A Quantile-Conserving Ensemble Filter Framework. Part III: Data Assimilation for Mixed Distributions with Application to a Low-Order Tracer Advection Model

The uncertainty associated with many observed and modeled quantities of interest in Earth system prediction can be represented by mixed probability distributions that are neither discrete nor continuous. For instance, a forecast probability of precipitation can have a finite probability of zero precipitation, consistent with a discrete distribution. However, nonzero values are not discrete and are represented by a continuous distribution; the same is true for rainfall rate. Other examples include snow depth, sea ice concentration, the amount of a tracer, or the source rate of a tracer. Some Earth system model parameters may also have discrete or mixed distributions. Most ensemble data assimilation methods do not explicitly consider the possibility of mixed distributions. The quantile-conserving ensemble filter framework is extended to explicitly deal with discrete or mixed distributions. An example is given using bounded normal rank histogram probability distributions applied to observing system simulation experiments in a low-order tracer advection model. Analyses of tracer concentration and tracer source are shown to be improved when using the extended methods. A key feature of the resulting ensembles is that there can be ensemble members with duplicate values. An extension of the rank histogram diagnostic method to deal with potential duplicates shows that the ensemble distributions from the extended assimilation methods are more consistent with the truth.

Significance Statement
Data assimilation is a statistical method that is used to combine information from computer forecasts with measurements of the Earth system. The result is a better estimate of what is occurring in the physical system. As an example, data assimilation is used for making weather predictions. Some Earth system quantities, like precipitation, have special values that can occur very frequently. For instance, zero rainfall is quite common, while any other specific amount of rainfall, say, 0.42 in., is unusual. New data assimilation tools that work well for quantities like this are introduced and should lead to better estimates and predictions of the Earth system.