Calculating sensitivity or gradient for geophysical inverse problems using automatic and implicit differentiation

Automatic differentiation (AD) is a valuable computing technique that can automatically calculate the derivative of a function. Using the chain rule and algebraic manipulations, AD can save significant human effort by eliminating the need for theoretical derivations, coding, and debugging. Most importantly, it guarantees accurate derivatives, making it a popular choice for many non-linear optimization problems. However, its use in the geophysical inversion has been limited due to difficulties in differentiating the linear-equations solution, which cannot be explicitly defined as an elementary function. To address this issue, we employ an improved AD scheme using implicit differentiation (ADID) that creates a new AD operator that customizes the standard AD scheme to function more efficiently. We demonstrate the effectiveness and validity of ADID using a toy example and compare it with the widely used adjoint equation (AE) approach in a synthetic 2D magnetotelluric (MT) problem. ADID is highly versatile and compatible and can be easily implemented for similar geophysical problems. Finally, we show how ADID can be integrated into 3D MT and 3D direct current resistivity (DC) inversions.