Inversion of shallow-seismic wavefields: I. Wavefield transformation

Abstract

SUMMARY I calculate Fourier–Bessel expansion coefficients for recorded shallow‐seismic wavefields using a discrete approximation to the Bessel transformation. This is the first stage of a full‐wavefield inversion. The transform is a complete representation of the data, recorded waveforms can be reconstructed from the expansion coefficients obtained. In a second stage (described in a companion paper) I infer a 1‐D model of the subsurface from these transforms and P‐wave arrival times by fitting them with their synthetic counterpart. The whole procedure avoids dealing with dispersion in terms of normal modes, but exploits the full signal‐content, including the dispersion of higher modes, leaky modes and their true amplitudes. It is robust even in the absence of a priori information. I successfully apply it to the near field of the source. And it is more efficient than direct inversion of seismograms. I have developed this new approach because the inversion of shallow‐seismic Rayleigh waves suffers from the interference of multiple modes that are present in the majority of our field data sets. Since even the fundamental‐mode signal cannot be isolated in the time domain, conventional phase‐difference techniques are not applicable. The potential to reconstruct the full waveform from the transform is confirmed by two field‐data examples, which are recorded with 10 Hz geophones at effective intervals of about 1 m and spreads of less than 70 m length and are excited by a hammer source. Their transforms are discussed in detail, regarding aliasing and resolution. They reveal typical properties of shallow surface waves that are at variance with assumptions inherent to conventional inversion techniques: multiple modes contribute to the wavefield and overtones may dominate over the fundamental mode. The total wavefield may bear the signature of inverse or anomalous dispersion, although the excited modes have regular and normal dispersion. The resolution at long wavelengths (and thus the penetration depth of the survey) is limited by the length of the profile rather than by the signal‐to‐noise ratio at low frequencies. Finally, this approach is compared with conventional techniques of dispersion analysis. This illustrates the advantage of conserving the full wavefield in contrast to the reduction to one dispersion curve.

Rayleigh-to-shear wave conversion at the tunnel face — From 3D-FD modeling to ahead-of-drill exploration

“…To correct the latter two effects, we apply a frequency-dependent traveltime correction in the frequency domain by multiplying the phase spectra of the seismograms with ‫2מ‬ f͑␦ p n ͑f͒ ‫ם‬ ␦ p m ͑f͒͒x s ‫מ‬ p m ͑f͒͑x͒͒. The dispersion curves that are required for this deconvolution may be derived by a dispersion analysis of the direct wavefield ͑McMechan and Yedlin, 1981;Forbriger, 2003͒. If the EDZ exhibits lateral variations of the elastic properties, the functions p n ͑f͒ and p m ͑f͒ represent effective slowness that is averaged over the travel path of the Rayleigh wave.…”

A Review and Some New Issues on the Theory of the H/V Technique for Ambient Vibrations

“…An interesting special and less-known feature of the ellipticity is the role of so-called osculation points, which are those points (see, e.g., Forbriger 2003) where two dispersion curves of surface waves (especially Rayleigh waves) come very near to each other and eventually even cross under certain circumstances (see Kausel et al 2015). For sake of simplicity, just a stratigraphic profile constituted by a single horizontal layer over an half-space (LOH) is used to describe the special behaviour of the ellipticity at these points.…”

On the Significance of Viscoelasticity in a 2D Full Waveform Inversion of Shallow Seismic Surface Waves

“…We infer a 1D model of the subsurface by an inversion of wavefield spectra (Forbriger, 2003). This model is used to calculate purely elastic and viscoelastic wavefields with a 2D Finite Difference algorithm in the time domain (Bohlen, 2002).…”

“…This is done with a transformation suggested by Amundsen and Reitan (1994) which is exact for a 1D subsurface structure. Furthermore, we determine an optimized source time function by a deconvolution in the frequency domain (Forbriger, 2003) which is convolved with the modeled data.…”

“…Furthermore, surface waves have a better signal to noise ratio compared to body waves and can be used to investigate sites with low velocity zones which cannot be done with refracted body waves. There are established methods to invert surface waves (e. g. inversion of dispersion curves or wavefield spectra (Forbriger, 2003)) but all these methods assume 1D subsurface structures. To overcome this limitation we apply FWI to shallow seismic surface waves.…”