Oceanographic observations exist with different spatio-temporal resolutions and can be assimilated at various precision. The availability of numerous numerical simulations like ocean re-analysis make supervised machine learning appealing to deal with scale-related inverse problems. However data assimilation at finest resolutions using detailed oceanographic models is computationally intensive and building an exhaustive database may not be practical. Here we investigate the deep image prior method to downscale sea surface height observation and characterize estimation uncertainty in a fully-unsupervised manner. To do so, we set up a twin experiment using high resolution simulation from the NEMO Ocean engine and up-scale degraded data with multiple factors. Finally we give further perspectives of the method and make the link with data assimilation.