The estimation of Q values and/or source corner frequencies fc from single-station narrow-band recordings of microearthquake spectra is a strongly nonunique problem. This is due to the fact that the spectra can be equally well fitted with low-Q/high-fc or a high-Q/low-fc spectra models. Here, a method is proposed to constrain this ambiguity by inverting a set of microearthquake spectra for a three-dimensional Q model structure and model source parameters seismic moment (M0) and corner frequency (fc) simultaneously. The inversion of whole path Q can be stated as a linear problem in the attenuation operator t* and solved using a tomographic reconstruction of the three-dimensional Q structure. This Q structure is then used as a ''geometrical constraint'' for a nonlinear Marquardt-Levenberg inversion of M0 and fc and a new Q value. The first step of the method consists of interactively fitting the observed microearthquake spectra by spectral models consisting of a source spectrum with an assumed high-frequency decay, a single-layer resonance filter to account for local site effects, and additional ''whole path attenuation'' along the ray path. From the obtained Q values, a three-dimensional Q model is calculated using a tomographic reconstruction technique (SIRT). The individual Q values along each ray path are then used as Q starting values for a nonlinear iterative Marquardt-Levenberg inversion of M0 and fc and a ''new'' Q value. Subsequently, the ''new'' Q values are used to reconstruct the next Q model which again provides starting values for the ''next'' nonlinear inversion of M0, fc, and Q. This process is repeated until the ''goodness of fit measure'' indicates no further improvement of the results.

The method has been testedon a set of approximately 2800 P wave spectra (0.9<M<2.0) from the recordings of 635 microearthquakes from the Kaoiki seismic zone in Hawaii (Big Island) which were recorded at up to six stations. The hypocenters are distributed within a volume of approximately 18¿18¿15 km (depth). The Q model uncertainties have been estimated on the basis of several different tests: Self-consistency, constraining the corner frequencies, and additionally splitting of the data set. The standard deviation of the final Q model which used a grid size of 1.5¿1.5¿2.0 km (depth) was less than 3% for the depth range 0--5 km, less than 5% between 5 and 7 km, and 7% between 7 and 9 km. The simulation of strong attenuation effects close to the surface shows that site effects may cause a corruption of the resulting Q model at shallow depths. For the given data set and depths below 3--5 km, the method is believed to be able to resolve the model dependent attenuation structure on a scale down to 1--2 km with a resolution of a few percent. ¿ American Geophysical Union 1990