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

Abstract

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.

1. Introduction

The airborne transient electromagnetic method (ATEM) has been widely used in mineral exploration [1], groundwater investigation [2], engineering geophysics [3], and regional geological mapping [4] because of its high efficiency, operational flexibility, and broad spatial coverage. With increasingly complex exploration targets and greater investigation depths, conventional apparent-resistivity interpretation and one-dimensional inversion are often insufficient for high-resolution quantitative interpretation. Inversion provides one of the most effective means of extracting subsurface electrical structures from electromagnetic data. In particular, three-dimensional inversion can explicitly account for complex topography and three-dimensional geological variations, thereby reducing interpretation artifacts caused by lateral heterogeneity and providing a more realistic image of subsurface conductivity distributions than one- or two-dimensional approaches.

However, three-dimensional inversion of ATEM data remains computationally demanding, especially for surveys with dense receiver coverage and many time channels. Conventional three-dimensional time-domain electromagnetic forward modeling commonly relies on implicit time-stepping schemes, in which large sparse linear systems must be solved at successive time levels [4,5,6,7,8]. During inversion, sensitivity-related calculations further amplify this computational burden. To improve efficiency, footprint-based approaches [9] and local mesh techniques [4,5] have been used to reduce the computational domain, while approximate sensitivity calculations have been developed to reduce the cost of Jacobian construction [8,10]. Related open-source frameworks have also advanced large-scale electromagnetic modeling. In particular, PETGEM combines parallel edge-based finite elements, unstructured tetrahedral discretization, and high-performance computing [11,12,13], while other representative frameworks include custEM and emg3d [14,15]. These frameworks mainly improve computational efficiency through spatial discretization, large-scale linear-system solution, and parallel execution. However, this does not alter the underlying reliance of existing three-dimensional TEM inversion algorithms on time stepping for forward modeling, which still requires repeated solutions of large-scale linear systems. In contrast, Krylov subspace methods fundamentally depart from this sequential time-marching strategy, providing an alternative means of computing three-dimensional transient electromagnetic responses.

Druskin and co-workers first introduced polynomial Krylov subspace methods for electromagnetic diffusion problems [16], overcoming the stability restriction associated with explicit finite-difference time-domain schemes. Nevertheless, the convergence of polynomial Krylov methods is strongly affected by conductivity contrast and by the time or frequency range of interest. For stiff diffusion problems, a large subspace dimension may be required. Rational Krylov subspace (RKS) methods were later developed to address this limitation [17,18]. Compared with polynomial Krylov methods, RKS methods generally require much smaller subspaces and are less sensitive to strong conductivity contrasts. Their efficiency, however, depends strongly on the choice of interpolation poles, and different poles usually require additional expensive matrix factorizations.

To reduce the cost associated with pole selection and repeated factorizations, Börner proposed a simplified pole-optimization strategy and used a small number of repeated poles to construct the rational Krylov subspace [19]. Building on this idea, Zhou et al. [20] further developed a single-pole rational Krylov strategy. By allowing a modest increase in the subspace dimension, this approach requires only one LU factorization while retaining accurate responses over the full time range, thereby improving the efficiency of three-dimensional TEM forward modeling. These developments indicate that Krylov subspace methods have become increasingly mature for three-dimensional TEM forward modeling. In contrast, their application to three-dimensional inversion remains relatively limited because RKS-based forward modeling is not readily compatible with conventional adjoint-equation sensitivity calculations, making construction of the Jacobian matrix challenging [21,22,23]. Recently, Cao et al. [24] developed a model order reduction (MOR)-based sensitivity formulation for three-dimensional central-loop ATEM inversion by reusing the reduced Ritz information obtained during rational Krylov forward modeling. This development enabled efficient three-dimensional inversion within the rational Krylov framework. However, the formulation was derived for the source-free initial-value problem associated with an ideal step-off excitation and did not directly account for finite-duration transmitter waveforms.

In many theoretical studies of TEM, the ideal step-off waveform is adopted as the commonly adopted simplified excitation. This simplification is attractive because a step-off source transforms the driven Maxwell system into a source-free initial-value problem, thereby reducing the complexity of forward modeling. In practical airborne electromagnetic surveys, however, transmitter systems usually employ finite-duration current waveforms, such as half-sine, triangular, trapezoidal, or VTEM-like waveforms, rather than an ideal instantaneous turn-off. These waveforms are used to increase the transmitter moment, extend the investigation depth, and maintain a sufficient signal-to-noise ratio. Ignoring the true transmitter waveform in inversion may therefore introduce errors in both the timing and amplitude of the predicted responses, ultimately affecting the reliability of the recovered conductivity model.

For arbitrary-waveform forward modeling, conventional full-time Krylov-subspace approaches can accurately compute full-waveform responses, but the time-dependent on-time source may require repeated construction of Krylov subspaces, limiting their computational advantage over implicit time stepping. To address this issue, source-decoupled Krylov-subspace methods have been proposed to project the governing equation onto a low-dimensional subspace, thereby substantially reducing the cost of three-dimensional full-time TEM modeling [25]. Although effective for forward modeling, its driven reduced system differs from the source-free initial-value formulation used by the MOR inversion method of Cao et al. [24]. Directly incorporating this time-dependent source formulation into the existing MOR inversion framework would require a substantial reformulation of the sensitivity derivation, increasing both theoretical and implementation complexity.

An alternative is to retain the source-free step-off formulation used by the MOR inversion framework and obtain arbitrary-waveform responses through convolution. This strategy avoids introducing a time-dependent source term into the reduced system and therefore preserves the existing MOR sensitivity formulation. Convolution has previously been used to model full-time TEM responses and to analyze the effects of realistic transmitter-current waveforms on TEM responses and interpretation [26,27,28]. Li et al. [29] further incorporated the complete transmitter waveform into one-dimensional inversion of ground-based TEM magnetic-induction data. However, these studies did not extend convolution to the sensitivity calculation required by three-dimensional rational Krylov MOR inversion. Accordingly, this study extends the three-dimensional MOR inversion framework of Cao et al. [24] from an ideal step-off source to arbitrary transmitter waveforms. The step-off responses and MOR-based sensitivities are first computed using the existing rational Krylov formulation; the same model-independent convolution operator is then applied to both quantities to obtain the corresponding arbitrary-waveform responses and sensitivities. The main contribution is therefore an arbitrary-waveform extension of the existing MOR inversion framework that requires no reformulation of its source-free sensitivity derivation. As a secondary contribution, Gaussian quadrature and global interpolation are assembled into a reusable convolution matrix to reduce repeated calculations for multiple receivers, model parameters, and inversion iterations. Synthetic and field examples are used to evaluate the accuracy and inversion performance of the proposed extension.

The remainder of this paper is organized as follows. Section 2 presents the rational Krylov formulation for step-off responses, the three-dimensional inversion algorithm, the computation of the sensitivity matrix based on model order reduction, and the convolution transformation operator. Section 3.1 analyzes the influence of the convolution parameters on computational accuracy. Section 3.2 validates the accuracy of the proposed forward modeling approach for several representative transmitter waveforms. Section 3.3 evaluates the convolution-based MOR sensitivity calculation through comparison with the conventional time-stepping method. Section 3.4 and Section 4 present synthetic and field examples of three-dimensional full-waveform inversion, respectively.

