We analyze surface wave data taken in Currituck Sound, North Carolina, during a storm on 4 February 2002. Our focus is on the application of
nonlinear Fourier analysis
(NLFA) methods (Osborne
2010
) ...to analyze the data set: The approach spectrally decomposes a nonlinear wave field into
sine waves
,
Stokes waves
, and
phase-locked Stokes waves
otherwise known as
breather trains
. Breathers are nonlinear beats, or packets which “breathe” up and down smoothly over
cycle times
of minutes to hours. The maximum amplitudes of the packets during the cycle have a largest central wave whose properties are often associated with the study of “rogue waves.” The mathematical physics of the nonlinear Schrödinger (NLS) equation is assumed and the methods of algebraic geometry are applied to give the
nonlinear spectral representation
. The distinguishing characteristic of the NLFA method is its ability to spectrally decompose a time series into its
nonlinear coherent structures
(Stokes waves and breathers) rather than just sine waves. This is done by the implementation of
multidimensional, quasi-periodic Fourier series
, rather than ordinary Fourier series. To determine preliminary estimates of nonlinearity, we use the significant wave height
H
s
, the peak period
T
p
, and the length of the time series
T
. The time series analyzed here have 8192 points and
T
=1677.72 s = 27.96 min. Near the peak of the storm, we find
H
s
≈ 0.55 m,
T
p
≈ 2.4 s so that for the wave steepness of a near Gaussian process,
S
=
π
5
/
2
/
g
H
s
/
T
p
2
, we find
S
≈ 0.17, quite high for ocean waves. Likewise, we estimate the Benjamin-Feir (BF) parameter for a near Gaussian process,
I
B
F
=
π
5
/
2
/
g
H
s
T
/
T
p
3
, and we find
I
B
F
≈ 119. Since the BF parameter describes the nonlinear behavior of the
modulational instability
, leading to the formation of breather packets in a measured wave train, we find the
I
B
F
for these storm waves to be a surprisingly high number. This is because
I
B
F
, as derived here, roughly estimates the number of breather trains in a near Gaussian time series. The BF parameter suggests that there are roughly 119 breather trains in a time series of length 28 min near the peak of the storm, meaning that we would have average breather packets of about 14 s each with about 5-6 waves in each packet. Can these surprising results, estimated from simple parameters, be true from the point of view of the complex nonlinear wave dynamics of the BF instability and the NLS equation? We analyze the data set with the NLFA to verify, from a
nonlinear spectral point of view
, the presence of large numbers of breather trains and we determine many of their properties, including the
rise time
for the breathers to grow to their maximum amplitudes from a quiescent initial state. Energetically, about 95% of the NLFA components are found to consist of breather trains; the remaining small amplitude components are sine and Stokes waves. The presence of a large number of densely packed breather trains suggests an interpretation of the data in terms of
breather turbulence
, highly nonlinear
integrable turbulence
theoretically predicted for the NLS equation, providing an interesting paradigm for the nonlinear wave motion, in contrast to the random phase Gaussian approximation often considered in the analysis of data.
Waves in Oceanic and Coastal Waters describes the observation, analysis and prediction of wind-generated waves in the open ocean, in shelf seas, and in coastal regions with islands, channels, tidal ...flats and inlets, estuaries, fjords and lagoons. Most of this richly illustrated book is devoted to the physical aspects of waves. After introducing observation techniques for waves, both at sea and from space, the book defines the parameters that characterise waves. Using basic statistical and physical concepts, the author discusses the prediction of waves in oceanic and coastal waters, first in terms of generalised observations, and then in terms of the more theoretical framework of the spectral energy balance. He gives the results of established theories and also the direction in which research is developing. The book ends with a description of SWAN (Simulating Waves Nearshore), the preferred computer model of the engineering community for predicting waves in coastal waters.
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, over the last decade several schemes for inversion of the full HVSRN curve for near surface surveying have been developed.
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.
•H/V spectral ratios of ambient seismic noise are modeled by using full wavefield.•The theoretical framework is consistent with ambient noise interferometry.•The method provides separate calculations of different wave modes.•The software is also suitable for coda waves and any other diffuse-like wavefields.•It supports joint inversion of seismic velocity models from H/V and dispersion curves.
