Ensemble data assimilation with an adjusted forecast spread

Ensemble data assimilation typically evolves an ensemble of model states whose spread is intended to represent the algorithm's uncertainty about the state of the physical system that produces the data. The analysis phase treats the forecast ensemble as a random sample from a background distribution, and it transforms the ensemble according to the background and observation error statistics to provide an appropriate sample for the next forecast phase. We find that in the presence of model nonlinearity and model error, it can be fruitful to rescale the ensemble spread prior to the forecast and then reverse this rescaling after the forecast. We call this approach forecast spread adjustment, which we discuss and test in this article using an ensemble Kalman filter and a 2005 model due to Lorenz. We argue that forecast spread adjustment provides a tunable parameter, that is, complementary to covariance inflation, which cumulatively increases ensemble spread to compensate for underestimation of uncertainty. We also show that as the adjustment parameter approaches zero, the filter approaches the extended Kalman filter if the ensemble size is sufficiently large. We find that varying the adjustment parameter can significantly reduce analysis and forecast errors in some cases. We evaluate how the improvement provided by forecast spread adjustment depends on ensemble size, observation error and model error. Our results indicate that the technique is most effective for small ensembles, small observation error and large model error, though the effectiveness depends significantly on the nature of the model error.