Solution of the inverse problem of estimating particle size distributions


  • Milton Alejandro Escobar Vázquez Universidad Tecnológica de la Mixteca
  • Silvia Reyes Mora Universidad Tecnológica de la Mixteca
  • Angel Sinue Cruz Felix Instituto Nacional de Astrofísica, Óptica y Electrónica



Turbidimetry, Mie Theory, Inverse Penrose matrix, Tikhonov regularization, light extinction


In this work, we describe two alternative methods for solving the ill-conditioned inverse problem that allows estimating the particle size distribution (PSD) from turbidimetry measurements. The first method uses the inverse Penrose matrix to solve the inverse problem in its discrete form. The second method consists of replacing an ill-posed problem with a collection of well-posed problems, penalizing the norm of the solution, and it is known as the Tikhonov regularization. Both methods are used to solve a synthetic application of the inverse problem by solving the direct problem using a theoretical expression of the distribution of particles sizes function f(D) and considering soft industrial latex particles (NBR), with average particle diameters of: 80.4, 82.8, 83.6, and 84.5 nm; and three illumination wavelengths in the UV-Vis region: 300, 450, and 600 nm. The estimated solution obtained by the inverse Penrose matrix is different from the original solution due to the inverse problem is ill-conditioned. In contrast, when using Tikhonov’s regularization, the estimate obtained is close to the original solution, which proves that the particle size distribution is adequate.


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How to Cite

M. A. Escobar Vázquez, S. Reyes Mora, and A. S. Cruz Felix, “Solution of the inverse problem of estimating particle size distributions”, Rev. Mex. Fís., vol. 70, no. 2 Mar-Apr, pp. 021304 1–, Mar. 2024.