Physics-Informed Neural Networks for Seismic Noise Attenuation: When the Wave Equation Is Your Only Label

In oil-and-gas reflection seismics the hardest step between a raw shot gather and an interpretable image is separating reflected signal from noise: random ambient noise, coherent ground roll, swell and cable noise, and multiples, all removed without eroding the weak reflectors that mark a reservoir. Supervised denoisers need clean labels the field never provides. This note describes physics-informed neural networks (PINNs) as an unsupervised alternative. Because the primary reflection wavefield satisfies the acoustic (or elastic) wave equation for the subsurface velocity, a network penalised by the PDE residual reconstructs only the physics-consistent part of the record and leaves the rest behind: random noise that has no coherent phase, and ground roll that is a dispersive surface wave rather than a body-wave reflection. The wave equation is the only label.

We frame the method as a before-and-after on a synthetic shot gather where the clean reflection is known by construction, compare a PINN denoiser against conventional bandpass / f-k filtering across signal-to-noise ratio, and are explicit about where the physics prior helps (random and surface-wave noise, trace reconstruction) and where it does not (multiples, which satisfy the same equation along a longer path). The same PINN wavefield doubles as a mesh-free forward-modelling and full-waveform-inversion engine, so noise attenuation, imaging, and velocity building sit on one differentiable physics core.

Background: the noise problem O&G actually has

A land or marine reflection survey fires a controlled source and records body-wave reflections off impedance contrasts at reservoir depth. The raw common-shot gather is dominated not by those reflections but by noise, and every processing shop spends the bulk of its effort suppressing it:

• Random / ambient noise. Incoherent, broadband, worst on weak deep reflectors.
• Ground roll. Rayleigh surface waves: high amplitude, low apparent velocity, dispersive, and the single largest coherent-noise problem on land data. It fans across near and mid offsets and buries shallow reflections.
• Swell noise and cable strumming. Low-frequency marine noise.
• Multiples. Reverberations that are themselves valid solutions of the wave equation but arrive along the wrong path, mimicking deeper structure.
• Acquisition footprint. Geometry imprint in the stacked volume.

The conventional toolkit is well understood: bandpass, f-k velocity fans to reject ground roll, f-x deconvolution for random noise, predictive deconvolution and surface-related multiple elimination (SRME) for multiples, Radon transforms. Every one of them trades signal preservation against noise removal. Push an f-k fan too hard and you notch out the reflection's own bandwidth; over-smooth and a weak reservoir reflector disappears into the background, which is the exact failure mode asset teams have learned to distrust.

Method: the wave equation as an unsupervised label

Represent the wavefield as a network $u_\theta(x, t)$ that takes space and time and returns amplitude. Train it with two terms. The data term pulls $u_\theta$ toward the recorded traces. The physics term penalises the residual of the wave equation:

$$r = \frac{\partial^2 u_\theta}{\partial t^2} - v^2 \nabla^2 u_\theta$$

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so the total loss is

$$\mathcal{L} = \mathcal{L}_{\text{data}} + \lambda \, \lVert r \rVert^2$$

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A primary reflection is a solution of that equation for the medium velocity $v$, so fitting the data while driving $r$ to zero captures it. Random noise carries no coherent phase and cannot satisfy the PDE; ground roll is a surface wave whose slow, dispersive moveout is not a solution of the body-wave equation at reservoir velocities. Both are pushed into the residual and excluded from $u_\theta$. Nothing in this loop needs a clean reference trace. The physics supplies the supervision, which is why it works on field data where labels do not exist.

The scalar $\lambda$ is the whole tuning story, and it is the same kind of hidden regularisation knob as the water level in spectral deconvolution or the Gaussian width in stacking: too small and the network fits noise; too large and it imposes physics the acquisition never recorded and erodes genuine amplitude variation. It should be set against the data's own noise level, not inherited from a script.

The same $u_\theta$ that denoises is also a mesh-free forward solver: evaluate it on a clean model and it emulates the wave-equation response, as in physics-informed Helmholtz and VTI wavefield solvers [3]. Add the velocity $v$ to the trainable parameters and the identical loss becomes full-waveform inversion; freeze the field and solve for traveltimes and it becomes an eikonal solver [2]. Noise attenuation, imaging, and velocity building end up on one differentiable core.

