Recently, the trend in understanding the radial mantle viscosity structure has been toward forming and solving inverse problems rather than repeatedly solving the forward problem. In theory, the inverse problem provides not only a model, but also estimates of the resolving power of the data and of the trade-offs between model parameters; however, the resolution and trade-off analyses are not always straight-forward.
The approach used in obtaining information about mantle viscosity from the geoid or plate motion data is to solve the equations of motion for a viscous, self-gravitating spherical shell, using separation of variables and assuming a radially symmetric viscosity [ Richards and Hager, 1984; Ricard et al. , 1984; Forte and Peltier, 1987]. The resulting ordinary differential equations can be solved for the Green's functions (often referred to as response functions or response kernels) that depend only on the viscosity structure. The response functions can then be convolved with a distribution of internal density anomalies to calculate the geoid.
Seismology provides a model of the internal structure of the mantle. Shear waves, or S-waves, are internal body waves whose motion is perpendicular to the direction of propagation of the wave. Using earthquakes as the source of shear waves and a global network of seismometers, seismologists can construct three-dimensional images of the variation of S-wave velocity throughout the mantle. This 3-D image of structure is refered to as a tomographic model. All of the mantle viscosity studies that use a density anomaly model to drive flow in the mantle, use these tomographic models, or a density model based on past history of plate tectonics. This includes all mantle viscosity models based on geoid an plate velocities. One way in which some of the assumptions about seismic models can be tested is to develop a tomogram of density anomalies based on the location of subducted slab material from plate reconstruction histories. Ricard and co-workers found that using a reconstruction of plate histories, much of the structure in the mantle observed in seismic tomography models could be explained as the remnants of subducted slabs [ Ricard et al. , 1993].
Density anomalies are calculated from the tomographic model using a simple estimate of the effect of temperature on seismic shear velocity. In a homogeneous fluid, this is a reasonable assumption but if there are significant lateral variations in chemical composition in the mantle, this is problematic because, the effect of temperature on shear velocity and density is different from the effect of composition on shear velocity and density. Most geoid and plate velocity studies use a simple radial ratio of seismic velocity to density. Due to changes in the phase of the major mantle minerals, the ratio of seismic velocity to density may be a strong function of radius, even if the mantle is chemically homogeneous. If the anomalies in the tomographic models represent both chemical and thermal anomalies in the mantle, then the simple scaling used in most geoid and plate velocity studies should be questioned. An alternate approach is to take a viscosity structure from one data set and invert for the radial function of the ratio of seismic velocity to density needed to fit another data set [e.g., Forte et al. , 1993].