We present an iterative method to constrain lateral variations in surf
ace wave attenuation using long-period surface wave amplitude anomalie
s. The method acts to isolate the anelastic signal from elastic focusi
ng effects, yielding largely unbiased estimates of lateral variations
in the inverse seismic quality factor, q(omega) = Q-1(omega), of the s
urface wave. In the zeroth iteration, linearized ray theory motivates
the construction of a reduced datum, using measurements from four cons
ecutive surface wave orbits, which is insensitive to elastic heterogen
eity, an operation which requires no a priori knowledge of elastic str
ucture. Synthetic experiments using both ray theoretic formalisms and
normal-mode calculations reveal that significant levels of elastic bia
s remain in the reduced data due to deviations from linearized ray the
ory. In further efforts to eliminate elastic bias, the remaining elast
ic signal in each reduced datum is predicted and subtracted using accu
rate forward theory and existing aspherical elastic mantle models. Thi
s operation is the first iteration of a non-linear inversion in which
the data at each iteration are the residuals between the observed anom
alies and the anomalies predicted for a fixed phase velocity model and
updated attenuation model. The zeroth and first iterations are perfor
med with 1610 vertical and 790 longitudinal component seismograms from
144 events. Heterogeneity maps of Rayleigh wave attenuation deltaq(om
ega, theta, phi) are retrieved for the even degrees 2, 4 and 6 of a sp
herical harmonic expansion in the period range of 150-300 s (S-0(25)-
S-0(60)). These surface wave attenuation maps are analogous to elastic
phase velocity in their radial averaging of, and linear relationship
to, intrinsic heterogeneity and are inverted for upper mantle anelasti
c structure. Under the physically plausible assumption that intrinsic
elastic and anelastic structure are correlated at every depth, the sur
face wave attenuation maps are well explained by a source region of an
elastic heterogeneity that is radially localized in the shallow mantle
(100-300 km) in a region we infer to be near the solidus temperature.