Combined inversion for the three‐dimensional Q structure and source parameters using microearthquake spectra

The estimation of Q values and/or source corner frequencies f c 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‐ f c or a high‐ Q /low‐ f c spectral 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 ( M o ) and corner frequency ( f c ) 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 M o and f c 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 M o and f c 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 M o , f c , and Q . This process is repeated until the “goodness of fit measure” indicates no further improvement of the results. The method has been tested on a set of approximately 2800 P wave spectra (0.9 < M < 2.0) from the recordings of 635 microearth‐quakes 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×l8×l5km (depth). The Q model uncertainties have been estimated on the basis of several different tests: Self‐consistency, constraining the comer frequencies, and additionally splitting 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.