**2.1 Background**

With limited range of predictability at the convective scale and short timescale and the complex variation of nature, predicting tropical cyclone (TC) tracks and intensity is one specific example that demonstrates vividly the sensitivity of numerical models to uncertainties in the atmosphere [26, 38, 39]. The inherent uncertainties associated with our current incomplete understanding of model physical processes or numerical approximations often lead to large errors in track and intensity forecast, especially at the lead times longer 3 days or under circumstances interacting with uneven terrain or complicated vortex mergers [40–42]. Currently, the US Joint Typhoon Warning Center (JTWC) showed that the official track errors in the North Western Pacific (WPAC) basin are as high as 220 km at 3-day estimation and 450 km at 5-day estimation. Likewise, the intensity forecast errors make no headway since no significant update was taken at all forecast ranges during the last 30 years. The recent effort to calculate uncertainty in TC forecasts is based on the ensemble prediction systems. Generally, there are 3 major special techniques to develop an ensemble forecast system include: (1) use the different initial conditions obtained from a posterior analysis error distribution (the Monte-Carlo ensembles) for one specific model, (2) Use a single initial condition for multiple different prediction models; and (3) use combine both dissimilar initial conditions and different prediction models.

The breeding ensemble approach in the first direction was first implemented in the operational Global Forecasting System (GFS) at the National Center for Environmental Prediction (NCEP, by Toth and Kalnay in 1993 and 1997 [43, 44], hereinafter TK93 and TK97, respectively) in 1993, and then became more popular and more applied in practice. The breeding method continuously employed previous cycles to calculate the fastest growing instabilities and then normalized these errors vectors into the so-called the bred vectors. This procedure could allow projecting the fastest growing modes onto the calculated bred vectors in a shade of perturbations in each breeding cycle. Likewise, a similar ensemble forecasting technique generating singular vectors instead of bred vectors is implemented in the European Center for medium-term weather forecast (ECMWF) in early 1992 [45, 46]. Although theoretically, the fastest growing modes should be projected onto bred vectors (at the far limit of the backward Lyapunov vectors), the experimental results retrieved from the TK93's breeding method indicate that the produced TC ensemble tracks could be very similar to each other, i.e., the spread of the system was relatively narrow (**Figure 4**). One possible explanation for such small ensemble dispersion is because the bred vectors collapsed into a similar dominant direction after several cycles, which is not an uncommon issue (e.g., see [48, 49]). The singular vectors display the fastest growing modes in terms of orthogonal directions within a short-range interval (via a tangential linear model). In contrast, the bred vectors are some extent equivalent to the leading Lyapunov vectors in a nonlinear finite-amplitude method [43, 50]. This method allows the bred vectors collapsing afterwards, and becoming linearly independent (non-orthogonal) in the presence of the lower dimension attractor [48, 51].

By consider both the spatial–temporal variations of the scaling vector at each cycle, the bred vectors could capture the local growing directions and thus allow for

#### **Figure 4.**

*Schematic design of the TC-breeding ensemble technique: a) illustration of generating environmental bred vectors and TC bred vectors during a warm start cycle (from 24 h to 18 h before the target forecast date) b) illustration of making six pairs of lagged-averaged forecast (LAF) vectors for the first cold start cycle used in the TC-breeding ensemble [47].*

larger ensemble spread [44, 52]. However, fast convective instabilities still quickly saturate after several breeding cycles, especially within the region where the atmospheric dynamics are complicated [53, 54]. Unfortunately, the TCs system act as such complicated phenomenon with multi-scale interactions. It is expected that the instability within the storm's inner-core should behave differently as compared to the outer environmental region. Previous studies (see [55– 57] indicated that perturbations inside the TC inner-core area often develop and propagate rapidly in the manner of vortex Rossby waves with typical time scale of 12–24 h. Contrarily, the large-scale environmental related-perturbations propagate in a much smaller time scale, often manifested in terms of gravity waves and mesoscale clustering along the most unstable regions [58]. Representing the interaction between the faster storm-scale instabilities and slower large-scale environment is a big challenge in constructing an ensemble breeding system for TC forecasts. Hence, this section presented a new TC breeding approach that could help improve this challenge.

## **2.2 TC breeding method**

Though the first breeding method presented by TK93 could capture the trend of fastest-growing during a finite time window, the real world TCs have a finite life cycle. Due to the high-resolution regional modeling required large computation, TC predicting models are typically spark off only when their TCs are already first reported in the warning centers, because it is a challenge to conserve a continuous ensemble of breeding cycles with taking much computational capacity for a long time. Therefore, TK93's breeding scheme could not instantaneously acquire

*Application of Kalman Filter and Breeding Ensemble Technique to Forecast the Tropical… DOI: http://dx.doi.org/10.5772/intechopen.97783*

directions of most unstable modes during the earliest cycles. Moreover, utilizing only single re-scaling factor for both storm inner-core region and ambient environment with distinguished spatio-temporal scales does not enclose all mesoscale unstable nodes associated with TC vortex dynamics for which perturbations at different spatial–temporal scales grow at different rates [59]. Hence, it is necessary to change the rescaling factors following both the flow and the scales of instabilities [13].

