Method for determining the thickness of a casing in a wellbore by signal processing pulse-echo data from an acoustic pulse-echo imaging tool

A method of determining the thickness of a casing installed in a wellbore penetrating an earth formation comprising the steps of:

• a) lowering an acoustic pulse-echo imaging tool into said wellbore, said acoustic pulse-echo imaging tool comprising an acoustic transducer;
• b) causing said acoustic transducer to emit an acoustic energy pulse whereby said casing and said wellbore become acoustically energized;
• c) causing said acoustic transducer to receive an at least partial reflection of said acoustic energy pulse from said casing and said wellbore and causing said acoustic transducer to generate an electrical analog of said at least partial reflection;
• d) digitizing said electrical analog whereby a first plurality of numbers is generated, each number of said plurality of numbers representing an amplitude of said electrical analog sampled at spaced-apart time intervals and having an index value representing an ordinal position of each number in said first plurality of numbers;
• e) calculating a first Fourier transform whereby said first plurality of numbers is used to generate a first plurality of Fourier coefficient pairs comprising real coefficients and imaginary coefficients resulting from calculating said first Fourier transform;
• f) calculating a second plurality of numbers by multiplying each number of said first plurality of numbers by said index value which represents said ordinal position of each number in said first plurality of numbers;
• g) calculating a third plurality of numbers representing magnitudes of a combination of said real coefficients and said imaginary coefficients from said first Fourier coefficient series, said magnitudes sampled as a function of frequency;
• h) calculating a second Fourier transform whereby said third plurality of numbers is used to generate a second plurality of Fourier coefficient pairs representing real coefficients and imaginary coefficients resulting from calculating said second Fourier transform;
• i) calculating a third plurality of coefficient pairs representing a complex group delay series comprising real components sampled as a function of frequency and imaginary components sampled as a function of frequency, whereby said third plurality of coefficient pairs is calculated from said first plurality of Fourier coefficient pairs and said second plurality of Fourier coefficient pairs;
• j) searching said second plurality of numbers and determining a first frequency at which a peak value occurs, and selecting a lowest frequency and a highest frequency between which said magnitude remains above a predetermined portion of said peak value;
• k) searching said real components of said third plurality of coefficient pairs between said lowest frequency and said highest frequency and determining a second frequency at which a maximum value occurs;
• and
• l) calculating the thickness of said casing from said second frequency whereby said second frequency is determined to be a resonant frequency of said casing.