3-D seismic endoscopy: multiscale analysis and dynamic azimuthal filtering of borehole waves

Abstract

SUMMARY We previously proposed a method of seismic endoscopy and a related prototype tool to acquire directional information and to produce three‐dimensional (3‐D) seismic images in a cylindrical volume surrounding a borehole, with an investigation radius of several metres. Basic imaging algorithms were developed where the azimuthal move out (AMO) process is combined to a timescale method to refocus directional information and analyse the surrounding medium. Further processing tools, however, were necessary for separating the various types of waves recorded. The present paper describes a multiscale and dynamic azimuthal filtering to separate the far‐field waves from borehole or tube waves, generated by the pipe. We call this filter Stoneley move out (SMO), because it characterizes Stoneley waves associated with the tube. It is then possible to reconstruct details of the far field by an inverse wavelet transform. The comparison with classical methods such as the covariance method is discussed. Applications on noisy synthetic and experimental data are presented.

Extraction of non-continuous orbital frequencies from noisy insolation data and from palaeoproxy records of geomagnetic intensity using the phase of continuous wavelet transforms

“…The progressivity is a necessary condition to eliminate interferences due to spectral components on the negative axis (Saracco 1989(Saracco , 1994Valero & Saracco 2005). This condition is deduced from the requirement of isometry of the transform; it allows us to interpret the square modulus of wavelet coefficients as a density of energy in the time-scale half-plane.…”

“…This method is robust in the presence of a high level of noise, Gaussian white noise (see Section 3; Valero & Saracco 2005). The restriction of the modulus of wavelet transform along ridges gives an estimation of the amplitude modulation laws.…”

“…Moreover g is chosen to be progressive [i.e.ĝ(ω) = 0, for ω < 0]. The progressivity is a necessary condition to eliminate interferences due to spectral components on the negative axis (Saracco 1989(Saracco , 1994Valero & Saracco 2005).…”

“…The intuitive way is to define trajectories or spectral lines (ridges), representative of frequency modulation laws of the signal along which the energy is concentrated (Kronland-Martinet 1989;Saracco 1989;Saracco et al 1990a (trajectories with respect to dilation parameters); Saracco et al 1990bSaracco et al , 1991Guillemain et al 1991 (trajectories with respect to translation parameters). From the reproducing equation and the restriction of the wavelet transform along ridges, we can rebuilt the signal or some contributions of the signal (Valero & Saracco 2005). The developments that follow gives one way to construct such subsets.…”

“…The first methods for extracting frequencies and amplitude modulation laws were developed in the area of acoustics (Kronland-Martinet 1989;Saracco 1989;Saracco et al 1990aSaracco et al ,b, 1991Guillemain et al 1991;Delprat et al 1991;Valero & Saracco 2005). The interest of these methods, based on the phase of the CCWT, is that they can be applied to regions where the modulus of the transform is small.…”