promising method for both targeted observation operations and
700 research purposes, given the large number of ensemble members. In TyPOS,
701 an additional assumption is introduced to fully determine n-elements of the
702 sensitivity field from m realizations as explained in section 2a. Thus, the
703 sensitivity field is not likely to reflect the relationship between input
704 physical variables and the output scalar as the number of degrees-of-
705 freedom increases. Recent developments in large parallel computer systems
706 with thousands of processors make it possible to obtain sensitivity fields
707 from a very large number of nonlinear forward model runs with full physics
708 (Martin and Xue 2006). However, without enough computational resources,
709 this method provides less reliable features or is merely useful for capturing
710 some features in the further reduced-dimension space. In terms of the
711 operational ensemble forecast, it would be possible to use a basis function
712 corresponding to the fast developing mode as discussed below or to shrink
39
713 the domain size of initial perturbations along with the reduction of
714 ensemble size.
715
716 b. The use of EnKF-based perturbations
717 So far, large-scale random initial perturbations are used to illustrate
718 the feasibility of TyPOS. However, the use of faster developing
719 perturbations may help efficiently capture the sensitivity field with a
720 smaller number of ensemble members. Its importance in practical
721 application prompted us to conduct another sensitivity analysis based on
722 100 ensemble members initialized from EnKF-based perturbations for
723 Shanshan. The settings are the same as in section 4 except that the
724 sensitivity field is derived from Eqs. (3) and (4) by using EnKF-based
725 perturbations instead of large-scale uncorrelated perturbations.
726 Figure 16 shows the TC forecast tracks initialized from EnKF-based
727 perturbations. Ensemble spreads of the forecasted TC central longitude and
728 latitude at 0000 UTC 17 September are 1.22 and 1.40 degree (a distance of
729 114 km and 156 km), respectively. They are larger than those in section 4
730 presumably due to the faster development of perturbations and the larger
40
731 displacement at the initial time. Figure 17 shows the sensitivity of TC
732 forecast latitude with respect to the vorticity field. The values at =
733 0.8445 and = 0.5060 are larger than those at = 0.2879. Generally
734 speaking, the sensitivity is relatively strong north of 25N. In particular,
735 there is a pair pattern in the north of the initial TC position which is
736 inclined to the north with increasing altitude. These sensitivity signals are
737 quite different from those obtained in section 4, likely due to the property
738 of perturbations. Detailed discussion on this topic will be covered in future
739 works.
740 In order to verify that these sensitivity signals are relevant to the TC
741 motion, two ensemble experiments are performed as in sections 4c and 5c.
LAT 742 Initial vorticity perturbation with the magnitude of is added
743 in the “POSITIVE” experiment, while the initial vorticity perturbation with
LAT 744 the magnitude of is introduced to the other experiment
745 termed “NEGATIVE”. Here, is the ensemble spread of the vorticity at
746 the initial time, and is set to the constant value of 0.01 so that the
747 maximum change is about the size of the ensemble spread.
748 Fig. 18 shows that the changes of TC positions are consistent with
41
749 those obtained from the sensitivity field. The vortex displacement as a
750 response to the initial vorticity changes was located in the north-northeast
751 (south-southwest) direction as expected, while the distance of displacement
752 is slightly overestimated. It exhibits that 100 EnKF-based perturbations can
753 be utilized to detect the sensitivity of typhoon forecast positions through
754 TyPOS.
755
756 7. Concluding Remarks
757 Selection of the forecast metric is one of the critical issues in
758 sensitivity analyses associated with studies of TC motion. In this study, we
759 proposed a typhoon-position-oriented sensitivity analysis (TyPOS) for
760 ensemble forecast system. The sensitivity is interpreted as the slope of the
761 regression line (or its approximation) based on the ensemble simulation
762 between the initial perturbations and the resulted forecast scalar metric
763 representing the TC position.
