**1. Introduction**

## **1.1 The forecast of TCs formation using the ensemble Kalman filter**

Among several approaches for real-time monitoring and forecasting of TC formation, direct numerical products from global and regional weather prediction models appear to be the most reliable at present, despite their inherent limitations and uncertainties (e.g., see [1, 2]). The skillful performance of TC formation forecasts by numerical models has been well documented in many previous studies [3–8]. This achievement of numerical models is attributed to a variety of advanced research on upgrading parameterizations of physics, resolution, computational resources, and data assimilation schemes [1, 9]. Among several different assimilation schemes, the ensemble Kalman filter (EnKF) has been extensively applied to many practical problems in recent years due to its straightforward implementation for TC forecast applications [10–16]. The use of EnKF for TC forecasting applications is increasingly

popular, given the current availability of real-time flight reconnaissance data that allows direct assimilation of airborne observations without the need of a bogus vortex (e.g., see [10, 14–16]). In essence, the development of EnKF addressed the problem when using variational assimilation schemes in which the background covariance matrix is allowed to be time-dependent. Hence, the model would adapt better with fast-evolving and complicated dynamical systems such as TCs or mesoscale convective systems [12, 17–19]. There is an efficient method of implementing an Ensemble Kalman Filter (EnKF), which was called a Local Ensemble Transform Kalman Filter (LETKF) scheme.

#### **1.2 Data assimilation system**

In this section, the LETKF algorithm proposed by Ott et al. [20] and Hunt et al. [21] is adopted and implemented for the WRF Model. The primary usage of the LETKF algorithm is utilizing the background ensemble matrix as an operator to transform state vectors from a model space spanned by the model grid points within a local patch to an ensemble space spanned by ensemble members. The procedures for calculating matrix and generating the ensemble analyses are executed in this low dimension ensemble space at every single grid point. In this sense, the LETKF scheme allows the ensemble space to be performed locally and in parallel efficiently for practical problems, especially when carrying out a large-volume of data (e.g., see [8, 11, 12, 22–25]).

With its promising capability, LETKF has been implemented in the WRF Model (V3.6, hereafter referred to as the WRF-LETKF system). With an aim to practical forecasting applications, all the observations utilizing in the WRF – LETKF scheme are preprocessing in a quality control taken by the WRF data assimilation (WRFDA) component. In addition, the WRFDA component also generates lateral boundary conditions for each ensemble member once obtained the analysis update. Hence, each ensemble member possesses its own boundary dynamically consistent with its own updated initial conditions. More details in the WRF-LETKF design can be found in [12, 24]. The focal point here is how the ensembles with and without augmented observations perform. In this regard, the relative differences in the output among these ensembles can derive the main effects of additional augmented observations.

To begin the ensemble system, a first-guess background is generated in a coldstart ensemble by first using 3DVAR scheme to produce an analysis from a GFS initial condition. Random perturbations with standard deviations of 1 ms<sup>1</sup> for the wind field, 1 K for temperature, and 1 <sup>10</sup><sup>3</sup> kgkg<sup>1</sup> for specific humidity at all model grid points are then added to the 3DVAR-generated analyses for the coldstart ensemble. The 3DVAR-generated analyses as initial conditions for 12-h running in a manner that the outputs from these 12-h integrations can be subsequently used as a *warm-start* background for the LETKF ensemble assimilation in the next cycle. Note that these random perturbations are added only for the first cold-start cycle to create a background ensemble. All subsequent warm-run cycles use the WRF-LETKF 12-h forecasts as a background ensemble and so no additional random noises are necessary. The newly generated analysis perturbation ensemble at each cycle is then added to the GFS analysis to produce the next ensemble initial conditions when run in the cycling mode as described in [26].

#### **1.3 The LETKF algorithm**

To get a better understanding of the LETKF algorithm mentioned in the previous sub-section. A brief description of this LETKF algorithm that developed by Kieu et al. [12] has been presented below:

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

Assume that give a background ensemble {**x**b ið Þ: i = 1, 2, … , k}, where k is the number of ensemble members (assuming that the analysis is taken one at a time, so the time index is not included). According to Hunt et al. [21], an ensemble mean **x**� <sup>b</sup> and an ensemble perturbation matrix **X**<sup>b</sup> are defined respectively as:

$$
\bar{\mathbf{x}}^{\mathbf{b}} = \frac{1}{k} \sum\_{i=1}^{k} \mathbf{x}^{\mathbf{b}(i)}.
$$

$$
\mathbf{X}^{\mathbf{b}} = \left\{ \mathbf{x}^{\mathbf{b}(1)} - \bar{\mathbf{x}}^{\mathbf{b}}, \mathbf{x}^{\mathbf{b}(2)} - \bar{\mathbf{x}}^{\mathbf{b}}, \dots, \mathbf{x}^{\mathbf{b}(k)} - \bar{\mathbf{x}}^{\mathbf{b}} \right\}.\tag{1}
$$

