Full Waveform Inversion vs. Dispersion Curve Inversion: A Comparative Study

Surface wave analysis serves as a primary tool for characterizing the shallow subsurface in both geophysical exploration and civil engineering. The discipline utilizes the dispersive nature of seismic surface waves—primarily Rayleigh and Love waves—to infer the elastic properties of layered or heterogeneous media. Traditionally, researchers and practitioners have relied on dispersion curve inversion, a method that extracts phase or group velocities from seismic records to build a depth-velocity profile. However, the advancement of high-performance computing has facilitated the rise of Full Waveform Inversion (FWI), a technique that utilizes the entire information content of the seismic trace, including phase, amplitude, and travel time, to reconstruct high-resolution images of the subsurface.

Surface Wave Hub focuses on the empirical study of these acoustic wave propagation characteristics within complex geological and engineered materials. By calibrating geophones and accelerometers to capture ground-motion signatures, research at the hub addresses the generation and attenuation of waves across stratigraphies. The choice between dispersion-based methods and FWI often depends on the scale of the target, the complexity of the heterogeneity, and the availability of computational resources. While dispersion curve inversion provides a strong and computationally efficient framework for 1D and 2D profiling, FWI offers the potential for unprecedented detail in 3D imaging, provided the initial constraints are sufficiently accurate to avoid mathematical pitfalls.

In brief

Methodological Core:Dispersion curve inversion utilizes picked velocity data; Full Waveform Inversion (FWI) utilizes the entire seismic record.
Resolution:FWI typically provides higher spatial resolution, often reaching sub-wavelength scales, whereas dispersion-based methods are limited by the wavelengths of the extracted modes.
Computational Demand:FWI requires iterative solutions to the wave equation, making it several orders of magnitude more computationally expensive than dispersion curve analysis.
Data Requirements:Dispersion analysis can function with lower signal-to-noise ratios by focusing on coherent energy; FWI requires dense receiver arrays and high-quality amplitude preservation.
Primary Limitation:FWI is highly susceptible to the 'cycle-skipping' phenomenon, where the inversion converges to a local minimum instead of the global solution.

Background

The study of surface waves in heterogeneous solid-state media has evolved from basic structural assessments to sophisticated lithological characterization. Surface waves, which travel along the interface between the earth and the atmosphere, are particularly sensitive to the shear-wave velocity (Vs) of the shallow subsurface. Because Vs is directly related to the shear modulus of a material, these waves are invaluable for identifying the mechanical properties of foundations, tunnels, and bridge abutments. In the mid-20th century, the spectral analysis of surface waves (SASW) and later the multi-channel analysis of surface waves (MASW) became standard protocols for non-destructive testing and geotechnical engineering.

Dispersion curve inversion relies on the physical principle that in a layered medium, different frequencies of surface waves travel at different speeds. Higher frequencies penetrate only shallow depths, while lower frequencies reach deeper layers. By measuring these velocities, an inversion algorithm can estimate a vertical velocity profile. In contrast, Full Waveform Inversion was originally conceptualized for deep crustal imaging using body waves. Its application to surface waves is a more recent development, driven by the need to map complex anomalies like buried utilities or voids that traditional ray-theory-based methods often fail to resolve accurately.

Theoretical Foundations of Full Waveform Inversion

The theoretical foundations of FWI in crustal and near-surface imaging were significantly advanced byVirieux and Operto (2009). They established that FWI is essentially a non-linear data-fitting procedure that seeks to minimize the difference between observed seismic waveforms and those predicted by a forward model. Unlike traditional methods that treat seismic data as a collection of arrival times or dispersion points, FWI treats the entire wavefield as a continuous signal. This approach leverages the adjoint-state method to compute gradients of the objective function, allowing for the update of physical parameters such as density, P-wave velocity, and S-wave velocity simultaneously.

Virieux and Operto (2009) highlighted that the success of FWI depends on the accuracy of the forward modeling engine, which must account for the full physics of wave propagation, including scattering, diffraction, and multi-pathing. In heterogeneous media, where material properties vary abruptly at interfaces, the full wave equation provides a more complete description than the high-frequency approximations used in ray theory. However, the high sensitivity of the method means that any discrepancy in source characterization or instrument calibration can lead to significant artifacts in the resulting image.

