The Ensemble Kalman Filter

For high-dimensional field states the twin uses an ensemble of members to represent uncertainty, making Kalman updating tractable at plant scale.

Covariance without a covariance matrix

The KRONOS-CTRL twin's full state - fields on a grid - can have very high dimension, where storing and propagating a dense covariance is infeasible. The ensemble Kalman filter (EnKF) represents the belief by an ensemble of state realizations and estimates covariance from their spread, so the Kalman update works with matrices sized by ensemble count, not state dimension.

EnKF with N members {x_i}:

  Forecast:  x_i^f = f(x_i, u) + w_i
  Ensemble mean and anomalies:
    xbar = (1/N) sum x_i^f ,  A = [x_i^f - xbar]
  Sample covariance:  P^f = A A' / (N-1)

  Update each member with perturbed observations:
    x_i^a = x_i^f + K ( z + eps_i - H x_i^f )
    K = P^f H' (H P^f H' + R)^-1     (via ensemble, low-rank)

Why ensembles scale

The gain is computed through the ensemble anomalies, so cost scales with the number of members (tens to hundreds), not the millions of state variables. The ensemble also propagates through the true nonlinear model, capturing non-Gaussian structure the EKF misses. This is the practical estimator for a field-scale digital twin.

# EnKF update (schematic, low-rank via ensemble)
Xf = [model(xi, u) for xi in ensemble]      # forecast members
Xbar = mean(Xf); A = stack(Xf) - Xbar
HX = [H(xi) for xi in Xf]; Hbar = mean(HX); Ah = stack(HX)-Hbar
S = (Ah.T@Ah)/(N-1) + R
for i in range(N):
    ensemble[i] = Xf[i] + (A.T@Ah)/(N-1) @ inv(S) @ (z+pert[i]-HX[i])

Localization and inflation

Finite ensembles produce spurious long-range correlations and tend to under-spread. The stack applies localization (damping covariance between physically distant points using the flux-coordinate metric) and covariance inflation (modestly enlarging spread) to keep the filter consistent - standard, necessary corrections for a real high-dimensional twin.

• Cost scales with ensemble size, not state dimension.
• Nonlinear propagation captures non-Gaussian structure.
• Localization removes spurious distant correlations.
• Inflation prevents ensemble collapse / filter divergence.

The EnKF also naturally supplies the twin's per-region confidence: ensemble spread is the uncertainty MPC reads to widen margins where the estimate is weak, and on the burner it carries the regime-extrapolation caveat.