To take advantage of complementary strengths of earthquake body wave data and surface wave data from both earthquake and ambient noise sources, we have jointly used them to determine a ...high-resolution 3D Vs model of the lithosphere in South China by the joint inversion algorithm of Zhang et al. (2014). For body wave data, we assembled P- and S-wave arrival times for 40,028 earthquakes during the period of 2008/10–2018/06 recorded by 676 stations in South China. For surface wave dispersion dataset, it includes Rayleigh wave phase velocity maps at periods of 8–70 s and group velocity maps at periods of 8–50 s from Shen et al. (2016). Due to the complementary strengths of the two data types, the resolution of the jointly inverted Vs model has improved compared to the separately inverted models. The joint inversion model can also fit both data types at a similar level compared to separate inversions. In comparison with previous models, the new velocity model better delineates the sedimentary basins in the shallow crust and high velocity root in the lithosphere beneath the Sichuan basin, as well as better resolves the velocity variations between western and eastern Yangtze craton, thin lithosphere beneath eastern Yangtze craton and Cathaysia block, and helps us to better understand the mechanism of the lithosphere thinning. From the correlation between high velocity anomalies in the middle crust and the distribution of Neoproterozoic basalt samples, we propose the amalgamation suture between the Yangtze craton and Cathaysia block is along the eastern segment of the Jiangshan-Shaoxing fault and the west margin of the Jiangnan orogen volcanic rocks. Our new joint inversion model is useful for constructing a community velocity model and for better understanding the complex tectonics of the South China block.
•A new Vs model of South China Block is determined from joint inversion of body wave and surface wave data.•New Vs model better resolves the velocity variations between western and eastern Yangtze craton.•The amalgamation suture between Yangtze craton and Cathaysia block is imaged as a high Vs belt.
Primary microseisms are background seismic oscillations recorded everywhere on Earth with typical frequencies 0.05 < f < 0.1 Hz. They appear to be generated by ocean waves of the same frequency f, ...propagating over shallow bottom topography. Previous quantitative models for the generation of primary microseisms considered wave propagation over topographic features with either large scales, equivalent to a vertical point force, or small scales matching ocean wave wavelengths, equivalent to a horizontal force. While the first requires unrealistic bottom slopes to explain measured Rayleigh wave amplitudes, the second produced Love waves and not enough Rayleigh waves. Here we show how the small scales actually produce comparable horizontal and vertical forces. For example, a realistic rough bottom over an area of 100 km2 with depths around 15 m is enough to explain the vertical ground motion observed at a seismic station located 150 km away. Ocean waves propagating over small‐scale topography is thus a plausible explanation for the observed microseisms at frequencies around 0.07 Hz.
Plain Language Summary
Microseisms are background oscillations of the solid Earth. Most of these oscillations are caused by ocean waves and can thus be used to study their source, the ocean waves, or the medium in which they propagate, the solid Earth. Several theories have been proposed for how ocean waves going over shallow ocean topography make microseisms in the band of periods 10 to 20 s, but they are not satisfactory because they either require unrealistic large slopes of the ocean floor or they produce a ratio of different types of seismic waves, Love and Rayleigh waves, that is too large. We thus revise these theories to show that a plausible seismic source is the propagation ocean of waves over a wavy bottom, when the bottom has wavelengths that match those of ocean wave. We particularly verify that the predicted Rayleigh wave amplitude is of the order of what is measured at a particular seismometer located in Ireland. Because the necessary details in bottom topography vary a lot between different ocean regions, the new theory suggests that the spatial distribution of seismic sources is more heterogeneous than previously thought.
Key Points
Ocean waves over small‐scale topography produce both horizontal and vertical equivalent point forces
Small‐scale wavy bottoms are stronger sources of primary microseism than constant slopes
The vertical force, caused by pressure modulations, is generally weaker than the horizontal force
We estimate crustal and uppermost mantle shear velocity structure beneath 30 stations in North America by jointly inverting the high frequency scattered wavefield observed in the P wave coda, ...together with long period surface wave phase and group dispersion data. Several features distinguish our approach from previous such joint inversions. 1) We apply a cross-convolution method, rather than more standard deconvolution approaches used in receiver function studies, and consider both Love and Rayleigh wave dispersion, allowing us to infer profiles of radial anisotropy. 2) We generate probabilistic 1D radially anisotropic depth profiles across the whole uppermost mantle, down to ∼350 km depth. 3) The inverse problem is cast in a trans-dimensional Bayesian formalism, where the number of isotropic and anisotropic layers is treated as unknown, allowing us to obtain models described with the least number of parameters. Results show that the tectonically active region west of the Rocky Mountain Front is marked by a Lithospheric Asthenosphere Boundary and a Lehmann Discontinuity occurring at relatively shallow depths (60–150 km and 100–200 km, respectively), whereas further east, in the stable craton, these discontinuities are deeper (170–200 km and 200–250 km, respectively). In addition, in the stable part of the continent, at least two Mid-Lithospheric Discontinuities are present at intermediate depths, suggesting the existence of strong lithospheric layering, and a mechanism for lithospheric thickening by underplating of additional layers as cratonic age increases. The Moho across the continent as well as mid-crustal discontinuities in the craton are also imaged, in agreement with independent studies.
