  Speckle filtering is generally a prerequisite to the analysis of synthetic
aperture radar (SAR) images. Tremendous progress has been achieved in the
domain of single-image despeckling. Latest techniques rely on deep neural
networks to restore the various structures and textures peculiar to SAR images.
The availability of time series of SAR images offers the possibility of
improving speckle filtering by combining different speckle realizations over
the same area. The supervised training of deep neural networks requires
ground-truth speckle-free images. Such images can only be obtained indirectly
through some form of averaging, by spatial or temporal integration, and are
imperfect. Given the potential of very high quality restoration reachable by
multi-temporal speckle filtering, the limitations of ground-truth images need
to be circumvented. We extend a recent self-supervised training strategy for
single-look complex SAR images, called MERLIN, to the case of multi-temporal
filtering. This requires modeling the sources of statistical dependencies in
the spatial and temporal dimensions as well as between the real and imaginary
components of the complex amplitudes. Quantitative analysis on datasets with
simulated speckle indicates a clear improvement of speckle reduction when
additional SAR images are included. Our method is then applied to stacks of
TerraSAR-X images and shown to outperform competing multi-temporal speckle
filtering approaches. The code of the trained models is made freely available
on the Gitlab of the IMAGES team of the LTCI Lab, T\'el\'ecom Paris Institut
Polytechnique de Paris
(https://gitlab.telecom-paris.fr/ring/multi-temporal-merlin/).
