Notation
Conventions used throughout the Mathematical Reference. The single
most important one: the ensemble matrix stacks members as rows,
\(X \in \mathbb{R}^{N_e \times N_x}\), so axis 0 of every ensemble
array is the ensemble axis — matching the (N_e, N_x) shape of every
filterax particles argument.
Dimensions
| Symbol |
Description |
| \(N_e\) |
Ensemble size (number of members) |
| \(N_x\) |
State dimension |
| \(N_y\) |
Observation dimension (per assimilation window) |
| \(N_z\) |
Latent dimension (chapter 15) |
| \(T\) |
Number of assimilation windows / time steps |
| \(J\), \(N_p\), \(N_d\) |
Ensemble size, parameter dim, data dim for EKP processes (chapter 9) |
Ensemble quantities
| Symbol |
Description |
| \(X \in \mathbb{R}^{N_e \times N_x}\) |
Ensemble matrix; members as rows, axis 0 = ensemble |
| \(x^{(j)}\) |
The \(j\)-th ensemble member (a row of \(X\)) |
| \(\bar{x} = \frac{1}{N_e} \sum_j x^{(j)}\) |
Ensemble mean |
| \(X' = X - \mathbf{1}\bar{x}^\top\) |
Anomaly (perturbation) matrix |
| \(P = \frac{1}{N_e - 1} X'^\top X'\) |
Sample (forecast-error) covariance, Bessel-corrected |
| \(Y = \mathcal{H}(X)\) |
Ensemble mapped to observation space, \((N_e, N_y)\) |
| \(Y'\) |
Observation-space anomaly matrix |
| \(C^{xH}\) |
State–observation cross-covariance \(\frac{1}{N_e-1} X'^\top Y'\) |
| \(C^{HH}\) |
Observation-space covariance \(\frac{1}{N_e-1} Y'^\top Y'\) |
| \(\cdot^f\), \(\cdot^a\) |
Forecast (prior) / analysis (posterior) superscripts |
| \(\sigma_i\) |
Per-variable ensemble standard deviation (spread) |
| \(w_j\) |
Importance weight of member \(j\) |
Observation model
| Symbol |
Description |
| \(y \in \mathbb{R}^{N_y}\) |
Observation vector |
| \(\mathcal{H}\) |
Observation operator (possibly nonlinear); \(H\) its linear(ised) matrix form |
| \(R\) |
Observation error covariance |
| \(d = y - \mathcal{H}(\bar{x})\) |
Innovation (also \(v\) in some L1 docstrings) |
| \(S = C^{HH} + R\) |
Innovation covariance |
| \(K = C^{xH} S^{-1}\) |
(Ensemble) Kalman gain |
| \(U = (HX)'^\top / \sqrt{N_e - 1}\) |
Low-rank factor of the ensemble term in \(S\) |
| Symbol |
Description |
| \(\mathcal{M}\), \(M\) |
Forecast model (dynamics); \(M_z\) a latent surrogate |
| \(\tilde{C} = (N_e - 1) I + Y' R^{-1} Y'^\top\) |
ETKF ensemble-space transform precision |
| \(W\) (also \(W_a\)) |
Ensemble transform / perturbation-weight matrix |
| \(\bar{w}\) |
Mean-update weight vector in ensemble space |
| \(\Theta\) |
Mean-preserving random rotation (ETKF-Livings) |
| \(E\), \(D\) |
Encoder / decoder of a latent map; \(z = E(x)\), \(x \approx D(z)\) |
| \(\theta\) |
Trainable parameters (chapter 14) |
| \(\mathcal{L}\), \(L_t\) |
Training loss; per-step local loss (ROAD-EnKF) |
Localization and inflation
| Symbol |
Description |
| \(\rho\) |
Localization taper (function or matrix); entries in \([0, 1]\) |
| \(r\) |
Localization half-width; compactly supported tapers vanish at \(2r\) |
| \(\circ\) |
Schur (Hadamard, element-wise) matrix product |
| \(\rho^{xy}\), \(\rho^{yy}\) |
State–obs and obs–obs taper matrices |
| \(d_{ij}\) |
Physical distance between variables \(i\) and \(j\) |
| \(\lambda\) |
Multiplicative inflation factor (\(X' \leftarrow \lambda X'\)) |
| \(\alpha\) |
Relaxation coefficient (RTPS / RTPP), in \([0, 1]\) |
| \(Q_{\text{add}}\) |
Additive-inflation (model-error) covariance |
| \(\lambda^*\), \(\mu\) |
Ledoit-Wolf shrinkage intensity; average sample eigenvalue \(\operatorname{tr}(P)/N_x\) |
Diagnostics
| Symbol |
Description |
| \(\chi^2 = d^\top S^{-1} d\) |
Innovation consistency statistic; \(E[\chi^2] = N_y\) |
| \(d_f\), \(d_a\) |
Forecast / analysis departures (Desroziers) |
| \(\mathrm{SSR}\) |
Spread-skill ratio (RMS spread / RMSE of the mean) |
| \(\mathrm{DFS} = \operatorname{tr}(KH)\) |
Degrees of freedom for signal |
| \(\mathrm{CRPS}\) |
Continuous Ranked Probability Score |
| \(N_{\mathrm{eff}} = 1 / \sum_j w_j^2\) |
Effective ensemble size |
| \(A\) |
Analysis-error covariance (in Desroziers identities) |
General
| Symbol |
Description |
| \(\mathbf{1}\) |
Vector of ones, \((N_e,)\) unless stated otherwise |
| \(I\) |
Identity matrix |
| \(\mathcal{N}(\mu, \Sigma)\) |
Gaussian distribution |
| \(\varepsilon\) |
Random perturbation / noise draw |
| \(\lVert \cdot \rVert_F\) |
Frobenius norm |
| \(\operatorname{tr}(\cdot)\) |
Matrix trace |
| \(E[\cdot]\) |
Expectation |
| \(\mathbb{1}\{\cdot\}\) |
Indicator function |
| \(x^*\) |
True state (twin / OSSE experiments) |
filterax shape conventions
| Array |
Shape |
Ensemble (particles) |
(N_e, N_x) |
| Observation-space ensemble |
(N_e, N_y) |
| Observation vector |
(N_y,) |
| Kalman gain |
(N_x, N_y) |
Taper matrices rho_xy / rho_yy |
(N_x, N_y) / (N_y, N_y) |
Stacked histories (*_history) |
(T, N_e, N_x) |
| Stacked observations (differentiable loop) |
(T, N_y) |
| Latent ensemble |
(N_e, N_z) |