764 As a first step, Typhoon Shanshan (2006) and Typhoon Dolphin
765 (2008) are used to demonstrate the applicability of TyPOS as a response to
766 large-scale initial perturbations. The sensitivity field of the TC central
42
767 latitude with respect to the vorticity field is maximized in the middle
768 troposphere. Interestingly, the sensitivity field is characterized by a dipole
769 pattern and the horizontally tilted patterns centered at the initial TC
770 position. The sensitivity signals related to the vorticity field in the middle
771 troposphere are stronger than those in other altitudes, while the sensitivity
772 related to the divergence field is less significant. This indicates that the
773 sensitivity signals shown by TyPOS are fairly consistent with the signals
774 obtained from existing methods, although there are still a few differences.
775 The sensitivity signals can be utilized to evaluate the changes in the
776 ensemble-mean TC position as a response to the changes in the initial
777 condition. They are also useful for determining which observations more
778 likely to objectively contribute to improving ensemble track forecasts. The
779 verification experiments confirm that sensitivity signals in TyPOS reflect
780 the displacement of the TC forecast position even at the verification time of
781 96 h.
782 As mentioned above, TyPOS is a useful method to resolve problems
783 in the choice of metrics. However, further improvements are needed
784 because TyPOS is still subject to a variety of uncertainties in specifying
43
785 initial perturbation magnitudes and patterns. In particular, limitations
786 resulting from the large-scale perturbations would make it difficult to
787 address the finer structure of the sensitivity field. Some noise arises as the
788 number of ensemble members decreases and also with a finer horizontal
789 grid spacing, although the t-test results serve as a tool to evaluate whether
790 the acquired signals are reliable or not.
791 The use of the EnKF-based perturbations may help efficiently
792 capture the sensitivity field with a smaller number of ensemble members
793 and reflect the sophisticated treatment of perturbation development through
794 a data assimilation system. Another sensitivity analysis based on 100
795 members initialized from EnKF-based perturbations is conducted for
796 Shanshan. It is encouraging to find that the ensemble-mean displacement of
797 TC forecast position is consistent with that expected by the sensitivity field.
798 However, the sensitivity for EnKF-based perturbations is clearly different
799 from that obtained for large-scale perturbations. Further study is required to
800 explain the factors leading to such a difference.
801 Other issues to be further investigated include the physical
802 interpretation of the signals, detailed comparison with other methodologies
44
803 and applications to real cases in which a bimodal distribution of TC
804 positions is obtained from ensemble runs. Despite these remaining
805 problems, the results shown here are meaningful because a distinctly
806 objective sensitivity analysis method for ensemble TC track forecasts is
807 proposed and verified. This method has potentials to contribute to further
808 reductions of ensemble forecast track errors in connection with targeted
809 observations, and to improved understanding of physical processes
810 associated with TC motion.
811
812 Acknowledgments.
813 Special thanks to J.-H. Chen, S.-G. Chen, G.-Y. Lien, M. Ueno, Y.-H.
814 Huang, N. Sugiura, and T. Unuma for their useful comments in developing
815 the current system. This work was funded by National Science Council
816 (NSC) grant 98-2111-M-002-008-MY3 and NSC100-2119-M-002-009-
817 MY3 in Taiwan. The figures were produced by the GFD-DENNOU Library.
45
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983
52
984 Figure captions
985 1. Schematic illustration of the sensitivity with respect to the variable of single-
986 degree-of-freedom. Angle brackets denote the ensemble-mean values.
987 2. The domains used in this experiment. The dashed line in (a) indicates TC
988 position from JMA best track for typhoon Shanshan. Closed circles indicate TC
989 positions at time intervals of 24 hours from 0000Z 11 September. (b) Same as (a) but
990 for typhoon Dolphin. Closed circles indicate the TC positions every 24 hours from
991 0000 UTC 12 December 2008.
992 3. Time schedule of the sensitivity analysis for (a) typhoon Shanshan and (b)
993 typhoon Dolphin.
994 4. TC position from JMA best track for typhoon Shanshan plotted over simulated
995 tracks. Circles indicate the TC positions every 6 hours from 2100 UTC 10 September,
996 while solid ones indicate the TC positions every 24 hours from 0000 UTC 11
997 September. The TC symbols and thick solid lines indicate the ensemble-mean
998 simulated track from 0000 UTC 15 September to 0000 UTC 17 September. The gray
999 lines correspond to the tracks of each member. The cross marks (x) indicate the
1000 simulated positions of TC center at 0000 UTC 17 September.
