Seismic: Wave Analysis and Inversion

• Seismic is the study of Earth’s and planetary vibrations using wavefield analysis, source inversion, and controlled experiments to decode internal structures.
• Advanced monitoring architectures and machine learning techniques improve event detection, missing-trace reconstruction, and subsurface imaging with high precision.
• Innovations in seismic metamaterials, inversion frameworks, and planetary seismology extend applications from earthquake analysis to extraterrestrial exploration.

Seismic denotes the physics, observation, and analysis of wavefields generated by Earth’s vibrations and, in planetary contexts, by analogous vibrations of other bodies. In current research, the term spans earthquake-source characterization from far-field $P$- and $S$-waves, continuous monitoring with station networks and unconventional sensors, reconstruction and interpretation of incomplete or noisy seismic records, subsurface inversion and semantic annotation, engineered control of near-surface wave propagation, and extrapolation of seismological methods to icy ocean worlds and other planetary bodies (Apostol, 2018, Moore et al., 2017, Stähler et al., 2022).

1. Source physics, wave types, and inverse formulations

A central seismic problem is the reconstruction of source properties from observed wavefields. One explicit formulation treats the source as a localized tensorial force density,

$$F_i = M_{ij}\,\partial_j \delta(\mathbf R-\mathbf R_0),$$

with a symmetric seismic-moment tensor $M_{ij}$. In a homogeneous isotropic body, the displacement field is decomposed into near-field and far-field parts, and the far field further separates into longitudinal and transverse components,

$$\mathbf u=\mathbf u^n+\mathbf u^f, \qquad \mathbf u^f=\mathbf u_l^f+\mathbf u_t^f.$$

The far-field amplitudes yield three observational quantities through the vectors $\mathbf v_l$ and $\mathbf v_t$, but a general symmetric moment tensor has six components, so additional physical structure is required for inversion (Apostol, 2018).

One such structure is the Kostrov vectorial representation,

$M_{ij}=2\mu S u^0\left(s_i a_j+a_i s_j\right),$

where $\mathbf s$ is the fault-normal direction, $\mathbf a$ is the slip direction, $S$0 is slip magnitude, $S$1 is fault area, and $S$2 is shear modulus. This reduces the unknowns to four and makes the tensor traceless, $S$3. The remaining closure is obtained from energy conservation together with a covariance condition that constrains $S$4, $S$5, and the source-to-receiver direction $S$6 to lie in the same plane. The resulting framework yields an explicit resolved form of the moment tensor,

$S$7

and connects source observables to radiated energy, source duration, focal volume, and focal strain (Apostol, 2018).

The same formulation introduces a geometric representation of the source mechanism through the quadratic form $S$8. In coordinates aligned with $S$9 and $$F_i = M_{ij}\,\partial_j \delta(\mathbf R-\mathbf R_0),$$0, the level set is a rectangular hyperbola, termed the seismic hyperbola: $$F_i = M_{ij}\,\partial_j \delta(\mathbf R-\mathbf R_0),$$1 Its asymptotes align with the fault normal and slip direction, providing a compact geometric image of focal-region structure (Apostol, 2018).

At smaller scales, source complexity can be resolved through explicit rupture and damage simulation. A hybrid finite-discrete element model of a rough laboratory fault shows that quasi-static shear loading still produces local dynamic seismic activities associated with stress concentration on interlocking asperities. In that study, 7,557 broken CCEs were clustered into 1,561 seismic events, with event magnitudes ranging from $$F_i = M_{ij}\,\partial_j \delta(\mathbf R-\mathbf R_0),$$2 to $$F_i = M_{ij}\,\partial_j \delta(\mathbf R-\mathbf R_0),$$3. Larger events were mostly shear-mode, smaller events mostly tensile-mode, and damage was spatially heterogeneous, with gouge-forming regions adjacent to minimally damaged patches (Zhao et al., 2023). This suggests that even nominally simple fault slip can embed a strongly intermittent seismic source field.

2. Monitoring architectures and the ambient seismic environment

Modern seismic monitoring increasingly treats continuous waveform streams, rather than thresholded detections, as the primary observable. In SIGVISA, seismic monitoring is posed as Bayesian inference over a generative model,

$$F_i = M_{ij}\,\partial_j \delta(\mathbf R-\mathbf R_0),$$4

where events $$F_i = M_{ij}\,\partial_j \delta(\mathbf R-\mathbf R_0),$$5 and station signals $$F_i = M_{ij}\,\partial_j \delta(\mathbf R-\mathbf R_0),$$6 are jointly modeled. The event prior uses a homogeneous Poisson process, while each station waveform is represented as a sum of phase-wise envelopes, wavelet-based modulation, and autoregressive background noise. Gaussian processes over arrival-time, amplitude, rise-time, decay parameters, and wavelet coefficients allow the model to exploit waveform similarity from historical seismicity while degrading gracefully to parametric envelopes in regions with no nearby training data. On western United States data, this system recovered three times as many events as previous work and reduced mean location errors by a factor of four (Moore et al., 2017).

