The envelope shape around the arrival of the direct waves of seismic signals traveling in a randomly inhomogeneous medium has been predicted, based on the parabolic approximation, to be a measure of the long-wavelength components of the randomness as well as the attenuation properties of the medium. We use a nonlinear Marquardt-Levenberg inversion technique in order to model the SH wave envelopes of 119 earthquakes in the frequency band 2--16 Hz for lapse times less than 1.5 times the shear wave travel time. We attempted to obtain the ratio of the mean square fractional velocity fluctuation to the correlation length (&egr;2V/a) estimates as well as estimates of attenuation Q-1s. For the majority of the events we found a good correlation between the envelope shape and the hypocentral distances. The resultant &egr;2V/a of 10-3.27¿0.32 km-1 is independent of frequency. It agrees well with the choice of the Gaussian autocorrelation function for the long-wavelength components of the random velocity fluctuations. The resultant attenuation Q-1s is roughly proportional to the reciprocal of frequency. We may interpret it as either the scattering loss due to short-wavelength components of randomness or intrinsic loss. We have performed numerical simulations of the inversion process to quantify the model parameter uncertainties and to obtain a better understanding of the model parameter resolution. By modeling the wave envelopes as a superposition of noise-free wave envelopes and band-pass-filtered Gaussian noise we were able to reproduce the visual appearance of the observed envelopes as well as the observed features in the model parameter dependency. We find that for long hypocentral distances the envelope shape is controlled by the attenuation coefficient, while for short hypocentral distances the velocity fluctuations contribute dominantly. ¿1991 American Geophysical Union