The accurate determination of seismic attenuation, or , is useful for signal enhancement and reservoir characterization. To arrive at the necessary accuracy however, a number of issues must be addressed in the measurement technique. Specifically, spectral interference from closely spaced reflections is a major concern, in addition to the assumptions and errors associated with the raypath geometries of the reference and measured reflections. We have developed a robust method for measuring attenuation from prestack surface seismic gathers that helps minimize these issues. In our prestack -inversion technique; the presence of spectral interference was first reduced by making use of a variable-window time-frequency transform. To minimize the effects of the remaining interference, we then made use of an inversion scheme operating simultaneously in the frequency and traveltime-difference coordinates. A by-product of this inversion was a collection of the frequency-independent amplitude changes, which in the absence of geometric spreading, contains valuable amplitude variation with angle information, free from attenuation amplitude losses. Furthermore, under the assumption of locally 1D velocity and attenuation distributions, we made use of the transform to operate on traces of constant horizontal slowness. This allowed angle-dependent effects in the overburden such as attenuation anisotropy and source or receiver directivity to be eliminated. In the second part of our study, published separately, this technique was also demonstrated upon a shallow 3D seismic survey, and the measurements compared to another Q-estimation technique, as well as measurements from a vertical seismic profile.