  We introduce a `double-difference' method for the inversion for seismic
wavespeed structure based on adjoint tomography. Differences between seismic
observations and model predictions at individual stations may arise from
factors other than structural heterogeneity, such as errors in the assumed
source-time function, inaccurate timings, and systematic uncertainties. To
alleviate the corresponding nonuniqueness in the inverse problem, we construct
differential measurements between stations, thereby reducing the influence of
the source signature and systematic errors. We minimize the discrepancy between
observations and simulations in terms of the differential measurements made on
station pairs. We show how to implement the double-difference concept in
adjoint tomography, both theoretically and in practice. We compare the
sensitivities of absolute and differential measurements. The former provide
absolute information on structure along the ray paths between stations and
sources, whereas the latter explain relative (and thus higher-resolution)
structural variations in areas close to the stations. Whereas in conventional
tomography a measurement made on a single earthquake-station pair provides very
limited structural information, in double-difference tomography one earthquake
can actually resolve significant details of the structure. The
double-difference methodology can be incorporated into the usual adjoint
tomography workflow by simply pairing up all conventional measurements; the
computational cost of the necessary adjoint simulations is largely unaffected.
Rather than adding to the computational burden, the inversion of
double-difference measurements merely modifies the construction of the adjoint
sources for data assimilation.
