  This paper presents a seamless algorithm for the application of the
multilevel Monte Carlo (MLMC) method to the ensemble transform particle filter
(ETPF). The algorithm uses a combination of optimal coupling transformations
between coarse and fine ensembles in difference estimators within a multilevel
framework, to minimise estimator variance. It differs from that of Gregory et
al. (2016) in that strong coupling between the coarse and fine ensembles is
seamlessly maintained during all stages of the assimilation algorithm, instead
of using independent transformations to equal weights followed by recoupling
with an assignment problem. This modification is found to lead to an increased
rate in variance decay between coarse and fine ensembles with level in the
hierarchy, a key component of MLMC. This offers the potential for greater
computational cost reductions. This is shown, alongside evidence of asymptotic
consistency, in numerical examples.
