First order models for the combined depth and frequency dependence of Q are derived and tested using several independent constraints. (1) Using a microphysics approach, the adoption of an absorption band as a first-order model for the frequency dependence of Q is justified, and the expected depth behavior of relaxation times in the earth is derived. The significant new parameter in this model of Q is &tgr;2, the period at the half-amplitude point of the high frequency end of the absorption band. (2) Using observed body-wave spectra, the existence of a frequency dependence in Q is proved, and the average location of that frequency dependence (i.e., &tgr;2) is estimated to be in the range 1 to 2.5 Hz. (3) Under the constraints of Q model ratios, the dept dependence of &tgr;2 is estimated by assuming that a free-oscillation and a body-wave Q model both measure Q from the same absorption band. The resulting &tgr;2 is about 0.04 s in the upper 200 km and then increases exponentially with depth in the mantle to about 1.9 at the core mantle boundary. The Q model ratios are better satisfied if a second absorption band is hypothesized to operate in the depth range of the asthenosphere. In that case, &tgr;2 for the mantle absorption band varies from about 0.09 s in the first 200 km to 4.0 s at 2886 km, and &tgr;2 for the asthenosphere absorption band is about 0.005 s in the depth range 35-220 km. (4) Both classes of Q models are tested in the time domain using synthetic seismograms of Russian and American nuclear explosions. Although trade-offs between source and mantle transfer functions preclude further refinement of the models at this time, a compatibility is demonstrated between the double absorption band model and time domain constraints, including arrival time and pulse shape.