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A computer code for forward calculation and inversion of the H/V spectral ratio under the diffuse field assumption
Antonio García-Jerez, José Piña-Flores, Francisco J. Sánchez-Sesma, Francisco Luzón, Mathieu Perton
TL;DR
Existing H/V spectral-ratio inversion approaches use differing theoretical assumptions, and some are limited for raw microtremor mixtures or computationally demanding full-wavefield calculations. The paper presents and tests a diffuse-field forward code using Green’s functions, contour integration, wavefield decomposition, and stabilized high-frequency computation; it is incorporated into an inversion program for passive seismic surveying.
Problem
Existing H/V spectral-ratio approaches rely on diverse theoretical models, while some methods are unsuitable for raw microtremor energy ratios and full-wavefield methods have not been incorporated into inversion tools.
Method
The paper develops a diffuse-field forward code that computes Green’s-function contributions using complex-wavenumber contour integration, separates SH, P-SV, Rayleigh, and Love waves, and applies Wang’s orthonormalization.
Results
The code restricts body-wave integration to [0, ω/β_N], which can significantly speed computation relative to the conventional discrete-wavenumber method.
Takeaways & Limitations
The code is incorporated into an H/V spectral-ratio inversion program supporting passive seismic surveying.
Takeaways & Limitations
Full-wavefield methods had not been incorporated into ground-inversion tools, probably because of intensive computing requirements.
Abstract
from arXiv · showhide
During a quarter of a century, the main characteristics of the horizontal-to-vertical spectral ratio of ambient noise HVSRN have been extensively used for site effect assessment. In spite of the uncertainties about the optimum theoretical model to describe these observations, several schemes for inversion of the full HVSRN curve for near surface surveying have been developed over the last decade. In this work, a computer code for forward calculation of H/V spectra based on the diffuse field assumption (DFA) is presented and tested.It takes advantage of the recently stated connection between the HVSRN and the elastodynamic Green's function which arises from the ambient noise interferometry theory. The algorithm allows for (1) a natural calculation of the Green's functions imaginary parts by using suitable contour integrals in the complex wavenumber plane, and (2) separate calculation of the contributions of Rayleigh, Love, P-SV and SH waves as well. The stability of the algorithm at high frequencies is preserved by means of an adaptation of the Wang's orthonormalization method to the calculation of dispersion curves, surface-waves medium responses and contributions of body waves. This code has been combined with a variety of inversion methods to make up a powerful tool for passive seismic surveying.
1. Introduction
The introduction reviews competing approaches for interpreting and inverting ambient-noise H/V spectral ratios. It motivates a faster diffuse-field code that separates wave contributions and supports inversion of ground structures.
- Ambient-noise spectral ratios have motivated diverse, sometimes opposing, theoretical approaches for seismic exploration.
- Rayleigh-wave ellipticity inversions were not intended for energy ratios measured directly from raw microtremor records.
- Surface-wave approximations provide suitable power-ratio estimates when surface waves dominate the relevant models and frequency bands.
- Full-wavefield methods had not been incorporated into ground-inversion tools, probably because of intensive computing requirements.
- The approach builds on the connection between ambient-noise interferometry, elastodynamic Green’s functions, and HVSRN forward calculation.
- The paper presents a faster code that separately computes SH, P-SV, Rayleigh, and Love contributions and supports ground-structure inversion.
2. Forward calculation of the H/V spectral ratio
The forward algorithm derives H/V spectra from Green’s-function imaginary parts in a horizontally layered medium. Complex-wavenumber contour integration separates body- and surface-wave contributions, while orthonormalization stabilizes high-frequency calculations.
- The H/V ratio is obtained from horizontal and vertical directional power spectral densities derived from Green’s-function components.
- The model represents a horizontally layered structure over a half-space with homogeneous, elastic, isotropic layers and plane interfaces.
- Contour integration in the complex wavenumber plane isolates body-wave branch-cut integrals from surface-wave pole residues.
- The body-wave integration is restricted to [0, ω/β_N], which can significantly speed computation relative to direct discrete-wavenumber integration.
- Wang’s orthonormalization adaptation preserves the 4 x 4 propagator structure for P-SV and Rayleigh waves while avoiding high-frequency numerical instability.
N N PSV
The method stabilizes dispersion-curve and wavefield calculations through orthonormalization, then evaluates surface- and body-wave contributions to the H/V response. Tests show improved short-wavelength stability, accurate modal shapes, and distinct wave contributions to HVSRN.
- Rayleigh-wave dispersion: Orthonormalization improves numerical stability in the short-wavelength region when locating Rayleigh-wave dispersion curves.The frequency–velocity-plane signs are evaluated after stabilization; the improvement is evident at short wavelengths.
- Surface-wave validation: The stabilized surface-wave calculation matches global-matrix reference curves for the fundamental and higher Rayleigh modes.The calculation was tested across multiple models and showed accuracy and stability, including perfect matching in the reported comparison.
- Surface-wave response: Surface-wave medium responses use modal energy integrals and avoid numerical derivatives of phase-velocity dispersion curves when calculating group velocity.The same framework treats Rayleigh and Love waves, with analogous but simpler calculations for Love-wave modal shapes.
