Efficient 3D Transient Electromagnetic Model Order Reduction Inversion for Arbitrary Transmitter Waveforms Based on Convolution

Three-dimensional inversion of airborne transient electromagnetic (ATEM) data is computationally demanding because it requires repeated forward simulations and sensitivity calculations. Previous Krylov-subspace studies have substantially improved the efficiency of three-dimensional transient electromagnetic forward modeling, while a recently developed model order reduction (MOR)-based sensitivity framework has enabled efficient three-dimensional ATEM inversion for an ideal step-off excitation. However, this MOR inversion framework is derived from a source-free step-off initial-value problem and cannot readily incorporate the time-dependent source associated with a realistic transmitter waveform. Directly introducing such a source term would substantially complicate the corresponding MOR sensitivity derivation and implementation. In this study, we extend the existing MOR inversion framework to arbitrary transmitter waveforms using a convolution-based transformation. The step-off responses and sensitivities are first computed within the rational-Krylov MOR framework and are then transformed to those corresponding to the actual transmitter waveform. Gaussian quadrature and interpolation operations are assembled into a precomputed full-waveform transformation matrix. Because this matrix is independent of the subsurface conductivity model, it can be reused for all receivers, model parameters, and inversion iterations, thereby avoiding repeated interpolation and convolution operations. Forward modeling tests for trapezoidal, half-sine, triangular, and VTEM-type waveforms and show that the proposed method accurately reproduces the full-waveform responses, with off-time relative errors generally below 5%. Synthetic inversions demonstrate that the step-off approximation may cause conductivity overestimation, boundary distortion, spurious anomalies, and degraded convergence for long turn-off waveforms, whereas the waveform-corrected MOR inversion provides more stable and reliable results. A field ATEM example further provides an initial field-scale assessment of the feasibility of the method, yielding a conductivity model consistent with the known fault-controlled geological setting. The field inversion requires approximately 4.96 h for 14 iterations.