Radon migration of acoustic data

An apparatus for imaging a formation penetrated by a borehole, the apparatus comprising:

• an acoustic downhole tool configured to be disposed in the borehole, the tool comprising an acoustic source configured to emit acoustic energy into the formation and an acoustic receiver configured to receive acoustic energy reflected by the formation;
• and
• a processor coupled to the downhole tool and configured to implement a method comprising:
• obtaining acoustic data in a depth-time domain from the acoustic downhole tool disposed at a depth in the borehole;
• transforming the acoustic data having waveform w(z,t) where z is a depth of the acoustic receiver and t is time in the depth-time domain into a Radon domain {tilde over (w)}(p,τ) that is a function of p and tau (τ) using a Radon transform where {tilde over (w)}(p,τ)=∫w(z,τ+pz)dz;
• filtering the acoustic data in the Radon domain to increase a signal of interest in the acoustic data in the Radon domain by removing unwanted signals in the Radon domain;
• determining a location of a point in the formation that reflected acoustic energy emitted from the acoustic source to the acoustic receiver, the location of the point being represented in the Radon domain;
• and
• inverting the location of the point represented in the Radon domain into a radius-depth, r-z, domain to image the formation, the depth being a distance into the borehole and the radius being a radial distance from the borehole.

A non-transitory computer-readable medium comprising computer-executable instructions for imaging a formation penetrated by a borehole, when executed by a computer, cause the computer to implement a method comprising:

• obtaining acoustic data in a depth-time domain from an acoustic downhole tool disposed at a depth in the borehole, the acoustic downhole tool comprising an acoustic source and an acoustic receiver;
• transforming the acoustic data having waveform w(z,t) where z is a depth of the acoustic receiver and t is time in the depth-time domain {tilde over (w)}(p,τ) into a Radon domain that is a function of p and tau (τ) using a Radon transform where {tilde over (w)}(p,τ)=∫w(z,τ+pz)dz;
• filtering the acoustic data in the Radon domain to increase a signal of interest in the acoustic data in the Radon domain by removing unwanted signals in the Radon domain;
• determining a location of a point in the formation that reflected acoustic energy emitted from the acoustic source to the acoustic receiver, the location of the point being represented in the Radon domain;
• and
• inverting the location of the point represented in the Radon domain into a radius-depth, r-z, domain to image the formation, the depth being a distance into the borehole and the radius being a radial distance from the borehole.