Dispersion Curve Inversion: Principles and Constraints

Dispersion-based inversion remains the workhorse of shallow subsurface imaging due to its stability and lower data requirements. The process begins with the transformation of time-domain seismic data into the frequency-velocity or frequency-wavenumber domain. In this space, the dominant energy of Rayleigh or Love waves appears as distinct modes. Analysts pick these modes to create a dispersion curve, which serves as the input for the inversion algorithm. This reduction of data—from millions of samples in a trace to a few dozen points on a curve—is both a strength and a weakness.

The primary constraint of dispersion curve inversion is its reliance on the assumption of a locally 1D or slowly varying 2D medium. In highly complex environments where lateral heterogeneity is significant, such as an urban environment with buried concrete structures and voids, the 1D assumption can lead to smoothing of small-scale features. Furthermore, the identification of higher-order modes is often difficult due to modal interference, yet these modes are essential for resolving deep velocity reversals or stiff layers overlying soft sediments. Despite these limitations, the method is less sensitive to phase noise and amplitude variations than FWI, making it more practical for noisy field conditions.

The Cycle-Skipping Phenomenon

A major obstacle in the practical application of FWI is 'cycle-skipping.' This phenomenon occurs when the predicted waveform and the observed waveform are out of phase by more than half a cycle. Because most FWI implementations use a local optimization strategy based on least-squares residuals, the algorithm may incorrectly align the peaks of one cycle with the troughs of the next. This results in an inversion that converges to a local minimum, producing a model that is geophysically nonsensical.

In heterogeneous media, cycle-skipping is exacerbated by the presence of strong velocity contrasts and complex scattering. To mitigate this, practitioners often use a multi-scale approach, beginning the inversion with low-frequency data and gradually introducing higher frequencies as the model improves. This requires seismic sources capable of generating low-frequency energy (typically below 5 Hz), which can be challenging to achieve with traditional impulsive sources like hammers or small explosives. Additionally, a highly accurate starting model—often derived from dispersion curve inversion—is frequently required to ensure the waveforms are within the 'half-cycle' limit before FWI begins.

Comparative Analysis of Data Requirements

The data requirements for FWI and dispersion curve inversion differ significantly in both quantity and quality. These differences dictate the types of equipment and survey geometries utilized at sites like Surface Wave Hub. Below is a comparison of the typical requirements for each method:

Requirement	Dispersion Curve Inversion	Full Waveform Inversion
Receiver Density	Moderate; requires enough geophones to sample the spatial wavefield without aliasing.	High; requires dense, often 3D arrays to capture the full scattered wavefield.
Source Knowledge	Minimal; often uses a generic source wavelet as the phase is less critical.	Critical; the source time function must be precisely known or inverted for.
Frequency Range	Flexible; depends on the target depth and available modes.	Requires low frequencies to mitigate cycle-skipping and high frequencies for resolution.
Signal-to-Noise (SNR)	Tolerant; stacking and transform techniques improve SNR.	Stringent; amplitude and phase distortions from noise can lead to inversion failure.
Initial Model	Minimal; often starts from a half-space or simple linear gradient.	High; requires a starting model that predicts arrivals within half a cycle of observations.

Applications in Infrastructure and Shallow Subsurface

The practical application of these methods extends to the non-destructive testing (NDT) of critical infrastructure. For bridges and foundations, dispersion curve analysis provides a rapid assessment of stiffness and the detection of large-scale delamination. By analyzing the dispersion curves of induced surface waves, engineers can monitor changes in elastic moduli over time, identifying degradation before visible cracks appear. In these scenarios, the relative simplicity of dispersion inversion allows for real-time or near-real-time monitoring on-site.

For more detailed forensic investigations, such as detecting buried utilities, voids, or shallow anomalies in engineered soil, FWI is increasingly preferred despite its complexity. The ability of FWI to account for diffraction allows it to image small objects that are smaller than the Fresnel zone of the seismic waves. Research into microtremor or controlled source wavefield data often combines these two techniques: using microtremor data to provide a low-resolution starting model via dispersion analysis, followed by FWI using controlled sources to sharpen the image of the anomaly. This hybrid approach leverages the stability of one method with the resolution of the other, advancing the capability of subsurface imaging in diverse geological contexts.