- Wave contributions: The horizontal component is dominated by surface waves from 0.7 Hz, with Love waves generally providing the main contribution and SH waves dominating lower frequencies.The dominant Love mode changes with model and frequency, while Rayleigh waves dominate a narrow transition band around 14.5 Hz in Model 2.
- Wave contributions: The vertical component is dominated by the fundamental Rayleigh mode, and surface waves control the overall HVSRN shape except below the fundamental SH resonance.Body-wave resonances can add bumps, including the visible SH resonance at 4.9 Hz for Model 2.
- Attenuation: Anelastic attenuation decreases the main H/V peak amplitude as the quality factor decreases.This attenuation effect is reported as a principal influence on the H/V decay.
3. Inversion of HVSRN and of surface wave velocities
The software frames inversion as estimating ground models that fit HVSRN and surface-wave observations under prior constraints. It combines global and local search methods, supports joint inversion, and provides model statistics and uncertainty measures.
- Inversion framework: The Matlab software implements Monte Carlo, simulated annealing, interior-point, random-search, and downhill-simplex inversion methods.These methods can be applied sequentially through the program flow chart.
- Inversion framework: The interface supports independent or joint inversion of HVSRN with Rayleigh- or Love-wave phase and group velocities.The assumed subsurface structure is locally horizontally layered.
- Objective function: Misfit functions compare observed and theoretical data, with an adjustable weight controlling the relative contribution of dispersion curves in joint inversion.The weighting can also compensate for different numbers of HVSRN and velocity samples.
- Global methods: Simulated annealing samples the model space while gradually lowering temperature toward a minimum-energy, maximum-likelihood model.The software also computes a best-fitting model, mean model, parameter uncertainties, and normalized covariance matrix.
- Local methods: Downhill simplex can converge faster than simulated annealing but may reach local minima, making it most suitable when prior information strongly restricts the model space.It also performs well for joint HVSRN and dispersion-curve inversion because joint data reduce equivalent solutions.
4. Test with real data
A Campo de Dalías test demonstrates that the inversion can fit a complex HVSRN curve with multiple peaks. The resulting model agrees with independent borehole and active-seismic information.
- Observed HVSRN: The measured HVSRN has a clear main peak at 0.65–0.7 Hz, a broad secondary bump from approximately 1.3 to 8.0 Hz, and at least two minor peaks.The data were obtained using a broadband seismometer and processed from overlapping 40.48 s windows.
- Inversion: The inversion combines simulated annealing with local methods and then uses 4000 Monte Carlo models to estimate the mean model, standard deviations, and covariance matrix.Parameter ranges were based on previous work at Campo de Dalías.
- Validation: The obtained model is consistent with previous studies, including a 418 ms two-way P-wave travel time to the bottom of the interpreted Messinian marls.The basement depth also matches results from earlier investigations.
5. Concluding remarks
The presented DFA code provides full-wavefield HVSRN calculations and supports inversion with surface-wave data. Tests show stable computation, good fitting of complex real data, and consistency with independent seismic information.
- Concluding remarks: The code calculates HVSRN under the diffuse field approximation and separately evaluates Rayleigh, Love, P-SV, SH, and body-wave contributions.Separate wavefield contributions support interpretation of the observable.
- Concluding remarks: An adapted orthonormalization technique preserves numerical stability at high frequencies during dispersion, surface-wave response, and body-wave calculations.It prevents underflow and overflow in susceptible algorithmic steps.
- Concluding remarks: The code is incorporated into an inversion program that jointly handles HVSRN and surface-wave dispersion curves.Joint inversion reduces thickness–velocity tradeoffs and improves sensitivity to velocity contrasts, especially basement velocity.
- Concluding remarks: Body-wave integrals are often the most time-consuming part of forward computation even though they frequently have minor effects on HVSRN curves.Preliminary inversions can therefore use surface-wave components before full-wavefield refinement.
- Concluding remarks: Real-data application produced a good fit to the experimental curve, including secondary peaks, while agreeing with borehole and conventional seismic results.The example concerns a moderately deep sedimentary structure.
Appendix A. Algorithm for generation of models with velocity increasing downwards.
The appendix introduces a semi-analytical sampler for velocity profiles that increase with depth. It reproduces the reference model statistics while avoiding inefficient rejection as layer count grows.
- Efficiency: The appendix presents an alternative to simply discarding random models that violate monotonicity, because rejection becomes very inefficient as layer number increases.The approach is designed for user-imposed increasing-velocity constraints.
- Sampling procedure: The method uses piecewise-polynomial probability and cumulative-distribution functions with numerical inversion to draw admissible velocities.The same procedure is iterated from the upper layer to the halfspace.
- Sampling procedure: The sampler generates layer velocities sequentially, using the previously sampled velocity as the lower bound for each deeper layer.This directly enforces downward-increasing velocity profiles.
- Validation: The generated models have statistics comparable to a reference population produced by uniform sampling followed by rejection of invalid profiles.The reference approach retained only 0.02% of generated models in the illustrated case.