Two equivalent expressions of the Kramers-Kronig relations

Natural and man-made materials show viscoelastic behaviour, by which stress and strain are related by a relaxation function or a complex stiffness modulus in the time and frequency domains, respectively. Viscoelasticity led to significant research in material science and seismology (Christensen, 1982). Real-world systems should satisfy the Kramers-Kronig relations (KKRs), known from the beginning of the 20th century from the works of de Laer Kronig (1926) and Kramers (1927) on electromagnetism, showing the interrelation between the real and imaginary parts of the complex susceptibility. Electrical and mechanical representations include the Debye model, used to describe the behaviour of dielectric materials, and the Zener viscoelastic model, respectively, both being mathematically equivalent (Carcione, 2014). In viscoelasticity, the KKRs connect the real and imaginary parts of the stiffness modulus. Carcione et al. (2019) provide a complete derivation of the relations using the Sokhotski-Plemelj equation, showing explicitly what are the conditions for the relations to hold. There are many forms of the relations. In geophysics, the book of Mavko et al. (2009) provides the most popular expression, based on the relaxed modulus. In this note, we show that this expression is equivalent to a simpler one involving the unrelaxed modulus. Two different demonstrations are given, illustrating the eclectic mathematical apparatus available to obtain the relations. Moreover, we develop the KKRs relations for the creep function (creep compliance) and derive them for seismological applications, i.e. based on the seismic velocity and attenuation factor.