3.1. Training process

The vectors values represent individual grid points for a single variable with a correspondent observation value on model point localization. The grid points is considered where observation value exists, see Figure 3. In the training algorithm, the MLP-DA computes the output and compared it with the "input" analysis vector of LETKF results (the target data), but it is not a node for the MLP generalization. The output vectors represent the analysis values for one grid point too. Care must be taken in specifying the number of neurons. Too many neurons can lead the NN to memorize the training data (over fitting), instead of extracting the general features that allow the generalization. Too few neurons may force the NN to spend too much time

trying to find an optimal representation and thus wasting valuable computation time.

MLP-DA; the LETKF procedures are not modified.

274 Advanced Applications for Artificial Neural Networks

back-propagation algorithm.

for Miyoshi's experiments.

six regions.

One strategy used to collect data and to accelerate the processing of the MLP-DA training was to divide the entire globe into six regions: for the Northern Hemisphere, 90 N and three longitudinal regions of 120 each; for the Southern Hemisphere, 90 S and three longitudinal regions of 120 each. This division provides the same size for each region, but the number of observations is distinct, as illustrated by Figure 3. This regional division is applied only for the

The MLP-DA scheme was developed with a set of 30 NN (six regions with five prognostic variables (ps, u, v, T, and q)). Each grid point has all vertical layers values for the model. One MLP with characteristics described above was designed for each meteorological variable of the SPEEDY model and each region. Each MLP has two inputs (model and observation vectors), one output neuron which is the analysis vector, and the training scheme is the

The MLP-DA is designed to emulate the global LETKF analysis for SPEEDY initial condition. The LETKF analysis is the mean field of an ensemble of analyses. Fortran codes for SPEEDY and LETKF [32] were adapted to create the training data set for that period. The upper levels and the surface covariance error matrices to run the LETKF system, as well as the SPEEDY model boundary conditions data and physical parameterizations, are the same as those used

The initial process is the implementation of the model, it assumes that it is perfect (initialization = 0); and the SPEEDY model T30 L7 was integrated for 1 year of spin-up, i.e., the period required for a model to reach steady state and obtain the simulated atmosphere. The model

Figure 3. Observations localizations in global area. The dot points represent radiosonde stations (about 415) divided in

The training process for the experiment is conducted with data obtained from the SPEEDY model and the LETKF analyses. The LETKF analyses are executed with synthetic observations: upper levels wind, temperature and humidity, and surface pressure to 0000 and 1200 UTC and 0600 and 1800 UTC with surface observations only. The LETKF runs generate the analyses target vectors, the input observations vectors, and analyses field to run the SPEEDY model, which generates the input forecasts vectors for training the MLP-DA. The training is made with back-propagation algorithm.

Executions of the model with the LETKF data assimilation are made for the same period mentioned for the true model: from 01 January 1982 through 31 December 1984. The ensemble forecasts/analyses of LETKF have 30 members. The ensemble average of the forecasts and analyses fields, to this training process, is obtained by running SPEEDY model with the LETKF scheme.

These data are collected, initially, by dividing the globe into two regions (northern and southern hemispheres), but the computational cost was high because the training process took 1 day for the performance to converge. Next, the two regions were divided each into three regions, for a total of six regions. Then, we use this division strategy to collect the 30 input vectors (observations, mean forecasts, and mean analyses) at chosen grid points by the observation mask during LETKF process. The NN training process begin after collecting the input vectors for whole period (3 years), the training took about 15 min, for a set of 30 NN.

The MLP-DA data assimilation scheme has no error covariance matrices to spread the observation influence. Therefore, it is necessary to capture the influence of observations from the neighboring region around a grid point considered as a "new" observation. This calculation is based on the distance from the grid point related to observations inside a determined neighborhood (initially: γ = 0)

$$\begin{cases} \hat{y}^{\rho}\_{i \pm m, j \pm m, k \pm m} = \frac{y^{\rho}\_{ijk}}{\left(\left(\delta - \gamma\right)r\_{ijk}^{2}\right)} + \sum\_{l=1}^{6} \alpha\_{l} \frac{y^{\rho}\_{i \pm m, j \pm m, k \pm m}}{r\_{ijk}^{2}}\\\\ (m = 1, 2, \dots, M) \\\\ \alpha\_{l} = \begin{cases} 0 & \text{(if there is no observation)} \\\\ 1 & \text{(if there is observation, } \text{ and } \cdot \text{)}, \text{ and } \cdot \text{)} \gamma\_{\text{new}} = \gamma\_{\text{old}} + 1 \end{cases} \end{cases} \tag{10}$$

where <sup>b</sup>y<sup>o</sup> is the weighted observation, <sup>M</sup> is the number of discrete layers considered around observation,

r2 ijk <sup>¼</sup> xp � yo i � �<sup>2</sup> þ ðyp � yo jÞ <sup>2</sup> <sup>þ</sup> zp � yo k � �<sup>2</sup> , where (xp, yp, zp) is coordinate of the grid point, and the <sup>ð</sup>yo <sup>i</sup> , yo <sup>j</sup> , yo <sup>k</sup>Þ is the coordinate of the observation, and γ is a counter of grid points with observations around that grid point <sup>ð</sup>yo <sup>i</sup> , yo <sup>j</sup> , yo <sup>k</sup>Þ. If γ = 6, there is no influence to be considered. Each observation's influence computed on a certain grid point is a new location, hereafter referred to as pseudo-observation, which adds values to the three input vectors to NN training process and also adds positive flags in the observation mask.

Then, the grid points to be considered in MLP-DA analysis are greater than grid points considered to LETKF analysis, although these calculations are made without interference on LETKF system. The back-propagation algorithm stops the training process using the criteria cited at item 6 above (Section 3), after obtaining the best set of weights; it is a function of smallest error between the MPL-NN analysis and the target analysis (e.g., when the root mean square error between the calculated output and the input target vector is less than 10�<sup>5</sup> ). The learning process is the same for each MLP of the set of 30 NN and takes about 15 min to get the fixed weights before the MLP-DA data assimilation cycle or generalization process of MLP-DA.

#### 3.2. Generalization process

The training is performed with combined data from January, February, and March of 1982, 1983, and 1984, and in generalization process, MLP-DA is able to perform analyses similar to the LETKF analyses.

The generalization process is indeed the data assimilation process. The MLP-DA results a global analysis field. The MLP-DA activation is entering by input values (only 6 hours forecast and observations) at each grid point once, with no data used in the training process. The input vectors are done at grid model point, where it is marked (by positive flag mask) with observation or pseudo-observation (Eq. 10). The procedure is the same for all NN but one NN for each region, and each prognostic variable has own connection weights. All NNs have one hidden layer, with the same number of neurons for all regions. The regional grid points are put in the global domain to make the analysis field after generalization process of the MLP-DA, e.g., the activation of 30 NN results a global analysis. The regional division is only for inputting each NN activation.

The MLP-DA data assimilation is performed for 1-month cycle. It starts at 0000 UTC 01 January 1985, generating the initial condition to SPEEDY model. Running the SPEEDY model starting at 31 December 1984 1800 UTC carried out the previous model prediction. There were observations available at 0000 UTC 01 January 1985. Therefore, an analysis was computed for the SPEEDY model at 0000 UTC 01 January 1985. The SPEEDY model is re-executed with the former analysis, producing a new 6 hours forecast. The process is repeated at each 6 hours.

In this experiment, the MLP-DA begins the activation at 01 January 1985 0000 UTC and generates analyses and 6 hours forecasts up through 31 January 1985 1800 UTC.
