Numerical Inversion of the Sumudu Integral Transform in the Simulation of Electromagnetic Sounding of the Earth’s Interior

Abstract

––The integral Sumudu transform was established as an alternative to the Laplace transform in the 1990s. The Sumudu transform fundamental properties include preservation of the dimensionality of a function, ensuring measurement units both in the function and its image to be equivalent. Among its disadvantages is the absence of an explicit formula for calculating the inverse transform. The transforms are inverted by solving the corresponding Fredholm integral equation of the first kind, which is reduced to solving an ill-conditioned system of linear algebraic equations. We apply Tikhonov’s method for regularization of this type system. The paper proposes a technique for constructing a parametrized regularizing matrix that takes into account the properties of the Sumudu images obtained by modeling the electromagnetic sounding process of the Earth’s interior. A method for choosing the Tikhonov regularization parameters and the regularizing matrix is considered. The effectiveness of the proposed method for the Sumudu transform inversion is examined on a model problem of electromagnetic sounding of the Earth’s interior by a measurement system consisting of two loops spaced apart.