Data Assimilation
Daniel Wheeler
04/09/2025
Why DA?
- DA is one of the underpinnings for DT
- First DT: dynamic data assimilation enabled weather forecasting
- DA + ROM (physics-driven surrogate models)
Why DA?
- FNO + DA
- Melt pool dimensions for experimental observations
- Laser absorption is free parameter
DA Methods
Perfect System
X_{k+1} = \Phi X_k + w_k
z_k = H X_{k} + v_k
State vector mapping
Observation mapping
with noise
w_k \sim N(0, Q)
v_k \sim N(0, R)
Q and R are stationary over time!
Note on covariance matrices
R = E[v v^T]
R_{ij} = \frac{1}{n} \sum_{l=1}^n v_{i,l} v_{j,l}
Covariance matrices are important here (dropping k)
What does that mean? Sampe n times for element of v
Imperfect System
\hat{X}_{k+1}^{\prime} = \Phi \hat{X}_k + w_k
z_k - H \hat{X}_k^{\prime} \ne 0
The imperfect state vector
It's imperfect so we have a measurment residual or innovation
We need to improve the imperfect state vector with the innovation
\hat{X}_k = \hat{X}_k^{\prime} + K_k \left( z_k - H \hat{X}_k^{\prime} \right)
Kalman Gain
How to find the Kalman gain?
Write down the error covariance matrix
P_k = E[(X_k - \hat{X}_k)(X_k - \hat{X}_k)^T]
Note that the trace of P is the mean square error. Minimize with respect to K.
P_k = \left(I + K_k H\right) P_k^{\prime} \left(I + K_k H\right)^T + K_k R K_k^T
Using
using independence of v and w as well as
P_k^{\prime} = E[(X_k - \hat{X}_k^{\prime})(X_k - \hat{X}_k^{\prime})^T]
\hat{X}_k = \hat{X}_k^{\prime} + K_k \left(H X_k + v - H \hat{X}_k^{\prime} \right)
sub into error matrix
How to find the Kalman gain?
Minimize Tr(P) w.r.t K
\frac{\partial \text{Tr}[P_k]}{\partial K_k} = -2 \left(H P_k^{\prime}\right)^T + 2 K_k \left(H P_k^{\prime} H^T + R \right)
K_k = P_k^{\prime} H^T \left(H P_k^{\prime} H^T + R \right)^{-1}
\frac{\partial \text{Tr}[P_k]}{\partial K_k} = -2 \left(H P_k^{\prime}\right)^T + 2 K_k \left(H P_k^{\prime} H^T + R \right)
Set to 0 and find K
Putting it together
Forward Model
Correction
\hat{X}_{k+1}^{\prime} = \Phi \hat{X}_k + w_k
\hat{X}_k = \hat{X}_k^{\prime} + K_k \left( z_k - H \hat{X}_k^{\prime} \right)
Can also show that
P_k = \left(I - K_k H\right) P_k^{\prime}
P_{k+1}^{\prime} = \Phi P_k^{\prime} \Phi^T + Q
K_k = P_k^{\prime} H^T \left(H P_k^{\prime} H^T + R \right)^{-1}
EnKF
\hat{X}_{k+1}^{(i)\prime} = \mathcal{\Phi} \left( \hat{X}_k^{(i)} \right) + w_k^{(i)}
EnKF
Non-linear forward model
P prime represents the covariance error of the mean of the ensemble
\hat{X}_{k+1}^{(i)\prime} = \mathcal{\Phi} \left( \hat{X}_k^{(i)} \right) + w_k^{(i)}
P_k^{\prime} = E[(X_k - \overline{\hat{X}_k^{(i)\prime}})(X_k - \overline{\hat{X}_k^{(i)\prime}})^T]
\hat{X}_k^{(i)} = \hat{X}_k^{(i)\prime} + K_k \left( z_k^{(i)} - \mathcal{H} \left( \hat{X}_k^{(i)\prime} \right) \right)
K_k = P_k^{\prime} H^T \left(H P_k^{\prime} H^T + R \right)^{-1}
Parameter Estimation
Forward model for joint state-parameter estimation
\hat{X}_{k+1}^{(i)\prime} = \mathcal{\Phi} \left( \hat{X}_k^{(i)}; \theta_k \right) + w_k^{(i)}
- Joint state-parameter estimation
- Dual estimation (separate EnKFs)
- Augmented
- Iterative
- Adapative
\hat{Z}_{k+1}^{(i)\prime} = \mathcal{\Phi} \left( \hat{Z}_k^{(i)} \right) + w_k^{(i)}
Z_{k+1} = \left[ X_k, \theta_k \right]^T
becomes
TorchDA
deck
By Daniel Wheeler
