# Presentation: 570290

## Introducing Error Co-variances in the ARM Variational Analysis 

**Introducing Error Co-variances in the ARM Variational Analysis **

- Minghua Zhang (Stony Brook University/SUNY) and Shaocheng Xie (Lawrence Livermore National Laboratory)

_**1. Introduction**_
- Any optimization algorithm involves the minimization of a cost function.  For multi-variable analysis, such as analysis of ARM data with several stations, many levels and time steps, the cost function contains a error covariance matrix.  Elements of the matrix determines how observations are weighted to produce the analysis. In NWP community, it is well known that the error covariance has a major impact on the quality of the optimal analysis. For example, if several measurements are highly correlated, each individual data entry should be given small weight relative to an independent data entry.
- Because of the complexities to derive the error covariance matrix from actual data, the current ARM variational analysis assumes that errors are independent.  Since most of the errors are due to sampling rather than random instrument error, this assumption needs to be improved.

_**4. Error Structures and Correlation Matrices **_
- Analysis increments or errors in observations relative to the first iteration of the variational analysis for the TWP-ICE temperature and u wind are shown in Figure 1.  The correlation matrices in the vertical direction for the two variables are shown in Figure 2.  The matrices derived from the AR1 model are shown in Figure 3.    The AR(1) model captures the general features of the correlations.

_**5. Summary**_**  **
- An AR(1) model is used to characterize the error covariance in the vertical direction in the ARM variational analysis that allows inversion of the covariance matrix for the minimization of the cost function.  The model captures the de-correlation lengths and the different matrix structures for different variables. The numerical algorithm is being tested and implemented into the variational analysis of TWP-ICE data.

**ITPA**

_**2. The Problem**_
- For a field experiment such as TWP-ICE, the atmospheric state variables of winds (u,v), temperature () and specific humidity (q) at S stations, K levels, and N time steps are written as

- Similarly

- and

- With truth & observations as:

- We write errors:

- and

- and

- Maximum likelihood or minimum variance leads to cost function:

- is populated by covariance among all stations/levels/variables/ time steps.

- The dimension of this matrix is

- In TWP-ICE, this is *(4X45X6X201)^2 = 217080**2*

- Not only this matrix is too large to invert, but also the covariance cannot be easily obtained from data.

_**3. A New Method**_
- Since the constraints are vertically integrated, we first assume errors to be vertically correlated.  This reduces the cost function to:

- The minimization of I is subject to the five constraints of column integrated conservations at each time step:

- Where

- These matrices are *KXK* in dimension.

- To obtain

- an AR(1) model is used such that:

- Covariance between two levels for other variables can be similarly calculated.

- It can be also shown that:

- can be similarly obtained.

- This is a symmetric polynomial matrix that can be inverted using the Cholesky decomposition.  When the correlation length scale is short, it is a narrow diagonal matrix.

- The analysis is then calculated from:

- with constraints:

- Figure 3

- Figure 2

- Figure 1

- The merit of the above matrix structure is that it yields an explicit solution from the cost function term in the E-L equation.