Let **x** = **x**�<sup>b</sup> + **X**<sup>b</sup>**w**, where w is a local vector in the ensemble space, the local cost function to be minimized in the ensemble space is given by:

$$\hat{j}(\mathbf{w}) = (\mathbf{k} - \mathbf{1})\mathbf{w}^{T} \left\{ I - \left(\mathbf{X}^{b}\right)^{T} \left[\mathbf{X}^{b} \left(\mathbf{X}^{b}\right)^{T}\right]^{-1} \mathbf{X}^{b} \right\} \mathbf{w} + J \left[\mathbf{x}^{\mathrm{b}} + \mathbf{X}^{\mathrm{b}} \mathbf{w}\right],\tag{2}$$

Where *<sup>J</sup>*[**x**<sup>b</sup> <sup>þ</sup> **<sup>X</sup>**<sup>b</sup>**w**] is the cost function in the model space. If one defines the null space of **<sup>X</sup>***<sup>b</sup>* as *<sup>N</sup>* = {**v|X***<sup>b</sup>***<sup>v</sup>** <sup>¼</sup> 0}, then the cost function ^*J*(**w**) is divided into two parts: one containing the component of **w** in *N* (the first term in Eq. (2)), and the second depending on the components of **w** that are orthogonal to *N*. By requiring that the mean analysis state **w**� <sup>a</sup> is orthogonal to *N* such that the cost function ^*J*(**w**) is minimized, the mean analysis state and its corresponding analysis error covariance matrix in the ensemble space can be found as:

$$\bar{\mathbf{w}}^{\mathfrak{a}} = \hat{\mathbf{P}}^{\mathfrak{a}} \left(\mathbf{Y}^{\mathfrak{b}}\right)^{\mathrm{T}} \mathbf{R}^{-1} \left[\mathbf{y}^{\mathfrak{0}} \mathbf{-} \boldsymbol{\mathsf{H}} (\bar{\mathbf{x}}^{\mathfrak{b}})\right] \tag{3}$$

$$
\hat{\mathbf{P}}^{\mathbf{a}} = \left[ (\mathbf{k} - \mathbf{1}) \mathbf{I} + \left( \mathbf{Y}^{\mathbf{b}} \right)^{\mathbf{T}} \mathbf{R}^{-1} \mathbf{Y}^{\mathbf{b}} \right]^{-1}, \tag{4}
$$

Where **Y**<sup>b</sup> � *<sup>H</sup>*(**x**b ið Þ � **<sup>x</sup>**�<sup>b</sup> ) is the ensemble matrix of background perturbations valid at the observation locations, and **R** is the observational error covariance matrix. By noting that the analysis error covariance matrix **P**<sup>a</sup> in the model space and **<sup>P</sup>**^<sup>a</sup> in the ensemble space have a simple connection of **<sup>P</sup>**<sup>a</sup> <sup>=</sup> **<sup>X</sup>**<sup>b</sup>**P**^<sup>a</sup> **<sup>X</sup>**<sup>b</sup> � �<sup>T</sup> , the analysis ensemble perturbation matrix **X**<sup>a</sup> can be chosen as follows:

$$\mathbf{X^{a}} = \mathbf{X^{b}} \left[ (\mathbf{k} - \mathbf{1}) \hat{\mathbf{P}}^{\mathbf{a}} \right]^{1/2}. \tag{5}$$

The analysis ensemble **x**<sup>a</sup> is finally obtained as:

$$\mathbf{x}^{\mathbf{a}(i)} = \bar{\mathbf{x}}^{\mathbf{b}} + \mathbf{X}^{\mathbf{b}} \left\{ \bar{\mathbf{w}}^{\mathbf{a}} + \left[ (\mathbf{k} - \mathbf{1}) \hat{\mathbf{P}}^{\mathbf{a}(i)} \right]^{1/2} \right\}. \tag{6}$$

Detailed handling of more general nonlinear and synchronous observations in LETKF can be found in [21]. It should be noticed that the above formulas are only valid without model errors. To take into account the model errors, Hunt et al. [21] suggested that a multiplicative factor should be introduced in Eq. (4) (specifically, the first factor on the right hand side of Eq. (4)). This simple additional multiplicative inflation is easy to implement in the scheme, and has been shown to be efficient in many applications of the LETKF (e.g., see [25, 27, 28]).
