NWP | DA | Optimal weight matrix W

• Data assimilation (DA)
• NWP | DA | Kalman Filter
• NWP | DA | 4D-Var

---

To derive the optimal weight matrix $ $in DA, we follow a minimum variance approach to combine background (model) and observation data. Here’s a step-by-step mathematical formulation:

---

Step 1: Define Analysis and Errors

• Analysis: \(\mathbf{x}^a = \mathbf{x}^b + \mathbf{W}(\mathbf{y}^o - \mathbf{H}\mathbf{x}^b)\)
• Analysis error: \(\mathbf{\epsilon}^a = \mathbf{x}^a - \mathbf{x}^t = \mathbf{\epsilon}^b - \mathbf{W}(\mathbf{H}\mathbf{\epsilon}^b + \mathbf{\epsilon}^o)\)
  • \(\mathbf{x}^t\): True state
  • \(\mathbf{\epsilon}^b\): Background error (\(\mathbf{x}^b - \mathbf{x}^t\))
  • \(\mathbf{\epsilon}^o\): Observation error (\(\mathbf{y}^o - \mathbf{H}\mathbf{x}^t\))

---

Step 2: Analysis Error Covariance Matrix

The covariance matrix of the analysis error is: \[ \mathbf{P}^a = \mathbb{E}[\mathbf{\epsilon}^a (\mathbf{\epsilon}^a)^T] = (\mathbf{I} - \mathbf{W}\mathbf{H})\mathbf{B}(\mathbf{I} - \mathbf{W}\mathbf{H})^T + \mathbf{W}\mathbf{R}\mathbf{W}^T \] - \(\mathbf{B} = \mathbb{E}[\mathbf{\epsilon}^b (\mathbf{\epsilon}^b)^T]\): Background error covariance - \(\mathbf{R} = \mathbb{E}[\mathbf{\epsilon}^o (\mathbf{\epsilon}^o)^T]\): Observation error covariance

---

Step 3: Minimize Analysis Error Variance

Minimize the trace of $^a $(total analysis error variance) with respect to \(\mathbf{W}\): \[ \frac{\partial \, \text{tr}(\mathbf{P}^a)}{\partial \mathbf{W}} = -2(\mathbf{I} - \mathbf{W}\mathbf{H})\mathbf{B}\mathbf{H}^T + 2\mathbf{W}\mathbf{R} = 0 \]

---

Step 4: Solve for \(\mathbf{W}\)

Rearrange the derivative equation: \[ (\mathbf{I} - \mathbf{W}\mathbf{H})\mathbf{B}\mathbf{H}^T = \mathbf{W}\mathbf{R} \] Expand and solve: \[ \mathbf{B}\mathbf{H}^T = \mathbf{W}(\mathbf{H}\mathbf{B}\mathbf{H}^T + \mathbf{R}) \] Optimal Weight Matrix: \[ \mathbf{W} = \mathbf{B}\mathbf{H}^T (\mathbf{H}\mathbf{B}\mathbf{H}^T + \mathbf{R})^{-1} \]

---

Key Insights

1. Structure of \(\mathbf{W}\):
   • \(\mathbf{B}\mathbf{H}^T\): Covariance between background and observation spaces.
   • \((\mathbf{H}\mathbf{B}\mathbf{H}^T + \mathbf{R})^{-1}\): Normalizes by total uncertainty (background + observation errors).
2. Assumptions:
   • Background and observation errors are uncorrelated: \(\mathbb{E}[\mathbf{\epsilon}^b (\mathbf{\epsilon}^o)^T] = 0\).
   • Linear observation operator \(\mathbf{H}\).

---

Comparison to 3D-Var and Kalman Filter

Method	Weight Equation	Covariance Treatment
OI	\(\mathbf{W} = \mathbf{B}\mathbf{H}^T (\mathbf{H}\mathbf{B}\mathbf{H}^T + \mathbf{R})^{-1}\)	Static \(\mathbf{B}\), \(\mathbf{R}\)
3D-Var	Implicit via cost function minimization	Static \(\mathbf{B}\)
Kalman	\(\mathbf{K}_k = \mathbf{P}_{k\|k-1}\mathbf{H}^T (\mathbf{H}\mathbf{P}_{k\|k-1}\mathbf{H}^T + \mathbf{R})^{-1}\)	Dynamic \(\mathbf{P}_{k\|k}\)

---

Practical Example

For a 2-observation system (simplified from[3]): - Background error variance: \((\sigma^b)^2\) - Observation error variance: \((\sigma^o)^2\) - Correlation coefficients: \(\rho_{10}, \rho_{20}, \rho_{12}\) - Weights: \[ w_1 = \frac{\rho_{10}(1 + \alpha) - \rho_{12}\rho_{20}}{(1 + \alpha)^2 - \rho_{12}^2}, \quad w_2 = \frac{\rho_{20}(1 + \alpha) - \rho_{12}\rho_{10}}{(1 + \alpha)^2 - \rho_{12}^2} \] where \(\alpha = (\sigma^o)^2 / (\sigma^b)^2\).

This ensures observations with smaller errors (\(\sigma^o\)) or higher correlation (\(\rho\)) receive greater weight.

Rerferences

1. https://dspace.library.uu.nl/bitstream/1874/580/16/c4.pdf
2. http://psc.apl.washington.edu/nonwp_projects/PHC/oi.html
3. https://www.atmosp.physics.utoronto.ca/PHY2509/ch3.pdf
4. https://twister.caps.ou.edu/OBAN2019/3DVAR.pdf
5. https://maths.ucd.ie/~plynch/LECTURE-NOTES/NWP-2004/NWP-CH05-4-1.pdf
6. https://people.duke.edu/~hpgavin/Risk/interpolation.pdf
7. https://pro.arcgis.com/en/pro-app/latest/tool-reference/image-analyst/learn-more-about-optimal-interpolation.htm
8. https://twister.caps.ou.edu/OBAN2019/METR5303_Lecture14.pdf