We improve marine low-frequency (1-100 mHz) vertical seismometer data by subtracting noise generated by tilting under fluid flow and by seafloor deformation under ocean-surface gravity waves. We model the noise from the coherency and transfer functions between the vertical channel and other data channels that are more sensitive to the noise sources: the horizontal seismometer components for tilting and a differential pressure gauge for ocean waves. We subtract noise from three adjacent seafloor broadband seismometer stations at the OSN-1 deep-ocean test site: one sitting on the seafloor, another buried 1 m deep in sediments, and the third clamped in a borehole 248 m beneath the seafloor. Seafloor currents generate the seafloor sensor tilt noise, whereas tidally driven fluid pumping generates the borehole sensor tilt noise. Subtracting the tilt noise reduces the vertical channel noise levels by 35-40 dB between 1 and 60 mHz on the seafloor sensor and by 15-20 dB between 1 and 10 mHz on the borehole sensor. Subtracting the ocean-wave noise further reduces the noise level on all instruments by 5-15 dB between 4 and 20 mHz. After subtracting tilt and ocean-wave noise, the seafloor vertical channel is 5-10 dB quieter than the buried sensor vertical channel at frequencies below 30 mHz. The corrected borehole vertical channel has a similar noise level to the seafloor and buried sites above 10 mHz, but noise increases rapidly at lower frequencies, probably because of vertical strumming under tidally driven fluid flow.