Observation systems have also broadened beyond standard seismometer networks. A subsea telecom cable from Iceland to Ireland was converted into a distributed seismic sensor using per-span laser interferometry with 17 monitored spans over about 1770 km. The system detected clear $$F_i = M_{ij}\,\partial_j \delta(\mathbf R-\mathbf R_0),$$7-waves, $$F_i = M_{ij}\,\partial_j \delta(\mathbf R-\mathbf R_0),$$8-waves, and surface waves from multiple worldwide earthquakes, grouped arrivals into 13 phase families within the same minute, and reported about 20 earthquakes detected on several spans, with 15 detected across all spans. The Japan example showed the expected sequence of arrivals, with the $$F_i = M_{ij}\,\partial_j \delta(\mathbf R-\mathbf R_0),$$9-wave after about 12.3 minutes, the $M_{ij}$0-wave at 22.5 minutes total, and the surface wave after about 43.5 minutes, consistent with global travel-time physics (Mazur et al., 2024).

Seismic background and site characterization remain equally important. At the Sanford Underground Laboratory, an array of broadband instruments established the 4100-ft level as a world-class low seismic-noise environment. Above about 1 Hz, seismic noise at 4100 ft is about a factor of 10 weaker in amplitude than at 300 ft. The study linked the primary and secondary microseismic peaks in the 50 mHz–0.3 Hz band to ocean-wave activity and showed that low-frequency wind correlation present at 300 ft disappeared at 2000 ft and 4100 ft, indicating that surface wind-driven seismicity is not well described by simple homogeneous-Rayleigh-wave propagation (Harms et al., 2010).

Precision infrastructure can itself function as a seismic observer. In the European X-ray Free-Electron Laser, controller I/O data from phase-locked loops in link stabilization units were analyzed as a proxy for ground motion. The study separated earthquakes, ocean-generated microseism, and civilization noise using spectral fingerprints and external data. Distant earthquakes were detectable mainly below roughly 5 Hz, ocean-generated microseism occupied $M_{ij}$1, and civilization noise occupied $M_{ij}$2. Even earthquakes about 5000 km away produced measurable fluctuations, with controller-output jitter spans up to $M_{ij}$3 and controller-input spans exceeding $M_{ij}$4. The phase-locked loops eliminated more than 99% of interference, and the conclusion describes attenuation of 99.99% of disturbances caused by seismic activity (Grünhagen et al., 4 Feb 2025).

3. Computational seismic analysis, machine learning, and multimodal models

Seismic data processing has become a major application area for modern machine learning. For missing-trace reconstruction, DSPRecon formulates recovery of complete data $M_{ij}$5 from masked observations $M_{ij}$6 through an untrained U-net,

$M_{ij}$7

with parameters optimized by

$M_{ij}$8

The method uses one undersampled seismic record, a fixed random noise tensor, and the architecture itself as a prior. Its reported signal-to-noise ratios were 32.68 dB versus 19.11 dB for SSA on synthetic single-shot pre-stack data with 50% randomly missing traces, 33.09 dB versus 24.27 dB for post-stack data, and 35.91 dB versus 15.32 dB for regularly missing field traces compared with de-aliased Cadzow (Liu et al., 2019).

For seismic event detection, Seismic-Net casts continuous monitoring as sliding-window binary classification on 18,000-timestamp windows with a 6,000-timestamp offset at test time. The architecture is a deep densely connected 1D CNN with growth rate $M_{ij}$9 and about 800K parameters. Trained on Chimayó geyser data as a natural analog for $$\mathbf u=\mathbf u^n+\mathbf u^f, \qquad \mathbf u^f=\mathbf u_l^f+\mathbf u_t^f.$$0 leakage, it achieved precision 0.889 and recall 0.923, outperforming a Kernel SVM, a VGG-based CNN, and a ResNet-based CNN (Wu et al., 2018).

Synthetic waveform generation addresses label scarcity. SeismoGen uses a conditional GAN for 40 s, 3-channel waveform windows at 40 Hz, with three separate generator pipelines for BHE, BHN, and BHZ. The best synthetic-only classifier $$\mathbf u=\mathbf u^n+\mathbf u^f, \qquad \mathbf u^f=\mathbf u_l^f+\mathbf u_t^f.$$1 achieved 97.11% accuracy on station V34A and 96.96% on V35A, compared with 98.56% and 98.48% for a real-trained classifier. When synthetic augmentation was added to limited real training sets, classification accuracy improved in 23 out of 30 cases on V35A and 27 out of 30 cases on V34A, with largest gains over 14% and over 17%, respectively (Wang et al., 2019).

