Ensemble Kalman Filter with the Lorenz (1963) model

Stefano Migliorini,

Here is presented the analysis trajectory in the x-z plane resulting from assimilating simulated observations in the Lorenz (1963) model, at the end of the assimilation window. The assimilation technique used here is the Ensemble Kalman Filter (EnKF) and makes use of the square-root algorithm described in Evensen (2004) with 1000 ensemble members.

The reference case (the "truth") has initial conditions given by xt(0) = 1.508870; yt[0] = -1.531271; zt(0) = 25.46091. The initial conditions for the central forecast are given by x(0) = xt(0) + 0.1; y[0] = yt(0) + 0.1; z(0) = zt(0) + 0.2 with normally distributed initial error having zero mean and diagonal covariance, being the variances all equal to 1.0. Observations of x, y and z are simulated by adding to the reference trajectory a normally distributed error with zero mean and diagonal covariance with variances equal to 1.0. In this experiment observations of x, y and z are assimilated every 40 time steps (for a total of 20 observations within the assimilation window).

The three coupled nonlinear ordinary differential equations of the Lorenz (1963) model were integrated by means of the Runge-Kutta method (fourth-order formula) with a time step of 0.01. A model error term was added to each equation, assumed normally distributed with zero mean and variance in x equal to 0.1491, in y equal to 0.9048 and in z equal to 0.9180.

The blue ellipsoid defines the 1-sigma uncertainty region on the x-z plane calculated from the analysis error covariance.

Evensen, G., 2004: Sampling strategies and square root analysis schemes for the EnKF, Ocean Dynamics, 54, 539-560.
Lorenz , E., N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci., 20, 130-141.

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