Nuclear Magnetic Resonance Spectrum Inversion Based on the Residual Hybrid <inline-formula> <tex-math notation="LaTeX">$l_{1}$ </tex-math> </inline-formula>/<inline-formula> <tex-math notation="LaTeX">$l_{2}$ </tex-math> </inline-formula> Norm

Abstract

The residual l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm is commonly used to measure the fitting error for a nuclear magnetic resonance (NMR) spectrum inversion. However, the solution of the objective function with the residual l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm is sensitive to noise. When 1 ≤ p <; 2, the solution of the objective function with the residual l p norm is less sensitive to noise. In this letter, we develop a method for NMR spectrum inversion based on the residual hybrid l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> /l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm, which behaves like l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm for small residuals and like l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm for large residuals. The numerical results of the 1-D T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> inversion and the 2-D D-T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> inversion show that the residual hybrid l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> /l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm method provides an inversion result with a smaller porosity error and exhibits a better ability to distinguish the fluid spectrum peaks.