During geomagnetic storms, charge exchange between neutral hydrogen (H) atoms in the terrestrial exosphere and H+ and O+ ions in the ring current serves to dissipate magnetospheric energy through the generation of energetic neutral atoms (ENAs), which escape Earth’s gravity on ballistic trajectories. Imaging of the resulting ENA flux is a well-known technique to infer the ring current ion distribution, but its accuracy depends critically on the specification of the exospheric H density distribution. Although measurements of H airglow emission exhibit storm-time variations, the H density distributions used in ENA image inversion are typically assumed to be static, and the current lack of knowledge regarding global exospheric evolution during storms represents an important source of error in investigations of ring current dynamics. In this work, we present a new technique to reconstruct the global, 3D, and time-dependent H density distribution from observations of its optically thin emission at 121.6 nm (Lyman-α) acquired from the Lyman-alpha detectors onboard the NASA TWINS satellites. The technique is based on our recent development of a robust tomographic inversion algorithm, which is modified to incorporate the temporal dimension via Kalman filtering. We present the first time-dependent reconstructions of exospheric structure during geomagnetic storms, which exhibit pronounced dayside density enhancements and a strong anti-correlation with the DST index.