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Lorenz96s⚓︎

A perfect-random version of Lorenz-96.

Used by grudzien2020a to study the precision of stochastic integration schemes.

Both the model and truth are to be integrated by the same random model (with almost surely different outcomes). For simplicity, this case should be used with Q = 0, i.e. with no model error (as perceived by the DA schemes). Inflation, localisation, and other auxiliary techiques may be used to handle sampling error and perform regularization.

The truth twin should be generated by the order 2.0 Taylor scheme below, for the accuracy with respect to convergence in the strong sense. See grudzien2020a for a full discussion of benchmarks on this model and statistically robust configurations.

This study uses no multiplicative inflation / localization or other regularization instead using a large ensemble size in the perturbed observation EnKF as a simple estimator to study the asymptotic filtering statistics under different model scenarios.

The purpose of the study in grudzien2020a was to explore the relationships between:

  • numerical discretization error in truth twins;
  • numerical discretization error in model twins;
  • model uncertainty in perfect-random models;
  • filter divergence and / or bias in filtering forecast statistics;

Numerical discretization error increases with dt, with the strong / weak order of convergence discussed in the refs. Although the orders of convergence of the stochastic Runge-Kutta and the Euler-Maruyama model match, it is shown that the step size configuration above keeps the discretization error for the model and truth twins bounded by approximately \(10^{-3}\) in expectation.

Model uncertainty increases with the diffusion, representing the "instantaneous" standard deviation of the model noise at any moment. Larger diffusion thus corresponds to a wider variance of the relizations of the diffeomorphsims that generate the model / truth twin between observation times.

It is demonstrated by grudzien2020a that the model error due to discretization of the SDE equations of motion is most detrimental to the filtering cycle when model uncertainty is low and observation precision is high. In other configurations, such as those with high model uncertainty, the differences between ensembles with low discretization error (those using the Runge-Kutta scheme) and high discretization error (those using the Euler-Maruyama scheme) tend to be relaxed.

Set-up with three different step functions, using different SDE integrators. The truth twin is generated by the order 2.0 Taylor scheme, for accuracy with respect to convergence in the strong sense for generating the observation sequence. The model simulation step sizes are varied in the settings below to demonstrate the differences between the commonly uses Euler-Maruyama and the more statistically robust Runge-Kutta method for SDE integration. See README in mods.Lorenz96s.

Modules:

Name Description
grudzien2020

Settings as in grudzien2020a.

l96s_tay2_step(x, t, dt, s) ⚓︎

Advance state of L96s model using order-2.0 Taylor scheme.

This is the method that should be used to generate the truth twin for this model due to the high-accuracy with respect to convergence in the strong sense. The ensemble model twin will be generated by on of the wrappers below. The order 2.0 Taylor-Stratonovich discretization scheme implemented here is the basic formulation which makes a Fourier truncation at p=1 for the Brownian bridge process. See grudzien2020a for full details of the scheme and other versions.

steppers(kind) ⚓︎

Wrapper around the different model integrators / time steppers.

Note that they all forward (i.e. use) the diffusion parameter.