Time Lag RTM: Advanced Seismic Imaging

• Time Lag RTM is an advanced seismic imaging method that integrates a time-shift parameter to capture kinematic and dynamic misalignments in velocity models.
• It improves subsurface imaging by revealing energy shifts through extended imaging conditions, enabling enhanced recovery of mid- and high-wavenumber features.
• Neural operator surrogates and diffusion priors accelerate the workflow, offering up to 300x speed improvements and robust, high-resolution velocity model updates.

Time Lag Reverse Time Migration (RTM) is an advanced seismic imaging and inversion methodology that extends the conventional reverse time migration by incorporating a time-lag (or time-shift) parameter into the imaging condition. This technique computes extended images as a function of both spatial position and lag, explicitly revealing kinematic and dynamic misalignments caused by inaccuracies in the subsurface velocity model. Recent developments leverage neural operator surrogates and generative diffusion priors to accelerate and regularize velocity model building workflows using time-lag RTM output, enabling high-resolution, geologically plausible updates far faster than traditional methods (Ma et al., 29 Dec 2025, Ma et al., 2 Oct 2025).

1. Mathematical Foundations of Time Lag RTM

The core of time-lag RTM is the extended imaging condition, which generalizes the zero-lag cross-correlation of source and receiver wavefields. Let $p^+(x,t)$ and $p^-(x,t)$ denote the forward-propagated (source) and back-propagated (receiver) wavefields, respectively, in a given migration velocity $v_{\text{mig}}(x)$. The time-lag extended image $R(x,\tau)$ is defined as

$$R(x,\tau) = \int_{-\infty}^{+\infty} p^+(x, t + \tau)\;p^-(x, t - \tau)\;dt,$$

where $\tau$ is the temporal shift, or lag, parameter (Ma et al., 29 Dec 2025, Ma et al., 2 Oct 2025). For $\tau = 0$, this reduces to the conventional imaging condition. Nonzero $\tau$ explicitly displays the misalignment between wavefields induced by velocity model errors.

Taylor expansion around $\tau = 0$ indicates that $R(x,\tau)$ encodes higher-order time-derivative mismatches between wavefields. The energy ridge position in the $\tau$-dimension directly diagnoses the kinematic error in the current velocity estimate (Ma et al., 2 Oct 2025).

2. Algorithmic Implementation of Time-Lag RTM

Practical implementation of time-lag RTM for each shot follows these steps:

1. Source Wavefield Propagation: Solve the forward wave equation for $p^+(x, t)$, storing all—or a windowed subset—of time levels needed for the desired $\{\tau_i\}$ values.
2. Receiver Wavefield Propagation: Inject observed shot records as time-reversed boundary conditions and propagate $p^-(x, t)$ backward in time.
3. Imaging Condition Evaluation: For each $\tau_i$, compute $R(x,\tau_i)$ by cross-correlating appropriately shifted wavefields.
4. Stacking and Output: The complete set $\{R(x,\tau_i)\}$ may be collapsed for a traditional image or passed in its extended form as input to downstream machine learning models (Ma et al., 29 Dec 2025, Ma et al., 2 Oct 2025).

Memory requirements are dictated by storing sufficient time levels of $p^+(x, t)$ for all $N_\tau$ chosen lags, potentially mitigated via checkpointing or interpolation. The computational overhead relative to zero-lag RTM is proportional to $N_\tau$, since $N_\tau$ cross-correlations are performed per spatial location and time step; however, the cost remains dominated by wave propagation for typical choices of $N_\tau$ (often $N_\tau=3$ suffices for neural-operator workflows) (Ma et al., 2 Oct 2025).

3. Role of Time-Lag RTM in Velocity Model Building

Time-lag gathers $R(x,\tau)$ provide additional information beyond the zero-lag image by sampling non-normal incidence angles. In spectral terms, varying $\tau$ selects different components of reflectivity, allowing improved illumination and imaging of mid- and high-wavenumber velocity contrasts. The relation

$k \approx \frac{\omega}{v}\sin\theta$

links the local incidence angle $\theta$ to the horizontal wavenumber content. Thus, the stack of $R(x,\tau)$ over multiple lags enables the recovery of fine-scale and complex subsurface features critical for high-fidelity velocity inversion (Ma et al., 2 Oct 2025).

In neural-operator-based velocity model building, several $R(x,\tau_i)$ images, along with the migration velocity $v_{\text{mig}}(x)$, are presented as multichannel input to a deep learning model tasked with reconstructing the true velocity $v_{\text{true}}(x)$. This design leverages the diverse angle and frequency coverage afforded by time-lag RTM to drive high-resolution updates (Ma et al., 29 Dec 2025, Ma et al., 2 Oct 2025).

4. Neural Operator Surrogates and Diffusion Regularization

Time-lag RTM's computational cost motivates the use of surrogate models to approximate the forward mapping:

$$G_\theta: \{v_{\text{true}}(x), v_{\text{mig}}(x)\} \mapsto R(x,\tau)$$

where $G_\theta$ is a trainable neural operator. The recent Fourier Neural Operator–U-Net hybrid architecture takes as input the true and migration velocity fields, lifts these to a high-dimensional feature space with a Fourier Neural Operator layer, processes spatial features via a ResNet-based encoder-decoder, and projects back to the extended image $R(x,\tau)$ using a final FNO layer (Ma et al., 29 Dec 2025). Training is supervised using an $L^2$ loss on the predicted vs. reference $R(x,\tau)$.

