SUMMARY
The retrieval of high-frequency seismic source–time functions (STFs) of similar earthquakes tends to be an ill-posed problem, causing unstable solutions. This is particularly true when waveforms are complex and band-limited, such as the regional phase Lg. We present a new procedure implementing the multichannel deconvolution (MCD) method to retrieve robust and objective STF solutions. The procedure relies on well-developed geophysical inverse theory to obtain stable STF solutions that jointly minimize the residual misfit, model roughness and data underfitting. MCD is formulated as a least-squares inverse problem with a Tikhonov regularization. The problem is solved using a convex optimization algorithm which rapidly converges to the global minimum while accommodating physical solution constraints including positivity, causality, finiteness and known seismic moments. We construct two L-shaped curves showing how the solution residual and roughness vary with trial solution durations. The optimal damping is chosen when the curves have acceptable levels while exhibiting no oscillations caused by solution instability. The optimal solution duration is chosen to avoid a rapidly decaying segment of the residual curve caused by parameter underfitting. We apply the MCD method to synthetic Lg data constructed by convolving a real Lg waveform with five pairs of simulated STFs. Four pairs consist of single triangular or parabolic pulses. The remaining pair consists of multipulse STFs with a complex, four-spike large STF. Without noise, the larger STFs in all single-pulse cases are well-recovered with Tikhonov regularization. Shape distortions are minor and duration errors are within 5 per cent. The multipulse case is a rare well-posed problem for which the true STFs are recovered without regularization. When a noise of ∼20 per cent is added to the synthetic data, the MCD method retrieves large single-pulse STFs with minor shape distortions and small duration errors (from 0 to 18 per cent). For the multipulse case, the retrieved large STF is overly smeared, losing details in the later portion. The small STF solutions for all cases are less resilient. Finally, we apply the MCD method to Lg data from two pairs of moderate earthquakes in central Asia. The problem becomes more ill-posed owing to lower signal-to-noise ratios (as low as 3) and non-identical Green's functions. A shape constraint of the small STF is needed. For the larger events with M5.7 and 5.8, the retrieved STFs are asymmetric, rising sharply and lasting about 2.0 and 2.5 s. We estimate radiated energies of 2.47 × 1013 and 2.53 × 1013 J and apparent stresses of 1.4 and 1.9 MPa, which are very reasonable. Our results are very consistent with those obtained in a previous study that used a very different, less objective ‘Landweber deconvolution’ method and a pre-fixed small STF duration. Novel improvements made by our new procedure include the application of a convex algorithm rather than a Newton-like method, a procedure for simultaneously optimizing regularization and solution duration parameters, a shape constraint for the smaller STF, and application to the complex Lg wave.
A Novel Fractional Tikhonov Regularization Coupled with an Improved Super-Memory Gradient Method and Application to Dynamic Force Identification Problems
