Sensitivity kernels in seismic wave propagation: a simplified explanation for the banana-doughnut paradox

Ray theory, a high-frequency approximation to describe wave propagation, has been a cornerstone in seismology for over a hundred years. Despite its simplicity and wide range of applications, some limitations combined with the ever-increasing computational power motivated the development of finite-frequency theory, a better model to describe how the Earth's inner structure affects seismic waves. Finite-frequency theory has matured a lot in the last decades, and it is now widely applied in many geophysical problems. However, most students and even some experienced researchers face difficulties understanding it. An appropriate theoretical comprehension is paramount to making the most out of the methods a theory underpins, avoiding pushing it beyond its limits, and further developing it. With that problem in mind, this paper shows a simplified formulation of the sensitivity kernels, which are the generalization of rays in the finite-frequency regime. The resultant model, despite its limitations, correctly predicts the main features of finite-frequency theory, including the zero sensitivity in the middle of the travel-time kernels, known as the banana-doughnut paradox, shedding new light on that intriguing phenomenon. The step-by-step derivation and relatively easy equations should be understandable by an undergraduate student with a reasonable knowledge of classical physics and calculus. A Colab Notebook implementing the main formulas accompanies the paper, allowing readers to interact and play with the results. (AU)