A multichannel deconvolution method to retrieve source–time functions: application to the regional Lg wave

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.

Extracting the Response of the Bay Bridge, California, from the Application of Multichannel Deconvolution to Earthquake-Induced Shaking

ABSTRACT Engineered structures, such as bridges, are excited by earthquakes at the base of the towers and the endpoint of decks. The different structural units of bridges, such as the towers and decks, are coupled. We extract the response of the towers and decks of the Bay Bridge in California from the motion of the bridge that is caused by earthquakes. This constitutes a multichannel deconvolution problem, which is, in general, ill-posed. We use the redundancy of the western half of the Bay Bridge, with near-identical towers and decks, to estimate the response of the upper towers, lower towers, and decks, from the transverse motion recorded in the bridge after four earthquakes. The extracted response functions for the four earthquakes show consistent wave arrivals that correspond to the waves that propagate through the towers and the decks. This method can, in principle, be used to monitor changes in the structural response.

3-D imaging of subsurface magnetic permeability/susceptibility with portable frequency domain electromagnetic sensors for near surface exploration

SUMMARY The in-phase response collected by portable loop–loop electromagnetic induction (EMI) sensors operating at low and moderate induction numbers (≤1) is typically used for sensing the magnetic permeability (or susceptibility) of the subsurface. This is due to the fact that the in-phase response contains a small induction fraction and a preponderant induced magnetization fraction. The magnetization fraction follows the magneto-static equations similarly to the magnetic method but with an active magnetic source. The use of an active source offers the possibility to collect data with several loop–loop configurations, which illuminate the subsurface with different sensitivity patterns. Such multiconfiguration soundings thereby allows the imaging of subsurface magnetic permeability/susceptibility variations through an inversion procedure. This method is not affected by the remnant magnetization and theoretically overcomes the classical depth ambiguity generally encountered with passive geomagnetic data. To invert multiconfiguration in-phase data sets, we propose a novel methodology based on a full-grid 3-D multichannel deconvolution (MCD) procedure. This method allows us to invert large data sets (e.g. consisting of more than a hundred thousand of data points) for a dense voxel-based 3-D model of magnetic susceptibility subject to smoothness constraints. In this study, we first present and discuss synthetic examples of our imaging procedure, which aim at simulating realistic conditions. Finally, we demonstrate the applicability of our method to field data collected across an archaeological site in Auvergne (France) to image the foundations of a Gallo-Roman villa built with basalt rock material. Our synthetic and field data examples demonstrate the potential of the proposed inversion procedure offering new and complementary ways to interpret data sets collected with modern EMI instruments.