**8. Remarks and conclusions**

( ) 1 1 <sup>1</sup> ( /) ( ) *fT fT fT fT K P H HP H R P H HP H R*

gg

1( )( ) *f f i i <sup>i</sup> v x K y Hx* =+ g

1 1 <sup>1</sup> () ()*<sup>T</sup> Q K RK* g

for each updated member

; ,1

*<sup>R</sup> w y Hv HQH*

g

/∑ = 1 

( ) 1,

ò

g

g

is a random observation error drawn from the Gaussian (0, ).

( ) 2,

=+ - + - ê ú

*γ* can be determined recursively to match the optimal performance of EnKPF. More details of

Previous sections have introduced the localization technique in EnKF, which greatly improves the performance of EnKF in high-dimensional models. The advantages of localization motivate

*a u i u i i <sup>i</sup> x x K y Hx* g

1

g

ê ú - ë û ò

é ù

() 1 *u i i si xv K*

= +

**4.** Compute 2(1 − ) = (1 − )[(1 − ) + ]−1, and generate 2, *<sup>i</sup>*

<sup>2</sup> 1

EnKF with the inflated observation error again as follows:

the search for a localization procedure in particle filtering.

æ ö = + ç ÷

*i i*

f

**3.** Calculate the resampling index () for each member

and normalize the weights by =

g

 g

 g- - = += + (126)

*T*

(127)

<sup>=</sup> (128)

as follows:

è ø - (129)

, in which *ϕ* is the probability density

using the SIR

from (0, ) and

(130)

(131)

according to

g

186 Nonlinear Systems - Design, Analysis, Estimation and Control

**2.** Compute the weights

function of a Gaussian.

algorithm, then set

EnKPF can be found in [45, 46].

*7.3.2. Localization in PF*

where 1, *<sup>i</sup>*

Data assimilation is the process by which observations of the actual system are incorporated into a numerical model to optimally estimate the system states. In this chapter, we introduced several ensemble-based data assimilation methods that are widely used in the earth sciences. One can read it as an introduction to ensemble-based data assimilation methods, but also can view it as a brief review of the application of these ensemble-based assimilation methods on the earth sciences. It is author's effort to write such a 'review' chapter with introductory language, making it more readable. As found in the chapter, many discussions, derivations and analyses are actually very thoughtful, not only introducing these methods, but also deepening the understanding to them. This is emphasized by the analysis of the rationale behind each method, including: i). the principle for deriving the algorithm; ii) basic assumptions of each method; iii). the connection and relation of different methods (e.g., EKF and EnKF, EnKF and SPKF etc.); iv). the advantages and deficiencies of each method. Especially we put rather weights to discuss potential concerns, challenges and possible solutions when these methods are applied to high-dimensional systems in the earth sciences. This chapter can be a "textbook" for the beginners to learn these data assimilation algorithms, and also a good reference for researchers for better understanding and applying these methods.