•Joint inversion of short and long period seismic data images upper mantle layering.•North America exhibits LAB and Lehmann Discontinuity at variable depths.•The North American craton presents at least two Mid-Lithosphere Discontinuities.
We have observed both minor‐arc (R1) and major‐arc (R2) Rayleigh waves for the largest marsquake (magnitude of 4.7 ± 0.2) ever recorded. Along the R1 path (in the lowlands), inversion results show ...that a simple, two‐layer model with an interface located at 21–29 km and an upper crustal shear‐wave velocity of 3.05–3.17 km/s can fit the group velocity measurements. Along the R2 path, observations can be explained by upper crustal thickness models constrained from gravity data and upper crustal shear‐wave velocities of 2.61–3.27 and 3.28–3.52 km/s in the lowlands and highlands, respectively. The shear‐wave velocity being faster in the highlands than in the lowlands indicates the possible existence of sedimentary rocks, and relatively higher porosity in the lowlands.
Plain Language Summary
The largest marsquake ever recorded occurred recently and waves propagating at the surface, called surface waves, have been observed. Owing to the relatively large magnitude (i.e., 4.7 ± 0.2) of this event, surface wave energy is still clearly visible after one orbit around the red planet. The shortest path taken by the wave propagating between the source and the receiver is located in the northern lowlands, near the boundary with the southern highlands (called dichotomy). The surface wave traveling in the opposite direction, following the longer distance between the quake and the seismic station, mostly passes through the highlands. Analyses of these two paths reveal that the average shear‐wave velocity is faster in the highlands than in the lowlands near the dichotomy boundary. This lower velocity in the lowlands may be due to the presence of thick accumulations of sedimentary rocks and relatively higher porosity.
Key Points
Analyses of the minor‐arc and major‐arc Rayleigh waves reveal different Martian crustal structures across the dichotomy boundary
The average shear‐wave velocity is faster in the highlands than in the lowlands near the dichotomy boundary
The lower shear‐wave velocity in the lowlands may be due to the presence of sedimentary rocks and relatively higher porosity
The effect of stress on wave propagation in fluid-saturated porous thermoelastic media is poorly understood. To fill this gap, we propose the dynamical equations for stressed fluid-saturated porous ...thermoelastic media based on the poroacoustoelasticity model and porothermoelasticity model to describe the effect of stress on the wave dispersion and attenuation. A plane-wave analysis for dynamical equations formulates stress-dependent velocities of five wave propagation modes, including three longitudinal (P) waves, namely fast P wave, slow P wave and thermal (T) wave, and two shear (S) waves, namely fast S wave and slow S wave. Additional slow P wave and T wave arise due to the Biot and thermal loss mechanisms in porothermoelastic media. The stress-induced rock anisotropy accounts for the S wave splitting phenomenon. Modelling results show that energy dissipations of fast P wave and T wave are induced by the coupling between Biot and thermal loss mechanisms, while the fast and slow S waves, slow P wave are only affected by Biot loss mechanism. The rock permeability and fluid viscosity are mainly related to Biot mechanism, while the thermal conductivity and thermal expansion coefficient for solid phase are related to Biot and thermal mechanisms. In addition, the triaxial stress and confining stress have remarkable effects on the wave velocities as well as attenuation peaks. The predicted wave velocities in water-saturated sandstone and granite behave a reasonable agreement with the laboratory measurements. Our results help to provide better understanding of wave propagation in high-stress high-temperature fields.
Article Highlights
We propose the dynamical equations for fluid-saturated porous thermoelastic media with the effect of stress.
Our model predicts five wave propagation modes, namely fast P wave, slow P wave, thermal wave, fast S wave and slow S wave.
Biot and Thermal loss mechanisms are coupled to describe the stress-dependent dispersions and attenuations for these five wave modes.
PDF