1001 5. Ensemble-mean geopotential height (shadings) and horizontal wind (vectors) at
1002 (a)-(c) 300 hPa and (d)-(f) 500 hPa based on 800 member forecasts at (a)(d) 0000
1003 UTC 15 September, (b)(e) 0000 UTC 16 September and (c)(f) 0000 UTC 17
1004 September. Unit vectors for horizontal wind fields (m s-1) are shown on the rhs of each
1005 panel. The TC symbol indicates the ensemble-mean simulated position at each time.
1006 6. Ensemble-based sensitivity of typhoon Shanshan's central latitude with respect
1007 to the vorticity field (Unit in degree s). Circles in the panels indicate the results of
53
1008 significance t-test at each reduced grid point. The open TC symbols indicate the
1009 ensemble-mean TC positions at the initial time and closed TC symbols indicate those
1010 at the verification time. Left color bar indicates the values in the left column and right
1011 color bar indicates the values in the middle and right columns.
1012 7. Horizontal velocity perturbations relative to CTRL in (a) WTILT, (b) ETILT and
1013 (c) TROUGH. Unit vectors for horizontal wind fields (m s-1) are shown on the rhs of
1014 each panel. Vectors that are too small are not plotted. The open TC symbol indicates
1015 the position of typhoon Shanshan at the initial time, while the closed one indicates the
1016 ensemble-mean position at the verification time in the CTRL run.
1017 8. Time evolution of horizontal velocity perturbations on 500 hPa in ETILT_m001
1018 relative to CTRL_m001 at (a) 0, (b) 12, (c) 24 and (d) 48 h. Unit vectors for horizontal
1019 wind fields (m s-1) are shown on the rhs of each panel. Shadings indicate the
1020 geopotential height in CTRL_m001. The open TC symbol indicates the TC position in
1021 ETILT_m001, while the closed one indicates that of CTRL_m001.
1022 9. (a) Ensemble-mean TC track for CTRL (black), WTILT (red), ETILT (blue) and
1023 TROUGH (green). (b) Frequency distributions of the TC central latitude at the
1024 verification time of 48 h for the results of the CTRL, WTILT, ETILT and TROUGH
1025 runs. Vertical lines denote the ensemble-mean TC central latitude corresponding to the
1026 same line-color. (c) Ensemble-mean displacement of TC center position (relative to the
1027 CTRL run) at the verification time obtained from the nonlinear numerical model run.
1028 Horizontal and vertical axes indicate the displacement in longitude and latitude (Units
1029 in arc minute), respectively. (d) Same as (c), but for those evaluated by using TyPOS.
1030 10. TC position from JMA best track for Dolphin plotted over simulated tracks.
1031 Circles indicate the TC positions every 6 hours from 0600 UTC 11 December, while
54
1032 solid ones indicate the TC positions every 24 hours from 0000 UTC 12 December. The
1033 TC symbols and thick solid lines indicate the ensemble-mean simulated track from
1034 1200 UTC 11 December to 1200 UTC 15 December. The gray lines correspond to the
1035 tracks of each member. The cross marks (x) indicate the simulated positions of TC
1036 center at 1200 UTC 15 September.
1037 11. Same as Fig. 5 but for ensemble-mean geopotential height and horizontal wind
1038 around Dolphin on (a)(d) 1200 UTC 11 December, (b)(e) 1200 UTC 13 December and
1039 (c)(f) 1200 UTC 15 December.
1040 12. Same as Fig. 6 but for the sensitivity of Dolphin's longitude at the verification
1041 time of 24, 48 and 96 h.
1042 13. Same as Fig. 7 but for the verification experiment on the sensitivity for Dolphin:
1043 (a) P1, (b) P2, (c) N1, and (d) N2.