Representation learning has expanded from task-specific models to foundation models. SeisLM pretrains on a union of eight seismic datasets—ETHZ, INSTANCE, Iquique, STEAD, GEOFON, MLAAPDE, PNW, and OBST2024—using a masked contrastive objective with $$\mathbf u=\mathbf u^n+\mathbf u^f, \qquad \mathbf u^f=\mathbf u_l^f+\mathbf u_t^f.$$2 negatives and $$\mathbf u=\mathbf u^n+\mathbf u^f, \qquad \mathbf u^f=\mathbf u_l^f+\mathbf u_t^f.$$3. Two variants are reported: SeisLM-base with about 11.4M parameters and SeisLM-large with about 90.7M parameters. Fine-tuning improves event detection, phase identification, onset time regression, and foreshock–aftershock classification, with the largest gains in low-label regimes (Liu et al., 2024).

Seismic analysis is also becoming explicitly multimodal. MultiSeismo assembles over 16K seismic events spanning 2010 to 2023, integrating waveform recordings, intensity maps, population exposure visualizations, and text in a standardized JSON format. The image archive contains $$\mathbf u=\mathbf u^n+\mathbf u^f, \qquad \mathbf u^f=\mathbf u_l^f+\mathbf u_t^f.$$4 images, and the instruction set MISCE/MICSE includes 17 instruction templates and more than 500K total instructions. SeisModal, built by augmenting Unified-IO 2 with a pretrained Chronos-T5 time-series encoder, outperformed Phi 4 and Unified-IO on text, image, and especially time-series reasoning; in the reported instruction evaluation, its time-series scores were BLEU 0.675, ROUGE 0.664, and BERT 0.824 (Munikoti et al., 25 May 2026).

4. Imaging, inversion, and interpretation of the subsurface

Seismic inversion addresses the recovery of quantitative subsurface properties from band-limited wavefields. In the post-stack setting considered by IntraSeismic, the forward model is

$$\mathbf u=\mathbf u^n+\mathbf u^f, \qquad \mathbf u^f=\mathbf u_l^f+\mathbf u_t^f.$$5

or

$$\mathbf u=\mathbf u^n+\mathbf u^f, \qquad \mathbf u^f=\mathbf u_l^f+\mathbf u_t^f.$$6

leading to the inverse problem

$$\mathbf u=\mathbf u^n+\mathbf u^f, \qquad \mathbf u^f=\mathbf u_l^f+\mathbf u_t^f.$$7

The method represents the impedance field implicitly as $$\mathbf u=\mathbf u^n+\mathbf u^f, \qquad \mathbf u^f=\mathbf u_l^f+\mathbf u_t^f.$$8, using multiresolution hash encoding and a small MLP. On Marmousi, reported SNRs were 27.34 dB, 25.00 dB, and 22.93 dB for noise levels $$\mathbf u=\mathbf u^n+\mathbf u^f, \qquad \mathbf u^f=\mathbf u_l^f+\mathbf u_t^f.$$9, $\mathbf v_l$0, and $\mathbf v_l$1; on a $\mathbf v_l$2 SEAM Phase I subvolume, the method achieved 40.84 dB, 36.92 dB, and 34.5 dB, converging in under 400 iterations in each case. The coordinate-based parameterization also reduced storage from about 5.83 million voxel values to about 1.16 million parameters, giving an approximate compression ratio of 5 (Romero et al., 2023).

ContextSeisNet extends seismic processing with in-context learning. Instead of a fixed mapping $\mathbf v_l$3, it conditions predictions on a support set of neighboring gather-label pairs,

$\mathbf v_l$4

Applied to demultiple processing, the method uses support examples from spatially related CDPs on the same line and performs inference in a single forward pass without retraining. The reported field-data result is comparable performance with 90% less training data than the U-Net baseline, together with improved lateral consistency, improved near-offset performance, and more complete multiple removal (Fuchs et al., 12 Dec 2025).

Semantic interpretation has likewise moved toward dense annotation. SpiNet defines 12 commonly observed seismic patterns grouped into 7 horizons, 2 stratigraphic sequences, and 3 structures, and learns pixel-wise annotations with a deconvolutional encoder–decoder network. Training on 51×51 patches, augmented to 76,950 training images, the model reached about 0.8 training accuracy and below 0.2 loss, and achieved 78% pixel-wise annotation accuracy on held-out inline #390. Applied to the F3 cube of 651 inlines, 951 crosslines, and 463 samples per trace, SpiNet annotated the full volume in about 10 minutes on one NVIDIA P4000 GPU (Di, 2018).