2. Theory

2.1. Forward Modeling

The detailed forward discretization based on the mimetic finite volume (MFV) method can be found in previous studies and in the work of Zhou et al. [20] In transient electromagnetic modeling, the governing equations are commonly derived from Maxwell’s equations under the quasi-static assumption, where the displacement current is neglected. For a classical step-off excitation waveform, the effect of the source can be represented by the initial magnetic field at the turn-off time. As a result, the driven diffusion problem is transformed into a source-free initial-value problem. After MFV spatial discretization, the governing equation can be written as

$$\begin{array}{c}{\displaystyle \frac{\partial \mathbf{b}}{\partial t}}+\mathbf{C}{\mathbf{M}}_{e\sigma }^{-1}{\mathbf{C}}^{\mathrm{T}}{\mathbf{M}}_{f\mu }\mathbf{b}=\mathbf{0}\\ \mathbf{b}(t=0)={\mathbf{b}}_0\end{array}$$

(1)

Here, $\mathbf{b}$ denotes the discrete magnetic flux density, $\mathbf{C}$ is the discrete curl operator, ${\mathbf{M}}_{e\sigma }$ is the edge inner-product matrix containing the conductivity information, ${\mathbf{M}}_{f\mu }$ is the face inner-product matrix associated with the magnetic permeability, ${\mathbf{M}}_f$ is the geometric face inner-product matrix, and ${\mathbf{b}}_0$ represents the initial magnetic flux density at the step-off time.

To obtain a symmetric coefficient matrix, the matrix ${\mathbf{M}}_{f\mu }$ is factorized as

$${\mathbf{M}}_{f\mu }={\mathbf{M}}_{f\mu }^{{\displaystyle \frac{1}{2}}}{\mathbf{M}}_{f\mu }^{{\displaystyle \frac{1}{2}}}$$

(2)

Multiplying the governing equation by ${\mathbf{M}}_{f\mu }^{{\displaystyle \frac{1}{2}}}$ from the left gives:

$$\begin{array}{c}{\mathbf{M}}_{f\mu }^{{\displaystyle \frac{1}{2}}}\mathbf{C}{\mathbf{M}}_{e\sigma }^{-1}{\mathbf{C}}^{\mathrm{T}}{\mathbf{M}}_{f\mu }^{{\displaystyle \frac{1}{2}}}{\mathbf{M}}_{f\mu }^{{\displaystyle \frac{1}{2}}}\mathbf{b}+{\displaystyle \frac{\partial {\mathbf{M}}_{f\mu }^{\frac{1}{2}}\mathbf{b}}{\partial t}}=\mathbf{0}\\ {\mathbf{M}}_{f\mu }^{{\displaystyle \frac{1}{2}}}\mathbf{b}(t=0)={\mathbf{M}}_{f\mu }^{{\displaystyle \frac{1}{2}}}{\mathbf{b}}_0\end{array}$$

(3)

By defining ${\mathbf{A}}_u={\mathbf{M}}_{f\mu }^{{\displaystyle \frac{1}{2}}}\mathbf{C}{\mathbf{M}}_{e\sigma }^{-1}{\mathbf{C}}^{\mathrm{T}}{\mathbf{M}}_{f\mu }^{{\displaystyle \frac{1}{2}}}$ and $\mathbf{u}={\mathbf{M}}_{f\mu }^{{\displaystyle \frac{1}{2}}}\mathbf{b}$, the above equation can be simplified as:

$$\begin{array}{c}{\displaystyle \frac{\partial \mathbf{u}}{\partial t}}+{\mathbf{A}}_u\mathbf{u}=\mathbf{0}\\ \mathbf{u}(t=0)={\mathbf{u}}_0\end{array}$$

(4)

Traditional time-stepping schemes require the repeated solution of large sparse linear systems at many time levels. When the simulation spans a wide time range or involves dense temporal sampling, such schemes may require multiple LU factorizations and a large number of back substitutions, leading to a rapidly increasing computational cost. Krylov subspace methods provide an efficient alternative by approximating the action of the matrix exponential in a low-dimensional subspace, thereby avoiding explicit time marching and reducing the number of large-scale linear solvers.

The solution of the initial-value problem can be expressed in terms of a matrix exponential as

$$\mathbf{u}\left(t\right)=e^{-t{\mathbf{A}}_u}{\mathbf{u}}_0=f^t\left({\mathbf{A}}_u\right){\mathbf{u}}_0$$

(5)

Since ${\mathbf{A}}_u$ is generally a very large sparse matrix, direct evaluation of the matrix exponential $exp(-{\mathbf{A}}_{u}t)$ is computationally infeasible. Instead, Krylov subspace methods approximate the action of the matrix exponential on the initial field through projection onto a low-dimensional subspace. For transient electromagnetic diffusion problems, the rational Krylov subspace method is particularly attractive because of its favorable convergence and stability properties. In this study, the rational Krylov subspace is defined as

$${\mathcal{K}}_m\left({\mathbf{A}}_u,{\mathbf{u}}_0\right)=span\left\{{\mathbf{u}}_0,{\left({\mathbf{A}}_u-\xi \mathbf{I}\right)}^{-1}{\mathbf{u}}_0,{\left({\mathbf{A}}_u-\xi \mathbf{I}\right)}^{-2}{\mathbf{u}}_0,\dots ,{\left({\mathbf{A}}_u-\xi \mathbf{I}\right)}^{-m}{\mathbf{u}}_0\right\}$$

(6)

Here, $\xi$ is the pole parameter of the rational Krylov subspace and m is the subspace dimension. The Arnoldi orthogonalization process is used to generate an orthonormal basis for this subspace, and the detailed implementation is summarized in Algorithm 1.

Algorithm 1 Rational Krylov Subspace Model Order Reduction Algorithm for $f^t\left(\mathbf{A}\right)\mathbf{b}$.

Input: ${\mathbf{A}}_u$, ${\mathbf{u}}_0$, m, $\xi$ 1. set ${\mathbf{v}}_1:={\mathbf{u}}_0/{\parallel {\mathbf{u}}_0\parallel }_2$ 2. Do $j=1,2,\dots ,m$ 3.    compute ${\mathbf{w}}_j:={({\mathbf{A}}_u-\xi \mathbf{I})}^{-1}{\mathbf{v}}_j$ 4.    Do $i=1,2,\dots ,j$ 5.       $h_{i,j}:={\mathbf{v}}_i^{\mathrm{T}}{\mathbf{w}}_j$ 6.       ${\mathbf{w}}_j:={\mathbf{w}}_j-h_{i,j}{\mathbf{v}}_i$ 7.    End do 8.    $h_{j+1,j}:={\parallel {\mathbf{w}}_j\parallel }_2$       If $h_{j+1,j}=0$ Stop 9.    ${\mathbf{v}}_{j+1}:={\mathbf{w}}_j/h_{j+1,j}$ 10. End do 11. ${\mathbf{V}}_{m+1}:=[{\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_{m+1}]$,   ${\mathbf{T}}_{m+1}:={\mathbf{V}}_{m+1}^{\mathrm{T}}{\mathbf{A}}_u{\mathbf{V}}_{m+1}$ Output: ${\mathbf{V}}_{m+1}$, ${\mathbf{T}}_{m+1}$

According to the Rayleigh–Ritz projection principle, the original high-dimensional operator can be projected onto the reduced subspace as

$${\mathbf{T}}_{m+1}:={\mathbf{V}}_{m+1}^{\mathrm{T}}{\mathbf{A}}_u{\mathbf{V}}_{m+1}$$

(7)

The matrix exponential solution can then be approximated by

$$\mathbf{u}\left(t\right)\approx {\mathbf{V}}_{m+1}{e}^{-t{\mathbf{T}}_{m+1}}{\mathbf{V}}_{m+1}^{\mathrm{T}}{\mathbf{u}}_0$$

(8)

2.2. Inversion

Transient electromagnetic inversion is a typical ill-posed problem. In most practical cases, the number of observed data is much smaller than the number of unknown model parameters, and the data are inevitably contaminated by noise. Therefore, minimizing the data misfit alone usually cannot provide a stable and geologically meaningful model. To alleviate this difficulty, we adopt a classical Tikhonov regularization framework, in which the inverse problem is formulated as the minimization of an objective function consisting of a data misfit term and a model regularization term:

$$\varphi \left(\mathbf{m}\right)={\varphi }_d\left(\mathbf{m}\right)+\lambda {\varphi }_m\left(\mathbf{m}\right)$$

(9)

Here, ${\varphi }_d\left(\mathbf{m}\right)$ measures the discrepancy between the predicted and observed data, ${\varphi }_m\left(\mathbf{m}\right)$ penalizes undesired model structures and stabilizes the inversion, $\lambda$ is the regularization parameter that balances data fitting and model regularization, and $\mathbf{m}$ denotes the model parameter vector to be recovered.