1044 14. (a) Ensemble-mean TC tracks of CTRL (black), P1 (red) and P2 (orange). (b)
1045 Same as (a) but for CTRL, N1 (cyan) and N2 (blue) experiments. (c) Frequency
1046 distributions of the TC central longitude at the verification time for CTRL, P1 and P2
1047 runs. (d) Same as (c) but for CTRL, N1 and N2 experiments. (e)(f) Same as Figs. 9c,d
1048 but for Dolphin at the verification time of 96 h.
1049 15. Sensitivity with respect to the vorticity field at = 0.5060 using (a)-(c) 800
1050 (d)-(f) 400 (g)-(i) 200 (j)-(l) 100 ensemble members. The metric represents the TC
1051 central latitude at (a)(d)(g)(j) 12, (b)(e)(h)(k) 24 and (c)(f)(i)(l) 48 hours.
1052 16. Same as Fig. 6 but for the sensitivity analysis with a reduced-grid spacing of 270
1053 km.
1054 17. Same as Fig. 4 except that the ensemble simulations are initialized from EnKF-
1055 based perturbations.
55
1056 18. Same as Fig. 6 except that the ensemble simulations are initialized from EnKF-
1057 based perturbations.
1058 19. Same as Fig. 9 but for the POSITIVE and NEGATIVE experiments.
56
1059 1060 1061 Fig. 1. Schematic illustration of the sensitivity with respect to the variable 1062 of single-degree-of-freedom. Angle brackets denote the ensemble-mean 1063 values. 1064 1065
57
1066 1067
1068 1069 Fig. 2. The domains used in this experiment. The dashed line in (a) 1070 indicates TC position from JMA best track for typhoon Shanshan. Closed 1071 circles indicate TC positions at time intervals of 24 hours from 0000Z 11 1072 September. (b) Same as (a) but for typhoon Dolphin. Closed circles indicate 1073 the TC positions every 24 hours from 0000 UTC 12 December 2008. 1074
58
1075 1076 Fig. 3. Time schedule of the sensitivity analysis for (a) typhoon Shanshan 1077 and (b) typhoon Dolphin. 1078 1079 1080 1081
59
1082 1083 Fig. 4. TC position from JMA best track for typhoon Shanshan plotted over 1084 simulated tracks. Circles indicate the TC positions every 6 hours from 2100 1085 UTC 10 September, while solid ones indicate the TC positions every 24 1086 hours from 0000 UTC 11 September. The TC symbols and thick solid lines 1087 indicate the ensemble-mean simulated track from 0000 UTC 15 September 1088 to 0000 UTC 17 September. The gray lines correspond to the tracks of each 1089 member. The cross marks (x) indicate the simulated positions of TC center 1090 at 0000 UTC 17 September. 1091
60
1092 1093 1094 Fig. 5. Ensemble-mean geopotential height (shadings) and horizontal wind 1095 (vectors) at (a)-(c) 300 hPa and (d)-(f) 500 hPa based on 800 member 1096 forecasts at (a)(d) 0000 UTC 15 September, (b)(e) 0000 UTC 16 September 1097 and (c)(f) 0000 UTC 17 September. Unit vectors for horizontal wind fields 1098 (m s-1) are shown on the rhs of upper and lower panels. The TC symbol 1099 indicates the ensemble-mean simulated position at each time.
61
1100 1101 1102 Fig. 6. Ensemble-based sensitivity of typhoon Shanshan's central latitude 1103 with respect to the vorticity field (Unit in degree s). Circles in the panels 1104 indicate the results of significance t-test at each reduced grid point. The 1105 open TC symbols indicate the ensemble-mean TC positions at the initial 1106 time and closed TC symbols indicate those at the verification time. Left 1107 color bar indicates the values in the left column and right color bar 1108 indicates the values in the middle and right columns. 1109 1110
62
1111 1112 Fig. 7. Horizontal velocity perturbations relative to CTRL in (a) WTILT, 1113 (b) ETILT and (c) TROUGH. Unit vectors for horizontal wind fields (m s- 1114 1) are shown on the rhs of each panel. Vectors that are too small are not 1115 plotted. The open TC symbol indicates the position of typhoon Shanshan at 1116 the initial time, while the closed one indicates the ensemble-mean position 1117 at the verification time in the CTRL run. 1118
63
1119 1120 Fig. 8. Time evolution of horizontal velocity perturbations on 500 hPa in 1121 ETILT_m001 relative to CTRL_m001 at (a) 0, (b) 12, (c) 24 and (d) 48 h. 1122 Unit vectors for horizontal wind fields (m s-1) are shown on the rhs of each 1123 panel. Shadings indicate the geopotential height in CTRL_m001. The open 1124 TC symbol indicates the TC position in ETILT_m001, while the closed one 1125 indicates that of CTRL_m001.
