7. Assimilation in high-dimensional ocean model MICOM

#### 7.1. Ocean model MICOM

To see the impact of optimal SchVs in the design of filtering algorithm for HdS, in this section, we present the results of the experiment on the Hd ocean model MICOM (Miami Isopycnal Ocean Model). This numerical experiment is identical to that described in Hoang and Baraille [15]. The model configuration is a domain situated in the North Atlantic from 30�N to 60�N and 80�W to 44�W; for the exact model domain and some main features of the ocean current produced by the model, see and Baraille [15]. The system state x ¼ ð Þ h; u; v where h ¼ h ið Þ ; j; k is a layer thickness and u ¼ u ið Þ ; j; k , v ¼ v ið Þ ; j; k are two velocity components. Mention that after discretization, the dimension of the system state is n ¼ 302400. The observations available at each assimilation instant are the sea surface height (SSH) with dimension p ¼ 221.

#### 7.1.1. Data matrix based on dominant Sch-Ops

The filter is a reduced-order filter (ROF) with the variable h as a reduced state and u, v are calculated on the basis of the geostrophy hypothesis. To obtain the gain in the ROF, first the Algorithm 3.1 has been implemented to generate an ensemble of dominant SchVs (totally 72 SchVs, denoted as En SCH ð Þ). The sample ECM Md ð Þ SCH is computed on the basis of the En SCH ð Þ. Due to rank deficiency, the sample Md ð Þ SCH is considered only as a data matrix. The optimization procedure is applied to minimize the distance between the data matrix M<sup>d</sup> ð Þ SCH and the structured parametrized ECM M SCH ð Þ¼ Mv ⊗ Mh which is written in the form of the Schur product of two matrices M SCH ð Þ¼ Mvð Þ θ ⊗ Mhð Þ θ . Here, Mv is the vertical ECM, Mh is the horizontal ECM [18]), ð Þ θ is a vector of unknown parameters. Mention that the hypothesis on separability of the vertical and horizontal variables in the ECM is not new in the meteorology [20]. The gain is computed according to Eq. (3) with R ¼ αI, α > 0 is a small positive value. The ROF is denoted as PEF (SSP).

## 7.1.2. Data matrix based on SSP approach

(Sampling-P). As to <sup>b</sup><sup>c</sup>

of <sup>b</sup><sup>c</sup> ð Þi .

<sup>ϕ</sup>bð Þ<sup>2</sup>

ð Þi

76 Perturbation Methods with Applications in Science and Engineering

The second xschð Þ<sup>2</sup> is much less opimal than xr

and Baraille [19]. Mention that only <sup>b</sup>xschð Þ<sup>i</sup> and <sup>b</sup><sup>c</sup>

<sup>Φ</sup><sup>b</sup> <sup>¼</sup> <sup>Φ</sup><sup>b</sup> ð Þ<sup>1</sup>

After the first iteration, <sup>Φ</sup>ð Þ<sup>2</sup> <sup>≔</sup> <sup>Φ</sup> � <sup>Φ</sup><sup>b</sup> ð Þ<sup>1</sup>

7.1. Ocean model MICOM

tion captures the two biggest elements of Φð Þ<sup>2</sup> .

7.1.1. Data matrix based on dominant Sch-Ops

SchVs, denoted as En SCH ð Þ). The sample ECM Md

Looking at the first OPs, one sees that xr

<sup>n</sup> , they are the results of normalization (with the unit Euclidean norm)

ð Þ2

<sup>≔</sup> <sup>ϕ</sup>bð Þ<sup>i</sup> ij h i

<sup>n</sup> can be calculated for HdS.

ð Þ1

<sup>n</sup> . By comparing xr

h i

¼ bb ð Þi bc ð Þi ,T

<sup>¼</sup> <sup>b</sup>ð Þ<sup>2</sup> <sup>c</sup>ð Þ<sup>2</sup> ,T and the optimization yields the estimates

ð Þ SCH is computed on the basis of the

<sup>n</sup> produce almost the same

svð Þ<sup>i</sup> with <sup>b</sup><sup>c</sup>

are displayed in the

(Table 1, column 4).

svð Þ1 , xschð Þ1 .