In this study, both the data misfit and the model regularization terms are constructed using the $L_2$ norm. Following Oldenburg and Cockett et al. [30,31], these terms are written as

$$\begin{array}{c}{\varphi }_d\left(\mathbf{m}\right)={\displaystyle \frac{1}{2}}{\parallel {\mathbf{W}}_d\left({\mathbf{d}}_{pred}-{\mathbf{d}}_{obs}\right)\parallel }_2^2\\ {\varphi }_m\left(\mathbf{m}\right)={\displaystyle \frac{1}{2}}{\parallel {\mathbf{W}}_m\left(\mathbf{m}-{\mathbf{m}}_{ref}\right)\parallel }_2^2\end{array}$$

(10)

In these expressions, ${\mathbf{d}}_{obs}$ is the observed data vector, ${\mathbf{d}}_{pred}$ is the predicted data vector computed from the current model, and ${\mathbf{W}}_d$ and ${\mathbf{W}}_m$ are the data and model weighting matrices, respectively. The reference model ${\mathbf{m}}_{ref}$ can be defined using prior geological or geophysical information; when such information is unavailable, a homogeneous background model is commonly used.

The data weighting matrix ${\mathbf{W}}_d$ assigns different weights to data at different receivers and time channels according to their uncertainties. Following a strategy similar to that of Cox et al. [32], the data weighting matrix is defined as

$${\mathbf{W}}_d=\mathrm{diag}\left({\displaystyle \frac{1}{|d_{obs}^i|r_i+\eta }}\right),i=1,2,\dots ,N_d$$

(11)

Here, $r_i$ is the relative error of the i-th datum, $\eta$ denotes the noise floor, which prevents the inversion from overfitting data with very small amplitudes, and $N_d$ is the total number of observed data. For a single transmitter, if the data contain $N_t$ time channels and $N_r$ receivers, then $N_d=N_t{N}_r$.

The objective function is minimized using the Gauss–Newton method:

$$\begin{array}{c}\left({\mathbf{J}}^{\mathrm{T}}{\mathbf{W}}_d^{\mathrm{T}}{\mathbf{W}}_d\mathbf{J}+\lambda {\mathbf{W}}_m^{\mathrm{T}}{\mathbf{W}}_m\right)\delta \mathbf{m}=-\mathbf{g}\left(\mathbf{m}\right)\\ \mathbf{g}\left(\mathbf{m}\right)={\mathbf{J}}^{\mathrm{T}}{\mathbf{W}}_d^{\mathrm{T}}{\mathbf{W}}_d\left({\mathbf{d}}_{pred}-{\mathbf{d}}_{obs}\right)+\lambda {\mathbf{W}}_m^{\mathrm{T}}{\mathbf{W}}_m\left(\mathbf{m}-{\mathbf{m}}_{ref}\right)\end{array}$$

(12)

Here, $\delta \mathbf{m}$ is the model update and $\mathbf{J}$ is the sensitivity matrix, also called the Jacobian matrix, whose entries are the partial derivatives of the predicted data with respect to the model parameters. In practical large-scale inversions, the normal matrix is not explicitly formed. Instead, the model update $\delta \mathbf{m}$ is commonly solved using a preconditioned conjugate-gradient method to reduce memory requirements and improve computational efficiency.

Once the model update is obtained, the model is updated according to

$${\mathbf{m}}^{k+1}={\mathbf{m}}^k+\alpha \delta \mathbf{m}$$

(13)

where $\alpha$ is the step length. It is initially set to unity. If the updated model does not satisfy the descent condition of the objective function, a line-search procedure is used to determine an appropriate step length [6], ensuring stable convergence of the inversion.

The choice of the regularization parameter $\lambda$ has an important influence on the inversion result. In this study, $\lambda$ is updated using a cooling strategy. A relatively large regularization parameter is used during the early iterations to maintain model smoothness and stability. It is then gradually reduced as the inversion proceeds, allowing the model constraints to be relaxed and the data fit to improve. The initial value of $\lambda$ is selected following the relative weighting strategy used in SimPEG [31]. Specifically, power iteration is used to estimate the largest eigenvalues of the approximate Hessian of the data misfit term and that of the model regularization term. Their ratio, multiplied by an empirical scaling factor, is then used as the initial value of $\lambda$. This strategy provides a reasonable initial balance between data fitting and model regularization.

2.3. MOR-Based Sensitivity Calculation

The MOR-based sensitivity formulation follows Cao et al. [24]. Only the main expressions required for the subsequent full-waveform transformation are summarized here. The method represents the transient response as a finite sum of Ritz modes and evaluates its derivatives with respect to the model parameters using eigenvalue perturbation theory.

From the rational Krylov projection introduced above, the reduced-order approximation of the step-off response is written as

$$\mathbf{b}\left(t\right)\approx {\mathbf{M}}_{f\mu }^{-{\displaystyle \frac{1}{2}}}{\mathbf{V}}_{m+1}{e}^{-t{\mathbf{T}}_{m+1}}{\mathbf{V}}_{m+1}^{\mathrm{T}}{\mathbf{u}}_0$$

(14)

where ${\mathbf{V}}_{m+1}$ is the orthonormal basis of the rational Krylov subspace and ${\mathbf{T}}_{m+1}={\mathbf{V}}_{m+1}^{\mathrm{T}}{\mathbf{A}}_u{\mathbf{V}}_{m+1}$ is the reduced matrix.

Let ${\mathbf{T}}_{m+1}=\mathbf{\Psi }\mathbf{\Theta }{\mathbf{\Psi }}^{\mathrm{T}}$, where $\mathbf{\Theta }=diag({\theta }_1,{\theta }_2,\dots ,{\theta }_{m+1})$, and define ${\mathit{\phi }}_i={\mathbf{V}}_{m+1}{\mathit{\psi }}_i$. The pair $({\theta }_i,{\mathit{\phi }}_i)$ is the corresponding Ritz pair. The response at the receiver locations can then be expressed as

$$\mathbf{b}\left(t\right)\approx \mathbf{P}{\mathbf{M}}_{f\mu }^{-{\displaystyle \frac{1}{2}}}{\displaystyle \sum _{i=1}^{m+1}}{\mathit{\phi }}_i{e}^{-{\theta }_{i}t}{\mathit{\phi }}_i^{\top }{\mathbf{u}}_0$$

(15)

where $\mathbf{P}$ maps the face-based magnetic fields to the receiver locations.

Differentiating Equation (15) with respect to the jth conductivity parameter gives

$${\displaystyle \frac{\partial \mathbf{b}\left(t\right)}{\partial {\sigma }_j}}\approx \mathbf{P}{\mathbf{M}}_{f\mu }^{-{\displaystyle \frac{1}{2}}}{\displaystyle \sum _{i=1}^{m+1}}\left[-t{e}^{-{\theta }_{i}t}{\displaystyle \frac{\partial {\theta }_i}{\partial {\sigma }_j}}{\mathit{\phi }}_i{\mathit{\phi }}_i^{\mathrm{T}}{\mathbf{u}}_0+e^{-{\theta }_{i}t}{\displaystyle \frac{\partial }{\partial {\sigma }_j}}\left({\mathit{\phi }}_i{\mathit{\phi }}_i^{\mathrm{T}}{\mathbf{u}}_0\right)\right]$$

(16)

For the central-loop configuration considered here, the initial field is determined by the transmitter geometry and is assumed to be independent of the conductivity model; hence, $\partial {\mathbf{u}}_0/\partial {\sigma }_j=\mathbf{0}$. According to eigenvalue perturbation theory for symmetric matrices, the required derivatives are

$${\displaystyle \frac{\partial {\theta }_i}{\partial {\sigma }_j}}={\mathit{\phi }}_i^{\mathrm{T}}{\displaystyle \frac{\partial {\mathbf{A}}_u}{\partial {\sigma }_j}}{\mathit{\phi }}_i$$

(17)

and

$${\displaystyle \frac{\partial {\mathit{\phi }}_i}{\partial {\sigma }_j}}={\displaystyle \sum _{\begin{array}{c}k=1\\ k\ne i\end{array}}^{m+1}}{\displaystyle \frac{{\mathit{\phi }}_k^{\mathrm{T}}\frac{\partial {\mathbf{A}}_u}{\partial {\sigma }_j}{\mathit{\phi }}_i}{{\theta }_i-{\theta }_k}}{\mathit{\phi }}_k$$

(18)

Substituting these derivatives into Equation (16) gives the jth column of the step-off sensitivity matrix, ${\mathbf{J}}_j^{\mathrm{step}}\left(t\right)$. The full derivation is provided by Cao et al. [24].