64
1126 1127 Fig. 9. (a) Ensemble-mean TC track for CTRL (black), WTILT (red), 1128 ETILT (blue) and TROUGH (green). (b) Frequency distributions of the TC 1129 central latitude at the verification time of 48 h for the results of the CTRL, 1130 WTILT, ETILT and TROUGH runs. Vertical lines denote the ensemble- 1131 mean TC central latitude corresponding to the same line-color. (c) 1132 Ensemble-mean displacement of TC center position (relative to the CTRL 1133 run) at the verification time obtained from the nonlinear numerical model 1134 run. Horizontal and vertical axes indicate the displacement in longitude and 1135 latitude (Units in arc minute), respectively. (d) Same as (c), but for those 1136 evaluated by using TyPOS.
65
1137 1138 Fig. 10. TC position from JMA best track for Dolphin plotted over 1139 simulated tracks. Circles indicate the TC positions every 6 hours from 0600 1140 UTC 11 December, while solid ones indicate the TC positions every 24 1141 hours from 0000 UTC 12 December. The TC symbols and thick solid lines 1142 indicate the ensemble-mean simulated track from 1200 UTC 11 December 1143 to 1200 UTC 15 December. The gray lines correspond to the tracks of each 1144 member. The cross marks (x) indicate the simulated positions of TC center 1145 at 1200 UTC 15 September.
66
1146 1147 Fig. 11. Same as Fig. 5 but for ensemble-mean geopotential height and 1148 horizontal wind around Dolphin on (a)(d) 1200 UTC 11 December, (b)(e) 1149 1200 UTC 13 December and (c)(f) 1200 UTC 15 December.
67
1150 1151 Fig. 12. Same as Fig. 6 but for the sensitivity of Dolphin's longitude at the 1152 verification time of 24, 48 and 96 h. 1153 1154
68
1155 1156 Fig. 13. Same as Fig. 9 but for the verification experiment on the sensitivity 1157 for Dolphin: (a) P1, (b) P2, (c) N1, and (d) N2.
69
1158 1159 Fig. 14. (a) Ensemble-mean TC tracks of CTRL (black), P1 (red) and P2 1160 (orange). (b) Same as (a) but for CTRL, N1 (cyan) and N2 (blue) 1161 experiments. (c) Frequency distributions of the TC central longitude at the 1162 verification time for CTRL, P1 and P2 runs. (d) Same as (c) but for CTRL, 1163 N1 and N2 experiments. (e)(f) Same as Figs. 9c,d but for Dolphin at the 1164 verification time of 96 h.
70
1165 1166 Fig. 15. Sensitivity with respect to the vorticity field at = 0.5060 using 1167 (a)-(c) 800 (d)-(f) 400 (g)-(i) 200 (j)-(l) 100 ensemble members. The metric 1168 represents the TC central latitude at (a)(d)(g)(j) 12, (b)(e)(h)(k) 24 and 1169 (c)(f)(i)(l) 48 hours.
71
1170 1171 Fig. 16. Same as Fig. 6 but for the sensitivity analysis with a reduced-grid 1172 spacing of 270 km. 1173 1174
72
1175 1176 1177 Fig. 17. Same as Fig. 4 except that the ensemble simulations are 1178 initialized from EnKF-based perturbations. 1179
73
1180 1181 Fig. 18. Same as Fig. 6 except that the ensemble simulations are 1182 initialized from EnKF-based perturbations. 1183 1184 1185 1186
74
1187 1188 1189 Fig. 19. Same as Fig. 9 but for the POSITIVE and NEGATIVE experiments. 1190
75