Since our TC-breeding approach is focus on characterizing not only the storm-scale but also the large-scale unstable modes and their mutual interaction, there are two different scaling factors for these scale modes separately.

In the TK93's breeding extended design for TC predicting (hereafter known as the TC-breeding method or TCB), steps to make the TC-bred seeds as follows:

Step 1. Remove the GFS original vortex and insert a bogus vortex into the GFS initial condition to obtain a new first guess *x<sup>a</sup>*. In which, the bogus vortex is dynamical constructed based on the observed minimum sea-level pressure and maximum surface wind, using the Australian Bureau of Meteorology's Tropical Cyclone Limited Area Prediction System (TC-LAPS) package. This step is essential due to the weaknesses of the original GFS vortex in coarse resolution;

Step 2: adding and subtract a bred seeds di (i = 1,2, … ,6), then we have 6 first guess x1<sup>a</sup> <sup>i</sup> = xa + di (positive sector) and x2<sup>a</sup> <sup>i</sup> = x<sup>a</sup> - di (negative sector)

Step 3: Run 6-hour lead time forecasts for both positive and negative sectors Step 4: Separate 6-h forecasts (operators **Sm** and **S<sup>v</sup>** ) of positive sector (x1<sup>f</sup> i) and negative sector (x2f i) from the previous breeding ensemble forecasts into an environmental component **S**m**x1**<sup>f</sup> <sup>i</sup> and **S**<sup>m</sup> **x2**<sup>f</sup> i) and a vortex component (**S**<sup>v</sup> **x1**<sup>f</sup> <sup>i</sup> and **S**<sup>v</sup> **x2**<sup>f</sup> i, **Figure 4a**).

Step 5: Find difference (operator H, **Figure 4a**) of each set of bred vector pairs (or seeds) from previous 6-h cycles to obtain environmental bred vectors

$$\mathfrak{m}\_{i} = \mathop{\mathbf{S}^{\mathbf{m}}}\limits\_{\mathbf{n}} \left( \boldsymbol{p}\_{i}^{f} - \boldsymbol{n}\_{i}^{f} \right) \\
\text{and the TC bed vectors } \boldsymbol{\nu}\_{i} = \boldsymbol{\mathcal{S}}^{\boldsymbol{\nu}} \left( \boldsymbol{p}\_{i}^{f} - \boldsymbol{n}\_{i}^{f} \right);$$

Step 6: normalize the environmental bred vectors (by using a normalizing operator **Cm**) to obtain a new set of normalized bred vector Cmmi., then use an orthogonal operator T to obtain an orthogonal set of environmental bred vectors **TCmmi.** Here, the environmental re-scaling operator **Cm** acting on a vector **v** is defined as:

$$\mathbf{C}^{m}\mathbf{v} \equiv \Lambda \frac{\mathbf{v}}{||\mathbf{v}||},\tag{7}$$

With the scaling factor for the environmental perturbations given by

$$
\Lambda = \left[\frac{1}{2\Gamma} \int\_D \left[ U'^2 + V'^2 + \frac{C\_p}{T} T'^2 \right] dz dS \right]^{\frac{1}{2}},\tag{8}
$$

And the norm ||.|| taken to be the energy norm as follows:

$$\left\|\left|\mathbf{v}\right|\right\|^2 = \left[\frac{1}{2\Gamma}\int\_{D} \left[u'^2 + v'^2 + \frac{C\_p}{T\_0}T'^2\right]dzdS\right]^{\frac{1}{2}},$$

where *Γ* is the normalized factor proportional to the model domain volume, Cp = 1006 J kg�<sup>1</sup> K�<sup>1</sup> ; T0 = 300 K), D is the model domain area after the model vortex was filtered, *<sup>U</sup>*<sup>0</sup> <sup>¼</sup> *<sup>V</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup>*:*8 ms�<sup>1</sup> , and *T*<sup>0</sup> ¼ 0*:*7K. These values are established in the study of Saito et al. [13], which are also consistent with the

previous estimation by Wang and Bishop [60]; Repeat step 6 for the TC bred vectors with operator Cx to obtain a set of orthogonal TC bred vectors **TCx**vi.