This formulation reuses the Ritz information obtained during forward modeling and separates the temporal terms from the spatial terms. It therefore avoids repeated solutions of large-scale linear systems at different time channels.

2.4. Convolution-Based Full-Waveform Modeling and Sensitivity

To account for finite-duration transmitter waveforms in practical ATEM systems without directly solving the driven Maxwell system with time-dependent source terms, we transform arbitrary-waveform responses into linear combinations of step-off responses using a convolution strategy. According to linear time-invariant system theory, the response to an arbitrary transmitter current waveform can be expressed as the convolution of the step-off response or sensitivity with the time derivative of the transmitter current:

$$f_{\mathrm{waveform}}\left(t\right)=-{\int }_0^{t_{\mathrm{on}}}{f}_{\mathrm{step}}(t-\tau ){\displaystyle \frac{\mathrm{d}I\left(\tau \right)}{\mathrm{d}\tau }}\mathrm{d}\tau$$

(19)

Here, $f_{step}\left(t\right)$ is the step-off response or sensitivity, $I\left(t\right)$ is the transmitter current waveform, and $t_{\mathrm{on}}$ denotes the duration of the transmitter waveform.

For discrete receiver time channels, the above expression becomes

$$f_{\mathrm{waveform}}\left(t_j\right)=-{\int }_0^{t_{\mathrm{on}}}{f}_{\mathrm{step}}(t_j-\tau )I^{\prime }\left(\tau \right)\mathrm{d}\tau ,j=1,2,\dots ,N_t$$

(20)

where $N_t$ is the number of receiver time channels, and $I^{\prime }\left(\tau \right)$ is the derivative of the transmitter current waveform.

As shown in Figure 1, the construction of the reusable waveform-transformation matrix starts from resampling the step-off response onto a logarithmic time grid:

$$N_s=⌈N_{\mathrm{dec}}{log}_{10}\left({\displaystyle \frac{t_{max}}{t_{min}}}\right)⌉,{\mathbf{t}}_s=LogSpace\left(t_{min},t_{max},N_s\right)$$

(21)

Here, $N_{\mathrm{dec}}$ is the number of step-response samples per decade. The input current record is then reduced to a rate-weighted numerical integration grid. For adjacent current samples,

$$s_i={\displaystyle \frac{I_{i+1}-I_i}{{\tau }_{i+1}-{\tau }_i}},a_i=a_0+{\displaystyle \frac{|s_i|}{{max}_k\left|s_k\right|}},u=u+a_i({\tau }_{i+1}-{\tau }_i)$$

(22)

The waveform is resampled uniformly in the cumulative coordinate u (u initial value is 0). Consequently, intervals in which the current changes rapidly retain more samples, whereas slowly varying intervals are thinned; samples at sharp slope changes are retained explicitly. This procedure partitions the entire integration domain into $N_{\mathrm{seg}}$ numerical integration intervals.

Gaussian quadrature is applied segmentwise on these integration intervals. Collecting all Gaussian nodes and weights from the waveform segments gives

$$f_{\mathrm{waveform}}\left(t_j\right)\approx -{\displaystyle \sum _{k=1}^{N_g}}{H}_k{F}_{kj},H_k=I^{\prime }\left(x_k\right)w_k,F_{kj}=f_{\mathrm{step}}(t_j-x_k)$$

(23)

Here, $x_k$ and $w_k$ are the collected Gaussian node and weight, $N_g$ is the total number of Gaussian nodes, and $H_k$ contains the waveform derivative and quadrature weight.

The delayed values $F_{kj}=f_{\mathrm{step}}(t_j-x_k)$ are generally not located on the resampled time grid ${\mathbf{t}}_s$. Therefore, as illustrated in Figure 1, these values are not obtained by repeatedly calling an interpolation routine. Instead, for each receiver time channel, the interpolation operation is expressed as a matrix mapping from the resampled step-response grid to the delayed Gaussian nodes:

$$\begin{array}{c}{\mathbf{F}}_j={\mathbf{P}}_j{\mathbf{f}}_{\mathrm{step}},{\mathbf{F}}_j={\left[F_{1j},F_{2j},\dots ,F_{N_g,j}\right]}^{\mathrm{T}}\end{array}$$

(24)

Here, ${\mathbf{P}}_j$ is the interpolation matrix for the jth receiver time channel, and ${\mathbf{f}}_{\mathrm{step}}=f_{\mathrm{step}}\left({\mathbf{t}}_s\right)$ is the step-off response vector on the resampled time grid.

In summary, the otherwise complicated interpolation–convolution procedure can be unified into a compact matrix operation. Combining the convolution matrix $\mathbf{H}$ and the global interpolation matrix $\mathbf{C}$, the time-domain response for an arbitrary transmitter waveform can be written as

$$\begin{array}{c}{\mathbf{T}}_{j,:}=-{\mathbf{H}}^{\mathrm{T}}{\mathbf{P}}_j,j=1,2,\dots ,N_t\end{array}$$

(25)

Stacking all rows ${\mathbf{T}}_{j,:}$ gives the reusable waveform-transformation matrix

$$\mathbf{T}=\left[\begin{array}{c}{\mathbf{T}}_{1,:}\\ {\mathbf{T}}_{2,:}\\ ⋮\\ {\mathbf{T}}_{N_t,:}\end{array}\right],{\mathbf{f}}_{\mathrm{waveform}}=\mathbf{T}{\mathbf{f}}_{\mathrm{step}}$$

(26)

The matrix $\mathbf{T}$ is defined as the full-waveform transformation matrix. The physical and computational significance of this formulation is that the operations required for complex waveform treatment, including Gaussian quadrature, convolution with the current derivative, and cubic spline interpolation, are compressed into a small precomputed matrix $\mathbf{T}$. During inversion, for the step-off response ${\mathbf{f}}_{step}$ associated with any spatial location or model parameter, the corresponding full-waveform response can be obtained through a single low-dimensional matrix-vector multiplication. This strategy avoids reconstructing a driven Krylov subspace or deriving complicated sensitivity expressions for complex waveforms. At the same time, it preserves numerical accuracy and substantially improves the efficiency of waveform transformation in both forward modeling and inversion.

Because the transformation is linear, the same matrix is applied to the step-off sensitivity matrix:

$${\mathbf{J}}_{\mathrm{waveform}}=\mathbf{T}{\mathbf{J}}_{\mathrm{step}}$$

(27)

Thus, once $\mathbf{T}$ has been constructed, both full-waveform responses and sensitivities can be obtained without repeated waveform convolution or repeated interpolation calls.

In terms of computational cost, for each receiver time channel, the interpolation matrix ${\mathbf{P}}_j$ has dimensions $N_g\times N_s$, and forming ${\mathbf{H}}^{\mathrm{T}}{\mathbf{P}}_j$ requires approximately $𝒪\left(N_s{N}_g\right)$ operations. Repeating this operation for all $N_t$ time channels gives a one-time construction complexity of $𝒪\left(N_t{N}_s{N}_g\right)$. Because the rows of $\mathbf{T}$ are constructed sequentially, the temporary storage required during construction is $𝒪\left(N_s{N}_g\right)$, whereas the final matrix requires $𝒪\left(N_t{N}_s\right)$ storage. Once $\mathbf{T}$ has been constructed, transforming the response of one receiver requires $𝒪\left(N_t{N}_s\right)$ operations. For $N_r$ receivers and $N_m$ model parameters, the response and full-sensitivity transformations require $𝒪\left(N_r{N}_t{N}_s\right)$ and $𝒪\left(N_r{N}_m{N}_t{N}_s\right)$ operations, respectively.

