We consider a non-linear filtering problem, whereby the signal obeys the stochastic Navier-Stokes equations and is observed through a linear mapping with additive noise. The setup is relevant to data assimilation for numerical weather prediction and climate modelling, where similar models are used for unknown ocean or wind velocities. We present a particle filtering methodology that uses likelihood informed importance proposals, adaptive tempering, and a small number of appropriate Markov Chain Monte Carlo steps. We provide a detailed design for each of these steps and show in our numerical examples that they are all crucial in terms of achieving good performance and efficiency.

Biases in expendable bathythermograph (XBT) instruments have emerged as a leading uncertainty in reconstructions of historical ocean heat content change and therefore climate change. Corrections for these biases depend on the type of XBT used; however, this is unspecified for 52% of the historical XBT profiles in the World Ocean Database. Here, we use profiles of known XBT type to train a neural network that can classify probe type based on three covariates: profile date, maximum recorded depth, and country of origin. Whereas previous studies have shown an average classification skill of 77%, falling below 50% for some periods, our new algorithm maintains an average skill of 90%, with a minimum of 70%. Our study illustrates the potential for successfully applying machine learning approaches in a wide variety of instrument classification problems in order to promote more homogeneous climate data records.

No abstract