5. Structured media, forecasting, and seismic engineering

Seismic research also includes deliberate modification and anticipation of wave propagation. Seismic metamaterials translate ideas from electromagnetic metamaterials to surface seismic wave control in sedimentary soils structured at the meter scale. Large-scale in-situ experiments near Grenoble and Lyon demonstrated that borehole arrays can redistribute energy, reduce wave amplitude along some directions, change polarization, and act as a seismic filter or lens. The Lyon experiment used a grid of 23 boreholes, about 2 m diameter and 5 m depth, with spacing about 5–7 m, while a 17 ton mass drop generated the source. A later reinterpretation emphasized energy corridors: some paths between boreholes acted as guiding channels rather than simple barriers (Brule et al., 2019).

The corresponding transfer function was defined as

$\mathbf v_l$5

and the homogenized effective-medium description used Willis-type constitutive relations,

$\mathbf v_l$6

In that framework, ambient seismic noise can be interpreted as a time modulation that makes structured soils behave like moving media. The reported application domain includes shielding, lensing, cloaking, Rayleigh-wave control, energy harvesting, and analog computation using ambient seismic noise (Brule et al., 2019).

Forecasting is a complementary engineering direction. SeismoGPT is a transformer-based autoregressive model for forecasting three-component seismic waveforms for future gravitational-wave detectors. It uses token length 16 samples, context window 64 tokens, 6 layers, and 8 attention heads per layer, and is trained with mean squared error on synthetic waveforms generated with Instaseis from the ak135f_2s database. The paper reports good performance in the immediate prediction window, gradual degradation at longer horizons due to autoregressive error accumulation, and more robust late-forecast behavior for the array-based model than for the single-station version. The intended applications are Newtonian-noise mitigation, active seismic isolation, and real-time observatory control (Esmail et al., 25 Sep 2025).

6. Planetary and icy-ocean-world seismology

Seismic methods have expanded from Earth to the Solar System. A broad review argues that seismic experiments constrain crust, mantle, and core structure, tectonic activity, thermal state, ocean depth, ice-shell thickness, and habitability-relevant layering. InSight on Mars demonstrated that even a single seismometer can constrain crustal thickness, mantle properties, and core radius while identifying active source regions such as Cerberus Fossae. The same review treats the Moon, Europa, Titan, Enceladus, Io, Mercury, Venus, Ceres, and the giant planets as potential seismic targets, each with distinct mission and environmental constraints (Stähler et al., 2022).

Icy ocean worlds require a revised seismic lexicon because an ice shell overlies a global ocean. A dedicated framework introduces $\mathbf v_l$7 and $\mathbf v_l$8 for compressional and shear waves in ice or mantle, $\mathbf v_l$9 for ocean legs, and $\mathbf v_t$0 for compressional waves in the core, together with boundary markers such as $\mathbf v_t$1 for the bottom of the ice shell and $\mathbf v_t$2 for the bottom of the ocean. It also emphasizes guided phases such as flexural waves, longitudinal waves, Love waves, and the Crary phase. For a floating ice shell, the longitudinal-wave speed is

$\mathbf v_t$3

and the Crary-phase frequency is

$\mathbf v_t$4

These relations make guided-wave observations direct constraints on ice thickness and elastic structure (Stähler et al., 2017).

Europa-specific seismicity models use a Gutenberg–Richter framework with cumulative seismic moment release between

$\mathbf v_t$5

Across the explored models, ambient noise levels vary by about 80 dB. Most mean noise estimates are below the self-noise floor of a typical high-frequency geophone, but larger events are expected to stand about $\mathbf v_t$6–50 dB above the mean background. The study also shows that autocorrelation of simulated noise can recover reflection-like arrivals, including an ocean-floor reflection at about 175 s in one example (Panning et al., 2017).

Titan introduces a different microseismic regime because its methane–ethane seas can generate ambient seismic noise analogous to Earth’s ocean microseisms. Using wind-wave and loading models, one study concludes that storms of more than 2 m/s wind speed would create a signal that is globally observable with a high-quality broadband sensor and observable to a thousand kilometer distance with an InSight-class space-ready seismometer. The modeled Titan microseisms lie mainly between about 0.05 and 0.2 Hz, and are expected to be episodic rather than constant (Stähler et al., 2019).

Taken together, these results show that seismic research is no longer confined to terrestrial earthquake seismology. It includes analytic source inversion, ambient-noise characterization, distributed sensing on cables and precision timing systems, data-driven reconstruction and interpretation, structured control of wave propagation, and planetary interior probing in environments where oceans, ice shells, and nonstandard guided phases fundamentally reshape the wavefield.