Results

The exhibit below is a before/after on a synthetic shot gather. The before panel carries the true reflection hyperbola, a steep low-velocity ground-roll cone, and random noise; the after panel is the reconstruction. Slide the SNR and the ground-roll strength, and toggle between conventional f-k / bandpass filtering and the PINN.

At high SNR both recover the reflection cleanly. As noise rises the separation appears: the f-k fan either leaves a residual of ground roll that has leaked through the reject zone or, tightened, notches part of the reflection's bandwidth and flattens the wavelet; the PINN keeps the reflection intact because it is the only component consistent with the wave equation, and the ground roll and random noise fall into the residual. The ground-truth reflection is overlaid so the recovery can be scored rather than eyeballed.

The honest boundary is multiples. A multiple is a legitimate solution of the same wave equation, just along a longer reverberation path, so a PDE-consistency prior does not reject it. SRME or predictive deconvolution still owns that job. The PINN's edge is over random and surface-wave noise and over reconstructing missing or dead traces, where "must satisfy the wave equation" is a strong, label-free constraint.

Discussion

The reason this matters for an O&G asset team is labels. Supervised denoising and deep-learning ground-roll removal need clean/noisy pairs, and outside a handful of benchmark lines those pairs do not exist for proprietary field data under NDA. A physics-informed loss replaces the missing label with the wave equation, which is available everywhere and is not confidential.

The cost is that the prior is only as right as the velocity: feed the wrong $v$ and the wave equation becomes the wrong constraint, so the denoiser should be coupled to the migration velocity model or co-trained with FWI, and out-of-distribution acquisition geometries must be flagged. Used inside its competence (random noise, ground-roll separation, trace reconstruction, and as the forward engine for imaging) the wave equation earns its keep as the one label the field always gives you.

By the numbers

Noise type	Physics status	Conventional tool	PINN wave-equation prior
Random / ambient	violates the PDE	f-x deconvolution	rejected into the residual
Ground roll (surface wave)	wrong velocity / dispersion	f-k velocity fan	separated by physics, no fan tuning
Swell / cable	violates the PDE	median / despike	rejected into the residual
Multiples	satisfies the PDE, longer path	SRME / predictive deconv	NOT removed, still needs SRME
Missing / dead traces	must satisfy the PDE	interpolation	wave-equation reconstruction
Forward cost after training	n/a	FD modelling per shot	mesh-free network evaluation

Takeaways

1. 1The primary reflection satisfies the wave equation; random noise and ground roll do not, so a PDE-residual penalty separates them without any clean-trace label.
2. 2The physics weight lambda is a regularisation knob and should be set against the data's own noise level, not inherited from a script.
3. 3Multiples satisfy the same equation along a longer path: SRME and predictive deconvolution still own that job.
4. 4The identical differentiable core serves denoising, mesh-free forward modelling, eikonal solving, and full-waveform inversion.

References

[1] Raissi, M., Perdikaris, P., and Karniadakis, G. E. (2019). Physics-informed neural networks. J. Comput. Phys., 378, 686-707. doi:10.1016/j.jcp.2018.10.045

[2] bin Waheed, U., Haghighat, E., Alkhalifah, T., Song, C., and Hao, Q. (2021). PINNeik: Eikonal solution using physics-informed neural networks. Computers & Geosciences, 155, 104833. doi:10.1016/j.cageo.2021.104833

[3] Song, C., Alkhalifah, T., and bin Waheed, U. (2021). Solving the frequency-domain acoustic VTI wave equation using physics-informed neural networks. Geophys. J. Int., 225, 846-859. doi:10.1093/gji/ggab010

[4] Verschuur, D. J., Berkhout, A. J., and Wapenaar, C. P. A. (1992). Adaptive surface-related multiple elimination. Geophysics, 57, 1166-1177. doi:10.1190/1.1443330

[5] Karniadakis, G. E., et al. (2021). Physics-informed machine learning. Nature Reviews Physics, 3, 422-440. doi:10.1038/s42254-021-00314-5

[6] Yilmaz, O. (2001). Seismic Data Analysis. Society of Exploration Geophysicists. doi:10.1190/1.9781560801580