3. Numerical Experiments

3.1. Analysis of Convolution Parameters

The derivation of the full-waveform transformation matrix in Section 2.4 shows that the convolution accuracy is mainly affected by the temporal sampling density of the step-off response, $N_{\mathrm{dec}}$, defined as the number of samples per decade, and the number of integration intervals, $N_{\mathrm{seg}}$. Because transmitter waveforms have different levels of complexity, the number of intervals required to achieve comparable integration accuracy is waveform dependent. Therefore, the influence of $N_{\mathrm{seg}}$ on the convolution accuracy differs among transmitter waveforms. In addition, once the integration intervals are sufficiently refined, further increasing the number of Gaussian points has little effect on the calculated results. Accordingly, five Gaussian points are used in each integration interval, and the following tests mainly examine the effects of $N_{\mathrm{dec}}$ and $N_{\mathrm{seg}}$ on the convolution accuracy for different transmitter waveforms.

A homogeneous half-space with a conductivity of $0.1\mathrm{S}/\mathrm{m}$ is used for the parameter tests. A typical central-loop airborne electromagnetic configuration is simulated. The transmitter is modeled as a circular loop with a radius of $8.256\mathrm{m}$, and the flight height is set to $30\mathrm{m}$. Four transmitter waveforms are considered, including trapezoidal, half-sine, triangular, and VTEM waveforms, as shown in Figure 2. Figure 3 presents the maximum relative errors for the four waveforms under different parameter combinations. The results calculated with $N_{\mathrm{dec}}=30$ and $N_{\mathrm{seg}}=2000$ are used as the reference. The error distributions differ considerably among the four waveforms, and complex transmitter waveforms require more integration intervals than simple waveforms to achieve comparable accuracy. When $N_{\mathrm{seg}}$ exceeds 50, the errors become less sensitive to further increases in the number of intervals, and the calculation accuracy is mainly controlled by the step-off response sampling density.

Based on the results for the four transmitter waveforms, $N_{\mathrm{dec}}$ should be kept as small as possible while satisfying the accuracy requirement. Increasing $N_{\mathrm{dec}}$ increases the number of step-off response time samples required in every forward modeling and sensitivity calculation, thereby increasing the computational cost. In contrast, $N_{\mathrm{seg}}$ only affects the one-time construction of the full-waveform transformation matrix, whose cost is small compared with the forward modeling and sensitivity calculations. Therefore, as a conservative compromise between computational accuracy and efficiency, $N_{\mathrm{dec}}=10$ and $N_{\mathrm{seg}}=960$ are adopted for the subsequent calculations.

In addition to the parameter-dependent accuracy, the propagation of perturbations through the convolution transformation is considered. To remove the influence of transmitter-current amplitude, let

$$\overline{I}=I/I_{max},\overline{\mathbf{T}}=\mathbf{T}/I_{max},and{\overline{\mathbf{f}}}_{\mathrm{waveform}}={\mathbf{f}}_{\mathrm{waveform}}/I_{max}$$

(28)

For a perturbation in the sampled step-off response, the corresponding perturbation in the normalized full-waveform response satisfies

$$\delta {\overline{\mathbf{f}}}_{\mathrm{waveform}}=\overline{\mathbf{T}}\delta {\mathbf{f}}_{\mathrm{step}},{∥\delta {\overline{\mathbf{f}}}_{\mathrm{waveform}}∥}_{\infty }\le {∥\overline{\mathbf{T}}∥}_{\infty }{∥\delta {\mathbf{f}}_{\mathrm{step}}∥}_{\infty }.$$

(29)

Thus, ${∥\overline{\mathbf{T}}∥}_{\infty }$ provides a direct upper bound on the amplification of absolute perturbations during the forward transformation. As shown in

Table 1. Infinity norms of the normalized off-time transformation matrices and total variations of the normalized transmitter currents.

Transmitter Waveform	${∥\overline{\mathbf{T}}∥}_{\mathit{\infty }}$	$TV\left(\overline{\mathit{I}}\right)$
Trapezoidal	2.804	2.000
Half-sine	2.193	2.000
Triangular	1.968	2.000
VTEM	2.505	2.363

, the infinity norms of the four off-time transformation matrices are all below 3, indicating that perturbations remain bounded by a moderate factor during the convolution transformation.

Interpolation errors are also introduced during construction of the transformation matrix. Let ${\mathbf{e}}_j$ denote the interpolation error at the Gaussian quadrature nodes for the jth receiver time channel. According to Equation (25),

$$\left|\delta {\overline{f}}_j^{\mathrm{waveform}}\right|=\left|{\overline{\mathbf{H}}}^{\mathrm{T}}{\mathbf{e}}_j\right|\le {∥\overline{\mathbf{H}}∥}_1{∥{\mathbf{e}}_j∥}_{\infty },$$

(30)

where $\overline{\mathbf{H}}=\mathbf{H}/I_{max}$. When the transmitter waveform is divided into sufficiently fine integration intervals, the current derivative within each interval can be approximated by the constant slope of the corresponding piecewise-linear segment. Because the Gaussian weights sum to the interval width, the following relation holds for this discrete waveform representation:

$${∥\overline{\mathbf{H}}∥}_1=\sum _k\left|{\overline{H}}_k\right|=\sum _i\left|{\overline{I}}_{i+1}-{\overline{I}}_i\right|=TV\left(\overline{I}\right),$$

(31)

and consequently

$$\left|\delta {\overline{f}}_j^{\mathrm{waveform}}\right|\le TV\left(\overline{I}\right){∥{\mathbf{e}}_j∥}_{\infty }.$$

(32)

This relation shows that the amplification of interpolation errors is controlled by the total variation of the transmitter current waveform. The parameter $N_{\mathrm{dec}}$ mainly affects the interpolation error ${∥{\mathbf{e}}_j∥}_{\infty }$ through the temporal sampling density of the step-off response, whereas $N_{\mathrm{seg}}$ controls the representation of the waveform derivative in the convolution integral. Consequently, waveforms with different levels of complexity exhibit different sensitivities to $N_{\mathrm{seg}}$. Table 1 shows that the normalized total variation is approximately 2 for the three simple waveforms and remains below 3 for the more complex VTEM waveform. This indicates that the interpolation-error amplification is also bounded by a moderate factor for the transmitter waveforms considered in this study.

3.2. Three-Dimensional Forward Modeling Based on Convolution

This section presents numerical experiments to verify the accuracy and numerical stability of the proposed convolution-based rational Krylov subspace forward modeling method for complex transmitter waveforms. The results obtained using the proposed method are compared with those computed by the conventional time-stepping method in SimPEG [31,33], in which the transmitter current waveform is directly imposed in the time domain. In contrast, the proposed method first computes the step-off response using the rational Krylov subspace method and then synthesizes the full-waveform response through the precomputed full-waveform transformation matrix.

3.2.1. Trapezoidal Transmitter Waveform

To verify the reliability of the proposed algorithm for trapezoidal transmitter waveforms, a homogeneous half-space model with an electrical conductivity of 0.1 S/m is used. The survey configuration mimics a typical central-loop airborne electromagnetic system. The transmitter is modeled as a circular loop with a radius of 8.256 m, and the flight height is set to 30 m. The computational domain is discretized using an Octree mesh, with a minimum cell size of 12.5 m. The final mesh contains 36,275 cells.