ð Þi <sup>n</sup> for

svð Þ<sup>1</sup> , xschð Þ<sup>1</sup> , <sup>b</sup>xschð Þ<sup>1</sup> , and <sup>b</sup><sup>c</sup>

svð Þ<sup>2</sup> and <sup>b</sup><sup>c</sup>

i ¼ 1, 2, one concludes that the obtained results justify the correctness of Theorem 3.1 of Hoang

In Table 1, we show the results obtained by Algorithm 5.2 after two consecutive iterations

ij displayed in the column 5. From the columns 4–5, one sees that the first iteration allows to well estimate the two biggest elements Φ<sup>11</sup> ¼ 5, Φ<sup>12</sup> ¼ 7. In the similar way, the second itera-

To see the impact of optimal SchVs in the design of filtering algorithm for HdS, in this section, we present the results of the experiment on the Hd ocean model MICOM (Miami Isopycnal Ocean Model). This numerical experiment is identical to that described in Hoang and Baraille [15]. The model configuration is a domain situated in the North Atlantic from 30�N to 60�N and 80�W to 44�W; for the exact model domain and some main features of the ocean current produced by the model, see and Baraille [15]. The system state x ¼ ð Þ h; u; v where h ¼ h ið Þ ; j; k is a layer thickness and u ¼ u ið Þ ; j; k , v ¼ v ið Þ ; j; k are two velocity components. Mention that after discretization, the dimension of the system state is n ¼ 302400. The observations available

at each assimilation instant are the sea surface height (SSH) with dimension p ¼ 221.

The filter is a reduced-order filter (ROF) with the variable h as a reduced state and u, v are calculated on the basis of the geostrophy hypothesis. To obtain the gain in the ROF, first the Algorithm 3.1 has been implemented to generate an ensemble of dominant SchVs (totally 72

ð Þi

amplification. The first xeið Þ<sup>1</sup> has the amplification three times less than those of <sup>x</sup><sup>r</sup>

(matrix estimation in <sup>R</sup><sup>1</sup> subspace). The elements of the true <sup>Φ</sup> <sup>¼</sup> <sup>ϕ</sup>ij

The estimates, resulting from the first iteration, are the elements of <sup>Φ</sup><sup>b</sup> ð Þ<sup>1</sup>

7. Assimilation in high-dimensional ocean model MICOM

<sup>¼</sup> <sup>X</sup><sup>2</sup> i¼1 bb ð Þi bc ð Þ<sup>i</sup> ,TΦ<sup>b</sup> ð Þ<sup>i</sup>

second column, whereas their estimates—in the third column,

<sup>þ</sup> <sup>Φ</sup><sup>b</sup> ð Þ<sup>2</sup>

The second data matrix Md ð Þ SSP is obtained by perturbing the system state according to the SSP method. The SSP samples are simulated in the way similar to that described above for generating En SCH ð Þ, with the difference that the perturbation components <sup>δ</sup>hð Þ<sup>l</sup> ð Þ <sup>i</sup>; <sup>j</sup>; <sup>k</sup> are the i. i.d. random Bernoulli variables assuming two values �1 with the equal probability 1/2. The same optimization procedure has been applied to estimate M SSP ð Þ. The obtained ROF is denoted as PEF (SSP).

Figure 2 shows the evolution of estimates for the gain coefficients kð Þ1 computed from the estimated coefficients of <sup>b</sup>ckl of Md ð Þ SCH and Md ð Þ SSP on the basis of En SCH ð Þ (curve" schur") and En SSP ð Þ (curve "random"), during model integration. It is seen that two coefficients are evolved in nearly the same manner, of nearly the same magnitude as that of kð Þ1 in the CHF (Cooper-Haines filter, Cooper and Haines [21]). Mention that the CHF is a filter widely used in the oceanic data assimilation, which projects the PE of the surface height data by lifting or lowering of water columns.

