In geophysics, including gravimetry, the term free-air anomaly (or free-air gravity anomaly) is the gravity anomaly (of some point on a planet) after compensating for altitude: after measuring gravity at some place and then making an adjustment to your measurement to remove the gravity-contribution of altitude differences, the free-air anomaly is whatever gravity anomaly remains. Given a difference in altitude, the gravity of a spherically symmetric body (planet or moon) will differ by a certain amount solely due to the difference in distance from the body's center. Actual bodies are not spherically symmetric: they rotate and tend to be oblate, but this type of difference is reflected in the altitude above some reference surface, e.g., Earth's sea level. The compensation for altitude is termed the free-air correction. (The phrase free-air correction is also used for a particular constant used for this correction on Earth, the typically quoted number being that of the equator, 0.3086 mGal per meter of altitude). The term free-air gravity refers to the gravity across the surface of the body after the free-air correction has been applied. A free-air gravity map is a map of the free-air gravity.
The free-air correction specifically does not include differences in gravity due to the mass of the landforms that are defining surface's altitude, i.e., for gravity on top of a mountain, it takes no account of the mountain's mass or density, but only reflects the measurement on top of the mountain corrected for the height itself (The phrase "free air" is supposed to reflect this), so the free-air anomaly includes the effects of these contributors. The location's remaining gravity anomaly after the additional correction for the landform itself and its mass is termed the Bouguer anomaly. The free-air anomaly is easier to obtain because it can be determined even if you have no information the density of the landforms.