Campi Flegrei is a volcanic field located west of Naples (Italy) in a densely populated area. Since 2005, its ground has been rising steadily due to the accumulation of fluids at shallow depths. The inflation of volcanic edifices is a possible precursor of an impending eruption. The uplift is accompanied by increasing seismic activity. This raises concerns about the possibility that the volcano may be on the verge of an eruption. To track the fluid movement, it is possible to monitor subtle changes of velocities of seismic waves by exploring ambient seismic noise. By examining different frequency bands, we can observe velocity changes at different depths. We interpret these changes as a monitoring of depth-dependent deformation in addition to the standard monitoring of surface deformation. We observe a velocity decrease in the long-term trend, presumably due to the extension of the hydrothermal system at shallow depths. To explain the long-term changes, we model a spherical pressure source to simulate volumetric strain changes induced by recent fluid activity. The model explains both, surface and subsurface deformation which leads to the opening of microcracks and pores, resulting in the observed velocity decrease. The short-term velocity changes are mainly driven by temperature or groundwater level changes. Once velocity changes are corrected for seasonal effects, remaining short term velocity changes can be associated with volcanic activity and earthquake swarms.