Improvement and assessment of a damped least-square solution of Rayleigh-wave inversion

摘要

High-frequency (≥ 2 Hz) Rayleigh-wave phase velocities have been utilized to determine shear (S)-wave velocities in near-surface geophysics since the early 1980s.For a given near-surface geophysical problem,it is essential to understand how well the data,calculated according to a layered-earth model,might match the observed data.It is also important to recognize that a match may only be possible for data within a certain frequency range or at specific frequencies because the sensitivity of Rayleigh-wave phase velocities due to changes in S-wave velocities varies with frequency.A data-resolution matrix is a function of the data kernel (the Jacobian matrix,determined by a geophysical model and a priori information applied to the problem) not the data.A data-resolution matrix of high-frequency Rayleigh-wave phase velocities,therefore,offers a quantitative tool for designing of field surveys and predicting the match between calculated and observed data.The resulting discussion provides insights into the process of inverting Rayleigh-wave phase velocities to estimate S-wave velocity structure.Because of restrictions on the data kernel for the inversion system,each near-surface geophysical target can only be resolved using Rayleigh-wave phase velocities within specific frequency ranges,and higher mode data are normally more accurately predicted than fundamental mode data.In a real-world example,we evaluated how well phase velocities can be predicted at different frequencies and the advantages of incorporating higher modes in inversion with the data-resolution matrix.Inversion with selected surface-wave data determined by data-resolution matrix provided better results in terms of model resolution.We determined an optimal damping vector in a vicinity of an inverted model with the singular value decomposition of a trade-off function of model resolution and variance.With these vectors,we can assess an inverted model obtained using a damped least-square method.