5.2. Inflation Schemes

In order to account for an underdispersive ensemble, several multiplicative inflation schemes have been implemented in UMD-LETKF (with hopefully more to be implemented eventually). If you’re not sure which one to pick, it is usually safest to choose Relaxation to Prior Spread (RTPS) with a value between 0.0 and 1.0.

5.2.1. Multiplicative

The inflation factor \(\alpha\), which is greater than or equal to 1.0, increases the magnitude of the analysis perturbations.

\[\mathbf{x}_i^{'a} \leftarrow \alpha \mathbf{x}_i^{'a}\]

This method works sufficiently for domain that are regularly sampled by observations. (e.g. the atmosphere). If a domain is not sufficiently sampled (such as the deep ocean), this method may result in the ensemble spread growing far too rapidly and the filter ultimately diverging.

5.2.2. Relaxation to Prior Perturbations (RTPP)

The perturbations of the analysis, \(\mathbf{x}_i^{'a}\) are relaxed a percentage, \(\alpha\), back to the background perturbations, \(\mathbf{x}_i^{'b}\) [Zhang2004]. This has the benefit of effectively being a combination of both multiplicative, and additive inflation.

\[\mathbf{x}_i^{'a} \leftarrow \left ( 1 - \alpha \right ) \mathbf{x}_i^{'a} + \alpha \mathbf{x}_i^{'b}\]

5.2.3. Relaxation to Prior Spread (RTPS)

The spread of the analysis, \(\sigma^a\), is relaxed a percentage of the way, \(\alpha\), back to the spread of the background, \(\sigma^b\) [Whitaker2012].

\[\mathbf{x}_i^{'a} \leftarrow \mathbf{x}_i^{'a} \left ( \alpha \frac{\sigma^b - \sigma^a}{\sigma_b} +1 \right )\]