Two-Stage Inversion Based on PDFNO and FDFD for Logging-While-Drilling Electromagnetic Ultradeep Detection

The inversion of Logging-While-Drilling (LWD) electromagnetic ultra-deep detection plays a critical role in geological guidance and reservoir evaluation. However, the computational speed of forward modeling for complex formations hinders the realization of real-time inversion. The application of deep learning (DL) has demonstrated potential to significantly enhance inversion efficiency. In this paper, we develop a two-stage method based on physics-driven Fourier neural operator (PDFNO) and frequency domain finite difference (FDFD) to achieve efficient two-dimensional formation inversion. In the first stage inversion, the PDFNO method significantly accelerates both forward modeling and Jacobian matrix computation, reducing computational time. During the second stage inversion, the FDFD method precisely inverts formation details, thereby ensuring inversion accuracy. The two crucial practical switching criteria are proposed for transitioning from PDFNO stage inversion to FDFD stage inversion. Both switching criteria can identify the appropriate switching iteration, thereby ensuring the accuracy and efficiency of the inversion. Three typical geological structure numerical examples are designed to evaluate the proposed method and the inversion results are compared against conventional FDFD based inversion. In numerical examples, the two-stage inversion method achieves comparable accuracy to FDFD based inversion while reducing computational time by approximately 50%, demonstrating the effectiveness of the two-stage inversion method.