The transmitter current is defined as the trapezoidal waveform shown in Figure 2a. The total pulse duration is 4 ms, and both the ramp-up and ramp-down durations are set to 0.5 ms to mimic the current ramping and turn-off processes in practical transmitter systems. The observed data include the vertical magnetic field component ${\mathrm{B}}_z$ and its time derivative $\partial {\mathrm{B}}_z/\partial t$. During the on-time interval, 100 time channels are uniformly sampled with a time interval of $4\times {10}^{-6}$ s. During the off-time interval, 41 logarithmically spaced time channels are selected from ${10}^{-6}$ s to ${10}^{-2}$ s after turn-off.

Figure 4 compares the results obtained by the two methods. Figure 4a and Figure 4b show the ${\mathrm{B}}_z$ response and its relative error, respectively, while Figure 4c and Figure 4d show the $\partial {\mathrm{B}}_z/\partial t$ response and its relative error. The two methods show excellent agreement for the full-waveform responses. The relative errors for most time channels are within 5%, demonstrating the accuracy of the proposed convolution-based full-waveform forward modeling method. Slightly larger local errors occur mainly near the current switching times, which can be attributed to the discontinuity of the current derivative at the corners of the trapezoidal waveform.

3.2.2. Half-Sine Transmitter Waveform

To further examine the applicability of the proposed method to smooth transmitter waveforms, the same model parameters, survey configuration, and time-channel settings as those in Section 3.2.1 are used, while the transmitter current is replaced by a half-sine waveform. As shown in Figure 2b, the total pulse duration is 4 ms, and the peak current occurs at 2 ms.

The comparison results for the half-sine transmitter waveform are shown in Figure 5. The ${\mathrm{B}}_z$ and $\partial {\mathrm{B}}_z/\partial t$ responses computed by the proposed method agree well with those obtained using the conventional time-stepping method. The relative errors are below 5% for most time channels. Compared with the trapezoidal waveform, the half-sine waveform varies more smoothly, leading to a more stable error distribution. This result further confirms the applicability of the proposed convolution-based full-waveform forward modeling method to smooth complex waveforms.

3.2.3. Triangular Transmitter Waveform

The same homogeneous half-space model, survey configuration, and sampling settings are used again, while the transmitter current is changed to a triangular waveform. As shown in Figure 2c, the total pulse duration is 4 ms, and the peak current occurs at 2 ms.

Figure 6 shows the full-waveform responses computed by the two methods for the triangular transmitter waveform. The results demonstrate that the proposed method and the conventional time-stepping method produce highly consistent ${\mathrm{B}}_z$ and $\partial {\mathrm{B}}_z/\partial t$ responses. The relative errors for most time channels remain within 5%. Similar to the trapezoidal case, relatively larger errors occur mainly near the time when the slope of the transmitter current changes abruptly. This indicates that the proposed method can stably handle piecewise linear transmitter current waveforms.

3.2.4. VTEM Transmitter Waveform

The VTEM system is a widely used helicopter-borne time-domain electromagnetic system, and its transmitter current generally has a complex finite-duration waveform. To further evaluate the performance of the proposed algorithm for practical engineering waveforms, a VTEM-type transmitter waveform is considered in this example. As shown in Figure 2d, the total pulse duration is 7.4 ms. The model parameters, flight height, transmitter geometry, and spatial discretization are kept the same as in the previous examples.

Figure 7 compares the results obtained by the proposed method and the conventional time-stepping method for the VTEM transmitter waveform. For the ${\mathrm{B}}_z$ response, the two methods show good overall agreement, with only local error increases near abrupt changes in the transmitter current. This behavior is broadly consistent with the results obtained for the simpler waveforms. For the $\partial {\mathrm{B}}_z/\partial t$ response, the VTEM waveform varies more frequently during the on-time interval and exhibits a sawtooth-like feature. As a result, the time-derivative response contains stronger high-frequency oscillations, and the relative errors during the on-time interval are generally larger. Nevertheless, during the off-time interval, the relative errors between the two methods remain mostly within 5%. These results demonstrate that the proposed convolution-based full-waveform forward modeling technique can accurately and stably reproduce the main response characteristics even for complex practical transmitter waveforms, especially during the off-time interval, which is often of primary interest in inversion.

Table 2. Relative errors of the off-time $B_z$ responses calculated by the proposed method with respect to the SimPEG time-stepping results.

Error Measure	Trapezoidal	Half-Sine	Triangular	VTEM
Maximum relative error, $E_{max}$ (%)	1.52	2.45	2.30	1.26
$L_2$ norm of relative error, $E_{L_2}$ (%)	3.75	3.37	2.60	2.70
RMS relative error, $E_{\mathrm{RMS}}$ (%)	0.59	0.53	0.41	0.42

summarizes the maximum, $L_2$-norm, and RMS relative errors for the 41 off-time $B_z$ channels, using the SimPEG time-stepping results as the reference.

The maximum relative error does not exceed 2.45% for any of the four waveforms, and the RMS relative error does not exceed 0.59%, showing good agreement during the off-time interval. The runtime and memory usage of the proposed rational Krylov/convolution method and the conventional SimPEG time-stepping method are compared in

Table 3. Runtime and memory usage of the rational Krylov/convolution method and the conventional time-stepping method in SimPEG.

Method	Metric	Step-Off	Trapezoidal	Half-Sine	Triangular	VTEM
RKS/conv.	Runtime (s)	6.31	6.67	6.73	6.87	7.02
	Memory (GB)	1.66	1.70	1.72	1.72	1.71
	$\mathbf{T}$ setup (s)	–	$<0.1$	$<0.1$	$<0.1$	$<0.1$
SimPEG	Runtime (s)	58.59	66.17	66.71	66.27	74.34
	Memory (GB)	2.55	2.76	2.72	2.74	2.91

.

For the same model, mesh, and off-time channels, the rational Krylov/convolution method is approximately 10 times faster than SimPEG and uses less memory. Construction of $\mathbf{T}$ takes less than 0.1 s for all four waveforms and adds little to the total runtime.

Overall, the numerical results for the four transmitter waveforms demonstrate that the proposed convolution-based rational Krylov subspace method can accurately reproduce full-waveform TEM responses for both smooth and piecewise linear current waveforms. The discrepancies are mainly localized near current switching or slope-changing times, where the waveform derivative is discontinuous or rapidly varying. The off-time errors remain small for all four waveforms. These results confirm that the proposed full-waveform transformation provides an efficient and reliable way to incorporate complex transmitter waveforms into RKS-based three-dimensional forward modeling.

3.3. MOR Sensitivity Analysis Based on Convolution

The preceding forward modeling tests have verified the accuracy of constructing arbitrary-waveform responses from step-off responses through the full-waveform transformation matrix. In this section, the same strategy is further applied to sensitivity matrix calculation, which is a central component of inversion. The basic procedure is broadly consistent with that used in forward modeling. The MOR method is first used to efficiently calculate the sensitivity corresponding to an ideal step-off excitation, after which the full-waveform transformation matrix is applied to obtain the sensitivity corresponding to the actual transmitter waveform. To examine the reliability of this approach, the sensitivity calculated by the proposed convolution-based MOR method is compared with that obtained using the conventional time-stepping method in SimPEG.

A typical single conductive block model is used for this test. The background is a homogeneous half-space with a conductivity of $0.01\mathrm{S}/\mathrm{m}$. A conductive anomalous body with a conductivity of $0.1\mathrm{S}/\mathrm{m}$ is embedded in the half-space. The anomalous body has dimensions of $100\mathrm{m}\times 100\mathrm{m}\times 200\mathrm{m}$, and its geometric center is located at $(0,0,-150)\mathrm{m}$. The observation system simulates a VTEM airborne electromagnetic system with a central-loop configuration. The transmitter is modeled as a circular loop with a radius of $8.256\mathrm{m}$. The transmitter current uses the VTEM waveform described above, with a pulse duration of $7.4\mathrm{ms}$, as shown in Figure 2d. The observed component is the vertical magnetic flux density $B_z$. The receiver time window ranges from ${10}^{-6}\mathrm{s}$ to ${10}^{-2}\mathrm{s}$ after turn-off and contains 41 logarithmically distributed time channels. An Octree mesh is used for spatial discretization, resulting in 34,819 mesh cells and 97,980 electromagnetic field unknowns.