Figure 2. Evolution of estimates for the gain coefficients <sup>k</sup>ð Þ<sup>1</sup> computed from <sup>b</sup>ckl on the basis of En SCH ð Þ (curve "Schur") and En SSP ð Þ (curve "random"), during model integration. It is seen that two coefficients are evolved in nearly the same manner, of nearly the same magnitude as that of kð Þ1 in the CHF. The same picture is obtained for other ck, k ¼ 2; 3; 4.

The same pictures are obtained for the estimates <sup>b</sup>ck, k <sup>¼</sup> <sup>2</sup>; <sup>3</sup>; 4. Mention that in the CHF, c<sup>2</sup> ¼ c<sup>3</sup> ¼ 0.

To illustrate the efficiency of adaptation, in Figure 3, we show the cost functions (variances of innovation) resulting from the three filters PEF (SCH), PEF (SSP), and AF (i.e., APEF based on PEF (SCH); the same performance is observed for the AF based on PEF (SSP)). Undoubtedly, the adaptation allows to improve considerably the performances of nonadaptive filters.

On Optimal and Simultaneous Stochastic Perturbations with Application to Estimation of High-Dimensional…

http://dx.doi.org/10.5772/intechopen.77273

79

We have presented in this chapter the different types of OPs—deterministic, stochastic, or optimal, the invariant subspaces of the system dynamics. The ODPs and OSPs play an important role in the study on the predictability of the system dynamics as well as in construction of

One other class of perturbation known as SSP is found to be a very efficient tool for solving optimization and estimation problems, especially with Hd matrices and in computing the

The numerical experiments presented in this chapter confirm the important role of the differ-

[1] Myhrvold N. Moore's Law Corollary: Pixel Power. New York Times. June 7, 2006

[2] Kalman RE. A new approach to linear filtering and prediction problems. Transactions of

[3] Lorenz EN. Deterministic non-periodic flow. Journal of the Atmospheric Sciences. 1963;20:

[4] Lorenz EN. Atmospheric predictability experiments with a large numerical model. Tellus.

[5] Palmer TN, Barkmeijer J, Buizza R, Petroliagis T. The ECMWF ensemble prediction sys-

tem. Quarterly Journal of the Royal Meteorological Society. 1997;4(4):301-304

8. Conclusion remarks

optimal perturbations.

Author details

References

130-141

1982;34:505-513

Hong Son Hoang\* and Remy Baraille

SHOM/HOM/REC, Toulouse, France

[Retrieved: 2011-11-27]

\*Address all correspondence to: hhoang@shom.fr

optimal OFS for environmental geophysical systems.

ent types of OPs in the numerical study of Hd assimilation systems.

the ASME-Journal of Basic Engineering. 1960;82:35-45

#### 7.2. Performance of different filters

In Table 3, the performances of the three filters are displayed. The errors are the averaged (spatially and temporally) rms of PE for the SSH and for the two velocity components u and v.

The results in Table 3 show that two filters PEF (SCH) and PEF (SSP) are practically of the same performance, and their estimates are much better compared to those of the CHF, with a slightly better performance for the PEF (SSP). We note that as the PEF (SCH) is constructed on the basis of an ensemble of samples tending to the first dominant SchV, its performance must be theoretically better than that of the PEF (SSP). The slightly better performance of PEF (SSP) (compared to that of PEF (SCH)) may be explained by the fact that the best theoretical performance of PEF (SCH) can be obtained only if the model is linear, stationary, and the number of PE samples in En SCH ð Þ at each iteration must be large enough. The ensemble size of En SCH ð Þ in the present experiment is too small compared with the dimension of the MICOM model.


Table 3. rms of PE for ssh, and u, v velocity components.

Figure 3. Variance of PE resulting from three filters PEF (SCH), PEF (SSP), and AF. It is seen that the AF yields much better performance compared to the two other nonadaptive filters PEF (SCH) and PEF (SSP).

To illustrate the efficiency of adaptation, in Figure 3, we show the cost functions (variances of innovation) resulting from the three filters PEF (SCH), PEF (SSP), and AF (i.e., APEF based on PEF (SCH); the same performance is observed for the AF based on PEF (SSP)). Undoubtedly, the adaptation allows to improve considerably the performances of nonadaptive filters.