Step 7. Make a new pairs of breeding members by adding/subtracting the environmental and TC bred components into the analysis **x<sup>a</sup>** , i.e.,

**x1**<sup>a</sup> <sup>i</sup> = **x**<sup>a</sup> + **TC**m**m**<sup>i</sup> + **TC**x**v**<sup>i</sup> and **x2**<sup>a</sup> <sup>i</sup> = **x**<sup>a</sup> - **TC**m**m**<sup>i</sup> - **TC**x**v**i. Step 8: Run 6-hour lead time forecasts of all positive and negative pair to serve as the first guess for the next analysis cycle;

Step 9. Repeat step 1–9 for the next analysis cycle;

Noted that the above steps are taken only for the warm-start mode in which the ensemble breeding forecasts in the analysis procedure has been available since the previous. For the cold start cycle at which the "INVEST" information for a tropical depression is first issued, it is apparent that the bred vectors are unknown yet, therefore the ensemble initialization requires a different procedure.

One can do the cold-start in countless ways, for example using a random Gaussian noise with a prescribed error distribution, or directly use of the global GFS ensemble forecasts. For simplicity, the approach uses the 6-h difference from previous GFS short-range forecasts for all of the cold-start ensembles. This approach, known as lagged-averaged forecasts (LAF) from Kalnay [58], can quickly capture the most unstable modes in the model, thus allowing the breeding ensemble to speed up the dynamically representation to the environment. The combination of these short-range forecasts can generate a predefined number of seeds from which the breeding ensemble can be obtained. Consider, for example, a configuration of the breeding ensemble experiments requires a total of six bred vectors. Those bred vectors are initialized by taking six 6-h differences of the previous -36 h, 24 h, 18 h, 12 h, and -6 h forecasts that are all taken from the cold start ensemble (**Figure 4b**). The control forecast preprocessed directly from the GFS analysis then adds/subtracts the given bred vectors to create an ensemble of total 13 members for subsequent ensemble forecasts.

### *2.2.1 Example 2*

The TC Breeding method has been implemented the Regional Atmospheric Modeling System (RAMS, version 6.0) model to forecast the TC track in the WPAC basin. In this study, the model domain is a region limited by 5°S–35°N and 100–150°E. This domain is sufficiently large to cover most of the tropical cyclone that formed in the WPAC basin and part of the Tibetan plateau that affects the large-scale steering flow of the TC tracks in the WPAC basin. The model integration time is 60s, and the experimental maximum lead times were up to 5 days (120 h). The convection parameterization schemes used among all experiments included a Kuo scheme, a Kain–Fristch scheme (original) and the new Kain–Fristch scheme (modified version). Initial data for model input were taken from the National Center for Environmental Prediction (NCEP) Global Forecast System (GFS) operational forecast with resolution of 1° 1°. A set of 14 tropical cyclones between 2009 and 2011 in the WPAC basin were chosen for testing the TCB method (**Table 1**).

A series of 120 h forecasts for all storms in **Table 1** were conducted, using the aforementioned TC-breeding technique. The retrospective experiments include six positive/negative pairs and a control forecast (total 13 members). Here, the control forecasts are just the integrated results from the RAMS model with initial conditions where the original GFS forecasts adding a bogus vortex to make sure the model storm intensity was equivalent to the reality. The experiments used the default mode of the TC-LAPS package in which the constructed bogus vortex that had the horizontal resolution of 1° 1°, and the isobaric vertical coordinates with 26


*Application of Kalman Filter and Breeding Ensemble Technique to Forecast the Tropical… DOI: http://dx.doi.org/10.5772/intechopen.97783*

**Table 1.**

*List of storms between 2009 and 2011 in the WPAC basin used in this study.*

pressure levels as 1000, 975, 950, 925, 850, 800, 750, 700, 650, 600, 550, 500, 450, 400, 350, 300, 250, 200, 150, 100, 70, 50, 30, 20, and 10 h Pa. The variables to characterize the bogus vortex included sea level pressure (P), horizontal wind components (U, V), temperature (T), geopotential height (H), and relative humidity (RH). The cycles of all breeding ensemble were every 12 h, using the TCB method described above. Besides, for convenience, the domain of storm-scale perturbation was also fixed with enclosing area of 1000 km 1000 km, centered at the vortex location. Three perturbed state variables in the model at the breeding cycles included the horizontal winds and potential temperature at all pressure levels. These cycles (12-h interval) are suitable enough to capture both the fast-growing weather signals at the micro- to meso- scale and the slower baroclinic modes at larger scales.

Results indicated that TCB method helps reduce the track errors. The improvement is approximately 10% reduction in the track forecast errors at the 4- to 5- day lead times as compared to deterministic forecasts integrated from GFS derivedinitial conditions. While the improvement is not significant at shorter lead time (1–3 lead times, **Figure 5**).