Figure 8 compares the spatial distributions of the model-space vector ${\mathbf{J}}^{\mathrm{T}}\mathbf{w}$ calculated by the two methods at $t={10}^{-4}\mathrm{s}$ after turn-off. Here, $\mathbf{w}$ denotes the data-space vector used in the transpose sensitivity matrix–vector product. Figure 8a shows the result obtained using the conventional time-stepping method, whereas Figure 8b shows the result obtained using the proposed convolution-based MOR method. The main sensitive regions obtained by the two methods have nearly the same shape and spatial distribution, and the two results agree well overall. This agreement shows that the convolution-based MOR method accurately reproduces the spatial distribution of ${\mathbf{J}}^{\mathrm{T}}\mathbf{w}$ at the selected time channel.

To further provide a quantitative validation of the Jacobian, a subsurface cell centered at $(135,45,-165)\mathrm{m}$ was randomly selected, and its sensitivity was estimated using the perturbation method. The resulting sensitivity was then compared with the corresponding $\mathbf{J}\mathbf{v}$ product obtained using the convolution-based MOR method. As shown in Figure 9, the two results agree closely over the entire off-time interval. The maximum relative error, the $L_2$ norm of the relative error, and the RMS relative error are approximately 1.46%, 3.10%, and 0.08%, respectively, and all three error measures remain below 5%. These quantitative results further validate the accuracy of the proposed MOR sensitivity calculation.

3.4. MOR Inversion Based on Convolution

In this section, synthetic inversion examples are used to further examine the accuracy and computational efficiency of the proposed convolution-based MOR method in three-dimensional inversion. To investigate the influence of transmitter waveform treatment on the inversion results, two inversion strategies are compared. The first one is the convolution-based MOR method, in which the effect of the actual transmitter waveform is explicitly included. The second one is the conventional MOR method without waveform convolution, where the step-off response is directly used as an approximation. The comparison between these two inversion results provides a direct assessment of the importance of transmitter waveform correction in practical airborne electromagnetic data processing.

To ensure a fair comparison, all inversions are performed using the same numerical discretization and inversion framework. The spatial discretization is based on the octree finite-volume method, and the three-dimensional inversion is carried out using MOR-based sensitivity calculation. The only essential difference between the two inversion strategies is whether the actual transmitter current waveform is taken into account. All numerical computations are performed in a Python 3.12 environment on a Windows-based personal computer equipped with an Intel Core i9-12900K CPU, an NVIDIA GeForce RTX 3060 Ti GPU, and 128 GB of memory.

3.4.1. Synthetic Model 1

The first synthetic example is a simple synclinal model. The anomalous body has a finite extension in the y-direction. The background conductivity is set to $0.005\mathrm{S}/\mathrm{m}$, and the conductivity of the anomalous body is set to $0.05\mathrm{S}/\mathrm{m}$.

A central-loop airborne electromagnetic configuration is used. The transmitter is modeled as a circular loop with a radius of $5\mathrm{m}$. The transmitter current is a trapezoidal waveform with a pulse duration of $4\mathrm{ms}$, and both the ramp-up and ramp-down durations are $0.2\mathrm{ms}$. The transmitter–receiver locations are arranged over a rectangular survey area. Seventeen stations are uniformly distributed along the x-direction, and seven survey lines are uniformly arranged along the y-direction, giving a total of 119 measurement locations. The observed component is $\partial B_z/\partial t$, and 31 time channels are logarithmically sampled within the receiver time window. The model domain is discretized using an Octree mesh, resulting in 52,963 mesh cells and 150,786 unknowns. The model parameters and mesh discretization are illustrated in Figure 10.

The initial model for inversion is a homogeneous half-space with a conductivity of $0.005\mathrm{S}/\mathrm{m}$. The convolution-based MOR method and the conventional MOR method without transmitter waveform correction are then applied separately to the same data set. Both inversions are run for 12 iterations. The MOR method with waveform convolution requires 1.23 h, whereas the MOR inversion using the step-off approximation takes 2.61 h. The longer runtime of the latter is mainly due to the additional time spent in searching for a suitable descent direction and step length during the inversion.

Figure 11 shows the three-dimensional inversion results for the synclinal model. Figure 11a,b show the results obtained using the convolution-based MOR method with transmitter waveform correction. Figure 11c,d show the results obtained using the MOR method with the step-off approximation. Figure 11a,c are the $xz$ sections at $y=0\mathrm{m}$, whereas Figure 11b,d show the three-dimensional anomalous bodies with conductivity greater than $0.02\mathrm{S}/\mathrm{m}$.

As shown in the inversion results, when the actual trapezoidal transmitter waveform is included, the recovered location, burial depth, and spatial geometry of the anomalous body agree well with the true model. The background conductivity remains close to the true value of $5\times {10}^{-3}\mathrm{S}/\mathrm{m}$, and the recovered conductivity of the anomalous body is also close to the true value of $5\times {10}^{-2}\mathrm{S}/\mathrm{m}$. In contrast, when the ideal step-off approximation is used, the inversion result exhibits clear systematic bias. The conductivity of the anomalous body is significantly overestimated, reaching locally up to ${10}^{-1}\mathrm{S}/\mathrm{m}$, which is about twice the true value. Meanwhile, layered spurious resistive anomalies appear in the shallow part, while the conductivity in the deep background region is generally elevated, producing large-scale false conductive anomalies.

The three-dimensional result further shows that, in the inversion using the step-off approximation, the recovered conductive anomaly is no longer confined within the true target boundary but spreads outward, leading to distortion of both the target geometry and its boundary. This indicates that although the difference between the trapezoidal and step-off responses is mainly concentrated in the early time channels, the waveform-related error can propagate through the entire model space during three-dimensional inversion through data fitting, model regularization, and electromagnetic diffusion coupling. Therefore, using a step-off approximation to process data generated by an actual trapezoidal turn-off waveform can affect not only the recovery of shallow structures, but also the estimated conductivity amplitude of the anomalous body, the background conductivity level, and the reconstructed three-dimensional geometry.

Figure 12 shows the variation of the root-mean-square error (RMS) with iteration number during the MOR inversions. Here, CMOR denotes the MOR inversion with the actual transmitter waveform included, whereas MOR denotes the inversion using the step-off approximation. The convergence behavior of the two methods is clearly different. The initial RMS of CMOR is approximately 20, which is much lower than that of the MOR inversion using the step-off approximation. As the inversion proceeds, the RMS of CMOR decreases rapidly during the first few iterations, then gradually levels off and finally stabilizes close to 1. This indicates that the method can explain the observed data well and achieves a satisfactory convergence behavior. By contrast, the initial RMS of the MOR inversion using the step-off approximation is greater than 500, suggesting a large systematic discrepancy between the step-off approximation response and the response generated by the actual transmitter waveform. Although this method also shows a rapid decrease in data misfit during the early iterations, its final RMS only converges to approximately 2–3, which is still significantly higher than that obtained by CMOR. This result indicates that when the actual transmitter waveform differs substantially from the ideal step-off waveform, the step-off approximation cannot adequately represent the observed response, thereby limiting both the data fitting accuracy and the reliability of the inversion result. Overall, the CMOR method with transmitter waveform correction not only starts from a much lower initial data misfit, but also achieves a better final convergence level, demonstrating the importance of transmitter waveform correction for improving the accuracy of MOR-based three-dimensional inversion.

3.4.2. Synthetic Model 2

To further evaluate the effectiveness and practicality of the proposed convolution-based MOR method for complex geometries, we reuse an irregular complex ore-body model from [24] as synthetic model 2.

