The anelastic structure of the region surrounding the Tonga slab and Lau back arc spreading center in the southwest Pacific is studied using data from 12 broadband island stations and 30 ocean bottom seismographs. Two differential attenuation methods determine deltat* over the frequency band 0.1 to 3.5 Hz for earthquakes in the Tonga slab. The S-P method measures the difference in spectral decay between P and S waves arriving at the same station. The P-P method measures the difference in spectral decay for P waves with different paths through the upper mantle. Eight hundred sixty phase pairs are used to invert for two-dimensional 1/Qalpha structure using a nonnegative least squares algorithm. A grid search method determines the Qalpha/Qbeta ratio most compatible with both the S-P and P-P differential measurements. The highest attenuation (Qalpha=90) is found within the upper 100 km beneath the active portions of the Lau Basin extending westward to the Lau Ridge. These regions probably delineate the source region for the back arc spreading center magmas, expected to be within the upper 100 km based on petrological considerations. The high attenuation regions also correlate well with zones of low P wave velocity determined by regional velocity tomography. Somewhat lower attenuation is found beneath the Fiji Plateau than beneath the Lau Basin. The entire back arc is characterized by a gradual decrease in attenuation to a depth of 300 to 400 km. The slab is imaged as a region of low attenuation (Qalpha>900) material. A Qalpha/Qbeta ratio of 1.75 provides the best fit between the S-P and P-P data sets upon inversion. Spectral stacking shows no frequency dependence within the frequency band analyzed.