There is a lot of confusion around the use of the gravity anomaly in a geophysical context. I have been discussing this topic with Vanderlei for many years now. It's been in our minds ever since our first gravity and magnetics undergraduate course back in 2007. I'm sure we were the source of many headaches for Professor Eder Molina, who was teaching the course at the time.
This short paper is our attempt at bringing a geophysical modeling point of view to the debate.
The gravity anomaly is defined as the difference between the Earth's gravity on the geoid and the normal gravity on the reference ellipsoid. Because these quantities are not at the same point, the anomaly contains centrifugal accelerations and cannot be considered a harmonic function. The gravity disturbance is the difference between gravity and normal gravity at the same point. Consequently, the centrifugal effects can be neglected and the disturbance can be considered a harmonic function. This is the premise behind most potential-field data processing techniques (e.g., upward/downward continuation). Unlike the anomaly, the disturbance is due solely to the gravitational effects of geologic sources, making it the most appropriate for geophysical purposes. Use of the gravity anomaly in geophysics carries with it the implicit assumption that it is a good approximation for the gravity disturbance. However, bear in mind that the difference between the gravity disturbance and the free-air anomaly can be larger than 10 mGal worldwide. In fact, we argue that the assumptions made during gravity forward and inverse modeling imply that the quantity being modelled is the disturbance, not the anomaly.
Map of the difference between the gravity disturbance and the free-air anomaly worldwide. This is the error committed when assuming that the free-anomaly is a good approximation for the disturbance.