The forward simulation is carried out using a central-loop configuration. The transmitter is modeled as a circular loop with a radius of $10\mathrm{m}$. A VTEM transmitter waveform with a duration of $7.4\mathrm{ms}$ is used, as shown in Figure 2d. The receiver grid covers a rectangular area of $x=-300$ to $300\mathrm{m}$ and $y=-200$ to $200\mathrm{m}$. The receivers are placed at a height of $z=30\mathrm{m}$ with a uniform spacing of $50\mathrm{m}$, giving a total of 117 receiver locations. The measured component is $\partial B_z/\partial t$. The time channels range from ${10}^{-5}$ to ${10}^{-3}\mathrm{s}$, and 21 channels are logarithmically sampled within this interval. The model domain is discretized using Octree mesh, resulting in a total of 44,619 cells and 126,834 unknowns. The schematic illustration of the model is shown in Figure 13.

The initial model for inversion is a homogeneous half-space with a conductivity of $0.002\mathrm{S}/\mathrm{m}$. Figure 14 shows the three-dimensional MOR inversion result obtained by including the actual VTEM transmitter waveform. The recovered model captures the overall spatial geometry of the complex ore body reasonably well. In particular, the anomalous boundary in the southeastern part agrees well with the true model, and the main location, burial depth, and lateral extent of the target are effectively recovered. The recovery of several small blocks in the central and northwestern parts is relatively weaker, showing locally underestimated anomaly amplitudes or slightly smoothed boundaries. Overall, the MOR inversion with waveform correction provides a reasonable reconstruction of the main outline of the complex ore body, with relatively clear anomaly boundaries and a plausible conductivity distribution. This demonstrates that the proposed method is applicable to transient electromagnetic inversion for complex three-dimensional targets.

In contrast, when the ideal step-off waveform is used as an approximation to the VTEM transmitter waveform and directly applied in the MOR three-dimensional inversion, the inversion fails to converge stably, and the model cannot be effectively updated. The main reason is that the VTEM transmitter current has a relatively long ramp-down time, so the actual turn-off process deviates significantly from the ideal instantaneous turn-off assumption. The ideal step-off waveform assumes an abrupt current change within zero time, which is equivalent to introducing strong high-frequency excitation components. By comparison, the finite turn-off process of the VTEM waveform introduces pronounced smoothing and delay effects in the early-time and part of the intermediate-time responses. Therefore, when a step-off forward operator is used to fit data generated by the actual VTEM waveform, a significant systematic response discrepancy is introduced.

For trapezoidal waveforms with a short ramp-down time, the difference caused by the step-off approximation is usually concentrated mainly in the earliest time channels, while the late-time responses may still remain partly similar. However, the VTEM waveform has a much longer ramp-down time, and its waveform effect is no longer limited to the earliest data. Instead, it affects the response amplitude and decay behavior over a broader portion of the effective observation window. In this case, the step-off response can no longer serve as a valid approximation to the VTEM response. Since the forward operator used in the inversion is inconsistent with the physical excitation process corresponding to the observed data, the objective function contains structural residuals that cannot be eliminated simply by adjusting the subsurface conductivity model. As a result, the data misfit cannot decrease stably, leading to difficult model updates or even non-convergence.

These results indicate that, for transmitter systems with a long turn-off time such as VTEM, the transmitter waveform effect is an essential factor in three-dimensional transient electromagnetic inversion. Ignoring the actual waveform and using a step-off approximation not only reduces the quantitative reliability of the inversion result, but may also directly undermine the stability of the inversion iterations. Therefore, in fast three-dimensional inversion based on MOR, the actual transmitter current waveform must be incorporated into the forward response calculation to maintain the physical consistency between data fitting and model updating.

Figure 15 shows the RMS convergence curve for model 2. The RMS of CMOR gradually decreases as the iteration proceeds and eventually approaches 1, indicating that the inversion result provides a good fit to the observed data.

In terms of computational efficiency, the MOR inversion with the actual VTEM transmitter waveform is completed in 8 iterations, with a total runtime of only 1.07 h. This result shows that the proposed convolution-based MOR method still maintains high computational efficiency for complex three-dimensional models, further confirming the effectiveness of the acceleration strategy in practical applications involving complex targets.

4. Field Data Application

The proposed method is further tested using a field airborne TEM dataset from Inner Mongolia, China, which was also studied by Qi et al. [4]. This example provides a useful test case because the survey area has a clear structural background and has been independently interpreted in previous work. The local geology is mainly controlled by two northwest-trending faults, F101 and F102. These faults form the principal structural boundaries in the area and separate the Lower Proterozoic dolomitic marble in the hanging wall from the Caledonian medium-grained granite in the footwall.

The dataset consists of 11 flight lines with 294 measurement stations. Ten of the lines are oriented southwest–northeast, while the remaining line is arranged southeast–northwest and intersects the other lines. The station spacing is $20\mathrm{m}$, and the average terrain-clearance height of the receiver is approximately $53\mathrm{m}$. The survey uses a central-loop airborne TEM system with an equivalent transmitter area of $530.75{\mathrm{m}}^2$ and 4 turns. A VTEM waveform with a pulse duration of $7.4\mathrm{ms}$ is adopted, as illustrated in Figure 2d. The recorded component is $\partial B_z/\partial t$, and 35 processed time channels are used in the inversion.

The inversion domain is discretized using an Octree mesh, resulting in 61,699 mesh cells and 172,485 unknowns. The station distribution and mesh discretization are shown in Figure 16. The initial model is set as a homogeneous half-space with a conductivity of $5\times {10}^{-3}\mathrm{S}/\mathrm{m}$.

After 14 iterations, the RMS data misfit decreases to a value close to 1, indicating that the field observations are well fitted. The RMS convergence curve, and the comparison between observed and predicted data are shown in Figure 17 and Figure 18, respectively.

Figure 19 shows the three-dimensional MOR inversion result obtained by including the actual VTEM transmitter waveform. The inversion result displays a continuous conductive structure with a southeast–northwest trend. The conductor deepens gradually and extends through the main part of the survey area. This feature is in good agreement with the known geological framework and is spatially consistent with the strike of faults F101 and F102. The resistive zones adjacent to the conductor are also similar to those obtained by Qi et al. [4]. The comparison between observed and predicted responses shows that the main data features are well reproduced, suggesting that the proposed method is capable of fitting practical airborne TEM data with sufficient accuracy.

The total runtime for the field inversion is 4.96 h for 14 iterations. In total, the inversion involves 18 forward modeling runs and 14 sensitivity matrix computations. This field example confirms that the proposed MOR-based inversion strategy remains computationally efficient for realistic airborne TEM surveys while producing geologically reasonable conductivity images.

5. Conclusions

In this study, the rational Krylov subspace method is used to calculate the step-off responses, while the MOR formulation of Cao et al. is used to obtain the corresponding step-off sensitivities. A model-independent full-waveform transformation matrix is then applied to both quantities. This treatment incorporates prescribed transmitter waveforms without introducing a time-dependent source term into the MOR sensitivity derivation. In addition, Gaussian quadrature and interpolation operations are assembled into a reusable matrix, thereby avoiding repeated convolution calculations. The method is evaluated through convolution-parameter tests, forward and sensitivity comparisons, synthetic inversions, and one field example.

Forward modeling comparisons show that the proposed method can accurately reconstruct transient electromagnetic responses for trapezoidal, half-sine, triangular, and VTEM waveforms. The results agree well with those obtained by the conventional time-stepping method, indicating that the proposed approach provides sufficient accuracy for three-dimensional inversion. Synthetic inversion results demonstrate that waveform treatment has a clear influence on inversion stability and imaging quality. The step-off approximation may introduce systematic errors, including overestimated conductivity, inaccurate boundaries, and spurious shallow anomalies, and may even lead to unstable or non-convergent inversion for long turn-off waveforms such as VTEM. In contrast, the convolution-based MOR method with the actual transmitter waveform included can steadily reduce the data misfit and recover the main geometry of anomalous bodies more reliably. The field example provides an initial field-scale assessment of the proposed method. The recovered conductivity structure is broadly consistent with the known fault-controlled geological setting and delineates the main conductive anomaly in the survey area. The complete field inversion takes approximately 4.96 h, suggesting that the proposed convolution-based MOR method has the potential to provide an efficient and stable tool for practical three-dimensional airborne transient electromagnetic inversion.