Besides, the major difference between this TCB method and the original approach of TK93 is the dissimilarity in treatment of perturbations between largescale environments and storm-scale inner-core, which are then orthogonalized in different manners. For the environmental perturbations in all experiments, a volume limited by [100–150°E] [5°S–35°N] [1000–10 h Pa] is chosen. In facts, the domain size does not have significant impact on the magnitude of EBVs, when it is important for the TBVs in some aspects. That is because storms do not always have a fixed size, thus the use of a predefined domain with a constant radius of 1000 km may not fully characterize the storm-scale TC-like vortex. One can design a suitable

**Figure 5.**

*Track forecast distance errors between the TK93's original breeding ensemble (striped column) forecasts and the deterministic control forecasts (gray columns) for 2009–2011 seasons using the RAMS model [47].*

adaptive storm domain to optimize the effectiveness of the TCB method. However, with a coarse resolution of 30 km, the adaptive approach cannot capture the true detailed TC inner-core structure. For more simplicity of the experiment design in this study, the filtering domain has a fixed horizontal radius of 1000 km in all experiments with a warning that this constant size could be a caveat for very broad TCs. It should be noted also that the control analysis would add or subtract the bred vectors, and a potential drift of the control run from the actual states may shift the entire ensemble further from the truth after several cycles. However, with the

*Application of Kalman Filter and Breeding Ensemble Technique to Forecast the Tropical… DOI: http://dx.doi.org/10.5772/intechopen.97783*

#### **Figure 6.**

*Rate of track forecast error base on the number of the ensemble members for forecast ranges: The 24-h (diamond), 48-h (circle), 72-h (triangle), 96-h (times), and 120-h (square). The reducing rate is determined as the difference of the track errors when adding newly member to the system at each lead time [56].*

integration lead time of only 12 h at each cycle from the control forecast using the GFS forecasts, such a drift is not a big issue and the ensemble thus always maintains their close trajectory to the truth at every initial time.

Sensitivity experiments showed that the best results with 30 ensemble members are adequate to construct a TCB technique. By gradually increasing the number of ensemble members, the rate of reducing track error per newly added member becomes saturated after reaching the number of 30 ensemble members (**Figure 6**). This saturation of the track errors could link to the maximum information that the orthogonalization of the bred vectors to be obtained after the system reaches its noise level. Otherwise, adding more ensemble member could provide no further benefit to the system, it could even slow down the computation. It should be noted that the 30-km resolution of all ensemble experiments does not fully verify the necessity of separating the treatments for the storm-scale bred vectors and the large-scale bred vectors in distinguished manners. Theoretically, one could design the experiments with higher resolution to further assess the sensitivity of the breeding ensemble technique for more precise experiments, but this would require a large amount of computational and storage resources beyond our current capability. Although this minor problem about resolution, the overall track forecast improvement with the TCB approach suggests that this approach could somehow shed light on ensemble TC track forecast, especially under the circumstances where the observational information is not enough to execute more complex data assimilation steps in real-time forecasting systems.

## **3. Conclusions**

This chapter has presented several techniques to improve the predictive quality of tropical cyclone formation and trajectory. For the forecast of TCs formation, the LETKF algorithm and its implementation in the WRF model and the Vortex tracking method have been introduced. Results in example 1 show that due to a better approach in capturing the real world monsoon trough by assimilating augmented

observations available during the early stages of TC Wutip, the WRF-LETKF model had provided better forecasts about the formation location and timing of typhoon Wutip in comparison to the forecasts that used initial conditions directly from GFS global model. Besides, the results from this study also show the CIMSS-AMV data played a vital role in improving the information of the large-scale environment required for TC formation that one should consider for real-time TC forecasts. For the tropical cyclone track forecasts, a breeding ensemble technique is introduced. This technique is developed based on the original breeding method (TK93). Experiments with 14 TCs (**Table 1**) in **example** showed a promising reduction of track forecast errors by using the TCB technique, especially at 4–5 days forecast range.

However, both the track forecasts by TCB method and the control forecasts are similar in the patterns of cross- and along track forecast errors. This indicated that model inherent errors also are a significant contributor to the track forecast errors that the TCB method is unable to eliminate. Sensitivity experiments of adding gradually each ensemble members exhibit further that the increasing number of members could reduce the track forecast errors, but reduction rate saturates when the number reaches 30 dues to the inefficiency of the TCB method in orthogonalizing bred vectors. However, while the TCB method cannot eliminate model inherent errors related to inadequate representation of sub-grid scales when using only parameterizations of physical processes in the RAMS model or the inefficient model resolution, this method could somehow optimize the use of the breeding ensemble technique for tropical cyclone track forecasts in real-time forecasting systems which do not require high computational resources.