Automatic differentiation enables use of $G_\theta$ as a fully differentiable forward operator in inversion, allowing gradient-based updates of the velocity model. The misfit between predicted and observed extended images,

$$\mathcal{L}_{\text{inv}}(v) = \big\| G_\theta(v; v_{\text{mig}}) - I_{\text{obs}}\big\|_2^2$$

is minimized via steepest descent or Adam in the input space $v$ (Ma et al., 29 Dec 2025).

To regularize inversion and inject geologically plausible details, a Conditional Denoising Diffusion Probabilistic Model (DDPM) is introduced. The learned prior enforces consistency with true velocity field distributions and suppresses artifacts, with the diffusion denoising step alternated with neural-operator-based inversion. The resulting workflow is algorithmically equivalent to solving a regularized problem

$$\min_v \;\mathcal{L}_{\text{inv}}(v) + \lambda R(v)$$

where $R(\cdot)$ is the implicit regularizer embodied by the DDPM (Ma et al., 29 Dec 2025).

5. Comparative Performance and Applications

Empirical assessments on synthetic and real (field) data show that neural-operator-based time-lag RTM surrogates achieve forward prediction of $R(x,\tau)$ within an $L^2$ error $\lesssim 1\%$ of peak amplitude compared to conventional RTM, while yielding a $\sim300\times$ acceleration (runtime $\sim$0.9s per example vs $4$min $50$s on the same A100 GPU) (Ma et al., 29 Dec 2025).

Velocity inversion using time-lag images outperforms zero-lag (conventional) inversion, particularly in recovering mid- and high-wavenumber features. However, artifacts may arise from unstable gradients in the neural inversion. Augmenting the workflow with a diffusion prior suppresses such artifacts and enhances the recovery of high-frequency geological structure. In practice, inference with the diffusion prior requires $\sim113$s (vs $\sim28$s without) for typical synthetic models, whereas conventional full waveform inversion (FWI) or wave-equation MVA can require hours (Ma et al., 29 Dec 2025).

Field data applications (CGG NW Australia) use patch-based inference to map large-scale domains (e.g., $30$km $\times$ $4$km), achieving inversion and imaging with continuous and geologically consistent reflectors in RTM output, corroborated by spectral comparisons with well-log data and observed shot gathers (Ma et al., 29 Dec 2025, Ma et al., 2 Oct 2025).

6. Advantages, Limitations, and Practical Considerations

Advantages:

• Resolution: Multichannel $R(x,\tau_i)$ imaging enhances high-wavenumber recovery, enabling detailed reconstruction of complex geologies (e.g., salt domes, layered media).
• Velocity Diagnosis: Time-lag gathers directly reveal the magnitude and sign of velocity errors via shifts in the $\tau$-domain, facilitating robust tomographic updates.
• Speed: Neural operator surrogates accelerate both the forward and inversion workflows by up to $\sim300\times$ compared to classical algorithms.
• Robustness: Correlations across $\tau$ mitigate random focusing artifacts found in zero-lag images, improving the stability of machine learning inversions.

Limitations:

• Storage Overhead: The need to store or checkpoint sufficient wavefield time levels for the desired range of $\tau$ increases memory demands by $(1 + 2\,\mathrm{max}|\tau|/\Delta t)$ relative to zero-lag imaging.
• Computational Overhead: Forming multiple cross-correlations per time step is linearly more expensive in $N_\tau$. Dominant cost remains wave propagation unless $N_\tau$ is large.
• Sensitivity: For highly inaccurate migration velocities, energy may distribute over wide $\tau$ ranges, making extended images less interpretable and increasing requirements for $\tau$ coverage and thus storage.
• Regularization Need: Unregularized neural inversion may exhibit artifacts (“noise stripes”), motivating the integration of learned diffusion priors (Ma et al., 29 Dec 2025, Ma et al., 2 Oct 2025).

7. Summary Table: Key Aspects of Time-Lag RTM Workflows

Aspect	Time-Lag RTM	Zero-Lag RTM
Imaging Condition	$R(x,\tau):$ parametrized by lag $\tau$	$I_0(x): \tau=0$ only
Memory Requirement	High (stores window for all $\tau_i$)	Moderate
Input to Neural Operator	$\{v_{\text{mig}}(x), R(x,\tau_1), ...\}$	$\{v_{\text{mig}}(x), I_0(x)\}$
Frequency/Wavenumber Content	Low, mid, and high (from multiple $\tau_i$)	Primarily low (near $\tau=0$)
Velocity Sensitivity	Explicit diagnosis via $\tau$-domain shifts	Implicit, less diagnostic
Computational Cost	$N_\tau$ cross-correlations per shot	Single cross-correlation per shot

Time-lag RTM, especially when integrated with neural operator surrogates and diffusion-based regularization, constitutes a state-of-the-art framework for rapid, high-resolution velocity model building, exhibiting scalability and robustness previously unattainable with conventional FWI or wave-equation MVA methods (Ma et al., 29 Dec 2025, Ma et al., 2 Oct 2025).