The first thermal images of Phobos' shadow on the surface of Mars, in addition to simultaneous visible images, were obtained by the Phobos '88 Termoskan instrument. The best observed shadow occurrence was on the flanks of Arsia Mons. For this occurrence, we combined the observed decrease in visible illumination of the surface with the observed decrease in brightness temperature to calculate thermal inertias of the Martian surface. The most realistic of our three models of eclipse cooling improves upon our preliminary model by including nonisothermal initial conditions and downward atmospheric flux. Most of our derived inertias fall within the range 38 to 59 J m-2 s-1/2 K-1 (0.9 to 1.4¿10-3 cal cm-2 s-1/2 K-1), corresponding to dust-size particles (for a homogeneous surface), consistent with previous theories of Tharsis as a current area of dust deposition. Viking infrared thermal mapper (IRTM) inertias are diurnally derived and are sensitive to centimeter depths, whereas the shadow-derived inertias sample the upper tenths of a millimeter of the surface. The shadow-derived inertias are lower than those derived from Viking IRTM measurements (84 to 147), however, uncertainties in both sets of derived inertias make conclusions about layering tenuous. Thus, near-surface millimeter versus centimeter layering may exist in this region, but if it does, it is likely not very significant. Both eclipse and diurnal inertias appear to increase near the eastern end of the shadow occurrence. We also analyzed a shadow occurrence near the crater Herschel that showed no observed cooling. This analysis was limited by cool morning temperatures and instrument sensitivity, but yield a lower bound of 80 on eclipse inertias in that region. Based upon our results, we strongly recommend future spacecraft thermal observations of Phobos' shadow, and suggest that they will be most useful if they improve upon Termoskan's geographic and temporal coverage and its accuracy. ¿ American Geophysical Union 1995